{"text":"\\section{Introduction}\n\nClustering studies provide insights into the formation and evolution\nof galaxies that complement those coming from the joint distribution\nof intrinsic properties -- mass, size, morphology, gas content,\nstar-formation rate, nuclear activity, characteristic velocity and\nmetallicity. In particular, clustering studies connect galaxies to\ntheir unseen dark matter haloes and indicate how these were\nassembled. According to the current standard\n$\\mathrm{\\Lambda}\\text{CDM}$ paradigm, galaxies form as gas cools and\ncondenses at the centres of a hierarchically aggregating population of\ndark matter haloes, as originally outlined by\n\\cite{1978MNRAS.183..341W}.  As smaller haloes fall into more massive\nones, their central galaxies become satellites of these new hosts,\noccasionally merging at some later time into the new central galaxies\nin their cores.  Thus, each halo contains a dominant galaxy at the\nbottom of its potential well, and a set of satellites which were the\ncentral galaxies of smaller progenitors.  Observational studies of\nsatellite populations provide a check on this picture, indicating how\ncentral galaxy properties relate to halo mass, and how these\nproperties are modified when a halo falls into a bigger system.\n\nThe abundance of satellites, their spatial distribution and their\nintrinsic properties are thus intimately bound up with halo merger\nhistories, which are themselves closely related to the underlying\ncosmology. For example, the evolution of merger rates is sensitive to\nthe cosmic matter density, while the mass distribution of merging\nobjects depends on the linear power spectrum of initial density\nfluctuations \\citep{1993MNRAS.262..627L}. Thus, satellite properties\ncan, in principle, be used to constrain cosmological parameters. On\nthe other hand, the physical processes driving galaxy evolution have\nstrong effects on satellites. For example, their colours are affected\nby stripping of the gas reservoirs which supply star formation, and\nboth gravitational and hydrodynamical processes can modify their\nstructure, changing their morphology and partially or even totally\ndisrupting them. Modern evolutionary models for the galaxy population\nattempt to include such processes and can be tested by comparison with\nthe abundances, colours and spatial distributions of satellites.\n\nThe ``missing satellite problem'' is a particularly striking example\nof how satellite galaxy studies can constrain cosmology and galaxy\nevolution. The problem highlights an apparent mismatch between the large\nnumber of self-bound subhaloes found in $\\mathrm{\\Lambda}\\text{CDM}$\nsimulations of the formation of haloes like those of the Milky Way and\nM~31, and the much smaller number of satellite galaxies observed\naround these two Local Group galaxies\n\\citep[]{1999ApJ...522...82K,1999ApJ...524L..19M,\n  2004ApJ...609..482K}. This discrepancy has traditionally been\naddressed by claiming that photoionisation and supernova feedback\nsuppress cooling and star formation in low-mass haloes, so that only\nthe most massive Milky Way subhaloes were able to make stars, the rest\nremaining dark \\citep[e.g.][]{1993MNRAS.264..201K,\n  2000ApJ...539..517B, 2002MNRAS.333..177B, 2002ApJ...572L..23S,\n  2010MNRAS.402.1995M, 2011MNRAS.413..101G}. Recent analyses of the\nkinematics of Galactic satellites suggest, however, that their dark\nmatter haloes are not dense enough to correspond to the most massive\nsubhaloes in a $\\mathrm{\\Lambda}\\text{CDM}$ universe\n\\citep{2012MNRAS.422.1203B, 2011arXiv1111.6609F}. Such problems have\nled a number of authors to invoke warm dark matter (WDM) to eliminate\nsubhaloes less massive than a few billion solar masses, and to reduce\nthe central density of subhaloes above this cut-off\n\\citep[e.g.][]{2000ApJ...535L..21M, 2000PhRvL..84.3760S,\n  2000ApJ...544L..87Y,2001ApJ...556...93B,2009ApJ...700.1779Z,\n  2012MNRAS.420.2318L}. Others claim that part of the problem may be\nincompleteness of the observed satellite population\n\\citep[e.g.,][]{2004MNRAS.353..639W,2007ApJ...670..313S,2008ApJ...686..279K,\n  2008ApJ...688..277T,2009AJ....137..450W}.\n\nTwo kinds of methods are now commonly used to compare galaxy clustering in\nlarge redshift surveys to the predictions of high-resolution simulations of\ncosmic structure formation. The Halo Occupation Distribution \\citep[HOD;\n  e.g.,][]{ 1998ApJ...494....1J, 1998ApJ...503...37J, 2000MNRAS.318.1144P,\n  2000ApJ...543..503M, 2000MNRAS.318..203S, 2002ApJ...575..587B,\n  2002PhR...372....1C, 2005ApJ...633..791Z} and the closely related\nConditional Luminosity Function \\citep[CLF;][]{2003MNRAS.339.1057Y} approaches\ndetermine the central and satellite galaxy populations of haloes as a function\nof mass by optimizing the fit to abundance and clustering observations,\ntypically luminosity and correlation functions. In a non-parametric variant,\nthe observed abundance of central galaxies is matched directly to the\nsimulated abundance of haloes to obtain a monotonic relation between galaxy\nluminosity and halo mass \\citep[Abundance Matching,\n  AM:][]{2004ApJ...614..533T, 2004MNRAS.353..189V, 2006ApJ...647..201C,\n  2010ApJ...710..903M, 2010MNRAS.404.1111G}. This relation can be used to\npopulate both main and satellite subhaloes, if it is assumed to hold when\nsatellites first fall into more massive systems. By construction, these\nmethods fit observed luminosity functions perfectly. The same is true for\nobserved correlation functions for HOD and CLF, whereas these serve as a test\nfor AM. None of these schemes explains {\\it why} haloes of given mass have\ncentral galaxies with specific properties.\n\nIn contrast, semi-analytic models use simplified representations of the\nrelevant astrophysics to follow galaxy growth within the evolving\ndark halo population, and so attempt to predict the detailed\nproperties of both central and satellite galaxies \\citep[SAM;\n  e.g.,][]{1991ApJ...379...52W, 1993MNRAS.264..201K, 1994MNRAS.271..781C,\n  1999MNRAS.303..188K, 1999MNRAS.310.1087S, 2001MNRAS.328..726S,\n  2005ApJ...631...21K}. Here, adjustable parameters correspond to the\nefficiencies of poorly understood processes like star formation or AGN\nfeedback, so that the values derived from fitting to observation are\ninteresting in their own right. In recent years, ever more detailed\nastrophysical models have been incorporated into ever larger and higher\nresolution simulations of dark matter evolution, leading to increasingly\nfaithful representation of the observed galaxy population\n\\citep[e.g.][]{2005Natur.435..629S,2006MNRAS.365...11C,\n  2006MNRAS.370..645B,2007MNRAS.375....2D,2011MNRAS.413..101G}.\n\nMany studies of satellite galaxy populations have focused on the Local Group\n(LG) because of the greater depth and detail with which nearby galaxies can be\nstudied (see \\cite{2000ESASP.445...87G, 2001ApSSS.277..231G,\n  2007ggnu.conf....3G,2011EAS....48..315G} and references therein).  Recent\nwork has been particularly concerned with comparing\n$\\mathrm{\\Lambda}\\text{CDM}$ predictions to the abundance and internal\nstructure of dwarf spheroidal galaxies and to the orbital and internal\nproperties of the Magellanic Clouds \\citep[e.g.][]{2009ApJ...696.2179K,\n  2011ApJ...733...62L, 2011MNRAS.417.1260F,  2011ApJ...738..102T, \n2011MNRAS.418..648S,2011MNRAS.415L..40B}. Such studies are \nlimited by the fact that the LG contains only two large spirals, since\nconsiderable scatter is expected among the satellite populations of similar\nmass haloes \\citep[e.g.][]{2010MNRAS.406..896B,2011MNRAS.413..101G}.\n\nBeyond the LG, many observational studies of satellite populations have\nestimated their luminosity functions and their radial\ndistribution around their primaries, but rather few have compared directly\nwith theoretical expectations.  Some studies have used redshift surveys to\ninvestigate the projected number density profiles of satellites\n\\citep[e.g.][]{1991ApJ...379L...1V, 2005MNRAS.356.1045S,2006ApJ...647...86C}\nusually fitting power laws $\\Sigma(r)\\sim r^\\alpha$, and obtaining slopes\n$\\alpha$ between $-0.5$ and $-1.2$. The availability of redshifts for all\ngalaxies facilitates discrimination between satellites and background objects,\nbut, for most objects, satellites are detectable only one or two magnitudes\nfainter than their primaries. An important application made possible by the\nredshifts is the measurement of mean dynamical masses for haloes as a function\nof central galaxy luminosity and colour \\citep{1993ApJ...405..464Z,\n  1997ApJ...478...39Z, 2003ApJ...598..260P,\n  2005ApJ...635..982C,2007ApJ...654..153C, 2011MNRAS.410..210M}. The last of\nthese papers finds that, at given luminosity, red central galaxies have more\nmassive haloes than blue ones, but that this difference goes away if red and\nblue primaries are compared at the same stellar mass.  We will return to this\nissue below.\n\nThe abundance of satellites at magnitudes much fainter than their primaries is\nmost easily studied by combining a redshift survey with a photometric survey\nwhich catalogues galaxies several magnitudes below the spectroscopic limit. In\nthe absence of redshifts, it is not possible, of course, to distinguish true\nsatellites from background galaxies. Results can therefore be obtained only by\nstacking large samples of primaries so that a statistical substraction of the\nbackground population is possible \\citep{1969ArA.....5..305H,\n  1987MNRAS.229..621P, 1994MNRAS.269..696L, 2004ApJ...617.1017S, 2012ApJ...746..138T}.  In\nparticular, \\cite{1994MNRAS.269..696L} counted the number of faint images on\nSchmidt survey plates around primaries of known redshift, using a\n``bootstrap'' method to remove the background and fitting the projected\nsurface density to a power law $\\Sigma(r_p)\\sim r_p^{-\\alpha}$, finding\n$\\alpha\\sim 0.9$.  They showed that satellites are more abundant and are\nconcentrated to smaller radii around early-type primaries than around\nlate-types.\n\nMost recent work has taken advantage of the enormous increase in data provided\nby the Sloan Digital Sky Survey  \\citep[SDSS;][]{2000AJ....120.1579Y}. \\cite{2006MNRAS.372.1161W} used the\ngroup catalogue of \\cite{2007ApJ...671..153Y}, constructed from the SDSS\nspectroscopic data, to study in detail how the properties of satellite\ngalaxies depend on the colour, luminosity, and morphology of the central\ngalaxy and on their inferred dark halo mass. They compared their observational\nresults with a mock redshift survey based on the SAM of\n\\cite{2006MNRAS.365...11C}, finding significant discrepancies. In particular,\nthe model overpredicted the number of faint satellites in massive haloes and\nproduced too many red satellites.  The fraction of blue central galaxies was\nalso too high at high luminosities.  \\cite{2006MNRAS.372.1161W} argued that\nthe satellite problems most likely reflect an improper treatment of tidal\nstripping or of the truncation of star formation, while the central problem\nmay reflect an overly simple treatment of dust or of AGN feedback. In\n\\cite{2011MNRAS.416.1197W} a mock catalog based on the more recent model of\n\\cite{2011MNRAS.413..101G} was compared with several nearby galaxy clusters as\nwell as with the group catalogue of \\cite{2007ApJ...671..153Y}.  Discrepancies\nwere weaker than for the earlier model, but the predicted fraction of red\ndwarf satellites remains higher than in the Virgo cluster or in the group\ncatalogue of \\cite{2007ApJ...671..153Y}, although agreeing with the fractions\nfound in the Coma and Perseus clusters.\n\nStudies of satellite galaxies based on both spectroscopic and\nphotometric data from the SDSS have been published recently by\n\\cite{2011AJ....142...13L}, by \\cite{2011MNRAS.417..370G} and by\n\\cite{2012ApJ...746..138T,2012ApJ...751L...5T}. The first of these\ninvestigated how the luminosity function and number density profile of\nsatellites depend on the colour and luminosity of their central\ngalaxy, finding the abundance of satellites to depend strongly on\nprimary luminosity, and the faint-end slope of their luminosity\nfunction to be consistent with that of the field.  Using similar\ndatasets, \\cite{2011MNRAS.417..370G} investigated the satellite\nluminosity function and its dependence on primary luminosity, colour\nand concentration.  Their satellite luminosity function estimates are\nnot well fit by Schechter functions, tending to be flat at bright\nluminosities but very steep at faint luminosities, apparently at odds\nwith the conclusions of \\cite{2011AJ....142...13L}. For the primary\nmagnitude range ($M_V=-21.25\\pm 0.5$) the mean luminosity function of\n\\cite{2011MNRAS.417..370G}is similar in shape to that of the MW and\nM31, but the abundance of satellites is about a factor two\nlower. \\cite{2012ApJ...746..138T} studied satellites of SDSS Luminous\nRed Galaxies using, in particular, the deep Stripe 82 data finding a\nluminosity function with a shallow faint end slope and a very\ndifferent shape from those of \\cite{2011MNRAS.417..370G}. \n\\cite{2012ApJ...751L...5T}  constructed radial number density\nprofiles for these same systems, concluding that they are well fitted \nby a NFW model \\citep{1996ApJ...462..563N,1997ApJ...490..493N} on large scales \nwhile at small radii there is an excess  of satellites compared with the NFW profile, \nwhich can be well dscribed by a Sersic model.  Using data from the Galaxy and\nMass Assembly Survey \\citep[GAMA;][]{2009A&G....50e..12D,\n  2011MNRAS.413..971D}, \\cite{2011MNRAS.417.1374P} also studied\nsatellite number density profiles and red fractions as functions of\nprojected separation and the masses of both satellite and primary,\narguing that their results favour removal of gas reservoirs as the\nmain mechanism quenching star formation in satellites. Finally,\n\\cite{2012ApJ...752...99N} use HST data from the Cosmological Evolution Survey (COSMOS) to\nstudy similar issues for smaller samples of galaxies, but out to\n$z\\sim0.8$.\n\nIn the present paper we return to many of these questions, using the full SDSS\nspectroscopic and photometric databases in conjunction with the galaxy\npopulation simulations of \\citet[][hereafter G11]{2011MNRAS.413..101G}. The\nsimulations allow us to compare expectations based on our current\nunderstanding of galaxy formation in a $\\Lambda$CDM universe with the observed\ndependences of satellite luminosity, stellar mass and colour on primary galaxy\nproperties.  By using mock catalogues from the simulations, we are able to\nexplore how satellite populations relate to the dark matter haloes in which\nthey are embedded, to gain insight into the effect of physical processes\nlike quenching and tidal stripping on their observable properties, and to explore\nhow the observational criteria defining isolated primary galaxies impact the\nclustering of other galaxies around them (see \\cite{1976ApJ...205L.121F} for an\nold example of the potential strength of such effects).\n\n\nWe describe the datasets we use and the selection criteria which\ndefine our primary and satellite galaxy samples in\nsection~\\ref{sec:data}.  In section~\\ref{sec:method} we introduce our\nbackground subtraction method.  We present our SDSS results and\ncompare them directly with the G11 simulation in\nsections~\\ref{sec:LFMF} and~\\ref{sec:colour}. Further discussion and\ncomparison with previous work is given in our concluding section. An\nappendix describes a variety of tests for systematics in the SDSS\nphotometric data and in the techniques we use to correct satellite\ncounts for contamination by foreground and background\ngalaxies. Throughout this paper, we convert observational to intrinsic\nproperties assuming a cosmology with $\\Omega_m=0.25$,\n$\\Omega_\\Lambda=0.75$ and $h=0.73$. We quote all masses in units of\n$M_\\odot$ rather than $h^{-1}M_\\odot$.\n\n\n\\section{Data and Sample Selection}\n\\label{sec:data}\n\\subsection{Primary Selection}\n\\label{subsec:select} \nWe wish to study the satellite populations of bright isolated galaxies out to\ndistances $\\sim 0.5$~Mpc.  We begin by considering all galaxies brighter than\n$r=16.6$ ($r$-band extinction corrected Petrosian magnitude) in the\nspectroscopic galaxy catalogue of the New York University Value Added Galaxy\nCatalog (NYU-VAGC)\\footnote{http:\/\/sdss.physics.nyu.edu\/vagc\/} \\citep{2011ApJS..193...29A}. \nThis catalogue was built by \\cite{2005AJ....129.2562B} on the basis of the \nseventh Data Release of the Sloan Digital Sky Survey\n\\citep[SDSS\/DR7;][]{2009ApJS..182..543A}. This apparent magnitude limit provides\nus with a parent catalogue of 145070 objects.  We select isolated galaxies\nfrom this sample by requiring that there should be no companion in the\nspectroscopic sample at $r_p<0.5$~Mpc and $|\\Delta{z}|<1000$~km\/s that is less\nthan a magnitude fainter in $r$ than the central object, and no companion at\n$r_p<1$~Mpc and $|\\Delta{z}|<1000$~km\/s that is brighter than it. These\nisolation criteria reduce our sample to 66285 objects.\n\nThe SDSS spectroscopic sample is incomplete, because observing efficiency\nconstraints made it impossible to put a fibre on all candidates or to\nre-observe objects where an initial spectrum yielded an unreliable\nredshift. The completeness varies with position on the sky and has a mean of\n$91.5\\%$ for our parent sample. Thus $\\sim 10\\%$ of our ``isolated'' galaxies\nwill not, in fact, be isolated according to our criteria, because their\ncompanion was missed by the spectroscopic survey. To eliminate such systems we\napply an additional cut using the SDSS photometric data.  The photometric\nredshift 2 catalogue \\citep[photoz2;][]{2009MNRAS.396.2379C} on the SDSS\nwebsite provides redshift probability distributions for all galaxies in the\nSDSS footprint down to apparent magnitude limits much fainter than we require.\nThese distributions are tabulated for 100 redshift bins, centered from\n$z_1=0.03$ to $z_2=1.47$ with spacing $dz=1.44\/99$. We find all the objects in\nour candidate isolated galaxy list which have a companion in the photoz2\ncatalogue satisfying the above projected separation and magnitude difference\ncriteria, and we discard those where the companion has a photometrically\nestimated redshift distribution compatible with the spectroscopic redshift of\nthe primary.  Our definition of ``compatible'' is that the probability for the\ncompanion to have a redshift equal to or less than that of the primary exceeds\n0.1. Apparent companions which fail this test usually do so because their\ncolours are too red to be consistent with a redshift as low as that of the\nprimary. After applying this additional cut, 41883 objects remain in our\nisolated galaxy sample.\n\n\nFinally, we exclude any object for which more than $20\\%$ of a surrounding\ndisc of radius $r_p=1$Mpc lies outside the survey footprint.  Such objects\ncould have bright companions which are not included in the SDSS databases. To\nevaluate these overlaps we made use of the set of ``spherical polygons''\nprovided on the NYU-VAGC website. These account both for the survey boundary\nand for masked areas around bright stars. We generate a large number of points\nuniformly and randomly over the 1~Mpc disc surrounding each galaxy and discard\nany which lie outside the .survey boundary. A galaxy is eliminated from the\nsample if more than 20\\% of its points are discarded in this way. This last\ncut removes about 1.5\\% of our objects, leaving a final sample of 41271 bright\nisolated galaxies. The set of randomly generated points surrounding each of\nthese ``primaries'' is kept for later use when estimating background\ncorrections to the counts of its faint companions (see below).\n\n\nFigure~\\ref{fig:prop_dis} compares the distributions of colour,\nPetrosian half-light radius $R_{50}$, concentration $C=R_{90}\/R_{50}$\nand stellar surface mass density $\\mu_\\star= M_\\star\/2\\pi R_{50}^2$\nfor our parent galaxy sample (all 145070 galaxies with $r<16.6$) and\nfor our 41271 isolated primaries, separated into bins of galaxy stellar \nmass.  These quantities were taken directly from the NYU-VAGC \ncatalogue. The stellar masses were estimated by fitting stellar population \nsynthesis models to the K-corrected galaxy colours assuming a \n\\cite{2003PASP..115..763C} initial mass function as in \n\\cite{2007AJ....133..734B}. The sensitivity of the stellar masses to \nassumptions underlying the estimation technique is explored in the \nAppendix of \\cite{2009MNRAS.398.2177L}.\nEach stellar mass bin is a factor of two wide, and we show data for\nthe five mass bins on which we will concentrate our analysis\nthroughout the rest of this paper. The numbers in the lower right of\nthe $R_{50}$ plots indicate the number of isolated primaries in each\nmass bin.  Volume corrections have been applied to all the histograms\nin this figure by calculating the total volume $V_{\\rm max}$ of the\nsurvey over which each individual galaxy would be brighter than the\nflux limit, $r=16.6$, and accumulating counts weighted by $1\/V_{\\rm\n  max}$. It turns out that our isolated primaries are slightly bluer\nthan the parent sample, particularly at low masses.  In addition, they\nare slightly more concentrated than the parent sample.  Our selection\nprocedure appears to have no significant effect on the distributions\nof the other two properties, and the same is true for the redshift\ndistributions which we do not show. To a very good approximation our\nisolated primary galaxies are typical objects of their stellar mass,\nalthough as we will see below, our selection has a strong influence on\ntheir relation to their environment.  Vertical dashed lines in the\ncolour plots indicate the split we adopt when separating our primaries\ninto red and blue populations.  This split is slightly dependent on\nstellar mass.\n\n\n\\begin{figure*}\n\\centerline{             \\epsfig{figure=fig01a.ps,width=0.45\\textwidth}\n  \\epsfig{figure=fig01b.ps,width=0.45\\textwidth}}          \\centerline{\n  \\epsfig{figure=fig01c.ps,width=0.45\\textwidth}\n  \\epsfig{figure=fig01d.ps,width=0.45\\textwidth} }\n\\caption{Volume weighted distributions for the parent sample of SDSS\/DR7\n  galaxies with $r<16.6$ (black curves) and for our sample of isolated\n  galaxies (red curves). Different panels within each set refer to different\n  ranges of $\\log M_\\star\/M_\\odot$ as labelled.  Each set corresponds to a\n  different property. {\\bf Top left:} $^{0.1}(g-r)$ colour -- the vertical\n  dashed line in each panel shows the colour separating red and blue\n  populations. {\\bf Top right:} Concentration $C=R_{90}\/R_{50}$.  {\\bf Bottom\n    left:} Petrosian half-light radius $R_{50}$ (in kpc) -- the numbers at\n  bottom right of each panel indicate the numbers of galaxies in the parent\n  (black) and isolated samples (red) shown. {\\bf Bottom right:}\n  Stellar surface mass density $\\mu_\\star= M_\\star\/2\\pi R_{50}^2$.\n\\label{fig:prop_dis}\n}\n\\end{figure*}\n\n\n\n\\subsection{The mock catalogue}\n\nIn the analysis and interpretation of our results for satellite galaxy\npopulations we will make considerable use of mock catalogues built from the\ngalaxy formation simulations of \\citet[][hereafter\n  G11]{2011MNRAS.413..101G}. These are implemented on two very large dark\nmatter simulations, the Millennium Simulation \\citep[MS;][]{2005Natur.435..629S} and\nthe Millennium-II Simulation \\citep[MS-II;][]{2009MNRAS.398.1150B}. The MS follows the\nevolution of structure within a cube of side $500h^{-1}\\rm{Mpc}$ (comoving)\nand its merger trees are complete for subhaloes above a mass resolution limit\nof $1.7\\times10^{10}h^{-1}\\rm{M}_{\\sun}$. The MS-II follows a cube of side\n$100h^{-1}\\rm{Mpc}$ but with 125 times better mass resolution (subhalo masses\ngreater than $1.4\\times10^{8}h^{-1}\\rm{M}_{\\sun}$).  Both adopt the same\nWMAP1-based $\\Lambda$CDM cosmology \\citep{2003ApJS..148..175S} with parameters\n$h=0.73, \\Omega_m=0.25, \\Omega_\\Lambda=0.75, n=1$ and $\\sigma_8=0.9$. These\nare outside the region preferred by more recent analyses (in particular,\n$\\sigma_8$ appears too high) but this is of no consequence for the issues we\nstudy in this paper. For consistency, we will adopt this cosmology\nwhen quoting numbers in the rest of this paper. \n\nIn G11's galaxy formation model, the uncertain star formation and feedback\nefficiencies are tuned to produce close fits to the stellar mass, luminosity\nand autocorrelation functions of low redshift galaxies as inferred from the\nSDSS.  Together with their high resolution (particularly for the MS-II) and\nlarge volume (for the MS) this makes them ideal for our purposes in this\npaper. Here we use the publicly available data from\nhttp:\/\/www.mpa-garching.mpg.de\/millennium.  We project the simulation boxes in\nthree orthogonal directions parallel to their $x$, $y$ and $z$ axes.  In each\nprojection we can assign each galaxy a redshift based on its ``line-of-sight''\ndistance and peculiar velocity, and we can select isolated primaries using\ncriteria which are directly analogous to those used for the SDSS (though we do\nnot need to worry about the complications due to completeness and boundary\nissues).  In addition to observables like luminosities, colours, sizes and\nmorphologies, the simulation databases provide information which is not\ndirectly accessible for real galaxies (e.g. halo mass, environment type and\nfull 3D position and peculiar velocity).  These can give insight into the\nnature of the isolated primary sample we have selected.  \n\nAll the SDSS luminosities and colours we use in this paper are rest-frame\nquantities K-corrected to the $^{0.1}r$ band.  The absolute magnitudes in our\nmock catalogues are in the true $z=0$ $r$ band, because this is what is\ndirectly provided by the database and the difference between the $^{0.1}r$\nand $r$ bands is too small to significantly affect luminosities. We do,\nhowever, transform the database $(g-r)$ colours to the $^{0.1}(g-r)$ band\nusing the empirical fitting formula of \\cite{2007AJ....133..734B} because here\nthe shifts seem large enough to cause (minor) differences.\n\nStructure in the MS and MS-II is characterized using Friends-of\nFriends (FoF) groups partitioned into sets of disjoint self-bound\nsubhaloes. The subhalo populations at neighboring output times are\nlinked to build merger trees which record the assembly history of all\nnonlinear structures and provide the framework for the galaxy\nformation simulations. In these simulations galaxy evolution is\naffected by environment in several ways. The galaxy at the centre of\nthe most massive subhalo of each FoF group (which usually contains\nmost of its mass) is considered the ``central galaxy'' and is the only\none to accrete material from the diffuse gas associated with the\ngroup. When evolution joins two FoF groups, G11 continue to treat the\ngalaxy at the centre of the less massive subhalo as a central galaxy\nuntil it falls within the nominal virial radius\\footnote{We define\n  this as $r_{200}$ the radius of the sphere centred on the\n  gravitational potential minimum of the FoF group within which the\n  mean density is 200 times the critical value. $M_{200}$ is then the\n  mass within this sphere.} of the new FoF group. After this point the\ninfalling galaxy is considered as a ``satellite'', the mass of its\ndark halo starts dropping as a result of tidal stripping, and its\ndiffuse gas is assumed to be removed in proportion to the subhalo dark\nmatter. Such satellites may later lose their subhaloes entirely\nthrough tidal disruption.  At this point they are either disrupted\nthemselves or (more commonly) they become ``orphan satellites'' which\ncontinue to orbit until dynamical friction causes a merger with their\ncentral galaxy. In this paper we will follow G11, considering together\nthe two kinds of satellites (with and without a dark matter subhalo)\nand the two kinds of centrals (in dominant and in newly accreted,\ndistant subhaloes).\n\nThe left panels of figure~\\ref{fig:prop_dis2} compare halo mass distributions\nas a function of stellar mass for isolated simulation galaxies to those of\ntheir parent population. Here, halo mass is $M_{200}$ of the FoF group for the\ndominant central galaxy and all the satellites, and $M_{\\rm inf}$ for the\nother ``centrals'', where $M_{\\rm inf}$ is the $M_{200}$ of the old FoF group\nof a newly accreted central just prior to infall. Our isolation criteria have\na substantial effect on these mass distributions, eliminating a high-mass tail\nwhich is particularly evident for lower stellar mass galaxies. This tail is\ndue to the satellites which, as we will show more explicitly below, are very\neffectively excluded by our criteria.  \n\nThe right panels of figure~\\ref{fig:prop_dis2} compare colour\ndistributions for these same simulated galaxy populations. Just as for\nthe real SDSS galaxies (figure~\\ref{fig:prop_dis}) our isolation\ncriteria bias the distributions to bluer galaxies because central\ngalaxies have ongoing gas accretion and so are more actively\nstar-forming than satellites. Selection induces larger shifts for the\nsimulated distributions than for the real ones, however, and the\nbimodal nature of the colour distributions is more obvious in the\nsimulation.  As already discussed by G11 and\n\\citet{2011MNRAS.416.1197W} this reflects the facts that the red and\nblue sequences are more sharply defined in the simulation than in\nreality and that satellite galaxies appear to be too uniformly\nred. Note also that the red and blue populations separate at bluer\ncolours (indicated by dashed vertical lines) in the simulation than in\nthe SDSS data, particularly at high mass. This appears to be a\nconsequence of the stellar population synthesis models used in the\nsimulation, together with the fact that stellar metallicities are too\nlow for high-mass galaxies \\citep[see][]{2010MNRAS.403..768H}.\n\n\n\\begin{figure*}\n\\centerline{             \\epsfig{figure=fig02a.ps,width=0.45\\textwidth}\n  \\epsfig{figure=fig02b.ps,width=0.45\\textwidth} }\n\\caption{{\\bf Left:} Halo mass distributions for isolated galaxies (red\n  curves) in five disjoint stellar mass bins and for the parent populations in\n  the simulations of G11 from which they were drawn (black curves). {\\bf\n    Right:} Colour distributions for these same sets of simulated\n  galaxies. Vertical dashed lines indicate the colour at which we separate red\n  and blue populations.}\n\\label{fig:prop_dis2}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{             \\epsfig{figure=fig03a.ps,width=0.45\\textwidth}\n  \\epsfig{figure=fig03b.ps,width=0.45\\textwidth} }\n\\caption{{\\bf Left:} The fraction of isolated galaxies (bottom panel) and of\n  their parent sample of all galaxies (top panel) in the galaxy formation\n  simulation of G11 which are classified as centrals rather than satellites\n  according to the G11 criteria which we also adopt here (see the text for\n  details). Black, red and blue lines indicate the fractions as a function of\n  stellar mass for all, for red and for blue galaxies respectively.  {\\bf\n    Right:} The fractions of galaxies which are red (rather than blue) are\n  shown as a function of stellar mass for isolated galaxies (red curves) and\n  for the parent sample of all galaxies (black curves). The solid curves in\n  each case represent observed results for the SDSS (after weighting to\n  represent values for volume-limited samples) while the dashed curves are for\n  the galaxies in the simulation of G11, Note the excellent agreement for\n  $\\log M_\\star\/M_\\odot > 10.2$ the range of interest for this paper.}\n\\label{fig:frac}\n\\end{figure*}\n\nThe left panels of figure~\\ref{fig:frac} illustrate how well our\nisolation criteria select central galaxies, at least in the\nsimulation.  The fraction of centrals according to the definitions of\nG11 is plotted as a function of stellar mass in the lower panel, with\nthe black curve referring to all isolated galaxies and the red and\nblue curves referring to the red and blue subpopulations. For the\nisolated population as a whole and for its red subpopulation, the\ncontamination by satellites {\\it maximizes} at just over 2\\%. Slightly\nlarger contamination occurs at high mass in the blue subpopulation,\nbut such massive blue galaxies are in any case very rare (see\nbelow). In contrast, the curves in the upper panel show that only 65\\%\nof the parent population in the G11 model are centrals at $\\log\nM_\\star\/M_\\odot = 10$, about 80\\% are centrals at $\\log\nM_\\star\/M_\\odot = 11$ and 88\\% are centrals at $\\log M_\\star\/M_\\odot =\n11.5$.  For blue galaxies the central fraction is above 90\\% at all\nmasses, while for $\\log M_\\star\/M_\\odot < 10.3$ most red galaxies are\nsatellites. The application of our isolation criteria to the simulated\ngalaxy population of G11 thus results in an extremely pure sample of\ncentral galaxies.\n\nWe cannot be sure, of course, that the elimination of satellite galaxies is as\neffective in the SDSS samples as in the simulation. We can, however, check\nthat the separation into red and blue subpopulations matches as a function of\nstellar mass, both for isolated galaxies and for their parent population.\nThis comparison is shown in the right panel of figure~\\ref{fig:frac}. Here red\ncurves indicate the red fraction as a function of stellar mass for isolated\ngalaxies, while black curves indicate the same quantity for the parent\nsample. In each case, solid curves are the observational result from SDSS and\ndashed curves are for the simulation of G11. For the observations we have\nagain used $1\/V_{\\rm max}$ weighting to ensure that the plotted quantity is\nappropriate for a volume-limited sample. For the primary stellar masses\nrelevant for this paper ( $\\log M_\\star\/M_\\odot > 10.2$) the agreement between\nobservation and simulation is almost perfect. At lower stellar masses there\nare too many red galaxies in the simulation, again reflecting the problem\nnoted above and discussed in detail by \\citet{2011MNRAS.416.1197W}: at low\nmasses the simulated satellite galaxies are too uniformly red.\n\n\nAnother comparison of the effects of our isolation criteria on the\nobserved and simulated galaxy samples is shown in\nFigure~\\ref{fig:select}. The thick black solid line here indicates, as\na function of stellar mass, the fraction of galaxies from the parent\nsample which remain in our final sample of isolated SDSS galaxies. The\nthick black dashed line shows the result when analogous isolation\ncriteria are applied to our G11 mock catalogues. The agreement of\nobservation and simulation is again quite good, though not as perfect\nas in the right panel of figure~\\ref{fig:frac}. Note, however, that\nperfect agreement is not expected, since the underlying SDSS sample is\nmagnitude-limited whereas the G11 sample is volume-limited.