Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — Revision Notes

How to read these notes

Coverage: Chapters 3–12 plus the Epilogue
Every bullet ends with a citation like [R 4 81] = Ryden, Ch 4, PDF page 81.
Formulas relevant to a bullet are included inline, tagged with the book's own equation number, e.g. (Eq. 4.20) — the same numbering used in the formulas sheet, which remains the master list.

Quick reference: ★ Essentials Cheat Sheet, Ch 2-6 Formulas, Ch 7-12 Formulas, Ch 8-12 Notes

Chapter 3 — Newton versus Einstein
Chapter 4 — Cosmic Dynamics
Chapter 5 — Model Universes
Chapter 6 — Measuring Cosmological Parameters
Chapter 7 — Dark Matter

Chapter 3 — Newton versus Einstein

3.4 Describing Curvature

Einstein's goal: a field equation relating spacetime curvature to mass-energy density, analogous to Poisson's equation relating $\Phi$ to $\rho$; en route he needed a mathematical description of curvature — built up here from 2D examples. [R 3 61]
Flat plane: geodesics are straight lines; a geodesic triangle satisfies $\alpha + \beta + \gamma = \pi$ (Eq. 3.22), angles in radians. [R 3 61]
Flat 2D metric: $d\ell^2 = dx^2 + dy^2$ (Eq. 3.23, cartesian) or $d\ell^2 = dr^2 + r^2 d\theta^2$ (Eq. 3.24, polar); saying Eq. 3.23 holds everywhere is equivalent to saying the space is a plane, and the two forms are the same geometry (substitute $x = r\cos\theta$, $y = r\sin\theta$). [R 3 62]
Sphere of radius $R$: geodesics are portions of great circles (circles centered on the sphere's center); a geodesic triangle satisfies $\alpha + \beta + \gamma = \pi + A/R^2$ (Eq. 3.25) — angle excess proportional to triangle area $A$. Any space with $\alpha+\beta+\gamma > \pi$ is positively curved. [R 3 62]
Uniform curvature = curvature that is homogeneous and isotropic (same $R$ everywhere, in all directions); the sphere is a 2D space of uniform positive curvature, with metric $d\ell^2 = dr^2 + R^2 \sin^2(r/R), d\theta^2$ (Eq. 3.26), $r$ = geodesic distance from the pole. [R 3 63]
Sphere global properties: finite area $4\pi R^2$; maximum possible separation $\ell_{max} = \pi R$ (antipodal points); the plane has infinite area and unbounded separations. (In a non-Euclidean space, distance between points = length of the connecting geodesic.) [R 3 64]
Negative curvature (saddle/hyperboloid shape): Hilbert proved that a 2D surface of uniform negative curvature cannot be constructed in 3D Euclidean space — the saddle is uniformly curved only near its "seat" — but the properties of such a surface are easily written down. [R 3 64]
Uniform negative curvature: $\alpha + \beta + \gamma = \pi - A/R^2$ (Eq. 3.27, angle deficit ∝ area); metric $d\ell^2 = dr^2 + R^2 \sinh^2(r/R), d\theta^2$ (Eq. 3.28); infinite area, no maximum separation. [R 3 65]
Metric: a relation giving the distance $d\ell$ between nearby points (Eqs. 3.24, 3.26, 3.28 are examples). Curvature is in general a local property (a rumpled tablecloth; a bagel is positively curved in places and negatively in others). [R 3 65]
Demanding homogeneity + isotropy leaves only three possible 2D geometries, fully specified by the curvature constant $\kappa$ ($0$ flat, $+1$ positive, $-1$ negative) and, if curved, the radius of curvature $R$ (dimensions of length). [R 3 65]
The trichotomy extends directly to 3D. Flat ($\kappa=0$): $d\ell^2 = dx^2 + dy^2 + dz^2$ (Eq. 3.29), or in spherical coordinates $d\ell^2 = dr^2 + r^2[d\theta^2 + \sin^2\theta, d\phi^2]$ (Eq. 3.30). [R 3 65]
Uniform positive curvature ($\kappa=+1$): $d\ell^2 = dr^2 + R^2 \sin^2(r/R)[d\theta^2 + \sin^2\theta, d\phi^2]$ (Eq. 3.31); finite volume; the point $r = \pi R$ is antipodal to the origin; the space can be "circumnavigated" by traveling $C = 2\pi R$. [R 3 66]
Uniform negative curvature ($\kappa=-1$): $d\ell^2 = dr^2 + R^2 \sinh^2(r/R)[d\theta^2 + \sin^2\theta, d\phi^2]$ (Eq. 3.32); infinite volume, like flat space. [R 3 66]
Compact unified form for all three homogeneous isotropic 3D spaces: $d\ell^2 = dr^2 + S_\kappa(r)^2, d\Omega^2$ (Eq. 3.33), with $d\Omega^2 \equiv d\theta^2 + \sin^2\theta, d\phi^2$ (Eq. 3.34) and $$ S_\kappa(r) = \left{ \begin{array}{ll} R\sin(r/R) & (\kappa = +1) \ r & (\kappa = 0) \ R\sinh(r/R) & (\kappa = -1) \end{array} \right. $$ (Eq. 3.35). [R 3 66]
Behavior of $S_\kappa$: for $r \ll R$, $S_\kappa \approx r$ for every $\kappa$; for $\kappa = 0, -1$ it increases monotonically with $S_\kappa \to \infty$; for $\kappa = +1$ it peaks at $S_{max} = R$ at $r/R = \pi/2$, then falls back to 0 at $r/R = \pi$ (the antipodal point). [R 3 67]
Switching radial coordinate to $x \equiv S_\kappa(r)$ gives $d\ell^2 = \frac{dx^2}{1 - \kappa x^2/R^2} + x^2 d\Omega^2$ (Eq. 3.36) — same spaces as Eq. 3.33, merely a different coordinate choice. [R 3 67]

