CAMB Notes. Antony Lewis 1, 1 DAMTP/CITA/IoA

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1 CAMB Notes Antony Lewis 1, 1 DAMTP/CITA/IoA (Dated: March 13, 8) Brief summary of some theory reating to CAMB. Incudes definitions for the initia power spectra, quintessence equations, series soutions for the reguar scaar modes, variabe definitions and reationships, and random notes. I. INITIAL CONDITIONS For adiabatic modes we choose the initia conditions are set by the ampitude of χ, the comoving curvature perturbation (conserved on super-hubbe scaes for the adiabatic mode). In genera this is defined via the perturbation in the 3-Ricci scaar, η a 1 SD ar (3) with harmonic coefficient η, on co-moving hypersurfaces (tota heat fux q =, denoted by a bar), so that [ η = β Φ + 3 Ω 1 H 1 Φ ] Ψ β χ. 1 + w where (for cosed modes) β = (ν 4)/(ν 1). Here Φ and Ψ are defined as in [1, ], Ψ is the Newtonian potentia, and Φ = Φ H is the potentia satisfying the Poisson equation β Φ = 1 κs X. In genera Φ = Ψ S κπ/. In fat modes β = Ω = 1 and [ χ = η/ = Φ + 3 H 1 Φ ] Ψ. 1 + w For isocurvature modes the initia conditions are set for the corresponding non-zero mode (see ater section). We assume statistica isotropy, and the initia power spectrum for the primordia variabe X (= χ for adiabatic) on super-hubbe scaes is defined so that X = ν ν + 1 P X(ν). ν Here P χ gives the power in the 3-Ricci curvature by D a R(3) = ν ν 16 6 ν 1 S 6 ( ν ) 4 ν P χ (ν). (1) 1 During infation (no anisotropic stress, Π = ) χ = ψ + H φ δφ i = Ψ + i 3 Ω 1 H 1 Ψ + Ψ 1 + w. where φ i is is the infaton fied, and at the end of infation χ = 3/Ψ. In terms of the Wey tensor variabe φ (defined in [3], differing by a sign from Φ in [4]) Φ = φ 1 S κπ Ψ = φ 1 S κπ. In the eary radiation dominated era, assuming purey adiabatic perturbations, χ = 1 4R ν + 15 R ν + 5 φ = 4R ν + 15 Ψ 1 where R ν = ρ ν /(ρ ν + ρ γ ) (.4). CMBFAST used to set the initia condition Ψ = 1, where Ψ is the metric perturbation ψ as defined in [5] in the conforma Newtonian gauge. In CAMB we set χ = 1, same as the new CMBFAST defaut, so that the CAMB transfer functions are reated to those of the od CMBFAST (in fat modes) by (4R ν + 15)/1. URL:

2 II. CAMB VARIABLES CAMB propagates the covariant equations in the zero-acceeration frame, in which the CDM veocity is zero. This is equivaent to the synchronous gauge. Note that CAMB uses the variabe eta = η s, where η s = η/ (in the CDM frame) is the usua synchronous gauge variabe. Here η is the curvature perturbation variabe as in the previous section. The fractiona density perturbation variabes are caed cxc ( c ), cxg ( γ ), cxr ( ν ), cxb ( b ) etc. The synchronous gauge variabe h s appears in the CDM frame as h s = 6h = Z. The = 1 moments are handed in terms of the heat fuxes q i where ρ i q i = (ρ i + p i )v i. The tota heat fux appears as dgq = κa i ρ iq i in the code, and the tota matter perturbation is dgrho = κa i ρ i i. Other variabes are the shear σ and perturbation to the expansion rate Z. They are reated by 3 (β σ Z) = κa i ρ i q i = dgq η = κa i ρ i i HZ = dgrho HZ. III. MULTIPOLE EQUATIONS, HARMONIC EXPANSION AND C Here we summarize the equations of the covariant approach. The photon mutipoe evoution is governed by the geodesic equation and Thomson scattering, giving [6] I A ΘI A + D b I ba + 1 D ai A IA a 1 δ Iσ a 1 a δ = n e σ T ( I A Iδ 4 3 Iv a 1 δ 1 15 ζ a 1 a δ ) () where I A is taen to be zero for < and ζ ab 3 4 I ab + 9 E ab (3) is a source from the anisotropic stress and E-poarization. The equation for the density perturbation D a I is obtained by taing the spatia derivative of the above equation for =. The corresponding evoution equations for the poarization mutipoe tensors are [6] E A ΘE ( + 3)( 1) A + ( + 1) D b E ba + 1 D a E A cur B A = n e σ T (E A 15 ζ a 1 a δ ) B A ΘB A + ( + 3)( 1) ( + 1) D b B ba + 1 D a B A cur E A =. (4) For numerica soution we expand the covariant equations into scaar, vector and tensor harmonics. The resuting equations for the modes at each wavenumber can be studied easiy and aso integrated numericay. A. Scaar, vector, tensor decomposition It is usefu to do a decomposition into m-type tensors, scaar (m = ), vector (m = 1) and -tensor (m = ) modes. They describe respectivey density perturbations, vorticity and gravitationa waves. In genera a ran PSTF tensor X A can be written as a sum of m type tensors X A = m= X (m) A. (5)

