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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nHigh-energy neutrino astronomy has finally begun. \nThe detection of PeV-energy neutrinos~\\cite{icecubePeV2013} and the follow-up\nanalyses~\\cite{icecubeHESE2013} by the IceCube Collaboration revealed the existence\nof astrophysical ``on-source'' neutrinos at energies ranging from TeV to PeV.\nThese neutrinos are expected to be produced by the interactions\nof ultrahigh-energy cosmic-ray (UHECR) protons via $pp$ collisions\nor $\\gamma p$ collisions. The bulk intensity of these neutrinos,\n$E_\\nu^2 \\phi_{\\nu_e+\\nu_\\mu+\\nu_\\tau}\\simeq \n3.6\\times 10^{-8} {\\rm GeV} {\\rm cm^{-2}} \\sec^{-1} {\\rm sr^{-1}}$,\nprovides an important clue to understanding the general characteristics\nof UHECR sources through the connection between the observed cosmic-ray\nand neutrino intensities. \n\nIn the even higher-energy region from EeV to 100 EeV (EeV $=10^9 {\\rm GeV}$),\nthe highest-energy cosmic-ray (HECR) protons generate EeV-energy neutrinos\nvia interactions with cosmic microwave background (CMB) photons~\\cite{GZK} and extragalactic background light (EBL)\nduring their propagation in intergalactic space. The intensity of these\n``GZK cosmogenic'' neutrinos~\\cite{BZ} averaged over the sky is a consequence\nof the integral of the HECR emission over cosmic time, as neutrinos are\nstrongly penetrating particles that can travel cosmological\ndistances. It is, therefore, an observational probe to trace\nthe HECR source evolution. In particular, the cosmogenic neutrino intensity\nfrom 100 PeV to 10 EeV is highly sensitive to the evolution of the HECR emission rate\nand less dependent on other uncertain factors such as the highest energy\nof accelerated cosmic rays at their sources. As this energy range coincides with\nthe central region covered by the IceCube ultrahigh-energy neutrino searches,\nthe flux sensitivity achieved by IceCube has started to constrain a sizable\nparameter space of HECR source evolution, revealing the general\ncharacteristics of UHECR and HECR sources independent of the cosmic-ray acceleration model.\n\nIn this article, we review the new knowledge of UHECR\/HECR sources\nprovided by neutrino observations by IceCube. \nThe standard $\\Lambda$CDM cosmology with $H_0 = 73.5$ km s$^{-1}$ Mpc$^{-1}$, $\\Omega_{\\rm M} = 0.3$, \nand $\\Omega_{\\Lambda}=0.7$ is assumed throughout this article.\n\n\\section{The cosmic neutrino spectrum: Overview}\n\\label{sec:overview}\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{.\/NeutrinoFluxOverviewWzCaption.pdf}\n\\caption{Illustrative display of the differential fluxes of diffuse neutrinos having various origins. \nSupernova relic neutrinos are expected to appear in the MeV sky. Atmospheric\nneutrinos originating in extensive cosmic-ray airshowers dominate the background\nfor high-energy cosmic neutrino detection. Astrophysical neutrinos produced\n{\\it in situ} at cosmic-ray sources emerge at TeV and PeV energies, which have now been\ndetected by IceCube. GZK cosmogenic neutrinos, whose flux has yet to be measured, fill the highest-energy universe. Shown here is the possible range of fluxes\nfrom various source evolution models. The proton-dominated HECR composition is assumed.}\n\\label{fig:neut_fluxes}\n\\end{figure}\n\nFigure~\\ref{fig:neut_fluxes} displays the spectrum of neutrinos coming from\nall over the sky, {\\it i.e.}, diffuse neutrino fluxes. The massive background\nof atmospheric neutrinos ranging in energy over many orders of magnitude\nhad masked the astrophysical neutrinos until the IceCube observatory finally\nrevealed their existence. The spectrum shown here, taken from a model~\\cite{YoshidaTakami2014},\nhas a cutoff feature at $\\sim{\\rm PeV}$, but it is not clear yet whether the IceCube neutrino\nfluxes have a spectral cutoff. It is not even obvious that the spectrum can be well\ndescribed by a single power law formula. An analysis with enhanced sensitivity\nin the 10 TeV region seems to exhibit a trend toward a softer spectrum~\\cite{MESE} than\nthe up-going diffuse muon neutrino analysis, which is sensitive\nat energies above $\\sim 100\\ {\\rm TeV}$~\\cite{diffuse_nu}.\nOn-going efforts in the IceCube Collaboration will ultimately resolve these \nissues. Nevertheless, the intensity, \n$E_\\nu^2 \\phi_{\\nu_e+\\nu_\\mu+\\nu_\\tau}\\simeq \n3.6\\times 10^{-8} {\\rm GeV} {\\rm cm^{-2}} \\sec^{-1} {\\rm sr^{-1}}$, \nhas been well determined within a factor of two, and the implications\nfor the origin of UHECRs based on the intensity are not affected by these\ndetails of the spectral structure. If the neutrino emitters are also\nsources of the cosmic rays we are observing (which is very likely but not an undeniable assumption),\nwe can associate the neutrino flux with their parent cosmic-ray proton flux, \nand its comparison to the {\\it observed} cosmic-ray spectrum places\nsome constraints on the source characteristics.\n\nThe GZK cosmogenic neutrinos are expected to emerge in the 100 PeV--EeV sky.\nTheir intensity at the highest-energy end ($\\sim50-100\\ {\\rm EeV}$) depends mainly on\nthe maximal accelerated energy of cosmic rays at their sources\nand is not relevant to the ultrahigh-energy neutrino search by IceCube,\nas it is most sensitive at energies below 10 EeV.\nThe intensity at the lowest-energy tail ($\\sim10-100\\ {\\rm PeV}$)\nis determined by the EBL density\nand its evolution, which is EBL-model-dependent and could vary\nthe flux by a factor of $\\sim5$ at $\\sim10\\ {\\rm PeV}$.\nThe EeV-energy intensity is decided primarily by the HECR source evolution\nin redshift space.\nThe integral intensity of cosmogenic neutrinos above 100 PeV ranges from $10^{-17}$ to \n$\\sim3\\times 10^{-16} {\\rm cm^{-2}} \\sec^{-1} {\\rm sr^{-1}}$\ndepending on these factors. The IceCube detection exposure\nfor UHE neutrinos has now reached $\\sim3\\times 10^{16} {\\rm cm^{2}} \\sec {\\rm sr}$,\nand one can see that the IceCube sensitivity enables access to\na significant parameter space of the cosmogenic neutrino production models.\n\n\\section{The constraints on PeV- and EeV-energy UHECR sources}\n\\label{sec:PeV-EeV}\n\nThe flux of astrophysical neutrinos produced by UHECR protons at\ntheir sources is related to their parent cosmic-ray intensity\nvia the proton-to-neutrino conversion efficiency. The efficiency is usually\nparameterized in the form of the optical depth of the proton interactions.\nFor neutrinos produced through photomeson production ($\\gamma p$),\ntheir diffuse flux integrating emitted neutrinos over all the sources\nof UHECR protons with a spectrum in the source\nframe, $\\sim \\kappa_{\\rm CR}(E_{\\rm CR}\/E_0)^{-\\alpha}$ (where $E_0$ is the reference energy,\nwhich is conveniently set to $\\sim10$ PeV), is described as~\\cite{YoshidaTakami2014}\n\\begin{widetext}\n\\begin{eqnarray}\n\\phi_{\\nu_e + \\nu_\\mu + \\nu_\\tau}(E_\\nu) &\\simeq& \n\\frac{2 n_0 \\kappa_{\\rm CR}}{\\alpha^2} \\frac{c}{H_0} \n\\frac{s_{\\rm R}}{\\sqrt{(s_{\\rm R} + m_{\\pi}^2 - m_p^2)^2 - 4 s_{\\rm R} m_{\\pi}^2}} \n\\frac{3}{1 - r_{\\pi}} \n\\frac{(1 - e^{-\\tau_0})}{2 (m - \\alpha) - 1} \\Omega_{\\rm M}^{- \\frac{m - \\alpha + 1}{3}} \\nonumber \\\\\n&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\times \n\\left[ \\left\\{ \\Omega_{\\rm M} (1 + z_{\\rm max})^3 + \\Omega_{\\Lambda} \\right\\}^{\\frac{m - \\alpha}{3} - \\frac{1}{6}} - 1 \\right] \n\\left( \\frac{E_{\\nu}}{E_0 x_{\\rm R}^+ (1 - r_{\\pi})} \\right)^{-\\alpha}. \n\\label{eq:onsource_simple}\n\\end{eqnarray}\n\\end{widetext}\nHere $n_0$ is the source number density at the present epoch,\nand the source evolution is parameterized as $\\psi(z)=(1+z)^m$\nextending to the maximal redshift $z_{\\rm max}$\nsuch that the parameter $m$ represents the scale of the cosmological\nevolution often used in the literature. Further, $s_R\\ (\\simeq 1.5 {\\rm GeV^2})$ \nis the squared collision energy at the $\\Delta$ resonance of photopion production,\n$r_{\\pi} \\equiv m_{\\mu}^2 \/ m_{\\pi}^2 \\simeq 0.57$ is the muon-to-pion mass-squared ratio,\n$m_p$ is the proton mass, and $x_{\\rm R}^+\\ (\\simeq 0.36)$ \nis the kinematically maximal bound \nof the relative energy of emitted pions normalized by the parent cosmic-ray energy.\n$\\tau_0$, the optical depth of $\\gamma p$ interactions at the reference energy $E_0$,\nlinks the neutrino flux to the parent cosmic-ray intensity determined by $\\kappa_{\\rm CR}$.\n\nThe cosmic-ray flux integrating a UHECR spectrum over all\nthe sources in the redshift space, which corresponds to the UHECR spectrum\nwe {\\it observe}, is given by\n\\begin{eqnarray}\n\\phi_{\\rm CR}(E_{\\rm CR}) &=& \nn_0 c \\kappa_{\\rm CR} \\int_0^{z_{\\rm max}} dz \\nonumber \\\\\n&& (1 + z)^{1 - \\alpha} \\psi(z) \\left| \\frac{dt}{dz} \\right| \ne^{- \\tau_0} \n\\left( \\frac{E_{\\rm CR}}{E_0} \\right)^{-\\alpha},\\nonumber\\\\\n&&\n\\end{eqnarray}\nneglecting intergalactic magnetic fields and the energy loss\nin the CMB field during UHECR propagation.\nIntroducing some analytical approximations leads to the following simple formula:\n\\begin{eqnarray}\n\\phi_{\\rm CR}(E_{\\rm CR}) &\\simeq& \n2n_0 \\kappa_{\\rm CR} \\frac{H_0}{c} e^{-\\tau_0}\\left( \\frac{E_{\\rm CR}}{E_0} \\right)^{-\\alpha}\\nonumber\\\\\n&& \\frac{1}{2(m-\\alpha)-1} \\Omega_{\\rm M}^{- \\frac{m-\\alpha+1}{3}}\\nonumber\\\\\n&&\\left[ \\left\\{ \\Omega_{\\rm M} (1 + z_{\\rm max})^3 + \\Omega_{\\Lambda} \\right\\}^{\\frac{m-\\alpha}{3} - \\frac{1}{6}} - 1 \\right].\\nonumber\\\\\n&& \n\\label{eq:UHECR_flux_approx}\n\\end{eqnarray}\n\nComparing this formula to Equation~(\\ref{eq:onsource_simple}),\none can find that the source evolution effect represented by the evolution\nparameter $m$ is canceled in the ratio of the neutrino flux\nto the parent UHECR flux. This is because both the secondary produced neutrinos\nand emitted UHECRs originate in the same sources with the same evolution history.\nConsequently, the optical depth $\\tau_0$ is a deciding factor\nin the relation between these two fluxes. The TeV--PeV neutrino observation\nby IceCube that determines the neutrino flux $\\phi_\\nu$ thus\nassociates the UHECR source optical depth with the UHECR flux.\n\n\\begin{figure}[tb]\n  \\includegraphics[width=0.4\\textwidth]{.\/CRFluxWzDumpOpticalDepthContourHESE_SFR.pdf}\n  \\caption{Constraints on the optical depth of UHECR sources \nfor PeV-energy neutrino production and the energy flux of \nextragalactic UHECRs, $E_{\\rm CR}^2\\phi_{\\rm CR}$, at an energy of 10 PeV~\\cite{YoshidaTakami2014}. \nThe regions between the two blue solid curves ($\\alpha=2.5$), \ngreen dashed curves ($\\alpha=2.7$), and light blue dot-dashed curves ($\\alpha=2.3$) \nare allowed by the present IceCube observations~\\cite{icecubePeV2013,icecubeHESE2013}.\nThe unshaded region highlights the allowed region for $\\alpha=2.5$ \ntaking into account the observed intensity of UHECRs measured by the IceTop experiment~\\cite{icetop2013}.}\n\\label{fig:constraints_10PeV} \n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[width=0.4\\textwidth]{.\/CRFluxWzDumpOpticalDepthContourHESE1EeV_SFR.pdf}\n  \\caption{Same as Figure~\\ref{fig:constraints_10PeV}, but constraints against the cosmic-ray flux\n    at an energy of 1 EeV, showing the constraints when the UHECR spectrum from PeV neutrino\n    sources extends to higher energies.}\n\\label{fig:constraints_1EeV} \n\\end{figure}\n\nFigure~\\ref{fig:constraints_10PeV}\ndisplays the relations between the optical depth and the UHECR flux for several values of $\\alpha$,\nall of which are consistent with the IceCube observation\nat the present statistics~\\cite{icecubeHESE2013}. Star-formation-like source evolution\nis assumed, but the other assumptions regarding the evolution \nwould not change the main results, as explained above.\nThe smaller optical depth, implying a lower neutrino production efficiency, would require\nmore UHECR protons to be compatible with the neutrino intensity measured by IceCube.\nThe optical depth of the UHECR sources must be larger than 0.01,\nas the parent UHECR flux would exceed the observed cosmic-ray flux otherwise.\nThe proton flux from the neutrino sources contributes more than at least a few percent\nof all the UHECRs in the 10-PeV energy range. The magnetic horizon effect\nwould not change these constraints unless the sources are very rare, for example, \nif their number density is much smaller than $\\sim10^{-6} {\\rm Mpc}^{-3}$~\\cite{YoshidaTakami2014}.\nNote that the lower bound of the source density set by the small-scale UHECR anisotropy study\nconducted by the Auger observatory~\\cite{auger_density} is $6\\times 10^{-6} {\\rm Mpc}^{-3}$.\n\nGamma-ray bursters (GRBs) are strong candidates for UHECR acceleration sites\nand therefore high-energy neutrino production sites. \nInternal shocks are the most popular sites to produce\nhigh-energy neutrinos. An optical depth of $0.1-10^{-2}$ can be achieved,\ndepending on the dissipation radius, which satisfies the optical depth condition\nshown in Figure~\\ref{fig:constraints_10PeV}.\nHowever, their energetics may be problematic.\nThe typical gamma-ray energy output of a\nregular GRB is $10^{52}$ erg in gamma rays, and the local\noccurrence rate of long-duration GRBs is $\\sim1 {\\rm Gpc^{-3}}\\ {\\rm yr^{-1}}$.\nThese data indicate that the luminosity of local cosmic rays generated from GRBs is\n$10^{44}(\\eta_p\/10) {\\rm erg}\\ {\\rm Mpc^{-3}}\\ {\\rm yr^{-1}}$,\nwhere $\\eta_p$ is the ratio of\nthe UHECR output and $\\gamma$-ray output, which is known as the\nbaryon loading factor. This luminosity is 2 orders\nof magnitude smaller than that of UHECRs at 10 PeV and\nthus is too low to meet the requirement shown in Figure~\\ref{fig:constraints_10PeV}\nthat the UHECR flux from the sources must account for $\\geq0.1$ of the total UHECR flux.\nWe conclude that \nGRBs are unlikely to be major sources of both PeV-energy UHECRs and neutrinos.\n\nAmong known astronomical objects,\nonly flat-spectrum radio quasars (FSRQs)\ncan realize a large $\\gamma p$ optical depth ($\\geq0.01$)\nand large energetics. The typical $\\gamma$-ray luminosity density\nof FSRQs is $\\sim10^{46} {\\rm erg}\\ {\\rm Mpc^{-3}}\\ {\\rm yr^{-1}}$\nin our local universe. This is comparable to the local density\nof UHECRs at $\\sim$10 PeV.\n\nFigure~\\ref{fig:constraints_1EeV} shows the constraints on the $\\gamma p$ optical depth\nand UHECR flux when the energy spectrum of UHECR protons emitted from the neutrino sources\nextends to much higher energies than the observed neutrino energies.\nThe allowed regions in the parameter space become much smaller than those\nfor the constraints for $E_0 = 10$ PeV because the spectrum of the observed cosmic rays\nis steeper than the observed neutrino spectrum.\nNote that the optical depth constrained here is not at the EeV level, but at the PeV level,\nbecause PeV-energy protons are responsible for the neutrinos detected by IceCube.\nThe constraints suggest that the optical depth of protons for PeV-energy\nneutrinos is rather high, $\\tau_0\\geq 0.2$, and also that a major\nfraction of UHECRs in the EeV region is extragalactic\nprotons. This supports the ``dip'' transition model~\\cite{dip_model} of UHECR protons, \nwhere the ankle structure of the cosmic-ray spectrum, which\nappears at 3 to 10 EeV, is caused by the energy loss of\nextragalactic UHECR protons by Bethe--Heitler pair production with\nCMB photons. This model predicts a high GZK cosmogenic neutrino flux \nat 10--100 PeV.\nHowever, as we see in the next section,\nthe null detection of cosmogenic neutrino candidates in the IceCube seven year dataset\nexcludes the dip transition scenario if HECRs are proton-dominated.\nThis suggests that the source of neutrinos\nseen by IceCube is not the main source of cosmic rays at EeV energies or higher.\nNo known class of astronomical objects can \nmeet the stringent requirement of the $\\gamma p$ optical depth, $\\tau_0\\geq 0.1$,\nand the UHECR energetics. Only bright FSRQs such as those with \n$L_\\gamma\\sim 10^{50} {\\rm erg}\\ \\sec^{-1}$ {\\it could} realize\n$\\tau_0\\sim 0.1$, but such FSRQs are too rare to energetically reproduce\nthe observed UHECR flux at 1 EeV.\n\nAccording to these considerations, none of the known extragalactic objects\ncan be found to function as an origin of both PeV-energy\nneutrinos and HECRs. \n\n\\section{The constraints on the highest-energy cosmic-ray sources}\n\\label{sec:EeV}\n\n\\begin{figure*}\n  \\includegraphics[width=0.4\\textwidth]{.\/GZK2DHistogram.pdf}\n  \\includegraphics[width=0.4\\textwidth]{.\/AstrophysE2_58_2DHistogram.pdf}\n  \\caption{Expected event distributions on the plane of the energy proxy\nand cosine of the reconstructed zenith angle seen by IceCube. \nSimulation of the GZK cosmogenic model~\\cite{Ahlers2010}\n(left panel) and astrophysical neutrino models with a spectrum following $E_\\nu^{-2.5}$\nare shown with the intensity measured by IceCube~\\cite{IceCubeHESE2014}. The $z$ axis displays the number of events seen by the IceCube extremely high-energy analysis\nbased on the seven year data.}\n\\label{fig:energy_cosZenith}\n\\end{figure*}\n\nThe analysis of seven years of IceCube data obtained in the search for ultrahigh-energy\nneutrinos (energies larger than 10 PeV) has been reported~\\cite{EHE2016}.\nThe analysis was optimized in particular for neutrinos with energies above 100 PeV.\nThe exposure has reached $\\sim10^{17}{\\rm cm^2} \\sec\\ {\\rm sr}$ at 1 EeV,\nwhich makes it possible to probe an important region of the parameter space \nin the GZK cosmogenic neutrino models. \nTwo events with estimated deposited energies of\n2.6 and 0.77 PeV, respectively, were identified in this analysis, but no events were found\nin the higher-energy region. This observation presents a serious challenge\nto the standard baseline candidates of HECR sources discussed in the literature.\n\nAn IceCube simulation was used\nto predict the number of events IceCube would detect on the plane of the reconstructed\nenergy and zenith angle for each model of ultrahigh-energy neutrinos (including GZK cosmogenic neutrinos). \nFigure~\\ref{fig:energy_cosZenith} shows examples\nfrom the models. The resolution of the reconstructed deposited energies of energetic\nevents is less than excellent owing to the stochastic nature of the muon energy loss profile\nat PeV energies. Furthermore, IceCube's ability to associate the estimated energy deposit\nof an event with its parent neutrino energy is rather limited because only a small fraction\nof the neutrino energy is converted to the visible form ({\\it i.e.}, the deposited muon energy)\nby the IceCube detectors. Nevertheless, Figure~\\ref{fig:energy_cosZenith} exhibits\nclear differences between two different models. The GZK model yields an event distribution\nwith an energy peak higher than the softer astrophysical neutrino models.\nThe events from the GZK model are also distributed more sharply in the horizontal\ndirection {\\it i.e.}, $\\cos({\\rm zenith})=0$. This is because neutrinos with\nenergies at the EeV level experience strong absorption effects \nas they propagate through the Earth. These features, as well as\nthe total event rate (which is equivalent to the normalization of the event distributions\nshown in Figure~\\ref{fig:energy_cosZenith}), makes it possible to determine which models\nare compatible with the observation. The binned Poisson log-likelihood ratio\ntest was performed. The simulated event distributions \non the energy--zenith angle plane, such as those shown in Figure~\\ref{fig:energy_cosZenith},\ngive the expected number of events in each bin of the energy proxy and\ncosine of the zenith, which were used to construct the binned Poisson likelihood. A log-likelihood ratio was\nused as a test statistic, and an ensemble of pseudo-experiments to derive\nthe test statistical distribution was used to calculate the p-values.\n\n\\begin{figure}\n  \\includegraphics[width=0.4\\textwidth]{.\/GZKFluxConstraints.pdf}\n  \\caption{Energy spectra of various cosmogenic neutrino\n    models~\\cite{Ahlers2010, Kotera2010,Aloisio2015, Yoshida1993}. The spectra shown\nby red (brown) curves are rejected (disfavored), with p-values less than 10\\% (32\\%).\nAll these models assume a proton-dominated HECR composition.}\n\\label{fig:gzk_flux}\n\\end{figure}\n\nThe hypothesis that the two observed {\\it PeV-ish} events are of GZK cosmogenic origin\nis rejected, with a p-value of 0.3\\% (which implies that it is incompatible \nwith the event distribution shown in the left panel of Figure~\\ref{fig:energy_cosZenith})\nbut is consistent with a generic astrophysical power-law flux such\nas $E_\\nu^{-2}$ or $E_\\nu^{-2.5}$~\\cite{EHE2016}. \nThis result makes it possible to set an upper limit\non the ultrahigh-energy neutrino flux extending above 10 PeV and thus to also test\nthe GZK cosmogenic neutrino models using the binned Poisson log-likelihood ratio method.\n\nThe various cosmogenic neutrino energy spectra are displayed in\nFigure~\\ref{fig:gzk_flux}. Many of them are rejected or disfavored by the IceCube\nobservation. Regardless of where the HECR sources are\nand how they accelerate cosmic rays, the emitted HECR protons {\\it must} produce\nsecondary neutrinos by the GZK mechanism as they travel through space.\nIn this sense, any consequences of these bounds on GZK neutrinos\nare considered as robust and model-independent arguments.\n\nWe summarize the findings below.\n\n\\begin{itemize}\n\\item Cosmogenic models with the maximal flux allowed\n  by the Fermi-LAT measurement~\\cite{FermiDiffuse} of \n  the diffuse extragalactic $\\gamma-$ray background are rejected.\n  This finding implies that the present limits imposed by the neutrino observation are\n  at least as stringent as those imposed by the $\\gamma-$ray observation.\n\n\\item HECR source evolution comparable to the star formation rate (SFR) is beginning to\nbe constrained. Sources evolving more strongly than the SFR, such as FSRQs and GRBs, are unlikely\nto be HECR sources; otherwise, IceCube would have detected cosmogenic-neutrino-induced events already.\n\n\\item Any GZK cosmogenic type of energy spectrum must have an intensity below\n$E_\\nu^2\\phi_{\\nu_e + \\nu_\\mu + \\nu_\\tau}(E_\\nu) = 3\\times 10^{-9} \n{\\rm GeV}\\ {\\rm cm}^{-2}\\ \\sec^{-1}\\ {\\rm sr}^{-1}$ at 100 PeV. This limit rejects the dip transition\nmodel of UHECRs.\n\n\\end{itemize}\n\n\\begin{figure}\n  \\includegraphics[width=0.45\\textwidth]{.\/Plot_2DConstraint_m_zmax_0909.pdf}\n  \\caption{Constraints on HECR source evolution parameters. The emission rate per co-moving volume\n    is parameterized as $\\psi(z)=(1+z)^m$ with redshifts up to $Z_{\\rm max}$. GZK cosmogenic neutrino\n    fluxes for various $m$ and $Z_{\\rm max}$ values are calculated by the approximated analytical\n    formulation~\\cite{YoshidaIshihara2012} and used for the likelihood calculation to derive\n    the confidence levels. The boxes indicate approximate parameter regions for the SFR~\\cite{Beacom} and\n    FR-II-A~\\cite{FR2-A} and -B~\\cite{FR2-B} radio galaxies.