Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — All Formulas Ch 7-12 How to read this list
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Quick reference: ★ Essentials Cheat Sheet , Ch 2-6 Formulas , Ch 3-7 Notes , Ch 8-12 Notes
Dark Matter The Cosmic Microwave Background Nucleosynthesis and the Early Universe Inflation and the Very Early Universe Structure Formation: Gravitational Instability Structure Formation: Baryons and Photons
Eq (7.1) — luminosity density of galaxies in the V band in the local universe. (p. 182)
$$
\Psi_V = 1.1 \times 10^8 , L_{\odot,V},\mathrm{Mpc^{-3}}.
$$
Eq (7.2) — initial mass function fitted as a power law for high-mass stars. (p. 183)
$$
\chi(M) \propto M^{-\beta} \qquad [M > 1,\mathrm{M_\odot}].
$$
Eq (7.3) — initial mass function fitted as a log-normal distribution for low-mass stars. (p. 183)
$$
\chi(M) \propto \frac{1}{M} \exp\left(-\frac{(\log M - \log M_c)^2}{2\sigma^2}\right) \qquad [M < 1,\mathrm{M_\odot}].
$$
Eq (7.4) — present-day mass density of stars from luminosity density times mass-to-light ratio. (p. 184)
$$
\rho_{\star,0} = \langle M/L_V \rangle , \Psi_V \approx 4 \times 10^8 , \mathrm{M_\odot,Mpc^{-3}}.
$$
Eq (7.5) — present-day density parameter in stars (stars = 0.3% of critical density). (p. 185)
$$
\Omega_{\star,0} = \frac{\rho_{\star,0}}{\rho_{c,0}} \approx \frac{4 \times 10^8 , \mathrm{M_\odot,Mpc^{-3}}}{1.28 \times 10^{11} , \mathrm{M_\odot,Mpc^{-3}}} \approx 0.003.
$$
Eq (7.6) — present-day density parameter of all baryonic matter (from CMB + BBN). (p. 187)
$$
\Omega_{\mathrm{bary},0} = 0.048 \pm 0.003,
$$
7.2 Dark Matter in Galaxies Eq (7.7) — centripetal acceleration of a star on a circular orbit of radius R, speed v. (p. 188)
$$
a = \frac{v^2}{R},
$$
Eq (7.8) — gravitational acceleration from the galactic mass M(R) enclosed within radius R. (p. 188)
$$
a = \frac{GM(R)}{R^2},
$$
Eq (7.9) — equating centripetal and gravitational accelerations. (p. 188)
$$
\frac{v^2}{R} = \frac{GM(R)}{R^2},
$$
Eq (7.10) ★ — orbital speed as a function of enclosed mass (circular-orbit rotation curve). (p. 188)
$$
v = \sqrt{\frac{GM(R)}{R}}.
$$
Eq (7.11) — surface brightness of a spiral-galaxy disk falls off exponentially with radius. (p. 188)
$$
I(R) = I(0)\exp\left(-\frac{R}{R_s}\right),
$$
Eq (7.12) ★ — mass of a spiral galaxy (disk + dark halo) from a flat rotation curve, scaled to the Sun's orbit. (p. 189)
$$
M(R) = \frac{v^2 R}{G} = 1.05 \times 10^{11},\mathrm{M_\odot} \left(\frac{v}{235,\mathrm{km,s^{-1}}}\right)^{2} \left(\frac{R}{8.2,\mathrm{kpc}}\right).
$$
Eq (7.13) — total mass-to-light ratio of our galaxy as a function of dark-halo radius. (p. 190)
$$
\langle M/L_V \rangle_{\mathrm{gal}} \approx 64,\mathrm{M_\odot/L_{\odot,V}} \left(\frac{R_{\mathrm{halo}}}{100,\mathrm{kpc}}\right),
$$
7.3 Dark Matter in Clusters Eq (7.14) — Newtonian acceleration of the ith galaxy in an isolated cluster of point masses. (p. 191)
$$
\ddot{\vec{x}}_i = G \sum_{j \neq i} m_j \frac{\vec{x}_j - \vec{x}_i}{|\vec{x}_j - \vec{x}_i|^3}.
$$
Eq (7.15) — total gravitational potential energy of the N-galaxy system. (p. 191)
$$
W = -\frac{G}{2} \sum_{\substack{i,j \ j \neq i}} \frac{m_i m_j}{|\vec{x}_j - \vec{x}_i|}.
$$
Eq (7.16) — potential energy written via total mass and half-mass radius (α ≈ 0.45). (p. 191)
$$
W = -\alpha \frac{GM^2}{r_h},
$$
Eq (7.17) — total kinetic energy of the galaxies in the cluster. (p. 192)
$$
K = \frac{1}{2} \sum_i m_i |\dot{\vec{x}}_i|^2.
$$
Eq (7.18) — kinetic energy written via total mass and mass-weighted mean square velocity. (p. 192)
$$
K = \frac{1}{2} M \langle v^2 \rangle,
$$
Eq (7.19) — definition of the mass-weighted mean square velocity of the cluster galaxies. (p. 192)
$$
\langle v^2 \rangle \equiv \frac{1}{M} \sum_i m_i |\dot{\vec{x}}_i|^2
$$
Eq (7.20) — definition of the moment of inertia of the cluster. (p. 192)
$$
I \equiv \sum_i m_i |\vec{x}_i|^2.
$$
Eq (7.21) — second time derivative of the moment of inertia. (p. 192)
$$
\ddot{I} = 2 \sum_i m_i \left(\vec{x}_i \cdot \ddot{\vec{x}}_i + \dot{\vec{x}}_i \cdot \dot{\vec{x}}_i\right).
$$
Eq (7.22) — the second derivative of I rewritten using the kinetic energy. (p. 192)
$$
\ddot{I} = 2 \sum_i m_i \left(\vec{x}_i \cdot \ddot{\vec{x}}_i\right) + 4K.
$$
Eq (7.23) — the mass-weighted term expanded using the equation of motion (7.14). (p. 193)
$$
\sum_i m_i \left(\vec{x}_i \cdot \ddot{\vec{x}}_i\right) = G \sum_{\substack{i,j \ j \neq i}} m_i m_j \frac{\vec{x}_i \cdot (\vec{x}_j - \vec{x}_i)}{|\vec{x}_j - \vec{x}_i|^3}.
$$
Eq (7.24) — the same expression with i and j subscripts interchanged. (p. 193)
$$
\sum_j m_j \left(\vec{x}_j \cdot \ddot{\vec{x}}_j\right) = G \sum_{\substack{j,i \ i \neq j}} m_j m_i \frac{\vec{x}_j \cdot (\vec{x}_i - \vec{x}_j)}{|\vec{x}_i - \vec{x}_j|^3}.
$$
Eq (7.25) — the summation index is arbitrary, so the two sums are equal. (p. 193)
$$
\sum_i m_i \left(\vec{x}_i \cdot \ddot{\vec{x}}_i\right) = \sum_j m_j \left(\vec{x}_j \cdot \ddot{\vec{x}}_j\right)
$$
Eq (7.26) — averaging (7.23) and (7.24) shows the term equals the potential energy W. (p. 193)
$$
\begin{aligned}
\sum_i m_i \left(\vec{x}_i \cdot \ddot{\vec{x}}_i\right) &= \frac{1}{2}\left[\sum_i m_i \left(\vec{x}_i \cdot \ddot{\vec{x}}_i\right) + \sum_j m_j \left(\vec{x}_j \cdot \ddot{\vec{x}}_j\right)\right] \\
&= -\frac{G}{2} \sum_{\substack{i,j \ j \neq i}} \frac{m_i m_j}{|\vec{x}_j - \vec{x}_i|} = W.
\end{aligned}
$$
Eq (7.27) ★ — the virial theorem (general dynamical form). (p. 193)
$$
\ddot{I} = 2W + 4K.
$$
Eq (7.28) ★ — steady-state virial theorem (constant moment of inertia; the 2K + W = 0 form). (p. 194)
$$
0 = W + 2K,
$$
Eq (7.29) — steady-state virial theorem solved for kinetic energy. (p. 194)
$$
K = -\frac{W}{2}.
$$
Eq (7.30) — virial theorem with K and W substituted from (7.18) and (7.16). (p. 194)
$$
\frac{1}{2} M \langle v^2 \rangle = \frac{\alpha}{2} \frac{GM^2}{r_h}.
$$
Eq (7.31) ★ — virial mass of a self-gravitating steady-state system (e.g. a galaxy cluster). (p. 194)
$$
M = \frac{\langle v^2 \rangle r_h}{\alpha G}.
$$
Eq (7.32) — mean redshift of the Coma cluster from hundreds of galaxy redshifts. (p. 195)
$$
\langle z \rangle = 0.0232,
$$
Eq (7.33) — distance to the Coma cluster from its mean redshift. (p. 195)
$$
d_{\mathrm{Coma}} = (c/H_0)\langle z \rangle = 102,\mathrm{Mpc}.
$$
Eq (7.34) ★ — line-of-sight velocity dispersion of the Coma cluster. (p. 195)
$$
\sigma_r = \left\langle (v_r - \langle v_r \rangle)^2 \right\rangle^{1/2} = 880,\mathrm{km,s^{-1}}.
$$
Eq (7.35) — 3-D mean square velocity assuming isotropic dispersion (three times the 1-D value). (p. 195)
$$
\langle v^2 \rangle = 3(880,\mathrm{km,s^{-1}})^2 = 2.32 \times 10^{12},\mathrm{m^2,s^{-2}}.
$$
Eq (7.36) — estimated half-mass radius of the Coma cluster. (p. 196)
$$
r_h \approx 1.5,\mathrm{Mpc} \approx 4.6 \times 10^{22},\mathrm{m}.
$$
Eq (7.37) — virial mass estimate of the Coma cluster. (p. 196)
$$
\begin{aligned}
M_{\mathrm{Coma}} = \frac{\langle v^2 \rangle r_h}{\alpha G} &\approx \frac{(2.32 \times 10^{12},\mathrm{m^2,s^{-2}})(4.6 \times 10^{22},\mathrm{m})}{(0.45)(6.67 \times 10^{-11},\mathrm{m^3,s^{-2},kg^{-1}})} \\
&\approx 4 \times 10^{45},\mathrm{kg} \approx 2 \times 10^{15},\mathrm{M_\odot}.
\end{aligned}
$$
Eq (7.38) — mass-to-light ratio of the Coma cluster (far larger than that of galaxies). (p. 196)
$$
\langle M/L_V \rangle_{\mathrm{Coma}} \sim 400,\mathrm{M_\odot/L_{\odot,V}},
$$
Eq (7.39) — equation of hydrostatic equilibrium for the hot intracluster gas. (p. 196)
$$
\frac{dP_{\mathrm{gas}}}{dr} = -\frac{GM(r)\rho_{\mathrm{gas}}(r)}{r^2},
$$
Eq (7.40) — pressure of the intracluster gas from the perfect gas law. (p. 197)
$$
P_{\mathrm{gas}} = \frac{\rho_{\mathrm{gas}}, k T_{\mathrm{gas}}}{\mu},
$$
Eq (7.41) — total cluster mass within radius r from the gas temperature and density gradients (X-ray method). (p. 197)
$$
M(r) = \frac{k T_{\mathrm{gas}}(r), r}{G\mu} \left[-\frac{d\ln\rho_{\mathrm{gas}}}{d\ln r} - \frac{d\ln T_{\mathrm{gas}}}{d\ln r}\right].
$$
Eq (7.42) — density parameter of all clusters of galaxies (a lower limit on Ω_m). (p. 197)
$$
\Omega_{\mathrm{clus},0} \approx 0.2.
$$
7.4 Gravitational Lensing Eq (7.43) ★ — deflection angle of light passing a compact mass M at impact parameter b. (p. 198)
$$
\alpha = \frac{4GM}{c^2 b},
$$
Eq (7.44) — deflection of starlight just grazing the Sun's surface (the 1919 eclipse test). (p. 198)
$$
\alpha = \frac{4G,\mathrm{M_\odot}}{c^2,\mathrm{R_\odot}} = 1.7,\mathrm{arcsec}.
$$
Eq (7.45) ★ — Einstein radius (angular radius of the ring image for perfect alignment). (p. 199)
$$
\theta_E = \left(\frac{4GM}{c^2 d},\frac{1-x}{x}\right)^{1/2},
$$
Eq (7.46) — Einstein radius for a MACHO lensing an LMC star, evaluated at x ≈ 0.5. (p. 199)
$$
\theta_E \approx 4 \times 10^{-4},\mathrm{arcsec} \left(\frac{M}{1,\mathrm{M_\odot}}\right)^{1/2} \left(\frac{d}{50,\mathrm{kpc}}\right)^{-1/2}.
