189 APPENDIX B. USEFUL QUANTITIES AND RELATIONS 190
1s =9.7157 × 10−15 Mpc 1yr =3.1558 × 107 s 1Mpc =3.0856 × 1024 cm 1AU =1.4960 × 1013 cm 1K =8.6170 × 10−5 eV 33 Appendix B 1M =1.989 × 10 g 1GeV =1.6022 × 10−3 erg =1.7827 × 10−24 g =(1.9733 × 10−14 cm)−1 Useful Quantities and Relations =(6.5821 × 10−25 s)−1 Planck’s constant ¯h =1.0546 × 10−27 cm2 gs−1 Speed of light c =2.9979 × 1010 cm s−1 −16 −1 Boltzmann’s constant kB =1.3807 × 10 erg K B.1 FRW Parameters Fine structure constant α =1/137.036 Gravitational constant G =6.6720 × 10−8 cm3 g−1 s−2 2 4 3 2 The expansion rate is given by the Hubble parameter Stefan-Boltzmann constant σ = ac/4=π kB/60¯h c × −15 −3 −4 2 2 a =7.5646 10 erg cm K 2 1 da a˙ a0 2 2 × −25 2 H ≡ = Thomson cross section σT =8πα /3me =6.6524 10 cm a dt a a Electron mass me =0.5110 MeV a 4 a + a a 2 0 eq 2 − 0 2 Neutron mass mn = 939.566 MeV = Ω0H0 K +ΩΛH0 , (B.1) a aeq + a0 a Proton mass mp = 938.272 MeV − 2 − − where the curvature is K = H0 (1 Ω0 ΩΛ). The value of the Hubble parameter today, for different choices of the fundamental units (see Tab. B.1), is expressed as Table B.1: Physical Constants and Conversion factors −1 −1 H0 = 100h kms Mpc −34 4 −3 − ρ =4.4738 × 10 Θ gcm , =2.1331 × 10 42h GeV γ0 2.7 −5 −2 4 − − Ω =2.3812 × 10 h Θ , (B.4) = (2997.9) 1h Mpc 1 γ 2.7 − − =(3.0857 × 1017) 1h s 1 and for the neutrinos −1 −1 =(9.7778) h Gyr . (B.2) −1 ρν0 =[(1− fν) − 1]ργ0, −1 Present day densities in a given particle species X are measured in units of the critical Ων =[(1− fν) − 1]Ωγ, (B.5) density ρX(a0)=ΩXρcrit,where with (1 − f )−1 =1.68 for the standard model, or for the total radiation 2 × −29 2 −3 ν ρcrit =3H0 /8πG =1.8788 10 h gcm −47 2 4 −1 =8.0980 × 10 h GeV ρr0 =(1− fν ) ργ0, 4 2 −3 −1 =1.0539 × 10 h eV cm Ωr =(1− fν ) Ωγ. (B.6) =1.1233 × 10−5h2protons cm−3 11 2 −3 =2.7754 × 10 h M Mpc . (B.3) B.2 Time Variables For the CMB, Throughout the text we use four time variables interchangeably, they are a the 3 −3 nγ0 = 399.3Θ2.7 cm , scale factor, z the redshift, η the conformal time, and t the coordinate time. In addition, APPENDIX B. USEFUL QUANTITIES AND RELATIONS 191 APPENDIX B. USEFUL QUANTITIES AND RELATIONS 192 three dimensionless time parameterizations are useful to consider: χ the development angle in an open universe, D the relative amplitude of pressureless matter fluctuations, and τ the optical depth to Compton scattering.