}\n\\label{fig:evolution}\n\\end{figure}\n\nMore generic constraints obtained by the IceCube Collaboration~\\cite{EHE2016} by scanning the parameter space for the\nsource evolution function, $\\psi(z)=(1+z)^m$,\nextending to the maximal redshift $Z_{\\rm max}$ \nare shown in Figure~\\ref{fig:evolution}.\nThe parameterized analytical formula for the cosmogenic fluxes~\\cite{YoshidaIshihara2012} is used here.\nBecause only the CMB is assumed as the target photon field in the parameterization,\nthe limits are systematically weaker than those on the models that include EBLs.\nApproximate regions for the SFR and the evolution\nof Fanaroff--Riley type II (FR-II) galaxies are also shown for comparison.\nNote that neutrinos yielded at redshifts larger than 2\nrepresent only a minor portion of the total cosmogenic fluxes owing to redshift\ndilution~\\cite{Kotera2010, YoshidaIshihara2012}.\nThis is especially true for the cosmogenic neutrino component created\nby interactions with the CMB (not the EBL). Considering this fact,\ntogether with the estimation that\nthe luminosity function of FR-II-type AGNs or FSRQs falls off rapidly at redshifts beyond\n$z\\simeq 2$, and that the evolution of the SFR becomes more or less constant or falls off\nat redshifts beyond $z\\sim2.5$, the boxes representing SFR and FR-II evolution\nin Figure~\\ref{fig:evolution} approximate well their representation by\nthe generic evolution function $\\psi(z)=(1+z)^m$ used in the plot.\nOne can find that HECR source evolution stronger than the SFR is unlikely.\n\n\\begin{figure}\n  \\includegraphics[width=0.45\\textwidth]{.\/AstroMuraseAGNConstraints.pdf}\n  \\caption{Constraints on the fluxes of astrophysical neutrinos produced\nin the inner jets of radio-loud AGNs~\\cite{murase2014}. Two bounds for\nthe UHECR spectral index, $\\alpha=2.0$ (thin) and $2.3$ (thick),\nare shown.}\n\\label{fig:agn_nu}\n\\end{figure}\n\nAll the constraints on the HECR origins described so far \nrely on one critical assumption, that HECRs are proton-dominated.\nIf HECRs are of mixed- or heavy-nuclei composition, the resultant GZK cosmogenic\nflux is lower than the proton UHECR case by more than an order of magnitude,\nand the present IceCube detection sensitivity cannot reach this low intensity.\nIt is expected, however, that neutrinos with energies from the PeV level to the EeV level and beyond\nmay be produced {\\it in situ} at the HECR acceleration site. \nThe AGN neutrino models are good examples. A recent theoretical study\nof ultrahigh-energy neutrino generation in the inner jets \nof radio-loud AGNs~\\cite{murase2014} found that, \ntaking into account the blazar sequence,\nFSRQs can emit PeV--EeV neutrinos, and BL Lac objects\ncan be HECR (heavy) {\\it nuclei} sources~\\cite{murase2012}.\nThe predicted PeV--EeV neutrino intensity is proportional\nto the baryon loading factor, that is, the ratio of the UHECR luminosity\nto the electromagnetic radiation luminosity $L_{\\rm CR}\/L_\\gamma$.\nThe null detection of 100 PeV--EeV neutrinos by IceCube thus\nbounds this factor.\n\nFigure~\\ref{fig:agn_nu} shows the present bound\non the fluxes of neutrinos from radio-loud AGNs\nby IceCube~\\cite{EHE2016}.\nThe observed HECR generation rate, $\\sim$10 EeV \n($10^{44}\\ {\\rm erg}\\ {\\rm Mpc}^{-3}\\ {\\rm yr}^{-1}$), requires loading factors\nof around 3 and 100 for UHECR spectral indices of $\\alpha = 2$ and 2.3, respectively.\nThe present constraints are comparable to or slightly below the values\nrequired for radio-loud AGN inner jets to be responsible for\nthe majority of UHECRs\/HECRs. \nThe neutrino observation has started to exclude a sizable parameter\nspace in the models of AGNs as an origin of HECRs even if\nHECRs are composed of heavy nuclei, although this is a model-dependent\nargument.\n\nFast-spinning newborn pulsars are also proposed as candidate sources\nof HECRs~\\cite{olinto1997}. This proposal predicts a heavy-nuclei-dominated composition\nat the highest energies and thus would yield GZK neutrinos too rare\nto be detected. However, in this model, the accelerated particles traveling through\nthe expanding supernova ejecta surrounding the star\nproduce neutrinos with energies of 100 PeV to $\\sim$EeV. The predicted\ndiffuse neutrino flux from fast pulsars is \n$E_\\nu^2 \\phi_{\\nu_e+\\nu_\\mu+\\nu_\\tau}\\simeq \n1.1\\times 10^{-8} {\\rm GeV} {\\rm cm^{-2}} \\sec^{-1} {\\rm sr^{-1}}$,\ndepending on the source emission evolution, and\nis accessible at the IceCube detection sensitivity~\\cite{fang2014}.\n\nA binned Poisson log-likelihood test of this model was performed by\nthe IceCube Collaboration~\\cite{EHE2016}. The model is rejected\nif the evolution of the source emission history traces the standard SFR, although\nit is not ruled out if the emission rate evolves more slowly than the SFR.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThe detection of TeV--PeV neutrinos and the null detection\nof EeV neutrinos by IceCube have yielded many insights on the origin of UHECRs, which cannot be probed by other cosmic messengers.\nThe most popular candidates for UHECR\/HECR sources, GRBs and radio-loud AGNs,\nhave now faced serious challenges from recent results in neutrino astronomy.\nGRBs and AGNs can still contribute to the observed bulk\nof the highest-energy cosmic rays but are unlikely to be their {\\it dominant} sources.\nAstronomical objects tracing the standard SFR or evolving much more slowly\nare needed to explain the observation without fine-tuning.\nIf the highest-energy cosmic rays are not proton-dominated,\nthese constraints are certainly relaxed. The model-dependent tests\ndescribed here, however, have already placed limits on some of\nthe parameter space of the AGN\/pulsar scenarios.\n\nThere is a loophole: a hypothesis that any high-energy neutrino emission\ndoes not involve cosmic-ray emission. A good example is the GRB choked jet\nmodel~\\cite{chokedGRB}. In dense environments, the optical depth $\\tau_0\\gg 1$,\nwhich implies that all the proton\u5c08 energy is\nconverted into neutrinos; {\\it i.e.}, the observed UHECRs and\nneutrinos are not directly connected.\n\nHow can we identify UHECR\/HECR sources that evolve\nat the usual SFR, or even more slowly?\nReal-time multi-messenger observation\ntriggered by high-energy neutrinos is a possible answer. IceCube has launched\nthe Gamma-ray Coordinates Network-based alert delivery system~\\cite{icecubeGCN}.\nThe search algorithms for Extremely High-Energy  neutrinos~\\cite{icecubePeV2013, EHE2016}\nand High-Energy Starting Events~\\cite{icecubeHESE2013, IceCubeHESE2014},\nthe analysis channels that discovered the high-energy cosmic neutrinos, are now running\nin real time at IceCube's South Pole data servers.\nOnce a high-energy-neutrino-induced event is detected, an alert is sent immediately\nto trigger follow-up observations by other astronomical instruments.\nIf UHECR\/HECR sources are transient neutrino sources,\nwe may be able to identify them by follow-up detection with optical\/X-ray\/$\\gamma$-ray\/radio\ntelescopes. This is probably a promising way to approach identification of\nthe yet-unknown origins of high-energy cosmic rays.\n\n\\section*{Acknowledgments}\nI am grateful to the CRIS 2016 organizers for\ntheir warm hospitality.\nI acknowledge my colleagues in the IceCube Collaboration\nfor useful discussions and suggestions.\nI also appreciate the input of Kunihito Ioka, Kumiko Kotera, Kohta Murase, \nand Hajime Takami on the theoretical arguments.\nSpecial thanks go to Aya Ishihara, who has worked together\non the analyses of extremely high-energy neutrinos \nfor many years. This work is supported by JSPS\nGrants-in-Aid for Scientific Research (Project \\#25105005 and \\#25220706).\n\n\n\n\n\\nocite{*}\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Sun is the best known star to astronomers, and is commonly used as a template in the study of other similar objects. Yet, there are still some of its aspects that are not well understood and that are crucial for a better understanding of how stars, and consequently how planetary systems and life evolve: how do the more complex physical parameters of a Sun-like star, such as rotation and magnetic activity, change with time? Is the Sun unique or typical (i.e., an average Sun-like star)? If the Sun is common, it would mean that life does not require a special star for it to flourish, eliminating the need to evoke an anthropic reasoning to explain it.\n\nIn an effort to assess how typical the Sun is, \\citet{2008ApJ...684..691R} compared 11 of its physical parameters with nearby stars, and concluded that the Sun is, in general, typical. Although they found it to be a slow-rotator against 276 F8 -- K2 (within $\\pm 0.1$ M$_\\odot$) nearby stars, this result may be rendered inconclusive owing to unnacounted noise caused by different masses and ages in their sample. Other studies have suggested that the Sun rotates either unusually slow \\citep{1979PASP...91..737S, 2015A&A...582A..85L} or regularly for its age \\citep{1983ApJS...53....1S, 1985AJ.....90.2103S, 1984ApJ...281..719G, 1998SSRv...85..419G, 2003ApJ...586..464B}, but none of them comprised stars that are very similar to the Sun, therefore preventing a reliable comparison. In fact, with \\textit{Kepler} and \\textit{CoRoT}, it is now possible to obtain precise measurements of rotation periods, masses and ages of stars in a very homogeneous way \\citep[e.g.,][]{2015EPJWC.10106016C, 2012A&A...548L...1D, 2014ApJS..210....1C}, but they generally lack high precision stellar parameters, which are accessible through spectroscopy. The challenging nature of these observations limited ground-based efforts to smaller, but key stellar samples \\citep[e.g.,][]{2003A&A...397..147P, 2012AN....333..663S}.\n\nThe rotational evolution of a star plays a crucial role in stellar interior physics and habitability. Previous studies proposed that rotation can produce extra mixing that is responsible for depleting the light elements Li and Be in their atmospheres \\citep{1989ApJ...338..424P, 1994A&A...283..155C, 2015A&A...576L..10T}, which could explain the disconnection between meteoritic and solar abundances of Li \\citep{2010A&A...519A..87B}. Moreover, rotation is highly correlated with magnetic activity \\citep[e.g.,][]{1984ApJ...279..763N, 1993ApJS...85..315S, 1995ApJ...438..269B, 2008ApJ...687.1264M}, and this trend is key to understand how planetary systems and life evolve in face of varying magnetic activity and energy outputs by solar-like stars during the main sequence \\citep{2009IAUS..258..395G, 2005ApJ...622..680R, 2016ApJ...820L..15D}.\n\nA theoretical treatment of rotational evolution from first principles is missing, so we often rely on empirical studies to infer about it. One of the pioneer efforts in this endeavor produced the well known Skumanich relation $v \\propto t^{-1\/2}$, where $v$ is the rotational velocity and $t$ is the stellar age \\citep{1972ApJ...171..565S}, which describes the rotational evolution of solar-type stars in the main sequence, and can be derived from the loss of angular momentum due to magnetized stellar winds \\citep[e.g.,][]{1988ApJ...333..236K, 1992ASPC...26..416C, 2003ApJ...586..464B, 2013A&A...556A..36G}. This relation sparked the development of gyrochronology, which consists in estimating stellar ages based on their rotation, and it was shown to provide a stellar clock as good as chromospheric ages \\citep{2007ApJ...669.1167B}. However, in Skumanich-like relations, the Sun generally falls on the curve (or plane, if we consider dependence on mass) defined by the rotational braking law by design. Thus it is of utmost importance to assess how common the Sun is in order to correctly calibrate it.\n\nSubsequent studies have proposed modifications to this paradigm of rotation and chromospheric activity evolution \\citep[e.g.,][]{1991ApJ...375..722S, 2004A&A...426.1021P}, exploring rotational braking laws of the form $v \\propto t^{-b}$. The formalism by \\citet{1988ApJ...333..236K} shows that this index $b$ can be related to the geometry of the stellar magnetic field, and that Skumanich's index ($b = 1\/2$) corresponds to a geometry that is slightly more complex than a simple radial field. It also dictates the dependence of the angular momentum on the rotation rate, and in practice, it determines how early the effects of braking are felt by a model. Such prescriptions for rotational evolution have a general agreement for young ages up to the solar age \\citep[see][and references therein]{2016arXiv160507125S, 2016A&A...587A.105A}, but the evolution for older ages still poses an open question. In particular, \\citet{2016Natur.529..181V} suggested that stars undergo a weakened magnetic braking after they reach a critical value of the Rossby number, thus explaining the stagnation trend observed on the rotational periods of older Kepler stars.\n\nIn order to assess how typical the Sun is in its rotation, our study aims to verify if it follows the rotational evolution of stars that are very similar to it, an objective that is achieved by precisely measuring their rotational velocities and ages. We take advantadge of an unprecedented large sample of solar twins \\citep{2014A&A...572A..48R} using high signal-to-noise ($S\/N > 500$) and high resolution ($R > 10^5$) spectra, which provides us with precise stellar parameters and is essential for the analysis that we perform (see Fig. \\ref{widths} for an illustration of the subtle effects of rotation in stellar spectra of Sun-like stars).\n\n\n\\section{Working sample}\n\nOur sample consists of bright solar twins in the Southern Hemisphere, which were mostly observed in our HARPS Large Program (ID: 188.C-0265) that aimed to search for planetary systems around stars very similar to the Sun \\citep[][Papers I, II and III, respectively, of the series The Solar Twin Planet Search]{2014A&A...572A..48R, 2015A&A...581A..34B, 2016A&A...590A..32T}. These stars are loosely defined as those that have T$\\mathrm{_{eff}}$, $\\log{g}$ and [Fe\/H] inside the intervals $\\pm 100$ K, $\\pm 0.1$ [cgs] and $0.1$ dex, respectively, around the solar values. It has been shown that these limits guarantee $\\sim$0.01 dex precision in the relative abundances derived using standard model atmosphere methods amd that the systematic uncertainties of that analysis are negligible within those ranges \\citep{2014ApJ...795...23B, 2015A&A...583A.135B, 2015A&A...582A..17S, 2016A&A...589A..17Y}. In total, we obtained high precision spectra for 73 stars and used data from 9 more targets observed in other programs, all of them overlapping the sample of 88 stars from Paper I. We used the spectrum of the Sun (reflected light from the Vesta asteroid) from the ESO program 088.C-0323, which was obtained with the same instrument and configuration as the solar twins.\n\nThe ages of the solar twin sample span between $0-10$ Gyr and are presented in the online material (Table \\ref{params}). They were obtained by \\citet{2016A&A...590A..32T} using Yonsei-Yale isochrones \\citep{2001ApJS..136..417Y} and probability distribution functions as described in \\citet{2013ApJ...764...78R,2014A&A...572A..48R}. Uncertainties are assumed to be symmetric. These ages are in excellent agreement with the ones obtained in Paper I, with a mean difference of $-0.1 \\pm 0.2$ Gyr (see footnote 5 in Paper III). We adopted 4.56 Gyr for the solar age \\citep{1995RvMP...67..781B}. The other stellar parameters ($T\\rm_{eff}$, $\\log{g}$, [Fe\/H] and microturbulence velocities $v\\rm_t$) were obtained by \\citet{2014A&A...572A..48R}. The stellar parameters of HIP 68468 and HIP 108158 were updated by \\citet{2016A&A...590A..32T}.\n\nOur targets were observed at the HARPS spectrograph\\footnote{\\footnotesize{The initial plan was to use the observations from the MIKE spectrograph, as described by the Paper I. However, we decided to use the HARPS spectra due to its higher spectral resolving power.}} \\citep{2003Msngr.114...20M} which is fed by ESO's 3.6 m telescope at La Silla Observatory. When available publicly, we also included all observations from other programs in our analysis in order to increase the signal to noise ratio ($S\/N$) of our spectra. However, we did not use observations for 18 Sco (HIP 79672) from May 2009\\footnote{\\footnotesize{These observations have instrumental artifacts.}} and we did not include observations post-HARPS upgrade (June 2015) when combining the spectra\\footnote{\\footnotesize{The spectra had a different shape in the red side, and since there were few observations, we chose not to use them to eliminate eventual problems with combination and normalization.}}.\n\nThe wavelength coverage for the observations ranged from 3780 to 6910 \\AA, with a spectral resolving power of $R = \\lambda\/\\Delta\\lambda = 115000$. Data reduction was performed automatically with the HARPS Data Reduction Software (DRS). Each spectrum was divided in two halves, corresponding to the mosaic of two detectors (one optimized for the blue and other for the red wavelengths). In this study we only worked with the red part (from 5330 to 6910 \\AA) due to its higher $S\/N$ and the presence of cleaner lines. The correction for radial velocities was performed with the task \\texttt{dopcor} from IRAF\\footnote{\\footnotesize{IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.}}, using the values obtained from the pipeline's cross-correlation function (CCF) data. The different observations were combined with IRAF's \\texttt{scombine}. The resulting average (of the sample) signal to noise ratio was 500 around 6070 \\AA. The red regions of the spectra were normalized with $\\sim$30th order polynomial fits to the upper envelopes of the entire red range, using the task \\texttt{continuum} on IRAF. We made sure that the continuum of the stars were consistent with the Sun's. Additionally, we verified that errors in the continuum determination introduce uncertainties in $v \\sin{i}$ lower than $0.1$ km s$^{-1}$.\n\n   \\begin{figure}[!t]\n   \\centering\n   \\includegraphics[width=\\hsize]{line_comp.pdf}\n   \\caption{Comparison of the spectral line broadening between two solar twins with different projected rotational velocities. The wider line correspond to HIP 19911, with $v \\sin i \\approx 4.1$ km s$^{-1}$, and the narrower one comes from HIP 8507, with $v \\sin i \\approx 0.8$ km s$^{-1}$.}\n   \\label{widths}\n   \\end{figure}\n\n\n\\section{Methods}\\label{methods}\n\nWe analyze five spectral lines, four due to Fe I and one to Ni I (see Table \\ref{lines}; equivalent widths were measured using the task \\texttt{splot} in IRAF.), that were selected for having low level of contamination by blending lines. The rotational velocity of a star can be measured by estimating the spectral line broadening that is due to rotation. The rotation axes of the stars are randomly oriented, thus the spectroscopic measurements of rotational velocity are a function of the inclination angle ($v \\sin{i}$).\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Line list used in the projected stellar rotation measurements.}\n\\begin{tabular}{cccccc}\n\\hline \\hline\\\\[-2ex]\nWavelength & Z & Exc. pot. & $\\log{(gf)}$ & $v_{\\mathrm{macro}}^\\odot$ & EW$^\\odot$\\\\\n(\\AA) &  & (eV) & & (km s$^{-1}$) & (\\AA)\\\\ \\hline\n6027.050 & 26 & 4.076 & -1.09  & 3.0 &\t0.064 \\\\\n6151.618 & 26 & 2.176 & -3.30 & 3.2 &\t0.051 \\\\\n6165.360 & 26 & 4.143 & -1.46  & 3.1 &\t0.045 \\\\\n6705.102 & 26 & 4.607  & -0.98  & 3.6 &\t0.047 \\\\\n6767.772 & 28 & 1.826 & -2.17  & 2.9 &\t0.079 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{EW are the equivalent widths and $v_{\\mathrm{macro}}$ are the macroturbulence velocities measured as in Sect. \\ref{vmacro_det}.}\n\\label{lines}\n\\end{center}\n\\end{table}\n\nWe estimate $v \\sin{i}$ for our sample of solar twins using the 2014 version of MOOG Synth \\citep{1973PhDT.......180S}, adopting stellar atmosphere models by \\citet{2004astro.ph..5087C}, with interpolations between models performed automatically by the Python package qoyllur-quipu\\footnote{\\footnotesize{Available at \\url{https:\/\/github.com\/astroChasqui\/q2}}} \\citep[see][]{2014A&A...572A..48R}. The instrumental broadening is taken into account by the spectral synthesis. We used the stellar parameters from \\citet{2016A&A...590A..32T} and microturbulence velocities from \\citet{2014A&A...572A..48R}. Macroturbulence velocities ($v_{\\mathrm{macro}}$) were calculated by scaling the solar values, line by line (see Sect. \\ref{vmacro_det}). Estimation of the rotational velocities was performed with our own algorithm\\footnote{\\footnotesize{Available at \\url{https:\/\/github.com\/RogueAstro\/PoWeRS}}} that makes automatic measurements for all spectral lines for each star. We applied fine tuning corrections by eye for the non-satisfactory automatic line profile fittings, and quote $v \\sin{i}$ as the mean of the values measured for the five lines. See Sects. \\ref{vmacro_det} and \\ref{vsini_det} for a detailed description on rotational velocities estimation as well as their uncertainties. Fig. \\ref{sun_fit} shows an example of spectral line fitting for one feature in the Sun.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\hsize]{sun_line.pdf}\n\\caption{Example of line profile fitting for the Fe I feature at 6151.62 \\AA\\ in the spectrum of the Sun. The continuous curve is the synthetic spectrum, and the open circles are the observed data.}\n\\label{sun_fit}\n\\end{figure}\n\n\\subsection{Macroturbulence velocities}\\label{vmacro_det}\n\nWe tested the possibility of measuring $v_{\\mathrm{macro}}$ (radial-tangential profile) simultaneously with $v \\sin{i}$, but even when using the extremely high-resolution spectra of HARPS, it is difficult to disentangle these two spectral line broadening processes, which is probably due to the low values of these velocities. Macroturbulence has a stronger effect on the wings of the spectral lines, but our selection of clean lines still has some contamination that requires this high-precision work to be done by eye. Some stars show more contamination than others, complicating the disentaglement. Fortunately, the variation of macroturbulence with effective temperature and luminosity is smooth \\citep{2005oasp.book.....G}, so that precise values of $v_{\\mathrm{macro}}$ could be obtained by a calibration. Thus we adopted a relation that fixes macroturbulence velocities in order to measure $v \\sin{i}$ with high precision using an automatic code, which provides the additional benefits of reproducibility and lower subjectivity.\n\nThe macroturbulence velocity is known to vary for different spectral lines \\citep{2005oasp.book.....G}, so for our high-precision analysis, we do not adopt a single value for each star. Instead, we measure the $v_{\\mathrm{macro}}$ for the Sun in each of the spectral lines from Table \\ref{lines}, and use these values to scale the $v_{\\mathrm{macro}}$ for all stars in our sample using the following equation\\footnote{\\footnotesize{In the future, it should be possible to calibrate macroturbulence velocities using 3D hydrodynamical stellar atmosphere models \\citep[e.g.,][]{2013A&A...557A..26M} by using predicted 3D line profiles (without rotational broadening) as observations and determine which value of $v_{\\rm{macro}}$ is needed to reproduce them with 1D model atmospheres.