3 3 Each X (m) A can be written in terms of m derivatives of a transverse tensor X (m) A = D A m Σ Am (6) where D A D a1 D a... D a and Σ Am is first order, PSTF and transverse D a m Σ m A m 1 a m =. The scaar component is X (), the vector component is X a (1), etc. Since GR gives no sources for X Am with m > usuay ony scaars, vectors and (-)tensors are considered. At inear order they evove independenty. B. Harmonic expansion For numerica wor we perform a harmonic expansion in terms of zero order eigenfunctions of the Lapacian Q m A m, D Q m A m = S Qm A m, (7) where Q m A m is transverse on a its indices, D a m Q m A m 1 a m =. So a scaar woud be expanded in terms of Q, vectors in terms of Q 1 a, etc. We usuay suppress the abeing of the different harmonics with the same eigenvaue, but when a function depends ony on the eigenvaue we write the argument expicity, e.g. f(). For m > there are eigenfunctions with positive and negative parity, which we can write expicity as Q m± A m when required. Since they are reated by the cur operation. Using the resut D ( cur Q Am ) = cur (D Q Am ) = S cur Q A m (8) cur cur Q m A m = S we can choose to normaize the ± harmonics the same way so that cur Q m± A m = S A ran- PSTF tensor of either parity may be constructed from Q m± A m Q m A [1 + (m + 1) K ] Q m A m (9) 1 + (m + 1) K Qm A m. (1) as ( ) m S D Qm A m A m (11) and a X (m) A component of X A may be expanded in terms of these tensors. They satisfy D Q m A = S D a Q m A 1 a = β m cur Q m± A = β m where β m 1 { (m + 1) } K/ and m. (1 [( + 1) m(m + 1)] K ) Q m A ( m ) S ( 1) Qm A 1 m S Qm A (1)

4 4 Dimensioness harmonic coefficients are defined by σ (m) ab =,± S σ(m)± Q m± ab H (m) ab =,± S H(m)± Q m± ab q (m) a π (m) ab =,± q (m)± Q m± a = Π (m)± Q m± ab,± Ω a = S Ω± Q 1± a,± E (m) ab I (m) A =,± = ρ γ S E(m)± Q m± ab,± I (m)± Q m± A A (m) a = S A(m)± Q m± a,± (D a X) (m) = S (δx)(m)± Q m± a (13),± where the dependence of the harmonic coefficients is suppressed. We aso often suppress m and ± indices for carity. The other mutipoes are expanded in anaogy with I A. The heat fux for each fuid component is given by q i = (ρ i + p i )v i, where v i is the veocity, and the tota heat fux is given by q = i q i. In the frame in which Ω a = gradients are purey scaar (δx) (1) =. C. Harmonic mutipoe equations Expanded into harmonics, the photon mutipoe equations () become I + [ β m ( + 1) m ] +1 I +1 I 1 = Sn e σ T (I δ I 4 3 δ 1v 15 ζδ ) σδ 4h δ 4 3 Aδ 1 (14) where m, I = δρ γ /ρ γ, I = for < m, and m superscripts are impicit. The scaar source is h = (δs/s). The equation for the neutrino mutipoes (after neutrino decouping) is the same but without the Thomson scattering terms (for massive neutrinos see Ref. [7]). The poarization mutipoe equations (4) become E m± + B m± + [ β+1 m ( + 3)( 1) ( + 1) m ( + 1) 3 ( + 1) [ β+1 m ( + 3)( 1) ( + 1) m ( + 1) 3 ( + 1) E m± E m± 1 B m± Bm± 1 + m ( + 1) m ( + 1) β m B m β m E m ] = Sn e σ T (E m± 15 ζm± δ ) ] =. (15) D. Integra soutions Soutions to the Botzmann hierarchies can be found in terms of ine of sight integras. The fat vector and tensor resuts are given in Ref. [8]. Genera scaar and tensor resuts are in [6] (though I in that paper differs by a curvature factor). E. Power spectra Using the harmonic expansion of I A the contribution to the C from type-m tensors becomes C T T (m) = π ()! 4 ( ) (!) I ±, I±, Q ± A QA ±. (16),,± The mutipoes I can be reated to some primordia variabe X Am = ( X + Q m+ A m + X Q m ) A m via a transfer function T X defined by I = T X X. Statistica isotropy and orthogonaity of the harmonics impies that X ± X± = f X ()δ (17)

5 where δ Y = Y and f X () is some function of the eigenvaue. The normaization of the Q m A is given by dv Q m A Q ma = dv ( ) m S Q m A m D A m Q m A = α m ( ) m ( + m)!( m)! N (18) ()! where we have integrated by parts repeatedy, then repeatedy appied the identity for the divergence (1). The normaization is N dv Q Am Q Am and αm n=m+1 βm n. By statistica isotropy C = (1/V ) dv C and hence C T T (m) = π ( + m)!( m)! N 4 ( ) m (!) V αm T X () f X () (19),± We choose to define a power spectrum P X () so that the rea space isotropic variance is given by X Am X A m = N V f X() d n P X () (),± 5 so the CMB power spectrum becomes C T T (m) = π ( + m)!( m)! 4 m (!) d n P X ()α m T X (). (1) For a non-fat universe d n is repaced by some some other measure which shoud be specified when defining the power spectrum. In a cosed universe the integra becomes a sum over the discrete modes. Note that we have not had to choose a specific representation of Q Am or. The poarization C are obtained simiary [6] and in genera we have C JK(m) = π [ ] p/ ( + 1)( + ) ()! J A K A 4 ( 1) ( ) (!) ρ () γ = π [ ] p/ ( + 1)( + ) ( + m)!( m)! 4 ( 1) m (!) d n P X () α m J X K X (3) where JK is T T (p = ), EE or BB (p = ) or T E (p = 1). We have assumed a parity symmetric ensembe, so C T B = C EB =. For tensors we use H T where h ij =,± H T Q ij and h ij is the transverse traceess part of the metric tensor. This introduces an additiona factor of 1/4 into the resut for the C in terms of P h and T H T. The numerica factors in the hierarchy and C equations depend on the choice of normaization for the and expansions. Neither e A nor Q A are normaized, so there are compensating numerica factors in the expression for the C. If desired one can do normaized expansions, corresponding to an - and -dependent re-scaing of the I and other harmonic coefficients, giving expressions in more manifest agreement with Ref. [9]. IV. QUINTESSENCE The bacground fied equation for a scaar fied ψ can be written d ( S ψ ) + S3 V,ψ ds H =. This aows ψ(s) and ψ (S) to be evauated, and hence the bacground evoution since H reates the time and S derivatives. The density and pressure are S ρ ψ = 1 ψ + S V S p ψ = 1 ψ S V The inearized exact fied equation is ψ + Hψ + D a D a ψ = S V,ψ.

