Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — Essential Formulas How to read this list
Equation numbers (N.m) are the book's own numbering (stable across printings).
Page numbers refer to this PDF's page index , not the printed Cambridge page numbers
(the file is a calibre EPUB→PDF conversion, so the two differ).
Quick reference: Ch 2-6 Formulas , Ch 7-12 Formulas , Ch 3-7 Notes , Ch 8-12 Notes
Table of Useful Constants Transcribed from the book's end-matter (PDF pp. 362–363) .
Fundamental constants
Quantity
Value
Gravitational constant $G$
$6.673\times10^{-11}\ \mathrm{m^3,kg^{-1},s^{-2}}$
Speed of light $c$
$2.998\times10^{8}\ \mathrm{m,s^{-1}}$
Reduced Planck constant $\hbar$
$1.055\times10^{-34}\ \mathrm{J,s} = 6.582\times10^{-16}\ \mathrm{eV,s}$
Boltzmann constant $k$
$1.381\times10^{-23}\ \mathrm{J,K^{-1}} = 8.617\times10^{-5}\ \mathrm{eV,K^{-1}}$
Electron rest energy $m_e c^2$
$0.5110\ \mathrm{MeV}$
Proton rest energy $m_p c^2$
$938.272\ \mathrm{MeV}$
Neutron rest energy $m_n c^2$
$939.566\ \mathrm{MeV}$
Planck units
Quantity
Value
Planck length $\ell_P=(G\hbar/c^3)^{1/2}$
$1.616\times10^{-35}\ \mathrm{m}$
Planck mass $M_P=(\hbar c/G)^{1/2}$
$2.177\times10^{-8}\ \mathrm{kg}$
Planck time $t_P=(G\hbar/c^5)^{1/2}$
$5.391\times10^{-44}\ \mathrm{s}$
Planck energy $E_P=(\hbar c^5/G)^{1/2}$
$1.956\times10^{9}\ \mathrm{J} = 1.221\times10^{28}\ \mathrm{eV}$
Planck temperature $T_P=E_P/k$
$1.417\times10^{32}\ \mathrm{K}$
Conversion of units
Quantity
Value
Astronomical unit
$1\ \mathrm{AU} = 1.496\times10^{11}\ \mathrm{m}$
Megaparsec
$1\ \mathrm{Mpc} = 3.086\times10^{22}\ \mathrm{m}$
Solar mass
$1\ M_\odot = 1.989\times10^{30}\ \mathrm{kg}$
Solar luminosity
$1\ L_\odot = 3.828\times10^{26}\ \mathrm{J,s^{-1}}$
Gigayear
$1\ \mathrm{Gyr} = 3.156\times10^{16}\ \mathrm{s}$
Electron volt
$1\ \mathrm{eV} = 1.602\times10^{-19}\ \mathrm{J}$
Cosmological parameters (the book's adopted values)
Quantity
Value
Hubble constant $H_0$
$68 \pm 2\ \mathrm{km,s^{-1},Mpc^{-1}}$
Hubble time $H_0^{-1}$
$(4.54\pm0.13)\times10^{17}\ \mathrm{s} = 14.4\pm0.4\ \mathrm{Gyr}$
Hubble distance $c/H_0$
$(1.35\pm0.04)\times10^{26}\ \mathrm{m} = 4380\pm130\ \mathrm{Mpc}$
Critical energy density $\varepsilon_{c,0}$
$4870 \pm 290\ \mathrm{MeV,m^{-3}}$
Critical mass density $\rho_{c,0}=\varepsilon_{c,0}/c^2$
$(8.7\pm0.5)\times10^{-27}\ \mathrm{kg,m^{-3}}$
The 160 landmark (★) equations, condensed for fast revision — equation number, a one-line tag, the page, and the formula. Full context for each is in the chapter sections below (same number).
(1.1) Planck length: the unique length scale built from $G$ — p. 16
$$
\ell_P \equiv \left(\frac{G\hbar}{c^3}\right)^{1/2} = 1.62\times10^{-35},\mathrm{m}.
$$
(1.2) Planck mass: the mass scale formed from the same constants — p. 16
$$
M_P \equiv \left(\frac{\hbar c}{G}\right)^{1/2} = 2.18\times10^{-8},\mathrm{kg}.
$$
(1.3) Planck time: the time for light to cross a Planck length — p. 16
$$
t_P \equiv \left(\frac{G\hbar}{c^5}\right)^{1/2} = 5.39\times10^{-44},\mathrm{s}.
