diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzoixp" "b/data_all_eng_slimpj/shuffled/split2/finalzzoixp" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzoixp" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nModels with multiple Higgs bosons provide one of the simplest\npossibilities for physics beyond the standard model. Indeed, two\nHiggs doublet models in particular have received a great deal of\nattention, and have arisen in a wide variety of contexts, including\nsupersymmetry or extra dimensions, as well as axion models. More\ngenerally, given our current lack of experimental data concerning the\nHiggs sector, it is natural to suppose that there may be more than a\nsingle Higgs boson waiting to be discovered at the weak scale.\n\nThe fundamental constraint on multi-Higgs doublet models comes from\nflavor physics. After diagonalizing the quark masses, the Yukawa\ncouplings of the neutral component of any extra Higgs boson will\ngenerically lead to tree-level flavor-changing neutral current (FCNC)\nprocesses, which are highly constrained by data. There are only a few\noptions generally considered for avoiding these difficulties. The\nfirst is simply to assume that the Yukawa couplings of any new Higgs boson\nto the standard model fermions are sufficiently small so as to be\nsafe. Generically, for Higgs boson masses of order the weak scale, this\nrequires Yukawa couplings of order $10^{-4}$ or less. The second\noption is to demand that only a single Higgs boson couples to standard\nmodel fermions of a given electric charge. This leads to two commonly\nconsidered scenarios, referred to as the Type I and II two Higgs\ndoublet models (2HDMs). In the Type I 2HDM, it is assumed that an\nadditional Higgs does not couple to any of the standard model\nfermions, while in the Type II model, a first Higgs couples only to\nthe up-type quarks, while a second couples only to the down-type\nquarks, as in supersymmetry. A third option often considered is that\nof ``minimal flavor violation\" \\cite{MFV}. In this scenario, it is assumed that\nthe full $U(3)^5$ flavor symmetry of the standard model is broken only\nby the Yukawa couplings of a single Higgs boson responsible for\ngenerating the fermion masses. The Yukawas are assumed to come from\nvacuum expectation values (vevs) of some set of fields transforming as\nbifundamentals under the flavor group. This results in all flavor\nviolation being of a size set by the ordinary standard model Cabbibo--Kobayashi--Maskawa (CKM)\nmatrix.\n\nOne feature which all of these scenarios have in common is that of\nvery small Yukawa couplings of the Higgs bosons to the first generation\nquarks and leptons. In the Type I and II 2HDMs this is required in\norder to avoid giving large masses to the first generation fermions,\nwhile in minimal flavor violation, this is required by virtue of the\nsmallness of the ordinary first-generation Yukawa couplings. The smallness of these Yukawa couplings can\nmake it difficult, for example, to explain a recent $Wjj$ anomaly at\nthe Tevatron \\cite{cdf} by using an extended Higgs sector. In this paper we will\nconsider a simple alternative scenario which is able to avoid this\nrequirement, address the $Wjj$ anomaly, and simultaneously suggest a mechanism for a\nstraightforward solution to the strong CP problem.\n\nOur setup assumes that a single Higgs boson $H$ dominates in providing the\nmasses for the standard model fermions. Beyond this, we will make\ntwo assumptions:\n\\begin{enumerate}\n\\item There is an $SU(3)_f$ flavor symmetry broken only by the $H$ Yukawa\n couplings. The 3 right-handed up quarks, 3 right-handed down quarks, and 3\nleft-handed doublet quarks are all taken to transform in triplet representations\nof the $SU(3)_f$ symmetry. In particular, a second Higgs doublet,\n $\\Phi=\\left(\\begin{array}{c} \\phi^+ \\\\ \\phi^0\\end{array}\\right)$ ,\n is assumed to have Yukawa couplings proportional to the identity\n matrix in a given canonical basis,\\footnote{A canonical basis is defined to be the basis of quark fields where the flavor symmetry $SU(3)_f$ is manifest.} preserving the $SU(3)_f$. Any further Higgs doublets beyond this should also have couplings of the same form.\\footnote{Large diagonal Yukawa couplings for $H$ may\nbe suppressed in an appropriate UV completion. See section 3 for an example.}\n\\item In the canonical basis, the $H$ Yukawa couplings are Hermitian matrices.\n\\end{enumerate}\n\nThe reason that such a scenario can be associated with a solution to the strong\nCP problem is straightforward; if CP is broken only\nspontaneously, then the strong CP parameter, $\\theta$, may be equal to zero\nin the original canonical basis. Diagonalizing the quark masses,\n$\\theta$ then {\\it remains} equal to zero due to the Hermiticity\nassumed for the Yukawa couplings. Spontaneous breaking of CP could take place through a variety\nof mechanisms already appearing in the literature, such as, for example, the Nelson\/Barr mechanism \\cite{NelsonCP, Barr}. We must simply require that our \"Hermitian Flavor Violation\" (HFV) structure emerges in the effective theory at low energies. This may be most easily accomplished if the $SU(3)_f$ flavor symmetry is respected by the sector of the UV theory responsible for CP violation. It may also be possible to have the same fields simultaneously break both the flavor $SU(3)_f$ and CP symmetries together; a candidate for this type of theory will be discussed in section 3.\n\nDue to the above assumed structure, the safety of the scenario from flavor-changing neutral currents\nis simple to understand at the qualitative level. After diagonalizing the quark masses,\nthe couplings of the neutral Higgs $\\phi^0$ remain unchanged, while\nthe charged Higgs $\\phi^+$ has flavor-changing interactions\nproportional to the corresponding CKM elements. In this way, the\nstructure of FCNC's is the same as in the standard model, with\nanalogous suppressions by the Glashow--Iliopoulos--Maiani (GIM) mechanism; we need only assume that\nthe $\\Phi$ Yukawa is somewhat smaller than the gauge\ncoupling of the weak interaction, depending on the $\\Phi$ mass. The only difference here is that, in the presence of the $H$ Yukawa couplings, there is no symmetry fixing the universality of the\n$\\phi^0$ interactions. In the standard model, the $Z^0$ couplings\nremain universal due to gauge invariance. This will lead to some\nsmall loop suppressed FCNCs, but not at a dangerous level. We will\ndemonstrate the safety of the flavor structure in more detail in\nsection 2, as well as discuss the limits on the $\\Phi$ Yukawa\ncouplings.\\footnote{An analogous 2HDM flavor scenario was discussed in reference \\cite{Argentina} but the motivation and underlying structure were different than what we consider here.} Section 3 will contain a proposal for a possible UV completion of our scenario, demonstrating a mechanism for realizing the required hierarchical, Hermitian structure of the $H$ Yukawa couplings, as well as addressing the strong CP problem.\n\nAs noted above, the key phenomenological difference between this model\nand more standard two Higgs doublet constructions is the presence of\nallowed large couplings of $\\Phi$ to the first generation fermions.\nAs a result, in section 4 we will discuss an explanation in this scenario for the excess in $Wjj$ events at CDF through resonant production of a heavy component of the new doublet.\\footnote{For other thoughts in this direction, see \\cite{Wang, Chen}} We will conclude in section 5.\n\n\\section{Flavor Constraints}\n\nIn the previous section, we presented a flavor structure which allows us to couple an additional Higgs boson to the standard model quarks, without having extremely suppressed couplings to the first generation. In this section we examine the major constraints on this model and place limits on the couplings of the additional Higgs boson. The Yukawa sector for this model is as follows:\n\n\\begin{equation}\n-{\\cal L}\\supset \\tilde{H} {\\bar Q_L}Y^U u_R + H {\\bar Q_L}Y^D d_R\n +\\tilde{\\Phi} {\\bar Q_L}G^U u_R + \\Phi {\\bar Q_L}G^D d_R,\n\\end{equation}\nwith $\\tilde{H} = i \\sigma_2 H^*$ and similarly for $\\Phi$.\nHere, the Yukawa couplings, $Y^U,Y^D,G^U,G^D$ are $3\\times 3$ matrices.\nWe assume that $Y^U$ and $Y^D$ are Hermitian matrices and $G^U, G^D$ are proportional to the identity matrix ${\\rm diag}(1,1,1)$ at some high-energy scale $\\Lambda_{UV}$, with constants of proportionality $g^U$ and $g^D$ respectively.\n\nOne of the most important constraints on the model comes from the up and down quark masses. Any term in the potential with an odd number of $\\Phi$ fields (and a corresponding odd number of $H$ fields) will lead to a $\\Phi$ vacuum expectation value and hence a contribution to the quark masses. While such terms may be taken to be absent at tree level in appropriate UV completions (see the next section for an example), they will still be generated at loop level due to the $\\Phi$ and $H$ Yukawa couplings. Indeed, the most important radiative corrections to the Higgs potential will come from top and bottom loops. These lead to contributions\n\\begin{equation}\n{\\cal L}_{mix}=-3\\left(\\frac{g^Dy_b+g^Uy_t}{8\\pi^2}\\right)\\Lambda_{UV}^2\\Phi^\\dagger H +h.c.\n\\end{equation}\nThe quark mass contributions induced by this operator are\n\\begin{eqnarray}\n\\Delta m_d =-3 g^D\\frac{g^Dm_b+g^Um_t}{8\\pi^2}\\frac{\\Lambda_{UV}^2}{m_\\phi^2}\\\\\n\\Delta m_u =-3 g^U\\frac{g^Dm_b+g^Um_t}{8\\pi^2}\\frac{\\Lambda_{UV}^2}{m_\\phi^2},\n\\end{eqnarray}\nwhere $m_\\phi$ is the mass of the CP-even scalar component of $\\Phi$. If we assume that these contributions are less than or equal to the values of the physical quark masses, then we may place the following approximate upper limits on $g^U$ and $g^D$:\n\\begin{eqnarray}\n g^U \\lesssim .007\\left(\\frac{m_\\phi}{300 {\\rm GeV}}\\right)\\left(\\frac{{\\rm TeV}}{\\Lambda_{UV}}\\right) \\;\\;\\;\\; \\\\\n g^D\\lesssim .06\\left(\\frac{m_\\phi}{300 {\\rm GeV}}\\right)\\left(\\frac{{\\rm TeV}}{\\Lambda_{UV}}\\right)\\;\\;\\;\\;\n\\end{eqnarray}\nwith the more severe constraint on $g^U$ due to $m_t\/m_b\\gg 1$. Although no tuning is needed when these constraints are satisfied, they can be relaxed if there is some degree of cancellation between the $\\Phi$ and $H$ contributions to the quark masses.\n\nWe next examine flavor-changing neutral current constraints. There are essentially two types of constraints we must consider. The first type come from loop corrections to FCNC processes which are present even in the strict limit that the $\\Phi$ couplings are proportional to the identity matrix. By construction, these types of corrections have the same structure as in the standard model, with $W^+$ or $H^+$ propagators replaced by $\\phi^+$, and generally obtain similar GIM style suppressions. In addition, we also have FCNC constraints coming from renormalization group (RG) running causing breaking of the perfect ${\\rm diag}(1, 1, 1)$ forms of the $\\Phi$ Yukawa couplings. We consider these in turn. Given the constraints already described above from the $\\Phi$ vev requirement, the most important FCNC process we must consider is $K^0-\\bar K^0$ mixing, and we will generally not discuss FCNC effects induced by the small coupling $g^U$.\n\n\nFor $K^0-\\bar K^0$ mixing, the experimental limits may be summarized as follows:\n In the below analysis, we will find three types of induced operators,\n\\begin{eqnarray}\n{\\cal O}_{LL}= \\bar s_R d_L \\bar s_R d_L \\;\\;\\;\\;\\;\\;\\;\\;\\;\\; & {\\cal O}_{RR}=\\bar s_L d_R \\bar s_L d_R \\;\\;\\;\\;\\;\\;\\;\\;\\;\\; & {\\cal O}_{LR}= \\bar s_R d_L \\bar s_L d_R,\n\\end{eqnarray}\nwhich all contribute to the $K^0-\\bar K^0$ mixing.\nThe current 95\\%\nconfidence level bounds on these operators, expressed in terms of the required mass scale suppressing the real ($\\Lambda_{\\rm Re}$) and imaginary ($\\Lambda_{\\rm Im}$) parts, are \\cite{constraints}\n\\begin{eqnarray}\n{\\cal O}_{LL},{\\cal O}_{RR}: \\;\\;\\;\\;\\;\\;\\;\\; &\\Lambda_{\\rm Re}\\geq 7.3\\times10^3 {\\rm TeV}, &\\Lambda_{\\rm Im}\\geq 10^5 {\\rm TeV}\\label{LLRRbound}\\\\\n{\\cal O}_{LR}: \\;\\;\\;\\;\\;\\;\\;\\; & \\Lambda_{\\rm Re}\\geq 1.7\\times10^4 {\\rm TeV}, &\\Lambda_{\\rm Im}\\geq 2.4 \\times 10^5 {\\rm TeV} \\label{LRbound}\n\\end{eqnarray}\n\nWe now consider the contribution to $K^0-\\bar K^0$ mixing coming from replacing one or both of the $W^\\pm$ lines with $\\phi^\\pm$ in the SM box diagrams. The expressions coming from these diagrams can be found in the Appendix. The box diagrams contribute dominantly to $K^0-\\bar K^0$ mixing through an induced operator of type ${\\cal O}_{LR}$. In Fig. (\\ref{KKMix}), we show the sizes of the mass scales in the real and imaginary parts of this operator in comparison with the experimental constraints, for a coupling $g^D$ of 0.06. \nExamining Fig. (\\ref{KKMix}), we see that the constraint coming from $\\Lambda_{\\rm Im}$ is somewhat similar in severity to that coming from the size of the induced down quark mass.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.47\\columnwidth]{ReKbarMix2.pdf}\\includegraphics[width=0.47\\columnwidth]{ImKbarMix2.pdf}\n\\caption{We plot the mass scales appearing in the real (left) and imaginary (right) parts of the effective operator of type ${\\cal O}_{LR}$ resulting from the $\\phi^+$ box diagrams for a Yukawa coupling $g^D$ of 0.06. The red lines show the constraints. More generally, the values of $\\Lambda_{\\rm Re}$ and $\\Lambda_{\\rm Im}$ scale inversely with the coupling $g^D$. \\label{KKMix} }\n\\end{figure}\n\n\nAs an aside, let us make a quick comment about the constraint coming from $b\\to s\\gamma$ decays. This constraint has been analyzed in \\cite{Mahmoudi:2009zx} for a two Higgs doublet model where the additional Higgs boson couples to the SM fermions diagonally. This analysis can be applied to our scenario. Taking $g^U$ small, we fall safely in the allowed parameter space for $g^D\\lesssim 0.1$.\\footnote{In \\cite{Mahmoudi:2009zx} they state that for $\\lambda_{tt}=0$ ($G^U_{33}$ in our notation, with $\\lambda_{bb} = G^D_{33}$),\n$b\\to s\\gamma$ decays are always safe. This is an artifact of an approximation they make which allows them to neglect the $\\lambda_{bb}^2$ contribution, which is not applicable to our case. However, the $b\\to s\\gamma$ contribution from the charged Higgs is invariant under an exchange $\\lambda_{tt} \\leftrightarrow \\lambda_{bb}$ (with the dominant diagram undergoing a parity transformation). In this way we may extract the limit for our case.}\n\nWe next consider the contribution to $K^0-\\bar K^0$ mixing from RG running breaking the universality of the $\\Phi$ couplings. The dominant effect comes from the wave-function renormalization of the $Q_L$ fields, due to the large top Yukawa coupling to the $H$ doublet.\nCalculating the wave function renormalization of $Q_L$, we find\n\\begin{equation}\n\\beta_{G^D}\\supset \\frac{G^D}{32\\pi^2}\\left(Y^UY^U\\right),\n\\end{equation}\nwhere we have neglected terms that are universal (since we only care about the breaking of universality here) and also smaller terms which are proportional to $Y^D$. We take the $\\Phi$ Yukawa couplings to be universal at a UV scale $\\Lambda_{UV}$, and then RG run down to the weak scale. Without loss of generality, we are free to diagonalize the up quark mass matrix $Y^U$ at the UV scale, before performing the running. Since we already know that we must take $\\Lambda_{UV}$ close to the TeV scale, we analyze the RG corrections using the leading-log approximation. The $G^D$ coupling at the EW scale is then\n\n\\begin{equation}\nG^D(M_{EW})=G^D(\\Lambda_{UV})+\\frac{G^D}{32\\pi^2}\\left(Y^UY^U\\right)\\ln\\left(\\frac{\\Lambda_{UV}}{M_{EW}}\\right).\n\\end{equation}\nAfter running the couplings to the weak scale, we then diagonalize $Y^D$ by redefining $d_{L_i}$ and $d_{R_j}$. Although the corrections to $G^D$ are diagonal in the basis we did the RG running, they are not universal. This non-universality gives family mixing when we rotate to the down quark mass eigenstates\n\n\\begin{equation}\nV^\\dagger G^D(M_{EW})V=G^D(\\Lambda_{UV})+\\frac{G^D}{32\\pi^2}V^{\\dagger}\\left(Y^UY^U\\right)V\\ln\\left(\\frac{\\Lambda_{UV}}{M_{EW}}\\right).\n\\end{equation}\nwhere $V$ is the CKM matrix. We will now show that these radiative corrections are small enough to be less constraining than earlier bounds presented in this section. We neglect all of the present contributions to $K^0-\\bar K^0$ mixing except those due to the top Yukawa coupling. The leading order contribution to the Lagrangian density resulting from scalar exchange is then\n\\begin{eqnarray}\n\\left(\\frac{g^Dy_t^2V^*_{ts}V_{td}}{64\\pi^2} \\ln\\left(\\frac{\\Lambda_{UV}}{M_{EW}}\\right)\\right)^2\\left[\\left(\\frac{1}{m_\\phi^2}-\\frac{1}{m_A^2} \\right)({\\cal O}_{LL}+{\\cal O}_{RR}) + 2\\left(\\frac{1}{m_\\phi^2}+\\frac{1}{m_A^2} \\right){\\cal O}_{LR} \\right],\n\\end{eqnarray}\nwhere $A$ is the CP-odd scalar, with $m_A$ being its mass. The coefficient of the ${\\cal O}_{LL}+{\\cal O}_{RR}$ operator then becomes\n\\begin{eqnarray}\n\\nonumber \\left(\\frac{g^D}{.1}\\right)^2\\left( \\frac{(150 {\\rm GeV})^2}{m_\\phi^2}-\\frac{(150 {\\rm GeV})^2}{m_A^2}\\right) &&\\!\\!\\!\\!\\!\\!\\!\\!\\! \\left(\\left(\\frac{\\ln (\\frac{\\Lambda_{UV}}{M_{EW}})}{3\\times 10^6{\\rm TeV}}\\right)^2-i\\left(\\frac{\\ln(\\frac{\\Lambda_{UV}}{M_{EW}})}{3\\times 10^6{\\rm TeV}}\\right)^2\\right),\n\\end{eqnarray}\nwhile that of the ${\\cal O}_{LR}$ operator is\n\\begin{eqnarray}\n\\nonumber \\left(\\frac{g^D}{.1}\\right)^2\\left( \\frac{(150 {\\rm GeV})^2}{m_\\phi^2}+\\frac{(150 {\\rm GeV})^2}{m_A^2}\\right) &&\\!\\!\\!\\!\\!\\!\\!\\!\\! \\left(\\left(\\frac{\\ln (\\frac{\\Lambda_{UV}}{M_{EW}})}{2\\times 10^6{\\rm TeV}}\\right)^2+i\\left(\\frac{\\ln(\\frac{\\Lambda_{UV}}{M_{EW}})}{2\\times 10^6{\\rm TeV}}\\right)^2\\right).\n\\end{eqnarray}\nThe strongest bounds on the above operators come from their imaginary parts, but by inspection of equations \\ref{LLRRbound} and \\ref{LRbound}, it is clear that they are quite a bit less constraining than the bounds coming from the scalar box diagram, or from the size of the induced down quark mass.\n\n\\section{A Possible UV Completion}\n\nHere we present a UV completion which can realize our scenario in the IR. To get a Hermitian Flavor Violation model, the couplings of the SM Higgs doublet $H$ and the extra doublet $\\Phi$ to the SM quarks must be very different; $H$ obviously needs to have a very hierarchical Yukawa\nmatrix, while $\\Phi$ must have an identity-like matrix. In\naddition, the $H$ Yukawa matrices are both required to be Hermitian.\n\nThe option we will consider here is to introduce a $Z_2$ symmetry to distinguish the two Higgs bosons, in addition to the $SU(3)_f$ flavor symmetry, with $Q_L$, $u_R$, and $d_R$ all taken as triplets under $SU(3)_f$ and even under $Z_2$. We\nassume $\\Phi$ is even while $H$ is odd under $Z_2$. In the symmetric\nlimit, $\\Phi$ has Yukawa matrices with both the up- and down-sectors\nproportional to the identity matrix, while $H$ does not have any\nYukawa couplings. The Yukawa matrices $Y^U$ and $Y^D$ need to be\ngenerated by exchange of heavy particles picking up\nsymmetry-breaking spurions. We assume that the $SU(3)_f$ symmetry is\nbroken by vevs of triplet spurions $v_1$, $v_2$, and $v_3$, while the\n$Z_2$ symmetry by a spurion $\\sigma$. CP is spontaneously broken in the\ntriplet VEVs, with the vev of $\\sigma$ assumed real. In order to communicate these symmetry breakings to the standard model sector, we introduce a set of Dirac fermions ${\\cal U}$,\n${\\cal U}'_i$, ${\\cal U}''_i$, ${\\cal D}$, ${\\cal D}'_i$, ${\\cal D}''_i$, where ${\\cal U}$ and ${\\cal D}$ are flavor triplets, and where fields of type $i$, including $v_i$, are charged under separate $U(1)_i$ abelian symmetries.\\footnote{We assume for simplicity that the flavor symmetries are gauged so that we don't have to worry about any light Goldstone modes, or possible Planck suppressed breaking effects.} The ${\\cal U}''_i$ and ${\\cal D}''_i$ fields are assumed even under the $Z_2$ symmetry, with all other new heavy quarks being odd. We will label the left and right handed components of these Dirac fermions with $L$ and $R$ subscripts, as usual. The full set of charges of the new fields are shown in Table 1\n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{|c||c|c|c|c|}\n \\hline\n Field & $SU(3)_f$ & $U(1)^3$ & ${\\mathbb Z}_2$ & $SU(2)_W \\times\n U(1)_Y $ \\\\ \\hline\n ${\\cal U}$ & 3 & 0 & $-$ & $1_{2\/3}$ \\\\\n ${\\cal U}'$ & 1 & 3 & $-$ & $1_{2\/3}$ \\\\\n ${\\cal U}''$ & 1 & 3 & $+$ & $1_{2\/3}$ \\\\\n ${\\cal D}$ & 3 & 0 & $-$ & $1_{-1\/3}$ \\\\\n ${\\cal D}'$ & 1 & 3 & $-$ & $1_{-1\/3}$ \\\\\n ${\\cal D}''$ & 1 & 3 & $+$ & $1_{-1\/3}$ \\\\ \\hline\n $H$ & 1 & 0 & $-$ & $2_{1\/2}$ \\\\\n $\\Phi$ & 1 & 0 & $+$ & $2_{1\/2}$ \\\\\n $\\sigma$ & 1 & 0 & $-$ & $1_0$ \\\\\n $v$ & 3 & 3 & $+$ & $1_0$ \\\\ \\hline\n \\end{tabular}\n\\label{tab:charges}\n\\caption{Charges of fields in the example UV completion. Here a $U(1)^3$ charge of ``3\" means that there are three such fields with separate $U(1)$ charges of the form $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. }\n\\end{table}\n\nThe following set of interactions are permitted by the symmetries, and will be used to generate the appropriate $H$ Yukawa structure (showing only the up sector for brevity):\\footnote{Allowed heavy quark interactions with flipped chiralities may also be included and do not pose any difficulty.}\n\\begin{equation}\n {\\cal O}_1 = \\tilde{H} \\bar{Q}_L {\\cal U}_R, \\quad\n {\\cal O}_2 = \\bar{{\\cal U}}_L v_i {\\cal U}'_{R i}, \\quad\n\n {\\cal O}_3 = \\sigma \\bar{\\cal U}'_{L i} {\\cal U}''_{R i}, \\quad\n {\\cal O}_4 = \\bar{\\cal U}''_{L i} v^\\dagger_i u_R.\n\\end{equation}\nThe resulting $H$ Yukawa couplings then have the form\n\\begin{equation}\n Y_u = \\sum_i \\frac{\\langle v_i\\rangle \\langle \\sigma\\rangle \\langle\n v_i\\rangle^\\dagger}{M_{{\\cal U}} M_{{\\cal\n U}'_i} M_{{\\cal U}''_i}}\\ , \\qquad\n Y_d = \\sum_i \\frac{\\langle v_i\\rangle \\langle \\sigma\\rangle \\langle\n v_i\\rangle^\\dagger}{M_{{\\cal D}} M_{{\\cal\n D}'_i} M_{{\\cal D}''_i}}\\ .\n\\end{equation}\nNote that there is no contribution that mixes up different $i$'s thanks to the $U(1)_i$ flavor symmetries. This is crucial to ensure the Hermiticity of the Yukawa matrices.\nWithout loss of generality, we may make $SU(3)_f$ flavor rotations to put the $v$ vevs in form $v_1=(a,b,c)$, $v_2 =(d, e, 0)$, and $v_3 =(f, 0, 0)$. Assuming an inverse hierarchy among the heavy fermion masses, we may then\nobtain both Hermitian and hierarchical Yukawa matrices. In this construction we may take the $\\sigma$, $Z_2$ breaking vev to be of order TeV, along with one or more of the heavy quark masses, providing the effective $\\Lambda_{UV}$ cutoff on the dangerous loop diagrams discussed in section 2. The schematic form\nof the Yukawa matrices likely from this construction is\n\\begin{equation}\n Y^U \\approx \\left(\n \\begin{array}{ccc}\n \\lambda^8 & \\lambda^8 & \\lambda^8 \\\\\n \\lambda^8 & \\lambda^4 & \\lambda^4 \\\\\n \\lambda^8 & \\lambda^4 & 1\n \\end{array} \\right), \\qquad\n Y^D \\approx y_b \\left(\n \\begin{array}{ccc}\n \\lambda^3 & \\lambda^3 & \\lambda^3 \\\\\n \\lambda^3 & \\lambda^2 & \\lambda^2 \\\\\n \\lambda^3 & \\lambda^2 & 1\n \\end{array} \\right),\n\\end{equation}\nwith $y_b$ being roughly the ratio of the bottom and top masses.\n\nThere are a few operators which are allowed by all of the symmetries of Table 1, but which are nevertheless dangerous to our construction. These are\n\\begin{equation}\n \\bar{Q}_L v^i i{\\not\\!\\! D} v_i^\\dagger Q_L, \\qquad\n \\sigma H^\\dagger \\Phi, \\qquad\n \\sigma \\tilde{H} \\bar{Q}_L u_R, \\qquad\n \\sigma \\tilde{H} \\bar{Q}_L v_i v_i^\\dagger v_j v_j^\\dagger u_R.\n\\end{equation}\nThe first one leads to a non-universal Yukawa coupling to $\\Phi$ and\nhence flavor-changing neutral currents; the second one induces a\nvacuum expectation value for $\\Phi$, too-large fermion masses and\nhence fine-tuning; the third leads directly to too-large fermion masses; the last one destroys the Hermiticity of the\nYukawa matrix if $i \\neq j$. Taking sufficiently suppressed coefficients for these operators is technically natural, so long as the $v_i$ vevs and masses are taken to be at least a few orders of magnitude larger than the TeV scale, with corresponding small coefficients for the ${\\cal O}_2$ and ${\\cal O}_4$ operators. In general, the masses of the various fields, and coefficients of operators in this construction are somewhat flexible, and we will not discuss them in further detail.\n\nIt is a straightforward exercise to check that this model leads to no strong CP parameter at tree level by considering the phase of the determinant of the full quark mass matrix. At loop level, a highly suppressed contribution might arise after taking into account radiatively induced breaking of Hermiticity\/universality, as well as the small induced $\\Phi$ vev. A full calculation of such loop corrections in this particular UV completion is however beyond the scope of the present work.\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\\section{$Wjj$ events at the Tevatron}\n\nThe CDF collaboration recently reported on the production of $Wjj$\nwith an integrated luminosity of $4.3 {\\rm ~fb}^{-1}$ \\cite{cdf}.\nInvestigating the invariant mass distribution of the jet pair, they\nfound an excess of 253 events ($156\\pm 42$ electrons, $97\\pm 38$ muons)\nin the $120-160$ GeV range, which is well fit by a Gaussian peak\ncentered at $144\\pm 5$ GeV. Additionally, it has been reported that analysis of an additional $3{\\rm~fb}^{-1}$ sample collected by CDF shows this same feature,\ngiving a total significance of $4.1\\sigma$ for the combined $7.3{\\rm~fb}^{-1}$ data set \\cite{cdfnote}.\n\nHermitian Flavor violation provides a perfect setting for explaining the $Wjj$ anomaly\nwith a new $SU(2)$ doublet scalar.\\footnote{Several explanations for the $Wjj$ anomaly have been presented in the literature, including other $SU(2)$ doublet scalars \\cite{Wang, Babu, Dutta:2011kg}, a $Z$-prime \\cite{Hooper, Cheung}, new colored states \\cite{XPWang,Dobrescu,Carpenter:2011yj}, in supersymmetry \\cite{Kilic, Sato}, technicolor \\cite{Lane}, or string theory \\cite{stringy}, and within the context of the Standard model \\cite{He, Sullivan:2011hu,Plehn:2011nx}.} Indeed, what seems to be required is a large coupling of the scalar to the first generation quarks. However, as noted in the introduction, such couplings usually go hand in hand with\nexcessive flavor changing neutral currents. Hermitian flavor violation addresses this problem.\n\nProducing $Wjj$ events at the Tevatron through a new Higgs doublet can proceed via two primary mechanisms. The first\nis to simply have $t$-channel production of a $W$-boson along with the new scalar, followed by decay of the scalar to two jets. This scenario, however, typically requires large couplings which run into difficulty with the constraints discussed in section 2 as well as collider constraints, and we will not discuss it further here. The second option is to split the masses of the charged and neutral Higgs components, and consider resonant production of the heavier state. This state will then primarily decay into a $W$-boson plus the lighter state, with the lighter state then decaying to jets, as in Fig. (2) (in this case there will also be a small additional $t$-channel contribution).\nThe CDF collaboration's background-subtracted invariant mass distribution of the $l\\nu jj$ system,\\footnote{The neutrino momentum may be reconstructed up to a two-fold ambiguity using the on-shell mass condition for the $W$.} $M_{l\\nu jj}$, shows a peak in the $250\\sim 300{\\rm~GeV}$ range \\cite{cdfnote}, which one would expect from a heavy resonance at around this mass range.\nIn general, we may consider some flexibility in the permitted scalar spectrum, pertaining to\nthe choice of which state is producing the final state jets, as well as whether or not the scalar and pseudo-scalar components are split from one another. Hereafter we refer to the field(s) with mass $\\sim 150{\\rm~GeV}$ contributing to the excess as $\\Phi_l$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Wjj_resonant}\n\\caption{Production of $W$ boson and two jets via resonant production. \\label{fig:diagrams}}\n\\end{figure}\n\nSince the $\\Phi$ coupling to the $u_R$ quarks is relatively constrained by the requirement of not over-contributing to the up quark mass, as discussed in section 2, we will focus on the case where $\\Phi$ couples dominantly to the $d_R$ quarks.\\footnote{Though we focus on the case where $g^D$ is the dominant coupling, the results for coupling size in table 2 are essentially unchanged for the case of dominant $g^U$, with the exception of case 5, where the necessary initial partons required to produce a neutral resonance differ. The $\\sigma(Z\\Phi_l)$, $\\sigma(\\gamma\\Phi_l)$, and $\\sigma(jj)$ entries in table 2, however, differ for the dominant $g^U$ case (due to different initial partons).} Since the proton contains twice as many up quarks as down quarks, the best case for our model is to take $\\phi^+$ to be the heavier, resonantly produced state. At least one of the neutral components must then have a mass of $\\sim 150$GeV in order to explain the CDF excess.