$$
Eq (7.47) — characteristic time scale of a MACHO microlensing event. (p. 200)
$$
\Delta t = \frac{d,\theta_E}{2v} \approx 90,\mathrm{days} \left(\frac{M}{1,\mathrm{M_\odot}}\right)^{1/2} \left(\frac{v}{200,\mathrm{km,s^{-1}}}\right)^{-1},
$$
Eq (7.48) — Einstein radius for a galaxy cluster lensing a background galaxy. (p. 201)
$$
\theta_E \approx 0.5,\mathrm{arcmin} \left(\frac{M}{10^{14},\mathrm{M_\odot}}\right)^{1/2} \left(\frac{d}{1000,\mathrm{Mpc}}\right)^{-1/2}.
$$
Eq (7.49) — present-day number density of cosmic background neutrinos (all three flavors). (p. 203)
$$
n_\nu = 3\left(\frac{3}{11}\right) n_\gamma = \left(\frac{9}{11}\right)(4.108 \times 10^8,\mathrm{m^{-3}}) = 3.36 \times 10^8,\mathrm{m^{-3}}.
$$
Eq (7.50) — average neutrino mass needed to account for all nonbaryonic dark matter. (p. 203)
$$
m_\nu c^2 = \frac{\Omega_{\mathrm{dm},0}, \varepsilon_{c,0}}{n_\nu}.
$$
Eq (7.51) — numerical evaluation of the required neutrino mass. (p. 203)
$$
m_\nu c^2 \approx \frac{0.262(4870,\mathrm{MeV,m^{-3}})}{3.36 \times 10^8,\mathrm{m^{-3}}} \approx 3.8,\mathrm{eV}
$$
Eq (7.52) — observationally allowed range of the average neutrino mass. (p. 203)
$$
0.019,\mathrm{eV} < m_\nu c^2 < 0.1,\mathrm{eV}.
$$
Eq (7.53) — implied present-day density parameter in massive neutrinos (< 3% of dark matter). (p. 204)
$$
0.0013 < \Omega_{\nu,0} < 0.007,
$$
Chapter 8 — The Cosmic Microwave Background Eq (8.1) — present-day energy density of the CMB radiation. (p. 207)
$$
\varepsilon_{\gamma,0} = \alpha T_0^4 = 0.2606,\mathrm{MeV,m^{-3}}
$$
Eq (8.2) — present-day number density of CMB photons. (p. 207)
$$
n_{\gamma,0} = 4.107 \times 10^{8},\mathrm{m^{-3}}
$$
Eq (8.3) — present-day energy density of baryons in the Benchmark Model. (p. 207)
$$
\varepsilon_{\mathrm{bary},0} = \Omega_{\mathrm{bary},0},\varepsilon_{c,0} \approx 234,\mathrm{MeV,m^{-3}}
$$
Eq (8.4) — present-day number density of baryons (energy density over rest energy). (p. 208)
$$
n_{\mathrm{bary},0} = \frac{\varepsilon_{\mathrm{bary},0}}{E_{\mathrm{bary}}} \approx \frac{234,\mathrm{MeV,m^{-3}}}{939,\mathrm{MeV}} \approx 0.25,\mathrm{m^{-3}}
$$
Eq (8.5) ★ — baryon-to-photon ratio η. (p. 208)
$$
\eta = \frac{n_{\mathrm{bary},0}}{n_{\gamma,0}} \approx \frac{0.25,\mathrm{m^{-3}}}{4.107 \times 10^{8},\mathrm{m^{-3}}} \approx 6.1 \times 10^{-10}
$$
Eq (8.6) ★ — mean (all-sky averaged) CMB temperature. (p. 212)
$$
\langle T \rangle = \frac{1}{4\pi} \int T(\theta,\phi),\sin\theta,d\theta,d\phi = 2.7255,\mathrm{K}
$$
Eq (8.7) ★ — definition of the dimensionless temperature fluctuation. (p. 212)
$$
\frac{\delta T}{T}(\theta,\phi) \equiv \frac{T(\theta,\phi) - \langle T \rangle}{\langle T \rangle}
$$
Eq (8.8) — rms temperature fluctuation measured by COBE (dipole removed). (p. 212)
$$
\left\langle \left(\frac{\delta T}{T}\right)^{2} \right\rangle^{1/2} = 1.1 \times 10^{-5}
$$
8.2 Recombination and Decoupling Eq (8.9) ★ — definition of the fractional ionization X (hydrogen-only universe). (p. 214)
$$
X \equiv \frac{n_p}{n_p + n_{\mathrm{H}}} = \frac{n_p}{n_{\mathrm{bary}}} = \frac{n_e}{n_{\mathrm{bary}}}
$$
Eq (8.10) — photoionization of hydrogen. (p. 215)
$$
\mathrm{H} + \gamma \rightarrow p + e^-
$$
Eq (8.11) — radiative recombination. (p. 215)
$$
p + e^- \rightarrow \mathrm{H} + \gamma
$$
Eq (8.12) — Thomson scattering of photons off electrons. (p. 215)
$$
\gamma + e^- \rightarrow \gamma + e^-
$$
Eq (8.13) — photon mean free path for Thomson scattering. (p. 216)
$$
\lambda = \frac{1}{n_e \sigma_e}
$$
Eq (8.14) — photon scattering rate. (p. 216)
$$
\Gamma = \frac{c}{\lambda} = n_e \sigma_e c
$$
Eq (8.15) — free-electron density when fully ionized, scaling as $1/a^3$ . (p. 216)
$$
n_e = n_{\mathrm{bary}} = \frac{n_{\mathrm{bary},0}}{a^3}
$$
Eq (8.16) — numerical photon scattering rate when fully ionized. (p. 216)
$$
\Gamma = \frac{n_{\mathrm{bary},0},\sigma_e c}{a^3} = \frac{5.0 \times 10^{-21},\mathrm{s^{-1}}}{a^3}
$$
Eq (8.17) — Friedmann equation in the radiation-dominated era. (p. 217)
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{r,0}}{a^4}
$$
Eq (8.18) — Hubble parameter in the radiation-dominated era. (p. 217)
$$
H = \frac{H_0 \Omega_{r,0}^{1/2}}{a^2} = \frac{2.1 \times 10^{-20},\mathrm{s^{-1}}}{a^2}
$$
8.3 The Physics of Recombination Eq (8.19) — crude estimate of the recombination temperature. (p. 218)
$$
T_{\mathrm{rec}} \sim \frac{Q}{2.7k} \sim \frac{13.6,\mathrm{eV}}{2.7,(8.6 \times 10^{-5},\mathrm{eV,K^{-1}})} \sim 60,000,\mathrm{K}
$$
Eq (8.20) — the reaction determining X (photoionization ⇌ recombination). (p. 218)
$$
\mathrm{H} + \gamma \rightleftharpoons p + e^-
$$
Eq (8.21) — number density of species x in kinetic equilibrium (Fermi–Dirac +, Bose–Einstein −). (p. 219)
$$
n_x(p),dp = g_x \frac{4\pi}{h^3}, \frac{p^2,dp}{\exp\left([E - \mu_x]/kT\right) \pm 1}
$$
Eq (8.22) — photon number density vs frequency (blackbody / Planck form). (p. 220)
$$
n_\gamma(f),df = \frac{8\pi}{c^3}, \frac{f^2,df}{\exp\left(hf/kT\right) - 1}
$$
Eq (8.23) — total photon number density from integrating the blackbody spectrum. (p. 220)
$$
n_\gamma = \frac{2.4041}{\pi^2}\left(\frac{kT}{\hbar c}\right)^{3} = 0.2436\left(\frac{kT}{\hbar c}\right)^{3}
$$
Eq (8.24) — nonrelativistic energy approximation for massive particles. (p. 220)
$$
E \approx m_x c^2 + \frac{1}{2}m_x v^2 \approx m_x c^2 + \frac{p^2}{2 m_x}
$$
Eq (8.25) — Maxwell–Boltzmann momentum distribution for nonrelativistic species x. (p. 220 — corrected; the printed book has a spurious $c^3$, see note at end.)
$$
n_x(p),dp = g_x \frac{4\pi}{h^3}, \exp\left(\frac{-m_x c^2 + \mu_x}{kT}\right) \exp\left(-\frac{p^2}{2 m_x kT}\right) p^2,dp
$$
Eq (8.26) — total number density of a nonrelativistic species x. (p. 221)
$$
n_x = g_x \left(\frac{m_x kT}{2\pi \hbar^2}\right)^{3/2} \exp\left(\frac{-m_x c^2 + \mu_x}{kT}\right)
$$
Eq (8.27) — recombination reaction assumed in chemical equilibrium. (p. 221)
$$
\mathrm{H} + \gamma \rightleftharpoons p + e^-
$$
Eq (8.28) — number-density relation from chemical equilibrium ($\mu_{\mathrm{H}}=\mu_p+\mu_e$ ). (p. 221)
$$
\frac{n_{\mathrm{H}}}{n_p n_e} = \frac{g_{\mathrm{H}}}{g_p g_e} \left(\frac{m_{\mathrm{H}}}{m_p m_e}\right)^{3/2} \left(\frac{kT}{2\pi \hbar^2}\right)^{-3/2} \exp\left(\frac{[m_p + m_e - m_{\mathrm{H}}]c^2}{kT}\right)
$$
Eq (8.29) ★ — the Saha equation (simplified form). (p. 222)
$$
\frac{n_{\mathrm{H}}}{n_p n_e} = \left(\frac{m_e kT}{2\pi \hbar^2}\right)^{-3/2} \exp\left(\frac{Q}{kT}\right)
$$
Eq (8.30) — hydrogen density in terms of X and proton density. (p. 222)
$$
n_{\mathrm{H}} = \frac{1-X}{X},n_p
$$
Eq (8.31) — Saha equation rewritten using X and charge neutrality ($n_e=n_p$ ). (p. 222)
$$
\frac{1-X}{X} = n_p \left(\frac{m_e kT}{2\pi \hbar^2}\right)^{-3/2} \exp\left(\frac{Q}{kT}\right)
$$
Eq (8.32) — baryon-to-photon ratio expressed via proton density and X. (p. 222)
$$
\eta = \frac{n_p}{X n_\gamma}
$$
Eq (8.33) — proton density from combining η with the photon density. (p. 222)
$$
n_p = 0.2436,X\eta \left(\frac{kT}{\hbar c}\right)^{3}
$$
Eq (8.34) — Saha equation as a quadratic in X (in terms of T and η). (p. 223)
$$
\frac{1-X}{X^2} = 3.84,\eta \left(\frac{kT}{m_e c^2}\right)^{3/2} \exp\left(\frac{Q}{kT}\right)
$$
Eq (8.35) — positive root of the quadratic for X. (p. 223)
$$
X = \frac{-1 + \sqrt{1 + 4S}}{2S}
$$
Eq (8.36) — definition of the function S(T, η). (p. 223)
$$
S(T,\eta) = 3.84,\eta \left(\frac{kT}{m_e c^2}\right)^{3/2} \exp\left(\frac{Q}{kT}\right)
$$
Eq (8.37) ★ — recombination temperature, defined at $X=\tfrac12$ . (p. 223)
$$
kT_{\mathrm{rec}} = 0.324,\mathrm{eV} = \frac{Q}{42}
$$
Eq (8.38) — photon scattering rate for partially ionized hydrogen. (p. 224)
$$
\Gamma(z) = n_e(z)\sigma_e c = X(z)(1+z)^3 n_{\mathrm{bary},0}\sigma_e c
$$
Eq (8.39) — numerical scattering rate (using $\Omega_{\mathrm{bary},0}=0.048$ ). (p. 224)
$$
\Gamma(z) = 5.0 \times 10^{-21},\mathrm{s^{-1}},X(z)(1+z)^3
$$
Eq (8.40) — Friedmann equation in the matter-dominated era. (p. 225)
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{m,0}}{a^3} = \Omega_{m,0}(1+z)^3
$$
Eq (8.41) — numerical Hubble parameter during recombination (using $\Omega_{m,0}=0.31$ ). (p. 225)
$$
H(z) = 1.23 \times 10^{-18},\mathrm{s^{-1}},(1+z)^{3/2}
$$
Eq (8.42) — redshift of photon decoupling, from setting $\Gamma = H$ . (p. 225)
$$
1 + z_{\mathrm{dec}} = \frac{39.3}{X(z_{\mathrm{dec}})^{2/3}}
$$
Eq (8.43) — expected number of scatterings (optical depth) as an integral over time. (p. 225)
$$
\tau(t) = \int_t^{t_0} \Gamma(t),dt