3.5 The Robertson–Walker Metric

Minkowski metric: $ds^2 = -c^2 dt^2 + dr^2 + r^2 d\Omega^2$ (Eq. 3.37) — the special-relativistic spacetime separation between two events; its spatial component is Euclidean; the spacetime is flat and static, valid only when gravity is absent. [R 3 68]
Null geodesic: a photon's path through spacetime is a geodesic with $ds = 0$ along every infinitesimal segment; radially in Minkowski spacetime this gives $dr/dt = \pm c$ (Eqs. 3.38–3.40). [R 3 68]
Robertson and Walker (1930s, working independently) asked: what form can the metric take if the universe is spatially homogeneous and isotropic at all times, with distances allowed to expand or contract with time? [R 3 68]
Robertson–Walker metric: $ds^2 = -c^2 dt^2 + a(t)^2\left[dr^2 + S_\kappa(r)^2, d\Omega^2\right]$ (Eq. 3.41), with $S_\kappa$ from Eq. 3.35 using $R = R_0$ — the spatial metric of a uniformly curved space of radius $R_0$, scaled by $a(t)^2$. (Footnote: also called the FRW or FLRW metric. [R 3 76]) [R 3 69]
Cosmic time $t$: the proper time measured by an observer who sees the universe expanding uniformly around them. Comoving coordinates $(r, \theta, \phi)$: constant in time for every point, if the expansion is perfectly homogeneous and isotropic. [R 3 69]
Homogeneity + isotropy is extremely powerful: the entire geometry reduces to $a(t)$, $\kappa \in {+1, 0, -1}$, and (if $\kappa \neq 0$) $R_0$ — much of modern cosmology is devoted to finding these. The assumption was adopted (Einstein, Friedmann, Lemaître, Robertson, Walker) long before observations supported it. [R 3 69]
The RW metric is an approximation valid only on scales ≳ 100 Mpc: small dense lumps (humans, teddy bears, dust grains) are held together electromagnetically, larger lumps (galaxies such as the Milky Way; clusters such as the Local Group) by their own gravity — bound systems do not expand. [R 3 69]

3.6 Proper Distance

Proper distance $d_p(t)$: the length of the spatial geodesic between two points when the scale factor is fixed at $a(t)$ — in an expanding universe a distance must carry a time stamp. For an observer at the origin and a galaxy at comoving $(r, \theta, \phi)$, the fixed-time radial geodesic has $ds = a(t), dr$ (Eqs. 3.42–3.43). [R 3 70]
Integrating over $r$: $d_p(t) = a(t)\int_0^r dr = a(t), r$ (Eq. 3.44) — proper distance is proportional to the scale factor, with $r$ constant in time. [R 3 71]
Differentiating: $\dot{d}p = \frac{\dot a}{a} d_p$ (Eq. 3.45), so at $t_0$ there is a linear velocity–distance relation $v_p(t_0) = H_0 d_p(t_0)$ (Eq. 3.46) with $H_0 = \left(\frac{\dot a}{a}\right){t=t_0}$ (Eq. 3.48) — Hubble's law derived from the RW metric; it echoes Section 2.3, but the growing separations are now interpreted as the expansion of space. [R 3 71]
The radius of curvature of the universe expands at the same rate as galaxy separations: $R(t) = a(t) R_0$. [R 3 71]
"Expanding space drags galaxies apart" and "receding galaxies drag space along" are both misleading — GR says spacetime and mass-energy are intimately linked: curvature tells mass-energy how to move, and mass-energy tells spacetime how to curve. [R 3 72]
Hubble distance: $d_H(t_0) \equiv c/H_0$ (Eq. 3.49); with $H_0 = 68 \pm 2\ \mathrm{km,s^{-1},Mpc^{-1}}$, $d_H(t_0) = 4380 \pm 130\ \mathrm{Mpc}$ (Eq. 3.51); galaxies beyond ~4400 Mpc currently recede at $v_p > c$ (Eq. 3.50). [R 3 72]
No violation of special relativity: the $v < c$ speed limit applies to the relative motion of objects within a static space; GR raises no objection to two points separating superluminally due to the expansion of space. [R 3 72]
Redshift does not give a galaxy's current proper distance, but it does give the scale factor $a(t_e)$ when the light was emitted. Light travels on a null geodesic with $\theta, \phi$ constant, so $c,\frac{dt}{a(t)} = dr$ (Eqs. 3.52–3.53) — the left side depends only on $t$, the right side only on $r$. [R 3 73]
Integrating for one wave crest (emitted $t_e$, observed $t_0$) and for the next (emitted $t_e + \lambda_e/c$, observed $t_0 + \lambda_0/c$) gives the same comoving $r$ (Eqs. 3.54–3.55), so $\int dt/a(t)$ is identical for every crest (Eq. 3.56); subtracting the overlapping integral yields $\int_{t_e}^{t_e + \lambda_e/c} \frac{dt}{a(t)} = \int_{t_0}^{t_0 + \lambda_0/c} \frac{dt}{a(t)}$ (Eq. 3.58). [R 3 73]
The universe cannot expand appreciably between successive wave crests (the expansion timescale — the Hubble time — vastly exceeds the wave period of visible light), so $a(t)$ is effectively constant in each integral, giving $\frac{\lambda_e}{a(t_e)} = \frac{\lambda_0}{a(t_0)}$ (Eq. 3.60): wavelength stretches in proportion to the scale factor. [R 3 74]
Redshift–scale-factor relation: with $z = (\lambda_0 - \lambda_e)/\lambda_e$ and the convention $a(t_0) = 1$, $$1 + z = \frac{a(t_0)}{a(t_e)} = \frac{1}{a(t_e)}$$ (Eq. 3.61); e.g. a galaxy at $z = 2$ is seen as it was when $a(t_e) = 1/3$. [R 3 75]
The observed redshift depends only on the scale factors at emission and observation — not on how the expansion proceeded in between (gradual or abrupt, monotonic or oscillatory). [R 3 75]

Chapter 4 — Cosmic Dynamics

Chapter introduction — geometric limits on curvature

Triangle test for curvature (proposed by Lobachevski as early as 1829): the angle sum of a big triangle obeys $\alpha + \beta + \gamma = \pi + \frac{\kappa A}{R_0^2}$ (Eq. 4.1); sum $> \pi$ means positive curvature, $< \pi$ negative; measuring the area $A$ too would give $R_0$. In practice any drawable triangle is far too small for the deviation from $\pi$ to be measurable. [R 4 77]
Angular-size test: in a flat universe a galaxy of diameter $D$ at distance $r$ subtends $\alpha = D/r$ (Eq. 4.2); curvature modifies this, so galaxy angular sizes probe geometry. [R 4 77]
Positive curvature acts as a magnifying lens: $\alpha_{+} = \frac{D}{R_0 \sin(r/R_0)}$ (Eq. 4.3) exceeds $D/r$; $\alpha_+$ blows up at $r = \pi R_0$, where a galaxy at half the circumference of the universe fills the entire sky. No such bloated, magnified galaxies are seen out to $r \sim c/H_0$, so if $\kappa = +1$ then $\pi R_0 > c/H_0$. [R 4 78]
Negative curvature shrinks images: $\alpha_{-} = \frac{D}{R_0 \sinh(r/R_0)}$ (Eq. 4.4); for $r \gg R_0$, $\alpha_{-} \approx \frac{2D}{R_0}\exp(-r/R_0)$ (Eq. 4.5) — galaxies far beyond $R_0$ would look exponentially tiny. Since galaxies are resolved out to $r \sim c/H_0$, if $\kappa = -1$ then $R_0 > c/H_0$. [R 4 78]
Net geometric conclusion: if the universe is curved at all, its radius of curvature $R_0$ cannot be significantly smaller than the current Hubble distance $c/H_0 \approx 4380$ Mpc. [R 4 77]