\nTo get an idea of the size of the coupling needed to account for the $Wjj$ excess, we generate $p\\bar{p}\\to\\Phi_l W^\\pm\\to l\\nu jj$ events with Madgraph\/MadEvent \\cite{Alwall:2007st}, which are then showered with Pythia \\cite{Sjostrand:2006za}, with detector simulation by PGS \\cite{pgs} using CDF parameters. We implement the cuts described in \\cite{cdf}, and require a total of $\\sim 250$ events to pass with a luminosity of $4.3{\\rm~fb}^{-1}$. We find good agreement between a $WW+WZ$ background created in this manner and the distribution in \\cite{cdf}, suggesting that this provides a reasonable estimate. The results for several scenarios are shown in table 2.\nFor a variety of mass spectra, we see that the size of the required $\\Phi$ Yukawa coupling is around $g^D \\sim .06$.\\footnote{This finding is consistent with the results of \\cite{Wang}, who considered a phenomenologically similar model.} As shown in section 2, a coupling of this size can evade all flavor constraints, as well as the constraint from the induced down quark mass, although the masses of $m_\\phi$ and $m_\\phi^+$ are preferred to be somewhat heavy, $ \\mathop{}_{\\textstyle \\sim}^{\\textstyle >} 300 {\\rm GeV}$. These constraints might be relaxed somewhat after taking into account QCD uncertainties, or if we didn't require producing the central value of the CDF excess. In particular, if we were to require producing only one standard deviation below the central value of the excess, then cases which previously required a coupling of .06 would instead require couplings of about .05.\\footnote{The constraints from flavor could also be weakened if one were to adopt case ``5u\" from Table 2. In that case, $\\Phi$ couples dominantly to the up sector, and fine tuning is required in order to keep the up quark mass small. Such a fine tuning might be considered acceptable depending on one's perspective on the origin of the fermion mass hierarchy. }\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c||c|c||c|c|c|c|}\n\\hline\n{} & $m_{\\phi}$, $m_{A}$, $m_{\\phi^\\pm}$ (GeV) & $g^D$ &$\\sigma(W^\\pm \\Phi_l)$ & $\\sigma(Z \\Phi_l)$ & $\\sigma(\\gamma \\Phi_l)$ & UA2 $\\sigma(jj)$\\\\ \\hline\n1 & 150 , 150 , 250 & 0.075 &4.1 pb & .032 pb & .008 pb& 2.0 pb\\\\ \\hline\n2 & 150 , 150 , 300 & 0.06 &1.7 pb & .020 pb & .005 pb& 1.3 pb\\\\ \\hline\n3 & 300 , 150 , 300 & 0.06 &1.6 pb& .430 pb & .003 pb& 0.6 pb\\\\ \\hline\n4 & 230 , 150 , 300 & 0.06 &1.7 pb& .016 pb & .003 pb& 0.6 pb\\\\ \\hline\n5d & 300 , 300 , 150 & 0.08 &1.6 pb& .028 pb & .016 pb& 5.2 pb\\\\ \\hline\n5u & 300 , 300 , 150 & (0.04) &1.5 pb& .008 pb & .004 pb& 1.3 pb\\\\ \\hline\n\\end{tabular}\n\\label{tab:couplings}\n\\caption{Size of Yukawa couplings which explain the CDF $Wjj$ excess. The cross sections $\\sigma(W^\\pm \\Phi_l)$, $\\sigma(Z \\Phi_l)$, and $\\sigma(\\gamma \\Phi_l)$ are calculated at Tevatron energy, with no cuts apart from requiring $p_T>30$ for the photon. $\\Phi_l$ refers to all fields with masses of $150{\\rm~GeV}$. $\\sigma(jj)$ refers to the dijet cross section for the process $p\\bar{p}\\to\\Phi_l\\to jj$ at $\\sqrt{s}=630{\\rm~GeV}$ and should be compared with the limit of ${\\mathcal O}(100{\\rm~GeV})$ \\cite{Alitti:1993pn}. The parentheses for model 5u indicate the value for the coupling $g^U$ rather than $g^D$.}\n\\end{center}\n\\end{table}\n\nWhile table 2 presents only a few benchmark points, the behavior suggests that various mass spectra could in principle be able to explain the $Wjj$ excess. There is some small variation in the required coupling with changes to the mass of the heavy resonance, as seen by comparison of scenarios 1 and 2. This reflects both a larger branching ratio $BR(\\phi^\\pm\\to W^\\pm\\phi^0)$ as well as a greater acceptance of events for the heavier resonance. \n\nIf the scalar and pseudo-scalar masses are split, then the required coupling may change slightly, but not significantly, compared with the degenerate case. As an example, consider taking the CP-even scalar component heavy, so that it is no longer within kinematic reach of the $\\phi^+$ decays. In that case, the size of the required $g^D$ coupling will remain essentially unchanged, at $\\sim .06$, as seen by comparison of scenarios 2, 3, and 4 in table 2. This follows because the width of the $\\phi^+$ resonance, $\\Gamma$, is cut in half.\\footnote{For these scenarios, $\\phi^\\pm$ decays dominantly to $W^\\pm\\phi^0$, with $BR(\\phi^\\pm\\to W^\\pm\\phi^0)\\approx 96\\%$ for scenario 2.} Indeed, in the tree level production diagram, we obtain an increased resonant enhancement from a $1\/\\Gamma^2$ in the propagator, yielding a factor of 4. There are half as many final states for the $\\phi^+$ decay, yielding a suppression by a factor of 2. Finally, due to the smaller width, there is half as much phase space volume for the initial quarks which can successfully hit the resonance. Taking into account the fact that the kinematics of the produced $Wjj$ events are unchanged from the degenerate case, and multiplying these factors together, we see that the overall event rate is essentially unchanged.\n\nAside from FCNC considerations, there are also direct collider constraints on two Higgs doublet models.\nOne might expect evidence of our additional Higgs sector in $\\gamma jj$ and $Zjj$ events. However, note that with resonant production, such events are quite suppressed, as shown in table 2, since the $\\gamma$ and $Z$ cannot be produced in $\\phi^+$ decays. Scenario 3 has the largest $Zjj$ cross section because the CP-even scalar is heavy enough for the resonant process $d\\bar{d}\\to \\phi\\to A+Z$. In contrast, the mass of $\\phi$ in scenario 4 lies below the threshold for decay to $A+Z$, so it does not receive such an enhancement.\n\n\nAdditionally, a new scalar with\na coupling to first generation quarks could be produced as an $s$-channel\nresonance and appear in dijet searches. Because of the large QCD\nbackgrounds, Tevatron dijet bounds are only significant for resonances\nheavier than those we consider here \\cite{Aaltonen:2008dn}. However,\nthe lower energy $p\\bar{p}$ collisions ($\\sqrt{s}=630 {\\rm ~GeV}$)\nobserved by the UA2 collaboration provide an opportunity for\nconstraining ${\\cal O}(100{\\rm ~GeV})$ dijet resonances. A search for\n$W_R'$ resonances using a $10.9 {\\rm~pb}^{-1}$ data sample places\nconstraints of ${\\mathcal O}(100\\rm~pb)$ for $\\sigma\\times BR(W'\\to jj)$ at\nthe 90\\% confidence level for a mass of $\\sim 150{\\rm\n ~GeV}$ \\cite{Alitti:1993pn}. Although we are considering a scalar\nresonance, they provide a guideline for our extended Higgs\nsector. Our scenarios are very safe from this bound, as shown in table 2.\n\n\\section{Discussions and Conclusions}\n\nIn this paper, we presented a novel type of two-doublet Higgs model\nthat allows for new $O(.1)$ Yukawa couplings to the light generations while\nnaturally suppressing FCNCs via a GIM-like mechanism. We also\ndiscussed phenomenological consequences at colliders. Thanks\nto the allowed large couplings of the up- and down-quarks to the extra\ndoublet, the production of the doublet states can be significant. In\nparticular, the bump in the $W+jj$ mass distribution reported by the\nCDF collaboration may be explained straightforwardly in this setup, while remaining consistent with phenomenological constraints.\nIn addition, Hermiticity of the Yukawa couplings in our scenario suggests\na possible solution to the strong CP problem through spontaneous CP breaking.\n\n\n\\section*{Acknowledgements}\n\nH.M. was supported in part by the U.S. DOE under Contract\nDE-AC03-76SF00098, in part by the NSF under grant PHY-04-57315, and in\npart by the Grant in-Aid for scientific research (C) 23540289 from\nJapan Society for Promotion of Science (JSPS). The work of T.T.Y was supported by JSPS Grant-in-Aid for Scientific Research (A) (22244021). This work was also\nsupported by the World Premier International Center Initiative (WPI\nProgram), MEXT, Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nA possible candidate for an extra hot thermal relic component is the\naxion particle produced thermally in the early universe. Axions\ntherefore can contribute to the hot dark matter component together\nwith the standard relic neutrino background. \nAxions may be produced in the early universe via thermal or non\nthermal processes, and arise as the solution to solve the strong CP\nproblem~\\cite{PecceiQuinn,Weinberg:1977ma,Wilczek:1977pj}. Axions are the Pseudo-Nambu-Goldstone\nbosons of a new global $U(1)_{PQ}$ (Peccei-Quinn) symmetry that is \nspontaneously broken at an energy scale $f_a$.\n The axion mass is given by\n\\begin{equation}\nm_a = \\frac{f_\\pi m_\\pi}{ f_a } \\frac{\\sqrt{R}}{1 + R}=\n0.6\\ {\\rm eV}\\ \\frac{10^7\\, {\\rm GeV}}{f_a}~,\n\\end{equation}\nwhere $f_a$ is the axion coupling constant, $R=0.553 \\pm 0.043 $ is the up-to-down quark masses\nratio and $f_\\pi = 93$ MeV is the pion decay constant. Non-thermal axions, as those produced by the misalignment mechanism, \nwhile being a negligible hot dark matter candidate, may constitute a fraction or the total cold dark matter component of the universe.\nWe do not explore such a possibility here. \nThermal axions will affect the cosmological observables in a very\nsimilar way to that induced by the presence of neutrino masses and\/or\nextra sterile neutrino species. \nMassive thermal axions as hot relics affect large scale structure,\nsince they will only cluster at scales larger than\ntheir free-streaming scale when they become non-relativistic,\nsuppressing therefore structure formation at small scales (large\nwavenumbers $k$). \nConcerning Cosmic Microwave Background (CMB) physics, \nan axion mass will also lead to a signature in the CMB photon\ntemperature anisotropies via the early integrated Sachs-Wolfe effect.\nIn addition, extra light species as thermal axions\nwill contribute to the dark radiation content of the universe, or, in\nother words, will lead to an increase of the effective number of relativistic degrees of freedom $N_{\\textrm{eff}}$, defined via\n\\begin{equation}\n \\rho_{rad} = \\left[1 + \\frac{7}{8} \\left(\\frac{4}{11}\\right)^{4\/3}N_{\\textrm{eff}}\\right]\\rho_{\\gamma} \\, ,\n\\end{equation}\nwhere $\\rho_{\\gamma}$ refers to the present photon energy density. \nIn the standard cosmological model in which a thermal axion content is\nabsent, the three active neutrino contribution leads to the canonical\nvalue of $N_{\\textrm{eff}}=3.046$ \\cite{Mangano:2005cc}. The extra contribution\nto $N_{\\textrm{eff}}$ arising from thermal axions can modify both the\nCMB anisotropies (via Silk damping) and\nthe light element primordial abundances predicted by Big Bang\nNucleosynthesis. \nThe former cosmological signatures of thermal axions have been extensively\nexploited in the literature to derive bounds on the thermal axion\nmass, see Refs.~\\cite{Melchiorri:2007cd,Hannestad:2007dd,Hannestad:2008js,Hannestad:2010yi,Archidiacono:2013cha,Giusarma:2014zza}.\n\nHowever, all the cosmological axion mass limits to date have assumed the usual simple power-law\ndescription for the primordial perturbations. The aim of this paper\nis to constrain the mass of the thermal axion using a non-parametric\ndescription of the Primordial Power Spectrum (PPS hereinafter) of the\nscalar perturbations, as introduced in Ref.~\\cite{Gariazzo:2014dla}.\nWhile in the simplest models of\ninflation~\\cite{\nGuth:1980zm,Linde:1981mu,Starobinsky:1982ee,Hawking:1982cz,Albrecht:1982wi,Mukhanov:1990me,Mukhanov:1981xt,Lucchin:1984yf, Lyth:1998xn,Bassett:2005xm,Baumann:2008bn}\nthe PPS has a scale-free power-law form, the PPS could be more complicated, presenting various features or a scale dependence.\nSeveral methods have been proposed in the literature to reconstruct\nthe shape of the PPS (see the recent work of Ref.~\\cite{Ade:2015lrj}). It has been\nshown~\\cite{Hunt:2013bha,Hazra:2014jwa} that there are small hints for deviations from the power-law form,\neven when using different methods and different data sets.\n\n\nThe energy scales at which the PPS was produced during inflation\ncan not be directly tested. We can only infer the PPS by measuring the\ncurrent matter power spectrum in the galaxy distribution and the power\nspectrum of the CMB fluctuations. The latter one, measured with exquisite \nprecision by the Planck experiment \\cite{Planck:2015xua,Ade:2013zuv,Ade:2013ktc}, \nis the convolution of the PPS with the transfer function. \nTherefore, in order to reconstruct the PPS, the assumption of an underlying cosmological\nmodel is a mandatory first step in order to compute the transfer function.\n\nHere we rather exploit a non-standard PPS approach, which can allow for a good fit to\nexperimental data even in models that deviates from the standard\ncosmological picture. In particular, we consider a thermal axion\nscenario, allowing the PPS to assume a more general shape than the\nusual power-law description. This will allow us to test the robustness of the cosmological thermal axion mass bounds (see Ref.~\\cite{Giusarma:2014zza} for a recent standard thermal axion analysis), as first performed in Ref.~\\cite{dePutter:2014hza} for the neutrino mass case. \n\nThe structure of the paper is as follows. Section \\ref{sec:method}\ndescribes the PPS modeling used in this study, as well as the\ndescription of the thermal axion model explored here and the\ncosmological data sets exploited to constrain such a model. In\nSec. \\ref{sec:results} we present and discuss the results arising from\nour bayesian analysis, performed through the Monte Carlo Markov Chains (MCMC) package \\texttt{CosmoMC} \\cite{Lewis:2002ah}, \nwhile the calculation of the theoretical observables is done through the Boltzman equations solver \n\\texttt{CAMB} (Code for Anisotropies in the Microwave Background) \\cite{Lewis:1999bs}.\nWe draw our conclusions in Sec.~\\ref{sec:concl}.\n\n\n\n\\section{Method}\n\\label{sec:method}\nIn this section we focus on the tools used in the numerical analyses performed here.\nSubsection~\\ref{sub:pps} describes the alternative model for the PPS\nof scalar perturbations used for the analyses here (see also Ref.~\\cite{Gariazzo:2014dla}),\nwhile in Subsection~\\ref{sub:model} we introduce the cosmological\nmodel and the thermal axion treatment followed in this study. Finally,\nwe shall present in Subsection~\\ref{sub:data}\n the cosmological data sets used in the MCMC analyses.\n\n\\subsection{Primordial Power Spectrum Model}\n\\label{sub:pps}\nThe primordial fluctuations in scalar and tensor modes are generated during the inflationary phase in the early universe.\nThe simplest models of inflation predict a power-law form for the PPS of scalar and tensor perturbations \n(see e.g. ~\\cite{Guth:1980zm,Linde:1981mu,Starobinsky:1982ee,Hawking:1982cz,Albrecht:1982wi,Mukhanov:1990me,Mukhanov:1981xt,Lucchin:1984yf,Lyth:1998xn,Bassett:2005xm,Baumann:2008bn}\nand references therein),\nbut in principle inflation can be generated by more complicated mechanisms, thus giving a different shape for the PPS \n(see Refs.~\\cite{Martin:2014vha,Kitazawa:2014dya} and references therein). In order to study how the cosmological constraints on the parameters change in more general inflationary scenarios, we assume a non-parametric form for the PPS.\n\nAmong the large number of possibilities, we decided to describe the\nPPS of scalar perturbations using a function to interpolate the PPS\nvalues in a series of nodes at fixed position. The interpolating\nfunction we used is named ``piecewise cubic Hermite interpolating\npolynomial'' (\\texttt{PCHIP}\\xspace) \\cite{Fritsch:1980} and it is a modified spline\nfunction, defined to preserve the original monotonicity of the point series that is interpolated.\nWe use a modified version of the original \\texttt{PCHIP}\\xspace algorithm \\cite{Fritsch:1984}, detailed in Appendix~A of Ref.~\\cite{Gariazzo:2014dla}.\n\nTo describe the scalar PPS with the \\texttt{PCHIP}\\xspace model, we only need to give the values of the PPS in a discrete number of nodes and to interpolate among them.\nWe use 12 nodes\nwhich span a wide range of $k$-values:\n\\begin{align}\nk_1 &= 5\\e{-6} \\, \\text{Mpc}^{-1} , \\nonumber\\\\\nk_2 &= 10^{-3} \\, \\text{Mpc}^{-1} , \\nonumber\\\\\nk_j &= k_2 (k_{11}\/k_2)^{(j-2)\/9} \\quad \\text{for} \\quad j\\in[3,10] , \\nonumber\\\\\nk_{11} &= 0.35 \\, \\text{Mpc}^{-1} , \\nonumber\\\\\nk_{12} &= 10\\, \\text{Mpc}^{-1} .\n\\label{eq:nodesspacing}\n\\end{align}\nWe choose equally spaced nodes in the logarithmic scale in the range $(k_2, k_{11})$, that is well constrained from the data \\cite{dePutter:2014hza},\nwhile the first and the last nodes are useful to allow for a non-constant behaviour of the PPS outside the well-constrained range.\n\nThe \\texttt{PCHIP}\\xspace PPS is described by\n\\begin{equation}\nP_{s}(k)=P_0 \\times \\texttt{PCHIP}\\xspace(k; P_{s,1}, \\ldots, P_{s,12})\n,\n\\label{eq:pchip}\n\\end{equation}\nwhere $P_{s,j}$ is the value of the PPS at the node $k_j$ divided by $P_0=2.36\\e{-9}$ \\cite{Larson:2010gs}.\n\\subsection{Cosmological and Axion Model}\n\\label{sub:model}\n\nThe baseline scenario we consider here is the $\\Lambda$CDM model, extended with hot thermal relics (the axions), together\nwith the PPS approach outlined in the previous section. For the numerical analyses \nwe use the following set of parameters, for which we assume flat priors in the intervals listed in Tab.~\\ref{tab:priors}:\n\\begin{equation}\\label{parameterPPS}\n\\{\\omega_b,\\omega_c, \\Theta_s, \\tau, m_a, \\sum m_\\nu, P_{s,1}, \\ldots, P_{s,12}\\}~,\n\\end{equation}\nwhere $\\omega_b\\equiv\\Omega_bh^{2}$ and $\\omega_c\\equiv\\Omega_ch^{2}$ \nare, respectively, the physical baryon and cold dark matter energy densities,\n$\\Theta_{s}$ is the ratio between the sound horizon and the angular\ndiameter distance at decoupling, $\\tau$ is the reionization optical depth, $m_a$ and $\\sum m_\\nu$ are the axion and the sum of three active neutrino masses (both in eV) and $P_{s,1}, \\ldots,\nP_{s,12}$ are the parameters of the \\texttt{PCHIP}\\xspace PPS. We shall also consider a scenario in which massive neutrinos are also present, to explore the expected degeneracy between the sum of the neutrino masses and the thermal axion mass~\\cite{Giusarma:2014zza}. \n\nIn order to compare the results obtained with the \\texttt{PCHIP}\\xspace PPS to the results obtained with the usual power-law PPS model, we describe the latter case with the following set of parameters:\n\\begin{equation}\\label{parameterPL}\n\\{\\omega_b,\\omega_c, \\Theta_s, \\tau, m_a, n_s, \\log[10^{10}A_{s}]\\}~,\n\\end{equation}\nwhere $n_s$ is the scalar spectral index, $A_{s}$ the amplitude of the primordial spectrum\nand the other parameters are the same ones described above. The case of several hot thermal relics for the standard scenario will not be carried out here, as it has been done in the past by several authors (see e.g.~\\cite{Giusarma:2014zza}).\nThe flat priors we use are listed in Tab.~\\ref{tab:priors}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{c|c}\nParameter & Prior\\\\\n\\hline\n$\\Omega_{\\rm b} h^2$ & $[0.005,0.1]$\\\\\n$\\Omega_{\\rm cdm} h^2$ & $[0.001,0.99]$\\\\\n$\\Theta_{\\rm s}$ & $[0.5,10]$\\\\\n$\\tau$ & $[0.01,0.8]$\\\\\n$m_a$ (eV) & $[0.1,3]$\\\\\n$\\sum m_\\nu$ (eV) & $[0.06,3]$\\\\\n$P_{s,1}, \\ldots, P_{s,12}$ & $[0.01,10]$\\\\\n$n_s$ & $[0.9, 1.1]$\\\\\n$\\log[10^{10}A_{s}]$ & $[2.7,4]$\\\\\n\\end{tabular}\n\\end{center}\n\\caption{\nPriors for the parameters used in the MCMC analyses.\n}\n\\label{tab:priors}\n\\end{table}\n\n\n\\begin{figure*}[!t]\n\\includegraphics[width=15cm]{ma_tot_loglog.pdf}\n\\caption{The left upper panel shows the temperature of decoupling as a function of the axion mass (solid curve), as well as the Big Bang Nucleosynthesis temperature, $T_{\\textrm{BBN}}\\simeq 1$~MeV (dashed curve). The right upper panel shows the axion contribution to the extra dark radiation content of the universe, while the bottom right plot depicts the free-streaming scale of an axion (solid curve) or a neutrino (dashed curve) versus the axion\/neutrino mass, in eV. The left bottom panel shows the current axion mass-energy density as a function of the axion mass.}\n\\label{fig:maref}\n\\end{figure*}\n\n\nConcerning the contribution of the axion mass-energy density to\nthe universe's expansion rate, we briefly summarize our treatment in the following.\nAxions decoupled in the early universe at a temperature $T_D$ given by\nthe usual freeze out condition for a thermal relic:\n\\begin{eqnarray}\n\\Gamma (T_D) = H (T_D)~,\n\\label{eq:decouplinga}\n\\end{eqnarray} \nwhere the thermally averaged interaction rate $\\Gamma$ refers to the $\\pi + \\pi \\rightarrow \\pi\n+a$ process:\n\\begin{eqnarray}\n\\Gamma = \\frac{3}{1024\\pi^5}\\frac{1}{f_a^2f_{\\pi}^2}C_{a\\pi}^2 I_a~,\n\\end{eqnarray}\nwith $C_{a\\pi} = \\frac{1-R}{3(1+R)}$ representing the axion-pion coupling constant and the integral $I_a$ reads as follows\n\\begin{eqnarray}\nI_a &=&n_a^{-1}T^8\\int dx_1dx_2\\frac{x_1^2x_2^2}{y_1y_2}\nf(y_1)f(y_2) \\nonumber \\\\\n&\\times&\\int^{1}_{-1}\nd\\omega\\frac{(s-m_{\\pi}^2)^3(5s-2m_{\\pi}^2)}{s^2T^4}~,\n\\end{eqnarray}\nin which $n_a=(\\zeta_{3}\/\\pi^2) T^3$ refers to the number density for axions in thermal equilibrium. The function $f(y)=1\/(e^y-1)$ is the pion thermal distribution and there are three different kinematical variables ($x_i=|\\vec{p}_i|\/T$, $y_i=E_i\/T$ ($i=1,2$) and $s=2(m_{\\pi}^2+T^2(y_1y_2-x_1x_2\\omega))$). \nThe freeze out equation above, Eq.~(\\ref{eq:decouplinga}), can be numerically solved, obtaining the axion decoupling temperature\n$T_D$ as a function of the axion mass $m_a$. Figure~\\ref{fig:maref} shows, in the left upper panel, the axion decoupling temperature as a function of the axion mass, in eV units. Notice that, the higher the axion mass, the lower the temperature of decoupling is. From the axion decoupling temperature it is possible to infer the present axion number density,\nrelated to the current photon density $n_\\gamma$ by \n\\begin{eqnarray}\nn_a=\\frac{g_{\\star S}(T_0)}{g_{\\star S}(T_D)} \\times \\frac{n_\\gamma}{2}~, \n\\label{eq:numberdens}\n\\end{eqnarray} \nwhere $g_{\\star S}$ represents the number of \\emph{entropic} degrees of\nfreedom, with $g_{\\star S}(T_0) = 3.91$. The contribution of the relic axion to the total mass-energy density of the universe will be given by the product of the axion mass times the axion number density. The quantity $\\Omega_a h^2$ at the present epoch is depicted in the bottom left panel of Fig.~\\ref{fig:maref}. Notice that, currently, a $1$~eV axion will give rise to $\\Omega_a h^2\\simeq 0.005$, while a neutrino of the same mass will contribute to the total mass-energy density of the universe with $\\Omega_\\nu h^2\\simeq 0.01$. Notice however that $\\Omega_a h^2$ represents the contribution from relic, thermal axion states. Non-thermal processes, as the misalignment production, could also produce a non-thermal axion population which we do not consider here, see the work of \\cite{DiValentino:2014zna} for the most recent cosmological constraints on such scenario. As previously stated, the\npresence of a thermal axion will also imply an extra radiation component at the BBN period:\n\\begin{equation}\n\\Delta N_{\\textrm{eff}} =\\frac{ 4}{7}\\left(\\frac{3}{2}\\frac{n_a}{n_\\nu}\\right)^{4\/3}~,\n\\end{equation}\nwhere $n_a$ is given by Eq.~(\\ref{eq:numberdens}) and $n_\\nu$ refers\nto the present neutrino plus antineutrino number density per flavour. The top right panel of Fig.~\\ref{fig:maref} shows the axion contribution to the radiation component of the universe as a function of the axion mass. Notice that the extra dark radiation arising from a $1$~eV axion is still compatible (at $95\\%$~CL) with the most recent measurements of $N_{\\textrm{eff}}$ from the Planck mission~\\cite{Planck:2015xua}. The last crucial cosmological axion quantity is the axion free streaming scale, i.e. the wavenumber $k_{\\rm {fs}}$ below which axion density perturbations will contribute to clustering once the axion is a non-relativistic particle. This scale is illustrated in Fig.~\\ref{fig:maref}, in the bottom right panel, together with that corresponding to a neutrino of the same mass. Notice that they cover the same scales for our choice of priors for $m_a$ and $\\sum m_\\nu$ and therefore one can expect a large correlation between these two quantities in measurements of galaxy clustering. We will explore this degeneracy in the following sections. We summarize the axion parameters in Tab.~\\ref{tab:axionparams}, where we specify the values of the decoupling temperature, $\\Delta N_{\\textrm{eff}}$, $\\Omega_a h^2$ and $k_{\\rm {fs}}$ for the range of axion masses considered here, $(0.1, 3)$~eV.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nAxion parameter & & \\\\\n\\hline\\hline\n$m_a$ (eV) & $0.1$ &$3$ \\\\\n$T_D$ (MeV) & $245.6$ &$43.2$ \\\\\n$\\Omega_a h^2$ & $0.0003$ &$0.016$ \\\\\n$\\Delta N_{\\textrm{eff}}$ & $0.18$ &$0.43$ \\\\ \n$k_{\\rm {fs}}$ ($h$\/Mpc) &$0.06$ &$1.46$\\\\\n\\hline\n\\end{tabular}\n\n\\end{center}\n\\caption{Values for the axion parameters, $T_D$, $\\Delta N_{\\textrm{eff}}$, $\\Omega_a h^2$ and $k_{\\rm {fs}}$ for the lower and upper prior choice of $m_a$ explored here. }\n\\label{tab:axionparams}\n\\end{table}\n\n\\subsection{Cosmological measurements}\n\\label{sub:data}\n\nOur baseline data set consists of CMB measurements. These include the\ntemperature data from the Planck satellite, see\nRefs.~\\cite{Ade:2013ktc,Planck:2013kta}, together with the\nWMAP 9-year polarization measurements, following\n\\cite{Bennett:2012fp}. We also consider high multipole data from the\nSouth Pole Telescope (SPT) \\cite{Reichardt:2011yv} as well as from the Atacama Cosmology Telescope (ACT) \\cite{Das:2013zf}. \nThe combination of all the above CMB data is referred to as the \\emph{CMB} data set.\n\n\nGalaxy clusters represent an independent tool to probe the cosmological\nparameters. Cluster surveys usually report their measurements by means\n of the so-called cluster normalization condition, $\\sigma_8\n \\Omega^\\gamma_m$, where $\\gamma \\sim \n 0.4$~\\cite{Allen:2011zs,Weinberg:2012es,Rozo:2013hha}.\nWe shall use here the cluster normalization condition as measured by\nthe Planck Sunyaev-Zeldovich (PSZ) 2013 catalogue~\\cite{Ade:2013lmv},\nreferring to it as the \\emph{PSZ} data set. The PSZ measurements of\nthe cluster mass function provide the constraint $\\sigma_8\n(\\Omega_m\/0.27)^{0.3}=0.764\\pm 0.025$. \nAs there exists a strong degeneracy between the value of the\n$\\sigma_8$ parameter and the cluster mass bias, it is possible to fix the value of the\nbias parameter accordingly to the results arising from numerical\nsimulations. In this last case, the error on the cluster\nnormalization condition from the PSZ catalogue is considerably reduced:\n$\\sigma_8 (\\Omega_m\/0.27)^{0.3}=0.78\\pm 0.01$. In our analyses, we shall consider the two PSZ measurements of the cluster normalization condition, to illustrate the impact of the cluster mass bias in the thermal axion mass bounds, as recently explored in Ref.~\\cite{Ade:2015fva} for the neutrino mass case. Figure \\ref{fig:sigma8} illustrates the prediction for the cluster normalisation condition, $\\sigma_8\n(\\Omega_m\/0.27)^{0.3}$, as a function of the thermal axion mass. We also show the current PSZ measurements with their associated $95\\%$~CL uncertainties, including those in which the cluster mass bias parameter is fixed. Notice that the normalisation condition decreases as the axion mass increases, due to the decrease induced in the $\\sigma_8$ parameter in the presence of axion masses: the larger the axion mass is, the larger the reduction in the matter power spectra will be. \n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=12.cm]{s8om03_ma.pdf}\n\\end{center}\n \\caption{Cluster normalisation condition, $\\sigma_8\n(\\Omega_m\/0.27)^{0.3}$, as a function of the thermal axion mass. We also show the current PSZ measurements with their associated $95\\%$~CL uncertainties.}\n\\label{fig:sigma8}\n\\end{figure*}\n\n\n\nTomographic weak lensing surveys are sensitive to the overall\namplitude of the matter power spectrum by measuring the correlations in the observed shape of distant \ngalaxies induced by the intervening large scale structure. \nThe matter power spectrum amplitude depends on both the $\\sigma_8$\nclustering parameter and the matter density $\\Omega_m$. Consequently, tomographic lensing surveys, via\nmeasurements of the galaxy power shear spectra, provide additional\nand independent constraints in the ($\\sigma_8$, $\\Omega_m$) plane. \nThe Canada-France-Hawaii Telescope Lensing Survey, CFHTLenS, with six tomographic redshift bins (from $z=0.28$ to\n$z=1.12$), provides a constraint on the relationship between $\\sigma_8$\nand $\\Omega_m$ of $\\sigma_8 (\\Omega_m\/0.27)^{0.46}=0.774\\pm\n0.040$~\\cite{Heymans:2013fya}. We shall refer to this data set as \\emph{CFHT}.\n\nWe also address here the impact of a gaussian prior on the Hubble constant\n$H_0=70.6\\pm3.3$ km\/s\/Mpc from an independent reanalysis of Cepheid\ndata~\\cite{Efstathiou:2013via}, referring to this prior as the \n\\emph{HST} data set. \n\nWe have also included measurements of the large scale structure of the\nuniverse in their geometrical form, i.e., in the form of Baryon Acoustic\nOscillations (BAO). Although previous studies in the literature have shown that, \nfor constraining hot thermal relics, the shape information contained in the galaxy power spectrum is more\npowerful when dealing simultaneously with extra relativistic species\nand hot thermal relic masses~\\cite{Hamann:2010pw,Giusarma:2012ph}, we exploit here the BAO signature, as the\ncontribution from the thermal axions to the relativistic number of\nspecies is not very large (see Tab.