B.2.1 Scale Factor and Redshift Epoch Definition
The scale factor a(t) describes the state of expansion and is the fundamental −0.027/(1+0.11ln Ωb ) z∗ =103Ω Ω = 1 Last scattering (recomb.) measure of time in the Hubble equation (B.1) since it controls the energy density of the b 0 2 2 1/3 2 −2/3 universe. In this Appendix, we leave the normalization of a free to preserve generality. =10 (Ω0h /0.25) (xeΩbh /0.0125) Last scattering (reion.) However, the normalization applied in §4, §5, §6, and Appendix A is a = 1. The conversion 2 1/5 −2/5 eq zd = 160(Ω0h ) xe Drag epoch factor between the more commonly employed normalization a0 =1is × 4 2 − −4 zeq =4.20 10 Ω0h (1 fν )Θ2.7 Matter-radiation equality − aeq Ωr × 4 2 4 = zbγ =3.17 10 Ωbh Θ2.7 Baryon-photon equality a Ω − Ω 0 0 r z =(1+z ){4(k/k )2/[1 + (1 + 8(k/k )2)1/2]}−1 Hubble length crossing × −5 2 −1 − −1 4 H eq eq eq =2.38 10 (Ω0h ) (1 fν ) Θ2.7. (B.7) z =(1− Ω0 − ΩΛ)/Ω0 − 1 Matter-curvature equality 1/3 The redshift z is defined by (1 + z)=a0/a and serves the same role as the scale factor z =(ΩΛ/Ω0) − 1 Matter-Λ equality normalized to the present. We give the scale factor normalized to 3/4 at baryon-photon 1/2 z =[ΩΛ/(1 − Ω0 − ΩΛ)] − 1 Curvature-Λ equality equality a special symbol R given the frequency of its appearance in equations related to − z =9.08Θ 16/5f 2/5 (Ω h2)1/5 − 1 Compton cooling era Compton scattering. More explicitly, cool √ 2.7 cool 0 z>4 2z Bose-Einstein era 3 ρ 3 Ω a K R = b =(1− f )−1 b ν z
is the Compton mean free time between scatterings. B.2.3CoordinateTime Thecoordinatetimeisdefinedintermsofthescalefactoras, Z B.2.4 Growth Function da 1 t= .(B.15) The amplitude of matter fluctuations undergoing pressureless growth is another a H useful parameterization of time. It is given by equation (5.9) as Italsotakesonsimpleasymptoticforms,e.g. Z 2 5 a0 da 1 a0 2 − −3/2 D = Ω g(a) , (B.24) t= (Ω H2 ) 1/2a [(a+a )1/2(a−2a )+2a3/2].RD/MD(B.16) 0 3 3 0 0 0 eq eq eq 2 aeq a g (a) a Explicitly,thisbecomes where the dimensionless, “pressureless” Hubble parameter is 1 2 −1/2 1/2 2 a 3 a 2 t= (Ω0H0) (a0/aeq) (a/a0) RD g2(a)= 0 Ω + 0 (1 − Ω − Ω )+Ω . (B.25) 2 a 0 a 0 Λ Λ × 19 −2 −2 =2.4358 10 Θ2.7(1+z) s.(B.17) In the matter or radiation-dominated epoch, D = a/aeq by construction. In a Λ = 0 and universe, D becomes 2 2 -1/2 3/2 " # t= (Ω0H0) (a/a0) MD 1/2 3 5 3 3(1 + x) 1/2 − 1/2 17 2 −1/2 −3/2 D = 1+ + 3/2 ln[(1 + x) x ] , (B.26) =2.0571×10 (Ω0h ) (1+z) s.(B.18) 2xeq x x Theexpansiontime,definedasH−1scalessimilarly −1 − where x =(Ω0 1)a/a0. Fitting formulae for the growth factor, valid for the general case, − 1/2 − are occasionally useful [26]: t =(Ω H2 ) 1/2(a/a )2a (a+a ) 1/2 exp 0 0 0 0 eq 19 −1/2 −2 −3/2 −1 =4.88 × 10 (z + zeq +2) Θ (1 + z) s. (B.19) D 5 1 1 2.7 0 ' Ω Ω4/7 − Ω + 1+ Ω 1 − Ω , (B.27) a 2 0 0 Λ 2 0 70 Λ For Λ = 0 universes, the coordinate time at late epochs when radiation can be neglected is 0 " # 3 4/7 given by d lnD Ω0(1 + z) ' . (B.28) " # d ln a Ω (1 + z)3 − (Ω +Ω − 1)(1 + z)2 +Ω 1/2 − 0 0 Λ Λ −1 (1 + Ω0z) − Ω0 −1 2(1 Ω0) t = H0 3/2 cosh +1 . MD/CD (B.20) (1 − Ω0)(1 + z) 2(1 − Ω0) Ω0(1 + z) The latter relation is often employed to relate the velocity to the density field. APPENDIX B. USEFUL QUANTITIES AND RELATIONS 195 APPENDIX B. USEFUL QUANTITIES AND RELATIONS 196
