Optimal polarisation equations in FLRW universes

Transcription

1 Prepared for submission to JCAP CERN-PH-TH/01-099, LAPTH-05/1 arxiv:105.61v1 astro-ph.co 1 May 01 Optima poarisation equations in FLRW universes Thomas Tram a and Juien Lesgourgues a,b,c a Institut de Théorie des Phénomènes Physiques, Écoe Poytechnique Fédérae de Lausanne, CH-1015, Lausanne, Switzerand b CERN, Theory Division, CH-111 Geneva, Switzerand c LAPTh CNRS - Université de Savoie, BP 110, F-791 Annecy-e-Vieux Cedex, France E-mai: thomas.tram@epf.ch, Juien.Lesgourgues@cern.ch Abstract. This paper presents the inearised Botzmann equation for photons for scaar, vector and tensor perturbations in fat, open and cosed FLRW cosmoogies. We show that E- and B-mode poarisation for a types can be computed using ony a singe hierarchy. This was previousy shown expicity for tensor modes in fat cosmoogies but not for vectors, and not for non-fat cosmoogies.

2 Contents 1 Preude Introduction 1 1. Conventions 1. Metric 1. Perturbation types and norma modes Botzmann equation.1 Temperature and poarisation. Reducing the system of equations 6. Change of variabes 6. Scaar, vector and tensor Botzmann equations 7.5 The metric source term 8.6 Free-streaming 9.7 Botzmann hierarchies 10.8 The ine of sight integras 11 Concusion 1 A Scattering terms 1 A.1 Scaar perturbations 1 A. Vector perturbations 1 A. Tensor perturbations 1 B Correspondence between expansions 1 B.1 Scaar modes 1 B. Vector modes 17 B. Tensor modes 19 C Reativistic Botzmann equation in an arbitrary gauge 1 Preude 1.1 Introduction The Botzmann equation for inear cosmoogica perturbations constitutes a set of roughy one hundred couped, ordinary differentia equations ODE, depending on the assumed cosmoogica mode and the requested accuracy. This system of ODE s are soved severa thousand times a day, which easiy makes it the most often soved system of differentia equations in cosmoogy. The equations were derived in a series of papers with contributions from many authors. The effect of photon poarisation for scaar perturbations was first correcty incuded by Kaiser 1 and Bond&Efstathiou. The equations for tensor perturbations were given by Crittenden et. a. buiding on previous work by Ponarev. Kosowsky 5 gave a quantum mechanica re-derivation of the scaar and tensor photon Botzmann equation. Sejak&Zadarriaga invented the ine-of-sight method 6 for cacuating CMB-anisotropies which drasticay reduced the number of differentia equations necessary for cacuating the 1

3 power-spectrum of anisotropies. They were aso the first to reaise that the Stokes parameters Q and U were not optima for an a-sky anaysis, but that the parity eigenstates E and B shoud be used instead 7. Hu&White finay gave a unified treatment of a modes in their semina paper 8. Their approach is caed the tota anguar momentum method, and it is based upon expansions in spin-weighted spherica harmonics. In this formaism E and B are the natura expansion coefficients in a spin-weighted expansion of Q ± U. Due to the reation between spinweighted spherica harmonics and rotation matrices, this formuation gave a cean derivation of the genera scattering term in the photon Botzmann equation. At the same time, Sejak, Zadarriaga and Bertschinger had derived the scaar equations in non-fat cosmoogies 9. Hu&White, in coaboration with Sejak and Zadarriaga, finay extended the tota anguar momentum method to non-fat cosmoogies 10, thereby giving equations vaid for a modes in cosmoogies with constant curvature. However, the reduction from poarisation hierarchies to 1 for tensor modes, which was previousy used in 5, 7 was ost in this process. These equations are aso not easy to compare with the ones of Ma&Bertschinger 11 which for many cosmoogists continue to be the primary reference on cosmoogica perturbation theory. Thus, the purpose of this paper is to show how the number of poarisation hierarchies can be reduced in a cases, and aso to connect the modern paper of Hu et. a. to the cassic paper of Ma&Bertschinger. There is another formaism that we did not mention so far, namey the covariant and gauge invariant approach of Chainor&Lasenby, see1 and references herein. This approach was further expanded and refined by Chainor 1, 1 and ater aso by Lewis 15. This approach shares many of the advantages of the tota anguar momentum method, and it can be easiy generaised to non-inear perturbations. The equations of the popuar Botzmann code CAMB is derived in this formaism16. However, this method aso shares the disadvantage of requiring two poarisation hierarchies for vectors and tensors. 1. Conventions We are using the +++ sign-convention for the metric, and greek indices are running from 0 to whie atin indices are running from 1 to. For consistency with Hu et a. 8, we are omitting the Condon-Shortey phase 1 m in the definition of the spin-weighted spherica harmonics and in the definition of the associated Legendre poynomias. Note that this is contrary to the conventions of both Wikipedia and Mathematica, which incudes the phase for both. Under this convention, the foowing equations hod: P m µ 1µ Y m θ,φ m/ dm dµ m P µ, m! π +m! Pm µe mφ, 1. where µ cosθ. Throughout this paper we wi not write the arguments of these functions expicity. Instead we empoy the foowing notation: s Y m s Y m θ,φ 1. s Y m s Y m θ,φ 1. P m P m µ 1.5 P m P m µ 1.6

