The large-scale curvature perturbations induced by spectator anisotropic
stresses are analyzed across the matter-radiation transition. It is assumed
that the anisotropic stress is associated with a plasma component whose energy
density is subdominant both today and prior to photon decoupling. The
enforcement of the momentum constraint and the interplay with the neutrino
anisotropic stress determine the regular initial conditions of the
Einstein-Boltzmann hierarchy. The Cosmic Microwave Background observables have
shapes and phases which differ both from the ones of the conventional adiabatic
mode as well as from their non-adiabatic counterparts.Comment: 9 pages, 4 included figure

Giovannini, Massimo

English

arXiv

The large-scale curvature perturbations induced by spectator anisotropic stresses are analyzed across the matter-radiation transition. It is assumed that the anisotropic stress is associated with a plasma component whose energy density is subdominant both today and prior to photon decoupling. The enforcement of the momentum constraint and the interplay with the neutrino anisotropic stress determine the regular initial conditions of the Einstein-Boltzmann hierarchy. The Cosmic Microwave Background observables have shapes and phases which differ both from the ones of the conventional adiabatic mode as well as from their non-adiabatic counterparts.The large-scale curvature perturbations induced by spectator anisotropic stresses are analyzed across the matter-radiation transition. It is assumed that the anisotropic stress is associated with a plasma component whose energy density is subdominant both today and prior to photon decoupling. The enforcement of the momentum constraint and the interplay with the neutrino anisotropic stress determine the regular initial conditions of the Einstein-Boltzmann hierarchy. The Cosmic Microwave Background observables have shapes and phases which differ both from the ones of the conventional adiabatic mode as well as from their non-adiabatic counterparts

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Spectator stresses and CMB observablesMassimo Giovannini*Department of Physics, Theory Division, CERN, 1211 Geneva 23, Switzerland,and INFN, Section of Milan-Bicocca, 20126 Milan, Italy(Received 22 December 2009; published 23 June 2010)The large-scale curvature perturbations induced by spectator anisotropic stresses are analyzed acrossthe matter-radiation transition. It is assumed that the anisotropic stress is associated with a plasmacomponent whose energy density is subdominant both today and prior to photon decoupling. Theenforcement of the momentum constraint and the interplay with the neutrino anisotropic stress determinethe regular initial conditions of the Einstein-Boltzmann hierarchy. The cosmic microwave backgroundobservables have shapes and phases which differ both from the ones of the conventional adiabatic mode aswell as from their nonadiabatic counterparts.DOI: 10.1103/PhysRevD.81.127302 PACS numbers: 98.80.Jk, 98.70.Vc, 98.80.Cq, 98.80.EsSupplementary anisotropic stresses may arise beforematter-radiation equality for diverse physical reasons. Forinstance, on a phenomenological ground, one is often led toconsider a putative anisotropic stress associated with thedark-matter component [1]. Along a similar line it is alsolegitimate to consider the possibility that the dark-energycomponent is endowed with an appropriate anisotropicstress [2]. The two possibilities mentioned so far are char-acterized by a common feature: in both cases the aniso-tropic stress is attributed to species which are dominant (orjust slightly subdominant) today. This is the case for thedark-matter and for the dark-energy components in theCDM paradigm