Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — All Formulas How to read this list
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Quick reference: ★ Essentials Cheat Sheet , Ch 7-12 Formulas , Ch 3-7 Notes , Ch 8-12 Notes
Fundamental Observations Newton versus Einstein Cosmic Dynamics Model Universes Measuring Cosmological Parameters
Chapter 2 — Fundamental Observations 2.1 The Night Sky is Dark Eq (2.1) — average number of stars whose centers lie inside a cylinder of radius $R_\star$ and length $\lambda$ drawn around the line of sight. (p. 21)
$$
N = n_\star V = n_\star \lambda \pi R_\star^2 .
$$
Eq (2.2) — mean free path: the distance you can see before the line of sight is blocked, set by $N=1$ . (p. 21)
$$
\lambda = \frac{1}{n_\star \pi R_\star^2} .
$$
Eq (2.3) — numerical evaluation of the sight distance in an infinite universe of stars. (p. 22)
$$
\lambda \sim \frac{1}{(10^9,\mathrm{Mpc^{-3}})(10^{-27},\mathrm{Mpc^2})} \sim 10^{18},\mathrm{Mpc}
$$
Eq (2.4) — angular area (in steradians) subtended by a star of radius $R_\star$ at distance $r \gg R_\star$ . (p. 22)
$$
\Omega = \frac{\pi R_\star^2}{4\pi r^2} = \frac{R_\star^2}{4 r^2} .
$$
Eq (2.5) — flux (inverse-square law) of a star of luminosity $L_\star$ measured at distance $r$ . (p. 22)
$$
f = \frac{L_\star}{4\pi r^2} .
$$
Eq (2.6) ★ — surface brightness of a star is independent of distance (crux of Olbers' paradox). (p. 22)
$$
\Sigma_\star = \frac{f}{\Omega} = \frac{L_\star}{\pi R_\star^2} ,
$$
2.2 The Universe is Isotropic and Homogeneous (No numbered displayed equations in this section.)
2.3 Redshift is Proportional to Distance Eq (2.7) ★ — definition of redshift $z$ in terms of observed and emitted wavelengths. (p. 28)
$$
z \equiv \frac{\lambda_{\mathrm{ob}} - \lambda_{\mathrm{em}}}{\lambda_{\mathrm{em}}} .
$$
Eq (2.8) ★ — Hubble's law, redshift form. (p. 29)
$$
z = \frac{H_0}{c}, r ,
$$
Eq (2.9) ★ — Hubble's law, velocity form (redshift interpreted as a Doppler shift). (p. 29)
$$
v = H_0, r .
$$
Eq (2.10) ★ — best current value of the Hubble constant, used throughout the book. (p. 30)
$$
H_0 = 68 \pm 2\ \mathrm{km,s^{-1},Mpc^{-1}} .
$$
Eq (2.11) — side length of the galaxy triangle between galaxies 1 and 2. (p. 31)
$$
r_{12} \equiv |\vec{r}_1 - \vec{r}_2| .
$$
Eq (2.12) — side length between galaxies 2 and 3. (p. 31)
$$
r_{23} \equiv |\vec{r}_2 - \vec{r}_3| .
$$
Eq (2.13) — side length between galaxies 3 and 1. (p. 31)
$$
r_{31} \equiv |\vec{r}_3 - \vec{r}_1| .
$$
Eq (2.14) — expansion law for triangle side 1–2 via the scale factor $a(t)$ . (p. 32)
$$
r_{12}(t) = a(t), r_{12}(t_0) .
$$
Eq (2.15) — expansion law for triangle side 2–3. (p. 32)
$$
r_{23}(t) = a(t), r_{23}(t_0) .
$$
Eq (2.16) — expansion law for triangle side 3–1. (p. 32)
$$
r_{31}(t) = a(t), r_{31}(t_0) .
$$
Eq (2.17) — recession speed seen from galaxy 1 toward galaxy 2, giving $H = \dot a/a$ . (p. 32)
$$
v_{12}(t) = \frac{dr_{12}}{dt} = \dot{a}, r_{12}(t_0) = \frac{\dot{a}}{a}, r_{12}(t) .
$$
Eq (2.18) — recession speed from galaxy 1 toward galaxy 3 (same linear relation). (p. 32)
$$
v_{31}(t) = \frac{dr_{31}}{dt} = \dot{a}, r_{31}(t_0) = \frac{\dot{a}}{a}, r_{31}(t) .
$$
Eq (2.19) ★ — Hubble time: elapsed time since galaxies were in contact, assuming constant velocity. (p. 33)
$$
t_0 = \frac{r}{v} = \frac{r}{H_0 r} = H_0^{-1} ,
$$
Eq (2.20) — velocity–distance relation in a Steady State universe. (p. 34)
$$
\frac{dr}{dt} = H_0 r
$$
Eq (2.21) — exponential expansion law resulting from integrating Eq (2.20). (p. 34)
$$
r(t) \propto e^{H_0 t} .
$$
Eq (2.22) — exponential growth of the volume of a spherical region in the Steady State model. (p. 35)
$$
V = \frac{4\pi}{3} r^3 \propto e^{3 H_0 t} .
$$
Eq (2.23) — rate of continuous matter creation needed to keep density constant. (p. 35)
$$
\dot{M}_{\mathrm{ss}} = \rho_0 \dot{V} = \rho_0, 3 H_0 V .
$$
Eq (2.24) — matter-creation rate per unit volume for our universe's mean density. (p. 35)
$$
\frac{\dot{M}_{\mathrm{ss}}}{V} = 3 H_0 \rho_0 \approx 5.6 \times 10^{-28}\ \mathrm{kg,m^{-3},Gyr^{-1}} .
$$
2.4 Different Types of Particles Eq (2.25) — lower limit on the summed neutrino masses from oscillation data. (p. 38)
$$
(m_1 + m_2 + m_3)c^2 = [m(\nu_e) + m(\nu_\mu) + m(\nu_\tau)]c^2 \geq 0.057\ \mathrm{eV} .
$$
Eq (2.26) — upper limit on the summed neutrino masses from large-scale structure. (p. 38)
$$
(m_1 + m_2 + m_3)c^2 = [m(\nu_e) + m(\nu_\mu) + m(\nu_\tau)]c^2 \leq 0.3\ \mathrm{eV} .
$$
Eq (2.27) ★ — Planck blackbody spectrum: energy density of photons in the frequency range $f \to f+df$ . (p. 39)
$$
\varepsilon(f),df = \frac{8\pi h}{c^3}, \frac{f^3,df}{\exp(hf/kT) - 1} ,
$$
Eq (2.28) ★ — total energy density of blackbody radiation (Stefan–Boltzmann, $\propto T^4$ ). (p. 40)
$$
\varepsilon_\gamma = \alpha T^4 ,
$$
Eq (2.29) — the radiation constant $\alpha$ . (p. 40)
$$
\alpha = \frac{\pi^2}{15}, \frac{k^4}{\hbar^3 c^3} = 7.566 \times 10^{-16}\ \mathrm{J,m^{-3},K^{-4}} .
$$
Eq (2.30) — number density of photons in the frequency range $f \to f+df$ . (p. 40)
$$
n(f),df = \frac{\varepsilon(f),df}{hf} = \frac{8\pi}{c^3}, \frac{f^2,df}{\exp(hf/kT) - 1} .
$$
Eq (2.31) ★ — total number density of blackbody photons ($\propto T^3$ ). (p. 41)
$$
n_\gamma = \beta T^3 ,
$$
Eq (2.32) — the constant $\beta$ . (p. 41)
$$
\beta = \frac{2.4041}{\pi^2}, \frac{k^3}{\hbar^3 c^3} = 2.029 \times 10^7\ \mathrm{m^{-3},K^{-3}} .
$$
2.5 Cosmic Microwave Background Eq (2.33) ★ — present-day CMB blackbody temperature. (p. 42)
$$
T_0 = 2.7255 \pm 0.0006\ \mathrm{K} .
$$
Eq (2.34) ★ — present-day energy density of the CMB (Eq 2.28 evaluated at $T_0$ ). (p. 42)
$$
\varepsilon_\gamma = 4.175 \times 10^{-14}\ \mathrm{J,m^{-3}} = 0.2606\ \mathrm{MeV,m^{-3}} .
$$
Eq (2.35) ★ — present-day number density of CMB photons (Eq 2.31 evaluated at $T_0$ ; ~411 cm⁻³). (p. 42)
$$
n_\gamma = 4.107 \times 10^8\ \mathrm{m^{-3}} .
$$
Eq (2.36) — mean energy of a CMB photon (in the microwave region). (p. 43)
$$
E_{\mathrm{mean}} = 6.344 \times 10^{-4}\ \mathrm{eV} .
$$
Eq (2.37) — first law of thermodynamics for the photon gas. (p. 44)
$$
dQ = dE + P,dV ,
$$
Eq (2.38) — first law for adiabatic ($dQ=0$ ) expansion of a homogeneous universe. (p. 44)
$$
\frac{dE}{dt} = -P(t), \frac{dV}{dt} .
$$
Eq (2.39) — Eq (2.38) written out for a photon gas with $E=\alpha T^4 V$ , $P=\alpha T^4/3$ . (p. 44)
$$
\alpha \left( 4 T^3 \frac{dT}{dt} V + T^4 \frac{dV}{dt} \right) = -\frac{1}{3} \alpha T^4 \frac{dV}{dt} ,
$$
Eq (2.40) — simplified logarithmic-rate form of Eq (2.39). (p. 44)
$$
\frac{1}{T} \frac{dT}{dt} = -\frac{1}{3V} \frac{dV}{dt} .
$$
Eq (2.41) ★ — with $V \propto a(t)^3$ , gives the CMB temperature–scale-factor relation $T \propto a^{-1}$ . (p. 44)
$$
\frac{d}{dt}(\ln T) = -\frac{d}{dt}(\ln a) .
$$
Eq (2.42) — "tired light" hypothesis: photon energy loss per unit distance travelled (Exercise 2.4). (p. 45)
$$
\frac{dE}{dr} = -k E ,
$$
Eq (2.43) — fraction of blackbody photons with $hf > E_0$ for a threshold $E_0 \gg kT$ (Exercise 2.5). (p. 46)
$$
\frac{n(hf > E_0)}{n_\gamma} \approx 0.42 \left( \frac{E_0}{kT} \right)^2 \exp!\left( -\frac{E_0}{kT} \right) .
$$
Eq (2.44) — fraction of blackbody photons with $hf < E_0$ for a threshold $E_0 \ll kT$ (Exercise 2.6). (p. 46)
$$
\frac{n(hf < E_0)}{n_\gamma} \approx 0.21 \left( \frac{E_0}{kT} \right)^2 .
$$
Chapter 3 — Newton versus Einstein Eq (3.1) — Newton's law of gravity: attractive force between two spherical masses. (p. 49)
$$
F = -\frac{GM_g m_g}{r^2}.
$$
Eq (3.2) — Newton's second law relating force and acceleration via inertial mass. (p. 50)
$$
F = m_i a.
$$
Eq (3.3) — Equivalence principle: gravitational mass equals inertial mass. (p. 50)
$$
m_g = m_i.
$$
Eq (3.4) — Gravitational acceleration toward mass $M_g$ if the equivalence principle did not hold. (p. 50)
$$
a = -\frac{GM_g}{r^2}\left(\frac{m_g}{m_i}\right),
$$
Eq (3.5) — Poisson's equation relating gravitational potential to mass density. (p. 51)
$$
\nabla^2 \Phi = 4\pi G \rho.
$$
Eq (3.6) — Integral form of Poisson's equation giving the potential from a known density distribution. (p. 51)
$$
\Phi(\vec{r}) = -G \int \frac{\rho(\vec{x})}{|\vec{x} - \vec{r}|}, d^3x.
$$
3.2 The Special Way of Einstein Eq (3.7) — Newton's second law in the form defining an inertial reference frame. (p. 52)
$$
\frac{d^2\vec{r}}{dt^2} = \frac{1}{m}\vec{F},
$$
Eq (3.8) — Galilean transformation between two inertial frames in relative motion at speed $v$ along $x$ . (p. 53)
$$
\begin{aligned}
x' &= x - vt \\
y' &= y \\
z' &= z \\
t' &= t.