~\\ref{tab:axionparams}), and current measurements from galaxy surveys are\nmostly reported in the geometrical (BAO) form.\n\nThe BAO wiggles, imprinted in the power spectrum of the galaxy\ndistribution, result from the competition in the coupled photon-baryon\nfluid between radiation pressure and gravity. The BAO measurements\nthat have been considered in our numerical analyses include the\nresults from the WiggleZ~\\cite{Blake:2011en}, the\n6dF~\\cite{Beutler:2011hx} and the SDSS II surveys~\\cite{Percival:2009xn,Padmanabhan:2012hf}, at redshifts of $z=0.44,\n0.6, 0.73$, $z=0.106$ and $z=0.35$, respectively. We also include in our analyses as well the Data Release 11\n(DR11) BAO signal of the BOSS experiment~\\cite{Dawson:2012va}, which provides the most precise distant\nconstraints~\\cite{Anderson:2013zyy} measuring both the Hubble parameter and the angular diameter distance at an effective redshift of $0.57$. \nFigure \\ref{fig:dvboss} illustrates the spherically averaged BAO distance, $D_V(z) \\propto D^2_A(z)\/H(z)$, as a function of the axion mass, at a redshift of $z=0.57$, as well as the measurement from the BOSS experiment with $95\\%$~CL error bars~\\cite{Anderson:2013zyy}. Notice that, from background measurements only, there exists a strong degeneracy between the cold dark matter mass-energy density and the axion one. The solid black line in Fig.~ \\ref{fig:dvboss} shows the spherically averaged BAO distance if all the cosmological parameters are fixed, including $\\omega_c$. The spherically averaged BAO distance deviates strongly from the $\\Lambda$CDM prediction. However, if $\\omega_c$ is varied while $m_a$ is changed (in order to keep the total matter mass-energy density constant, see the dotted blue line in Fig.~\\ref{fig:dvboss}), the spherically averaged BAO distance approaches to its expected value in a $\\Lambda$CDM cosmology.\n\n\\begin{figure*}[!t]\n\\includegraphics[width=10.cm]{dv_ma.pdf}\n \\caption{The solid black line depicts the spherically averaged BAO distance $D_V(z)$, as a function of the axion mass, at a redshift of $z=0.57$, after keeping fixed all the remaining cosmological parameters, the cold dark matter included. The dashed blue line depicts the equivalent but keeping fixed the total matter mass-energy density (and consequently changing the cold dark matter $\\omega_c$). The bands show the measurement from the BOSS experiment (DR11) with its associated $95\\%$~CL error.}\n\\label{fig:dvboss}\n\\end{figure*}\n\n\n\\subsection{Compatibility of data}\nIt has been pointed out (see Sec. 5.5 of Ref.~\\cite{Ade:2013zuv} and also Refs.~\\cite{Giusarma:2014zza,Leistedt:2014sia}) that the value of $\\sigma_8$ reported by cluster measurements and the value estimated from Planck CMB measurements show a tension at the $\\sim 2\\sigma$ level. These discrepancies may arise due to the lack of a full understanding of the cluster mass calibrations. Although some studies in the literature, including the present one, show that in extended cosmological models with non-zero neutrino masses the discrepancies previously mentioned could be alleviated, the results from Ref.~\\cite{Leistedt:2014sia} show, using also Bayesian evidence, that a canonical $\\Lambda$CDM scenario with no massive neutrinos is preferred over its neutrino extensions by several combinations of cosmological datasets. Therefore, the results presented here and obtained when considering cluster data depend strongly on the reliability of low-redshift cluster data. If future data confirm current low-redshift cluster measurements, one could further test some of the possible beyond the $\\Lambda$CDM models using particle physics experiments. For instance, the existence of a full thermal sterile neutrino could be tested with neutrino oscillation experiments, and the active neutrino mass could also be tested by tritium experiments or, if the neutrino is a Majorana particle, by neutrinoless double beta decay searches.\n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{table*}\n\\begin{center}\\footnotesize\n\\begin{tabular}{lcccccccc}\n\\hline \\hline\n & CMB & CMB+HST & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO\\\\\n & & & & +HST & HST+CFHT & +HST+PSZ (fixed bias) & +HST+PSZ\\\\ \n\\hline\n\\hspace{1mm}\\\\\n\n$\\Omega_{\\textrm{c}}h^2$ & $0.127\\,^{+0.007}_{-0.007}$ & $0.122\\,^{+0.006}_{-0.006}$ & $0.122\\,^{+0.003}_{-0.003}$ & $0.121\\,^{+0.003}_{-0.003}$ & $0.120\\,^{+0.003}_{-0.003}$ & $0.118\\,^{+0.002}_{-0.002}$ & $0.119\\,^{+0.003}_{-0.004}$ \\\\\n\\hspace{1mm}\\\\\n\n$m_a$ [eV] & {\\rm {Unconstrained}} & $<1.31$ & $<0.89$ & $<0.91$ & $<1.29$ & $1.00\\,^{+0.50}_{-0.48}$ & $0.93\\,^{+0.70}_{-0.71}$ \\\\\n\\hspace{1mm}\\\\\n\n$\\sigma_8$ & $0.788\\,^{+0.079}_{-0.086}$ & $0.821\\,^{+0.052}_{-0.074}$ & $0.827\\,^{+0.044}_{-0.057}$ & $0.825\\,^{+0.045}_{-0.059}$ & $0.793\\,^{+0.049}_{-0.058}$ & $0.760\\,^{+0.023}_{-0.022}$ & $0.767\\,^{+0.046}_{-0.044}$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\Omega_{\\textrm{m}}$ & $0.369\\,^{+0.070}_{-0.065}$ & $0.314\\,^{+0.045}_{-0.039}$ & $0.308\\,^{+0.016}_{-0.015}$ & $0.304\\,^{+0.016}_{-0.014}$ & $0.302\\,^{+0.016}_{-0.015}$ & $0.304\\,^{+0.016}_{-0.015}$ & $0.304\\,^{+0.016}_{-0.016}$ \\\\\n\\hspace{1mm}\\\\ \n\n$P_{s,1}$ & $<8.13$ & $<8.17$ & $<7.91$ & $<8.06$ & $<7.85$ & $<8.09$ & $<8.11$ \\\\\n\\hspace{1mm}\\\\ \n\n$P_{s,2}$ & $1.09\\,^{+0.42}_{-0.35}$ & $1.01\\,^{+0.43}_{-0.35}$ & $1.01\\,^{+0.40}_{-0.32}$ & $0.99\\,^{+0.42}_{-0.33}$ & $1.02\\,^{+0.43}_{-0.34}$ & $1.01\\,^{+0.42}_{-0.33}$ & $1.05\\,^{+0.43}_{-0.38}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,3}$ & $0.68\\,^{+0.39}_{-0.36}$ & $0.71\\,^{+0.39}_{-0.39}$ & $0.71\\,^{+0.39}_{-0.37}$ & $0.72\\,^{+0.39}_{-0.38}$ & $0.69\\,^{+0.39}_{-0.37}$ & $0.70\\,^{+0.40}_{-0.38}$ & $0.69\\,^{+0.40}_{-0.39}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,4}$ & $1.14\\,^{+0.24}_{-0.22}$ & $1.15\\,^{+0.24}_{-0.22}$ & $1.15\\,^{+0.23}_{-0.21}$ & $1.15\\,^{+0.23}_{-0.20}$ & $1.15\\,^{+0.23}_{-0.21}$ & $1.15\\,^{+0.23}_{-0.21}$ & $1.15\\,^{+0.22}_{-0.21}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,5}$ & $1.02\\,^{+0.11}_{-0.10}$ & $1.01\\,^{+0.11}_{-0.11}$ & $1.00\\,^{+0.11}_{-0.10}$ & $1.00\\,^{+0.11}_{-0.10}$ & $0.99\\,^{+0.11}_{-0.10}$ & $0.99\\,^{+0.11}_{-0.10}$ & $0.99\\,^{+0.11}_{-0.11}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,6}$ & $1.03\\,^{+0.08}_{-0.07}$ & $1.00\\,^{+0.08}_{-0.07}$ & $1.00\\,^{+0.08}_{-0.07}$ & $1.00\\,^{+0.08}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.06}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.08}_{-0.07}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,7}$ & $0.99\\,^{+0.07}_{-0.06}$ & $0.98\\,^{+0.08}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.08}_{-0.07}$ & $0.96\\,^{+0.07}_{-0.06}$ & $0.95\\,^{+0.07}_{-0.06}$ & $0.96\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,8}$ & $0.94\\,^{+0.06}_{-0.06}$ & $0.95\\,^{+0.08}_{-0.07}$ & $0.95\\,^{+0.07}_{-0.06}$ & $0.95\\,^{+0.08}_{-0.07}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,9}$ & $0.92\\,^{+0.06}_{-0.05}$ & $0.94\\,^{+0.08}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.08}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,10}$ & $0.90\\,^{+0.06}_{-0.06}$ & $0.91\\,^{+0.08}_{-0.07}$ & $0.91\\,^{+0.07}_{-0.06}$ & $0.91\\,^{+0.08}_{-0.06}$ & $0.90\\,^{+0.07}_{-0.06}$ & $0.90\\,^{+0.07}_{-0.06}$ & $0.90\\,^{+0.07}_{-0.07}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,11}$ & $1.25\\,^{+0.30}_{-0.28}$ & $1.24\\,^{+0.32}_{-0.31}$ & $1.23\\,^{+0.31}_{-0.31}$ & $1.24\\,^{+0.31}_{-0.31}$ & $1.22\\,^{+0.30}_{-0.31}$ & $1.22\\,^{+0.32}_{-0.28}$ & $1.23\\,^{+0.31}_{-0.30}$ \\\\\n\\hspace{1mm}\\\\ \n\n$P_{s,12}$ & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{$95\\%$~CL constraints on the physical cold dark matter density $\\Omega_{\\textrm{c}}h^2$, \nthe axion mass $m_a$ (in eV), the clustering parameter $\\sigma_8$, the relative matter energy density $\\Omega_{\\textrm{m}}$ and the $P_{s,j}$ parameters for the PPS nodes from the different combinations of data sets explored here in the $\\Lambda$CDM+$m_a$ model, considering the \\texttt{PCHIP}\\xspace PPS modeling.}\n\\label{tab:lcdm+ma+pchip}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table*}\n\\begin{center}\\footnotesize\n\\begin{tabular}{lcccccccc}\n\\hline \\hline\n & CMB & CMB+HST & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO\\\\\n & & & & +HST & HST+CFHT & +HST+PSZ (fixed bias) & +HST+PSZ\\\\ \n\\hline\n\\hspace{1mm}\\\\\n\n$\\Omega_{\\textrm{c}}h^2$ & $0.124\\,^{+0.006}_{-0.005}$ & $0.124\\,^{+0.005}_{-0.005}$ & $0.122\\,^{+0.004}_{-0.004}$ & $0.121\\,^{+0.004}_{-0.004}$ & $0.120\\,^{+0.003}_{-0.003}$ & $0.119\\,^{+0.003}_{-0.003}$ & $0.120\\,^{+0.003}_{-0.003}$ \\\\\n\\hspace{1mm}\\\\ \n \n$m_a$ [eV] & $<1.83$ & $<1.56$ & $<0.84$ & $<0.83$ & $<1.16$ & $0.80\\,^{+0.53}_{-0.50}$ & $<1.26$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\sigma_8$ & $0.785\\,^{+0.064}_{-0.083}$ & $0.791\\,^{+0.057}_{-0.076}$ & $0.803\\,^{+0.041}_{-0.048}$ & $0.803\\,^{+0.041}_{-0.048}$ & $0.783\\,^{+0.047}_{-0.054}$ & $0.758\\,^{+0.028}_{-0.029}$ & $0.767\\,^{+0.045}_{-0.045}$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\Omega_{\\textrm{m}}$ & $0.337\\,^{+0.048}_{-0.044}$ & $0.328\\,^{+0.041}_{-0.039}$ & $0.310\\,^{+0.025}_{-0.023}$ & $0.308\\,^{+0.024}_{-0.023}$ & $0.305\\,^{+0.025}_{-0.024}$ & $0.307\\,^{+0.027}_{-0.026}$ & $0.306\\,^{+0.027}_{-0.025}$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\log[10^{10} A_s]$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.09\\,^{+0.05}_{-0.05}$ & $3.09\\,^{+0.05}_{-0.05}$ \\\\\n\\hspace{1mm}\\\\ \n \n$n_s$ & $0.961\\,^{+0.014}_{-0.015}$ & $0.963\\,^{+0.013}_{-0.014}$ & $0.968\\,^{+0.011}_{-0.011}$ & $0.969\\,^{+0.011}_{-0.011}$ & $0.971\\,^{+0.011}_{-0.011}$ & $0.973\\,^{+0.011}_{-0.011}$ & $0.972\\,^{+0.011}_{-0.011}$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{$95\\%$~CL constraints on \n$\\Omega_{\\textrm{c}}h^2$, \nthe axion mass $m_a$ (in eV), \n$\\sigma_8$, $\\Omega_{\\textrm{m}}$, ${\\rm{log}}(10^{10} A_s)$ and $n_s$\nfrom the different combinations of data sets explored here in the $\\Lambda$CDM+$m_a$ model, assuming the standard power-law PPS.}\n\\label{tab:lcdm+ma+pl}\n\\end{center}\n\\end{table*}\n\n\nTable \\ref{tab:lcdm+ma+pchip} depicts our results in the first scenario explored\nhere, in which the axion mass is a free parameter and the PPS is\ndescribed by the approach specified in Sec.~\\ref{sub:pps}. Concerning CMB measurements only, the\nbounds on the thermal axion masses are largely relaxed in the case in which the PPS\nis not described by a simple power-law, as can be noticed after comparing the results depicted in Tab.~\\ref{tab:lcdm+ma+pchip} with those shown in Tab.~\\ref{tab:lcdm+ma+pl}. This can be understood in terms of Fig.~\\ref{fig:plottt}, which\nillustrates the degeneracies in the temperature anisotropies between the\nthermal axion mass and the \\texttt{PCHIP}\\xspace PPS.\nFigure~\\ref{fig:plottt} shows the temperature anisotropies for a $\\Lambda$CDM model and a power-law PPS (solid red line), for a $2$~eV thermal axion\nmass and a power-law PPS (dashed blue line) and for a $\\Lambda$CDM model\nbut the PPS described by the \\texttt{PCHIP}\\xspace model explored here (dotted black line), with values\nfor the $P_{s,j}$ chosen to match the non-zero thermal axion mass curve, accordingly to their current allowed regions \n(see Tab.~\\ref{tab:lcdm+ma+pchip}). More concretely, we have used the following values for the PPS parameters: \n$P_{s,1}=1.15$, $P_{s,2}=1.073$, $P_{s,3}=1.058$, $P_{s,4}=1.03$, $P_{s,5}=0.99$, $P_{s,6}=0.97$,\n$P_{s,7}=0.966$, $P_{s,8}=0.932$, $P_{s,9}=0.91$, $P_{s,10}=0.86$, $P_{s,11}=0.84$ and $P_{s,12}=0.77$.\nNotice that the case of a $2$~eV thermal axion can be easily mimicked\nby a simple $\\Lambda$CDM model if the assumptions concerning the PPS \nshape are relaxed. We also add in this figure the measurements of the photon temperature anisotropies from the Planck 2013 data release~\\cite{Ade:2013zuv}. \n\n\n\n\\begin{figure*}[!t]\n\\includegraphics[width=11.1cm]{cl_compare-2.pdf}\n \\caption{Temperature anisotropies for the pure $\\Lambda$CDM model and a power-law PPS (solid red line), for a $2$~eV thermal axion\nmass and a power-law PPS (dashed blue line) and for the standard $\\Lambda$CDM model but the PPS described by the \\texttt{PCHIP}\\xspace model (dotted black line). The data points and the error bars in the left panel show the measurements of the photon temperature anisotropies arising from the Planck 2013 data release~\\cite{Ade:2013zuv}.}\n\\label{fig:plottt}\n\\end{figure*}\n\n\n\nThe addition to the CMB data of the HST prior on the Hubble constant\nprovides a $95\\%$~CL upper limit on the thermal axion mass of\n$1.31$~eV~\\footnote{There exists a very large degeneracy between $H_0$\n and the neutrino masses when restricting the numerical analyses to\n CMB measurements. The addition of the HST prior on the Hubble\n constant helps enormously in breaking this degeneracy,\n see~\\cite{Giusarma:2012ph}.}, while the further addition of the BAO\nmeasurements brings this constraint down to $0.91$~eV, as these last\ndata sets are directly sensitive to the free-streaming nature of the\nthermal axion. Notice that these two $95\\%$~CL upper bounds are very\nsimilar to the ones obtained when considering the standard power-law\npower spectrum, which are $1.56$~eV and $0.83$~eV for the CMB+HST and\nCMB+HST+BAO data combinations, respectively. \n\n\nInterestingly, when adding the CFHT bounds on the $\\sigma_8$-$\\Omega_m$ relationship, the bounds on the thermal axion mass become weaker. \nThe reason for that is due to the lower $\\sigma_8$ values preferred by weak lensing measurements, \nvalues that can be achieved by allowing for higher axion masses. \nThe larger the axion mass, the larger is the reduction of the matter power spectrum at small (i.e. cluster) scales, \nleading consequently to a smaller value of the clustering parameter $\\sigma_8$. \n\nIf we instead consider now the PSZ data set with fixed cluster mass\nbias, together with the CMB, BAO and HST measurements, a non-zero\nvalue of the thermal axion mass of $\\sim 1$~eV ($\\sim 0.80$~eV) is\nfavoured at $\\sim4\\sigma$ ($\\sim3\\sigma$) level, when considering the\n\\texttt{PCHIP}\\xspace (standard power-law) PPS approach~\\footnote{A similar effect when considering PSZ data for constraining either thermal axion or neutrino masses has also been found in Refs.~\\cite{Hamann:2013iba,Wyman:2013lza,Giusarma:2014zza,Dvorkin:2014lea,Archidiacono:2014apa}.}. However, these results must be regarded as an illustration of what could be achieved with future cluster mass calibrations, as the Planck collaboration has recently shown in their analyses of the 2015 Planck cluster catalogue~\\cite{Ade:2015fva}. When more realistic approaches for the cluster mass bias are used, the errors on the so-called cluster normalization condition are larger, and, consequently, the preference for a non-zero axion mass of $1$~eV is only mild in the \\texttt{PCHIP}\\xspace PPS case, while in the case of a standard power-law PPS such an evidence completely disappears.\n\n\nFigure~\\ref{fig:ma_pchip} (left panel) shows the $68\\%$ and $95\\%$~CL\nallowed regions in the ($m_a$, $\\Omega_c h^2$) plane for some of the\npossible data combinations explored in this study, and assuming the \\texttt{PCHIP}\\xspace PPS modeling. Notice that, when\nadding BAO measurements, lower values of the physical cold dark matter density are\npreferred. This is due to the fact that large scale structure allows\nfor lower axion masses than CMB data alone. \nThe lower is the thermal axion mass,\nthe lower is the amount of hot dark matter and consequently the lower should be\nthe cold dark matter component. \nThis effect is clear \nfrom the results shown in Tab.~\\ref{tab:lcdm+ma+pchip} and Tab.~\\ref{tab:lcdm+ma+pl}, where the values of the\nphysical cold dark matter density $\\Omega_c h^2$ and of the relative\ncurrent matter density $\\Omega_m$ arising from our numerical fits\nare shown, for the different data combinations considered here. \n\nThe right panel of Fig.~\\ref{fig:ma_pchip} shows the $68\\%$ and $95\\%$~CL\nallowed regions in the ($m_a$, $\\sigma_8$) plane in the \\texttt{PCHIP}\\xspace PPS scenario. The lower values of\nthe $\\sigma_8$ clustering parameter preferred by PSZ data (see the results shown in Tab.~\\ref{tab:lcdm+ma+pchip} and Tab.~\\ref{tab:lcdm+ma+pl}) are translated into a preference for non-zero thermal axion masses. Larger values of $m_a$ will enhance the matter power spectrum suppression at scales below the axion free-streaming scale, leading to smaller values of the $\\sigma_8$ clustering parameter, as preferred by PSZ measurements. The evidence for non-zero axion masses is more significant when fixing the cluster mass bias in the PSZ data analyses. \n\nFigure \\ref{fig:ma_pl} shows the equivalent to Fig.~\\ref{fig:ma_pchip} but for a standard power-law PPS. Notice that, except for the case in which CMB measurements are considered alone, the thermal axion mass constraints do not change significantly, if they are compared to the \\texttt{PCHIP}\\xspace PPS modeling. \nThis fact clearly states the robustness of the cosmological bounds on thermal axion masses and it is applicable to the remaining cosmological parameters, see Tabs.~\\ref{tab:lcdm+ma+pchip} and \\ref{tab:lcdm+ma+pl}. Note that, for the standard case of a power-law PPS, the preference for non-zero axion masses appears only when considering the (unrealistic) PSZ analysis with a fixed cluster mass bias. When more realistic PSZ measurements of the cluster normalization condition are exploited, there is no preference for a non-zero thermal axion mass. \n\n\n\\begin{figure*}[!t]\n\\begin{tabular}{c c}\n\\includegraphics[width=8.3cm]{lcdm_ma_pchip_omegac-ma.pdf}&\\includegraphics[width=8.3cm]{lcdm_ma_pchip_sigma8-ma.pdf}\\\\\n\\end{tabular}\n \\caption{The left panel depicts the $68\\%$ and $95\\%$~CL allowed\n regions in the ($m_a$, $\\Omega_c h^2$) plane for different possible\n data combinations, when a \\texttt{PCHIP}\\xspace{} PPS is assumed. The right panel shows the equivalent but in the\n ($m_a$, $\\sigma_8$) plane.}\n\\label{fig:ma_pchip}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\begin{tabular}{c c}\n\\includegraphics[width=8.3cm]{standard_ma_omegac-ma.pdf}&\\includegraphics[width=8.3cm]{standard_ma_sigma8-ma.pdf}\\\\\n\\end{tabular}\n \\caption{The left panel depicts the $68\\%$ and $95\\%$~CL allowed\n regions in the ($m_a$, $\\Omega_c h^2$) plane for different possible\n data combinations, when a power-law PPS is assumed. The right panel shows the equivalent but in the\n ($m_a$, $\\sigma_8$) plane.}\n\\label{fig:ma_pl}\n\\end{figure*}\n\n\\begin{figure}[!t]\n\\includegraphics[width=9cm]{ma_mnu.pdf}\n\\caption{$68\\%$ and $95\\%$~CL allowed\n regions in the ($\\sum m_\\nu$, $m_a$) plane, both in eV, for three different possible\n data combinations, when a \\texttt{PCHIP}\\xspace{} PPS is assumed.}\n\\label{fig:mamnu}\n\\end{figure}\nThe last scenario we explore here is a $\\Lambda$CDM+$m_a$+$\\sum m_\\nu$ universe, in which we consider two coexisting hot dark matter species: thermal axions and three active (massive) neutrinos.\nTable~\\ref{tab:lcdm+ma+mnu+pchip} illustrates the equivalent of Tab.~\\ref{tab:lcdm+ma+pchip} but including the active neutrino masses in the MCMC parameters. We do not perform here the analysis for the hot mixed dark matter model with the standard power-law matter power spectrum, as it was already presented previously in Ref.~\\cite{Giusarma:2014zza}. If we compare to the standard power-law case, we find that the bounds on the axion and neutrino masses presented here are very similar. Furthermore, no evidence for neutrino masses nor for a non-zero axion mass appears in this mixed hot dark matter scenario (except for the axion case and only if considering PSZ clusters with the bias fixed). The reason for that is due to the strong degeneracy between $m_a$ and $\\sum m_\\nu$, see Fig.~\\ref{fig:mamnu}, where one can notice that that these two parameters are negatively correlated: an increase in the axion mass will increase the amount of the hot dark matter component. In order to compensate the changes in both the CMB temperature anisotropies (via the early ISW effect) and in the power spectrum (via the suppression at small scales of galaxy clustering), the contribution to the hot dark matter from the neutrinos should be reduced. We have shown in Fig.~\\ref{fig:mamnu} three possible data combinations. Notice that for the case in which PSZ cluster measurements (with the bias fixed) are included the strong degeneracy between $m_a$ and $\\sum m_\\nu$ is partially broken, due to the smaller value of $\\sigma_8$ preferred by the former data set. However, these results strongly rely on the numerical results concerning the cluster mass bias and therefore the evidence for $m_a\\neq 0$ should be regarded as what could be obtained in if these measurements are further supported by independent data from future cluster surveys.\n\n\n\n\\begin{table*}\n\\begin{center}\\footnotesize\n\\begin{tabular}{lcccccccc}\n\\hline \\hline\n & CMB & CMB+HST & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO \\\\\n & & & & +HST & +HST+CFHT & +HST+PSZ(fixed bias) & +HST+PSZ \\\\ \n\\hline\n\\hspace{1mm}\\\\\n\n$\\Omega_{\\textrm{c}}h^2$ & $0.130\\,^{+0.008}_{-0.007}$ & $0.125\\,^{+0.006}_{-0.007}$ & $0.121\\,^{+0.003}_{-0.003}$ & $0.121\\,^{+0.003}_{-0.003}$ & $0.119\\,^{+0.003}_{-0.003}$ & $0.118\\,^{+0.003}_{-0.003}$ & $0.118\\,^{+0.003}_{-0.003}$ \\\\\n\\hspace{1mm}\\\\ \n \n$m_a$ [eV] & $<2.48$ & $<1.64$ & $<0.81$ & $<0.86$ & $<1.23$ & $0.81\\,^{+0.59}_{-0.69}$ & $<1.46$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\sum m_\\nu$ [eV] & $<2.11$ & $<0.43$ & $<0.22$ & $<0.21$ & $<0.27$ & $<0.32$ & $<0.35$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\sigma_8$ & $0.700\\,^{+0.172}_{-0.202}$ & $0.803\\,^{+0.082}_{-0.091}$ & $0.833\\,^{+0.055}_{-0.058}$ & $0.834\\,^{+0.058}_{-0.064}$ & $0.787\\,^{+0.052}_{-0.055}$ & $0.766\\,^{+0.043}_{-0.044}$ & $0.757\\,^{+0.023}_{-0.022}$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\Omega_{\\textrm{m}}$ & $0.486\\,^{+0.277}_{-0.193}$ & $0.356\\,^{+0.064}_{-0.062}$ & $0.309\\,^{+0.016}_{-0.015}$ & $0.308\\,^{+0.016}_{-0.015}$ & $0.306\\,^{+0.015}_{-0.015}$ & $0.308\\,^{+0.016}_{-0.016}$ & $0.308\\,^{+0.017}_{-0.016}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,1}$ & $<8.01$ & $<8.13$ & $<7.00$ & $<8.17$ & $<7.59$ & $<8.29$ & $<8.18$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,2}$ & $1.17\\,^{+0.42}_{-0.38}$ & $1.09\\,^{+0.41}_{-0.37}$ & $1.03\\,^{+0.40}_{-0.35}$ & $1.02\\,^{+0.39}_{-0.34}$ & $1.02\\,^{+0.40}_{-0.32}$ & $1.03\\,^{+0.36}_{-0.34}$ & $1.05\\,^{+0.40}_{-0.36}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,3}$ & $0.66\\,^{+0.37}_{-0.35}$ & $0.69\\,^{+0.38}_{-0.37}$ & $0.70\\,^{+0.38}_{-0.38}$ & $0.72\\,^{+0.38}_{-0.37}$ & $0.68\\,^{+0.37}_{-0.33}$ & $0.71\\,^{+0.40}_{-0.39}$ & $0.69\\,^{+0.39}_{-0.37}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,4}$ & $1.17\\,^{+0.23}_{-0.23}$ & $1.15\\,^{+0.23}_{-0.22}$ & $1.15\\,^{+0.22}_{-0.21}$ & $1.15\\,^{+0.21}_{-0.21}$ & $1.15\\,^{+0.20}_{-0.19}$ & $1.14\\,^{+0.21}_{-0.20}$ & $1.16\\,^{+0.22}_{-0.21}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,5}$ & $1.05\\,^{+0.15}_{-0.14}$ & $1.01\\,^{+0.11}_{-0.10}$ & $1.00\\,^{+0.11}_{-0.10}$ & $1.00\\,^{+0.11}_{-0.10}$ & $0.98\\,^{+0.11}_{-0.10}$ & $0.99\\,^{+0.11}_{-0.10}$ & $0.98\\,^{+0.11}_{-0.10}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,6}$ & $1.04\\,^{+0.09}_{-0.08}$ & $1.01\\,^{+0.08}_{-0.07}$ & $1.00\\,^{+0.07}_{-0.07}$ & $1.00\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.06}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.07}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,7}$ & $0.99\\,^{+0.06}_{-0.06}$ & $0.98\\,^{+0.07}_{-0.06}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.95\\,^{+0.07}_{-0.06}$ & $0.95\\,^{+0.06}_{-0.06}$ & $0.95\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,8}$ & $0.93\\,^{+0.06}_{-0.05}$ & $0.94\\,^{+0.06}_{-0.06}$ & $0.95\\,^{+0.07}_{-0.07}$ & $0.95\\,^{+0.07}_{-0.07}$ & $0.93\\,^{+0.07}_{-0.05}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,9}$ & $0.91\\,^{+0.06}_{-0.05}$ & $0.93\\,^{+0.06}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ & $0.93\\,^{+0.06}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,10}$ & $0.90\\,^{+0.06}_{-0.06}$ & $0.90\\,^{+0.07}_{-0.06}$ & $0.91\\,^{+0.07}_{-0.07}$ & $0.91\\,^{+0.08}_{-0.07}$ & $0.88\\,^{+0.07}_{-0.06}$ & $0.89\\,^{+0.07}_{-0.07}$ & $0.90\\,^{+0.07}_{-0.07}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,11}$ & $2.18\\,^{+0.85}_{-0.77}$ & $2.07\\,^{+0.81}_{-0.80}$ & $2.12\\,^{+0.90}_{-0.86}$ & $2.15\\,^{+0.95}_{-0.94}$ & $1.64\\,^{+0.79}_{-0.75}$ & $1.83\\,^{+0.87}_{-0.86}$ & $1.84\\,^{+0.86}_{-0.87}$ \\\\\n\\hspace{1mm}\\\\ \n\n$P_{s,12}$ & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\caption{$95\\%$~CL constraints on the physical cold dark matter density $\\Omega_{\\textrm{c}}h^2$, \nthe axion mass $m_a$, the sum of the active neutrino masses $\\sum m_\\nu$ (both in eV), the clustering parameter $\\sigma_8$, the relative matter energy density $\\Omega_{\\textrm{m}}$ and the $P_{s,j}$ parameters for the PPS nodes from the different combinations of data sets explored here in the $\\Lambda$CDM+$m_a$+$\\sum m_\\nu$ model, considering the \\texttt{PCHIP}\\xspace PPS modeling.}\n\\label{tab:lcdm+ma+mnu+pchip}\n\\end{center}\n\\end{table*}\n\n\nBesides the results concerning the thermal axion mass and the standard $\\Lambda$CDM parameters,\nwe also obtain constraints on the form of the PPS when modeled accordingly to the \\texttt{PCHIP}\\xspace scenario.\nThe 95\\% CL limits for the $P_{s,j}$ parameters are shown in Tab.~\\ref{tab:lcdm+ma+pchip}, \nwhile an example of the reconstructed PPS is given in Fig.~\\ref{fig:outputPPS}, \nwhere we show the 68\\%, 95\\% and 99\\% CL allowed regions arising from a fit to CMB data of the \\texttt{PCHIP}\\xspace PPS scale dependence, \n in the context of a $\\Lambda$CDM+$m_a$ model.\nWe do not show the corresponding figures obtained from all the other data combinations \nsince they are equivalent to Fig.~\\ref{fig:outputPPS}, as one can infer from the very small \ndifferences in the $95\\%$~CL allowed ranges for the $P_{s,j}$ parameters arising from different data sets, see Tab.~\\ref{tab:lcdm+ma+pchip}.\nNote that both $P_{s,1}$ and $P_{s,12}$ are poorly constrained at this confidence level:\nthe reason for that is the absence of measurements at their corresponding wavenumbers.\nAll the remaining $P_{s,j}$, with $j=2,\\ldots,11$ are\nwell-constrained. In particular, in the range between $k_5$\nand $k_{10}$ (see Eq.~(\\ref{eq:nodesspacing})), the $P_{s,j}$ are determined with few percent accuracy. Indeed, in the range covered between these nodes, the PPS does not present features and can be perfectly described by a power-law parametrization.\nAmong the interesting features outside the former range, we can notice in Fig.~\\ref{fig:outputPPS}\na significant dip at wavenumbers around $k=0.002\\, \\text{Mpc}^{-1}$, that comes from the dip at $\\ell=20-30$ in the CMB temperature power spectrum and a small bump around $k=0.0035\\, \\text{Mpc}^{-1}$, corresponding to the increase at $\\ell\\simeq40$. These features have been obtained in previous works \\cite{Hunt:2013bha,Hazra:2014jwa,Gariazzo:2014dla} using different methods and data sets. In addition, we obtain an increase of power at $k\\simeq0.2\\, \\text{Mpc}^{-1}$, necessary to compensate the effects of the thermal axion mass in both the temperature anisotropies and the large scale structure of the universe.\n\n\n\\begin{figure*}[!t]\n\\includegraphics[width=12cm]{lcdm_ma_pchip_PPSbands.pdf}\n\\caption{$68\\%$, $95\\%$ and $99\\%$~CL allowed regions for the \\texttt{PCHIP}\\xspace PPS scale dependence in the $\\Lambda$CDM+$m_a$ model, using CMB data only.\nThe bands are obtained with a marginalization of the posterior distribution for each different value of the wavenumber $k$ in a fine grid.\nThe black line represents the peak of the posterior distribution at each value of $k$.}\n\\label{fig:outputPPS}\n\\end{figure*}\n\n\\section{Conclusions}\n\\label{sec:concl}\nAxions provide the most elegant scenario to solve the strong CP problem, and may be produced in the early universe via both thermal and non-thermal processes. While non thermal axions are highly promising cold dark matter candidates, their thermal companions will contribute to the hot dark matter component of the universe, together with the (light) three active neutrinos of the standard model of elementary particles. Therefore, the cosmological consequences of light massive thermal axions are very much alike those associated with neutrinos, as axions also have a free-streaming nature, suppressing structure formation at small scales. Furthermore, these light thermal axions will also contribute to the dark radiation background, leading to deviations of the relativistic degrees of freedom $N_{\\textrm{eff}}$ from its canonically expected value of $N_{\\textrm{eff}}=3.046$. Based on these signatures, several studies have been carried out in the literature deriving bounds on the thermal axion mass~\\cite{Melchiorri:2007cd,Hannestad:2007dd,Hannestad:2008js,Hannestad:2010yi,Archidiacono:2013cha,Giusarma:2014zza}. \n\nNevertheless, these previous constraints assumed that the underlying\nprimordial perturbation power spectrum follows the usual power-law\ndescription governed, in its most economical form, by an amplitude\nand a scalar spectral index. Here we have relaxed such an assumption, \nin order to test the robustness of the cosmological axion mass\nbounds. Using an alternative, non-parametric description of the\nprimordial power spectrum of the scalar perturbations,\nnamed \\texttt{PCHIP}\\xspace and introduced in Ref.