4 where the conventions for ordinary spherica harmonics and ordinary Legendre poynomias foows by putting s 0 and m 0 respectivey. Note the impicit compex conjugation for Y m and s Y m. We wi use Ma&Bertschinger s convention for Legendre expansions Xµ +1X P, and it wi be understood that X Xµ. 1. Metric Foowing 10, we write the metric as g µν a γ µν +h µν, 1.8 where h µν is a perturbation and the spatia part of the background metric can be written as γ ij 1 sinh ξ K < 0 dξ + ξ dθ +sin θdφ, K K sin ξ K > 0 This metric can be constructed by embedding a -dimensiona sphere pseudo-sphere of positive negative curvature K in dimensiona Eucidean Lorentzian space. The radia coordinate r of the -dimensiona space is then mapped to ξ by arcsinh K r, K < 0, ξ K r, K 0, arcsin 1.10 K r, K > 0. The covariant derivative of X with respect to the spatia background metric γ ij wi be denoted X i. We wi use conforma time τ, and the derivative with respect to τ is denoted by a dot. K is constant and given by K H 0 1Ω tot. 1. Perturbation types and norma modes Cosmoogica perturbations are usuay divided into three types: scaar-, vector- and tensorperturbations. This decomposition is based on how the perturbation behaves under spatia rotations, and it is usefu because the Botzmann equations does not coupe the different types at inear order. More formay, we can write the eigentensor equation for the Lapacian: Q m i 1 i i m γ jk Q m i 1 i i m jk k Q m i 1 i i m Thevector modeshaszerodivergence, i.e. γ ij Q ±1 i j 0 whie the tensor modes are transverse andtraceess: γ ik Q ± ij k γij Q ± ij 0. These eigentensors, together with the three auxiiary tensors Q 0 i 1 k Q0 i, Q 0 ij 1 k Q0 ij + 1 γ ijq 0, Q ±1 ij 1 k Q ±1 i j +Q ±1 j i, 1.1 can be used to decompose a genera perturbation such as the perturbed metric or the baryon veocity. In fat space, these objects have simpe expicit representations 8. In the Botzmann equation, a Q m -tensors are fuy contracted with the propagation unit vector for the

5 photons, ˆn, or with ê 1 ± ê in the case of poarisation 1. Thesefuy contracted Q m -tensors are caed norma modes and can be used to expand any function of any type m and spin s. In fat space, the s 0 and s± norma modes can be written as M m π +1 Y m e k x, 1.1 ±M m π +1 ± Y m e k x. 1.1 We aways assume that the direction of propagation ˆn is expressed in spherica coordinates θ,φ where θ is the ange between ˆn and the wave vector k of each Fourier mode. It was a key insight by Hu et. a. to reaise that one can construct modes with the same anguar structure in a non-fat space. They found that the norma modes can be written in the form M m π +1 Y m e δ x, k, 1.15 ± Mm π +1 ±Y m e δ x, k, 1.16 where e δ x, k can in principe be cacuated, aong with a genera formua for the spatia derivative of a mode: n i s M m i q +1 s κ m sm1 m s κ m +1 sm+1 m qms +1 s M m, 1.17 where sκ m m s 1K q and q K ν k + m +1K. We sha aso define the wavevector q which is parae to k and has ength q. Note that equation 1.17 tes us everything we need to know about the generaised pane wave e δ x, k : it wi just cance out at each side of the equations just ike the usua pane wave. Because of this, we wi amost never write it expicity in our equations. Botzmann equation.1 Temperature and poarisation The CMB radiation can be described by Stokes parameters I, Q and U. The th Stokes parameter V represents circuar poarisation but it is irreevant since it is not generated by Thomsonscattering. FoowingHu&White8,weformthevectoroffirstorderperturbations T x,ˆn,τ 1 δi T δq+ δu,, δq δu Θ,Q+ U,Q U,.1 T T 1ê 1 and ê form an orthonorma basis in the pane perpendicuar to the wave vector k. Two different conventions exist for writing the perturbations, which differ by a factor. The first convention is using intensity fuctuation δi/t,... and is being used by Ma&Bertschinger 11, Crittenden et. a. and Kosowsky 5. The second convention uses the equivaent of temperature fuctuations Θ,... and is being used by Hu&White 8, 10. Sejak&Zadarriaga 6, 7, 9 are using the second convention for tensors and the first convention for scaars. We wi make use of both.

6 where ˆn is the propagation direction of the photons and Θ is the temperature fuctuation Θ δt/t. We start from the Botzmann equation for T, equation 5 in Hu&White 8 dt dτ T τ +ni it C T+ Dhµν,. where D D Θ,0,0 is the source term reated to the metric, and the coision term can be written as C T κ T dω + κ π Θ + ˆn v B,0,0 + κ dω P m Ω,Ω T.. 10 m Here v B is the baryon veocity, κ n e σ T a is the differentia optica depth, and the scattering matrix can be written in the foowing form: P m Ω,Ω Y m Y m Y m Y m Y m Y m 6Y m Y m Y m Y m Y m Y m.. 6Y m Y m Y m Y m Y m Y m Evauating the matrix product yieds: { } Y m P m T Y m Θ Y m Q + U Y m Q U Y m { 6Y m Θ+ Y m Q + U + Y m Q U } Y m { 6Y m Θ+ Y m Q + U + Y m Q U } { } Y m Y m Θ Y m + Y m Q Y m Y m U Y m { 6Y m Θ+ Y m + Y m Q + Y m Y m U } Y m { 6Y m Θ+ Y m + Y m Q + Y m Y m U } { } Y m Y m Θ E Q B U Y m { 6Y m Θ+E Q +B U }. Y m { 6Y m Θ+E Q +B U } where we defined the symbos E m Y m + Y m, E m Y m + Y m,.5 B m Y m Y m, B m Y m Y m..6 The expicit representations of these symbos for m 0,1 and are given in tabe 1. By formingthesumandthedifferenceofrowandofthebotzmann equation, wefindseparate evoution equations for Q and U: d dτ Θ Θ 1 π Q + κ U κ 10 m dω Θ + ˆn v B Q U D Θ 0 0 { } Y m dω Y m Θ Em Q Bm U 1 Em{ 6Y m Θ +E m Q +B m U } 1 Bm{..7 6Y m Θ +E m Q +B m U } Here we used the fact that the second and third entry in Dh µν vanish. 5