6 6 where D a is the spatia covariant derivative. Defining the first order perturbation V a SD a ψ and fixing to the zero-acceeration frame (the CDM frame) this impies the foowing equation for the perturbations V a + HV a + SZ a ψ + S D a D b V b = S V a V,ψψ. This corrects the equation in [3], where other definitions are given. Performing the harmonic expansion one gets the scaar equation V + HV + ψ Z + V = S VV,ψψ. This is what tes you that you can t consistenty have V = in an evoving bacground if there are other perturbations present (Z = ). The heat fux and density perturbation X a (ψ) = SD a ρ ψ give the scaar reations S q = ψ V S X ψ = ψ V + S VV,ψ. There is no contribution to the anisotropic stress. CMBFAST/CAMB propagate quantities ie κs ρ, hence it is usefu to wor with φ κψ and simiary for V. Constant w A usefu parameterization is w p ψ /ρ ψ = constant, which impies ρ ψ = ρ S 3(1+w) This potentia and ψ are easiy obtained in terms of S V = 1 ψ 1 w 1 + w. S V = 1 w S ρ ψ ψ = 1 + w S ρ ψ. For w < 1 we tae the action to have a negative inetic term, so ψ + Hψ + D a D a ψ ± S V,ψ = etc, where ± is the sign of 1 + w. The derivatives needed are then V,ψ = [ 3(1 w) V,ψψ = Ḣ 3 ] H (1 + w) 3(1 w) H ψ S 3(1 w) V,ψ = Hψ, [ S 3(1 w) V,ψψ = H 1 ] H (5 + 3w). In many cases constant w is actuay a very good approximation as far as the CMB is concerned, with [1] daωψ (a)w(a) w eff. daωψ (a) In quintessence domination ρ ρ ψ and V,ψψ ±9(1 w )H /. Fuid equations Foowing [11] the defaut CAMB modue actuay uses the fuid equations. For varying w these are δ i + 3H(ĉ s,i w i )(δ i + 3H(1 + w i )v i /) + (1 + w i )v i + 3Hw iv i / = 3(1 + w i )h (4) v i + H(1 3ĉ s,i)v i + A = ĉ s,iδ i /(1 + w i ), (5) where hatted quantities are evauated in the frame co-moving with the dar energy. These equations are impemented in the code with w =, and are equivaent to using the above, with the additiona possibiity of using the rest-frame sound speed ĉ different from one for generaized dar energy modes (ĉ = 1 for quintessence)

7 7 V. TIGHT COUPLING At eary times the baryons and photons are tighty couped; the opacity τc 1 Sn e σ T is arge. This means δ q γ 4/3v b, and propagating the fu photon hierarchy is impossibe due to numerica probems in the source term proportiona to τc 1 δ. The soution it to do an expansion in τ c which is vaid for ɛ max(τ c, Hτ c ) 1. To first order in ɛ, assuming c s 1/S and dropping tiny terms in c s, v b 1 = (c s b + 4 ) R( γ β π γ ) Hv b (6) 1 + R ( ) H (n τc ) 3R 1 + R 4(1 + R) δ Rτ c τ c ( δ = γ 4c ) 3(1 + R) s b + 4Hv b ( 4(H 4(1 + R) + H )v b + [ H γ + γ 4c s ]) b π γ = 3 45 τ c(v b + σ) (9) E = π γ 4 (3) where R 4ρ γ /3ρ b. Note that here I use the sign for E consistent with that in the code, which is reated to the poarization tensor by a minus sign. To a good, but not good enough, approximation n e 1/S 3 over the region of interest, so (n τc 1 ) H. However using this resut does ead, for certain, to a region in time in which neither the tight couping approximation is accurate nor the fu equations are stabe. This is probaby exacerbated by the RECFAST reionization history which doesn t change so abrupty at recombination. Using a numerica vaue for (n τc 1 ) avoids the probem. Incuding the vaues for π γ and E maes the switch from tight couping more robust, and aows it to be pushed to ater times thereby speeding up the evoution on sma scaes. (7) (8) VI. REGULAR INITIAL CONDITIONS Foowing 1 [13] we define ω Ω m H / Ω R, where Ω R = Ω γ + Ω ν. The Friedmann equation gives S = Ω mh ω ( ωτ ω τ 1 6 Kωτ 3 1 ) 48 Kω τ 4 + O(τ 5 ). Taing the owest order in the tight couping approximation we have the drag term Sn e σ T (4/3v b q γ ) = ( γ + 4Hv b ). 3ρ b + 4ρ γ Higher orders corrections in 1/(n e σ T ) are negected (and hence π γ and higher moments are zero, v b = 3q γ /4), and aso assume c s = p b =. We define R ν = Ω ν /Ω R, R γ = Ω γ /Ω R, R b = Ω b /Ω m, R c = Ω c /Ω m. The adiabatic mode (in ρ b 1 The ony difference here is that we give resuts for additiona variabes, and define the isocurvature modes foowing [1].