\n\nThis same plot provides a convenient way to summaries the relative effects of\nthe spectroscopic and photometric catalogues in defining our observed sample\nof isolated galaxies. The thin red solid line shows the fraction of objects\nwhich are isolated relative to the spectroscopic sample, but not necessarily\nrelative to the photometric sample.  At most masses, use of the photometric\ncatalogue increases the number of objects with an identified ``companion'' by\n30 to 50\\%. This has a relatively small effect on the isolated fraction at\nhigh mass, since the great majority of massive galaxies are isolated by our\ncriteria. At $\\log M_\\star\/M_\\odot = 10.3$ it reduces the number of isolated\ngalaxies by almost a factor of two, however, because the majority of such SDSS\ngalaxies have a companion by these same criteria. These numbers suggest (and\nwe have checked in more detail) that our photometric rejection step is\nconservative, in that a significant fraction of the photometric ``companions''\n(about one third) are, in fact, galaxies at a significantly different redshift\nwhich are projected on top of the primary. This does not matter for the\nanalysis of this paper -- it just causes us to end up with a slightly smaller\nsample of isolated primaries than if we had been less conservative.\n\nFinally we can use the simulated catalogue to see how our isolation criteria\naffect the numbers of satellite and central galaxies in our samples. The green\nand blue dashed lines in Figure~\\ref{fig:select} show as a function of stellar\nmass the fractions of centrals and satellites which pass our isolation\ncriteria. At high mass, most centrals are indeed isolated, but for $\\log\nM_\\star\/M_\\odot < 10.6$ more than half of them are rejected because of\napparent companions. In most cases these companions are actually the central\ngalaxies of other haloes. On the other hand, for $\\log M_\\star\/M_\\odot < 11$\nalmost all satellite galaxies are found to have companions according to our\ncriteria. With increasing mass the number of apparently isolated satellites\ngrows, reaching 30\\% at the highest mass. This is because these objects lie in\nvery massive galaxy clusters and so can be projected more than 1~Mpc from\ntheir associated central galaxies. Note, however, that the actual number of\nsuch massive satellites is very small (see figure~\\ref{fig:frac}). It is the\ninterplay of these different effects which produces the very high purity\n(i.e. central galaxy fraction) at all masses in our final isolated galaxy\nsample.\n\n\n\\begin{figure}\n\\epsfig{figure=fig04.ps,width=0.45\\textwidth}\n\\caption{The fraction of galaxies that are selected by our isolation\n  criteria as a function of stellar mass. The thick black solid line\n  gives the result for our final sample of isolated SDSS galaxies,\n  while the thick black dashed line is the corresponding result for\n  simulated galaxies in the G11 model. These agree moderately well\n  over the full mass range shown. The thin solid red line shows the\n  fraction of retained SDSS galaxies after demanding isolation\n  relative to the spectroscopic sample but before additionally\n  requiring isolation relative to the photometric sample. Green and\n  blue dashed curves indicate the fractions of central and satellite\n  galaxies which remain after applying our isolation criteria to the\n  G11 simulated galaxy sample.}\n\\label{fig:select}\n\\end{figure}\n\n\n\\subsection{The photometric catalogue for satellites}\n\\label{subsec:photo}\n\nBoth the spectroscopic catalogue and the photoz2 photometric catalogue used in\nthe last section are based on DR7, the seventh release of SDSS data.  When\nidentifying and verifying the isolation of our isolated primary galaxies, only\nobjects with apparent magnitude brighter than $r=17.6$ had to be\nconsidered. This is far above the magnitude limit of the SDSS photometry, and\nthe accuracy of the DR7 magnitudes\/colours and of the derived photo-$z$\ndistributions is well tested at these magnitudes. When compiling\ncounts of faint satellite galaxies around these primaries, we need to go to\nthe SDSS limit for reliable photometry, however, which we take to be $r=21$.\nAt this limit, improvements to the photometry pipeline have continually\nenhanced its reliability, identifying (and correcting when possible)\nsystematic artifacts which can have a significant influence on our\nanalysis. We therefore use the photometric catalogue from\nSDSS\/DR8\\footnote{http:\/\/skyservice.pha.jhu.edu\/casjobs\/} \\citep{2011ApJS..193...29A} when compiling\nsatellite counts. To be specific we created a reference photometric catalogue\nby downloading objects that are classified as galaxies in the survey's primary\nobject list, and that do not have any of the flags BRIGHT, SATURATED,\nSATUR\\_CENTER or NOPETRO\\_BIG set. This follows the selection criteria\nfor the DR7 photoz2 catalogue used above,\n\nIt is important to note that the magnitude we use for satellites is the\nso-called SDSS model magnitude, which, at faint apparent magnitudes, is\nclaimed to have the highest signal-to-noise of all the alternatives\ncatalogued. In contrast, when defining our primary sample and checking its\nisolation we used Petrosian magnitudes as listed on the NYU-VAGC website. We\ncontinue to use these magnitudes for the primaries below. When we quote\ncolours, these are always measured within a well-defined aperture related to\nthe Petrosian radius, and are rest-frame quantities K-corrected to the\n$^{0.1}(g-r)$ band for both primaries and satellites. All absolute magnitudes\nquoted for SDSS galaxies are also in this same rest-frame band.\n\nWe have carried out a variety of tests for systematics in the SDSS\nphotometry, checking completeness and the quality of star-galaxy\nseparation by comparing with much deeper HST data, and quantifying\nsystematic biases in the magnitudes of faint images (e.g. the\nsatellites) in the neighborhood of substantially brighter images\n(e.g. the primaries). Detailed descriptions of these tests are given\nin an Appendix, along with details of tests of our procedures\n(outlined in the next section) for counting apparent companions around\nour primary galaxies and correcting for the (usually dominant)\ncontribution from unrelated foreground and background objects.\n\n\\section{Satellite counting methodology}\n\\label{sec:method}\nWe want to study the abundance of satellites as a function of their\nluminosity, stellar mass and colour around our sample of isolated bright\nprimaries, and to see how these abundances depend on the stellar mass and\ncolour of the primary.  We will need to use the SDSS data down to their\nreliable photometric limit (which we take to be $r=21$) and as a result the\ngreat majority of the potential satellites do not have a spectroscopically\nmeasured redshift.  Hence it is necessary to count {\\it all} apparent\nneighbours around our primaries and to correct statistically for unassociated\nobjects which happen to be projected near them. The number of such (mainly)\nbackground objects can substantially exceed the number of true satellites, so\nit is important to take considerable care in making these corrections.  We\nadopt the following procedure.\n\n\nFor each isolated bright galaxy, we identify all photometric galaxies with\napparent projected separation $r_p<0.5$~Mpc and we accumulate counts in bins\nof projected separation $r_p$, apparent magnitude $r$ and observed colour\n$g-r$.  From the count in each ($r_p$,$r$,$g-r$) bin, we subtract the expected\nnumber of background galaxies which we take to be $N(r,g-r) A(r_p,z)f\/A_{\\rm\n  tot}$, where $N(r,g-r)$ is the total number of galaxies in the ($r$,$g-r$)\nbin in the full photometric catalogue, $A_{\\rm tot}$ is the solid angle of the\nsurvey footprint, $A(r_p,z_{\\mathrm{pri}})$ is the solid angle corresponding\nto the annular $r_p$ bin at the redshift $z_{\\mathrm{pri}}$ of the primary\ngalaxy, and $f$ is the incompleteness factor, the fraction of this annulus\nwhich lies within the survey footprint (we estimate this using the random\npoints generated around the position of each primary during the selection\nprocess -- see above). We then use the redshift of the primary galaxy to\nconvert observed apparent magnitudes and colours into rest-frame luminosities\nand colours (in the $^{0.1}r$ and $^{0.1}(g-r)$ system)\\footnote{We use the\n  empirical fitting formula of \\cite{2010PASP..122.1258W} which gives the\n  K-correction as a function of redshift and observed colour.} and we transfer\nthe background-subtracted satellite counts from our narrow bins of $r$ and\n$g-r$ into substantially broader bins of the rest-frame quantities.  Finally\nwe average these counts for each $r_p$ bin over the set of all primaries in\nthe desired range of stellar mass (and sometimes colour) and we sum the result\nover the desired range in $r_p$.  Uncertainties in the resulting numbers are\nestimated from the scatter among results for 100 bootstrap resamplings of the\nset of primaries.\n\nSome apparent companions are too red to be at the redshift of the primary galaxy. \nIt is useful to exclude them when accumulating counts since they add noise without\nadding signal.  Hence, we exclude all bins redder than\n$^{0.1}(g-r)=0.032\\mathrm{log}_{10}M_\\star+0.73$, a fit to the upper envelope\nof the distribution of rest-frame colour against stellar mass for galaxies of\nmeasured redshift. The stellar mass $M_\\star$ of the apparent companion is\nestimated by assuming it to be at the primary's redshift and adopting\n\\begin{equation}\n(M\/L)_r=-1.0819^{0.1}(g-r)^2+4.1183^{0.1}(g-r)-0.7837\n\\label{eqn:masstolight}\n\\end{equation}\n\nThis empirical relation is a fit to a flux-limited ($r<17.6$) galaxy\nsample from the NYU-VAGC website for which stellar masses were \nestimated from the K-corrected galaxy colours by fitting stellar population \nsynthesis models assuming a \\cite{2003PASP..115..763C} initial mass function \n\\citep{2007AJ....133..734B}. For this sample the 1-$\\sigma$ scatter in $(M\/L)_r$ \nof this simple relation is about 0.1.\n\nThe photometric catalogue we use is complete down to an $r$-band apparent\nmodel magnitude of 21. This limit corresponds, of course, to different\nsatellite luminosities and stellar masses for different primary redshifts and\ndifferent satellite colours. In order to ensure that our samples are complete\nwhen compiling satellite luminosity functions, we allow a particular primary\nto contribute counts to a particular luminosity bin only if the K-corrected\nabsolute luminosity corresponding to $r=21$ for a galaxy at the redshift of\nthe primary and lying on the red envelope of the intrinsic colour distribution\nis fainter than the lower luminosity limit of the bin. Thus only the nearest\nprimaries will contribute to the faintest luminosity bins of our satellite\nluminosity functions, and different numbers of primaries will contribute to\neach bin.  We follow an exactly analogous procedure when compiling stellar mass\nfunctions for satellites. \n\nFor each individual satellite luminosity\/stellar mass bin, this\ntreatment is equivalent to imposing an upper limit on primary\nredshift.  Thus there is a maximum volume which is surveyed for\nsatellites in the $j$th luminosity\/stellar mass bin which we\ndenote $V_{max,bin,j}$.\n\nOn the other hand, our primary sample is flux-limited at $r=16.6$, so\nfor brighter satellite bins where $V_{max,bin,j}$ is large,\nintrinsically faint primaries will not be visible to the redshift\nlimit. The effective volume surveyed is then $V_{max,pri,i}$, the\ntotal survey volume over which the $i$th primary would lie above the\nflux limit.  Note that because of K-corrections, this volume depends\non the intrinsic colour of the primary as well as its intrinsic\nluminosity. We present our final results in the form of the mean\nnumber of satellites per primary for a volume-limited primary sample\n\\begin{equation}\nN_{sat,j}=\\frac{\\Sigma_i N_{sat,i,j}\/V_{max,ij}}{\\Sigma_i 1\/V_{max,ij}}, \n\\label{eqn:numsat}\n\\end{equation}\nwhere \n\\begin{equation}\\label{eqn:error}\nV_{max,ij}= {\\rm min}\\big[V_{max,pri,i}, V_{max,bin,j}\\big].\n\\end{equation}\nThus, for satellite luminosity\/stellar mass bin $j$, we sum satellite\ncounts over all primaries $i$ that are within $V_{max,bin,j}$. At the\nbright end of the satellite luminosity function, we expect\n$V_{max,ij}=V_{max,pri,i}$ because satellites are less than 4.4\nmagnitudes fainter than their primaries and so can be seen around all\nprimaries.  For intrinsically faint satellites, however,\n$V_{max,ij}=V_{max,bin,j}$ because primaries can be seen to well\nbeyond the distance at which $r=21$ for the satellites.\n  \nWe apply similar selection criteria to our mock catalogues based on\nG11. Since we know the absolute magnitude, rest-frame colour and\nstellar mass of all galaxies, the background subtraction can be\ncarried out directly using rest-frame quantities. In addition, the\neffective depth of the satellite catalogue is the same for all\nprimaries so that all primaries can contribute to all satellite\nluminosity or stellar mass bins and we do not need any weighting in\norder to obtain proper volume-weighted statistics. Note that we do not\nuse any information about the redshift difference between primary and\napparent companion, so projection effects occur over $\\Delta z =\n50,000$~km\/s corresponding to the side of the Millennium Simulation\n``box''. Both for the simulation and for the real SDSS data we use a\nglobal background estimate based on the full survey in order to\nminimize the statistical uncertainty in the correction.  \n\nPrevious work often estimated a background density locally for each\nprimary \\citep[e.g.][]{1994MNRAS.269..696L,2011MNRAS.417..370G}. This not only\nsubstantially increases the noise, it also introduces a significant\nbias since galaxies are correlated on all scales and the\n``background'' fields are, in fact, expected to have faint galaxy\ndensities significantly above the mean. The extent of the bias is, in fact,\nstrongly dependent on the isolation criteria and can be of either sign\n\\citep[see, for example,][]{1976ApJ...205L.121F}. The large-scale uniformity of\nthe SDSS photometry is such that there appears to be no advantage to\nadopt a local estimate, provided all photometric quantities are\nproperly corrected for Galactic extinction. The accuracy of our method\nis confirmed by the accurate convergence to unity at large angular\nscale in the left panel of figure~\\ref{fig:photo} and by a variety of tests\nwhich we describe in detail in the Appendix.\n\n\\section{Luminosity and Mass Functions of Satellite Galaxies}\n\\label{sec:LFMF}\n\n\\subsection{Luminosity functions}\n\\begin{figure*}\n\\epsfig{figure=fig05.ps,width=0.9\\textwidth}\n\\caption{Luminosity functions in the $^{0.1}r$ band for satellites of primary\n  galaxies in five disjoint ranges of $\\log M_\\star\/M_\\odot$, as indicated in\n  the legend. Satellites are counted within a projected radius of 300~kpc,\n  except for the lowest mass range where we count within 170~kpc. In the left\n  panel, the data points connected by solid lines give observational results\n  for SDSS\/DR8 with error bars estimated by bootstrap resampling of the\n  primary sample. The solid black line is a fit of a\n  \\citet{1976ApJ...203..297S} function to the most massive bin, with the\n  characteristic luminosity and faint-end slope constrained to match those of\n  the SDSS field luminosity function. Dashed black lines are power-law fits\n  excluding one or two of the brightest points for each mass range, as\n  indicated by the line extent (see the text).  In the right panel, the small\n  symbols connected by solid lines show corresponding results for the G11\n  simulated galaxy populations.  Black dashed lines here show the mean\n  luminosity function for all simulated galaxies, renormalized to fit the\n  satellite data as in the left panel. The SDSS results are overplotted as\n  filled triangles for ease of comparison.}\n\\label{fig:LFall}\n\\end{figure*}\n\nIn figure~\\ref{fig:LFall} we present $^{0.1}r$-band luminosity functions for\nsatellites projected within 300~kpc of their primaries, except for the\nfaintest bin, where the halo virial radius is much smaller than 300~kpc and we\nestimate the luminosity function within 170~kpc in order to increase the\nsignal-to-noise. In the left panel, data points connected by solid lines show\nour observational results for SDSS\/DR8.  As indicated in the legend, colours\nencode the range in $\\log M_\\star\/M_\\odot$ of the primaries contributing to\neach luminosity function estimate.  Error bars are derived by bootstrap\nresampling the primary sample. At the faint end we lose higher redshift\nprimaries because of the apparent magnitude limit of our photometric\ncatalogue. We do not plot data for bins with fewer than eight primaries.  It\nis evident that satellite numbers increase strongly with primary mass.\n\nThe black solid line in the left panel shows a \\cite{1976ApJ...203..297S}\nfunction fit to the data for the most massive primaries. We have fixed the\ncharacteristic luminosity and faint-end slope to be those of the SDSS field\nluminosity function in $^{0.1}r$ \\citep{2009MNRAS.399.1106M}. The result is a\nmoderately good fit to the satellite data, although these appear steeper than\nthe field luminosity function at the faintest magnitudes. The satellite\nluminosity functions clearly become steeper for lower mass\nprimaries. Power-law fits to the faint-end data are shown as dashed lines in\nthe figure and give the slopes $\\alpha$ listed in the first row of\ntable~\\ref{tbl:slope}. The bright end of each function is cut off by our\nrequirement that every satellite be at least one magnitude fainter than its\nprimary. We therefore exclude one or two of the brightest points when making\nthese fits. Specifically, we find the median absolute magnitude\n$M_{\\mathrm{med}}$ for primaries in each mass range, and we include only\npoints for which the corresponding absolute magnitude bin lies entirely below\n$M_{\\mathrm{med}}+1$. The ranges fitted in each case are indicated by the\nextents of the dashed black straight lines.\\footnote{We have checked that the\nremaining points are indeed unaffected by rebinning our data as a function\nof the $r$-band absolute magnitude of the primaries. In this case we know\nexactly which bins are unaffected by our isolation criterion. When\nprimary absolute magnitude and stellar mass are matched appropriately, the\nresulting satellite luminosity functions match those of\nfigure~~\\ref{fig:LFall} very closely over the full range used to determine\nthe faint-end slope and are unaffected by the isolation criterion over this\nrange.}  The faint-end slope decreases from $\\alpha\\sim-1.2$ for $\\log\nM_{\\star,p}\/M_\\odot \\sim 11.5$ to $\\alpha\\sim-1.6$ for $\\log\nM_{\\star,p}\/M_\\odot \\sim 10.3$. \n\n\\cite{2011MNRAS.417..370G} divided their primaries into three\nluminosity ranges ($M_r=-23.0\\pm0.5$, $-23.0\\pm0.5$ and $-23.0\\pm0.5$)\nand compiled satellite luminosity functions in bins of\nsatellite-primary magnitude difference rather than satellite absolute\nmagnitude. They quote faint-end slopes of -1.45, -1.725 and -1.96 for\nthese three sets of primaries, with the fainter primaries having\nsteeper luminosity functions. For the largest measured magnitude\ndifferences ($\\Delta m\\sim 8$) they found similar numbers of\nsatellites independent of primary luminosity. These $\\alpha$ values\nare substantially more negative than ours, particularly for the\nfaintest primaries. In order to compare with their results, we adopt\nsimilar isolation criteria, we take the same ranges of primary\nluminosity, and we also accumulate satellite number as a function of\nmagnitude difference. Fitting the faint-end slope over the same\nsatellite magnitude range as in figure 7 of\n\\cite{2011MNRAS.417..370G}, we find $\\alpha$ values of -1.189, -1.376\nand -1.588, substantially shallower than those of\n\\cite{2011MNRAS.417..370G} and quite compatible with those we quote in\nTable~\\ref{tbl:slope}. Detailed tests show this inconsistency to be due \npartly to the local background subtraction scheme of \\cite{2011MNRAS.417..370G}\nwhich removes part of the signal {\\footnote{\\cite{2011MNRAS.417..370G} used \nphotometry from SDSS\/DR7  while our own tests, similar to those discussed in section \nA1 of the  Appendix, showed to suffer from substantially more serious systematics for \nfaint images close to brighter ones than is the case for the SDSS\/DR8 catalogues used \nhere.}, but mainly to the fact that they use model magnitudes K-corrected to $z=0$ for \ntheir primaries, rather than the $^{0.1}r$ Petrosian magnitudes which we use here.\n\n\n\\begin{table*}\n\\caption{Exponent $\\alpha$ of the  faint-end-slope for SDSS luminosity and mass functions}\n\\begin{center}\n\\begin{tabular}{lrrrrr}\\hline\\hline\nRange in primary $\\log M_\\star\/M_\\odot$ & \\multicolumn{1}{c}{11.7-11.4} & \\multicolumn{1}{c}{11.1-11.4} & \\multicolumn{1}{c}{10.8-11.1} & \\multicolumn{1}{c}{10.5-10.8} & \\multicolumn{1}{c}{10.2-10.5} \\\\ \\hline\nLuminosity function & -1.170 & -1.295 & -1.424 & -1.587 & -1.622 \\\\\nMass function       & -1.231 & -1.297 & -1.455 & -1.800 & -1.748 \\\\\n\\hline\n\\label{tbl:slope}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nIn the right panel of figure~\\ref{fig:LFall}, the small symbols joined by\nsolid lines show analogous results for the galaxy formation model of G11,\nbased on the Millennium and Millennium-II simulations.  Points brighter than\n$M_r=-18$ are MS data. At fainter magnitudes, resolution effects cause the MS\nto underestimate galaxy abundances and we take our data from the MS-II.  (The\ntwo simulations agree very well in the range $-18>M_r>-20$.) To facilitate\ncomparison, we replot the SDSS data from the left panel as filled\ntriangles. Agreement of model and observation is fair but far from\nperfect. The simulation overpredicts the number of satellites around the most\nmassive primaries by 25 to 50\\% for $M_r>-19.5$. In the two lower primary\nstellar mass ranges, the simulation underpredicts the number of satellites by\n20 to 30\\% . As we will see below, the latter discrepancy reflects a problem\nwith the colours of the simulated satellites rather than with their stellar\nmasses. The black dashed lines in this panel are the ``field'' luminosity\nfunction for the full simulations renormalized to fit the satellite data for each\nprimary mass range. Here also there is a trend for the faint-end slope to be\nsteeper for satellites than in the field, but the effect is much less marked\nthan for the SDSS data. Furthermore the variation of faint-end slope with\nprimary mass seen in the SDSS is weak or absent in the simulation data.\n\nMany previous papers have investigated the shape and faint-end slope of\ngroup\/cluster luminosity functions \\citep[e.g.][]{2001A&A...367...59P,\n  2002AJ....123.1807G,2003ApJ...591..764C,2003MNRAS.342..725D,2005MNRAS.360..727A,\n2005A&A...433..415P,2006A&A...445...29P,2006ApJ...650..137Z,\n2009ApJ...699.1333H,2010MNRAS.401..941A,2011MNRAS.414.2771D}. Unfortunately\nthere is little consensus. Some authors found large differences between\ncluster and field luminosity functions, while others found the two to be quite\nsimilar. For example, by stacking SDSS data around the centres of clusters\ndetected in the Rosat All Sky Survey, \\cite{2006A&A...445...29P} obtained\ncluster luminosity functions with a very obvious steepening at faint\nmagnitudes.  This faint-end upturn appeared to be contributed primarily by\nearly-type galaxies.  In our most massive primary stellar mass bin\n($11.7>\\mathrm{log}_{10}M_\\star>11.4$), there is qualitatively similar\nbehaviour with an upturn at $M_r\\sim-18$ as in \\cite{2006A&A...445...29P} but\nthe steepening is much less dramatic in our data than in theirs. \nIn contrast, no evidence of an upturn at the faint end was found by \n\\cite{2010MNRAS.401..941A} and \\cite{2011MNRAS.414.2771D} in galaxy clusters.\n\n\\subsection{Stellar mass functions}\n\n\\begin{figure*}\n\\epsfig{figure=fig06.ps,width=0.9\\textwidth}\n\\caption{Similar to figure~\\ref{fig:LFall} but showing stellar mass functions\n  for satellites projected within 300~kpc (or 170~kpc) of their primaries,\n  both for the SDSS (left panel) and for the G11 simulations (right panel with\n  the SDSS data repeated as filled triangles). As in figure~\\ref{fig:LFall},\n  the primaries are grouped into five disjoint ranges of $\\log\n  M_\\star\/M_\\odot$, as indicated by the colours.  A black solid line in the\n  left panel is a renormalized version of the ``field'' stellar mass function\n  of \\citet{2009MNRAS.398.2177L} overplotted on the data for the highest mass\n  primaries, while dashed black lines show a power law fit to each mass\n  function estimate. In the right panel the dashed black lines are fits of the\n  satellite data to renormalized versions of the stellar mass function of the\n  simulation as a whole.}\n\\label{fig:MFall}\n\\end{figure*}\n\nFigure~\\ref{fig:MFall} is similar to figure~\\ref{fig:LFall}, but shows\nstellar mass functions for satellites both in the SDSS (left panel)\nand in the G11 simulations (right panel with the SDSS data repeated as\nfilled triangles). The strong dependence of satellite number on\nprimary mass is again evident. The black solid line overplotted on the\nmost massive bin is a Schechter function fit with characteristic mass\nand low-mass slope fixed to the ``field'' values of\n\\cite{2009MNRAS.398.2177L}. The observed satellite stellar mass\nfunctions are again steeper than the corresponding field function at\nthe low-mass end. The black dashed lines in the left panel are\npower-law fits to the observational data. The corresponding\nfaint-end-slopes are given in the second row of table~\\ref{tbl:slope}.\nHere also we have ignored the few brightest points in each\nestimate. As before the extent of each dashed black line indicates the\npoints actually used in the fit.  The steepening of the satellite\nstellar mass functions with decreasing primary mass is even stronger\nthan was the case for the satellite luminosity functions.\n\n\nIn the right panel of figure~\\ref{fig:MFall}, the G11 points are\ntaken from the MS at $\\log M_\\star\/M_\\odot >9.5$ and from the MS-II at\nlower mass. Fits to the field stellar mass function from the\nsimulations are shown as dashed lines and indicate no significant\nshape difference between the two. Comparison with the SDSS data shows\nthat the overprediction of the satellite luminosity function for the\nmost massive primaries persists in very similar form in their stellar\nmass function, The underprediction found for the two lower primary\nmass ranges has gone away, however, indicating that the discrepancy\nwas due primarily to the colours of the simulated satellites, rather\nthan to their stellar masses. The simulation does not reproduce the\nclear steepening of the SDSS satellite mass functions with decreasing\nprimary mass.  Any effect in this direction is very weak.\n\nTo investigate further the origin of the observed steepening with\ndecreasing primary mass, we take all spectroscopic galaxies brighter\nthan 16.6 in the $r$-band, without applying any isolation criterion,\nand calculate the mass function of surrounding satellites within 300\n(or 170)~kpc in exactly the same way as for our isolated primaries.\nResults are shown as solid curves in figure~\\ref{fig:background}. In\nthis case there is no steepening with decreasing primary mass -- the\ndifferent colour curves are more or less parallel to each other and to\nthe field stellar mass function.  Apparently, the steepening is caused\nby our isolation criteria -- isolated galaxies have fewer lower mass\nneighbours than typical galaxies.  The suppression is stronger for\nmore massive neighbours, steepening the satellite mass functions of\nisolated primaries. Notice that for primaries with $\\log\nM_\\star\/M_\\odot > 10.8$, the mean numbers of low-mass companions are\nsimilar for isolated and for typical objects, but that lower mass\nprimaries have substantially fewer such companions if they are\nisolated.  This is because many of the lower mass non-isolated\n''primaries'' are, in fact, satellites themselves, and their low-mass\n``companions'' are fellow satellites within the larger system.  It is\nquite interesting that the abundance of relatively small satellites is\nstrongly affected by the presence or absence of a nearby galaxy\ncomparable in luminosity to the primary, particularly since the\nobserved trends are not fully reproduced by the simulation.  This suggests\nthat dwarf galaxy formation may be influenced by nearby giants in a\nway which the simulation does not represent.\n\n\n\\begin{figure*}\n\\epsfig{figure=fig07.ps,width=0.9\\textwidth}\n\\caption{Similar to figure~\\ref{fig:MFall}, but for red primaries.\n  See figures \\ref{fig:prop_dis} and \\ref{fig:prop_dis2} for the\n  colour cuts separating red and blue primaries.}\n\\label{fig:MFred}\n\\end{figure*}\n\n\\begin{figure*}\n\\epsfig{figure=fig08.ps,width=0.9\\textwidth}\n\\caption{Similar to figure~\\ref{fig:MFall}, but for blue primaries. In the\n  right panel, the red dots connected by a curve stop at\n  $\\mathrm{log}_{10}M_\\star=9.5$, because the Millennium-II Simulation has\n  fewer than eight blue primaries more massive than $\\log M_\\star\/M_\\odot=11.4$.}\n\\label{fig:MFblue}\n\\end{figure*}\n\nIt is interesting to see whether satellite galaxy populations depend on the\ncolour of the primary galaxy as well as on its stellar\nmass. Figures~\\ref{fig:MFred} and \\ref{fig:MFblue} show the satellite mass\nfunctions surrounding red and blue primaries respectively, where the primary\npopulations have been split at the colours indicated in figures\n\\ref{fig:prop_dis} and \\ref{fig:prop_dis2}. The black dashed lines in the left\npanels are power-law fits, but in each case the slope is fixed to be that\nfound for all primaries in the relevant stellar mass bin\n(table~\\ref{tbl:slope}). In the right panel of figure~\\ref{fig:MFblue} the red\ncurve stops at $\\log M_\\star\/M_\\odot=9.5$ because the MS-II contains fewer\nthan eight blue primaries more massive than $\\log M_\\star\/M_\\odot=11.4$ and\nthe MS population is affected by numerical resolution at lower satellite mass.\nFor red primaries the SDSS and G11 data agree quite well, apart from the slope\ndiscrepancy and a residual overprediction of the abundance of faint satellites\naround high-mass primaries.  For blue primaries, the differences are bigger\nbut in the highest mass bin this could reflect the small number of SDSS\nprimaries and the correspondingly large observational error bars. The\ndiscrepancy for primaries in the stellar mass range $10.1 > \\log\nM_\\star\/M_\\odot>10.8$ is smaller but more significant, particularly since\nthere is good agreement in this mass range for red primaries. Interestingly, a\ncomparison of figures \\ref{fig:MFred} and \\ref{fig:MFblue} shows that the\namplitude of the satellite luminosity function is higher around red primaries\nthan around blue primaries of the same stellar mass.  We analyze this result\nin more detail in the following subsections.\n\n\\subsection{Satellite abundance as a function of primary\nstellar mass and colour}\n\n\\begin{figure*}\n\\centerline{             \\epsfig{figure=fig09a.ps,width=0.45\\textwidth}\n  \\epsfig{figure=fig09b.ps,width=0.45\\textwidth} }\n\\caption{{\\bf Left:} Mean number of satellites in the stellar mass range $10.0\n  >\\log M_\\star\/M_\\odot>9.0$ as a function of primary stellar mass.  Black,\n  red and blue points refer to all, to red and to blue primaries,\n  respectively. For the points at lowest primary mass, satellite counts were\n  accumulated within 170~kpc, whereas for all other primary masses they were\n  accumulated within 300~kpc.  The top panel gives observational results for\n  the SDSS while the bottom panel gives corresponding results for the G11\n  galaxy formation simulations.  The SDSS result for all primaries is\n  re-plotted as a dashed curve in the bottom panel in order to facilitate\n  comparison.  {\\bf Right:} Host halo mass distributions for red and blue\n  primaries in the simulated catalogues, split into the same primary stellar\n  mass ranges as in the left-hand plots.  }\n\\label{fig:number}\n\\end{figure*}\n\nIn order to better display how the abundance of satellites depends on\nthe stellar mass and colour of the primary, we use the power-law fits\nshown as dashed lines in figures~\\ref{fig:MFall}, ~\\ref{fig:MFred}\nand~\\ref{fig:MFblue} to predict the mean number of satellites per\nprimary in the stellar mass range $10.0>\\log M_\\star \/ M_\\odot >9.0$\nand at projected separation $r_p<300$~kpc ($r_p<170$~kpc for the\nlowest mass primaries).  This has the advantage of producing a robust\nmeasure of satellite abundance which is little affected either by\nselection-induced cut-offs (most important for low-mass and red\nprimaries) or by incompleteness (most important for high-mass and blue\nprimaries). The results are shown in the left panels of\nfigure~\\ref{fig:number}, where black dots and lines give results for\nall primaries, while red and blue dots and lines give results for red\nand blue primaries, respectively.  The top panel presents results for\nthe SDSS and the lower panel results for the G11 simulations.  The\nSDSS result for all primaries is repeated in the lower panel, showing\nthat the simulation overpredicts the number of satellites in this mass\nand projected radius range both for the highest mass and for the\nlowest mass primaries. Note that because the low-mass slopes differ in\nsimulation and observation, the result for low-mass primaries depends\non the satellite mass range chosen for the comparison.