4.1 Einstein's Field Equation

Field equation as GR's Poisson equation: Poisson's equation $\nabla^2 \Phi = 4\pi G \rho$ (Eq. 4.6) gives the potential from mass density, and its gradient gives accelerations; analogously the field equation gives spacetime curvature from energy density $\varepsilon$ and pressure $P$, and trajectories follow as geodesics of the curved spacetime. [R 4 79]
Einstein's field equation: $G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$ (Eq. 4.7), where $G_{\mu\nu}$ is the Einstein tensor (4×4, symmetric, 10 independent components, describing curvature at each spacetime point) and $T_{\mu\nu}$ is the stress-energy (energy-momentum) tensor. [R 4 79]
Deceptive simplicity: the field equation is really a set of ten nonlinear second-order differential equations; being second order, spacetime can be curved even where $T_{\mu\nu} = 0$ (just as Newtonian gravity is nonzero where $\rho = 0$). [R 4 79]
Gravitational waves: second-order equations in space and time admit propagating wave solutions — a time-varying mass-energy quadrupole creates gravitational waves, analogous to a time-varying electric dipole creating electromagnetic waves. [R 4 80]
Footnote: Einstein predicted gravitational waves in 1916, "un-predicted" them in 1936, and they were finally detected by LIGO on 14 Sept 2015. [R 4 103]
Perfect-gas simplification: for a homogeneous, isotropic perfect gas, $T_{\mu\nu}$ depends only on $\varepsilon(t)$ and $P(t)$ (no bulk velocity — that would break isotropy), and the metric is the Robertson–Walker metric $ds^2 = -c^2 dt^2 + a(t)^2[dr^2 + S_\kappa(r)^2 d\Omega^2]$ (Eq. 4.8) with $S_\kappa(r) = R_0\sin(r/R_0),\ r,\ R_0\sinh(r/R_0)$ for $\kappa = +1, 0, -1$ (Eq. 4.9). The task: link the curvature parameters $a(t)$, $\kappa$, $R_0$ to the contents $\varepsilon(t)$, $P(t)$. [R 4 80]

4.2 The Friedmann Equation

Historical priority: Alexander Friedmann (originally a meteorologist) derived his equation from Einstein's field equation in 1922 — five years before Lemaître interpreted galaxy redshifts as expansion and seven years before Hubble's law. [R 4 81]
Newtonian derivation setup: a homogeneous sphere of constant mass $M_s$, expanding or contracting isotropically, with a test mass on its surface; Newton gives $F = -\frac{G M_s m}{R_s(t)^2}$ (Eq. 4.10) and hence $\ddot{R}_s = -G M_s / R_s^2$ (Eq. 4.11). [R 4 81]
Energy integral: integrating gives $\frac{1}{2}\dot{R}_s^2 = \frac{G M_s}{R_s} + U$ (Eq. 4.12) — kinetic plus gravitational potential energy per unit mass is a constant $U$ for matter at the sphere's surface. [R 4 82]
Newtonian Friedmann equation: substituting $M_s = \frac{4\pi}{3}\rho R_s^3$ (Eq. 4.15) and $R_s(t) = a(t) r_s$ (Eq. 4.16) yields $\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho(t) + \frac{2U}{r_s^2}\frac{1}{a(t)^2}$ (Eq. 4.18); only $\dot{a}^2$ enters, so contraction is the time reversal of expansion. [R 4 83]
Three fates by sign of U: $U > 0$ — expansion never stops; $U < 0$ — expansion halts at $a_{\max} = -\frac{G M_s}{U r_s}$ (Eq. 4.19), then contraction; $U = 0$ — boundary case, $\dot a \to 0$ as $t \to \infty$ and $\rho \to 0$. Analogous to a ball thrown above, below, or exactly at escape speed. [R 4 83]
Why the Newtonian derivation must be distrusted: a finite sphere has a special center and special directions (violating homogeneity and isotropy), and carving the sphere from an infinite universe uses a shell argument that assumes perfectly Euclidean space; the correct derivation must start from Einstein's field equation. [R 4 84]
Relativistic Friedmann equation: $\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon(t) - \frac{\kappa c^2}{R_0^2}\frac{1}{a(t)^2}$ (Eq. 4.20) — a Very Important Equation. [R 4 85]
First change from Newtonian form: $\rho \to \varepsilon/c^2$ — what gravitates is energy, $E = (m^2 c^4 + p^2 c^2)^{1/2}$ (Eq. 4.21); nonrelativistic particles have $E \approx mc^2 + \frac{1}{2}mv^2$ (Eq. 4.22) so $\varepsilon \approx \rho c^2$, but massless photons with $E = pc = hf$ (Eq. 4.23) also contribute: photons both respond to and contribute to spacetime curvature. [R 4 85]
Peculiar motion: any motion of a particle over and above the motion associated with the expansion or contraction of the universe; $p$ in Eq. 4.21 is the peculiar momentum, measured by an observer at the particle's location who sees isotropic expansion. [R 4 85]
Second change: the substitution $\frac{2U}{r_s^2} = -\frac{\kappa c^2}{R_0^2}$ (Eq. 4.24); Newtonian $U < 0 \leftrightarrow \kappa = +1$, $U > 0 \leftrightarrow \kappa = -1$, $U = 0 \leftrightarrow \kappa = 0$ (flat). [R 4 86]
Hubble-parameter form: since $v = H(t),d$ (Eq. 4.25) with $H \equiv \dot a/a$, the Friedmann equation reads $H(t)^2 = \frac{8\pi G}{3c^2}\varepsilon(t) - \frac{\kappa c^2}{R_0^2 a(t)^2}$ (Eq. 4.26). [R 4 86]
Hubble constant vs parameter: $H(t)$ is the "Hubble parameter"; its present value is the "Hubble constant" $H_0 = 68 \pm 2\ \mathrm{km,s^{-1},Mpc^{-1}}$ (Eq. 4.27); today the Friedmann equation ties together $H_0$, $\varepsilon_0$ and the curvature: $H_0^2 = \frac{8\pi G}{3c^2}\varepsilon_0 - \frac{\kappa c^2}{R_0^2}$ (Eq. 4.28). [R 4 87]
Critical density: the density giving a spatially flat universe (where $H^2 = \frac{8\pi G}{3c^2}\varepsilon$, Eq. 4.29): $\varepsilon_c(t) \equiv \frac{3c^2}{8\pi G} H(t)^2$ (Eq. 4.30); $\varepsilon > \varepsilon_c$ means $\kappa = +1$, $\varepsilon < \varepsilon_c$ means $\kappa = -1$. [R 4 87]
Present critical density: knowing $H_0$ to ~3% gives $\varepsilon_c$ to ~6%: $\varepsilon_{c,0} = (7.8 \pm 0.5)\times 10^{-10}\ \mathrm{J,m^{-3}} = 4870 \pm 290\ \mathrm{MeV,m^{-3}}$ (Eq. 4.31). [R 4 87]
Equivalent mass density: $\rho_{c,0} \equiv \varepsilon_{c,0}/c^2 = (8.7 \pm 0.5)\times 10^{-27}\ \mathrm{kg,m^{-3}} = (1.28 \pm 0.08)\times 10^{11}\ M_\odot,\mathrm{Mpc^{-3}}$ (Eq. 4.32) — roughly one proton per 200 liters, lower even than the hottest, most tenuous interstellar gas; yet averaged over scales $\gtrsim 100$ Mpc (dominated by intergalactic voids) the mean density of the universe is very close to critical. [R 4 88]
Density parameter: dimensionless $\Omega(t) \equiv \frac{\varepsilon(t)}{\varepsilon_c(t)}$ (Eq. 4.33); observations constrain the present value to $0.995 < \Omega_0 < 1.005$. [R 4 88]
Ω never crosses unity: the Friedmann equation in the form $1 - \Omega(t) = -\frac{\kappa c^2}{R_0^2 a(t)^2 H(t)^2}$ (Eq. 4.34) has a right-hand side that cannot change sign, so $\Omega < 1$ stays $< 1$ forever, $\Omega > 1$ stays $> 1$, and $\Omega = 1$ stays exactly 1 at all times. [R 4 88]
Curvature from observables: today $\frac{\kappa}{R_0^2} = \frac{H_0^2}{c^2}(\Omega_0 - 1)$ (Eq. 4.36) — $\Omega_0$ fixes the sign of $\kappa$, and adding the Hubble distance $c/H_0$ fixes the radius of curvature $R_0$. [R 4 89]