\end{aligned}
$$
Eq (3.9) — Size of an expanding spherical light shell in the unprimed frame. (p. 54)
$$
c^2 t^2 = x^2 + y^2 + z^2.
$$
Eq (3.10) — Size of the same light shell in the primed frame. (p. 54)
$$
c^2 (t')^2 = (x')^2 + (y')^2 + (z')^2.
$$
Eq (3.11) ★ — Lorentz transformation between two inertial frames in relative motion. (p. 55)
$$
\begin{aligned}
x' &= \gamma(x - vt) \\
y' &= y \\
z' &= z \\
t' &= \gamma!\left(t - vx/c^2\right),
\end{aligned}
$$
Eq (3.12) ★ — Definition of the Lorentz factor. (p. 55)
$$
\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}.
$$
Eq (3.13) — Spatial distance between two events in the unprimed frame (Euclidean). (p. 55)
$$
(\Delta\ell)^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2.
$$
Eq (3.14) — Time elapsed between the two events in the unprimed frame. (p. 55)
$$
\Delta t = t_1 - t_2.
$$
Eq (3.15) — Spatial distance between the two events in the primed frame, via the Lorentz transformation. (p. 56)
$$
\begin{aligned}
(\Delta\ell')^2 &= (x_1' - x_2')^2 + (y_1' - y_2')^2 + (z_1' - z_2')^2 \\
&= \gamma^2 \left[x_1 - x_2 - v(t_1 - t_2)\right]^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2.
\end{aligned}
$$
Eq (3.16) — Time elapsed between the two events in the primed frame. (p. 56)
$$
\Delta t' = t_1' - t_2' = \gamma\left[t_1 - t_2 - \frac{v}{c^2}(x_1 - x_2)\right].
$$
Eq (3.17) ★ — Spacetime separation (interval) between two events in the unprimed frame; frame-invariant. (p. 56)
$$
(\Delta s)^2 = -c^2(t_1 - t_2)^2 + (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2.
$$
Eq (3.18) ★ — Spacetime interval in compact form. (p. 56)
$$
(\Delta s)^2 = -c^2(\Delta t)^2 + (\Delta\ell)^2.
$$
Eq (3.19) — Spacetime separation in the primed frame (equal to the unprimed value). (p. 56)
$$
(\Delta s')^2 = -c^2(\Delta t')^2 + (\Delta\ell')^2,
$$
Eq (3.20) — Spacetime separation in the primed frame after substituting the Lorentz transformation. (p. 57 — corrected; the printed book has a typo, see note at end.)
$$
\begin{aligned}
(\Delta s')^2 = -\gamma^2 &\left[c(t_1 - t_2) - \frac{v}{c}(x_1 - x_2)\right]^2 \\
&+ \gamma^2 \left[x_1 - x_2 - v(t_1 - t_2)\right]^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2.
\end{aligned}
$$
Eq (3.21) — Simplified result showing the spacetime separation is frame-invariant: $(\Delta s)^2 = (\Delta s')^2$ . (p. 57)
$$
(\Delta s')^2 = -c^2(t_1 - t_2)^2 + (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2,
$$
3.3 The General Way of Einstein (No numbered displayed equations in this section. The only equation mentioned, $E = mc^2$, appears inline in the text, and the section is otherwise the equivalence-principle thought experiments — Figures 3.2 and 3.3 — and the geodesic viewpoint.)
Eq (3.22) — On a flat (Euclidean) plane, the interior angles of a geodesic triangle sum to $\pi$ . (p. 61)
$$
\alpha + \beta + \gamma = \pi,
$$
Eq (3.23) — Metric of a flat 2D plane in cartesian coordinates (Pythagorean line element). (p. 62)
$$
d\ell^2 = dx^2 + dy^2.
$$
Eq (3.24) — Same flat 2D metric written in polar coordinates. (p. 62)
$$
d\ell^2 = dr^2 + r^2 d\theta^2.
$$
Eq (3.25) ★ — Angle sum of a geodesic triangle on a sphere of radius $R$ (positive curvature); angle excess is proportional to area. (p. 63)
$$
\alpha + \beta + \gamma = \pi + A/R^2,
$$
Eq (3.26) — Metric of a 2D sphere of radius $R$ (uniform positive curvature), in geodesic polar coordinates. (p. 63)
$$
d\ell^2 = dr^2 + R^2 \sin^2(r/R), d\theta^2.
$$
Eq (3.27) — Angle sum of a geodesic triangle on a uniformly negatively curved (hyperbolic) 2D surface; angle deficit proportional to area. (p. 65)
$$
\alpha + \beta + \gamma = \pi - A/R^2,
$$
Eq (3.28) — Metric of a 2D surface of uniform negative curvature, in geodesic polar coordinates. (p. 65)
$$
d\ell^2 = dr^2 + R^2 \sinh^2(r/R), d\theta^2.
$$
Eq (3.29) — Metric of a flat 3D space ($\kappa = 0$ ) in cartesian coordinates. (p. 65)
$$
d\ell^2 = dx^2 + dy^2 + dz^2,
$$
Eq (3.30) — Metric of a flat 3D space in spherical coordinates. (p. 66)
$$
d\ell^2 = dr^2 + r^2\left[d\theta^2 + \sin^2\theta, d\phi^2\right],
$$
Eq (3.31) — Metric of a 3D space of uniform positive curvature ($\kappa = +1$ ). (p. 66)
$$
d\ell^2 = dr^2 + R^2 \sin^2(r/R)\left[d\theta^2 + \sin^2\theta, d\phi^2\right].
$$
Eq (3.32) — Metric of a 3D space of uniform negative curvature ($\kappa = -1$ ). (p. 66)
$$
d\ell^2 = dr^2 + R^2 \sinh^2(r/R)\left[d\theta^2 + \sin^2\theta, d\phi^2\right].
$$
Eq (3.33) ★ — Compact unified metric for a homogeneous, isotropic 3D space (all three curvatures). (p. 66)
$$
d\ell^2 = dr^2 + S_\kappa(r)^2, d\Omega^2,
$$
Eq (3.34) — Definition of the solid-angle element $d\Omega^2$ . (p. 66)
$$
d\Omega^2 \equiv d\theta^2 + \sin^2\theta, d\phi^2
$$
Eq (3.35) ★ — The function $S_\kappa(r)$ encoding the three curvature cases. (p. 67)
$$
S_\kappa(r) =
\begin{cases}
R\sin(r/R) & (\kappa = +1) \\
r & (\kappa = 0) \\
R\sinh(r/R) & (\kappa = -1).
\end{cases}
$$
Eq (3.36) — Same homogeneous, isotropic 3D metric with radial coordinate $x \equiv S_\kappa(r)$ . (p. 67)
$$
d\ell^2 = \frac{dx^2}{1 - \kappa x^2/R^2} + x^2 d\Omega^2.
$$
3.5 The Robertson–Walker Metric Eq (3.37) ★ — The Minkowski metric: spacetime interval in flat, static (special-relativistic) spacetime. (p. 68)
$$
ds^2 = -c^2 dt^2 + dr^2 + r^2 d\Omega^2.
$$
Eq (3.38) — Null-geodesic condition ($ds = 0$ ) for a photon in Minkowski spacetime. (p. 68)
$$
ds^2 = 0 = -c^2 dt^2 + dr^2 + r^2 d\Omega^2.
$$
Eq (3.39) — Radial null geodesic ($\theta,\phi$ constant). (p. 68)
$$
c^2 dt^2 = dr^2,
$$
Eq (3.40) — Radial photon moves at the speed of light. (p. 68)
$$
\frac{dr}{dt} = \pm c.
$$
Eq (3.41) ★ — The Robertson–Walker metric: spacetime interval for a spatially homogeneous, isotropic, expanding (or contracting) universe. (p. 69)
$$
ds^2 = -c^2 dt^2 + a(t)^2\left[dr^2 + S_\kappa(r)^2, d\Omega^2\right],
$$
Eq (3.42) — Robertson–Walker metric evaluated at a fixed time $t$ (temporal part vanishes), used to measure proper distance. (p. 70)
$$
ds^2 = a(t)^2\left[dr^2 + S_\kappa(r)^2, d\Omega^2\right].
$$
Eq (3.43) — Line element along the radial spatial geodesic ($\theta,\phi$ constant). (p. 70)
$$
ds = a(t), dr.
$$
Eq (3.44) ★ — Proper distance to a galaxy at comoving coordinate $r$ ; proportional to the scale factor. (p. 71)
$$
d_p(t) = a(t)\int_0^r dr = a(t), r.
$$
Eq (3.45) — Rate of change of proper distance (recession due to expansion). (p. 71)
$$
\dot{d}_p = \dot{a}, r = \frac{\dot{a}}{a}, d_p.
$$
Eq (3.46) ★ — Hubble's law: linear relation between recession speed and proper distance at the present time. (p. 71)
$$
v_p(t_0) = H_0, d_p(t_0),
$$
Eq (3.47) — Definition of the recession speed as the time derivative of proper distance. (p. 71)
$$
v_p(t_0) \equiv \dot{d}_p(t_0)
$$
Eq (3.48) ★ — Definition of the Hubble constant as the present-day value of $\dot{a}/a$ . (p. 71)
$$
H_0 = \left(\frac{\dot{a}}{a}\right)_{t=t_0}.
$$
Eq (3.49) ★ — Definition of the Hubble distance. (p. 72)
$$
d_H(t_0) \equiv c/H_0,
$$
Eq (3.50) — Points beyond the Hubble distance recede faster than light (allowed in GR). (p. 72)
$$
v_p = \dot{d}_p > c.
