CMB anisotropies: Total angular momentum method

PHYSICAL REVIEW D VOLUME 56, NUMBER 2 15 JULY 1997 CMB anisotropies: Total angular oentu ethod Wayne Hu Institute for Advanced Study, School of Natural Sciences, Princeton, New Jersey 08540 Martin White Enrico Feri Institute, University of Chicago, Chicago, Illinois 60637 Received 20 February 1997 A total angular oentu representation siplifies the radiation transport proble for teperature and polarization anisotropy in the cosic icrowave background CMB. Scattering ters couple only the quadrupole oents of the distributions and each oent corresponds directly to the observable angular pattern on the sky. We develop and eploy these techniques to study the general properties of anisotropy generation fro scalar, vector, and tensor perturbations to the etric and the atter, both in the cosological fluids and fro any seed perturbations e.g., defects that ay be present. The sipler, ore transparent for and derivation of the Boltzann equations brings out the geoetric and odel-independent aspects of teperature and polarization anisotropy foration. Large angle scalar polarization provides a robust eans to distinguish between isocurvature and adiabatic odels for structure foration in principle. Vector odes have the unique property that the CMB polarization is doinated by agnetic-type parity at sall angles a factor of 6 in power copared with 0 for the scalars and 8/13 for the tensors and hence potentially distinguishable independent of the odel for the seed. The tensor odes produce a different sign fro the scalars and vectors for the teperature-polarization correlations at large angles. We explore conditions under which one perturbation type ay doinate over the others including a detailed treatent of the photon-baryon fluid before recobination. S0556-28219700614-0 PACS nubers: 98.70.Vc, 98.80.Es I. INTRODUCTION The cosic icrowave background CMB is fast becoing the preier laboratory for early universe and classical cosology. With the flood of high-quality data expected in the coing years, ost notably fro the new MAP 1 and Planck Surveyor 2 satellite issions, it is iperative that theoretical tools for their interpretation be developed. The corresponding techniques involved should be as physically transparent as possible so that the iplications for cosology will be readily apparent fro the data. Toward this end, we reconsider the general proble of teperature and polarization anisotropy foration in the CMB. These anisotropies arise fro gravitational perturbations which separate into scalar copressional, vector vortical, and tensor gravity wave odes. In previous treatents, the siple underlying geoetrical distinctions and physical processes involved in their appearance as CMB anisotropies has been obscured by the choice of representation for the angular distribution of the CMB. In this paper, we systeatically develop a new representation, the total angular oentu representation, which puts vector and tensor odes for the teperature and all polarization odes on an equal footing with the failiar scalar teperature odes. For polarization, this copletes and substantially siplifies the ground-breaking work of 3,4. Although we consider only flat geoetries here for siplicity, the fraework we establish allows for straightforward generalization to open geoetries 5 8 unlike previous treatents. The central idea of this ethod is to eploy only observable quantities, i.e., those which involve the total angular dependence of the teperature and polarization distributions. By applying this principle fro beginning to end, we obtain a substantial siplification of the radiation transport proble underlying anisotropy foration. Scattering ters couple only the quadrupole oents of the teperature and polarization distributions. Each oent of the distribution corresponds to angular oents on the sky which allows a direct relation between the fundaental scattering and gravitational sources and the observable anisotropy through their integral solutions. We study the eans by which gravitational perturbations of the scalar, vector, or tensor type, originating in either the cosological fluids or seed sources such as defects, for teperature, and polarization anisotropies in the CMB. As is well established 3,4, scalar perturbations generate only the so-called electric parity ode of the polarization. Here we show that conversely the ratio of agnetic to electric parity power is a factor of 6 for vectors, copared with 8/13 for tensors, independent of their source. Furtherore, the large angle liits of polarization ust obey siple geoetrical constraints for its aplitude that differ between scalars, vectors, and tensors. The sense of the teperature-polarization cross correlation at large angle is also deterined by geoetric considerations which separate the scalars and vectors fro the tensors 9. These constraints are iportant since large-angle polarization unlike large-angle teperature anisotropies allow one to see directly scales above the horizon at last scattering. Cobined with causal constraints, they provide robust signatures of causal isocurvature odels for structure foration such as cosological defects. In Sec. II we develop the foralis of the total angular oentu representation and lay the groundwork for the geoetric interpretation of the radiation transport proble 0556-2821/97/562/59620/$10.00 56 596 1997 The Aerican Physical Society

56 CMB ANISOTROPIES: TOTAL ANGULAR MOMENTUM METHOD 597 and its integral solutions. We further establish the relationship between scalars, vectors, and tensors and the orthogonal angular odes on the sphere. In Sec. III, we treat the radiation transport proble fro first principles. The total angular oentu representation siplifies both the derivation and the for of the evolution equations for the radiation. We present the differential for of these equations, their integral solutions, and their geoetric interpretation. In Sec. IV we specialize the treatent to the tight-coupling liit for the photon-baryon fluid before recobination and show how acoustic waves and vorticity are generated fro etric perturbations and dissipated through the action of viscosity, polarization, and heat conduction. In Sec. V, we provide specific exaples inspired by seeded odels such as cosological defects. We trace the full process that transfers seed fluctuations in the atter through etric perturbations to observable anisotropies in the teperature and polarization distributions. For reference, Table I provides a list of coonly used sybols. II. NORMAL MODES In this section, we introduce the total angular oentu representation for the noral odes of fluctuations in a flat universe that are used to describe the CMB teperature and polarization as well as the etric and atter fluctuations. This representation greatly siplifies the derivation and for of the evolution equations for fluctuations in Sec. III. In particular, the angular structure of odes corresponds directly to