\n\nAt high mass the black and red curves in figure~\\ref{fig:number} are close to\neach other, reflecting the fact that the fraction of red primaries is large\n(see figure~\\ref{fig:frac}). At the highest mass, the blue curve indicates a\nconsistent number of satellites around blue primaries, although with\nconsiderable uncertainty because such primaries are rare. At somewhat lower\nmass, however, blue primaries have significantly fewer satellites than red\nprimaries {\\it of the same stellar mass}, both in the SDSS data and in the\nsimulation. This is a primary result of our paper. The effect is a factor of\ntwo to three in satellite abundance for primaries with $\\log\nM_\\star\/M_\\odot\\sim 11$. In the SDSS data there is some indication that the\nthe colour dependence may get smaller again for lower mass primaries, but this\ndoes not happen in the simulation, where there is still more than a factor of\ntwo difference for $M_\\star\\sim 2\\times 10^{10}M_\\odot$. Overall, the\ndifferences appear somewhat larger in the model than in the real data.  \n\nThe cause of this effect in the G11 simulation is easy to track down. In the\nright-hand panels of figure~\\ref{fig:number} we plot histograms of host halo\nmass for isolated galaxies as a function of their stellar mass and colour. (As\nshown in figure~\\ref{fig:frac}, almost all isolated galaxies in the simulation\nare the central galaxies of their haloes.)  For all except the highest stellar\nmass range, red primaries have significantly more massive dark haloes than\nblue ones. The shift between the peaks of the two distribution is an order of\nmagnitude for primaries with $11.4>\\log M_\\star\/M_\\odot\\sim 11.1$, dropping to\na factor of two for $10.5>\\log M_\\star\/M_\\odot\\sim 10.2$. In the simulation\nred primaries have more satellites because they live in more massive haloes. A\ndirect indication that the same may hold for real galaxies comes from the\ngalaxy-galaxy lensing study of \\cite{2006MNRAS.368..715M}. By combining their\nSDSS lensing data with HOD modeling, these authors concluded that red\ngalaxies have more massive haloes than blue ones for $\\log M_\\star\/M_\\odot >\n11$. At lower central galaxy masses their results appear consistent with no\noffset, although the error bars are large (see their figure 4).\n\nThe fact that both the number of satellites and the mass of the\nassociated dark halo depend not only on the stellar mass of the\nprimary galaxy but also on its colour contradicts the assumptions\nunderlying many HOD or abundance matching schemes for interpreting\nlarge-scale galaxy clustering. Such a dependence could be included in\nmore complex versions of at least the former, but would require\nadditional parameters and additional observational data to constrain\nthem\n\\citep[e.g.][]{2009MNRAS.398..807S,2009MNRAS.399..878R,2009MNRAS.392.1080S}.\n\n\n\\begin{figure}\n\\epsfig{figure=fig10,width=0.45\\textwidth}\n\\caption{Satellite mass functions within $r_p=300$~kpc, split according to\n  {\\it halo} mass, for dark haloes in the G11 galaxy formation simulation. The\n  colours refer to different ranges of $\\log M_{\\rm halo}\/M_\\odot$ as indicted\n  in the legend. For each halo mass range, satellite stellar mass\n  functions are shown separately for haloes with red (solid lines) and blue\n  (dashed lines) central galaxies.}\n\\label{fig:halo}\n\\end{figure}\n\nWithin the simulation it is possible to check whether halo mass is the only\nfactor responsible for the difference in satellite abundance between red and\nblue primaries. In figure~\\ref{fig:halo} we present stellar mass functions for\nsatellites of isolated galaxies as a function of the {\\it halo} mass of the\nprimary. As before these are compiled for satellites projected within\n$r_p=300$~kpc, The halo mass ranges for this plot are chosen to correspond\nroughly to the primary stellar mass ranges in previous figures.  For each halo\nmass range, the mass functions are also split according to the colour of the\nprimary galaxy, with solid and dashed lines referring to results for red and\nblue primaries respectively.  At low satellite mass there is excellent\nagreement between the solid and dashed curves, indicating that satellite\nabundance does not depend on central galaxy colour at fixed halo mass.  For the\nmost massive haloes there are few blue primaries in the MS-II, so the red\ndashed curve is quite noisy below $\\log M_\\star\/M_\\odot=9.5$. For massive\nsatellites there are obvious discrepancies between the dashed and solid\ncurves, but these result from our sample definition. At given halo mass, red\nprimaries have smaller stellar masses and so substantially lower luminosities\nthan blue ones. Our isolation criteria then imply a correspondingly lower\nupper limit on the stellar mass of satellites for the red primaries. The\neffect is largest for the lowest mass haloes.\n\n\\cite{2007MNRAS.382.1901S} used the galaxy formation simulation of\n\\cite{2006MNRAS.365...11C}, also based on the Millennium Simulation,\nto study the relation between primary luminosity\/stellar mass and\nsatellite velocity dispersion, which should be a good diagnostic of\nhalo mass. Although they found a strong dependence of velocity\ndispersion on galaxy colour at fixed primary luminosity, this\ndependence almost vanished at fixed primary stellar mass. This appears\nto contradict our results from the G11 simulation. At $\\log\nM_\\star\/M_\\odot\\sim 11$, where we find the biggest difference in\nsatellite abundance between red and blue primaries, almost a factor of\nthree, the difference in velocity dispersion in their figure 13 is at\nmost 20\\%, corresponding to a factor of at most 1.7 in halo mass\n(since $M_h\\propto \\sigma^3$).  The discrepancy could result from the\ndifferent isolation criteria adopted in the two studies, from\ndepartures from a straightforward relation between 3-D velocity\ndispersion within $r_{200}$ (the quantity considered by\n\\cite{2007MNRAS.382.1901S}), halo mass and projected satellite count\nwithin 300~kpc (the quantity considered here), or from differences\nbetween the two galaxy formation models.\n\nSo far we have characterized the abundance of satellites by the count\nwithin $r_p = 300$~kpc (or $r_p=170$~kpc for the lowest mass bin in\nfigure~\\ref{fig:number}) In order to better understand the relation\nwith halo mass, it is useful instead to consider the count within the\nvirial radius of the haloes, which we define as $r_{200}$, the radius\nof a sphere within which the mean mass density is 200 times the\ncritical value.  We obtain such a measure using the equation,\n\\begin{equation}\nN(r<r_{200})=N_p(r_p < 300~{\\rm kpc})\\times\n\\frac{M(r<r_{200})}{M_p(r_p< 300~{\\rm kpc})},\n\\label{eqn:abunvir}\n\\end{equation}\nwhere $M_p$ and $N_p$ are projected quantities and $M$ and $N$ the\ncorresponding 3-dimensional quantities. The projected abundance $N_p(r_p <\n300~{\\rm kpc})$ is taken directly from figure~\\ref{fig:number}, and we\ncalculate the mass ratio on the {\\it rhs} of equation~\\ref{eqn:abunvir}\nassuming an NFW profile . The\nhalo radius $r_{200}$ is known for each simulated galaxy, but can only be\ninferred indirectly for the SDSS objects. We assume that the mean halo mass at\ngiven stellar mass is the same in the observations as in the G11\nsimulations. The mass ratio also depends weakly on halo concentration which we\ntake from the model of \\cite{2009ApJ...707..354Z}.\n\nFor each primary stellar mass bin in figure~\\ref{fig:number} we can\nthen estimate a mean halo mass $\\langle M_{200}\\rangle$. We take the\nmean satellite number per primary given by equation~\\ref{eqn:abunvir}\nand divide by this mean halo mass to obtain the abundance per unit\nhalo mass of satellites in the stellar mass range $10.0>\\log\nM_\\star\/M_\\odot > 9.0$. Finally, we divide this quantity by the\nabundance per unit (total) mass of galaxies in this same stellar mass\nrange in the Universe as a whole, taken from\n\\cite{2009MNRAS.398.2177L} for the SDSS data and from the G11\nsimulation as a whole for the mock data. The result is a measure of\nthe formation efficiency of low-mass galaxies as a function of their\npresent environment, as characterized by mass of the halo in which\nthey live.  Figure~\\ref{fig:num_vir} shows this efficiency as a\nfunction of mean halo mass for all primaries in the five stellar mass\nbins of figure~\\ref{fig:number}. There are three significant points to\ntake from this plot: (i) the formation efficiency of low-mass\nsatellite galaxies varies rather little with the mass of the halo in\nwhich the galaxies are found today; (ii) in massive haloes this\nefficiency is about 50\\% larger than the efficiency for forming such\ngalaxies in the universe as a whole (remember that, globally, about\nhalf the galaxies in this stellar mass range are satellites); (iii)\nfinally there is fair agreement between the formation efficiencies in\nthe simulation and in the real universe, although this is in part due\nto our use of the simulation to assign halo masses to the observed\ngalaxies.\n\n\n\n\\begin{figure}\n\\epsfig{figure=fig11.ps,width=0.45\\textwidth}\n\\caption{Mean number of satellites in the stellar mass range\n  $10.0>\\log M_\\star\/M_\\odot>9.0$ per unit total halo mass, relative\n  to the mean abundance per unit total mass of such galaxies in the\n  Universe as a whole. Isolated primary galaxies are grouped into the\n  same five stellar mass bins used in figure~\\ref{fig:number}. Results\n  for the G11 simulations and for the SDSS are shown by solid and\n  dashed curves respectively. The mean halo mass has been calculated\n  directly for each bin in the simulation data. Each SDSS primary is\n  assigned a halo mass using the simulation relation between mean halo\n  mass and primary stellar mass for all primaries. This plot hence\n  shows the efficiency of low-mass galaxy formation as a function of\n  present-day halo mass in units of the overall efficiency in the\n  Universe as a whole.}\n\\label{fig:num_vir}\n\\end{figure}\n\n\n\n\\section{Satellite colour distributions}\n\\label{sec:colour}\nSo far we have studied the abundance of satellites as a function of the\nstellar mass and colour of their primary and as a function of their own\nluminosity and stellar mass. In this section we study how the {\\it colours} of\nsatellite galaxies depend on the properties of their primaries.\nFigures~\\ref{fig:colour_obs} and~\\ref{fig:colour_mock1} show cumulative colour\ndistributions for satellites in the SDSS and in the G11 catalogues\nrespectively, as a function of the stellar mass and colour of their\nprimary. The distributions are for satellites in two different stellar mass\nranges, as indicated by the labels above the relevant panels, and refer to all\nsatellites projected within 300~kpc.  The top, middle and bottom panels in\neach column refer to satellites of all, of red and of blue primaries\nrespectively, while the different colours of the curves in each panel encode\nthe stellar mass range of the primary galaxies. The black curves which repeat\nin all the panels of each column give the colour distributions for field\ngalaxies in the same stellar mass range as the satellites (calculated for SDSS\nfrom all galaxies in the NYU-VAGC with $r<17.6$, and for G11 from all galaxies\nwithin the simulation volume). The dashed horizontal line is merely a\nreference to facilitate identification of the median colour.\n\nA number of systematic trends are evident in these plots. Concentrating first\non the observational results in figure~\\ref{fig:colour_obs}, we see that more\nmassive primaries have redder satellites within 300~kpc (in every panel the\ncurves are ordered cyan-blue-green-red from top to bottom), that low-mass\nsatellites are bluer than high-mass ones (the curves in the right panels are\nalways bluer than the corresponding curves in the left panels), satellites are\nsystematically redder than field galaxies of the same mass, except possibly\nfor the lowest mass primaries (the coloured curves almost always lie below the\ncorresponding black curves), and red primaries have redder satellites than\nblue primaries of the same stellar mass (every coloured curve in the lowest\npanels is bluer than the corresponding curve in the middle panel). This last\ntrend is the ``galactic conformity'' effect pointed out by\n\\cite{2006MNRAS.366....2W}.\n\nIf we now compare with the simulation results in figure~\\ref{fig:colour_mock1}\nwe see that the same four systematic trends are present. More massive\nprimaries have redder satellites; lower mass satellites are bluer; satellite\ngalaxies are redder than field galaxies of the same stellar mass; and red\nprimaries have redder satellites than blue primaries of the same stellar mass.\nHowever, there is an obvious discrepancy in that simulated satellites are\nsystematically redder than observed satellites. This is true for all primary\nand satellite masses, but is particularly marked for lower mass and red\nprimaries, and for lower mass satellites. Clearly, the theoretical model of\nG11 suppresses star formation much more effectively in such satellites than is\nthe case in the real universe. This echoes the conclusions of\n\\cite{2006MNRAS.366....2W} about the earlier models of \\cite{2007MNRAS.375....2D}.  \nThe excessive reddening of the simulated satellite population reduces but does\nnot eliminate all the other trends mentioned above.\n\nThe galactic conformity phenomenon has been discussed in a number of previous\npublications\n\\citep[e.g.][]{2006MNRAS.366....2W,2008MNRAS.389...86A,2010MNRAS.409..491K,\n2011MNRAS.417.1374P}. \\cite{2006MNRAS.366....2W} showed that, among groups of\ngiven luminosity (which they considered a proxy for halo mass), those with an\nearly-type central galaxy have a larger fraction of early-type satellites.\nThey considered several physical processes which might be responsible for this\n(halo and\/or galaxy mergers, ram-pressure stripping, strangulation,\nharassment...), focusing on whether these processes could alter galaxy\nmorphology. However, as discussed in some detail by \\cite{2004MNRAS.353..713K}\nand re-emphasized in the context of galactic conformity by\n\\cite{2010MNRAS.409..491K}, it is important to separate star formation\nactivity, stellar mass and galaxy structure when analyzing the influence of\nenvironment on galaxy properties. Typical classifications into ``early'' and\n``late'' types mix aspects of all of these.  The conformity effects we see\nhere are for given central galaxy stellar mass (rather than luminosity), are\nwithin a fixed projected radius (300~kpc), and refer specifically to the {\\it\n  colours} of satellites and primaries. It seems possible that they could be\ndue at least in part to the tendency for red centrals to have more massive\nhaloes than blue ones, together with the trends for satellites to get redder\nwith increasing halo mass and decreasing $r\/r_{200}$.\n\nThis can be checked directly for the simulated galaxy catalogues of G11.  In\nfigure~\\ref{fig:colour_halo} we again show cumulative satellite colour\ndistributions for two different satellite mass ranges and for all, for red and\nfor blue primaries, but now in bins of host halo mass rather than of primary\nstellar mass.  In this plot, the colour distributions are calculated for all\nsatellites projected within the halo virial radius rather than for a fixed\nprojected radius of 300~kpc. The halo mass ranges have been chosen to\ncorrespond approximately to the primary stellar mass ranges we have been using\nin previous plots. For the two high-mass bins, the colour distributions show\nno significant dependence on the colour of the central galaxy, but for the two\nlow-mass bins the dependence on primary colour, while smaller than in\nfigure~\\ref{fig:colour_mock1}, is clearly still present.  Thus there must be\nphysical processes in the model which contribute to the ``galactic\nconformity'' phenomenon in addition to those which result in red primaries\nhaving more massive haloes than blue primaries of the same stellar mass.\n\nWe have analyzed our simulation to identify candidates for these additional \nprocesses. In figures~\\ref{fig:bh} and~\\ref{fig:time} we plot distributions \nfor the two which show the strongest trends. \nFigure~\\ref{fig:bh} shows the distributions of black hole\nmass and of hot gas mass for haloes in these same four mass ranges, split\naccording to the colour of the central galaxy. Figure~\\ref{fig:time} shows\nsimilar plots for the cumulative distributions of infall redshift (defined as\nthe redshift when a satellite last entered the virial radius $r_{200}$ of the\nmain progenitor of its halo) for satellites with stellar mass above\n$10^9M_\\odot$. In all cases the distributions depend strongly on central\ngalaxy colour for haloes in the two lower mass ranges where galactic conformity\neffects are substantial, but at most weakly for higher mass haloes where these\neffects are absent.  For $\\log M_{\\rm h}\/M_\\odot < 12.8$, haloes with red\ncentral galaxies have more hot gas, more massive central black holes and\nearlier satellite infall redshifts than haloes of the same mass with blue\ncentral galaxies. Notice that the black hole mass distribution for blue\ncentrals is bimodal in the lower left panel of figure~\\ref{fig:bh}.  Clearly,\nthere is a transition in this halo mass range between blue centrals with\nhigh-mass black holes (about 0.3\\% of their stellar mass, as in the upper\npanels) and blue centrals with low-mass black holes (roughly an order of\nmagnitude smaller as a fraction of stellar mass, as in the lower right\npanel). The hot gas fractions are similar to the overall cosmic baryon\nfraction ($\\sim 17\\%$) for haloes with red central galaxies and about a factor\nof two smaller in the lower mass haloes with blue centrals. The transition\nhalo mass is approximately the mass at which both observational and simulated\nsamples of isolated galaxies become dominated by red objects (see\nfigure~\\ref{fig:frac}).\n\nGiven the modeling assumptions of G11, these systematic differences mean\nthat, for $\\log M_{\\rm h}\/M_\\odot < 12.8$, haloes with red central galaxies\nhave gas cooling rates which are twice but ``radio mode'' heating rates which\nare 20 times those of similar mass haloes with blue central galaxies. Thus,\ntheir central galaxies are clearly red because feedback has quenched their\ngrowth, and this is why their stellar masses are smaller than those of the\nblue centrals which have continued to grow to the present day. The redness of\nthe satellite population in haloes with red centrals can be traced to the\nfacts that the satellites are accreted earlier and orbit through a denser hot\ngas medium.  Thus both tidal and ram pressure stripping processes are more\neffective, and the satellites have had longer to exhaust any remaining\nstar-forming gas. Finally, the earlier assembly indicated by the higher\nsatellite infall redshifts presumably explains why these particular haloes\nhave red centrals, bigger black holes and more hot gas.\n\nThus, at least in the models, it is clear why there is galactic conformity at\ngiven halo mass. The more easily observable galactic conformity at given\ncentral stellar mass is predicted to be stronger as a result of the\ncombination of these effects with the tendencies for for red centrals to have\nmore massive haloes at given stellar mass than blue ones, and for more massive\nhaloes to have redder satellites. Yet stronger conformity effects are\nexpected at given central galaxy luminosity, since blue central galaxies have\nsmaller stellar masses (and thus even smaller halo masses) than red central\ngalaxies of the same luminosity.\n\n\\begin{figure}\n\\epsfig{figure=fig12.ps,width=0.45\\textwidth}\n\\caption{Cumulative colour distributions for satellites projected within\n  300~kpc of their primary as a function of primary stellar mass (indicated by\n  line colour) for two ranges of satellite mass (left and right\n  columns with the range indicated by the label above each column) and for\n  all, for red and for blue primaries (upper middle and lower panels in each\n  column).  Black lines show the cumulative colour distribution for a\n  volume-limited sample of field galaxies derived from the SDSS DR7\n  spectroscopic sample with $r<17.6$.}\n\\label{fig:colour_obs}\n\\end{figure}\n\n\n\\begin{figure}\n\\epsfig{figure=fig13.ps,width=0.45\\textwidth}\n\\caption{Similar  to figure~\\ref{fig:colour_obs},  but for the simulated\n  galaxy catalogues of G11. The black lines here are the colour distributions\n  for all galaxies in the specified stellar mass ranges within the full\n  simulation volume.}\n\\label{fig:colour_mock1}\n\\end{figure}\n\n\\begin{figure}\n\\epsfig{figure=fig14.ps,width=0.45\\textwidth}\n\\caption{Colour distribution of satellites projected within the halo virial\n  radius $r_{200}$ of all primaries, of red primaries and of blue primaries\n  (top, middle and bottom rows) as a function of host halo mass (indicated by\n  line colours corresponding to the ranges of $\\log M_{\\rm h}\/M_\\odot$ given\n  in the legend) in the simulated galaxy catalogues of G11. Results are again\n  shown for two ranges of satellite stellar mass.}\n\\label{fig:colour_halo}\n\\end{figure}\n\n\n\\section{Summary and Conclusions}\n\\label{sec:conclusion}\nWe have used a photometric catalogue of SDSS\/DR8 galaxies brighter\nthan $r=21$ to study the satellite populations of 41271 \n isolated galaxies with $r<16.6$ selected from the\nSDSS\/DR7 spectroscopic catalogue.  In particular, we have studied how\nthe abundance of satellites as a function of luminosity, stellar mass\nand colour depends on the stellar mass and colour of the central\ngalaxy.  Our study differs from other recent SDSS-based studies of\nsatellite galaxies \\citep{2011AJ....142...13L,2011MNRAS.413..101G} in\nthe size of the sample analyzed, in our primary focus on systematics\nas a function of stellar mass, and in our detailed comparison with the\npredictions of simulations of the evolution of the galaxy population\nin the concordance $\\Lambda$CDM cosmology. In general, our results\nconfirm and extend those obtained earlier, but the comparison with\nsimulations allows us to identify the likely physical cause of most of\nthe effects we see, and to isolate those which do not have a natural\nexplanation within our current theory of galaxy formation.\n\nOur observational samples and analysis procedures allow us to measure the\nproperties of the satellite population in an unbiased way down to absolute\nmagnitudes $M_{^{0.1}r}\\sim-14$ and stellar masses $\\log M_\\star\/M_\\odot\n\\sim8$. Our main observational conclusions are as follows:\n\n\\begin{itemize}\n\n\\item Satellite luminosity and stellar mass functions have shapes consistent\n  with those of the general field galaxy population only around the highest\n  stellar mass primaries $\\log M_\\star\/M_\\odot > 11.4$. These are all\n  brightest cluster galaxies. For lower mass primaries, these functions become\n  progressively steeper, even after accounting for the bright-end cut-off\n  induced by our isolation criteria. This steepening is more marked for the\n  stellar mass functions than for the luminosity functions because observed\n  satellites get bluer as their stellar mass decreases. \n\n\\item The mean abundance of satellites increases strongly with primary \n  stellar mass, approximately as expected if the number of satellites is \n  proportional to dark halo mass.\n\n\\item For $\\log M_\\star\/M_\\odot > 10.8$, red primaries have more\n  satellites than blue primaries of the same stellar mass. The effect exceeds\n  a factor of two for $\\log M_\\star\/M_\\odot \\sim 11.2$. This is reminiscent of\n  the result of \\cite{2006MNRAS.368..715M} who showed that at high stellar\n  mass, red central galaxies have more massive haloes than blue ones.  This \n  trend could in part be due to colour dependent errors in deriving \n  stellar masses from the photometry, but such errors would need to be quite \n  large and to depend on primary mass.\n\n\\item Satellite galaxies are systematically redder than field galaxies of the\n  same stellar mass except around blue primaries with $\\log M_\\star\/M_\\odot <10.8$\n  where the satellites can have similar colours or even be systematically\n  bluer than the field (i.e. the galaxy population within a large\n  representative volume).\n\n\\item The satellite population is systematically redder around more massive\n  primaries, for more massive satellites and around red primaries. The first\n  effect reflects the fact that cluster galaxies are systematically redder\n  than field galaxies, the second echoes the trend found in the general\n  field, and the third is the galactic conformity effect pointed out by\n  \\cite{2006MNRAS.366....2W} but measured here for fixed central stellar mass\n  rather than fixed central luminosity.\n\n\\end{itemize}\n\nWe used criteria directly analogous to those employed on the SDSS to construct\nan isolated galaxy sample from the $z=0$ output of the publicly available\ngalaxy formation simulations of \\citet[][G11]{2011MNRAS.413..101G}. These are based on the\nMillennium and Millennium-II Simulations. The mock catalogue contains similar\nmagnitude, stellar mass, colour and position\/velocity information to the real\ncatalogue, but also contains information about dark haloes and the location of\nthe galaxies within them. Based on the mock sample, we conclude that $\\sim\n98\\%$ of our isolated galaxies are the central objects of their dark\nhaloes. Both in the SDSS and in the mock catalogue, the distributions of\nintrinsic properties for the isolated and parent populations are very similar.\nOnly the colour distributions shift slightly, with the isolated galaxies being\nsystematically bluer than the full population. A detailed comparison of the\nmock and real samples leads to the following conclusions\n\n\\begin{itemize}\n\n\\item At all primary masses, the luminosity and stellar mass functions of G11\n  satellites are quite similar both in shape and in normalization to those\n  measured for SDSS.  However, in the simulation there is only a weak tendency\n  for the satellite functions to be steeper than those of the field, or to be\n  steeper for lower mass primaries. This disagrees with the SDSS where the\n  steepening with decreasing primary mass is quite marked.\n\n\\item In the mock catalogues the abundance of satellites increases with\n  primary stellar mass almost in proportion to mean halo mass. For high-mass\n  haloes, the abundance per unit mass of satellite galaxies is about 1.5 times\n  the value for the universe as a whole.\n \n\\item For $\\log M_\\star\/M_\\odot <11.4$, red simulated primaries have more\n  satellites then blue ones of the same stellar mass. The effect is similar in\n  strength to that seen in SDSS but continues to lower primary mass. In the\n  simulation it is due entirely to red primaries having more massive haloes\n  than blue ones. At fixed {\\it halo} mass the abundance of faint satellites is\n  independent of the colour of the primary, but red primaries have lower\n  stellar masses because of their truncated star formation histories.\n\n\\item Satellite galaxies in the G11 simulations are systematically redder than\n  in the SDSS. The effect is particularly marked for lower mass\n  satellites and around lower mass primaries. The modeling improvements\n  introduced by G11 to address this issue are apparently insufficient to\n  fully solve it. Star formation is still terminated too early when galaxies\n  become satellites.\n\n\\item Despite this overall shift, the colours of simulated satellites depend\n  on the colour and stellar mass of their primary and on their own stellar\n  mass in very similar ways as in the SDSS. Satellites are systematically\n  redder if they are more massive, if their primary is more massive, and if\n  their primary is red. The first trend echoes that found for field galaxies,\n  while the second reflects the fact that more massive haloes contain redder\n  satellites. The third trend, the galactic conformity effect, is caused by\n  redder primaries having more massive haloes at fixed stellar mass, and\n  denser hot gas atmospheres (hence more effective ram-pressure stripping) at\n  fixed halo mass.\n\n\\end{itemize}\n\nSatellites go red in the G11 simulations because they lose their source of new\ncold gas. Once they fall within the virial radius of their host, their hot gas\nreservoirs are gradually removed by tidal and ram-pressure stripping and no\nnew material is added by infall. Star formation uses up the remaining cold\ngas and then switches off. Clearly this happens too quickly (see\n\\cite{2011MNRAS.416.1197W} and also \\cite{2007MNRAS.377.1419W} for an explicit\ndemonstration of how lengthening the relevant timescales can cure the problem)\nso improving the model will require changing the star formation assumptions to\nincrease the time to gas exhaustion (for example, by removing the threshold\ngas surface density for star formation) or providing new sources of fuel (for\nexample, by including the gas return from stellar evolution). Once this is\nfixed, it seems likely that the other trends of satellite colour with\nenvironment will be well matched.\n\nThe other clear discrepancy between the SDSS data and the simulation is the\nsteepening of satellite mass and luminosity functions as one goes to fainter\nprimaries. This is a relatively strong effect in the real data (and is visible\nalso in the analysis of \\cite{2011MNRAS.413..101G}) but does not occur at a significant\nlevel in the simulations. Thus, it must reflect not merely the statistics of\nhierarchical clustering in a $\\Lambda$CDM cosmology, but in addition some\ndifference in the star-formation histories of satellite galaxies living in\ndifferent mass host haloes. This may be related to the discrepancy noted by\nG11 -- comparisons with high-redshift data suggest that low-mass galaxies form\ntoo early in their simulations -- but this can only be checked by further\nsimulation work. It is clear that the detailed and relatively precise\nstatistical information provided by large-sample studies of satellite galaxies\nis useful for testing and refining our understanding of how galaxies form.\nIn a follow-up paper we will extend our current study by considering \nsatellite galaxy properties as a function of distance from the primary,\n\n\n\\begin{figure*}\n\\centerline{                \\epsfig{figure=fig15a,width=0.45\\textwidth}\n  \\epsfig{figure=fig15b,width=0.45\\textwidth} }\n\\caption{{\\bf Left:} Central black hole mass distributions for the same four\n  ranges of halo mass used in figure~\\ref{fig:colour_halo} and split according\n  to the colour of the central galaxy.  {\\bf Right:} Hot gas mass\n  distributions for the same four sets of haloes and again split according\n  to the colour of the central galaxy.}\n\\label{fig:bh}\n\\end{figure*}\n\n\n\\begin{figure}\n\\epsfig{figure=fig16.ps,width=0.45\\textwidth}\n\\caption{Infall time distributions for satellites in the same four ranges of\n  halo mass used in figure~\\ref{fig:colour_halo} and again split according to\n  the colour of the central galaxy.}\n\\label{fig:time}\n\\end{figure}\n\n\n\n\\section*{Acknowledgements}\nWe gratefully thank Rachel Mandelbaum for useful discussions about SDSS systematics, Xu Kong for supplying \nthe COSMOS mask and Cheng Li for discussions about details of the NYU-VAGC and photoz2 catalogues. \nWenting Wang is partially supported by NSFC (11121062, 10878001, 11033006,11003035), and by the CAS\/SAFEA \nInternational Partnership Program for Creative Research Teams (KJCX2-YW-T23). \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nIn stationary gravitational fields, a timelike Killing vector field exists such that the projection of the $4$-velocity of a free test particle on the Killing vector is a constant of the motion along the particle world line \\cite{N1}. This circumstance can be interpreted to mean that there is no net exchange of energy between the particle and the gravitational field.\nIt is therefore a problem of basic interest whether free test particles can gain or lose energy in dynamic, i.e. time-dependent, gravitational fields. We note that the exchange of energy between charges and the electromagnetic field is a fundamental feature of electrodynamics and leads to Joule's law \\cite{N2}.\n\nTo determine the energy of a test particle in the context of general relativity theory, it is necessary to refer the motion of free test particles to a set of reference observers. We take these fiducial observers to be the fundamental observers in spacetime, namely, those that are at rest in space.\nWe are thus interested in the peculiar velocities of free test particles relative to the class of comoving observers.\n\nThe issue of energy exchange and the nature of peculiar velocities has thus far been investigated mainly in a physical context that essentially corresponds to the modern version of the Kant-Laplace nebular hypothesis, in which the formation of elementary structure in the universe is due to the collapse of a spinning cloud of gas and dust. \nIn these studies one considers exact solutions of general relativity involving certain physically significant spacetimes in which the proper distance along one spatial axis---henceforth designated\n as the $z$ axis---decreases to zero as $t\\to \\infty$, while the proper distances along the corresponding $x$ and $y$ axes tend asymptotically to infinity. It has been demonstrated that in such spacetimes, the timelike geodesics have a universal behavior: relative to the reference observers, free test particles asymptotically (i.e., as $t\\to \\infty$) form a double-jet structure along the axis of collapse and the speeds of such bulk flows tend asymptotically to the speed of light \\cite{N3,N4,N5,N6}.\n\nThese {\\it cosmic jets} are idealized mathematical constructs and must be clearly distinguished from astrophysical jets that are persistent high-energy magnetohydrodynamic (MHD) bipolar outflows that are generally associated with configurations that have already undergone gravitational collapse. One may hypothesize that\nthe collapse process is accompained by a rather mild form of the cosmic double-jet pattern, a part of which is then confined and sustained over time by various MHD mechanisms characteristic of the particular astrophysical environment.\n\nTo summarize the results of previous investigations \\cite{N3,N4,N5,N6}, one may say that in a dynamic spacetime region in which asymmetric collapse\/expansion is taking place, free test particles are {\\it accelerated} relative to comoving observers along the collapsing direction, while they are {\\it decelerated} along the expanding direction. This is in agreement with the behavior of peculiar velocities in the standard cosmological models.