$$
Eq (3.51) — Present-day Hubble distance for $H_0 = 68 \pm 2,\mathrm{km,s^{-1},Mpc^{-1}}$ . (p. 72)
$$
d_H(t_0) = c/H_0 = 4380 \pm 130,\mathrm{Mpc}.
$$
Eq (3.52) — Radial null geodesic in the Robertson–Walker metric for light from a distant galaxy. (p. 73)
$$
c^2 dt^2 = a(t)^2 dr^2.
$$
Eq (3.53) — Rearranged null-geodesic relation (LHS depends only on $t$ , RHS only on $r$ ). (p. 73)
$$
c,\frac{dt}{a(t)} = dr.
$$
Eq (3.54) — Integrated null geodesic for the first wave crest (emitted at $t_e$ , observed at $t_0$ ). (p. 73)
$$
c\int_{t_e}^{t_0} \frac{dt}{a(t)} = \int_0^r dr = r.
$$
Eq (3.55) — Integrated null geodesic for the next wave crest (emitted at $t_e + \lambda_e/c$ , observed at $t_0 + \lambda_0/c$ ). (p. 73)
$$
c\int_{t_e + \lambda_e/c}^{t_0 + \lambda_0/c} \frac{dt}{a(t)} = \int_0^r dr = r.
$$
Eq (3.56) — The integral of $dt/a(t)$ from emission to observation is the same for every wave crest. (p. 74)
$$
\int_{t_e}^{t_0} \frac{dt}{a(t)} = \int_{t_e + \lambda_e/c}^{t_0 + \lambda_0/c} \frac{dt}{a(t)}.
$$
Eq (3.57) — The overlapping integral subtracted from each side of Eq (3.56). (p. 74)
$$
\int_{t_e + \lambda_e/c}^{t_0} \frac{dt}{a(t)}
$$
Eq (3.58) — Integral between emission of two successive wave crests equals the integral between their observations. (p. 74)
$$
\int_{t_e}^{t_e + \lambda_e/c} \frac{dt}{a(t)} = \int_{t_0}^{t_0 + \lambda_0/c} \frac{dt}{a(t)}.
$$
Eq (3.59) — Same relation with $a(t)$ taken constant over the short wave-crest interval. (p. 74)
$$
\frac{1}{a(t_e)}\int_{t_e}^{t_e + \lambda_e/c} dt = \frac{1}{a(t_0)}\int_{t_0}^{t_0 + \lambda_0/c} dt,
$$
Eq (3.60) — Emitted and observed wavelengths scale with the scale factor. (p. 74)
$$
\frac{\lambda_e}{a(t_e)} = \frac{\lambda_0}{a(t_0)}.
$$
Eq (3.61) ★ — Redshift–scale-factor relation (with the convention $a(t_0)=1$ ). (p. 75)
$$
1 + z = \frac{a(t_0)}{a(t_e)} = \frac{1}{a(t_e)}.
$$
Eq (3.62) — (Exercise 3.3) Circumference of a circle of radius $r$ on a sphere of radius $R$ . (p. 75)
$$
C = 2\pi R \sin(r/R).
$$
Chapter 4 — Cosmic Dynamics 4.0 Chapter Introduction (curvature and observable angular size) Eq (4.1) — sum of a triangle's angles in a curved space, generalizing the flat-space result. (p. 77)
$$
\alpha + \beta + \gamma = \pi + \frac{\kappa A}{R_0^2},
$$
Eq (4.2) — small-angle formula for the observed angular size of a galaxy in a flat universe. (p. 77)
$$
\alpha = \frac{D}{r}.
$$
Eq (4.3) — observed angular size in a positively curved universe (curvature magnifies). (p. 78)
$$
\alpha_{+} = \frac{D}{R_0 \sin(r/R_0)}.
$$
Eq (4.4) — observed angular size in a negatively curved universe (always smaller than flat). (p. 78)
$$
\alpha_{-} = \frac{D}{R_0 \sinh(r/R_0)} < \frac{D}{r}.
$$
Eq (4.5) — far-field limit ($r \gg R_0$ ) of the negatively-curved angular size. (p. 78)
$$
\alpha_{-} \approx \frac{2D}{R_0} \exp!\left(-\frac{r}{R_0}\right).
$$
4.1 Einstein's Field Equation Eq (4.6) — Poisson's equation for the Newtonian gravitational potential. (p. 79)
$$
\nabla^2 \Phi = 4\pi G \rho,
$$
Eq (4.7) ★ — Einstein's field equation relating spacetime curvature to the stress-energy of matter. (p. 79)
$$
G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}.
$$
Eq (4.8) ★ — the homogeneous, isotropic Robertson–Walker metric. (p. 80)
$$
ds^2 = -c^2 dt^2 + a(t)^2\left[dr^2 + S_\kappa(r)^2, d\Omega^2\right],
$$
Eq (4.9) — the curvature function $S_\kappa(r)$ for the three curvature cases. (p. 80)
$$
S_\kappa(r) =
\begin{cases}
R_0 \sin(r/R_0) & (\kappa = +1) \\
r & (\kappa = 0) \\
R_0 \sinh(r/R_0) & (\kappa = -1).
\end{cases}
$$
4.2 The Friedmann Equation Eq (4.10) — Newton's law of gravity on a test mass at the surface of the sphere. (p. 81)
$$
F = -\frac{G M_s m}{R_s(t)^2}.
$$
Eq (4.11) — gravitational acceleration of the sphere's surface (Newton's second law). (p. 82)
$$
\frac{d^2 R_s}{dt^2} = -\frac{G M_s}{R_s(t)^2}.
$$
Eq (4.12) — energy conservation after integrating, with constant of integration $U$ . (p. 82)
$$
\frac{1}{2}\left(\frac{dR_s}{dt}\right)^2 = \frac{G M_s}{R_s(t)} + U,
$$
Eq (4.13) — kinetic energy per unit mass of the sphere's surface. (p. 82)
$$
\epsilon_{\mathrm{kin}} = \frac{1}{2}\left(\frac{dR_s}{dt}\right)^2,
$$
Eq (4.14) — gravitational potential energy per unit mass. (p. 82)
$$
\epsilon_{\mathrm{pot}} = -\frac{G M_s}{R_s(t)},
$$
Eq (4.15) — constant total mass of the sphere in terms of mass density. (p. 83)
$$
M_s = \frac{4\pi}{3}\rho(t) R_s(t)^3.
$$
Eq (4.16) — sphere's proper radius written via the scale factor and comoving radius. (p. 83)
$$
R_s(t) = a(t) r_s,
$$
Eq (4.17) — energy-conservation relation rewritten in terms of $\rho(t)$ and $a(t)$ . (p. 83)
$$
\frac{1}{2} r_s^2 \dot{a}^2 = \frac{4\pi}{3} G r_s^2 \rho(t) a(t)^2 + U.
$$
Eq (4.18) ★ — the Friedmann equation in its Newtonian form. (p. 83)
$$
\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho(t) + \frac{2U}{r_s^2}\frac{1}{a(t)^2}.
$$
Eq (4.19) — maximum scale factor for the bound ($U<0$ ) case. (p. 84)
$$
a_{\max} = -\frac{G M_s}{U r_s},
$$
Eq (4.20) ★ — the Friedmann equation in its correct relativistic form. (p. 85)
$$
\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon(t) - \frac{\kappa c^2}{R_0^2}\frac{1}{a(t)^2}.
$$
Eq (4.21) — relativistic energy of a particle (why energy density, not mass, gravitates). (p. 85)
$$
E = (m^2 c^4 + p^2 c^2)^{1/2}.
$$
Eq (4.22) — nonrelativistic limit: rest energy plus kinetic correction. (p. 85)
$$
E_{\mathrm{nonrel}} \approx mc^2(1 + v^2/c^2)^{1/2} \approx mc^2 + \frac{1}{2}mv^2.
$$
Eq (4.23) — energy of a photon / massless particle. (p. 85)
$$
E_{\mathrm{rel}} = pc = hf,
$$
Eq (4.24) — substitution linking the Newtonian constant $U$ to the relativistic curvature term. (p. 86)
$$
\frac{2U}{r_s^2} = -\frac{\kappa c^2}{R_0^2}.
$$
Eq (4.25) — linear velocity–distance (Hubble) relation for a scale-factor universe. (p. 86)
$$
v(t) = H(t) d(t),
$$
Eq (4.26) — Friedmann equation written with the Hubble parameter $H(t)\equiv\dot a/a$ . (p. 86)
$$
H(t)^2 = \frac{8\pi G}{3c^2}\varepsilon(t) - \frac{\kappa c^2}{R_0^2 a(t)^2}.
$$
Eq (4.27) — present-day value of the Hubble parameter (the Hubble constant). (p. 87)
$$
H_0 = H(t_0) = \left(\frac{\dot a}{a}\right)_{t=t_0} = 68 \pm 2\ \mathrm{km,s^{-1},Mpc^{-1}}.
$$
Eq (4.28) — Friedmann equation evaluated at the present moment. (p. 87)
$$
H_0^2 = \frac{8\pi G}{3c^2}\varepsilon_0 - \frac{\kappa c^2}{R_0^2},
$$
Eq (4.29) — Friedmann equation in a spatially flat universe ($\kappa=0$ ). (p. 87)
$$
H(t)^2 = \frac{8\pi G}{3c^2}\varepsilon(t).
$$
Eq (4.30) ★ — definition of the critical density. (p. 87)
$$
\varepsilon_c(t) \equiv \frac{3c^2}{8\pi G} H(t)^2.
$$
Eq (4.31) — present-day critical (energy) density, numerical value. (p. 87–88)
$$
\varepsilon_{c,0} = \frac{3c^2}{8\pi G} H_0^2 = (7.8 \pm 0.5)\times 10^{-10}\ \mathrm{J,m^{-3}} = 4870 \pm 290\ \mathrm{MeV,m^{-3}}.
$$
Eq (4.32) — critical density as an equivalent mass density. (p. 88)
$$
\rho_{c,0} \equiv \frac{\varepsilon_{c,0}}{c^2} = (8.7 \pm 0.5)\times 10^{-27}\ \mathrm{kg,m^{-3}} = (1.28 \pm 0.08)\times 10^{11}\ M_\odot,\mathrm{Mpc^{-3}}.
$$
Eq (4.33) ★ — definition of the dimensionless density parameter $\Omega$ . (p. 88)
$$
\Omega(t) \equiv \frac{\varepsilon(t)}{\varepsilon_c(t)}.
$$
Eq (4.34) — Friedmann equation recast in terms of $\Omega$ (curvature vs. density). (p. 88)
$$
1 - \Omega(t) = -\frac{\kappa c^2}{R_0^2 a(t)^2 H(t)^2}.
$$
Eq (4.35) — present-day relation between density parameter and curvature. (p. 89)
$$
1 - \Omega_0 = -\frac{\kappa c^2}{R_0^2 H_0^2},
$$
Eq (4.36) — solving for the present curvature $\kappa/R_0^2$ . (p. 89)
$$
\frac{\kappa}{R_0^2} = \frac{H_0^2}{c^2}(\Omega_0 - 1).