8 the CDM frame remember χ is defined in terms of η in the comoving frame, η = η + O( τ )) is, for χ = 1, ( [ ]) η = β 1 (τ) 1 β (31) 1 4R ν + 15 γ = ν = β 3 (τ) β 15 ω τ 3 (3) c = b = β 4 (τ) β ω τ 3 (33) q γ = 4 3 v b = β (τ) 3 7 (34) q ν = β (τ) 3 4R ν R ν + 15 (35) π ν = 4 (τ) 3 4R ν + 15 ω τ 3 4R ν 5 3 (4R ν + 15)(R ν + 15) (36) G 3 = 4 (τ) 3 1 4R ν + 15 Further quantities not needed in the code are χ = 1 + (τ) 1 Ψ = ( β + 4R ) ν 5 4R ν R ν ωτ (8R ν 3) 8 (4R ν + 15)(R ν + 15) κs Π = R v 4R ν σ = ωτr ν (R ν + 15)(4R ν + 15) 5τ 4R ν ωτ 4R ν 5 8 (4R ν + 15)(R ν + 15) 8 (37) (38) (39) (4) (41) Z = 3h = β τ + 3β ωτ. (4) Note that η and hence χ are constant to inear order, whereas the potentia is ony constant at zeroth order due to the matter changing the bacground equation of state. CDM isocurvature mode: η = β 3 R cωτ β 8 R c(ωτ) (43) c = 1 1 R cωτ R c(ωτ) (44) b = 1 R cωτ R c(ωτ) (45) γ = ν = 3 R cωτ R c(ωτ) (46) q γ = q ν + O(τ 3 ) = 1 9 R cωτ (47) π ν = 1 R c ω τ 3 3 R ν + 15 Φ = 1 R c (4R ν + 15)ωτ 8 (R ν + 15) σ = 1 R c (4R ν 15)ωτ 4 R ν + 15 There is a soution R c c = R b b with everything ese zero, since in this case there is no density perturbation and hence the dynamics is that of the bacground. The baryon isocurvature mode is given by subtracting ( c = 1, b = R c /R b ) from the above mode, and is equivaent as far as the CMB is concerned so that C (CDM iso, c = 1) = R c/r b C (baryon iso, b = 1). The CDM isocurvature mode has a particuary simpe form using the gauge invariant (48) (49) (5)

9 variabes η= i (the perturbation in the frame in which η =, see ater), with η= c = 1, η= γ,b,ν = to the above order. We choose the neutrino isocurvature modes to have zero initia perturbation to the 3-Ricci scaar a fairy natura definition of isocurvature. This impies that the density perturbation in the comoving frame (proportiona to Φ) is non-zero, which maes an aternative definition of isocurvature, in which the resut is a sum of the mode beow and the adiabatic mode. Neutrino isocurvature density mode: γ = R ν R γ + R ν 6R γ (τ) (51) ν = 1 (τ) (5) 6 c = ω τ 3 R ν R b (53) 8 R γ b = 1 R ν (τ) (54) 8 R γ q γ q ν = R ν τ + ωτ 3R γ 4 = τ 3 (τ)3 54 The weird neutrino veocity isocurvature mode is γ ν R ν R b R γ ( 1 + 1β 4R ν + 15 π ν = (τ) 4R ν + 15 η = β (τ) R ν 3 4R ν + 15 Φ = R ν 4R ν + 15 ωτ R ν (R ν 15) 4 (4R ν + 15)(R ν + 15) = τ R ν 3ωτ R b ( + R γ ) R γ 16 Rγ ) = τ 3ωτ R b 16R γ (61) c = 9ωτ 64 b 9 (55) (56) (57) (58) (59) (6) R ν R b R γ (6) = 3R ν τ 9ωτ R b ( + R γ ) 4R γ 64 Rγ q γ = R ν + 3R νr b R γ 4Rγ ωτ + (τ) R ν + 3(ωτ) R ν R b 6 R γ 16 Rγ 3 (R γ 3R b ) (64) ( q ν = 1 (τ) 1 + 4β ) (65) 6 4R ν + 5 τ π ν = (4R ν + 5) + ωτ 6 (66) (4R ν + 5)(4R ν + 15) G 3 = 3 (τ) (67) 7 4R ν + 5 R ν η = β τ 4R ν ωτ 3β ( ) R ν Rb 8 (68) 3 R γ (4R ν + 5)(4R ν + 15) Φ = Ψ + O(1) = 3R ν 4R ν + 5 (τ) 1 (69) φ = 45 R ν ω 4 (4R ν + 15)(4R ν + 5) (7) R ν σ = 3 + φτ (71) 4R ν + 5 (63)

10 Note that the variabes Φ and Ψ are singuar. However these are ony natura variabes in the zero shear frame (the curvature and acceeration respectivey), whereas φ is frame invariant and is reguar. The resuts above generaize those of [1] for non-fat modes (note that the adiabatic anisotropic stress term in [1] is wrong, as is the neutrino octopoe in the neutrino veocity mode). The quintessence fied ψ is strongy damped by the high expansion rate in the eary universe, and thus rapidy obtains very sma veocity. Choosing the origin of the fied ψ() = we have ψ(τ) = H Ω R V,ψ τ 4 / + O(τ 5 ), w ψ = 1 + H Ω R V,ψ τ 4 /(5V ), and the above modes are unchanged on adding a quintessence fied. There is an additiona quintessence isocurvature mode with ψ V = V V,ψ = 1 (τ) 1 (1 (τ) 6 + ω τ 3 ) 7 and the other perturbations zero to this order. Note that fixing w = const is inconsistent with these assumptions. However in the eary universe the evoution is independent of the potentia, so using the above initia conditions with w = const is potentiay not a bad approximation for various casses of quintessence modes which start with w 1, then evove to have an effective constant w = w eff. The reguar vector mode soution is given in Ref. [14] and the mode with magnetic fieds in Ref. [8]. 1 (7) (73) Frame invariant series The isocurvature modes tae a particuary simpe form in the frame of identicay vanishing curvature, in other words using the frame invariant variabes ˆ i = i + 3 β (1 + w i )η, e.g. CDM isocurvature: ˆ c = 1 1 R c (4R ν 15)ω τ 3 7 R ν + 15 ˆ γ = ˆ ν = 5 R c ω τ 3 6 (R ν + 15) ˆ b = 5 R c ω τ 3 8 (R ν + 15) v b + σ = 15 R c ωτ 8 R ν + 15 v c + σ = 1 R c (4R ν 15)ωτ 4 R ν + 15 q ν σ = 5 R c ωτ R ν + 15 Ψ = 1 R c (4R ν 15)ωτ 8 R ν + 15 (equa signs at this order ony), consistent with comments beow about conservation of the density perturbations in the zero curvature frame. See aso Ref. [15]. (74) (75) (76) (77) (78) (79) (8) VII. TENSORS The power spectrum defined by the transverse traceess part of the metric tensor so that h ab h ab = ν ν ν 1 ν 4 ν P h (ν) where the ν 4 factor is the mode sum =... ν 1 of + 1. Note that ν /(ν 1) is no onger the measure over n for tensor modes. The CMB resut is then C = π ( + 1)( + ) 3 ( 1) ν ν ν 1 ν 3 ν P h (ν)[t () I (ν)].