}}:\n\n\\begin{eqnarray}\nv_{\\mathrm{macro},\\lambda}^{*} = v_{\\mathrm{macro},\\lambda}^{\\odot} - 0.00707\\ T_{\\mathrm{eff}} + 9.2422 \\times 10^{-7}\\ T_{\\mathrm{eff}}^2 \\nonumber \\\\ + 10.0 + k_1 \\left(\\log{g} - 4.44\\right) + k_2 \\\\\n\\equiv f(T_{\\mathrm{eff}}) + k_1\\left(\\log{g} - 4.44\\right) + k_2 \\nonumber\n\\label{vmacro_eq}\n\\end{eqnarray}where $v\\rm_{macro,\\lambda}^{\\odot}$ is the macroturbulence velocity of the Sun for a given spectral line, $T_{\\mathrm{eff}}$ and $\\log{g}$ are, respectively, the effective temperature and gravity of a given star, $k_1$ is a proportionality factor for $\\log{g}$ and $k_2$ is a small correction constant.\n\nThis formula is partly based on the relation derived by \\citet{2012A&A...543A..29M} (Eq. E.1 in their paper) from the trend of macroturbulence with effective temperature in solar-type stars described by \\citet{2005oasp.book.....G}. The $\\log{g}$-dependent term (a proxy for luminosity) comes from the empirical relation derived by \\citet{2014MNRAS.444.3592D} (Eq. 8 in their paper), and is based on spectroscopic measurements of $v_{\\mathrm{macro}}$ of \\textit{Kepler} stars, which were disentangled from $v \\sin{i}$ using asteroseismic estimates of the projected rotational velocities. Doyle et al. obtained a value for the proportionality factor $k_1$ of -$2.0$. However their uncertainties on $v_{\\mathrm{macro}}$ were of the order of 1.0 km s$^{-1}$. Thus, we decided to derive our own values of $k_1$ and $k_2$ by simultaneously measuring $v_{\\mathrm{macro}}$ and $v \\sin{i}$ of a sub-sample of solar twins.\n\nThis sub-sample was chosen to contain only single stars or visual binaries mostly in the extremes of $\\log{g}$ ($4.25$ -- $4.52$) in our entire sample. We assume these values to have a linear relationship with $v_{\\mathrm{macro}}$ inside this short interval of $\\log{g}$. We used as a first guess the values of $v \\sin{i}$ and $v_{\\mathrm{macro}}$ from a previous, cruder estimation we made, and performed line profile fits by eye using MOOG Synth. The velocities in Table \\ref{vmacro_stars} are the median of the values measured for each line and their standard error. Note that these $v \\sin{i}$ are not consistently measured in the same way that the final results are. The rotational velocity broadening was calculated by our own code (see Sect. \\ref{vsini_det} for details). By performing a linear fit in the $v_{\\mathrm{macro}} - f(T_{\\mathrm{eff}})$ vs. $\\log{g}-4.44$ relation ($f$ comprises all the $T_{\\mathrm{eff}}$-dependent terms, the macroturbulence velocity of the Sun and the known constant on Eq. \\ref{vmacro_eq}), we obtain that $k_1 = -1.81 \\pm 0.26$ and $k_2 = -0.05 \\pm 0.03$ (see Fig. \\ref{vmacro_logg}). For the stars farthest from the Sun in $\\log{g}$ from our sample, these values of $k_1$ and $k_2$ would amount to differences of up to $\\pm 0.4$ km s$^{-1}$ in their macroturbulence velocities, therefore it is essential to consider the luminosity effect on $v\\rm_{macro}$ for accurate $v \\sin{i}$ determinations.\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Simultaneous measurements of rotational and macroturbulence velocities of stars in the extremes of $\\log{g}$ from our sample of solar twins.}\n\\begin{tabular}{lcccc}\n\\hline \\hline\\\\[-2ex]\nStar & $v \\sin{i}$ & $v_{\\mathrm{macro}}$ & $T\\rm_{eff}$ & $\\log{g}$ \\\\\n & (km s$^{-1}$) & (km s$^{-1}$) & & \\\\ \\hline\nHIP 115577 & $0.95 \\pm 0.05$ & $3.35 \\pm 0.09$ & 5699 & 4.25 \\\\\nHIP 65708 & $1.20 \\pm 0.09$ & $3.55 \\pm 0.08$ & 5755 & 4.25 \\\\\nHIP 74432 & $1.40 \\pm 0.03$ & $3.35 \\pm 0.08$ & 5684 & 4.25 \\\\\nHIP 118115 & $1.40 \\pm 0.10$ & $3.43 \\pm 0.09$ & 5808 & 4.28 \\\\\nHIP 68468 & $1.75 \\pm 0.07$ & $3.70 \\pm 0.08$ & 5857 & 4.32 \\\\\nHIP 41317 & $1.55 \\pm 0.03$ & $3.10 \\pm 0.06$ & 5700 & 4.38 \\\\\nSun & $1.75 \\pm 0.07$ & $3.30 \\pm 0.06$ & 5777 & 4.44 \\\\\nHIP 105184 & $2.50 \\pm 0.03$ & $3.21 \\pm 0.08$ & 5833 & 4.50 \\\\\nHIP 10175 & $1.55 \\pm 0.06$ & $3.05 \\pm 0.08$ & 5738 & 4.51 \\\\\nHIP 114615 & $2.20 \\pm 0.03$ & $3.25 \\pm 0.08$ & 5816 & 4.52 \\\\\nHIP 3203 & $3.90 \\pm 0.03$ & $3.40 \\pm 0.10$ & 5850 & 4.52 \\\\\n\\hline\n\\end{tabular}\n\\label{vmacro_stars}\n\\end{center}\n\\end{table}\n\n   \\begin{figure}[!t]\n   \\centering\n   \\includegraphics[width=\\hsize]{vmacro_logg.pdf}\n   \\caption{Linear relation between $v_{\\mathrm{macro}}$ and $\\log{g}$ (a proxy for luminosity) for the stars on Table \\ref{vmacro_stars}. See the definition of $f(T_{\\mathrm{eff}})$ in Sect. \\ref{vmacro_det}. The orange continuous line represents our determination of a proportionality coefficient of -1.81 and a vertical shift of -0.05 km s$^{-1}$. The black dashed line is the coefficient found by \\citet{2014MNRAS.444.3592D}. The light grey region is a composition of 200 curves with parameters drawn from a multivariate gaussian distribution. The Sun is located at the origin.}\n   \\label{vmacro_logg}\n   \\end{figure}\n\nTo obtain the macroturbulence velocities for the Sun to use in Eq. \\ref{vmacro_eq}, we forced the rotational velocity of the Sun to 1.9 km s$^{-1}$ \\citep{1970SoPh...12...23H}, and then estimated values of $v\\rm_{macro, \\lambda}^{\\odot}$ by fitting each line profile using MOOG Synth, and the results are shown in Table \\ref{lines}. We estimate the error in determining $v\\rm_{macro,\\lambda}^{\\odot}$ to be $\\pm 0.1$ km s$^{-1}$. Since Eq. \\ref{vmacro_eq} is an additive scaling, the error for $v\\rm_{macro}$ of all stars is the same as in the Sun\\footnote{\\footnotesize{The uncertainties in stellar parameters have contributions that are negligible compared to the ones introduced by the error in $v_{\\mathrm{macro}}$.}}.\n\n\\subsection{Rotational velocities}\\label{vsini_det}\n\nOur code takes as input the list of stars and their parameters (effective temperature, surface gravity, metallicity and microturbulence velocities obtained on Paper I), their spectra and the spectral line list in MOOG-readable format. For each line in a given star, the code automatically corrects the spectral line shift and the continuum. The first is done by fitting a second order polynomial to the kernel of a line and estimating what distance the observed line center is from the laboratory value. Usually, the spectral line shift corrections were of the order of $10^{-2}$ \\AA, corresponding to 0.5 km s$^{-1}$ in the wavelength range we worked on. This is a reasonable shift that likely arises from a combination of granulation and gravitational redshift effects, which are of similar magnitude. The continuum correction for each line is defined as the value of a multiplicative factor that sets the highest flux inside a radius of 2.5 \\AA\\ around the line center to 1.0. The multiplicative factor usually has a value inside the range $1.000 \\pm 0.002$.\n\nThe code starts with a range of $v \\sin{i}$ and abundances and optimizes these two parameters through a series of iterations that measure the least squares difference between the observed line and the synthetic line (generated with MOOG synth). Convergence is achieved when the difference between the best solution and the previous one, for both $v \\sin{i}$ and abundance, is less than 1\\%. Additionally, the code also forces at least 10 iterations in order to avoid falling into local minima.\n\nOne of the main limitations of MOOG Synth for our analysis is that it has a \"quantized\" behavior for $v \\sin{i}$: the changes in the synthetic spectra occur most strongly in steps of 0.5 km s$^{-1}$. This behavior is not observed in varying the macroturbulence velocities. Therefore, we had to incorporate a rotational broadening routine in our code that was separated from MOOG. We used the Eq. 18.14 from \\citet{2005oasp.book.....G}, in velocity space, to compute the rotational profile\\footnote{\\footnotesize{This is the same recipe adopted by the radiative transfer code MOOG.}}:\n\n\\begin{equation}\nG(v) = \\frac{2(1-\\epsilon)\\left[ 1-(v\/v\\rm_L)^2 \\right]^{1\/2} + \\frac{1}{2} \\pi \\epsilon \\left[ 1-(v\/v\\rm_L)^2 \\right]}{\\pi v\\rm_L (1-\\epsilon\/3)}\\mathrm{,}\n\\end{equation}where $v\\rm_L$ is the projected rotational velocity and $\\epsilon$ is the limb darkening coefficient (for which we adopt the value 0.6). The rotational profile $G(v)$ is then convolved with MOOG's synthetic profiles (which were generated with $v \\sin{i}$ = 0).\n\nThe total uncertainties in rotational velocities are obtained from the quadratic sum of the standard error of the five measurements and an uncertainty of 0.1 km s$^{-1}$ introduced by the error in macroturbulence velocities. Systematic errors in the calculation of $v\\rm_{macro,\\lambda}$ for the stars do not significantly contribute to the $v \\sin{i}$ uncertainties.\n\nSome of the stars in the sample show very low rotational velocities, most probably due to the effect of projection (see left panel of Fig. \\ref{sku}). The achieved precision is validated by comparison with the values of the full-width at half maximum (FWHM) measured by the cross-correlation function (CCF) from the data reduction pipeline, with the effects of macroturbulence subtracted (see Fig. \\ref{ccf}). The spectroscopic binary star HIP 103983 has an unusually high $v \\sin{i}$ when compared to the CCF FWHM, and a verification of its spectral line profiles reveals the presence of distortions that are the most probably caused by mis-measurement of rotational velocity (contamination of the combined spectrum by a companion -- observations range from October 2011 to August 2012). We obtained a curve fit for the $v \\sin{i}$ vs. CFF FWHM (km s$^{-1}$) using a similar relation as used by \\citet{2001A&A...375..851M, 2004A&A...426.1021P, 2007A&A...475.1003H}, which resulted in the following calibration: $v \\sin{i} = \\sqrt{(0.73 \\pm 0.02) \\left[\\mathrm{FWHM}^2 - v_{\\mathrm{macro}}^2 - (5.97 \\pm 0.01)^2\\right]}$ km s$^{-1}$ \\citep[estimation performed with the MCMC code \\texttt{emcee}\\footnote{\\footnotesize{Available at \\url{http:\/\/dan.iel.fm\/emcee\/current\/}}}][]{2013PASP..125..306F}. The scatter between the measured $v \\sin{i}$ and the ones estimated from CCF is $\\sigma = 0.20$ km s$^{-1}$ (excluding the outlier HIP 103983). The typical uncertainty in the rotational velocities we obtain with our method -- line profile fitting with extreme high resolution spectra -- is 0.12 km s$^{-1}$, which implies that the average error of the CCF FWHM $v \\sin{i}$ scaling is 0.16 km s$^{-1}$, which could be significantly higher if the broadening by $v_\\mathrm{macro}$ is not accounted for.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\hsize]{ccf_vs_moog.pdf}\n\\caption{Comparison between our estimated values of $v \\sin{i}$ (y-axis) and the ones inferred from the cross-correlation funcion FWHM (x-axis). The spread around the 1:1 relation (black line) is $\\sigma = 0.20$ km s$^{-1}$.}\n\\label{ccf}\n\\end{figure}\n\n\n\\section{Binary stars}\n\nWe identified 16 spectroscopic binaries (SB) in our sample of 82 solar twins by analyzing their radial velocities; some of these stars are reported as binaries by \\citet{2014AJ....147...86T, 2014AJ....147...87T, 2001AJ....121.3224M, 2015ApJ...802...37B}. We did not find previous reports of multiplicity for the stars HIP 30037, HIP 62039 and HIP 64673 in the literature. Our analysis of variation in the HARPS radial velocities suggest that the first two are probable SBs, while the latter is a candidate. No binary shows a double-lined spectrum, but HIP 103983 has distortions that could be from contamination by a companion. The star HIP 64150 is a Sirius-like system with a directly observed white dwarf companion \\citep{2013ApJ...774....1C,2014ApJ...783L..25M}. The sample from Paper I contains another SB, HIP 109110, for which we could not reliably determine the $v \\sin{i}$ due to strong contamination in the spectra, possibly caused by a relatively bright companion. Thus, we did not include this star in our sample.\n\n\\begin{figure*}[!ht]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.48\\textwidth]{all_stars.pdf} & \\includegraphics[width=0.48\\textwidth]{sku_final.pdf} \\\\\n\\end{tabular}\n\\caption{Projected rotational velocity of solar twins in function of their age. The Sun is represented by the symbol $\\odot$. Left panel: all stars of our sample; the orange triangles are spectroscopic binaries, blue circles are the \\textit{selected sample} and the blue dots are the remaining non-spectroscopic binaries. Right panel: the rotational braking law; the purple continous curve is our relation inferred from fitting the \\textit{selected sample} (blue circles) of solar twins with the form $v \\sin{i} = v_\\mathrm{f} + m\\ t^{-b}$, where $t$ is the stellar age, and the fit parameters are $v_\\mathrm{f} = 1.224 \\pm 0.447$, $m = 1.932 \\pm 0.431$ and $b = 0.622 \\pm 0.354$, with $v_\\mathrm{f}$ and $b$ highly and positively correlated. The light grey region is composed of 300 curves that are created with parameters drawn from a multivariate gaussian distribution defined by the mean values of the fit parameters and their covariance matrix. Skumanich's law (red $\\times$ symbols, calibrated for $v^\\odot_{\\mathrm{rot}} = 1.9$ km s$^{-1}$) and the rotational braking curves proposed by \\citet[][black dashed curve, smoothed]{2014ApJ...790L..23D} and \\citet[][black dot-dashed curve]{2004A&A...426.1021P} are plotted for comparison.}\n\\label{sku}\n\\end{figure*}\n\nOf these 16 spectroscopic binaries, at least four of them (HIP 19911, 43297, 67620 and 73241) show unusually high $v \\sin{i}$ (see the left panel of Fig. \\ref{sku}). These stars also present other anormalities, such as their [Y\/Mg] abundances \\citep{2016A&A...590A..32T} and magnetic activity \\citep{2014A&A...572A..48R, F16inprep}. The solar twin blue straggler HIP 10725 \\citep{2015A&A...584A.116S}, which is not included in our sample, also shows a high $v \\sin{i}$ for its age. We find that five of the binaries have rotational velocities below the expected for Sun-like stars, but this is most likely an effect of projection of the stars' rotational axes. For the remaining binaries, which follow the rotational braking law, it is again difficult to disentangle this behavior from the $\\sin{i}$, and a statistical analysis is precluded by the low numbers involved. Tidal interactions between companions that could potentially enhance rotation depend on binary separation, which is unknown for most of these stars. They should be regular rotators, since they do not show anormalities in chromospheric activity \\citep{F16inprep} or [Y\/Mg] abundances \\citep{2016A&A...590A..32T}.\n\nBased on the information that at least 25\\% of the spectroscopic binaries in our sample show higher rotational velocities than expected for single stars, we conclude that stellar multiplicity is an important enhancer of rotation in Sun-like stars. Blue stragglers are expected to have a strong enhancement on rotation due to injection of angular momentum from the donor companion.\n\n\n\\section{The rotational braking law}\\label{br_law}\n\nIn order to correctly constrain the rotational braking, we removed from this analysis all the spectroscopic binaries. The non-SB HIP 29525 displays a $v \\sin{i}$ much higher than expected ($3.85 \\pm 0.13$ km s$^{-1}$), but it is likely that this is due to an overestimated isochronal age ($2.83 \\pm 1.06$ Gyr). Because it is a clear outlier in our results, we decided to not include HIP 29525 in the rotational braking determination. \\citet{2010A&A...521A..12M} found X-ray and chromospheric ages of 0.55 and 0.17 Gyr, respectively, for HIP 29525. We then divided the remaining 65 stars and the Sun in bins of 2 Gyr, and removed from this sample all the stars which were below the 70th percentile of $v \\sin{i}$ in each bin\\footnote{\\footnotesize{By doing a simple simulation with angles $i$ drawn from a flat distribution between 0 and $\\pi\/2$, we verify that 30\\% of the stars should have $\\sin{i}$ above 0.9.}}. This allowed us to select the stars that had the highest chance of having $\\sin{i}$ above $0.9$. In total, 21 solar twins and the Sun compose what we hereafter reference as the \\textit{selected sample}. Albeit this sub-sample is smaller, it has the advantage of mostly removing uncertainties on the inclination angle of the stellar rotation axes\\footnote{\\footnotesize{This procedure can also allow for unusually fast-rotating stars (although rare) with $\\sin {i}$ below 0.9 to leak into our sample.}}. We stress that the only reason we can select the most probable edge-on rotating stars ($i = \\pi\/2$) is because we have a large sample of solar twins in the first place.\n\nWe then proceeded to fit a general curve to the \\textit{selected sample} (see Fig. \\ref{sku}) using the method of orthogonal distance regression \\citep[ODR,][]{boggs1990orthogonal}, which takes into account the uncertainties on both $v \\sin{i}$ and ages. This curve is a power law plus constant of the form $v = v_\\mathrm{f} + m\\ t^{-b}$ \\citep[the same chromospheric activity and $v \\sin{i}$ vs. age relation used by][]{2004A&A...426.1021P, 2009IAUS..258..395G}, with $v$ (rotational velocity) and $v\\rm_f$ (asymptotic velocity) in km s$^{-1}$ and $t$ (age) in Gyr.\n\nWe find that the best fit parameters are $v_\\mathrm{f} = 1.224 \\pm 0.447$, $m = 1.932 \\pm 0.431$ and $b = 0.622 \\pm 0.354$ (see right panel of Fig. \\ref{sku}). These large uncertainties are likely due to: i) the strong correlation between $v_\\mathrm{f}$ and $b$; and ii) the relatively limited number of datapoints between 1 and 4 Gyr, where the parameter is most effective in changing the values of $v$. This limitation is also present in past studies \\citep[e.g.,][]{2016Natur.529..181V, 2003ApJ...586..464B, 2004A&A...426.1021P, 2008ApJ...687.1264M, 2014A&A...572A..34G, 2016A&A...587A.105A}. On the other hand, our sample is the largest comprising solar twins, and therefore should produce more reliable results. With more datapoints, we could be able to use 1 Gyr bins instead of 2 Gyr in order to select the fastest rotating stars, which would result in a better sub-sample for constraining the rotational evolution for young stars.\n\nThe relation we obtain is in contrast with some previous studies on modelling the rotational braking \\citep{2001ApJ...561.1095B, 2003ApJ...586..464B, 2015A&A...584A..30L} which either found or assumed that the Skumanich's law explains well the rotational braking of Sun-like stars. The conclusions by \\citet{2016Natur.529..181V} limit the range of validation up to approximately the solar age (4 Gyr) for stars with solar mass. When we enforce the Skumanich's power law index $b = 1\/2$, we obtain a worse fit between the ages 2 and 4 Gyr (and, not surprisingly, also after the solar age).\n\nOur data and the rotational braking law that results from them show that the Sun is a normal star regarding its rotational velocity when compared to solar twins. However, they do not agree with a regular Skumanich's law \\citep[][red $\\times$ symbols in Fig. \\ref{sku}]{2007ApJ...669.1167B}. We find a better agreement with the model proposed by \\citet[][black dashed curve in Fig. \\ref{sku}]{2014ApJ...790L..23D}, especially for stars older than 2 Gyr. This model is thoroughly described in Appendix A of \\citet{2012A&A...548L...1D}. In summary, it uses an updated treatment of the instabilities relevant to the transport of angular momentum according to \\citet{1992A&A...265..115Z} and \\citet{1997A&A...317..749T}, with an initial angular momentum for the Sun $J_0 = 1.63 \\times 10^{50}$ g cm$^2$ s$^{-1}$. Its corresponding rotational braking curve is computed using the output radii of the model, which vary from $\\sim$1 R$_\\odot$ at the current solar age to 1.57 R$_\\odot$ at the age of 11 Gyr, and it changes significantly if we use a constant radius $R = 1$ R$_\\odot$, resulting in a more Skumanich-like rotational braking.\n\nOur result agrees with the chromospheric activity vs. age behavior for solar twins obtained by \\citet{2014A&A...572A..48R}, in which a steep decay of the $R'\\rm_{HK}$ index during the first 4 Gyr was deduced (see Fig. 11 in their paper). The study by \\citet{2004A&A...426.1021P} also suggests a steeper power-law index ($b = 1.47$) than Skumanich's ($b_\\mathrm{S} = 1\/2$) in the rotational braking law derived from young open clusters, the Sun and M 67. However, as seen in Fig. \\ref{sku}, their relation significantly overestimates the rotational velocities of stars, especially for those older than 2 Gyr. This is most probably caused by other line broadening processes, mainly the macroturbulence, which were not considered in that study. As we saw in Sect. \\ref{vmacro_det}, those introduce important effects that are sometimes larger than the rotational broadening. Moreover, a CCF-only analysis tends to produce more spread in the $v \\sin{i}$ than the more detailed analysis we used.\n\nThe rotational braking law we obtain produces a similar outcome to that achieved by \\citet{2016Natur.529..181V} for stars older than the Sun (a weaker rotational braking law after solar age than previously suggested). Our data also requires a different power law index than Skumanich's index for stars younger than the Sun, one that accounts for an earlier decay of rotational velocities up to 2 Gyr.\n\nThe main sequence spin-down model by \\citet{1988ApJ...333..236K} states that, for constant moment of inertia and radius during the main sequence, we would have\n\n\\begin{equation}\n v_{\\mathrm{eq}} \\propto t^{-3\/(4an)} \\mathrm{,}\n \\label{kawaler}\n\\end{equation}where $v_{\\mathrm{eq}}$ is the rotational velocity at the equator and $a$ and $n$ are parameters that measure the dependence on rotation rate and radius, respectively (see Eqs. 7, 8 and 12 in their paper). If we assume a dipole geometry for the stellar magnetic field ($B_\\mathrm{r} \\propto B_0 r^3$), then $n = 3\/7$. Furthermore, assuming that $a = 1$, then Eq. \\ref{kawaler} results in $v_{\\mathrm{eq}} \\propto t^{-7\/4} = t^{-1.75}$. Skumanich's law ($v_{\\mathrm{eq}} \\propto t^{-0.5}$) is recovered for $n = 3\/2$, which is close to the case of a purely radial field ($n = 2$, $v_{\\mathrm{eq}} \\propto t^{-0.38}$). A more extensive exploration of the configuration and evolution of magnetic fields of solar twins is outside the scope of this paper, but our results suggest that the rotational rotational braking we observe on this sample of solar twins stems from a magnetic field with an intermediate geometry between dipole and purely radial.