4.3 The Fluid and Acceleration Equations

Friedmann equation alone is insufficient: it is one equation in two unknowns, $a(t)$ and $\varepsilon(t)$; another relation between $a$ and $\varepsilon$ is needed. [R 4 89]
First law of thermodynamics: $dQ = dE + P,dV$ (Eq. 4.37), applied to a comoving volume of any fluid. [R 4 89]
Expansion is adiabatic: perfect homogeneity means no bulk heat flow, $dQ = 0$; since $dS = dQ/T$, a homogeneous, isotropic expansion does not increase the universe's entropy. [R 4 90]
Fluid equation: applying $\dot E + P\dot V = 0$ (Eq. 4.38) to a comoving sphere ($V \propto a^3$, $E = V\varepsilon$) gives $\dot{\varepsilon} + 3\frac{\dot a}{a}(\varepsilon + P) = 0$ (Eq. 4.44) — the second key equation; unlike the Friedmann equation, it is unchanged in going from Newtonian physics to general relativity. [R 4 90]
Acceleration equation derivation: multiply the Friedmann equation by $a^2$, take the time derivative, divide by $2\dot a a$, and substitute $\dot{\varepsilon}\frac{a}{\dot a} = -3(\varepsilon + P)$ (Eq. 4.48) from the fluid equation. [R 4 91]
Acceleration equation: $\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}(\varepsilon + 3P)$ (Eq. 4.49) — positive energy density produces negative acceleration, and pressure itself gravitates: baryonic gas, photons, neutrinos and WIMPs all have positive pressure, which slows the expansion. [R 4 92]
Condition for acceleration: a component with $\varepsilon > 0$ but $P < -\frac{1}{3}\varepsilon$ (Eq. 4.50) makes the expansion speed up rather than slow down. [R 4 92]
Footnote: $\varepsilon$ and $P$ share the same dimensionality — $1\ \mathrm{J,m^{-3}} = 1\ \mathrm{N,m^{-2}} = 1\ \mathrm{kg,m^{-1},s^{-2}}$. [R 4 103]

4.4 Equations of State

Counting equations: of the Friedmann (Eq. 4.51), fluid (Eq. 4.52), and acceleration (Eq. 4.53) equations only two are independent (the acceleration equation is derivable from the other two); with three unknowns $a(t)$, $\varepsilon(t)$, $P(t)$, a closing relation — an equation of state $P = P(\varepsilon)$ (Eq. 4.54) — is required. [R 4 93]
Linear equation of state: cosmology deals with dilute gases, so the equation of state takes the simple linear form $P = w\varepsilon$ (Eq. 4.55), with $w$ a dimensionless number. [R 4 94]
Nonrelativistic gas: obeys the perfect gas law $P = \frac{\rho}{\mu}kT$ (Eq. 4.56), and since $\varepsilon \approx \rho c^2$ and $3kT = \mu\langle v^2\rangle$ (Eq. 4.58), its equation-of-state parameter is $w \approx \frac{\langle v^2 \rangle}{3c^2} \ll 1$ (Eq. 4.60). [R 4 94]
Nonrelativistic in practice: room-temperature nitrogen ($v_{\rm rms} \sim 500\ \mathrm{m,s^{-1}}$) has $w \sim 10^{-12}$; in ionized hydrogen, electrons stay nonrelativistic for $T \ll 6\times 10^9$ K, protons for $T \ll 10^{13}$ K. [R 4 95]
Relativistic gas: photons are massless but carry momentum and hence exert pressure; any relativistic gas has $P = \frac{1}{3}\varepsilon$ (Eq. 4.61); highly relativistic massive particles ($\langle v^2\rangle \sim c^2$) also have $w = 1/3$, mildly relativistic ones $0 < w < 1/3$. [R 4 95]
Naming conventions: "matter" = the nonrelativistic component ($w \approx 0$); "radiation" = photons and other relativistic particles ($w = 1/3$). [R 4 95]
Dark energy: any component with $w < -\frac{1}{3}$, which by the acceleration equation drives positive $\ddot a$; the phrase was coined by cosmologist Michael Turner. [R 4 95]
Cosmological constant defined by its equation of state: a component with $w = -1$, hence $P = -\varepsilon$. [R 4 96]

