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<zK/8 Compton-y era 4 ργ 4 Ω0 aeq − − 3 −1/2 2 −1/2 1/2 2 4 3 1 zK =7.09 × 10 (1 − Yp/2) (xeΩbh ) Θ Comptonization epoch =31.5Ωbh Θ2.7(z/10 ) . (B.8) 2.7 × 5 − −2/5 2 −2/5 1/5 zµ,dc =4.09 10 (1 Yp/2) (xeΩbh ) Θ2.7 Dbl. Compton therm. epoch Epochs of interest for the CMB are listed in Tab. B.2 by their redshifts. × 4 − −4/5 2 −6/5 13/5 zµ,br =5.60 10 (1 Yp/2) (xeΩbh ) Θ2.7 Bremss. therm. epoch Θ = T /2.7K' 1.01 Temperature Scaling B.2.2 Conformal Time 2.7 0 R Yp =4nHe/nb ' 0.23 Helium mass fraction By definition, the conformal time η = dt/a is related to the scale factor as −1 (1 − fν ) =1+ρν/ργ → 1.68132 Neutrino density correction Z da 1 a 2 1/2 η = 0 . (B.9) keq =(2Ω0H0 a0/aeq ) Equality Hubble wavenumber a H a × −2 2 − 1/2 −2 −1 =9.67 10 Ω0h (1 fν ) Θ2.7Mpc Note that in these c = 1 units, the conformal time doubles as the comoving size of the −1 − − −1 fcool = xe [(1 + xe)/2 (3 + 2xe)Yp/8](1 Yp/2) Cooling correction factor horizon. In an open universe, it is also related to the development angle by √ χ = −Kη. (B.10) Table B.2: Critical Redshifts Critical epochs are also denoted as the corresponding value in the coordinate time t,scale Asymptotic relations are often useful for converting values. Before curvature or Λ domina- factor a, and conformal time η. The neutrino fraction fν is given for three families of tion, the conformal time massless neutrinos and the standard thermal history. The Hubble crossing redshift zH is given for the matter and radiation dominated epochs. √ q 2 2 η = 1+a/aeq − 1 keq q 2 −1/2 1/2 − =2(Ω0H0 ) (aeq/a0) 1+a/aeq 1 , (B.11) APPENDIXB.USEFULQUANTITIESANDRELATIONS193 APPENDIX B. USEFUL QUANTITIES AND RELATIONS 194 andreducesto ( In particular, the age of the universe today is 2 −1/2 (ΩrH0) a/a0RD η= (B.12) − − Ω0 − 2(Ω H2 )−1/2(a/a )1/2,MD t = H 1(1 − Ω ) 1 1 − (1 − Ω ) 1/2 cosh(2/Ω − 1) , Ω = 0 (B.21) 0 0 0 0 0 0 2 0 0 Λ whereΩr/Ω0'aeq/a0.InaΛ=0universe,italsohasanasymptoticsolutionforaaeq where the factor in square brackets goes to unity as Ω0 → 0. This should be compared with 1 − 2(1−Ω0) a η= √ cosh 1 1+ MD/CD the flat Ω0 +ΩΛ =1result −K Ω0 a0 " √ # −1/2 2 − − 1+ 1 − Ω0 lim η0→(−K) ln(4/Ω0),(B.13) 1 − 1/2 √ Ω0→0 t0 = H0 (1 Ω0) ln , Ω0 +ΩΛ =1, (B.22) 3 Ω0 andthusthehorizonscaleislargerthanthecurvaturescale(−K)−1/2forlowΩ universes. 0 which diverges logarithmically as Ω → 0. Finally a microphysical time scale of interest for Inaflatuniverse, 0 the CMB, ' 2 −1/2 0.085 η0 2(Ω0H0) (1+lnΩ0 ),Ω0+ΩΛ=1(B.14) −1 −1 tC =(dτ/dt) =(xeneσT ) − −1 1/3 − −1/6 →∞ 18 −1 2 −1 −3 andthehorizongoestoaconstantη=2.8 H0 Ω0 (1 Ω0) asa/a0 . =4.4674 × 10 (1 − Yp/2) (xeΩbh ) (1 + z) s, (B.23) 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.