the angular distribution of the teperature and polarization, whereas the radial structure deterines how distant sources contribute to this angular distribution. The new aspect of this approach is the isolation of the total angular dependence of the odes by cobining the intrinsic angular structure with that of the plane-wave spatial dependence. This property iplies that the noral odes correspond directly to angular structures on the sky as opposed to the coonly eployed technique that isolates portions of the intrinsic angular dependence and hence a linear cobination of observable odes 10. Eleents of this approach can be found in earlier works e.g., 6,7,11 for the teperature and 3 for the scalar and tensor polarization. We provide here a systeatic study of this technique which also provides for a substantial siplification of the evolution equations and their integral solution in Sec. III C, including the ters involving the radiation transport of the CMB. We discuss in detail how the onopole, dipole, and quadrupole sources that enter into the radiation transport proble project as anisotropies on the sky today. Readers not interested in the foral details ay skip this section on first reading and siply note that the teperature and polarization distribution is decoposed into the odes Y l exp(ik x) and 2 Y l exp(ik x) with 0,1,2 for scalar, vectors, and tensor etric perturbations, respectively. In this representation, the geoetric distinction between scalar, vector, and tensor contributions to the anisotropies is clear as is the reason why they do not ix. Here the 2 Y l are the spin-2 spherical haronics 12 and were introduced to the study of CMB polarization by 3. The radial decopositions of the odes Y (l l j ) l (kr) and 2Y () l l (kr) TABLE I. Coonly used sybols. 0,1,2 for the scalars, vectors, and tensors. For the fluid variables f for the photons, f B for the baryons, and B for the photon-baryon fluid. X, E, B for the teperature-polarization power spectra. Sybol Definition Eq., Scalar etric 36 () l T/T oents 55,, Euler angles 7 () () l, l Radial B,E function 16 f Fluid density perturbation 39 Conforal tie 35 s l Clebsch-Gordan coefficient 59 f, s Fluid, seed density 39 () () f, f Fluid, seed anisotropic stress 39, Spherical coordinates in kˆ frae 10 Thoson optical depth 49 () B l B-pol. oents 55 C Collision ter 50 XX () C l XX power spectru fro 56 () E l E-pol. oents 55 G Gravitational redshift ter 54 G l Teperature basis 10 2G l Polarization basis 11 H Tensor etric 38 M Pauli atrix basis 1 P () Anisotropic scattering source 62 Q (0) Scalar basis 26 (1) Q i Vector basis 28 (2) Q ij Tensor basis 32 R B/ oentu density 67 () S l Teperature source 61 V Vector etric 37 sy l Spin-s haronics 2 (l ) j l Radial tep function 15 k Wavenuber 10 () k D Daping wavenuber 99 l Multipole 2 eff Effective ass 1R 81 nˆ Propagation direction 12 p f,p s Fluid, seed pressure 39 () v f Fluid velocity 39 () v s Seed oentu density 39 w f p f / f 66 i () l (kr)] for l 2) isolate the total angular dependence by cobining the intrinsic and plane wave angular oenta. A. Angular odes In this section, we derive the basic properties of the angular odes of the teperature and polarization distributions that will be useful in Sec. III to describe their evolution. In particular, the Clebsch-Gordan relation for the addition of angular oentu plays a central role in exposing the siplicity of the total angular oentu representation. A scalar, or spin-0 field on the sky such as the teperature can be decoposed into spherical haronics Y l. Like-

598 WAYNE HU AND MARTIN WHITE 56 TABLE II. Quadrupole (l 2) haronics for spin-0 and 2. Y 2 2Y 2 2 1 1 4 15/2sin2 e 2i 8 5/1cos2 e 2i 1 15/8sincose i 1 4 5/sin1cosei 0 1 3 2 5/43cos2 1 4 5/6sin2-1 15/8sincose i 1 4 5/sin1cosei -2 1 1 4 15/2sin2 e 2i 8 5/1cos2 e 2i wise a spin-s field on the sky can be decoposed into the spin-weighted spherical haronics s Y l and a tensor constructed out of the basis vectors ê iê, ê r 12. The basis for a spin-2 field such as the polarization is 2 Y l M 3,4, where M 1 2 ê iê ê iê, since it transfors under rotations as a 22 syetric traceless tensor. This property is ore easily seen through the relation to the Pauli atrices M 3 i 1 in spherical coordinates (,). The spin-s haronics are expressed in ters of rotation atrices 1 as 12 sy l, The rotation atrix 2l 1 4 1/2 l D s,,,0 2l 1 l!l! 4 l s!l s! sin/2 2l r l s r 1/2 l rs 1 l rs e i cot/2 2rs. l D s,,,4/2l 1 s Y l,e is represents rotations by the Euler angles (,,). Since the spin-2 haronics will be useful in the following sections, we give their explicit for in Table II for l 2; the higher l haronics are related to the ordinary spherical haronics as l 2Y 2! l l 2! 1/2 2 cot 2i sin cot 1 See e.g., Sakurai 13, but note that our conventions differ fro those of Jackson 14 for Y l by (1). The correspondence to 4 is 2 Y l (l 2)!/(l 2)! 1/2 W (l ) ix (l ). 1 2 FIG. 1. Addition theore and scattering geoetry. The addition theore for spin-s haronics Eq. 7 is established by their relation to rotations Eq. 2 and by noting that a rotation fro (,) through the origin pole to (,) is equivalent to a direct rotation by the Euler angles (,,). For the scattering proble of Eq. 48, these angles represent the rotation by fro the kˆ ê 3 frae to the scattering frae, by the scattering angle, and by back into the kˆ frae. 1 sin 2 2 Y l. By virtue of their relation to the rotation atrices, the spin haronics satisfy the copatibility relation with spherical haronics 0Y l Y l, the conjugation relation s Y l * (1) s sy l, the orthonorality relation d s Y l * s Y l l,l,, 4 the copleteness relation l, the parity relation s Y l *, s Y l, coscos, sy l 1 l sy l, the generalized addition relation s1 Y l *, s2 Y l, 3 5 6 2l 1 4 s s 2 Y 1 l,e is 2, 7 which follows fro the group ultiplication property of rotation atrices which relates a rotation fro (,) through the origin to (,) with a direct rotation in ters of the Euler angles (,,) defined in Fig. 1, and the Clebsch- Gordan relation s1 Y 1 l s2 Y 2 2l 1 12l 2 1 1 l 2 4 l,,s l 1,l 2 ; 1, 2 l 1,l 2 ;l,