7 m Y m E m B m π cos 15 θ1 8π sin θ 0 π cosθsinθe φ 1 15 π sin θe φ 1 5 π 5 5 π sinθcosθe φ 1 π sinθe φ 1+cos θ e φ 1 5 π cosθe φ Tabe 1. Expicit representations of Y m, Em and B m.. Reducing the system of equations We wi decompose a quantities on the eft hand side of equation.7 into scaar m 0, vector m ±1 and tensor m ± components. Each component coupes ony to the corresponding term in the sum over coision terms. By considering the equations for Q m and U m, it is cear that U m and Q m B m /E m satisfy the same differentia equation, and since the initia condition of both Q and U is zero, we must aways have U m Bm E mqm,.8 which aso covers the specia case of B 0 0. U-type poarisation vanishes for scaar modes. This enabes us to reduce the system of equations, a fact which has been used in the past for scaar modes and for tensor modes in fat space. We find d dτ Θ m Q m + κ Θ m 1 π κ { dω Y m Y m Θ 10 1 Em{ 6Y m { κ dω Y m Y m Θ 10 Y m { Em dω m Θ + ˆn v m B Q m } Em Q Bm U Θ +E m Q +B m U } Θ E m D m Θ 0 } E m + Bm E Q E m }..9 m + Bm Q Equation.9 shows that a singe poarisation hierarchy is aways enough, and this is our main resut. Physicay, this comes from the axia symmetry of the Thomson scattering term combined with the fact that the metric perturbations do not source poarisation directy.. Change of variabes If we do the change of variabes Θ m f m θe mφ F m,.10 Q m g m θe mφ G m,.11 where f m and g m are two arbitrary functions to be specified ater, F m and G m can be expanded in ordinary Legendre poynomias since they no onger depend on φ. The form of the function f m is constrained by the requirement that the θ-dependence of the foowing three terms can be written as a finite sum of Legendre poynomias: the metric terms, e mφ f 1 m D m Θ, 6

8 the Dopper term, e mφ f 1 m ˆn v m B, the factor in front of the F m scattering term, e mφ f 1 m Y m. One can check that this requirement singes out an optima f m up to a normaisation factor for a cases, and we give them in equation.1. The function g m is ess constrained due to the absence of metric sources. The terms that must be representabe by a finite number of Legendre poynomias are the coefficient in front of G m : E m + Bm E m e mφ g m θ, the factor in front of the G m scattering term, e mφ g 1 m E m. When m 0, the simpest function that satisfies both requirements is g m E m. For m 0 we have two choices: g m 1 or g m sin θ. The atter greaty simpifies the correspondence between E 0 and G 0, whie simpifying the equations at the same time. However, the first choice is the one commony empoyed in the iterature on scaar poarisation e.g. Ma&Bertschinger 11, so we stick with this convention. In summary we have: Θ 0 1 F0, Q 0 1 G0,.1a Θ 1 1 sinθe φ F 1, Q 1 1 sinθcosθe φ G 1,.1b Θ 1 sin θe φ F, Q 1 1+cos θe φ G..1c The constant has been chosen such that F 0 Fγ M&B and G M&B γ are the scaar temperature and poarisation perturbations of Ma&Bertschinger. For tensor modes, our proposed substitution is equivaent to the one introduced by Ponarev and used impicit, if not expicit by subsequent workers,, 5 7. However, in these papers the substitution was aways imposed before reducing the hierarchy. We woud ike to emphasise that the reduction in the hierarchies has nothing to do with the variabe substitution, but is a direct consequence of the structure of the Thomson scattering term in the Botzmann equation for photons.. Scaar, vector and tensor Botzmann equations and G 0 G M&B γ where F M&B γ After the change of variabes in the previous section, the Botzmann equation.9 becomes d F 0 F dτ G 0 + κ 0 dω π F0 +ˆn v 0 B D 0 P P G 0 Θ κ 0 0 P 0 P P 0,.1a d F 1 F dτ G 1 + κ 1 sinθˆn v1 e φ B G 1 e φ sinθ D1 Θ 6 P κ 1 P 1 0 6P 1,.1b d F F e φ dτ G + κ G sin θ D Θ 6P κ 0 6P..1c 7