B.2.5 Optical Depth spatial metric. Moreover, in an open universe, it is not equivalent to the distance measured in conformal time. For an observer at the present, it is given by For the CMB, the optical depth τ to Compton scattering is a useful lookback time parameterization, r (η)=(−K)−1/2 sinh[(η − η)(−K)1/2]. (B.32) Z θ 0 η0 0 0 τ(a, a0)= dη xeneσT a Note that the argument of sinh is the difference in development angle χ in an open universe. η Z a0 0 3 Of particular interest is the angular diameter distance to the horizon rθ(0) since many × −2 − da H0 a0 =6.91 10 (1 Yp/2)xeΩbh 0 0 , (B.29) features in the CMB are generated early a a H a ( 2(Ω H )−1 Ω =0 for constant ionization fraction. If a aeq , this has closed form solution, 0 0 Λ rθ(0) ' (B.33) 2(Ω H2)−1/2(1 + ln Ω0.085). Ω +Ω =1 Ω h 0 0 0 Λ 0 τ(a, a )=4.61 × 10−2(1 − Y /2)x b 0 p e Ω2 ( 0 In the flat case, rθ(0) = η0. 1/2 2 − 3Ω0 +(1+Ω0z) (Ω0z +3Ω0 − 2) ΩΛ =0 A microphysical scale, the mean free path of a photon to Compton scattering, is × (B.30) 3 1/2 also of interest for the CMB, Ω0[1 − Ω0 +Ω0(1 + z) ] − Ω0. Ω0 +ΩΛ =1 −1 4 −1 2 −1 −2 Furthermore, since the optical depth is dominated by early contributions the distinction λC =(xeneσT a/a0) =4.3404 × 10 (1 − Yp/2) (xeΩbh ) (1 + z) Mpc. (B.34) between open and Λ universes for τ ∼> 1 is negligible. The diffusion length is roughly the geometric mean of λC and the horizon η. More precisely, it is given by equation (A.55) as B.3 Critical Scales Z 1 1 R2 +4f −1(1 + R)/5 λ2 ∼ k−2 = dη 2 . (B.35) D D 6 τ˙ (1 + R)2 B.3.1 Physical Scales where Several physical scales are also of interest. We always use comoving measures 1 isotropic, unpolarized when quoting distances. The most critical quantity is the horizon scale η given in the last − −1/2 − − 1/2 f2 = 9/10 unpolarized (B.36) section and the curvature scale ( K) = 2997.9h(1 Ω0 ΩΛ) Mpc. There are two related quantities of interest, the Hubble scale and the conformal angular diameter distance 3/4 polarized to the horizon. The Hubble scale is often employed instead of the horizon scale because it where isotropic means that the angular dependence of Compton scattering has been ne- is independent of the past evolution of the universe. The wavenumber corresponding to the glected, and the polarization case accounts for feedback from scattering induced polariza- Hubble scale is tion. Throughout the main text, we have used f = 1 for simplicity. If the diffusion scale 2 1/2 2 (ΩrH ) (a0/a)RD 0 is smaller than the sound horizon, acoustic oscillations will be present in the CMB. The 2 1/2 1/2 a˙ (Ω0H0 ) (a0/a) MD sound horizon is given by kH = = (B.31) s a (−K)1/2 CD Z √ p η 0 2 1 6 1+R +p R + Req (Ω H2)1/2a/a . ΛD rs = csdη = ln , (B.37) Λ 0 0 0 3 keq Req 1+ Req Comparison with the relations for η shows that kH η ∼ 1 during radiation and matter √ − −1 domination but not curvature or Λ domination. Indeed, due to the exponential expansion, which relates it to the horizon at equality ηeq =(4 2 2)keq ,where the Hubble scale goes to zero as a/a0 →∞, reflecting the fact that regions which were once 2 1/2 in causal contact can no longer communicate. This is of course how inflation solves the keq =(2Ω0H0 a0/aeq) × −2 2 − 1/2 −2 −1 horizon problem. Throughout the main text we have blurred the distinction between the =9.67 10 Ω0h (1 fν ) Θ2.7Mpc , Hubble scale and the horizon scale when discussing the radiation- and matter-dominated 2 −1 − 1/2 2 ηeq=12.1(Ω0h ) (1 fν) Θ2.7Mpc,(B.38) epochs. The distance inferred for an object of known spatial extent by its angular diameter with keq as the wavenumber that passes the Hubble scale at equality. is known as the conformal angular diameter distance. It multiplies the angular part of the APPENDIX B. USEFUL QUANTITIES AND RELATIONS 197 APPENDIX B. USEFUL QUANTITIES AND RELATIONS 198