56 CMB ANISOTROPIES: TOTAL ANGULAR MOMENTUM METHOD 599 l 1,l 2 ;s 1,s 2 l 1,l 2 ;l,s 4 2l 1 sy l. 8 It is worthwhile to exaine the iplications of these properties. Note that the orthogonality and copleteness relations Eqs. 4 and 5 do not extend to different spin states. Orthogonality between s2 states is established by the Pauli basis of Eq. 1 M *M 1 and M *M 0. The parity equation 6 tells us that the spin flips under a parity transforation so that, unlike the s0 spherical haronics, the higher spin haronics are not parity eigenstates. Orthonoral parity states can be constructed as 3,4 1 2 2Y l M 2 Y l M, 9 which have electric-type (1) l and agnetic-type (1) l 1 parity for the () states, respectively. We shall see in Sec. III C that the polarization evolution naturally separates into parity eigenstates. The addition property will be useful in relating the scattering angle to coordinates on the sphere in Sec. III B. Finally the Clebsch-Gordan relation Eq. 8 is central to the following discussion and will be used to derive the total angular oentu representation in Sec. II B and evolution equations for angular oents of the radiation in Sec. III C. B. Radial odes We now coplete the foralis needed to describe the teperature and polarization fields by adding a spatial dependence to the odes. By further separating the radial dependence of the odes, we gain insight on their full angular structure. This decoposition will be useful in constructing the foral integral solutions of the perturbation equations in Sec. III C. We begin with its derivation and then proceed to its geoetric interpretation. 1. Derivation The teperature and polarization distribution of the radiation is in general a function of both spatial position x and angle n defining the propagation direction. In flat space, we know that plane waves for a coplete basis for the spatial dependence. Thus a spin-0 field, as with the teperature, ay be expanded in FIG. 2. Projection effects. A plane wave exp(ik x) can be decoposed into j l (kr)y 0 l and hence carries an orbital angular dependence. A plane wave source at distance r thus contributes angular power to l kr at /2 but also to larger angles l kr at 0 which is encapsulated into the structure of j l see Fig. 3. If the source has an intrinsic angular dependence, the distribution of power is altered. For an aligned dipole Y 0 1 cos figure eights power at /2 or l kr is suppressed. These arguents are generalized for other intrinsic angular dependences in the text. expik x l i l 42l 1j l kry l 0 nˆ, 12 where ê 3 kˆ and xrnˆ see Fig. 2. The sign convention for the direction is opposite to direction on the sky to be in accord with the direction of propagation of the radiation to the observer. Thus the extra factor of (1) l coes fro the parity relation Eq. 6. The separation of the ode functions into an intrinsic angular dependence and plane-wave spatial dependence is essentially a division into spin ( s Y l ) and orbital (Y 0 l ) angular oentu. Since only the total angular dependence is observable, it is instructive to eploy the Clebsch-Gordan relation of Eq. 8 to add the angular oenta. In general this couples the states between l l and l l. Correspondingly a state of definite total l will correspond to a weighted su of j l l to j l l in its radial dependence. This can be reexpressed in ters of the j l using the recursion relations of spherical Bessel functions: j l x x 1 2l 1 j l 1x j l 1 x, G l i l 4 2l 1 Y l nˆ expik x, 10 where the noralization is chosen to agree with the standard Legendre polynoial conventions for 0. Likewise a spin-2 field, as with the polarization, ay be expanded in j l x 1 2l 1 l j l 1xl 1j l 1 x. We can then rewrite 13 2G l i l 4 2l 1 2Y l nˆ expik x. 11 G l i l l 42l 1j l kry l nˆ, 14 l The plane wave itself also carries an angular dependence, of course, where the lowest (l,) radial functions are

600 WAYNE HU AND MARTIN WHITE 56 j l 00 x j l x, j l 10 x j l x, j l 20 x 1 2 3 j l x j l x, l l j 11 1 l x 2 j l x, x 3l l j 21 1 l x 2 j l x x, j l 22 x 3 8 l 2! j l x l 2! x 2 15 with pries representing derivatives with respect to the arguent of the radial function xkr. These odes are shown in Fig. 3. Siilarly for the spin 2 functions with 0 see Fig. 4, 2G 2 l i l 42l 1 l kr i l kr 2 Y l nˆ, 16 where l 0 x 3 8 l 2! j l x l 2! x 2, l 1 x 1 2 l 1l 2 j l x x 2 j l x, x 2 l x 4 1 j l x j l x2 j l x x 2 4 j l x, 17 x which corresponds to the l l,l 2 coupling and l 0 x0, 1 l x 1 2 l 1l 2 j l x, x l 2 x 1 2j l x2 j l x x, 18 which corresponds to the l l 1 coupling. The corresponding relation for negative involves a reversal in sign of the functions: l l, l l. 19 These functions are plotted in Fig. 4. Note that l (0) j l (2) is displayed in Fig. 3. 2. Interpretation The structure of these functions is readily apparent fro geoetrical considerations. A single plane wave contributes FIG. 3. Radial spin-0 teperature odes. The angular power in a plane wave left panel, top is odified due to the intrinsic angular structure of the source as discussed in the text. The left panel corresponds to the power in scalar (0) onopole G 0 0, dipole G 1 0, and quadrupole G 2 0 sources top to botto; the right panel to that in vector (1) dipole G 1 1 and quadrupole G 2 1 sources and a tensor (2) quadrupole G 2 2 source top to botto. Note the differences in how sharply peaked the power is at l kr and how fast power falls as l kr. The arguent of the radial functions kr100 here. to a range of angular scales fro l kr at /2 to larger angles l kr as (0,), where kˆ nˆ cos see Fig. 1. The power in l of a single plane wave shown in Fig. 3a top panel drops to zero l kr, has a concentration of power around l kr, and an extended low aplitude tail to l kr. Now if the plane wave is ultiplied by an intrinsic angular dependence, the projected power changes. The key to understanding this effect is to note that the intrinsic angular behavior is related to power in l as 0, l kr, /2 l kr. 20

56 CMB ANISOTROPIES: TOTAL ANGULAR MOMENTUM METHOD 601 Secondly, even if there are no contributions fro long wavelength sources with kl /r, there will still be large angle anisotropies at l kr which scale as l j l l 2 l 22. 22 FIG. 4. Radial spin-2 polarization odes. Displayed is the angular power in a plane-wave spin-2 source. The top panel shows that vector (1, upper panel sources are doinated by B-parity contributions, whereas tensor (2, lower panel sources have coparable but less power in the B parity. Note that the power is strongly peaked at l kr for the B-parity vectors and E-parity tensors. The arguent of the radial functions kr100 here. Thus factors of sin in the intrinsic angular dependence suppress power at l kr aliasing suppression, whereas factors of cos suppress power at l kr projection suppression. Let us consider first a 0 dipole contribution Y 1 0 cos see Fig. 2. The cos dependence suppresses power in j (10) l at the peak in the plane-wave spectru l kr copare Fig. 3a top and iddle panels. The reaining power is broadly distributed for l kr. The sae reasoning applies for Y 0 2 quadrupole sources which have an intrinsic angular dependence of 3cos 2 1. Now the iniu falls at cos 1 (1/3) causing the double peaked for of the power in j (20) l shown in Fig. 3a botto panel. This series can be continued to higher G 0 l and such techniques have been used in the free streaing liit for teperature anisotropies 11. Siilarly, the structures of j (11) l, j (21) l, and j (22) l are apparent fro the intrinsic angular dependences of the G 1 1, G 1 2, and G 2 2 sources: Y 1 1 sine i, Y 2 1 sincose i, Y 2 2 sin 2 e 2i, 21 respectively. The sin factors iply that as increases, low l power in the source decreases copare Figs. 3a and 1 3b top panels. G 2 suffers a further suppression at /2 (l kr) fro its cos factor. There are two interesting consequences of this behavior. The sharpness