9 H 0 L 1 6 h Synchronous gauge Newtonian gauge H0 T η + 1 h H 0 L φ A0 ψ H 1 T h V H T H B1 V H T H Tabe. Metric perturbations in the synchronous gauge and conforma Newtonian gauge. Nonspecified components in each gauge are zero. We have adopted Ma&Bertschingers conventions for the scaar metric perturbations. where P m are given by P P 1 F 0 +G 0 0 +G 6 0,.1a,.1b..1c F 1 1 +F 1 +G G1 7 G1 P F F + 70 F 5 G G 70 G The Botzmann equations.1 are derived in detai in appendix A. Equations.1 can a priori be taken as a definition of P m but in appendix B we show that they are actuay identica to the P m of Hu&White. The inear combination of mutipoes in P is usuay denoted Ψ, 6, 17, whie the inear combination in P 0 is sometimes denoted by Π 6. However, we find the notation of Hu&White more systematic..5 The metric source term We spit the perturbed part of the metric into scaar, vector and tensor perturbations using the eigentensors defined in equation 1.11 and 1.1. In the notation of Hu et. a., the perturbed part of the metric is written as h 00 A 0 Q 0, h 0i B 0 Q 0 i B 1 Q 1 i, h ij H 0 L Q0 γ ij + m0 H m T Q m ij..15a.15b.15c Bychoosingagauge, wecaneiminateoutofthescaarperturbationsandoneofthevector perturbations. In tabe we have defined the synchronous and conforma Newtonian gauge. We now turn to the metric part of the Botzmann equation.1. Ony the Θ-component of Dh µν is non-zero, but it is not trivia to derive in the genera gauge defined by.15. Our metric term differ from the one given in 8, 10 by a few signs, so we have incuded our derivation in appendix C. We find D Θ 1 ni n j ḣ ij n i ḣ 0i + 1 ni h 00 i..16 Note that a the source terms given in both 8 and 10 agree with our expression for D Θ. 8

10 We need the foowing identities for contraction of Q m... by n i s: Q 0 M 0 0, ni Q 0 i M 0 1, ni Q 0 i km 0 1, ni n j Q 0 ij 1K/k M 0 n i Q 1 i M 1 1, ni n j Q 1 ij 1 1K/k M 1, ni n j Q ij M, where the norma modes M m have been defined in equation 1.1. Some of these reations foow directy from the definition of the norma modes of Hu et. a. 10, whie the rest foow from using equation 1.17 in the definition of the auxiiary tensors 1.1. The metric part then spits into three parts, D 0 Θ Ḣ0 L M0 0 ka + 0 +Ḃ0 M1 0 1K/k Ḣ 0 T M0,.17 D 1 Θ Ḃ1 M K/k Ḣ 1 T M1,.18 D Θ Ḣ M..19 The vector and tensor metric terms appearing in the Botzmann equations.1b and.1c read expicity: e φ sinθ 1 D 1 Θ Ḃ1 M K/k Ḣ1 T M0 1,.0 e φ sinθ D Θ 6Ḣ M 0 0,.1 where we have used the three reations e φ sinθ 1 M M 0 0,.6 Free-streaming e φ sinθ 1 M 1 M0 1, e φ sinθ M 1 M0 0. Any arbitrary φ-independent quantity Xτ, x,ˆn can be expanded in generaised Fourier modes and Legendre mutipoes as Xτ, x,ˆn 1 π d q +1X τ, qp µe δ x, k. 1 π d q +1X τ, qm 0,. and the free-streaming equation of such a function reads By using equation 1.17, we find dx dτ X τ +ni X i 0.. Ẋ k +1 s X 1 +1s +1 X +1, s 1K 1 k..5 9

11 Noting that s 1 for K 0, we can easiy recover the fat imit. The hierarchy needs to be cosed at some finite max, and for this Ma&Bertschinger suggested to use the recurrence reation for spherica Besse functions. The equivaent of spherica Besse functions in non-fat space are hyperspherica Besse functions. They satisfy the recurrence reation 18 ν K K Φν x 1cot KxΦ ν 1 x { cot K xφ ν x 1 +1 and their derivatives can be expressed as d dx Φν x cot KxΦ ν x ν { 1 +1 where ν K1 Φ ν x,.6 K } ν K+1 Φ ν +1 K x+ ν K K Φν 1 x ν K K Φν 1 x+1,.7 K +1 Φ ν +1 x,.8 K } ν K+1 Φ ν +1 K x,.9 cothx K < 0 1 cot K x x K 0..0 cotx K > 0 Mutipying equation.9 by K and using K ν k +K vaid for m 0, we find d K dx Φν x k { s Φ ν 1 +1 x +1s +1Φ ν +1 x},.1 showing that Φ ν x with x K τ satisfies the free-streaming hierarchy, equation.5. Using the recurrence reation.6 then eads to the foowing ansatz for cosing the hierarchy: Ẋ max ks X 1 +1 K cot K K τ X...7 Botzmann hierarchies After variabe substitution and Legendre expansion, the Botzmann equation becomes a hierarchy of mutipoes: F m k s F m s +1F m +1 κf m +u m,. where the u m Ġ m u 0 0 κf 0 0 Ḣ0 L, u 1 k +1 source terms are 0 1 Ḃ1 +v 1 B u 0 T Ḣ s G m 1 +1s +1G m +1 κg m +v m,. u0 1 ka0 +Ḃ0 + κ θ b k, u 0 5 κp K k Ḣ0 T,, u1 1 1K/k Ḣ 1 T κp1, κp, 10