4.5 Learning to Love Lambda

Einstein's 1917 starting assumptions: unaware of the CMB, he took the universe's radiation to be starlight, whose energy density is far below the stars' rest energy — so he modeled a pressureless, matter-dominated universe; and since stellar motions in our galaxy showed no net expansion or contraction (external galaxies not yet established), he believed the universe static. [R 4 96]
A matter-only static universe is impossible: statically, $\vec{a} = -\vec{\nabla}\Phi$ (Eq. 4.63) must vanish everywhere, so $\Phi$ is constant and Poisson's equation forces $\rho = \frac{1}{4\pi G}\nabla^2\Phi = 0$ (Eq. 4.64) — only an empty universe can be static; a static matter-filled universe is like a thrown ball expected to hover in mid-air. [R 4 97]
Einstein's fudge factor: in Newtonian terms he modified Poisson's equation to $\nabla^2 \Phi + \Lambda = 4\pi G \rho$ (Eq. 4.65), which permits a static universe if $\Lambda = 4\pi G\rho$; in GR he added a $\Lambda$ term to the field equation. [R 4 97]
Friedmann equation with Λ: $\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon - \frac{\kappa c^2}{R_0^2 a^2} + \frac{\Lambda}{3}$ (Eq. 4.66); the fluid equation is unaffected (Eq. 4.67), and the acceleration equation becomes $\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}(\varepsilon + 3P) + \frac{\Lambda}{3}$ (Eq. 4.68). [R 4 98]
Λ as an energy component: adding Λ is equivalent to adding a component with energy density $\varepsilon_\Lambda \equiv \frac{c^2}{8\pi G}\Lambda$ (Eq. 4.69), constant in time if Λ is constant. [R 4 98]
Λ requires negative pressure: the fluid equation demands that a constant $\varepsilon_\Lambda$ be accompanied by $P_\Lambda = -\varepsilon_\Lambda = -\frac{c^2}{8\pi G}\Lambda$ (Eq. 4.70) — i.e. $w = -1$. [R 4 98]
Einstein's static model: $\ddot a = 0$ requires $\Lambda = 4\pi G\rho$ (Eq. 4.71); $\dot a = 0$ in the Friedmann equation then forces positive curvature $\kappa = +1$ (Eq. 4.72) with radius $R_0 = \frac{c}{2(\pi G \rho)^{1/2}} = \frac{c}{\Lambda^{1/2}}$ (Eq. 4.73); published 1917. [R 4 99]
Einstein disliked his own fix: he considered the cosmological constant "gravely detrimental to the formal beauty of the theory." [R 4 99]
Instability of the static model: the balance of Λ-repulsion against matter attraction is an unstable equilibrium — expand it slightly and $\varepsilon_\Lambda$ stays fixed while the matter density drops, so repulsion wins and expansion runs away; compress it slightly and collapse runs away. [R 4 99]
Λ discarded: Hubble's 1929 redshift–distance paper gave Einstein the excuse to drop Λ; according to Gamow's memoirs, Einstein called introducing it "the biggest blunder of his life." [R 4 100]
Λ ironically revived by the same paper: Hubble's distance underestimate gave $H_0 = 500\ \mathrm{km,s^{-1},Mpc^{-1}}$, hence a Hubble time $H_0^{-1} \approx 2$ Gyr — shorter than the ~3 Gyr radiometric age of the Earth (Arthur Holmes); a Λ large enough to make $\ddot a > 0$ means $\dot a$ was smaller in the past, so the universe is older than $H_0^{-1}$. [R 4 100]
Λ's fashion cycles: with $\Lambda > 4\pi G\rho_0$ the expansion accelerates and the universe can be arbitrarily old for a given $H_0^{-1}$; Λ has gone in and out of fashion since 1917, favored whenever the Hubble time looked embarrassingly short next to the ages of astronomical objects — and is currently popular because observations (Section 6.5) indicate accelerating expansion. [R 4 100]
Physical cause of Λ: requires a component whose energy density stays constant as the universe expands or contracts; the leading candidate is vacuum energy. [R 4 100]
Quantum vacuum energy: classically a vacuum has no energy, but the Heisenberg uncertainty principle allows virtual particle–antiparticle pairs satisfying $\Delta E, \Delta t \lesssim h$ (Eq. 4.74) to appear and annihilate; the resulting $\varepsilon_{\rm vac}$ is a quantum phenomenon independent of the universe's expansion — exactly the behavior Λ needs. [R 4 101]
The vacuum energy problem: computing $\varepsilon_{\rm vac}$ in quantum field theory has not been successfully completed; the suggested "natural" value is the Planck energy density $\varepsilon_{\mathrm{vac}} \sim \frac{E_P}{\ell_P^3}$ (Eq. 4.75), with $E_P = 1.22\times 10^{28}$ eV (= 540 kilowatt-hours) and $\ell_P = 1.62\times 10^{-35}$ m, giving $\varepsilon_{\mathrm{vac}} \sim 3 \times 10^{132}\ \mathrm{eV,m^{-3}}$ (Eq. 4.76) — 123 orders of magnitude above the current critical density, a spectacularly bad match between theory and observation. [R 4 101]
Astronomy probing particle physics: deducing $\varepsilon_\Lambda$ from the observed expansion means that by studying the universe on the largest scales, we indirectly examine the structure of the vacuum on the smallest scales. [R 4 102]

Chapter 5 — Model Universes

Governing equations: a homogeneous, isotropic universe is fully specified by the Friedmann equation $\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3c^2}\varepsilon - \frac{\kappa c^2}{R_0^2 a^2}$ (Eq. 5.1), the fluid equation $\dot{\varepsilon} + 3\frac{\dot{a}}{a}(\varepsilon + P) = 0$ (Eq. 5.2), and the equation of state $P = w\varepsilon$ (Eq. 5.3); with boundary conditions these yield $\varepsilon(t)$, $P(t)$, $a(t)$ — complicated in practice because the real universe has several components with different $w$. [R 5 105]