$$
Eq (8.44) — optical depth as an integral over scale factor. (p. 226)
$$
\tau(a) = \int_a^1 \Gamma(a)\frac{da}{\dot{a}} = \int_a^1 \frac{\Gamma(a)}{H(a)}\frac{da}{a}
$$
Eq (8.45) — optical depth as an integral over redshift. (p. 226)
$$
\tau(z) = \int_0^z \frac{\Gamma(z)}{H(z)}\frac{dz}{1+z} = 0.0041 \int_0^z X(z)(1+z)^{1/2},dz
$$
8.4 Temperature Fluctuations Eq (8.46) — angular-diameter distance relating physical size ℓ to angular size δθ. (p. 227)
$$
d_A = \frac{\ell}{\delta\theta}
$$
Eq (8.47) — approximate angular-diameter distance to the last scattering surface. (p. 227)
$$
d_A \approx \frac{d_{\mathrm{hor}}(t_0)}{z_{\mathrm{ls}}}
$$
Eq (8.48) — numerical angular-diameter distance to last scattering. (p. 228)
$$
d_A \approx \frac{14,000,\mathrm{Mpc}}{1090} \approx 12.8,\mathrm{Mpc}
$$
Eq (8.49) — physical size of a fluctuation vs its observed angular size. (p. 228)
$$
\ell = d_A \cdot \delta\theta = 12.8,\mathrm{Mpc}\left(\frac{\delta\theta}{1,\mathrm{rad}}\right) = 3.7,\mathrm{kpc}\left(\frac{\delta\theta}{1,\mathrm{arcmin}}\right)
$$
Eq (8.50) ★ — spherical-harmonic expansion of the temperature fluctuation. (p. 228)
$$
\frac{\delta T}{T}(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} a_{lm} Y_{lm}(\theta,\phi)
$$
Eq (8.51) — definition of the correlation function C(θ). (p. 229)
$$
C(\theta) = \left\langle \frac{\delta T}{T}(\hat{n}),\frac{\delta T}{T}(\hat{n}') \right\rangle_{\hat{n}\cdot\hat{n}' = \cos\theta}
$$
Eq (8.52) ★ — correlation function as a multipole (angular power spectrum) expansion. (p. 229)
$$
C(\theta) = \frac{1}{4\pi}\sum_{l=0}^{\infty}(2l+1),C_l, P_l(\cos\theta)
$$
Eq (8.53) — the first few Legendre polynomials. (p. 229)
$$
\begin{aligned}
P_0(x) &= 1 \\
P_1(x) &= x \\
P_2(x) &= \frac{1}{2}(3x^2 - 1)
\end{aligned}
$$
Eq (8.54) — the plotted CMB power-spectrum quantity $\Delta_T$ . (p. 229)
$$
\Delta_T \equiv \left(\frac{l(l+1)}{2\pi},C_l\right)^{1/2} \langle T \rangle
$$
8.5 What Causes the Fluctuations? Eq (8.55) — horizon distance at the time of last scattering. (p. 231)
$$
d_{\mathrm{hor}}(t_{\mathrm{ls}}) = a(t_{\mathrm{ls}})c \int_0^{t_{\mathrm{ls}}} \frac{dt}{a(t)}
$$
Eq (8.56) — numerical horizon distance at last scattering. (p. 231)
$$
d_{\mathrm{hor}}(t_{\mathrm{ls}}) = 2.24,c,t_{\mathrm{ls}} = 0.251,\mathrm{Mpc}
$$
Eq (8.57) — angular size of the horizon at last scattering. (p. 231)
$$
\theta_{\mathrm{hor}} = \frac{d_{\mathrm{hor}}(t_{\mathrm{ls}})}{d_A} = \frac{0.251,\mathrm{Mpc}}{12.8,\mathrm{Mpc}} \approx 0.020,\mathrm{rad} \approx 1.1^\circ
$$
Eq (8.58) — dark-matter energy density at last scattering. (p. 232)
$$
\varepsilon_{\mathrm{dm}}(z_{\mathrm{ls}}) = \Omega_{\mathrm{dm},0},\varepsilon_{c,0}(1+z_{\mathrm{ls}})^3 \approx 1.7 \times 10^{12},\mathrm{MeV,m^{-3}}
$$
Eq (8.59) — baryonic energy density at last scattering. (p. 232)
$$
\varepsilon_{\mathrm{bary}}(z_{\mathrm{ls}}) = \Omega_{\mathrm{bary},0},\varepsilon_{c,0}(1+z_{\mathrm{ls}})^3 \approx 3.1 \times 10^{11},\mathrm{MeV,m^{-3}}
$$
Eq (8.60) — photon energy density at last scattering (scales as $(1+z)^4$ ). (p. 232)
$$
\varepsilon_\gamma(z_{\mathrm{ls}}) = \Omega_{\gamma,0},\varepsilon_{c,0}(1+z_{\mathrm{ls}})^4 \approx 3.9 \times 10^{11},\mathrm{MeV,m^{-3}}
$$
Eq (8.61) — dark-matter energy density split into mean and fluctuation. (p. 232)
$$
\varepsilon(\vec{r}) = \bar{\varepsilon} + \delta\varepsilon(\vec{r})
$$
Eq (8.62) — Poisson's equation linking density and potential fluctuations. (p. 232)
$$
\nabla^2(\delta\Phi) = \frac{4\pi G}{c^2},\delta\varepsilon
$$
Eq (8.63) ★ — the Sachs–Wolfe effect (large-scale temperature fluctuations). (p. 233)
$$
\frac{\delta T}{T} = \frac{1}{3}\frac{\delta\Phi}{c^2}
$$
Eq (8.64) ★ — sound horizon distance at last scattering. (p. 234)
$$
d_s(t_{\mathrm{ls}}) = a(t_{\mathrm{ls}}) \int_0^{t_{\mathrm{ls}}} \frac{c_s(t),dt}{a(t)}
$$
Eq (8.65) — numerical sound horizon distance (using $c_s \approx c/\sqrt{3}$ ). (p. 234)
$$
d_s(t_{\mathrm{ls}}) \approx \frac{1}{\sqrt{3}},d_{\mathrm{hor}}(t_{\mathrm{ls}}) \approx 0.145,\mathrm{Mpc}
$$
Eq (8.66) — angular size of the sound horizon (first acoustic peak). (p. 235)
$$
\theta_s \approx \frac{d_s(t_{\mathrm{ls}})}{d_A} \approx \frac{0.145,\mathrm{Mpc}}{12.8,\mathrm{Mpc}} \approx 0.011,\mathrm{rad} \approx 0.7^\circ
$$
Eq (8.67) — baryon-to-photon ratio η from detailed analysis of the $\Delta_T$ curve. (p. 236)
$$
\eta = (6.10 \pm 0.06) \times 10^{-10}
$$
Eq (8.68) — present-day baryon number density from η. (p. 237)
$$
n_{\mathrm{bary},0} = \eta, n_{\gamma,0} = 0.251 \pm 0.003,\mathrm{m^{-3}}
$$
Eq (8.69) — present-day baryonic energy density (protons dominate). (p. 237)
$$
\varepsilon_{\mathrm{bary},0} = (m_p c^2),n_{\mathrm{bary},0} = 235 \pm 3,\mathrm{MeV,m^{-3}}
$$
Eq (8.70) — present-day baryon density parameter. (p. 237)
$$
\Omega_{\mathrm{bary},0} = \frac{\varepsilon_{\mathrm{bary},0}}{\varepsilon_{c,0}} = 0.048 \pm 0.003
$$
Chapter 9 — Nucleosynthesis and the Early Universe 9.1 Nuclear Physics and Cosmology Eq (9.1) — blackbody photon temperature in the radiation-dominated early universe as a function of cosmic time. (p. 239)
$$
T(t) \approx 10^{10},\mathrm{K}\left(\frac{t}{1,\mathrm{s}}\right)^{-1/2},
$$
Eq (9.2) — the same relation expressed as a thermal energy scale $kT$ . (p. 239)
$$
kT(t) \approx 1,\mathrm{MeV}\left(\frac{t}{1,\mathrm{s}}\right)^{-1/2}.
$$
Eq (9.3) — mean energy per photon in the early universe. (p. 239)
$$
E_{\mathrm{mean}}(t) \approx 2.7,kT(t) \approx 3,\mathrm{MeV}\left(\frac{t}{1,\mathrm{s}}\right)^{-1/2}.
$$
Eq (9.4) ★ — binding energy of deuterium: fusing a proton and neutron into a deuteron releases 2.22 MeV. (p. 241)
$$
p + n \rightleftharpoons \mathrm{D} + 2.22,\mathrm{MeV}.
$$
Eq (9.5) ★ — definition of the primordial helium mass fraction $Y_p$ . (p. 243)
$$
Y_p \equiv \frac{\rho(^4\mathrm{He})}{\rho_{\mathrm{bary}}}.
$$
Eq (9.6) ★ — neutron–proton rest-energy difference $Q_n$ . (p. 244)
$$
Q_n = m_n c^2 - m_p c^2 = 1.29,\mathrm{MeV}.
$$
Eq (9.7) ★ — free-neutron beta decay into a proton, electron, and electron antineutrino. (p. 244)
$$
n \rightarrow p + e^- + \bar{\nu}_e,
$$
Eq (9.8) — electron–positron pair production from two photons. (p. 244)
$$
\gamma + \gamma \rightleftharpoons e^- + e^+.
$$
Eq (9.9) — neutron–proton interconversion via neutrino capture. (p. 244)
$$
n + \nu_e \rightleftharpoons p + e^-
$$
Eq (9.10) — neutron–proton interconversion via positron capture. (p. 245)
$$
n + e^+ \rightleftharpoons p + \bar{\nu}_e.
$$
Eq (9.11) — equilibrium number density of neutrons (nonrelativistic Maxwell–Boltzmann). (p. 245)
$$
n_n = g_n \left(\frac{m_n kT}{2\pi\hbar^2}\right)^{3/2} \exp\left(-\frac{m_n c^2}{kT}\right),
$$
Eq (9.12) — equilibrium number density of protons. (p. 245)
$$
n_p = g_p \left(\frac{m_p kT}{2\pi\hbar^2}\right)^{3/2} \exp\left(-\frac{m_p c^2}{kT}\right).
$$
Eq (9.13) — equilibrium neutron-to-proton ratio before simplification. (p. 245)
$$
\frac{n_n}{n_p} = \left(\frac{m_n}{m_p}\right)^{3/2} \exp\left(-\frac{(m_n - m_p)c^2}{kT}\right).
$$
Eq (9.14) ★ — equilibrium neutron-to-proton ratio in simple form. (p. 245)
$$
\frac{n_n}{n_p} = \exp\left(-\frac{Q_n}{kT}\right),
$$
Eq (9.15) — weak-interaction cross-section (neutrino with proton or neutron) near $kT\sim 1$ MeV. (p. 247)
$$
\sigma_w \sim 10^{-47},\mathrm{m}^2 \left(\frac{kT}{1,\mathrm{MeV}}\right)^2.
$$
Eq (9.16) — weak-interaction rate $\Gamma$ falls steeply with time as neutrino density and cross-section drop. (p. 247)
$$
\Gamma = n_\nu c \sigma_w \propto t^{-5/2}.
$$
Eq (9.17) ★ — frozen-out neutron-to-proton ratio at $kT_{\mathrm{freeze}} = 0.8$ MeV. (p. 247)
$$
\frac{n_n}{n_p} = \exp\left(-\frac{Q_n}{kT_{\mathrm{freeze}}}\right) \approx \exp\left(-\frac{1.29,\mathrm{MeV}}{0.8,\mathrm{MeV}}\right) \approx 0.2.
$$
Eq (9.18) ★ — deuteron formation by neutron–proton fusion (strong force, no Coulomb barrier). (p. 248)
$$
p + n \rightleftharpoons \mathrm{D} + \gamma.
$$
Eq (9.19) — proton–proton fusion first forms an unstable diproton (helium-2). (p. 248)
$$
p + p \rightleftharpoons {}^2\mathrm{He}.
$$
Eq (9.20) — diproton decay to a deuteron via the weak force (second step of pp fusion). (p. 248)
$$
{}^2\mathrm{He} \rightarrow \mathrm{D} + e^+ + \nu_e,
$$
Eq (9.21) ★ — maximum possible primordial helium mass fraction for $n_n/n_p = 1/5$ . (p. 249)
$$
Y_{\max} = \frac{4}{12} = \frac{1}{3}.
$$
(The general form $Y_{\max} = 2f/(1+f)$ with $f \equiv n_n/n_p$ is stated inline here on p. 249 and appears as the numbered Eq (9.48) below.)