$$
(1.4) Planck energy: rest energy of the Planck mass — p. 16
$$
E_P = M_P c^2 = 1.96\times10^{9},\mathrm{J} = 1.22\times10^{28},\mathrm{eV}.
$$
(1.5) Planck temperature: temperature whose thermal energy… — p. 16
$$
T_P = E_P/k = 1.42\times10^{32},\mathrm{K}.
$$
Ch 2 · Fundamental Observations (2.6) surface brightness of a star is independent of distance — p. 22
$$
\Sigma_\star = \frac{f}{\Omega} = \frac{L_\star}{\pi R_\star^2} ,
$$
(2.7) definition of redshift $z$ in terms of observed and… — p. 28
$$
z \equiv \frac{\lambda_{\mathrm{ob}} - \lambda_{\mathrm{em}}}{\lambda_{\mathrm{em}}} .
$$
(2.8) Hubble's law, redshift form — p. 29
$$
z = \frac{H_0}{c}, r ,
$$
(2.9) Hubble's law, velocity form — p. 29
$$
v = H_0, r .
$$
(2.10) best current value of the Hubble constant — p. 30
$$
H_0 = 68 \pm 2\ \mathrm{km,s^{-1},Mpc^{-1}} .
$$
(2.19) Hubble time: elapsed time since galaxies were in contact — p. 33
$$
t_0 = \frac{r}{v} = \frac{r}{H_0 r} = H_0^{-1} ,
$$
(2.27) Planck blackbody spectrum: energy density of photons in… — p. 39
$$
\varepsilon(f),df = \frac{8\pi h}{c^3}, \frac{f^3,df}{\exp(hf/kT) - 1} ,
$$
(2.28) total energy density of blackbody radiation — p. 40
$$
\varepsilon_\gamma = \alpha T^4 ,
$$
(2.31) total number density of blackbody photons — p. 41
$$
n_\gamma = \beta T^3 ,
$$
(2.33) present-day CMB blackbody temperature — p. 42
$$
T_0 = 2.7255 \pm 0.0006\ \mathrm{K} .
$$
(2.34) present-day energy density of the CMB — p. 42
$$
\varepsilon_\gamma = 4.175 \times 10^{-14}\ \mathrm{J,m^{-3}} = 0.2606\ \mathrm{MeV,m^{-3}} .
$$
(2.35) present-day number density of CMB photons — p. 42
$$
n_\gamma = 4.107 \times 10^8\ \mathrm{m^{-3}} .
$$
(2.41) with… — p. 44
$$
\frac{d}{dt}(\ln T) = -\frac{d}{dt}(\ln a) .
$$
Ch 3 · Newton versus Einstein (3.11) Lorentz transformation between two inertial frames in… — p. 55
$$
\begin{aligned}
x' &= \gamma(x - vt) \\
y' &= y \\
z' &= z \\
t' &= \gamma!\left(t - vx/c^2\right),
\end{aligned}
$$
(3.12) Definition of the Lorentz factor — p. 55
$$
\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}.
$$
(3.17) Spacetime separation — p. 56
$$
(\Delta s)^2 = -c^2(t_1 - t_2)^2 + (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2.
$$
(3.18) Spacetime interval in compact form — p. 56
$$
(\Delta s)^2 = -c^2(\Delta t)^2 + (\Delta\ell)^2.
$$
(3.25) Angle sum of a geodesic triangle on a sphere of radius $R$ — p. 63
$$
\alpha + \beta + \gamma = \pi + A/R^2,
$$
(3.33) Compact unified metric for a homogeneous — p. 66
$$
d\ell^2 = dr^2 + S_\kappa(r)^2, d\Omega^2,
$$
(3.35) The function… — p. 67
$$
S_\kappa(r) =
\begin{cases}
R\sin(r/R) & (\kappa = +1) \\
r & (\kappa = 0) \\
R\sinh(r/R) & (\kappa = -1).
\end{cases}
$$
(3.37) The Minkowski metric: spacetime interval in flat, static — p. 68
$$
ds^2 = -c^2 dt^2 + dr^2 + r^2 d\Omega^2.
$$
(3.41) The Robertson–Walker metric: spacetime interval for a… — p. 69
$$
ds^2 = -c^2 dt^2 + a(t)^2\left[dr^2 + S_\kappa(r)^2, d\Omega^2\right],
$$
(3.44) Proper distance to a galaxy at comoving coordinate $r$ — p. 71
$$
d_p(t) = a(t)\int_0^r dr = a(t), r.
$$
(3.46) Hubble's law: linear relation between recession speed and… — p. 71
$$
v_p(t_0) = H_0, d_p(t_0),
$$
(3.48) Definition of the Hubble constant as the present-day… — p. 71
$$
H_0 = \left(\frac{\dot{a}}{a}\right)_{t=t_0}.