\n\nIt might appear that the general behavior described above is the only one that is possible in general relativity. The purpose of the present work is to show that the behavior described above is {\\it not} unique. We elucidate a different  type of dynamic behavior involving a single-jet structure that is characteristic of certain propagating plane-wave spacetimes.\n\nThe plan of this paper is as follows. In Sec. II, we illustrate the nature of cosmic jets via a certain \\lq\\lq white-hole\" interpretation of the interior Schwarzschild-Droste black hole. Sections III and IV discuss the general behavior of timelike geodesics in different plane wave spacetimes. Section V contains a discussion of our results and the possibility of a connection between these exact solutions of the gravitational field equations.\n\n\n\n\n\\section {Geodesics of an axially collapsing cylindrical spacetime}\n\nConsider the standard form of the exterior Schwarzschild-Droste solution\n\\begin{equation}\n\\label{II1}\nds^2=-c^2 \\left( 1-2\\frac{GM}{c^2{\\mathcal R}} \\right)d {\\mathcal T}^2+\\frac{d{\\mathcal R}^2}{\\left( 1-2\\displaystyle\\frac{GM}{c^2{\\mathcal R}} \\right)}\n+{\\mathcal R}^2 (d\\theta^2+\\sin^2\\theta d\\phi^2)\\,,\n\\end{equation}\nwhere $M$ is the mass of the source. We assume that the spacetime metric has signature $+2$ and henceforth we set $c=1$. The timelike Killing vector $\\partial_{\\mathcal T}$ becomes null at ${\\mathcal R}=2GM$ and spacelike for ${\\mathcal R}<2GM$.\nIn this latter region of spacetime, let us introduce a new coordinate system $(t,r,\\theta , \\phi)$, where $t={\\mathcal R}$ and $r={\\mathcal T}$; moreover, we introduce a constant $T=2GM>0$. Then, metric~\\eqref{II1} inside the horizon takes the form\n\\begin{equation}\n\\label{II2}\nds^2=- \\frac{t}{T-t}  dt^2 +\\frac{T-t}{t}dr^2+t^2 (d\\theta^2+\\sin^2\\theta d\\phi^2)\\,,\n\\end{equation}\nwhich is usually ignored in favor of the complete analytic extension of the Schwarzschild solution \\cite{PK}.\nNext, we introduce cylindrical coordinates $(\\rho, \\phi, z)$ such that\n\\begin{equation}\n\\label{II3}\n\\rho=L\\sin \\theta\\,,\\qquad z=r\\,,\n\\end{equation}\nand $\\phi$ is the azimuthal angular coordinate as before.\nHere $L>0$ is a constant length. Thus we express metric~\\eqref{II2} as\n\\begin{equation}\n\\label{II4}\nds^2=- \\frac{t}{T-t}  dt^2 +\\frac{t^2}{L^2}\\left( \\frac{d\\rho^2}{1-\\rho^2\/L^2}+\\rho^2 d\\phi^2 \\right)+\\frac{T-t}{t}dz^2\\,.\n\\end{equation} \nThis is an axially collapsing cylindrical solution of the vacuum gravitational field equations. The cylindrical axis is elementary flat and the spacetime coordinates are admissible for $0<t<T$ and $0<\\rho<L$ \\cite{BCM}. This solution admits two spacelike commuting Killing vector fields ($\\eta=\\partial_\\phi$, $\\zeta=\\partial_z$) that are hypersurface orthogonal. There are in general four algebraic invariants of the curvature tensor for this type of Ricci-flat spacetime. In the case under study here, the only nonzero invariant is given by the Kretschmann scalar\n\\begin{equation}\n\\label{II5}\nK=R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}=12~\\frac{T^2}{t^6}\\,.\n\\end{equation}\nThus $t=0$ is a curvature singularity and the solution is clearly related to the Schwarzschild-Droste white hole \\cite{PK}.\n\nTo study the geodesics of the cylindrical spacetime under consideration here, it proves useful to introduce standard Cartesian coordinates $(x,y,z)$, where $x=\\rho \\cos \\phi$ and $y=\\rho \\sin \\phi$. Thus we work with\nthe following metric\n\\begin{equation}\n\\label{II6}\nds^2=-\\frac{t}{T-t}dt^2+\\frac{t^2}{D}[(L^2-y^2)dx^2+2xydx dy+(L^2-x^2)dy^2]+\\frac{T-t}{t}dz^2\\,,\n\\end{equation}\nwhere the coordinates are denoted by $x^\\alpha=(t,x,y,z)$, $T$ and $L$ are two arbitrary constant parameters and \n\\begin{equation}\n\\label{II7}\nD=L^2[L^2-(x^2+y^2)]\\,.\n\\end{equation}\nIn this case, $\\sqrt{-g}=t^2\/D$ and the coordinates are admissible for $0<t<T$, $|x|<L$, $|y|<L$ and $x^2+y^2<L^2$. Let us note here that the $(x, y)$ part of the metric in Eq.~\\eqref{II6} can be written as $(t\/L)^2 d\\ell^2$, where $d\\ell^2$ is the flat Euclidean 3D metric restricted to the surface of a sphere of radius $L$; see the Appendix. \nMoreover, \n metric~\\eqref{II6} is related to the Schwarzschild-Droste  {\\it white hole}  \nand admits two Killing vectors\n\\begin{equation}\n\\label{II8}\n \\eta=-y\\partial_x +x\\partial_y, \\qquad  \\zeta=\\partial_z\\,,\n\\end{equation}\nassociated with its cylindrical symmetry.\n\nWe fix our set of fiducial observers to be at rest with respect to the spatial coordinates $(x, y, z)$. Thus these observers have 4-velocity\n\\begin{equation}\n\\label{II9}\ne_{\\hat 0}=\\frac{1}{\\sqrt{-g_{tt}}}\\partial_t=\\sqrt{\\frac{T-t}{t}}\\partial_t\\,.\n\\end{equation}\nThe following three spatial vectors form the spatial frame of the orthonormal tetrad $e_{\\hat{\\alpha}}$ of our reference observers \n\\begin{eqnarray}\n\\label{II10}\ne_{\\hat 1} &=& \\frac{L}{t}\\sqrt{\\frac{L^2-x^2-y^2}{L^2-y^2}}\\partial_x\\,,\\nonumber\\\\\ne_{\\hat 2} &=& -\\frac{xy}{t\\sqrt{L^2-y^2}}\\partial_x +\\frac{\\sqrt{L^2-y^2}}{t} \\partial_y\\,, \\nonumber\\\\\ne_{\\hat 3} &=& \\sqrt{\\frac{t}{T-t}}\\partial_z\\,.\n\\end{eqnarray}\n\nLet us study the timelike geodesics of this metric, i.e. the curves with parametric equations $x^\\alpha=x^\\alpha (\\tau)$ and with unit (timelike) tangent vector\n$U^\\alpha=dx^\\alpha\/d\\tau := \\dot x^\\alpha$, where $\\tau$ is the proper time parameter.\nDecomposing $U=U^\\alpha \\partial_\\alpha $ on the tetrad frame $e_{\\hat \\alpha}$, i.e., $U=U^{\\hat \\alpha}e_{\\hat \\alpha}$,  \nleads to the identification of the (spatial) relative velocity  vector, $v^{\\hat a}$, $a=1,2,3$, and the associated Lorentz $\\gamma$ factor\n\\begin{equation}\n\\label{II11}\nU=\\gamma (e_{\\hat{0}}+v^{\\hat  a}e_{\\hat  a})\\,.\n\\end{equation}\nMoreover, the relation between the coordinate and frame components of $U$ are \n\\begin{eqnarray}\n\\label{II12}\nU^{\\hat 0}&=&\\gamma=\\sqrt{\\frac{t}{T-t}}~\\dot t\\,,\\nonumber\\\\\nU^{\\hat 1}&=&\\gamma v^{\\hat  1}=\\frac{t}{\\sqrt{D(L^2-y^2)}}[ (L^2-y^2)\\dot x+xy \\dot y]\\,, \\nonumber\\\\\nU^{\\hat 2}&=&\\gamma v^{\\hat  2}=\\frac{t}{\\sqrt{L^2-y^2}}\\dot y\\,,  \\nonumber\\\\\nU^{\\hat 3}&=&\\gamma v^{\\hat 3}=\\sqrt{\\frac{T-t}{t}}~\\dot z\\,.\n\\end{eqnarray}\nThe Killing vectors $\\eta$ and $\\zeta$ ensure that  $-yU_x+xU_y$ and $U_z$ are constants of the motion, namely,\n\\begin{equation}\n\\label{II13}\n\\frac{t^2}{L^2}(-y\\dot x+x\\dot y)=C_\\eta\\,, \\qquad \\frac{T-t}{t}\\dot z=C_\\zeta\\,.\n\\end{equation}\nTherefore, one easily finds\n\\begin{equation}\n\\label{II14}\nz(\\tau) = z_0+C_\\zeta  \\int_0^\\tau \\frac{t(\\sigma)}{T-t(\\sigma)}d\\sigma\\,,\n\\end{equation}\nand the remaining equations for geodesic motion, $U^\\alpha{}_{;\\mu}~U^\\mu=0$, read\n\\begin{eqnarray}\n\\label{II15}\n\\ddot t &=& \\frac{T}{2t(t-T)}(\\dot t^2-C_\\zeta^2)+\\frac{t-T}{D}[(L^2-y^2)\\dot x^2 +2xy \\dot x \\dot y+(L^2-x^2)\\dot y^2 ], \\nonumber\\\\\n\\ddot x &=& -\\frac{(L^2-y^2)x\\dot x^2 +(L^2-x^2)x\\dot y^2}{D} -\\frac{2}{t }\\frac{(D\\dot t+tx^2 y \\dot y)\\dot x}{D}\\,,\\nonumber\\\\\n\\ddot y &=& -\\frac{(L^2-x^2)y\\dot y^2 +(L^2-y^2)y\\dot x^2}{D} -\\frac{2}{t }\\frac{(D\\dot t+tx  y^2 \\dot x)\\dot y}{D}\\,.\n\\end{eqnarray}\nFrom $U^\\alpha U_\\alpha=-1$ and Eq.~\\eqref{II14}, we get\n\\begin{equation}\n\\label{II16}\n\\ddot t=\\frac{3T-2t}{2t(T-t)}(C_\\zeta^2-\\dot t^2)+\\frac{T-t}{t^2}\\,,\n\\end{equation}\nwhich implies that\n\\begin{equation}\n\\label{II16a}\n\\dot t^2 = \\frac{(t^2+C_0)(T-t)}{t^3}+C_\\zeta^2\\,,\n\\end{equation}\nwhere $C_0$ is an integration constant. \n\nThese equations imply that our fiducial observers, for example, are {\\it geodesic} observers; in fact, $x=x_0$, $y=y_0$ and $z=z_0$ with $C_\\eta=C_\\zeta=0$ are simple solutions of Eqs.~\\eqref{II14} and~\\eqref{II15}. The corresponding equation for $t$, which simply follows from the normalization condition, \ncan be explicitly integrated; that is, for the geodesic reference observers at rest, $C_0=0$ and\n\\begin{equation}\n\\label{II17}\n\\dot t=\\sqrt{\\frac{T-t}{t}}\\,.\n\\end{equation}\nHence,\n\\begin{equation}\n\\label{II18}\n\\tau= \\frac{\\pi}{2}T-\\sqrt{t(T-t)}-T\\sin^{-1} \\left(1-\\frac{t}{T}  \\right)\\,,\n\\end{equation} \nso that for $t=0$, $\\tau=0$, and as $t\\to T$, $\\tau \\to \\frac{\\pi}{2}T$.\nMoreover, the spatial frame of the fiducial observers given by Eq.~\\eqref{II10} is parallel propagated along their geodesic world lines.\n\n\nWe now integrate Eqs.~\\eqref{II15} and~\\eqref{II16} numerically from $t=\\epsilon T$, where $\\epsilon$,  $0<\\epsilon\\ll 1$, is a constant, to  $t$ approaching $T$.\nAs time $t$ approaches $T$, the proper distance along the $z$ axis decreases to zero, while the proper distance in the $(x,y)$ plane increases in accordance with our spacetime metric~\\eqref{II6}. Our numerical experiments support the general result that a double-jet structure invariably emerges along the $z$ axis with the corresponding Lorentz $\\gamma$ factor approaching infinity.\nThat is, $(v^{\\hat 1},v^{\\hat 2},v^{\\hat 3})\\to (0,0,\\pm 1)$ as $t \\to T$.\nThe numerical results in a typical case are illustrated in Fig. 1.\n\n\\begin{figure}\n\\[\n\\begin{array}{cc}\n\\includegraphics[scale=0.3]{fig1b.eps} &  \\qquad  \\includegraphics[scale=0.3]{fig1a.eps} \\cr\n\\end{array}\n\\]\n\\caption{The \\emph{collapse scenario}. Left panel:  Plots of the  components of the spatial velocity $v^{\\hat a}$ relative to comoving observers, given in Eq.~\\eqref{II12}, versus proper time $\\tau$ for the choice of metric parameters $L=T=1$.\nInitial conditions at $\\tau=0$ are $x(0)=y(0)=0.5$, $z(0)=0.3$, $\\dot x(0)=-~\\dot y(0)=1$, $\\dot z_{\\pm}(0)=\\pm\\, 0.01$ and $\\dot t(0)=3$. The value of $t(0)$ is obtained from the normalization condition and is given by $t(0)=0.1019457965$.\nRight panel: The initial segments of geodesic curves $x=x(\\tau)$, $y=y(\\tau)$ and $z=z_{\\pm}(\\tau)$  are plotted for the same choice of initial conditions as in the left panel.}\n\\end{figure}\n\nThe mathematical properties of these cosmic jets that are characteristic of the scenario for the formation of elementary astrophysical structure through gravitational collapse have been illustrated here by means of a Ricci-flat solution of general relativity that is related to the Schwarzschild-Droste white hole.\nIt is known that the white hole region of the extended Schwarzschild-Droste spacetime is locally isometric to colliding plane gravitational waves \\cite{FI,HA}.\nThis circumstance provides the motivation to investigate peculiar velocities and cosmic jets in plane-wave spacetimes.\n\n\\section{Gravitational plane waves}\n\nConsider an exact plane wave propagating along the positive $z$ direction. Introducing retarded and advanced null coordinates $u$ and $v$, respectively, by\n\\begin{equation}\n\\label{III1}\nu=\\frac{1}{\\sqrt{2}}(t-z)\\,,\\qquad v=\\frac{1}{\\sqrt{2}}(t+z)\\,,\n\\end{equation}\nthe spacetime metric can be expressed as\n\\begin{equation}\n\\label{III2}\nds^2=-2du dv +{\\cal W}^2 (e^{2h}dx^2+e^{-2h}dy^2)\\,,\n\\end{equation}\nwhere $dt^2-dz^2=2du dv$.\nHenceforth, we use the $(t,x,y,z)$ coordinate system in our discussion of plane waves. \nHere ${\\cal W}(u)$ and $h(u)$ are functions of the retarded time $t-z=\\sqrt{2}u$. Einstein's field equations in vacuum ($R_{\\mu\\nu}=0$) are satisfied provided\n\\begin{equation}\n\\label{III3}\n{\\cal W}_{,uu}+h_{,u}^2~{\\cal W}=0\\,,\n\\end{equation}\nwhere ${\\cal W}_{,u}=d{\\cal W}\/du$, etc. This is a Petrov type N gravitational field representing a linearly polarized (\\lq\\lq $\\oplus$\") plane wave \\cite{BPR}. For $|h|\\ll 1$ and ${\\cal W}=1$, Eq.~\\eqref{III3} is satisfied to linear order in $h$ and we recover linearized gravitational plane waves with $\\oplus$ polarization in the transverse-traceless (TT) gauge.\n\nWe are interested in the peculiar velocities of timelike geodesics with respect to the class of comoving observers. It turns out that observers at rest follow timelike geodesic world lines and their natural tetrad frames, namely,\n\\begin{equation}\n\\label{III4}\ne_{\\hat 0}=\\partial_t\\,,\\quad e_{\\hat 1}=({\\cal W}e^h)^{-1}\\partial_x\\,,\\quad\ne_{\\hat 2}=({\\cal W}e^{-h})^{-1}\\partial_y\\,,\\quad e_{\\hat 3}=\\partial_z\\,,\n\\end{equation}\nare parallel propagated along their world lines.\n\nThe Riemann curvature tensor as measured by the reference observers can be expressed in any Ricci-flat spacetime in terms of $3\\times 3$ symmetric and traceless matrices ${\\mathcal E}$  and ${\\mathcal B}$ in the standard manner as\n\\begin{equation}\n\\label{III5}\n{\\mathfrak R}=\\left[\n\\begin{array}{cc}\n{\\mathcal E} & {\\mathcal B}\\cr\n{\\mathcal B} & -{\\mathcal E}\\cr\n\\end{array}\n\\right]\\,,\n\\end{equation}\nwhere ${\\mathcal E}$  and ${\\mathcal B}$ are the \\lq\\lq electric\" and \\lq\\lq magnetic\" components of the Weyl tensor. More specifically, we consider $R_{\\hat\\alpha\\hat\\beta \\hat \\gamma \\hat \\delta}$ and the mapping $(R_{\\hat\\alpha\\hat\\beta \\hat \\gamma \\hat \\delta})\\mapsto (R_{IJ})$, where $I$ and $J$ range over the set $\\{01,02,03,23,31,12\\}$. In the case under consideration here, we find\n\\begin{equation}\n\\label{III6}\n{\\mathcal E}={\\mathcal K}(u)\\left[\n\\begin{array}{ccc}\n0&0 & 0\\cr\n0&-1 & 0\\cr\n0&0 & 1\\cr\n\\end{array}\n\\right]\\,,\\qquad\n{\\mathcal B}={\\mathcal K}(u)\\left[\n\\begin{array}{ccc}\n0&0 & 0\\cr\n0&0 & 1\\cr\n0&1 & 0\\cr\n\\end{array}\n\\right]\\,,\n\\end{equation}\nwhere~\\cite{CM}\n\\begin{equation}\n\\label{III7}\n{\\mathcal K}(u)=h_{,uu}+2h_{,u}\\frac{{\\cal W}_{,u}}{{\\cal W}}\\,.\n\\end{equation}\n\nThis plane wave spacetime admits a null Killing vector $\\partial_v=(\\partial_t+\\partial_z)\/\\sqrt{2}$ characteristic of the wave propagation in the $z$ direction at the speed of light and two spacelike Killing vectors $\\partial_x$ and $\\partial_y$ characteristic of the planar symmetry of the wave front. \nThese Killing vectors are generators of an Abelian group $G_3$, which acts in  null hypersurfaces that are wave fronts given by $u=$ constant.\nMoreover, the plane waves under consideration here admit two additional Killing vectors given by\n\\begin{equation}\n\\label{III8}\nx\\partial_v +\\hat f(u) \\partial_x\\,,\\qquad y\\partial_v +\\hat g(u) \\partial_y\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{III9}\n\\hat f(u)= \\int^u \\frac{du'}{{\\mathcal F}^2(u')}\\,,\\qquad \\hat g(u)= \\int^u \\frac{du'}{{\\mathcal G}^2(u')}\\,,\n\\end{equation}\nwhere ${\\mathcal F}={\\cal W}{\\rm exp}(h)$ and ${\\mathcal G}={\\cal W}{\\rm exp}(-h)$. The five Killing vectors are generators of a group $G_5$, which has as its subgroup the Abelian group $G_3$; see the discussion of pp waves in Section 24.5 of Ref. \\cite{N1}.\nIt follows that the projections of the $4$-velocity of free test particles $U^\\alpha=dx^\\alpha\/d\\tau$ on these Killing vectors are constants of the motion. Hence, we can write\n\\begin{eqnarray}\n\\label{III10}\n&& \\frac{dt}{d\\tau}-\\frac{dz}{d\\tau}=C_v\\,, \\\\\n{}\\label{III11}\n&& {\\cal F}^2~ \\frac{dx}{d\\tau}=C_x\\,,\\qquad {\\cal G}^2~ \\frac{dy}{d\\tau}=C_y\\,,\n\\end{eqnarray}\nwhere $C_v$, $C_x$ and $C_y$ are constants.\nMoreover, $U^\\alpha U_\\alpha=-1$ implies that $C_v \\not =0$ and\n\\begin{eqnarray}\n\\label{III12}\n&& \\frac{dt}{d\\tau}=\\frac{1+C_v^2}{2C_v}+\\frac{1}{2C_v}\\left(\\frac{C_x^2}{{\\mathcal F}^2}+\\frac{C_y^2}{{\\mathcal G}^2}  \\right) \\,,\\quad \n\\frac{dx}{d\\tau}=\\frac{C_x}{{\\mathcal F}^2} \\,,\\qquad \\frac{dy}{d\\tau}=\\frac{C_y}{{\\mathcal G}^2}\\,, \\\\\n{}\\label{III13}\n&&  \\frac{dz}{d\\tau}=\\frac{1-C_v^2}{2C_v}+\\frac{1}{2C_v}\\left(\\frac{C_x^2}{{\\mathcal F}^2}+\\frac{C_y^2}{{\\mathcal G}^2}  \\right) \\,.\n\\end{eqnarray}\nHere $u$ is a linear function of $\\tau$; that is, $\\sqrt{2}~u=C_v~ \\tau$ + constant. For the comoving observers $C_v=1$, $C_x=C_y=0$.\nThe projection of $U^\\alpha$ on the tetrad frame of the fiducial observers~\\eqref{III4} results in\n\\begin{equation}\n\\label{III14}\nU^{\\hat 0}=\\gamma=\\frac{dt}{d\\tau}\\,, \\quad\nU^{\\hat 1}=\\gamma v^{\\hat 1}=\\frac{C_x}{\\mathcal F}\\,, \\quad\nU^{\\hat 2}=\\gamma v^{\\hat 2}=\\frac{C_y}{\\mathcal G}\\,, \\quad\nU^{\\hat 3}=\\gamma v^{\\hat 3}=\\frac{dz}{d\\tau}\\,.\n\\end{equation}\n\nTo proceed further, we need an explicit solution of Eq.~\\eqref{III3}. To this end, let us consider~\\cite{GR}\n\\begin{equation}\n\\label{III15}\n{\\cal W}=[\\cos (\\beta u)  \\cosh (\\beta u)]^{1\/2}\\,,\\quad e^h=\\left[ \\frac{\\cos (\\beta u) }{\\cosh (\\beta u)}  \\right]^{1\/2}\\,,\n\\end{equation}\nfor which Eq.~\\eqref{III3} is satisfied. Positive square roots are assumed throughout in our convention. Here, $\\beta$ is a constant parameter that can be chosen to be positive, $\\beta >0$, with no loss in generality, and $\\beta$ is related to the frequency of the gravitational wave.\nMoreover, Eq.~\\eqref{III15} is meaningful provided $\\cos (\\beta u)>0$. The curvature as measured by the reference observers is given by Eqs.~\\eqref{III6} and~\\eqref{III7}, where ${\\mathcal K}=-\\beta^2$ in this case. We note that in $(t,x,y,z)$ coordinates, $\\sqrt{-g}={\\cal W}^2=\\cos (\\beta u)  \\cosh (\\beta u)$. \nFor the sake of definiteness, we assume $\\beta u \\in (-\\pi\/2, \\pi\/2)$. Let us note that as $u$ increases from, say, $u=0$ and approaches $\\pi\/(2\\beta)$, the proper spatial distance along the $x$ direction decreases to zero, while the corresponding distance along the $y$ direction increases and $\\sqrt{-g}\\to 0$, since in this case\n\\begin{equation}\n\\label{III16}\n{\\mathcal F}(u)=\\cos (\\beta u)\\,,\\qquad {\\mathcal G}(u)=\\cosh (\\beta u)\\,.\n\\end{equation}\nIt is then interesting to investigate the behavior of free test particles with $C_x\\not =0$ with respect to the fiducial observers.\nAs $u\\to \\pi\/(2\\beta)$, the asymptotic expressions for $\\gamma$ and $v^{\\hat a}$ are\n\\begin{equation}\n\\label{III17}\n\\gamma \\sim \\frac{C_x^2}{2C_v}\\frac{1}{\\cos^2(\\beta u)}\\,, \\quad\nv^{\\hat 1}\\sim \\frac{2C_v}{C_x}\\cos (\\beta u)\\,, \\quad\nv^{\\hat 2}\\sim \\frac{2C_v C_y}{C_x^2}\\frac{\\cos^2(\\beta u)}{\\cosh (\\beta u)}\\,, \\quad\nv^{\\hat 3}\\sim 1\\,.\n\\end{equation}\nThus the free test particles with $C_x\\not=0$ in this spacetime line up asymptotically with $\\gamma \\to \\infty$ along the direction of propagation of the wave; that is, we have a single-jet pattern with  $(v^{\\hat 1},v^{\\hat 2},v^{\\hat 3})\\to (0,0,1)$ as $u\\to \\pi\/(2\\beta)$.\nIt is important to remark here that the null hypersurface $u=\\pi\/(2\\beta)$ is simply a coordinate singularity, as it occurs at the limit of admissibility of the $(t,x,y,z)$ coordinate system; nevertheless, the asymptotic jet structure has been invariantly characterized and it is therefore physically meaningful.\n\nIn a gravitational plane-wave spacetime, if the proper spatial distance tends to zero in a direction transverse to the direction of propagation of the wave, most of the free test particles in this gravitational field form a single-jet structure parallel to the direction of propagation such that the speed of the jet asymptotically approaches the speed of light. This plane-wave scenario for cosmic jet formation is entirely different from the collapse scenario. We will explore this scenario further in the next section.\n\n\n\\section{Electromagnetic plane-wave spacetimes}\n\nLet us next consider the metric~\\cite{BS, GR2}\n\\begin{equation}\n\\label{IV1}\nds^2 = -dt^2+ \\Psi^2(u)(d x^2+d y^2)+dz^2\\,,\n\\end{equation}\nwhich satisfies the gravitational field equations,  $G_{\\mu\\nu}=8\\pi G~ T_{\\mu\\nu}$,  with \n\\begin{equation}\n\\label{IV2}\nT_{\\mu\\nu}= \\Phi^2(u) k_{\\mu} k_{\\nu}\\,, \\qquad k=\\partial_v\\,,\n\\end{equation}\nwhere $\\Phi(u)$ represents the flux of the electromagnetic radiation field and is related to $\\Psi(u)$ via\n\\begin{equation}\n\\label{IV3}\n\\Psi_{,uu}+4 \\pi G~  \\Phi^2(u) \\Psi =0\\,.\n\\end{equation}\nWe note that the spacetime metric here is of the same general form as the metric of the plane wave of the previous section with $\\mathcal{F}^2=\\mathcal{G}^2=\\Psi^2$; therefore, in addition to the isometries of the previous section, metric~\\eqref{IV1} is also invariant under Euclidean rotations in the $(x, y)$ plane.  The traceless energy-momentum tensor~\\eqref{IV2} can be interpreted as representing either null dust moving along the $z$ direction or a pure electromagnetic radiation field with a wave vector parallel to $k$. Adopting the latter interpretation, we assume that the potential 1-form $A^\\flat$ of the null electromagnetic field is aligned with a transverse spatial direction, say the $x$ axis, so that \n\\begin{equation}\n\\label{IV4}\nA^\\flat=\\psi(u)\\, dx\\,.\n\\end{equation}\nHence, the transverse gauge condition, $\\partial_\\mu (\\sqrt{-g}A^\\mu)=0$, is satisfied and the  Faraday 2-form $F^\\flat =d A^\\flat $ is given by\n\\begin{equation}\n\\label{IV5}\nF=\\psi_{,u}(u)\\, du \\wedge d x\\,.\n\\end{equation}\nMaxwell's equations are satisfied in this case and we have\n\\begin{equation}\n\\label{IV6}\n\\Phi=\\frac{1}{\\sqrt{4 \\pi}}\\frac{\\psi_{,u}}{\\Psi}\\,.\n\\end{equation}\nThus the source of the gravitational field under consideration is a linearly polarized plane electromagnetic null field $(F_{\\mu\\nu}F^{\\mu\\nu}=0)$ propagating along the $z$ direction with its electric field along the $x$ direction and its magnetic field along the $y$ direction. \n\nAs before, it is useful to introduce a family of fiducial observers that are all at rest in space with $4$-velocity vector $e_{\\hat 0}=\\partial_t$. It turns out that the congruence of these fiducial observer world lines is geodesic and vorticity free, but has  nonzero expansion. Moreover,  \nthe natural orthonormal spatial triad adapted to such an observer is given by\n\\begin{equation}\n\\label{IV7}\ne_{\\hat 1}=\\frac{1}{\\Psi}\\partial_x\\,,\\qquad\ne_{\\hat 2}=\\frac{1}{\\Psi}\\partial_y\\,, \\qquad\ne_{\\hat 3}=\\partial_z\\,.\n\\end{equation}\nThis spatial frame is parallel propagated along the world line of the reference observer. \n\nThe electromagnetic field, as measured by these fiducial observers, is given by the projection of the Faraday tensor onto the observers' orthonormal tetrads, namely, \n\\begin{equation}\n\\label{IV8}\nF_{\\hat{\\alpha}\\hat{\\beta}}=\\frac{1}{\\sqrt{2}}\\frac{\\psi_{,u}}{\\Psi}\\left[\n\\begin{array}{cccc}\n0&1&0 & 0\\cr\n-1&0&0& 1\\cr\n0&0&0&0\\cr\n0&-1&0& 0\\cr\n\\end{array}\n\\right]\\,,\n\\end{equation}\nso that the measured electric and magnetic fields are each given by $-\\sqrt{2\\pi} ~\\Phi$ along the $x$ and $y$ axes, respectively. \n\nTurning now to the measurement of the gravitational field, it is clear from Eqs.~\\eqref{III6} and~\\eqref{III7} that the Weyl curvature tensor vanishes identically in this case and the Riemann curvature tensor is then given by\n\\begin{equation}\n\\label{IV9}\nR_{\\mu \\nu \\rho \\sigma} = \\frac{1}{2} (R_{\\mu \\rho}g_{\\nu \\sigma}+ R_{\\nu \\sigma}g_{\\mu \\rho}-R_{\\mu \\sigma}g_{\\nu \\rho}-R_{\\nu \\rho}g_{\\mu \\sigma})\\,, \n\\end{equation}\nsince the scalar curvature vanishes as well ($R=0$). In general, the measured components of the Riemann tensor can be represented as a $6\\times 6$ matrix \n\\begin{equation}\n\\label{IV10}\n{\\mathfrak R}=\\left[\n\\begin{array}{cc}\n{\\mathcal E} & {\\mathcal B}\\cr\n{\\mathcal B^{\\dagger}} & {\\mathcal S}\\cr\n\\end{array}\n\\right]\\,,\n\\end{equation}\nwhere $\\mathcal{E}$ and $\\mathcal{S}$ are symmetric $3 \\times 3$ matrices and $\\mathcal{B}$ is traceless. In the present case, we find that the electric and magnetic components are given by\n\\begin{equation}\n\\label{IV11}\n{\\mathcal E}=\\kappa (u)\\left[\n\\begin{array}{ccc}\n1&0 & 0\\cr\n0&1 & 0\\cr\n0&0 & 0\\cr\n\\end{array}\n\\right]\\,,\\qquad\n{\\mathcal B}=\\kappa (u)\\left[\n\\begin{array}{ccc}\n0&-1 & 0\\cr\n1&0 & 0\\cr\n0&0 & 0\\cr\n\\end{array}\n\\right]\\,,\n\\end{equation}\nwhile the spatial components are given by $\\mathcal{S}=\\mathcal{E}$. Here,\n\\begin{equation}\n\\label{IV12}\n\\kappa(u)=2 \\pi G~ \\Phi^2(u)\\,.\n\\end{equation}\nThis gravitational field is algebraically special and of Petrov type $O$. Furthermore, as mentioned before, the Weyl tensor vanishes in this case and the metric is thus conformally flat, as can be simply verified via the transformation of the null coordinate $u$, $u \\mapsto u'$, where $u'={\\hat f}(u)={\\hat g}(u)$ with ${\\hat f}$ and ${\\hat g}$ that were defined in Eq.~\\eqref{III9}.\n\nTo study the general behavior of test particles in this solution of the Einstein-Maxwell equations, we need an explicit solution of Eq.~\\eqref{IV3}. To this end, we consider in the rest of this section~\\cite{GR2}\n\\begin{equation}\n\\label{IV13}\n \\Psi=\\sqrt{G}~\\cos{(bu)}\\,, \\quad \\Phi=\\frac{b}{\\sqrt{4 \\pi G}}\\,, \\quad \\psi=\\sin{(bu)}\\,, \n\\end{equation}\nwhere $b>0$ is a constant parameter. In this case, the measured electric and magnetic fields are each of constant magnitude $b\/\\sqrt{2G}$ and the measured spacetime curvature is constant as well, since $\\kappa=b^2\/2$. To simplify matters, we assume henceforth that $G=1$. \n\nThe spacetime under consideration admits the following Killing vectors \n\\begin{eqnarray}\n\\label{IV14}\n\\xi_{(1)}&=&\\partial_v\\,,\\quad\n\\xi_{(2)}=\\partial_x\\,,\\quad\n\\xi_{(3)}=\\partial_y\\,,\\nonumber\\\\\n\\xi_{(4)}&=&-y\\partial_x+x\\partial_y\\,,\\nonumber\\\\\t\n\\xi_{(5)}&=&x\\partial_v+\\frac{\\tan{(bu)}}{b}\\partial_x\\,,\\nonumber\\\\\t\n\\xi_{(6)}&=&y\\partial_v+\\frac{\\tan{(bu)}}{b}\\partial_y\\,,\n\\end{eqnarray}\nsince ${\\hat f}={\\hat g}=b^{-1} \\tan{(bu)}$ in this case. The solution of the geodesic equations of motion is essentially the same as that given in the previous section and, repeating the same analysis as before, we recover the asymptotic single-jet pattern characteristic of the \\emph{wave scenario}.  \n\nIt is interesting to examine the motion of charged particles in this Einstein-Maxwell field. Consider a test particle of inertial mass $m$ and electric charge $q$ moving in this field in accordance with the Lorentz force law \n\\begin{equation}\n\\label{IV15}\n U^\\alpha{}_{;\\mu} U^\\mu = {\\hat q}~ F^\\alpha{}_{\\nu} U^\\nu\\,,\n \\end{equation}\nwhere ${\\hat q}:= q\/m$. In components, the equations of motion are given by\n\\begin{equation}\n\\label{IV16}\n \\frac{dU^0}{d\\tau} -\\frac{b}{2\\sqrt{2}}\\sin{(2bu)}~\\Sigma=-\\frac{{\\hat q}b}{\\sqrt{2}}\\cos{(bu)}~ U^1\\,, \\end{equation}\n\\begin{equation}\n\\label{IV17}\n \\frac{dU^1}{d\\tau} -2 b \\tan{(bu)}\\frac{du}{d\\tau}~ U^1=-\\frac{{\\hat q}b}{\\cos{(bu)}}~\\frac{du}{d\\tau}\\,, \n \\end{equation}\n\\begin{equation}\n\\label{IV18}\n \\frac{dU^2}{d\\tau} -2 b \\tan{(bu)}\\frac{du}{d\\tau}~ U^2=0\\,, \n \\end{equation}\n\\begin{equation}\n\\label{IV19}\n \\frac{dU^3}{d\\tau} -\\frac{b}{2\\sqrt{2}}\\sin{(2bu)}~\\Sigma=-\\frac{{\\hat q}b}{\\sqrt{2}}\\cos{(bu)}~ U^1\\,, \\end{equation}\nwhere $\\Sigma := (U^1)^2+(U^2)^2$. It follows from Eqs.~\\eqref{IV16} and~\\eqref{IV19} that the quantity \n\\begin{equation}\n\\label{IV20}\n \\sqrt{2}~\\frac{du}{d\\tau}=U^0-U^3\\, \n\\end{equation}\nis a constant of the motion $C_v$. Hence, as before, we have $\\sqrt{2}~u = C_v~\\tau$ + constant; moreover, it follows from $U_\\alpha U^\\alpha = - 1$ that $C_v\\ne0$.  The above system of equations can then be easily solved and the result is \n\\begin{equation}\n\\label{IV21}\n \\frac{dt}{d\\tau}=\\frac{1+C_v^2}{2C_v}+\\frac{1}{2C_v}\\frac{\\left[C_x-{\\hat q}\\sin{(bu)}\\right]^2+C_y^2}{\\cos^2{(bu)}}\\,\n\\end{equation}\n\\begin{equation}\n\\label{IV22} \n\\frac{dx}{d\\tau}=\\frac{C_x-{\\hat q}\\sin{(bu)}}{\\cos^2{(bu)}}\\,, \\quad \\frac{dy}{d\\tau}=\\frac{C_y}{\\cos^2{(bu)}}\\,,\n\\end{equation}\n\\begin{equation}\n\\label{IV23}\n \\frac{dz}{d\\tau}=\\frac{1-C_v^2}{2C_v}+\\frac{1}{2C_v}\\frac{\\left[C_x-{\\hat q}\\sin{(bu)}\\right]^2+C_y^2}{\\cos^2{(bu)}}\\,.\n\\end{equation}\nThus when ${\\hat q}=0$, we recover the geodesic equations of motion as would be expected from the analysis of the previous section. It is interesting to mention here that certain other physical aspects of this Einstein-Maxwell field, such as the motion of spinning test particles, have recently received attention~\\cite{BG, BGHJ, BGHO}.\n \n To describe the motion of charged test particles relative to the comoving reference observers, we consider the projection of the 4-velocity $U^\\mu$ on the tetrad frame $e_{\\hat \\alpha}$ and find\n\\begin{equation}\n\\label{IV24}\n\\gamma=\\frac{dt}{d\\tau}\\,, \\quad\nv^{\\hat 1}=\\Psi \\frac{dx}{dt}\\,, \\quad\nv^{\\hat 2}=\\Psi \\frac{dy}{dt}\\,, \\quad\nv^{\\hat 3}=\\frac{dz}{dt}\\,.\n\\end{equation}\nLet us assume that $b u \\in (-\\pi\/2, \\pi\/2)$ and note that as $u\\to \\pi\/(2b)$, all proper lengths in the transverse direction---i.e., in the $(x, y)$ plane---tend to zero, and $\\sqrt{-g}=\\cos^2{(bu)}$ tends to zero as well. It follows from the inspection of Eqs.~\\eqref{IV21}--\\eqref{IV24} that all charged or uncharged test particles with either $C_x\\ne {\\hat q}$ or $C_y\\ne0$ asymptotically form a single-jet structure with  $(v^{\\hat 1},v^{\\hat 2},v^{\\hat 3})\\to (0,0,1)$ as $u\\to \\pi\/(2b)$. On the other hand, if $C_x =  {\\hat q}$ and $C_y = 0$, then in this special case as $u\\to \\pi\/(2b)$, we have $v^{\\hat 1}=v^{\\hat 2}= 0$ and $v^{\\hat 3}=(1-C_v^2)\/(1+C_v^2)$, so that $0<v^{\\hat 3}<1$.  We conclude that in this \\emph{wave scenario}, test particles generally line up asymptotically along the direction of wave propagation and form a single-jet pattern with Lorentz factor $\\gamma \\to \\infty$. \n\n\n\n\\section{Concluding Remarks}\n\nA \\emph{cosmic jet} is a significant asymptotic pattern of test-particle motion in time-dependent gravitational fields such that the associated peculiar velocities generally line up in a flow with asymptotic Lorentz factor $\\gamma \\to \\infty$.\n\nIt has been shown that cosmic double-jet patterns occur in certain exact solutions of general relativity involving gravitational collapse~\\cite{N3,N4,N5,N6}. The precise relationship between these mathematical structures and the bipolar flow patterns of astrophysical jets is unknown, though in this connection certain apparently reasonable conjectures have been formulated~\\cite{N4, N5, N6}.\n\nThe main purpose of this paper has been to demonstrate the existence of a second independent scenario for cosmic jet formation. That is, in addition to the \\emph{collapse scenario} for the formation of \\emph{double-jet} patterns~\\cite{N3,N4,N5,N6}, we have presented, in the previous two sections of this paper, instances of the \\emph{wave scenario}  that clearly demonstrate the generation of \\emph{single-jet} patterns  in plane-wave spacetimes.  \n\nIs there a connection between the two scenarios for cosmic jet formation? Intuitively, it is tempting to think of a time-dependent gravitational field in the collapse scenario as representing the collision of two nonlinear gravitational wave packets moving in opposite directions.  However, as discussed in the Appendix, there are severe difficulties with such a heuristic interpretation. This issue and the question of the possibility of existence of other scenarios for jet formation require further investigation. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nJets in \\pp\\ and heavy-ion collisions arise from hard-scattered (high-$Q^{2}$) quarks and gluons of the incoming beams. In vacuum, a highly virtual parton generated in such interaction comes on-shell by radiating gluons, resulting in a jet shower. Studying jet properties in \\pp\\ collisions provides the opportunity to explore the perturbative and non-perturbative QCD effects in vacuum. In addition, the comparison between data and different QCD-based Monte Carlo (MC) event generators helps to constrain model parameters. In heavy-ion collisions, a highly energetic parton---while traversing through the Quark-Gluon Plasma (QGP)---interacts with the colored medium and loses its energy via medium-induced gluon radiation. This phenomenon is known as the jet quenching\\cite{Cunqueiro:2021wls}. Suppression of jet yield, modification of jet shape and substructure, and jet deflection in the QGP are the manifestations of jet quenching in heavy-ion collisions. \\\\\n\n In these proceedings we present recent jet measurements by the STAR experiment at RHIC, addressing jet substructure in \\pp\\ collisions and jet quenching in Au+Au collisions.\n \n \n\\section{Jet measurements in \\pp\\ collisions}\n\\label{Sect:pp}\n The fragmentation and evolution of a hard-scattered parton is described by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting kernels~\\cite{Dokshitzer:1977sg,Gribov:1972ri,Altarelli:1977zs}. This splitting probability depends on the momentum fraction of the split and the opening angle, and can be studied in \\pp\\ collisions. In QCD, higher order corrections contribute to the jet mass and substructure observables. To study these observables in\u00a0STAR, jets are studied by clustering charged tracks from the time projection chamber and neutral particles from the barrel electromagnetic calorimeter towers using  the anti-\\ensuremath{k_\\mathrm{T}}\\ jet reconstruction algorithm~\\cite{Salam:2010nqg} with different resolution parameters (\\ensuremath{R}) between 0.2 and 0.6. Furthermore, such measurements in \\pp\\ collisions provide a baseline for the similar measurements in heavy-ion collisions to study the modification of parton shower in the finite-temperature QCD medium. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{Zg.eps}\n    \\caption{The $z_{\\mathrm g}$\\ distributions for different $p_{\\rm T,jet}$ in \\pp\\ collisions at \\ensuremath{\\sqrt{s}}\\ = 200 GeV\\cite{STAR:2020ejj}.}\n    \\label{fig:zg}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n            \\includegraphics[width=0.7\\textwidth]{Rg.eps}\n    \\caption{The $R_{\\mathrm g}$\\ distributions for different $p_{\\rm T,jet}$ in \\pp\\ collisions at \\ensuremath{\\sqrt{s}}\\ = 200 GeV\\cite{STAR:2020ejj}.}\n    \\label{fig:Rg}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{JetMassA.pdf}\n        \\includegraphics[width=0.7\\textwidth]{JetMassb.pdf}\t\n    \\caption{Jet mass distributions for different $p_{\\rm T,jet}$ and jet \\ensuremath{R}\\ in \\pp\\ collisions at \\ensuremath{\\sqrt{s}}\\ = 200 GeV\\cite{STAR:2021lvw}.