Eq (9.22) — the essential first step of BBN: proton–neutron fusion to a deuteron plus a photon. (p. 250)
$$
p + n \rightleftharpoons \mathrm{D} + \gamma.
$$
Eq (9.23) ★ — binding energy of the deuteron as a mass defect. (p. 250)
$$
B_{\mathrm{D}} = (m_n + m_p - m_{\mathrm{D}})c^2 = 2.22,\mathrm{MeV}.
$$
Eq (9.24) — analogous recombination reaction (proton + electron → hydrogen + photon). (p. 251)
$$
p + e^- \rightleftharpoons \mathrm{H} + \gamma.
$$
Eq (9.25) ★ — Saha equation for hydrogen recombination (relative numbers of H, protons, electrons). (p. 251)
$$
\frac{n_H}{n_p n_e} = \left(\frac{m_e kT}{2\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{Q}{kT}\right),
$$
Eq (9.26) ★ — nucleosynthetic (Saha-like) equilibrium for deuterons, protons, and neutrons. (p. 251)
$$
\frac{n_{\mathrm{D}}}{n_p n_n} = \frac{g_{\mathrm{D}}}{g_p g_n} \left(\frac{m_{\mathrm{D}}}{m_p m_n}\right)^{3/2} \left(\frac{kT}{2\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{[m_p + m_n - m_{\mathrm{D}}]c^2}{kT}\right).
$$
Eq (9.27) ★ — simplified nucleosynthetic Saha equation using $g_{\mathrm{D}}=3$ , $g_p=g_n=2$ , $m_p=m_n=m_{\mathrm{D}}/2$ , and $B_{\mathrm{D}}$ . (p. 252)
$$
\frac{n_{\mathrm{D}}}{n_p n_n} = 6 \left(\frac{m_n kT}{\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{B_{\mathrm{D}}}{kT}\right),
$$
Eq (9.28) — deuteron-to-neutron ratio (multiply 9.27 by $n_p$ ). (p. 252)
$$
\frac{n_{\mathrm{D}}}{n_n} = 6 n_p \left(\frac{m_n kT}{\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{B_{\mathrm{D}}}{kT}\right).
$$
Eq (9.29) — proton number density in terms of the baryon-to-photon ratio $\eta$ and photon density. (p. 252)
$$
n_p \approx 0.8, n_{\mathrm{bary}} = 0.8, \eta, n_\gamma = 0.8, \eta \left[0.2436\left(\frac{kT}{\hbar c}\right)^3\right],
$$
Eq (9.30) ★ — deuteron-to-neutron ratio as a function of temperature and $\eta$ . (p. 253)
$$
\frac{n_{\mathrm{D}}}{n_n} \approx 6.5, \eta \left(\frac{kT}{m_n c^2}\right)^{3/2} \exp\left(\frac{B_{\mathrm{D}}}{kT}\right).
$$
Eq (9.31) — defining relation for the deuterium nucleosynthesis temperature $T_{\mathrm{nuc}}$ (where $n_{\mathrm{D}}/n_n = 1$ ). (p. 253)
$$
1 \approx 6.5, \eta \left(\frac{kT_{\mathrm{nuc}}}{m_n c^2}\right)^{3/2} \exp\left(\frac{B_{\mathrm{D}}}{kT_{\mathrm{nuc}}}\right).
$$
Eq (9.32) ★ — neutron-to-proton ratio reduced by neutron decay during the delay $t_{\mathrm{nuc}}\approx200$ s before nucleosynthesis. (p. 254)
$$
\frac{n_n}{n_p} \approx \frac{\exp(-200/880)}{5 + [1 - \exp(-200/880)]} \approx \frac{0.80}{5.20} \approx 0.15.
$$
Eq (9.33) — deuteron + proton fusion to helium-3. (p. 254)
$$
\mathrm{D} + p \rightleftharpoons {}^3\mathrm{He} + \gamma.
$$
Eq (9.34) — deuteron + neutron fusion to tritium ($^3$H). (p. 254)
$$
\mathrm{D} + n \rightleftharpoons {}^3\mathrm{H} + \gamma.
$$
Eq (9.35) — deuteron–deuteron fusion directly to helium-4. (p. 255)
$$
\mathrm{D} + \mathrm{D} \rightleftharpoons {}^4\mathrm{He} + \gamma.
$$
Eq (9.36) — more likely deuteron–deuteron outcome: tritium plus a proton. (p. 255)
$$
\mathrm{D} + \mathrm{D} \rightleftharpoons {}^3\mathrm{H} + p,
$$
Eq (9.37) — alternate deuteron–deuteron outcome: helium-3 plus a neutron. (p. 255)
$$
\mathrm{D} + \mathrm{D} \rightleftharpoons {}^3\mathrm{He} + n.
$$
Eq (9.38) — reactions converting $^3$H and $^3$He into tightly bound $^4$He. (p. 255)
$$
\begin{aligned}
{}^3\mathrm{H} + p &\rightleftharpoons {}^4\mathrm{He} + \gamma \\
{}^3\mathrm{He} + n &\rightleftharpoons {}^4\mathrm{He} + \gamma \\
{}^3\mathrm{H} + \mathrm{D} &\rightleftharpoons {}^4\mathrm{He} + n \\
{}^3\mathrm{He} + \mathrm{D} &\rightleftharpoons {}^4\mathrm{He} + p.
\end{aligned}
$$
Eq (9.39) — production of $^6$Li by $^4$He + deuteron fusion. (p. 256)
$$
{}^4\mathrm{He} + \mathrm{D} \rightleftharpoons {}^6\mathrm{Li} + \gamma
$$
Eq (9.40) — production of $^7$Li by $^4$He + tritium fusion. (p. 256)
$$
{}^4\mathrm{He} + {}^3\mathrm{H} \rightleftharpoons {}^7\mathrm{Li} + \gamma.
$$
Eq (9.41) — production of $^7$Be by $^4$He + $^3$He fusion. (p. 256)
$$
{}^4\mathrm{He} + {}^3\mathrm{He} \rightleftharpoons {}^7\mathrm{Be} + \gamma.
$$
Eq (9.42) — attempted formation of (unstable) $^8$Be by $^4$He + $^4$He fusion. (p. 256)
$$
{}^4\mathrm{He} + {}^4\mathrm{He} \rightarrow {}^8\mathrm{Be},
$$
9.5 Baryon–Antibaryon Asymmetry Eq (9.43) — radiation energy density at the epoch of deuterium nucleosynthesis. (p. 260)
$$
\varepsilon_{\mathrm{nuc}} \approx \alpha T_{\mathrm{nuc}}^4 \approx 1.6 \times 10^{33},\mathrm{MeV,m^{-3}}.
$$
Eq (9.44) — baryon mass density at the time of BBN. (p. 260)
$$
\rho_{\mathrm{bary}}(t_{\mathrm{nuc}}) = \Omega_{\mathrm{bary},0}, \rho_{c,0} \left(\frac{T_{\mathrm{nuc}}}{T_0}\right)^3 \approx 0.009,\mathrm{kg,m^{-3}}.
$$
Eq (9.45) — quark–antiquark pair production/annihilation with photons in the early quark soup. (p. 261)
$$
\gamma + \gamma \rightleftharpoons q + \bar{q},
$$
Eq (9.46) ★ — definition of the quark–antiquark asymmetry $\delta_q$ . (p. 261)
$$
\delta_q \equiv \frac{n_q - n_{\bar{q}}}{n_q + n_{\bar{q}}} \ll 1.
$$
Eq (9.47) ★ — residual quark-to-photon ratio set by the asymmetry (origin of $\eta$ ). (p. 261)
$$
\frac{n_q}{n_\gamma} \sim \delta_q.
$$
Eq (9.48) ★ — general maximum primordial helium mass fraction as a function of $f \equiv n_n/n_p$ (with $f \le 1$ ). (p. 263)
$$
Y_{\max} = \frac{2f}{1 + f},
$$
Chapter 10 — Inflation and the Very Early Universe 10.1 The Flatness Problem Eq (10.1) ★ — Friedmann equation cast as the link between curvature and the density parameter (from Eq 4.34). (p. 265)
$$
1 - \Omega(t) = -\kappa \left( \frac{c/H(t)}{a(t)R_0} \right)^{2}.
$$
Eq (10.2) — the present-day version linking curvature and $\Omega_0$ . (p. 265)
$$
1 - \Omega_0 = -\kappa \left( \frac{c/H_0}{R_0} \right)^{2}.
$$
Eq (10.3) — observational constraint on the present-day flatness. (p. 265)
$$
|1 - \Omega_0| \leq 0.005.
$$
Eq (10.4) — density-parameter deviation as a function of time, from combining (10.1) and (10.2). (p. 265)
$$
1 - \Omega(t) = \frac{H_0^{2}(1 - \Omega_0)}{H(t)^{2}a(t)^{2}}.
$$
Eq (10.5) — radiation-plus-matter Hubble parameter (from Eq 5.108). (p. 266)
$$
\frac{H(t)^{2}}{H_0^{2}} = \frac{\Omega_{r,0}}{a^{4}} + \frac{\Omega_{m,0}}{a^{3}}.
$$
Eq (10.6) — deviation of $\Omega$ from one during the radiation+matter era. (p. 266)
$$
1 - \Omega(t) = \frac{(1 - \Omega_0)a^{2}}{\Omega_{r,0} + a\Omega_{m,0}}.
$$
Eq (10.7) ★ — flatness problem: the deviation grows in the radiation-dominated phase. (p. 266)
$$
|1 - \Omega|_r \propto a^{2} \propto t.
$$
Eq (10.8) — flatness problem: the deviation grows in the matter-dominated phase. (p. 266)
$$
|1 - \Omega|_m \propto a \propto t^{2/3}.
$$
Eq (10.9) — flatness at radiation–matter equality. (p. 266)
$$
|1 - \Omega_{rm}| \leq 2 \times 10^{-6}.
$$
Eq (10.10) — flatness extrapolated back to Big Bang nucleosynthesis. (p. 267)
$$
|1 - \Omega_{\mathrm{nuc}}| \leq 7 \times 10^{-16}.
$$
Eq (10.11) — flatness extrapolated back to the Planck time (the fine-tuning problem). (p. 267)
$$
|1 - \Omega_P| \leq 2 \times 10^{-62}.
$$
Eq (10.12) — current proper distance to the last scattering surface. (p. 268)
$$
d_p(t_0) = c \int_{t_{\mathrm{ls}}}^{t_0} \frac{dt}{a(t)}.
$$
Eq (10.13) — angular separation today of two points a horizon distance apart at last scattering. (p. 269)
$$
\theta_{\mathrm{hor}} = \frac{d_{\mathrm{hor}}(t_{\mathrm{ls}})}{d_A} \approx \frac{0.251,\mathrm{Mpc}}{12.8,\mathrm{Mpc}} \approx 0.020,\mathrm{rad} \approx 1.1^{\circ}.
$$
10.3 The Monopole Problem Eq (10.14) — number density of magnetic monopoles at creation (≈ one per horizon volume). (p. 272)
$$
n_{\mathrm{M}}(t_{\mathrm{GUT}}) \sim \frac{1}{(2ct_{\mathrm{GUT}})^{3}} \sim 10^{82},\mathrm{m^{-3}}.
$$
Eq (10.15) — energy density of monopoles at the GUT phase transition. (p. 273)
$$
\varepsilon_{\mathrm{M}}(t_{\mathrm{GUT}}) \sim (m_{\mathrm{M}}c^{2}),n_{\mathrm{M}} \sim 10^{94},\mathrm{TeV,m^{-3}}.
$$
Eq (10.16) — radiation energy density at the GUT phase transition (for comparison). (p. 273)
$$
\varepsilon_{\gamma}(t_{\mathrm{GUT}}) \approx \alpha T_{\mathrm{GUT}}^{4} \sim 10^{104},\mathrm{TeV,m^{-3}}.
$$
10.4 The Inflation Solution Eq (10.17) ★ — acceleration equation; inflation ($\ddot a>0$ ) requires $P<-\varepsilon/3$ . (p. 274)
$$
\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^{2}}(\varepsilon + 3P).
$$
Eq (10.18) — acceleration equation when a positive cosmological constant $\Lambda_i$ dominates. (p. 274)
$$
\frac{\ddot{a}}{a} = \frac{\Lambda_i}{3} > 0.
$$
Eq (10.19) — Friedmann equation during the inflationary ($\Lambda_i$ -dominated) phase. (p. 274)
$$
\left(\frac{\dot{a}}{a}\right)^{2} = \frac{\Lambda_i}{3}.
$$
Eq (10.20) ★ — de Sitter exponential expansion during inflation, $H_i=(\Lambda_i/3)^{1/2}$ constant. (p. 274)
$$
a(t) \propto e^{H_i t}.
$$
Eq (10.21) — piecewise scale factor: radiation → exponential inflation → radiation. (p. 275)
$$
a(t) = \begin{cases}
a_i (t/t_i)^{1/2} & t < t_i \\
a_i, e^{H_i(t - t_i)} & t_i < t < t_f \\
a_i, e^{H_i(t_f - t_i)}(t/t_f)^{1/2} & t > t_f.