$$
(3.49) Definition of the Hubble distance — p. 72
$$
d_H(t_0) \equiv c/H_0,
$$
(3.61) Redshift–scale-factor relation — p. 75
$$
1 + z = \frac{a(t_0)}{a(t_e)} = \frac{1}{a(t_e)}.
$$
(4.7) Einstein's field equation relating spacetime curvature to… — p. 79
$$
G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}.
$$
(4.8) the homogeneous, isotropic Robertson–Walker metric — p. 80
$$
ds^2 = -c^2 dt^2 + a(t)^2\left[dr^2 + S_\kappa(r)^2, d\Omega^2\right],
$$
(4.18) the Friedmann equation in its Newtonian form — p. 83
$$
\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho(t) + \frac{2U}{r_s^2}\frac{1}{a(t)^2}.
$$
(4.20) the Friedmann equation in its correct relativistic form — p. 85
$$
\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon(t) - \frac{\kappa c^2}{R_0^2}\frac{1}{a(t)^2}.
$$
(4.30) definition of the critical density — p. 87
$$
\varepsilon_c(t) \equiv \frac{3c^2}{8\pi G} H(t)^2.
$$
(4.33) definition of the dimensionless density parameter $\Omega$ — p. 88
$$
\Omega(t) \equiv \frac{\varepsilon(t)}{\varepsilon_c(t)}.
$$
(4.44) the fluid equation — p. 91
$$
\dot{\varepsilon} + 3\frac{\dot a}{a}(\varepsilon + P) = 0.
$$
(4.49) the acceleration equation — p. 92
$$
\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}(\varepsilon + 3P).
$$
(4.55) the linear equation of state — p. 94
$$
P = w\varepsilon,
$$
(4.61) equation of state for photons / any relativistic gas — p. 95
$$
P_{\mathrm{rel}} = \frac{1}{3}\varepsilon_{\mathrm{rel}}.
$$
(4.66) the Friedmann equation with the cosmological constant… — p. 98
$$
\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon - \frac{\kappa c^2}{R_0^2 a^2} + \frac{\Lambda}{3}.
$$
(4.68) the acceleration equation with the $\Lambda$ term — p. 98
$$
\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}(\varepsilon + 3P) + \frac{\Lambda}{3}.
$$
(4.69) energy density associated with the cosmological constant — p. 98
$$
\varepsilon_\Lambda \equiv \frac{c^2}{8\pi G}\Lambda.
$$
(4.70) pressure of the cosmological constant — p. 99
$$
P_\Lambda = -\varepsilon_\Lambda = -\frac{c^2}{8\pi G}\Lambda.
$$
(5.1) Friedmann equation relating expansion rate to energy… — p. 105
$$
\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon - \frac{\kappa c^2}{R_0^2 a^2},
$$
(5.2) Fluid equation governing evolution of energy density — p. 105
$$
\dot{\varepsilon} + 3\frac{\dot{a}}{a}(\varepsilon + P) = 0,
$$
(5.3) Equation of state relating pressure to energy density — p. 105
$$
P = w\varepsilon.
$$
(5.9) General energy-density scaling for a component with… — p. 107
$$
\varepsilon_i(a) = \varepsilon_{i,0}, a^{-3(1+w_i)}.
$$
(5.10) Matter energy density scales as $a^{-3}$ — p. 107
$$
\varepsilon_m(a) = \varepsilon_{m,0}/a^3.
$$
(5.11) Radiation energy density scales as $a^{-4}$ — p. 107
$$
\varepsilon_r(a) = \varepsilon_{r,0}/a^4.
$$
(5.21) Scale factor at matter–$\Lambda$ equality — p. 111
$$
a_{m\Lambda} = \left(\frac{\Omega_{m,0}}{\Omega_{\Lambda,0}}\right)^{1/3} \approx \left(\frac{0.31}{0.69}\right)^{1/3} \approx 0.766.
$$
(5.24) Scale factor at radiation–matter equality — p. 112
$$
a_{rm} = \frac{\varepsilon_{r,0}}{\varepsilon_{m,0}} \approx \frac{1}{3400} \approx 2.9 \times 10^{-4}.
$$
(5.25) Friedmann equation for a multiple-component universe — p. 113
$$
\dot{a}^2 = \frac{8\pi G}{3c^2}\sum_i \varepsilon_{i,0}, a^{-1-3w_i} - \frac{\kappa c^2}{R_0^2}.