11 Internay CAMB uses the H, the metric tensor variabe, and the shear σ. The reation between H, E (the eectric part of the Wey tensor) and the shear σ is H = ν 3 ν 1 (E + σ /) and H = σ. Using the series soution the reation between the initia E and H is E = R ν + 1 ν 1 4R ν + 15 ν 3 H where R ν is the ratio of the neutrino and tota radiation densities. 11 VIII. MATTER POWER SPECTRUM σ 8 is defined by σ 8 = dv 3 (x) dv 3 where is the fractiona tota matter density perturbation (frame irreevant at ate times), and the integra is over a 8h 1 Mpc sphere. In spherica poar harmonics we have = d n m Y m j (r), m then, since Y = 1/ 4π, dv3 (x) = 3 dv3 4πr 3 d n r = 3 r cos r sin r 4π 3 r 3 dr4πr j (r) 4π where r = 8h 1 Mpc. Now by assumed statistica isotropy (x) = () = 1 d n d n 4π = d n P It foows that and hence m m = 4πδ δ mm δ( )P () σ8 = where T is the transfer function. We define the matter power spectrum so that (x) = 1 (π) 3 and is usuay expressed in units of h 1 Mpc. Therefore [ ] r cos r sin r d n 3 3 r 3 T P χ d 3 P = d n P = π P χ 3 T. ( 3 ) P π For COBE-normaized runs this can be compared with the output from CMBFAST using P = 8π 3 h 3 (T F ) dnorm where T F is the quantity in the transfer function output fie, and is just (the first coumn of the output fie is /h).

12 scae factor FIG. 1: Absoute vaue =.1Mpc 1 mode evoution of c (top), γ and ν (bottom), q γ and q ν (midde), together with the dotted ines which are the approximations used after matter domination. IX. THE CAMB CODE The physics is contained in the equations.f9 fie. Everything is in conforma time and units of megaparsecs. The dtauda routine returns 1/a (scae factor a = S), where the conforma Hubbe rate is a /a. This is used to compute the conforma age of the universe, and bacground evoution for the pre-computed ionization history and shoud aways be finite. Bacground density variabes incude a factor of a, ie grho κa ρ. The initia conditions for the scaars and tensors are in the initia and initiat routines (see above for detais). The derivs and derivst functions return the differentia equations - a vector y of derivatives of the variabes being evoved. The fderivs and fderivst routines are very simiar and used for fat modes. The output and outputt routines cacuate the sources for integration against the Besse functions. X. LATE TIME EVOLUTION To compute the matter transfer functions at ate time on sma scaes many osciations of the photon and neutrino mutipoes have to be integrated. The integration is in fact very inaccurate except on high accuracy settings because of the ow- truncation of the mutipoe equations. The effect after matter domination is rather sma so this is a waste of time. However even though the osciations are inaccurate, the mean vaues are non-zero (and about correct), so for accurate resuts it is important to capture the non-osciatory behaviour of the veocities when the radiation density fraction is not entirey negigibe. Ref. [16] suggest one scheme for cutting off this evoution. In CAMB we are using the synchronous gauge, and instead use the foowing anaytic soutions for the non-osciatory zero-acceeration frame evoution γ ν κs 3 (ρ c c + ρ b b ) (81) q γ q ν γ 3 κs [ρ b + ρ c ]/3 (8) ones the modes are we inside the horizon, after recombination, and we in to matter domination. The higher moments are set to zero. The particuar representation of q γ here has been chosen so that it wors quite we through