$$
4.3 The Fluid and Acceleration Equations Eq (4.37) — first law of thermodynamics. (p. 89)
$$
dQ = dE + P,dV,
$$
Eq (4.38) — adiabatic ($dQ=0$ ) first law for a comoving volume. (p. 90)
$$
\dot{E} + P\dot{V} = 0.
$$
Eq (4.39) — volume of a comoving sphere. (p. 90)
$$
V(t) = \frac{4\pi}{3} r_s^3 a(t)^3,
$$
Eq (4.40) — time rate of change of the comoving volume. (p. 90)
$$
\dot{V} = \frac{4\pi}{3} r_s^3 (3 a^2 \dot{a}) = V\left(3\frac{\dot a}{a}\right).
$$
Eq (4.41) — internal energy of the sphere. (p. 90)
$$
E(t) = V(t)\varepsilon(t),
$$
Eq (4.42) — rate of change of internal energy. (p. 91)
$$
\dot{E} = V\dot{\varepsilon} + \dot{V}\varepsilon = V\left(\dot{\varepsilon} + 3\frac{\dot a}{a}\varepsilon\right).
$$
Eq (4.43) — first law combining Eqs 4.38, 4.40, 4.42. (p. 91)
$$
V\left(\dot{\varepsilon} + 3\frac{\dot a}{a}\varepsilon + 3\frac{\dot a}{a}P\right) = 0,
$$
Eq (4.44) ★ — the fluid equation. (p. 91)
$$
\dot{\varepsilon} + 3\frac{\dot a}{a}(\varepsilon + P) = 0.
$$
Eq (4.45) — Friedmann equation (Eq 4.20) multiplied by $a^2$ . (p. 91)
$$
\dot{a}^2 = \frac{8\pi G}{3c^2}\varepsilon a^2 - \frac{\kappa c^2}{R_0^2}.
$$
Eq (4.46) — time derivative of Eq 4.45. (p. 92)
$$
2\dot{a}\ddot{a} = \frac{8\pi G}{3c^2}\left(\dot{\varepsilon} a^2 + 2\varepsilon a \dot{a}\right).
$$
Eq (4.47) — dividing by $2\dot{a}a$ . (p. 92)
$$
\frac{\ddot a}{a} = \frac{4\pi G}{3c^2}\left(\dot{\varepsilon}\frac{a}{\dot a} + 2\varepsilon\right).
$$
Eq (4.48) — substitution from the fluid equation. (p. 92)
$$
\dot{\varepsilon}\frac{a}{\dot a} = -3(\varepsilon + P)
$$
Eq (4.49) ★ — the acceleration equation. (p. 92)
$$
\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}(\varepsilon + 3P).
$$
Eq (4.50) — condition on pressure for a component to drive accelerated expansion. (p. 92)
$$
P < -\frac{1}{3}\varepsilon.
$$
Eq (4.51) — the Friedmann equation (recap of the three key equations). (p. 93)
$$
\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon - \frac{\kappa c^2}{R_0^2 a^2},
$$
Eq (4.52) — the fluid equation (recap). (p. 93)
$$
\dot{\varepsilon} + 3\frac{\dot a}{a}(\varepsilon + P) = 0,
$$
Eq (4.53) — the acceleration equation (recap). (p. 93)
$$
\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}(\varepsilon + 3P).
$$
Eq (4.54) — the general form of an equation of state (numbered, absent from manifest). (p. 93)
$$
P = P(\varepsilon),
$$
Eq (4.55) ★ — the linear equation of state (numbered, absent from manifest). (p. 94)
$$
P = w\varepsilon,
$$
Eq (4.56) — the perfect (ideal) gas law. (p. 94)
$$
P = \frac{\rho}{\mu} kT,
$$
Eq (4.57) — perfect gas law expressed in terms of energy density. (p. 94)
$$
P \approx \frac{kT}{\mu c^2}\varepsilon.
$$
Eq (4.58) — relation between temperature and rms thermal velocity. (p. 94)
$$
3kT = \mu \langle v^2 \rangle.
$$
Eq (4.59) — equation of state for a nonrelativistic gas. (p. 95)
$$
P_{\mathrm{nonrel}} = w \varepsilon_{\mathrm{nonrel}},
$$
Eq (4.60) — the tiny equation-of-state parameter of a nonrelativistic gas. (p. 95)
$$
w \approx \frac{\langle v^2 \rangle}{3c^2} \ll 1.
$$
Eq (4.61) ★ — equation of state for photons / any relativistic gas ($w=1/3$ ). (p. 95)
$$
P_{\mathrm{rel}} = \frac{1}{3}\varepsilon_{\mathrm{rel}}.
$$
4.5 Learning to Love Lambda Eq (4.62) — Poisson's equation for the potential of a matter density. (p. 97)
$$
\nabla^2 \Phi = 4\pi G \rho.
$$
Eq (4.63) — gravitational acceleration as the gradient of the potential. (p. 97)
$$
\vec{a} = -\vec{\nabla}\Phi.
$$
Eq (4.64) — a static Newtonian universe requires zero density. (p. 97)
$$
\rho = \frac{1}{4\pi G}\nabla^2\Phi = 0.
$$
Eq (4.65) — Poisson's equation modified with a cosmological-constant term $\Lambda$ . (p. 98)
$$
\nabla^2 \Phi + \Lambda = 4\pi G \rho.
$$
Eq (4.66) ★ — the Friedmann equation with the cosmological constant $\Lambda$ . (p. 98)
$$
\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon - \frac{\kappa c^2}{R_0^2 a^2} + \frac{\Lambda}{3}.
$$
Eq (4.67) — the fluid equation (unaffected by $\Lambda$ ). (p. 98)
$$
\dot{\varepsilon} + 3\frac{\dot a}{a}(\varepsilon + P) = 0.
$$
Eq (4.68) ★ — the acceleration equation with the $\Lambda$ term. (p. 98)
$$
\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}(\varepsilon + 3P) + \frac{\Lambda}{3}.
$$
Eq (4.69) ★ — energy density associated with the cosmological constant. (p. 98)
$$
\varepsilon_\Lambda \equiv \frac{c^2}{8\pi G}\Lambda.
$$
Eq (4.70) ★ — pressure of the cosmological constant ($w=-1$ ). (p. 99)
$$
P_\Lambda = -\varepsilon_\Lambda = -\frac{c^2}{8\pi G}\Lambda.
$$
Eq (4.71) — static-universe condition ($\ddot a = 0$ ) from the acceleration equation. (p. 99)
$$
0 = -\frac{4\pi G}{3}\rho + \frac{\Lambda}{3}.
$$
Eq (4.72) — static-universe condition ($\dot a = 0$ ) from the Friedmann equation. (p. 99)
$$
0 = \frac{8\pi G}{3}\rho - \frac{\kappa c^2}{R_0^2} + \frac{\Lambda}{3} = 4\pi G \rho - \frac{\kappa c^2}{R_0^2}.
$$
Eq (4.73) — radius of curvature of Einstein's static universe. (p. 99)
$$
R_0 = \frac{c}{2(\pi G \rho)^{1/2}} = \frac{c}{\Lambda^{1/2}}.
$$
Eq (4.74) — energy–time uncertainty relation for virtual particle pairs. (p. 101)
$$
\Delta E, \Delta t \lesssim h.
$$
Eq (4.75) — proposed "natural" vacuum energy density (the Planck energy density). (p. 101)
$$
\varepsilon_{\mathrm{vac}} \sim \frac{E_P}{\ell_P^3} \qquad (\text{???}).
$$
Eq (4.76) — numerical value of the predicted vacuum energy density. (p. 101)
$$
\varepsilon_{\mathrm{vac}} \sim 3 \times 10^{132}\ \mathrm{eV,m^{-3}} \qquad (!!!).
$$
Eq (4.77) — energy per particle in Exercise 4.5 (de Broglie form). (p. 103)
$$
E = (m^2 c^4 + p^2 c^2)^{1/2} = (m^2 c^4 + h^2 c^2/\lambda^2)^{1/2}.
$$
Chapter 5 — Model Universes Introductory equations (governing relations restated at the chapter opening):
Eq (5.1) ★ — Friedmann equation relating expansion rate to energy density and curvature. (p. 105)
$$
\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon - \frac{\kappa c^2}{R_0^2 a^2},
$$
Eq (5.2) ★ — Fluid equation governing evolution of energy density. (p. 105)
$$
\dot{\varepsilon} + 3\frac{\dot{a}}{a}(\varepsilon + P) = 0,
$$
Eq (5.3) ★ — Equation of state relating pressure to energy density. (p. 105)
$$
P = w\varepsilon.
$$
5.1 Evolution of Energy Density Eq (5.4) — Total energy density as the sum over components. (p. 106)
$$
\varepsilon = \sum_i \varepsilon_i.
$$
Eq (5.5) — Total pressure as the sum over components. (p. 106)
$$
P = \sum_i w_i \varepsilon_i.
$$
Eq (5.6) — Fluid equation holds separately for each non-interacting component. (p. 106)
$$
\dot{\varepsilon}_i + 3\frac{\dot{a}}{a}(\varepsilon_i + P_i) = 0
$$
Eq (5.7) — Fluid equation for component $i$ using its equation-of-state parameter. (p. 106)
$$
\dot{\varepsilon}_i + 3\frac{\dot{a}}{a}(1 + w_i)\varepsilon_i = 0.
$$
Eq (5.8) — Fluid equation rewritten as a separable differential relation. (p. 107)
$$
\frac{d\varepsilon_i}{\varepsilon_i} = -3(1 + w_i)\frac{da}{a}.
$$
Eq (5.9) ★ — General energy-density scaling for a component with constant $w_i$ . (p. 107)
$$
\varepsilon_i(a) = \varepsilon_{i,0}, a^{-3(1+w_i)}.
$$
Eq (5.10) ★ — Matter energy density scales as $a^{-3}$ . (p. 107)
$$
\varepsilon_m(a) = \varepsilon_{m,0}/a^3.
$$
Eq (5.11) ★ — Radiation energy density scales as $a^{-4}$ . (p. 107)
$$
\varepsilon_r(a) = \varepsilon_{r,0}/a^4.
$$
Eq (5.12) — Present-day energy density of the CMB. (p. 108)
$$
\varepsilon_{\mathrm{CMB},0} = \alpha T_0^4 = 4.175 \times 10^{-14},\mathrm{J,m^{-3}} = 0.2606,\mathrm{MeV,m^{-3}}.
$$
Eq (5.13) — Present-day density parameter of the CMB. (p. 108)
$$
\Omega_{\mathrm{CMB},0} = \frac{\varepsilon_{\mathrm{CMB},0}}{\varepsilon_{c,0}} = \frac{0.2606,\mathrm{MeV,m^{-3}}}{4870,\mathrm{MeV,m^{-3}}} = 5.35 \times 10^{-5}.
$$
Eq (5.14) — Present luminosity density of galaxies. (p. 109)
$$
\Psi \approx 1.7 \times 10^8,\mathrm{L_\odot,Mpc^{-3}} \approx 2.2 \times 10^{-33},\mathrm{watts,m^{-3}}.
$$
Eq (5.15) — Rough estimate of the present energy density in starlight. (p. 109)
$$
\begin{aligned}
\varepsilon_{\mathrm{starlight},0} &\sim \Psi t_0 \sim (2.2 \times 10^{-33},\mathrm{J,s^{-1},m^{-3}})(4.5 \times 10^{17},\mathrm{s})\\
&\sim 10^{-15},\mathrm{J,m^{-3}} \sim 0.006,\mathrm{MeV,m^{-3}}.