whose basic parameters will be takento coincide, for the purposes of the present analysis, withthe ones determined by analyzing the WMAP 5 yr dataalone1 [3,4]. The purpose of the present paper is to scruti-nize the situation where the source of large-scale inhomo-geneity resides in a spectator anisotropic stress whoseassociated energy density is subdominant today. It will beargued that the ‘‘stressed’’ initial conditions for the CMBanisotropies constitute an intermediate case between thepurely adiabatic and the purely nonadiabatic initial con-ditions (see, e.g., [5]). The energy-momentum tensor of thespectator fluid can be parametrized asT 00 ¼ sx; T ji ¼spxþji ; T i0 ¼ 0; (1)where spx ¼ wxsx and @i@jij ¼ ðp þ Þr2x;p and  are the pressure and energy density of thephotons; the ð0iÞ components of the energy-momentumtensor of the spectator field are set to zero since it is ofhigher order when the energy density and pressure of thesame field are fully inhomogeneous (i.e. they do not havehomogeneous background contribution). The barotropicindex wx will be taken to be constant for the purposes ofthe present investigation. In the present paper, only thescalar fluctuations of the metric will be considered: thesymbol s appearing in Eq. (1) denotes the scalar fluctua-tions of the corresponding quantity. The time evolution ofthe system2 will be parametrized in terms of the scalefactor aðÞ normalized at equality, i.e.  ¼ a=aeq; withinthis parametrization the evolution equations for the fluctu-ations of the geometry and of the sources shall be written inFourier space and in terms of the rescaled wave number ¼ k1; if  < 1 the given wavelength exceeds theHubble radius right before the equality time. In the oppo-site case (i.e.  > 1) the given wavelength is smaller thanthe Hubble radius at the same time 1 ¼ ðﬃﬃﬃ2p þ 1Þeq. Inthe synchronous coordinate system the perturbed entry ofthe geometry reads3sgijð;Þ ¼ a2eq2ðÞ^i^jhþ 6^i^j ij3; (2)where hð;Þ and ð;Þ describe the scalar fluctuationsof the metric and enter also the contravariant componentsof the energy-momentum tensor, i.e.*massimo.giovannini@cern.ch1For sake of concreteness the values of the cosmologicalparameters used in the present paper will be taken tocoincide with the parameters arising from the 5 yr best fit tothe WMAP data alone, i.e., using standard notations,ðb0;c0;; h0; ns; 	reÞ ¼ (0.0441, 0.214, 0.742, 0.719,0.963, 0.087), where 	re denotes the optical depth to reionizationand ns the spectral index of (adiabatic) curvature perturbations.2Without loss of generality the space-time geometry will betaken to be conformally flat implying that the backgroundgeometry can be written as g ¼ a2ðÞ where  diagð1;1;1;1Þ is the Minkowski metric with signaturemostly minus.3The conventions employed in Refs. [6,7] differ from the onesemployed here (and match the ones of [8] and referencestherein).PHYSICAL REVIEW D 81, 127302 (2010)1550-7998=2010=81(12)=127302(4) 127302-1  2010 The American Physical SocietysT00 ¼ sta2eq2; sT0i ¼ 1a2eq2ðpt þ tÞvit;sTij ¼ 1a2eq2sptij þ 2ptij þ ^i^j2ðhþ 6Þijt; (3)whereijt denotes the total anisotropic stress. Introducingthe density contrasts for neutrinos, photons, CDM particle,and baryons (i.e., respectively, , , c, and b) as wellas x ¼ sx=, the fluctuation of the total energy den-sity and the total anisotropic stress are4st ¼ tRðR þ RÞ þM!c0!M0c þ !b0!M0 bþ RRx; (4)ijijt ¼ 2ðp þ Þ þXaðpa þ aÞaþ ðp þ Þx; (5)where RðÞ ¼ 1=ðþ 1Þ and MðÞ ¼ =ðþ 1Þ;ð;Þ denotes the neutrino anisotropic stress; R ¼0:4052, R ¼ ð1 RÞ, and RbðÞ ¼ ð3=4Þb= ’0:215 denote, respectively, the neutrino fraction, thephoton fraction, and the ratio of the baryon and photonenergy densities (weighted by a factor 3=4 which arises inthe photon-baryon sound