12 whie the poarisation source terms v m are given in terms of the quantities P m of equation.1: v 0 0 κp 0, v 0 5 κp0, v 1 0 κp1, v 0 κp. In the synchronous and conforma gauge, u m Newtonian gauge reduce to u 0 0 φ+ κf 0 0, u 0 1 ψ + κ k θ b, u 0 5 κp0, u 1 0 V κv 1 B, u 0 Ḣ κp. Synchronous gauge u1 1 κp1, u 0 0 ḣ+ κf0 0, u 0 1 κ k θ b, u K/k 6η +h+ 5 κp0, u 1 0 κv 1 u 0 B, Ḣ κp. u1 1.8 The ine of sight integras 1K/k h V κp1, The ine of sight integras are most easiy derived in the origina variabes of Hu et. a. 10. We wi just quote their genera resuts here: Θ m +1 E m +1 B m +1 τ0 dτe κ 0 j τ0 0 τ0 0 S m j φ jm,.5 d κτe κ 6P m ǫ m,.6 d κτe κ 6P m β m,.7 where φ, ǫ and β are the radia functions of Hu et. a. 10. In a fat Universe, they are given in terms of spherica Besse functions, and in the non-fat case they are given in terms of hyperspherica Besse functions. The source for poarisation, P m, are given by equation.1, whie the S m j are given by S 0 j j +1 u 0 j, S u 1 0, S1 8 8 u1 1, S 8 u 0. 11

13 Concusion In this paper we have showed that cacuating the CMB poarisation by evoving the E- and B-mode as it is usuay done is not optima. Instead, evoving a singe quantity, G m, is enough, and the mutipoes of E m and B m can then be recovered from the mutipoes of G m. This was previousy known ony for scaar perturbations and tensor perturbations in fat space. In addition to the obvious computationa advantage of having one ess hierarchy to evove in time, the free-streaming soution is aso simpified in our approach since a perturbations are expanded in ordinary Legendre poynomias. These equations wi soon be impemented in the pubic code CLASS. Acknowedgments This project is supported by the Swiss Nationa Foundation. We wish to thank Simon Prunet for very stimuating discussions. A Scattering terms In this appendix we show how to cacuate the scattering terms in equations.1. A.1 Scaar perturbations After the substitution Θ 0 1 F0, Q 0 1 G0, A.1 A. equation.9 for m 0 reads d F 0 dτ G 0 + κ F 0 1 π dω F 0 +ˆn v 0 B D 0 Θ G 0 0 { } κ dω Y 0 Y 0 F0 E 0 + B0 G 0 E { 0 } 10 E0 Y 0 F0 E 0 + B0 G 0 E 0 } κ 1 5 dω π { P 1 5 π P F0 5 6π P 0 P G0 } π P 0 P { 1 5 π P F0 5 6π P 0 P G0 κ 1 { dµ 5P 1 P F0 1 P 0 P G0 } P 0 P { 1 P F0 1 P 0 P G0 } κ 1 { dµ 5P 1 P F0 1 P 0 P G0 } P 0 P { 1 P F0 1 P 0 P { } G0 } κ 5P { 10 5P 0 P P P κ 0 P 0 P P 0 Avaiabe at F 0 +G 0 0 +G 0 F 0 +G 0 0 +G 0 }. A. 1

14 A. Vector perturbations For the vector perturbations, we do the foowing change of variabes: Θ 1 1 sinθe φ F 1 1 π 1 15 cosθ Y 1, A. Q 1 1 sinθcosθe φ G 1 1 π 5 E1. A.5 In these variabes, the Botzmann equation.9 takes the form d F 1 F dτ G 1 + κ 1 sinθˆn v1 e φ B G 1 e φ sinθ D1 Θ 0 { } κ 1 15 dω π cosθ Y 1 sinθ e φ F 1 E 1 + B1 E sinθcosθe 1 φ G 1 { } π Y 1 sinθ e φ F 1 E 1 + B1 sinθcosθe E 1 φ G 1 { } κ 1 15 dω π µ 1 15 π µ 1µ F π 1µ G 1 { } π π µ 1µ F π 1µ G 1 κ 1 dω 15 π µ{ µ 1µ F 1 + 1µ G 1 } 10 1 { 15 π µ 1µ F 1 + 1µ G 1 } κ 1 dω 15 π P 1{ 5 P 1 5 P F P 0 7 P 8 5 P G 1 } 10 1 { 15 π 5 P 1 5 P F P 0 7 P 8 5 P G 1 } { } κ 15 P 1 5 F F1 + 5 G G1 8 5 G1 { } κ P 1 { 10 6 P κ 1 P 1 6P 1 5 F F1 + 5 G G1 8 5 G1 { F 1 1 +F 1 +G G1 7 G1 } F 1 1 +F 1 +G G1 7 G1. A. Tensor perturbations } A.6 For the tensor perturbations, we do the change of variabes Θ 1 π sin θe φ F 15 Y F, A.7 Q 1 π 1+cos θe φ G 15 E G. A.8 1