\end{cases}
$$
Eq (10.22) — growth of the scale factor during inflation expressed via e-foldings. (p. 275)
$$
\frac{a(t_f)}{a(t_i)} = e^{N}.
$$
Eq (10.23) ★ — definition of the number of e-foldings of inflation. (p. 275)
$$
N \equiv H_i(t_f - t_i).
$$
Eq (10.24) — energy density of the cosmological constant during inflation. (p. 276)
$$
\varepsilon_{\Lambda_i} = \frac{c^{2}}{8\pi G}\Lambda_i = \frac{3c^{2}}{8\pi G}H_i^{2} \sim 10^{105},\mathrm{TeV,m^{-3}}.
$$
Eq (10.25) — density-parameter deviation rewritten for any non-flat universe. (p. 276)
$$
|1 - \Omega(t)| = \frac{c^{2}}{R_0^{2},a(t)^{2},H(t)^{2}}.
$$
Eq (10.26) — during inflation the deviation from flatness plummets exponentially. (p. 276)
$$
|1 - \Omega(t)| \propto e^{-2H_i t}.
$$
Eq (10.27) — flattening from the start to the end of inflation. (p. 276)
$$
|1 - \Omega(t_f)| = e^{-2N}|1 - \Omega(t_i)|.
$$
Eq (10.28) — assumed strong pre-inflation curvature. (p. 276)
$$
|1 - \Omega(t_i)| \sim 1.
$$
Eq (10.29) — flatness after $N$ e-foldings of inflation. (p. 276)
$$
|1 - \Omega(t_f)| \sim e^{-2N}.
$$
Eq (10.30) — scale factor at the end of inflation (extrapolated from today). (p. 277)
$$
a(t_f) \approx 2 \times 10^{-28}\sqrt{N + 1}.
$$
Eq (10.31) — required flatness immediately after inflation (from observations). (p. 277)
$$
|1 - \Omega(t_f)| \leq 2 \times 10^{-54}(N + 1).
$$
Eq (10.32) ★ — horizon distance at any time $t$ . (p. 277)
$$
d_{\mathrm{hor}}(t) = a(t),c \int_0^{t} \frac{dt}{a(t)}.
$$
Eq (10.33) — horizon distance at the beginning of inflation (radiation-dominated). (p. 277)
$$
d_{\mathrm{hor}}(t_i) = a_i,c \int_0^{t_i} \frac{dt}{a_i (t/t_i)^{1/2}} = 2ct_i.
$$
Eq (10.34) — horizon distance at the end of inflation (full integral). (p. 278)
$$
d_{\mathrm{hor}}(t_f) = a_i e^{N} c \left( \int_0^{t_i} \frac{dt}{a_i (t/t_i)^{1/2}} + \int_{t_i}^{t_f} \frac{dt}{a_i \exp[H_i(t - t_i)]} \right).
$$
Eq (10.35) — horizon size at the end of inflation (large-$N$ limit). (p. 278)
$$
d_{\mathrm{hor}}(t_f) = e^{N} c,(2t_i + H_i^{-1}).
$$
Eq (10.36) — horizon size immediately before inflation. (p. 278)
$$
d_{\mathrm{hor}}(t_i) = 2ct_i \approx 6 \times 10^{-28},\mathrm{m}.
$$
Eq (10.37) — horizon size immediately after inflation ($N=65$ , $H_i\approx t_i^{-1}$ ). (p. 278)
$$
d_{\mathrm{hor}}(t_f) \approx e^{N} 3ct_i \sim 15,\mathrm{m}.
$$
Eq (10.38) — proper radius at end of inflation of the currently visible universe. (p. 280)
$$
d_p(t_f) = a_f,d_p(t_0) \sim 3 \times 10^{-23},\mathrm{Mpc} \sim 0.9,\mathrm{m}.
$$
Eq (10.39) — proper radius just before inflation of the currently visible universe. (p. 280)
$$
d_p(t_i) = e^{-N}d_p(t_f) \sim 4 \times 10^{-29},\mathrm{m}.
$$
10.5 The Physics of Inflation Eq (10.40) ★ — energy density of a homogeneous inflaton field. (p. 281)
$$
\varepsilon_{\phi} = \frac{1}{2}\frac{1}{\hbar c^{3}}\dot{\phi}^{2} + V(\phi).
$$
Eq (10.41) — pressure of the inflaton field. (p. 282)
$$
P_{\phi} = \frac{1}{2}\frac{1}{\hbar c^{3}}\dot{\phi}^{2} - V(\phi).
$$
Eq (10.42) ★ — slow-roll condition (kinetic term negligible vs. potential). (p. 282)
$$
\dot{\phi}^{2} \ll \hbar c^{3} V(\phi).
$$
Eq (10.43) — in slow-roll the inflaton acts like a cosmological constant. (p. 282)
$$
\varepsilon_{\phi} \approx -P_{\phi} \approx V(\phi).
$$
Eq (10.44) — fluid equation for the inflaton energy density. (p. 282)
$$
\dot{\varepsilon}_{\phi} + 3H(t)(\varepsilon_{\phi} + P_{\phi}) = 0.
$$
Eq (10.45) ★ — equation of motion of the inflaton field (with Hubble friction). (p. 282)
$$
\ddot{\phi} + 3H(t)\dot{\phi} = -\hbar c^{3}\frac{dV}{d\phi}.
$$
Eq (10.46) — terminal-velocity balance ($\ddot\phi=0$ ). (p. 283)
$$
3H\dot{\phi} = -\hbar c^{3}\frac{dV}{d\phi}.
$$
Eq (10.47) — "terminal velocity" of the inflaton field. (p. 283)
$$
\dot{\phi} = -\frac{\hbar c^{3}}{3H}\frac{dV}{d\phi}.
$$
Eq (10.48) — slow-roll condition recast in terms of the potential slope. (p. 283)
$$
\left(\frac{dV}{d\phi}\right)^{2} \ll \frac{9H^{2}V}{\hbar c^{3}}.
$$
Eq (10.49) — Hubble parameter set by the inflaton potential. (p. 283)
$$
H = \left(\frac{8\pi G \varepsilon_{\phi}}{3c^{2}}\right)^{1/2} = \left(\frac{8\pi G V}{3c^{2}}\right)^{1/2}.
$$
Eq (10.50) — slow-roll condition after substituting $H$ . (p. 283)
$$
\left(\frac{dV}{d\phi}\right)^{2} \ll \frac{24\pi G V^{2}}{\hbar c^{5}}.
$$
Eq (10.51) — slow-roll condition written with the Planck energy $E_P$ . (p. 283)
$$
\left(\frac{E_P}{V}\frac{dV}{d\phi}\right)^{2} \ll 1.
$$
Eq (10.52) — slow-roll condition on the plateau ($V\approx V_0$ ). (p. 284)
$$
\left(\frac{dV}{d\phi}\right)^{2} \ll \frac{V_0^{2}}{E_P^{2}}.
$$
Eq (10.53) — temperature at which inflaton-driven inflation begins. (p. 285)
$$
T_i \approx \left(\frac{V_0}{\alpha}\right)^{1/4} \approx 2 \times 10^{28},\mathrm{K} \left(\frac{V_0}{10^{105},\mathrm{TeV,m^{-3}}}\right)^{1/4}.
$$
Eq (10.54) — same onset temperature expressed as an energy $kT_i$ . (p. 285)
$$
kT_i \approx (\hbar^{3} c^{3} V_0)^{1/4} \approx 2 \times 10^{12},\mathrm{TeV} \left(\frac{V_0}{10^{105},\mathrm{TeV,m^{-3}}}\right)^{1/4}.
$$
Eq (10.55) — cosmic time at the onset of inflation. (p. 285)
$$
t_i \approx \left(\frac{c^{2}}{GV_0}\right)^{1/2} \approx 3 \times 10^{-36},\mathrm{s} \left(\frac{V_0}{10^{105},\mathrm{TeV,m^{-3}}}\right)^{-1/2}.
$$
Eq (10.56) — Hubble parameter during inflaton-driven inflation. (p. 286)
$$
H_i \approx \left(\frac{8\pi G V_0}{3c^{2}}\right)^{1/2} \approx t_i^{-1}.
$$
Eq (10.57) — number of e-foldings for the plateau potential of Figure 10.4. (p. 286)
$$
N \sim H_i \frac{\phi_0}{\dot{\phi}} \sim \left(\frac{E_P}{V_0}\frac{dV}{d\phi}\right)^{-1}\left(\frac{\phi_0}{E_P}\right).
$$
Eq (10.58) — growth of the scale factor during inflation. (p. 286)
$$
\frac{a(t_f)}{a(t_i)} = e^{N}.
$$
Eq (10.59) — density fluctuations after 65 e-foldings (inflation over-smooths). (p. 287)
$$
\frac{\delta\varepsilon}{\bar{\varepsilon}} \sim e^{-65} \sim 10^{-28}.
$$
Chapter 11 — Structure Formation: Gravitational Instability Eq (11.1) — spatially averaged energy density over a large volume V. (p. 293)
$$
\bar\varepsilon(t) = \frac{1}{V}\int_V \varepsilon(\vec r,t),d^3r.
$$
Eq (11.2) ★ — definition of the dimensionless density fluctuation (density contrast) δ. (p. 293)
$$
\delta(\vec r,t) \equiv \frac{\varepsilon(\vec r,t) - \bar\varepsilon(t)}{\bar\varepsilon(t)}.
$$
Eq (11.3) — gravitational acceleration at the surface of an overdense sphere from its excess mass. (p. 294)
$$
\ddot R = -\frac{G(\Delta M)}{R^2} = -\frac{G}{R^2}\left(\frac{4\pi}{3}R^3\bar\rho,\delta\right),
$$
Eq (11.4) — acceleration equation for the sphere in terms of the density fluctuation. (p. 294)
$$
\frac{\ddot R}{R} = -\frac{4\pi G\bar\rho}{3},\delta(t).
$$
Eq (11.5) — mass of the sphere (conserved during collapse). (p. 295)
$$
M = \frac{4\pi}{3}\bar\rho,[1+\delta(t)],R(t)^3,
$$
Eq (11.6) — radius as a function of the density fluctuation from mass conservation. (p. 295)
$$
R(t) = R_0[1+\delta(t)]^{-1/3},
$$
Eq (11.7) — the constant reference radius R₀. (p. 295)
$$
R_0 \equiv \left(\frac{3M}{4\pi\bar\rho}\right)^{1/3} = \text{constant}.
$$
Eq (11.8) — linearized radius for small δ. (p. 295)
$$
R(t) \approx R_0\left[1 - \frac{1}{3}\delta(t)\right].
$$
Eq (11.9) — second time derivative of the linearized radius. (p. 295)
$$
\ddot R \approx -\frac{1}{3}R_0\ddot\delta \approx -\frac{1}{3}R\ddot\delta.
$$
Eq (11.10) — acceleration ratio from mass conservation (δ ≪ 1). (p. 295)
$$
\frac{\ddot R}{R} \approx -\frac{1}{3}\ddot\delta.
$$
Eq (11.11) — governing equation for growth of the density fluctuation in a static pressureless medium. (p. 296)
$$
\ddot\delta = 4\pi G\bar\rho,\delta.
$$
Eq (11.12) — general solution: growing + decaying exponential modes. (p. 296)
$$
\delta(t) = A_1 e^{t/t_{\rm dyn}} + A_2 e^{-t/t_{\rm dyn}},
$$
Eq (11.13) ★ — dynamical time for gravitational collapse (depends only on ρ̄). (p. 296)
$$
t_{\rm dyn} = \frac{1}{(4\pi G\bar\rho)^{1/2}} \approx 9.6,\text{hours}\left(\frac{\bar\rho}{1,\mathrm{kg,m^{-3}}}\right)^{-1/2}.
$$
Eq (11.14) — equation-of-state parameter for a nonrelativistic gas. (p. 296)
$$
w \approx \frac{kT}{\mu c^2},
$$
Eq (11.15) — order-of-magnitude dynamical (collapse) time. (p. 297)
$$
t_{\rm dyn} \sim \frac{1}{(G\bar\rho)^{1/2}} \sim \left(\frac{c^2}{G\bar\varepsilon}\right)^{1/2}.
$$
Eq (11.16) — time for a pressure gradient to build up across a region of radius R. (p. 297)
$$
t_{\rm pre} \sim \frac{R}{c_s},
$$
Eq (11.17) — sound speed in a medium with equation-of-state parameter w > 0. (p. 297)
$$
c_s = c\left(\frac{dP}{d\varepsilon}\right)^{1/2} = \sqrt{w},c.