of the radial function around l kr quantifies how faithfully features in the k-space spectru are preserved in l space. If all else is equal, this faithfulness increases with for G due to aliasing suppression fro sin.onthe other hand, features in G 1 are washed out in coparison due projection suppression fro the cos factor. This scaling puts an upper bound on how steeply the power can rise with l that increases with and hence a lower bound on the aount of large relative to sall angle power that decreases with. The sae arguents apply to the spin-2 functions with the added coplication of the appearance of two radial functions l and l. The addition of spin-2 angular oenta introduces a contribution fro e i except for 0. For 1, the contribution strongly doinates over the contributions; whereas for 2, contributions are slightly larger than contributions see Fig. 4. The ratios reach the asyptotic values of l l l 2 6, 1, l l l 2 8/13, 2, 23 for fixed kr1. These considerations are closely related to the parity of the ultipole expansion. Although the orbital angular oentu does not ix states of different spin, it does ix states of different parity since the plane wave itself does not have definite parity. A state with electric parity in the intrinsic angular dependence see Eq. 9 becoes 2G 2 M 2 G 2 M l i l 42l 1 l 2 Y l M 2 Y l M i l 2 Y l M 2 Y l M. 24 Thus the addition of angular oentu of the plane wave generates agnetic B-type parity of aplitude l out of an intrinsically electric E-type source as well as E-type parity of aplitude l. Thus the behavior of the two radial functions has significant consequences for the polarization calculation in Sec. III C and iplies that B-parity polarization is absent for scalars, doinant for vectors, and coparable to but slightly saller than the E parity for tensors. Now let us consider the low l kr tail of the spin-2 radial functions. Unlike the spin-0 projection, the spin-2 projection allows increasingly ore power at 0 and/or, i.e., l kr,asincreases see Table I and note the factors of sin). In this liit, the power in a plane wave fluctuation goes as l l 2 l 62, l l 2 l 62. 25 Coparing these expressions with Eq. 22, we note that the spin-0 and spin-2 functions have an opposite dependence on. The consequence is that the relative power in large vs sall angle polarization tends to decrease fro the 2 tensors to the 0 scalars. Finally it is interesting to consider the cross power between spin-0 and spin-2 sources because it will be used to represent the teperature-polarization cross correlation.

602 WAYNE HU AND MARTIN WHITE 56 As is well known see, e.g., 7,15, a general syetric tensor such as the etric and stress-energy perturbations can be separated into scalar, vector, and tensor pieces through their coordinate transforation properties. We now review the properties of their noral odes so that they ay be related to those of the radiation. We find that the 0,1,2 odes of the radiation couple to the scalar, vector, and tensor odes of the etric. Although we consider flat geoetries here, we preserve a covariant notation that ensures straightforward generalization to open geoetries through the replaceent of ij with the curved three etric and ordinary derivatives with covariant derivatives 6,7. 1. Scalar perturbations Scalar perturbations in a flat universe are represented by plane waves Q (0) exp(ik x), which are the eigenfunctions of the Laplacian operator 2 Q 0 k 2 Q 0 26 and their spatial derivatives. For exaple, vector and syetric tensor quantities such as velocities and stresses based on scalar perturbations can be constructed as Q i 0 k 1 i Q 0, FIG. 5. Spin-0 Spin-2 teperature polarization odes. Displayed is the cross angular power in plane wave spin-0 and spin-2 sources. The top panel shows that a scalar onopole (0) source correlates with a scalar spin-2 polarization quadrupole source, whereas the tensor quadrupole (2) anticorrelates with a tensor spin-2 source. Vector dipole (1) sources oscillate in their correlation with vector spin-2 sources and contribute negligible once odes are superiposed. One ust go to vector quadrupole sources lower panel for a strong correlation. The arguent of the radial functions kr100 here. Again interesting geoetric effects can be identified see Fig. 5. For 0, the power in j (00) l (0) l correlates Fig. 5, top panel solid line, positive definite; for 1, j (11) (1) l l oscillates short dashed line, and for 2, j (22) l (2) l anticorrelates long dashed line, negative definite. The cross power involves only () (l ) l j l due to the opposite parity of the () l odes. These properties will becoe iportant in Secs. III and IV B and translates into cross power contributions with opposite sign between the scalar onopole teperature cross polarization sources and tensor quadrupole teperature cross polarization sources 9. Vector dipole teperature and polarization sources do not contribute strongly to the cross power since correlations and anticorrelations in j (11) l (1) l will cancel when odes are superiposed. The sae is true of the scalar dipole teperature cross polarization j (10) l (0) l as is apparent fro Figs. 3 and 4. The vector cross power is doinated by quadrupole teperature and polarization sources j (21) l (1) l Fig. 5, lower panel. C. Perturbation classification Q 0 ij k 2 i j 1 3 ijq 0. 27 Since Q (0) 0, velocity fields based on scalar perturbations are irrotational. Notice that Q (0) G 0 0, n i Q (0) i G 0 1, and n i n j Q (0) ij G 0 2, where the coordinate syste is defined by ê 3 kˆ. Fro the orthogonality of the spherical haronics, it follows that scalars generate only 0 fluctuations in the radiation. 2. Vector perturbations Vector perturbations can be decoposed into haronic (1) odes Q i of the Laplacian in the sae anner as the scalars, 2 Q i 1 k 2 Q i 1, which satisfy a divergenceless condition i Q i 1 0. 28 29 A velocity field based on vector perturbations thus represents vorticity, whereas scalar objects such as density perturbations are entirely absent. The corresponding syetric tensor is constructed out of derivatives as Q 1 ij 1 2k iq 1 j j Q 1 i. A convenient representation is Q i 1 i 2 ê 1iê 2 i expik x. 30 31 Notice that n i Q (1) i G 1 1 and n i n j Q (1) ij G 1 2. Thus vector perturbations stiulate the 1 odes in the radiation. 3. Tensor perturbations Tensor perturbations are represented by Laplacian eigenfunctions 2 Q 2 ij k 2 Q 2 ij, 32