speed). In Eq. (5) a sum overother potential components has been added: these compo-nents could account for the possible stresses arising eitherin the dark-matter or in the dark-energy sectors (see, e.g.,[1]) but will not be explicitly considered hereunder. Thetimelike and the spacelike components of the covariantconservation of the energy-momentum tensor of Eq. (1)demand, respectively, that@x@þ 3wx  1x ¼ 0; xð;Þ ¼ 43wx xð;Þ:(6)In the  parametrization the Hamiltonian and the momen-tum constraints read, respectively,@h@¼ 22ðþ 1Þ3RðR þ RÞþM!c0!M0c þ !B0!M0 bþ RRx; (7)22@@¼  4ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1pR þ R½1þ RbðÞbþ 34!c0!M0c; (8)where b ¼  ’ b represents the common value of thephoton and baryon velocities; the same identification willbe made hereunder [see Eqs. (14) and (15)] and it is fullyjustified to lowest order in the tight-coupling approxima-tion. Defining the variable Q ¼ ðhþ 6Þ, the remainingtwo (perturbed) Einstein equations can be written as@2h@2þ 5þ 42ðþ 1Þ@h@ 22ðþ 1Þ¼ 32ðþ 1Þ ½R þ R þ 3wxRx; (9)@2Q@2þ 5þ 42ðþ 1Þ@Q@¼ 22ðþ 1Þ þ122ðþ 1Þ ðR þ RxÞ: (10)The evolution equations of the neutrinos obey@@¼  23ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p þ23@h@;@@¼ 28ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p  22ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p ;(11)@@¼ 215ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p 215@Q@ 320F 3ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p : (12)The evolution equations of the dark-matter sector obeyinstead@c@¼  c2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p þ12@h@;@c@þ c¼ 0: (13)The system of photons and baryons can be treated, withinan excellent approximation, to lowest order in the tight-coupling expansion where the quadrupole of the photonsvanishes and the governing equations are given by@b@þ RbbðRb þ 1Þ ¼28ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p ðRb þ 1Þ; (14)@@¼  23bﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p þ23@h@;@b@¼  b2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃþ 1p þ12@h@:(15)From Eq. (6), it follows that Pð; Þ ¼½4=ð3wxÞPðÞð=iÞ13wx , where Pð;Þ denotesthe power spectrum of x while PðkÞ denotes the powerspectrum of x whose explicit expression is assigned asPðkÞ ¼ Bðk=kpÞnx1; kp ¼ 0:002 Mpc1 is the pivot4Using standard notations !c0 ¼ h20c0, !b0 ¼ h20b0, and!M0 ¼ h20M0.BRIEF REPORTS PHYSICAL REVIEW D 81, 127302 (2010)127302-2scale at which the power spectrum coincides with itsamplitude B. Diverse initial conditions can be imposedto the whole system of equations. They are customarilydivided into adiabatic and nonadiabatic [5,9] (see also[6,7]). The adiabaticity condition together with the con-straints on the evolution of the fluctuations implies thefollowing set of initial conditions:ð;iÞ ’ ð;iÞ ’ 34bð;iÞ ’ 34c ¼Rxð;iÞ;ð;iÞ ¼ 0; (16)ð;iÞ ’ bð;iÞ ’ cð;iÞ ’ 0;xð;iÞ ¼ 3wx4 xð;iÞ;(17)where i denotes the initial integration time when theneutrino anisotropic stress vanishes exactly. We are nowinterested to see what happens to the neutrino anisotropicstress when the initial conditions of the system obeyEqs. (16) and (17). The results of the numerical integrationare illustrated in Fig. 1 for different values of wx anddifferent values of . The initial conditions are set in thelimit  dec, where dec ¼ adec=aeq denotes the valueof  at photon decoupling (for the best fit to the 5 yrWMAP data alone dec ’ 2:92 as indicated in Fig. 1 withthe vertical line in the plot at the left). If wx > 1=3 theanisotropic stress is driven to zero; if  1, thenð;Þwill not oscillate for   1 while in the opposite case (i.e. > 1) the neutrino anisotropic stress will be oscillating forthe same range of . If wx ¼ 1=3 the asymptotic value ofð;Þ will be, approximately, R=Rxðk; decÞ for < 1; in the opposite case (i.e.  > 1)  will oscillatearound the same asymptote reached in the  < 1 case. Asimilar phenomenon has been discussed, in the past, in thecase of large-scale magnetic fields which affect the aniso-tropic stress but which also interact with the chargedparticles [10]. Finally notice that, if wx < 1=3, xð;Þgrows with  and might even get dominant. This situationis more similar to the one treated in [1] and will not bespecifically addressed here. From the numerical solution ofthe system in terms of hð; Þ and ð; Þ one can alsocompute other fluctuations with relevant gauge-invariantinterpretation such as the curvature perturbations on co-moving orthogonal hypersurfaces (i.e. R), the densitycontrast on comoving orthogonal hypersurfaces (i.e. 	m),and also  the curvature perturbations on constant densityhypersurfaces (see, e.g. [11]). Denoting with i  1 thevalue of  at the onset of the numerical integration,ð;Þ / lnð=iÞ provided ð; iÞ ! 0 since  andb are both proportional to 2.It is interesting to assume now that the whole source oflarge-scale inhomogeneity resides in the spectator aniso-tropic stresses. Could we get reasonable shapes of theCMB observables in the absence of the conventional adia-batic mode? The answer is negative and it is contained inFig. 2 where the temperature autocorrelations (i.e. TTpower spectra) and the temperature-polarization cross cor-relations (i.e. TE power spectra) are computed in the mostfavorable case, i.e. wx ¼ 1=3. In both plots of Fig. 2 thefull line illustrates the WMAP 5 yr best fit (in terms of theWMAP data alone and in the case of the conventionaladiabatic mode) while the dashed and dot-dashed linesrefer to the case of the spectator stresses with differentspectral indices (one of them coinciding with the one of thestandard adiabatic mode). For the purposes of Fig. 2, theamplitude of the power spectrum of the anisotropic stresshas been taken to beB ¼ 2:41 109. In spite of possibleadjustments in the amplitude as well as in the pivot scale,the shapes of the TTand TE correlations cannot be made tocoincide either with the ones of the conventional adiabaticmode nor with the typical patterns of the isocurvaturemodes.The numerical integration has been carried on by using,as initial conditions of the Boltzmann solver, the result ofthe numerical integration of the explicit system discussedabove. The matching regime between the two regimescoincides with  ’ 104 when the tight coupling betweenbaryons and photons is valid. The first regime of evolutiondictated by the equations derived here avoids a potentiallystiff problem if  is initially very small. The Boltzmannsolvers is a modified version of [8,12] which, in turn, is−8 −6 −4 −2 0 2 4−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.0500.05log ασν(κ,α)ωb0 = 0.02273, ωc0 = 0.1099, ΩΛ = 0.742κ = 10−4, wx − 1/3 =0κ = 10−4, wx  − 1/3 = 0.1 κ = 10−4, wx − 1/3 = 0.3αdec = 2.927−8 −6 −4 −2 0 2 4−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.0500.05log ασν(κ,α)ωb0 =0.02273, ωc0 =0.1099, ΩΛ =0.742wx  −1/3 =0, κ =0.1wx −1/3 =0, κ =0.01wx − 1/3 =0, κ=0.001wx − 1/3 =0.3, κ =1wx − 1/3 = 0.3, κ = 0.1wx −1/3 =  0.3, κ =0.01FIG. 1 (color online). The relaxation of the neutrino anisotropic stress for different values of wx and .BRIEF REPORTS PHYSICAL REVIEW D 81, 127302 (2010)127302-3based on [6,7]. From Fig. 2 it appears that, as expected, theshapes and phases of the TT and TE correlations obtainedin the case of the adiabatic mode (full lines in both plots)differ from the ones induced by the spectator stresses(dashed and dot-dashed lines in both plots of Fig. 2).Spectator stresses can be treated and discussed in conjunc-tion with a dominant adiabatic