$$
(5.28) Scale factor grows linearly with time in an expanding… — p. 114
$$
a(t) = \frac{t}{t_0},
$$
(5.37) Friedmann equation for a spatially flat — p. 117
$$
\dot{a}^2 = \frac{8\pi G \varepsilon_0}{3c^2},a^{-(1+3w)}.
$$
(5.39) Scale factor in a flat, single-component universe — p. 118
$$
a(t) = \left(\frac{t}{t_0}\right)^{2/(3+3w)}.
$$
(5.52) Horizon distance in a flat, single-component universe — p. 120
$$
d_{\mathrm{hor}}(t_0) = ct_0\frac{3(1+w)}{1+3w} = \frac{c}{H_0}\frac{2}{1+3w}.
$$
(5.53) Age of a flat, matter-only — p. 121
$$
t_0 = \frac{2}{3H_0},
$$
(5.55) Scale factor in a flat, matter-only — p. 121
$$
a_m(t) = \left(\frac{t}{t_0}\right)^{2/3},
$$
(5.58) Age of a flat, radiation-only universe — p. 122
$$
t_0 = \frac{1}{2H_0},
$$
(5.60) Scale factor in a flat, radiation-only universe — p. 123
$$
a(t) = \left(\frac{t}{t_0}\right)^{1/2},
$$
(5.70) Friedmann equation for a flat, $\Lambda$ -only universe — p. 125
$$
\dot{a}^2 = \frac{8\pi G \varepsilon_\Lambda}{3c^2},a^2,
$$
(5.73) Scale factor grows exponentially in a flat, $\Lambda$ -only — p. 126
$$
a(t) = e^{H_0(t - t_0)}.
$$
(5.76) Friedmann equation written with the Hubble parameter and… — p. 127
$$
H(t)^2 = \frac{8\pi G}{3c^2}\varepsilon(t) - \frac{\kappa c^2}{R_0^2 a(t)^2},
$$
(5.79) Dimensionless Friedmann equation — p. 128
$$
\frac{H(t)^2}{H_0^2} = \frac{\varepsilon(t)}{\varepsilon_{c,0}} + \frac{1 - \Omega_0}{a(t)^2},
$$
(5.81) Friedmann equation for our universe in terms of density… — p. 128
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{r,0}}{a^4} + \frac{\Omega_{m,0}}{a^3} + \Omega_{\Lambda,0} + \frac{1 - \Omega_0}{a^2},
$$
(5.83) Cosmic time as a function of scale factor — p. 129
$$
\int_0^a \frac{da}{\left[\Omega_{r,0}/a^2 + \Omega_{m,0}/a + \Omega_{\Lambda,0}a^2 + (1 - \Omega_0)\right]^{1/2}} = H_0 t.
$$
(5.87) Scale factor at maximum expansion — p. 131
$$
a_{\mathrm{max}} = \frac{\Omega_0}{\Omega_0 - 1},
$$
(5.92) Time until the Big Crunch for a closed — p. 133
$$
t_{\mathrm{crunch}} = \frac{\pi}{H_0}\frac{\Omega_0}{(\Omega_0 - 1)^{3/2}}.
$$
(5.96) Friedmann equation for a flat matter+$\Lambda$ universe — p. 135
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{m,0}}{a^3} + (1 - \Omega_{m,0}).
$$
(5.100) Scale factor at matter–$\Lambda$ equality in a flat… — p. 137
$$
a_{m\Lambda} = \left(\frac{\Omega_{m,0}}{\Omega_{\Lambda,0}}\right)^{1/3} = \left(\frac{\Omega_{m,0}}{1 - \Omega_{m,0}}\right)^{1/3}.
$$
(5.101) Analytic solution… — p. 137
$$
H_0 t = \frac{2}{3\sqrt{1 - \Omega_{m,0}}}\ln\left[\left(\frac{a}{a_{m\Lambda}}\right)^{3/2} + \sqrt{1 + \left(\frac{a}{a_{m\Lambda}}\right)^3}\right].
$$
(5.104) Age of a flat matter+$\Lambda$ universe — p. 138
$$
t_0 = \frac{2H_0^{-1}}{3\sqrt{1 - \Omega_{m,0}}}\ln\left[\frac{\sqrt{1 - \Omega_{m,0}} + 1}{\sqrt{\Omega_{m,0}}}\right].
$$
(5.105) Age of the Benchmark Model universe — p. 138
$$
t_0 = 0.955 H_0^{-1} = 13.74 \pm 0.40,\mathrm{Gyr},
$$
(5.107) Friedmann equation for a curved universe with matter and… — p. 139
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{m,0}}{a^3} + \frac{1 - \Omega_{m,0} - \Omega_{\Lambda,0}}{a^2} + \Omega_{\Lambda,0}.