~\\cite{Gariazzo:2014dla}, we have shown that, in practice, when combining CMB measurements with low redshift cosmological probes, the axion mass constraints are only mildly sensitive to the primordial\npower spectrum choice and therefore are not strongly dependent on the\nparticular details of the underlying inflationary model. \nThese results agree with the findings of Ref.~\\cite{dePutter:2014hza} for the neutrino mass case. The tightest\n bound we find in the \\texttt{PCHIP}\\xspace primordial power spectrum approach is obtained when considering BAO measurements together with CMB data, with $m_a<\n 0.89$~eV at $95\\%$~CL. In the standard power-law primordial power\n spectrum modeling, the tightest bound is $m_a<\n 0.83$~eV at $95\\%$~CL, obtained when combining BAO, CMB and HST measurements. Notice that these\n bounds are very similar, confirming the robustness of the cosmological\n axion mass measurements versus the primordial power spectrum\n modeling. \n\nInterestingly, both weak lensing measurements and cluster number\ncounts weaken the thermal axion mass bounds. The reason for that is\ndue to the lower $\\sigma_8$ values preferred by \nthese measurements, which could be generated by a larger axion\nmass. More concretely, Planck cluster measurements provide a\nmeasurement of the so-called cluster normalization condition,\nwhich establishes a relationship between the clustering parameter\n$\\sigma_8$ and the current matter mass-energy density $\\Omega_m$. \nHowever, the errors on this relationship depend crucially on the\nknowledge of the cluster mass bias. A conservative approach for the\ncluster mass calibration results in a mild (zero) evidence for a\nnon-zero axion mass of $1$~eV in the \\texttt{PCHIP}\\xspace (power-law) PPS case.\nWe also illustrate a case in which the cluster mass bias is fixed, to forecast the expected results from future cosmological\nmeasurements. In this case, a non-zero\nvalue of the thermal axion mass of $\\sim 1$~eV ($\\sim 0.80$~eV) is\nfavoured at $\\sim4\\sigma$ ($\\sim3\\sigma$) level, when considering the\n\\texttt{PCHIP}\\xspace (power-law) PPS approach. \nWhen considering additional hot relics in our analyses, as the sum of the three active neutrino masses, the evidence for \na $\\sim 1$~eV thermal axion mass disappears almost completely. Furthermore, these values of axion masses correspond to an axion coupling constant \n$f_a= 6\\times 10^6$~GeV, which seems to be in tension with the limits extracted from the neutrino signal duration from SN 1987A~\\cite{Raffelt:2006cw,Raffelt:1990yz} (albeit these limits depend strongly on the precise axion emission rate and still remain rough estimates). Precise cluster mass calibration measurements are therefore mandatory to assess whether there exists a cosmological indication for non-zero axion masses, as the cluster mass bias is highly correlated with the clustering parameter $\\sigma_8$, which, in turn, is highly affected by the free-streaming nature of a hot dark matter component, as thermal axions.\n\n\\section{Acknowledgments}\nOM is supported by PROMETEO II\/2014\/050, by the Spanish Grant FPA2011--29678 of the MINECO and by PITN-GA-2011-289442-INVISIBLES. This work has been done within the Labex ILP (reference ANR-10-LABX-63) part of the Idex SUPER, and received financial state aid managed by the Agence Nationale de la Recherche, as part of the programme Investissements d'avenir under the reference ANR-11-IDEX-0004-02. EDV acknowledges the support of the European Research Council via the Grant number 267117 (DARK, P.I. Joseph Silk).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSince the seminal paper \\cite{GPV}, the theory of hydrodynamic limit of interacting particle systems has evolved into a powerful tool in the study of non-equilibrium properties of statistical systems of many components (see the book \\cite{KL} for a comprehensive exposition). Recently, and due to the infuence of physical and mathematical works about random walks in random environment, an increasing attention has been posed into particle systems evolving in random environments. Despite the early works \\cite{Fri}, \\cite{Qua}, \\cite{Kou}, we mention \\cite{FM}, \\cite{Qua} \\cite{Nag}, \\cite{JL}, \\cite{Fag}, \\cite{GJ}, \\cite{FJL}, \\cite{FL}, \\cite{Fag2}, \\cite{GJ2}. In \\cite{GJ}, \\cite{JL} the {\\em corrected empirical density} was introduced, which is nothing but a microscopic version of the compensated compactness lemma of Tartar \\cite{Tar}. Roughly speaking, when the inhomogeneous environment (random or not) has a divergence form and has a $\\Gamma$-limit, space homogenization of the environment and time homogenization of the interaction decouples, and the standard tools from the theory of hydrodynamic limit can be used to obtain the asymptotic behavior of the density of particles in a family of models, including the exclusion process and the zero-range process. \n\nIn this review, we give an unified approach to this problem, recovering previous results in \\cite{Nag}, \\cite{JL}, \\cite{Fag}, \\cite{FJL}, \\cite{Fag2}, in a simple way. In order to concentrate our efforts in the influence of the inhomogeneous environment on the asymptotics of the density of particles, we consider the simplest model of interacting particle systems, which is the symmetric exclusion process $\\eta_t^n$ in an unoriented graph. In this process, particles perform symmetric random walks on a graph $\\{X_n\\}_n$ with some rates $\\omega^n=\\{\\omega_{x,y}^n; x,y \\in X_n\\}$, conditioned to have at most one particle per site. We think of $\\{X_n\\}_n$ as a sequence of graphs embedding in some metric space $X$, and we are interested in the evolution of the measure $\\pi_t^n(dx)$ in $X$, obtained by giving a mass $a_n^{-1}$ to each particle. \n\nThis article is organized as follows. In Section \\ref{s1} we give precise definitions of the exclusion process, the inhomogeneous environment and we state our main result. We also define what we mean by an approximation $\\{X_n\\}_n$ of $X$ and by $\\Gamma$-convegence of the environment. In Section \\ref{s2} we introduce the corrected empirical density and we prove our main theorem. In Section \\ref{s3} we introduce the concept of energy solutions of the hydrodynamic equation, we prove uniqueness of such solutions and we obtain a substantial improvement of the main Theorem. The material of this Section is new and it gives a better understanding of the relation between $\\Gamma$-convergence of the environment and hydrodynamic limit of the particle system. In Section \\ref{s4} we discuss how to reobtain previous results in the literature relying in our main Theorem.\n\n\n\n\\section{Definitions and results}\n\\label{s1}\nIn this section we define the exclusion process in inhomogeneous environment and we recall some notions of $\\Gamma$-convergence that will be necessary in order to obtain the hydrodynamic limit of this process.\n\n\\subsection{Partitions of the unity and approximating sequences}\n\\label{s1.1}\nIn this section we fix some notation and we define some objects which will be useful in the sequel. \nLet $(X,\\mc B)$ be a Polish space. We assume that $X$ is $\\sigma$-compact.\nWe say that a sequence of functions $\\{\\mc U_i; i \\in I\\}$ is a {\\em partition of the unity} if:\n\\begin{description}\n \\item[i)] for any $i \\in I$, $\\mc U_i:X \\to [0,1]$ is a continuous function, \n \\item[ii)] for any $x \\in X$, $\\sum_{i \\in I} \\mc U_i(x)=1$,\n \\item[iii)] for any $x \\in X$, the set $\\{i \\in I; \\mc U_i(x)>0\\}$ is finite.\n\\end{description}\n\nWe say that the partition of the unity $\\{\\mc U_i;i \\in I\\}$ is {\\em regular} if $\\text{supp } \\mc U_i$ is compact for any $i \\in I$, and additionally $\\mc U_i(X) = [0,1]$. \nWe denote by $\\mc M_+(X)$ the set of Radon, positive measures in $X$. The symbol $\\{x_n\\}_n$ will denote a sequence of elements $x_n$ in some space, indexed by the set $\\bb N$ of positive integers. \n\n\nLet $\\{\\mc U_i\\}_i$ be a regular partition of the unity. We say that a sequence $\\{x_i ; i \\in I\\}$ in $X$ is a {\\em representative} of $\\{\\mc U_i\\}_i$ if $\\mc U_i(x_i)=1$ for any $i \\in I$. Notice that we have $x_i \\neq x_j$ for $i \\neq j$. \n\nLet $\\{\\mc U_i^n; i \\in I_n\\}_n$ be a sequence of regular partitions of the unity. We say that a measure $\\mu \\in \\mc M_+(X)$ is the scaling limit of the sequence $\\{\\mc U_i^n\\}_n$ if there exists a sequence $\\{a_n\\}_n$ of positive numbers such that for any sequence $\\{x_i^n;i \\in I_n\\}$ of representatives of $\\{\\mc U_i^n\\}_n$ we have\n\\[\n\\lim_{n \\to \\infty} \\frac{1}{a_n} \\sum_{i \\in I_n} \\delta_{x_i^n} = \\mu\n\\]\nwith respect to the vague topology, where $\\delta_x$ is the Dirac mass at $x \\in X$. We call $\\{a_n\\}_n$ the {\\em scaling} sequence. \n\nFrom now on, we fix a sequence $\\{\\mc U_i^n\\}_n$ of regular partitions of the unity with scaling limit $\\mu$, scaling sequence $\\{a_n\\}_n$ and we assume that $\\mu(A)>0$ for any non-empty, open set $A \\subseteq X$. \nFix a sequence $\\{x_i^n;i \\in I_n\\}$ of representatives of $\\{\\mc U_i^n\\}_n$. Define $X_n = \\{x_i^n; i \\in I_n\\}$. \nSince $\\{\\mc U_i^n\\}$ is a partition of the unity, the induced topology in $X_n$ coincides with the discrete topology. For $x=x_i^n$, we will denote $\\mc U_x^n= \\mc U_i^n$. Define\n\\[\n\\mu_n(dx) = \\frac{1}{a_n} \\sum_{x \\in X_n} \\delta_x(dx).\n\\]\n\nBy definition, $\\mu_n \\to \\mu$ in the vague topology. We denote by $\\mc L^2(\\mu_n)$ the Hilbert space of functions $f: X_n \\to \\bb R$ such that $\\sum_{x \\in X_n} f(x)^2 <+\\infty$, equipped with the inner product\n\\[\n\\< f, g\\>_n = \\frac{1}{a_n} \\sum_{x \\in X_n} f(x)g(x).\n\\]\n\nWe define $\\mc L^2(\\mu)$, $\\mc L^1(X_n)$ and $\\mc L^1(\\mu)$ in the analogous way and we denote $\\ = \\int fg d\\mu$.\nWe denote by $\\mc C_c(X)$ the set of continuous functions $f: X \\to \\bb R$ with compact support. In the same spirit, we denote by $\\mc C_c(X_n)$ the set of functions $f: X_n \\to \\bb R$ with finite support. We define the projection $S_n: \\mc C_c(X) \\to \\mc C_c(X_n)$ by taking\n\\[\n\\big( S_n G\\big)(x) = a_n \\int G \\mc U_x^n d\\mu.\n\\]\n\nThis operator, under suitable conditions, can be extended to a bounded operator from $\\mc L^2(X)$ to $\\mc L^2(X_n)$. Notice that $\\int S_n G d\\mu_n = \\int G d\\mu$. Therefore $S_n$ is continuous from $\\mc L^1(\\mu)$ to $\\mc L^1(X_n)$.\n\n\\subsection{$\\Gamma$-convergence}\n\n\nDefine $\\bar{ \\bb R} = [-\\infty,+\\infty]$. Let $(Y,\\mc F)$ be a topological space, and let $F_n,F: Y \\to \\bar{ \\bb R}$. We say that $F_n$ is $\\Gamma$-convergent to $F$ if:\n\\begin{description}\n \\item[i)] For any sequence $\\{y_n\\}_n$ in $Y$ converging to $y \\in Y$,\n \\[\n F(y) \\leq \\liminf_{n \\to \\infty} F_n(y_n).\n \\]\n \\item[ii)] For any $y \\in Y$ there exists a sequence $\\{y_n\\}_n$ converging to $y$ such that\n \\[\n \\limsup_{n \\to \\infty} F_n(y_n) \\leq F(y).\n \\]\n\\end{description}\n\nAn important property of $\\Gamma$-convergence is that it implies {\\em convergence of minimizers} in the following sense:\n\n\\begin{proposition}\n\\label{p1}\nLet $F_n, F: Y \\to \\bar{\\bb R}$ be such that $F_n$ is $\\Gamma$-convergent to $F$. Assume that there exists a relatively compact set $K \\subseteq Y$ such that for any $n$,\n\\[\n\\inf_{y \\in Y} F_n(y) =\\inf_{y \\in K} F_n(y).\n\\]\n\nThen,\n\\[\n\\lim_{n \\to \\infty} \\inf_{y \\in K} F_n(y) = \\min_{y \\in Y} F(y).\n\\]\nMoreover, if $\\{y_n\\}_n$ is a sequence in $K$ such that $\\lim_n (F_n(y_n)-\\inf_K F_n)=0$, then any limit point $y$ of $\\{y_n\\}_n$ satisfies $F(y) = \\min_Y F$.\n\\end{proposition}\n\nA useful property that follows easily from the definition is the stability of $\\Gamma$-convergence under continuous perturbations:\n\n\\begin{proposition}\n\\label{p2}\nLet $F_n,F: Y \\to \\bar{\\bb R}$ be such that $F_n$ is $\\Gamma$-convergent to $F$. Let $G_n: Y \\to \\bb R$ be such that $G_n$ converges uniformly to a continuous limit $G$. Then, $F_n+G_n$ is $\\Gamma$-convergent to $F+G$.\n\\end{proposition}\n\n\\subsection{The exclusion process in inhomogeneous environment}\n\nIn this section we define the exclusion process in inhomogeneous environment as a system of particles evolving in the set $X_n$. \nLet $\\omega^n = \\{\\omega_{x,y}^n;x,y \\in X_n\\}$ be a sequence of non-negative numbers such that $\\omega^n_{x,x}=0$ and $\\omega^n_{x,y}=\\omega^n_{y,x}$ for any $x,y \\in X_n$. We call $\\omega^n$ the {\\em environment}. We define the exclusion process $\\eta_t^n$ with environment $\\omega^n$ as a continuous-time Markov chain of state space $\\Omega_n = \\{0,1\\}^{X_n}$ and generated by the operator\n\\[\nL_n f(\\eta) = \\sum_{x,y \\in X_n} \\omega_{x,y}^n \\big[f(\\eta^{x,y}) - f(\\eta)\\big],\n\\]\nwhere $\\eta$ is a generic element of $\\Omega_n$, $f: \\Omega_n \\to \\bb R$ is a function which depends on $\\eta(x)$ for a finite number of elements $x \\in X_n$ (that is, $f$ is a {\\em local function}) and $\\eta^{x,y} \\in \\Omega_n$ is defined by\n\\[\n\\eta^{x,y}(z) = \n\\begin{cases}\n\\eta(y), & \\text{if } z=x\\\\\n\\eta(x), & \\text{if } z=y\\\\\n\\eta(z), & \\text{if } z \\neq x,y.\\\\\n\\end{cases}\n\\]\n\nIn order to have a well-defined Markovian evolution for any initial distribution $\\eta_0^n$, we assume that $\\sup_x \\sum_{y \\in X_n} \\omega^n_{x,y} <+\\infty$.\nWe interpret $X_n$ as a set of sites and $\\eta_t^n(x)$ as the number of particles at site $x \\in X_n$ at time $t$. Since $\\eta_t^n(x) \\in \\{0,1\\}$, there is at most one particle per site at any given time: this is the so-called {\\em exclusion rule}. Notice that the dynamics is conservative in the sense that no particles are annihilated or destroyed.\n\nOur interest is to study the collective behavior of particles for the sequence of processes $\\{\\eta_\\cdot^n\\}_n$. In order to do this, we introduce the {\\em empirical density of particles} as the measure-valued process $\\pi_t^n$ defined by\n\\[\n\\pi_t^n(G) = \\frac{1}{a_n} \\sum_{x \\in X_n} \\eta_t^n(x) S_n G(x)\n\\]\nfor any $G \\in \\mc C_c(X)$. Using Riesz's theorem, it is not difficult to check that $\\pi_t^n$ is effectively a positive Radon measure in $X$. \nObserve that when $\\eta_0^n(x)=1$ for any $x \\in X_n$, then $\\eta_t^n(x)=1$ for any $x \\in X_n$ and any $t \\geq 0$. In this situation, the empirical process $\\pi_t^n$ is identically equal to the measure $\\mu$. Notice that the random variable $\\pi_t^n$ defined in this way corresponds to a process defined in the space $\\mc D([0,\\infty), \\mc M_+(X))$ of c\\`adl\\`ag paths with values in $\\mc M_+(X)$. For functions $G:X_n \\to \\bb R$, we define $\\pi_t^n(G) = a_n^{-1} \\sum_x \\eta_t^n(x) G(x)$.\n\n\\subsection{$\\Gamma$-convergence of the environment}\n\nIn this section we will make a set of assumptions on the environment $\\{\\omega^n\\}_n$ which will allows us to obtain an asymptotic result for the sequence $\\{\\pi_\\cdot^n\\}_n$. We start with two assumptions about the sequence of partitions of the unity $\\{\\mc U_x^n\\}_n$. Our first assumption corresponds to a sort of ellipticity condition on the partitions of the unity $\\{\\mc U_x^n\\}_n$: \n\n\\begin{description}\n\\item[\\bf (H1)] There exists $\\Theta <+\\infty$ such that \n\\[\n\\sup_{x \\in X_n} a_n \\int \\mc U_x^n d\\mu \\leq\\Theta \\text{ for any $n>0$.}\n\\]\n\n\\end{description}\n\nUnder this condition, the projection $S_n$ satisfies $||S_n G||_\\infty \\leq \\theta ||G||_\\infty$, and by interpolation $S_n$ can be extended to a continuous operator from $\\mc L^2(\\mu)$ to $\\mc L^2(X_n)$. Our second condition states that $S_n$ is close to an isometry when $n \\to \\infty$:\n\\begin{description}\n \\item[\\bf{(H2)}] For any $F \\in \\mc L^2(\\mu)$, we have\n \\[\n \\lim_{n \\to \\infty} \\_n = \\.\n \\] \n\\end{description}\n\nNow we are ready to discuss on which sense we will say that the environment $\\omega^n$ converges. \nFor a given function $F: X_n \\to \\bb R$ of finite support, we define $\\mc L_n F$ by\n\\[\n\\mc L_n F(x) = \\sum_{y \\in X_n} \\omega_{x,y}^n \\big(F(y)-F(x)\\big).\n\\]\n\nIt turns out that $\\mc L_n$ can be extended to a non-positive operator in $\\mc L^2(X_n)$. In fact, for any function $F$ of finite support, the {\\em Dirichlet form}\n\\[\n\\_n = \\frac{1}{2 a_n} \\sum_{x, y \\in X_n} \\omega_{x,y}^n \\big(F(y)-F(x)\\big)^2\n\\]\nis clearly non-negative. For a function $G \\in \\mc L^2(\\mu)$, define $\\mc E_n(G) = \\$. Notice that $\\mc E_n: \\mc L^2(\\mu) \\to \\bar{\\bb R}$ is a quadratic form. Now we are ready to state our first hypothesis about the environment:\n\\begin{description}\n \\item[{\\bf (H3)}] There exists a non-negative, symmetric operator $\\mc L: D(\\mc L) \\subseteq \\mc L^2(\\mu) \\to \\mc L^2(\\mu)$ such that $\\mc E_n$ is $\\Gamma$-convergent to $\\mc E$, where $\\mc E(G) = -\\int G \\mc L G d\\mu$. \n\\end{description}\n\nOur second hypothesis about the environment $\\omega^n$ concerns to its $\\Gamma$-limit $\\mc L$:\n\\begin{description}\n \\item[{\\bf (H4)}] There exists a dense set $\\mc K \\subseteq \\mc C_c(X)$ such that $\\mc K$ is a kernel for the operator $\\mc L$, and for any $G \\in \\mc K$, $\\mc LG$ is continuous and $\\int |\\mc L G| d\\mu <+\\infty$.\n\\end{description}\n\n\\subsection{Hydrodynamic limit of $\\eta_t^n$}\n\nIn this section we explain what we understand as the hydrodynamic limit of $\\eta_t^n$. We say that a sequence $\\{\\nu_n\\}_n$ of distributions in $\\Omega_n$ is {\\em associated } to a function $u:X \\to \\bb R$ if for any function $G \\in \\mc C_c(X)$ and any $\\epsilon >0$ we have\n\\[\n\\lim_{n \\to \\infty} \\nu_n \\Big\\{ \\Big|\\frac{1}{a_n} \\sum_{x \\in X_n} \\eta(x) G(x) - \\int G(x)u(x)\\mu(dx)\\Big|>\\epsilon\\Big\\}=0. \n\\]\n\nNotice that we necessarily have $0 \\leq u(x) \\leq 1$ for any $x \\in X$, since $\\eta(x) \\in \\{0,1\\}$.\nFix an initial profile $u_0:X \\to [0,1]$ and take a sequence of distributions $\\{\\nu_n\\}$ associated to $u_0$. Let $\\eta_t^n$ be the exclusion process with initial distribution $\\nu_n$. We denote by $\\bb P_n$ the law of $\\eta_t^n$ in $\\mc D([0,\\infty),\\Omega_n)$ and by $\\bb E_n$ the expectation with respect to $\\bb P_n$. The fact that $\\{\\nu_n\\}_n$ is associated to $u_0$ can be interpreted as a law of large numbers for the empirical measure $\\pi_0^n$: $\\pi_0^n(dx)$ converges in probability to the deterministic measure $u_0(x) \\mu(dx)$. We say that the hydrodynamic limit of $\\eta_t^n$ is given by the equation $\\partial_t u = \\mc L u$ if for any $t>0$, the empirical measure $\\pi_t^n(dx)$ converges in probability to the measure $u(t,x) \\mu(dx)$, where $u(t,x)$ is the solution of the equation $\\partial_t u = \\mc L u$ with initial condition $u_0$. Before stating our main result in a more precise way, we need some definitions. \n\nFor $F, G \\in D(\\mc L)$, define the bilinear form $\\mc E(F,G) = - \\int F \\mc L G d\\mu$. Notice that $\\mc E(F,G)$ is still well defined if only $G \\in D(\\mc L)$. \nWe say that a function $u:[0,T] \\times X \\to [0,1]$ is a weak solution of (\\ref{echid}) with initial condition $u_0$ if $\\int_0^T \\int u_t^2 d\\mu dt <+\\infty$ and for any differentiable path $G: [0,T] \\to \\mc K$ such that $G_T \\equiv 0$ we have\n\\[\n\\ +\\int_0^T \\Big\\{ \\<\\partial_t G_t,u_t\\> -\\mc E(G_t,u_t)\\Big\\}dt=0.\n\\]\n\n\n\\begin{theorem}\n\\label{t1}\nLet $\\{\\nu_n\\}_n$ be associated to $u_0$ and consider the exclusion process $\\eta_t^n$ with initial distribution $\\nu_n$. Assume that $\\int \\pi_0^n(dx)$ is uniformly finite:\n\\begin{description}\n \\item[{\\bf (H5)}] \n \\[\n \\lim_{M \\to \\infty} \\sup_n \\nu_n\\Big\\{\\frac{1}{a_n} \\sum_{x \\in X_n} \\eta(x) >M \\Big\\} =0.\n \\]\n\\end{description}\n\nThen, the sequence of processes $\\{\\pi_\\cdot^n(dx)\\}_n$ is tight and the limit points are concentrated on measures of the form $u(t,x)\\mu(dx)$, where $u(t,x)$ is a weak solution of the {\\em hydrodynamic equation}\n\\begin{equation}\n\\label{echid}\n\\left\\{\n\\begin{array}{rcl}\n\\partial_t u & = & \\mc L u, \\\\\nu(0,\\cdot) & = & u_0(\\cdot).\\\\\n\\end{array}\n\\right.\n\\end{equation}\n\nIf such solution is unique, the process $\\pi_\\cdot^n(dx)$ converges in probability with respect to the Skorohod topology of $\\mc D([0,\\infty),\\mc M_+(X))$ to the deterministic trajectory $u(t,x)\\mu(dx)$.\n\\end{theorem}\n\n\nUsually in the literature, hydrodynamic limits are obtained in finite volume, since the pass from finite to infinite volume is non-trivial. Assumption {\\bf (H5)} is in this spirit: it is automatically satisfied when the cardinality of $X_n$ is of the order of $a_n$ (on which case $\\mu(X)<+\\infty$), and it is very restrictive when $X_n$ is infinite. For simplicity, we restrict ourselves to the case on which {\\bf (H5)} is satisfied. \n\n\n\\section{Hydrodynamic limit of $\\eta_t^n$: proofs}\n\\label{s2}\nIn this section we obtain the hydrodynamic limit of the process $\\eta_t^n$. The strategy of proof of this result is the usual one for convergence of stochastic processes. First we prove tightness of the sequence of processes $\\{\\pi_\\cdot^n\\}_n$. Then we prove that any limit point of this sequence is concentrated on solutions of the hydrodynamic equation. Finally, a uniqueness result for such solutions allows us to conclude the proof. However, the strategy outlined above will not be carried out for $\\{\\pi_\\cdot^n\\}_n$ directly, but for another process $\\hat \\pi_\\cdot^n$, which we call the {\\em corrected} empirical process.\n\n\\subsection{The corrected empirical measure}\n\\label{s2.1}\n\nIn this section we define the so-called corrected empirical measure, relying on the $\\Gamma$-convergence of the environment. First we need to extract some information about convergence of the operators $\\mc L_n$ to $\\mc L$ from the $\\Gamma$-convergence of the associated Dirichlet forms.\n\nTake a general Hilbert space $\\mc H$ and let $\\mc A$ be a non-negative, symmetric operator defined in $\\mc H$. By Lax-Milgram theorem, we know that for any $\\lambda>0$ and any $g \\in \\mc H$, the equation $(\\lambda+\\mc A)f =g$ has a unique solution in $\\mc H$. Moreover, the solution $f$ is the minimizer of the functional $f \\mapsto \\+\\lambda||f||^2-2\\$. Fix $\\lambda>0$. For a given function $G \\in \\mc L^2(\\mu)$, define the functionals\n\\[\n\\mc E_n^G(F) = \\mc E_n(F) +\\lambda\\_n -2 \\_n,\n\\]\n\\[\n\\mc E^G(F) = \\mc E(F) +\\lambda\\-2 \\.\n\\]\n\nBy Proposition \\ref{p2}, $\\mc E_n^G$ is $\\Gamma$-convergent to $\\mc E^G$. In particular, a sequence of minimizers $F_n$ of $\\mc E_n^G$ converge to the minimizer $F$ of $\\mc E^G$. Notice that $F_n$ is not uniquely defined in general, although $S_n F_n$ it is. By the discussion above, $(\\lambda-\\mc L_n) S_n F_n = S_n G$ and $(\\lambda -\\mc L) F =G$. Since the operator norm of $S_n$ is bounded by $\\Theta$, we conclude that the $\\mc L^2(X_n)$-norm of $S_n F_n-S_n F$ converges to 0 as $n \\to \\infty$.\nBy {\\bf (H2)}, we conclude that $\\mc E_n(F_n)$ converges to $\\mc E(F)$. \n\n\n\nNow we are ready to define the corrected empirical measure $\\hat \\pi_t^n$. Take a function $G \\in \\mc K$ and define $H=(\\lambda-\\mc L) G$. Define $G_n$ as a minimizer of $\\mc E_n^H$. Notice that in this way $S_n G_n$ is uniquely defined. Then we define \n\\[\n\\hat \\pi_t^n(G) = \\frac{1}{a_n} \\sum_{x \\in X_n} \\eta_t^n(x) S_n G_n(x).\n\\]\n\nIn order to prove that $\\hat \\pi_t^n(G)$ is well defined, we need to prove that $\\sum_x S_n G_n(x)$ is finite. Remember that $(\\lambda - \\mc L_n) S_n G_n = S_n H$. Consider the continuous-time random walk with jump rates $\\omega_{x,y}^n$. Remember that the condition $\\sup_x \\sum_y \\omega_{x,y}^n$ ensures that this random walk is well defined. Let $p_t^n(x,y)$ be its transition probability function. An explicit formula for $S_n G_n$ in terms of $p_t^n(x,y)$ is\n\\[\nS_n G_n (x) = \\int_0^\\infty e^{-\\lambda t} \\sum_{y \\in X_n} p_t^n(x,y) S_n H(y) dt.\n\\]\n\nSince $\\sum_x p_t(x,y)=1$ for any $y \\in X_t$, we conclude that\n\\[\n\\frac{1}{a_n} \\sum_{x \\in X_n} S_n G_n(x) = \\frac{1}{\\lambda} \\int H d\\mu\n\\]\nand in particular $S_n G_n$ is summable. We conclude that $\\hat \\pi_t^n(G)$ is well defined. Notice that it is not clear at all if $\\hat \\pi_t^n$ is well defined as a measure in $X$. \n\n\\subsection{Tightness of $\\{\\pi_\\cdot^n\\}_n$ and proof of Theorem \\ref{t1}}\n\nIn this section we prove tightness of $\\{\\pi_\\cdot^n\\}_n$ and we prove Theorem \\ref{t1}. As we will see, we rely on the corrected empirical measure, which turns out to be the right object to be studied. \nBy {\\bf (H5)}, we have\n\\[\n\\lim_{n \\to \\infty} \\bb P_n \\Big( \\sup_{0 \\leq t <+\\infty} \\big|\\pi_t^n(G) - \\hat \\pi_t^n(G)\\big|>\\epsilon\\Big) =0.\n\\]\n\nNotice that {\\bf (H5)} can be substituted by the following condition, which can be sometimes proved directly.\n\\begin{description}\n \\item[{\\bf (H5')}] For any $G \\in \\mc K$,\n \\[\n \\lim_{n \\to \\infty} \\frac{1}{a_n} \\sum_{x \\in X_n} \\big| S_n G_n(x) - S_n G(x)\\big| =0.\n \\]\n\\end{description}\n\nIn particular, $\\{\\pi_\\cdot^n(G)\\}_n$ is tight if and only if $\\{\\hat \\pi_\\cdot^n(G)\\}_n$ is tight. The usual way of proving tightness of $\\{\\hat \\pi_\\cdot^n(G)\\}_n$ is to use a proper martingale decomposition. A simple computation based on Dynkin's formula shows that\n\\begin{equation}\n\\label{ec1}\n\\mc M_t^n(G) = \\hat \\pi_t^n(G) - \\hat \\pi_0^n(G) - \\int_0^t \\pi_s^n(\\mc L_n S_n G_n) ds\n\\end{equation}\nis a martingale. The quadratic variation of$\\mc M_t^n(G)$ is given by\n\\[\n\\<\\mc M_t^n(G)\\> = \\int_0^t \\frac{1}{a_n^2} \\sum_{x,y \\in X_n} \\big(\\eta_s^n(y)-\\eta_s^n(x)\\big)^2 \\omega^n_{x,y} \\big( S_n G_n(y) -S_n G_n(x)\\big)^2 ds.\n\\]\nIn particular, $\\<\\mc M_t^n(G)\\> \\leq t a_n^{-1} \\mc E_n(G_n)$. At this point, the convenience of introducing the corrected empirical process becomes evident. By definition, $\\mc L_n S_n G_n = S_n \\mc L G +\\lambda(S_n G_n - S_n G)$. Since $H= (\\lambda-\\mc L)G$, the function $G$ is the minimizer of $\\mc E^H$. Therefore, $G_n$ converges to $G$ in $\\mc L^2(X)$. By {\\bf (H2)}, the $\\mc L^2(X_n)$-norm of $S_n G_n -S_n G$ goes to 0 and $\\mc E_n(G_n)$ converges to $\\mc E(G)$. \n\nWe conclude that $\\mc M_t^n(G)$ converges to 0 as $n \\to \\infty$, and in particular the sequence $\\{\\mc M_\\cdot^n(G)\\}_n$ is tight. In the other hand, the integral term in (\\ref{ec1}) is equal to\n$\\int_0^t \\pi_s^n(\\mc L G)ds$.\n\nNotice that $\\pi_s^n(\\mc L G) \\leq \\int |\\mc L G| d \\mu$ for any $t \\geq 0$, from where we conclude that the integral term is of bounded variation, uniformly in $n$. Tightness follows at once. Since $\\{\\hat \\pi_0^n(G)\\}_n$ is tight by comparison with $\\{\\pi_0^n(G)\\}_n$, we conclude that $\\{\\hat \\pi_\\cdot^n(G)\\}_n$ is tight, which proves the first part of Theorem \\ref{t1}. As a by-product, we have obtained tightness for $\\{\\pi_\\cdot^n\\}_n$ as well, and the convergence result\n\\[\n\\lim_{n \\to \\infty}\\Big\\{ \\pi_t^n(G) -\\pi_0^n(G) - \\int_0^t \\pi_s^n(\\mc L G) ds\\Big\\} =0\n\\]\nfor any $G \\in \\mc K$. Notice that we have exchanged $\\hat \\pi_t^n(G)$ by $\\pi_t^n(G)$. Let $\\pi_\\cdot$ be a limit point of $\\{\\pi_\\cdot^n\\}_n$. Then, $\\pi_\\cdot$ satisfies the identity\n\\[\n\\pi_t(G) -\\pi_0(G) - \\int_0^t \\pi_s(\\mc L G) ds =0\n\\] \nfor any function $G \\in \\mc K$. By hypothesis, $\\pi_0(dx) = u_0(x)\\mu(dx)$. Repeating the arguments for a function $G_t(x) = G_0(x) + t G_1(x)$ with $G_0,G_1 \\in \\mc K$, we can prove that\n\\[\n\\pi_t(G_t) - \\pi_0(G_0) -\\int_0^t \\pi_s((\\partial_t+\\mc L)G_s)ds=0\n\\] \nfor any piecewise-linear trajectory $G_\\cdot:[0,T] \\to \\mc K$. The same identity holds by approximation for any smooth path $G_\\cdot: [0,T] \\to \\mc C_c(X)$, which proves that the process $\\pi_\\cdot$ is concentrated on weak solutions of the hydrodynamic equation. When such solutions are unique, the process $\\pi$ is just a $\\delta$-distribution concentrated on the path $u(t,x)\\mu(dx)$. Since compactness plus uniqueness of limit points imply convergence, Theorem \\ref{t1} is proved. \n\n\\section{Energy solutions and energy estimate}\n\\label{s3}\nIn this section we define what we mean by {\\em energy solutions} of Equation (\\ref{echid}), we prove that any limit point of the empirical measure $\\{\\pi_\\cdot^n\\}$ is concentrated on energy solutions of (\\ref{echid}) and we give a simple criterion for uniqueness of such solutions. \n\n\\subsection{Energy solutions} \n\nLet $\\mc E : H \\to \\bar{\\bb R}$ be a quadratic form defined over a Hilbert space $H$ of inner product $\\<\\cdot,\\cdot\\>$. We say that $\\mc E$ is {\\em closable} if for any sequence $\\{f_n\\}_n$ converging in $H$ to some limit $f$ such that $\\mc E(f_n-f_m)$ goes to $0$ as $n, m \\to \\infty$, we have $f=0$. \nLet $\\mc E: H \\to \\bar{\\bb R}$ be closable. We define $\\mc H_1=\\mc H_1(\\mc E)$ as the closure of the set $\\{f\\in H; \\mc E(f)<+\\infty\\}$ under the norm $||f||_1 = (\\mc E(f)+ \\)^{1\/2}$. \n\nWe say that a dense set $K \\subseteq H$ is a {\\em kernel} of $\\mc E$ if $\\mc H_1$ is equal to the closure of $K$ under the norm $||\\cdot||_1$. We say that a symmetric operator $\\mc L: D(\\mc L) \\subseteq H \\to H$ generates $\\mc E$ if $\\mc E(f)=\\$ for $f \\in D(\\mc L)$ and $D(\\mc L)$ is a kernel of $\\mc E$. \n\nFix $T >0$. For a function $u: [0,T] \\to H$ we define the norm\n\\[\n||u||_{1,T} = \\Big( \\int_0^T ||u_t||_1^2 dt \\Big)^{1\/2}\n\\]\nand we define $\\mc H_{1,T}$ as the Hilbert space generated by this norm. \nGiven a closable form $\\mc E$ generated by the operator $\\mc L$, we say that a trajectory $u: [0,T] \\to H$ is an {\\em energy solution} of (\\ref{echid}) if $u \\in \\mc H_{1,T}$ and for any differentiable trajectory $G: [0,T] \\to \\mc H_1$ with $G(T)=0$ we have\n\\[\n\\ + \\int_0^T \\Big\\{ \\<\\partial_t G_t, u_t\\> - \\mc E(G_t,u_t)\\Big\\} dt =0.