$$
Eq (11.18) — condition for pressure to prevent collapse (hydrostatic equilibrium attainable). (p. 298)
$$
t_{\rm pre} < t_{\rm dyn}.
$$
Eq (11.19) — order-of-magnitude Jeans length from comparing timescales. (p. 298)
$$
\lambda_J \sim c_s t_{\rm dyn} \sim c_s\left(\frac{c^2}{G\bar\varepsilon}\right)^{1/2}.
$$
Eq (11.20) ★ — Jeans length (precise, with correct factors of π). (p. 298)
$$
\lambda_J = c_s\left(\frac{\pi c^2}{G\bar\varepsilon}\right)^{1/2} = 2\pi c_s t_{\rm dyn}.
$$
Eq (11.21) — Hubble time expressed via the mean energy density. (p. 298)
$$
H^{-1} = \left(\frac{3c^2}{8\pi G\bar\varepsilon}\right)^{1/2}.
$$
Eq (11.22) — Hubble time is comparable to the dynamical time. (p. 299)
$$
H^{-1} = \left(\frac{3}{2}\right)^{1/2} t_{\rm dyn} \approx 1.22,t_{\rm dyn}.
$$
Eq (11.23) — Jeans length in an expanding flat universe. (p. 299)
$$
\lambda_J = 2\pi c_s t_{\rm dyn} = 2\pi\left(\frac{2}{3}\right)^{1/2}\frac{c_s}{H}.
$$
Eq (11.24) — Jeans length for a single component with EoS parameter w. (p. 299)
$$
\lambda_J = 2\pi\left(\frac{2}{3}\right)^{1/2}\sqrt{w},\frac{c}{H}.
$$
Eq (11.25) — Jeans length for the radiation component (w = 1/3). (p. 299)
$$
\lambda_J = \frac{2\pi\sqrt{2}}{3}\frac{c}{H} \approx 3.0,\frac{c}{H}.
$$
Eq (11.26) — Jeans length of the photon–baryon fluid just before decoupling. (p. 300)
$$
\lambda_J(\text{before}) \approx 3c/H(z_{\rm dec}) \approx 0.66,\mathrm{Mpc} \approx 2.0\times 10^{22},\mathrm{m}.
$$
Eq (11.27) ★ — definition of the baryonic Jeans mass (baryons within a sphere of radius λ_J). (p. 300)
$$
M_J \equiv \rho_{\rm bary}\left(\frac{4\pi}{3}\lambda_J^3\right).
$$
Eq (11.28) — baryonic Jeans mass just before decoupling. (p. 300)
$$
M_J(\text{before}) = \rho_{\rm bary}\frac{4\pi}{3}\lambda_J(\text{before})^3 \approx 2\times 10^{49},\mathrm{kg} \approx 10^{19},\mathrm{M_\odot}.
$$
Eq (11.29) — sound speed in the photon gas after decoupling. (p. 300)
$$
c_s(\text{photon}) = c/\sqrt{3} \approx 0.58c.
$$
Eq (11.30) — sound speed in the baryonic gas after decoupling. (p. 301)
$$
c_s(\text{baryon}) = \left(\frac{kT}{mc^2}\right)^{1/2}c.
$$
Eq (11.31) — numerical baryonic sound speed immediately after decoupling. (p. 301)
$$
c_s(\text{baryon}) \approx \left(\frac{0.26,\mathrm{eV}}{1140\times 10^6,\mathrm{eV}}\right)^{1/2}c \approx 1.5\times 10^{-5}c,
$$
Eq (11.32) — factor by which the baryonic Jeans length dropped at decoupling. (p. 301)
$$
F = \frac{c_s(\text{baryon})}{c_s(\text{photon})} \approx \frac{1.5\times 10^{-5}}{0.58} \approx 2.6\times 10^{-5}.
$$
Eq (11.33) — baryonic Jeans mass immediately after decoupling. (p. 301)
$$
M_J(\text{after}) = F^3 M_J(\text{before}) \approx 2\times 10^5,\mathrm{M_\odot}.
$$
11.3 Instability in an Expanding Universe Eq (11.34) — density inside a spherical region as a mean-plus-perturbation. (p. 302)
$$
\rho(t) = \bar\rho(t)[1+\delta(t)],
$$
Eq (11.35) — total gravitational acceleration at the sphere's surface (Newtonian). (p. 303)
$$
\ddot R = -\frac{GM}{R^2} = -\frac{G}{R^2}\left(\frac{4\pi}{3}\rho R^3\right) = -\frac{4\pi}{3}G\bar\rho R - \frac{4\pi}{3}G(\bar\rho\delta)R.
$$
Eq (11.36) — equation of motion for a point on the sphere's surface. (p. 303)
$$
\frac{\ddot R}{R} = -\frac{4\pi}{3}G\bar\rho - \frac{4\pi}{3}G\bar\rho,\delta.
$$
Eq (11.37) — mass conservation inside the expanding sphere. (p. 303)
$$
M = \frac{4\pi}{3}\bar\rho(t)[1+\delta(t)]R(t)^3,
$$
Eq (11.38) — radius scaling from mass conservation. (p. 303)
$$
R(t) \propto \bar\rho(t)^{-1/3}[1+\delta(t)]^{-1/3},
$$
Eq (11.39) — radius tracks the scale factor, modulated by the perturbation (since ρ̄ ∝ a⁻³). (p. 303)
$$
R(t) \propto a(t)[1+\delta(t)]^{-1/3}.
$$
Eq (11.40) — second time derivative of the radius (δ ≪ 1). (p. 303)
$$
\frac{\ddot R}{R} = \frac{\ddot a}{a} - \frac{1}{3}\ddot\delta - \frac{2}{3}\frac{\dot a}{a}\dot\delta,
$$
Eq (11.41) — combining the equation of motion with mass conservation. (p. 304)
$$
\frac{\ddot a}{a} - \frac{1}{3}\ddot\delta - \frac{2}{3}\frac{\dot a}{a}\dot\delta = -\frac{4\pi}{3}G\bar\rho - \frac{4\pi}{3}G\bar\rho,\delta.
$$
Eq (11.42) — acceleration equation for the unperturbed matter-only universe (δ = 0). (p. 304)
$$
\frac{\ddot a}{a} = -\frac{4\pi}{3}G\bar\rho,
$$
Eq (11.43) — linearized perturbation equation (after subtracting the background). (p. 304)
$$
-\frac{1}{3}\ddot\delta - \frac{2}{3}\frac{\dot a}{a}\dot\delta = -\frac{4\pi}{3}G\bar\rho,\delta,
$$
Eq (11.44) ★ — growth equation for small perturbations in an expanding universe (with Hubble-friction term 2Hδ̇). (p. 304)
$$
\ddot\delta + 2H\dot\delta = 4\pi G\bar\rho,\delta,
$$
Eq (11.45) — static-universe limit (H = 0) recovers Eq (11.11). (p. 304)
$$
\ddot\delta = 4\pi G\bar\rho,\delta.
$$
Eq (11.46) — fully relativistic growth equation (matter energy density drives growth). (p. 304)
$$
\ddot\delta + 2H\dot\delta = \frac{4\pi G}{c^2}\bar\varepsilon_m,\delta.
$$
Eq (11.47) — δ is the fluctuation in the matter density specifically. (p. 305)
$$
\delta = \frac{\varepsilon_m - \bar\varepsilon_m}{\bar\varepsilon_m},
$$
Eq (11.48) — matter density parameter Ωₘ. (p. 305)
$$
\Omega_m = \frac{\bar\varepsilon_m}{\varepsilon_c} = \frac{8\pi G\bar\varepsilon_m}{3c^2H^2},
$$
Eq (11.49) ★ — growth equation for matter perturbations in terms of Ωₘ. (p. 305)
$$
\ddot\delta + 2H\dot\delta - \frac{3}{2}\Omega_m H^2\delta = 0.
$$
Eq (11.50) — growth equation in the radiation-dominated era (Ωₘ ≪ 1, H = 1/(2t)). (p. 305)
$$
\ddot\delta + \frac{1}{t}\dot\delta \approx 0,
$$
Eq (11.51) — solution in the radiation era: only logarithmic growth. (p. 305)
$$
\delta(t) \approx B_1 + B_2\ln t.
$$
Eq (11.52) — growth equation in a Λ-dominated era (H = H_Λ = const). (p. 306)
$$
\ddot\delta + 2H_\Lambda\dot\delta \approx 0,
$$
Eq (11.53) — solution in the Λ-dominated era: perturbations freeze to constant amplitude. (p. 306)
$$
\delta(t) \approx C_1 + C_2 e^{-2H_\Lambda t}.
$$
Eq (11.54) — growth equation in a flat matter-dominated universe (Ωₘ = 1, H = 2/(3t)). (p. 306)
$$
\ddot\delta + \frac{4}{3t}\dot\delta - \frac{2}{3t^2}\delta = 0.
$$
Eq (11.55) — substituting the power-law ansatz δ = Dtⁿ. (p. 306)
$$
n(n-1)Dt^{n-2} + \frac{4}{3t}nDt^{n-1} - \frac{2}{3t^2}Dt^n = 0,
$$
Eq (11.56) — resulting indicial (characteristic) equation for n. (p. 306)
$$
n(n-1) + \frac{4}{3}n - \frac{2}{3} = 0.
$$
Eq (11.57) — general solution in a flat matter-only universe (growing + decaying modes). (p. 307)
$$
\delta(t) \approx D_1 t^{2/3} + D_2 t^{-1}.
$$
Eq (11.58) ★ — growing mode: perturbations grow as the scale factor in the matter era. (p. 307)
$$
\delta \propto t^{2/3} \propto a(t) \propto \frac{1}{1+z}.
$$
Eq (11.59) — Robertson–Walker metric (geometry still valid while |δ| ≪ 1). (p. 309)
$$
ds^2 = -c^2dt^2 + a(t)^2\left[dr^2 + S_\kappa(r)^2,d\Omega^2\right].
$$
Eq (11.60) — proper distance from origin in comoving coordinates. (p. 309)
$$
d_p(t_i) = a(t_i),r,
$$
Eq (11.61) — Fourier expansion of the density fluctuation field. (p. 309)
$$
\delta(\vec r) = \frac{V}{(2\pi)^3}\int \delta_{\vec k},e^{-i\vec k\cdot\vec r},d^3k,
$$
Eq (11.62) — individual Fourier component (inverse transform). (p. 310)
$$
\delta_{\vec k} = \frac{1}{V}\int \delta(\vec r),e^{i\vec k\cdot\vec r},d^3r.
$$
Eq (11.63) — each Fourier component as a complex number (amplitude and phase). (p. 310)
$$
\delta_{\vec k} = |\delta_{\vec k}|,e^{i\phi_{\vec k}}.
$$
Eq (11.64) — each Fourier mode obeys the linear growth equation. (p. 310)
$$
\ddot\delta_{\vec k} + 2H\dot\delta_{\vec k} - \frac{3}{2}\Omega_m H^2\delta_{\vec k} = 0,
$$
Eq (11.65) ★ — definition of the power spectrum (mean square Fourier amplitude). (p. 311)
$$
P(k) = \langle|\delta_{\vec k}|^2\rangle,
$$
Eq (11.66) — Gaussian probability distribution for δ at a random point. (p. 311)
$$
p(\delta) = \frac{1}{\sqrt{2\pi},\sigma_\delta}\exp\left(-\frac{\delta^2}{2\sigma_\delta^2}\right),
$$
Eq (11.67) — variance of δ computed from the power spectrum. (p. 311)
$$
\sigma_\delta^2 = \frac{V}{(2\pi)^3}\int P(k),d^3k = \frac{V}{2\pi^2}\int_0^\infty P(k),k^2,dk.
$$
Eq (11.68) ★ — power-law (near scale-invariant) primordial power spectrum with spectral index n. (p. 311)
$$
P(k) \propto k^n.
$$
Eq (11.69) — mean mass of nonrelativistic matter in a comoving sphere of radius r. (p. 312)
$$
\langle M\rangle = \frac{4\pi}{3}r^3\rho_{m,0} = 1.67\times 10^{11},\mathrm{M_\odot}\left(\frac{r}{1,\mathrm{Mpc}}\right)^3.
$$
Eq (11.70) — mean square mass fluctuation in spheres of radius r (top-hat window, spherical Bessel function j₁). (p. 312)
$$
\begin{aligned}
\left\langle\left(\frac{M-\langle M\rangle}{\langle M\rangle}\right)^2\right\rangle &= \frac{V}{2\pi^2}\int P(k)\left[\frac{3j_1(kr)}{kr}\right]^2 k^2,dk\\
&= \frac{9V}{2\pi^2 r^2}\int P(k),j_1(kr)^2,dk,
\end{aligned}
$$
Eq (11.71) — mean square mass fluctuation for a power-law spectrum P(k) ∝ kⁿ (substitution u = kr). (p. 312)
$$
\left\langle\left(\frac{M-\langle M\rangle}{\langle M\rangle}\right)^2\right\rangle = \frac{9V}{2\pi^2},r^{-3-n}\int_0^\infty u^n j_1(u)^2,du.