5.1 Evolution of Energy Density

Additivity of components: total energy density and pressure are sums over components, $\varepsilon = \sum_i \varepsilon_i$ (Eq. 5.4), $P = \sum_i w_i\varepsilon_i$ (Eq. 5.5); consequently the fluid equation holds for each component separately, as long as the components do not interact. [R 5 106]
Density scaling law: for constant $w_i$, $\varepsilon_i(a) = \varepsilon_{i,0},a^{-3(1+w_i)}$ (Eq. 5.9) — derived from the fluid equation and equation of state alone; the Friedmann equation doesn't enter. [R 5 107]
Why matter and radiation dilute differently: write $\varepsilon = nE$ with $n \propto a^{-3}$ for both (particles neither created nor destroyed); nonrelativistic matter has constant $E = mc^2$, so $\varepsilon_m \propto a^{-3}$ (Eq. 5.10), while photons have $E = hc/\lambda \propto a^{-1}$ since wavelengths stretch with expansion, so $\varepsilon_r \propto a^{-4}$ (Eq. 5.11). [R 5 107]
CMB energy density: at $T_0 = 2.7255$ K, $\varepsilon_{\mathrm{CMB},0} = \alpha T_0^4 = 0.2606\ \mathrm{MeV,m^{-3}}$ (Eq. 5.12), giving density parameter $\Omega_{\mathrm{CMB},0} = 0.2606/4870 = 5.35\times10^{-5}$ (Eq. 5.13). [R 5 108]
Photon non-conservation is negligible: stars do create photons, but from the galaxy luminosity density $\Psi \approx 1.7\times10^8\ L_\odot,\mathrm{Mpc^{-3}}$ (Eq. 5.14) a rough estimate gives $\varepsilon_{\mathrm{starlight},0} \sim \Psi t_0 \sim 0.006\ \mathrm{MeV,m^{-3}}$ (Eq. 5.15); measured (with dust-reprocessed light) $\varepsilon_{\mathrm{starlight}}/\varepsilon_{\mathrm{CMB}} \approx 0.1$, so ignoring non-CMB photons is an acceptable approximation. [R 5 109]
Cosmic neutrino background: relic of the epoch when the universe was hot and dense enough to be opaque to neutrinos; per flavor (while relativistic) $\varepsilon = \frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\varepsilon_{\mathrm{CMB}} = 0.227,\varepsilon_{\mathrm{CMB}}$ (Eq. 5.16), so all three flavors give $\Omega_\nu = 0.681,\Omega_{\mathrm{CMB}}$. [R 5 109]
Neutrino radiation-to-matter transition: mean energy per neutrino $E_\nu \approx \frac{5\times10^{-4}\ \mathrm{eV}}{a}$ (Eq. 5.17); when this drops to $\sim m_\nu c^2$, that species stops being "radiation" and becomes "matter". [R 5 110]
Total radiation today: if all neutrinos were still effectively massless, $\Omega_{r,0} = \Omega_{\mathrm{CMB},0} + \Omega_{\nu,0} = 5.35\times10^{-5} + 3.65\times10^{-5} = 9.0\times10^{-5}$ (Eq. 5.18). [R 5 110]
Benchmark Model introduced: the working model matching observation — $\Omega_{r,0} = 9.0\times10^{-5}$, $\Omega_{m,0} = 0.31$, $\Omega_{\Lambda,0} = 1 - \Omega_{r,0} - \Omega_{m,0} \approx 0.69$; defined to be spatially flat. [R 5 110]
Matter–lambda equality: today $\varepsilon_{\Lambda,0}/\varepsilon_{m,0} = 0.69/0.31 \approx 2.23$ (Eq. 5.19) — Λ is "dominant"; the ratio scales as $\frac{\varepsilon_\Lambda}{\varepsilon_m} = \frac{\Omega_{\Lambda,0}}{\Omega_{m,0}}a^3$ (Eq. 5.20), so equality occurred at $a_{m\Lambda} = \left(\frac{\Omega_{m,0}}{\Omega_{\Lambda,0}}\right)^{1/3} \approx 0.766$ (Eq. 5.21). [R 5 111]
Radiation–matter equality: today $\varepsilon_{m,0}/\varepsilon_{r,0} = \Omega_{m,0}/\Omega_{r,0} \approx 3400$ (Eq. 5.22, assuming all three neutrino flavors relativistic); the ratio scales $\propto a$ (Eq. 5.23), so equality occurred at $a_{rm} = \varepsilon_{r,0}/\varepsilon_{m,0} \approx 1/3400 \approx 2.9\times10^{-4}$ (Eq. 5.24); any neutrino with $m_\nu c^2 \ll 2$ eV was still relativistic ("radiation") then. [R 5 111]
Dominance ordering set by $w$: as $a \to 0$ the component with the largest $w$ dominates; as $a \to \infty$, the smallest $w$; our history runs radiation ($w = 1/3$) → matter ($w = 0$) → cosmological constant ($w = -1$). [R 5 112]
Scale factor and redshift as time surrogates: in a continuously expanding universe $a(t)$ is monotonic, and $1+z = 1/a$, so epochs are labeled by $a$ or $z$ (e.g. matter–lambda equality at $z_{m\Lambda} \approx 0.31$); this is convenient because converting $a$ to $t$ is not simple in a multi-component universe. [R 5 112]
Component-wise Friedmann equation: $\dot{a}^2 = \frac{8\pi G}{3c^2}\sum_i \varepsilon_{i,0},a^{-1-3w_i} - \frac{\kappa c^2}{R_0^2}$ (Eq. 5.25); the terms scale as $a^{-2}$ (radiation), $a^{-1}$ (matter), $a^0$ (curvature), $a^{2}$ (Λ); the multi-term case has no simple analytic $a(t)$, but single-component universes are instructive. [R 5 113]

5.2 Empty Universes

Empty-universe solutions: with $\varepsilon = 0$ the Friedmann equation is $\dot{a}^2 = -\kappa c^2/R_0^2$ (Eq. 5.26); solutions are either static and flat ($\kappa = 0$, Minkowski space where special relativity holds) or negatively curved with $\dot{a} = \pm c/R_0$ (Eq. 5.27); positively curved empty universes are forbidden (imaginary $\dot a$). [R 5 113]
Milne universe: the expanding empty universe has $a(t) = t/t_0$ (Eq. 5.28) with $t_0 = R_0/c$ — Newtonian reading: no gravity, so relative velocities stay constant and $a$ grows linearly; its age exactly equals the Hubble time (named "Milne universe" in footnote, [R 5 152]). [R 5 114]
Low-density limit: any universe with $\Omega \ll 1$ is well approximated by the empty, negatively curved model with $a = t/t_0$. [R 5 114]
Emission time in an empty universe: $1+z = t_0/t_e$ (Eq. 5.29), so $t_e = \frac{H_0^{-1}}{1+z}$ (Eq. 5.30). [R 5 115]
Proper distance, general RW result: for light emitted at $t_e$ and observed at $t_0$, the current proper distance in any Robertson–Walker universe is $d_p(t_0) = c\int_{t_e}^{t_0}\frac{dt}{a(t)}$ (Eq. 5.33). [R 5 116]
Empty-universe distance–redshift relation: $d_p(t_0) = \frac{c}{H_0}\ln(1+z)$ (Eq. 5.35) — linear in $z$ for $z \ll 1$, logarithmic for $z \gg 1$; objects at arbitrarily large current proper distance can be seen. [R 5 116]
Emission-distance maximum (empty): $d_p(t_e) = \frac{c}{H_0}\frac{\ln(1+z)}{1+z}$ (Eq. 5.36) peaks at $z = e - 1 \approx 1.72$, where $d_p(t_e) = (1/e),c/H_0 \approx 0.37,c/H_0$; higher-$z$ objects are seen from a time when they were very close. [R 5 117]