}\n    \\label{fig:Jetmass}\n\\end{figure}\n\n\\subsection{SoftDrop jet grooming}\nThe SoftDrop jet-grooming\\cite{Larkoski:2015lea,Dasgupta:2013ihk,Larkoski:2014wba} algorithm helps to study jet substructure by suppressing soft large-angle radiations. In this procedure, the soft and wide-angle radiations are removed sequentially from the jet de-clustering tree. This is achieved using the Cambridge\/Aachen (C\/A) clustering algorithm\\cite{Salam:2010nqg} by de-clustering jet branching history with\u00a0removing\u00a0the\u00a0soft branch until it satisfies the condition: \n\n\\begin{equation}\n z_{g} = \\frac{ {\\rm min} (p_{\\rm T,1}, p_{\\rm T,2})}{p_{\\rm T,1}+p_{\\rm T,2}} > z_{\\rm cut} \\big( \\frac{R_{\\rm g}}{R}\\big)^{\\beta}.   \n\\end{equation}\nIn Eq. (1), indices 1 and 2 represent the two sub-jets of splitting. The radius ($R$) is the distance in pseudorapidity and azimuthal angle space between two sub-jets, and $R_{\\mathrm g}$\\ is the groomed jet radius. The SoftDrop threshold $z_{\\rm cut}=0.1$, and angular exponent $\\beta=0$ are used for this de-clustering procedure for infrared and collinear (IRC) safety\\cite{Larkoski:2014wba}.\n\\\\\n \nAt RHIC, the first fully corrected SoftDrop observables, $z_{\\mathrm g}$\\ and $R_{\\mathrm g}$ , are measured in \\pp\\ collisions at $\\sqrt{s}$ = 200 GeV by the STAR experiment for inclusive jets with \\ensuremath{R}\\ = 0.2, 0.4, and 0.6, and 15 $<$ \\ensuremath{p_\\mathrm{T,jet}} $<$ 60 \\ensuremath{\\mathrm{GeV\/}c} \\cite{STAR:2020ejj}. Figures~\\ref{fig:zg} and \\ref{fig:Rg} show the distributions of $z_{\\mathrm g}$\\ and $R_{\\mathrm g}$, respectively. The shape of $z_{\\mathrm g}$\\ distributions indicates no \\ensuremath{p_\\mathrm{T,jet}}\\ dependence above 30 \\ensuremath{\\mathrm{GeV\/}c}, and they are more asymmetric than the DGLAP splitting function for a leading order quark emitting a gluon. $R_{\\mathrm g}$\\ distributions reveal a narrowing with increasing \\ensuremath{p_\\mathrm{T,jet}}, and the splitting is asymmetric at high \\ensuremath{p_\\mathrm{T,jet}}.  The STAR-tuned\\cite{STAR:2019yqm} PYTHIA-6 with Perugia 2012 well describes the jet substructure observables at this energy. The comparisons with MC event generator predictions help further study different hadronization models for the higher-order QCD corrections at RHIC energy.\n\n\\subsection{Jet mass}\nThe mass of quark or gluon jets is sensitive to the fragmentation of highly virtual parent partons. The SoftDrop grooming procedure removes soft and wide-angle radiations from jets making the groomed jets less sensitive to the higher order QCD corrections. Jet mass is defined as the four-momentum sum of jet constituents, $M= \\large| \\sum_{i \\in \\rm jet}  p_{i}\\large| =  \\sqrt{E^{2}-{  {\\textbf {\\it p}}}^{2}}$. Here $E$ and ${\\textbf {\\it p}}$ are the energy and three-momentum of the jet, respectively.\n Studying both ungroomed and groomed jets, and comparing to different MC event generators can provide information on different pQCD effects and fragmentation. The STAR experiment has reported the first fully corrected ungroomed ($M$) and groomed ($M_{\\rm g}$) mass distributions of inclusive jets for several values of \\ensuremath{R}\\ at $\\sqrt{s}=200$ GeV as shown in Fig~\\ref{fig:Jetmass}\\cite{STAR:2021lvw}. These jets are selected within the range of $30 < \\ensuremath{p_\\mathrm{T,jet}} < 40 $ \\ensuremath{\\mathrm{GeV\/}c}. It is observed that the mean and width of the jet mass increases with increasing \\ensuremath{R}\\ due to the inclusion of wide-angle radiation. The same trend is also seen with growing \\ensuremath{p_\\mathrm{T,jet}}\\ that increases the radiation phase space. The groomed jet mass distribution gets shifted to a smaller value than that of ungroomed mass due to the reduction of soft radiation in the SoftDrop grooming procedure. The LHC-tuned PYTHIA-8 and HERWIG-7 EE4C MC event generators over- and under-predicts the jet mass at RHIC, respectively, whereas the STAR-tuned PYTHIA-6 quantitatively describes the data. This observation is similar to that for the $R_{\\mathrm g}$\\ observable as discussed in the previous subsection. These measurements serve as a reference for future jet mass measurements in heavy-ion collisions at RHIC.\n\n\\section{Jet quenching measurements in Au+Au collisions}\n\\label{Sect:AuAu}\nThe STAR experiment has reported measurements of jet quenching using observable high-\\ensuremath{p_\\mathrm{T}}\\ hadron suppression\\cite{STAR:2002ggv} and dihadron correlations\\cite{STAR:2002svs,STAR:2006vcp}.The hadronic measurements have limited information on the underlying mechanism of jet quenching due to the final-state effects in heavy-ion collisions.  Over the last few years, the application of jet reconstruction algorithms and the development of methods for rigorous correction of uncorrelated background in heavy-ion collisions enable us to study jet quenching in more detail using fully reconstructed jets.\\\\\n \nThe first measurements of inclusive jet, semi-inclusive hadron+jet, and preliminary results of \\ensuremath{\\gamma_\\mathrm{dir}}+jet and \\ensuremath{\\pi^0}+jet measurements have been reported by the STAR experiment. The measurement techniques and their results are discussed in this section.\n\n\\subsection{Inclusive jet suppression}\n Jet measurements in heavy-ion collisions are complicated due to the presence of large uncorrelated background. For the inclusive jet spectrum measurements in STAR, jets are reconstructed using anti-\\ensuremath{k_\\mathrm{T}}\\ algorithm\\cite{Salam:2010nqg} with an additional requirement of a high-\\ensuremath{p_\\mathrm{T}}\\ hadronic constituent ($p_{\\rm T,lead}^{\\rm min}$), in order to identify jets from hard scattering processes. The selection of $p_{\\rm T,lead}^{\\rm min}$ should satisfy the following criteria: i) it must be sufficiently high so that contributions from combinatorial jets are negligible; ii) the probability of multiple constituents with \\ensuremath{p_\\mathrm{T}}\\ $\\geq p_{\\rm T,lead}^{\\rm min}$ is negligible; iii) this $p_{\\rm T,lead}^{\\rm min}$ cut does not introduce a selection bias on the jet population within the considered $p_{\\rm T,jet}$ range. Using this technique, the first fully corrected inclusive jet spectrea in central and peripheral Au+Au collisions at \\ensuremath{\\sqrt{s_\\mathrm{NN}}}\\ = 200 GeV have been reported with $p_{\\rm T,lead}^{\\rm min}$=5 \\ensuremath{\\mathrm{GeV\/}c}\\ \\cite{STAR:2020xiv}. \\\\\n \n The nuclear modification factor (\\ensuremath{R_\\mathrm{AA}}) is defined as the ratio of inclusive charged jet yield in central Au+Au collisions to its cross sections in \\pp\\ collisions scaled by the nuclear thickness factor $\\langle\u00a0T_{\\mathrm{AA}}\u00a0\\rangle$ of central collisions. Similarly, \\ensuremath{R_\\mathrm{CP}}\\ is defined considering 60-80\\% peripheral collisions as a reference instead of \\pp\\ collisions. For \\ensuremath{R_\\mathrm{AA}},  PYTHIA is used as a vacuum reference, hence it is labeled as \\ensuremath{R^{\\rm Pythia}_\\mathrm{AA}}. Figure~\\ref{fig:InclJetRAa} shows the \\ensuremath{R^{\\rm Pythia}_\\mathrm{AA}}\\ as a function of $p_{\\rm T,jet}^{\\rm ch}$ for inclusive jets with $\\ensuremath{R}=0.2, 0.3$ and 0.4 within 15 $<p_{\\rm T,jet}^{\\rm ch}<$ 30 \\ensuremath{\\mathrm{GeV\/}c}. Strong suppression is observed in 0-10\\% central \\ensuremath{{\\rm Au+Au}}\\ collisions, and no jet \\ensuremath{R}\\ dependence of the suppression is seen. Different theory calculations\\cite{Vitev:2009rd,Chien:2015hda,Casalderrey-Solana:2016jvj,He:2015pra,Ke:2018jem}are consistent with the data. The \\ensuremath{R_\\mathrm{CP}}\\ shows strong and similar suppressions for $\\ensuremath{R}=0.2$ and 0.3 jets as shown in Fig~\\ref{fig:InclJetRCP}. The yields of inclusive charged hadrons and jets show a comparable level of suppression within the same \\ensuremath{p_\\mathrm{T}}\\ interval in central Au+Au collisions at \\ensuremath{\\sqrt{s_\\mathrm{NN}}}\\ = 200 GeV. In addition, the comparison between central Au+Au collisions at RHIC and central Pb+Pb collisions at the LHC---although within different \\ensuremath{p_\\mathrm{T}}\\ intervals---show a similar magnitude of suppression for charged hadrons and jets yields.\n The medium-induced broadening of the inclusive jet is measured by taking the ratio of inclusive jet yields for $\\ensuremath{R}=0.2$ and 0.4. No significant modification of the transverse jet profile due to jet quenching is observed for the inclusive jet population in central Au+Au collisions at $\\ensuremath{\\sqrt{s_\\mathrm{NN}}}\\ =  200$ GeV and is consistent with the LHC data. The ongoing full jet analysis will access the inclusive jet suppression measurement at higher \\ensuremath{p_\\mathrm{T,jet}}\\ at RHIC energies. \n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Fig19.eps}\n    \\caption{\\ensuremath{R^{\\rm Pythia}_\\mathrm{AA}}\\ as a function of \\ensuremath{p_\\mathrm{T,jet}}\\ in 0-10\\% central Au+Au collisions and for different jet \\ensuremath{R}\\ at \\ensuremath{\\sqrt{s_\\mathrm{NN}}}\\ = 200 GeV\\cite{STAR:2020xiv}.}\n    \\label{fig:InclJetRAa}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Fig16.eps}\n    \\caption{\\ensuremath{R_\\mathrm{CP}}\\ as a function of \\ensuremath{p_\\mathrm{T,jet}}\\ in Au+Au collisions and for different jet \\ensuremath{R}\\  at \\ensuremath{\\sqrt{s_\\mathrm{NN}}}\\ = 200 GeV\\cite{STAR:2020xiv}.}\n    \\label{fig:InclJetRCP}\n\\end{figure}\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1\\textwidth]{DirPhoJet_1520_R0205_Pythia6_WTheory_HP2020.pdf}\n    \\caption{\\ensuremath{\\gamma_\\mathrm{dir}}+jet \\ensuremath{{I}_{AA}}\\ as a function of \\ensuremath{p_\\mathrm{T,jet}}\\ for \\ensuremath{R}\\ = 0.2 and 0.5 at \\ensuremath{\\sqrt{s_\\mathrm{NN}}}\\ = 200 GeV\\cite{Sahoo:2020kwh}.}\n    \\label{fig:DirPhoIAA}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{PTShift_All_HP2020_Pythia6and8_HP2020_Aug12.pdf}\n    \\caption{$-\\Delta p_{\\rm T,jet}$ for different observables measured at RHIC and the LHC\\cite{Sahoo:2020kwh}.}\n    \\label{fig:PtShift}\n\\end{figure}\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1.1\\textwidth]{STARLHCHardProbes_v5}\n    \\caption{The kinematic coverages of the STAR hard probe measurements (past, current, and future projection) are shown and compared to the LHC (published) measurements\\cite{STARBUR2225}.}\n    \\label{fig:STARBUR}\n\\end{figure}\n\n\n\\subsection{Semi-inclusive \\ensuremath{\\gamma_\\mathrm{dir}}+jet and hadron+jet suppression}\n\nThe STAR experiment published\\cite{STAR:2017hhs} the measurement of semi-inclusive distribution of reconstructed recoil jets from a high-\\ensuremath{p_\\mathrm{T}}\\ trigger hadron (h+jet) in Au+Au collisions. In this measurement, the uncorrelated background contribution is mitigated by using a novel mixed-event (ME) technique. It is found that the contributions from multi-parton interactions in the recoil acceptance are negligible. \\\\\n\nThe semi-inclusive h+jet measurement enables us to perform similar \\ensuremath{\\gamma_\\mathrm{dir}}+jet and \\ensuremath{\\pi^0}+jet measurements in STAR by combining the method used in the previous \\ensuremath{\\gamma_\\mathrm{dir}}+hadron and \\ensuremath{\\pi^0}+hadron correlation measurements\\cite{STAR:2016jdz}. In this measurement, \\ensuremath{\\gamma_\\mathrm{dir}}\\ and \\ensuremath{\\pi^0}\\ are within the trigger energy ranges of 9 $<$ \\ensuremath{E_\\mathrm{T}^\\mathrm{trig}} $<$ 20 GeV and 9 $<$ \\ensuremath{E_\\mathrm{T}^\\mathrm{trig}} $<$ 15 GeV, respectively. However, in these proceedings only \\ensuremath{\\gamma_\\mathrm{dir}}+jet results with 15 $<$ \\ensuremath{E_\\mathrm{T}^\\mathrm{trig}} $<$ 20 GeV range is discussed, which is the highest \\ensuremath{E_\\mathrm{T}^\\mathrm{trig}}\\ range measured in this analysis. Recoil jets are measured using anti-\\ensuremath{k_\\mathrm{T}}\\ algorithm with \\ensuremath{R}\\ = 0.2 and 0.5. The same ME technique as in h+jet paper is applied to subtract uncorrelated jet background. The unfolding procedure, as in h+jet paper, is applied to correct for the detector effects and heavy-ion background fluctuations. Finally, the ratio of recoil jet yield in central Au+Au collisions to that of PYTHIA-8, \\ensuremath{I^\\mathrm{PYTHIA}_{AA}}, is presented for the aforementioned jet radii. The \\ensuremath{I^\\mathrm{PYTHIA}_{AA}}\\ of \\ensuremath{\\gamma_\\mathrm{dir}}+jet for \\ensuremath{R}\\ = 0.2, within 15 $<$ \\ensuremath{E_\\mathrm{T}^\\mathrm{trig}} $<$ 20 GeV, shows a stronger suppression than that of \\ensuremath{R}\\ = 0.5 as shown in Fig.~\\ref{fig:DirPhoIAA}, which hints at a potential \\ensuremath{R}\\ dependence of recoil jet suppression at RHIC. The upcoming results with the \\pp\\ data as a reference will be reported in the final publication of this measurement. Precision measurement with extended recoil jet \\ensuremath{p_\\mathrm{T}^\\mathrm{jet,ch}}\\ range for \\ensuremath{\\gamma_\\mathrm{dir}}+jet is planned with the upcoming RHIC runs.\n \n\n\n\\subsection{Charged jet \\ensuremath{p_\\mathrm{T,jet}}\\ spectrum shift: RHIC vs. LHC}\nJet suppression is commonly reported via \\ensuremath{R_\\mathrm{AA}}\\ and \\ensuremath{{I}_{AA}}\\ as a function of \\ensuremath{p_\\mathrm{T,jet}}. These observables convolute the effect of energy loss with the shape of the jet \\ensuremath{p_\\mathrm{T,jet}}\\ spectrum. In order to deconvolute this, a \\ensuremath{p_\\mathrm{T,jet}}\\ shift ($-\\Delta p_{\\rm T,jet}$ in Fig.~\\ref{fig:PtShift}) is measured for a quantitative comparison of jet energy loss from different observables.  The STAR h+jet\\cite{STAR:2017hhs}, inclusive jet, preliminary \\ensuremath{\\gamma_\\mathrm{dir}}+jet, and \\ensuremath{\\pi^0}+jet measurements, and ALICE h+jet\\cite{ALICE:2015mdb} measurements report values of $-\\Delta p_{\\rm T,jet}$, although within different kinematic ranges. While comparing these values as shown in Fig.~\\ref{fig:PtShift}, an indication of smaller in-medium energy loss at RHIC than the LHC is seen. \n\n \n\\section{STAR ongoing measurements and upcoming data-taking plan}\n\\label{Sect:Future}\n There are several ongoing jet measurements in STAR to study the QGP medium properties in heavy-ion collisions, such as full jet reconstruction to extend the \\ensuremath{p_\\mathrm{T,jet}}\\ reach, jet fragmentation function, jet shape, heavy-flavor jet, and recoil jet azimuthal angular correlation with trigger particles.\n\nThe STAR experiment plans to take high statistics data of Au+Au collisions at \\ensuremath{\\sqrt{s_\\mathrm{NN}}}\\ = 200 GeV in 2023 and 2025, and \\pp\\ collision data at $\\sqrt{s} = 200$ GeV along with $p$+A collisions in 2024 with 28 cryo-weeks for each year\\cite{STARBUR2225}. \nThe kinematic coverages of various measurements related to hard probes using these datasets are presented in Fig~\\ref{fig:STARBUR}.\nThese datasets are crucial for studying the inner-workings of QGP utilizing precision jet measurements.\n\n\n \n\\section{Summary}\n\\label{Sect:Summary}\n The STAR experiment has recently reported important results on jet substructure observables in \\pp\\ collisions, and various jet quenching observables in heavy-ion collisions to study QGP properties. Several other jet measurements are ongoing and will be presented in the near future. Besides, STAR's upcoming data-taking (during 2023-2025 RHIC runs) is crucial for the precision jet measurements with large kinematic coverages, whereas high statistics \\pp\\ data (2024 RHIC run) are important for providing high precision references. \n\n\n\\section*{Acknowledgement}\nNRS is supported by the Fundamental Research Funds of Shandong University and NNSF of China:12050410235.\n\n\n\\bibliographystyle{ws-ijprai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nWith the successful launch of the {\\em Swift} Gamma-ray Burst\nExplorer, an era of systematic, multi-wavelength observations of\nGRB early afterglows has been ushered in. Very early optical\/IR\ndetections have been made with ground-based telescopes (Akerlof et\nal. 1999; Fox et al. 2003; Li et al. 2003; Blake et al. 2005;\nVestrand et al. 2005) before and during the initial operation of\n{\\em Swift}. In the X-ray band, some evidence of the early afterglows\nhas been collected earlier (e.g. Piro et al. 1998; Giblin et al. 1998;\nBurenin et al. 1999; Piro et al. 2005). However, it is the operation\nof the {\\em Swift} X-Ray Telescope (XRT) that offers the possibility\nto unveil the final gap between the prompt emission and the late\nafterglow stage.  \n\nThere has been widespread expectation that the early X-ray\nobservations could answer a series of core questions in GRB\nstudies. What is the \nconnection between the GRB prompt emission and the afterglow? Are\nprompt emission and afterglow both from the external shock\n(M\\'esz\\'aros \\& Rees 1993; Dermer \\& Mitman 1999) or do they come from\ndifferent locations [i.e. prompt emission from the internal shocks\n(Rees \\& M\\'esz\\'aros 1994; Paczynski \\& Xu 1994), while the\nafterglow comes from the external shock (M\\'esz\\'aros \\& Rees 1997a;\nSari et al. 1998)]? Does the central engine become\ndormant after the burst is over? What is the immediate environment of\nthe burst, an ISM or a wind? Are there density clumps surrounding the\nGRB progenitor? What is the role of\nthe reverse shock? What is the initial Lorentz factor of the fireball?\n\nAll these questions can be at least partially answered with the early\nX-ray afterglow data, sometimes in combination with the prompt\ngamma-ray data and the early optical\/IR afterglow data. \nAlthough early afterglow lightcurves have been extensively modeled in\nthe optical band (mainly driven by the observations and by the\ntheoretical argument that the reverse shock emission component plays\nan important role in the optical band, e.g. M\\'esz\\'aros \\& Rees\n1997a; Sari \\& \nPiran 1999; M\\'esz\\'aros \\& Rees 1999; Kobayashi 2000; Kobayashi \\&\nZhang 2003a,b; Zhang et al. 2003; Wei 2003; Wu et al. 2003; Li et\nal. 2003b; Fan et al. 2004a; Zhang \\& Kobayashi 2005; Fan et\nal. 2005a,b; Nakar \\& Piran 2004; McMahon et al. 2004), possible early\nX-ray afterglow signatures have been only sparsely studied (e.g. Kumar\n\\& Panaitescu 2000a; Kobayashi et al. 2005a; Fan \\& Wei 2005).\n\nIn its first six months of operations, the {\\em Swift} X-Ray Telescope\n(XRT) has already accumulated a rich collection of early afterglow\nfeatures in the X-ray band. The XRT is a sensitive broad-band\n(0.2-10 keV) imager, which can be promptly slewed to GRB targets\ntriggered by the Burst Alert Telescope (BAT) within 1-2 minutes \n(Burrows et al. 2005a). It is therefore an ideal instrument\nto study the transition between the GRB prompt emission and the very \nearly X-ray afterglow.\nThe following features are all detected by XRT in a good sample of\nbursts (see, e.g. Chincarini et al. 2005; Nousek et al. 2005 for a\ncollection of data), reflecting some common underlying physics of GRBs.\n\n1. In most cases (e.g. GRB 050126 and GRB 050219A), a steep decay is\nobserved up to several hundred seconds after the burst trigger, which\nis followed by a more conventional shallower decay (Tagliaferri et\nal. 2005; Goad et al. 2005). This conclusion is drawn by choosing\nthe GRB trigger time as the zero time point ($t_0$). At later times \n(e.g. $(t-t_0) \\gg T_{90}$, where $T_{90}$ is the duration of the GRB), \nthe afterglow decay slope ${\\rm d} \\ln F_\\nu\/ {\\rm d} \\ln (t-t_0)$ is\nessentially independent on the adopted $t_0$. However, at early times\n(e.g. $(t-t_0)$ not much larger than $T_{90}$, the measured\ndecay slope could be very sensitive to the assumed $t_0$. \nTagliaferri et al. (2005) explored the $t_0$ effect and concluded that\nthe two distinct lightcurve segments are likely intrinsic rather than\ndue to a poor choice of $t_0$. Furthermore, in some cases, the steep decay\nsegment also has a different spectral index (e.g. for GRB 050319,\nCusumano et al. 2005). Usually it also connects to the spectral\nextrapolation of the BAT prompt emission lightcurve smoothly\n(Barthelmy et al. 2005a).  All these facts indicate that the steep decay\ncomponent is a distinct new component that may be unrelated to the\nconventional afterglow component.\n\n2. In a good fraction of GRBs (e.g. GRB 050128, Campana et al. 2005;\nGRB 050315, Vaughan et al. 2005; GRB 050319, Cusumano et al. 2005; and\nGRB 050401, de Pasquale et al. 2005), the early X-ray afterglow is\ncharacterized by a shallow to ``normal'' transition. During the\ntransition the spectral index is not changed. The decay slope after the\nbreak (e.g. $\\sim -1.2$) is consistent with the standard ISM afterglow\nmodel, while the decay slope before the break is usually much\nshallower (e.g. $\\sim -0.5$).\n\n3. In some cases (e.g. GRB 050315, Vaughan et al. 2005), a further\nsteepening is detected after the ``normal'' decay phase, which is\nconsistent with a jet break.\n\n4. Bright X-ray flares have been detected in the early X-ray\nlightcurves of nearly one half of the burst population \n(e.g. GRB 050406, GRB 050202B, Burrows et al. 2005; Romano et\nal. 2005; Falcone et al. 2005). In particular, the X-ray afterglow of\nthe short-hard burst GRB 050724 also shows at least three flares \n(Barthelmy et al. 2005b). The flares typically happen hundreds of\nseconds after the trigger or earlier, but in some cases they\noccur around a day (e.g. GRB 050502B, Falcone et al. 2005; GRB 050724,\nBarthelmy et al. 2005b). The amplitudes of the flares are usually\nlarger than the underlying afterglow component by a factor of several \n(e.g. a factor of 6 in GRB 050406, Burrows et al. 2005; Romano et\nal. 2005), but can be much larger (e.g. $\\sim 500$ in the case of GRB\n050202B, Burrows et al. 2005; Falcone et al. 2005). A similar\nfeature was evident for GRB 011121 detected by BeppoSAX (Piro et\nal. 2005). \n\nIn summarizing the current X-ray afterglow data, one can tentatively\ndraw a synthetic cartoon lightcurve in the X-ray band, which consists\nof 5 components (see Figure \\ref{XRTlc}): I. an initial steep decay\n(with a typical slope $\\sim -3$ or steeper); II. a shallower-than-normal\ndecay (with a typical slope $\\sim -0.5$); III. a normal decay (with a\ntypical slope $\\sim -1.2$); IV. a late steeper decay (with a typical\nslope $\\sim -2$); and V. one or more X-ray flares. We note that\nNousek et al (2005) also arrived at the similar schematic diagram that\nincludes the segments I, II and III in our cartoon picture (see their \nFig.3). Limited by the \nquality of the data, the current analyses indicate that the spectral \nindices remain unchanged in the segments II, III and IV, with a\ntypical value of $\\beta_X \\sim 1$ ($F_X \\propto\n\\nu^{-\\beta_X}$, Nousek et al. 2005). In some bursts, the segments I\nand II have different spectral indices (e.g. GRB 050319, Cusumano\net al. 2005). In some cases, a time-evolution of the spectral index\nhas been detected (e.g. in the giant flare of GRB 050502B, Falcone et\nal. 2005). In Fig.1, we have indicated the typical temporal index for\neach segment. Throughout the paper\nthe transition times between the adjacent segments for \nthe four lightcurve segments are denoted as $t_{b1}$, $t_{b2}$, and\n$t_{b3}$, respectively.\n\\begin{figure}\n\\includegraphics[angle=-90,scale=0.35]{f1.eps}\n\\caption{A synthetic cartoon X-ray lightcurve based on the\nobservational data from the {\\em Swift} XRT. The phase ``0'' denotes\nthe prompt emission. Four power law lightcurve segments together with\na flaring component are identified in the afterglow phase.  \nThe segments I and III are most common, and they are marked with solid\nlines. Other three components are only observed in a fraction of\nbursts, so they are marked as dashed lines. Typical temporal\nindices in the four segments are indicated in the figure. The spectral\nindices remain unchanged for segments II, III and IV, with a typical \nvalue of $\\beta_X \\sim 1$ ($F_X \\propto \\nu^{-\\beta_X}$). The segment\nI sometimes has a softer spectrum (e.g. $\\beta_X \\sim 1.5$),\nbut in some other cases it has a similar spectral index as the other\nthree segments. The flares (segment V) have similar spectra as the\nsegment I, and time evolution of the spectral index during the flares\nhas been observed in some bursts (e.g. GRB 050502B).} \n\\label{XRTlc}\n\\end{figure}\nIn this paper, we systematically study the physical processes\nthat shape an early X-ray lightcurve, and discuss possible\ntheoretical interpretations to the above phenomena. In\n\\S{\\ref{sec:Curvature}}, we discuss the GRB tail emission\narising from high angular latitude relative to the viewing direction,\nwhich takes longer to reach the observer due to the extra\ndistance it travels, as a conical (or spherical) shell suddenly stops\nshining. This is the so-called ``curvature effect\" (Fenimore et al. 1996;\nKumar \\& Panaitescu 2000a; Dermer 2004; Dyks et al. 2005).  In\n\\S{\\ref{sec:FS}}, we review the main \nemission properties from the external forward shock region in the\nX-ray band, summarizing the temporal and spectral indices expected in\nthe X-ray band for both the ISM and the wind models. Furthermore, we\nalso discuss the case of a continuously refreshed shock as well as its\nthree possible physical mechanisms. Several case studies are\ninvestigated to reveal an intriguing refreshed shock phase commonly\npresent in many \nbursts. In \\S{\\ref{sec:RS}}, we briefly discuss whether and how the\nreverse shock emission would influence the X-ray band emission. \nIn \\S{\\ref{sec:flare}}, we explore \nvarious mechanisms that might give rise to the X-ray flares observed\nin many bursts (e.g. GRB 050406 and GRB 050202b), and conclude that\nthe phenomenon is best interpreted as due to a late central engine\nactivity. Our conclusions are summarized in \\S{\\ref{sec:conclusions}}.\n\n\n\\section{GRB tail emission and the curvature effect}\n\\label{sec:Curvature}  \n\n\\subsection{GRB tail emission}\n\nThe temporal bridge between the GRB prompt emission and the afterglow\nemission is essential for revealing whether the prompt emission and\nthe afterglow originate from the same component. The earliest GRB\nrelativistic shock model invoked the external shock as the site for\nprompt $\\gamma$-ray emission (Rees \\& M\\'esz\\'aros 1992; M\\'esz\\'aros\n\\& Rees 1993). The rapid variability observed in many GRBs is in\ngreat contrast with the intuitive expectations in the external shock\nmodel, which generally predicts a smooth burst temporal profile, and\nit has been argued that the radiative efficiency is too low for the\nmodel so that a much larger energy budget is required (Sari \\& Piran\n1997). Dermer \\& Mitman (1999, 2003) argued that if\nthe GRB ambient medium is sufficiently clumpy, an external shock GRB\nmodel could reproduce the observed variability with a high energy\nefficiency. Within such a picture, the prompt emission and the\nafterglow originate from the same component (i.e. the external\nshock), and it is expected that the two emission components are likely\nsmoothly connected in the early phase.\n\nOn the other hand, it is now commonly believed that GRB prompt\nemission originates from some ``internal processes'', i.e. the\n$\\gamma$-rays are emitted before the fireball is decelerated by the\nambient medium.  The most widely discussed\nmodel is the internal shock model (Rees \\& M\\'esz\\'aros 1994;\nPaczynski \\& Xu 1994; Kobayashi et al. 1997; Daigne \\& Mochkovitch\n1998; Fan et al. 2004b). Alternatively, the internal emission may be\ncaused by dissipation of strong magnetic fields (e.g. Drenkhahn \\&\nSpruit 2002), or Comptonization of the photospheric emission (Rees \\&\nM\\'esz\\'aros 2005). Within such scenarios, there exist two distinct\ntemporal episodes dominated by the prompt emission and the afterglow,\nrespectively, since the latter is emitted at a much larger distance\nfrom the central engine when the fireball is decelerated. Generally\none should expect a flux contrast between these two episodes. \n\nBefore the {\\em Swift} era, no solid observation was available to\nfinally differentiate both scenarios, and evidence in favor of each\nscenario had been collected (see e.g. Zhang \\& M\\'esz\\'aros 2004 for a\nreview). It is one of the major tasks of {\\em Swift} to pin\ndown the emission site of the GRB prompt emission.\n\nIf the prompt emission and the afterglow arise from different emission\nsites, as is expected in the internal shock (or similar) scenario, and\nif the prompt emission flux level is much higher than \nthe afterglow emission flux level, one expects to see a steeply decaying\nlightcurve during the transition from the prompt emission phase to the \nafterglow phase. Such a steep decay is due to the so-called\n``curvature effect'' (e.g. Kumar \\& Panaitescu 2000a; Dermer 2004;\nDyks et al. 2005; Panaitescu et al. 2005).  In principle,\nsuch an effect also applies to the tail emission of the X-ray flares \n(Burrows et al. 2005).  Hereafter we generally define such an emission\ncomponent as ``GRB tail emission''.\n \n\n\\subsection{Curvature effect}\n\n\\subsubsection{The simplest case}\n\nFor a conical jet with an opening angle $\\theta_{\\rm j}$, emission\nfrom the same radius $R_{cr}$ but from different viewing latitudes\n$\\theta$ ($\\theta<\\theta_{\\rm j}$) would reach the observer at\ndifferent times. Even if the emission is turned off instantaneously,\ndue to the propagation effect the observer would receive the emitted\nphotons at the angle $\\theta$ at $t = (1+z) (R_{cr}\/c)\n(\\theta^2\/2)$. Such a tail emission thus lasts for a duration of \n\\begin{equation}\nt_{tail} = (1+z) (R_{cr}\/c) (\\theta_{\\rm j}^2\/2) \\simeq 330~{\\rm s}\n\\left(\\frac{R_{cr} \\theta_j^2}{10^{13}}\\right) \\left(\\frac{1+z}{2}\n\\right). \n\\end{equation} \nif the line of sight is not too\nclose to the jet edge.  \n\nWe consider the simplest case of a jet moving with a constant bulk\nLorentz factor $\\Gamma$ (or a constant velocity $v$). The electrons\nare shock-heated up to a radius $R_{cr}$, beyond which no fresh\nshocked electrons are injected, and the already heated electrons cool\nrapidly. The comoving emission frequency $\\nu'$ is boosted to $\\nu =\n{\\cal D} \\nu'$ in the observer's frame, where ${\\cal D}=[\\Gamma\n(1-v\\cos\\theta\/c)]^{-1}$ is the Doppler factor, which is ${\\cal D}\n\\sim 2 \\Gamma$ for $\\theta \\ll 1\/\\Gamma$, and ${\\cal D} \\sim 2\/(\\Gamma\n\\theta^2)$ for $\\theta \\gg 1\/\\Gamma$. Since $t \\propto \\theta^2$, one\ngets ${\\cal D} \\propto t^{-1}$ for $\\theta \\gg 1\/\\Gamma$. \n\nThe observed flux $F_\\nu$ is related to the comoving surface\nbrightness $L'_{\\nu'}$ by \n\\begin{equation}\nF_\\nu \\propto L'_{\\nu'} {\\cal D}^2 \\propto (\\nu')^{-\\beta} {\\cal D}^2\n\\propto \\nu^{-\\beta} {\\cal D}^{2+\\beta} \\propto \\nu^{-\\beta}\nt^{-2-\\beta}~, \n\\end{equation}\nwhere $\\beta$ is the observed spectral index around the observed\nfrequency $\\nu$, and the last proportionality is valid for $1\/\\Gamma\n\\ll \\theta < \\theta_j$. With the standard convention $F_\\nu \\propto \n\\nu^{-\\beta} t^{-\\alpha}$, one has the well-known result for the\ncurvature effect (e.g. Kumar \\& Panaitescu 2000a; Dermer 2004; Fan \\&\nWel 2005; Dyks et al. 2005; Panaitescu et al. 2005)\n\\begin{equation}\n\\alpha=2+\\beta~.\n\\end{equation}\n\n\\bigskip\n\\subsubsection{Emission from a decelerating fireball}\nIn reality, the Lorentz factor of the shell could be decreasing\nright before the sudden cessation of the emission. This is valid for\nthe external shock case in the deceleration phase, or even in the\ninternal shock case. We perform numerical calculations to investigate\nsuch an \neffect\\footnote{In this paper, we use two codes to perform numerical \ncalculations. The first code was developed by J. Dyks (Dyks et\nal. 2005). The code can deal with the afterglow emission of an outflow\nwith an arbitrary axisymmetric structure and an arbitrary observer's\nviewing direction. The dynamics of the radial outflow \nis modeled similar to Granot \\& Kumar (2003). Only synchrotron\nradiation is taken into account at the moment and synchrotron\nself-absorption (which is irrelevant for X-ray emission) is\nignored. The synchrotron \nemissivity in the comoving frame is calculated by integration of the\nsynchrotron spectrum over the electron energy distribution. The latter\nis calculated by solving the continuity equation with the power-law\nsource function $Q=K\\gamma^{-p}$, normalized by a local injection rate\n(Moderski et al. 2000). The electrons cool down through synchrotron\nradiation and \nadiabatic expansion. All kinematic effects which affect the observed\nflux (e.g. Doppler boost, propagation time effects) have been taken\ninto account rigorously, following Salmonson (2003). The second code\nwas developed by Y. Z. Fan. This is an afterglow code developed from\nthe dynamical model of Huang et al. (2000), and has been used in\nseveral previous studies (e.g. Fan et al. 2004a; Fan et\nal. 2005a). The latest addition is to also include the kinetic\nevolution of the electron distribution (Moderski et al. 2000). We have\nused both codes in various calculations in this paper, and the results\nare consistent with each other. The figures presented in this paper\nare all generated from the second code. }.\nThe curvature effect for the sudden switching-off of radiation in\na decelerating outflow is presented Figure \\ref{Fig:Cur-ISM}, in which\nwe plot the X-ray lightcurve from an expanding jet blastwave with\n$E_{iso}=10^{52}$ergs, $\\Gamma_0=240$, $\\theta_j = 0.1$, $n=1{\\rm\ncm^{-3}}$, and $z=1$. We manually turn off the radiation at radii\n($R_{cut}$) of $10^{16}$ cm, $3\\times 10^{16}$ cm, and $6\\times\n10^{16}$ cm, respectively, and investigate the subsequent curvature\neffect. The radiation from fluid elements at $R<R_{cut}$ has been \nrigorously integrated. The result indicates that the $\\alpha =\n2+\\beta$ conclusion is essentially unchanged.\n\\begin{figure}\n\\epsscale{1.0}\n\\plotone{f2.eps}\n\\caption{The curvature effect for a decelerating ejecta. In the\ncalculation, the radiation from the fluid elements on the equal\narrival time surface for $R>R_{cut}$ are cut out, and the\nradiation from all other fluid elements is integrated. Here\n$R_{cut}=10^{16}, 3\\times 10^{16}, 6\\times 10^{16}$ cm are the radii\nat which the radiation is assumed to terminate abruptly. Notice\nthe insensitivity of the decay slope ($\\alpha=2+\\beta$) on the\ncut-off radius. The following parameters are adopted: the isotropic\nkinetic energy $E_{iso}=10^{52}$ergs, the initial Lorentz factor\n$\\Gamma_0=240$, the jet half-opening angle $\\theta_{\\rm j}=0.1$,  \nthe ambient density $n=1{\\rm cm^{-3}}$, the redshift $z=1$ (the\nluminosity distance $D_L=2.2 \\times 10^{28}$cm), the electron spectral\nindex $p=2.3$, and the electron and magnetic equipartition parameters\n$\\epsilon_e=0.1$ and $\\epsilon_B=0.01$, respectively.