15 After this substitution, the Botzmann equation.9 reads d F F e φ dτ G + κ G sin θ D Θ 0 κ 10 κ 10 κ κ 10 dω dω 1 κ 10 κ π 15 π π π { } Y Θ E + B E Q { } Y Θ E + B Q E { 1µ F 1+6µ +µ } G { 1µ F 1+6µ +µ } G 1 { dµ P P P F 16 5 P P P G } 1 { P P P F 16 5 P P P G } 1 { dµ P P + 7 P F 6P P + 7 P G } 1 { P P + 7 P F 6P P + 7 P G } { } F F + 7 F 6G G 7 G F + 7 F 6G G 7 G { F 6P 6P. } A.9 B Correspondence between expansions HerewederivethecorrespondencebetweentheordinaryLegendreexpansioncoefficients F m and G m and the coefficients Θ m, E m and B m in the spin-weighted spherica harmonics expansions of Hu et. a. B.1 Scaar modes The scaar temperature expansions can be directy compared. Hu et. a. s expansion is Θ 0 Θ 0 π +1 Y 0 Θ 0 P, B.1 whie ours is Θ 0 1 F F0 P, B. so we find Θ 0 +1 F 0. The E 0 reation is more compicated. Remember that B 0 foowing expicit formua for ± Y 0: ± Y 0! +1 +! π P, B. is zero. We use the B. 1

16 which is easiy derived from equation in 8. Hu et. a. s expansion can then be written as Q 0 Q 0 ± U 0! +! E0 P, B.5 which eads to E 0! +!! +! dµq 0 P, kk +1 k G 0 k 1 1 dµp k P. B.6 We coud not find the genera formua for the needed integra 5 anywhere in the iterature, so we have incuded its derivation here. The first step is to use the definition of the associated Legendre poynomia, equation 1.1: 1 dµp k P dµ 1 1 dµ1µ P k P kk 1P k k1k +1 + k +k 1P k k 1k + k +1k +P k+ k +1k + P. B.7 The second derivative of a Legendre poynomia can be recast into a sum over Legendre poynomias. To derive this sum, we start from the basic reation n+1p n P n+1 P n1, B.8 which can be iterated to give n1 P n 11 j+n j +1 P j. j0 B.9 The second derivative can then be expressed as n1 P n j0 n1 j1 j0 k0 n 11 j+n j +1 n1 k0 jk+1 j1 11 k+j k +1 k0 11 k+j 11 j+n j k+j 11 j+n j +1 n 1+1 k+n k +1 nkn+k+1p k. k0 P k k +1 P k k+1 P k 5 However, had we instead chosen Q 0 sin θg 0 as discussed in section., this integra woud have been given directy by Gaunt s formua. 15

17 This eads to 1 1 dµp k P 1+1 j+ j +1 j +j +1 k +1 δ j j0 { k+n k +k +1, k, 0, k >. We now insert equation B.10 that we just derived in equation B.7. We find 1 1 { dµp k P 1+1+k kk 1 k +k +11 k + k 1k +1 + k +k 1 k +k+11 k k 1k + k+1k + k + +k ++11 } k k +1k + if k and k+ is even, 1 +1 for k,. B.10 0 otherwise. We can now evauate the integras in equation B.6. We find that the correspondence between E 0 and G 0 reads: E 0! +1 1G 0 +! + k 1+1 +k k +1G 0 k, k0! +1 1G 0 +! + k 1+1 +k k +1G 0 k. B.11 k0 We can now cacuate the quantity P 0 of Hu&White starting from their definition We have Θ 0 5 F0 and 6E 0 P m 1 Θ m 6E m. B G 0 G G 0 0 +G 0,! so we find P F 0 0 +G 0 0 +G 0, B.1 in accordance with our definition of P 0, equation.1. In the notation of Sejak&Zadarriaga 6, we have P 0 1 Π where we have used the correspondence F 0 S T, and G 0 S P, between our mutipoes and their mutipoes. 16

18 B. Vector modes For connecting F 1 and Θ 1, we use the formua 1µ P 1 P P1 1. B.1 Note that this equation depends on the convention for the Condon-Shortey phase. We rewrite our expansion of F 1 in the foowing way: Θ 1 1 sinθe φ +1F 1 P, 11 F1 e φ P 1 +1 P1 1, 1 F 1 1 F 1 +1 P 1 e φ, 1 π +1! F ! 1 F1 +1 Y 1. Comparing this to the expansion of Hu&White yieds Θ ! F 1 1! 1 +F1 +1. B.15 We now turn to the E 1 and B 1 reations. We have The product P Y 1 Q 1 + U 1 Q 1 + B1 E 1Q1, sinθcosθe φ G 1, cosθ π5 Y 1 +1G 1 P. can be expanded in s,m 1 spin-weighted spherica harmonics: P Y 1 j γ j Y 1 j, B.16 with expansion coefficients given by γ j π +1 5j +1 dω Y 1 0 Y 0 Y 1 j π j dω +1 Y 1 0 Y 0 Y j 1 j. B.17 0 The seection rues for the Wigner -j symbos tes us that γ j is non-zero if and ony if j and j +. In this range, γ j can be written as 6 γ j 1 j+6 5j +1j +j jj +1j +j +1 j++! j+!. B.18 j j + 6 We have used equation B.5 to re-express the quantity +j!j!. The Wigner -j symbos can be simpified by appying equation B.1. 17

19 Since j 0 and j 1 are roots in this formua, we do not need to consider the restriction j. We can now rewrite our Legendre expansion as an expansion in Y 1 : π Q 1 + U 1 5 π 5 π 5 π 5 { +1G 1 { + j γ j Y 1 j +1G 1 γ Y G 1 + γ +Y } +1G 1 γ + Y G 1 γ Y 1 { γ G1 + +1γ G1 +5γ+ G1 + + } 1γ1 G1 1 +γ +1 G1 +1 Y 1. } We can finay compare with Hu et a., and we find E γ 5 G1 +1γ G1 + +5γ+ G1 +, B.19 B γ1 5 G γ +1 G1 +1. B.0 In order to cacuate the P 1 of Hu&White, we need Θ 1 and E 1. They are Θ 1 1! F 1 1 +F 1 E 1 1 γ0 G1 0 5γ G1 +9γ G1, B.1, 1 G G1 1 7 G1, B. eading to P 1 1 Θ 1 6E F 1 1 +F 1 +G G1 7 G1, B. which matches our definition of P 1 in equation.1. 18