\n\n\n\\section{Conclusions}\n\nWe analyzed the rotational velocities of 82 bright solar twins in the Southern Hemisphere and the Sun using extremely high resolution spectra. Radial velocities revealed that our sample contained 16 spectroscopic binaries, three of which (HIP 30037, 62039, 64673) were not listed as so in the literature. At least five of these stars show an enhancement on their measured $v \\sin{i}$, which is probably caused by interaction with their close-by companions. They also present other anomalies in chemical abundances and chromospheric activities. We did not clearly identify non-spectroscopic binary stars with unusually high rotational velocities for their age.\n\nIn order to better constrain the rotational evolution of the solar twins, we selected a subsample of stars with higher chances of having their rotational axis inclination close to $\\pi\/2$ (almost edge-on). We opted to use carefully measured isochronal ages for these stars because it is the most reliable method available for this sample. We finally conclude that the Sun seems to be a common rotator, within our uncertainties, when compared to solar twins, therefore it can be used to calibrate stellar models.\n\nMoreover, we have found that Skumanich's law does not describe well the rotation evolution for solar twins observed in our data, a discrepancy that is stronger after the solar age. Therefore, we propose a new rotational braking law that supports the weakened braking after the age of the Sun, and comes with a earlier decay in rotational velocities up to 2 Gyr than the classical Skumanich's law. Interestingly, it also reveals an evolution that is more similar to the magnetic activity evolution observed in Sun-like stars, which sees a steep decay in the first 3 Gyr and flattens near the solar age. Additionally, we suggest that more high-precision spectroscopic observations of solar twins younger and much older than the Sun could help us better constrain the rotational evolution of solar-like stars.\n\\begin{acknowledgements}\n      LdS thanks CAPES and FAPESP, grants no. 2014\/26908-1 and 2016\/01684-9 for support. JM thanks for support by FAPESP (2012\/24392-2). LS acknowledges support by FAPESP (2014\/15706-9). We also would like to thank the anonymous referee for the valuable comments that significantly improved this manuscript.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\renewcommand{\\thefootnote}{\\arabic{footnote}}\n\\setcounter{footnote}{0}\n\n\nMore than a decade ago, matrix model was expected to realize the non-perturbative definition of string field theories through the double scaling limit\\cite{DS}.\nIn the matrix models, the interaction of the spin cluster domain wall (I-K type interaction) plays an important role to obtain the non-critical string field theory\\cite{IK}.\nHermitian matrix models formulate the Dynamical Triangulation of the discrete 2D surface, in which orientable strings propagate or interact\\cite{JR}\\cite{AJ}\\cite{Mog}. \nThe Loop gas model is a description of the non-critical string field theory, in which every string is located in 1-dimensional discrete target-space point $x$ and interacts with another one in the same point or neighboring points in each time evolution\\cite{KK}.\nIt is formulated by the matrix possessing the additional index $x$\\cite{Kos}\\cite{EKN}.\nOn the other hand, it is well known that the stochastic quantization of the matrix models is effective to deduce the string field theories, and the stochastic time plays the role of the geodesic distance on the 2D random surfaces\\cite{JR}\\cite{Nak}.\nHowever, one of the problems in the Dynamical Triangulation is that the probability of splitting interaction becomes too large to construct the realistic space-time.\nThis problem becomes more severe in higher dimension.\nEven in 2D model, the whole surface of the world-sheet is covered with many projections of infinitesimal baby universe.\n\nThe Causal Dynamical Triangulation (CDT) model is proposed to improve the above problem\\cite{AL}.\nIn this model, the triangulation is severely restricted because of the time-foliation structure.\nThe most characteristic feature of the CDT model is that the causality forbid the splitting and merging interaction.\nThere appears no baby universe and a string propagator becomes a torus with smooth surface.\nThe CDT model is generalized to include only the splitting interaction but not the merging interaction.\nIt is the Generalized Causal Dynamical Triangulation (GCDT) model and baby universes make the world surface not be smooth\\cite{ALWZ}.\nA string field theory is constructed from the CDT and its Schwinger-Dyson (S-D) equations are investigated\\cite{ALWWZ1}.\nIt is also formulated by a matrix model further\\cite{ALWWZ2}.\nRecently, the GCDT model is extended to include additional I-K type interaction and its matrix model formulation is also proposed\\cite{FSW}.\nThe S-D equation of this model has features of the non-critical string field theory.\n\nIn this note, we construct the CDT model from the matrix model for the loop gas model.\nWe assign the matrix an additional discrete index, which is interpreted as discrete time or geodesic distance.\nIn the original loop gas model, it is interpreted as space.\nThen, we apply the stochastic quantization method to this model in order to realize a string field theory of the GCDT model.\nThe main difference from other matrix model formulation is that the stochastic time does not have relation to the geodesic distance in our model.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics [width=100mm, height=20mm]{CDT1.eps}\n\\caption{A configulation of the CDT triangulation of the 1-step propagation}\n\\label{fig:CDT1}\n\\end{center}\n\\end{figure}\n\nAt the beginning, we briefly review some fundamental nature about the CDT in the 2D space-time.\nIn this model, a torus of loop propagation is sliced to many rings with small width $a$, the minimal discrete time.\nEach ring corresponds to the 1-step time propagation of a loop, from an edge to another edge.\nLoops of the edges are composed of the links with length $a$ so that the length of loop is also discretized.\nA ring is constructed with triangles, one of whose three edges has to be a component of the loop on one side and the other two edges are connected to other triangles.\nTherefore the 1-step propagation from a loop with the length $k$ to another one with the length $m$ is composed of $k+m$ triangles, with $k$ upward ones and $m$ downward ones (Fig.\\ref{fig:CDT1}).\nSince the number of the configuration corresponds to the amplitude of this 1-step propagation, we define the 1-step \"two-loop function\" as\n\\bea\n\\label{eq:amplitude}\nG^{(0)} (k,m; 1) \\equiv {g^{k+m} \\over k+m}~_{k+m}{\\rm C}_k ,\n\\ena\nwhere $g$ is associated with each triangle and $_{k+m}{\\rm C}_k$ expresses the binomial coefficient.\nIt is rather convenient to define the 1-step marked propagator of the loop with length $k$ as $G^{(1)} (k,m;1) \\equiv k G^{(0)}(k,m;1)$.\nWe can construct $t$-step unmarked (and marked) propagators by piling up the 1-step unmarked (and marked) propagators\n\\bea\n\\label{eq:tstep}\nG^{(0)} (n,m;t) & = & \\sum_{k=1}^{\\infty} G^{(0)}(n,k;t-1) k G^{(0)} (k,m;1), \\nonumber \\\\\nG^{(1)} (n,m;t) & = & \\sum_{k=1}^{\\infty} G^{(1)}(n,k;t-1) G^{(1)} (k,m;1),\n\\ena\nrespectively.\nThe disc amplitude $W(n)$ is the summation of the amplitudes such that the loop with the length $n$ becomes to zero in some future time, and it is expressed as\n\\bea\n\\label{eq:discamp}\nW(n) \\equiv \\sum_{t}^{\\infty} G^{(1)} (n,0;t).\n\\ena\nThen, we expect the superposing relation,\n\\bea\n\\label{eq:discsum}\nW(n) \\equiv \\sum_{k=1}^{\\infty} G^{(1)} (n,k;1) W(k)+G^{(1)}(n,0;1).\n\\ena\n\nOriginally, the CDT model does not contain splitting interaction nor merging interaction, because these processes violate the causality.\nThis means that a saddle point on the world-sheet causes to two distinct light cones.\nHowever, we can include the splitting interaction if we impose the condition such that any separated baby loop shrinks to length zero and the mother loop propagates without interacting with it.\nIt is the GCDT model, in which the \"causality\" in a broad sense is respected.\n \nWe propose a matrix model of the modified version of the loop gas model, with a fundamental matrix $(M_{tt'})_{ij}$, where the indices $i,~j$ run from 1 to $N$.\nThe $N \\times N$ matrix $M_{tt'}$ corresponds to a link variable which connects two sites on the discrete times $t$ and $t'$ with the direction from $t$ to $t'$.\nThen we start with the action of $U(N)$ gauge invariant form,  \n\\bea\n\\label{eq:actionM}\nS[M] = -g\\sqrt{N} {\\rm tr} \\sum_t M_{tt'} + {1 \\over 2}{\\rm tr} \\sum_{t,t'} M_{tt'} M_{t't} - {g \\over 3\\sqrt{N}} {\\rm tr} \\sum_{t,t',t''} M_{tt'} M_{t't''} M_{t''t},\n\\ena\nwith the partition function $Z = \\int {\\cal D} M e^{-S[M]}$.\n$M_{tt} \\equiv A_t$ is a hermitian matrix, which corresponds to a link of discrete string element soaked in one time $t$.\n$M_{t,t+1} \\equiv B_t$ and $M_{t+1,t} \\equiv B^{\\dagger}_t$ are associated with a link connecting sites on the nearest neighboring times $t$ and $t+1$.\nOtherwise $M_{tt'} = 0$ (for $t' \\neq t, t \\pm 1$).\nHence we can rewrite the partition function as $Z = \\int {\\cal D} A {\\cal D} B {\\cal D} B^{\\dagger} e^{-S[A,B,B^{\\dagger} ] } $ with the action \n\\bea\n\\label{eq:actionAB}\nS[A, B, B^{\\dagger}] &=& -g\\sqrt{N} {\\rm tr} \\sum_t A_{t} + {1 \\over 2}{\\rm tr} \\sum_t A_t^2 + {\\rm tr} \\sum_t B_t B_t^{\\dagger}  \\nonumber \\\\\n & & - {g \\over 3\\sqrt{N}} {\\rm tr} \\sum_t A_t^3 - {g \\over \\sqrt{N}} {\\rm tr} \\sum_t (A_t B_t B_t^{\\dagger} + A_{t+1} B_t^{\\dagger} B_t).\n\\ena\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics [width=100mm, height=35mm]{CDT2.eps}\n\\caption{The triangulation assignment for the cubic terms and the square term}\n\\label{fig:CDT2}\n\\end{center}\n\\end{figure}\nTwo square terms in the first line play the role of gluing the links of two triangles, and three cubic terms in the second line express the triangles which are the elements of surface (Fig.\\ref{fig:CDT2}).\nWhile the last two terms composed of $A, B$ and $B^{\\dagger}$ correspond to the triangles on the surfaces of string propagation ring, the cubic term of $A$ produces a triangle soaked in one time-slice, which does not exist in the CDT model.\nThrough integrating out the non-hermitian matrices $B_t$ and $B^{\\dagger}_t$, we obtain the effective theory only with equi-temporal link matrices $A_t$.\nThe partition function becomes $Z \\equiv \\int {\\cal D} A e^{-S_{\\rm eff} [A]}$, where\n\\begin{eqnarray*}\nS_{\\rm eff} [A] = {\\rm tr} \\sum_t \\left[ -g \\sqrt{N} A_t + {1 \\over 2} A_t^2  - {g \\over 3\\sqrt{N}} A_t ^3 + {\\rm log} \\Bigl\\{ {\\bf 1} - {g \\over \\sqrt{N}}  \\left( A_t {\\bf 1}_{t+1} + {\\bf 1}_t A_{t+1}  \\right) \\Bigr\\} \\right].\n\\end{eqnarray*}\nIf we define a loop variable $\\phi _t (n) \\equiv {1 \\over N}{\\rm tr}({A_t \\over \\sqrt{N}})^n$ as the discrete closed string of length $n$ in time $t$, the effective action of the loop variables is written as\n\\bea\n\\label{eq:actioneff}\nS_{\\rm eff} [\\phi , A] & = & - N^2 \\sum_t \\left[ {g \\over N} {\\rm tr}  {A_t \\over \\sqrt{N}} - {1 \\over 2N}{\\rm tr}  \\left( {A_t \\over \\sqrt{N}} \\right) ^2  + { g \\over 3N} {\\rm tr}  \\left( {A_t \\over \\sqrt{N}} \\right) ^3 \\right. \\nonumber \\\\\n & &  \\left. +  \\sum_{k=0}^{\\infty} \\sum_{m=0}^{\\infty} G^{(0)}(k,m;1) \\phi _t (k) \\phi _{t+1}(m) \\right],\n\\ena\nwhere $G^{(0)}(k,m;1)$ is the two-loop function of the 1-step time appeared in the CDT model.\nThe last term of eq.(\\ref{eq:actioneff}) is expressed graphically in Fig.\\ref{fig:CDT1}.\nWe may construct the $t$-step propagator as \n\\begin{eqnarray*}\nnmG^{(0)}(n,m;t)  = \\langle \\phi_0 (n) \\phi _t (m) \\rangle  = {1 \\over Z} \\int {\\cal D} A \\phi _0 (n) \\phi _t (m) e^{-S_{\\rm eff}[A]}, \n\\end{eqnarray*}\nThis expresses the sum over all ways of connecting $\\phi _0 (n)$ with $\\phi _t (m)$ by $t$ times piling of 1-step two-loop functions.\nThus we realize the matrix model formulation of the CDT model.\n\nNow, we extend this model to the GCDT with loop interactions by applying the stochastic quantization method to the above model.\nThe Langevin equation for a matrix variable and white noise correlation\n\\bea\n\\label{eq:langevinA}\n\\v (A_t)_{ij} = - {{\\partial S_{\\rm eff}} \\over {\\partial (A_t)_{ji}}} \\v \\t + \\v (\\xi _t)_{ij}, ~~~~~~\n\\langle \\v (\\xi _t)_{ij} \\v (\\xi _t')_{kl} \\rangle _\\xi = 2\\v \\t \\d _{tt'} \\d _{il} \\d _{jk},\n\\ena\ndescribe the evolution of the matrices on the step of the unit stochastic time $\\v \\t$.\nThey generate the Langevin equation for a loop variable \n\\bea\n\\label{eq:langevin}\n\\v \\phi _t (n) &=& \\v \\t n \\left[ g \\phi _t (n-1) - \\phi _t (n) + g \\phi _t (n+1) \n + \\sum_{k =0}^{n-2} \\phi _t ( k ) \\phi_t (n - k - 2) \\right. \\nonumber \\\\\n & & + \\sum_{k=1}^{\\infty} \\sum_{m=0}^{\\infty} G^{(1)} (k,m;1) \\phi _t (n+k-2) \\phi _{t+1} (m)  \\nonumber \\\\\n & & \\left. + \\sum_{k=1}^{\\infty} \\sum_{\\ell =0}^{\\infty} G^{(2)} (\\ell ,k;1) \\phi _t (n+k-2) \\phi _{t-1} (\\ell) \\right] + \\v \\zeta _t (n),\n\\ena\nwhere $G^{(1)} (k,m;1) \\equiv kG^{(0)} (k,m,t=1) $ and $G^{(2)} (k,m;1) \\equiv mG^{(0)} (k,m,t=1) $ are 1-step marked propagators with a mark on the entrance loop and the exit loop, respectively.\n$\\v \\zeta _t (n) \\equiv N^{-1-{n \\over 2}} n {\\rm tr} ( \\v \\xi _t A_t^{n-1} )$ is the constructive noise term which satisfies the correlation\n\\bea\n\\label{eq:noisecorrelation}\n\\langle \\v \\zeta _t (n) \\v \\zeta _{t'} (m) \\rangle _\\xi = 2 \\v \\t \\d _{tt'} {1 \\over N^2} nm \\langle \\phi _t (n+m-2) \\rangle _\\xi .\n\\ena\nAny observable $O(\\phi)$ composed of loop variables is deformed, under the stochastic time 1-step progress $\\v \\t$, following the variation of $\\phi _t (n)$ with the Langevin equation (\\ref{eq:langevin}) and the noise correlation (\\ref{eq:noisecorrelation}).\nThe generator of this $\\v \\t$ evolution corresponds to the Fokker-Planck (F-P) Hamiltonian $H_{\\rm FP}$,\n\\bea\n\\label{eq:FPHdef}\n\\langle \\v O(\\phi ) \\rangle _\\xi &=& \\langle \\sum_m \\v \\phi _t (m) {\\partial \\over \\partial \\phi _t (m) } O(\\phi ) + {1 \\over 2} \\sum_{m,n} \\v \\phi _t (m) \\v \\phi _t (n) {\\partial ^2 \\over \\partial \\phi _t (m) \\partial \\phi _t (n)} O(\\phi ) \\rangle _\\xi \\nonumber \\\\\n & & +{\\rm O}(\\v \\t^{3 \\over 2}) \\nonumber \\\\\n & \\equiv & - \\v \\t \\langle H_{\\rm FP} O(\\phi ) \\rangle _{\\xi}.\n\\ena\nWe interpret $\\phi _t (n)$ and $\\pi _t (n) \\equiv {\\partial \\over \\partial \\phi _t (n) }$ as the operators for creation and annihilation of the loop with length $n$ at time $t$, respectively.\nThey fulfill the following commutation relation:\n\\bea\n\\label{eq:commutation}\n[\\pi _t (n), \\phi _{t'} (m) ] = \\d _{tt'} \\d _{nm}.\n\\ena\nThen the F-P Hamiltonian is expressed as\n\\bea\n\\label{eq:FP1}\nH_{\\rm FP} = -{1 \\over N^2} \\sum_t \\sum_{n=1}^{\\infty} n L_t (n-2) \\pi _t (n),\n\\ena\nwhere $L_t (n)$ is defined by\n\\bea\n\\label{eq:generator}\nL_t (n) &=& -N^2 \\Biggl[ g \\phi _t (n+1) - \\phi _t (n+2) + g \\phi _t (n+3) \\Biggr. \\nonumber \\\\\n& & +\\sum_{k=1}^{\\infty} \\sum_{m=0}^{\\infty} G^{(1)} (k,m;1) \\phi _t (n+k) \\phi _{t+1} (m) \n + \\sum_{k=1}^{\\infty} \\sum_{\\ell =0}^{\\infty} G^{(2)} (\\ell ,k;1) \\phi _t (n+k) \\phi _{t-1} (\\ell) \\nonumber \\\\\n& & \\left. + \\sum_{k=0}^n \\phi _t (k) \\phi _t (n-k) + {1 \\over N^2} \\sum_{k=1}^{\\infty} k \\phi _t (n+k) \\pi _t (k) \\right],\n\\ena\nand it satisfies the Virasoro algebra\n\\bea\n\\label{eq:virasoro}\n[L _t (n), L _{t'} (m) ] = (n-m)\\d _{tt'} L _t (n+m).\n\\ena\n\nIn order to take the continuum limit we introduce the minimum length of this matrix model $a$.\nThe continuum limit is realized by taking $a$ to zero simultaneously with $N$ to infinity.\nIt is called the double scaling limit.\nAccording to the CDT model, we set the continuum loop length as $ L \\equiv an $ and time as $T \\equiv at $\\cite{AL}.\nWe also define the cosmological constant $\\Lambda$ by ${1 \\over 2}e^{-{1 \\over 2}a^2 \\Lambda} = g$.\nHere we introduce two scaling parameters $D$ and $D_N$.\nSince the commutation relation of the continuum field operators satisfy,\n\\bea\n\\label{eq:commutator}\n[\\Pi  (L;T), \\Phi  (L';T') ] = \\d (T-T') \\d (L-L'),\n\\ena\nthe scaling of the loop field operators can be described as $\\Phi  (L;T) \\equiv a^{-{1 \\over 2}D} \\phi _t (n) $ and $\\Pi (L;T) \\equiv a^{{1 \\over 2}D-2} \\pi _t (n)$ by using $D$.\nTo keep the effect of the first term of the last line in eq.(\\ref{eq:generator}), the splitting interaction, we fix the scaling of the infinitesimal stochastic time as\n\\bea\n\\label{eq:stochastic}\nd \\t \\equiv a^{{1 \\over 2}D-2} \\v \\t.\n\\ena\nHence the existence of the continuum stochastic time requires $D>4$.\nThe terms in the second line of eq.(\\ref{eq:generator}) express the characteristic interaction of this model, which corresponds to the I-K type interaction.\nThus we maintain these terms by redefining the 1-step propagator as $\\tilde{G}^{(1)}(L_1, L_2 ; a) \\equiv a^{-1} G^{(1)} (k,m;1)$, which gives the expression using the continuum lengths.\nWe define the string coupling constant as $G_{\\rm st} \\equiv {1 \\over N^2} a^{D_N}$ with $D_N$.\nThen we obtain the continuum limit of the F-P Hamiltonian ${\\cal H}_{\\rm FP}$ by $\\v \\t H_{\\rm FP} \\equiv d \\t {\\cal H}_{\\rm FP}$, and it is written as\n\\bea\n\\label{eq:FPhamiltonian}\n{\\cal H}_{\\rm FP}  & = & \\int dT \\int_0^{\\infty} dL L \\left[ a^{-{1 \\over 2}D+3} {1 \\over 2} \\left( {{\\partial ^2} \\over {\\partial L ^2}} - \\Lambda \\right) \\Phi (L;T)  \\right. \\nonumber \\\\\n & & +  \\int _0^{\\infty} dL_1 \\int _0^{\\infty} dL_2 \\tilde G ^{(1)} (L_1, L_2; a) \\Phi (L+L_1;T) \\Phi (L_2; T+a)  \\nonumber \\\\\n & & +  \\int _0^{\\infty} dL_1 \\int _0^{\\infty} dL_2 \\tilde G ^{(2)} (L_2, L_1; a) \\Phi (L+L_1;T) \\Phi (L_2; T-a)  \\nonumber \\\\\n & & + \\int _0^L dL_1 \\Phi (L_1 ; T) \\Phi (L - L_1 ;T) \\nonumber \\\\\n & & \\left.  + a^{-D_N -D+1} G_{\\rm st} \\int _0^{\\infty} dL_1 L_1 \\Phi (L+L_1; T) \\Pi (L_1; T) \\right]  \\Pi(L ; T).\n\\ena\nThe first term on the r.h.s is the potential term, which means the propagation of a loop in an equi-temporal slice.\nWe have to remember that any propagation of the loop in one equi-temporal slice is not contained in the GCDT model.\nHence we expect this term to scale out in the continuum limit.\nThis fact requires $D<6$.\nIt should be noted that, thanks to the scaling of the cosmological constant, after summing up the first three terms in eq.(\\ref{eq:generator}) the scaling of the propagation terms are enhanced two order higher compared with that of the original terms.\nThis enhancement makes the above restriction $D<6$ for $D$ consistent with another restriction $D>4$ from eq.(\\ref{eq:stochastic}).\n\nThe next two terms are I-K type interactions which are similar terms appeared in the non-critical string field theory model (Fig.\\ref{fig:CDT3}(b))\\cite{IK}.\nThe second and the third terms cause the annihilation of a loop with the length $L$ and creation of a loop with the length $L+L_1$ at the same time $T$.\nThey also create another loop with the length $L_2$ at the infinitesimal future time $T+a$ and infinitesimal past time $T-a$, respectively.\nThen the lengths $L_1$ and $L_2$ are connected by the infinitesimal 1-step marked propagator.\nThe fourth term is the splitting interaction, which annihilates a loop with length $L$ and create two loops with the sum of their lengths $L$, simultaneously.\nThe last term expresses the merging interaction, which annihilates two loops and create one loop whose length is equal to the sum of the two annihilated loops.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics [width=100mm, height=25mm]{CDT3.eps}\n\\caption{(a) Splitting interaction and (b) I-K type interaction. The entrance loop propagates upward to the exit loop as the mother universe, while the baby universe must disappear to the vacuum before long. On the surface of propagating world-sheet, many baby universes of two types are attached as quantum effect.}\n\\label{fig:CDT3}\n\\end{center}\n\\end{figure}\n\nWhile the splitting interaction is permitted in the GCDT model, the merging interaction should be forbidden because of the \"causality\" in a broad sense.\nFor this purpose we restrict the scaling parameter $D_N$ to satisfy\n\\bea\n\\label{eq:DN}\n D_N < -D+1.\n\\ena\nCombining this condition and $4<D<6$, the value of $D_N$ can be fixed.\nIf $D_N<-5$ the causality is respected, while if $D_N>-3$ it is violated.\nWhen $-5 \\le D_N \\le -3$, the condition of eq.(\\ref{eq:DN}) must be taken into account.\n\nThe interpretation of the stochastic process is following.\nIf the stochastic process is not included, we have the exact CDT model.\nEach stochastic time step $\\v \\t$ produces one baby universe in either of two ways, that is the ordinary splitting interaction and the I-K type interaction (Fig.\\ref{fig:CDT3}).\nWhile infinitesimal time or geodesic distance scales as $dT = a$, infinitesimal stochastic time scales by eq.(\\ref{eq:stochastic}) with $4<D<6$.\nThis may be expressed as $d\\t = a^{D_{\\t}} \\v \\t $, with $0<D_{\\t} <1 $.\nThis expression shows that $d \\t$ goes to zero more slowly than $dT$.\nFrom this we give a conjecture that the stochastic process, or the quantum effect occurs less frequently than the ordinary propagation in time.\nTherefore we can expect the processes creating baby universes may not become dominant in the propagation.\n\nIt is interesting to suppose the change of the scaling parameter $D$.\nThe GCDT model is realized for $4<D<6$, and then the world-sheet might have reasonable number of baby universes of two types on its surface.\nWhen $D>6$, the loop propagator contains too many stochastic processes, dominated by the propagation in the equi-temporal slice.\nThe string interactions are suppressed except for the merging interaction, whose probability depends on the scaling parameter $D_N$.\nHence, we can imagine that the loop propagator has too much length-fluctuation in the time evolution.\nIn $D=6$, each equi-temporal slice contains fertile stochastic processes including propagation, as well as splitting and I-K type interactions.\n\\footnote{In the S-D equation, as we will discuss later, an additional potential term concerning propagation $L ({\\partial ^2 \\over \\partial L^2}-\\Lambda ){W}(L)$, that is characteristic in the CDT model, survives in the scaling limit.}\\\nHowever, it is not desirable from the viewpoint of our model.\n\nFinally, we derive the S-D equation from the continuum version of the Langevin equation.\nThe disc amplitude is the expectation value of a loop variable, which propagates and shrink to nothing eventually.\nIt is expressed in the continuum expression as\n\\begin{eqnarray*}\nW(L) \\equiv \\langle \\Phi (L;T) \\rangle \\equiv \\int _0^{\\infty}  dT \\tilde{G}^{(1)} (L,0;T).\n\\end{eqnarray*}\nWith the help of eq.(\\ref{eq:discsum}), at the level of the expectation value we can expect the relation \n\\bea\n\\label{eq:disc}\n\\langle \\int _0^{\\infty} dL_2 \\tilde{G}^{(1)}(L_1,L_2;a) \\Phi (L_2; T+a) \\rangle = W(L_1).