$$
(5.108) Friedmann equation near radiation–matter equality — p. 142
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{r,0}}{a^4} + \frac{\Omega_{m,0}}{a^3}.
$$
(5.114) Time of radiation–matter equality in the Benchmark Model — p. 143
$$
t_{rm} = 3.47 \times 10^{-6},H_0^{-1} = 50,000,\mathrm{yr}.
$$
(5.115) Horizon distance of the Benchmark Model — p. 147
$$
d_{\mathrm{hor}}(t_0) = 3.20 c/H_0 = 3.35 ct_0 = 14,000,\mathrm{Mpc}.
$$
Ch 6 · Measuring Cosmological Parameters (6.4) the "two-number" expansion of… — p. 154
$$
a(t) \approx 1 + H_0(t-t_0) - \frac{1}{2}q_0 H_0^2 (t-t_0)^2.
$$
(6.5) definition of the Hubble constant — p. 154
$$
H_0 \equiv \frac{\dot a}{a}\bigg|_{t=t_0},
$$
(6.6) definition of the deceleration parameter $q_0$ — p. 154
$$
q_0 \equiv -\left(\frac{\ddot a, a}{\dot a^2}\right)_{t=t_0} = -\left(\frac{\ddot a}{aH^2}\right)_{t=t_0}.
$$
(6.11) $q_0$ for a universe of radiation — p. 156
$$
q_0 = \Omega_{r,0} + \frac{1}{2}\Omega_{m,0} - \Omega_{\Lambda,0}.
$$
(6.12) Hubble's law: the linear distance–redshift relation at… — p. 156
$$
cz = H_0 d.
$$
(6.16) redshift in terms of the scale factor at emission — p. 158
$$
z = \frac{1}{a(t_e)} - 1.
$$
(6.21) definition of the luminosity distance from luminosity $L$ … — p. 160
$$
d_L \equiv \left(\frac{L}{4\pi f}\right)^{1/2}.
$$
(6.27) observed flux–luminosity relation in an expanding — p. 162
$$
f = \frac{L}{4\pi S_\kappa(r)^2(1+z)^2},
$$
(6.28) luminosity distance in terms of the curvature function… — p. 163
$$
d_L = S_\kappa(r)(1+z).
$$
(6.31) low-$z$ approximation of the luminosity distance in a… — p. 164
$$
d_L \approx \frac{c}{H_0}z\left(1 - \frac{1+q_0}{2}z\right)(1+z) \approx \frac{c}{H_0}z\left(1 + \frac{1-q_0}{2}z\right).
$$
(6.32) definition of the angular-diameter distance from proper… — p. 165
$$
d_A \equiv \frac{\ell}{\delta\theta}.
$$
(6.35) angular-diameter distance in terms of the curvature… — p. 166
$$
d_A \equiv \frac{\ell}{\delta\theta} = \frac{S_\kappa(r)}{1+z}.
$$
(6.36) relation between angular-diameter distance and luminosity… — p. 166
$$
d_A = \frac{d_L}{(1+z)^2}.
$$
(6.45) definition of bolometric apparent magnitude — p. 174
$$
m \equiv -2.5\log_{10}(f/f_x),
$$
(6.46) definition of bolometric absolute magnitude — p. 174
$$
M \equiv -2.5\log_{10}(L/L_x),
$$
(6.49) distance modulus $m-M$ in terms of the luminosity distance — p. 175
$$
m - M = 5\log_{10}\left(\frac{d_L}{1,\mathrm{Mpc}}\right) + 25.
$$
(7.10) orbital speed as a function of enclosed mass — p. 188
$$
v = \sqrt{\frac{GM(R)}{R}}.
$$
(7.12) mass of a spiral galaxy — p. 189
$$
M(R) = \frac{v^2 R}{G} = 1.05 \times 10^{11},\mathrm{M_\odot} \left(\frac{v}{235,\mathrm{km,s^{-1}}}\right)^{2} \left(\frac{R}{8.2,\mathrm{kpc}}\right).
$$
(7.27) the virial theorem — p. 193
$$
\ddot{I} = 2W + 4K.
$$
(7.28) steady-state virial theorem — p. 194
$$
0 = W + 2K,
$$
(7.31) virial mass of a self-gravitating steady-state system — p. 194
$$
M = \frac{\langle v^2 \rangle r_h}{\alpha G}.