\n\\]\n\nIn other words, an energy solution of (\\ref{echid}) is basically a weak solution belonging to $\\mc H_{1,T}$. In fact, by taking suitable approximations of $G$, it is enough to prove this identity for trajectories $G$ such that $G_t \\in K$ for any $t \\in [0,T]$, where $K$ is any kernel of $\\mc E$ contained in $D(\\mc L)$. Notice that the norm in $\\mc H_{1,T}$ is stronger than the norm $\\int_0^T u_t^2 dt$, and therefore a weak solution is effectively weaker than an energy solution of (\\ref{echid}).\n\n\\subsection{The energy estimate}\n\nIn this section we prove that the limit points of the empirical measure are concentrated on energy solutions of (\\ref{echid}). For simplicity, we work on finite volume. From now on we assume that $X$ is compact. Therefore, there exists a constant $\\kappa$ such that the cadinality of $X_n$ is bounded by $\\kappa a_n$. We have the following estimate.\n\n\\begin{theorem}\n\\label{t2}\nFix $T >0$. Let $\\{H^i: X_n \\times X_n \\times [0,T] \\to \\bb R; i=1,\\dots,l\\}$ be a finite sequence of functions. There exists a constant $C=C(T)$ such that\n\\begin{multline}\n\\label{en.est}\n\\bb E_n \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T \\Big\\{ \\frac{2}{a_n} \\sum_{x,y \\in X_n} \\omega^n_{x,y} H_{x,y}^i(t) \\big( \\eta_t^n(y) -\\eta_t^n(x)\\big) \\\\\n\t-\\frac{1}{a_n} \\sum_{x,y \\in X_n} \\omega_{x,y}^n (H_{x,y}^i)^2 \\eta_t^n(x)\\Big\\} dt \\Big]\n\\leq C + \\frac{\\log l}{a_n}.\n\\end{multline}\n\\end{theorem}\n\n\\begin{proof}\n\nBefore starting the proof of this theorem, we need some definitions. Fix $\\rho >0$. Denote by $\\nu^\\rho$ the product measure in $\\Omega_n$ defined by \n\\[\n\\nu^\\rho\\big(\\eta(x_1)=1,\\dots,\\eta(x_k)=1\\big) = \\rho^k.\n\\]\n\nIt is not difficult to check that the measure $\\nu^\\rho$ is left invariant under the evolution of $\\eta_t$. For two given probability measures $P_1$, $P_2$, we define the entropy $H(P_1|P_2)$ of $P_1$ with respect to $P_2$ as\n\\[\nH(P_1|P_2)= \n\\begin{cases}\n+ \\infty, & \\text{ if $P_1$ is not absolutely continuous with respect to $P_2$}\\\\\n\\int \\log \\frac{dP_1}{dP_2} d P_1 & \\text{ otherwise. }\\\\\n\\end{cases}\n\\]\n\nFor $\\eta \\in \\Omega_n$, denote by $\\delta_\\eta$ the Dirac measure at $\\eta$. It is not difficult to see that $H(\\delta_\\eta|\\nu^\\rho) \\leq C(\\rho) a_n$ for any $\\eta \\in \\Omega_n$, where $C(\\rho)$ is a constant that can be chosen independently from $n$. Let us denote by $\\bb P^\\rho$ the distribution in $D([0,T],\\Omega_n)$ of the process $\\eta_t^n$ with initial distribution $\\nu^\\rho$. By the convexity of the entropy, $H(\\bb P_n|\\bb P^\\rho) \\leq C(\\rho,T)a_n$ for a constant $C(\\rho,T)$ not depending on $n$. The following arguments are standard and can be found in full rigor in \\cite{KL}. Let us denote by $F^i(s)$ the function (depending on $H^i(s)$ and $\\eta_s^n$) under the time integral in \\eqref{en.est}. By the entropy estimate,\n\\begin{align*}\n\\bb E_n \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T F^i(t)dt\\Big]\n\t\\leq \\frac{H(\\bb P_n|\\bb P^\\rho)}{a_n} +\n\t\\frac{1}{a_n} \\log \\bb E^\\rho \\Big[ \\exp\\big\\{ \\sup_{i=1,\\dots,l} a_n \\int_0^T F^i(t)dt\\big\\}\\Big].\n\\end{align*}\n\nIn order to take the supremum out of the expectation, we use the inequalities $\\exp\\{\\sup_i b_i\\} \\leq \\sum_i\\exp\\{b_i\\}$ and $\\log\\{\\sum_i b_i\\} \\leq \\log l + \\sup_i \\log b_i$, valid for any real numbers $\\{b_i, i=1,\\dots,l\\}$. In this way we obtain the bound\n\\begin{equation}\n \\label{ec2}\n\\begin{split}\n\\bb E_n \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T F^i(t)dt\\Big]\n\t&\\leq C(\\rho,T) +\\frac{\\log l}{a_n} \\\\\n\t&\\quad+\\sup_{i=1,\\dots,l} \\frac{1}{a_n} \\log \\bb E^\\rho \\Big[ \\exp\\big\\{ a_n\\int_0^T F^i(t)dt\\big\\}\\Big].\n\\end{split}\n\\end{equation}\n\n\nTherefore, it is left to prove that the last supremum is not positive. It is enough to prove that the expectation $ \\bb E^\\rho \\big[ \\exp\\big\\{ \\int_0^T F^i(t)dt\\big\\}\\big]$ is less or equal than $1$ for any function $F^i$. From now on we drop the index $i$. By Feynman-Kac's formula plus the variational formula for the largest eigenvalue of the operator $F(t)+L_n$, we have\n\\[\n\\frac{1}{a_n} \\log \\bb E^\\rho \\Big[ \\exp\\big\\{ a_n\\int_0^T F(t)dt\\big\\}\\Big]\n\t\\leq \\int_0^T \\sup_f \\big\\{ \\_\\rho -\\_\\rho\\},\n\\]\nwhere we have denoted by $\\<\\cdot,\\cdot\\>_\\rho$ the inner product in $\\mc L^2(\\nu_\\rho)$ and the supremum is over functions $f \\in \\mc L^2(\\nu_\\rho)$. A simple computation using the invariance of $\\nu_\\rho$ shows that\n\\[\n\\_\\rho = \\sum_{x,y \\in X_n} \\omega_{x,y}^n \\int \\big[f(\\eta^{x,y})-f(\\eta)\\big]^2 \\nu_\\rho(d\\eta).\n\\]\nRecall the expression for $F(t)$ in terms of $H$. We will estimate each term of the form $2 a_n^{-1} \\_\\rho$ separatedly:\n\\begin{align*}\n\\frac{2}{a_n} \\_\\rho\n\t&= \\frac{2}{a_n} H_{x,y} \\<\\eta(x), f(\\eta^{x,y})^2-f(\\eta)^2\\>_\\rho \\\\\n\t& \\leq \\frac{2}{a_n} \\Big\\{ \\frac{(H_{x,y})^2\\beta_{x,y}^n}{2} \\<\\eta(x), (f(\\eta^{x,y})+f(\\eta))^2\\>_\\rho \\\\\n\t&\\quad+ \\frac{1}{2 \\beta_{x,y}^n} \\<\\eta(x),(f(\\eta^{x,y})-f(\\eta))^2\\>_\\rho \\Big\\}.\n\\end{align*}\n\nChoosing $\\beta_{x,y}^n = 1\/\\omega_{x,y}^n$ and putting this estimate back into (\\ref{ec2}), we obtain the desired estimate.\n\\qed\n\\end{proof}\n\nTake $G^i \\in \\mc K$ and take $H_{x,y}^i = S_n G_n^i(y) - S_n G_n^i(x)$, with $G_n^i$ defined as in Section \\ref{s2.1}. Recall the identity $\\mc L_n S_n G_n^i= S_n \\mc L G^i + \\lambda(S_n G_n^i -S_n G^i)$. The energy estimate (\\ref{en.est}) gives\n\\[\n \\bb E_n \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T \\big(2 \\hat \\pi_t^n(\\mc L G^i) - \\mc E_n(G_n^i) \\big)dt \\Big] \\leq C(\\rho,T) + C_1(l,n),\n\\]\nwhere $C_1(l,n)$ is a constant that goes to 0 when $l$ is fixed and $n \\to \\infty$. Take a limit point of the sequence $\\{\\pi_\\cdot^n\\}_n$. We have already seen that $\\hat \\pi_t^n(\\mc L G^i)$ converges to $\\pi_t(\\mc L G)$. Therefore, the process $\\pi_\\cdot$ satisfies\n\\[\n E \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T \\big(2 \\pi_s(\\mc L G^i) - \\mc E(G^i)\\big)dt\\Big] \\leq C(\\rho,T).\n\\]\n\nSimilar arguments prove that for piecewise linear trajectories $\\{G^i_t; i=1,\\dots,l\\}$ in $\\mc K$, we have\n\\[\n E \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T \\big(2 \\pi_s(\\mc L G^i(t)) - \\mc E(G^i(t))\\big)dt\\Big] \\leq C(\\rho,T).\n\\]\n\nSince $l$ is arbitrary and piecewise linear trajectories with values in $\\mc K$ are dense in $\\mc H_{1,T}$, we conclude that $E[||\\pi_\\cdot||_{1,T}^2] <+\\infty$, from where we conclude that $||\\pi_\\cdot||_{1,T}$ is finite $a.s.$\nWe establish this result as a theorem.\n\n\\begin{theorem}\n \\label{t3}\n Let $\\eta_t^n$ an exclusion process as in Theorem \\ref{t1}.\n If one of the following conditions is satisfied,\n \\begin{enumerate}\n \\item[i)]\n $X$ is compact,\n \\item[ii)]\n Assumption {\\bf (H5')} holds and the entropy density is finite:\n \\[\n \\sup_{n} \\frac{H(\\bb P_n| \\bb P^\\rho)}{a_n} <+\\infty,\n \\]\n\\end{enumerate}\nthen any limit point of the sequence $\\{\\pi^n_\\cdot(dx)\\}_n$ is concentrated on energy solutions of the hydrodynamic equation (\\ref{echid}). In particular, since such energy solutions are unique, the sequence $\\{\\pi_\\cdot^n(dx)\\}_n$ is convergent.\n\\end{theorem}\n\n\\subsection{Uniqueness of energy solutions}\n\nIn this section we prove uniqueness of energy solutions for (\\ref{echid}). Since the equation is linear, it is enough to prove uniqueness for the case $u_0 \\equiv 0$. Let $u_t$ be a solution of (\\ref{echid}) with $u_0 \\equiv 0$. Then,\n\\[\n \\int_0^T \\big\\{ \\<\\partial_t G_t , u_t\\> - \\mc E(G_t,u_t)\\big\\}dt =0\n\\]\nfor any differentiable trajectory in $\\mc H_{1,T}$ with $G_T=0$. Take $G_t = - \\int_t^T u_s ds$. Then $\\partial_t G_t = u_t$ and the first term above is equal to $\\int_0^T\\dt$. An approximation procedure and Fubini's theorem shows that the second term above is equal to\n\\[\n \\frac{1}{2} \\mc E\\Big(\\int_0^T u_t dt\\Big).\n\\]\n\nBoth terms are non-negative, so we conclude that $\\int_0^T \\dt =0$ and $u_t \\equiv 0$.\n\n\\section{Applications}\n\\label{s4}\nIn this section we give some examples of systems on which Theorems \\ref{t1} and \\ref{t3} apply. In the literature, the sequence $\\omega^n$ is often referred as the set of {\\em conductances} of the model. Unless stated explicitely, in these examples, $X$ will be equal to $\\bb R^d$ or the torus $\\bb T^d = \\bb R^d \/ \\bb Z^d$. The set $X_n$ will be equal to $n^{-1} \\bb Z^d$ and we construct the partitions $\\{\\mc U_x^n\\}$ in the canonical way, taking $\\mc U_x^n$ as a continuous, piecewise linear function with $\\mc U_x^n(x)=1$ and $\\mc U_x^n(y)=0$ for $y\\in X_n$, $y \\neq x$.\n\n\\subsection{Homogenization of ergodic, elliptic environments}\n\nLet $(\\Omega,\\mc F,P)$ be a probability space. Let $\\{\\tau_x; x \\in \\bb Z^d\\}$ be a family of $\\mc F$-mesurable maps $\\tau_x:\\Omega \\to \\Omega$ such that\n\\begin{description}\n\\item[i)] $P(\\tau_x^{-1} A) = P(A)$ for any $A \\in \\mc F$, $x \\in \\bb Z^d$.\n\\item[ii)] $\\tau_x \\tau_{x'} = \\tau_{x+x'}$ for any $x,x' \\in \\bb Z^d$.\n\\item[iii)] If $\\tau_x A =A$ for any $x \\in \\bb Z^d$, then $P(A)=0$ or $1$.\n\\end{description}\n\nIn this case, we say that the family $\\{\\tau_x\\}_{x \\in \\bb Z^d}$ is ergodic and invariant under $P$. Let $a=(a_1,\\dots,a_d): \\Omega \\to \\bb R^d$ be an $\\mc F$-measurable function. Assume that there exists $\\epsilon_0>0$ such that\n\\[\n\\epsilon_0 \\leq a_i(\\omega) \\leq \\epsilon_0^{-1} \\text{ for all } \\omega \\in \\Omega \\text{ and } i=1,\\dots,d.\n\\] \n\nWe say in this situation that the environment satisfies the {\\em ellipticity condition}. Fix $\\omega \\in \\Omega$. Define $\\omega^n$ by $\\omega^n_{x,x+e_i\/n}=\\omega^n_{x+e_i\/n,x}= n^2 a_i(\\tau_nx \\omega)$, $\\omega^n_{x,y}=0$ if $|y-x| \\neq 1\/n$. Here $\\{e_i\\}_i$ is the canonical basis of $\\bb Z^d$. In this case, $a_n = n^d$ and $\\mu$ is the Lebesgue measure in $\\bb R^d$.\nIn \\cite{PV}, it is proved that there is a positive definite matrix $A$ such that the quadratic form $\\mc E_n$ associated to $\\omega^n$ is $\\Gamma$-convergent to $\\mc E(f)=\\int \\nabla f \\cdot A \\nabla f dx$, $P-a.s.$ In particular, Theorem \\ref{t1} applies with $\\mc L f= \\text{div}(A \\nabla f)$. This result was first obtained in \\cite{GJ}. \n\n\\subsection{The percolation cluster}\n\nLet $e=\\{e^i_x; x \\in \\bb Z^d, i=1,\\cdots,d\\}$ be a sequence of i.i.d. random variables, with $P(e_x^i=1) = 1- P(e_x^i=0) =p$ for some $p=(0,1)$. Define for $x,y \\in X_n$, $\\omega^n_{x,x+e_i\/n}=\\omega^n_{x+e_i\/n,x}= n^2 e_{nx}^i$, $\\omega^n_{x,y}=0$ if $|y-x| \\neq 1\/n$. Fix a realization of $e$. We say that two points $x, y \\in X_n$ are connected if there is a finite sequence $\\{x_0=x,\\dots,x_l=y\\} \\subseteq X_n$ such that $|x_{i-1}-x_i|=1\/n$ and $\\omega_{x_{i-1},i}^n =1$ for any $i$. Denote by $\\mc C_0$ the set of points connected to the origin. It is well known that there exists $p_c \\in (0,1)$ such that $\\theta(p)= P(\\mc C_0 \\text{ is infinite })$ is 0 for $pp_c$. Fix $p>p_c$. Define $a_n = n^d$ and $\\mu_0(dx) = \\theta(p) dx$. In \\cite{F2}, it is proved that there exists a constant $D$ such that, $P-a.s$ in the set $\\{\\mc C_0 \\text{ is infinite }\\}$, the quadratic form $\\mc E_n$ associated to the environment $\\omega^n$ restricted to $\\mc C_0$ is $\\Gamma$-convergent to $\\mc E(f) = \\theta(p) D \\int (\\nabla f)^2 dx$. Theorem \\ref{t1} applies with $\\mc L = D \\Delta$, assuming that the initial measures $\\nu_n$ put mass zero in configurations with particles outside $\\mc C_0$.\nThis result was first obtained in \\cite{Fag}, relying on a duality representation of the simple exclusion process.\n\n\\subsection{One-dimensional, inhomogeneous environments} \n\nIn dimension $d=1$, the $\\Gamma$-convergence of $\\mc E_n$ can be studied explicitely. For nearest-neighbors environments ($\\omega^n_{x,y} =0$ if $|x-y|=1$), $\\Gamma$-convergence of $\\mc E_n$ is equivalent to convergence in distribution of the measures\n\\[\nW_n(dx) = \\frac{1}{n} \\sum_{x \\in \\bb Z} (\\omega_{x,x+1}^n)^{-1} \\delta_{x\/n}(dx).\n\\] \n\nLet $W(dx)$ be the limit. We assume that $W(dx)$ gives positive mass to any open set. For simplicity, suppose that $W(\\{0\\})=0$. Otherwise, we simply change the origin to another point with mass zero. For two functions $f,g: \\bb R \\to \\bb R$ we say that $g = df\/dW$ if\n\\[\nf(x) = f(0) + \\int_0^x g(y) W(dy).\n\\]\nThen $\\mc E_n$ is $\\Gamma$-convergent to the quadratic form defined by $\\mc E(f) = \\int (df\/dW)^2 dW$. In this case, $\\mc L = d\/dx d\/dW$. A technical difficulty appears if $W(dx)$ has atoms. In that case, there is no kernel $\\mc K$ for $\\mc L$ contained in $\\mc C_c(\\bb R)$. To overcome this point, we define for $x \\leq y$, $d_W(x,y)=d_W(y,x)= W((x,y])$. The function $d_W$ is a metric in $\\bb R$, and in general $\\bb R$ is {\\em not} complete under this metric: an increasing sequence $x_n$ converging to $x$ is always a Cauchy sequence with respect to $d_W$, but $d_W(x_n,x)\\geq W(\\{x\\})$, which is non-zero if $x$ is an atom of $W$. Define $\\bb R_W = \\bb R \\cup \\{x-; W(\\{x\\})>0\\}$. It is easy to see that $\\bb R_W$ is a complete, separable space under the natural extension of $d_W$, and that continuous functions in $\\bb R_W$ are in bijection with c\\`adl\\`ag functions in $\\bb R$ with discontinuity points contained on the set of atoms of $W(dx)$. It is not difficult to see that the set of $W$-differentiable functions in $\\mc C_c(\\bb R_W)$ is a kernel for $\\mc L$ and that Theorems \\ref{t1} and \\ref{t3} apply to this setting. In \\cite{FJL}, the remarkable case on which $W(dx)$ is a {\\em random}, self-similar measure (an $\\alpha$-stable subordinator) was studied in great detail. \n\n\\subsection{Finitely ramified fractals} \n\nLet us consider the following sequence of graphs in $\\bb R^2$. Define $a_0=(0,0)$, $a_1=(1\/2,\\sqrt 3\/2),$ and $a_2= (1,0)$ and define $\\varphi_i: \\bb R^2 \\to \\bb R^2$ by taking $\\varphi_i(x) = (x+a_i)\/2$. Define $X_0= \\{a_0,a_,a_2\\}$ and $X_{n+1}= \\cup_i \\varphi_i(X_n)$ for $n \\geq 0$. For $x,y \\in X_0$ we define $\\omega^0_{x,y}=1$, we put $\\omega_{x,y}^0=0$ if $\\{x,y\\} \\subsetneq X$ and inductively we define\n\\[\n\\omega_{x,y}^{n+1} = 5 \\sum_i \\omega^n_{\\varphi^{-1}_i(x), \\varphi^{-1}(y)}.\n\\]\n\nThe set $X_n$ is a discrete approximation of the Sierpinski gasket $X$ defined as the unique compact, non-empty set $X$ such that $X = \\cup_i \\varphi_i(X)$. Here we are just saying that $\\omega_{x,y}^n = 5^n$ if $x,y$ are neighbors in the canonical sense. In this case $a_n=3^n$ and $\\mu$ is the Hausdorff measure in $X$. It has been proved \\cite{Kig} that the quadratic forms $\\mc E_n$ converge to a certain Dirichlet form $\\mc E$ which is used to define an abstract Laplacian in $X$. In particular, Theorems \\ref{t1} and \\ref{t3} apply to this model. This result was obtained in \\cite{Jar3} in the context of a zero-range process. The same result can be proved for general {\\em finitely ramified fractals}, in the framework of \\cite{Kig}.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nStars with initial masses in the range 0.8--8~M$_{\\odot}$ \nend their life with a phase of catastrophic mass-loss.\nDuring the asymptotic giant branch (AGB) phase, they develop \na superwind leading to mass-loss rates up to \n10$^{-4}$M$_{\\odot}$\\,yr$^{-1}$. This superwind enriches the \nISM with newly synthesized elements.\n\nThe mass-loss mechanism of AGB stars is likely due to \npulsations from the star and radiation pressure on dust \ngrains. Shocks due to pulsation extend the atmosphere, so \nthat the material ejected by the star becomes dense and \ncold enough for dust to form. Due to its opacity, dust \nabsorbs the radiation from the star and is driven away by \nradiation pressure, carrying the gas along through friction.\nTheoretical models (Winters et al.\\ 2000), show that the \nmass loss evolves from a pulsation driven regime \ncharacterized by a low mass-loss rate and a slow expansion \nvelocity to a dust-driven regime with a high mass-loss rate \nand a high expansion velocity ($>$5km.s$^{-1}$).\n \nStudying the effect of metallicity on the mass-loss \nprocess is important to understand the formation of dust \naround AGB stars in the early Universe. At low metallicity, \nless seeds are present for dust formation, so one might \nexpect dust formation to be less efficient and thus the \nmass-loss rates to be lower.\n\nTheoretical work by Bowen \\& Willson (1991) predicts that for \nmetallicities below [Fe\/H]$=-1$ dust-driven winds fail, and \nthe wind must stay pulsation-driven. However, observational \nevidence for any metallicity dependence is still very limited \n(Zijlstra 2004). More recent observational (Groenewegen et \nal.\\ 2007) and theoretical works (Wachter et al.\\ 2008; \nMattsson et al. 2008) indicate that the mass-loss rates from \ncarbon stars in metal-poor environments are similar to our \nGalaxy.\n\nTo obtain the first observational evidence on mass-loss rates \nat low metallicity, we have carried out several surveys with \nthe {\\it Spitzer Space Telescope} of stars in nearby dwarf \ngalaxies. These show significant mass-loss rates down to \nZ=1\/25 Z${_\\odot}$ (Lagadec et al.\\ 2007b; Matsuura et al.\\ \n2007; Sloan et al.\\ 2009), but only for carbon-rich stars. \nThe current evidence indicates that oxygen-rich stars have \nlower mass-loss rates at lower metallicities. For carbon \nstars, no evidence for a dependency of mass-loss rate on\nmetallicity has yet been uncovered. Consequently, (Lagadec \n\\& Zijlstra 2008) have proposed that the carbon-rich dust \nplays an important role in triggering the AGB superwind.\n\nThe main uncertainty arises from the unknown expansion \nvelocity. This parameter is needed to convert the density\ndistribution to a mass-loss rate. The expansion velocity is \nalso in itself a powerful tool. Hydrodynamical simulations \n(Winters et al.\\ 2000), have shown that radiation-driven \nwinds have expansion velocity in excess of 5\\,km\\,s$^{-1}$, \nwhile pulsation-driven winds are slower.\n\nThere is some evidence that expansion velocities are lower \nat low metallicity, from measurement of OH masers (Marshall \net al.\\ 2004). However, OH masers are found only in \noxygen-rich stars, which appear to have suppressed mass-loss\nat low metallicities. We lack equivalent measurements for\nmetal-poor carbon-rich stars, which do reach substantial \nmass-loss rates. For these, the only available velocity \ntracer is CO. Currently, extra-galactic stars are too \ndistant for CO measurements. However, there are a number of \ncarbon stars in the Galactic Halo, which are believed to \nhave similarly low metallicity. One CO measurement exists\nfor a Galactic Halo star: its expansion velocity has been \nestimated to be $\\sim$3.2 km\\,s$^{-1}$ through the \n$^{12}$CO $J=2 \\rightarrow 1$ transition (Groenewegen et \nal.\\ 1997). This is much lower than typical expansion \nvelocities, which are in the range 10-40 km\\,s$^{-1}$ for stars with similar colours,\n and thus optical depths (Fig.\\ref{histo_vexp}).\n\nA large number of metal-poor carbon stars have recently \nbeen discovered in the Galactic Halo (Totten \\& Irwin 1998; \nMauron et al.\\ 2004, 2005, 2007). These stars may have been \nstripped from the Sagittarius Dwarf Spheroidal galaxy (Sgr\ndSph), which has a metallicity of [Fe\/H]$\\sim-1.1$ (Van de \nBergh 2000). These stars are the closest metal-poor carbon \nstars known, and they are bright enough to be detected in CO \nusing ground-based millimeter telescopes.\n\nWe have carried out observations of six Halo carbon stars in \nthe CO J $= 3 \\rightarrow 2$ transition. Here, we report the \nresults of these observations.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{halo_plot.ps}\n\\caption{\\label{histo_vexp} Distribution of the expansion \nvelocity for the observed Halo carbon stars compared \nwith stars from the disc with similar J$-$K colors.}\n\\end{center}\n\\end{figure}\n\n \n\n\n\n\n\\section{Sample selection}\n\nMauron et al.\\ (2004, 2005, 2007) and Mauron (2008) have \ndiscovered $\\sim$ 100 carbon stars in the Galactic Halo, \nadding to the sample of $\\sim$ 50 described by Totten \\& \nIrwin (1998). All of these stars are spectroscopically \nconfirmed carbon stars. Only the brightest can be detected \nin the sub-millimeter range. We selected the six stars with \nthe highest IRAS 12$\\mu$m flux observable with the James \nClerk Maxwell Telescope (JCMT, Mauna Kea, Hawaii). The \nemission from an AGB star at 12$\\mu$m is due to thermal \nemission from the dust in the envelope. Thus one expects \nthe stars with the largest 12$\\mu$m flux to be the brightest \nin CO. All the observed stars have 3$$\\\\\n &&&days&kpc&kpc &kpc &kpc\\\\\n\\hline\n\n\nIRAS 04188+0122 & 192.1775&-31.9867&359 &7.3 &6.0 &6.4& 6.5$\\pm$0.6\\\\\nIRAS 08427+0338 & 223.4859&+26.8173&288 &6.6 &5.3 &5.1& 5.5$\\pm$0.8\\\\\nIRAS 11308-1020 & 273.6969&+47.7772&- &2.8 &2.1 &- & 2.5$\\pm$0.5\\\\\nIRAS 16339-0317 & 012.7346&+27.7944&- &5.6 &4.2 &- & 4.9$\\pm$1.0\\\\\nIRAS 12560+1656 & 312.2528&+79.4127&- &13.3&10.7&- & 12.0$\\pm$1.9\\\\\nIRAS 18120+4530 & 073.0530&+25.3482&408 &7.6 &5.7 &6.7& 6.7$\\pm$0.9\\\\ \n\n\n\n\\hline \\\\\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{fig_all_mc.ps}\n\\caption{\\label{fig_dist} Absolute K magnitude (M$_K$) of carbon stars as a fonction of their J-K infrared colours. \nDiamonds and squares represent LMC and SMC carbon stars respectively. The vertical dashed line represent the limit \nunder which relation Eq.\\ref{dist_lmc} is no longer valid.}\n\\end{center}\n\\end{figure}\n\n\nMost of the methods to determine mass-loss rates rely on an\naccurate distance to the observed stars. To determine \nthese distances, we applied three methods. Two are based on \nnear-infrared colours; the third is the period-luminosity \nrelationship.\n\n\nThe first method uses the infrared colours of the observed stars, \nusing the relation between M$_K$ and J$-$K determined by\nSloan et al. (2008). They found from a sample of carbon\nstars in the Small Magellanic Cloud (SMC) that:\n \n\\begin{equation} \n\\label{dist_smc}\nM_K = -9.18 + 0.395(J-K) \n\\end{equation} \n\nMauron et al.\\ (2008) compared the near-infrared 2MASS \nphotometry of stars in the Halo and the Large Magellanic\nCloud (LMC) and found a different relationship from the one \nabove between J$-$K and M$_K$. Their relation gives fainter \nM$_K$ values at a given J$-$K colour than that of Sloan et \nal., with the difference increasing with redder colours. We \nhave recalibrated the relation for the LMC, using the samples \nof stars confirmed to be carbon-rich with the Infrared \nSpectrograph on {\\it Spitzer} by Zijlstra et al.\\ (2006), \nBuchanan et al.\\ (2006), Leisenring et al.\\ (2008), and Sloan \net al.\\ (2008). We find that:\n\n\\begin{equation} \n\\label{dist_lmc}\nM_K = -9.94 + 0.754(J-K),\n\\end{equation} \n\n\\noindent for J$-$K colours greater than 2.0. \n This \ncalibration of the J$-$K relation is our second method.\nFig.\\, \\ref{fig_dist} compares the SMC and LMC calibrations of the relation\nbetween M$_K$ and J$-$K.\n\n\n\n\nThe scatter in the LMC sample is 0.81 magnitudes about the\nfitted line, compared to 0.51 magnitudes for the SMC sample.\nThe two fitted lines yield nearly identical distances at\nJ$-$K = 2, but as the colour grows redder, the samples\ndiverge from each other. At J$-$K = 5, the difference in\n$M_K$ is a full magnitude. For the colours in our sample,\nthe two methods yield results differening by 0.41 to 0.64\nmagnitudes, comparable to the spread in the SMC sample.\nThe different slopes in the two samples may result from\ntheir different metallicities, but that is only speculation\non our part.\n\n\nThe third method utilises the period-luminosity relation for \ncarbon Miras described by Feast et al.\\ (2006): \n \n\\begin{equation} \nM_{\\rm bol}=-2.54 \\log P+2.06, \n\\end{equation} \nWith an uncertainty of 0.24 magnitudes. \nWe derived the bolometric magnitudes using the equation for\nbolometric correction derived by Whitelock et al.\\ (2006), \nafter converting all of the photometry to the SAAO system as \ndescribed by Lagadec et al.\\ (2008):\n \n\\begin{eqnarray} \n\\nonumber {\\rm BC_K} & = & +\\, 0.972 + 2.9292\\times(J-K) \n -1.1144\\times(J-K)^2 \\\\ \n & & +0.1595\\times(J-K)^3 -9.5689\\,10^3(J-K)^4 \n\\end{eqnarray} \n\\noindent \n\nTable \\ref{gal} presents the obtained distances, D$_1$, D$_2$ \nand D$_3$ respectively. For those stars without periods, the \nfinal estimated distance is the average of D$_1$ and D$_2$. \nFor those stars with periods, the D$_1$ and D$_2$ values\nbracket D$_3$ in two of the three cases. Consequently, we \nfirst averaged D$_1$ and D$_2$, then averaged the result \nwith D$_3$ to arrive at our final estimate of the distance.\nThe uncertainties in the final estimated distances are the the standard deviation of the individual\ndistances. \n\n\\subsection{Galactic location of the stars}\n\\label{location}\n\nKnowing the distance to the stars and their $V_{\\rm lsr}$ \nallows us to study their location in the Galaxy. All six\nstars examined here have been classified previously as \nmembers of the Halo, based on their distances from the \nGalactic plane. Our CO observations give us velocity \ninformation for these stars, which we can compare to Galactic \nrotation models. Table \\ref{gal} lists the Galactic \ncoordinates l and b.\n\nFig.\\ref{aitoff} shows the location of our six carbon stars\non an Aitoff projection. The dashed line schematically \nrepresents the Sgr dSph orbit (Ibata et al. 2001). One of \nour stars, IRAS\\,12560+1656, lies very close to this orbit. Its \n$V_{\\rm lsr}$ is in the range of the observed $V_{\\rm lsr}$ \nfor stars in the Sgr dSph stream. IRAS\\,12560+1656 very likely belongs \nto this stream. Our observations are thus certainly the \nfirst CO observations of an extragalactic AGB star. Two \nother stars, IRAS\\,16339-0317 and IRAS\\,18120+4530, have large negative \n$V_{\\rm lsr}$, fully consistent with membership in the \nHalo. Finally, IRAS\\,04188+0122, IRAS\\,08427+0338, and IRAS\\,11308-1020 have a distance, \nlocation and $V_{\\rm lsr}$ consistent with membership in the \nthick disc. These three last stars thus have a metallicity \nbetween that of the thin Galactic disc and the Galactic \nHalo, the average metallicity of the thin disc being $\\sim$-0.17 while the one of the thick disc is \n$\\sim$-0.48 (Soubiran et al., 2003).\nOur sample thus contains three metal-poor AGB stars and three \nAGB stars with intermediate metallicity.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=13cm,angle=-90]{zz-myplot-last1.ps}\n\\caption{\\label{aitoff} The location of the six carbon\nstars on an Aitoff projection of the sky. Open circles \nrepresent Halo carbon stars from Mauron et al.\\ (2004, \n2005, 2007) and Totten \\& Irwin (1998) with J$-$K $>$1.2 \nand K$>$ 6 to eliminate carbon stars in the disc. Filled \nsymbols represent the six stars in our sample.}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Low expansion velocities in the Halo}\n\n\\label{lowvexp}\nThe present observations allow us to directly measure the \nexpansion velocity for a sample of carbon stars in the Halo.\nThe stars we observed are quite red (3$<$J$-$K$<$4), and have \nsubstantial circumstellar dusty envelopes responsible \nfor the observed reddening. Our measured expansion \nvelocities are in the range 3--16.5 km\\,s$^{-1}$. The \nexpansion velocity of carbon stars increases during the \nevolution on the AGB (Scho\\\"ier, 2007), i.e.\\ when the dusty \nenvelope becomes optically thicker. To compare the expansion \nvelocities we measured in Halo carbon stars with carbon stars\nin the disc, we took a sample of carbon stars in the disc \nwith colours similar to our sample. We selected all of the\ncarbon stars with 3$<$J$-$K$<$4 in the extensive catalogue of \nCO observations of evolved stars by Loup et al.\\ (1993). \nFig.\\ref{histo_vexp} compares the distribution of expansion \nvelocities in the two samples. The Halo carbon stars clearly\nhave a lower mean expansion velocity.