56 CMB ANISOTROPIES: TOTAL ANGULAR MOMENTUM METHOD 603 which satisfy a transverse-traceless condition and the tensors ij Q 2 ij i Q 2 ij 0, 33 h ij 2HQ 2 ij. 38 that forbids the construction of scalar and vector objects such as density and velocity fields. The odes take on an explicit representation of Q 2 ij 3 8 ê 1iê 2 i ê 1 iê 2 j expik x. 34 Notice that n i n j Q (2) ij G 2 2 and thus tensors stiulate the 2 odes in the radiation. In the following sections, we often only explicitly show the positive value with the understanding that its opposite takes on the sae for except where otherwise noted i.e., in the B-type polarization where a sign reversal occurs. III. PERTURBATION EVOLUTION We discuss here the evolution of perturbations in the noral odes of Sec. II. We first review the decoposition of perturbations in the etric and stress-energy tensor into scalar, vector, and tensor types Sec. III A. We further divide the stress-energy tensor into fluid contributions, applicable to the usual particle species, and seed perturbations, applicable to cosological defect odels. We then eploy the techniques developed in Sec. II to obtain a new, sipler derivation and for of the radiation transport of the CMB under Thoson scattering, including polarization Sec. III B, than that obtained first by 16. The coplete evolution equations, given in Sec. III C, are again substantially sipler in for than those of prior works where they overlap 3,4,10,11 and treats the case of vector perturbations. Finally in Sec. III D, we derive the foral integral solutions through the use of the radial functions of Sec. II B and discuss their geoetric interpretation. These solutions encapsulate any of the iportant results. A. Perturbations 1. Metric tensor The ultiate source of CMB anisotropies is the gravitational redshift induced by the etric fluctuation h : g a 2 h, 35 where the zeroth coponent represents conforal tie ddt/a and, in the flat universe considered here, is the Minkowski etric. The etric perturbation can be further broken up into the noral odes of scalar, vector, and tensor types as in Sec. II C. Scalar and vector odes exhibit gauge freedo which is fixed by an explicit choice of the coordinates that relate the perturbation to the background. For the scalars, we choose the Newtonian gauge see, e.g., 15,17 h 00 2Q 0, h ij 2Q 0 ij, 36 Note that tensor fluctuations do not exhibit gauge freedo of this type. 2. Stress energy tensor The stress energy tensor can be broken up into fluid ( f ) contributions and seed (s) contributions see e.g. 18. The latter is distinguished by the fact that the net effect can be viewed as a perturbation to the background. Specifically T T T, where T 0 0 f, T i 0 T 0 i 0, and T j i p f i j is given by the fluid alone. The fluctuations can be decoposed into the noral odes of Sec. II C as T 0 0 f f s Q 0, T i 0 f p f v f 0 v s 0 Q i 0, T 0 i f p f v f 0 v s 0 Q 0i, T i j p f p s i j Q 0 p f f p s Q 0 i j for the scalar coponents, T 0 i f p f v f 1 s Q 1i, T i 0 f p f v f 1 Vv s 1 Q i 1, T i j p f 1 f 1 s Q 1 i j for the vector coponents, and for the tensor coponents. T i j p f 2 f 2 s Q 2 i j B. Radiation transport 39 40 41 1. Stokes paraeters The Boltzann equation for the CMB describes the transport of the photons under Thoson scattering by the electrons. The radiation is described by the intensity atrix: the tie average of the electric field tensor E i *E j over a tie long copared to the frequency of the light or equivalently as the coponents of the photon density atrix see 19 for reviews. For radiation propagating radially Eê r, so that the intensity atrix exists on the ê ê subspace. The atrix can further be decoposed in ters of the 22 Pauli atrices i and the unit atrix 1 on this subspace. For our purposes, it is convenient to describe the polarization in teperature fluctuation units rather than intensity, where the analogous atrix becoes where the etric is shear free. For the vectors, we choose T1Q 3 U 1 V 2. 42 h 0i VQ i 1 37 Tr(T1)/2T/T is the teperature perturbation

604 WAYNE HU AND MARTIN WHITE 56 sued over polarization states. Since QTr(T 3 )/2, it is the difference in teperature fluctuations polarized in the ê and ê directions. Siilarly UTr(T 1 )/2 is the difference along axes rotated by 45, (ê ê )/2, and VTr(T 2 )/2 that between (ê iê )/2. Q and U thus represent linearly polarized light in the north/south-east/west and northeast/southwest-northwest/southeast directions on the sphere, respectively. V represents circularly polarized light in this section only, not to be confused with vector etric perturbations. Under a counterclockwise rotation of the axes through an angle the intensity T transfors as TRTR 1. and V reain distinct while Q and U transfor into one another. Since the Pauli atrices transfor as 3 i 1 e 2i ( 3 i 1 ) a ore convenient description is T1V 2 QiUM QiUM, 43 where recall that M ( 3 i 1 )/2 see Eq. 1, so that QiU transfors into itself under rotation. Thus Eq. 2 iplies that QiU should be decoposed into s2 spin haronics 3,4. Since circular polarization cannot be generated by Thoson scattering alone, we shall hereafter ignore V. It is then convenient to reexpress the atrix as a vector: T,QiU,QiU. 44 The Boltzann equation describes the evolution of the vector T under the Thoson collisional ter CT and gravitational redshifts in a perturbed etric Gh : d d T,x,nˆ T n i i T C T G h, 45 where we have used the fact that ẋ i n i and that in a flat universe photons propagate in straight lines ṅ0. We shall now evaluate the Thoson scattering and gravitational redshift ters. 2. Scattering atrix The calculation of Thoson scattering including polarization was first perfored by Chandrasekhar 16; here we show a uch sipler derivation eploying the spin haronics. The Thoson differential scattering cross section depends on angle as ˆ ˆ 2, where ˆ and ˆ are the incoing and outgoing polarization vectors, respectively, in the electron rest frae. Radiation polarized perpendicular to the scattering plane scatters isotropically, while that in the scattering plane picks up a factor of cos 2, where is the scattering angle. If the radiation has different intensities or teperatures at right angles, the radiation scattered into a given angle will be linearly polarized. Now let us evaluate the scattering ter explicitly. The angular dependence of the scattering gives U cos2 0 0 0 1 0 0 0 cos U, 46 where the U transforation follows fro its definition in ters of the difference in intensities polarized 45 fro the scattering plane. With the relations and QiU iu, the angular dependence in the T representation of Eq. 44 becoes 2 T ST 1 2 4cos21 sin2 1 2 sin2 3 1 2 sin2 1 1 2 cos12 2 cos12 cos12t, 1 2 sin2 1 1 2 cos12 2 47 where the overall noralization is fixed by photon conservation in the scattering. To relate these scattering frae quantities to those in the frae defined by kˆ ê 3, we ust first perfor a rotation fro the kˆ frae to the scattering frae. The geoetry is displayed in Fig. 1, where the angle separates the scattering plane fro the eridian plane at (,) spanned by ê r and ê. After scattering, we rotate by the angle between the scattering plane and the eridian plane at (,) to return to the kˆ frae. Equation 43 tells us these rotations transfor T as R()T diag(1,e 2i, e 2i )T. The net result is thus expressed as RSR 1 2 4 5 Y 0 2,25Y 0 0, 3 2 Y 2 2, 3 2 Y 2 2, 6 2 Y 0 2,e 2i 3 2 Y 2 2,e 2i 3 2 Y 2 2,e 2i 2i, 48 6 2 Y 0 2,e 2i 3 2 Y 2 2,e 2i 3 2 Y 2 2,e 2 Chandrasekhar eploys a different sign convention for U U.