\end{aligned}
$$
Eq (5.16) — Predicted energy density of one neutrino flavor relative to the CMB. (p. 110)
$$
\varepsilon = \frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\varepsilon_{\mathrm{CMB}} = 0.227,\varepsilon_{\mathrm{CMB}}.
$$
Eq (5.17) — Mean energy per neutrino as a function of scale factor. (p. 110)
$$
E_\nu \approx \frac{5 \times 10^{-4},\mathrm{eV}}{a},
$$
Eq (5.18) — Present-day radiation density parameter (CMB + neutrinos). (p. 110)
$$
\Omega_{r,0} = \Omega_{\mathrm{CMB},0} + \Omega_{\nu,0} = 5.35 \times 10^{-5} + 3.65 \times 10^{-5} = 9.00 \times 10^{-5}.
$$
Eq (5.19) — Present-day ratio of $\Lambda$ energy density to matter energy density (Benchmark Model). (p. 111)
$$
\frac{\varepsilon_{\Lambda,0}}{\varepsilon_{m,0}} = \frac{\Omega_{\Lambda,0}}{\Omega_{m,0}} \approx \frac{0.69}{0.31} \approx 2.23.
$$
Eq (5.20) — Ratio of $\Lambda$ to matter energy density as a function of scale factor. (p. 111)
$$
\frac{\varepsilon_\Lambda(a)}{\varepsilon_m(a)} = \frac{\varepsilon_{\Lambda,0}}{\varepsilon_{m,0}/a^3} = \frac{\Omega_{\Lambda,0}}{\Omega_{m,0}},a^3.
$$
Eq (5.21) ★ — Scale factor at matter–$\Lambda$ equality. (p. 111)
$$
a_{m\Lambda} = \left(\frac{\Omega_{m,0}}{\Omega_{\Lambda,0}}\right)^{1/3} \approx \left(\frac{0.31}{0.69}\right)^{1/3} \approx 0.766.
$$
Eq (5.22) — Present-day ratio of matter to radiation energy density (Benchmark Model). (p. 111)
$$
\frac{\varepsilon_{m,0}}{\varepsilon_{r,0}} = \frac{\Omega_{m,0}}{\Omega_{r,0}} \approx \frac{0.31}{9.0 \times 10^{-5}} \approx 3400,
$$
Eq (5.23) — Ratio of matter to radiation energy density as a function of scale factor. (p. 112)
$$
\frac{\varepsilon_m(a)}{\varepsilon_r(a)} = \frac{\varepsilon_{m,0}}{\varepsilon_{r,0}},a.
$$
Eq (5.24) ★ — Scale factor at radiation–matter equality. (p. 112)
$$
a_{rm} = \frac{\varepsilon_{r,0}}{\varepsilon_{m,0}} \approx \frac{1}{3400} \approx 2.9 \times 10^{-4}.
$$
Eq (5.25) ★ — Friedmann equation for a multiple-component universe, showing each component's scale-factor dependence. (p. 113)
$$
\dot{a}^2 = \frac{8\pi G}{3c^2}\sum_i \varepsilon_{i,0}, a^{-1-3w_i} - \frac{\kappa c^2}{R_0^2}.
$$
Eq (5.26) — Friedmann equation for an empty universe. (p. 113)
$$
\dot{a}^2 = -\frac{\kappa c^2}{R_0^2}.
$$
Eq (5.27) — Expansion rate of a negatively curved ($\kappa = -1$ ) empty universe. (p. 113)
$$
\dot{a} = \pm,\frac{c}{R_0}.
$$
Eq (5.28) ★ — Scale factor grows linearly with time in an expanding empty universe (Milne universe). (p. 114)
$$
a(t) = \frac{t}{t_0},
$$
Eq (5.29) — Redshift–scale-factor–time relation in an empty expanding universe. (p. 115)
$$
1 + z = \frac{1}{a(t_e)} = \frac{t_0}{t_e},
$$
Eq (5.30) — Emission time of observed light in an empty universe. (p. 115)
$$
t_e = \frac{t_0}{1 + z} = \frac{H_0^{-1}}{1 + z}.
$$
Eq (5.31) — Present proper distance to a galaxy in a Robertson–Walker metric. (p. 115)
$$
d_p(t_0) = a(t_0)\int_0^r dr = r.
$$
Eq (5.32) — Null geodesic condition relating light travel to comoving distance. (p. 115)
$$
c\int_{t_e}^{t_0} \frac{dt}{a(t)} = \int_0^r dr = r.
$$
Eq (5.33) — Present proper distance as an integral over cosmic time (any RW metric). (p. 116)
$$
d_p(t_0) = c\int_{t_e}^{t_0} \frac{dt}{a(t)}.
$$
Eq (5.34) — Proper distance in an empty expanding universe. (p. 116)
$$
d_p(t_0) = ct_0\int_{t_e}^{t_0} \frac{dt}{t} = ct_0 \ln\left(\frac{t_0}{t_e}\right).
$$
Eq (5.35) — Proper distance vs. redshift in an empty expanding universe. (p. 116)
$$
d_p(t_0) = \frac{c}{H_0}\ln(1 + z).
$$
Eq (5.36) — Proper distance at time of emission in an empty universe. (p. 117)
$$
d_p(t_e) = \frac{c}{H_0}\frac{\ln(1 + z)}{1 + z},
$$
5.3 Single-component Universes Eq (5.37) ★ — Friedmann equation for a spatially flat, single-component universe. (p. 117)
$$
\dot{a}^2 = \frac{8\pi G \varepsilon_0}{3c^2},a^{-(1+3w)}.
$$
Eq (5.38) — Power-law exponent for the scale factor. (p. 118)
$$
q = \frac{2}{3 + 3w},
$$
Eq (5.39) ★ — Scale factor in a flat, single-component universe. (p. 118)
$$
a(t) = \left(\frac{t}{t_0}\right)^{2/(3+3w)}.
$$
Eq (5.40) — Age of the universe linked to present energy density. (p. 118)
$$
t_0 = \frac{1}{1 + w}\left(\frac{c^2}{6\pi G \varepsilon_0}\right)^{1/2}.
$$
Eq (5.41) — Hubble constant in a flat, single-component universe. (p. 118)
$$
H_0 \equiv \left(\frac{\dot{a}}{a}\right)_{t=t_0} = \frac{2}{3(1 + w)},t_0^{-1}.
$$
Eq (5.42) — Age of the universe in terms of the Hubble time. (p. 118)
$$
t_0 = \frac{2}{3(1 + w)},H_0^{-1}.
$$
Eq (5.43) — Energy density vs. scale factor for a single component (restatement of 5.9). (p. 119)
$$
\varepsilon(a) = \varepsilon_0, a^{-3(1+w)},
$$
Eq (5.44) — Energy density as a function of time in a flat, single-component universe. (p. 119)
$$
\varepsilon(t) = \varepsilon_0\left(\frac{t}{t_0}\right)^{-2},
$$
Eq (5.45) — Present energy density expressed via the Hubble constant (critical density). (p. 119 — corrected; the printed book drops the $G$ in the last term, see note at end.)
$$
\varepsilon_0 = \varepsilon_{c,0} = \frac{3c^2}{8\pi G}H_0^2 = \frac{c^2}{6\pi G(1+w)^2},t_0^{-2},
$$
Eq (5.46) — Energy density expressed in Planck units. (p. 119)
$$
\varepsilon(t) = \frac{1}{6\pi(1+w)^2}\frac{c^2}{G}t^{-2} = \frac{1}{6\pi(1+w)^2}\frac{E_p}{\ell_p^3}\left(\frac{t}{t_P}\right)^{-2}.
$$
Eq (5.47) — Redshift–time relation in a flat, single-component universe. (p. 119)
$$
1 + z = \frac{a(t_0)}{a(t_e)} = \left(\frac{t_0}{t_e}\right)^{2/(3+3w)}
$$
Eq (5.48) — Emission time of observed light in a flat, single-component universe. (p. 119)
$$
t_e = \frac{t_0}{(1+z)^{3(1+w)/2}} = \frac{2}{3(1+w)H_0}\frac{1}{(1+z)^{3(1+w)/2}}.
$$
Eq (5.49) — Current proper distance to a galaxy (integral form). (p. 120)
$$
d_p(t_0) = c\int_{t_e}^{t_0}\frac{dt}{a(t)} = ct_0\frac{3(1+w)}{1+3w}\left[1 - (t_e/t_0)^{(1+3w)/(3+3w)}\right],
$$
Eq (5.50) — Current proper distance in terms of $H_0$ and $z$ . (p. 120)
$$
d_p(t_0) = \frac{c}{H_0}\frac{2}{1+3w}\left[1 - (1+z)^{-(1+3w)/2}\right].
$$
Eq (5.51) — Current horizon distance (integral definition). (p. 120)
$$
d_{\mathrm{hor}}(t_0) = c\int_0^{t_0}\frac{dt}{a(t)}.
$$
Eq (5.52) ★ — Horizon distance in a flat, single-component universe ($w>-1/3$ ). (p. 120)
$$
d_{\mathrm{hor}}(t_0) = ct_0\frac{3(1+w)}{1+3w} = \frac{c}{H_0}\frac{2}{1+3w}.
$$
Eq (5.53) ★ — Age of a flat, matter-only (Einstein–de Sitter) universe. (p. 121)
$$
t_0 = \frac{2}{3H_0},
$$
Eq (5.54) — Horizon distance in a flat, matter-only universe. (p. 121)
$$
d_{\mathrm{hor}}(t_0) = 3ct_0 = 2c/H_0.
$$
Eq (5.55) ★ — Scale factor in a flat, matter-only (Einstein–de Sitter) universe. (p. 121)
$$
a_m(t) = \left(\frac{t}{t_0}\right)^{2/3},
$$
Eq (5.56) — Current proper distance to a galaxy in a flat, matter-only universe. (p. 122)
$$
d_p(t_0) = c\int_{t_e}^{t_0}\frac{dt}{(t/t_0)^{2/3}} = 3ct_0\left[1 - \left(\frac{t_e}{t_0}\right)^{1/3}\right] = \frac{2c}{H_0}\left[1 - \frac{1}{\sqrt{1+z}}\right],
$$
Eq (5.57) — Proper distance at emission in a flat, matter-only universe. (p. 122)
$$
d_p(t_e) = \frac{2c}{H_0(1+z)}\left[1 - \frac{1}{\sqrt{1+z}}\right],
$$
Eq (5.58) ★ — Age of a flat, radiation-only universe. (p. 122)
$$
t_0 = \frac{1}{2H_0},
$$
Eq (5.59) — Horizon distance in a flat, radiation-only universe (equals the Hubble distance). (p. 122)
$$
d_{\mathrm{hor}}(t_0) = 2ct_0 = \frac{c}{H_0}.