\n\nThe three stars in the Halo and the Sgr dSph stream have \n$V_{\\rm exp}$ in the range 3--8.5 km\\,s$^{-1}$, while the \nthree stars associated with the thick disc have velocities \nranging from 11.5 to 16.5 km\\,s$^{-1}$. The latter range is\nat the low end of expansion velocities for AGB stars with\nsimilar near-infrared colours in the thin disc.\n\n\\subsection{Origin of the low expansion velocities}\n\nSection \\ref{location} and \\ref{lowvexp} have shown that the \nthree stars we observed in the Halo and Sgr dSph stream have \nlow expansion velocities. The stars we observed in the thick \ndisc have expansion velocities intermediate between those \ntypically observed in the Halo and the thin disc. This \ndifference could arise from differences in metallicity, with\nthe more metal-poor carbon stars having the slower winds.\n\nMattsson et al.\\ (2008) and Wachter et al.\\ (2008) have \nrecently conducted theoretical investigations of the winds \nfrom metal-poor carbon stars. Both studies show that \nmetal-poor carbon stars can develop high mass loss rates, \nleading to the formation of a large dusty envelope, in \nagreement with spectroscopic observations from {\\it Spitzer} \nof AGB stars in metal-poor galaxies (Zijlstra et al.\\ 2006; \nSloan et al.\\ 2006; Groenewegen et al.\\ 2007; Lagadec et \nal.\\ 2007; Matsuura et al.\\ 2007; Leisenring et al.\\ 2008;\nLagadec et al.\\ 2009; Sloan et al.\\ 2009). \n\nWachter et al.\\ (2008) predict that the outflow velocities \nfrom carbon stars should be lower in metal-poor environments,\nbecause of the lower gas-to-dust mass ratio and because the \nformation of less dust leads to less efficient acceleration \nof the wind outside of the sonic region. This interpretation\nis consistent with our interpretation that the low expansion \nvelocities we have observed in the Halo are due to their low \nmetallicities.\n\nHydrodynamical models (Winters et al.\\ 2000, Wachter et al.\\ \n2008) distinguish two types of models. Model A applies to\ncases where the radiation pressure on dust is efficient. \nMass-loss rates can exceed 10$^{-7}$M$_{\\odot}$\\,yr$^{-1}$, \nand expansion velocities can climb above $\\sim$5 km\\,$s^{-1}$.\nModel B applies to cases where pulsations drive the mass loss.\nIn these cases, the mass-loss rates and expansion velocities\nare smaller. The mass loss occurs in a two-step process, with\nstars first losing mass due to pulsations, followed by \nacceleration due to radiation pressure on the dust grains. The \nhigh mass-loss rate and low expansion velocity of TI~32 does not \nfit either of these models, possibly because metal-poor carbon\nstars can develop strong mass-loss from pulsation alone. \n\n\n\\section{Conclusions and perspectives}\n\nWe have detected the CO J\\,$=$\\,3$\\rightarrow$\\,2 in six carbon stars \nselected as Halo stars. Only one carbon star had been \ndetected in CO previously. Comparison of the infrared \nobservations and radiative transfer models indicates that \nthese stars are losing mass and producing dust. Their \nmass-loss rates are larger than their nuclear burning rates, \nso their final evolution will be driven by this mass-loss \nphenomenon.\n\nWe show that three of the observed stars are certainly \nmembers of the thick disc, while one is in the Sgr dSph stream \nand two are in the Halo. The CO observation of the Sgr dSph \nstream star are thus the first identified millimetre \nobservations of an extragalactic AGB star. The expansion \nvelocity we determined from our CO observations are lower \nthan those of carbon stars in the thin disc with similar \nnear-infrared colours. The observed carbon stars with the \nlowest expansion velocities are Halo or Sgr dSph stream carbon \nstars. There is a strong indication that the expansion winds \nare lower in metal-poor environments, which agrees with recent \ntheoretical models (Wachter et al.\\ 2008). \n\nSo far, the effect of metallicity on the mass-loss from \ncarbon-rich AGB stars has studied primarily from infrared \nobservations of AGB stars in Local Group galaxies, mostly \nwith the {\\it Spitzer Space Telescope}. Infrared \nobservations measure the infrared excess, which can \nbe converted to a mass-loss rates assuming an expansion \nvelocity for the circumstellar material. So far all of the \nmass-loss rates have been estimated using the assumptions \nthat the expansion velocity is independent of the \nmetallicity. The results presented here show that this \nassumption needs to be reconsidered.\n\nWe have recently obtained spectra of some of the present\nsample with the Infrared Spectrograph on {\\it Spitzer}.\nCombining our CO observations and these spectra together \nand comparing them to {\\it Spitzer} spectra from carbon \nstars in other galaxies in the Local Group will allow us \nto quantitatively study the mass-loss from carbon-rich AGB \nstars in the Local Group and its dependence on metallicity. \nThis will be the subject of a forthcoming paper.\n\n\n\n\n\\section*{Acknowledgments}\nEL wishes to thank the JAC staff for their great help \ncarrying out this program, and Rodrigo Ibata for useful \ndiscussions about membership of stars to the Sgr stream. \nwe thank the referee C. Loup for her useful comments that helped improving the quality of the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn~\\cite{Polyak} Polyak suggested a quantization $l_q(L)\\in\n\\frac{1}{2}{\\mathbb Z}[q,q^{-1}]$ of the Bennequin \ninvariant of a\ngeneric cooriented oriented wave front $L\\subset{\\mathbb R}^2$. \nIn this paper we construct an \ninvariant $S(L)$ which is, in a sense, \na generalization of $l_q(L)$ to the case of\na wave front on an arbitrary surface $F$. \n\nIn the same paper~\\cite{Polyak} Polyak \nintroduced Arnold's~\\cite{Arnold} $J^+$-type\ninvariant of a front $L$ on an oriented surface $F$. It takes values in\n$H_1(ST^*F,\\frac{1}{2}{\\mathbb Z})$. \nWe show that $S(L)\\in \\frac{1}{2}{\\mathbb Z}[H_1(ST^*F)]$ is a refinement of this\ninvariant in the sense that it is taken to \nPolyak's invariant under the natural mapping \n$\\frac{1}{2}{\\mathbb Z}[H_1(ST^*F)]\\rightarrow H_1(ST^*F,\\frac{1}{2}{\\mathbb Z})$. \n\nFurther we generalize $S(L)$ to the case where $L$ is a wave front\non an orbifold.\n\nInvariant $S(L)$ is constructed in two steps. The first one consists in \nlifting of $L$ to the Legendrian knot\n$\\lambda$ in the $S^1$-fibration\n$\\pi:ST^*F\\rightarrow F$. The second step can be applied to any knot in an\n$S^1$-fibration, and \nit involves the structure of the fibration in a\ncrucial way. This step allows us to define the\n$S_K$ invariant of a knot $K$ in the total space $N$ of an \n$S^1$-fibration. Since ordinary knots are considered up to a rougher\nequivalence relation (ordinary isotopy versus Legendrian isotopy), in order\nfor $S_K$ to be well defined it has to take values in a quotient of \n${\\mathbb Z}[H_1(N)]$. This invariant is\ngeneralized to the case of a knot in a Seifert fibration, and this allows\nus to\ndefine $S(L)$ for wave fronts on orbifolds.\n\nAll these invariants are Vassiliev invariants of order one in an\nappropriate sense. \n\nFor each of these invariants we introduce its version with values in \nthe group of formal linear combinations of the \nfree homotopy classes of oriented curves in the\ntotal space of the corresponding fibration.\n\nThe first invariants of this kind were constructed by\nFiedler~\\cite{Fiedler} in the case\nof a knot $K$ in a ${\\mathbb R}^1$-fibration over a surface and by Aicardi in the\ncase of\na generic oriented cooriented wave front $L\\subset {\\mathbb R}^2$. \nThe connection between\nthese invariants and $S_K$ is discussed in~\\cite{Tchernov}. \n\n\n\nThe space $ST^*F$ is naturally fibered over a surface $F$ with a fiber\n$S^1$. In~\\cite{Turaev} Turaev introduced a shadow description \nof a knot $K$ in\nan oriented three dimensional manifold $N$ fibered over an oriented surface \nwith a fiber \n$S^1$. A shadow presentation of a knot $K$ is a generic projection of $K$,\ntogether with an assignment of numbers to regions. It describes a knot type\nmodulo a natural action of $H_1(F)$.\nIt appeared to be a very useful\ntool. Many invariants of knots in $S^1$-fibrations, in particular quantum\nstate sums, can be expressed\nas state sums for their shadows. In this paper\nwe construct shadows of Legendrian liftings of wave fronts. \nThis allows one to use any\ninvariant already known for shadows in the case of wave fronts. \n\nHowever, in this paper shadows are used mainly for the purpose of depicting\nknots in $S^1$-fibrations.\n\n\n\n\n\\section{Shadows}\\label{sh-def}\n\\subsection{Preliminary constructions}\\label{prelim}\n\nWe say that a one-dimensional submanifold $L$ of a total space $N^3$ of \na fibration \n$\\pi:N^3\\rightarrow F^2$ is {\\em generic\nwith respect to $\\pi$\\\/ } if $\\pi\\big|_L$ is a generic immersion.\nRecall that an immersion of 1-manifold into a surface is said to\nbe {\\em generic\\\/} if it has neither self-intersection points with\nmultiplicity greater than $2$ nor self-tangency points, and at\neach double point its branches are transversal to each other. An\nimmersion of (a circle) $S^1$ to a surface is called a {\\em curve\\\/}.\n\n\n\nLet $\\pi$ be an oriented $S^1$-fibration of $N$ over an oriented\nclosed surface $F$.\n\n$N$ admits a fixed point free involution which preserves fibers.\nLet $\\tilde N$ be the quotient of $N$ by\nthis involution, and let $p:N\\rightarrow\\tilde{N}$ be the corresponding double\ncovering. Each fiber of $p$ (a pair of antipodal points) is\ncontained in a fiber of $\\pi$. Therefore, $\\pi$ factorizes through\n$p$ and we have a fibration $\\tilde{\\pi}:\\tilde{N}\\rightarrow F$.\nFibers of $\\tilde\\pi$ are projective lines. They are\nhomeomorphic to circles.\n\n\nAn isotopy of a link $L\\subset N$ is said to be vertical with respect to\n$\\pi$ if each point of $L$ moves along a fiber of $\\pi$. It is clear\nthat if two links are vertically isotopic, then their projections\ncoincide. Using vertical isotopy we can modify each generic\nlink $L$ in such a way that any two points of $L$ belonging to the\nsame fiber lie in the same orbit of the involution. \nDenote the obtained generic link by $L'$.\n\n\nLet $\\tilde{L}=p(L')$. It is obtained from $L'$ by gluing together\npoints lying over the same point of $F$. Hence $\\tilde{\\pi}$ maps\n$\\tilde{L}$ bijectively to $\\pi(L)=\\pi(L')$. Let $r:\\pi (L)\\rightarrow\n\\tilde{L}$ be an inverse bijection. It is a section of\n$\\tilde{\\pi}$ over $\\pi (L)$.\n\nFor a generic non-empty collection of curves on a surface by a {\\em region\\\/}\nwe mean the closure of a connected component of the complement of\nthis collection. Let $X$ be a region for $\\pi (L)$ on $F$. Then \n$\\tilde{\\pi}\\big|_X$ is a trivial fibration. Hence we can identify it with\nthe projection $S^1 \\times X\\rightarrow X$. Let $\\phi$ be a\ncomposition of the section $r\\big|_{\\partial X}$ with the projection to $S^1$.\nIt maps $\\partial X$ to $S^1$.\nDenote by $\\alpha_X$ the degree of $\\phi$. (This is actually an\nobstruction to an extension of $r\\big|_{\\partial X}$ to $X$.) One can\nsee that $\\alpha_X$ does not depend on the choice of the trivialization\nof $\\tilde{\\pi}$ and on the choice of $L'$.\n\n\\subsection{Basic definitions and properties}\n\\begin{defin}\\label{def-shadowlink}\nThe number $\\frac{1}{2}\\alpha _X$ corresponding to a region $X$ is\ncalled the {\\em gleam\\\/} of $X$ and is denoted by $\\protect\\operatorname{gl}(X)$.\nA {\\em shadow\\ $s(L)$ of a generic link $L\\subset N$\\\/} is a (generic)\ncollection of curves $\\pi (L)\\subset F$ with the gleams assigned to each\nregion $X$. The sum of\ngleams over all regions is said to be the {\\em total gleam\\\/} of the\nshadow. \n\\end{defin}\n\n\\begin{emf}\\label{prop1-gleam}\nOne can check that for any region $X$ the integer $\\alpha _X$ is congruent\nmodulo 2 to the number of corners of $X$. Therefore, $\\protect\\operatorname{gl}(X)$ is an integer if\nthe region $X$ has even number of corners and half-integer otherwise.\n\\end{emf}\n\n\\begin{emf}\\label{prop2-gleam}\nThe total gleam of the shadow is equal to the Euler number of $\\pi$.\n\\end{emf}\n\n\\begin{defin}\\label{def-shadow}\nA {\\em shadow \\\/} on $F$ is a generic collection of curves together\nwith the numbers $\\protect\\operatorname{gl}(X)$ assigned to each region $X$. These numbers\ncan be either integers or half-integers, and they should satisfy the\nconditions of~\\ref{prop1-gleam}~and~\\ref{prop2-gleam}.\n\\end{defin}\n\nThere are three local moves $S_1,S_2$, and $S_3$ of shadows shown in\nFigure~\\ref{shad3.fig}. They are similar to the well-known\nRiedemeister moves of planar knot diagrams.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{shad3.eps}\n \\end{center}\n\\caption{}\\label{shad3.fig}\n\\end{figure}\n\n\\begin{defin} Two shadows are said to be {\\em shadow equivalent\\\/} if\nthey can be transformed to each other by a finite sequence of moves\n$S_1,S_2,S_3$, and their inverses.\n\\end{defin}\n\n\\begin{emf}\n There are two more important shadow moves $\\bar S_1$ and $\\bar S_3$ \nshown in Figure~\\ref{shad5.fig}. They are similar to the previous versions of \nthe first and the third Riedemeister\nmoves and can be expressed \nin terms of $S_1,S_2$, $S_3$, and their inverses.\n\\end{emf}\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{shad5.eps}\n \\end{center}\n\\caption{}\\label{shad5.fig}\n\\end{figure}\n\n\n\\begin{emf}\\label{actionconstruct}\nIn~\\cite{Turaev} the action of \n$H_1(F)$ on the set of all isotopy\ntypes of links in $N$ is constructed as follows. \nLet $L$ be a generic link in $N$ and $\\beta$ an\noriented (possibly self-intersecting) curve on $F$ presenting\na homology class $[\\beta ]\\in H_1(F)$. Deforming $\\beta $ we can\nassume that $\\beta $ intersects $\\pi(K)$ transversally at a finite\nnumber of points different from the self-intersection points of $\\pi(K)$.\nDenote by $\\alpha =[a,b]$ a small segment of $L$ such that\n$\\pi(\\alpha )$ contains exactly one intersection point $c$ of $\\pi(L)$\nand $\\beta $. Assume that $\\pi(a)$ lies to the left, and\n$\\pi(b)$ to the right of $\\beta $. Replace $\\alpha $ by the arc\n$\\alpha '$ shown in Figure~\\ref{shad13.fig}.\n\\begin{figure}[htbp]\n\\begin{center}\n\\epsfxsize 5cm\n \\leavevmode\\epsffile{shad13.eps}\n \\end{center}\n\\caption{}\\label{shad13.fig}\n\\end{figure}\n We will call this\ntransformation of $L$ a {\\em fiber fusion\\\/} over the point $c$.\nAfter we apply fiber fusion to $L$ over all points of $\\pi(L)\\cap\\beta$\nwe get a new generic link $L'$ with $\\pi(L)=\\pi(L')$. One can notice that\nthe shadows of $K$ and $K'$ coincide. Indeed, each time $\\beta $\nenters a region $X$ of $s(L)$, it must leave it. \nHence the contributions of the newly inserted arcs to\nthe gleam of $X$ cancel out. Thus links belonging to one\n$H_1(F)$-orbit always produce the same shadow-link on $F$.\n\n\\end{emf}\n\n\\begin{thm}[Turaev \\cite{Turaev}]\\label{action}\nLet $N$ be an oriented closed manifold, $F$ an oriented surface, and \n$\\pi:N\\rightarrow F$ an\n$S^1$-fibration with the Euler number $\\chi (\\pi)$. The\nmapping that associates to each link $L\\subset N$ its shadow equivalence\nclass on $F$ establishes a bijective correspondence between the\nset of isotopy types of links in $N$ modulo the action of\n$H_1(F)$ and the set of all shadow equivalence classes on $F$ \nwith the total gleam $\\chi(\\pi)$. \n\\end{thm}\n\n\n\\begin{emf}\\label{homolfusion}\nIt is easy to see that all links whose projections represent \n$0\\in H_1(F)$ and whose shadows coincide are homologous \nto each other. To prove this,\none looks at the description of a fiber fusion and notices that to each\nfiber fusion where we add a positive fiber corresponds another where we\nadd a negative one. Thus the numbers of positively and negatively \noriented fibers we add are equal, and they cancel\nout.\n\\end{emf}\n\n\\begin{emf}\\label{shadowgeneral}\nAs it was remarked in~\\cite{Turaev} it is easy to transfer the construction \nof shadows and Theorem~\\ref{action} to the \ncase where $F$ is a non-closed oriented surface and $N$ is an oriented\nmanifold. In order\nto define the gleams of the regions that have a non-compact closure or\ncontain components of $\\partial F$, we have to choose a section of the fibration \nover all boundary components and ends of $F$. In the case of non-closed $F$\nthe total gleam of the shadow is equal to the obstruction \nto the extension of the section to the entire surface.\n\\end{emf}\n\n\n\\section{Invariants of knots in $S^1$-fibrations.}\n\\subsection{Main constructions}\n\nIn this section we deal with knots in an $S^1$-fibration\n$\\pi$ of an oriented three-dimensional manifold $N$ over an oriented \nsurface $F$. We do not assume $F$ and $N$ to be\nclosed. As it was said in~\\ref{shadowgeneral}, all theorems \nfrom the previous section are applicable in this case.\n\n\n\n\n\n\n\n\\begin{defin}[of $S_K$]\\label{SK} Orientations of $N$ and $F$ determine an \norientation of a fiber of the fibration. Denote by $f\\in H_1(N)$ \nthe homology class of a positively oriented fiber.\n\n\nLet $K\\subset N$ be an oriented knot which is generic with respect to $\\pi$.\nLet $v$ be a double point of $\\pi (K)$. The fiber $\\pi^{-1}(v)$ divides $K$ \ninto two arcs that inherit the orientation from $K$. Complete each arc\nof $K$ to an oriented knot by adding the arc of $\\pi^{-1}(v)$ \nsuch that the orientations of these\ntwo arcs define an orientation of their union. The orientations of $F$ and\n$\\pi(K)$ allow one to identify a small neighborhood of $v$ in $F$ with a\nmodel picture shown in Figure~\\ref{shad1.fig}a. Denote the knots\nobtained by the operation above by $\\mu^+_v$ and $\\mu^-_v$ as shown in\nFigure~\\ref{shad1.fig}. We will often call this construction a {\\em\nsplitting\\\/} of $K$ (with respect to the orientation of $K$).\n\nThis splitting can be described in terms of shadows as follows.\nNote that $\\mu ^+_v$ and $\\mu ^+_v$ are not in general position. We slightly \ndeform them in a neighborhood of $\\pi^{-1}(v)$, so that\n$\\pi(\\mu ^+_v)$ and $\\pi(\\mu ^+_v)$ \ndo not have double points in the neighborhood of $v$.\nLet $P$ be a neighborhood of $v$ in $F$\nhomeomorphic to a closed disk. \nFix a section over $\\partial P$ such that the intersection\npoints of $K\\cap\\pi^{-1}(\\partial P)$ belong to the section. Inside $P$ we can\nconstruct Turaev's shadow (see \\ref{shadowgeneral}). \nThe action of $H_1(\\protect\\operatorname{Int} P)=e$ \non the set of the isotopy types of links is trivial. \nThus the part of $K$ can be reconstructed in the unique way \n(up to an isotopy fixed on $\\partial P$) from the shadow over $P$\n(see~\\ref{shadowgeneral}). \nThe shadows for $\\mu^+_v$ and $\\mu^-_v$ are \nshown in Figures~\\ref{shad1.fig}a~and~\\ref{shad1.fig}b respectively. \n\n\n\\begin{figure}[htb]\n\\begin{center}\n\n \\epsfxsize 10 cm\n\\leavevmode\\epsffile{shad1.eps}\n \\end{center}\n\\caption{}\\label{shad1.fig}\n\\end{figure}\n\n\nRegions for the shadows $s(\\mu ^+_v)$ and \n$s(\\mu ^-_v)$ are, in fact, unions of regions for $s(K)$. One should think of \ngleams as of measure, so that the gleam of a region is the sum \nof all numbers inside.\n\n\nLet $H$ be the quotient of the group ring ${\\mathbb Z} [H_1(N)]$ (viewed as a\n${\\mathbb Z}$-module) \nby the submodule generated by $\\bigl\\{ [K]-f, [K]f-e \\bigr\\} $. \nHere by $[K]\\in H_1(N)$ we denote the homology class represented by the image\nof $K$.\n\nFinally define $S_K\\in H$ by the following\nformula, where the summation is taken over all double points $v$\nof $\\pi (K)$:\n\n\\begin{equation}\\label{eqSK}\nS_K=\\sum_v \\bigl([\\mu ^+_v]-[\\mu ^-_v]\\bigr).\n\\end{equation}\n\\end{defin}\n\n\\begin{emf}\\label{sum-property} Since $\\mu^+_v\\cup\\mu^-_v=K\\cup\\pi^{-1}(v)$\nwe have\n\\begin{equation}\n[\\mu ^+_v][\\mu ^-_v]= [K]f.\n\\end{equation}\n\\end{emf}\n\n\n\n\\begin{thm}\\label{correct2} $S_K$ is an isotopy invariant of the knot $K$.\n\\end{thm}\n\n\nFor the proof of Theorem~\\ref{correct2} see Subsection~\\ref{pfcorrect2}.\n\n\n\n\\begin{emf}\\label{stupid}\nIt follows from~\\ref{sum-property} \nthat $S_K$ can also be described as an element of ${\\mathbb Z}[H_1(N)]$ equal to the\nsum of $\\bigl([\\mu^+_v]-[\\mu^-_v]\\bigr)$ over all double points for \nwhich the sets $\\{ [\\mu ^+_v],[\\mu ^-_v] \\}$ and $\\{ e,f \\}$ are disjoint. \nNote\nthat in this case we do not need to factorize ${\\mathbb Z}[H_1(N)]$ to make $S_K$\nwell defined.\n\\end{emf}\n\n\n\\begin{emf}\\label{homotopvers}\nOne can obtain an invariant similar to $S_K$ with\nvalues in the free ${\\mathbb Z}$-module generated by the set of all free \nhomotopy classes of oriented curves in $N$. To do this one substitutes\nthe homology classes of $\\mu ^+_v$ and $\\mu ^-_v$ in~\\eqref{eqSK} \nwith their free homotopy\nclasses and takes the summation over the set of all double points $v$ of \n$\\pi (K)$ such that neither one of the knots $\\mu ^+_v$ and $\\mu ^-_v$ \nis homotopic to\na trivial loop and neither one of them is homotopic to a positively\noriented fiber\n(see~\\ref{stupid}).\n\nTo prove that this is indeed an invariant of $K$ one can easily modify the\nproof of Theorem~\\ref{correct2}. \n\\end{emf}\n\n\\subsection{$S_K$ is a Vassiliev invariant of order one}\n\n\\begin{emf} Let $\\pi:N\\rightarrow F$ be an $S^1$-fibration over a\nsurface. Let $K\\subset N$ be a knot generic with respect to $\\pi$ and $v$ a\ndouble point of $\\pi(K)$. A modification of pushing of one branch of $K$ \nthrough the other along the fiber $\\pi ^{-1}(v)$ \nis called a {\\em modification of $K$ along the fiber\\\/} $\\pi^{-1}(v)$.\n\\end{emf}\n\n\\begin{emf}\\label{helpdelta}\nIf a fiber fusion increases by one the gleam $\\gamma$ \nin Figure~\\ref{shad1.fig}b, then $[\\mu ^+_v]$ is multiplied by $f$. \nIf a fiber fusion increases by one the gleam $\\alpha$ \nin Figure~\\ref{shad1.fig}c, then $[\\mu ^-_v]$ is multiplied by $f^{-1}$. \nThese facts are easy to verify.\n\\end{emf} \n\n\n\\begin{emf}\\label{vassiliev2} \nLet us find out how $S_K$ changes under the modification along a fiber over a\ndouble point $v$. Consider a singular knot $K'$\n(whose only singularity is a point $v$ of transverse\nself-intersection). Let $\\xi_1$ and $\\xi_2$ be the homology classes of the\ntwo loops of $K'$ adjacent to $v$. The two\nresolutions of this double point correspond to adding $\\pm\\frac{1}{2}$ to\nthe gleams of the regions adjacent to $v$ in two ways shown in\nFigures~\\ref{shad14.fig}b and~\\ref{shad14.fig}c. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 12cm\n \\leavevmode\\epsffile{shad14.eps}\n \\end{center}\n\\caption{}\\label{shad14.fig}\n\\end{figure} \n\nUsing~\\ref{helpdelta} one verifies that under the corresponding modification \n$S_K$ changes by \n\\begin{equation}\\label{type2}\n(f-e)(\\xi_1+\\xi_2).\n\\end{equation}\n\n\nThis means that the first derivative of $S_K$ depends\nonly on the homology classes of the two loops adjacent to the singular\npoint. Hence the second derivative of $S_K$ is $0$. Thus it is a Vassiliev \ninvariant of order one in the usual sense.\n\n\n\nFor similar reasons\nthe version of $S_K$ with values in the free ${\\mathbb Z}$-module generated \nby all free homotopy classes of oriented curves in $N$ is also a Vassiliev\ninvariant of order one.\n\\end{emf}\n\n\n\\begin{thm}\\label{realization2}$ $\n\\begin{description}\n\\item[\\textrm{I}] If $K$ and $K'$ are two knots representing the same free\nhomotopy class, then $S_{K}$ and $S_{K'}$ are congruent modulo \nthe submodule generated by elements of form\n\\begin{equation}\\label{type}\n(f-e)(j+[K]j^{-1})\n\\end{equation}\nfor $j\\in H_1(N)$.\n\n\\item[\\textrm{II}] If $K$ is a knot, and $S\\in H$ is congruent to $S_{K}$ modulo the \nsubmodule generated by elements of \nform~\\eqref{type} (for $j\\in H_1(N)$), \nthen there exists a knot $K'$ such that:\n\n\\begin{description}\n\\item[a] $K$ and $K'$ represent the same free homotopy class;\n\n\n\\item[b] $S_{K'}=S$.\n\\end{description}\n\\end{description}\n\\end{thm}\n\nFor the proof of Theorem~\\ref{realization2} see\nsubsection~\\ref{pfrealization2}. \n\n\n\\subsection{Example.}\nIf $N$ is a solid torus $T$ fibered over a disk, \nthen we can calculate the value\nof $S_K$ directly from the shadow of $K$.\n\n\n\\begin{defin} Let $C$ be an oriented closed curve in ${\\mathbb R}^2$ and $X$ a\nregion for $C$. Take a point $x\\in\\protect\\operatorname{Int} X$ and connect it to a point near\ninfinity by a generic oriented path $D$. \nDefine the sign of an intersection point of $C$ and $D$ as \nshown in Figure~\\ref{shad12.fig}. Let $\\protect\\operatorname{ind}_C X$ be the sum over\nall intersection points of $C$ and $D$ of the signs of these points.\n\\end{defin}\n\n\\begin{figure}[htb]\n\\begin{center}\n \\epsfxsize\\hsize\\advance\\epsfxsize -0.5cm\n \\leavevmode\\epsffile{shad12.eps}\n \\end{center}\n\\caption{}\\label{shad12.fig}\n\\end{figure}\n\nIt is easy to see that $\\protect\\operatorname{ind}_C(X)$ is independent on the choices \nof $x$ and $D$.\n\n\\begin{defin} Let $K\\subset T$ be an oriented knot which is\ngeneric with respect to $\\pi$, and let $s(K)$ be its\nshadow. Define $\\sigma(s(K))\\in {\\mathbb Z}$ as the following sum \nover all regions $X$ for $\\pi (K)$: \n\\begin{equation}\n\\sigma(s(K))=\\sum_X\\protect\\operatorname{ind}_{\\pi(K)}(X)\\protect\\operatorname{gl}(X).\n\\end{equation}\n\\end{defin}\n\n\nDenote by $h\\in {\\mathbb Z}$ the image of $[K]$ under the natural identification of \n$H_1(T)$ with ${\\mathbb Z}$. \n\n\\begin{lem}\\label{homtor}\n$\\sigma(s(K))=h.$ \n\\end{lem}\n\n\\begin{emf}Put \n\n\\begin{equation}\nS'_K=\\sum t^{\\sigma(s(\\mu ^+_v))}-t^{\\sigma(s(\\mu ^-_v))},\n\\end{equation}\nwhere the sum is taken over all double points $v$ of $\\pi (K)$\nsuch that $\\{ 0,1\\}$ \nand $\\{ \\sigma(s(\\mu ^+_v)),\\sigma(s(\\mu ^-_v))\\}$ are disjoint\n(see~\\ref{stupid}).\n \nLemma~\\ref{homtor} implies that $S'_K$ is the image of $S_K$ under the\nnatural identification of ${\\mathbb Z}[H_1(T)]$ with the ring of\nfinite Laurent polynomials (see~\\ref{stupid}).\n\n\\end{emf}\n\nOne can show~\\cite{Tchernov} that $S'_K$ and Aicardi's partial\nlinking polynomial of $K$ (which was introduced in~\\cite{Aicardi})\ncan be explicitly expressed in terms of each other. \n\n\n\n\\subsection{Further generalizations of the $S_K$ invariant}\nOne can show that an invariant similar to $S_K$ can be introduced in the\ncase where $N$ is oriented and $F$ is non-orientable.\n\n\\begin{defin}[of $\\tilde S_K$]\\label{tildeSK}\nLet $N$ be oriented and $F$ non-orientable.\nLet $K\\subset N$ be an oriented knot generic with respect to $\\pi$, and \nlet $v$ be a double point of $\\pi(K)$. Fix an orientation of a small\nneighborhood of $v$ in $F$. Since $N$ is oriented\nthis induces an orientation\nof the fiber $\\pi^{-1}(v)$. Similarly to the definition of $S_K$\n(see~\\ref{SK}), we split our knot with respect to the \norientation and obtain two\nknots $\\mu_1^+(v)$ and $\\mu_1^-(v)$. Then we take the other orientation \nof the neighborhood of $v$ in $F$, and in the same way we obtain another pair\nof knots $\\mu_2^+(v)$ and $\\mu_2^-(v)$. The element \n$\\bigl( [\\mu_1^+(v)]-[\\mu_1^-(v)]+[\\mu_2^+(v)]-[\\mu_2^-(v)]\\bigr)\\in\n{\\mathbb Z}[H_1(N)]$ does not depend on which orientation of the neighborhood of $v$\nwe choose first. \n\nSimilarly to the definition of $S_K$, we can describe all\nthis in terms of shadows as it is shown in Figure~\\ref{shadun2.fig}.\nThese shadows are constructed with respect to the same orientation of the\nneighborhood of $v$.\n\n\n\n\nLet $f$ be the homology class of a fiber of $\\pi$ oriented in some way. \nAs one can easily prove $f^2=e$, so it\ndoes not matter which orientation we choose to define $f$.\nLet $\\tilde H$ be the quotient of ${\\mathbb Z} [H_1(N)]$ (viewed as a ${\\mathbb Z}$-module) by the \n${\\mathbb Z}$-submodule\ngenerated by $\\Bigl \\{ [K]-f+e-[K]f=(e-f)([K]+e) \\Bigr\\}$.\nFinally define $\\tilde S_K\\in \\tilde H$ by the\nfollowing formula, where the summation is taken \nover all double points\n$v$ of $\\pi (K)$: \n\\begin{equation}\\label{eqtildeSK}\n\\tilde \nS_K=\\sum_v \\Bigl([\\mu _1^+(v)]-[\\mu _1^-(v)]+[\\mu _2^+(v)]-[\\mu\n_2^-(v)]\\Bigr).\n\\end{equation}\n\\end{defin}\n\n\\begin{figure}[htb]\n \\begin{center}\n\n \\epsfxsize 12cm\n \\leavevmode\\epsffile{shadun2.eps}\n \\end{center}\n\\caption{}\\label{shadun2.fig}\n\\end{figure}\n\n\\begin{thm}\\label{correct3} $\\tilde \nS_K$ is an isotopy invariant of the knot $K$.\n\\end{thm}\n\nThe proof is essentially the same as the proof of Theorem~\\ref{correct2}.