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PHYSICAL REVIEW D VOLUME 47, NUMBER 4 15 FEBRUARY 1993 Residual fluctuations in the microwave background at large angular scales: Revision of the Sachs-Wolfe effect Joan Josep Ferrando, Juan Antonio Morales, and Miquel Portilla Departament de Física Tebrica, Universitat de Valencia, 46100 Burjassot, Valencia, Spain (Received 2 April 1992) In this paper we revise the Sachs-Wolfe (SW) computation of large-scale anisotropies of the mi- crowave background temperature, taking into account the properties of (he metrics admilling an isotro- pie distribution of collisionless photons. We show that the motric uscd by SW belongs Lo the aforemen- tioned class, and conclude that the microwave background (ones the dipolar anisotropy has been sub- tracted) should now be isotropic at large angular scales, provided that it was isotropic on the last scatter- ing surface and assuming that the growing mode of a pressureless Einstein—de Sitter perturbation is a good description of the metric. PACS number(s): 98.70.Vc, 98,80,Hw TL INTRODUCTION Up to date measurements of lhe cosmic microwave background (CMB) temperature show only dipolar an- isotropy; technical refinements are providing smaller upper limits for temperature anisotropies at different an- gular scales. Tn the framework of big bang models this highly isotropic CMB is interpreted as a relic of a hot past age of the Universe, when matter and radiation were strongly coupled. Tf the Universe was then exactly homo- geneous and isotropic (a Robertson-Walker universe), the CMB could have entered the recombination era as highly isotropic and with a blackbody spectrum (1]. However, to understand the origin of the structures observed at present, one needs to assume that in the early Universe inhomogeneities existed which were much bigger than statistical fluctuations. The problem is how these irregu- larities are imprinted in the CMB, and if these predic- tions are compatible with the upper limits observed. The physical processes involved in this problem depend on the angular scale of the observation. At large scales the physics becomes simpler. The horizon at redshift z sub- tends now an angle 0, =1/W1+z (we are considering an Ejnstein-de Sitter background) and, assuming recom- bination oceurs at z = 1000, onc gets 9, 1.8", Therefore angular scales greater than 6, correspond to structures that are bigger than the horizon at decoupling time, and so scattering with electrons can be neglected. On the oth- er hand, reionization may occur later in the process of galaxy formation, Using general arguments one can justi- fy (2] that if this process starts at z==30, scattering with electrons may erase primordial anisotropies at angular scales smaller than 9,6”, introducing new ones linked to the peculiar motion of the plasma. According to these considerations, anisotropies at angular scales bigger than 0, are unafíccted by electron scattering. For this reason One expects that measurements of CMB anisotropy at these angular scales (bigger than 6, ) will probe the distri- bution of matter on scales larger than the horizon at decoupling time. In this case, CMB anisotropies are ex- 47 pected because of gravitational potential fuctuations. This effect was first stated by Sachs and Wolfe (SW) [3] on considering the growing mode of a perturbed pres- sureless Einstein—de Sitter universe (k =0), and assum- ing that radiation, as measured by an observer comoving with matter, was isotropic on the last scattering surface 2. This was defined in physica] terms as a surface of constant temperature T =T.,, and expressed, in the coor- dinates used by them, as a surface of constant time Ma The aim of this paper is to revise this procedure. So, in Sec. Il, we recall well-known results concerning the ex- istence of isotropic solutions to the Liouville equation for massless particles in an inhomogeneous space-time. In Sec. III we show that the metric used in the SW paper is not compatible with the assumption of an isotropic distri- bution of collisionless photons with uniform temperature on a hypersurface 7 —const in comoving time. The con- clusion drawn is that radiation will remain isotropic if it was so at the recombination cpoch. TL. SPACE-TIMES ADMITTING ISOTROPIC DIATION Radiation can be described as a gas of photons with a distribution function f(x,p)p?=0. If scattering with electrons can be neglected (or if radiation is in equilibri- um with matter) the distribution function must satisfy the Liouville equation SGD, PLA) =comst , (1) x (2) being any null geodesic and p(1)=dx/dA. As one is interested in the temperature measured by a given ob- server, it is preferable to write the Liouville equation in terms of specific intensity. Let us consider a unit timelike vector field n, the frequency v and the specific intensity f,, measured by observer n are given by v==n:p and 1,=h*w f, respectively. From the Liouville equation one gets £,,/vó=const along any light ray, and denoting any 1308 (6,1993 The American Physical Society 47 RESIDUAL FLUCTUATIONS IN THE MICROWAVE ... 