B.3.2 Angular Scales B.4 Normalization Conventions A physical scale at η subtends an angle or equivalently a multipole on the sky ` B.4.1 Power Spectra −1 ` = krθ(η) ' θ ,` 1 (B.39) There are unfortunately a number of normalization conventions used in the litera- where the angle-distance relation rθ is given by equation (B.32). Three angular scales are ture and indeed several that run through the body of this work. Perhaps the most confusing of interest to the CMB. The sound horizon at last scattering determines the location of the conventions are associated with open universes. The power in fluctuations is expressed alter- acoustic peaks nately per√ logarithmic intervals of the Laplacian wavenumber k or the eigenfunction index ν = k/˜ −K, k˜ =(k2 + K)1/2. The relation between the two follows from the identity rθ(η∗) `A = π , kdk = kd˜ k˜, rs(η∗) ( ˜ k m` adiabatic P˜X(k)= PX(k), (B.47) A k˜ `p = 1 (B.40) (m − )`A, isocurvature 2 where PX is the power spectrum of fluctuations in X. For example, our power law spectra where ` is the location of the pth acoustic peak. If R∗ 1, ` takes on a simple form p A | |2 n−4 Φ(0,k) = Bk , 1/2 z∗ fG |S(0,k)|2 = Ckm, (B.48) `A = 172 3 , (B.41) 10 fR become where fR is the correction for the expansion during radiation domination 1/2 − 1/2 |˜ ˜ |2 − ˜2 (n−3)/2˜n−4 fR =(1+xR) xR , Φ(0, k) = B(1 K/k ) k , × −2 2 −1 − −1 4 3 | ˜ ˜ |2 − ˜2 m/2˜m xR = aeq /a∗ =2.38 10 (Ω0h ) (1 fν ) Θ2.7(z∗/10 ), (B.42) S(0, k) = C(1 K/k ) k . (B.49) and fG is the geometrical factor To add to the confusion, adiabatic fluctuations are often expressed in terms of the density ( 2 −1/2 power spectrum at present P(k)=|∆T (η0,k)| . The two conventions are related by the Ω0 ΩΛ =0 fG ' (B.43) Poisson equation, 0.085 3 a 1+lnΩ0 . ΩΛ +Ω0 =1 2 − 2 0 (k 3K)Φ = Ω0H0 (1 + aeq /a) ∆T . (B.50) The diffusion damping scale at last scattering subtends an angle given by 2 a To account for the growth between the initial conditions and the present, one notes that at `D = kD(η∗)rθ(η∗), (B.44) large scales (k → 0) the growth function is described by pressureless linear theory. From equations (A.8) and (A.9), where kD(η∗) is the effective damping scale at last scattering accounting for the recombi- § nation process. From 6.3.4, to order of magnitude it is 3 4 2 −1 D a ∆ (η ,k)= (Ω H2)−1 1+ f 1+ f (1 − 3K/k2) eq Φ(0,k). (B.51) ∼ 3 1/4 −1/4 −1/2 T 0 5 0 0 15 ν 5 ν D a `D 10 (Ωb/0.05) Ω0 fR fG, (B.45) eq 2 if Ωbh is low as required by nucleosynthesis. The scaling is only approximate since the If the neutrino anisotropic stress is neglected, drop the fν factors for consistency. Thus for n detailed physics of recombination complicates the calculation of kD (see Appendix A.2.2). a normalization convention of P(k)=Ak at large scales The curvature radius at the horizon distance (i.e. early times) subtends an angle given by ! √ 2 −2 2 9 2 −2 4 2 − 2 2 D aeq `K ' −Krθ(0) A = (Ω0H0 ) 1+ fν 1+ fν (1 3K/k ) B. (B.52) √ 25 15 5 Deq a 2 1 − Ω ' 0 . (B.46) Ω0 Notice that in an open universe, power law conditions for the potential do not imply power law conditions for the density, This relation is also not exact since for reasonable Ω0, the curvature scale subtends a large angle on the sky and the small angle approximation breaks down. Note also that at closer − P(k) ∝ (k2 − 3K)2kn 4, distances as is relevant for the late ISW effect, the curvature scale subtends an even larger ˜ ˜ ∝ ˜−1 ˜2 − −1 ˜2 − 2 ˜2 − (n−1)/2 angle on the sky than this relation predicts. P (k) k (k K) (k 4K) (k K) . (B.53) APPENDIX B. USEFUL QUANTITIES AND RELATIONS 199 APPENDIX B. USEFUL QUANTITIES AND RELATIONS 200