\n \n\n\n\\begin{emf} One can easily prove that $\\tilde S_K$ \ninvariant satisfies\nrelations similar to~\\eqref{type2}. In particular, $\\tilde S_K$ is also a\nVassiliev invariant of order one. \n\nOne can introduce a version of this invariant with values in the free\n${\\mathbb Z}$-module generated by all free homotopy classes of oriented curves in \n$N$. \nTo do this, we substitute the homology classes of $\\mu _1^+(v),$\n$\\mu _1^-(v),$ $\\mu _2^+(v)$, and $\\mu_2^-(v)$ with the corresponding \nfree homotopy classes. The summation should be taken over the set of all\ndouble points of $\\pi(K)$ for which neither one of $\\mu _1^+(v),\n\\mu _1^-(v),\\mu _2^+(v)$, and $\\mu_2^-(v)$ is homotopic to a trivial loop and\nneither one of them is homotopic to a fiber of $\\pi$. To prove that this is\nindeed an invariant of $K$, one easily modifies the proof of\nTheorem~\\ref{correct2}. \n\\end{emf} \n\n\n\n\n\n\n\n\\section{Invariants of knots in Seifert fibered spaces}\n\nLet $(\\mu, \\nu)$ be a pair of relatively prime integers. Let \n$$D^2=\\Bigl\\{ (r,\\theta); 0\\leq r\\leq 1, 0\\leq \\theta \\leq 2\\pi\\Bigr\\}\\subset\n{\\mathbb R} ^2$$\nbe the unit disk defined in polar coordinates. A fibered solid torus of \ntype $(\\mu, \\nu )$ is the quotient space of the cylinder $D^2\\times I$ via\nthe identification $\\bigl (\\bigl (r,\\theta \\bigr ),1\\bigr )=\n\\bigl (\\bigl (r,\\theta+\\frac{2\\pi \\nu}{\\mu}\\bigr\n),0\\bigr)$. The fibers are the images of the curves ${x}\\times I$. The\ninteger $\\mu$ is called the index or the multiplicity. For $|\\mu| >1$ the\nfibered solid torus is said to be {\\em exceptionally fibered,\\\/} and the\nfiber that is the image of $0\\times I$ is called the {\\em exceptional fiber\\\/}. \nOtherwise the\nfibered solid torus is said to be {\\em regularly fibered,\\\/} \nand each fiber is a {\\em regular fiber\\\/}.\n\n\\begin{defin}\\label{Seifert}\nAn orientable three manifold $S$ is said to be a \n{\\em Seifert fibered manifold\\\/}\nif it is a union of pairwise disjoint closed curves, called fibers, such\nthat each one has a closed neighborhood which is a union of fibers and\nis homeomorphic to a fibered solid torus by a fiber preserving\nhomeomorphism.\n\\end{defin}\n\nA fiber $h$ is called {\\em exceptional\\\/} if $h$ has a neighborhood \nhomeomorphic to\nan exceptionally fibered solid torus (by a fiber preserving homeomorphism), \nand $h$ corresponds via the\nhomeomorphism to the exceptional fiber of the solid torus. If $\\partial S\\neq\n\\emptyset$,\nthen $\\partial S$ should be a union of regular fibers. \n\nThe quotient space obtained from a Seifert fibered manifold $S$ by\nidentifying each fiber to a point is a 2-manifold. It is called the orbit\nspace and the images of the exceptional fibers are called\n{\\em the cone points.\\\/}\n\n\n\\begin{emf}\\label{numbers}\nFor an exceptional fiber $a$ of\nan oriented Seifert fibered manifold there is a unique pair of relatively\nprime integers $(\\mu_a, \\nu_a)$ such that $\\mu_a>0$, $|\\nu_a|<\\mu_a$, and a\nneighborhood of $a$ \nis homeomorphic (by a fiber preserving homeomorphism) to a\nfibered solid torus of type $(\\mu_a, \\nu_a)$. We call the pair $(\\mu_a,\n\\nu_a)$ {\\em the type of the exceptional fiber\\\/} $a$. We also call this\npair the type of the corresponding cone point. \n\\end{emf}\n\n\nWe can define an invariant of an oriented knot\nin a Seifert fibered manifold that is similar to the $S_K$ invariant.\n\nClearly any $S^{1}$-fibration can be viewed as a Seifert fibration without\ncone points. This justifies the notation in the definition below.\n\n\\begin{defin}[of $S_K$]\\label{SeifertSK}\nLet $N$ be an oriented Seifert fibered manifold with an oriented orbit space \n$F$. Let $\\pi :N\\rightarrow F$ be the corresponding fibration and $K\\subset\nN$ an oriented knot in general position with respect to $\\pi$. Assume\nalso that $K$ does not intersect the exceptional fibers. For each double\npoint $v$ of $\\pi (K)$ we split $K$ into $\\mu ^+_v$ and $\\mu ^-_v$ (see\n~\\ref{SK}). Let $A$ be the set of all exceptional\nfibers. Since $N$ and $F$ are oriented, we have an induced orientation of\neach exceptional fiber $a\\in A$. For $a\\in A$ set $f_a$ to be the homology \nclass of the fiber with this orientation. For $a\\in A$ of type $(\\mu_a,\n\\nu_a)$ (see~\\ref{numbers})\nset $N_1(a)=\\bigl\\{ k\\in \\{1,\\dots,\\mu_a\\}| \\frac{2\\pi k\\nu_a}{\\mu_a}\\text{ } mod\n\\text{ }2\\pi\\in (0,\\pi]\\bigr\\}$, \n$N_2(a)=\\bigl\\{ k\\in \\{1,\\dots,\\mu_a\\}| \\frac{2\\pi k\\nu_a}{\\mu_a}\\text{ } mod\n\\text{ }2\\pi\\in (0,\\pi)\\bigr\\}$. Define $R^1_a, R^2_a\\in {\\mathbb Z}[H_1(N)]$ by the\nfollowing formulas:\n\n\\begin{equation}\nR^1_a=\\sum_{k\\in N_1(a)}\\bigl([K]f_a^{\\mu_a-k}-f_a^k\\bigr)-\n\\sum_{k\\in N_2(a)}\\bigl(f_a^{\\mu_a-k}-[K]f_a^k\\bigr),\n\\end{equation}\n\\begin{equation}\nR^2_a=\\sum_{k\\in N_1(a)}\\bigl(f_a^{\\mu_a-k}-[K]f_a^k\\bigr)-\n\\sum_{k\\in N_2(a)}\\bigl([K]f_a^{\\mu_a-k}-f_a^k\\bigr).\n\\end{equation}\n\nLet $H$ be the quotient of ${\\mathbb Z} [(H_1(N)]$ (viewed as a ${\\mathbb Z}$-module) \nby the free ${\\mathbb Z}$-submodule \ngenerated by \n$\\Bigl \\{[K]f-e, [K]-f, \\bigl \\{R^1_a, R^2_a\\bigr\\} _{a\\in A}\\Bigr \\}$.\nFinally, define $S_K\\in H$ by the following formula, where the \nsummation is taken over all double points $v$ of\n$\\pi (K)$:\n\n\\begin{equation}\\label{SSK}\nS_K=\\sum_v\\bigl( [\\mu ^+_v]-[\\mu ^-_v]\\bigr).\n\\end{equation}\n \n\\end{defin}\n\n\\begin{thm}\\label{correctSSK}\n$S_K$ is an isotopy invariant of the knot $K$.\n\\end{thm}\n\nFor the proof of Theorem~\\ref{correctSSK} see Subsection~\\ref{correctSSK}.\n\nWe introduce a similar invariant in the case \nwhere $N$ is oriented and $F$ is non-orientable.\n\n\\begin{defin}[of $\\tilde S_K$]\\label{tilde1SK}\n\nLet $N$ be an oriented Seifert fibered manifold with a non-orientable orbit\nspace $F$. Let $\\pi:N\\rightarrow F$ be the corresponding fibration and\n$K\\subset N$ an oriented knot in general position with respect to $\\pi$.\nAssume also that $K$ does not intersect the exceptional fibers. For each\ndouble point $v$ of $\\pi(K)$ we split $K$ into \n$\\mu_1^+(v),$ $\\mu_1^-(v),$ $\\mu_2^+(v)$, and $\\mu_2^-(v)$ \nas in~\\ref{tildeSK}. The element \n$\\bigl([\\mu_1^+(v)]-[\\mu_1^-(v)]+[\\mu_2^+(v)]-[\\mu_2^-(v)]\\bigr)\\in\n{\\mathbb Z}[H_1(N)]$ \nis well\ndefined. \n\nDenote by $f$ the homology class of a regular fiber oriented in some way. \nNote that $f^2=e$, so the orientation we use to define $f$ does not matter. \nFor a cone point $a$ denote by $f_a$ \nthe homology class of the fiber $\\pi^{-1}(a)$ oriented in some\nway.\n \nFor $a\\in A$ of type $(\\mu_a,\\nu_a)$ set \n$N_1(a)=\\bigl\\{ k\\in \\{1,\\dots,\\mu_a\\}| \\frac{2\\pi k\\nu_a}{\\mu_a}\\text{ } \nmod\n\\text{ }2\\pi\\in (0,\\pi]\\bigr\\}$, \n$N_2(a)=\\bigl\\{ k\\in \\{1,\\dots,\\mu_a\\}| \\frac{2\\pi k\\nu_a}{\\mu_a}\\text{ } mod\n\\text{ }2\\pi\\in (0,\\pi)\\bigr\\}$. \n\nDefine $R_a\\in {\\mathbb Z}[H_1(N)]$ by the\nfollowing formula:\n\n\\begin{multline}\nR_a=\n\\sum_{k\\in N_1(a)}\\Bigl([K]f_a^{\\mu_a-k}-f_a^k+\nf_a^{k-\\mu_a}-[K]f_a^{-k}\\Bigr)\\\\\n-\\sum_{k\\in N_2(a)}\\Bigl(f_a^{\\mu_a-k}-[K]f_a^k+\n[K]f_a^{k-\\mu_a}-f_a^{-k}\\Bigr)\n\\end{multline}\n\n\n\n\n\n\n\nLet $\\tilde H$ be the quotient of\n${\\mathbb Z}[H_1(N)]$ (viewed as a ${\\mathbb Z}$-module) by the free\n${\\mathbb Z}$-submodule generated by $\\Bigl\\{(e-f)([K]+e),\\{R_a\\}_{a\\in A}\\Bigr\\}$.\n\nOne can prove that under the change of the orientation of $\\pi^{-1}(a)$ \n(used to define $f_a$) $R_a$ goes to \n$-R_a$. Thus $\\tilde H$ is well defined. \nTo show this, one verifies that if $\\mu_a$ is odd, then $N_1(a)=N_2(a)$. Under\nthis change each term from the first sum (used to define $R_a$) goes to minus\nthe corresponding term from the second sum and vice versa. (Note that\n$f^2=e$.) If $\\mu_a=2l$ is even, then $N_1(a)\\setminus\\{l\\}=N_2(a)$. \nUnder this change each term with $k\\in N_1(a)\\setminus \\{l\\}$ goes to minus the\ncorresponding term with $k\\in N_2(a)$ and vice versa. \nThe term in the first sum\nthat corresponds to $k=l$ goes to minus itself.\n\nFinally define $\\tilde S_K\\in \\tilde H$ as \nthe sum over all double points $v$ of $\\pi(K)$:\n\\begin{equation}\\tilde S_K=\n\\sum_v\\Bigl([\\mu^+_1(v)]-[\\mu^-_1(v)]+\n[\\mu^+_2(v)]-[\\mu^-_2(v)]\n\\Bigr).\n\\end{equation} \n\\end{defin}\n\n\n\\begin{thm}\\label{correcttildeSSK}\n$\\tilde S_K$ is an isotopy invariant of $K$.\n\\end{thm}\n\n\nThe proof is a straightforward generalization of the proof of\nTheorem~\\ref{correctSSK}. \n\n\n\n\n\n\n\\begin{emf} \nOne can easily verify that $S_K$ and $\\tilde S_K$ \nsatisfy relations similar to~\\eqref{type2}. \nHence both of them are Vassiliev invariants of order one\n(see~\\ref{vassiliev2}). \n\n\\end{emf}\n \n\\section{Wave fronts on surfaces}\n\\subsection{Definitions}\nLet $F$ be a two-dimensional manifold. A {\\em contact element\\\/}\nat a point in $F$ is a one-dimensional vector subspace of the tangent plane. \nThis subspace divides the tangent plane into two half-planes. A choice of\none of them is called a {\\em coorientation\\\/} of a contact element. \nThe space of all cooriented contact\nelements of $F$ is a spherical cotangent bundle $ST^*F$. \nWe will also denote it by $N$. It is an $S^1$-fibration over $F$. The\nnatural contact structure on $ST^*F$ is a\ndistribution of hyperplanes given by the condition that a velocity vector of\nan incidence point of a contact element belongs to the element. \nA {\\em Legendrian\\\/} curve $\\lambda$ in $N$ is an immersion of $S^1$ into\n$N$ such that for each $p\\in S^1$ \nthe velocity vector of $\\lambda$ at $\\lambda(p)$ \nlies in the contact plane. The naturally cooriented \nprojection $L\\subset F$ of a Legendrian curve $\\lambda\\subset N$ \nis called {\\em the wave front\\\/} of\n$\\lambda$. \nA cooriented wave front may be uniquely lifted to a Legendrian curve\n$\\lambda \\subset\nN$ by taking a coorienting normal direction as a contact element at each\npoint of the front.\nA wave front is said to be generic if it is an immersion\neverywhere except a finite number of points, where it has cusp\nsingularities,\nand all multiple points are double points with transversal self-intersection.\nA cusp is the projection of a point where the corresponding Legendrian\ncurve is tangent to the fiber of the bundle.\n\n\n\n\\subsection{Shadows of wave fronts}\n\n\n\\begin{emf}\\label{orientST*F}\nFor any surface $F$ the space $ST^*F$ is canonically oriented. The\norientation is constructed as follows. \nFor a point $x\\in F$ fix an\norientation of $T_xF$. It induces an orientation of the fiber over $x$. \nThese two orientations determine an orientation of three dimensional planes\ntangent to the points of the fiber over $x$. A straightforward \nverification shows \nthat this orientation is independent on the orientation of $T_xF$ we choose.\nHence the orientation of $ST^*F$ is well defined. \n\n\nThus for oriented $F$ the shadow of a generic knot in $ST^*F$ is well defined \n(see~\\ref{prelim} and~\\ref{shadowgeneral}). Theorem~\\ref{front-shadow}\ndescribes the shadow of a Legendrian lifting of a generic cooriented wave front\n$L\\subset F$. \n\\end{emf}\n\n\n\\begin{defin} Let $X$ be a connected component of $F\\setminus L$. We denote \nby $\\chi \\protect\\operatorname{Int} (X)$ the Euler characteristic of $\\protect\\operatorname{Int} (X)$, \nby $C^i_X$ the number of cusps in the boundary of the region $X$ pointing \ninside\n$X$ (as in Figure~\\ref{pic2.fig}a), by $C^o_X$ the number of cusps \nin the boundary of $X$\npointing outside (as in Figure~\\ref{pic2.fig}b),\nand by $V_X$ the number of corners of $X$ where locally the picture\nlooks in one the two ways shown in Figure~\\ref{pic2.fig}c. It\ncan happen that a cusp point is pointing both inside and outside of $X$. In\nthis case it contributes both in $C^i_X$ and in $C^o_X$. If the corner of the\ntype shown in Figure~\\ref{pic2.fig}c enters twice in $\\partial X$, \nthen it should be counted twice. \n\\end{defin}\n\n\\begin{figure}[htb]\n\\begin{center}\n \\epsfxsize\\hsize\\advance\\epsfxsize -0.5cm\n \\leavevmode\\epsffile{pic2.eps}\n \\end{center}\n\\caption{}\\label{pic2.fig}\n\\end{figure}\n\n\n\n\n\n\n\\begin{thm}\\label{front-shadow}\nLet $F$ be an oriented surface and\n$L$ a generic cooriented wave front on $F$ corresponding to a \nLegendrian curve $\\lambda$. \nThere exists a small deformation of $\\lambda$ in the class of all smooth \n(not \nonly Legendrian) curves such that the resulting curve is generic with respect\nto the projection, and the shadow of this curve can be constructed in the \nfollowing way. We \nreplace a small neighborhood of each cusp of $L$ with a smooth simple arc. \nThe gleam of an arbitrary region $X$ that has a compact closure and does not\ncontain boundary components of $F$ is calculated by the following formula:\n\\begin{equation}\n\\protect\\operatorname{gl}_X=\\chi \\protect\\operatorname{Int} (X)+\\frac {1}{2}(C^i_X-C^o_X-V_X).\n\\end{equation}\n\\end{thm}\n\nFor the proof of Theorem~\\ref{front-shadow} see \nSubsection~\\ref{pffront-shadow}.\n\n\\begin{rem}\nThe surface $F$ in the statement of Theorem~\\ref{front-shadow} is not\nassumed to be compact.\n\nNote that as we mentioned in~\\ref{shadowgeneral}, the gleam of a region $X$ \nthat does not have compact closure or contains boundary components is\nnot well defined unless we fix a section over all ends of $X$ and components of\n$\\partial F$ in $X$.\n\nThis theorem first appeared in~\\cite{Tchernov}. A similar result was\nindependently obtained by Polyak~\\cite{MPolyak}. \n\\end{rem}\n\n\\begin{emf} A self-tangency point $p$ of a wave front is said to be a\npoint of a {\\em dangerous self-tangency\\\/} if the coorienting normals of the two \nbranches coincide at $p$ (see Figure~\\ref{pic8.fig}). Dangerous\nself-tangency points correspond to self-intersection of the Legendrian curve. \nHence a generic deformation of the front $L$ \nnot involving {\\em dangerous\\\/} self-tangencies\ncorresponds to an isotopy of the Legendrian knot $\\lambda$.\n\n\\begin{figure}[htb]\n\\begin{center}\n \\epsfysize 1.1cm\n \\leavevmode\\epsffile{pic8.eps}\n \\end{center}\n\\caption{}\\label{pic8.fig}\n\\end{figure}\n\nAny generic deformation of a wave front $L$ corresponding to an isotopy\nin the class of the Legendrian knots \ncan be split into a sequence of modifications shown in \nFigure~\\ref{pic6.fig}. The construction of Theorem~\\ref{front-shadow}\ntransforms these generic modifications of wave fronts to shadow moves:\nIa and Ib in Figure~\\ref{pic6.fig} are transformed to the \n$\\bar S_1$ move for shadow diagrams, IIa, IIb, II'a, II'b, II'c, and\nII'd are transformed to the $S_2$ move, \nfinally IIIa and IIIb are transformed to $S_3$ and $\\bar S_3$ respectively.\n\\end{emf}\n\n\\begin{figure} [htbp]\n\\begin{center}\n \\epsfxsize 12.5cm\n \\leavevmode\\epsffile{pic6.eps}\n \\end{center}\n\\caption{}\\label{pic6.fig}\n\\end{figure}\n\n\n\n\\begin{emf} Thus for the Legendrian lifting of a wave front \nwe are able to calculate all invariants that we can calculate for shadows. \nThis includes the analogue of the linking number for the\nfronts on ${\\mathbb R}^2$ (see~\\cite{Turaev}), the second order Vassiliev invariant\n(see~\\cite{Shumakovitch}), \nand quantum state sums (see~\\cite{Turaev}).\n\\end{emf}\n\n\\subsection{Invariants of wave fronts on surfaces.}\nIn particular, the $S_K$ invariant gives rise to an invariant of a generic\nwave front. This invariant appears to be related to the formula for the\nBennequin invariant of a wave front introduced by Polyak in~\\cite{Polyak}.\n\nLet us recall the corresponding results and definitions of~\\cite{Polyak}. \n\nLet $L$ be a generic cooriented oriented wave front on an oriented surface\n$F$.\nA branch of a wave front is said to be positive (resp.\nnegative) if the frame of coorienting and orienting vectors defines\npositive (resp. negative) orientation of the surface $F$. \nDefine the {\\em sign\\\/} $\\epsilon (v)$ of a double point $v$ of $L$ to be\n$+1$ if the signs of both branches of the front intersecting at $v$\ncoincide and\n$-1$ otherwise. Similarly \nwe assign a positive (resp. negative) sign to a cusp point\nif the coorienting vector turns in a positive (resp. negative)\ndirection while traversing a small neighborhood of the cusp point along the\norientation. We denote half of the number of \npositive and negative cusp points by \n$C^+$ and $C^-$ respectively.\n\n\nLet $v$ be a double point of $L$. The orientations of $F$ and $L$ allow one\nto distinguish the two wave fronts $L^+_v$ and $L^-_v$ obtained by splitting \nof $L$ in $v$ with respect to orientation and\ncoorientation (see Figures~\\ref{pic9.fig}a.1 and~\\ref{pic9.fig}b.1).\n(Locally one of the two fronts lies to the left and another to the right of\n$v$.) \n\n\\begin{figure}[htb]\n\\begin{center}\n\n \\epsfxsize 10cm \n \\leavevmode\\epsffile{pic9.eps}\n \\end{center}\n\\caption{}\\label{pic9.fig}\n\\end{figure}\n\nFor a Legendrian curve $\\lambda$ in $ST^*{\\mathbb R} ^2$ denote by $l(\\lambda)$ its\nBennequin invariant described in the works of \nTabachnikov~\\cite{Tabachnikov} and Arnold~\\cite{Arnold} with the sign\nconvention of~\\cite{Arnold} and~\\cite{Polyak}. \n\n\\begin{thm}[Polyak~\\cite{Polyak}]\\label{BenR2}\nLet $L$ be a generic oriented cooriented wave front on ${\\mathbb R}^2$\nand $\\lambda$ the corresponding Legendrian curve. Denote by $\\protect\\operatorname{ind}(L)$ \nthe degree of the mapping taking a point $p\\in S^1$ \nto the point of $S^1$ corresponding to the direction of \nthe coorienting normal of $L$ at $L(p)$. Define $S$ as the \nfollowing sum over all double points of $L:$\n\\begin{equation}\nS=\\sum_v (\\protect\\operatorname{ind}(L^+_v)-\\protect\\operatorname{ind}(L^-_v)-\\epsilon (v)).\n\\end{equation}\nThen \n\\begin{equation}\nl(\\lambda )=S+(1-\\protect\\operatorname{ind}(L))C^++(\\protect\\operatorname{ind}(L)+1)C^- +\\protect\\operatorname{ind}(L)^2.\n\\end{equation}\n\\end{thm}\n\n\nIn~\\cite{Polyak} it is shown that the Bennequin invariant of a wave front on\nthe ${\\mathbb R}^2$ plane admits quantization. Consider a formal quantum parameter\n$q$. Recall that for any $n\\in {\\mathbb Z}$ the corresponding quantum number\n$[n]_q\\in {\\mathbb Z}[q,q^{-1}]$ is a finite Laurent polynomial in $q$ defined by \n\\begin{equation}\n[n]_q=\\frac{q^n-q^{-n}}{q-q^{-1}}.\n\\end{equation}\nSubstituting quantum integers instead of integers in~\\ref{BenR2}\nwe get the following theorem.\n\n\\begin{thm}[Polyak~\\cite{Polyak}]\\label{QBen}\nLet $L$ be a generic cooriented oriented wave \nfront on ${\\mathbb R}^2$ and $\\lambda$ the corresponding\nLegendrian curve. Define $S_q$ by the following\nformula, where the sum is taken over the set of all \ndouble points of $L:$\n\\begin{equation}\\label{Sq}\nS_q=\\sum_v[ind(L^+_v)-ind(L^-_v)-\\epsilon (v)]_q.\n\\end{equation}\nPut\n\\begin{equation}\nl_q(L)=S_q+[1-ind(L)]_qC^++[ind(L)+1]_qC^-+[ind(L)]_qind(L).\n\\end{equation} \nThen $l_q(\\lambda)=l_q(L)\\in\\frac{1}{2}{\\mathbb Z}[q,q^{-1}]$ is invariant \nunder isotopy in the class of the Legendrian knots. \n\\end{thm}\n \n\nThe $l_q(\\lambda)$ invariant can be expressed~\\cite{Aicpriv} \nin terms of the partial linking\npolynomial of a generic cooriented oriented wave front\nintroduced by Aicardi~\\cite{Aicardi}.\n\nThe reason why this invariant takes values in $\\frac{1}{2}{\\mathbb Z}[q,q^{-1}]$ and\nnot in ${\\mathbb Z}[q,q^{-1}]$ is that the number of positive (or negative) cusps can\nbe odd. This makes $C^+$ ($C^-$) a half-integer.\n\nLet $\\lambda ^{\\epsilon}_v$ with $\\epsilon =\\pm$ be the Legendrian lifting \nof the front $L^{\\epsilon}_v$. Let $f\\in \nH_1(ST^*F)$ be the homology class of a positively\noriented fiber.\n\n\n\\begin{thm}[Polyak~\\cite{Polyak}]\\label{Ben} Let $L$ be a generic oriented \ncooriented wave front on an oriented surface $F$. Let $\\lambda$ be the corresponding \nLegendrian curve. Define $l_F(\\lambda)\\in H_1(ST^*F,\\frac{1}{2}{\\mathbb Z})$ \nby the following formula: \n\\begin{equation}\nl_{F}(\\lambda)=\\Bigl(\\prod_v[\\lambda ^+_v][\\lambda ^-_v]^{-1}\nf^{-\\epsilon (v)}\\Bigr)(f[\\lambda]^{-1})^{C^+}([\\lambda]f)^{C^-}\n\\end{equation}\n(We use the multiplicative notation for the group operation in\n$H_1(ST^*F)$.)\n\n \nThen $l_{F}(\\lambda)$ is invariant under isotopy in the class of the\nLegendrian knots.\n\\end{thm}\n\nThe proof is straightforward. One verifies that $l_{F}(\\lambda)$ is \ninvariant under all oriented versions of non-dangerous self-tangency, \ntriple point, cusp crossing, and cusp birth moves of the wave front.\n\nIn~\\cite{Polyak} this invariant is denoted by $I_\\Sigma^+(\\lambda)$ and, in\na sense, it\nappears to be a natural generalization of Arnold's $J^+$\ninvariant~\\cite{Arnold} \nto the case of an\noriented cooriented wave front on an oriented surface.\n\n\nNote that in the situation of Theorem~\\ref{BenR2} the indices of all\nthe fronts involved are the images of the homology classes of their\nLegendrian liftings under the natural identification of $H_1(ST^*{\\mathbb R}\n^2)$ with ${\\mathbb Z}$. If one replaces everywhere in~\\ref{BenR2} indices with \nthe corresponding homology classes and puts $f$ instead of $1$, then the\nonly difference between the two formulas is the term $\\protect\\operatorname{ind} ^2(L)$. (One has\nto remember that we use the multiplicative notation for the operation \nin $H_1(ST^*F)$.)\n \n\\begin{emf} The splitting of a Legendrian knot $K$ into \n$\\mu^+_v$ and $\\mu^-_v$\n(see~\\ref{SK}) can be done up to an isotopy in the\nclass of the Legendrian knots. Although this can be done in many ways, there \nexists the simplest way. The projections $\\tilde L^+_v$ and $\\tilde L^-_v$\nof the Legendrian curves created by the splitting are shown in \nFigure~\\ref{pic5.fig}. (This fact follows from Theorem~\\ref{front-shadow}.) \n\nLet $\\tilde\\lambda ^{\\epsilon}_v$ with $\\epsilon =\\pm$ be the Legendrian lifting \nof the front $\\tilde L^{\\epsilon}_v$.\n\\end{emf}\n\n\\begin{figure}[htb]\n\\begin{center}\n\n \\epsfxsize 12 cm\n \\leavevmode\\epsffile{pic5.eps}\n \\end{center}\n\\caption{}\\label{pic5.fig}\n\\end{figure}\n\n\n\n\\begin{thm}\\label{splitwave} Let $L$ be a generic oriented cooriented wave\nfront on an oriented surface $F$. Let $\\lambda$ be the corresponding\nLegendrian curve. \nDefine $S(\\lambda)\\in \\frac{1}{2}{\\mathbb Z}\\bigl[ H_1(ST^*F)\\bigr]$ by the following \nformula: \n\\begin{equation}\\label{SFL}\nS(\\lambda)=\\sum_v\\Bigl([\\tilde \\lambda ^+_v]-[\\tilde \\lambda\n^-_v]\\Bigr)\n+(f-[\\lambda])C^+ +([\\lambda]f-e)C^-.\n\\end{equation} \nThen $S(\\lambda)$ is invariant under isotopy in the class of the Legendrian\nknots. \n\\end{thm}\n\nThe proof is straightforward. One verifies that $S(\\lambda)$ is indeed\ninvariant under all oriented versions of non-dangerous self-tangency, \ntriple point, cusp crossing, and cusp birth moves of the wave front.\n\n\\begin{emf}\\label{homvers}\nBy taking the free homotopy classes of $\\tilde \\lambda^+_v$ and $\\tilde\n\\lambda^-_v$ instead of the homology classes one obtains a different \nversion of the $S(\\lambda)$ invariant. It takes values in the group of\nformal half-integer linear combinations of \nthe free homotopy classes of oriented \ncurves in $ST^*F$.\nIn this case the terms $[\\lambda]$ and $f$ in~\\eqref{SFL} should be\nsubstituted with the free homotopy classes of $\\lambda$ and of a \npositively oriented fiber respectively. The terms $[\\lambda]f$ and $e$\nin~\\eqref{SFL} should be substituted with the free homotopy classes of \n$\\lambda$ with a positive fiber added to it and the class of a \ncontractible curve\nrespectively. Note that $f$ lies in the center of $\\pi_1(ST^*F)$, so that the\nclass of $\\lambda$ with a fiber added to it is well defined.\n\nA straightforward verification shows that this version of $S(\\lambda)$ is also\ninvariant under isotopy in the class of the Legendrian knots.\n\\end{emf}\n\n \n\n\\begin{thm}\\label{splitting} Let $L$ be a generic oriented cooriented wave\nfront on an oriented surface $F$. Let $\\lambda$ be the corresponding \nLegendrian curve. \nLet $S(\\lambda)$ and $l_F(\\lambda)$ be the invariants\nintroduced in~\\ref{splitwave} and~\\ref{Ben} respectively. Let\n\\begin{equation}\n\\protect\\operatorname{pr}:\\frac{1}{2}{\\mathbb Z}\\bigl [H_1(ST^*F)\\bigr]\\rightarrow \nH_1(ST^* F,\\frac{1}{2}{\\mathbb Z})\n\\end{equation}\nbe the mapping defined as\nfollows: for any $n_i\\in \\frac{1}{2}{\\mathbb Z}$ and $g_i\\in H_1(ST^*F),$\n\\begin{equation}\n\\sum n_i g_i\\mapsto \\prod g_i^{n_i}.\n\\end{equation}\nThen $\\protect\\operatorname{pr}(S(\\lambda))=l_F(\\lambda)$.\n\\end{thm}\n \nThe proof is straightforward: one must verify that \n\\begin{equation}\\label{svyaz}\n[\\lambda ^+_v][\\lambda ^-_v]^{-1}f^{-\\epsilon (v)}=\n[\\tilde \\lambda ^+_v][\\tilde \\lambda ^-_v]^{-1}\n\\text{ in } H_1(ST^*F). \n\\end{equation}\n(Recall that we use a multiplicative notation for the group operation\nin $H_1(ST^*F)$.)\n\nThis means that $S_F(\\lambda)$ is a refinement of Polyak's invariant \n$l_F(\\lambda)$.\n \n\n\\begin{emf}\\label{quant} \nOne can verify that there is a unique linear combination \n$\\sum_{m\\in {\\mathbb Z}}n_m[m]_q=l_q(\\lambda)$ with $n_m$ being non-negative\nhalf-integers such that $n_0=0$, and if $n_m>0$, then $n_{-m}=0$. \nTo prove this one must verify that\n$\\{\\frac{1}{2}[n]_q|0< n\\}$ is a basis for the ${\\mathbb Z}$-submodule of \n$\\frac{1}{2}{\\mathbb Z}[q,q^{-1}]$ \ngenerated by the quantum numbers and use the identity $n[m]_q=-n[-m]_q$. \n\\end{emf} \n\n \nThe following theorem shows that if $L\\subset {\\mathbb R}^2$, then $S(\\lambda)$ and\nPolyak's quantization $l_q(\\lambda)$ (see~\\ref{QBen}) of the Bennequin \ninvariant can be explicitly expressed in terms of each other. \n\n\\begin{thm}\\label{equivalence}\nLet $f\\in H_1(ST^*{\\mathbb R}^2)$ be the class of a positively oriented fiber. Let \n$L$ be a generic oriented cooriented wave front on ${\\mathbb R}^2$, \n$\\lambda$ the corresponding Legendrian\ncurve, and $f^h$ the homology class realized by it. Let\n$l_q(\\lambda)-[h]_qh=\\sum_{m\\in Z} n_m[m]_q$ be written in the form described \nin~\\ref{quant} and $S(\\lambda)=\\sum_{l\\in Z}k_l f^l$. \nThen\n\\begin{equation}\\label{rel1}\nl_q(\\lambda)=[h]_qh+\\sum_{k_l>0} k_l[2l-h-1]_q,\n\\end{equation}\nand\n\\begin{equation}\\label{rel2}\nS(\\lambda)=\\sum_{n_m > 0}\nn_m(f^{\\frac{h+1+m}{2}}-f^{\\frac{h+1-m}{2}}).\n\\end{equation}\n\\end{thm}\nFor the proof of Theorem~\\ref{equivalence} see Subsection~\\ref{pfequivalence}.\n\nOne can show that for $n_m>0$ both $\\frac{h+1+m}{2}$ and $\\frac{h+1-m}{2}$\nare integers, so that the sum~\\eqref{rel2} takes values \nin $\\frac{1}{2}{\\mathbb Z}[H_1(ST^*{\\mathbb R}^2)]$.\n\nNote that the $l_q(\\lambda)$ invariant was defined only for fronts on the\nplane ${\\mathbb R}^2$. Thus $S(\\lambda)$ is, in a sense, a generalization of\nPolyak's\n$l_q(\\lambda)$ to the case of wave fronts on an arbitrary oriented surface $F$.\n\n\\begin{emf}\nThe splitting of the Legendrian knot $K$ into $\\mu_1^+(v)$, $\\mu_1^-(v)$, \n$\\mu_2^+(v)$, and $\\mu_2^-(v)$ (which was used to define $\\tilde S(K)$, \nsee~\\ref{tildeSK}) can be done up to an isotopy in the class of the\nLegendrian knots. Although this can be done in many ways, there is the\nsimplest one. The projections $\\tilde L_1^+(v)$, $\\tilde L_1^-(v)$, \n$\\tilde L_2^+(v)$, and $\\tilde L_2^-(v)$ are shown in\nFigure~\\ref{front1.fig}. (This fact follows from \nTheorem~\\ref{front-shadow}.) \n\n\nThis allows us to introduce an invariant similar to $S(\\lambda)$ \nfor generic oriented cooriented \nwave fronts on a non-orientable surface $F$ in the\nfollowing way. \n\n\nLet $L$ be a generic wave front on a\nnon-orientable surface $F$. Let $v$ be a double point of $L$. Fix some \norientation of a small neighborhood of $v$ in $F$. The orientations of the\nneighborhood and $L$ allow one to distinguish the wave fronts $L^+_1$,\n$L^-_1$, $L^+_2$, and $L^-_2$ obtained by the two splittings of $L$ with\nrespect to the orientation and coorientation (see Figure~\\ref{front1.fig}).\nLocally the fronts with the upper indices plus and minus are located\nrespectively to the right and to the left of $v$.\nTo each double\npoint $v$ of $L$ we associate an element\n$\\Bigl([\\tilde\\lambda^+_1(v)]-[\\tilde\\lambda^-_1(v)]+[\\tilde\\lambda^+_2(v)]-\n[\\tilde\\lambda^-_2(v)]\\Bigr)\\in {\\mathbb Z}[H_1(ST^*(F)]$. Here we denote by lambdas\nthe Legendrian curves corresponding to the wave fronts appearing under\nthe splitting. Clearly \nthis element does not depend on the orientation of the\nneighborhood of $v$ we have chosen.\n\n\\begin{figure}[htb]\n\\begin{center}\n\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{front1.eps}\n \\end{center}\n\\caption{}\\label{front1.fig}\n\\end{figure}\n\n\nFor a wave front $L$ let $C$ be half of the number of cusps of $L$. Denote\nby $f$ the homology class of the fiber of $ST^*F$ oriented in some way. \nNote that $f^2=e$, so it does not matter which orientation of the fiber we\nuse to define $f$.\n\\end{emf}\n\n\\begin{thm}\\label{unorientsplit}\nLet $L$ be a generic cooriented oriented wave front on a non-orientable surface $F$\nand $\\lambda$ the corresponding Legendrian curve.\nDefine $\\tilde S(\\lambda)\\in\\frac{1}{2}{\\mathbb Z}\\bigl[ H_1(ST^*(F))\\bigr]$ by the following\nformula, where the summation is taken over the set of all double points of\n$L:$\n\\begin{equation}\\label{tildeSdef}\n\\tilde S(\\lambda)=\\sum_v\\Bigl([\\tilde \\lambda ^+_1(v)]-\n[\\tilde \\lambda ^-_1(v)]+[\\tilde \\lambda ^+_2(v)]-[\\tilde \\lambda\n^-_2(v)]\\Bigr)+C\\bigl([\\lambda]f-e+f-[\\lambda]\\bigr).\n\\end{equation}\nThen $\\tilde S(\\lambda)$ is invariant under isotopy in the class of \nthe Legendrian knots.\n\\end{thm}\n\nThe proof is straightforward. One verifies that $\\tilde S(\\lambda)$ is\nindeed invariant under all oriented versions of non-dangerous\nself-tangency, triple point passing, cusp crossing, and cusp birth moves of\nthe wave front.\n\nThe reason we have $\\tilde S(\\lambda)\\in \\frac{1}{2}{\\mathbb Z}[H_1(ST^*F)]$ is that\nif $L$ is an orientation reversing curve, then \nthe number of cusps of $L$ is odd. In this case $C$ is a\nhalf-integer.