$$
Eq (5.60) ★ — Scale factor in a flat, radiation-only universe. (p. 123)
$$
a(t) = \left(\frac{t}{t_0}\right)^{1/2},
$$
Eq (5.61) — Current proper distance to a source in a flat, radiation-only universe. (p. 123)
$$
d_p(t_0) = c\int_{t_e}^{t_0}\frac{dt}{(t/t_0)^{1/2}} = 2ct_0\left[1 - \left(\frac{t_e}{t_0}\right)^{1/2}\right] = \frac{c}{H_0}\frac{z}{1+z},
$$
Eq (5.62) — Proper distance at emission in a flat, radiation-only universe. (p. 123)
$$
d_p(t_e) = \frac{c}{H_0}\frac{z}{(1+z)^2},
$$
Eq (5.63) — Energy density in a flat, radiation-only universe (Planck units). (p. 123)
$$
\varepsilon_r(t) = \frac{3}{32\pi}\frac{E_P}{\ell_P^3}\left(\frac{t}{t_P}\right)^{-2} \approx 0.030\frac{E_P}{\ell_P^3}\left(\frac{t}{t_P}\right)^{-2}.
$$
Eq (5.64) — Temperature of a radiation-dominated universe as a function of time. (p. 124 — corrected ; the book prints $45/(32\pi^2)$ / $0.61$, a dropped factor of $\pi$. See note at end.)
$$
T(t) = \left(\frac{45}{32\pi^3}\right)^{1/4}T_P\left(\frac{t}{t_P}\right)^{-1/2} \approx 0.46,T_P\left(\frac{t}{t_P}\right)^{-1/2}.
$$
Eq (5.65) — Mean energy per photon in a radiation-dominated universe. (p. 124 — coefficient corrected downstream of 5.64; book prints $1.7$.)
$$
E_{\mathrm{mean}}(t) \approx 2.7, kT(t) \approx 1.2, E_P\left(\frac{t}{t_P}\right)^{-1/2},
$$
Eq (5.66) — Number density of photons in a radiation-dominated universe. (p. 124 — coefficient corrected downstream of 5.64; book prints $0.018$.)
$$
n(t) = \frac{\varepsilon_r(t)}{E_{\mathrm{mean}}(t)} \approx \frac{0.024}{\ell_P^3}\left(\frac{t}{t_P}\right)^{-3/2}.
$$
Eq (5.67) — Horizon distance vs. time in a flat, radiation-only universe. (p. 125)
$$
d_{\mathrm{hor}}(t) = 2ct = 2\ell_P\left(\frac{t}{t_P}\right),
$$
Eq (5.68) — Volume of the visible universe at time $t$ . (p. 125)
$$
V_{\mathrm{hor}}(t) = \frac{4\pi}{3}d_{\mathrm{hor}}^3 \approx 34\ell_P^3\left(\frac{t}{t_P}\right)^3.
$$
Eq (5.69) — Number of photons inside the horizon at time $t$ . (p. 125 — coefficient corrected downstream of 5.64; book prints $0.6$.)
$$
N(t) = V_{\mathrm{hor}}(t),n(t) \approx 0.8\left(\frac{t}{t_P}\right)^{3/2}.
$$
Eq (5.70) ★ — Friedmann equation for a flat, $\Lambda$ -only universe. (p. 125)
$$
\dot{a}^2 = \frac{8\pi G \varepsilon_\Lambda}{3c^2},a^2,
$$
Eq (5.71) — Friedmann equation rewritten for the $\Lambda$ -only case. (p. 126)
$$
\dot{a} = H_0 a,
$$
Eq (5.72) — Hubble constant for a flat, $\Lambda$ -only universe. (p. 126)
$$
H_0 = \left(\frac{8\pi G \varepsilon_\Lambda}{3c^2}\right)^{1/2}.
$$
Eq (5.73) ★ — Scale factor grows exponentially in a flat, $\Lambda$ -only (de Sitter) universe. (p. 126)
$$
a(t) = e^{H_0(t - t_0)}.
$$
Eq (5.74) — Current proper distance to a source in a flat, $\Lambda$ -only universe. (p. 126)
$$
d_p(t_0) = c\int_{t_e}^{t_0} e^{H_0(t_0 - t)},dt = \frac{c}{H_0}\left[e^{H_0(t_0 - t_e)} - 1\right] = \frac{c}{H_0}z,
$$
Eq (5.75) — Proper distance at emission in a flat, $\Lambda$ -only universe. (p. 127)
$$
d_p(t_e) = \frac{c}{H_0}\frac{z}{1+z},
$$
5.4 Multiple-component Universes Eq (5.76) ★ — Friedmann equation written with the Hubble parameter and total energy density. (p. 127)
$$
H(t)^2 = \frac{8\pi G}{3c^2}\varepsilon(t) - \frac{\kappa c^2}{R_0^2 a(t)^2},
$$
Eq (5.77) — Curvature expressed via present density parameter (from Eq. 4.36). (p. 128)
$$
\frac{\kappa}{R_0^2} = \frac{H_0^2}{c^2}(\Omega_0 - 1),
$$
Eq (5.78) — Friedmann equation with curvature replaced by $\Omega_0$ . (p. 128)
$$
H(t)^2 = \frac{8\pi G}{3c^2}\varepsilon(t) - \frac{H_0^2}{a(t)^2}(\Omega_0 - 1).
$$
Eq (5.79) ★ — Dimensionless Friedmann equation (divided by $H_0^2$ ). (p. 128)
$$
\frac{H(t)^2}{H_0^2} = \frac{\varepsilon(t)}{\varepsilon_{c,0}} + \frac{1 - \Omega_0}{a(t)^2},
$$
Eq (5.80) — Present-day critical density. (p. 128)
$$
\varepsilon_{c,0} \equiv \frac{3c^2 H_0^2}{8\pi G}.
$$
Eq (5.81) ★ — Friedmann equation for our universe in terms of density parameters. (p. 128)
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{r,0}}{a^4} + \frac{\Omega_{m,0}}{a^3} + \Omega_{\Lambda,0} + \frac{1 - \Omega_0}{a^2},
$$
Eq (5.82) — Friedmann equation solved for $\dot{a}$ (multiple components). (p. 129)
$$
H_0^{-1}\dot{a} = \left[\frac{\Omega_{r,0}}{a^2} + \frac{\Omega_{m,0}}{a} + \Omega_{\Lambda,0}a^2 + (1 - \Omega_0)\right]^{1/2}.
$$
Eq (5.83) ★ — Cosmic time as a function of scale factor (master integral). (p. 129)
$$
\int_0^a \frac{da}{\left[\Omega_{r,0}/a^2 + \Omega_{m,0}/a + \Omega_{\Lambda,0}a^2 + (1 - \Omega_0)\right]^{1/2}} = H_0 t.
$$
Eq (5.84) — Scale factor of a flat, matter-only universe (restatement of 5.55). (p. 130)
$$
a(t) = \left(\frac{t}{t_0}\right)^{2/3}.
$$
Eq (5.85) — Friedmann equation for a curved, matter-dominated universe. (p. 130)
$$
\frac{H(t)^2}{H_0^2} = \frac{\Omega_0}{a^3} + \frac{1 - \Omega_0}{a^2},
$$
Eq (5.86) — Condition $H=0$ at maximum expansion in a matter+curvature universe. (p. 131)
$$
0 = \frac{\Omega_0}{a_{\mathrm{max}}^3} + \frac{1 - \Omega_0}{a_{\mathrm{max}}^2}.
$$
Eq (5.87) ★ — Scale factor at maximum expansion (turnaround) for $\Omega_0 > 1$ . (p. 131)
$$
a_{\mathrm{max}} = \frac{\Omega_0}{\Omega_0 - 1},
$$
Eq (5.88) — Friedmann equation for a curved, matter-only universe solved for $\dot{a}$ . (p. 132)
$$
\frac{\dot{a}^2}{H_0^2} = \frac{\Omega_0}{a} + (1 - \Omega_0),
$$
Eq (5.89) — Age of the universe as an integral (curved, matter-only). (p. 132)
$$
H_0 t = \int_0^a \frac{da}{\left[\Omega_0/a + (1 - \Omega_0)\right]^{1/2}}.
$$
Eq (5.90) — Parametric solution for scale factor, curved matter-only universe with $\Omega_0 > 1$ (cycloid). (p. 133)
$$
a(\theta) = \frac{1}{2}\frac{\Omega_0}{\Omega_0 - 1}(1 - \cos\theta)
$$
Eq (5.91) — Parametric solution for time, curved matter-only universe with $\Omega_0 > 1$ . (p. 133)
$$
t(\theta) = \frac{1}{2H_0}\frac{\Omega_0}{(\Omega_0 - 1)^{3/2}}(\theta - \sin\theta),
$$
Eq (5.92) ★ — Time until the Big Crunch for a closed, matter-only universe. (p. 133)
$$
t_{\mathrm{crunch}} = \frac{\pi}{H_0}\frac{\Omega_0}{(\Omega_0 - 1)^{3/2}}.
$$
Eq (5.93) — Parametric solution for scale factor, curved matter-only universe with $\Omega_0 < 1$ . (p. 134)
$$
a(\eta) = \frac{1}{2}\frac{\Omega_0}{1 - \Omega_0}(\cosh\eta - 1)
$$
Eq (5.94) — Parametric solution for time, curved matter-only universe with $\Omega_0 < 1$ . (p. 134)
$$
t(\eta) = \frac{1}{2H_0}\frac{\Omega_0}{(1 - \Omega_0)^{3/2}}(\sinh\eta - \eta),
$$
Eq (5.95) — Flatness condition linking matter and $\Lambda$ density parameters. (p. 135)
$$
\Omega_{\Lambda,0} = 1 - \Omega_{m,0},
$$
Eq (5.96) ★ — Friedmann equation for a flat matter+$\Lambda$ universe. (p. 135)
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{m,0}}{a^3} + (1 - \Omega_{m,0}).
$$
Eq (5.97) — Maximum scale factor for a flat universe with $\Omega_{\Lambda,0} < 0$ . (p. 136)
$$
a_{\mathrm{max}} = \left(\frac{\Omega_{m,0}}{\Omega_{m,0} - 1}\right)^{1/3},
$$
Eq (5.98) — Big Crunch time for a flat matter + negative-$\Lambda$ universe. (p. 136)
$$
t_{\mathrm{crunch}} = \frac{2\pi}{3H_0}\frac{1}{\sqrt{\Omega_{m,0} - 1}}.