$$
(7.34) line-of-sight velocity dispersion of the Coma cluster — p. 195
$$
\sigma_r = \left\langle (v_r - \langle v_r \rangle)^2 \right\rangle^{1/2} = 880,\mathrm{km,s^{-1}}.
$$
(7.43) deflection angle of light passing a compact mass M at… — p. 198
$$
\alpha = \frac{4GM}{c^2 b},
$$
(7.45) Einstein radius — p. 199
$$
\theta_E = \left(\frac{4GM}{c^2 d},\frac{1-x}{x}\right)^{1/2},
$$
Ch 8 · The Cosmic Microwave Background (8.5) baryon-to-photon ratio η — p. 208
$$
\eta = \frac{n_{\mathrm{bary},0}}{n_{\gamma,0}} \approx \frac{0.25,\mathrm{m^{-3}}}{4.107 \times 10^{8},\mathrm{m^{-3}}} \approx 6.1 \times 10^{-10}
$$
(8.6) mean — p. 212
$$
\langle T \rangle = \frac{1}{4\pi} \int T(\theta,\phi),\sin\theta,d\theta,d\phi = 2.7255,\mathrm{K}
$$
(8.7) definition of the dimensionless temperature fluctuation — p. 212
$$
\frac{\delta T}{T}(\theta,\phi) \equiv \frac{T(\theta,\phi) - \langle T \rangle}{\langle T \rangle}
$$
(8.9) definition of the fractional ionization X — p. 214
$$
X \equiv \frac{n_p}{n_p + n_{\mathrm{H}}} = \frac{n_p}{n_{\mathrm{bary}}} = \frac{n_e}{n_{\mathrm{bary}}}
$$
(8.29) the Saha equation — p. 222
$$
\frac{n_{\mathrm{H}}}{n_p n_e} = \left(\frac{m_e kT}{2\pi \hbar^2}\right)^{-3/2} \exp\left(\frac{Q}{kT}\right)
$$
(8.37) recombination temperature, defined at $X=\tfrac12$ — p. 223
$$
kT_{\mathrm{rec}} = 0.324,\mathrm{eV} = \frac{Q}{42}
$$
(8.50) spherical-harmonic expansion of the temperature… — p. 228
$$
\frac{\delta T}{T}(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} a_{lm} Y_{lm}(\theta,\phi)
$$
(8.52) correlation function as a multipole — p. 229
$$
C(\theta) = \frac{1}{4\pi}\sum_{l=0}^{\infty}(2l+1),C_l, P_l(\cos\theta)
$$
(8.63) the Sachs–Wolfe effect — p. 233
$$
\frac{\delta T}{T} = \frac{1}{3}\frac{\delta\Phi}{c^2}
$$
(8.64) sound horizon distance at last scattering — p. 234
$$
d_s(t_{\mathrm{ls}}) = a(t_{\mathrm{ls}}) \int_0^{t_{\mathrm{ls}}} \frac{c_s(t),dt}{a(t)}
$$
Ch 9 · Nucleosynthesis and the Early Universe (9.4) binding energy of deuterium: fusing a proton and neutron… — p. 241
$$
p + n \rightleftharpoons \mathrm{D} + 2.22,\mathrm{MeV}.
$$
(9.5) definition of the primordial helium mass fraction $Y_p$ — p. 243
$$
Y_p \equiv \frac{\rho(^4\mathrm{He})}{\rho_{\mathrm{bary}}}.
$$
(9.6) neutron–proton rest-energy difference $Q_n$ — p. 244
$$
Q_n = m_n c^2 - m_p c^2 = 1.29,\mathrm{MeV}.
$$
(9.7) free-neutron beta decay into a proton — p. 244
$$
n \rightarrow p + e^- + \bar{\nu}_e,
$$
(9.14) equilibrium neutron-to-proton ratio in simple form — p. 245
$$
\frac{n_n}{n_p} = \exp\left(-\frac{Q_n}{kT}\right),
$$
(9.17) frozen-out neutron-to-proton ratio at… — p. 247
$$
\frac{n_n}{n_p} = \exp\left(-\frac{Q_n}{kT_{\mathrm{freeze}}}\right) \approx \exp\left(-\frac{1.29,\mathrm{MeV}}{0.8,\mathrm{MeV}}\right) \approx 0.2.
$$
(9.18) deuteron formation by neutron–proton fusion — p. 248
$$
p + n \rightleftharpoons \mathrm{D} + \gamma.
$$
(9.21) maximum possible primordial helium mass fraction for… — p. 249
$$
Y_{\max} = \frac{4}{12} = \frac{1}{3}.