\n\n\n\\section{Wave fronts on orbifolds}\n\\subsection{Definitions}\n\\begin{defin}\\label{orbifold}\n\nAn {\\em orbifold\\\/} is a surface $F$ with the additional structure\nconsisting of: \n\n1) set $A\\subset F$;\n\n2) smooth structure on $F\\setminus A$;\n\n3) set of homeomorphisms $\\phi_a$ of neighborhoods $U_a$ of $a$ in $F$ \nonto ${\\mathbb R}^2\/G_a$ such that $\\phi_a(a)=0$ and\n$\\phi_a\\Big|_{U_a\\setminus a}$ is a diffeomorphism.\nHere $G_a=\\bigl\\{e^{\\frac{2\\pi\nk}{\\mu_a}}\\big| k\\in\\{1,\\dots,\\mu_a\\}\\bigr\\}$ is a group\nacting on ${\\mathbb R}^2={\\mathbb C}$ by multiplication. ($\\mu _a\\in {\\mathbb Z}$ is assumed to be\npositive.) \n\\end{defin}\n\nThe points $a\\in A$ are called {\\em cone points\\\/}.\n\nThe action of $G$ on ${\\mathbb R}^2$ induces the action of $G$ on\n$ST^*{\\mathbb R}^2$. This makes $ST^*{\\mathbb R}^2\/G$ a Seifert fibration over\n${\\mathbb R}^2\/G$. Gluing together the pieces over neighborhoods of $F$ we obtain a\nSeifert fibration $\\pi:N\\rightarrow F$. The fiber over a cone point $a$ is an\nexceptional fiber of type $(\\mu_a,-1)$ (see~\\ref{numbers}). \n \nThe natural contact structure on $ST^*{\\mathbb R}^2$ is invariant under \nthe induced action of $G$. Since $G$ acts freely on $ST^*{\\mathbb R}^2$, this implies\nthat $N$ has an induced contact structure. As before, the naturally\ncooriented projection\n$L\\subset F$ of a generic Legendrian curve $\\lambda$ is called \n{\\em the front of\\\/} $\\lambda$.\n \n\\subsection{Invariants for fronts on orbifolds}\nFor oriented $F$ we construct an\ninvariant similar to $S(\\lambda)$. It corresponds to the $S_K$\ninvariant of a knot in a Seifert fibered space. \nFor non-orientable surface $F$\nwe construct an analogue of $\\tilde S(\\lambda)$. It corresponds to\nthe $\\tilde S_K$ invariant of a knot in a Seifert fibered space.\n\nNote that any surface $F$ can be viewed as an orbifold without cone\npoints. This justifies the notation below.\n\n \nLet $F$ be an oriented surface. The orientation of $F$ induces an \norientation of all fibers. Denote by $f$ \nthe homology class of a positively oriented fiber. For a cone point $a$\ndenote by $f_a$ the homology class of a positively oriented fiber\n$\\pi^{-1}(a)$. For a generic oriented cooriented wave front $L\\subset F$\ndenote by $C^+$ (resp. $C^-$) half of the number \nof positive (resp. negative) cusps of $L$.\nNote that for a double point $v$ of a generic front $L$\nthe splitting into $\\tilde L^+_v$ and $\\tilde L^-_v$ is well defined. \nThe corresponding Legendrian\ncurves $\\tilde \\lambda^+_v$ and $\\tilde \\lambda^-_v$ in \n$N$ are also well defined.\n\nFor $a\\in A$ of type $(\\mu_a,-1)$ \nput $N_1(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big|\\frac{-2k\\pi}{\\mu_a}\n\\protect\\operatorname{mod} 2\\pi\n\\in(0,\\pi]\\bigr\\}$, \n$N_2(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big|\\frac{2k\\pi}{\\mu_a}\n\\protect\\operatorname{mod} 2\\pi\n\\in(0,\\pi)\\bigr\\}$. \nDefine $R^1_a, R^2_a\\in\n{\\mathbb Z}[H_1(N)]$ by the following formulas: \n\n\\begin{equation}\nR^1_a=\\sum_{k\\in N_1(a)}\\bigl([\\lambda]f_a^{\\mu_a-k}-f_a^k\\bigr)-\n\\sum_{k\\in N_2(a)}\\bigl(f_a^{\\mu_a-k}-[\\lambda]f_a^k\\bigr),\n\\end{equation}\n\\begin{equation}\nR^2_a=\\sum_{k\\in N_1(a)}\\bigl(f_a^{\\mu_a-k}-[\\lambda]f_a^k\\bigr)-\n\\sum_{k\\in N_2(a)}\\bigl([\\lambda]f_a^{\\mu_a-k}-f_a^k\\bigr).\n\\end{equation}\n\n\n\nSet $J$ to be the quotient of\n$\\frac{1}{2}{\\mathbb Z}[H_1(N)]$ by the free Abelian \nsubgroup generated by \n$\\Bigl\\{\\{\\frac{1}{2}R_1(a),\\frac{1}{2}R_2(a)\\}_{a\\in A}\\Bigr\\}$. \n\n \n\\begin{thm}\\label{orbifold1}\nLet $L$ be a generic cooriented oriented wave front on $F$ and\n$\\lambda$ the corresponding Legendrian curve.\n\nThen $S(\\lambda)\\in J$ defined by the sum over \nall double points of $L$,\n\\begin{equation}\nS(\\lambda)=\n\\sum\\Bigl([\\tilde \\lambda^+(v)]-[\\tilde \\lambda^-(v)]\\Bigr)+(f-[\\lambda])C^++\n([\\lambda]f-e)C^-,\n\\end{equation}\nis invariant under isotopy in the class of the Legendrian knots.\n\\end{thm} \n\nFor the proof of Theorem~\\ref{orbifold1} see Subsection~\\ref{pforbifold1}.\n\n\nLet $F$ be a non-orientable surface. \nDenote by $f$ the homology class of a regular fiber oriented in some way. \nNote that $f^2=e$, so the orientation we use to define $f$ does not matter. \nFor a cone point $a$ denote by $f_a$ \nthe homology class of the fiber $\\pi^{-1}(a)$ oriented in some\nway. For a generic oriented cooriented wave front $L\\subset F$\ndenote by $C$ half of the number of cusps of $L$.\nNote that for a double point $v$ of a generic front $L$ \nthe element \n$\\bigl([\\tilde\\lambda^+_1(v)]-[\\tilde\\lambda^-_1(v)]+\n[\\tilde\\lambda^+_2(v)]-[\\tilde\\lambda^-_2(v)]\\bigr)\\in {\\mathbb Z}[H_1(N)]$ \nused to introduce\n$\\tilde S(\\lambda)$ is well defined.\n \nFor $a\\in A$ of type $(\\mu_a,-1)$ \nput $N_1(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big |\\frac{-2k\\pi}{\\mu_a}\n\\protect\\operatorname{mod} 2\\pi\n\\in(0,\\pi]\\bigr\\}$, and \n$N_2(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big |\\frac{-2k\\pi}{\\mu_a}\n\\protect\\operatorname{mod} 2\\pi\n\\in(0,\\pi)\\bigr\\}$. \nDefine $R_a\\in\n{\\mathbb Z}[H_1(N)]$ by the following formula:\n\\begin{multline}\nR_a=\n\\sum_{k\\in N_1(a)}\\Bigl([\\lambda]f_a^{\\mu_a-k}-f_a^k+\nf_a^{k-\\mu_a}-[\\lambda]f_a^{-k}\\Bigr)\\\\\n-\\sum_{k\\in N_2(a)}\\Bigl(f_a^{\\mu_a-k}-[\\lambda]f_a^k+\n[\\lambda]f_a^{k-\\mu_a}-f_a^{-k}\\Bigr).\n\\end{multline}\n\n\n\n\n\n\n\nPut $\\tilde J$ to be the quotient of\n$\\frac{1}{2}{\\mathbb Z}[H_1(N)]$ by a free Abelian subgroup generated by \n$\\Bigl\\{\\{\\frac{1}{2}R_a\\}_{a\\in A}\\Bigr\\}$.\n\nSimilarly to~\\ref{tilde1SK}, one \ncan prove that under the change of the orientation of $\\pi^{-1}(a)$ \n(used to define $f_a$) $R_a$ goes to \n$-R_a$. Thus $\\tilde J$ is well defined. \n\n\n\\begin{thm}\\label{orbifold2}\nLet $L$ be a generic cooriented oriented wave front on $F$ and\n$\\lambda$ the corresponding Legendrian curve.\n\n\nThen $\\tilde S(\\lambda)\\in \\tilde J$ defined by \nthe summation over all double points of $L$,\n\\begin{equation}\n\\tilde S(\\lambda)=\n\\sum\\Bigl([\\tilde \\lambda^+_1(v)]-[\\tilde \\lambda^-_1(v)]+\n[\\tilde \\lambda^+_2(v)]-[\\lambda^-_2(v)]\n\\Bigr)+\\bigl(([\\tilde \\lambda]f-e+f-[\\lambda]\\bigr)C,\n\\end{equation}\nis invariant under isotopy in the class of the Legendrian knots.\n\\end{thm} \n\nThe proof is a straightforward generalization of the proof of\nTheorem~\\ref{orbifold1}. \n\n\n\n\n\n\n\n\n\\section{Proofs}\n\n\\subsection{Proof of Theorem~\\ref{correct2}.}\\label{pfcorrect2}\nTo prove the theorem it suffices to show that $S_K$ is invariant\nunder the elementary isotopies. They project to: \na death of a double point, cancellation of two double points, and passing\nthrough a triple point. \n\nTo prove the invariance, we fix a homeomorphic to a closed disk \npart $P$ of $F$ containing the projection of one of the elementary\nisotopies.\nFix a section over the boundary of $P$ such that the \npoints of $K\\cap\\pi^{-1}(\\partial P)$ belong to the section. Inside $P$ we can\nconstruct the Turaev shadow (see \\ref{shadowgeneral}). \nThe action of $H_1(\\protect\\operatorname{Int} P)=e$ \non the set of isotopy types of links is trivial (see~\\ref{action}). \nThus the part of $K$ can be reconstructed in the unique way \nfrom the shadow over $P$. \nIn particular, one can compare the homology classes\nof the curves created by splitting at a double point inside $P$. \nHence to prove the theorem, it suffices to verify the invariance under the\noriented versions of the moves $S_1,S_2$, and $S_3$.\n\nThere are two versions of the oriented move $S_1$ shown in\nFigures~\\ref{shad11.fig}a and~\\ref{shad11.fig}b. \n\nFor Figure~\\ref{shad11.fig}a the term $[\\mu^+_v]$ appears to be equal to\n$f$. From~\\ref{sum-property} we know that \n$[\\mu^+_v][\\mu^-_v]=[K]f$, so that $[\\mu^-_v]=[K]$. Hence\n$[\\mu^+_v]-[\\mu^-_v]=f-[K]$ and is equal to zero in $H$. In the same way we\nverify that $[\\mu^+_v]-[\\mu^-_v]$ (for $v$ shown in\nFigure~\\ref{shad11.fig}b) is equal to $[K]f-e$. It is also zero in $H$. \nThe summands corresponding to other double points do not change under this move, \nsince it does\nnot change the homology classes of the knots created by the splittings.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 6cm\n \\leavevmode\\epsffile{shad11.eps}\n \\end{center}\n\\caption{}\\label{shad11.fig}\n\\end{figure}\n\n There are three oriented versions of the $S_2$ move. We show\nthat $S_K$ does not change under one of them. The proof for\nthe other two is the same or easier. We choose the version \ncorresponding to the upper part of Figure~\\ref{shad2.fig}. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 8cm\n \\leavevmode\\epsffile{shad2.eps}\n \\end{center}\n\\caption{}\\label{shad2.fig}\n\\end{figure}\nThe summands corresponding to the double points not in this figure are \npreserved under the move, since it does not change the homology classes \nof the corresponding knots. So it suffices to show that the terms \nproduced by \nthe double points $u$ and $v$ in this figure cancel out. \nNote that the shadow $\\mu ^-_v$ is transformed to $\\mu^+_u$\nby the $\\bar S_1$ move. It is known that \n$\\bar S_1$ can be expressed in terms of $S_1,S_2$, and \n$S_3$, thus it also does not change the homology classes of the \nknots created by the splittings. Hence $[\\mu^+_u]$ and $[\\mu^-_v]$ cancel out. \nIn the same way one \nproves that $[\\mu ^-_u]$ and $[\\mu ^+_v]$ also cancel out. \n\nThere are two oriented versions of the $S_3$ move: $S'_3$ and \n$S''_3$, shown in Figures~\\ref{shad7.fig}a and~\\ref{shad7.fig}b respectively.\n\\begin{figure}[htb]\n \\begin{center}\n \\epsfxsize 7cm\n \\leavevmode\\epsffile{shad7.eps}\n \\end{center}\n\\caption{}\\label{shad7.fig}\n\\end{figure}\nThe $S''_3$ move can be expressed in terms of $S'_3$ and \noriented versions of $S_2$ and $S_2^{-1}$. To prove this we use \nFigure~\\ref{shad6.fig}. There are two ways to get from \nFigure~\\ref{shad6.fig}a to Figure~\\ref{shad6.fig}b. One is to apply \n$S''_3$. Another way is to apply three times the oriented version of \n$S_2$ to obtain Figure~\\ref{shad6.fig}c, \nthen apply $S'_3$ to get Figure~\\ref{shad6.fig}d, \nand finally use three times the oriented \nversion of $S_2^{-1}$ to end up at Figure~\\ref{shad6.fig}b.\n\\begin{figure}[htb]\n \\begin{center}\n \\epsfxsize 9cm\n \\leavevmode\\epsffile{shad6.eps}\n \\end{center}\n\\caption{}\\label{shad6.fig}\n\\end{figure}\n\nThus it suffices to verify the invariance under $S'_3$. The terms\ncorresponding to the double points not in Figures~\\ref{shad8.fig}a \nand~\\ref{shad8.fig}b are preserved for the same reasons as above. \nThe terms coming from double points $u$ in \nFigure~\\ref{shad8.fig}a and $u$ in Figure~\\ref{shad8.fig}.b are the \nsame. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{shad8.eps}\n \\end{center}\n\\caption{}\\label{shad8.fig}\n\\end{figure}\nThis holds also for the $v$- and $w$-pairs of double points in these two\nfigures. We prove this statement only for the $u$-pair of double points. For\n$v$- and $w$-pairs the proof is the same or simpler. There is only\none possibility: either the dashed line belongs to both $\\pi(\\mu^+_u)$\nin Figures~\\ref{shad8.fig}a.1 and~\\ref{shad8.fig}b.1 respectively or to\nboth $\\pi(\\mu ^-_u)$ in Figures~\\ref{shad8.fig}a.2 \nand~\\ref{shad8.fig}b.2 respectively. \nWe choose the one to which it does not belong. \nSumming up gleams on each of the two sides of it we immediately see\nthat the corresponding shadows are the same on both pictures. \nThus the homology classes of the corresponding knots are equal. \nBut $[\\mu ^+_u][\\mu ^-_u]=[K]f$ (see~\\ref{sum-property}),\nthus the homology classes of the knots represented by the \nother shadows are also equal. \nThis completes the proof of Theorem~\\ref{correct2}.\\qed\n\n\\subsection{Proof of Theorem~\\ref{realization2}.}\\label{pfrealization2}\nI: $K'$ can be obtained from $K$ by a sequence of isotopies and\nmodifications along fibers. Isotopies do not change $S$. The modifications \nchange $S$ by\nelements of type~\\eqref{type2}. To complete the proof we use the\nidentity $\\xi_1 \\xi_2=[K]$.\n\nII: We prove that for any $i\\in H_1(N)$ there exist\ntwo knots $K_1$ and $K_2$ such that they represent the same free homotopy\nclass as $K$, \n\\begin{equation}\nS_{K_1}=S_K+(f-e)([K]i^{-1}+i),\\text{ and}\n\\end{equation}\n\\begin{equation} \nS_{K_2}=S_K-(f-e)([K]i^{-1}+i).\n\\end{equation}\nClearly this would imply the second statement of the theorem.\n\nTake $i\\in H_1(N)$. Let $K_i$ be an oriented knot in $N$ such that $[K_i]=i$. \nThe space $N$ is oriented, hence the tubular neighborhood $T_{K_i}$ of $K_i$ \nis homeomorphic to an oriented solid torus $T$. Deform $K_i$, so that\n$K_i\\cap T_{K_i}$ is a small arc (see Figure~\\ref{aicardi9.fig}). \nPull one part of the arc along $K_i$ in $T_{K_i}$ under the other part\nof the arc (see Figure~\\ref{aicardi1.fig}). This isotopy creates \ntwo new double points \n$u$ and $v$ of $\\pi(K)$. (Since $T_{K_i}$ may be knotted, it might happen that \nthere are other new double points, but we do not need them for\nour construction.) Making a fiber modification along the part of \n$\\pi^{-1}(u)$ that lies in $T$ one obtains $K_2$. Making a fiber\nmodification along the part of $\\pi^{-1}(v)$ that lies in $T$ one obtains\n$K_1$. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 7cm\n \\leavevmode\\epsffile{aicardi9.eps}\n \\end{center}\n\\caption{}\\label{aicardi9.fig}\n\\end{figure}\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 7cm\n \\leavevmode\\epsffile{aicardi1.eps}\n \\end{center}\n\\caption{}\\label{aicardi1.fig}\n\\end{figure}\nThis completes the proof of Theorem~\\ref{realization2}.\n\\qed\n\n\n\\subsection{Proof of Theorem~\\ref{homtor}.}\\label{pfhomtor}\nIt is easy to verify that any two shadows with the same projection can be \ntransformed to each other by a sequence of fiber fusions. \nOne can easily create a trivial knot with an ascending diagram \nsuch that its projection is any desired curve. \nThis implies that any two shadows \non ${\\mathbb R}^2$ can be transformed to each other by a sequence of fiber fusions, \nmovements $S_1,S_2,S_3$, and their inverses. \nA straightforward verification shows that $\\sigma(s(K))$ does \nnot change under the moves $S_1,S_2,S_3$, and their inverses. \nUnder fiber fusions the homology class of the knot and the element \n$\\sigma$ change in the same way. \nTo prove this, we use Figure~\\ref{shad16.fig}, where Figure~\\ref{shad16.fig}a shows the shadow before the application \nof the fiber fusion (that adds $1$ \nto the homology class of the knot) and \nFigure~\\ref{shad16.fig}b after. In this diagrams \nthe indices of the regions are denoted by Latin letters. Now one easily \nverifies that $\\sigma$ also increases by one. Finally, for \nthe trivial knot with a trivial shadow diagram its homology class \nand $\\sigma (s(K))$ are both equal to $0$.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 9cm\n \\leavevmode\\epsffile{shad16.eps}\n \\end{center}\n\\caption{}\\label{shad16.fig}\n\\end{figure}\nThis completes the proof of Theorem~\\ref{homtor}.\n\\qed\n\n\\subsection{Proof of Theorem~\\ref{correctSSK}.}\\label{pfcorrectSSK}\nIt suffices to show that $S_K$ does\nnot change under the elementary isotopies of the knot. \nThree of them correspond in the projection to: a \nbirth of a small loop, passing through a point of self-tangency, and passing\nthrough a triple point. The fourth one is \npassing through an exceptional fiber.\n\nFrom the proof of Theorem~\\ref{correct2} one gets that $S_K$\nis invariant under the first three of the elementary isotopies described above. \nThus it suffices to prove invariance under passing through an exceptional \nfiber $a$.\n\nLet $a$ be a singular fiber of type $(\\mu_a, \\nu_a)$ (see~\\ref{numbers}). \nLet $T_a$ be a neighborhood of $a$\nwhich is fiber-wise isomorphic to the standardly fibered solid torus with\nan exceptional fiber of type $(\\mu_a, \\nu_a)$. \n\nWe can assume that the move proceeds as follows. At the start \n$K$ and $T_a$\nintersect along a curve lying in the meridional disk $D$ of $T_a$. The part of\n$K$ close to $a$ in $D$ is an arc $C$ of a circle of a very large radius. This\narc is symmetric with respect to the $y$ axis passing through $a$ in $D$.\nDuring the move this arc slides along the $y$ axis through the fiber $a$\n(see Figure~\\ref{seif2.fig}). \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 11cm\n \\leavevmode\\epsffile{seif2.eps}\n \\end{center}\n\\caption{}\\label{seif2.fig}\n\\end{figure}\n\n\nClearly two points $u$ and $v$ of $C$ after this move are in the same \nfiber if and only if they are symmetric with respect to the $y$ axis, and the \nangle formed by $v,a,u$ in $D$ is less or equal to $\\pi$ and is equal \nto $\\frac{2l\\pi}{\\mu_a}$ for some $l\\in \\{1,\\dots,\\mu_a\\}$ \n(see Figure~\\ref{seif2.fig}).\nThey are in the same fiber \nbefore the move if and only if the angle formed by $u,a,v$ in $D$ \nis less than $\\pi$ and is equal to $\\frac{2l\\pi}{\\mu_a}$ for \nsome $l\\in \\{1,\\dots,\\mu_a\\}$ (see Figure~\\ref{seif2.fig}).\n \nConsider a double point $v$ of $\\pi\\big|_D(K)$ that appears \nafter the move and\ncorresponds to the angle $\\frac{2l\\pi}{\\mu_a}$. There is a unique\n$k\\in N_1(a)$ such that \n$\\frac{2\\pi\\nu_a k}{\\mu_a}\\protect\\operatorname{mod} 2\\pi=\\frac{2l\\pi}{\\mu}$. Note that to make\nthe splitting of $[K]$ into $[\\mu^+_v]$ and $[\\mu^-_v]$ \nwell defined, we do not need the two points of $K$ projecting to $v$ \nto be antipodal in $\\pi^{-1}(v)$. This allows one to compare \nthese homology classes with $f_a$. For the orientation of $C$ shown in \nFigure~\\ref{seif2.fig} one verifies that connecting $v$ to $u$ along the orientation of\nthe fiber we are adding $k$ fibers $f_a$. (Note that the\nfactorization we used to define the exceptionally fibered torus was\n$\\bigl(\\bigl(r,\\theta\\bigr),1\\bigr)=\n\\bigl(\\bigl(r,\\theta+\\frac{2\\pi\\nu}{\\mu} \\bigr),0\\bigr)$.) \nThus $[\\mu^-_v]=f_a^k$ (see\nFigure~\\ref{seif1.fig}). From~\\ref{sum-property} we know that\n$[\\mu^+_v][\\mu^-_v]=[K]f$. Hence $[\\mu^+_v]=[K]f_a^{\\mu_a-k}$. \n\n\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{seif1.eps}\n \\end{center}\n\\caption{}\\label{seif1.fig}\n\\end{figure} \n\nAs above, to each double point $v$ of $\\pi\\big|_D(K)$ before this move\nthere corresponds $k\\in N_2(a)$. For this double point $[\\mu^+_v]=f_a^{\\mu_a-k}$ and\n$[\\mu^-_v]=[K]f_a^k$. \n\nSumming up over the corresponding values of $k$ we see that \n$S_K$ changes by $R^1_a$ under this move. Recall that $R^1_a=0\\in H$. Thus \n$S_K$ is invariant under the move.\n\nFor the other choice of the orientation of $C$ the value of $S_K$ \nchanges by $R^2_a=0\\in H$.\n\nThus $S_K$ is invariant under all elementary isotopies, and this proves\nTheorem~\\ref{correctSSK}.\n\\qed \n\n\n\n\n\n\n\n\n\n \n\n\n\\subsection{Proof of Theorem~\\ref{front-shadow}.}\\label{pffront-shadow}\nDeform $L$ in the neighborhoods of all double points of $L$ \n(see Figure~\\ref{pic1.fig}), \nso that the two points of the Legendrian knot corresponding \nto the double point of $L$ are antipodal in the fiber. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{pic1.eps}\n \\end{center}\n\\caption{}\\label{pic1.fig}\n\\end{figure}\nAfter we make the quotient of the fibration by the ${\\mathbb Z}_2$-action, \nthe projection of the \ndeformed $\\lambda$ is not a cooriented front anymore but a front \nequipped with a normal field of lines. \n(This corresponds to the factorization $S^1\\rightarrow{\\mathbb R} P^1$.) \nUsing Figure~\\ref{pic3.fig} one calculates \nthe contributions of different cusps and double points to the total \nrotation number of the line field under traversing the boundary in the \ncounter clockwise direction.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 12cm\n \\leavevmode\\epsffile{pic3.eps}\n \\end{center}\n\\caption{}\\label{pic3.fig}\n\\end{figure} \n\n\nThese contributions are as follows:\n\\begin{equation}\n\\begin{cases}\n 1 & \\text{for every cusp point pointing inside $X$};\\\\\n -1 & \\text{for every cusp point pointing outside $X$};\\\\\n -1 & \\text{for every double point of the type shown in\nFigure~\\ref{pic2.fig}c};\\\\\n 0 & \\text{for the other types of double points.}\n\\end{cases}\n\\end{equation}\n\nTo get the contributions to gleams, we divide these numbers by $2$ \n(as we do in the construction of shadows, see~\\ref{prelim}).\n\nIf the region does not have cusps and double points in its boundary, \nthen the obstruction to an extension of the section over $\\partial X$ \nto $X$ is equal to $\\chi(\\protect\\operatorname{Int} X)$.\n\n\nThis completes the proof of Theorem~\\ref{front-shadow}.\n\\qed\n\n\\subsection{Proof of Theorem~\\ref{equivalence}.}\\label{pfequivalence}\nA straightforward verification shows that \n\\begin{equation}\\label{useful1}\n\\protect\\operatorname{ind} \\tilde L_u^+-\\protect\\operatorname{ind} \\tilde L_u^-=\n\\protect\\operatorname{ind} L^+_u-\\protect\\operatorname{ind} L^-_u-\\epsilon(u),\n\\end{equation} \n\\begin{equation}\\label{useful2}\n\\protect\\operatorname{ind} \\tilde L^+_u+ \\protect\\operatorname{ind} \\tilde L^-_u=\\protect\\operatorname{ind} L+1,\n\\end{equation}\nand\n\\begin{equation}\\label{useful3}\n\\protect\\operatorname{ind} L^+_u + \\protect\\operatorname{ind} L^-_u = \\protect\\operatorname{ind} L\n\\end{equation}\nfor any double point $u$ of $L$.\n\nLet us prove~\\eqref{rel1}. We write down the formal sums used to\ndefine $S(\\lambda)$ and $l_q(\\lambda)$ and start to reduce them in a\nparallel way as described below. \n\nWe say that a double point is essential if $[\\tilde\\lambda^+_u]\\neq \n[\\tilde\\lambda^-_u]$. \n\nFor non-essential $u$ we see that the term \n$\\bigl([\\tilde\\lambda^+_u]-[\\tilde\\lambda^-_u]\\bigr)$ in\n$S(\\lambda)$ is zero. Using~\\eqref{useful1} we get that the term\n$[\\protect\\operatorname{ind} L^+_v- \\protect\\operatorname{ind} L^-_v-\\epsilon (v)]_q$ in \n$l_q(\\lambda)$ is also zero. \n\nThe index of a wave front coincides with the homology class of its lifting\nunder the natural identification of $H_1(ST^*F)$ with ${\\mathbb Z}$. This fact\nand~\\eqref{useful2} imply that if we have \n$[\\tilde \\lambda^+_u]=[\\tilde \\lambda^-_v]$ for two double points $u$ and $v$, \nthen $[\\tilde \\lambda^-_u]=[\\tilde \\lambda^+_v]$. \nHence $\\bigl([\\tilde \\lambda^+_u]-\n[\\tilde \\lambda^-_u]\\bigr)=-\\bigl([\\tilde \\lambda^+_v]-\n[\\tilde \\lambda^-_v]\\bigr)$,\nand these two terms cancel out. Identity~\\eqref{useful1} implies that the \nterms $[\\protect\\operatorname{ind} L^+_u-\\protect\\operatorname{ind} L^-_u-\\epsilon(u)]_q$ and \n$[\\protect\\operatorname{ind} L^+_v-\\protect\\operatorname{ind} L^-_v-\\epsilon(v)]_q$ also cancel out. \n\nFor similar reasons, if for a double point $u$ the term \n$\\bigl ([\\tilde \n\\lambda^+_u]-[\\tilde \\lambda^-_u]\\bigr)$ is equal to \n$\\bigl([\\lambda]-f\\bigr)$, so that we can simplify $S(\\lambda)$ by crossing\nout the term and decreasing the coefficient $C^+$ by one. \nThen \n$[\\protect\\operatorname{ind} L^+_u-\\protect\\operatorname{ind} L^-_u-\\epsilon(u)]_q=[\\protect\\operatorname{ind} L-1]_q$, and we can simplify \n$l_q(\\lambda)$ by crossing out the term and decreasing \nthe coefficient $C^+$ by one.\n\n\nSimilarly if the input of double point $u$ into $S(\\lambda)$ is \n$\\bigl(e-[\\lambda]f\\bigr)$, then we reduce the two sums in the parallel way\nby crossing out the corresponding terms and decreasing by one the coefficients\n$C^-$. \n\nWe make the cancellations described above in both $S(\\lambda)$ and\n$(l_q(\\lambda)-[h]_qh)$ in a parallel way until we can \nnot reduce $S(\\lambda)$ any more. \nIn this reduced form the terms of the form $k_lf^l$ with\n$k_l>0$ correspond to \nthe terms of type $[\\tilde \\lambda^+_u]$ for some double points $u$ of $L$.\n(The case where a term of this type correspond to cusps is \ntreated separately below.) \nIdentities~\\eqref{useful1} \nand~\\eqref{useful2} imply that the contribution of the corresponding \ndouble points into $l_q(\\lambda)$ is $k_l[2l-h-1]_q$. \n\nIn the case where $k_lf^l$ term comes from the cusps and not from the double\npoints of $L$, one can easily verify that the corresponding input of cusps \ninto $(l_q(\\lambda)-[h]_qh)$ can still be written as $k_l[2l-h-1]q$.\n\nThus $l_q(\\lambda)=[h]_qh+\\sum_{k_l>0} k_l[2l-h-1]_q$, \nand we have proved~\\eqref{rel1}.\n\nLet us prove~\\eqref{rel2}. \nAs above we reduce $S(\\lambda)$ and\n$(l_q(\\lambda)-[h]_qh)$ in a parallel way. Note that the\ncoefficient at\neach $[m]_q$ was positive from the very beginning by the definition\nof $l_q(\\lambda)$, and it stays positive under the cancellations described \nabove. \nAfter this reduction each term $n_m[m]_q$ \nis a contribution of $n_m$ double points. (The case where it is a\ncontribution of \ncusps is treated separately as in the proof of~\\eqref{rel2}.) \nLet $u$ be one of these double points.\nThen from~\\eqref{useful1} and~\\eqref{useful2} we get\nthe following system of two equations in variables $\\protect\\operatorname{ind} \\tilde L^+_u$ and \n$\\protect\\operatorname{ind} \\tilde L^-_u$:\n\\begin{equation}\n\\begin{cases}\n\\protect\\operatorname{ind} \\tilde L^+_u-\\protect\\operatorname{ind} \\tilde L^-_u=m &\\\\\n\\protect\\operatorname{ind} \\tilde L^+_u+\\protect\\operatorname{ind} \\tilde L^-_u=\\protect\\operatorname{ind} L+1.&\n\\end{cases}\n\\end{equation}\n\nSolving the system we get that $[\\tilde \\lambda\n^+_u]=f^{\\frac{m+h+1}{2}}$ and $[\\tilde \\lambda\n^-_u]=f^{\\frac{h+1-m}{2}}$.\n\nThis proves identity~\\eqref{rel2} and Theorem~\\ref{equivalence}.\n\\qed \n \n\n\n\\subsection{Proof of Theorem~\\ref{orbifold1}.}\\label{pforbifold1}\nThere are five elementary isotopies of a generic front $L$ on an orbifold\n$F$. Four of them are: the birth of two cusps, passing through a\nnon-dangerous self-tangency point, passing through a triple point,\nand passing of a branch through a cusp point. For all possible oriented\nversions of these moves a straightforward calculation shows that \n$S(\\lambda)\\in \\frac{1}{2}{\\mathbb Z}[H_1(N)]$ is preserved.\n\nThe fifth move is more complicated. It corresponds to a generic passing of \na wave front lifted to ${\\mathbb R}^2$ \nthrough the preimage of a cone point $a$. \nWe can assume that this\nmove is a symmetrization by $G_a$ of the following move. \nThe lifted front in the neighborhood of $a$ is an arc $C$ of a circle\nof large radius with center at the $y$ axis, and during this move \nthis arc slides through $a$ along $y$ (see Figure~\\ref{orbi2.fig}).\n\nClearly after this move points $u$ and $v$ on the arc $C$ turn out to be \nin the same fiber if and only if they are symmetric with respect \nto the $y$ axis, and the angle formed by $v,a,u$ is less or equal to $\\pi$ \nand is equal to $\\frac{2k\\pi}{\\mu_a}$ for some $k\\in \\{1,\\dots,\\mu_a\\}$ \n(see Figure~\\ref{orbi2.fig}). \nWe denote the set of such numbers $k$ by \n$\\bar N_1(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big|\\frac{2k\\pi}{\\mu_a}\n\\in(0,\\pi]\\bigr\\}$. \n\nTwo points $u$ and $v$ on the arc $C$ are in the same fiber \nbefore the move if and only if they are symmetric with respect \nto the $y$ axis, and the angle formed by $u,a,v$ is less than\n$\\pi$ and is equal $\\frac{2k\\pi}{\\mu_a}$ for \nsome $k\\in \\{1,\\dots,\\mu_a\\}$ (see Figure~\\ref{orbi2.fig}).\nWe denote the set of such numbers $k$ by \n$\\bar N_2(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big|\\frac{2k\\pi}{\\mu_a}\n\\in(0,\\pi)\\bigr\\}$.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 11cm\n \\leavevmode\\epsffile{orbi2.eps}\n \\end{center}\n\\caption{}\\label{orbi2.fig}\n\\end{figure} \n\n\nThe projection of this move for the orientation of $L'$ drawn in\nFigure~\\ref{orbi2.fig} is shown in Figure~\\ref{orbi1.fig}. \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{orbi1.eps}\n \\end{center}\n\\caption{}\\label{orbi1.fig}\n\\end{figure} \n\n\nSplit the wave front in\nFigure~\\ref{orbi1.fig} at the double point $v$ (appearing after the move) \nthat corresponds to some\n$k\\in \\bar N_1(a)$. \nThen $\\tilde\\lambda^-_v$ is a front with two positive cusps that\nrotates $k$ times around $a$ in the clockwise direction. Hence\n$[\\tilde\\lambda^-_v]=ff_a^{-k}=f_a^{\\mu_a-k}$. We know that \n$[\\tilde\\lambda^+_v][\\tilde\\lambda^-_v]=[\\lambda]f$ and that\n$f_a^{\\mu_a}=f$. \nThus\n$[\\tilde\\lambda^+_v]=[\\tilde\\lambda]f_a^k$.\n \nIn the same way we verify that if we split the front \nat the double point $v$ \n(existing before the move) that corresponds to some $k\\in \\bar N_2(a)$,\nthen $[\\tilde \\lambda^+_v]=f_a^k$ and $[\\tilde \\lambda^-_v]=[K]f_a^{|\\mu_a|-k}$. \n\nNow making sums over all corresponding numbers $k\\in \\{1,\\dots,\\mu_a\\}$\nwe get\nthat under this move $S(\\lambda)$ changes by \n\\begin{equation}\n\\bar R^1_a=\\sum_{k\\in\\bar N_1(a)}\\bigl([\\lambda]f_a^k-f_a^{\\mu_a-k}\\bigr)-\n\\sum_{k\\in\\bar N_2(a)}\\bigl(f_a^k-[\\lambda]f_a^{\\mu_a-k}\\bigr).\n\\end{equation}\nA straightforward verification shows that $R^1_a=\\bar R^1_a$. (Note that the \nsets $N_1(a)$ and $N_2(a)$ are different from $\\bar N_1(a)$ and $\\bar N_2(a)$.)\n\nRecall that $R^1_a=0\\in J$. Thus $S(\\lambda)$ is invariant under the move. \n\nFor the other choice of the orientation of $C$ the value of \n$S(\\lambda)$ changes by $R^2_a=0\\in J$. \n\nHence $S(\\lambda)$ is invariant under all elementary\nisotopies, and we have proved Theorem~\\ref{orbifold1}.\n\\qed\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}