where T`(k) is the radiation transfer function from the solution to the Boltzmann equation. Experiment `0 `1 `2 Qflat(µK) Ref. Examples are given in §6. The power measured by a given experiment with a window COBE ––1819.9 1.6[62] function W` has an ensemble average value of FIRS ––3019 5[57] ∆T 2 1 X Ten. 20 13 30 26 6[70] = (2` +1)C W . (B.58) T 4π ` ` SP94 67 32 110 26 6[68] rms ` SK 69 42 100 29 6 [119] Pyth. 73 50 107 37 12 [49] Only if the whole sky is measured at high signal to noise does the variance follow the “cos- 2 ARGO 107 53 180 25 6[43] mic variance” prediction of a χ with 2` + 1 degrees of freedom. Real experiments make IAB 125 60 205 61 27 [131] noisy measurements of a fraction of the sky and therefore require a more detailed statistical MAX-2 (γUMi) 158 78 263 74 31 [2] treatment. To employ likelihood techniques, we must assume some underlying power spec- MAX-3 (γUMi) 158 78 263 50 11 [67] trum. In order to divorce the measurement from theoretical prejudice, experimental results MAX-4 (γUMi) 158 78 263 48 11 [44] are usually quoted with a model independent choice. The two most common conventions − 2 2 MAX-3 (µPeg) 158 78 263 19 8 [117] are the gaussian autocorrelation function Cgacf(θ)=C0exp( θ /2θc ) and the “flat” power MAX-4 (σHer) 158 78 263 39 8[32] spectrum motivated by the Sachs-Wolfe tail of adiabatic models (see e.g. [174]), MAX-4 (ιDra) 158 78 263 39 11 [32] 2 2 C`gacf =2πC0θ exp[−`(` +1)θ /2], MSAM2 143 69 234 40 14 [30] c c 2 MSAM3 249 152 362 39 12 [30] 24π Qflat −1 C`flat = [`(` +1)] . (B.59) 5 T0 Table B.3: Anisotropy Data Points The two power estimates are thus related by A compilation of anisotropy measurements from [146]. The experimental window function X X 2 6 2` +1 2 1 2 peaks at `0 and falls to half power at `1 and `2. Points are plotted in Fig. 1.3. Q W = C θ (2` + 1)exp[−`(` +1)θ /2]W . (B.60) flat 5 `(` +1) ` 0 c 2 c ` ` ` ˜ ˜ P(k) is the form most often quoted in the literature [175, 82, 134]. The current status of measurements is summarized in Tab. B.3 [146]. The power spectrum may also be expressed in terms of the bulk velocity field. At late times, pressure can be neglected and the total continuity equation (5.6) reduces to B.4.3 Large Scale Structure kVT = −∆˙ T a˙ d ln D Large scale structure measurements probe a smaller scale and have yet another set = − ∆ , (B.54) a d ln a T of normalization conventions based on the two point correlation function of astrophysical objects andinparticular 0 0 d ln D ξab(x)= δρa(x + x)δρb(x )/ρ¯aρ¯b . (B.61) kV (η ,k)=−H ∆ (η ,k), (B.55) T 0 0 d ln a T 0 η0 If all objects are clustered similarly, then all ξaa = ξ and the two-point correlation function or is related to the underlying power spectrum by d ln D 2 P (k) ≡|V (η ,k)|2 = H2 P(k), (B.56) Z V T 0 0 V dk √ d ln a η0 3 0 − ξ(r)= k P(k)Xν ( Kr) ' 0.6 2π2 k for the velocity power spectrum. Recall from equation (B.27) that d lnD/d ln a Ω0 in Z V dk sin(kr) an open universe. ' k3P(k) , (B.62) 2π2 k kr B.4.2 Anisotropies where the approximation assumes that scales of interest are well below the curvature scale. The normalization of theR power spectrum is often quoted by the Nth moment of the corre- r (N−1) The anisotropy power spectrum C` is given by lation function JN (r)= ξ(x)x dx which implies Z ( 0 k3|Φ(0,k)|2 adiabatic Z 2` +1 dk 2 × V dk C` = T` (k) (B.57) J (r)= P(k)(kr)2j (kr). (B.63) 4π k k3|S(0,k)|2, isocurvature 3 2π2 k 1 APPENDIX B. USEFUL QUANTITIES AND RELATIONS 201
−2 −1 For reference, j1(x)=x sinx − x cosx. Another normalization convention involves the rms density fluctuation in spheres of constant radii Z V dk 3j (kr) 2 σ2(r)= k3P(k) 1 . (B.64) 2π2 k kr The observed galaxy distribution implies that
−1 ' −3 3 J3(10h Mpc) (270h Mpc (B.65) 1.1 0.15 optical [109] −1 σ8 ≡ σ(8h Mpc) = (B.66) 0.69 0.04. IRAS [55]
The discrepancy between estimates of the normalization obtained by different populations of objects implies that they may all be biased tracers of the underlying mass. The simplest 2 model for bias assumes ξaa = baξ with constant b. Peacock & Dodds [122] find that the best fit to the Abell cluster (A), radio galaxy (R), optical galaxy (O), and IRAS galaxy (I) data sets yields bA : bR : bO : bI =4.5:1.9:1.3:1.