5.3 Single-component Universes

Flat single-component universe: setting $\kappa = 0$ with one component, $\dot{a}^2 = \frac{8\pi G\varepsilon_0}{3c^2}a^{-(1+3w)}$ (Eq. 5.37); a power-law ansatz gives $a(t) = \left(\frac{t}{t_0}\right)^{2/(3+3w)}$ (Eq. 5.39), valid for $w \neq -1$. [R 5 117]
Age vs. Hubble time: $t_0 = \frac{2}{3(1+w)}H_0^{-1}$ (Eq. 5.42); the universe is younger than the Hubble time if $w > -1/3$, older if $w < -1/3$. [R 5 118]
Energy density falls as $t^{-2}$ regardless of $w$: $\varepsilon(t) = \frac{1}{6\pi(1+w)^2}\frac{c^2}{G}t^{-2}$ (Eq. 5.46) in any flat single-component universe. [R 5 119]
Emission time and proper distance: $t_e = t_0/(1+z)^{3(1+w)/2}$ (Eq. 5.48); the current proper distance is $d_p(t_0) = \frac{c}{H_0}\frac{2}{1+3w}\left[1 - (1+z)^{-(1+3w)/2}\right]$ (Eq. 5.50), for $w \neq -1/3$. [R 5 119]
Horizon distance: the proper distance (at observation) to the most distant visible object, whose light was emitted at $t = 0$: $d_{\mathrm{hor}}(t_0) = c\int_0^{t_0}\frac{dt}{a(t)}$ (Eq. 5.51); finite in a flat universe when $w > -1/3$, with $d_{\mathrm{hor}}(t_0) = ct_0\frac{3(1+w)}{1+3w} = \frac{c}{H_0}\frac{2}{1+3w}$ (Eq. 5.52). [R 5 120]
Visible universe: the portion within the horizon — all points causally connected to the observer; for flat $w \le -1/3$ the horizon distance is infinite and every point in space is causally connected to every observer. [R 5 120]

5.3.1 Matter only

Flat matter-only universe: $a(t) = (t/t_0)^{2/3}$ (Eq. 5.55); age $t_0 = \frac{2}{3H_0}$ (Eq. 5.53); horizon $d_{\mathrm{hor}}(t_0) = 3ct_0 = 2c/H_0$ (Eq. 5.54) (called an "Einstein–de Sitter universe" in footnote, [R 5 152]). [R 5 121]
Matter-only distances: $d_p(t_0) = \frac{2c}{H_0}\left[1 - \frac{1}{\sqrt{1+z}}\right]$ (Eq. 5.56); the emission distance $d_p(t_e)$ (Eq. 5.57) peaks at $z = 5/4$, where $d_p(t_e) = (8/27),c/H_0 \approx 0.30,c/H_0$. [R 5 122]

5.3.2 Radiation only

Flat radiation-only universe: a good description of our own universe well before radiation–matter equality; age $t_0 = \frac{1}{2H_0}$ (Eq. 5.58); horizon $d_{\mathrm{hor}}(t_0) = 2ct_0 = c/H_0$ (Eq. 5.59) — exactly the Hubble distance, which is not generally the case. [R 5 122]
Radiation-only distances: $a(t) = (t/t_0)^{1/2}$ (Eq. 5.60); $d_p(t_0) = \frac{c}{H_0}\frac{z}{1+z}$ (Eq. 5.61); $d_p(t_e) = \frac{c}{H_0}\frac{z}{(1+z)^2}$ (Eq. 5.62), which peaks at $z = 1$ with $d_p(t_e) = 0.25,c/H_0$. [R 5 123]
Radiation density in Planck units: $\varepsilon_r(t) = \frac{3}{32\pi}\frac{E_P}{\ell_P^3}\left(\frac{t}{t_P}\right)^{-2}$ (Eq. 5.63). [R 5 123]
Temperature of a radiation universe: via the blackbody relation, $T(t) \approx 0.46,T_P,(t/t_P)^{-1/2}$ (Eq. 5.64, coefficient as corrected in the formulas doc — the book prints 0.61), where $T_P = 1.42\times10^{32}$ K is the Planck temperature; mean photon energy $E_{\mathrm{mean}} \approx 2.7,kT$ (Eq. 5.65). [R 5 124]
The $t = 0$ infinities aren't physical: formally $\varepsilon_r \to \infty$ as $t \to 0$ (infinite number density of infinite-energy photons), but general relativity is classical — it assumes smooth energy density — and breaks down at $t \approx t_P$, so the infinities need not be taken seriously. [R 5 124]
Planck-time cutoff: the number of photons inside the horizon is $N(t) = V_{\mathrm{hor}},n \sim (t/t_P)^{3/2}$ up to an $O(1)$ coefficient (Eq. 5.69, using $d_{\mathrm{hor}} = 2ct$, Eq. 5.67); quantization becomes non-negligible when $N \approx 1$, i.e. $t \sim t_P$ (book: $t \approx 1.4,t_P$); describing earlier times needs a still-nonexistent quantum gravity theory, so the book stops at $t \sim t_P \sim 10^{-43}$ s, when $E_{\mathrm{mean}} \sim E_P \sim 10^{28}$ eV. [R 5 125]