\n\\ena\nFrom this the S-D equation is derived as follows,\n\\bea\n\\label{eq:SD}\n\\int _0^L dL_1 W(L_1) W(L-L_1) +2 \\int _0^{\\infty} dL_1 W(L+L_1)W(L_1) =0. \n\\ena\nIn terms of the Laplace-transformed variable $\\tilde{W}(z) \\equiv \\int _0^{\\infty} dL e^{-Lz} W(L) $, eq.(\\ref{eq:SD}) is transformed to the expression in the Laplace space.\nWith the exchange of $z \\rightarrow -z$, we obtain another equation for $\\tilde{W}(-z)$.\nThen using above two equations, we obtain the S-D equation as\n\\bea\n\\label{eq:LSD}\n\\tilde{W}(z)^2 +2 \\tilde{W}(z) \\tilde{W}(-z) +\\tilde{W}(-z)^2 = \\mu ,\n\\ena\nwhere $\\mu$ is some constant.\nThis S-D equation expresses the same form as the one of the non-critical closed string field theory except for the coefficient of the second term of the l.h.s..\nIt is replaced with a coefficient $2{\\rm cos} \\pi p_0 $, where the background momentum $p_0 ={1 \\over m}$  corresponts to the central charge $c=1-{6 \\over m(m+1)}$.\nThe last term of the effective action eq.(\\ref{eq:actioneff}) is rewritten as $\\sum_{t,t'}  C_{tt'} \\sum_{k,m}^{\\infty} G^{(0)}(k,m;1) \\phi _t (k) \\phi _{t'}(m) $ with an adjacency matrix $C_{tt'} \\equiv \\d _{t',t+1} + \\d _{t',t-1} $.\nIf we adopt a twisted adjacency matrix $C^{(p_0)}_{tt'}= e^{i\\pi p_0}\\d _{t',t+1} + e^{-i\\pi p_0} \\d _{t',t-1}  $ as ref.\\cite{Kos}, instead of our choice $C_{tt'} $, we can express the loop bilinear term of the effective action eq.(\\ref{eq:actioneff}) as\n\\begin{eqnarray*}\n 2 {\\rm cos}\\pi p_0 \\sum _{t} \\sum_{k=0}^{\\infty} \\sum_{m=0}^{\\infty} G^{(0)}(k,m;1) \\phi _t (k) \\phi _{t+1}(m).\n\\end{eqnarray*}\nWe obtain the S-D equation of the non-critical string field theory exactly which has the right coefficient for the second term on the l.h.s. of eq.(\\ref{eq:LSD}).\n\nIn conclusion, we have proposed the matrix model formulation to construct the 2D GCDT model.\nUsing the stochastic quantization approach and taking a continuum limit, we obtain the non-critical string field theory.\nThe scaling parameter $D$ and $D_N$ on the double scaling limit are fixed as eq.(\\ref{eq:DN}) with $4<D<6$ by requiring the consistency to the CDT model.\nWe present one comment about a difference between the GCDT model and the discrete version of loop field theory formulated by our matrix model.\nIn our effective action eq.(\\ref{eq:actioneff}) the sum of the 1-step propagation terms contains the propagation whose length of the exit loop or the entrance loop equal zero.\nIt is not contained in the original CDT model\\cite{AL}.\nWe can cancel the $G^{(0)}(k,0;1)$ and $G^{(0)}(0,m;1)$ terms in the effective action by adding two fermionic degrees of freedom to the matrix model action.\nIt is done by assigning the Grassmann variables $C_t^{\\a} , \\overline{C}_t^{\\a} (\\a =1,2)$ to the link matrix connecting the sites in the same time slices, or $M_{tt} = A_t, C_t^{\\a}, \\overline{C}_t^{\\a}$.\n\\footnote{The fermionic matrices $C_t^{\\a}, \\overline{C}_t^{\\a}$ satisfy $(\\overline{C}_t^{\\a})_{ij}^* = - (C_t^{\\a})_{ji}$.}\\\nThen the fundamental action is modified as $S_1 \\equiv S - {\\rm tr} \\sum _t \\overline{C}_t C_t + {g \\over \\sqrt{N}}{\\rm tr} \\sum _t \\overline{C}_t A_t C_t $, where $S$ is the action of eq.(\\ref{eq:actionAB}).\nWhen we integrate out the fermionic degrees of freedom in $Z = \\int {\\cal D} A {\\cal D} B {\\cal D} B^{\\dagger} {\\cal D} C {\\cal D} {\\overline C} e^{-S_1 } $, the effective action has the additional term, $-N {\\rm tr} \\sum _t \\sum _{n=1}^{\\infty} {1 \\over n} ( {g \\over \\sqrt{N}} )^n A_t^n$, which cancels the terms concerning to the loop-to-vacuum 1-step processes.\nTherefore, we need not worry about whether the summation of the loop length begins from $n=0$ or $n=1$.\n\n\\section*{Acknowledgments}\nThe author would like to thank D. Suematsu for valuable comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe base $\\Lambda$CDM (Lambda cold dark matter) model is the simplest cosmological model that describes most of the data with remarkable accuracy~\\cite{Riess:1998cb,SupernovaCosmologyProject:1998vns,Planck:2018vyg,Alam:2020sor,DES:2021wwk}. However, even if we set aside its persistent theoretical issues associated with the cosmological constant $\\Lambda$~\\cite{Weinberg:1988cp,Sahni:1999gb,Peebles:2002gy,Padmanabhan:2002ji}, with the increase in the diversity and precision of observational measurements and also advances in data analysis and statistical methods, it has become increasingly clear that a more realistic alternative model is required to replace the six-parameter base $\\Lambda$CDM model as the new standard model of cosmology, and should phenomenologically exhibit non-trivial, if not significant, deviations from it~\\cite{DiValentino:2020vhf,DiValentino:2020zio,DiValentino:2020vvd,DiValentino:2020srs,DiValentino:2021izs,Perivolaropoulos:2021jda,Abdalla:2022yfr}. These deviations, encoded in modifications on top of the Hubble function of the standard cosmological model $H_{\\Lambda\\rm CDM}(z)$, when associated with dark energy (DE), as an effective or actual source, suggest features that are difficult to obtain within the canonical\/simple extensions of $\\Lambda$CDM and are hard to deal with within the established fundamental theories of physics; e.g., DE models that yield a density that attains negative values in the past which present at least one pole in their equation of state (EoS) parameters~\\cite{Delubac:2014aqe,Aubourg:2014yra,Sahni:2014ooa,DiValentino:2017rcr,Mortsell:2018mfj,Poulin:2018zxs,Capozziello:2018jya,Wang:2018fng,Akarsu:2019ygx,Dutta:2018vmq,Banihashemi:2018oxo,Banihashemi:2018has,Farhang:2020sij,Banihashemi:2020wtb,Visinelli:2019qqu,Akarsu:2019hmw,Ye:2020btb,Ye:2020oix,Perez:2020cwa,Calderon:2020hoc,Paliathanasis:2020sfe,Bonilla:2020wbn,Vazquez:2012ag,Akarsu:2020yqa,LinaresCedeno:2021aqk,Zhou:2021xov,LinaresCedeno:2020uxx,DiValentino:2020naf,Akarsu:2021fol,Ozulker:2022slu,DiGennaro:2022ykp}, and\/or present non-trivial characteristics such as an oscillatory EoS parameter that can even cross below the phantom divide line \\cite{Zhao:2017cud,Escamilla:2021uoj,Hojjati:2009ab,Xia:2006rr,Pace:2011kb,Cicoli:2018kdo,Ruchika:2020avj,Heisenberg:2022gqk,Tamayo:2019gqj}, and\/or an oscillatory density \\cite{Colgain:2021pmf,Pogosian:2021mcs,Raveri:2021dbu,Wang:2018fng,Escamilla:2021uoj,Bernardo:2021cxi,Tamayo:2019gqj}.\n\nWhile such deviations are highly non-trivial, the reason they appear in observational studies, as we will show, may be fairly simple. To begin with, let us describe the deviation of any alternative cosmological model from the $\\Lambda$CDM model by ${\\Delta H(z)\\equiv H(z)-H_{\\Lambda\\rm CDM}(z)}$; within the framework of spatially flat Robertson-Walker (RW) metric, fixing the comoving angular diameter distance $D_M(z)=c\\int_0^{z}\\dd{z'}H^{-1}(z')$ of the alternative model to that of $\\Lambda$CDM at any redshift $z=z_{\\rm s}$, requires that $\\int_0^{z_{\\rm s}}\\dd{z'}H^{-1}_{\\Lambda \\rm CDM}(z')= \\int_0^{z_{\\rm s}}\\dd{z'}H^{-1}(z')$. This is satisfied only if there exist at least two redshifts $z_1$ and $z_2$ in the interval $(0,z_{\\rm s})$ for which $\\Delta H(z_1)\/\\Delta H(z_2)<0$ unless $\\Delta H(z<z_{\\rm s})$ vanishes everywhere. Thus, a negative $\\Delta H(z)$ at any redshift (e.g., a dip as preferred by the Ly-$\\alpha$ data \\cite{Sahni:2014ooa,Aubourg:2014yra,Planck:2018vyg}) should be compensated by at least one positive $\\Delta H(z)$ somewhere else (e.g., the bump found in DE density reconstructions at $1.5\\lesssim z\\lesssim2$ \\cite{Wang:2018fng,Escamilla:2021uoj,Bernardo:2021cxi}). This compensatory behaviour implies an oscillation (not necessarily periodic) on top of $H_{\\Lambda\\rm CDM}(z)$ as suggested in the above mentioned observational analyses---which is in line with such behavior being favored by the baryon acoustic oscillations (BAO) data. In particular, we can choose $z_{\\rm s}$ to be the redshift of last scattering $z_*$ as $D_M(z_*)$ is strictly constrained by cosmic microwave background (CMB) observations almost model-independently for a given pre-recombination expansion history. This allows for oscillations up to $z_*$ with no constraints on their characteristics as long as they compensate each other so that the $D_M(z)$ integral is satisfied (they can be very frequent or just a single oscillation spread throughout the whole interval with an arbitrary shape etc.); however, it is conceivable that as cosmological observations other than CMB mostly probe redshifts $z\\lesssim3$, the shape and the place of the oscillations will be constrained (possibly amplified due to overfitting) by these local data.\n\n\nIn addition to the oscillations that arise from fixing the pre-recombination universe and hence $D_M(z_*)$ to that of $\\Lambda$CDM, if one also respects the success of $\\Lambda$CDM in the late universe ($z\\sim0$), we show that, the deviations from the Hubble radius, $H^{-1}(z)$, of $\\Lambda$CDM are described by localized oscillatory functions, namely, \\textit{wavelets}, and, these wavelets should satisfy the \\textit{admissibility condition}. Admissible wavelets are oscillatory functions with a vanishing integral over their whole range, and either have compact support or vanish approximately outside of a compact set of their parameters \\cite{Chui:1992}. They can generically be obtained from derivatives of probability distributions but are by no means limited to this method. In cosmology, the wavelets have been used in various contexts. The wavelet transforms have been used in analyzing the CMB signals \\cite{Sanz:1999xu,Tenorio:1999mf,McEwen:2007ni}, and analyzing the large-scale structure of the universe (to capture its non-Gaussian information content) \\cite{Einasto:2010bi,Allys:2020vld,Valogiannis:2021chp,Eickenberg:2022qvy,Wang:2021zse}; also, see Ref.~\\cite{Valogiannis:2022xwu} and references therein for some applications of wavelets in cosmology and astrophysics. Wavelets have also been considered for investigating possible oscillatory deviations in the DE EoS parameter from minus unity describing the cosmological constant~\\cite{Hojjati:2009ab}; however, note that, their approach of characterizing the oscillations of the EoS parameter with wavelets is fundamentally different from the central idea in this paper that deviations from $H^{-1}_{\\Lambda \\rm CDM}(z)$ should be described by admissible wavelets. Such deviations in the Hubble radius may also be described by wavelet oscillations in the EoS parameter, but, they may also correspond to much more violent behaviours with singularities, or even correspond to a cosmological constant if the wiggles in the Hubble radius are not attributed to the DE. Here, we show that models whose deviations from $\\Lambda$CDM are described by admissible wavelets on top of $H^{-1}_{\\Lambda \\rm CDM}(z)$ constitute a family of cosmological models that are in excellent agreement with the CMB measurements; and, discuss how even the simplest wavelets can lead to non-trivial behaviours in the Hubble parameter describing the BAO measurements without excessive number of free parameters. We also discuss that the same deviations from $H^{-1}_{\\Lambda\\rm CDM}(z)$ can originate from many different extensions in fundamental physics: modified theories of gravity, dynamical or non-minimally interacting DE etc.; and, we expose through examples, how the behaviour of some functions relevant to the source phenomena can be even more non-trivial. For example, if the deviations from $H^{-1}_{\\Lambda\\rm CDM}(z)$ are attributed to DE, the oscillations of the wavelet may cause the DE density to oscillate with a large enough amplitude so that the density attains negative values, resulting in divergences in its EoS parameter. Note that, while the same background dynamics may originate from different extensions, it may be possible to differentiate between these scenarios as we show by comparing the implications of attributing the deviations to the gravitational ``constant\", with attributing them to the DE.\n\n\\section{Wavelets on top of the standard cosmological model's Hubble radius}\n\nWe begin with the fact that the angular scale of the sound horizon at last scattering,\n\\begin{equation}\n\\theta_*=\\frac{r_*}{D_M(z_*)},\\label{eq:theta}\n\\end{equation}\nis measured almost model-independently, e.g., $100\\theta_*=1.04110\\pm 0.00031$ ($\\Lambda$CDM Planck18), with a precision of 0.03\\%, where $r_*$ is the comoving sound horizon at last scattering, and $D_M(z_*)$ is the comoving angular diameter distance out to last scattering~\\cite{Planck:2018vyg}. Then, fixing the pre-recombination physics to that of the standard cosmological model, i.e., $\\Lambda$CDM,\n\\begin{equation}\\label{eq:soundhorizon}\nr_*=\\int_{z_*}^\\infty \\frac{c_{\\rm s}(z)}{H_{\\Lambda \\rm CDM}(z)}\\dd{z}\n\\end{equation}\nis also determined, viz., $r_*=144.43\\pm 0.26\\,\\rm Mpc$ ($\\Lambda$CDM Planck18). Here $c_{\\rm s}(z)$ is the sound speed in the plasma and $z_{*}\\approx 1090$ is the redshift of last scattering (redshift for which the optical depth to Thomson scattering reaches unity) and $H_{\\Lambda \\rm CDM}(z)$ is the Hubble parameter of the standard cosmological model: \n\\begin{equation}\n3H^2_{\\Lambda \\rm CDM}(z)=\\rho_{\\rm m0}(1+z)^3+\\rho_{\\rm r0}(1+z)^4+\\rho_{\\Lambda},\n\\end{equation}\nwhere $\\rho_{\\rm m0}$, $\\rho_{\\rm r0}$, and $\\rho_{\\Lambda}$\nare the present-day energy densities corresponding to those of the pressureless matter (m), the radiation (r), and the cosmological constant ($\\Lambda$)---or, equivalently, the usual vacuum energy of the quantum field theory. We work, for convenience, in units for which the Newton's constant $G_{\\rm N}=1\/8\\pi$ and the speed of light $c=1$ unless they are shown explicitly. Here and in what follows, the subscript $0$ denotes the present-day ($z=0$) value of any quantity. While the values of these energy densities are subject to observational constraints, for the rest of this paper, we will assume them to be fixed (but unknown) values for which\n\\begin{equation}\nD_{M}(z_*)=c\\int_0^{z_*}\\frac{\\dd{z}}{H_{\\Lambda\\rm CDM}(z)}\n\\end{equation}\nis consistent with~\\cref{eq:theta,eq:soundhorizon}, and $\\rho_{\\rm m0}$ is compatible with the positions and relative heights of the peaks in the CMB power spectrum and $\\rho_{\\rm r0}$ is compatible with the observed CMB monopole temperature and standard model of particle physics. This ensures the basic consistency of $\\Lambda$CDM with the CMB data at the background level.\n\nAssuming $H_{\\Lambda \\rm CDM}(z)$ accurately describes the pre-recombination universe (hence $r_*$ is known), for a universe described by the spatially flat RW metric, the comoving angular diameter distance to $z_*$,\n\\begin{equation}\n      D_{M}(z_*)=c\\int_0^{z_*}\\frac{\\dd{z}}{H(z)},\\label{eq:DM}\n\\end{equation}\n of any model described by the Hubble parameter $H(z)$, is strictly constrained almost model-independently through the measurement of $\\theta_*$. Thus, for any $H(z)>0$ (expanding universe) deviating from $H_{\\Lambda \\rm CDM}(z)$, strict observational constraints from CMB still require\n\\begin{equation}\n\\int_0^{z_*}\\frac{\\dd{z}}{H_{\\Lambda \\rm CDM}(z)}\\approx \\int_0^{z_*}\\frac{\\dd{z}}{H(z)}, \\label{eq:approx}\n\\end{equation}\ncf., $D_M(z_*)=13872.83\\pm 25.31\\,\\rm Mpc$ ($\\Lambda$CDM Planck18).\nFor simplicity, we will assume\nthe approximation in Eq.~\\eqref{eq:approx} to be exact, and comment on the approximate case when necessary. Now, we define the deviation of a cosmological model from $\\Lambda$CDM in terms of its Hubble radius, $H(z)^{-1}$, as follows:\n\\begin{equation}\n\\psi(z)\\equiv\\frac{1}{H(z)}-\\frac{1}{H_{\\Lambda \\rm CDM}(z)}.\\label{eqn:devdef}\n\\end{equation}\nThen, we have\n\\begin{equation}\n D_{M}(z_*)=c\\int_0^{z_*}\\dd{z}\\qty[\\frac{1}{H_{\\Lambda \\rm CDM}(z)}+\\psi(z)],\\label{eq:exact}\n\\end{equation}\nand consequently, the exact version of Eq.~\\eqref{eq:approx} implies\n\\begin{equation}\n\\Psi(z_*)\\equiv\\int_0^{z_*}\\psi(z) \\dd{z}=0.\\label{eq:int}\n\\end{equation}\nOur assumption that the pre-recombination universe is accurately described by $H_{\\Lambda \\rm CDM}(z)$, viz., $H(z\\geq z_*)=H_{\\Lambda \\rm CDM}(z\\geq z_*)$, implies another condition on $\\psi(z)$, that is, \n\\begin{equation}\n\\psi(z\\geq z_*)=0. \\label{eq:prereccond}\n\\end{equation}\n\nThis mathematical framework allows one to naturally classify a family of $H(z)$ functions which can deviate, even significantly, from $H_{\\Lambda \\rm CDM}(z)$, but still have the same $D_M(z_*)$ the $\\Lambda$CDM model has, ensuring basic consistency with the CMB measurements at the background level (one might want to also consider the constraints on $\\rho_{\\rm m0}$ and $\\rho_{\\rm r0}$ from CMB). This family is described by \n\\begin{equation}\nH(z)=\\frac{H_{\\Lambda \\rm CDM}(z)}{1+\\psi(z)H_{\\Lambda \\rm CDM}(z)},\\label{eq:hz}\n\\end{equation}\nwhere $\\psi(z)$ satisfies the conditions introduced in~\\cref{eq:int,eq:prereccond}. We notice from this equation that introduction of the condition $-H^{-1}_{\\Lambda \\rm CDM}(z)<\\psi(z)<\\infty$ ensures that, in the past ($z>0$), $H(z)$ never diverges (except at the Big Bang) and the universe has always been expanding. Also, on top of all these conditions, let us demand \n\\begin{equation}\n\\psi(z=0)=0\\label{eq:present}\n\\end{equation}\nsince we know the universe at $z\\sim0$ is well described by the standard $\\Lambda$CDM model \\cite{Planck:2018vyg,Alam:2020sor,DES:2021wwk}.\n\nWe notice that \\cref{eq:int,eq:prereccond,eq:present} describe characteristic properties of functions that are known as \\textit{wavelets} where \\cref{eq:int} is true for wavelets that satisfy the \\textit{admissibility condition} \\cite{Chui:1992}. Wavelets are oscillatory (not necessarily periodic) functions that are well-localized, i.e., they have compact support or they vanish approximately outside of a compact set of its parameters. With such boundary conditions that the function should absolutely or approximately vanish outside of certain bounds, \\cref{eq:int} requires that the function oscillates at least once if it does not vanish everywhere; because, say $\\psi(z)<0$ for a certain value of $z$, this integral can vanish only if $\\psi(z)>0$ at another value of $z$, hence the oscillation. Note that, for a continuous $\\psi(z)$, this argument also implies that there exists at least one value of $z$ in the interval $(0,z_*)$ for which $\\psi=0$; this corresponds to the Rolle's theorem, which in our particular case states that the conditions $\\Psi(0)=0$ and $\\Psi(z_*)=0$ imply the existence of a $z_p\\in(0,z_*)$ for which $\\psi(z_p)=0$. \\textit{Thus, the deviations from the standard $\\Lambda$CDM model's Hubble radius, $\\psi(z)$, must be described by admissible wavelets, i.e., must have a wiggly (wave-like) behaviour characterized by the conditions given in \\cref{eq:int,eq:prereccond,eq:present}.}\n\nWe proceed with showing explicitly that the characteristics of $\\psi(z)$ described above, corresponds to a wiggly behaviour for $H(z)$ with respect to $H_{\\Lambda \\rm CDM}(z)$ \\textit{in a particular way}, namely, not necessarily wavelet type but such that $\\psi(z)$ is an admissible wavelet; to see this, we define a unitless parameter $\\delta(z)$, namely, the fractional deviation from $H_{\\Lambda\\rm CDM}(z)$, as follows;\n\\begin{equation}\n\\label{eqn:deltaH}\n\\delta(z)\\equiv \\frac{H(z)-H_{\\Lambda \\rm CDM}(z)}{H_{\\Lambda \\rm CDM}(z)}=-\\frac{\\psi(z)H_{\\Lambda \\rm CDM}(z)}{1+\\psi(z)H_{\\Lambda \\rm CDM}(z)}.\n\\end{equation}\nWe see that if we demand an ever-expanding universe $H(z)>0$, we should set $\\delta(z)>-1$. And, in what follows, unless otherwise is stated, we continue our discussions with the assumption that $\\delta(z)>-1$. For small deviations from $\\Lambda$CDM, i.e., $|\\delta(z)|\\ll1$, we can also write\n\\begin{equation}\n\\label{eqn:sdevH}\n\\delta(z)\\approx-\\psi(z)H_{\\Lambda \\rm CDM}(z).\n\\end{equation}\nThe small deviation region is quite important to study; because, despite its shortcomings, $\\Lambda$CDM is still the simplest model to explain the cosmological observations with remarkable accuracy. Particularly, in the late universe, the small deviation approximation is robustly imposed by many cosmological probes that require $|\\delta(z)|\\ll1$ for ${z\\lesssim2.5}$; even the largest discrepancies between the $H_{\\Lambda \\rm CDM}(z)$ of the Planck 2018 $\\Lambda$CDM~\\cite{Planck:2018vyg} and observed $H(z)$ values, viz., $H_0=73.04 \\pm 1.04$ km s${}^{-1}$ Mpc${}^{-1}$ (the SH0ES $H_0$ measurement~\\cite{Riess:2021jrx}) and $H(2.33)=224\\pm8{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$ (the Ly-$\\alpha$-quasar data~\\cite{eBOSS:2020yzd}) correspond to $\\delta_0\\sim0.08$ and $\\delta(z=2.33)\\sim-0.05$, respectively. The form of~\\cref{eqn:sdevH} makes it even easier to see that $H(z)$ will have wiggles; since $H_{\\Lambda \\rm CDM}(z)$ is a monotonically varying function of $z$ and strictly positive, when $\\psi(z)$ changes sign (as it must at least once), this sign change (around which the small deviation condition is clearly satisfied) is directly reflected on $\\delta(z)$, producing a wiggle. Furthermore, respecting the successes of the $\\Lambda$CDM model, one may even wish to impose $|\\delta(z)|\\ll1$ at all times. In this case, since $H_{\\Lambda \\rm CDM}(z)$ monotonically grows with increasing redshift, one should have $\\psi(z)\\to0$ fast enough with $z\\to z_*$, such that the small deviation condition $\\abs{\\psi(z)H_{\\Lambda \\rm CDM}(z)}\\ll1$ is not broken.\n\nHaving said that, note the interesting extra behaviours apparent from the full form of $\\delta(z)$ in~\\cref{eqn:deltaH}: first, as mentioned before, ${\\psi(z)=-H^{-1}_{\\Lambda \\rm CDM}(z)}$ results in a singular $H(z)$ function and is not allowed for finite $z$ values; second, while the previous condition might seem as it requires either one of the confinements ${\\psi(z)>-H^{-1}_{\\Lambda \\rm CDM}(z)}$ or ${\\psi(z)<-H^{-1}_{\\Lambda \\rm CDM}(z)}$ at all times, in principle, $\\psi(z)$ can be discontinuous and is not necessarily confined to one of these regions; third, \\cref{eqn:sdevH} indicates that $\\psi(z)<0$ corresponds to $\\delta(z)>0$, yet, for a region in which ${\\psi(z)<-H^{-1}_{\\Lambda \\rm CDM}(z)}$, we have $\\delta(z)<0$ despite having $\\psi(z)<0$, but, looking at~\\cref{eqn:devdef}, such a region also corresponds to an extreme case with $H(z)<0$ and the universe would have gone through a contracting phase.\n\n\nFinally, it is worth noting that due to the wiggly behaviour of the wavelets, similar to $H(z)$, the other important kinematical parameters in cosmology, the deceleration parameter $q=-1+\\frac{{\\rm d}}{{\\rm d}t}\\left[H^{-1}(z)\\right]$ and the jerk parameter $j=\\frac{{\\rm d}^3a\/{\\rm d}t^3}{aH^3(z)}$ (which is simply $j_{\\Lambda\\rm CDM}=1$ for $\\Lambda$CDM) will also exhibit wiggly behaviors; the deceleration parameter will oscillate around its usual evolution in $\\Lambda$CDM, $q_{\\Lambda\\rm CDM}(z)$, as can be immediately seen from\n\\begin{equation}\n    q(z)=q_{\\Lambda\\rm CDM}(z)+\\frac{{\\rm d}\\psi(z)}{{\\rm d}t},\n\\end{equation}\nobtained by using \\cref{eqn:devdef} in the definition of $q(z)$. And, the jerk parameter will oscillate around its constant $\\Lambda$CDM value of unity. These behaviours are reminiscent of the non-parametric reconstructions in Refs.