$$
Eq (5.99) — Analytic solution $t(a)$ for a flat matter + negative-$\Lambda$ universe. (p. 136)
$$
H_0 t = \frac{2}{3\sqrt{\Omega_{m,0} - 1}}\sin^{-1}\left[\left(\frac{a}{a_{\mathrm{max}}}\right)^{3/2}\right].
$$
Eq (5.100) ★ — Scale factor at matter–$\Lambda$ equality in a flat universe. (p. 137)
$$
a_{m\Lambda} = \left(\frac{\Omega_{m,0}}{\Omega_{\Lambda,0}}\right)^{1/3} = \left(\frac{\Omega_{m,0}}{1 - \Omega_{m,0}}\right)^{1/3}.
$$
Eq (5.101) ★ — Analytic solution $t(a)$ for a flat matter+$\Lambda$ universe ($\Omega_{\Lambda,0} > 0$ ). (p. 137)
$$
H_0 t = \frac{2}{3\sqrt{1 - \Omega_{m,0}}}\ln\left[\left(\frac{a}{a_{m\Lambda}}\right)^{3/2} + \sqrt{1 + \left(\frac{a}{a_{m\Lambda}}\right)^3}\right].
$$
Eq (5.102) — Early-time (matter-dominated) limit of the flat matter+$\Lambda$ solution. (p. 138)
$$
a(t) \approx \left(\frac{3}{2}\sqrt{\Omega_{m,0}},H_0 t\right)^{2/3},
$$
Eq (5.103) — Late-time ($\Lambda$ -dominated) limit of the flat matter+$\Lambda$ solution. (p. 138)
$$
a(t) \approx a_{m\Lambda}\exp\left(\sqrt{1 - \Omega_{m,0}},H_0 t\right),
$$
Eq (5.104) ★ — Age of a flat matter+$\Lambda$ universe. (p. 138)
$$
t_0 = \frac{2H_0^{-1}}{3\sqrt{1 - \Omega_{m,0}}}\ln\left[\frac{\sqrt{1 - \Omega_{m,0}} + 1}{\sqrt{\Omega_{m,0}}}\right].
$$
Eq (5.105) ★ — Age of the Benchmark Model universe. (p. 138)
$$
t_0 = 0.955 H_0^{-1} = 13.74 \pm 0.40,\mathrm{Gyr},
$$
Eq (5.106) — Age at matter–$\Lambda$ equality in the Benchmark Model. (p. 139)
$$
t_{m\Lambda} = \frac{2H_0^{-1}}{3\sqrt{1 - \Omega_{m,0}}}\ln[1 + \sqrt{2}] = 0.707 H_0^{-1} = 10.17 \pm 0.30,\mathrm{Gyr}.
$$
5.4.3 Matter + Curvature + Lambda Eq (5.107) ★ — Friedmann equation for a curved universe with matter and a cosmological constant. (p. 139)
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{m,0}}{a^3} + \frac{1 - \Omega_{m,0} - \Omega_{\Lambda,0}}{a^2} + \Omega_{\Lambda,0}.
$$
Eq (5.108) ★ — Friedmann equation near radiation–matter equality (flat radiation+matter model). (p. 142)
$$
\frac{H^2}{H_0^2} = \frac{\Omega_{r,0}}{a^4} + \frac{\Omega_{m,0}}{a^3}.
$$
Eq (5.109) — Differential time relation for a flat radiation+matter universe. (p. 143)
$$
H_0,dt = \frac{a,da}{\Omega_{r,0}^{1/2}}\left[1 + \frac{a}{a_{rm}}\right]^{-1/2}.
$$
Eq (5.110) — Cosmic time vs. scale factor in a flat radiation+matter universe. (p. 143)
$$
H_0 t = \frac{4a_{rm}^2}{3\sqrt{\Omega_{r,0}}}\left[1 - \left(1 - \frac{a}{2a_{rm}}\right)\left(1 + \frac{a}{a_{rm}}\right)^{1/2}\right].
$$
Eq (5.111) — Radiation-dominated early-time limit. (p. 143)
$$
a \approx \left(2\sqrt{\Omega_{r,0}},H_0 t\right)^{1/2} \qquad [a \ll a_{rm}].
$$
Eq (5.112) — Matter-dominated late-time limit. (p. 143)
$$
a \approx \left(\frac{3}{2}\sqrt{\Omega_{m,0}},H_0 t\right)^{2/3} \qquad [a \gg a_{rm}].
$$
Eq (5.113) — Time of radiation–matter equality. (p. 143)
$$
t_{rm} = \frac{4}{3}\left(1 - \frac{1}{\sqrt{2}}\right)\frac{a_{rm}^2}{\sqrt{\Omega_{r,0}}}H_0^{-1} \approx 0.391\frac{\Omega_{r,0}^{3/2}}{\Omega_{m,0}^2}H_0^{-1}.
$$
Eq (5.114) ★ — Time of radiation–matter equality in the Benchmark Model. (p. 143)
$$
t_{rm} = 3.47 \times 10^{-6},H_0^{-1} = 50,000,\mathrm{yr}.
$$
Benchmark Model parameters ★ (unnumbered — Table 5.2, pp. 144–145) — the $\Lambda$CDM fit adopted throughout the book. Hubble constant $H_0 = 68,\mathrm{km,s^{-1},Mpc^{-1}}$ ; present temperature $T_0 = 2.7255,\mathrm{K}$ ; spatially flat ($\Omega_0 = 1$ ).
$$
\begin{aligned}
&\text{Photons: } \Omega_{\gamma,0} = 5.35 \times 10^{-5}, \quad \text{Neutrinos: } \Omega_{\nu,0} = 3.65 \times 10^{-5}, \quad \textbf{Total radiation: } \Omega_{r,0} = 9.0 \times 10^{-5};\\
&\text{Baryonic matter: } \Omega_{\mathrm{bary},0} = 0.048, \quad \text{Nonbaryonic dark matter: } \Omega_{\mathrm{dm},0} = 0.262, \quad \textbf{Total matter: } \Omega_{m,0} = 0.31;\\
&\textbf{Cosmological constant: } \Omega_{\Lambda,0} \approx 0.69.\\
&\text{Radiation–matter equality: } a_{rm} = 2.9 \times 10^{-4},\ t_{rm} = 0.050,\mathrm{Myr};\\
&\text{Matter–}\Lambda\text{ equality: } a_{m\Lambda} = 0.77,\ t_{m\Lambda} = 10.2,\mathrm{Gyr};\\
&\text{Now: } a_0 = 1,\ t_0 = 13.7,\mathrm{Gyr}.
\end{aligned}
$$
Eq (5.115) ★ — Horizon distance of the Benchmark Model. (p. 147)
$$
d_{\mathrm{hor}}(t_0) = 3.20 c/H_0 = 3.35 ct_0 = 14,000,\mathrm{Mpc}.
$$
Eq (5.116) — Rate of change of observed redshift in a flat, single-component universe (Exercise 5.1). (p. 149)
$$
\frac{dz}{dt_0} = H_0(1 + z) - H_0(1 + z)^{3(1+w)/2}.
$$
Eq (5.117) — Present age of a positively curved, matter-only universe (Exercise 5.3). (p. 149)
$$
H_0 t_0 = \frac{\Omega_0}{2(\Omega_0 - 1)^{3/2}}\cos^{-1}\left(\frac{2 - \Omega_0}{\Omega_0}\right) - \frac{1}{\Omega_0 - 1}.
$$
Eq (5.118) — Present age of a negatively curved, matter-only universe (Exercise 5.4). (p. 150)
$$
H_0 t_0 = \frac{1}{1 - \Omega_0} - \frac{\Omega_0}{2(1 - \Omega_0)^{3/2}}\cosh^{-1}\left(\frac{2 - \Omega_0}{\Omega_0}\right).
$$
Eq (5.119) — Time until the "Big Rip" for a flat matter + phantom-energy universe (Exercise 5.5). (p. 150)
$$
H_0(t_{\mathrm{rip}} - t_0) \approx \frac{2}{3|1 + w_p|}(1 - \Omega_{m,0})^{-1/2}.
$$
Eq (5.120) — Scale factor at the "Big Bounce" for an expanding, positively curved $\Lambda$ -only universe (Exercise 5.8). (p. 151)
$$
a_{\mathrm{bounce}} = \left(\frac{\Omega_0 - 1}{\Omega_0}\right)^{1/2},
$$
Eq (5.121) — Scale factor vs. time for a "Big Bounce" $\Lambda$ -only universe (Exercise 5.8). (p. 151)
$$
a(t) = a_{\mathrm{bounce}}\cosh\left[\sqrt{\Omega_0},H_0(t - t_{\mathrm{bounce}})\right],
$$
Chapter 6 — Measuring Cosmological Parameters 6.1 "A Search for Two Numbers" Eq (6.1) — full Taylor series of the scale factor about the present moment $t_0$ . (p. 153)
$$
a(t) = a(t_0) + \frac{da}{dt}\bigg|_{t=t_0}(t-t_0) + \frac{1}{2}\frac{d^2a}{dt^2}\bigg|_{t=t_0}(t-t_0)^2 + \cdots
$$
Eq (6.2) — Taylor series truncated at the first three terms (recent past / near future). (p. 154)
$$
a(t) \approx a(t_0) + \frac{da}{dt}\bigg|_{t=t_0}(t-t_0) + \frac{1}{2}\frac{d^2a}{dt^2}\bigg|_{t=t_0}(t-t_0)^2.
$$
Eq (6.3) — same expansion divided by the current scale factor $a(t_0)$ . (p. 154)
$$
\frac{a(t)}{a(t_0)} \approx 1 + \frac{\dot a}{a}\bigg|_{t=t_0}(t-t_0) + \frac{1}{2}\frac{\ddot a}{a}\bigg|_{t=t_0}(t-t_0)^2.
$$
Eq (6.4) ★ — the "two-number" expansion of $a(t)$ in terms of $H_0$ and $q_0$ (with $a(t_0)=1$ ). (p. 154)
$$
a(t) \approx 1 + H_0(t-t_0) - \frac{1}{2}q_0 H_0^2 (t-t_0)^2.
$$
Eq (6.5) ★ — definition of the Hubble constant. (p. 154)
$$
H_0 \equiv \frac{\dot a}{a}\bigg|_{t=t_0},
$$
Eq (6.6) ★ — definition of the deceleration parameter $q_0$ . (p. 154)
$$
q_0 \equiv -\left(\frac{\ddot a, a}{\dot a^2}\right)_{t=t_0} = -\left(\frac{\ddot a}{aH^2}\right)_{t=t_0}.
$$
Eq (6.7) — acceleration equation for a universe of $N$ components. (p. 155)
$$
\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}\sum_{i=1}^{N}\varepsilon_i(1+3w_i).