$$
Eq (11.72) ★ — RMS mass fluctuation scales with sphere radius (defines δM/M). (p. 313)
$$
\frac{\delta M}{M} \equiv \left\langle\left(\frac{M-\langle M\rangle}{\langle M\rangle}\right)^2\right\rangle^{1/2} \propto r^{-(3+n)/2}.
$$
Eq (11.73) — Hubble parameter in the radiation-dominated era. (p. 314)
$$
H(a) = H_0\frac{\sqrt{\Omega_{r,0}}}{a^2}.
$$
Eq (11.74) — mass within a Hubble volume during the radiation era. (p. 314)
$$
M_H = \frac{4\pi}{3}\left(\frac{c}{H_0}\frac{a^2}{\sqrt{\Omega_{r,0}}}\right)^3\frac{\rho_{m,0}}{a^3} = 1.6\times 10^{28},\mathrm{M_\odot},a^3,
$$
Eq (11.75) — temperature at which hot dark matter of mass m_h becomes nonrelativistic. (p. 315)
$$
T_h \approx \frac{m_h c^2}{3k} \approx 12,000,\mathrm{K}\left(\frac{m_h c^2}{3,\mathrm{eV}}\right).
$$
Eq (11.76) — cosmic time when hot dark matter turns nonrelativistic. (p. 315)
$$
t_h \approx 42,000,\mathrm{yr}\left(\frac{m_h c^2}{3,\mathrm{eV}}\right)^{-2}.
$$
Eq (11.77) — physical (proper) free-streaming scale below which fluctuations are erased. (p. 316)
$$
d_{\min} \approx ct_h \approx 13,\mathrm{kpc}\left(\frac{m_h c^2}{3,\mathrm{eV}}\right)^{-2},
$$
Eq (11.78) ★ — comoving free-streaming length for hot dark matter. (p. 316)
$$
r_{\min} = \frac{d_{\min}}{a(t_h)} \approx \left(\frac{T_h}{2.7255,\mathrm{K}}\right)d_{\min} \approx 55,\mathrm{Mpc}\left(\frac{m_h c^2}{3,\mathrm{eV}}\right)^{-1}.
$$
Eq (11.79) — minimum mass scale surviving free streaming in hot dark matter. (p. 316)
$$
M_{\min} = \frac{4\pi}{3}r_{\min}^3\rho_{m,0} \approx 2.7\times 10^{16},\mathrm{M_\odot}\left(\frac{m_h c^2}{3,\mathrm{eV}}\right)^{-3}.
$$
Eq (11.80) — comoving Hubble length at the time of WIMP (cold dark matter) decoupling. (p. 318)
$$
r_d = \frac{2ct_d}{a_d} \sim 60,\mathrm{pc},
$$
Eq (11.81) — mass scale within that comoving length at WIMP decoupling. (p. 318)
$$
M_d = \frac{4\pi}{3}r_d^3\rho_{m,0} \sim 0.05,\mathrm{M_\odot},
$$
Eq (11.82) — comoving Hubble length at the time of radiation–matter equality. (p. 318)
$$
r_{rm} = \frac{1.8ct_{rm}}{a_{rm}} \approx 90,\mathrm{Mpc}.
$$
Eq (11.83) — upper limit on the summed neutrino masses (from hot dark matter constraint). (p. 320)
$$
[m(\nu_e) + m(\nu_\mu) + m(\nu_\tau)]c^2 \leq 0.3,\mathrm{eV}.
$$
11.6 Baryon Acoustic Oscillations Eq (11.84) — physical sound horizon distance at the time of last scattering. (p. 321)
$$
d_s(t_{\rm ls}) = 0.145,\mathrm{Mpc}.
$$
Eq (11.85) ★ — comoving acoustic scale (the BAO length scale). (p. 321)
$$
r_s = d_s(t_{\rm ls})(1+z_{\rm ls}) \approx 160,\mathrm{Mpc}.
$$
Eq (11.86) — mass scale associated with the acoustic scale. (p. 321)
$$
M_s = \frac{4\pi}{3}r_s^3\rho_{m,0} \approx 7\times 10^{17},\mathrm{M_\odot}.
$$
Eq (11.87) — definition of the galaxy two-point correlation function ξ(r). (p. 322)
$$
dN = n_{\rm gal}[1+\xi(r)],dV.
$$
Eq (11.88) — correlation function as the Fourier transform of the power spectrum. (p. 323)
$$
\xi(r) = \frac{V}{(2\pi)^3}\int P(k),e^{-i\vec k\cdot\vec r},d^3k.
$$
Eq (11.89) — correlation function for an isotropic power spectrum (spherical Bessel function j₀). (p. 323)
$$
\xi(r) = \frac{V}{2\pi^2}\int P(k),j_0(kr),k^2,dk,
$$
Chapter 12 — Structure Formation: Baryons and Photons 12.1 Baryonic Matter Today Eq (12.1) ★ — average mass density of baryonic matter today. (p. 328)
$$
\rho_{\mathrm{bary},0} = 4.2 \times 10^{-28},\mathrm{kg,m^{-3}} = 6.2 \times 10^{9},\mathrm{M_\odot,Mpc^{-3}}.
$$
Eq (12.2) — baryonic overdensity of the solar neighborhood relative to the cosmic average. (p. 328)
$$
\delta_{\mathrm{sn}} = \frac{\rho_{\mathrm{sn}} - \rho_{\mathrm{bary},0}}{\rho_{\mathrm{bary},0}} \sim 2 \times 10^{7}
$$
Eq (12.3) — mass density of stars today (from Equation 7.4), ~7% of baryonic mass. (p. 329)
$$
\rho_{\star,0} \approx 4 \times 10^{8},\mathrm{M_\odot,Mpc^{-3}} \approx 3 \times 10^{-29},\mathrm{kg,m^{-3}},
$$
12.2 Reionization of Hydrogen Eq (12.4) — rate at which a CMB photon scatters from free electrons in the reionized gas. (p. 331)
$$
\Gamma = n_e \sigma_e c,
$$
Eq (12.5) ★ — optical depth for scattering from the reionized gas (reionization begins at time $t_*$ ). (p. 332)
$$
\tau_* = \int_{t__}^{t_0} \Gamma(t),dt = c\sigma_e \int_{t__}^{t_0} n_e(t),dt.
$$
Eq (12.6) — baryons assumed to be pure hydrogen (neutral atoms plus free protons). (p. 332)
$$
n_{\mathrm{H}} + n_p = n_{\mathrm{bary}} = \frac{n_{\mathrm{bary},0}}{a^3}.
$$
Eq (12.7) — free-electron number density after complete instantaneous reionization ($t > t_*$ ). (p. 333)
$$
n_e = n_p = \frac{n_{\mathrm{bary},0}}{a^3}.
$$
Eq (12.8) — optical depth with the reionized electron density inserted. (p. 333)
$$
\tau_* = \Gamma_0 \int_{t_*}^{t_0} \frac{dt}{a(t)^3},
$$
Eq (12.9) — present-day scattering rate for a uniform, fully ionized hydrogen distribution. (p. 333)
$$
\Gamma_0 = c\sigma_e n_{\mathrm{bary},0} = 1.58 \times 10^{-4},\mathrm{Gyr^{-1}} \approx 0.0023,H_0
$$
Eq (12.10) — optical depth with integration variable changed from $t$ to $a$ (using $H = \dot a/a$ ). (p. 333)
$$
\tau_* = \Gamma_0 \int_{a(t__)}^{1} \frac{da}{\dot a, a^3} = \Gamma_0 \int_{a(t__)}^{1} \frac{da}{H(a),a^4},
$$
Eq (12.11) — optical depth expressed as an integral over redshift ($1+z = 1/a$ ). (p. 333)
$$
\tau_* = \Gamma_0 \int_{0}^{z_*} \frac{(1+z)^2,dz}{H(z)}.
$$
Eq (12.12) — Hubble parameter in the matter-plus-lambda universe (from Equation 5.96). (p. 334)
$$
H(z) = H_0 \left[ \Omega_{m,0}(1+z)^3 + \Omega_{\Lambda,0} \right]^{1/2}.
$$
Eq (12.13) — analytic solution for the reionization optical depth. (p. 334)
$$
\tau_* = \frac{2}{3\Omega_{m,0}} \frac{\Gamma_0}{H_0} \left( \left[ \Omega_{m,0}(1+z_*)^3 + \Omega_{\Lambda,0} \right]^{1/2} - 1 \right).
$$
Eq (12.14) — optical depth evaluated with Benchmark Model parameters. (p. 334)
$$
\tau_* = 0.00485 \left( \left[ 0.31(1+z_*)^3 + 0.69 \right]^{1/2} - 1 \right).
$$
12.3 The First Stars and Quasars Eq (12.15) — production rate of ionizing photons for a luminous AGN, scaled to $L_{\mathrm{AGN}} = 10^{13},\mathrm{L_\odot}$ . (p. 335)
$$
\dot N_* \approx 3 \times 10^{56},\mathrm{s^{-1}} \left( \frac{L_{\mathrm{AGN}}}{10^{13},\mathrm{L_\odot}} \right).
$$
Eq (12.16) — comoving number density of baryons in intergalactic space. (p. 337) (Rendered as inline text in the EPUB conversion, so absent from the equation-image manifest.)
$$
n_{\mathrm{bary}} = 0.25,\mathrm{m^{-3}} = 7.3 \times 10^{66},\mathrm{Mpc^{-3}}.
$$
Eq (12.17) ★ — comoving number density of ionizing photons that must be produced to reionize the hydrogen, given escape fraction $f$ (fiducial $f = f_{\mathrm{esc}} \approx 0.2$ ). (p. 337) (Book renders this inline and the EPUB conversion dropped both fraction bars; verified against a 300-dpi crop and by arithmetic, $7.3\times10^{66}/0.2 = 3.7\times10^{67}$ . $f$ is in the denominator — lower escape fraction ⇒ more photons must be produced.)
$$
n_* = \frac{n_{\mathrm{bary}}}{f} = 3.7 \times 10^{67},\mathrm{Mpc^{-3}} \left( \frac{0.2}{f} \right).
$$
Eq (12.18) — comoving star formation rate at the epoch of reionization ($z_* \approx 8$ ). (p. 339)
$$
\dot\rho_\star(z=8) \approx 20,000,\mathrm{M_\odot,Myr^{-1},Mpc^{-3}}.
$$
Eq (12.19) — total rate of ionizing-photon production per comoving cubic megaparsec (~400 O stars each emitting $\dot N_* \approx 5\times10^{48},\mathrm{s^{-1}}$ ). (p. 340)
$$
\begin{aligned}
\dot n_* &\approx (5 \times 10^{48},\mathrm{s^{-1}})(400,\mathrm{Mpc^{-3}}) \approx 2 \times 10^{51},\mathrm{s^{-1},Mpc^{-3}} \\
&\approx 6 \times 10^{64},\mathrm{Myr^{-1},Mpc^{-3}}.
\end{aligned}
$$
Eq (12.20) — time star formation must continue to reionize the intergalactic hydrogen. (p. 340)
$$
t = \frac{n__}{\dot n__} \approx \frac{3.7 \times 10^{67},\mathrm{Mpc^{-3}}}{6 \times 10^{64},\mathrm{Myr^{-1},Mpc^{-3}}} \left( \frac{0.2}{f_{\mathrm{esc}}} \right) \approx 600,\mathrm{Myr} \left( \frac{0.2}{f_{\mathrm{esc}}} \right)
$$
Eq (12.21) ★ — Schechter luminosity function fitting the observed galaxy number density in the range $L \to L+dL$ . (p. 341)
$$
\Phi(L),dL = \Phi^* \left( \frac{L}{L^_} \right)^{\alpha} \exp\left( -\frac{L}{L^_} \right) \frac{dL}{L^*}.
$$
Eq (12.22) — mass of a spherical overdense region in terms of its overdensity $\delta$ and radius $R$ (from Equation 11.5). (p. 343)
$$
M = \frac{4\pi}{3} \rho_m(t)[1 + \delta(t)]R(t)^3,
$$
Eq (12.23) — collapse redshift set by the initial overdensity at radiation–matter equality. (p. 343)
$$
1 + z_{\mathrm{coll}} \approx \delta_{rm}(1 + z_{rm}).
$$
Eq (12.24) — mean density of the sphere at the moment collapse begins. (p. 343)
$$
\bar\rho(t_{\mathrm{coll}}) \approx 2\rho_m(t_{\mathrm{coll}}) \approx 2\rho_{m,0}(1 + z_{\mathrm{coll}})^3.
$$
Eq (12.25) — mean density of the virialized halo (radius halved, density up by $\sim 8$ ). (p. 344)
$$
\bar\rho_{\mathrm{halo}} \approx 8\bar\rho(t_{\mathrm{coll}}) \approx 16\rho_{m,0}(1 + z_{\mathrm{coll}})^3.
$$
Eq (12.26) — mass of the Milky Way's dark halo as a function of halo radius (from Equation 7.12). (p. 344)
$$
M = 1.9 \times 10^{12},\mathrm{M_\odot} \left( \frac{R_{\mathrm{halo}}}{0.15,\mathrm{Mpc}} \right),
$$
Eq (12.27) — corresponding average halo density. (p. 344)
$$
\begin{aligned}
\bar\rho = \frac{3M}{4\pi R_{\mathrm{halo}}^3} &= 1.4 \times 10^{14},\mathrm{M_\odot,Mpc^{-3}} \left( \frac{R_{\mathrm{halo}}}{0.15,\mathrm{Mpc}} \right)^{-2} \\
&= 3400,\rho_{m,0} \left( \frac{R_{\mathrm{halo}}}{0.15,\mathrm{Mpc}} \right)^{-2}.