20 B. Tensor modes Our expansion for Θ reads Θ 1 e φ 1µ F 1 e φ 1µ F +1P 1 e φ 1µ F 1 d P +1 1 dµ P e φ F +1P P e φ F P 1 e φ F +1P e φ 1 1 F F 1 + F + 1 π +! +1! d P dµ e φ F P d P + dµ 1 1 F F F + P + + Y, B. where we used equation 1.1 and 1.. This expansion can be compared directy to that of Hu&White: Θ π +1 Θ Y, B.5 so we find Θ 1 +!! 1 1 F F F +. B.6 Thus, Θ F F F F F + 70 F. B.7 We can write our expansions of Q and U in a form which is easy to compare to the expansions of Hu and White: Q ± U 1+ B 1 E 1+cos θe φ G 1+cos θ cosθ 1 G π +1 5 P ±Y G. 0 B.8 19

21 Considering ony Q+ U, we expand P Y in terms of spin spherica harmonics: P Y j α j Y j, B.9 where the expansion coefficients can be found from the orthogonaity of the spin-weighted spherica harmonics: α j dωp Y Yj π dω +1 0 Y 0 Y Y 1 π 5 +1j +1 j j π 0 0 j 5j j. B.0 0 Using the seection rues for the Wigner -j symbos, we find that we ony have non-zero vaues for j +, so the sum in equation B.9 contains ony 5 terms. We can find a simpe anaytica expression for the Wigner -j symbo using the genera expression j1 j j 1 j 1j m j m 1 m m 1 j j s jk m k!j k +m k! k1 1 s 1 s!j 1 +j j s!j 1 m 1 s!j +m s! 1 j j +m 1 +s!j j 1 m +s!, B.1 where the sum is over a non-negative integers s such that a the arguments of the factorias are non-negative, and j 1 j j is defined by j 1 +j j!j 1 j +j!j 1 +j +j! j 1 j j. B. j 1 +j +j +1! It is easy to verify that in our case, ony s j + is aowed. We find j 1 j j +! 6 j 0 j! +j!j!1 1 j j +! +j! 1 6. j! +j +! +j!j! B. Note that the fractions of factorias just seect a finite number of terms which can be written expicity. The ast fraction can be written in the foowing way:!! j 0 χj +j!j! + j! j!! j 1, B.! j 0

22 which for a reevant vaues can be written compacty as + j! j! j j +. B.5 By combining these resuts we find j 1 j j 1jj +1j j 1 +j +j +1 +j + +j +χ j, B.6 eading to α j 1 5j +1j 1jj +1j + +j 1 +j +j +1 +j + +j + j j +. B.7 We can now insert the expansion B.9 into equation B.8: π + Q + U G j α j Y j π +1 5 G α Y π G α + Y + 0 π 5 Y 0 { π 5 Y 0 j +j+1g j +j α +j 1G 1 α 1 +G +1 α +1 + } + G α + +1G α +5G + α + This expansion can now be directy compared to Hu&Whites expansion E + B eading to the identification E +1 B Q + U π +1 Y G α + +1G α +5G + α +, B.8, B.9 1G 1 α 1 +G +1 α +1. B.0 Note that the square roots in α j combine with the square root in front, so a expressions wi be in terms of fractions. As an exampe we can cacuate E : E G G 1 1 G 5 G G 70 G. B.1 1

23 We can now cacuate P of Hu&White from the definition of P m, equation.1. Θ is given by B.7 and E by equation B.1. Thus P Θ 6E F F + 70 F G G F F + 70 F 5 G G 70 G 70 G, B. which is the same as our definition of P, equation.1. This expression can be compared to the Ψ used by different authors, but note that the exact expression depends on the definition of the Legendre expansion as we as the convention for the perturbation. Crittenden et. a. are using intensity fuctuation units 7 ike us, but they are omitting the in their convention for Legendre expansion. Taking this into account, we find Ψ 6P. C Reativistic Botzmann equation in an arbitrary gauge The reativistic Botzmann equation is often derived in the iterature in the synchronous gauge see e.g. 17 or ongitudina gauge see e.g The genera resut is derived in, and quoted in 8, 10, but with differences in the signs of a few terms, and even with a different expression for the contribution of the metric perturbation h 0i. This justifies the presentation a fu derivation in this Appendix. We stick to the metric choice g µν a γ µν +h µν C.1 such that x 0 τ represents conforma time, and γ In the Friedmann-Lemaître mode, for any set of comoving coordinates, the tensor γ µν must be diagona. Indices are raised using ḡ µν ḡ µρ ḡ νλ ḡ ρλ for the background, and δg µν ḡ µρ ḡ νλ δg ρλ for the perturbations. For photons traveing a aong a given geodesic, the four-momentum P µ dxµ dλ obeys as usua to P µ P µ 0. Instead the four-veocity u µ of a given observer is normaized to u µ u µ 1. When the trajectory of an observer crosses that of a photon, the observer measures a photon energy ω u µ P ν. If the observer happens to be at rest with respect to the coordinate system, u i vanishes. In that case, the previous reations impy u 0 1/ g 00 and ω g 0µ u 0 P µ g 00 P 0 ah 0i P i. C. It is trivia to show that at order one in perturbations, ω g ij P i P j the reation is even exact in any gauge where h 0i 0. The direction of propagation 8 of a photon is given by n i P i / γ ij P i P j, such that γ ij n i n j 1. Using P µ P µ 0, we see that at order zero in perturbations, n i is equa to P i /P 0. The phase-space distribution of a given species can be expressed as a function of coordinates x µ, of the energy ω measured by a comoving observer, and of the direction of propagation n i. In the specia case of massess partices, gravitationa interactions preserve the shape of the distribution, that can be expressed as: fx µ,ω,n i g yx µ,ω,n i C. 7 This can be verified by integrating their definition of the perturbed distribution function δf over momentum. 8 opposite to the direction of observation.