56 CMB ANISOTROPIES: TOTAL ANGULAR MOMENTUM METHOD 605 where we have eployed the explict spin-2, l 2 fors in Table II. Integrating over incoing angles, we obtain the collision ter in the electron rest frae C T rest T d 4 RSRT, 49 where the two ters on the right-hand side account for scattering out of and into a given angle, respectively. Here the differential optical depth n e T a sets the collision rate in conforal tie with n e as the free electron density and T as the Thoson cross section. The transforation fro the electron rest frae into the background frae yields a Doppler shift nˆ v B in the teperature of the scattered radiation. With the help of the generalized addition relation for the haronics Eq. 7, the full collision ter can be written as C T I 1 2 10 d 2 P,T. 50 The vector I describes the isotropization of distribution in the electron rest frae and is given by I T d 4 nˆ v B,0,0. 51 The atrix P () encapsulates the anisotropic nature of Thoson scattering and shows that as expected polarization is generated through quadrupole anisotropies in the teperature and vice versa Y 2 Y 2 P 6Y 2 2 Y 2 6Y 2 2 Y 2 3 2 2 Y 2 Y 2 3 2 Y 2 2 Y 2 3 2 Y 2 2 Y 2 3 2 2 Y 2 Y 2 3 2 Y 2 2 Y, 2 52 3 2 Y 2 2 Y 2 where Y l Y l *() and s Y l s Y l *() and the unpried haronics are with respect to. These 0,1,2 coponents correspond to the scalar, vector, and tensor scattering ters as discussed in Sec. II C and III C. 3. Gravitational redshift In a perturbed etric, gravitational interactions alter the teperature perturbation. The redshift properties ay be forally derived by eploying the equation of otion for the photon energy pu p, where u is the four-velocity of an observer at rest in the background frae and p is the photon four-oentu. The Euler-Lagrange equations of otion for the photon and the requireent that u 2 1 result in ṗ p a ȧ 1 2 ni n j ḣ ij n i ḣ 0i 1 2 ni i h 00, 53 which differs fro 20,7 since we take nˆ to be the photon propagation direction rather than the viewing direction of the observer. The first ter is the cosological reshift due to the expansion of the spatial etric; it does not affect teperature perturbations T/T. The second ter has a siilar origin and is due to stretching of the spatial etric. The third and fourth ters are the frae dragging and tie dilation effects. Since gravitational redshift affects the different polarization states alike, G h 1 2 ni n j ḣ ij n i ḣ 0i 1 2 ni i h 00,0,0 54 in the T basis. We now explicitly evaluate the Boltzann equation for scalar, vector, and tensor etric fluctuations of Eqs. 36 38. C. Evolution equations In this section, we derive the coplete set of evolution equations for the teperature and polarization distribution in the scalar, vector, and tensor decoposition of etric fluctuations. Though the scalar and tensor fluid results can be found elsewhere in the literature in a different for see, e.g., 11,10, the total angular oentu representation substantially siplifies the for and aids in the interpretation of the results. The vector derivation is new to this work. 1. Angular oents and power The teperature and polarization fluctuations are expanded into the noral odes defined in Sec. II B: 3 3 (S,T) Our conventions differ fro 3 as (2l 1) Tl 4 (0,2) l /(2) 3/2 (S,T) and siilarly for E,Bl with (0,2) l E (0,2) l,b (0,2) l and so C (S,T) Cl C E(0,2) l but with other power spectra the sae.

606 WAYNE HU AND MARTIN WHITE 56,x,n d3 k 2 3 l 2 2 l G l, 4 3 Y 1 0 s Y l s l 2l 12l 1 s Y l 1 QiU,x,n d3 k 2 3 l 2 2 E l ib l 2 G l. 55 () A coparison with Eqs. 9 and 43 shows that E l and () B l represent polarization with electric-type (1) l and agnetic-type (1) l 1 parities, respectively 3,4. Because () () the teperature l has electric-type parity, only E l couples directly to the teperature in the scattering sources. Note that B () l and E () l represent polarizations with Q and U interchanged and thus represent polarization patterns rotated by 45. A siple exaple is given by the 0 odes. In the kˆ frae, E (0) l represents a pure Q, or north/south-east/ west, polarization field whose aplitude depends on, e.g., sin 2 (0) for l 2. B l represents a pure U, or northwest/ southeast-northeast/southwest, polarization with the sae dependence. The power spectra of teperature and polarization anisotropies today are defined as, e.g., C l a l 2 for a l Y l with the average being over the (2l 1) values. Recalling the noralization of the ode functions fro Eqs. 10 and 11, we obtain 2l 1 2 C l XX 2 dk k 2 2 k 3 X l * 0,kX l 0,k, 56 where X takes on the values, E, and B. There is no cross correlation C B l or C EB l due to parity see Eqs. 6 and 24. We also eploy the notation C XX () l for the contributions individually. Note that B (0) l 0 here due to aziuthal syetry in the transport proble so that C BB(0) l 0. As we shall now show, the 0,1,2 odes are stiulated by scalar, vector, and tensor perturbations in the etric. The orthogonality of the spherical haronics assures us that these odes are independent, and we now discuss the contributions separately. 2. Free streaing As the radiation free streas, gradients in the distribution produce anisotropies. For exaple, as photons fro different teperature regions intersect on their trajectories, the teperature difference is reflected in the angular distribution. This effect is represented in the Boltzann equation 45 gradient ter, nˆ inˆ ki 4 3 ky 1 0, 57 which ultiplies the intrinsic angular dependence of the teperature and polarization distributions, Y l and 2 Y l, respectively, fro the expansion Eq. 55 and the angular basis of Eqs. 10 and 11. Free streaing thus involves the Clebsch-Gordan relation of Eq. 8 s l l 1 sy l s l 1 2l 12l 3 s Y l 1, 58 which couples the l to l 1 oents of the distribution. Here the coupling coefficient is s l l 2 2 l 2 s 2 /l 2. 59 As we shall now see, the result of this streaing effect is an infinite hierarchy of coupled l oents that passes power fro sources at low ultipoles up the l chain as tie progresses. 3. Boltzann equations The explicit for of the Boltzann equations for the teperature and polarization follows directly fro the Clebsch- Gordan relation of Eq. 58. For the teperature (s0), 0 k l l 2l 1 l 1 0 l 1 2l 3 l 1 l S l, l. 60 The ter in the square brackets is the free streaing effect that couples the l odes and tells us that in the absence of scattering power is transferred down the hierarchy when k1. This transferral erely represents geoetrical projection of fluctuations on the scale corresponding to k at distance which subtends an angle given by l k. The ain effect of scattering coes through the () l ter and iplies an exponential suppression of anisotropies with optical depth in the absence of sources. The source S l ac- () counts for the gravitational and residual scattering effects: S 0 0 0 0, S 1 0 vb 0 k, S 2 0 P 0, S 1 1 vb 1 V, S 2 1 P 1, 61 S 2 2 P 2 Ḣ. (0) The presence of 0 represents the fact that an isotropic teperature fluctuation is not destroyed by scattering. The Doppler effect enters the dipole (l 1) equation through the baryon velocity v () B ter. Finally the anisotropic nature of Copton scattering is expressed through P 1 10 2 6E 2, 62 and involves the quadrupole oents of the teperature and E-polarization distribution only.