}. \n\\label{Fig:Cur-ISM}\n\\end{figure}\n\\subsubsection{Jet structure effects}\nAnother interesting issue is the jet structure. In principle, GRB jets\ncould be structured (Zhang \\& M\\'esz\\'aros 2002a; Rossi et al. 2002;\nZhang et al. 2004a; Kumar \\& Granot 2003). Since the curvature effect\nallows one to see the high-latitude emission directly, an interesting\nquestion is whether the decay slope associated with the curvature\neffect depends on the unknown jet structure. We have investigated this\neffect with the first code, and find that for a relativistic outflow\nthe temporal slope of the curvature effect is largely insensitive to\nthe jet structure as long as the viewing angle is not far off the\nbright beam. The main reason is\nthat the decrease of flux because of the curvature effect occurs on\na much shorter timescale than that for the jet structure to take\neffect. For a spectral index $-\\beta$ ($f_\\nu \\propto \\nu^{-\\beta}$), \nthe flux decreases by $m$ orders of magnitude after a time of $t_{\\rm\ncrv}=10^{m\/(2+\\beta)}t_{cr}$, where $t_{cr}$ is the observer time at\nwhich the curvature effect began. For a typical $\\beta \\sim 1$,\nthe flux drops by one order of magnitude after a short time $t_{\\rm\ncrv}\\sim 2t_{cr}$. A drop of three orders of magnitude occurs in no\nmore than a decade in time. On the other hand, the observer can\nperceive the switch-off of emissivity at an angle $\\theta$ measured\nfrom the line of sight at a time $t_{\\rm \\theta} \\simeq [1 +\n(\\theta\\Gamma)^2]t_{cr}$. One can see that the structure of the outflow\nmust have a typical angular scale smaller than $3\/\\Gamma$ in order\nto affect the observed flux before $10t_{cr}$. For $\\Gamma > 10^2$, the\nparameters of the outflow would have to vary strongly on a scale\nsmaller than one degree. Nonetheless, the effect of the jet struture\nwould start to play a noticeable role if the line of sight is outside\nthe bright beam. Detailed calculations are presented elsewhere (Dyks\net al. 2005).\n\n\\subsubsection{Factors leading to deviations from the $\\alpha=2+\\beta$\nrelation} \n\nAlmost all the {\\em Swift} XRT early afterglow lightcurves are\ncategorized by a steep decay component followed by a more ``normal'' \ndecaying afterglow lightcurve (Tagliaferri et al. 2005; Nousek et\nal. 2005) - see Segment \n1 in Fig.\\ref{XRTlc}. In most of these cases, the measured $\\alpha$\nand $\\beta$ values in this rapidly decaying component are close to the\n$\\alpha=2+\\beta$ relation, but in most cases, they do not match\ncompletely. This does not invalidate the curvature effect\ninterpretation, however, since in principle the following factors\nwould lead to deviations from the simple $\\alpha=2+\\beta$ law. \n\n1. The time zero point ($t_0$) effect. In GRB studies, the afterglow \nlightcurves are plotted in the log-log scale, with $t_0=0$ defined as\nthe trigger time of the burst. When discussing the late afterglows,\nshifting $t_0$ by the order of the burst duration $T_{90}$ does not\nmake much difference. When discussing the early afterglow and its\nconnection to the prompt emission, however, the decay power law\nindex (${\\rm d} \\ln F_\\nu \/ {\\rm d} \\ln t$) is very sensitive to the\n$t_0$ value one chooses. Correctly \nchoosing $t_0$ is therefore essential to derive the correct temporal\ndecay index $\\alpha$ (see e.g. Kobayashi et al. 2005b\nfor a detailed discussion about the $t_0$ issue). In the case of the\ninternal shock \nmodel, the case is straightforward. The observed pulses essentially\ntrack the behavior of the central engine (Kobayashi et al. 1997). \nEach pulse marks another re-start of the central engine, so that\n$t_0$ should be re-defined for each pulse when the curvature effect of\nthat pulse is considered. Keeping the same $t_0$ as the beginning of the\nfirst pulse (i.e. the GRB trigger) would inevitably lead to a false,\nvery steep power law decay for later pulses. Figure \\ref{Fig:Internal}\ngives an example to show this point. When a series of shells are\nsuccessively ejected, each pulse will be followed by its tail emission\ndue to the curvature effect, but most of these tails are buried under the\nmain emission of the next pulse. The observed curvature effect is only\nthe tail emission of the very last pulse. As a result, properly\nshifting $t_0$ is essential to interpret the steep decay component\nobserved in XRT bursts.\n\\begin{figure}\n\\epsscale{1.0}\n\\plotone{f3.eps}\n\\caption{The effect of $t_0$ on the lightcurves. The same internal shock\n$\\gamma$-ray pulse is calculated, but is assigned to three different\nejection times $t_{ej}$. The dotted, solid, and dash-dotted lines are\nfor $t_{\\rm ej}=0.0,~1.0,~10.0$s, respectively. The following parameters\nare adopted to calculate the internal shock pulses: the pulse\nluminosity $L_{\\rm pulse}=10^{51}{\\rm ergs~s^{-1}}$, the variability\ntime scale $\\delta t=0.1$s, $\\theta_{\\rm j}=0.2$rad, $\\epsilon_{\\rm\ne}=0.5$, $\\epsilon_{\\rm B}=0.1$, $p=2.5$, and $z=1$.} \n\\label{Fig:Internal}\n\\end{figure}\n2. The superposition effect. The observed steep-to-shallow transition\nin the early phase of XRT bursts (Tagliaferri et al. 2005) suggests\nthat by the end of the tail emission, the fireball is already\ndecelerated, and the forward shock emission also contributes to the\nX-ray band. As a result, the observed steep decay should also include\nthe contribution from the forward shock. Assuming the later has a\ntemporal decay index $-w$, the X-ray flux at the early phase should\nread \n\\begin{equation}\nF_{\\nu}(t)=A\\left(\\frac{t-t_{0,i}}{t_{0,i}}\\right)^{-(2+{\\beta})}\n+B\\left(\\frac{t-t_{0,e}}{t_{0,e}}\\right)^{-w}, \n\\label{Eq:Fnu_obs}\n\\end{equation}\nwhere $A$ and $B$ are constants, and $t_{0,i}$, $t_{0,e}$ are the\ntime zero points for the steep decay component (presumably of the\ninternal origin) and for the shallow decay component (presumably of\nthe external origin), respectively. \nIn the intermediate regime between the two power-law segments, both\ncomponents are important, and the observed $\\alpha$ should be\nshallower than $2+\\beta$ during the steep decaying phase. This effect\nflattens the decay instead of steepening it. \n\n3. If with the above two adjustments, the observed $\\alpha$ is\nstill steeper than $2+\\beta$, one can draw the conclusion that the\nsolid angle of the emitting region is comparable to or smaller than\n$1\/\\Gamma$. This would correspond to a patchy shell (Kumar \\& Piran\n2000a) or a mini-jet (Yamazaki et al. 2004). A caveat on such an\ninterpretation is that the probability for the line of sight sitting\nright on top of such a very narrow patch\/mini-jet is very small.\nAs a result, this model can not interpret an effect that seems to be a\ngeneral property of X-ray afterglows.\n\n4. If with the first two adjustments, the observed $\\alpha$ is\nflatter than $2+\\beta$ but is still much steeper than that expected\nfrom a forward shock model, there could be two possibilities. One is\nthat the emission is still from the internal dissipation of energy but\nthe emission in the observational band does not cease abruptly. This\nis relevant when the observational band is below the cooling\nfrequency. The adiabatic cooling therefore gives a decay slope of\n$\\sim (1+3\\beta\/2)$ rather than $\\sim (2+\\beta)$ (e.g. Sari \\& Piran\n1999; Zhang et al. 2003). The second possibility is that one is\nlooking at a structured jet (Zhang \\& M\\'esz\\'aros 2002a; Rossi et\nal. 2002), with the line of sight significantly off-axis. The\ncurvature effect in such a configuration typically gives a flatter\ndecay slope than $2+\\beta$ (Dyks et al. 2005). This is particularly\nrelevant for X-ray rich GRBs or X-ray flashes for which a large\nviewing angle is usually expected (Zhang et al. 2004; Yamazaki et\nal. 2004). Further analyses of XRT data suggest that at least in some\nGRBs, the decay slope is shallower than $2+\\beta$ (O'Brien et\nal. 2005). The above two possibilities are in particular relevant for\nthese bursts.\n\nWe suggest that most of the rapid-decay lightcurves observed by the\n{\\em Swift} XRT may be interpreted as GRB (or X-ray flare) tail\nemission through the curvature effect, with the first two adjustments\ndiscussed above. In order to test this hypothesis, after the\nsubmission of this paper we have performed more detailed data analyses\non a large sample of XRT bursts (Liang et al. 2005). By assuming that\nthe decay slope should be $2+\\beta$, we search the appropriate $t_0$\nthat allows such an assumption to be satisfied. It is found that $t_0$\nis usually at the beginning of the last pulse (for the steep decay\nfollowing the prompt emission) or at the beginning of the X-ray flare\n(for steep decay following flares). This fact strongly suggests that\nthe curvature effect is likely to be the correct interpretation, and\nat the same time lends strong support to the internal origin of the\nprompt emission and X-ray flares (see \\S\\S2.3 \\& 5.7 for the arguments\nin favor of the internal models for both components).\n\nAnother potential test of the curvature effect is to search for a\ncorrelation between the spectral peak energy ($E_{pk}$) and the flux\nat the peak ($F_{pk}$) (Dermer 2004). This requires a well measured\n$E_{pk}$ in the XRT band. In most cases, the XRT spectrum is\nconsistent with a single power law. More detailed analyses on future\nbright X-ray flares are desirable to perform such a test.\n\n\n\\subsection{Theoretical implications}\n\nThe current {\\em Swift} XRT observations of the early rapid-to-shallow\ndecay transition of the X-ray lightcurves (Tagliaferri et al. 2005),\nwhen interpreted as the curvature effect, have profound\nimplications for the understanding of the GRB phenomenon. \n\n1. It gives a direct observational proof that the GRB prompt emission\nvery likely comes from a different site than the afterglow\nemission. This suggests that the emission either comes from the\ninternal shocks due to the collisions among many engine-ejected shells\n(Rees \\& M\\'esz\\'aros 1994; Paczynski \\& Xu 1994), or is due to\nmagnetic or other dissipation processes at a radius smaller than the\nfireball deceleration radius (e.g. Drenkhahn \\& Spruit 2002; Rees \\&\nM\\'esz\\'aros 2005). In both scenarios, the energy dissipation region\nis well inside the region where the deceleration of the whole fireball\noccurs. \n\n2. An interesting fact is that in most cases, after the\nprompt emission, the X-ray emission level (that spectrally\nextrapolated from the BAT data) drops by several orders of\nmagnitude (through the curvature effect, in our interpretation) before\n``landing'' on the  \nafterglow emission level. One could roughly estimate the expected\n``drop-off''. The flux level in the XRT band during the prompt phase\ncould be roughly estimated as $F_{\\nu,X}^{prompt} \\propto \n(E_{\\gamma,iso}\/T_{90}) (E_{\\rm XRT}\/E_p)^{\\hat\\alpha+2}$, where\n$E_{\\gamma,iso}$ is the isotropic energy of the emitted gamma-rays,\n$T_{90}$ is the duration of the burst, $E_{\\rm XRT} \\sim 5$ keV is the\ntypical energy in the XRT band, $E_p \\sim 100$ keV is the typical\npeak energy in the GRB spectrum, and $\\hat\\alpha \\sim -1$ is the\nlow-energy spectral index for a Band-spectrum (Band et al. 1993).\nAssuming that the X-ray band for the afterglow emission is above both\nthe typical synchrotron \nfrequency $\\nu_m$ and the cooling frequency $\\nu_c$ (which \nis usually the case for the ISM model, see eqs.[\\ref{tm}],[\\ref{tc}]),\nthe X-ray afterglow flux level can be estimated as (e.g. Freedman \\&\nWaxman 2001) $F_{\\nu,X}^{ag} \\propto \\epsilon_e E_{iso}\/t$, where\n$E_{iso}$ is the isotropic energy of the afterglow kinetic energy, and\n$\\epsilon_e$ is the electron equipartition parameter in the\nshock\\footnote{When the synchrotron self-Compton process dominates the\ncooling, the discussion could be more complicated.}.\nThe flux contrast can be estimated as\n\\begin{equation}\n\\frac{F_{\\nu,X}^{prompt}} {F_{\\nu,X}^{ag}} \\sim \\left(\\frac{E_{\\gamma,iso}}\n{E_{iso}}\\right) \\left(\\frac{t}{T_{90}}\\right) \\left[\\frac{(E_{\\rm\nXRT}\/E_p)^{\\hat\\alpha+2}}{\\epsilon_e}\\right]~.\n\\end{equation}\nFor typical parameters, one has $(E_{\\rm XRT}\/E_p)^{\\hat\\alpha+2} \\sim\n0.05$ and $\\epsilon_e \\sim 0.1$, so that the term in the bracket\n$\\lesssim 1$. Although $t>T_{90}$ would generally suggest that\n$F_{\\nu,X}^{prompt}$ should be higher than $F_{\\nu,X}^{ag}$, the\nlarge contrast between the two components observed in many bursts\nis usually not accounted for unless $E_{\\gamma,iso}$ is (much) larger\nthan $E_{iso}$. This refers to a very high apparent GRB radiation\nefficiency - even higher than the one estimated using the late X-ray\nafterglow data (Lloyd-Ronning \\& Zhang 2004)\\footnote{This could be\nattributed to the shallow decay injection phase (segment II in\nFig.\\ref{XRTlc} as discussed in \\S3.2. Because of the injection, the\neffective $E_{iso}$ in the early epochs is smaller than that in the\nlater epochs. As a result, a larger\n$F_{\\nu,X}^{prompt}-F_{\\nu,X}^{ag}$ contrast is expected.}. \nThe commonly-invoked internal shock model predicts a low emission\nefficiency (e.g. Panaitescu et al. 1999; Kumar 1999). Understanding\nsuch a high apparent radiation efficiency is therefore desirable (see\ne.g. Beloborodov 2000; Kobayashi \\& Sari 2001). \n\n3. The common steep-to-shallow transition feature indicates that the\nfireball has already been decelerated at the time when the GRB tail\nemission fades. Otherwise, one would see an initially rising\nlightcurve peaking at the fireball deceleration time. This fact alone\nsets a lower limit to the intial \nLorentz factor of the fireball, since the deceleration time $t_{dec}$\nmust be earlier than the transition time $t_{b1}$. The numerical\nexpression is\n\\begin{equation}\n\\Gamma_0 \\geq 125 \\left(\\frac{E_{\\gamma,iso,52}}{\\eta_\\gamma\nn}\\right)^{1\/8} t_{b1,2}^{-3\/8} \\left(\\frac{1+z}{2}\\right)^{3\/8}, \n\\label{Gam0}\n\\end{equation}\nwhere $E_{\\gamma,iso}$ is the isotropic gamma-ray energy (which is an\nobservable if the redshift $z$ is known), \n$\\eta_\\gamma = E_{\\gamma,iso} \/ E_{iso}$ is a conversion factor\nbetween the isotropic afterglow energy $E_{iso}$ and $E_{\\gamma,iso}$. \n{Throughout the paper, the convention $Q_x=Q\/10^x$ is\nadopted in cgs units.}\nApplying the method to the bursts with\nmeasured $z$ (Chincarini et al. 2005), we get the lower limits og\n$\\Gamma_0$ for several\nbursts (Table \\ref{Tab:Gamma}). Given the weak dependence on the\nunknown parameters (i.e. $(\\eta_\\gamma n)^{-1\/8}$), we conclude that\nthe data suggest that GRBs are highly relativistic events with typical\nLorentz factors higher than 100. This is an independent method, as\ncompared with previous ones using the high energy spectrum\n(e.g. Baring \\& Harding 1997; Lithwick \\& Sari 2001), the reverse shock\ndata (Sari \\& Piran 1999; Wang et al. 2000; Zhang et al. 2003), and\nthe superluminal expansion of the radio afterglow source image (Waxman\net al. 1998).\n\\begin{table}\n\\caption{\nConstraints on the initial Lorentz factors of several GRBs.}\n\n\\begin{tabular}{ccccc}\n\\hline\\hline  \n\nGRB &  $z$  &  $t_{b1}({\\rm s})$  &  $E_{\\gamma,iso,52}$\\tablenotemark{a} &\n$\\Gamma_0 $ \\\\\n\\hline\n050126...   & 1.290\\tablenotemark{b}  &  $\\sim 110$ &  0.77   & $> 120 (\\eta_\\gamma n)^{-1\/8}$ \\\\ \n050315...   & 1.949\\tablenotemark{c}  &  $\\sim 400$ &  2.77   & $> 100 (\\eta_\\gamma n)^{-1\/8}$ \\\\\n050319...   & 3.240\\tablenotemark{d}  &  $\\sim 400$ &  5.12   & $> 120 (\\eta_\\gamma n)^{-1\/8}$ \\\\\n050401...   & 2.900\\tablenotemark{e}  &  $\\sim 130$ & 27.49   & $> 220 (\\eta_\\gamma n)^{-1\/8}$ \\\\\n\\hline\n\\end{tabular}\n\\tablenotetext{a}{Chincarini et al. 2005}\n\\tablenotetext{b}{Berger et al. 2005}\n\\tablenotetext{c}{Kelson \\& Berger 2005}\n\\tablenotetext{d}{Fynbo et al. 2005a}\n\\tablenotetext{e}{Fynbo et al. 2005b}\n\\label{Tab:Gamma}\n\\end{table}\n\n\n\\section{Forward shock emission}\n\\label{sec:FS} \n\nAfter the rapid fading of the GRB tail emission, usually the forward\nshock emission component gives the main contribution to \nthe early X-ray afterglow lightcurves. The lightcurve shape depends on\nthe density profile of the ambient medium (i.e. ISM or wind).\nIn the ``standard'' case (i.e. adiabatic evolution with promp\ninjection of energy), the fireball energy is\nessentially constant during the deceleration phase. The bulk\nLorentz factor $\\Gamma \\propto R^{-3\/2}$ for the ISM case and \n$\\Gamma \\propto R^{-1\/2}$ for the wind case. When the\nbulk Lorentz factor $\\Gamma$ is larger than $\\theta_j^{-1}$, where\n$\\theta_j$ is the jet opening angle (or the viewing angle of a\nstructured jet), the system is simply determined by the ratio of the\nisotropic afterglow energy $E_{iso}$ and the ambient density $n$ (or\nthe $A$ parameter in the wind model). Such a ``normal'' decay phase\ncorresponds to the segment III in the synthetic lightcurve\n(Fig. \\ref{XRTlc}). When $\\Gamma$ becomes smaller\nthan $\\theta_j^{-1}$, the lightcurve steepens because of the\ncombination of the jet edge effect and the possible sideways expansion\neffect (Rhoads 1999; Sari et al. 1999; Panaitescu \\& M\\'esz\\'aros\n1999). The bulk Lorentz factor decreases exponentially with\nradius. This is known as a ``jet break'', and the post-break segment\ncorresponds to the segment IV in Fig. \\ref{XRTlc}.\n\nDuring the early evolution of the fireball, the forward shock may be\ncontinuously refreshed with additional energy. This could be either\nbecause of a continuous operation of the central engine (Dai \\& Lu\n1998a; Zhang \\& M\\'esz\\'aros 2001; Dai 2004), or because of a power\nlaw distribution of the Lorentz factors in the ejecta that results in\nslower ejecta catching up with the decelerated fireball at later times\n(Rees \\& M\\'esz\\'aros 1998; Panaitescu et al. 1998; Kumar \\& Piran\n2000b; Sari \\& M\\'esz\\'aros 2000; Zhang \\& M\\'esz\\'aros 2002b), or\nbecause of the transfering of the Poynting-flux energy to the medium\nwhen a Poynting-flux dominated flow is decelerated (e.g. Zhang \\&\nKobayashi 2005). The cannonical XRT lightcurve (Fig.\\ref{XRTlc})\nindeed shows a shallow decay phase (segment II), which we argue is\ndue to continuous energy injection.\n\n\\subsection{Standard afterglow models}\n\\label{sec:FS1}\n\nFor the convenience of the later discussion, we summarize\nthe ``standard'' early forward shock X-ray afterglow properties as\nfollows. \n\n{\\em The ISM model} (e.g. Sari et al. 1998). The typical synchrotron\nfrequency and the cooling frequency are $\\nu_m = 6.5\\times 10^{14}\n~{\\rm Hz} ~ \\epsilon_{B,-2}^{1\/2} \\epsilon_{e,-1}^2 E_{52}^{1\/2}\nt_3^{-3\/2} [(1+z)\/2]^{1\/2}$, and $\\nu_c = 2.5 \\times 10^{16} ~{\\rm Hz}\n~ (1+Y)^{-2} \\epsilon_{B,-2}^{-3\/2} E_{52}^{-1\/2} n^{-1} t_3^{-1\/2}\n[(1+z)\/2]^{-1\/2}$, respectively, where $E$ is the isotropic kinetic \nenergy of the fireball, $n$ is the ISM density, $\\epsilon_e$ and\n$\\epsilon_B$ are shock equipartition parameters for electrons and\nmagnetic fields, respectively, $Y$ is the energy ratio between the\ninverse Compton component and the synchrotron component, $z$ is the\nredshift, $t$ is the observer's time. Both frequencies decrease with\ntime. The time interval for $\\nu_m$ and $\\nu_c$ to cross the XRT\nenergy band (0.5-10 keV) from above can be expressed as\n\\begin{eqnarray}\nt_m & = & (4-30)~{\\rm s}~ \\epsilon_{B,-2}^{1\/3} \\epsilon_{e,-1}^{4\/3}\nE_{52}^{1\/3} \\left( \\frac{1+z}{2} \\right)^{1\/3} \\label{tm} \\\\\nt_c & = & (0.1-40) ~{\\rm s}~ (1+Y)^{-4} \\epsilon_{B,-2}^{-3}\nE_{52}^{-1} n^{-2} \\left( \\frac{1+z}{2} \\right)^{-1}~. \\label{tc}\n\\end{eqnarray} \nThe epoch when the fireball switches from fast cooling ($\\nu_c <\n\\nu_m$) to slow cooling ($\\nu_c > \\nu_m$) is defined by requiring\n$\\nu_m=\\nu_c$, which reads\n\\begin{equation}\nt_{mc} = 26 ~{\\rm s} ~ (1+Y)^2 \\epsilon_{B,-2}^2 \\epsilon_{e,-1}^2\nE_{52} n \\left( \\frac{1+z}{2} \\right)~.\n\\label{tmc}\n\\end{equation}\nFor comparison, the time when the fireball is decelerated (thin shell\ncase) is given by \n\\begin{eqnarray}\nt_{dec} & = & \\left( \\frac{3E} {4\\pi n m_p c^2 \\Gamma_0^2} \\right)^{1\/3}\n\\frac{1}{2 \\Gamma_0^2 c} \\nonumber \\\\\n& = & 180 ~{\\rm s}~ (E_{52}\/n)^{1\/3}\n\\Gamma_{0,2}^{-8\/3} \\left( \\frac{1+z}{2} \\right)~,\n\\label{tdec}\n\\end{eqnarray}\nwhere $\\Gamma_0$ is the initial Lorentz factor of the fireball.\nWe can see that for typical parameters, the XRT band is already in the\nregime of $\\nu_X > {\\rm max} (\\nu_m, \\nu_c)$ when deceleration\nstarts. Also, the blast wave evolution has usually entered the slow\ncooling regime where the radiative losses are not\nimportant\\footnote{In certain parameter regimes, the condition\n$t_{dec} < t_{mc}$ could be \nsatisfied, and in the temporal regime $t<t_{mc}$, the blastwave is in\nthe fast cooling ($\\nu_c < \\nu_m$) domain, so the radiative loss could \naffect the blastwave dynamics (e.g. B\\\"ottcher \\& Dermer\n2000; Wu et al. 2005).}.  Under such conditions, the temporal index\nis\\footnote{Here and below, the convention $F_\\nu(X) \\propto t^{-\\alpha_X} \n\\nu^{-\\beta_X}$ is adopted since in the X-ray band both the temporal\nand the spectral indices are negative. Also $p=2.2$ is adopted for\ntypical numerical values.} $\\alpha_X=(3p-2)\/4 \\sim\n1.15$, and the\nspectral index is $\\beta_X= p\/2 \\sim 1.1$ (photon index 2.1). The\nrelation between the two indices is $\\alpha_X=(3 \\beta_X-1)\/2$. \nIn view that $t_c$ is very sensitive to $\\epsilon_B$, one can \nstill get the regime $\\nu_m < \\nu_X < \\nu_c$ if $\\epsilon_B$ is large\n(say, around 0.1). In such a case, $\\alpha_X=3(p-1)\/4 \\sim 0.9$,\n$\\beta_X = (p-1)\/2 \\sim 0.6$, and $\\alpha_X = (3\/2) \\beta_X$. The\nabove two cases have been commonly observed in early X-ray afterglows\nof many Swift bursts (e.g. segment III in Fig. \\ref{XRTlc}),\nsuggesting that the fireball shock model can successfully interpret\nthe general properties of GRB afterglows, and that most GRBs are born\nin a constant density medium. This conclusion is consistent with\nprevious analyses (Panaitescu \\& Kumar 2002; Yost et al. 2003). \n\n{\\em The wind model} (e.g. Chevalier \\& Li 2000). The typical\nsynchrotron frequency and the cooling frequency are $\\nu_m = 1.3\\times\n10^{15} ~{\\rm Hz} ~ \\epsilon_{B,-2}^{1\/2} \\epsilon_{e,-1}^2 E_{52}^{1\/2}\nt_3^{-3\/2} [(1+z)\/2]^{1\/2}$ (notice that $\\nu_m$ in the wind case has\nthe same parameter dependences as the ISM case but is larger\nby a factor of $\\sim 2$, e.g. Dai \\& Lu 1998b), and $\\nu_c = 6.8 \\times\n10^{12} ~{\\rm Hz} ~ (1+Y)^{-2} \\epsilon_{B,-2}^{-3\/2} E_{52}^{1\/2}\nA_*^{-2} t_3^{1\/2} [(1+z)\/2]^{-3\/2}$, where $A_*= (\\dot M_W\/4\\pi V_W)\n\/ (5\\times 10^{11} ~{\\rm g~cm^{-1}})$ is the typical wind parameter, \n$\\dot M_W$ is the mass loss rate, and $V_W$ is the wind velocity. \nA distinguishing property of the wind model is that $\\nu_c$ increases\nwith time. Similarly, the time interval for $\\nu_m$ to cross the XRT\nband from above is\n\\begin{equation}\nt_m  =  (6.6-49)~{\\rm s}~ \\epsilon_{B,-2}^{1\/3} \\epsilon_{e,-1}^{4\/3}\nE_{52}^{1\/3} \\left( \\frac{1+z}{2} \\right)^{1\/3} \\label{tm2}~, \n\\end{equation}\nand the time interval for $\\nu_c$ to cross the band from below is\n\\begin{eqnarray}\nt_c & = & (3.1\\times 10^{11} - 1.2\\times 10^{14}) ~{\\rm s}~ (1+Y)^4\n\\epsilon_{B,-2}^3 E_{52}^{-1} A_*^4 \\left( \\frac{1+z}{2} \\right)^{3}\n\\nonumber \\\\ \n& = & (0.3 - 120) ~{\\rm s}~ (1+Y)^4 \\epsilon_{B,-2}^3 E_{52}^{-1}\nA_{*,-3}^4 \\left( \\frac{1+z}{2} \\right)^{3} ~. \n\\end{eqnarray}\nThe critical time for the fast-slow cooling transition is\n\\begin{eqnarray}\nt_{mc} & = & 1.4\\times 10^4 ~{\\rm s}~ (1+Y) \\epsilon_{e,-1}\n\\epsilon_{B,-2} A_* \\nonumber \\\\\n& = & 14 ~{\\rm s}~ (1+Y) \\epsilon_{e,-1}\n\\epsilon_{B,-2} A_{*,-3} ~.\n\\end{eqnarray}\nThe deceleration time is essentially the duration of the burst, i.e. \n$t_{dec} \\sim T = 20 ~{\\rm s}~ T_{1.3}$, since for typical parameters,\nthe wind model is the so-called ``thick shell'' case (e.g. Chevalier\n\\& Li 2000; Kobayashi \\& Zhang 2003b). For a typical\nwind parameter $A_* \\sim 1$, the X-ray lightcurve is very simple. For\n$t<t_m$, one has $\\nu_c < \\nu_X < \\nu_m$, so that $\\alpha_X = 1\/4$,\n$\\beta_X = 1\/2$ and $\\alpha_X = (1 - \\beta_X)\/2$. When $t > t_m$,\nduring most of the observational time of interest, one has $\\nu_X >\n{\\rm max} (\\nu_m, \\nu_c)$, so that $\\alpha_X = (3p-2)\/2 \\sim 1.15$, \n$\\beta_X = p\/2 = 1.1$ (photon index 2.1), and $\\alpha_X=(3 \\beta_X\n-1)\/2$.  The switching between the fast cooling and slow cooling\nregimes does not influence the temporal and spectral indices in the\nX-ray band. Only when $A_* < 0.01$, i.e. $t_c$ falls into the range of\nobservational interest, does a new temporal\/spectral domain appear.\nWhen $t>t_c$, one has $\\nu_m < \\nu_X < \\nu_c$, $\\alpha_X = (3p-1)\/4\n\\sim 1.4$, $\\beta_X = (p-1)\/2 \\sim 0.6$ (photon index 1.6), and \n$\\alpha_X=(3 \\beta_X + 1)\/2$. Such a feature has been used to\ninterpret GRB 050128 (Campana et al. 2005)\\footnote{As discussed in\n\\S\\ref{sec:FS2}, after collecting more data, we now believe that the\nshallow-to-normal decay observed in GRB 050128 is more likely due to\nthe transition from the energy injection phase to the standard phase\n(without injection).}. If $A_*$ is not much smaller than unity,\nthe blastwave is in the fast cooling regime, and radiative losses\ncould be substantial (B\\\"ottcher \\& Dermer 2000). A detailed analysis\nhas been presented in Wu et al. (2005).\n\n{\\em The jet model} (e.g. Rhoads 1999; Sari et al. 1999).\nAfter the jet break, the temporal decay index is predicted to be\n$\\alpha_X = p$. This is derived by assuming significant\nsideways expansion. This result is independent on whether the \nX-ray band is below or above $\\nu_c$, and whether the medium is an ISM\nor a stellar wind. For the latter, the time scale for the lightcurve\nto achieve the asymptotic $-p$ index is typically longer than that in\nthe ISM case (e.g. Kumar \\& Panaitescu 2000b; Gou et al. 2001). \n\nAll the above discussions apply for the case of $p>2$. For $p<2$, the\ncase could be different. Dai \\& Cheng (2001) proposed one scenario to\ndeal with the case of $p < 2$, while Panaitescu \\& Kumar (2002)\nextended the treatment of $p>2$ case to the $p<2$ regime.\n\n\n\\subsection{Refreshed shock models}\n\\label{sec:FS2}\n\nIf there is significant continuous energy injection into the fireball\nduring the deceleration phase, the forward shock keeps being\n``refreshed'', so that it decelerates less rapidly than in\nthe standard case. The bulk Lorentz factor of the fireball decays more\nslowly than $\\Gamma \\propto R^{-3\/2} (R^{-1\/2})$ for the ISM case \n(the wind case), respectively.\n\nThere are three possible physical origins for the refreshed shocks.\n\n1. The central engine itself is longer lasting, e.g. behaving as \n\\begin{equation}\nL(t) = L_0 (t\/t_b)^{-q}~.\n\\label{Lt}\n\\end{equation}\nThe dynamical evolution and the radiation signature of such a system\nhas been discussed in detail in Zhang \\& M\\'esz\\'aros (2001). A\nspecific model for such an injection case, i.e. the energy injection\nfrom the initial spin down from a millesecond pulsar (preferably a\nmillisecond magnetar) was discussed in that paper and earlier in\nDai \\& Lu (1998a). In such a specific model, $q=0$ is required\naccording to the spin-down law. Alternatively, the continued engine\nactivity could be due to continued infall onto a central black hole,\nresulting in the time dependence eq.(\\ref{Lt})\\footnote{The black hole\n- torus system typically has $q=5\/3$ at later times (MacFadyen et\nal. 2001; Janiuk et al. 2004), which has no effect on the blastwave\nevolution.}. In general, for an adiabatic \nfireball, the injection would modify the blastwave dynamics as long as\n$q<1$ (Zhang \\& M\\'esz\\'aros 2001). The energy in the fireball\nincreases with time as $E_{iso} \\propto\\ t^{1-q}$, so that \n\\begin{eqnarray}\n\\Gamma \\propto R^{-\\frac{2+q}{2(2-q)}} \\propto t^{-\\frac{2+q}{8}}, & R\\propto\nt^{\\frac{2-q}{4}}, & {\\rm\nISM} \\\\\n\\Gamma \\propto R^{-\\frac{q}{2(2-q)}} \\propto t^{-\\frac{q}{4}}, & R \\propto\nt^{\\frac{2-q}{2}}, & {\\rm wind} \n\\end{eqnarray}\nIt is then straightforward to work out the temporal indices for\nvarious temporal regimes.\n\n\n{\\it The ISM model}. The typical synchrotron\nfrequency $\\nu_m \\propto \n\\Gamma^2 \\gamma_e B \\propto \\Gamma^4 \\propto t^{-(2+q)\/2}$, the\nsynchrotron cooling frequency $\\nu_c \\propto \\Gamma^{-1} B^{-3} t^{-2}\n\\propto \\Gamma^{-4} t^{-2} \\propto t^{(q-2)\/2}$, and the peak flux\ndensity $F_{\\nu,max} \\propto N_e B \\Gamma \\propto t^{1-q}$,\nwhere $B \\propto \\Gamma$ is the comoving \nmagnetic field strength, $\\gamma_e \\propto \\Gamma$ is the typical\nelectron Lorentz factor in the shocked region, and $N_e \\propto R^3$\nis the total number of the emitting electrons. The temporal indices\n$\\alpha$ for various spectral regimes and their relationships with the\nspectral indices $\\alpha(\\beta)$ are listed in Table\n\\ref{Tab:alpha-beta}. \n\n{\\it The wind model}. In the wind case,\nthe ambient density is $n\\propto R^{-2}$, where $R$ is the radial\ndistance of the shock front to the central source. The typical\nsynchrotron frequency $\\nu_m \\propto \\Gamma^2 \\gamma_e B \\propto\n\\Gamma^3 B \\propto t^{-(2+q)\/2}$, the \nsynchrotron cooling frequency $\\nu_c \\propto \\Gamma^{-1} B^{-3} t^{-2}\n\\propto \\Gamma^{-4} t^{-2} \\propto t^{(2-q)\/2}$, and the peak flux\ndensity $F_{\\nu,max} \\propto N_e B \\Gamma \\propto \\Gamma^2 \\propto\nt^{-q\/2}$, where $B \\propto \\Gamma R^{-1}$ is the \ncomoving magnetic field strength, and $N_e \\propto R$ is the total\nnumber of the emitting electrons. The \ntemporal indices $\\alpha$ for various spectral regimes and their\nrelationships with the spectral indices $\\alpha(\\beta)$ are listed in\nTable \\ref{Tab:alpha-beta}. \n\nIn order for the central engine to continuously feed the\nblast wave, the Lorentz factor of the continuous flow must be  \n(much) larger than that of the blast wave. It could be a \nPoynting-flux-dominated flow. This is not difficult to satisfy \nsince the blast wave keeps decelerating. There could be a reverse\nshock propagating into the continuous ejecta, but the radiation\nsignature  of the reverse shock is typically not in the X-ray band\n(e.g. Zhang \\& M\\'esz\\'aros 2001).\n\n2. The central engine activity may be brief (e.g. as brief as the\nprompt emission itself, but at the end of the prompt phase, the ejecta\nhas a range of Lorentz factors, e.g., the amount of ejected mass\nmoving with Lorentz factors greater than $\\gamma$ is (Rees \\&\nM\\'esz\\'aros 1998; Panaitescu et al. 1998; Sari \\& M\\'esz\\'aros 2000)\n\\begin{equation}\nM(>\\gamma) \\propto \\gamma^{-s}~.\n\\end{equation}\nThe total energy in the fireball increases as $E_{iso} \\propto\n\\gamma^{1-s} \\propto \\Gamma^{1-s}$, so that\n\\begin{eqnarray}\n\\Gamma \\propto R^{-\\frac{3}{1+s}} \\propto t^{-\\frac{3}{7+s}}, & R\\propto\nt^{\\frac{1+s}{7+s}}, & {\\rm ISM} \\\\\n\\Gamma \\propto R^{-\\frac{1}{1+s}} \\propto t^{-\\frac{1}{3+s}}, & R \\propto\nt^{\\frac{1+s}{3+s}}, & {\\rm wind} \n\\end{eqnarray}\nOne can then work out the temporal decay indices in various spectral\nregimes (e.g. Rees \\& M\\'esz\\'aros 1998; Sari \\& M\\'esz\\'aros 2000). \nAlternatively, for each $s$ value, one can find an effective $q$\nvalue that mimics the $s$ effect, or vice versa. This gives\n\\begin{eqnarray}\ns =\\frac{10-7q}{2+q}, ~~ q=\\frac{10-2s}{7+s}, & {\\rm ISM} \\label{s-q-1}\\\\\ns =\\frac{4-3q}{q}, ~~ q=\\frac{4}{3+s}. & {\\rm wind} \\label{s-q-2}\n\\end{eqnarray}\nIn Table \\ref{Tab:alpha-beta}, the explicit $s$-dependences are not\nlisted, but they could be inferred from eqs.(\\ref{s-q-1}) and\n(\\ref{s-q-2}). \n\nIn this second scenario, the central engine need not last\nlong. All the material could be ejected promptly. The continuous\ninjection is due to the different velocities of the ejecta. Initially \nas the blast wave moves with high speed, the slower ejecta lag \nbehind and have no effect on the blastwave evlolution. They later \nprogressively pile up onto the blast wave as the latter decelerates. \nOnly when $s > 1$ does one expect a change in the fireball dynamics.\nThis corresponds to $q<1$. For $q=0.5$, one gets $s=2.6$ for the ISM case \nand $s=5$ for the wind case.\n\\clearpage\n\\begin{table}\n\\caption{Temporal index $\\alpha$ and spectral index\n$\\beta$ in various afterglow models.