24 where gy coud be a Bose-Einstein or Fermi-Dirac distribution, and the quantity y invoves the energy, the background temperature and the direction-dependent temperature fuctuation: yx µ,p,n i ω Tτ1+Θx µ,n i ω 1Θx µ,n i. C. Tτ The dependence of the temperature fuctuation Θ on direction appears when the tightcouping approximation breaks down. For massess species, the coisioness Botzmann equation or Liouvie equation expressing the conservation of the phase-space distribution aong geodesics reads df dλ g y y P 0 ogy +P i ogy τ x i + dω ogy + dni dλ ω dλ ogy n i 0. C.5 The ast term is of order two in perturbations, since in an unperturbed universe geodesics woud be straight ines, whie Θ and y woud be isotropic. Keeping at most first-order perturbations, we get T Θ+ T + Pi P 0 iθ 1 dogω P 0 0. C.6 dλ At order zero in perturbations we are eft with T T 1 dogap 0 P 0 dλ 0. C.7 By using the geodesic equation dp0 dλ Γ 0 µνp µ P ν, one can easiy show that T T + ȧ a 0, C.8 and that the temperature scaes ike the inverse of the scae factor. By subtracting this equation to the fu one, we get the first-order equation governing temperature fuctuations: Θ+ Pi 1 dogω P 0 iθ P 0 + ȧ 0 C.9 dλ a Since i Θ is a perturbation, we can repace its coefficient Pi by its zero-order approximation P 0 n i. Aso, at order one, we can use ω g 00 P 0 1h 0i n i, and expand the equation as: Θ+n i i Θ dog g00 dτ + 1 dogp 0 P 0 dh 0i dλ dτ ni + ȧ a 0 C.10 we reca that P 0 dτ/dλ. Using the reations g 00 1/g 00 and α g µν Γ λ αµg λν +Γ λ ανg λµ, we can express the first term between brackets as dog g 00 1 dx ġ i 00 + i g 00 dτ g 00 dτ Γ Γ 0 i0 ȧ P i a h 0i P 0. C.11 The second term between brackets can be expanded using the geodesics equation: 1 dogp 0 P 0 dλ Γ 0 P i 00 Γ0 i0 P 0 +Γ0 ij P i P j P 0 P 0. C.1

25 After some canceations, we are eft with a reduced expression: Θ+n i i Θ+Γ 0 P i P j P i ij P 0 +Γ0 0i P 0 + ȧ a h 0in i + dh 0i dτ ni ȧ a 0. C.1 The third term can be computed by substituting Γ 0 ij with its expicit expression, Pi P j by γ αβ P α P β n i n j, and γ αβ P α P β by a g 00 P 0 P 0 h 0α P α P 0 h αβ P α P β. After a few ines of somewhat tedious cacuations, we get: Γ 0 ij P i P j P 0 ȧ 1h0i n i + 1 i a ḣijn n j + 1 jh 0i + i h 0j + 1 h0i γ ii j γ ii +h 0j γ jj i γ jj n i n j. C.1 The second ine can be written in a more compact and intuitive way using the covariant derivative i definedforthespatia metricγ ij inotherwords, basedonthechristofesymbos γ k ij computedfromγ ij, whichareactuayequatothoseofthefuspatiametricg ij computed at order zero in perturbations: h 0j i i h 0j γ k ijh 0k. C.15 We notice that h 0i j n i n j h 0j i n i n j i h 0j h 0j γ jj i γ jj n i n j. C.16 Hence the second ine of equation C.1 is equa to h 0i j n i n j. The fourth term in equation C.1 is much easier to obtain: Γ 0 P i 0i P 0 1 ih 00 n i + ȧ a h 0in i. C.17 We can simpify equation C.1 using C.1, C.17. After severa canceations, we get: Θ+n i i Θ++ 1 ḣijn i n j h 0i j n i n j 1 ih 00 n i + dh 0i dτ ni 0. C.18 Severa authors define the energy ω as g 00 P 0, negecting the correction terms proportiona to h 0i, and obtain the above equation without the ast term see e.g.,. If instead we take this term into account and express it as dh 0i dτ ni we obtain as a fina resut ḣ 0i +h 0i j dx j dτ n i ḣ0in i +h 0i j n i n j, C.19 Θ+n i i Θ+ 1 ḣijn i n j 1 ih 00 n i +ḣ0in i 0. C.0 This expression differs from its counterpart in 8, 10 through the sign of the third and of the ast terms.

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