56 CMB ANISOTROPIES: TOTAL ANGULAR MOMENTUM METHOD 607 The polarization evolution follows a siilar pattern for l 2, 0 fro 4 Eq. 58 with s2: 2 Ė l k l 2l 1 E l 1 2 l l 1 B l 2 l 1 2l 3 E l 1 El 6P l,2, 63 2 Ḃ l k l 2l 1 B l 1 2 l l 1 E l 2 l 1 2l 3 B l 1 B l. 64 Notice that the source of polarization P () enters only in the E-ode quadrupole due to the opposite parity of 2 and B 2. However, as discussed in Sec. II B, free streaing or projection couples the two parities except for the 0 scalars. Thus B l (0) 0 by geoetry regardless of the source. It is unnecessary to solve separately for the relations since they satisfy the sae equations and solutions with B l () B l () and all other quantities equal. To coplete these equations, we need to express the evolution of the etric sources (,,V,H). It is to this subject we now turn. 4. Scalar Einstein equations The Einstein equations G 8GT express the etric evolution in ters of the atter sources. With the for of the scalar etric and stress energy tensor given in Eqs. 36 and 39, the Poisson equations becoe k 2 4Ga 2 f f s 3 ȧ a fp f v f 0 v s 0 /k, k 2 8Ga 2 p f f 0 s 0, 65 where the corresponding atter evolution is given by covariant conservation of the stress energy tensor T : f1w f kv f 0 3 3 ȧ a w f, d d 1w fv 0 f 1w f k ȧ a 13w fv 0 f w f kp f /p f 2 3 f 66 for the fluid part, where w f p f / f. These equations express energy and oentu density conservation, respectively. They reain true for each fluid individually in the absence of 4 The expressions above were all derived assuing a flat spatial geoetry. In this foralis, including the effects of spatial curvature is straightforward: the l 1 ters in the hierarchy are ultiplied by factors of 1(l 2 1)K/k 2 1/2 6,7, where the curvature is KH 0 2 (1 tot ). These factors account for geodesic deviation and alter the transfer of power through the hierarchy. A full treatent of such effects will be provided in 8. oentu exchange. Note that for the photons 4 (0) 0, v (0) (0) 1, and (0) 12 5 (0) 2. Massless neutrinos obey Eq. 60 without the Thoson coupling ter. Moentu exchange between the baryons and photons due to Thoson scattering follows by noting that for a given velocity perturbation the oentu density ratio between the two fluids is R Bp B p 3 B 4. 67 A coparison with photon Euler equation 60 with l 1, 0) gives the baryon equations as Bkv B 0 3, v B0 ȧ a v B 0 k R 1 0 v B 0. For a seed source, the conservation equations becoe s3 ȧ a sp s kv s 0, v s0 4 ȧ a v s 0 kp s 2 3 s 0, since the etric fluctuations produce higher order ters. 68 69 5. Vector Einstein equations The vector etric source evolution is siilarly constructed fro a Poisson equation V 2 ȧ a V8Ga2 p f f 1 s 1 /k, 70 and the oentu conservation equation for the stressenergy tensor or Euler equation v f1 V 13c 2 f ȧ a v 1 f V 1 2 k 1 1w f, f v s1 4 ȧ a v s 1 1 2 k s 1, w f 71 where we recall that c f 2 ṗ f / f is the sound speed. Again, the first of these equations reains true for each fluid individually save for oentu exchange ters. For the photons v (1) 1 (1) and (1) 8 53 2 (1). Thus with the photon Euler equation 60 with l 1, 1), the full baryon equation becoes v B1 V ȧ a v B 1 V see Eq. 68 for details. R 1 1 v B 1, 72

608 WAYNE HU AND MARTIN WHITE 56 6. Tensor Einstein equations The Einstein equations tell us that the tensor etric source is governed by Ḧ2 ȧ a Ḣk2 H8Ga 2 p f f 2 s 2, 73 where we note that the photon contribution is (2) 8 5 2 (2). D. Integral solutions The Boltzann equations have foral integral solutions that are siple to write down by considering the properties of source projection fro Sec. II B. The hierarchy equations for the teperature distribution Eq. 60 erely express the projection of the various plane wave teperature sources S l () G l on the sky today see Eq. 61. Fro the angular decoposition of G l in Eq. 14, the integral solution iediately follows: l 0,k 2l 1 Here 0 0de l 0 d S l l j l k 0. 74 75 is the optical depth between and the present. The cobination e is the visibility function and expresses the probability that a photon last scattered between d of and hence is sharply peaked at the last scattering epoch. Siilarly, the polarization solutions follow fro the radial decoposition of the 6 P 2 G 2 M 2 G 2 M 76 source. Fro Eq. 24, the solutions E l 0,k 2l 6 0d e P 1 l k 0, 0 B l 0,k 2l 6 0d e P 1 l k 0 0 77 iediately follow as well. (l Thus the structures of j ) l, () () l, and l shown in Figs. 3 and 4 directly reflect the angular power of the sources S () l and P (). There are several general results that can be read off the radial functions. Regardless of the source behavior in k, the B-parity polarization for scalars vanishes, doinates by a factor of 6 over the electric parity at l 2 for the vectors, and is reduced by a factor of 8/13 for the tensors at l 2 see Eq. 23. Furtherore, the power spectra in l can rise no faster than l 2 C l l 22, l 2 C l EE l 62, l 2 C l BB l 62, l 2 C l E l 4, 78 due to the aliasing of plane-wave power to l k( 0 ) see Eq. 25 which leads to interesting constraints on scalar teperature fluctuations 22 and polarization fluctuations see Sec. V C. Features in k space in the l oent at fixed tie are increasingly well preserved in l space as increases, but ay be washed out if the source is not well localized in tie. Only sources involving the visibility function e are required to be well localized at last scattering. However, even features in such sources will be washed out if they occur in the l 1 oent, such as the scalar dipole and the vector quadrupole see Fig. 3. Siilarly features in the vector E and tensor B odes are washed out. The geoetric properties of the teperature-polarization cross power spectru C E l can also be read off the integral solutions. It is first instructive, however, to rewrite the integral solutions as (l 2) 0 l 0,k 2l 0de 0 00 1 0 jl 0 vb 0 j l 10 P 0 j l 20, 1 l 0,k 2l 0de 1 0 1 11 vb Vj l P 1 1 3 kv j 21, l 2 l 