~\\cite{Mukherjee:2020vkx,Mukherjee:2020ytg}.\n\n\n\\section{Wiggles in dark energy density descended from the wavelets}\n\nIn the late universe where dust and DE are the only relevant components, we can treat $H(z)$ as an extension of ${H_{\\Lambda \\rm CDM}}(z)$ with the same matter density parameter $\\rho_{\\rm m}(z)$ but with a minimally interacting dynamical DE that explains the deviation of $\\delta(z)$ from zero; hereby, can write the DE density as $\\rho_{\\rm DE}(z)\\equiv3H^2(z)-\\rho_{\\rm m}(z)$, viz.,\n\\begin{equation}\n\\begin{aligned}\n\\label{eqn:drho}\n\\rho_{\\rm DE}(z)&=3H^2_{\\Lambda \\rm CDM}(z)[1+\\delta(z)]^2-\\rho_{\\rm m0}\\qty(1+z)^3\\\\\n&=\\rho_{\\rm DE0}+3H^2_{\\Lambda \\rm CDM}(z)\\delta(z)[2+\\delta(z)],\n\\end{aligned}\n\\end{equation}\nfrom which we can write the deviation of the DE density from $\\Lambda$, i.e., ${\\Delta\\rho_{\\rm DE}(z)\\equiv\\rho_{\\rm DE}(z)-\\rho_{\\Lambda}}$ (where we have ${\\rho_\\Lambda=\\rho_{\\rm DE0}}$), as follows:\n\\begin{equation}\n\\Delta\\rho_{\\rm DE}(z)=3H^2_{\\Lambda \\rm CDM}(z)\\delta(z)[2+\\delta(z)].\\label{eq:deltarho}\n\\end{equation}\nFor small deviations from $\\Lambda$CDM, these read\n\\begin{align}\n \\rho_{\\rm DE}(z)&\\approx \\rho_{\\rm DE0}+6\\delta(z)H^2_{\\Lambda \\rm CDM}(z),\\\\\n \\Delta\\rho_{\\rm DE}(z)&\\approx6\\delta(z)H^2_{\\Lambda \\rm CDM}(z),\n\\end{align}\ncorrespondingly. Thus, because $\\delta(z)$ is oscillatory around zero, $\\Delta\\rho_{\\rm DE}(z)$ will also be oscillatory around zero and its small oscillations will be scaled by $6H^2_{\\Lambda \\rm CDM}(z)$; in other words, the wiggles in $H(z)$ are implied by wiggles in $\\rho_{\\rm DE}(z)$ scaled by $6 H^2_{\\Lambda \\rm CDM}(z)$. That is, observational fitting\/non-parametric reconstruction procedures predicting wiggles in $H(z)$ will predict corresponding wiggles in $\\rho_{\\rm DE}(z)$ reconstructions.\n\nEven if our assumption that the pre-recombination universe is not modified with respect to the standard cosmology [implying~\\cref{eq:prereccond}], is taken to be approximate, for $z>z_*$, the fluctuations in the DE density should be much smaller than the matter energy density, ${\\abs{\\Delta\\rho_{\\rm DE}(z)\/\\rho_{\\rm m}(z)}\\ll1}$, in the matter dominated epoch, and much smaller than the radiation energy density, ${\\abs{\\Delta\\rho_{\\rm DE}(z)\/\\rho_{\\rm r}(z)}\\ll1}$, in the radiation dominated epoch. Since for both of these epochs the relevant energy densities can be well-approximated by the critical energy density of $\\Lambda$CDM $\\rho_{\\rm c}(z)\\equiv3H^2_{\\Lambda\\rm CDM}(z)$, in this approximate case for $z>z_*$, instead of $\\Delta \\rho_{\\rm DE}(z)=0$, we can write the more relaxed condition\n\\begin{equation}\n \\abs{\\frac{\\Delta\\rho_{\\rm DE}(z)}{\\rho_{\\rm c}(z)}}=\\abs{\\delta(z)[2+\\delta(z)]}\\ll1. \\label{eq:pert}\n\\end{equation}\nThis is satisfied for both $\\delta(z)\\sim 0$ (small deviation from $\\Lambda$CDM), and $\\delta(z)\\sim-2$ (corresponds to contracting universe), but only the former is of interest to us. Since~\\cref{eq:pert} requires small $|\\delta(z)|$ to be satisfied, it can be rewritten as\n\\begin{equation}\n\\abs{\\frac{\\Delta\\rho_{\\rm DE}(z)}{\\rho_{\\rm c}(z)}}\\approx2\\abs{\\delta(z)}\\approx2\\abs{-\\psi(z)H_{\\Lambda \\rm CDM}(z)}\\ll1, \\label{eq:rapidity}\n\\end{equation}\nfrom which we immediately see that $\\psi(z)$ should vanish rapid enough with increasing $z$ at large redshifts so that our assumption of almost unmodified pre-recombination physics holds.\n\nWe calculate from Eq.~\\eqref{eqn:drho} that the DE density passes below zero, $\\rho_{\\rm DE}(z)<0$, for\n\\begin{equation}\n\\delta(z)<-1+\\sqrt{1-\\frac{\\rho_{\\rm DE0}}{3H^2_{\\Lambda \\rm CDM}(z)}},\n\\end{equation}\nwhich can also be written as follows:\n\\begin{equation}\n\\delta(z)<-1+\\sqrt{1-\\frac{\\Omega_{\\rm DE0}}{\\Omega_{\\rm DE0}+(1-\\Omega_{\\rm DE0})(1+z)^3}}.\n\\end{equation}\nAccordingly, using Planck 2018 best fit $\\Lambda$CDM values $\\Omega_{\\rm m0}=0.3158$ and $H_0=67.32{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$~\\cite{Planck:2018vyg}, it turns out that $\\delta (2.33)<-0.028$, i.e., \n$\\Delta H(2.33)\\equiv{H(2.33)-H_{\\Lambda\\rm CDM}(2.33)<-6.65{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}}$ (corresponding to $H(2.33)\\lesssim230.536{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$), requires the DE density to yield negative values. Note that $H(2.33)=228\\pm7{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$ from the Ly-$\\alpha$-Ly$\\alpha$ and $H(2.33)=224\\pm8{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$ from the Ly-$\\alpha$-quasar data~\\cite{eBOSS:2020yzd}. \\textit{Thus, considering their mean values, fitting these data perfectly, requires a negative DE density at $z\\sim 2.3$.} In this sense, the function $\\delta(z)$ can also be used as a diagnostic to test for a negative DE density if all modifications to Planck $\\Lambda$CDM are attributed to DE.\n\nLastly, the continuity equation for the DE, viz., $\\dot\\rho_{\\rm DE}(z)+3H(z)[\\rho_{\\rm DE}(z)+p_{\\rm DE}(z)]=0$, implies $\\varrho_{\\rm DE}(z)=\\frac{1+z}{3} \\rho_{\\rm DE}'(z)$ for the inertial mass density, $\\varrho_{\\rm DE}(z)\\equiv\\rho_{\\rm DE}(z)+p_{\\rm DE}(z)$, and $w_{\\rm DE}(z)=-1+\\frac{1+z}{3} \\rho_{\\rm DE}'(z)\/\\rho_{\\rm DE}(z)$ for the corresponding EoS parameter $w_{\\rm DE}(z)\\equiv p_{\\rm DE}(z)\/\\rho_{\\rm DE}(z)$, where $\\rho_{\\rm DE}(z)$ is the DE density as defined in~\\eqref{eqn:drho}, $p_{\\rm DE}(z)$ is its pressure, and $'\\equiv\\dv{z}$. Accordingly, we have\n\\begin{equation}\n\\begin{aligned}\n    \\varrho_{\\rm DE}(z)&=2(1+z)H_{\\Lambda\\rm CDM}^2\\qty[\\frac{H'_{\\Lambda\\rm CDM}}{H_{\\Lambda\\rm CDM}}\\delta(\\delta+2)+\\delta'(\\delta+1)]\\\\\n    &\\approx2(1+z)H^2_{\\Lambda\\rm CDM}\\qty[2\\frac{H'_{\\Lambda\\rm CDM}}{H_{\\Lambda\\rm CDM}}\\delta+\\delta'],\n\\end{aligned}\n\\end{equation}\nfor the DE inertial mass density, and\n\\begin{equation}\n\\begin{aligned}\\label{eq:eos}\n    w_{\\rm DE}(z)&=-1+\\frac{2(1+z)\\qty[\\frac{H'_{\\Lambda\\rm CDM}}{H_{\\Lambda\\rm CDM}}\\delta(\\delta+2)+\\delta'(\\delta+1)]}{3\\qty[\\frac{\\rho_{\\rm DE0}}{\\rho_{\\rm c}}+\\delta(2+\\delta)]}\\\\\n    &\\approx-1+\\frac{2(1+z)\\qty[2\\frac{H'_{\\Lambda\\rm CDM}}{H_{\\Lambda\\rm CDM}}\\delta+\\delta']}{3\\qty[\\frac{\\rho_{\\rm DE0}}{\\rho_{\\rm c}}+2\\delta]},\n\\end{aligned}\n\\end{equation}\nfor the corresponding DE EoS parameter; in these two equations, the second lines are for small deviations from $\\Lambda$CDM. Notice that, in the exact form of~\\cref{eq:eos}, $w_{\\rm DE}(z)$ blows up if $\\rho_{\\rm DE0}\/\\rho_{\\rm c}(z)=-\\delta(z)[2+\\delta(z)]$ is satisfied for a redshift $z_{\\rm v}$. Comparing with \\cref{eqn:drho}, we see that this condition is equivalent to $\\rho_{\\rm DE}(z_{\\rm v})=0$; indeed, if the DE submits to the continuity equation as it does in this case, a vanishing energy density necessitates such a singularity~\\cite{Ozulker:2022slu}. Such infinities in the EoS parameter are not problematic from the fundamental physics point of view, instead, hints that the DE density is perhaps an effective one originating from a modified gravity model.\n\n\\section{Wiggles in Newton's ``constant\" descended from the wavelets}\n\nAlternatively, we can attribute the deviation of $H(z)$ from $H_{\\Lambda \\rm CDM}(z)$ to the deviations in the gravitational coupling strength, $G_{\\rm eff}(z)$, from the Newton's gravitational constant $G_{\\rm N}$ measured locally. We have, as usual,\n\\begin{equation}\n3H^2_{\\Lambda \\rm CDM}(z)=8\\pi G_{\\rm N}\\qty[\\rho_{\\rm m0}(1+z)^3+\\rho_{\\rm r0}(1+z)^4+\\rho_\\Lambda],\n\\end{equation}\nwhere the constant value $\\rho_\\Lambda$ is either the usual vacuum energy density or $\\rho_\\Lambda=\\frac{\\Lambda}{8\\pi G_{\\rm N}}$. We can write the Hubble parameter of the new model as\n\\begin{equation}\n3H^2(z)=8\\pi G_{\\rm eff}(z)\\qty[\\rho_{\\rm m0}(1+z)^3+\\rho_{\\rm r0}(1+z)^4+\\rho_\\Lambda],\n\\end{equation}\nfrom which, using the definition in~\\cref{eqn:deltaH},\n\\begin{equation}\n\\begin{aligned}\n\\label{eqn:Geff}\nG_{\\rm eff}(z)=\\qty[1+\\delta(z)]^{2}G_{\\rm N}\n\\end{aligned}\n\\end{equation}\ndirectly follows. Note that $G_{\\rm eff}(z)$ is also a wiggly function led by the wiggles of $\\delta(z)$, but, $G_{\\rm eff}(z)$ equals $G_{\\rm N}$ when $\\psi(z)=0$, and thereby, $G_{\\rm eff}(z=0)=G_{\\rm eff}(z>z_*)=G_{\\rm N}$ from~\\cref{eq:prereccond,eq:present}. And,\nfor small deviations from $\\Lambda$CDM, \\cref{eqn:Geff} reads\n\\begin{equation}\n\\begin{aligned}\nG_{\\rm eff}(z)\\approx [1+2\\delta(z)]G_{\\rm N}.\n\\end{aligned}\n\\end{equation}\nNote that, if we are to treat $\\rho_\\Lambda$ as the effective energy density of the cosmological ``constant\", i.e. ${\\Tilde{\\Lambda}(z)=8\\pi G_{\\rm eff}(z) \\rho_{\\Lambda}}$, this new cosmological term $\\Tilde{\\Lambda}(z)$ is not a constant anymore.\n\nIt is crucial to note that, while attributing the wiggles to the DE density or $G_{\\rm eff}(z)$ is indistinguishable in their background dynamics, this is not so for all physical observables. Particularly, a direct effect of the dynamical gravitational coupling strength would be observable, for instance, as this would promote the absolute magnitude $M_B=\\rm const$ of type Ia supernovae (SNIa) to a quantity varying with the redshift $M_B=M_B(z)$. Such an effect in the very late universe ($z\\lesssim0.1$) was recently suggested and investigated in a series of papers to address the so-called $M_B$ (and $H_0$) tension~\\cite{Alestas:2020zol,Marra:2021fvf,Perivolaropoulos:2021bds,Alestas:2022xxm,Perivolaropoulos:2022vql,Perivolaropoulos:2022txg}. Also, the idea that the supernovae absolute magnitudes are constant with redshift, has been questioned by observations and the question of whether or not this idea is valid has recently gained interest \\cite{Benisty:2022psx,DiValentino:2020evt,Rose:2020shp,Kang:2019azh,Kim:2019npy,Tutusaus:2017ibk,Linden:2009vh,Ferramacho:2008ap}. A possible variation of the $M_B(z)$ and equivalently of the SNIa luminosity $L(z)\\propto 10^{-\\frac{2}{5}M_B(z)}$ could be due to a variation of the Newton's ``constant\". Since the SNIa luminosity is proportional to the Chandrasekhar mass, which, in this case, is no longer a constant equal to $1.4\\,M_{\\odot}$, but a quantity that varies with $G_{\\rm eff}(z)$, we have $L(z)\\propto M_{\\rm Chandra}(z)$, so that\n$L(z)\\propto G_{\\rm eff}^{-3\/2}(z)$, which in turn leads, in this approach, to\n\\begin{equation}\n\\begin{aligned}\nM_{B}(z)-M_{B,G_{\\rm N}}=\\frac{15}{4}\\log \\frac{G_{\\rm eff}(z)}{G_{\\rm N}}=\\frac{15}{2}\\log[1+\\delta(z)],\\label{eq:mbwig}\n\\end{aligned}\n\\end{equation}\nwhere $M_{B,G_{\\rm N}}$ denotes the SNIa absolute magnitude when $G_{\\rm eff}(z)=G_{\\rm N}$, which satisfies ${M_{B,G_{\\rm N}}=M_{B,0}}$ due to~\\cref{eq:present}.\nThus, attributing wiggles to $G_{\\rm eff}(z)$ will have consequences not only on the expansion of the universe, but also on the absolute magnitudes of SNIa at different redshifts; and, as \\cref{eq:mbwig} shows, the wiggles of $G_{\\rm eff}(z)$ are directly manifested in the SNIa absolute magnitudes as a wiggly $M_B(z)$ reminiscent of the findings of Ref.~\\cite{Benisty:2022psx}. Investigating how this dual modification to the standard cosmology affects the cosmological parameter estimates from SNIa data and furthermore the so-called $M_B$ tension~\\cite{Efstathiou:2021ocp,Camarena:2021jlr}, is beyond the scope of this paper, and deserves a separate study.\n\n\\begin{figure*}[ht!]\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{wavelet.pdf}\n    \\includegraphics[width=0.46\\textwidth]{sin.pdf}\n      \\includegraphics[width=0.46\\textwidth]{hdot.pdf}\n      \\includegraphics[width=0.46\\textwidth]{sindm.pdf}\n    \\caption{The deviations from the $\\Lambda$CDM model in terms of some kinematical parameters for some wavelet examples of $\\psi(z)$ given in the top left panel where $\\Bar{\\alpha}$ and $\\alpha$ are in units of ${\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$, $\\Bar{\\beta}$ and $\\beta$ are unitless, $\\Bar{z}_\\dagger$ and $z_\\dagger$ are redshifts anchoring the wavelets. The dashed line corresponds to no deviation, i.e., $\\Lambda$CDM itself. The blue bars correspond to the TRGB $H_0$ measurement and various BAO measurements. See \\cref{sec:ex} for details.}\n    \\label{fig:kinematics}\n\\end{figure*}\n\n\\section{Employing some simplest wavelets}\n\\label{sec:ex}\n\n\n\n\\begin{figure*}[ht!]\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{dens.pdf}\n    \\includegraphics[width=0.45\\textwidth]{omegade.pdf}\n    \\includegraphics[width=0.46\\textwidth]{eoswig.pdf}\n    \\includegraphics[width=0.46\\textwidth]{inertial.pdf}\n    \\caption{The deviations from the cosmological constant, if the wavelet examples of $\\psi(z)$ are attributed to a dynamical DE, i.e., the wiggles in $H(z)$ are produced solely by a dynamical DE. The plots are matched by color to those in \\cref{fig:kinematics}.}\n    \\label{fig:dens}\n\\end{figure*}\n\n\\begin{figure}[t!]\n    \\begin{flushright}\n    \\includegraphics[width=0.45\\textwidth]{geff.pdf}\n    \\includegraphics[width=0.467\\textwidth]{mb.pdf}\n    \\end{flushright}\n    \\caption{ The deviation from $G_{\\rm N}$ if wiggles are produced solely by a varying Newton's ``constant\"; we also show the variation in the absolute magnitude $M_B$ of supernovae assuming the unmodified value to be the mean value of the measurement in Ref.~\\cite{Camarena:2021jlr}. The variation of $G_{\\rm eff}$ is less than $\\sim10\\%$ at all times, and there is practically no variation for $z\\sim0$ and $z\\gg0$.\n    The plots are matched by color to those in \\cref{fig:kinematics}.}\n    \\label{fig:ex}\n\\end{figure}\n\nWavelets constitute a wide family of functions that may or may not be smooth. They exhibit an oscillatory (not necessarily periodic) behaviour over a compact set of their parameters, and either vanish or quickly decay outside of this set. Even the superposition of arbitrarily many wavelets would describe another one. Here, we will consider some simplest examples: one discontinuous, namely, the Haar mother wavelet (\\cref{sec:haar}); and other smooth wavelets, namely, the Hermitian wavelets (\\cref{sec:hermitian}) that are acquired from the derivative\/s of a Gaussian distribution function. These examples have no inherent superiority to other possible wavelets; we provide them only because of their simplicity and to give a taste of how wavelets behave and their cosmological consequences.\n\nThese example wavelets, and their corresponding cosmologically relevant functions are plotted in~\\cref{fig:dens,fig:ex,fig:kinematics} for various values of their free parameters; matching colors in different figures indicate the same wavelet with the same choice of parameters. The dashed line corresponds to a vanishing wavelet, i.e., to the reference $\\Lambda$CDM model described with $H_{\\Lambda\\rm CDM}(z)$; for the figures, we neglected radiation for $z<z_*$, and used the mean values of the Planck 2018 TT,TE,EE+lowE+lensing results~\\cite{Planck:2018vyg} for $\\rho_{\\rm m0}\/H_0^2$, $r_*$, $\\theta_*$, and $z_*$. \\cref{fig:kinematics} shows the wavelets themselves scaled by $10^{3}$ on the top left panel, and the rest of the panels show some cosmological kinematics. The top right panel shows the comoving Hubble parameter, viz., the expansion speed $\\dot{a}=H(z)\/(1+z)$, where $a$ is the scale factor of the spatially flat RW metric, and dot denotes derivative with respect to cosmic time $t$. The lower right panel shows $D_M^{-1}(z)$ scaled by $c\\ln(1+z)$. The data points in these right panels are the TRGB ${H_0=69.8\\pm0.8\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$ measurement~\\cite{Freedman:2019jwv}, and the BAO measurements (see Ref.~\\cite{eBOSS:2020yzd} and references therein): BOSS DR12 consensus Galaxy (from $z_{\\rm eff}=0.38,\\,0.51$), eBOSS DR16 LRG (from $z_{\\rm eff}=0.70$), eBOSS DR16 Quasar (from $z_{\\rm eff}=1.48$), eBOSS DR16 Ly-$\\alpha$-Ly-$\\alpha$ (from $z_{\\rm eff}=2.33$), and eBOSS DR16 Ly-$\\alpha$-quasar (from $z_{\\rm eff}=2.33$ but shifted to $z=2.35$ in the figures for visual clarity). The lower left panel shows the derivative of the Hubble function with respect to the cosmic time normalized by $3H^2$, a crossing of the zero would indicate a non-monotonic behaviour in the Hubble function as suggested in~\\cite{Aubourg:2014yra,Akarsu:2019hmw,Akarsu:2021fol}. \\cref{fig:dens} shows various plots related to the DE dynamics when the wiggles are attributed to the DE. The top left panel shows the DE densities normalized by the present-day  critical energy density $\\rho_{\\rm c0}$, the top right panel shows the DE density parameters, $\\Omega_{\\rm DE}(z)\\equiv \\rho_{\\rm DE}(z)\/3H^2(z)$, the lower left panel shows the corresponding DE EoS parameters, and the lower right panel shows the DE inertial mass densities, $\\varrho_{\\rm DE}(z)\\equiv\\rho_{\\rm DE}(z)+p_{\\rm DE}(z)$, normalized by $\\rho_{\\rm c0}$. Finally, \\cref{fig:ex} shows the corresponding results when wiggles in $H(z)$ are attributed to $G_{\\rm eff}(z)$. The top panel shows $G_{\\rm eff}(z)$ normalized by $G_{\\rm N}$, say $G_{\\rm eff}(z=0)$, and the lower panel shows the variation of the absolute magnitude $M_B(z)$ for the mean value of the measurement $M_{B,G_{\\rm N}}=M_{B}(z=0)=-19.244\\pm0.037$ mag inferred in Ref.~\\cite{Camarena:2021jlr} (using Pantheon SnIa data set~\\cite{Pan-STARRS1:2017jku} along with Cepheid stars at $z<0.01$ for their calibration).\n\n\\subsection{Haar wavelet}\n\\label{sec:haar}\nTo begin with, we consider the simplest wavelet, the Haar mother wavelet \\cite{Chui:1992}; namely, $\\psi_{\\rm h}(z)=1$ for ${0\\leq z< 1\/2}$, ${\\psi_{\\rm h}(z)=-1}$ for $1\/2\\leq z<1$, and zero everywhere else; so that $\\Psi_{\\rm h}(z_*)=0$.  Shifting and scaling $\\psi_{\\rm h}$ with three parameters, i.e., defining \n\\begin{equation}\n   \\psi_{\\rm H}(z)\\equiv \\Bar{\\alpha} \\psi_{\\rm h}\\qty[\\Bar{\\beta}\\qty(z-\\Bar{z}_\\dagger)+\\frac{1}{2}],\n\\end{equation}\nwe can produce discontinuous wiggles on $H(z)$. For a good description of the Ly-$\\alpha$ data, we choose ${\\Bar{\\alpha}=0.00015}{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$, ${\\Bar{z}_\\dagger=2}$, and ${\\Bar{\\beta}=\\frac{1}{2}}$ for our example; see the green lines in the figures. It is clear that, for these values of its parameters, $\\psi_{\\rm H}(z)$ satisfies all the conditions we imposed on $\\psi(z)$, the major ones being~\\cref{eq:int,eq:prereccond,eq:present}. In the figures, $\\delta(z)<0$ for the interval $z\\in[2,3)$ leads to a dip in $H(z)$ for this interval that is in excellent agreement with the Ly-$\\alpha$ data. This is compensated by $\\delta(z)>0$ for the interval $z\\in[1,2)$ so that~\\cref{eq:int} is satisfied; this region constitutes a bump on $H(z)$. This bump presents itself in other functions such as $\\rho_{\\rm DE}(z)$ and $\\varrho_{\\rm DE}(z)$ [or $G_{\\rm eff}(z)$]; and, it is reminiscent of those that are found in non-parametric DE density reconstructions~\\cite{Escamilla:2021uoj,Bernardo:2021cxi} from observational data. For our particular example, we see in the figures that the bump results in slight disagreement with the eBOSS DR16 Quasar data at $z_{\\rm eff}=1.48$ for both $H(z)$ and $D_{M}(z)$. This can be mitigated by a different choice of parameters or more interestingly by adding more wiggles, for example, by superposing multiple Haar wavelets; however, this superposition would increase the number of free parameters. In the next subsection, we will increase the number of wiggles without increasing the number of free parameters. Note that, the $\\Dot{H}(z)\/3H^2(z)$ plot of the Haar example appears to never cross the zero line, implying monotonic behaviour for $H(z)$; however, this is not true. The discontinuities of the $H(z)$ function at $z=1,2,3$ results in spikes (Dirac delta distributions) that are not shown in~\\cref{fig:kinematics} for the $\\dot{H}(z)$ function at these redshifts, resulting in two crossings of the zero line at $z=1,2$ and a non-monotonic $H(z)$ that increases instantaneously in time at $z=2$. Similar spikes also exist for the $w_{\\rm DE}(z)$ and $\\varrho_{\\rm DE}(z)$ functions if the wiggles are attributed to the DE, but again are not shown in \\cref{fig:dens}. Additionally, if the deformations of the Hubble function described by $\\delta(z)$ is attributed to the DE density, the $w_{\\rm DE}(z)$ has a discontinuity at $z\\sim 2.2$ (as suggested in~\\cite{Akarsu:2019hmw,Akarsu:2021fol}) in addition to the obvious ones at $z={1,2,3}$. This discontinuity (present as a singularity) happens exactly at the redshift that $\\rho_{\\rm DE}(z)$ crosses from negative to positive values, and is characteristic of energy densities that have vanishing values in time and not problematic from the point of view of fundamental physics as discussed below~\\cref{eq:eos}. Of course, the discontinuities at $z={1,2,3}$ are not very compelling physically, but the Haar wavelet is the simplest example and shows what we should expect from the form of $H(z)$ for a minimal wavelet type deviation of the Hubble radius, $H(z)^{-1}$, from that of the standard cosmological model, $H_{\\Lambda \\rm CDM}(z)^{-1}$. A good alternative to the Haar wavelet can be the Beta wavelet~\\cite{betawave} derived from the derivative of the Beta distribution ${P_\\beta(z|\\gamma,\\lambda)\\equiv1\/B(\\gamma,\\lambda)z^{\\gamma-1}(1-z)^{\\lambda-1}}$ where $B(\\gamma,\\lambda)\\equiv\\int_0^1 k^{\\gamma-1}(1-k)^{\\lambda-1}\\dd{k}$ is the Euler beta function, $0\\leq z\\leq1$, and $1\\leq\\gamma,\\,\\lambda\\leq\\infty$. Beta wavelets are in some sense softened Haar wavelets as both have compact support and are unicycle (i.e., they have just one bump and one dip), however, unlike the Haar wavelet, the Beta wavelet is continuous~\\cite{betawave}. Thus, to describe more wiggles, one would need to superpose multiple Beta wavelets just like in case of the Haar wavelet, increasing the number of free parameters. While the Beta wavelets can satisfy \\cref{eq:int,eq:prereccond,eq:present} exactly without compromising continuity, they do not have a closed-form expression and are mathematically less tractable; thus, for simplicity, we will proceed with Hermitian wavelets that are also continuous\\footnote{One may also wish the wavelet satisfying~\\cref{eq:int,eq:prereccond,eq:present} exactly to have the stronger property of being smooth. However, since these conditions require that every derivative of the wavelet vanish outside of the interval $[0,z_*]$, but not inside, such a wavelet cannot be analytic. Non-analytic smooth functions can be constructed piecewise similar to splines but the pieces are not necessarily polynomial. These kind of functions are not compelling for the demonstrative purposes of this paper but may turn out to be useful in observational analyses.