$$
(9.23) binding energy of the deuteron as a mass defect — p. 250
$$
B_{\mathrm{D}} = (m_n + m_p - m_{\mathrm{D}})c^2 = 2.22,\mathrm{MeV}.
$$
(9.25) Saha equation for hydrogen recombination — p. 251
$$
\frac{n_H}{n_p n_e} = \left(\frac{m_e kT}{2\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{Q}{kT}\right),
$$
(9.26) nucleosynthetic — p. 251
$$
\frac{n_{\mathrm{D}}}{n_p n_n} = \frac{g_{\mathrm{D}}}{g_p g_n} \left(\frac{m_{\mathrm{D}}}{m_p m_n}\right)^{3/2} \left(\frac{kT}{2\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{[m_p + m_n - m_{\mathrm{D}}]c^2}{kT}\right).
$$
(9.27) simplified nucleosynthetic Saha equation using… — p. 252
$$
\frac{n_{\mathrm{D}}}{n_p n_n} = 6 \left(\frac{m_n kT}{\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{B_{\mathrm{D}}}{kT}\right),
$$
(9.30) deuteron-to-neutron ratio as a function of temperature… — p. 253
$$
\frac{n_{\mathrm{D}}}{n_n} \approx 6.5, \eta \left(\frac{kT}{m_n c^2}\right)^{3/2} \exp\left(\frac{B_{\mathrm{D}}}{kT}\right).
$$
(9.32) neutron-to-proton ratio reduced by neutron decay during… — p. 254
$$
\frac{n_n}{n_p} \approx \frac{\exp(-200/880)}{5 + [1 - \exp(-200/880)]} \approx \frac{0.80}{5.20} \approx 0.15.
$$
(9.46) definition of the quark–antiquark asymmetry $\delta_q$ — p. 261
$$
\delta_q \equiv \frac{n_q - n_{\bar{q}}}{n_q + n_{\bar{q}}} \ll 1.
$$
(9.47) residual quark-to-photon ratio set by the asymmetry — p. 261
$$
\frac{n_q}{n_\gamma} \sim \delta_q.
$$
(9.48) general maximum primordial helium mass fraction as a… — p. 263
$$
Y_{\max} = \frac{2f}{1 + f},
$$
Ch 10 · Inflation and the Very Early Universe (10.1) Friedmann equation cast as the link between curvature and… — p. 265
$$
1 - \Omega(t) = -\kappa \left( \frac{c/H(t)}{a(t)R_0} \right)^{2}.
$$
(10.7) flatness problem: the deviation grows in the… — p. 266
$$
|1 - \Omega|_r \propto a^{2} \propto t.
$$
(10.17) acceleration equation; inflation — p. 274
$$
\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^{2}}(\varepsilon + 3P).
$$
(10.20) de Sitter exponential expansion during inflation,… — p. 274
$$
a(t) \propto e^{H_i t}.
$$
(10.23) definition of the number of e-foldings of inflation — p. 275
$$
N \equiv H_i(t_f - t_i).
$$
(10.32) horizon distance at any time $t$ — p. 277
$$
d_{\mathrm{hor}}(t) = a(t),c \int_0^{t} \frac{dt}{a(t)}.
$$
(10.40) energy density of a homogeneous inflaton field — p. 281
$$
\varepsilon_{\phi} = \frac{1}{2}\frac{1}{\hbar c^{3}}\dot{\phi}^{2} + V(\phi).
$$
(10.42) slow-roll condition — p. 282
$$
\dot{\phi}^{2} \ll \hbar c^{3} V(\phi).
$$
(10.45) equation of motion of the inflaton field — p. 282
$$
\ddot{\phi} + 3H(t)\dot{\phi} = -\hbar c^{3}\frac{dV}{d\phi}.
$$
Ch 11 · Structure Formation: Gravitational Instability (11.2) definition of the dimensionless density fluctuation — p. 293
$$
\delta(\vec r,t) \equiv \frac{\varepsilon(\vec r,t) - \bar\varepsilon(t)}{\bar\varepsilon(t)}.
$$
(11.13) dynamical time for gravitational collapse — p. 296
$$
t_{\rm dyn} = \frac{1}{(4\pi G\bar\rho)^{1/2}} \approx 9.6,\text{hours}\left(\frac{\bar\rho}{1,\mathrm{kg,m^{-3}}}\right)^{-1/2}.
$$
(11.20) Jeans length — p. 298
$$
\lambda_J = c_s\left(\frac{\pi c^2}{G\bar\varepsilon}\right)^{1/2} = 2\pi c_s t_{\rm dyn}.