\end{aligned}
$$
Eq (12.28) — collapse redshift of the Milky Way inferred from its halo density. (p. 344)
$$
1 + z_{\mathrm{coll}} \approx 6 \left( \frac{R_{\mathrm{halo}}}{0.15,\mathrm{Mpc}} \right)^{-2/3}.
$$
Eq (12.29) — hydrostatic equilibrium relation for a sphere of gas (from Equation 7.41); $\mu$ = mean mass per gas particle. (p. 345)
$$
M(r) = \frac{kT_{\mathrm{gas}}(r),r}{G\mu} \left[ -\frac{d\ln\rho_{\mathrm{gas}}}{d\ln r} - \frac{d\ln T_{\mathrm{gas}}}{d\ln r} \right],
$$
Eq (12.30) ★ — virial temperature of gas in a virialized halo. (p. 345)
$$
kT_{\mathrm{gas}} = \frac{GM_{\mathrm{tot}}\mu}{\beta R_{\mathrm{halo}}}.
$$
Eq (12.31) — halo radius in terms of total mass and collapse redshift. (p. 345)
$$
R_{\mathrm{halo}} = \left( \frac{3M_{\mathrm{tot}}}{4\pi \bar\rho_{\mathrm{halo}}} \right)^{1/3} = \left( \frac{3M_{\mathrm{tot}}}{64\pi \rho_{m,0}(1 + z_{\mathrm{coll}})^3} \right)^{1/3},
$$
Eq (12.32) — virial temperature rewritten as a function of halo mass and collapse redshift. (p. 345)
$$
kT_{\mathrm{gas}} = \frac{4}{\beta} \left( \frac{\pi}{3} \right)^{1/3} G\mu,\rho_{m,0}^{1/3} M_{\mathrm{tot}}^{2/3}(1 + z_{\mathrm{coll}}).
$$
Eq (12.33) — numerical value of the virial temperature (ionized gas, $\beta \approx 2$ , $z_{\mathrm{coll}}$ scaled to 4). (p. 346)
$$
T_{\mathrm{gas}} \approx 1.0 \times 10^{6},\mathrm{K} \left( \frac{M_{\mathrm{tot}}}{10^{12},\mathrm{M_\odot}} \right)^{2/3} \left( \frac{1 + z_{\mathrm{coll}}}{5} \right).
$$
Eq (12.34) — luminosity density of bremsstrahlung (free–free) emission from a fully ionized primordial gas. (p. 347)
$$
\Psi = 5.3 \times 10^{-32},\mathrm{watts,m^{-3}} \left( \frac{\rho_{\mathrm{gas}}}{10^{-24},\mathrm{kg,m^{-3}}} \right)^{2} \left( \frac{T}{10^{6},\mathrm{K}} \right)^{1/2}.
$$
Eq (12.35) — thermal energy density of the gas. (p. 347)
$$
\varepsilon = \frac{3}{2}nkT = 2.1 \times 10^{-14},\mathrm{J,m^{-3}} \left( \frac{\rho_{\mathrm{gas}}}{10^{-24},\mathrm{kg,m^{-3}}} \right) \left( \frac{T}{10^{6},\mathrm{K}} \right).
$$
Eq (12.36) ★ — cooling time for ionized gas radiating by bremsstrahlung (cooling criterion for galaxy formation). (p. 347)
$$
t_{\mathrm{cool}} = \frac{\varepsilon}{\Psi} = 13,\mathrm{Gyr} \left( \frac{\rho_{\mathrm{gas}}}{10^{-24},\mathrm{kg,m^{-3}}} \right)^{-1} \left( \frac{T}{10^{6},\mathrm{K}} \right)^{1/2}.
$$
Eq (12.37) — average density of baryonic gas in a halo (baryon fraction $f$ ). (p. 348)
$$
\begin{aligned}
\bar\rho_{\mathrm{bary}} = f\bar\rho_{\mathrm{halo}} &\approx 16 f,\rho_{m,0}(1 + z_{\mathrm{coll}})^3 \\
&\approx 0.8 \times 10^{-24},\mathrm{kg,m^{-3}} \left( \frac{f}{0.15} \right) \left( \frac{1 + z_{\mathrm{coll}}}{5} \right)^{3}.
\end{aligned}
$$
Eq (12.38) — total mass inside the last scattering surface (out to $d_p(t_0) \approx 14,000$ Mpc). (p. 348)
$$
M = \rho_{m,0}\frac{4\pi}{3}d_p(t_0)^3 \approx 4.3 \times 10^{23},\mathrm{M_\odot}.
$$
Eq (12.39) — dynamical (free-fall) time in a molecular cloud core (from Equation 11.13). (p. 350)
$$
t_{\mathrm{dyn}} = \frac{1}{(4\pi G\rho_{\mathrm{core}})^{1/2}} \approx 1.1 \times 10^{12},\mathrm{s} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-1/2}.
$$
Eq (12.40) — isothermal sound speed in a molecular cloud ($\mu = 2.3,m_p$ ). (p. 350)
$$
c_s = \left( \frac{kT_{\mathrm{core}}}{\mu} \right)^{1/2} \approx 270,\mathrm{m,s^{-1}} \left( \frac{T_{\mathrm{core}}}{20,\mathrm{K}} \right)^{1/2},
$$
Eq (12.41) ★ — Jeans length in a molecular cloud core (from Equation 11.20). (p. 351)
$$
\lambda_J = 2\pi c_s t_{\mathrm{dyn}} \approx 1.9 \times 10^{15},\mathrm{m} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-1/2} \left( \frac{T_{\mathrm{core}}}{20,\mathrm{K}} \right)^{1/2}.
$$
Eq (12.42) ★ — baryonic Jeans mass in a dense molecular cloud core (~15 M_⊙). (p. 351)
$$
\begin{aligned}
M_J = \frac{4\pi}{3}\rho_{\mathrm{core}}\lambda_J^3 &\approx 3 \times 10^{31},\mathrm{kg} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-1/2} \left( \frac{T_{\mathrm{core}}}{20,\mathrm{K}} \right)^{3/2} \\
&\approx 15,\mathrm{M_\odot} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-1/2} \left( \frac{T_{\mathrm{core}}}{20,\mathrm{K}} \right)^{3/2}.
\end{aligned}
$$
Eq (12.43) — thermal energy of a molecular cloud core ($N = M_{\mathrm{core}}/\mu$ gas particles). (p. 351)
$$
E_{\mathrm{core}} = N\left( \frac{3}{2}kT_{\mathrm{core}} \right) = 3.2 \times 10^{36},\mathrm{J} \left( \frac{M_{\mathrm{core}}}{15,\mathrm{M_\odot}} \right) \left( \frac{T_{\mathrm{core}}}{20,\mathrm{K}} \right).
$$
Eq (12.44) — rate of change of thermal energy under adiabatic compression (first law). (p. 352)
$$
\frac{dE_{\mathrm{core}}}{dt} = -P_{\mathrm{core}}\frac{dV_{\mathrm{core}}}{dt} = -\frac{NkT_{\mathrm{core}}}{V_{\mathrm{core}}}\frac{dV_{\mathrm{core}}}{dt}.
$$
Eq (12.45) — same rate expressed via the core radius ($V_{\mathrm{core}} \propto R_{\mathrm{core}}^3$ ). (p. 352)
$$
\frac{dE_{\mathrm{core}}}{dt} = -3NkT_{\mathrm{core}}\left( \frac{1}{R_{\mathrm{core}}}\frac{dR_{\mathrm{core}}}{dt} \right) = -2E_{\mathrm{core}}\left( \frac{1}{R_{\mathrm{core}}}\frac{dR_{\mathrm{core}}}{dt} \right).
$$
Eq (12.46) — isothermal luminosity: rate energy must be radiated to hold $T_{\mathrm{core}}$ constant. (p. 352)
$$
L_{\mathrm{core}}^{\mathrm{iso}} = -\frac{dE_{\mathrm{core}}}{dt} = \frac{2E_{\mathrm{core}}}{R_{\mathrm{core}}}\frac{dR_{\mathrm{core}}}{dt}.
$$
Eq (12.47) — isothermal luminosity for a freely collapsing core at $T_{\mathrm{core}} = 20$ K. (p. 352)
$$
L_{\mathrm{core}}^{\mathrm{iso}} \approx \frac{2E_{\mathrm{core}}}{t_{\mathrm{dyn}}} \approx 0.015,\mathrm{L_\odot} \left( \frac{M_{\mathrm{core}}}{15,\mathrm{M_\odot}} \right) \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{1/2}.
$$
Eq (12.48) — actual luminosity of the (possibly non-opaque) core, with efficiency factor $f_e \le 1$ . (p. 354)
$$
L_{\mathrm{core}} = 4\pi R_{\mathrm{core}}^2 \cdot f_e,\sigma_{\mathrm{sb}}T_{\mathrm{core}}^4,
$$
Eq (12.49) — radius of a molecular cloud core scaled to standard values. (p. 354)
$$
R_{\mathrm{core}} = \left( \frac{3M_{\mathrm{core}}}{4\pi\rho_{\mathrm{core}}} \right)^{1/3} \approx 10^{4},\mathrm{AU} \left( \frac{M_{\mathrm{core}}}{15,\mathrm{M_\odot}} \right)^{1/3} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-1/3},
$$
Eq (12.50) — luminosity actually emitted from the core at $T_{\mathrm{core}} = 20$ K. (p. 354)
$$
L_{\mathrm{core}} \approx 1100,\mathrm{L_\odot},f_e \left( \frac{M_{\mathrm{core}}}{15,\mathrm{M_\odot}} \right)^{2/3} \left( \frac{\rho_{\mathrm{core}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-2/3}.
$$
Eq (12.51) — number of stars produced per logarithmic mass interval by hierarchical fragmentation. (p. 355)
$$
\frac{dN}{d\log M} \propto \frac{1}{M},
$$
Eq (12.52) — isothermal luminosity required to keep a fragment at $T_{\mathrm{core}} = 20$ K. (p. 355)
$$
L_{\mathrm{frag}}^{\mathrm{iso}} \approx 0.015,\mathrm{L_\odot} \left( \frac{M_{\mathrm{frag}}}{15,\mathrm{M_\odot}} \right) \left( \frac{\rho_{\mathrm{frag}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{1/2}.
$$
Eq (12.53) — ratio of a fragment's actual to required isothermal luminosity. (p. 356)
$$
\frac{L_{\mathrm{frag}}}{L_{\mathrm{frag}}^{\mathrm{iso}}} \approx 73,000,f_e \left( \frac{M_{\mathrm{frag}}}{15,\mathrm{M_\odot}} \right)^{-1/3} \left( \frac{\rho_{\mathrm{frag}}}{10^{-15},\mathrm{kg,m^{-3}}} \right)^{-7/6}.
$$
Eq (12.54) — luminosity ratio after $n$ rounds of fragmentation (fragmentation halts when it reaches 1). (p. 356)
$$
\frac{L_{\mathrm{frag}}}{L_{\mathrm{frag}}^{\mathrm{iso}}} \approx 73,000,f_e,2^{n/3},2^{-7n/3} \approx 73,000,f_e,2^{-2n}.
$$
Eq (12.55) — present-day density parameter of planets (estimated from the Solar System's planet-to-star mass ratio). (p. 357) (Rendered as inline text in the EPUB conversion, so absent from the equation-image manifest.)
$$
\Omega_{\mathrm{pl},0} \sim 0.0013,\Omega_{\star,0} \sim (0.0013)(0.003) \sim 4 \times 10^{-6}.
$$