\\label{Table1}} \n\\tabletypesize{\\scriptsize}\n\\begin{tabular}{llllll}\n\\hline\\hline  \n& & no injection & &  injection & \\\\\n\n& $\\beta$ & $\\alpha $  &  $\\alpha (\\beta)$ & $\\alpha$ & $\\alpha (\\beta)$  \\\\\n\\hline\nISM & slow cooling \\\\\n\\hline\n$\\nu<\\nu_m$   &  $-{1 \\over 3}$  &   $-{1\\over 2}$ & $\\alpha={3\\beta \\over 2}$ & ${5q-8 \\over 6}$ (-0.9) & $\\alpha=(q-1)+\\frac{(2+q)\\beta}{2}$\\\\\n$\\nu_m<\\nu<\\nu_c$   &  ${{p-1 \\over 2}}$ (0.65)  &  ${3(p-1)\\over 4}$\n(1.0)   &  $\\alpha={3\\beta \\over 2}$ & ${(2p-6)+(p+3)q \\over 4}$ (0.3) & $\\alpha=(q-1)+\\frac{(2+q)\\beta}{2}$\\\\\n$\\nu>\\nu_c$   &  ${{p\\over 2}}$ (1.15)  &   ${3p-2 \\over 4}$ (1.2) & $\\alpha={3\\beta-1 \\over 2}$ & ${(2p-4)+(p+2)q\\over 4}$ (0.7) & $\\alpha=\\frac{q-2}{2}+\\frac{(2+q)\\beta}{2}$ \\\\\n\n\\hline\nISM & fast cooling \\\\\n\\hline\n$\\nu<\\nu_c$   &  $-{1\\over 3}$  &   $-{1\\over 6}$ &  $\\alpha={\\beta \\over 2}$  &  ${7q-8 \\over 6}$ (-0.8) & $\\alpha=(q-1)+\\frac{(2-q)\\beta}{2}$\\\\\n$\\nu_c<\\nu<\\nu_m$   &  ${1\\over 2}$  &  ${1\\over 4}$  & $\\alpha={\\beta \\over 2}$ &  ${3q-2 \\over 4}$ (-0.1)  &  $\\alpha=(q-1)+\\frac{(2-q)\\beta}{2}$ \\\\\n$\\nu>\\nu_m$   &  ${p\\over 2}$ (1.15)  &   ${3p-2\\over 4}$ (1.2)     &  $\\alpha={3\\beta-1 \\over 2}$  &  ${(2p-4)+(p+2)q\\over 4}$ (0.7) & $\\alpha=\\frac{q-2}{2}+\\frac{(2+q)\\beta}{2}$ \\\\\n\n\\hline\nWind & slow cooling \\\\\n\\hline\n$\\nu<\\nu_m$   &  $-{1\\over 3}$   &   0  &  $\\alpha={3\\beta+1 \\over 2}$ & ${q-1 \\over 3}$ (-0.2) & $\\alpha=\\frac{q}{2}+\\frac{(2+q)\\beta}{2}$\\\\\n$\\nu_m<\\nu<\\nu_c$   &  ${p-1\\over 2}$ (0.65)  &   ${3p-1\\over 4}$ (1.5)   &   $\\alpha={3\\beta+1 \\over 2}$ & ${(2p-2)+(p+1)q \\over 4}$ (1.1)& $\\alpha=\\frac{q}{2}+\\frac{(2+q)\\beta}{2}$\\\\\n$\\nu>\\nu_c$   &  ${p\\over 2}$ (1.15)  &   ${3p-2\\over 4}$ (1.2)   &  $\\alpha={3\\beta-1 \\over 2}$   &   ${(2p-4)+(p+2)q\\over 4}$ (0.7) &  $\\alpha=\\frac{q-2}{2}+\\frac{(2+q)\\beta}{2}$\\\\\n\n\\hline\nWind & fast cooling  \\\\\n\\hline\n$\\nu<\\nu_c$   &  $-{1\\over 3}$   &   ${2\\over 3}$ & $\\alpha={1-\\beta \\over 2}$ &  ${(1+q) \\over 3}$ (0.5) & $\\alpha=\\frac{q}{2}-\\frac{(2-q)\\beta}{2}$  \\\\\n$\\nu_c<\\nu<\\nu_m$   &  ${1\\over 2}$  &   ${1\\over 4}$  & $\\alpha={1-\\beta \\over 2}$  & ${3q-2 \\over 4}$ (-0.1)& $\\alpha=\\frac{q}{2}-\\frac{(2-q)\\beta}{2}$\\\\\n$\\nu>\\nu_m$   &  ${p\\over 2}$ (1.15) &   ${3p-2\\over 4}$ (1.2)    & $\\alpha={3\\beta-1 \\over 2}$  &  ${(2p-4)+(p+2)q\\over 4}$ (0.7) & $\\alpha=\\frac{q-2}{2}+\\frac{(2+q)\\beta}{2}$\\\\\n\\hline\n\\end{tabular}\n\\label{Tab:alpha-beta}\n\\tablecomments{This is the extension of the\nTable 1 of Zhang \\& M\\'esz\\'aros (2004), with the inclusion of the\ncases of energy injection. The case of $p<2$ is not included, and\nthe self-absorption effect is not discussed. Notice\nthat a different convention $F_\\nu \\propto t^{-\\alpha} \\nu^{-\\beta}$\nis adopted here (in comparison to that used in Zhang \\& M\\'esz\\'aros\n2004), mainly because both the temporal index and the\nspectral index are generally negative in the X-ray band. The temporal\nindices with energy injection are valid only for $q < 1$, and they\nreduce to the standard case (without energy injection, e.g. Sari et\nal. 1998, Chevalier \\& Li \n2000) when $q=1$. For $q>1$ the expressions are no longer valid, and\nthe standard model applies. An injection case due to pulsar spindown\ncorresponds to $q=0$ (Dai \\& Lu 1998a; Zhang \\& M\\'esz\\'aros\n2001). Recent {\\em Swift} XRT data are generally consistent with $q\n\\sim 0.5$. The numerical values quoted in parentheses are for $p=2.3$\nand $q=0.5$. }\n\\end{table}\n\\clearpage\n\n3. The energy injection is also brief, but the outflow has a\nsignificant fraction of Poynting flux (e.g. Usov 1992; Thompson 1994;\nM\\'esz\\'aros \\& Rees 1997b; Lyutikov \\& Blandford 2005). Assigning \na parameter $\\sigma$ for the outflow, which is the ratio between the\nPoynting flux and baryonic kinetic energy flux, Zhang \\& Kobayashi\n(2005) modeled the reverse shock emission from ejecta with an\narbitrary $\\sigma$ value. They found that during the crossing of the\nreverse shock, the Poynting energy is not transferred to the ambient\nmedium.  The Poynting energy (roughly by a factor of $\\sigma$) is\nexpected to be transferred to the medium (and hence, to the afterglow\nemission) after the reverse shock disappears. Zhang \\& Kobayashi\n(2005) suggest that the transfer is delayed with respect to the\ntraditional case of $\\sigma=0$. The energy transfer process, however,\nis poorly studied so that one does not have a handy conversion\nrelation with the $q$ value derived in the first scenario. \n\n\\subsection{Case studies}\n\nIn this subsection, we discuss several {\\em Swift} GRBs with\nwell-monitored early afterglow data detected by XRT. The notations for\nthe break times and the temporal slopes are per those marked on\nFig. \\ref{XRTlc}. \n\n{\\bf GRB 050128} (Campana et al. 2005): The lightcurve can be fitted\nby a broken power law with the break time at $t_{b2} =\n1472^{+300}_{-290}$s. The temporal decay indices before and after the\nbreak are $\\alpha_2=0.27^{+0.12}_{-0.10}$ and\n$\\alpha_3=1.30^{+0.13}_{-0.18}$, respectively. The spectral indices\nbefore and after the break are essentially unchanged, i.e. $\\beta_2\n\\sim 0.59\\pm 0.08$, and $\\beta_3 = 0.79\\pm 0.11$. Campana et\nal. (2005) discussed two interpretations. A jet model requires a very\nflat electron spectral index, i.e. $p \\sim 1.3$, as well as a change\nof the spectral domain before and after the jet break. Alternatively,\nthe data may be accommodated in a wind model, but one has to assume\nthree switches of the spectral regimes during the observational gap\nfrom 400s to about 4000s. So neither explanation is completely\nsatisfactory. By comparing the predicted indices in Table\n\\ref{Tab:alpha-beta}, the observation may be well-interpreted within\nthe ISM model with an initial continuous energy injection episode. The\nsegment after \nthe break is consistent with a standard ISM model for $\\nu_m < \\nu_X <\n\\nu_c$, with $p \\sim 2.6$. The lightcurve before the break, on the\nother hand, is consistent with an injection model with $p \\sim 2.2$\nand $q \\sim 0.5$ in the same spectral regime. The break time is\nnaturally related to the cessation of the injection process, and a\nslight change of electron spectral index (from 2.2 to 2.6) is\nrequired. From the beginning of the observation (100s) to $t_{b2}$,\nthe total energy is increased by a factor of $(1472\/100)^{(1-0.5)}\n\\sim 2.8$. \n\n{\\bf GRB 050315} (Vaughan et al. 2005): After a steep decay ($\\alpha_1\n= 5.2^{+0.5}_{-0.4}$) up to $t_{b1}=308$s, the lightcurve shows a flat\n``plateau'' with a temporal index of $\\alpha_2\n=0.06^{+0.08}_{-0.13}$. It then turns to $\\alpha_3=0.71\\pm 0.04$ at\n$t_{b2}=1.2^{+0.5}_{-0.3}\\times 10^4$s. Finally there is a third break\nat $t_{b3}=2.5^{+1.1}_{-0.3}\\times 10^5$s, after which the temporal\ndecay index is $\\alpha_4=2.0^{+1.7}_{-0.3}$. So this burst displays\nall four segments presented in Fig.\\ref{XRTlc}. The spectral indices\nin segments II, III and IV are essentially constant,\ni.e. $\\beta_2=0.73\\pm 0.11$, $\\beta_3=0.79\\pm 0.13$ and\n$\\beta_4=0.7^{+0.5}_{-0.3}$, respectively. Segment III is\nconsistent with an ISM model with  \n$\\nu_X > \\nu_c$ and $p=1.6$, since in this model $\\beta=p\/2=0.8$,\n$\\alpha=(3p-2)\/4=0.7$, in perfect agreement with the data. The\nthird temporal break $t_{b3}$ is consistent with a jet\nbreak. According to Dai \\& Cheng (2001), the post-break temporal index\nfor $p<2$ is $\\alpha=(p+6)\/4=1.9$, which is also consistent with the\nobserved $\\alpha_4$. The plateau between $t_{b1}$ and $t_{b2}$ is then\ndue to an energy injection in the same ISM model ($\\nu_X > \\nu_c$),\nwith $p \\sim 1.5$ and $q \\sim 0.35$. The total injected energy is\nincreased by a factor of $(12000\/308)^{(1-0.35)} \\sim 11$.\n\n{\\bf GRB 050319} (Cusumano et al. 2005): After a\nsteep decay ($\\alpha_1 = 5.53\\pm 0.67$) up to $t_{b1}=(384\\pm 22)$s, the\nlightcurve shows a shallow decay with a temporal index of $\\alpha_2\n=0.54\\pm 0.04$. It steepens to $\\alpha_3=1.14\\pm 0.2$ at\n$t_{b2}=(2.60\\pm 0.70) \\times 10^4$s. The spectral indices in\nsegment II and III are $\\beta_2=0.69\\pm 0.06$ and $\\beta_3=0.8\\pm 0.08$,\nrespectively. Again segment III is well consistent with an ISM model\nfor $\\nu_m < \\nu_X < \\nu_c$ with $p=2.6$, which gives\n$\\beta=(p-1)\/2=0.8$ and $\\alpha=(3\/2)\\beta=1.2$, in excellent\nagreement with the data. Interpreting the segment II (the shallow decay\nphase) as the energy injection phase, for the same ISM model ($\\nu_m <\n\\nu_X < \\nu_c$), one gets $p\\sim 2.4$ and $q \\sim 0.6$. The total\ninjected energy is increased by a factor of $(26000\/384)^{(1-0.6)} \\sim\n5.4$. The UVOT observations are also consistent with such a picture\n(Mason et al. 2005). \n\n{\\bf GRB 050401} (de Pasquale et al. 2005): The early X-ray lightcurve\nis consistent with a broken power law, with $\\alpha_2=0.63 \\pm 0.02$,\n$\\alpha_3 = 1.41 \\pm 0.1$, and $t_{b2}=4480^{+520}_{-440}$s. The\nspectral indices before and after the break are all consistent with\n$\\beta_2 \\sim \\beta_3 = 0.90 \\pm 0.03$. The $\\alpha-\\beta$ relation\ndoes not fit into a simple $p<2$ jet model. \nOn the other hand, the energy injection model gives a natural\ninterpretation. After the \nbreak, the lightcurve is consistent with an ISM model for $\\nu_m <\n\\nu_X <\\nu_c$ with $p=2.8$. Before the break, it is consistent with\nthe same model with $q=0.5$. The total injected energy is increased by\na factor of $>(4480\/200)^{(1-0.5)} \\sim 4.7$.\n\nThe injection signature is also inferred in other bursts such as GRB\n050117 (Hill et al. 2005) and XRF 050416 (Sakamoto et al. 2005),\nwhere similar conclusions could be drawn. The injection model is\nsupported by an independent study of Panaitescu et al. (2005).\n\n\\subsection{Theoretical implications}\n\nThe following conclusions could be drawn from the above case studies.\n\n1. A common feature of the early X-ray afterglow lightcurves is a well\ndefined temporal steepening break. A crucial observational\nfact is that there is essentially no spectral variation before and \nafter the break. This suggests that the break is of hydrodynamic \norigin rather than due to the crossing of some typical frequencies of\nthe synchrotron spectrum in the band. It is worth mentioning that a\nlightcurve transition similar to the transition between segments II\nand III is expected in a radiative fireball (e.g. B\\\"ottcher \\& Dermer\n2000), see e.g. Figs. 1 \\& 2 of Wu et al. (2005). However, that\ntransition is due to the crossing of $\\nu_m$ in the observational\nband. One therefore expects a large spectral variation before and\nafter the break, which is inconsistent with the data.\nAnother straightforward interpretation would\nbe a jet break, but there are three reasons against such an\ninterpretation. First, in all the cases, $p<2$ has to be assumed. This\nis in stark contrast to the late jet breaks observed in the optical\nband, which typically have $p>2$. Furthermore, the $\\alpha-\\beta$\nrelation predicted in the jet model is usually not satisfied. Second,\nthe post-break $\\alpha-\\beta$ relation is usually satisified in a\nstandard slow cooling ISM model, with the X-ray band either below or\nabove the cooling frequency. In such a sense, this segment is quite\n``normal''. Third, in some cases (e.g. GRB 050315), another steepening\nbreak is observed after this normal segment, which is consistent with\nthe jet break interpretation. Since only one break could be attributed\nto a jet break, the ``shallow-to-normal'' break must be due to\nsomething else.\n\n2. A natural interpretation of the shallow decay phase is to attribute\nit to a continuous energy injection, so that the forward shock is\n``refreshed''. Three possibilities exist to account for the refreshed \nshock effect (\\S\\ref{sec:FS2}): a long-lived central engine with\nprogressively reduced activities, an instantaneous injection with a\nsteep power-law distribution of the shell Lorentz factors, and the\ndeceleration of an instantaneously-injected highly magnetized \n(high-$\\sigma$) flow. In terms of afterglow properties, these\npossibilities are degenerate (e.g. the connection between\n$q$ and $s$) and can not be differentiated. In principle, the first\nscenario may give rise to additional observational signatures\n(e.g. Rees \\& M\\'esz\\'aros 2000; Gao \\& Wei 2004, 2005), which may be\nused to differentiate the model from the others.  \n\n3. Two interesting characteristics during the injection phase are that\nthe injection process is rather smooth, and that the effective $q$\nvalue is around 0.5. This gives interesting constraints on the\npossible physical mechanisms. (1) For the scenario of a continuously\ninjecting central engine (Zhang \\& M\\'esz\\'aros 2001), the central\nengine luminosity must vary with time smoothly. This is in contrast\nto the conventional GRB central engine which injects energy\nerratically to allow the observed rapid variability in the\nlightcurves. This usually requires two different energy components,\ni.e. one ``hot'' fireball component that leads to the prompt emission\nand a ``cold'' Poynting flux component that gives to the smooth\ninjection. A natural Poynting flux component is due to the spin-down\nof a new-born millisecond pulsar (Dai \\& Lu 1998a; Zhang \\&\nM\\'esz\\'aros 2001). However, a straightforward prediction from such a\nmodel is $q=0$, not consistent with $q \\sim 0.5$ inferred from the\nobservations. Modifications to the simplest model are needed.\nAlternatively, the system may be a long-lived black\nhole torus system with a reducing accretion rate. However, at later\ntimes the long-term central engine power corresponds to $q=5\/3$\n(MacFadyen et al. 2001; Janiuk et al. 2004), too steep to give an\ninteresting injection signature. It is worth mentioning that in the\ncollapsar simulations (MacFadyen et al. 2001), an extended flat\ninjection episode sometimes lasts for $\\sim 1000$s, which could\npotentially interpret the short injection phase of some bursts, but is\ndifficult to account for some other bursts whose injection phase is\nmuch longer. (2) For the scenario\nof a power-law distribution of Lorentz factors (Rees \\& M\\'esz\\'aros\n1998), one should require that a smooth distribution of Lorentz\nfactors is produced after the internal shock phase. In the internal\nshock model, slow shells are indeed expected to follow the fast\nshells, but they tend to be discrete and give rise to bumpy\nlightcurves (e.g. Kumar \\& Piran 2000a) especially when the\ncontribution from the reverse shock is taken into account (Zhang \\&\nM\\'esz\\'aros 2002b). It is also unclear how an effective $q \\sim 0.5$\nis expected. (3) Deceleration of a promptly-ejected Poynting-flux\ndominated flow (e.g. Zhang \\& Kobayashi 2005) naturally gives a smooth\ninjection signature observed. Above case studies indicate that the\ninjected energy is by a factor of several to 10. Within such a\npicture, the unknown $\\sigma$ value is about several to 10. However,\nit is unclear how long the delay would be and whether one can account\nfor the shallow decay with $q \\sim 0.5$ extending for $10^4$ seconds. \nFurther more detailed theoretical modeling is needed to test this\nhypothesis.  \n\n4. Any model needs to interpret the sudden cessation of the injection\nat $t_{bs}$. This time has different meanings within the three\nscenarios discussed above. (1) Within the long-lived\ncentral engine model, this is simply the epoch when the injection\nprocess ceases. In the pulsar-injection model, there is a well-defined\ntime for injection to become insignificant (Dai \\& Lu 1998a; Zhang\n\\& M\\'esz\\'aros 2001), but within a black-hole-torus injection model,\nsuch a time is not straightforwardly defined. (2) In the varying\nLorentz factor scenario, \nthis time corresponds to a cut-off of the Lorentz factor distribution\nat the low end below which the distribution index $s$ is flatter than\n1 so that they are energetically unimportant. This lowest Lorentz\nfactor is defined by\n\\begin{equation}\n\\Gamma_m = 23 \\left(\\frac{E_{iso,52}}{n}\\right)^{1\/8}\nt_{b2,4}^{-3\/8} \\left(\\frac{1+z}{2}\\right)^{3\/8}, \n\\end{equation}\nA successful model must be able to address a well-defined\n$\\Gamma_m$ in this model. (3) Within the Poynting flux\ninjection model, a well-defined time cut-off\nis expected, which corresponds to the epoch when all the Poynting\nenergy is transferred to the blastwave. If the shallow decay is indeed\ndue to Poynting energy transfer, the cut-off time ($t_{b2}$) could be\nroughly defined by the $\\sigma$ parameter through\n$\\sigma \\sim ({t_{b2}}\/{t_{dec}})^{(1-q)}$,\nwhere $t_{dec}$ is the conventional deceleration time\ndefined by $E_{iso}\/(1+\\sigma)$, when only a fraction of\n$(1+\\sigma)^{-1}$ energy is transfered to the ISM (Zhang \\& Kobayashi\n2005). \n\n\n5. Although we have not tried hard to rule out a wind-model\ninterpretation, the case studies discussed above suggest that the\nearly afterglow data are consistent with an ISM model for essentially\nall the bursts. This conclusion also applies to other well-studied {\\em\nSwift} bursts (e.g. GRB 050525a, Blustin et al. 2005). This result is\nintriguing given that long GRBs are associated with the death of massive stars,\nfrom which a strong wind is expected. Previous analyses using late\ntime afterglow data (e.g. Panaitescu \\& Kumar 2002; Yost et al. 2003)\nhave also suggested that most afterglow data are consistent with an ISM\nmodel rather than a wind model. In order to accommodate the data,\nit has been suggested that the wind parameter may be small so that at\na late enough time the blastwave is already propagating in an ISM \n(e.g. Chevalier et al. 2004). The {\\em Swift} results push the ISM\nmodel to even earlier epochs (essentially right after the\ndeceleration), and indicate the need for a re-investigation of the\nproblem. The epoch shortly before the deaths of massive stars\nis not well studied (Woosley et al. 2003). One possibility is that the\nstellar wind ceases some time before the star collapses. Careful \nanalyses of early afterglows of a large sample of long GRBs may shed \nlight on the final stage of massive star evolution.\n\n\\section{Reverse shock emission}\n\\label{sec:RS} \n\n\\subsection{Synchrotron emission}\nIt is generally believed that a short-lived reverse shock exists\nduring the intial deceleration of the fireball and gives interesting\nemission signatures in the early afterglow phase. Given the same\ninternal energy in both the forward-shocked and the reverse-shocked\nregions, the typical synchrotron frequency for the reverse shock\nemission is typically much lower than that in the forward shock\nregion, since the ejecta is much denser than the medium. While the\nearly forward shock synchrotron emission peaks in X-rays at early\ntimes, the reverse shock synchrotron emission usually peaks in the\noptical\/IR band or even lower (e.g. M\\'esz\\'aros \\& Rees 1997a; Sari \\&\nPiran 1999; Kobayashi 2000; Zhang et al. 2003; Zhang \\& Kobayashi\n2005). This model has been succussful in interpreting the early\noptical emission from GRB 990123 (Akerlof et al. 1999; Sari \\&\nPiran 1999; M\\'esz\\'aros \\& Rees 1999), GRB 021211 (Fox et al. 2003; \nLi et al. 2003; Wei 2003), and GRB 041219a (Blake et al. 2005;\nVestrand et al. 2005; Fan et al. 2005b). As a result, it is expected\nthat the reverse shock component has a negligible contribution in the\nX-ray band.\n\nIn the above argument, it has been assumed that the shock parameters\n($\\epsilon_e$, $\\epsilon_B$ and $p$) are the same in both shocks. In\nreality this might not be the case. In particular, the GRB outflow may\nitself carry a dynamically important magnetic field component (or\nPoynting flux). This magnetic field would be shock-compressed and\namplified, giving a larger effective $\\epsilon_B$ (Fan et al. 2004a,b;\nZhang \\& Kobayashi 2005). Since the medium is generally not\nmagnetized, it is natural to expect different $\\epsilon_B$ values in\nboth regions, and a parameter ${\\cal R}_B \\equiv (\\epsilon_{B,r} \/\n\\epsilon_{B,f})^2$ have been used in the reverse shock analysis.\nIt has been found that ${\\cal R}_B$ is indeed larger than unity for\nGRB 990123 and GRB 021211 (Zhang et al. 2003; Fan et al. 2002; Kumar\n\\& Panaitescu 2003; Panaitescu \\& Kumar 2004; MaMahon et\nal. 2004). Hereafter the subscript\/superscript ``f'' and ``r''\nrepresent the forward shock and the reverse shock,\nrespectively. According to Zhang \\& Kobayashi (2005), the case of GRB\n990123 corresponds to the most optimized case with $\\sigma \\sim 1$, so\nthat ${\\cal R}_B$ is the largest. \n\nMore generally, $\\epsilon_e$ and $p$ may also vary in both shocks. \nFan et al. (2002) performed a detailed fit to the GRB 990123 data and\nobtained  $\\epsilon_e^{\\rm r}=4.7\\epsilon_e^{\\rm f}$ and \n$\\epsilon_B^{\\rm r}=400\\epsilon_B^{\\rm f}$. A general treatment\ntherefore requires that we introduce one more parameter, i.e. ${\\cal\nR}_{e}=[(p^{\\rm r}-2)\/(p^{\\rm r}-1)]\/[(p^{\\rm f}-2)\/(p^{\\rm\nf}-1)] (\\epsilon_{\\rm e}^{\\rm r}\/\\epsilon_{\\rm e}^{\\rm f})$. \nFollowing the treatment of Zhang et al. (2003), we have the following\nrelations in the thin-shell regime (see also Fan \\& Wei\n2005)\\footnote{For an\narbitrary $\\sigma$, the treatment becomes more complicated. The\ntreatment presented here is generally valid for $\\sigma \\lesssim 1$. For\n$\\sigma > 1$, the reverse shock emission starts to be suppressed\n(Zhang \\& Kobayashi 2005). Since we are investigating the most\noptimistic condition for the reverse shock contribution, in this paper\nwe adopt the standard hydrodynamic treatment which is valid for $\\sigma\n\\lesssim 1$.} \n\\begin{eqnarray}\n\\frac{\\nu_{\\rm m}^{\\rm r}(t_{\\rm \\times})}\n{\\nu_{\\rm m}^{\\rm f}(t_{\\rm \\times})}& \\sim &{\\cal R}_{\\rm B} \n{\\cal R}_{\\rm e}^2 \\left(\\frac{\\gamma_{34,\\times}-1} \n{\\Gamma_{\\times}-1}\\right)^2,\\label{Rnum}\\\\\n\\frac{\\nu_{\\rm c}^{\\rm r}(t_\\times)}{\\nu_{\\rm c}^{\\rm f}(t_\\times)}\n& \\sim & {\\cal R}_{\\rm B}^{-3} \\left(\\frac{1+Y^{\\rm f}}\n{1+Y^{\\rm r}}\\right)^2, \\label{Rnuc}\\\\\n\\frac{F_{\\rm \\nu, max}^{\\rm r}(t_{\\rm \\times})}\n{F_{\\rm \\nu, max}^{\\rm f}(t_{\\rm \\times})}\n& \\sim & {\\cal R}_{\\rm B} \\frac{\\Gamma_\\times^2}{\\Gamma_0} \\sim {\\cal\nR}_B \\Gamma_0~ \\label{RFnum},\n\\end{eqnarray}\nwhere $Y^{\\rm f}$ and $Y^{\\rm r}$ are the Compton parameters for the\nforward and the reverse shock emission components, respectively; \n$t_\\times$ is the reverse shock crossing time, which is essentially\n$t_{dec}$ (eq.[\\ref{tdec}]) for the thin shell case; $\\Gamma_\\times$\nis the bulk Lorentz factor of the outflow at $t_\\times$; \n$\\gamma_{34,\\times}\\approx\n(\\Gamma_0\/\\Gamma_\\times+\\Gamma_\\times\/\\Gamma_0)\/2$ is the Lorentz\nfactor of the shocked ejecta relative to the unshocked one. \n\nFor typical parameters, both $\\nu_m^{\\rm f}$ and \n $\\nu_c^{\\rm f}$ are below the XRT band (comparing eqs.[\\ref{tm}],\n[\\ref{tc}] with eq.[\\ref{tdec}]). According to eqs.(\\ref{Rnum}) and\n(\\ref{Rnuc}), $\\nu_c^{\\rm r}$, $\\nu_m^{\\rm r}$ should be also below\nthe XRT band. Following the standard synchrotron emission model, we\nthen derive the X-ray flux ratio of the reverse shock and the forward\nshock components at $t_\\times$:\n\\begin{equation}\n{F_{\\nu,X}^{\\rm r}(t_\\times) \\over F_{\\nu,X}^{\\rm f}(t_\\times)}\n\\approx  {\\cal R}_B^{\\frac{p-2}{2}} {\\cal R}_e^{p-1} \\Gamma_0\n\\left(\\frac{\\gamma_{34, \\times}-1}{\\Gamma_\\times -1} \\right)^{p-1}\n\\left({1+Y^{\\rm f}\\over 1+Y^{\\rm r}}\\right).\n\\end{equation} \nWe can see that for ${\\cal R}_B={\\cal R}_e=1$, $Y^{\\rm f}=Y^{\\rm\nr}$, and $p \\geq 2$, one has $F_{\\nu,X}^{\\rm r}(t_\\times) \\lesssim\nF_{\\nu,X}^{\\rm f}(t_\\times)$, since $\\gamma_{34,\\times} \\lesssim 1$,\nand $\\Gamma_\\times \\sim \\Gamma_0$ in the thin shell case. The reverse\nshock contamination in the X-ray band is therefore not important. The\nsituation changes if we allow higher ${\\cal R}_e$ and ${\\cal R}_B$\nvalues. Increasing ${\\cal R}_e$ directly increases the\nreverse-to-forward flux ratio. Although the dependence on ${\\cal R}_B$\nis only mild when $p$ is close to 2, a higher ${\\cal R}_B$\nsuppresses the IC process in the reverse shock region relative to that\nin the forward shock region, so that the ratio $(1+Y^{\\rm\nf})\/(1+Y^{\\rm r})$ also increases. As a result, as ${\\cal R}_B \\gg 1$\nand ${\\cal R}_e \\gg 1$, the reverse shock synchotron component would\nstick out above the forward shock synchrotron component, and an X-ray\nbump is likely to emerge (see also Fan \\& Wei 2005). As a numerical example,\ntaking $p=2.3$, $\\Gamma_\\times \\approx \\Gamma_0\/2=50$, ${\\cal R}_B=10$,\n${\\cal R}_e=5$, and $\\epsilon_e^{\\rm f}=30\\epsilon_B^{\\rm f}$, we get\n${F_{\\nu_X}^{\\rm r}(t_\\times) \/ F_{\\nu_X}^{\\rm f}(t_\\times)}\\approx\n6$. This could potentially explain the X-ray flare (by a factor of\n$\\sim 6$) detected in GRB 050406 (Burrows et al. 2005; P. Romano et\nal. 2005, in preparation). However, a big\ncaveat of such a model is that one expects a very bright UV\/Optical\nflash due to the large ${\\cal R}_B$ and ${\\cal R}_e$ involved - like\nthe case of GRB 990123. Unless this flash is completely suppressed by\nextinction, the \nnon-detection of such a flash in the UVOT band for GRB 050406 strongly\ndisfavors such an interpretation. \n\n\\subsection{Synchrotron self-Compton emission}\n\nThe synchrotron photons in the reverse shock region will be scattered\nby the same electrons that produce these photons. The characteristic\nenergy of this component is typically in the $\\gamma$-ray range.\nHowever, under some conditions, this\nsynchrotron self-Compton (SSC) component would also stand out in the\nX-ray band, giving rise to an X-ray bump in the lightcurve. A detailed\ndiscussion has been presented in Kobayashi et al. (2005), which we do\nnot repeat here. The general conclusion is that the SSC component could\naccount for an early X-ray flare bump by a factor of several under\ncertain optimized conditions. An advantage of this model over the\nreverse shock synchrotron model is that a bright UV-optical flash is\navoided. However, this model can not account for a flare with a very\nlarge contrast (e.g. by a factor of 500, as seen in GRB 050502B, Burrows\net al. 2005). \n\n\\section{Mechanisms to produce early X-ray flares}\n\\label{sec:flare} \n\nXRT observations indicate that X-ray flares are common features in the\nearly phase of X-ray afterglows (the component V in Figure\n\\ref{XRTlc}). After the report of the first two cases of flares in GRB\n050406 and GRB 050502B (Burrows et al. 2005), \nlater observations indicate that nearly half of long GRBs harbor early\nX-ray flares (e.g. O'Brien et al. 2005). More intriguingly, the early\nX-ray afterglow of the \nlatest localized short GRB 050724 (Barthelmy et al. 2005b) also\nrevealed flares similar to those in the long GRBs (e.g. 050502B). The common\nfeature of these flares is that the decay indices after the flares are\ntypically very steep, with a $\\delta t\/t$ much smaller than\nunity. In some cases (e.g. GRB050724, Barthelmy et al. 2005), the\npost-flare decay slopes are as steep as $\\leq -7$. In this\nsection we discuss various possible models to interpret the flares and\nconclude that the data require that the central engine is active and\nejecting these episodic flares at later times. \n\n\n\n\\subsection{Emission from the reverse shock?}\n\\label{subsec:model2}\n\nAs discussed in \\S\\ref{sec:RS}, synchrotron or SSC emission from the\nreverse shock region could dominate that from the forward shock\nemission in the X-ray band under \ncertain conditions. Because of the lack of strong UV-optical flares in\nthe UVOT observations, we tentatively rule out the reverse shock\nsynchrotron emission model. The prediction of the lightcurve in the\nSSC model could potentially interpret the X-ray flare seen in GRB\n050406 (Burrows et \nal. 2005), but the predicted amplitude is too low to interpret the\ncase of GRB 050502B (Burrows et al. 2005). In some bursts (e.g. GRB\n050607), more than one flare are seen in a burst. Although one of\nthese flares may be still interpreted as the reverse shock SSC\nemission, an elegant model should interpret these flares by a model\nwith the same underlying physics. We tentatively conclude that the\nreverse shock model cannot account for most of the X-ray flares\ndetected by the {\\em Swift} XRT in a unified manner.\n\n\n\n\\subsection{Density clouds surrounding the progenitor?}\n\\label{subsec:model3}\n\nLong-duration GRBs are believed to be associated with the deaths of\nmassive stars, such as Wolf-Rayet stars. According to Woosley et\nal. (2003), there are no observations to constrain the mass loss rate\nof a Wolf-Rayet star during the post-helium phase (100-1000 years\nbefore the explosion), No stability analyses have been carried out to\nassess whether such stars are stable. At the high end of a reasonable\nrange of the mass loss rate, dense clouds surrounding a GRB progenitor\nmay exist. For a wind velocity $v_{\\rm w}\\sim 100{\\rm\nkm~s^{-1}}$, the density clumps could occur at a radius $\\sim 3\\times\n10^{16}-3\\times10^{17}$cm. These density clumps have been invoked by\nDermer \\& Mitman (1999, 2003) to interpret the GRB prompt emission\nvariabilities. \n\nOne immediate question is whether these density clumps,\nif they exist, could give rise to the X-ray flares detected by XRT.\nIn order to check this possibility, we investigate the following toy\nmodel. For simplicity, we assume a dense clump extends from $R\\sim\n3\\times 10^{16}{\\rm cm}$ to $(R+\\Delta) \\sim 3.3\\times 10^{16}{\\rm cm}$\nwith a number \ndensity $n_{cloud}\\sim 100{\\rm cm^{-3}}$. The background ISM density\nis taken as $n\\sim 1~{\\rm cm^{-3}}$. Other parameters in the\ncalculation include: $E_{\\rm iso}=10^{52}$ergs, $z=1$,\n$\\Gamma_0=240$, $\\epsilon_{\\rm e}=0.1$, $\\epsilon_{\\rm B}=0.01$,\n$p=2.3$ and $\\theta_{\\rm j}=0.1$. No sideways expansion of the jet is\nincluded, which is consistent with previous simulations (Kumar \\&\nGranot 2003; Cannizzo et al. 2004). The X-ray lightcurve is shown in \nFig.\\ref{Fig:Cloud}, the general feature of which is reproduced in\nboth codes used in this paper. Although\nthe rising phase could be very sharp, the decaying slope is rather\nflat. This is because of two effects. First, after entering the\ndense cloud, the blastwave Lorentz factor falls off rapidly. The\nobserved time $t \\sim R\/2 \\Gamma^2$ is hence significantly\nstretched. The dashed line in Fig.\\ref{Fig:Cloud} represents the observed\nemission before the fireball exits the cloud. Second, after the\nfireball exits the cloud (dotted line in \\ref{Fig:Cloud}), the\nfireball does not decelerate immediately since the Lorentz factor is\nalready too low to be further decelerated in a medium with a low\ndensity. It is decelerated again when enough medium is swept up at a\nlarger radius. The interaction of a fireball with density bumps has\nbeen studied previously by many authors (e.g. Lazzati et al. 2002; Dai\n\\& Wu 2003). Our detailed calculations suggest that the variation\ncaused by density inhomogeneities is generally not very significant\n(see also Ramirez-Ruiz et al. 2005). The lightcurves for a blastwave\nsurfing on a density wave should be generally quite smooth\n(cf. Lazzati et al. 2002). \n\\begin{figure}\n\\plotone{f4.eps}\n\\caption{The X-ray flare powered by a relativistic fireball\ninteracting with a dense cloud. The cloud is located at  \n$R=(3.0, 3.3)\\times 10^{16}$ cm with a density $n_{cloud}=100~{\\rm\ncm^{-3}}$. The background ISM density is $n=1~{\\rm cm^{-3}}$.\nOther parameters include: $E_{\\rm iso}=10^{52}$ergs, $z=1$,\n$\\Gamma_0=240$,  $\\epsilon_{\\rm e}=0.1$, $\\epsilon_{\\rm B}=0.01$,\n$p=2.3$ and $\\theta_{\\rm j}=0.1$. The dashed line is the X-ray\nemission contributed by the electrons shocked at $R<3.3\\times\n10^{16}$cm, and the dotted line is the X-ray emission contributed by\nthe electrons shocked at $R>3.3\\times 10^{16}$cm. The solid line is\nthe sum of these two components.}\n\\label{Fig:Cloud}\n\\end{figure}\nThe one-dimensional model presented here effectively calculates\nthe feature when the fireball encounters a high density shell. A more \nrealistic model for density clumps should include the size of the \nclump ($\\Delta$). The lightcurve decay slope after the peak therefore \ndepends on the comparison between $\\Gamma^{-1}$ and the angle the \nclump extends from the central engine, $\\Delta \/ R$. If $\\Gamma^{-1}\n\\ll \\Delta\/R$, the above calculation is still valid since the observer\nwould not notice the edge of the clump. The resulting decay slope is\nrather shallow after the peak, which is in distinct contrast to the\nX-ray flare data. This is the case for the\nparameters we use in Fig. \\ref{Fig:Cloud}. For smaller clumps,\nthe lightcurve will steepen as $\\Gamma^{-1}$ becomes comparable to\n$\\Delta \/R$. Our numerical calculations indeed show such a feature.\nHowever, the flux does not fall rapidly right after the peak. The peak\ntime is typically shortly after the blastwave enters the density clump\nwhen the Lorentz factor is still very high. The epoch when\n$\\Gamma^{-1}=\\Delta \/R$ is satisfied happens much later. The\nlightcurve therefore still show a shallow decay \nsegment before the steep decay component. This is in contrast with the\nX-ray flare data that show rapid decays right after the flare peak\n(e.g. Burrows et al. 2005; Falcone et al. 2005; Romano et al. 2005;\nO'Brien et al. 2005). One could make the clouds small enough, so that \n$\\Gamma^{-1} > \\Delta\/R$ is satisfied from the very\nbeginning. However, in such a case it is very hard to achieve a large\ncontrast between the flares and the underlying afterglow level (Ioka\net al. 2005). The 500 time contrast observed in GRB 050502B is in any\ncase very difficult to achieve within all the density clump models we\nhave explored. We therefore tentatively conclude that the X-ray flares\ncommonly detected in the early X-ray afterglow lightcurves are likely\nnot caused by the putative density clouds surrounding the GRB\nprogenitors. The lack of these well-modeled features in the data also\nsuggest that the lumpiness of circumburst medium is, if any, rather\nmild. We note, however, Dermer (2005, in preparation) suggests that\nmore detailed 3-D modeling of the density clump problem could\npotentially reproduce the observed X-ray flares.\n\n\\subsection{Two-component jet?}\n\\label{subsec:model4}\n\nIn the collapsar progenitor models, it is expected that the ejecta\ngenerally has two components, one ultra-relativistic component\npowering the GRB, and another moderately relativistic cocoon component\n(e.g. Zhang. et al. 2004b; M\\'esz\\'aros \\& Rees 2001; Ramirez-Ruiz et\nal. 2002). This gives a physical motivation to the phenomenological\ntwo-component jet model (e.g. Lipunov et al. 2001; \nBerger et al. 2003; Huang et al. 2004). An interesting possibility\nwould be whether the X-ray flare following the prompt gamma-ray\nemission (which arises from the central relativistic component) is\ncaused by the deceleration of the wider mildly-relativistic cocoon\ncomponent as it interacts with the ambient medium. A straightforward\nconclusion is that the wide, off-beam component must contain more\nenergy than the narrow component in order to give noticeable features\nin the lightcurve. Also the decay after the lightcurve peak of the\nsecond component should follow the standard afterglow model, and\nthe variability time scale satisfies $\\delta t\/t \\sim 1$. The optical\nlightcurves of the two component jet model have been calculated by\nHuang et al. (2004) and Granot (2005). Fig.\\ref{Fig:TwoComp} shows a\nsample X-ray lightcurve in the two-component jet model. It is obvious\nthat this model cannot interpret the rapid fall-off observed in the\nXRT X-ray flares, and is therefore ruled out.