0,k 2l 0de P 2 Ḣ j 22 1 l, 0 79 where we have integrated the scalar and vector equations by parts noting that de /d e. Notice that (0) 0 acts as an effective teperature by accounting for the gravitational redshift fro the potential wells at last scattering. We shall see in Sec. IV that v (1) B V at last scattering which suppresses the first ter in the vector equation. Moreover, as discussed in Sec. II B and shown in Fig. 5, the vector dipole ters ( j (11) l ) do not correlate well with the polarization ( (1) l ), whereas the quadrupole ters ( j (21) l ) do. The cross power spectru contains two pieces: the relation between the teperature and polarization sources S l () and P (), respectively and the differences in their projection as anisotropies on the sky. The latter is independent of the odel and provides interesting consequences in conjunction with tight coupling and causal constraints on the sources. In particular, the sign of the correlation is deterined by 21 sgnc l E0 sgnp 0 0 0, sgnc l E1 sgnp 1 3 P 1 kv, sgnc l E2 sgnp 2 P 2 Ḣ, 80

56 CMB ANISOTROPIES: TOTAL ANGULAR MOMENTUM METHOD 609 where the sources are evaluated at last scattering and we have assued that 0 (0) P (0) as is the case for standard recobination see Sec. IV. The scalar Doppler effect couples only weakly to the polarization due to differences in the projection see Sec. II B. The iportant aspect is that relative to the sources, the tensor cross spectru has an opposite sign due to the projection see Fig. 5. These integral solutions are also useful in calculations. For exaple, they ay be eployed with approxiate solutions to the sources in the tight coupling regie to gain physical insight on anisotropy foration see Sec. IV and 22,23. Seljak and Zaldarriaga 24 have obtained exact solutions through nuerically tracking the evolution of the source by solving the truncated Boltzann hierarchy equations. Our expression agree with 3,4,24 where they overlap. IV. PHOTON-BARYON FLUID Before recobination, Thoson scattering between the photons and electrons and Coulob interactions between the electrons and baryons were sufficiently rapid that the photonbaryon syste behaves as a single tightly coupled fluid. Forally, one expands the evolution equations in powers of the Thoson ean-free path over the wavelength and horizon scale. Here we briefly review well-known results for the scalars see, e.g., 25,26 to show how vector or vorticity perturbations differ in their behavior Sec. IV A. In particular, the lack of pressure support for the vorticity changes the relation between the CMB and etric fluctuations. We then study the higher order effects of shear viscosity and polarization generation fro scalar, vector, and tensor perturbations Sec. IV B. We identify signatures in the teperaturepolarization power spectra that can help separate the types of perturbations. Entropy generation and heat conduction only occur for the scalars Sec. IV C and leads to differences in the dissipation rate for fluctuations Sec. IV D. A. Copression and vorticity For the (0) scalars, the well-known result of expanding the Boltzann equations 60 for l 0,1 and the baryon Euler equation 68 is 0 0 k 3 1 0, eff 1 0 k 0 0 eff, 81 which represent the photon fluid continuity and Euler equations and gives the baryon fluid quantities directly as B 1 3 0 0, v B 0 1 0, 82 to lowest order. Here eff 1R, where we recall that R is the baryon-photon oentu density ratio. We have dropped the viscosity ter (0) (0) 2 O(k/ ) 1 see Sec. IV B. The effect of the baryons is to introduce a Copton drag ter that slows the oscillation and enhances infall into gravitational potential wells. That these equations describe forced acoustic oscillations in the fluid is clear when we rewrite the equations as eff 0 0 k2 3 0 0 k2 3 eff eff, 83 whose solution in the absence of etric fluctuations is 0 0 A 1/4 eff cosks, 0 1 3A 3/4 eff sinks, 84 where sc B d(3 eff ) 1/2 d is the sound horizon, A is a constant aplitude, and is a constant phase shift. In the presence of potential perturbations, the redshift a photon experiences clibing out of a potential well akes the effective teperature 0 (0) see Eq. 79, which satisfies eff 0 0 k2 3 0 0 k2 3 R eff, 85 and shows that the effective force on the oscillator is due to baryon drag R and differential gravitational redshifts fro the tie dependence of the etric. As seen in Eqs. 79 and 84, the effective teperature at last scattering fors the ain contribution at last scattering with the Doppler effect v (0) (0) B 1 playing a secondary role for eff 1. Furtherore, because of the nature of the onopole versus dipole projection, features in l space are ainly created by the effective teperature see Fig. 3 and Sec. III D. If R1, then one expects contributions of O( )/k 2 to the oscillations in (0) 0 in addition to the initial fluctuations. These acoustic contributions should be copared with the O() contributions fro gravitational redshifts in a tie-dependent etric after last scattering. The stiulation of oscillations at k1 thus either requires large or rapidly varying etric fluctuations. In the case of the forer, acoustic oscillations would be sall copared to gravitational redshift contributions. Vector perturbations, on the other hand, lack pressure support and cannot generate acoustic or copressional waves. The tight coupling expansion of the photon (l 1,1) and baryon Euler equations 60 and 72 leads to eff 1 1 V 0 86 and v B (1) 1 (1). Thus the vorticity in the photon baryon fluid is of equal aplitude to the vector etric perturbation. In the absence of sources, it is constant in a photon-doinated fluid and decays as a 1 with the expansion in a baryon-doinated fluid. In the presence of sources, the solution is 1 1,kV,k 1 eff 1 0,kV0,k, 87 so that the photon dipole tracks the evolution of the etric fluctuation. With v B (1) 1 (1) in Eq. 61, vorticity leads to a Doppler effect in the CMB of agnitude on order the vector