13 the matter-radiation transition. On scaes with.mpc 1 we can t use this truncation because of reionization, but on smaer scaes it can save a significant amount of time, especiay when computing the matter power spectrum on sma scaes. Parameters are chosen to eep the matter power spectrum accurate to <.3% on standard accuracy settings. See Fig. 1. The approximation to γ is fairy irreevant, but the veocities are crucia to obtain sub-percent eve precision with a switch to the approximate soution at a > 1a eq for modes we inside the horizon. We switch off the higher mutipoes suddeny rather than using a smooth transition as in Ref. [16], so the mutipoe equations do not need to be propagated at a after the transition. Note the the synchronous (zero acceeration frame) photon density perturbations are not the same as the Newtonian one, γ 4Hσ/, amounting to a factor of three difference during matter domination. The photon veocities are aso quite different, with the Newtonian gauge q γ + 4/3σ. 13 XI. MASSIVE NEUTRINOS, LENSING ETC These are propery documented! The treatment of massive neutrinos is discussed in Ref. [7]. The CMB ensing method is described in Ref. [17]. XII. LIMBER APPROXIMATION AND LENSING This can be usefu for cacuating the ensing potentia power spectrum accuratey on sma scaes (it is not used on arge scaes). For introduction to ensing and the Limber approximation in the notation here see [18]. The fat universe fu inear theory resut for the ensing power spectrum is [ C ψ d χ = 4π P R() dχ S ψ (; η χ)j (χ)], (83) where the ensing source is given in terms of the transfer function for the Wey potentia by ( ) χ χ S ψ (; η χ) = T Ψ (; η χ) χ χ for χ < χ and zero otherwise. Note that in matter domination the potentias are neary constant, so T Ψ (; η χ) is neary constant. In this approximation, and maing the approximation that the integra goes to infinity, we can use ( ) [ χ χ π Γ[/] dχj (χ) = χ χ 4 Γ[( + 3)/] ] Γ[( + 1)/]. (85) χ Γ[( + )/] (can aso use χ imit, but resut is a mess.) The potentias wi of course change when dar energy becomes important, though the scae of variation may be arge compared to the Besse function frequency on sma scaes. The Limber approximation pics out /χ and the Besse functions vary much faster than the source on sma scaes. Using d j (χ)j (χ ) = π χ δ(χ χ ), (86) (84) we can Limber-approximate C ψ as C ψ 8π 3 χ χdχ P Ψ (/χ; η χ) ( ) χ χ. (87) χ χ Rather than using this resut directy it is convenient to write the integra in a form coser to the fu resut, approximating χ χ dχ S ψ (; η χ)j (χ) S ψ (; η χ s ) dχ j (χ) (88) where χ s = /. Using the resut dχj (χ) = π ( Γ ([ + 1]/) π Γ ([ + ]/) O ( )) 1 (89)

14 14 we have the arge fat universe approximation [ ] C ψ d π 4π P 1 R() S ψ(; η χ s ), (9) consistent with the previous resuts. For the non-fat case we have C ψ 8π 3 χ f K (χ)dχ P Ψ (/f K (χ); η χ) ( ) fk (χ χ), (91) f K (χ )f K (χ) which we can compute simiar to the fat case by using the sma scae source term given approximatey by π 1 1 (1 K / ) S ψ(; η 1/4 χ s ) (9) where f K (χ s ) = /. XIII. REIONIZATION The optica depth to reionization is defined by τ = η dηsn reion e σ T (93) where n reion e is the number density of free eectrons produced by reionization and η is the conforma time today. This is not quite the same as the number density of eectrons at ate times because there is a sma residua ionization fraction from recombination. At the eve of precision required we can negect this difference ( 1 3 ), though CAMB eeps the ionization history smooth my mapping smoothy onto the recombination-residua. Since n e (1 + z) 3 x e (z), and reionization is expected to happen during matter domination, τ dz x e 1 + z d[(1 + z) 3/ ]x e. (94) It is therefore handy to parameterize x e as a function of y (1 + z) 3/. As of March 8 CAMB s defaut parameterization uses a tanh-based fitting function x e (y) = f [ ( )] y y(zre ) 1 + tanh, (95) where y(z re ) = (1 + z re ) 3/ is where x e = f/: i.e. z re measures where the reionization fraction is haf of its maximum. Since the fitting function is antisymmetric about the mid point (and assuming it does not extend out of matter domination) η zre τ = dηsn reion e σ T athom f dz dη ds. (96) In other words, with this parameterization the optica depth agrees with that for an instantaneous reionization mode at the same z re for a (matter-dominated) vaues of y (CAMB s variabe athom is n p σ T today). Except in eary dar energy modes this resut is quite accurate for the expected range of z re. In practice the input parameter is z and y is taen to be z re z. To eep the impementation genera (e.g for eary dar energy modes), the defaut reionization modue actuay maps τ into z re by doing a binary search. This method is aso generay appicabe for other monotonic modes, e.g. the modue aso wi wor fine using a window function in a different power of (1 + z) (i.e. Rionization zexp 3/), though in this case there is in genera a more compicated reation between the optica depth and that of a sharp reionization mode. Changing the exponent aows fexibiity in reativey how quicy reionization starts and ends, with vaues.5.5 changing the EE power by a coupe of percent at fixed τ. If hydrogen fuy ionizes f = 1. However the first ionization energy of heium is simiar, and it is often assumed that heium first re-ionizes in roughy the same way. In this case f = 1 + f He, where f He = n He /n H is easiy cacuated from the input heium mass fraction Y He. This is CAMB s defaut vaue of f; typicay f 1.8 In addition at z 3.5 heium probaby gets douby ionized. Due to the ow redshift this ony affects the optica depth by.1, but for competeness this is incuded using a fixed tanh-ie fitting function (modifying the above resut for τ appropriatey). Some reionization histories are shown in Fig.. y

15 x e x e z z FIG. : Three recombination histories a with τ =.9. The dashed ine is the mode typicay used by CMBFAST and CAMB prior to March 8 with f = 1. The bac ine is the new mode with z =.5, the red ine with z = 1.5. XIV. DIFFERENT VARIABLES The foowing is not stricty reated to CAMB, but usefu maybe to understand the equations (and S ψ the obvious non-fat generaization). As in my thesis I use tota matter variabes, e.g. q = i ρ iq i, X = i ρ i i. In the co-moving q = (zero heat fux, denoted by a bar) frame are frame invariant, and hence Using the frame invariance of X X = X + 3Hq/ σ = σ + q ρ + p η = η β Hq (ρ + p). ˆq = q + X /3H where a hat denotes the X = frame, and hence ˆη = η + β ( X η 3(ρ + p) = H H β ) X ρ is aso frame invariant, so ˆη = η X = = β σ = Z β 3(ρ+p) X η=. In terms of the fat metric variabes as used by [19] 1 ˆη ζ = Hξ H ( ψ H + δρ ρ ) and η ψ. Since X / ρ transforms ie X i / ρ i the curvature perturbation in the frame X i = is the frame invariant quantity η β H X i. ρ i