$$
Eq (6.8) — acceleration equation divided by $H^2$ and sign-flipped. (p. 155)
$$
-\frac{\ddot a}{aH^2} = \frac{1}{2}\left[\frac{8\pi G}{3c^2 H^2}\right]\sum_{i=1}^{N}\varepsilon_i(1+3w_i).
$$
Eq (6.9) — same relation written with density parameters $\Omega_i$ . (p. 156)
$$
-\frac{\ddot a}{aH^2} = \frac{1}{2}\sum_{i=1}^{N}\Omega_i(1+3w_i).
$$
Eq (6.10) — deceleration parameter evaluated at $t_0$ in terms of present-day density parameters. (p. 156)
$$
q_0 = \frac{1}{2}\sum_{i=1}^{N}\Omega_{i,0}(1+3w_i).
$$
Eq (6.11) ★ — $q_0$ for a universe of radiation, matter, and a cosmological constant. (p. 156)
$$
q_0 = \Omega_{r,0} + \frac{1}{2}\Omega_{m,0} - \Omega_{\Lambda,0}.
$$
Eq (6.12) ★ — Hubble's law: the linear distance–redshift relation at small $z$ . (p. 156)
$$
cz = H_0 d.
$$
Eq (6.13) — current proper distance to a galaxy whose light was emitted at $t_e$ . (p. 157)
$$
d_p(t_0) = c\int_{t_e}^{t_0}\frac{dt}{a(t)}.
$$
Eq (6.14) — Taylor expansion of $1/a(t)$ in the lookback time. (p. 157)
$$
\frac{1}{a(t)} \approx 1 - H_0(t-t_0) + \left(1+\frac{q_0}{2}\right)H_0^2(t-t_0)^2
$$
Eq (6.15) — proper distance to lowest orders in the lookback time $t_0-t_e$ . (p. 157)
$$
d_p(t_0) \approx c(t_0-t_e) + \frac{cH_0}{2}(t_0-t_e)^2.
$$
Eq (6.16) ★ — redshift in terms of the scale factor at emission. (p. 158)
$$
z = \frac{1}{a(t_e)} - 1.
$$
Eq (6.17) — approximate relation between redshift and lookback time. (p. 158)
$$
z \approx H_0(t_0-t_e) + \left(1+\frac{q_0}{2}\right)H_0^2(t_0-t_e)^2.
$$
Eq (6.18) — lookback time as a function of redshift (inversion of 6.17). (p. 158)
$$
t_0 - t_e \approx H_0^{-1}\left[z - \left(1+\frac{q_0}{2}\right)z^2\right].
$$
Eq (6.19) — current proper distance to a galaxy of redshift $z$ (low-$z$ expansion). (p. 158)
$$
d_p(t_0) \approx \frac{c}{H_0}\left[z - \left(1+\frac{q_0}{2}\right)z^2\right] + \frac{cH_0}{2}\frac{z^2}{H_0^2} = \frac{c}{H_0}z\left[1 - \frac{1+q_0}{2}z\right].
$$
Eq (6.20) — parallax distance from a baseline $b$ and parallax angle $\theta$ . (p. 159)
$$
d_\pi = 1,\mathrm{pc}\left(\frac{b}{1,\mathrm{AU}}\right)\left(\frac{\theta}{1,\mathrm{arcsec}}\right)^{-1}.
$$
Eq (6.21) ★ — definition of the luminosity distance from luminosity $L$ and flux $f$ . (p. 160)
$$
d_L \equiv \left(\frac{L}{4\pi f}\right)^{1/2}.
$$
Eq (6.22) — Robertson–Walker metric. (p. 161)
$$
ds^2 = -c^2 dt^2 + a(t)^2\left[dr^2 + S_\kappa(r)^2, d\Omega^2\right],
$$
Eq (6.23) — the curvature function $S_\kappa(r)$ for the three curvature cases. (p. 161)
$$
S_\kappa(r) = \begin{cases} R_0\sin(r/R_0) & (\kappa = +1) \ r & (\kappa = 0) \ R_0\sinh(r/R_0) & (\kappa = -1). \end{cases}
$$
Eq (6.24) — proper surface area of a sphere of comoving radius $r$ . (p. 161)
$$
A_p(t_0) = 4\pi S_\kappa(r)^2.
$$
Eq (6.25) — observed (redshifted) wavelength of a photon. (p. 162)
$$
\lambda_0 = \frac{1}{a(t_e)}\lambda_e = (1+z)\lambda_e,
$$
Eq (6.26) — observed photon energy reduced by expansion. (p. 162)
$$
E_0 = \frac{E_e}{1+z}.
$$
Eq (6.27) ★ — observed flux–luminosity relation in an expanding, curved universe. (p. 162)
$$
f = \frac{L}{4\pi S_\kappa(r)^2(1+z)^2},
$$
Eq (6.28) ★ — luminosity distance in terms of the curvature function and redshift. (p. 163)
$$
d_L = S_\kappa(r)(1+z).
$$
Eq (6.29) — luminosity distance in a spatially flat universe. (p. 163)
$$
d_L = r(1+z) = d_p(t_0)(1+z) \qquad [\kappa = 0].
$$
Eq (6.30) — low-$z$ approximation of the current proper distance. (p. 163)
$$
d_p(t_0) \approx \frac{c}{H_0}z\left(1 - \frac{1+q_0}{2}z\right).
$$
Eq (6.31) ★ — low-$z$ approximation of the luminosity distance in a nearly flat universe. (p. 164)
$$
d_L \approx \frac{c}{H_0}z\left(1 - \frac{1+q_0}{2}z\right)(1+z) \approx \frac{c}{H_0}z\left(1 + \frac{1-q_0}{2}z\right).
$$
6.3 Angular-diameter Distance Eq (6.32) ★ — definition of the angular-diameter distance from proper length $\ell$ and angular size $\delta\theta$ . (p. 165)
$$
d_A \equiv \frac{\ell}{\delta\theta}.
$$
Eq (6.33) — proper separation of the ends of a yardstick from the Robertson–Walker metric. (p. 166)
$$
ds = a(t_e)S_\kappa(r),\delta\theta.
$$
Eq (6.34) — setting $ds = \ell$ for a standard yardstick of known length. (p. 166)
$$
\ell = a(t_e)S_\kappa(r),\delta\theta = \frac{S_\kappa(r),\delta\theta}{1+z}.
$$
Eq (6.35) ★ — angular-diameter distance in terms of the curvature function and redshift. (p. 166)
$$
d_A \equiv \frac{\ell}{\delta\theta} = \frac{S_\kappa(r)}{1+z}.
$$
Eq (6.36) ★ — relation between angular-diameter distance and luminosity distance. (p. 166)
$$
d_A = \frac{d_L}{(1+z)^2}.
$$
Eq (6.37) — in a flat universe, $d_A$ equals the proper distance at emission. (p. 166)
$$
d_A(1+z) = d_p(t_0) = \frac{d_L}{1+z} \qquad [\kappa = 0].
$$
Eq (6.38) — low-$z$ expansion of the angular-diameter distance. (p. 167)
$$
d_A \approx \frac{c}{H_0}z\left(1 - \frac{3+q_0}{2}z\right).
$$
Eq (6.39) — luminosity distance to highly redshifted objects. (p. 168)
$$
d_L(z\to\infty) \approx z, d_{\mathrm{hor}}(t_0).
$$
Eq (6.40) — angular-diameter distance approaches zero at high redshift. (p. 168)
$$
d_A(z\to\infty) \approx \frac{d_{\mathrm{hor}}(t_0)}{z}.
$$
Eq (6.41) — minimum angular size of a yardstick (at the critical redshift) in the Benchmark Model. (p. 168)
$$
\delta\theta(\mathrm{min}) = \frac{\ell}{d_A(\mathrm{max})} = \frac{\ell}{1770,\mathrm{Mpc}} \approx 0.1,\mathrm{arcsec}\left(\frac{\ell}{1,\mathrm{kpc}}\right).
$$
6.4 Standard Candles and H₀ Eq (6.42) — measured ratio of mean fluxes for equal-period Cepheids in the LMC and M31. (p. 170)
$$
\frac{\bar f_{\mathrm{LMC}}}{\bar f_{\mathrm{M31}}} = 230.
$$
Eq (6.43) — ratio of luminosity distances inferred from the flux ratio. (p. 170)
$$
\frac{d_L(\mathrm{M31})}{d_L(\mathrm{LMC})} = \left(\frac{\bar f_{\mathrm{LMC}}}{\bar f_{\mathrm{M31}}}\right)^{1/2} = \sqrt{230} = 15.2.
$$
6.5 Standard Candles and Acceleration Eq (6.44) — low-$z$ luminosity distance in terms of $H_0$ and $q_0$ . (p. 172)
$$
d_L \approx \frac{c}{H_0}z\left[1 + \frac{1-q_0}{2}z\right].
$$
Eq (6.45) ★ — definition of bolometric apparent magnitude. (p. 174)
$$
m \equiv -2.5\log_{10}(f/f_x),
$$
Eq (6.46) ★ — definition of bolometric absolute magnitude. (p. 174)
$$
M \equiv -2.5\log_{10}(L/L_x),
$$
Eq (6.47) — relation between absolute and apparent magnitude and luminosity distance (in pc). (p. 175)
$$
M = m - 5\log_{10}\left(\frac{d_L}{10,\mathrm{pc}}\right),
$$
Eq (6.48) — same relation with $d_L$ in megaparsecs. (p. 175)
$$
M = m - 5\log_{10}\left(\frac{d_L}{1,\mathrm{Mpc}}\right) - 25.
$$
Eq (6.49) ★ — distance modulus $m-M$ in terms of the luminosity distance. (p. 175)
$$
m - M = 5\log_{10}\left(\frac{d_L}{1,\mathrm{Mpc}}\right) + 25.
$$
Eq (6.50) — low-$z$ luminosity distance (repeat of 6.44/6.31). (p. 176)
$$
d_L \approx \frac{c}{H_0}z\left(1 + \frac{1-q_0}{2}z\right).
$$
Eq (6.51) — distance modulus versus redshift at low redshift. (p. 176)
$$
m - M \approx 43.23 - 5\log_{10}\left(\frac{H_0}{68,\mathrm{km,s^{-1},Mpc^{-1}}}\right) + 5\log_{10} z + 1.086(1-q_0)z.
$$
Eq (6.52) — observed flux from a standard candle at redshift $z$ in a flat, single-component ($w\neq -1/3$ ) universe (Exercise 6.8). (p. 180)
$$
f(z) = \frac{L(1+3w)^2}{16\pi(c/H_0)^2},\frac{1}{(1+z)^2}\left[1 - (1+z)^{-(1+3w)/2}\right]^{-2},
$$
Eq (6.53) — observed intensity contributed by standard candles in the redshift range $z \to z+dz$ (Exercise 6.8). (p. 180)
$$
dJ(z) = \frac{n_0 L(c/H_0)}{4\pi}(1+z)^{-(7+3w)/2},dz.
$$