$$
(11.27) definition of the baryonic Jeans mass — p. 300
$$
M_J \equiv \rho_{\rm bary}\left(\frac{4\pi}{3}\lambda_J^3\right).
$$
(11.44) growth equation for small perturbations in an expanding… — p. 304
$$
\ddot\delta + 2H\dot\delta = 4\pi G\bar\rho,\delta,
$$
(11.49) growth equation for matter perturbations in terms of Ωₘ — p. 305
$$
\ddot\delta + 2H\dot\delta - \frac{3}{2}\Omega_m H^2\delta = 0.
$$
(11.58) growing mode: perturbations grow as the scale factor in… — p. 307
$$
\delta \propto t^{2/3} \propto a(t) \propto \frac{1}{1+z}.
$$
(11.65) definition of the power spectrum — p. 311
$$
P(k) = \langle|\delta_{\vec k}|^2\rangle,
$$
(11.68) power-law — p. 311
$$
P(k) \propto k^n.
$$
(11.72) RMS mass fluctuation scales with sphere radius — p. 313
$$
\frac{\delta M}{M} \equiv \left\langle\left(\frac{M-\langle M\rangle}{\langle M\rangle}\right)^2\right\rangle^{1/2} \propto r^{-(3+n)/2}.
$$
(11.78) comoving free-streaming length for hot dark matter — p. 316
$$
r_{\min} = \frac{d_{\min}}{a(t_h)} \approx \left(\frac{T_h}{2.7255,\mathrm{K}}\right)d_{\min} \approx 55,\mathrm{Mpc}\left(\frac{m_h c^2}{3,\mathrm{eV}}\right)^{-1}.
$$
(11.85) comoving acoustic scale — p. 321
$$
r_s = d_s(t_{\rm ls})(1+z_{\rm ls}) \approx 160,\mathrm{Mpc}.
$$
Ch 12 · Structure Formation: Baryons and Photons (12.1) average mass density of baryonic matter today — p. 328
$$
\rho_{\mathrm{bary},0} = 4.2 \times 10^{-28},\mathrm{kg,m^{-3}} = 6.2 \times 10^{9},\mathrm{M_\odot,Mpc^{-3}}.
$$
(12.5) optical depth for scattering from the reionized gas — p. 332
$$
\tau_* = \int_{t__}^{t_0} \Gamma(t),dt = c\sigma_e \int_{t__}^{t_0} n_e(t),dt
$$
(12.17) comoving number density of ionizing photons that must be… — p. 337
$$
n_* = \frac{n_{\mathrm{bary}}}{f} = 3.7 \times 10^{67},\mathrm{Mpc^{-3}} \left( \frac{0.2}{f} \right).
$$
(12.21) Schechter luminosity function fitting the observed galaxy… — p. 341
$$
\Phi(L),dL = \Phi^* \left( \frac{L}{L^_} \right)^{\alpha} \exp\left( -\frac{L}{L^_} \right) \frac{dL}{L^*}.
$$
(12.30) virial temperature of gas in a virialized halo — p. 345
$$
kT_{\mathrm{gas}} = \frac{GM_{\mathrm{tot}}\mu}{\beta R_{\mathrm{halo}}}.
$$
(12.36) cooling time for ionized gas radiating by bremsstrahlung — p. 347
$$
t_{\mathrm{cool}} = \frac{\varepsilon}{\Psi} = 13,\mathrm{Gyr} \left( \frac{\rho_{\mathrm{gas}}}{10^{-24},\mathrm{kg,m^{-3}}} \right)^{-1} \left( \frac{T}{10^{6},\mathrm{K}} \right)^{1/2}.
$$
(12.41) Jeans length in a molecular cloud core — p. 351
$$
\lambda_J = 2\pi c_s t_{\mathrm{dyn}} \approx 1.9 \times 10^{15},\mathrm{m} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-1/2} \left( \frac{T_{\mathrm{core}}}{20,\mathrm{K}} \right)^{1/2}.
$$
(12.42) baryonic Jeans mass in a dense molecular cloud core — p. 351
$$
\begin{aligned}
M_J = \frac{4\pi}{3}\rho_{\mathrm{core}}\lambda_J^3 &\approx 3 \times 10^{31},\mathrm{kg} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-1/2} \left( \frac{T_{\mathrm{core}}}{20,\mathrm{K}} \right)^{3/2} \\
&\approx 15,\mathrm{M_\odot} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-1/2} \left( \frac{T_{\mathrm{core}}}{20,\mathrm{K}} \right)^{3/2}.
\end{aligned}
$$