16 16 The energy conservation equation gives 3ĥ a(ρ + p) = 3H ˆX p a S ˆDa ˆDbˆq b. which impies that ĥa is constant on arge scaes where the derivatives can be negected if the pressure perturbation in the uniform energy density frame is zero, i.e. ˆX p a = (D a p) nad X p a p /ρ X a =. As emphasised by [19] this foows purey from energy conservation, independenty of the fied equation (though in different theories the energy that is conserved may incude components from the different theory). Using the genera resut we have η = β 3 (σ 3h ) 1 β ˆη = H δp nad ρ + p σ 3. Hence on arge scaes ˆη is conserved if K = and δp nad =. This appies equay we if the tota densisties are repaced with those for an individua species as ong as they do not interact. For adiabatic modes ˆη = η + X β 3(ρ+p) η on arge scaes since X is suppressed by a factor of. The tota and individua resuts are reated by ˆη = i(ρ i + p i )ˆη i i (ρ i + p i ). Note that from the η equation above, in the gauge in which h =, η is conserved on arge scaes no matter what. Is this a usefu statement, or ie saying that in the h = gauge h is conserved!? Wands etc define this gauge by as the zero number density perturbation gauge, which amounts to the same thing as if h = the oca voume eement does not change with time and can be chosen to be zero initiay. See aso Ref. [15]. Newtonian Gauge The Newtonian gauge is the frame with zero shear: σ =. To construct Newtonian gauge quantities from those in any other frame you just need to identify the gauge-invariant combinations that give what you want when the shear vanishes. See the Change of Frame section ater for hep constructing frame invariant combinations. In particuar the Newtonian gauge veocity and density perturbations are given by where w = p/ρ. e.g. γ N = γ 4Hσ/. In genera v (i) N = v(i) + σ i (i) N = (i) 3H(1 + w (i) )σ/ Φ = η + Hσ β and hence β Φ = η N is the curvature in the zero shear frame (Newtonian gauge). From the σ equation A N = Ψ, and the η reation η N = β h N. A usefu reation is ( i + 3(1 + w ) i) η = (1 + w i )(v i + σ). β See aso the appendix of Ref. [15] and XV. NEUTRINO DAMPING ON TENSORS It s we nown that the effect of neutrino anisotropic stress can be negected when computing the arge scae temperature anisotropy. This is because the dominant arge scae contribution ( < 15) comes from the anisotropy generated by viewing an isotropic ast scattering surface through gravitationa waves aong the ine of sight. The gravitationa waves source photon anisotropies via their derivative H σ. On super-horizon scaes H = and there is no contribution. On sub-horizon scaes the waves decay, therefore the main contribution to the arge scae

17 17 (+1) C / π C / C FIG. 3: The BB power spectrum with and without incuding the effect of neutrino damping (CAMB s do tensor neutrinos parameter) anisotropy comes from when the waves come inside the horizon and start to decay, but before they start osciating at which point they become sma. After ast scattering the evoution is matter dominated, so the neutrinos have amost no contribution to the energy density and hence the effect of neutrino anisotropic stress is negigibe. For this reason CMBFAST, and by defaut CAMB, negect the neutrinos as they sow down the computation. For poarization things are different, as in this case the inear contribution to the anisotropy comes predominanty from the photon anisotropic stress sourced by H. The arge scae signa due to reionization is however insensitive to neutrinos as this is during matter domination, thus the poarization signa with highest signa to noise is aso unaffected. On smaer scaes the main contribution comes from the photon anisotropic stress at ast scattering, which is affected by H, itsef a function of the damping due to neutrinos. Ref. [] has anaysed the effect of neutrinos semi-anayticay. In particuar he finds a genera semi-anaytica resut for modes which enter the horizon deep in radiation domination. Since this is before ast scattering, and once inside the horizon the modes decay, this regime corresponds to the sma osciatory signa at high. This is of itte consequence for observations, though the effect is a substantia 35% reduction in power. The poarization power spectra pea at 1, corresponding to modes which have maxima veocity at the time of ast scattering. Therefore the most interesting region (apart from reionization) is on intermediate scaes. Ref. [] aso anayses the case of modes which enter the horizon after radiation domination, and caims a 1% damping of the power by anaysing the effect of neutrinos on H at ast scattering. This accounts for the damping of individua modes and the phase difference reative to the fixed time of recombination. See Figure. 3 for CAMB s output. The arge scae effect of 5% on < 1 is numericay smaer than that in Ref. [], and for 1 the neutrinos actuay increase the power sighty. My understanding is this: A mode which (without neutrinos) reaches maxima veocity at some time shorty after ast scattering may be sowed by the damping such that the veocity at the time of ast scattering is higher than without the damping, even though the overa ampitude of the mode is ower. Thus for modes on arger scaes than the pea < 1 there shoud be a damping effect, but for 1 there can be an increase in power at that (CAMB has aways supported tensor neutrinos by a fag in equationsxxx.f9; as of Dec 3 this can aso be set from the parameter input fie)