\n\\begin{figure}\n\\epsscale{1.0}\n\\plotone{f5.eps}\n\\caption{The X-ray lightcurve powered by a two-component jet. The\nangular range $0<\\theta<0.1$ is the central narrow component, which\nhas $E_{\\rm iso,n}=10^{52}$ergs and $\\Gamma_{0,n}=240$. The wide jet\ncomponent covers a range of $0.1<\\theta<0.3$, with $E_{\\rm iso,w} =\n5\\times 10^{52}$ergs and $\\Gamma_{0,w}=50$. The ISM density is \n$n=1{\\rm cm^{-3}}$. Other parameters: $\\epsilon_{\\rm e}=0.1$,\n$\\epsilon_{\\rm B}=0.01$, $p=2.3$, and $z=1$. The line of sight is at\n$\\theta=0$. The lightcurve peak of the second component corresponds to\nthe epoch when the off-beam wide component is decelerated, so that its\n$1\/\\Gamma_{w}$ beam enters the field of view.}\n\\label{Fig:TwoComp}\n\\end{figure}\n\n\\subsection{Patchy jets?}\n\\label{subsec:model5}\n\nA related model considers a jet with large energy fluctuations in\nthe angular direction, so that its energy distribution is patchy\n(Kumar \\& Piran \n2000b). This is a variation of the two-component jet, and could be\napproximated as a multi-component jet. When the $1\/\\Gamma$ cone of\ndifferent patches enter the field of view, the observed lightcurve may\npresent interesting signatures. However, the general feature of the\ntwo-component jet still applies: Only when a patch has a substantial\nenergy compared with the on-beam jet would it give a bump feature on\nthe lightcurve. After each bump, the afterglow level is boosted and\nwould not resume the previous level. The variability time scale is\nalso typically $\\delta t\/t \\sim 1$. Figure \\ref{Fig:Patchy} gives an\nexample. Apparently this model can not account for the observed X-ray\nflares. \n\\begin{figure}\n\\epsscale{1.0}\n\\plotone{f6.eps}\n\\caption{The X-ray lightcurve powered by a patchy jet. For simplicity,\nan annular patchy jet is simulated. Following parameters are adopted. \nPatch 1 (the on-beam jet): $0<\\theta<0.02$, $E_{\\rm iso,1}=10^{52}$ergs,\n$\\Gamma_{0,1}=240$; Patch 2: $0.02<\\theta<0.04$, $E_{\\rm iso,2}=5\n\\times 10^{52}$ergs, $\\Gamma_{0,2}=50$; Patch 3: $0.04<\\theta<0.06$,\n$E_{\\rm iso,3}=10^{52}$ergs, $\\Gamma_{0,3}=240$; Patch 4:\n$0.06<\\theta<0.08$, $E_{\\rm iso,4}=5 \\times 10^{52}$ergs,\n$\\Gamma_0=50$; Patch 5: $0.08<\\theta<0.10$, $E_{\\rm\niso,5}=10^{52}$ergs and $\\Gamma_{0,5}=240$. Other parameters are the\nsame as Fig.\\ref{Fig:TwoComp}.}\n\\label{Fig:Patchy}\n\\end{figure}\n\\subsection{Post energy injection into the blastwave?}\n\\label{subsec:model6}\n\nIn the internal shock model, it is expected that after the collisions are\nfinished, the shells are distributed such that a shell with a higher\nLorentz factor always leads a shell with a lower Lorentz factor. As the\nfast moving shell (blastwave) is decelerated, the trailing slow shell\nwill catch up with it and inject energy into the blastwave (Kumar \\&\nPiran 2000a). Such an injection also happens if the central engine\nfurther ejects high-$\\Gamma$ shells that catch up with the\ndecelerating blastwave (Zhang \\& M\\'esz\\'aros 2002b). Such a collision\nwould give rise to a bump signature on the lightcurve. A detailed\ntreatment suggests that the overall lightcurve should include emission\nfrom three components: a forward shock propagating into the medium, a\nsecond forward shock propagating into the blastwave, and a reverse\nshock propagating into the injected shell. In the X-ray band, the\ncontribution from the reverse shock is negligible. The lightcurve\ngenerally shows a step-like signature, due to the increase of the\ntotal energy in the blastwave (Zhang \\& M\\'esz\\'aros 2002b). \nAfter the peak, the flux level does not resume the previous level\nsince more energy has been injected into the fireball. The decay slope\nafter the injection peak follows the standard afterglow model, and\n$\\delta t\/t \\sim 1$ is expected. These are also inconsistent with the\ndata of X-ray flares.\n\n\\subsection{Neutron signature?}\n\nAnother interesting possibility is whether X-ray flashes are the\nsignature of the existence of free neutrons in the fireball. \nDerishev et al. (1999), Beloborodov (2003) and Rossi et al. (2004)\nsuggested that a baryonic fireball contains free neutrons, the decay\nof which would leave important imprints on the lightcurve. Fan et\nal. (2005a) modeled the process carefully and calculated the\nlightcurves. According to Fan et al. (2005), the neutron-feature is\nrather smooth in a wind model, and it is hard to detect. In the ISM\ncase, on the other hand, a bump does exist (in all bands). The\nphysical reason is that the trailing proton shell catches up with the\ndecelerated neutron-decay products. The physical process is analogous\nto the post energy injection effect discussed in \\S\\ref{subsec:model6}. \nThe amplitude of the flare is modest, at most a factor of\nseveral. Since the injection model is not favored, this possibility is\nalso disfavored.\n\n\\subsection{Late central engine activity}\n\\label{subsection:Model7}\n\nAfter ruling out various ``external-origin'' mechanisms, we are left\nonly with the possibility that involves the re-activation of the\ncentral engine. In this interpretation, the X-ray flares share\nessentially the same origin as the prompt gamma-ray emission, i.e. they\nare caused by some ``internal'' energy dissipation processes which\noccur before\nthe ejecta is decelerated by the ambient medium. The leading scenario\nis the ``late'' internal shock model, which suggests that the central\nengine ejects more energy in the form of an unsteady wind with varying\nLorentz factors at a late time. These discrete shells collide with each\nother and produce the observed emission. Alternatively, the late\ninjection could be mainly in the form of magnetic fields\/Poynting\nflux, and the X-ray flares are due to the intermittent dissipation of\nthe magnetic fields, likely through magnetic reconnection events.\nFan, Zhang \\& Proga (2005c) argued that at least for the flares\nfollowing short GRBs, the process that powers the flares has to be\nmagnetic-origin. \n\nThere are two advantages of the ``internal'' models over the\n``external'' models. \n\nFirst, re-starting the central engine equivalently re-set the time \nzero point. In this interpretation, the observed \nflare component and the underlying decaying component observed at the\nsame observer time $t$ originate from\ndifferent physical sites at different central engine times. Let us\nassume that the initial burst lasts $T_{90}$, that the central engine\nre-activates after a time interval $\\Delta t$, and that it ejects an\nunsteady wind with a typical variability time scale of $\\delta\nt$. At the observer time $t=T_{90}+\\Delta t \\sim 1000$ s, the\nunderlying decaying component (external afterglow) happens at a\ndistance $R_{ex} \\sim [\\Gamma(t)]^2 c t \\sim 5\\times 10^{16}$ cm,\nwhere $\\Gamma(t)$ is the Lorentz factor of the blastwave at the time\n$t$. The flare, on the other hand, happens at a distance of $R_{in}\n\\sim \\Gamma_0^2 c \\delta t \\sim (10^{13}-10^{14})$ cm, where\n$\\Gamma_0$ is the initial Lorentz factor of the late-time\nejecta. According to the clocks attached to \nthe central engine, the photons from the external afterglow component\nare emitted at $\\hat t_{ex} \\sim R_{ex} \/ c \\sim 1.7\\times 10^6$s,\nwhile the photons from late central engine activity are emitted at\n$\\hat t_{in} \\sim t+R_{in}\/c  \n\\sim 10^3 + R_{in}\/c \\sim (10^3 - 10^4)$s. Because of the relativistic\neffect, these photons reach the observer at exactly the same time $t$,\nand superpose onto the lightcurve detected by the observer. When\nplotted as a single $\\log-\\log$ lightcurve with the origin at the burst\ntrigger, a very steep apparent decay slope can be produced for a large\ntime shift $\\Delta t$ (Fig.\\ref{Fig:Internal}). This naturally\novercomes the $\\delta t\/t \\geq 1$ constraint encountered by the\nexternal models (Ioka et al. 2005 and references therein).\n\nSecond, invoking a late central engine activity greatly eases the\nrequired energy budget. In most of the external models, in order to\ngive rise to a significant bump on the lightcurve, the total\nnewly-added energy (either from the radial direction - late injection\ncase, or from the horizontal direction - patchy jets or\nmultiple-component jets) must be at least comparable to the energy\nthat defines the afterglow emission level. This model therefore\ndemands a very large energy budget. For the internal model, on the\nother hand, since the lightcurve is a superposition of two independent\nphysical components, the energy budget is greatly reduced, especially\nif the bump happens at later times when the background afterglow level\nis much lower. For example, for an X-ray lightcurve with the decay\nindex of -1 following a burst with duration 10 s, a significant flare\nat $\\sim 10^4$s only requires a luminosity slightly larger than\n$10^{-3}$ times of that of the prompt emission. \nThis model is therefore very ``economical'' in interpreting the very\nlate ($>10^4$s) flares detected in some bursts (e.g. GRB 050502B,\nFalcone et al. 2005; and GRB 050724, Barthelmy et al. 2005b).\n\nCan late central engine activity give rise to softer bursts\n(e.g. X-ray flares as compared with the prompt gamma-ray or hard X-ray\nemission)? According to the internal shock model, the peak energy\n$E_p$ of the synchrotron spectrum satisfies (Zhang \\&\nM\\'esz\\'aros 2002c)\n\\begin{equation}\nE_p \\propto \\Gamma B' \\propto L^{1\/2} r^{-1} \\propto L^{1\/2}\n\\Gamma^{-2} \\delta t^{-1}~, \n\\end{equation}\nwhere $L$ is the luminosity, $B'\\propto (L\/\\Gamma^2r^2)^{1\/2}$ is the\ncomoving magnetic field strength, $\\Gamma$ is the typical Lorentz\nfactor of the wind (for internal shock collisions, this $\\Gamma$ is\nfor slow shells), and $\\delta t$ is the variability time scale. We can\nsee that a smaller $L$, a higher $\\Gamma$ and a larger $\\delta t$ would be\nfavorable for a softer burst. The observed XRT X-ray flares generally\nhave a smaller luminosity $L$. The lightcurve is smoother with a\nlarger $\\delta t$ (Burrows et al. 2005; Barthelmy et al. 2005b). Also,\nat later times, the environment tends to be cleaner so that $\\Gamma$\ncould be larger. One therefore naturally expects softer flares at\nlater times. \n\nCan the central engine re-start after being quiescent for some time\n(e.g. $\\sim 10^3$s, but sometimes as late as $\\sim 10^5$s)? This is\nan interesting theoretical problem. The collapsar model predicts a\ncentral engine time scale of minutes to hours (MacFadyen et\nal. 2001). The star may also fragment into several pieces during the\ncollapse (King et al. 2005). The largest piece collapses first onto\nthe black hole, powering a prompt gamma-ray burst. Other fragments\nare intially ejected into elliptical orbits, but would eventually fall\ninto the black hole after some time, powering the late X-ray\nflares. Fragmentation could also happen within the accretion disk\nitself due to gravitational instabilities (Perna et al. 2005). A\nmagnetic-dominated accretion flow could also give rise to intermittent\naccretion flows due to interplay between the gravity and the magnetic\nbarrier (Proga \\& Begelman 2003; Proga \\& Zhang 2005). Flares could\nalso occur in other central engine scenarios (Dai et al. 2005). The rich\ninformation collected by the {\\em Swift} XRT suggests that we are\ngetting closer to unraveling of the details of the bursting mechanisms\nof GRBs. More detailed studies are called for to unveil the mystery of\nthese explosions.\n\nAn interesting fact (see next subsection for case studies) is that the\nduration of the flares are positively correlated with the time at\nwhich the flare occurs. The later the flare, the longer the flare\nduration. A successful central engine model must be able to address\nsuch a peculiar behavior. Perna et al. (2005) suggest that if the\naccretion disk is fragmented into blobs or otherwise has large\namplitude density fluctuation at large radii from the central engine,\nthe viscous disk evolution would cause more spread for blobs further\nout from the central engine. This gives a natural mechanism for the\nobserved correlation. Perna et al. (2005) suggest that gravitational\ninstability in the outer part of the disk is likely the origin of the\ndensity inhomogeneity within the disk.\n\n\\subsection{Case studies}\n\nIn this subsection, we briefly discuss the X-ray flares discovered in\nseveral GRBs.\n\n{\\bf GRB 050406 \\& GRB 050502B} (Burrows et al. 2005): These were the\nfirst two bursts with flares detected by XRT. For the case of GRB\n050406 whose \n$T_{90}$ is $\\sim 5$s, an X-ray flare starts at $\\sim 150$s and\nreaches a peak at $\\sim 230$s. The \nflux rebrightening at the flare peak is by a factor of 6. The rising\nand the decaying indices are $\\sim 4.9$ and $-5.7$, respectively. The\ntotal energy emitted during the flare is about 10\\% of that emitted in\nthe prompt emission. This suggests that the central engine becomes\nactive again after $\\sim 150$s, but with a reduced power. For the case\nof GRB 050502B whose $T_{90}$ is $\\sim 17.5$s, a giant flare starts at\n$\\sim 300$s and reaches a peak at $\\sim 740$s. The flux rebrightening\nis by a factor of $\\sim 500$. The decay index after the peak is\n$\\sim -6$. The total energy emitted during the\nflare is comparable to that emitted in the prompt emission. This\nsuggests that the central engine re-starts and ejects a substantial\namount of energy. The duration of the X-ray flare is much longer\nthat the duration of the prompt emission ($T_{90}$), so that the\nluminosity of the flare is much lower. In GRB 050502B, there is yet\nanother late flare-like event that peaks at $\\sim 7\\times 10^4$s. The\npost-peak decay index is $\\sim -3$. This is also consistent with the\ncase of a late-time central engine activity, and the decaying slope is\nconsistent with that expected from the curvature effect. The total\nenergy emitted in this flare is $\\sim 10\\%$ of that emitted in the\ngiant flare at $\\sim 740$s.\n\n{\\bf GRB 050724} (Barthelmy et al. 2005b): This is a short, hard burst\nwith $T_{90} \\sim 3$s whose host galaxy is an elliptical galaxy,\nsimilar to the case of the first {\\em Swift}-localized short burst\nGRB 050509B (Gehrels et al. 2005). The XRT lightcurve reveals rich\nfeatures which are quite similar to the case of GRB 050502B. \nThe XRT observation starts at $\\sim 74$s after the trigger, and the\nearly XRT lightcurve initially shows a steep decay with a slope \n$\\sim -2$. This component is connected to the extrapolated BAT\nlightcurve that shows a flare-like event around $(60-80)$s. This\nextended flare-like epoch (including the -2 \ndecay component) stops at $\\sim 200$s after which the lightcurve\ndecays even more rapidly (with an index $<-7$). A second, less\nenergetic flare peaks at $\\sim 300$s, which is followed by\nanother steep (with index $<-7$) decay. A third, significant\nflare starts at $\\sim 2\\times 10^4$s, and the decay index after the\npeak is $\\sim -2.8$. This quite similar to those the late flare \nseen in GRB 050502B. The energy emitted during the third flare is\ncomparable to that of the second one, and both are several times\nsmaller than the energy of the first flare. All the rapid-decay\ncomponents following the flares could be potentially interpreted as\nthe high-latitude emission, \ngiven a proper shift of the zero time point. Since the duration of the\nthird flare is very long, the $t_0$ effect does not affect the decay\nindex too much. This is why the decay index $-2.8$ is quite\n``normal'', i.e. consistent with the $-2-\\beta$ prediction. The\nearlier steeper decays (e.g. $<-7$) are all preceded by flares with\na sharp increasing phase. Shifting $t_0$ in these cases would lead to\nsignificant flattening of the decaying index, which could be still\nconsistent with the high-latitude emission. All these discussions also\napply to the case of GRB 050502B\\footnote{After submitting this paper,\ndetailed data analyses (Liang et al. 2005) confirm these speculations.\nThe late flares in both GRBs 050724 and 050502B are consistent\nwith the hypothesis that they are due to late time central engine\nactivity.}.\n\n{\\bf GRB 011121} (Piro et al. 2005). This {\\em BeppoSAX} burst also\nindicated a flare-like event around $\\sim 270$s. Piro et al. (2005)\ninterpreted the X-ray bump as the onset of the afterglow phase. In\nview of the fact that X-ray flares are commonly detected in {\\em\nSwift} bursts, it is natural to spectulate that the event is also an\nX-ray flare caused by late central engine activity. Fan \\& Wei (2005)\nhave suggested the late central engine activity and performed a\ndetailed case study on this event. \n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nDuring the past several months, the {\\em Swift} XRT has collected\na rich sample of early X-ray afterglow data. This, for the first time,\nallows us to peer at the final temporal gap left by the previous\nobservations and to \nexplore many interesting questions of GRB physics. In this paper,\nwe have systematically investigated various possible physical processes\nthat could give interesting contributions to the early X-ray afterglow\nobservations. This includes the tail emission of the prompt gamma-ray\nemission, both the forward shock and the reverse shock emission\ncomponents, refreshed shocks, post energy injection, medium\ndensity clumps near the burst, angular inhomogeneities of the\nfireball, emission component due to the presence of free neutrons, as\nwell as emission from late central engine activity. We discuss how\nthe above processes might leave interesting signatures on the early\nX-ray afterglow lightcurves.\n\nBased on the XRT data collected so far, we summarize the\nsalient features and suggest a\ntentative synthetic lightcurve for the X-ray afterglow\nlightcurves. As shown in Fig. \\ref{XRTlc}, the synthetic lightcurve\nincludes five components: I. an intial steep decay component;\nII. a shallow-than-normal decay component; III. a ``normal'' decay\ncomponent; IV. a post-jet break component; and V. X-ray flares. \nThe components I and III appear in almost all the \nbursts. Other three components also commonly appear in some bursts.\nFlares have been detected in nearly half of the XRT early\nlightcurves, and the shallow decay segment has also been discovered\nin a good fraction of {\\em Swift} GRBs. We therefore believe that they\nrepresent some common underlying physics for GRBs. After comparing\ndata with various physical models, we tentatively draw the\nfollowing conclusions.\n\n1. The rapid decay component (Tagliaferri et al. 2005) commonly\nobserved in the very early afterglow phase (which usually has a\ndifferent spectral slope than the late shallow decay components) is\nvery likely the tail \nemission of the prompt gamma-ray bursts or of the early X-ray flares.\nAllowing proper shifting of the time zero point and considering the\ncontribution of the underlying forward shock emission, we speculate\nthat essentially all the steep decay cases could be understood in\nterms of the ``curvature effect'' of the high-latitude emission as the\nemission ceases abruptly. More detailed data analyses (Liang et\nal. 2005) support such a speculation.\n\n2. The transition between the prompt emission and the afterglow\nemission appears to be universally represented by a rapid decay\nfollowed by a shallower decay, indicating that the GRB emission site\nis very likely different from the afterglow site, and that the\napparent gamma-ray efficiency is very high.\n\n3. In a good fraction of GRBs (e.g. GRBs 050128, 050315, 050319, 050401,\nCampana et al. 2005; Vaughan et al. 2005; Cusumano et al. 2005; de\nPasquale et al. 2005), a clear temporal break exists in the early\nX-ray lightcurves. There is \nno obvious spectral index change across the break. The temporal decay\nindex before the break is very flat, while that after the break is\nquite ``normal'', i.e. is consistent with the standard afterglow\nmodel for a fireball with constant energy expanding into an ISM. We\nsuggest that these breaks are likely not ``jet breaks''. Rather they\nmark the cessation of an early continuous energy injection phase\nduring which the external shock is refreshed. We suggest three\npossible physical mechanisms for the refreshed shocks, i.e. a\nlong-lived central engine with a decaying luminosity, a power law\ndistribution of the shell Lorentz factors before deceleration begins,\nand the deceleration of a high-$\\sigma$ flow. Further studies are\nneeded to better understand this phase.\n\n4. In the ``normal'' phase, the data for many bursts are consistent\nwith an ISM medium rather than a wind medium. This has important\nimplications for understanding the massive star progenitors of long\nGRBs, including their late evolution stage shortly before explosion.\n\n5. Given that most of the shallow-to-normal transitions are due to the\ncessation of the refreshed shock phase, the cases with a\nwell-identified jet break are not very common. Nonetheless, jet breaks\nare likely identified in some bursts, e.g. GRB 050315 (Vaughan et\nal. 2005), GRB 050525A (Blustin et al. 2005), XRF 050416 \n(Sakamoto et al. 2005), and some others.\n\n6. The X-ray flares detected in nearly half of the {\\em Swift} bursts\nare most likely due to late central engine activity, which results\nin internal shocks (or similar energy dissipation events) at later\ntimes. It seems to us that there is no evidence for the existence of\ndensity clumps in the GRB neighborhood, and there is no support for\nstrong angular inhomogeneities (e.g. two-component jet, patchy jets)\nfor the GRB fireball. However, their existence is not ruled out.\n\n7. The similar lightcurves for some long GRBs (e.g. GRB050502B,\nBurrows et al. 2005) and some short GRBs (e.g. GRB050724, Barthelmy et\nal. 2005) indicate that different progenitor systems may share some\nsimilarities in the central engine. This might be caused by the same\nunderlying physics that controls the hyper-accreting accretion disk -\na common agent in charge both types of systems (Perna et al. 2005).\n\nThe {\\em Swift} XRT is still rapidly accumulating data on the early\nX-ray afterglows. More careful statistical analyses of these X-ray data,\nas well as more detailed case studies, also including low frequency\ndata collected by UVOT and other ground-based telescopes, would greatly\nimporve our knowledge about GRB prompt emission and afterglows,\nadvancing the quest for the final answers to the core questions in the\nstudy of GRBs.\n\n\n\n\\acknowledgments \nWe thank the referee for helpful comments.\nB.Z. acknowledges useful discussions with T. Abel, P. Armitage,\nM. Begelman, Z. G. Dai, C. D. Dermer, B. Dingus, C. Fryer, L. J. Gou,\nJ. Granot, A. Heger, D. Lazzati, A. Panaitescu, R. Perna, D. Proga,\nS. Woosley, X. Y. Wang, X. F. Wu, and W. Zhang on various topics\ncovered in this paper. \nThis work is supported by NASA NNG05GB67G, NNG05GH91G (for BZ),\nNNG05GH92G (for BZ, SK and PM), Eberly Research Funds of Penn State\nand by the Center for Gravitational Wave Physics under grants\nPHY-01-14375 (for SK), NSF AST 0307376 and NASA NAG5-13286 (for PM),\nand NASA NAS5-00136 (DB \\& JN).  \n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{intro}\n\nJets, the collimated sprays of particles resulting from hard\nscattering processes, are an excellent probe for heavy ion\ncollisions. Since the hard scattering happens early in the collision,\nthe partons probe all stages of the collision while traversing the\nproduced medium. Suppression of jet production is quantified by comparing the fully\ncorrected jet spectrum measured in central Pb-Pb collisions to that measured in pp\nscaled by the average number of binary collisions, $N_{\\rm\n  coll}$ \\cite{alicePbPbchjets,alicePbPbjets}. This\nsuppression is attributed to partonic energy loss in the Quark Gluon\nPlasma (QGP) created in heavy ion collisions. However, cold nuclear matter (CNM)\neffects due to the initial state could also influence this\nmeasurement. Measurements of the jet cross section in p-Pb collisions, which experience CNM effects without the creation of the final state\nQGP are critical for disentangling initial and final state effects on\nthe observed Pb-Pb jet spectra.\n\n\n\\section{Analysis Details}\n\\label{analysis}\n\nThe present analysis uses minimum bias 5.02 TeV p-Pb events collected by the ALICE experiment\nduring the 2013 LHC run with an\nintegrated luminosity of 51 $\\mu b^{-1}$. Input to the jet reconstruction\nalgorithm are\nclusters, measured in the\nelectromagnetic calorimeter (EMCal) with $E_{T}>300$ MeV\/$c$ and\ncharged tracks with $p_{T} > 150$ MeV\/$c$ measured in the ALICE central tracking\nsystem, which consists of a time projection chamber (TPC) and a silicon inner\ntracking system (ITS). To correct the EMCal clusters for energy\ndeposited by charged tracks, 100\\% of the momentum of any tracks\ngeometrically matching that cluster is subtracted from the cluster \\cite{aliceppjets}.   \nJets are reconstructed with the anti-$k_{T}$ jet finding\nalgorithm with resolution parameters, $R$ = 0.2\nand $R$ = 0.4 using the FastJet package \\cite{fastjet}. To remove ``fake'' jets,\nclustered constituents that did not originate from a hard scattering, the area of the jet,  $A_{jet}$, is required to\nfulfill $A_{jet}>0.6\\pi R^{2}$. To ensure the jet is fully within the\ndetector acceptance, the jet axis must be at least $R$ away from\nthe edge of the detector. The detector boundaries are defined by the EMCal acceptance, $|\\eta|<0.7$ and $1.4< \\phi < \\pi$.\n\nEnergy from the underlying event is also clustered into the\njets by the algorithm and must be subtracted from the total raw jet\nenergy. An average energy density, $\\rho$, is determined on an event-by-event\nbasis and then subtracted from each jet in the event according to\n$p_{T,jet}^{reco}=p_{T,jet}^{raw}-\\rho_{scaled} \\times A_{jet}$. To\nreduce the effect of the limited EMCal acceptance, we base the\nbackground density on the charged track $p_{T}$ density, $\\rho$, which\nis measured in full azimuth and scale it up to $\\rho_{scaled}$, using a scale factor that is\ndetermined from measured data to include electromagnetic contributions, as done in the Pb-Pb jet spectra\nanalysis \\cite{alicePbPbjets}. While for Pb-Pb the median method was used to\ncalculate the charged track background density, a median occupancy\nmethod, which is a slightly modified implementation of that\npresented in \\cite{cmsrho}, is used here. The median occupancy method,\ndefined by\n\\begin{equation}\n\\label{eq:rho}\n\\rho=\\rm{median}\\left \\{\\frac{p_{T}^{i}}{A_{i}} \\right \\} \\times C,\n\\end{equation}\nis determined by running the $k_{T}$ algorithm over all tracks plus ``ghost\nparticles'' (unphysical particles with negligible momentum added\nartificially by FastJet\nfor the jet area calculation) in the event \\cite{fastjet}. Before determining the\nmedian of the physical jets, any $k_{T}$ jets overlapping with signal jets are\nexcluded. Signal jets are defined as anti-k$_T$ jets with $p_{T} > 5$ GeV\/$c$. The median is then scaled by an occupancy correction\nfactor, $C=A_{physical jets}\/A_{all jet}$, where $A_{physical jet}$ is the total area of all physical jets and\n$A_{all jet}$ is the area of all jets, including jets that only\ncontain ghost particles. This factor, $C$, accounts for the emptiness of\nthe p-Pb event. \n\n\n\n\\subsection{Unfolding}\n\\label{uf}\n\nThe measured spectra must be corrected for detector effects and the\ninfluence of fluctuations on the underlying event. The effect of the detector on the spectra is determined by\npassing PYTHIA events at $\\sqrt{s} =$ 5.02 TeV through a GEANT\nsimulation of the ALICE detector. Jets are reconstructed at the\ndetector level using the same cuts as applied on the data. Jets\nare also reconstructed at the particle level without cuts to\ngive the true jet $p_{T}$. All\ndetector level jets are geometrically matched to the nearest particle\nlevel jet. A\n2-dimensional histogram or response matrix (RM) maps\ndetector level jet $p_{T}$ to particle level jet $p_{T}$ and is used in the\nunfolding procedure. Figure \\ref{fig:RMdpt} shows an example of the detector\nRM for $R$ = 0.4. A correction\nis also applied to account for the jet reconstruction efficiency. This correction is determined by dividing the true\nPYTHIA jet spectrum by a projection of the\nRM onto the particle level jet $p_{T}$ axis.\n\n\\begin{figure}[htp]\n\\centerline\n{\n\\includegraphics[width=0.5\\textwidth]{.\/2014-May-14-R4_lhc13_semigood_DetectorRM.pdf}\n\\includegraphics[width=0.42\\textwidth]{.\/2014-May-14-R4_DeltaPt.pdf}\n  }\n  \\caption{(Left) Detector response matrix mapping the reconstructed detector level jet $p_{T}$ to\n    particle level jet $p_{T}$. (Right) The $\\delta p^{ch+em}_{T}$ determined\n    using Random Cones. The leading jet from each event were\n    removed with a\n    probability of $1\/N_{\\rm coll}$ (black astrics) for the analysis. To estimate the\n    systematic uncertainty,\n    all (red open squares) and no leading jets are removed (blue circles). Both plots are for anti-k$_{T}$ jets\n    with $R$ = 0.4.}\n\\label{fig:RMdpt}\n\\end{figure}\n\nThe underlying event energy density is determined on an event-by-event\nbasis, however, even within a single event  there are fluctuations\nwithin the background energy density.  The momentum of the jet may be over-\nor underestimated if it was positioned on an upward\nor downward fluctuation. These fluctuations can be quantified by\nmeasuring the $\\delta p^{ch+em}_{T}$ distribution using the method of\nRandom Cones (RC) according to\n\\begin{equation}\n\\delta p^{ch+em}_{T}=p_{T}^{RC}-\\pi R_{RC}^2 \\times \\rho, \n\\end{equation}\nwhere the $\\rho$ of the\nevent is compared to the total momentum, $p_{T}^{RC}$, within a cone of\nradius, $R_{RC}$, placed randomly in the event.  The right panel of Figure\n\\ref{fig:RMdpt} shows the $\\delta p^{ch+em}_{T}$ distribution for $R_{RC}$ = 0.4. The long tail on the right\nhand side of the distribution represents the probability of having two\noverlapping jets. Since the random cone can be thought of as introducing an\nadditional jet to the event, this probability is artificially enhanced. We account for this effect by\nexcluding the leading jet from the event at a rate of  $1\/N_{\\rm coll}$ (black\npoints). To estimate the systematic uncertainty on this procedure,\nthis exclusion probability can be varied between zero, where no jets are removed, and one,\nwhere all leading jets are removed. The $\\delta p^{ch+em}_{T}$ distribution with no jets removed and all\nleading jets removed from each event are shown as blue circles and\nred squares respectively in Figure \\ref{fig:RMdpt} and result in a  \n3\\% uncertainty on the final spectra.\n\nThe final RM, the multiplication of the detector RM and the $\\delta p^{ch+em}_{T}$ distribution, is input to the unfolding algorithm. The singular value decomposition (SVD) algorithm was chosen as the\ndefault for unfolding the spectrum \\cite{svd}. Unfolding with the Bayesian method was also performed\nand the difference used in the estimation of the systematic\nuncertainties \\cite{bayes}. A bin-by-bin correction procedure\nwas found to be in good agreement with the other methods. \n\n\\begin{figure}[htp]\n\\centerline\n{\n\\includegraphics[width=0.42\\textwidth]{.\/2014-May-14-R4_Data_MC_prelim1.pdf}\n\\includegraphics[width=0.42\\textwidth]{.\/2014-May-14-R2_Data_MC_prelim1.pdf}\n  }\n  \\caption{Fully corrected p-Pb jet spectrum for $R$ = 0.4 and $R$ = 0.2\n    scaled by $N_{\\rm coll}$\n    compared to PYTHIA and POWHEG simulations at 5.02 TeV.}\n\\label{fig:spectra}\n\\end{figure}\n\\section{Results}\n\nThe resulting spectra after unfolding for $R$ = 0.4 and $R$ = 0.2 are shown in\nFigure \\ref{fig:spectra}. The spectra were normalized per average number of\nbinary collisions, $N_{\\rm coll}$, to make a direct comparison to the pp\nreferences. Due to the lack of a measured pp reference at\n$\\sqrt{s}$ = 5.02 TeV, we compare the p-Pb data to simulations. The plot\nincludes PYTHIA8, PYTHA6 and POWHEG with 2 different parton\ndistribution functions (PDF). The POWHEG calculations include\nuncertainties on the factorization and renormalization scales (13\\%) and the uncertainty in the PDF\n(6\\% for CTEQ and 9\\% for EPS). The ratios between the data and the\ndifferent models are all consistent with one, indicating there are no\ncold nuclear matter effects to the jet spectrum. However, the spread and uncertainty from these\ndifferent references is significant and highlights the need for a data\nreference to better quantify this statement. Despite this uncertainty,\nthe p-Pb results clearly demonstrate that the strong suppression observed in the Pb-Pb is not\npurely due to initial state effects, but is rather a result of\nenergy loss in the produced medium. \n\nOne may question whether the fragmentation could be modified in p-Pb\ncollisions while the total jet cross section is not. A first\nindication of \nthe fragmentation behavior of these jets can be obtained by taking\nthe ratio of the spectra measured with different $R$. Figure\n\\ref{fig:JSR} shows the ratio of the $R$ = 0.2 spectrum to the $R$ = 0.4\nspectrum for 5.02 TeV p-Pb (red circles) and for 2.76 TeV pp (black\nsquares) collisions.  The agreement between the two collision systems\nsuggests that the fragmentation behavior for jets in p-Pb is very\nsimilar to that in pp collisions. Of course, more differential studies\nwill provide more detailed insight into the fragmentation properties. \n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[htp]\n\\centerline\n{\n\\includegraphics[width=0.6\\textwidth]{.\/2014-May-15-JSR_prelim2.pdf}\n  }\n  \\caption{Cross-section ratio between $R$ = 0.2 jet spectrum and $R$ =\n    0.4\n    jet spectrum for fully reconstructed jets in 5.02 TeV p-Pb\n    collisions (red) and 2.76 TeV pp collisions (black).}\n\\label{fig:JSR}\n\\end{figure}\n\n\\section{Conclusions}\n\nThe fully reconstructed jet spectra for $R$ = 0.2 and $R$ = 0.4 have been\nmeasured by ALICE in 5.02 TeV p-Pb collisions in the $p_{T}$ range 20-90 GeV\/$c$.  Comparisons to model\npredictions of the 5.02 TeV pp jet spectra indicate that the strong\nsuppression observed in Pb-Pb collisions is an effect of the medium and\nnot an initial state effect. To better quantify the CNM effects, if\nany, on the jet spectrum, systematic uncertainties on this measurement must be reduced. In\nparticular, the uncertainty on the reference can be reduced by measuring\npp collisions at 5 TeV at the LHC. The ratio between spectra\nreconstructed with different $R$ is consistent with the same ratio in pp\ncollisions. This also indicates no modification to the substructure of the\njet. Inclusion of EMCal triggered events will extend the\nkinematic reach of this measurement and allow for multiplicity\ndependent studies.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}