} and simpler, and satisfy~\\cref{eq:int,eq:prereccond,eq:present} to high precision.\n\n\n\\subsection{Hermitian wavelets}\n\\label{sec:hermitian}\nThe discontinuous features of the Haar wavelet can be considered as an approximate description of a rapidly varying smooth function which would be physically more relevant. A simple family of smooth wavelets can be acquired from the derivative\/s of a Gaussian distribution (cf., the Hermitian wavelets~\\cite{hermitianwave}). To do so, we consider the Gaussian distribution defined as follows;\n\\begin{equation}\n\\label{eqn:G0}\n{\\psi_{\\rm G0}(z)=-\\frac{\\alpha}{2\\beta}e^{-\\beta(z-z_\\dagger)^2}},\n\\end{equation}\nwhere $\\alpha$, $\\beta>0$, and $z_\\dagger>0$ are the three free parameters that will set, respectively, the amplitude, support, and center of the wiggles. The real part of the $n^{\\rm th}$ Hermitian wavelet can be obtained from the $n^{\\rm th}$ derivative of a Gaussian distribution ${\\psi_{\\text{G}n}(z)\\equiv\\dv[n]{\\psi_{\\rm G0}(z)}{z}}$; accordingly, utilizing~\\cref{eqn:G0} we obtain\n\\begin{equation}\n\\begin{aligned}\n\\psi_{\\rm G1}(z)=&-2\\beta(z-z_{\\dagger}) \\psi_{\\rm G0}(z),\\\\\n\\psi_{\\rm G2}(z)=&4\\beta \\qty[\\beta(z-z_{\\dagger})^2-\\frac{1}{2}]\n\\psi_{\\rm G0}(z), \\\\\n\\psi_{\\rm G3}(z)=&-8\\beta^2 \n\\qty[\\beta(z-z_{\\dagger})^3-\\frac{3}{2}(z-z_\\dagger)]\\psi_{\\rm G0}(z), \\\\\n\\psi_{\\rm G4}(z)=&16\\beta^2\\qty[\\frac{3}{4}+(z-z_{\\dagger})^4\\beta^2-3\\beta(z-z_{\\dagger})^2]\\psi_{\\rm G0}(z),\n\\end{aligned}\n\\end{equation}\netc., where only up to $4^{\\rm th}$ derivative are written explicitly. $\\psi_{\\rm G1}(z)$ and $\\psi_{\\rm G2}(z)$ are well-known wavelets and the latter is also known as the Ricker (Mexican hat) wavelet. $\\psi_{\\text{G}n}(z)$ are quasi-periodic functions, i.e., the redshift difference between consecutive peaks (whose amplitudes may differ) of the wave varies. We note that $\\psi_{\\rm G0}(z)$ itself is responsible for the fast damping of the wavelet function $\\psi_{\\text{G}n}(z)$ as $z$ moves away from $z_\\dagger$ and that $n^{\\rm th}$ derivative of $\\psi_{\\rm G0}(z)$ brings an $n^{\\rm th}$ degree polynomial as a factor to itself, which in turn implies that $n$ stands also for the number of nodes of the $\\psi_{\\text{G}n}(z)$ function, i.e., the number of times the function crosses zero. These $n$ nodes correspond to $n+1$ wiggles [total of $n+1$ dips and bumps of $\\psi(z)$]; the bumps of $\\psi(z)$ manifest themselves as dips, and dips of $\\psi(z)$ manifest themselves as bumps in $\\delta(z)$ and equivalently $H(z)$, cf., Eq.~\\eqref{eqn:sdevH}. These manifestations directly translate to wiggles on either $\\rho_{\\rm DE}(z)$ or $G_{\\rm eff}(z)$ depending on which function we attribute them to. The wiggly structure in these functions resemble the wiggles in their respective functions that are acquired from observational analyses utilizing parametric or non-parametric reconstructions~\\cite{Escamilla:2021uoj,Bernardo:2021cxi}. Wiggles acquired in observational reconstructions are no surprise even if the data set does not contain CMB, because, wiggles are necessary for $H(z)$ to fit the measurements of the Hubble parameter from the BAO data without spoiling the success of $\\Lambda$CDM in fitting the $D_M(z)$ values measured from the same BAO data (see Fig.~\\ref{fig:kinematics}); and the logic we used to show the necessity of bumps still apply when $z_*$ is swapped for the effective redshift of a BAO measurement. The necessity of the wiggles only when low redshift data ($z<3$) is considered, is the subject of an upcoming work. \n\nCoincidentally, the first derivative of the Gaussian distribution \\eqref{eqn:G0}, i.e., $\\psi_{\\rm G1}(z)$, can be used to roughly approximate the Haar wavelet smoothly. For $\\psi_{\\rm G1}(z)$, we pick $\\alpha=0.0005{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$, $z_\\dagger=2$, and $\\beta=2$, so that the wavelet approximates our previous Haar example. For rest of the examples, $\\psi_{\\rm G2}(z)$, $\\psi_{\\rm G3}(z)$, and $\\psi_{\\rm G4}(z)$, the values of the parameters are shown on the top left panel of~\\cref{fig:kinematics} and the increased number of wiggles for higher derivatives are clearly seen. The top right, and bottom right panels show how increasing the number of wiggles can provide a better description of the BAO data. Unlike the Haar and $\\psi_{\\rm G1}(z)$ examples, $\\psi_{\\rm G2}(z)$ and $\\psi_{\\rm G4}(z)$ examples better describe also the eBOSS DR16 Quasar data at $z_{\\rm eff}=1.48$ while retaining better agreement with the Ly-$\\alpha$ BAO data at $z_{\\rm eff}=2.33$; the $\\psi_{\\rm G4}(z)$ example even complies with the trend of the Galaxy BAO data (at $z_{\\rm eff}=0.38,\\,0.51,\\,0.70$) $H(z)\/(1+z)$ measurements that increase with redshift, which cannot be achieved within $\\Lambda$CDM (even though $\\Lambda$CDM is not in strong tension with any of these data points). Still, we emphasize that these wavelets are just illustrative examples and better wavelets can be looked for. Again, attributing the wiggles to the DE, the DE density also wiggles smoothly; however, for the $\\psi_{\\rm G1}(z)$ and $\\psi_{\\rm G2}(z)$ examples, two safe\/expected singularities are again present in $w_{\\rm DE}(z)$ at the redshifts that the DE density vanishes.\n\n\nNote that \\cref{eq:int,eq:prereccond,eq:present} are satisfied exactly only for admissible wavelets with compact support in the redshift interval $[0,z_*]$; thus, unlike the Haar wavelet, $\\psi_{\\text{G}n}(z)$ does not satisfy~\\cref{eq:int,eq:prereccond,eq:present} exactly, but rather approximately\\footnote{We emphasize that the Haar and Hermitian wavelets are just convenient examples we used to demonstrate various aspects of the wavelet framework. The previously mentioned Beta wavelets can satisfy these conditions exactly without compromising continuity (at the cost of simplicity due to their lack of closed-form expression); and working with wavelets generated by higher order derivatives of the Beta distribution, it should be possible to increase the number of wiggles without increasing the number of free parameters, but to our knowledge, there is no established literature on wavelets derived from their higher order derivatives. Another possibility is constructing wiggles out of splines that are piecewise polynomials which can have compact support, but these are likely to suffer from excessive number of free parameters. Also, a middle ground exists where some of the conditions are satisfied exactly and some approximately. For example, the $n^{\\rm th}$ Poisson wavelet, viz., $\\psi_{{\\rm P}n}(z)\\equiv\\frac{z-n}{n!}z^{n-1}e^{-z}$ for $z\\geq0$ and vanishing everywhere else, satisfies \\cref{eq:present} exactly but the other two equations approximately for $n>1$.} (yet, beyond a level that cannot be resolved by observation). These three conditions were imposed on $\\psi(z)$ through arguments relying on the robustness of certain observations; however, no matter how robust and model independent they are, the uncertainties of the measurements itself require only that \\cref{eq:int,eq:prereccond,eq:present} hold approximately. Reassuringly, for large redshifts, ${\\psi_{\\text{G}n}(z)H_{\\Lambda\\rm CDM}(z)\\propto z^{n+\\frac{3}{2}}e^{-\\beta z^2}}$ for matter dominated and $\\propto z^{n+2}e^{-\\beta z^2}$ for radiation dominated universes; both of these functions rapidly decay by virtue of the exponential term which eventually decays faster than any polynomial growth, ensuring $\\delta(z)\\to0$ at large redshifts, see \\cref{eqn:deltaH}. A similar argument can be made for $\\Delta\\rho_{\\rm DE}(z)\\to0$ through \\cref{eq:deltarho} at large redshifts. Finally, to demonstrate how successfully the $\\psi_{\\text{G}n}(z)$ examples approximate the conditions given in \\cref{eq:int,eq:prereccond,eq:present}, we examine our $\\psi_{\\rm G3}(z)$ example as it is the one that violates these conditions most strongly. The values we pick in our $\\psi_{\\rm G3}(z)$ example correspond to: $\\psi_{\\rm G3}(0)=(41.75\\times10^{-6})\\,{\\rm \\,km^{-1}\\, s\\, Mpc}$, which can be compared with ${H^{-1}_{\\Lambda\\rm CDM}(0)=(14.78\\times10^{-3})\\,{\\rm \\,km^{-1}\\, s\\, Mpc}}$ from Planck 2018 and corresponds to ${\\delta(0)\\sim3\\times10^{-3}}$; $\\psi_{\\rm G3}(z_*)\\sim 10^{-10^6}{\\rm \\,km^{-1}\\, s\\, Mpc}$ which can be compared with ${H^{-1}_{\\Lambda\\rm CDM}(z_*)=7.3\\times10^{-7}{\\rm \\,km^{-1}\\, s\\, Mpc}}$ from Planck 2018 and corresponds to ${\\delta(z_*)\\sim10^{-10^6}}$; and, ${c\\times\\Psi_{\\rm G3}(z_*)=-5.46\\,\\rm Mpc}$ which is extremely well within the $1\\sigma$ uncertainty of $D_M(z_*)=13872.83\\pm 25.31\\,\\rm Mpc$ measured in Planck 2018.\n\n\n\n\\section{Conclusion}\nIt is well-known that the comoving angular diameter distance to last scattering, $D_M(z_*)$, is strictly constrained by observations almost model-independently. Therefore, in a viable cosmological model, this distance should be the same with the one measured by assuming $\\Lambda$CDM, so that consistency with CMB data is ensured at the background level. We have shown that, assuming the pre-recombination and present-day universes are well described by $\\Lambda$CDM, this is satisfied only if the deviation of any model from $\\Lambda$CDM described by the function $\\psi(z)=H(z)^{-1}-H(z)_{\\Lambda\\rm CDM}^{-1}$, which is the deviation from the standard $\\Lambda$CDM model's Hubble radius, should be, or well approximated by, an admissible wavelet. \\textit{In other words, in a viable alternative cosmological model that leaves the pre-recombination and present-day universes as they are in the standard cosmological model, the modifications cannot be arbitrary but should satisfy (exactly or approximately at a precision level that can be absorbed within the precision of the available observational data) a Hubble radius function whose deviation from the one in the standard cosmological model is a member of the set of admissible wavelets.}\n\nThe admissible wavelets describing $\\psi(z)$ can be converted to modifications in various cosmological kinematic functions such as the Hubble and comoving Hubble parameters, $H(z)$ and $H(z)\/(1+z)$ as shown in \\cref{fig:kinematics}, as well as, the deceleration and jerk parameters, $q(z)$ and $j(z)$. The wiggly nature of wavelets describing $\\psi(z)$ leads to wiggles in these functions, but none of them are necessarily wavelets, moreover, even the ones that arise from the simplest wavelets have non-trivial behaviour that is highly unlikely to be constructed\/introduced by hand in the first place. Accordingly, while requiring $\\psi(z)$ to be an admissible wavelet ensures consistency with the CMB at the background level, the wiggly nature of the kinematic functions can be immensely effective in fitting the multitude of BAO data which have no clear common trend compared to $\\Lambda$CDM. Also, as the wavelets we used as examples show, the number of wiggles in $\\psi(z)$, hence also in cosmological kinematics, can be varied and then quite featured kinematics well fitting the observational data can be achieved without further increasing the number of free parameters; e.g., one may introduce any number of wiggles by taking a sufficient number of derivatives of the Gaussian distribution and have only three extra free parameters. These non-trivial modifications we have found in the cosmological kinematics can then be attributed to different physical origins. As the first examples that come to mind, we have attributed them either to a dynamical DE, viz., $\\rho_{\\rm DE}(z)$, or to a dynamical gravitational coupling strength, viz., $G_{\\rm eff}(z)$, and briefly discussed how these different approaches are, in principle, observationally distinguishable, even though they give rise to the same background kinematics, see \\cref{fig:dens,fig:ex}. We demonstrated also that the dynamics of the DE, or the gravitational ``constant'', led by the simplest wavelets, are even more non-tirivial compared to the kinematics; for instance, the DE density can change sign in the past, accompanied by singularities in its EoS parameter.\n\nA wiggly structure may be described as consecutive bumps and dips on a function. By using the simplest admissible wavelets, we encountered a common pattern that our toy examples, which well describe the BAO data, present a bump in the Hubble parameter (which can be attributed to a bump in the DE density) at $1.5\\lesssim z\\lesssim2$ just as found in various observational reconstructions \\cite{Wang:2018fng,Escamilla:2021uoj,Bernardo:2021cxi}. The existence of bumps is a natural outcome of our findings, because the dips in $H(z)$ required for better description of the data, e.g., at $z\\sim2.3$ relevant to the Ly-$\\alpha$ data, should be compensated by bumps elsewhere so that the comoving angular diameter distance to last scattering remains unaltered. This should raise serious concerns that the bumpy features in the non-parametric $H(z)$ and\/or $\\rho_{\\rm DE}(z)$ reconstructions may be fake and caused by overfitting to the BAO data; since various BAO data call for dips in the Hubble parameter, there will be compensatory bumps where there are no data points to oppose them. Although the redshift range devoid of data where these bumps may be present is arbitrary and can extend to very high redshifts (e.g., a plateau with a small amplitude over a large redshift range compensating a tight dip at $z\\lesssim3$), most observational analyses reconstruct the cosmological functions up to $z\\sim3$ where the most suitable redshift range for a fake bump appears to be at $1.5\\lesssim z\\lesssim2$. It is worth noting here that the wiggles in the DE density are not expected to be representative of an Effective Field Theory, more concretely any minimally coupled scalar model \\cite{Colgain:2021pmf}, and thus it is conceivable that the introduction of theoretical priors should smooth out the wiggles in the DE density \\cite{Pogosian:2021mcs,Raveri:2021dbu}. This may be implying that the origin of the wiggles in $H(z)$ must be sought in modified gravity theories. However, it may also be too hasty to completely ignore the possibility of finding highly wiggly (may be discreet) DE densities; see, for instance, the so-called Everpresent $\\Lambda$ model, which suggests the observed $\\Lambda$ fluctuates between positive and negative values with a magnitude comparable to the cosmological critical energy density about a vanishing mean, $\\braket{\\Lambda}=0$, in any epoch of the Universe, in accordance with a long-standing heuristic prediction of the causal set approach to quantum gravity \\cite{Ahmed:2002mj,Zwane:2017xbg,Surya:2019ndm}.\n\nUp until now we have avoided discussing the $H_0$ tension and assumed that any alternative cosmological model would not deviate from $\\Lambda$CDM at $z\\sim0$, based on the observational argument that $\\Lambda$CDM describes local observational data well and is also supported by non-parametric reconstructions. However, this no deviation condition, cf., \\cref{eq:present}, is stricter than necessary, as observational evidence suggests that it is essentially the functional form of ${3H^2_{\\Lambda\\rm CDM}(z)=\\rho_{\\rm m,0}(1+z)^3+\\rho_{\\Lambda}}$ that is favored by local data. This suggests that, the reference model from which the deviations are defined, can be taken to be any model that is compatible with CMB data while agreeing with the functional form of $\\Lambda$CDM exactly or approximately in the vicinity of the present-time of the universe, instead of the exact $\\Lambda$CDM model itself. Such models can be compatible with both CMB and local $H_0$ measurements at the same time, see e.g., Ref.~\\cite{Akarsu:2019hmw,Akarsu:2021fol}. Even the requirement of this functional form can be relaxed and the well-known CPL parametrization and $w$CDM model can be used for the reference model, in which case $\\psi(z)$ being an admissible wavelet is not a necessary condition but an analytically compelling case. However, it is possible that strict observational constraints from BAO data, prevent these models from occupying the part of their parameter space that allows them to simultaneously fit the CMB and $H_0$ measurements. If these models are taken to be the reference model, the $H_0$ tension may also be resolved within our wavelet framework; more importantly, if the observational success of these models were held back by the BAO data, the use of wavelets may resurrect them by letting them fit the BAO data without compromising their successful description of the CMB and $H_0$ observations.\n\nIn our discussions we basically allowed wavelets to have quite a bit of freedom, apart from requiring them to be admissible and vanish outside of the interval $z=[0,z_*]$, see \\cref{eq:int,eq:prereccond,eq:present}. However, it can also be very useful to focus on various subsets of these wavelets. Namely, using arguments based on the history of the expansion of the universe and\/or fundamental physics (also, these two can be related in a certain way through the putative theory of gravity), we can impose more conditions on them, thereby narrow down the extent of the family of cosmological models satisfying our conditions. For example, as we have already discussed to some extent, with regard to the kinematics of the universe, one may demand an ever expanding universe ($H(z)>0$) and\/or a monotonically decreasing Hubble parameter ($\\dot{H}(z)<0$) from beginning to the present, or, with regards to dynamics of the DE (supposing that GR is valid and the deviations are attributed to a dynamical DE fluid), one may demand a non-negative DE density ($\\rho_{\\rm DE}(z)\\geq0$) at all times, or a non-negative DE inertial mass density corresponding to the null energy condition ($\\varrho_{\\rm DE}(z)\\geq0$) at all times, or at least be cautious so that no instability problems are encountered. Indeed, DE fluids that leads to our example admissible wavelets, seem to easily violate the conventional energy conditions; namely, the EoS parameter crosses below minus unity and\/or plus unity and even exhibits a pole\/s in some cases; the DE inertial mass density, and even the DE density itself in some cases, cross below zero. Such violations are generally known to indicate possible instability issues in the DE fluid. One way out in this case, as we mentioned earlier, would be the possibility of deriving such dark energies from modified gravity theories as effective sources without causing some other instability problems. Employing the Parameterized Post-Friedmann (PPF) \\cite{Hu:2008zd, Fang:2008sn} approach may also provide us with another way out, namely, the PPF discussed in  \\cite{Hu:2008zd, Fang:2008sn} may be used to placate the violent behaviors of the DE source, particularly to solve the instability issues related to the DE EoS parameter or make them less severe by pulling it towards the safer interval $[-1,1]$. This approach that replaces the condition of DE pressure perturbation with a smooth transition scale will help us understand the momentum density of the DE and other components on the large scale structure. We leave advantages of considering such reconstruction methods in relevance with the family of the DE models introduced in this paper for future consideration.\n\nTo conclude with, the wavelet framework presented in this paper seems to have the potential to be a good guide to find new cosmological models, alternative to the base $\\Lambda$CDM model, that are consistent with the observational data and to analyze existing ones, but further observational and theoretical studies are required to uncover the full scope of the implications and applications of this framework.\n\n\\begin{acknowledgments}\n The authors thank to Bum-Hoon Lee and Kazuya Koyama for useful insights and discussions. \\\"{O}.A. acknowledges the support by the Turkish Academy of Sciences in the scheme of the Outstanding Young Scientist Award  (T\\\"{U}BA-GEB\\.{I}P). E.\\'O.C. was supported by the National Research Foundation of Korea grant funded by the Korea government (MSIT) (NRF-2020R1A2C1102899). E.\\\"{O}.~acknowledges the support by The Scientific and Technological Research Council of Turkey (T\\\"{U}B\\.{I}TAK) in scheme of 2211\/A National PhD Scholarship Program.  S.T. and L.Y. were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education through the Center for Quantum Spacetime (CQUeST) of Sogang University (NRF-2020R1A6A1A03047877).\n \\end{acknowledgments}\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\nIn recent years, supersymmetric quantum mechanics (SUSY QM) has provided a\ndeeper understanding of the exact solvability of several well known\npotentials in 1-dimensional QM. In particular, using the ideas of shape\ninvariance (SI), it provides a procedure for getting the spectrum, the\neigenfunctions and the S-matrix (i.e.  the reflection and transmission\ncoefficients) algebraically [1].  There also exist interesting connections\nbetween SUSY QM and soliton solutions. Despite these (and more)\ninteresting developments in SUSY QM for one particle system in one\ndimension, so far, not many of these results could be extended either for\n$N$-particle systems in one dimension or for one particle systems in more\nthan one dimension.\n\nIn recent times there is a revival of interest in the  $N$-body problems\nin one dimension with inverse square interaction which were introduced and\nstudied by Calogero [2] and developed by Sutherland [3] and others [4].\nThese models have several interesting features, like exact solvability,\nclassical and quantum integrability and also have interesting applications\nin several branches of physics [5,6]. Apart from the well known\ntranslational invariant inverse square interaction models, referred to as\n$A_{N - 1}$ Calogero-Sutherland Model (CSM), there also exist\ngeneralizations of this model, but without  the translational invariance,\nreferred to as $BC_{N}$, $B_N$, $D_N$ models. These nomenclatures refer to\nthe relationship of these models to the root system of the classical Lie\ngroup. It might be added here that these models also share with $A_{N -\n1}$ CSM, features like exact solvability, and integrability and have also\nfound application in certain physical systems.\n\nThe purpose of this note is to enquire if the ideas of one dimensional\nSUSY QM could be extended to the $N$-particle case. In particular, whether\nthe spectrum of the celebrated Calogero and other models could be obtained\nalgebraically by using the ideas of SI and SUSY QM.  The first step in\nthat direction was taken recently by Efthimiou and Spector [7] who showed\nthat the well known Calogero model (also termed as $A_{N-1}$ CSM) exhibits\nSI. However, they were unable to obtain the spectrum algebraically. This\nis because using SUSY they were unable to relate the eigenspectra of the\ntwo SUSY partner potentials.  