18 18 XVI. CHANGE OF FRAME In the frame ũ = u + v we have q = q (ρ + p)v X = X ρ v η = η β Hv σ = σ + v à = A + v + Hv h = h + v 3 (Hv). In terms of the covariant tensors we have (not checed for non-fat) η a = η a HD a D b v b h a = h a S D a D b v b (Hv a ) δρ a = δρ a ρ v a σ ab = σ ab + D a v b (note notation mess: δρ X ) so e.g. a frame invariant curvature perturbation is η a H D ad b δρ b ρ. (97) XVII. EXACT COVARIANT GR REFERENCE Using the u a u a = 1 signature. Energy conservation: ρ + (ρ + p)θ + D a q a A a q a π ab σ ab = q a θq a + (ρ + p)a a D a p + q b σ ab + q b ϖ ba + D b π ab A b π ab = Raychaudhuri equation: θ θ + κ (ρ + 3p) Da A a + ϖ ab ϖ ba + σ ab σ ab + A a A a =. Gauss-Codazzi (ϖ ab = to define (3) R) impies: (3) R = κρ 3 θ + σ ab σ ab. Scae factor perturbation: ḣ a = 1 3 (Z a SθA a ) (σ ab + ϖ ab )h b u a A b h b. Define ω a = 1 cur u a, so cur ω a = D b ϖ ba. Others: D b ϖ ba D b σ ab + 3 D aθ + κq a A b ϖ ba = D a ω a + A a ω a =

19 19 and the evoution equations σ ab + 3 θσ ab D a A b + E ab + 1 κπ ab + σ a c σ c b + A a A b + ω a ω b ω a + 3 θω a = 1 cur A a + ω b σ ab Here on the LHS is needed to project some time derivatives orthogona to u a. XVIII. ACTION (E.G. CLOSED INSTANTON) We perform a harmonic expansion of a quantity X in the form X = m X m Q m where Q m = Q and we choose Q m = Φ ν (χ)y m where we normaize the hyper-spherica Besse functions so that Φ ν () = 1 and = K(ν 1) (for cosed modes: everywhere ν 1 ν + 1 for open modes). For fat modes Φ ν (r) = j (r). We expand a variabe X in the form X = X Q where abes,m and ν and Q = Q. Mode expansion in the action gives S = 1 dv 4 XLX = N dτ X LX where dv 3 Q Q = Nδ where δ is defined by δ X = X. Spatia derivatives in L are converted to s. In a compact space the sum over modes is of the form = A ν νm where A ν is some arbitrary ν-dependent factor. The expectation vaue of the mode coefficients is therefore given by X (τ) = 1 NA ν G(τ, τ) where the Greens function is unambiguousy determined by the rea space operator and satisfies LG(τ, τ ) = δ(τ τ ). The expectation vaue of each ν mode wi be the same and X(x, τ) = K3/ π G(τ, τ)/a ν = ν ν ν 1 P(ν) where P(ν) K3/ ν(ν 1) π G(τ, τ)

20 is the power spectrum. The CMB power spectrum is given by C = 1 1 NK 3/ ν X (τ r ) [T () I (ν)] π where T () I (ν) is the transfer function giving I when X (τ r ) = 1 and r denotes the eary radiation dominated era when the mode is sti we outside the horizon. This gives C = 1 16 π ν K 3/ ν G(τ r, τ r )[T () I (ν)] = 4π 16 ν ν ν 1 P r(ν)[t () I (ν)]. For the CMB Anthony uses A ν = K 3/ ν and N = π/ where Q = Φ ν (χ)y m and Φ ν () = 1. In the tensor case we have an action of the form S = 1 dv 4 h ab Lh ab 8κ and write h ab = H Q ab where abes ν,,m and P, the parity. The power spectrum is given by P h (ν) 4 π K3/ ν(ν 1)G(τ, τ) where LG(τ, τ ) = δ(τ τ ). Note the additiona factor of two difference from the additiona sum over the two parities. The contribution to the CMB is C = 1 16 ( + 1)( + ) ( 1) 1 NK 3/ ν + 1 π (ν 3) ν H [T () I (ν)] where abes positive parity modes ony ( Q A = ) giving the previous resut. [1] W. Hu and N. Sugiyama, Astrophys. J. 444, 489 (1995), astro-ph/ [] W. Hu, U. Seja, M. White, and M. Zadarriaga, Phys. Rev. D57, 39 (1998), astro-ph/ [3] A. Lewis, Ph.D. thesis, Cambridge University (), [4] A. Chainor and A. Lasenby, Astrophys. J. 513, 1 (1999), astro-ph/ [5] C.-P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995), astro-ph/9567. [6] A. Chainor, Phys. Rev. D6, 434 (), astro-ph/ [7] A. Lewis and A. Chainor, Phys. Rev. D66, 3531 (), astro-ph/357. [8] A. Lewis, Phys. Rev. D7, 4311 (4), astro-ph/4696. [9] W. Hu and M. White, Phys. Rev. D56, 596 (1997), astro-ph/9717. [1] G. Huey, L. Wang, R. Dave, R. R. Cadwe, and P. J. Steinhardt, Phys. Rev. D59, 635 (1999), astro-ph/ [11] J. Weer and A. M. Lewis, Mon. Not. Roy. Astron. Soc. 346, 987 (3), astro-ph/3714. [1] M. Bucher, K. Moodey, and N. Turo, Phys. Rev. D6, 8358 (1), astro-ph/ [13] A. Chainor (1999), unpubished. [14] A. Lewis, Phys. Rev. D7, (4), astro-ph/ [15] C. Gordon and A. Lewis, Phys. Rev. D67, (3), astro-ph/148. [16] M. Doran, JCAP 56, 11 (5), astro-ph/5377. [17] A. Chainor and A. Lewis, Phys. Rev. D71, 131 (5), astro-ph/545. [18] A. Lewis and A. Chainor, Phys. Rept. 49, 1 (6), astro-ph/ [19] D. Wands, K. A. Mai, D. H. Lyth, and A. R. Lidde, Phys. Rev. D6, 4357 (), astro-ph/378. [] S. Weinberg, Phys. Rev. D69, 353 (4), astro-ph/3634.