1309 two events connected by a null geodesic as x, and xp, one has _ E (+2 Lo (2) where z is the redshift corresponding to the observers n, and 1 at x, and xp, respectively. Expressing 1, in terms of an effective temperature 7, one gets the relation T. T¿= Z" Tf the radiation is isotropic with respect to an observer n, the distribution function depends on p through the fre- quency measured by n, fix,p)=j(x,v). Tauber and Weinberg [4] and Eblers, Geren, and Sachs [5] have stud- ied the conditions imposed on the metric and the vector field n so that the solution to the Liouville equation is iso- tropic. We have recently treated the problem of getting inhomogeneous solutions to Einstein's equations admit- ting isotropic radiation [6]. The results concerning an isotropic distribution function of massless particles may be summarized by tbe equivalence of the following state- ments: (i) A distribution function of photons, f(x,p), p?=0, exists which is isotropic with respect to an ob- server n and verifies the Liouville equation (1); (ii) a unit timelike vector field a exists such that =0 and d(a —0n/3)=0, where a, a, and 0 are the shear, ac- celeration, and expansion of n, respectively; (iii) the space-time metric is conformally stationary, that is to say, there exists a timelike vector field £ and a scalar A such that L¿g = Ag. Moreover, the observer of condition (1) [which is also the vector field of (ii)] and the conformal Killing vector of (iii) are collinear. The distribution fune- tion is an arbitrary function of the first integral associated with the conformal Killing vector, f(x,p)=/(£*p). An important corollary for our discussion is the fol- lowing: when the vector field n admits a family of or- thogonal surfaces, the metric satisfying condition (hi) can be written in the form da NWE atada, where and y, depend only on the spatial coordinates. The observer measuring isotropic radiation will then be n=(aby a, (5) and the temperature of the isotropic radiation will be in- homogeneous r=—2—, aq) b being a positive constant. One can understand this result taking into account re- lation (3) Let us consider a spacelike surface (6) 2:9=3s(x!) in the past of the event xy. A null geodesic arriving at xp intersects E at x.=(10x.), with Ne Nxfx¿%). The temperature at this point, according to (6), is T,=b/(a,D,). On the other hand, the redshift of light emitted at rest with respect to 1, and received by ny is given by 1+2=a¿B,/(a,¿B,). So, taking into account relation (3) we get T¿=b/(a¿Dy) for the temperature at Xo: TIT. REVISION OF THE SACHS-WOLFE EFFECT The following hypotheses are appropriate to studying large angular scale anisotropies in the CMB produced by structures bigger than the horizon at decoupling time. (1) From matter radiation decoupling up to the present, space-time may be described as a p =0 perturba- tion of an Einstein—de Sitter universe. Only the density growing mode is relevant at this epoch. Sachs and Wolfe B] obtained this solution in comoving time-orthogonal coordinates (7,x*): anoto [ar -19 |-2s a a= il Hs where 7 runs from 0 to 1 at the present epoch, aud $ de- pends on the spatial coordinates only. (2) The Universe is inhomogeneous at scales bigger than the horizon. So density variations smaller than 10? light years are ignored lassuming the present value of the Hubble constant to be I¿=107% years” 1), (3) Isotropic CMB with a Planck spectrum dates from the recombination epoch. The first two hypotheses are the same as in the SW paper. The third one is less res- trictive because we are not assuming that decoupling occurs simultaneously for the comoving Observer. Recombination did not occur instantaneously, but lasted a short period of time; howover, it is represented by a last scattering surface 2:73 (x). We shall use this concept but we will not identify E with the surfaces 7 =const. Let us introduce a new coordinate system (9,x] by A=(1+44%m, +0. (8) x Then, neglecting second-order terms in $, the metric (7) turas out to be [7] ds*=a ql —[1+24bx 0d? +[1261x9]8 de dx? . (9 The vector field of matter in these coordinates is 2 [HITO o, 10) > ut 21 This space-time is of the type given by (4) with D=1+4, and y¿=(1—26),,. On the other hand, as a conse- quence of hypothesis (2), scattering with electrons may be neglected and the distribution function for CMB photons will be a solution of the Liouville equation. So, 1f we want isotropic CMB on the last scattering surface, as is 