In this paper we demonstrate that using SUSY\nQM, SI and exchange operator formalism [8], the spectrum of the rational\n$A_{N-1}$ CSM, and also of all its generalizations like $B_N$, $D_N$ and\n$BC_N$ can be obtained algebraically.  It is worth mentioning that the SI\nin our case is somewhat different from that of Efthimiou and Spector [7].\nSo far as we are aware off, this is the first instance when an\n$N$-particle quantum system has been solved using the techniques of SUSY\nQM and SI.\n\n\n\nThe plan of the paper is the following. We briefly review the ideas of one\ndimensional SUSY QM in Sec. II with the main emphasis on solvability using\nSI.. In Sec. II.A, we apply these ideas to the Calogero model, i.e., the\nrational $A_{N-1}$ model.  We show that the spectrum of such a model can\nbe derived using the ideas of SUSY QM, SI and the exchange operator\nformalism[8]. We also briefly discuss as to how to obtain the\ncorresponding eigen-functions. In Sec. II.B, we treat the rational $BC_N$\nmodel, a translationally non-invariant system, in the same spirit.  The\nfull spectrum is obtained and the method for obtaining the exact\neigen-functions is explicitly spelled out. It is also shown in this\nsection that the $BC_N$ model posses SI even if the exchange operator\nformalism is not employed. This is a generalization of Efthimiou et al's\nwork [7] on $A_{N-1}$ model to the $BC_{N}$ case.  Finally, in Sec. III,\nwe summarize our results and discuss the possible directions to be\nfollowed in order to have a viable formalism of many-body SUSY QM.  We\nalso point out the difficulties involved in extending these results to the\ntrigonometric case.  In Appendix we show that the $BC_N$ trigonometric\nmodel is also shape invariant.\n\n\\section{SUSY, SI and Solvability}\n\nIt may be worthwhile to first mention the key steps involved in obtaining\nthe eigen-spectrum of a one body problem by using the concepts of SUSY QM\nand SI.  One usually defines the SUSY partner potentials $H_1$ and $H_2$\nby\n\\be\\label{1.1} \nH_1= A^{\\dag} A \\, , \\ H_2 = A A^{\\dagger}~~,\n\\ee\n\\noindent where ($\\hbar = 2m =1$)\n\\be\\label{1.2}\nA = {d \\over dx} + W(x)~,~ \nA^{\\dag} = -{d \\over dx} + W(x)~.\n\\ee\nIn the case of unbroken SUSY, the ground state wave function is given in \nterms of the superpotential $W(x)$ by, \n\\be\\label{1.3} \n\\psi_0 (x) \\propto e^{-\\int^{x} W(y)dy}~~,\n\\ee\n\\noindent while the  \nenergy eigenvalues and the wave functions of $H_1$ and $H_2$\nare related by, \n$(n=0,1,2,...)$\n\\be\\label{1.4}\nE_n^{(2)} = E_{n+1}^{(1)}~~, \\hspace{.2in} E_0^{(1)} = 0~~,\n\\ee\n\\be \\label{1.5}\n\\psi_n^{(2)} = [E_{n+1}^{(1)}]^{-1\/2} A \\psi_{n+1}^{(1)}~~, \\ \\ \\ \\\n\\psi_{n+1}^{(1)} = [E_{n}^{(2)}]^{-1\/2} A^{\\dag}  \\psi_{n}^{(2)}~~.\n\\ee\n\nLet us now explain precisely what one means by SI.\nIf the pair of SUSY partner Hamiltonians $H_1,H_2$\ndefined above are similar in shape and differ only in the\nparameters that appear in them, then they are said to be SI.\nMore precisely, if the partner Hamiltonians $H_{1,2}(x;a_1)$ satisfy the\ncondition,\n\\be \\label{1.6}\nH_2(x;a_1) = H_1(x;a_2) + R(a_1),\n\\ee\nwhere $a_1$ is a set of parameters, $a_2$ is a function of $a_1$ (say\n$a_2=f(a_1)$) and the remainder $R(a_1)$ is independent of $x$, then\n$H_{1}(x;a_1)$ and $H_{2}(x;a_1)$ are said to be SI. \nThe property of SI permits an immediate analytic determination of the\nenergy eigenvalues, eigenfunctions and the scattering matrix [1]. In\nparticular the eigenvalues and the eigenfunctions of $H_1$ are given by\n($n = 1,2,...$)\n\\be \\label{1.7}\nE^{(1)}_n (a_1) = \\sum^n_{k=1} R(a_k)~, \\hspace {.2in}  E^{(1)}_0 (a_1)=0~~,\n\\ee\n\\be\\label{1.8}\n\\psi^{(1)}_n (x;a_1) \\propto A^{\\dag}(x;a_1)A^{\\dag}(x;a_2)....A^{\\dag}(x;a_n)\n\\psi^{(1)}_0 (x;a_{n+1})~,\n\\ee\n\\be\\label{1.9}\n\\psi^{(1)}_0 (x;a_1) \\propto e^{-\\int^{x} W(y;a_1) dy}~~.\n\\ee\n\t\n\\subsection{Rational $A_{N-1}$ Calogero Model}\n\nWe now apply the exchange operator formulation to the rational $A_{N-1}$ CSM.\nThe Hamiltonian of the rational $A_{N-1}$ CSM is given by \n\\be\\label{9}\nH_{CSM} =  {\\sum_{i}{\\frac{1}{2}} {p_{i}^2}}  + l(l \\mp 1) \\sum_{i < j} \n{(x_{i} - x_{j})}^{-2} + \\omega {\\sum_{i} {\\frac{1}{2}} {{x_{i}^2}}} \\, .\n\\ee\n\\noindent \nThe sign $\\mp$ in (\\ref{9})\nrefers to the fact that $H$ acts on completely anti-symmetric or symmetric\nfunctions, respectively.\nLet us now define an operator $D_i$\n\\be\\label{10}\nD_{i} = -i \\partial_{i} + il \\sum_{j}^\\prime {(x_i -x_j)}^{-1} M_{ij} ,\n\\ee\n\\noindent known as the Dunkl operator in the literature. Hereafter\n$\\prime$ means $i=j$ is excluded in the summation. The exchange operator\n$M_{ij}$\nhave the following properties [8]\n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray}\nM_{ij}^2=1~, \\ \\ \\ M_{ij}^\\dagger=M_{ij}~,\n\\ \\ \\ M_{ij} \\psi^{\\pm} = \\pm \\psi^{\\pm}~,\\nonumber \\\\\nM_{ij} D_{i} = D_{j} M_{ij}~, \\ \\\nM_{ij} D_k = D_k M_{ij}~, \\ \\ k \\neq i,j~~,\\nonumber \\\\\nM_{ijk}=M_{ij} M_{jk}~, \\ \\ M_{ijk}=M_{kij}=M_{jki}~,\n\\eea\n\\noindent where $\\psi^{\\pm}$ is a(an) symmetric(antisymmetric) function.\nNote that the Dunkl operator is hermitian by construction\nand $[D_i, D_j]=0$. If we now define\n\\be\\label{11}\na_i = D_i - i \\omega x_i \\, , \\ \n{a_i}^\\dagger = D_i +i \\omega x_i \\, ,\n\\ee\nthen it is easy to see that\n\\be\\label{12}\n{[ a_i, {a_j}^\\dagger]} = 2 w {\\delta_{ij}}(1 + l{\\sum_{k}^{\\prime}M_{ik}})\n\t     - 2 (1-{\\delta_{ij}}) l w M_{ij} \\, .\n\\ee\n\\noindent Let us now consider the SUSY partner potentials \n$H$ and $\\tilde {H}$ defined by\n\\be\\label{13}\nH = \\frac{1}{2}\\sum_{i}{a_i}^\\dagger a_i~,\\ \\ \\ \n\\tilde {H} =\\frac{1}{2} \\sum_{i}a_i {{a_i}^\\dagger} \\, . \n\\ee\n\\noindent Using eqs. (\\ref{10}) and (\\ref{11}) it is easily shown that \n\\be\\label{14}\nH_{CSM} = H + E_0^{CSM}~, \\ \\ \\\nE_0^{CSM} = [\\frac{N}{2} \\mp \\frac{l}{2} N (N-1)] \\omega~..\n\\ee\n\\noindent Thus, by construction, the ground state energy of $H$ is zero.\n\nUsing eqs. (\\ref{10}) and (\\ref{11}) it is easily shown that if  \n$\\psi$ is the eigenstate of $H$ with eigenvalue $E (>0)$, then \n$A_1 \\psi$ is the eigenstate of $\\tilde H$ with eigenvalue $E+ \\delta_1$ i.e. \n\\be\\label{16}\n\\tilde{H} (A_1 \\psi)=[E+ \\delta_1](A_1 \\psi),\n\\ee\nwhere,\n\\be\\label{17}\nA_1 ={\\sum_{i}} a_{i}~, \\ \\ \\\n\\delta_1 = [(N-1) \\pm l  N(N-1)] \\omega~~.\n\\ee\n\\noindent \nSimilarly, if $\\tilde {\\psi}$ is the eigenfunction of $\\tilde H$ with \neigenvalue $\\tilde E$, then $A_1^{\\dag}\\psi$ is the eigenfunction of $H$ \nwith eigenvalue $\\tilde{E} - \\delta_1$ i.e.\n\\be\\label{19}\nH({{A}_1^\\dagger}\\tilde {\\psi}) =[\\tilde {E}-\\delta_1] ({{A}_1^\\dagger} \n\\tilde{\\psi}).\n\\ee\n\\noindent This proves one to one correspondence between the non-zero\nenergy eigen values of $H$ and $\\tilde {H}$. \nThus it follows from here that the energy eigenvalues and eigenfunctions \nof the two partner Hamiltonians $H$ and $\\tilde H$ \nare related by \n\\be\\label{20}\n\\tilde{E}_n = E_{n+1} +\\delta_1 \\, ,~~ E_0 = 0 \\, , \\ \\ n = 0,1,2,...\n\\ee \n\\be\\label{21}\n\\tilde{\\psi}_n = {A_1 \\psi_{n+1} \\over \\sqrt {E_{n+1} +\\delta_1}}~~,\n~\\psi_{n+1} = {A_1^{\\dag} \\tilde{\\psi}_n \\over \\sqrt{E_{n+1}}}~~.\n\\ee\n\\noindent Note that $\\delta_1$ vanishes for $N=1$ and we\nrecover the usual results of SUSY QM with one degree of freedom.\nIt is worth noting that unlike the case of one dimensional QM, \nin this case the (positive) energy levels of $H$ and $\\tilde{H}$ \nare not degenerate. \n\nUsing eqs.. (\\ref{10}) and (\\ref{11}) it is also easily shown that $H$ and \n$\\tilde{H}$ satisfy the shape invariance condition \n\\be\\label{23}\n\\tilde{H} (\\{x_i\\} ,l)  = H (\\{x_i\\}, l) +R(l),\n\\ee\n\\noindent where\n\\be\\label{24}\nR(l) = [ N \\pm l N(N-1)] \\omega = \\omega + \\delta_1~~.\n\\ee\n\\noindent \nAs a result, using the formalism of SUSY QM, and the relation between\n$E_{n+1}$ and $\\tilde{E_n}$ as given by eq. (\\ref{20}), \nthe spectrum of H is given by\n\\be \\label{25}\n{E_{n}} =\\sum_{i}R(l_{i}) - n\\delta_1~.\n\\ee\n\\noindent Note that in this particular case all $l_i$ are \nidentical so that using $\\delta_1$ and $R(l)$ as given by eqs. (\\ref{17}) \nand (\\ref{24}), the spectrum turns out to be\n\\be\\label{26}\nE_{n} = n(R-\\delta_1)\n\t= n \\omega ~.\n\\ee\n\\noindent \nUsing eq. (\\ref{14}) we then get the correct spectrum of the Calogero $A_{N-1}$\nmodel.\n\nLet us now discuss as to how to obtain the eigenfunctions of CSM using the\nformalism of SUSY QM.\nWe have seen that $A_1$ and $A_1^\\dagger$ relate the non-zero eigen states\nof the partner Hamiltonians $H$ and $\\tilde{H}$. Once a particular state of\n$H (\\tilde{H})$ with non-zero eigen value is known, the use of eq.\n(\\ref{21}) enables us to find the corresponding state of \n$\\tilde{H}\n(H)$. In particular, using eq. (\\ref{1.8}) and the fact that in this case \nall the $l_i$ \nare identical, it follows that all\nthe eigen-functions can be obtained from the ground state wave\nfunctions $\\psi_0$ as, $\\psi_n=(A_1^\\dagger)^n \\psi_0$. Note that this\nis justified from the operator algebra also, since $A_1$ and $A_1^\\dagger$\ncan be identified as the annhilation and the creation operator respectively.\nIn particular, one can show\nusing eqs. (\\ref{16}), (\\ref{19}), (\\ref{23}) and (\\ref{24}) that\n$[H,A_1]=-A_1$ and $[H,A_1^\\dagger]=A_1$. \n\nThis procedure for obtaining the\neigen-functions is similar to that of Isikov {\\it et al.} [9].\nTo see this, define a set of operators,\n\\be\\label{27}\nA_n =\\sum_{i=1}^N a_i^n, \\ \\ \\ \\ n \\leq N, \\\n\\ee\n\\noindent which are symmetric in the particle indices. These operators\nsatisfy relations which are analogous to those given by eqs. (\\ref{16}) and \n(\\ref{19}) for any $n$\n(see the next paragraph).\nIt is easily checked that $[H,A_n]=-n A_n$ and\n$[H,A_n^\\dagger]=n A_n^\\dagger$. Following [9], \nthe $k$-th eigen-state is given by,\n\\be\\label{28}\n\\psi_{\\{n_i\\}} = \\prod_{i=1}^N \\left ( {A}^\\dagger_i \\right )^{n_i} \\psi_0,\n\\ \\ \\ \\ a_i \\psi_0 =0, \\ \\ \\ \\ k=\\sum_{i=1}^N n_i ~.\n\\ee\n\\noindent Note that $\\psi_{\\{n_i\\}}$ incorporates all the degenerate states\ncorresponding\nto a particular value of $k$ and all the corresponding states of $\\tilde{H}$\ncan be obtained by applying the same $A_1$ on $\\psi_{\\{n_i\\}}$.\n\n\nLet us now ask the question whether or not $A_1$ is the only operator which\nrelates the states with nonzero eigen values of the partner Hamiltonians. The\nanswer obviously is negative and in fact, any operator which is symmetric\nin the particle\nindices can be used to relate the non-zero eigenstates of the partner\nHamiltonians. However, none of these operators are useful\nin deriving the full spectrum of the $A_{N-1}$ CSM model.\nFor example, if $\\psi$ is an\neigen-function of $H$ with\nnon-zero energy eigen-value $E$, then,\n\\be\\label{29}\n\\tilde{H} ( A_n \\psi ) = \\left [ E + \\delta_n \\right ] ( A_n \\psi )~~,\n \\ \\ \\ \\ \\delta_n = [(N-n)  \\pm l N (N-1)] \\omega~.\n\\ee\n\\noindent Note that the above equation is valid only if $\\psi$ is at least\nthe $n$-th excited state, since $A_n(A_n^\\dagger)$ anhilates(creates)\n$n$ states.\nSimilarly, one can show that any state $\\tilde{\\psi}$ of $\\tilde{H}$,\nwhich represents at least the $(n-1)$-th excited state with energy\neigen value $\\tilde{E}$, is related to a state\n${A_n^\\dagger} \\tilde{\\psi}$ of $H$ with the eigen value \n$\\left ( \\tilde{E} - \\delta_n \\right )$. This again proves one to one\ncorrespondence between the $n$-th excited state of $H$ and the $(n-1)$-th\nexcited state of $\\tilde{H}$. As a result, the use of SI gives only\nthe spectrum beginning with the $n$'th excited state of $H$ and not the \nfull spectrum. \n\nIt is worth pointing out that for the $B_N$ type models, however, \nthe symmetry arguments force us to replace $A_1$ by $A_2$ in order\nto derive the full spectrum using SUSY QM.\nThis is discussed below in detail.\n\n\\subsection{Rational $BC_N$ Calogero Model}\n\n\nThe Hamiltonian for $BC_{N}$ Calogero model is given by\n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray}\\label{2.1}\nH_{BC_{N}} & = & {\\frac{1}{2}}[{\\sum_{i} {p_{i}^2}}  + \n l(l \\mp 1) \\sum_{i,j}^{'} \\left [(x_{i} - x_{j}) ^{-2} +\n{(x_{i} +x_{j})} ^{-2} \\right ]\\nonumber \\\\\n & + & (l_{1}-1)l_{1} {\\sum_{i}x_{i}}^{-2}\n + \\frac{(l_2)(l_2-1)}{2} \\sum_i { x_i}^{-2}+\n{\\frac{\\omega}{2}}  {\\sum_{i} x_i^2}] ~.\n\\eea\n\\noindent The sign $\\mp$ in front of the second term implies\nthat $H$ is restricted\nto act on the space of anti-symmetric (symmetric) wave-functions only.\nThis model reduces to CSM of $B_N$, $C_N$ and $D_N$\ntype in the limit $l_2=0$, $l_1=0$ and $l_2=l_1=0$, respectively.\nWithout loss of generality, in this section, we therefore only study \nthe $B_N$ type model,\ni.e. $l_2=0$. The other cases are easily obtained from here.\n\nIt is interesting to observe that this system also shares  \nthe property of SI\nas found in [7] in the case of the $A_{N-1}$ model. \nSuperpotential corresponding to this model is \n\\be\nW_{i} =\\frac{\\partial G(x_{1}...x_{N})}{ \\partial x_{i}}=\n\\frac{\\partial (ln \\psi_0)}{\\partial x_i}~,\n\\ee\n\\noindent where $\\psi_0$ is the ground state wave function of $H_{B_{N}}$\nand $G$ is given by \n\\be\nG = +l_{1} \\sum_{i} \\ln (x_{i}) +l\\sum_{i > j}\\ln (x_{i} - \nx_{j}){(x_{i} +x_{j})} -\\frac{\\omega}{2} \\sum_i x_i^2~. \n\\ee\n\\noindent Thus, the superpotential takes the form \n\\be\nW_i =+l \\sum_j^\\prime \\left [ (x_i - x_j)^{-1} +(x_i +x_j)^{-1} \\right ]+\nl_{1} x_i^{-1} - \\omega x_i~.\n\\ee\n\\noindent Following [7], define \n${\\cal A}_{i} ({\\cal A}_{i}^\\dagger)=\\pm \n\\partial_{i}+W_i $,from which Hamiltonian (\\ref{2.1}) with $l_2=0$\ncan be expressed as \n\\be\nH^{B_N}=\\sum_i {{\\cal A}_{i}}^\\dagger {\\cal A}_i \\ \\\n-\\left [ \\frac{N}{2} -l N (N-1) -l_{1} N \\right] \\omega~, \\ \\\n\\ee\n\\noindent\nShape invariance follows due to the\nidentity,\n\\be\n\\sum_{i}{\\cal{A}}_{i}{\\cal{A}}_{i}^\\dagger(l,l_{1}) =\n\\sum_i {\\cal{A}}_{i}^\\dagger\n{\\cal{A}}_{i}(l+1,l_{1} +1)~.\n\\ee\n\\noindent\nShape invariance as observed in [7] for $A_{N-1}$ CSM, is\npresent not only\nin the rational $B_N, D_N, BC_N$ models, but also in their trigonometric\ncounterparts. For trigonometric $BC_N$ models this is shown  \nin the Appendix .\n\nAs in the $A_{N-1}$ case, the SI condition does not help us in obtaining\nthe spectrum of the rational $BC_N$ models\nunless we employ the\nexchange operator formalism. Further, \nthe Hamiltonian in eq. (\\ref{2.1}) can also be cast in a diagonal form using\nexchange\noperator method. This however requires including\na reflection operator $(t_{i})$ where $t_{i}$ commutes with $x_{j}$ and \nanti-commutes\nwith $x_{i}$. The Dunkl derivative operator ( analogous to the $A_{N-1}$ case) \nis given by\n\\be\\label{2.2}\n{\\cal{D}}_{i}\n= -i \\partial_{i} + il \\sum_{j}^\\prime \\left [ {(x_i -x_j)}^{-1} M_{ij} +\n{(x_i +x_j)}^{-1} \\tilde {M_{ij}} \\right ] +i l_1 x_{i} ^{-1},\\ \\ \\\n\\tilde{M_{ij}} = t_{i}t_{j}M_{ij}~.\n\\ee\n\\noindent\nThe reflection operator $t_i$ satisfies the following relations \n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray}\n t_i^2=1~, \\ \\ t_i \\psi(x_1, \\dots, x_i, \\dots, x_N)=\n\\psi(x_1, \\dots, -x_i, \\dots, x_N)~,\\nonumber \\\\\nM_{ij} t_i = t_j M_{ij}~,\\ \\ \n\\tilde{M}_{ij}^\\dagger = \\tilde{M}_{ij}~,\\ \\\nt_i {\\cal{D}}_i = - {\\cal{D}}_{i} t_i~,\\ \\\nt_i {\\cal{D}}_j = {\\cal{D}}_j t_i~,\\ \\ j \\neq i,\\nonumber \\\\\n\\tilde {M_{ij}} {\\cal{D}}_i = -{\\cal{D}}_{i} \\tilde{M_{ij}} \\, .\n\\eea\n\\noindent It follows from eq. (\\ref{2.2}) that $[{\\cal{D}}_i,\n{\\cal{D}}_j] = 0$ and \n\\be\n[x_i,{\\cal{D}}_j]=i\\delta_{ij} \\left ( 1 + l \\sum_{k}^\\prime (M_{ik}+\n\\tilde{M_{ik}})+2l_1t_i \\right )\n-i(1-\\delta_{ij})l(M_{ij} -\\tilde{M_{ij}}) ~.\n\\ee\n\\noindent\nDefining, ${\\hat{a}}_i$  and ${\\hat{a}}_i^\\dagger$ \nwith the same defintion as in the previous case (see eq. (\\ref{11}))\nand using the above equations,\none finds\n\\be\n[{\\hat{a}}_i, {\\hat{a}}_j^\\dagger] = 2 \\omega \\delta_{ij} \\left (1+\nl\\sum_{k}^{\\prime} ( M_{ik}+\n\\tilde{M_{ik}}) +2l_{1}t_{i} \\right )\n- 2(1- \\delta_{ij}) l \\omega (M_{ij} - \\tilde{M_{ij}})~.      \n\\ee\n\n\\noindent As before, the SUSY partner Hamiltonians ${\\cal{H}}$\nand $\\tilde {{\\cal{H}}}$ for\nthe $B_{N}$ case \nare defined as \n\\be\n{\\cal{H}} = \\frac{1}{2}\\sum_{i}{\\hat{a}}_i^\\dagger {\\hat{a}}_i \\, ~, \\ \\ \n\\tilde{{\\cal{H}}} = \\frac{1}{2}\\sum_{i}{\\hat{a}}_i {\\hat{a}}_i^\\dagger \\, ~.\n\\ee\n\\noindent \nIt can be seen that,\n\\be\nH_{B_{N}} ={\\cal{H}} + E_{0}^{B_{N}}~, \\ \\ \\\nE_{0}^{B_{N}} = [\\frac{N}{2} \\mp \\frac{l}{2} N (N-1)+l_1 N] \\omega~.\n\\ee\n\\noindent\nThe operator which brings in a correspondence between the \neigenstates $\\psi$\nand $\\tilde{\\psi}$ are respectively\n\\be\n\\hat{A}_{2} = \\sum_{i} {\\hat{a}}_{i}^2 \\, ~, \\ \\\n\\hat{A}_{2}^\\dagger = \\sum_i (\\hat{a}_i^\\dagger)^2 \\, ~.\n\\ee\n\\noindent One can show that if ${{\\psi}} (\\tilde {{{\\psi}}})$ is the\neigenfunction of \n${\\cal{H}} (\\tilde{{\\cal {H}}})$ with eigenvalue ${\\cal{E}}\n(\\tilde {{\\cal{E}}})$ then \n\\be\n{\\cal{H}}(\\hat{A_{2}}^\\dagger {\\psi}) =\n(\\tilde{{\\cal{E}}}-\\hat{\\delta}_{2})(\\hat{A_{2}}^\\dagger \n\\tilde {\\psi})~,\\ \\\n\\tilde {{\\cal{H}}}(\\hat{A}_{2}\\psi) =\n({\\cal{E}}+\\hat{\\delta}_{2})(\\hat{A}_{2}\\psi)~~.\n\\ee\n\\noindent where,\n\\be\n\\hat{\\delta}_{2} = [N-2 \\pm 2lN(N-1) +2l_{1}N ] \\omega~.\n\\ee\n\\noindent Now the question is why we\nshould take $\\hat{A}_2$ instead of $\\hat{A}_1$ (note that \n$\\hat{A}_n=\\sum_i \\hat{a}_i^n$)?\nThe point is, unlike the $A_{N-1}$ case, the $BC_N$ Hamiltonian has the \nreflection \nsymmetry, $x_{i}{\\rightarrow}-x_{i}$. Such a symmetry on the wave-functions \nis ensured only if one uses\n$\\hat{A}_2$ and not $\\hat{A}_1$. \n\n\\noindent Following the treatment in the $A_{N-1}$ case, it is easy to show\nthat $H$ and $\\tilde {H}$ of the $B_N$ model also satisfy the SI condition i.e.\n\\be\n\\tilde {H} (\\{x_i\\},l,l_1) = H (\\{x_i\\},l,l_1) + R_2 (l,l_1)\n\\ee\nwhere\n\\be\nR_{2}(l,l_1) =[N  \\pm 2lN(N-1) +2l_{1}N] \\omega~~.\n\\ee \nSince in this case also all the $l_i$ are identical, hence it is easy to\nsee that the spectrum is given by\n\\be\nE_{n}=n(R_{2}-\\hat{\\delta}_{2})\n     =2n\\omega~.\n\\ee     \n\\noindent  Note that now the spectrum is given by $2n\\omega$, instead\nof $n\\omega$ as\nin the case of $\\hat{A}_{N-1}$. This spectrum was also obtained earlier in \n[10], but by different method.\n\nThus we have shown that for the N-body Calogero models,\nthe spectrum can also be obtained by using the ideas of SQM, SI and exchange \noperator formalism.\n\n\n\n\\section{Summary \\& Discussions }\n\nIn this paper, we have generalized the ideas of SUSY QM with one degree\nof freedom to the rational-CSM,\nwhich is a many-body problem.  In particular, we have shown that the exchange\noperator formalism is suitable for relating the non-zero eigen states\nof the partner Hamiltonians of CSM. The shape invariance in this formalism\nbecomes\ntrivial compared to the case discussed in [7]. In fact, the\npotentials\nof the partner Hamiltonians differ by a constant and\nthis is reminiscent of the usual harmonic oscillator case. \nAs a result, the operator method employed in [9] for solving\nthe rational-CSM algebraically and the SUSY method described here are \nnot very different from each other.\n\nOne of the nontrivial check of the applicability of the SUSY QM and the SI\nideas to the many-body problems lies in solving the trigonometric CSM,\nsince unlike the oscillator case, in this case  the energy spectrum is not\nlinear in the radial quantum number.  Unfortunately, the generalized\nmomentum operator $D_i$ for all types of models, rational as well as\ntrigonometric CSM associated with the root structure of $A_n$, $B_n$,\n$D_n$ and $BC_n$, are hermitian by construction. So, we can not talk of\npartner Hamiltonians in terms of $D_i$ alone. We can define the usual\ncreation and the anhilation operators, $a_i^\\dagger$ and $a_i$, in case we\nare dealing with the rational-CSM and construct partner Hamiltonians.\nUnfortunately, this can not be done for the trigonometric models. On the\nother hand, as described in [7], we can indeed introduce partner\nHamiltonians for $A_n$ type of trignometric models provided the exchange\noperator formalism has not been used. The SI is present in this formalism\nalso but the task of relating the eigenspectrum of the partner\nHamiltonians is unknown as yet.  We have shown in this paper that the SI\nis also present in the most general $BC_N$ type of trignometric models.\nHowever, the problem again lies in our inability to relate the the\nspectrum of the partner Hamiltonians.\n\n\n\n \n\n\\begin{appendix}\n\n\\section{SI in Trigonometric $BC_N$ CSM model}\n\nIn this Appendix, we present the SI conditions for the trigonometric\n$BC_N$ models.\nThe trigonometric $BC_N$ Hamiltonian is given by,\n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray} \nH_{BC_N} &=& \n- \\sum_i \\partial_i^2 + l (l - 1) \\sum_{i,j}^{\\prime} \\left [\n\\frac{1}{\\sin^2(x_i - x_j)} + \\frac{1}{\\sin^2(x_i + x_j)} \\right ]\\nonumber \\\\\n&& + \\sum_i \\frac{l_1 (l_1\n- 1)} {\\sin^2 x_i}\n+ l_2 (l_2 - 1) \\sum_i \\frac{1}{\\sin^2 2x_i}~.\n\\eea\n\\noindent This model reduces to $B_N$, $C_N$ and $D_N$ for\n(a) $l_2=0$, (b) $l_1=0$,\n(c) $l_1=l_2=0$, respectively. We define a superpotential\nof the form,\n\\be\nW_i = l \\sum_j \\left [ \\cot(x_i - x_j) + \\cot(x_i + x_j) \\right] + l_1 \\cot\nx_i + l_2 \\cot 2x_i~.\n\\ee\n\\noindent \nUsing this expression of $W_i$\nin the definition of the creation and the anhilation operators as defined\nin (\\ref{1.1}), one can construct partner Hamiltonians which are equivalent \nto $H_{BC_N}$ up to an overall constant. The SI condition for these partner \nHamiltonians is\n\\be \nH_2^{BC_N} (\\{x_i\\}, l , l_1 , l_2) = \nH_1^{BC_N} (\\{x_i\\}, l^\\prime , l_1^\\prime , l_2^\\prime) +\nR^{BC_N}~,\n\\ee\n\\noindent where\n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray}\nR^{BC_N} & = & \n2 ( l^\\prime l_1^\\prime - l l_1) N (N - 1) + \\frac{4}{3} N (N - 1) (N - 2)\n(l^{\\prime 2} - l^2) \n+ 4 N (l_2^\\prime l_1^\\prime - l_2 l_1)\\nonumber \\\\\n&& + 4 N (N - 1)\n(l_2^\\prime l^\\prime - l_2 l) + 4 N (l_2^{\\prime 2} - l_2)\n+ N (l_1^{\\prime 2} - l_1) + 2 N (N - 1) (l^{\\prime 2} - l^2)\n\\eea\n\\noindent and\n\\be \nl^\\prime=l-1~, \\ \\ l_1^\\prime = l_1 - 1 ~,\\ \\ l_2^\\prime=l_2 - 1~.\n\\ee\n\\noindent The SI condition for $B_N$, $C_N$ and $D_N$ can be obtained\nfrom the above equations by taking appropriate limits. In particular,\nby putting $l_2 = 0$, $l_1 = 0$ or $l_2 = l_1 = 0$, we obtain the corresponding\nresults for the $B_N$, $C_N$ and $D_N$ type models respectively.\\\\\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}