mode; absent of the adia-batic mode the shapes of the TT correlations haveintermediate features between the isocurvature humpsof the CDM-radiation mode [5]. The situation is alsodifferent from the case of the magnetized CMB observ-ables when the adiabatic mode is absent [8] (see also [12]).In [13] (first reference) it is claimed that large-scale mag-netic fields provide the same shapes for the TT correlationsobtainable in the case of an adiabatic mode with appropri-ate amplitude. The initial conditions derived in [10] havebeen used in [13] (with supplementary typos) and thensubsequently criticized by a subset of the authors (see,respectively, third and second references of [13]). Sincewe find the references [13] self-contradictory we do notwant to comment on the possibility that a fully inhomoge-neous magnetic field can seed structure formation in theabsence of an adiabatic mode (as, suggested in the firstreference of [13]). The present paper shows more modestlythat spectator stresses alone [as introduced in Eq. (1)] donot reproduce the observed patterns of the TT and TEcorrelations assuming the initial conditions given in (16)and (17).[1] W. Hu, Astrophys. J. 506, 485 (1998); W. Hu and D. J.Eisenstein, Phys. Rev. D 59, 083509 (1999); R. Maartens,J. Triginer, and D. Matravers, Phys. Rev. D 60, 103503(1999).[2] T. Koivisto and D. F. Mota, Phys. Rev. D 73, 083502(2006).[3] G. Hinshaw et al. (WMAP Collaboration), Astrophys. J.Suppl. Ser. 180, 225 (2009); E. Komatsu et al. (WMAPCollaboration), Astrophys. J. Suppl. Ser. 180, 330(2009).[4] J. Dunkley et al. (WMAP Collaboration), Astrophys. J.Suppl. Ser. 180, 306 (2009); M.R. Nolta et al. (WMAPCollaboration), Astrophys. J. Suppl. Ser. 180, 296 (2009).[5] H. Kurki-Suonio, V. Muhonen, and J. Valiviita, Phys. Rev.D 71, 063005 (2005); R. Keskitalo, H. Kurki-Suonio, V.Muhonen, and J. Valiviita, J. Cosmol. Astropart. Phys. 09(2007) 008; M. Giovannini and K. E. Kunze, Phys. Rev. D77, 123001 (2008).[6] C. P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995);E. Bertschinger, arXiv:astro-ph/9506070.[7] U. Seljak and M. Zaldarriaga, Astrophys. J. 469, 437(1996); M. Zaldarriaga, D.N. Spergel, and U. Seljak,Astrophys. J. 488, 1 (1997).[8] M. Giovannini, Phys. Rev. D 79, 121302 (2009); 79,103007 (2009).[9] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1(1984).[10] M. Giovannini, Phys. Rev. D 74, 063002 (2006); ClassicalQuantum Gravity 23, 4991 (2006); Phys. Rev. D 70,123507 (2004).[11] J. Hwang, Astrophys. J. 375, 443 (1991); J. Hwang and H.Noh, Phys. Rev. D 64, 103509 (2001); Classical QuantumGravity 19, 527 (2002).[12] M. Giovannini, PMC Phys. A 1, 5 (2007); M. Giovanniniand K. E. Kunze, Phys. Rev. D 77, 063003 (2008); 77,123001 (2008).[13] K. Kojima, T. Kajino, and G. J. Mathews, J. Cosmol.Astropart. Phys. 02 (2010) 018; K. Kojima and K.Ichiki, arXiv:0902.1367; D. G. Yamazaki et al., Phys.Rev. D 77, 043005 (2008).100 101 102 103 104102103104105ll(l+1)Cl(TT)/(2π) (µ K)2AR =2.41× 10−9,ωb0 =0.02273, ωc0 =0.1099, ΩΛ=0.742nx = 0.963, wx =1/3ns =0.963, 5yr best−fit WMAP data alonenx = 1.25, wx =1/30 500 1000 1500−800−600−400−2000200400ll(l+1)Cl(TE)/(2π) (µK)2AR =2.41× 10−9, ωb0 =0.02273, ωc0=0.1099, ΩΛ =0.742FIG. 2 (color online). The TT and TE power spectra induced by the spectator anisotropic stress in the absence of the conventionaladiabatic mode (dashed and dot-dashed lines in both plots). The WMAP 5 yr best fit is reported, for comparison, with the full lines inboth plots. Double logarithmic scale is used in the plot at the left while a linear scale is employed in the plot at the right.BRIEF REPORTS PHYSICAL REVIEW D 81, 127302 (2010)127302-4

Spectator stresses and CMB observables

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