5.3.3 Lambda only

de Sitter universe: flat with only Λ ($\varepsilon_\Lambda$ constant), the Friedmann equation becomes $\dot{a} = H_0 a$ (Eq. 5.71) with $H_0 = \left(\frac{8\pi G\varepsilon_\Lambda}{3c^2}\right)^{1/2}$ (Eq. 5.72), giving exponential expansion $a(t) = e^{H_0(t-t_0)}$ (Eq. 5.73) (called a "de Sitter universe" in footnote, [R 5 152]). [R 5 125]
Steady State analogy: exponential expansion also characterized the Steady State universe, where constant $\varepsilon$ was maintained by continuous creation of real particles; if Λ is vacuum energy, constant $\varepsilon$ is instead maintained by continuous creation and annihilation of virtual particle–antiparticle pairs. [R 5 126]
Λ-only properties: infinitely old, with infinite horizon distance; $d_p(t_0) = \frac{c}{H_0}z$ (Eq. 5.74) — the exponentially expanding universe is the only one in which $d_p(t_0) \propto z$ for all $z$ (other universes are linear only for $z \ll 1$). [R 5 126]
Superluminal recession: $d_p(t_e) = \frac{c}{H_0}\frac{z}{1+z}$ (Eq. 5.75) → $c/H_0$ as $z \to \infty$; once a source is more than a Hubble distance away, its recession speed exceeds $c$ and photons it emits thereafter can never reach the observer. [R 5 127]

5.4 Multiple-component Universes

Why go multi-component: all the empty and flat single-component models expand forever if expanding now; recollapsing universes, and $a(t)$ that isn't a power law or exponential, require multiple terms on the right-hand side of the Friedmann equation. [R 5 127]
Eliminating explicit curvature: using $\frac{\kappa}{R_0^2} = \frac{H_0^2}{c^2}(\Omega_0 - 1)$ (Eq. 5.77, from Eq. 4.36), the Friedmann equation becomes dimensionless: $\frac{H^2}{H_0^2} = \frac{\varepsilon(t)}{\varepsilon_{c,0}} + \frac{1-\Omega_0}{a^2}$ (Eq. 5.79), with $\varepsilon_{c,0} \equiv \frac{3c^2H_0^2}{8\pi G}$ (Eq. 5.80). [R 5 128]
Master equation for our universe (radiation + matter + Λ): $$\frac{H^2}{H_0^2} = \frac{\Omega_{r,0}}{a^4} + \frac{\Omega_{m,0}}{a^3} + \Omega_{\Lambda,0} + \frac{1-\Omega_0}{a^2}$$ (Eq. 5.81), with $\Omega_0 = \Omega_{r,0}+\Omega_{m,0}+\Omega_{\Lambda,0}$; the Benchmark Model has $\Omega_0 = 1$, but flatness is consistent with — not demanded by — the data, so the curvature term is retained. [R 5 128]
Master integral for $t(a)$: $\int_0^a \frac{da}{\left[\Omega_{r,0}/a^2 + \Omega_{m,0}/a + \Omega_{\Lambda,0}a^2 + (1-\Omega_0)\right]^{1/2}} = H_0 t$ (Eq. 5.83) — no simple analytic solution in general; integrate numerically for given density parameters. [R 5 129]
Epoch-by-epoch approximations: for $a \ll a_{rm} \approx 2.9\times10^{-4}$ the Benchmark Model behaves as flat radiation-only; for $a \gg a_{m\Lambda} \approx 0.77$ as lambda-only; near $a_{rm}$ and $a_{m\Lambda}$ two components are comparable and a two-component model (radiation+matter, matter+Λ) is required. [R 5 129]

5.4.1 Matter + Curvature

Historical importance: mid-twentieth-century cosmology, with Λ out of fashion, concentrated on curved matter-dominated universes; they illuminate the interplay of curvature, expansion, and density. [R 5 130]
Turnaround condition: for matter + curvature, $\frac{H^2}{H_0^2} = \frac{\Omega_0}{a^3} + \frac{1-\Omega_0}{a^2}$ (Eq. 5.85); $H = 0$ requires the curvature term negative, i.e. $\Omega_0 > 1$ ($\kappa = +1$), and maximum expansion occurs at $a_{\max} = \frac{\Omega_0}{\Omega_0 - 1}$ (Eq. 5.87). [R 5 130]
Contraction mirrors expansion: $H$ enters the Friedmann equation only as $H^2$, so the collapse phase is the time-reversal of expansion — strictly true only for a perfectly homogeneous, adiabatic universe; small-scale entropy-generating processes (stars emitting photons) do not run backward. [R 5 131]
Big Crunch: an $\Omega_0 > 1$ matter universe recollapses to $a = 0$ at finite $t_{\mathrm{crunch}}$ — finite in spatial extent and in duration, ending as it began in a hot dense state. [R 5 131]
$\Omega_0 < 1$ fate: both Friedmann terms positive, so expansion never stops; matter dominates early ($a \propto t^{2/3}$ while $a \ll \Omega_0/[1-\Omega_0]$), then dilution hands over to curvature and $a \propto t$ like the empty universe. [R 5 132]
Fate table for matter-only universes (Table 5.1): $\Omega_0 < 1$, $\kappa = -1$ → Big Chill with $a \propto t$; $\Omega_0 = 1$, $\kappa = 0$ → Big Chill with $a \propto t^{2/3}$; $\Omega_0 > 1$, $\kappa = +1$ → Big Crunch. [R 5 132]
Cycloid solution ($\Omega_0 > 1$): parametrically $a(\theta) = \frac{1}{2}\frac{\Omega_0}{\Omega_0-1}(1-\cos\theta)$, $t(\theta) = \frac{1}{2H_0}\frac{\Omega_0}{(\Omega_0-1)^{3/2}}(\theta - \sin\theta)$ (Eqs. 5.90–5.91), $\theta: 0 \to 2\pi$; the Big Bang–to–Big Crunch lifetime is $t_{\mathrm{crunch}} = \frac{\pi}{H_0}\frac{\Omega_0}{(\Omega_0-1)^{3/2}}$ (Eq. 5.92). [R 5 133]
Open counterpart ($\Omega_0 < 1$): $a(\eta) = \frac{1}{2}\frac{\Omega_0}{1-\Omega_0}(\cosh\eta - 1)$, $t(\eta) = \frac{1}{2H_0}\frac{\Omega_0}{(1-\Omega_0)^{3/2}}(\sinh\eta - \eta)$ (Eqs. 5.93–5.94), $\eta: 0 \to \infty$. [R 5 134]
Near-degeneracy today: universes with $\Omega_0 = 0.9$ and $\Omega_0 = 1.1$ have utterly different fates yet are very hard to tell apart at $t_0$; their scale factors diverge significantly only after a Hubble time or more. [R 5 134]
"Density is destiny!" — the matter-only T-shirt slogan: below critical density → Big Chill, above → Big Crunch; but it carries the essential footnote "*if Λ = 0" — with a cosmological constant (or any $w < -1/3$ component), density = destiny = curvature no longer holds. [R 5 134]

5.4.2 Matter + Lambda

Flat matter+Λ universe: flatness requires $\Omega_{\Lambda,0} = 1 - \Omega_{m,0}$ (Eq. 5.95) and $\frac{H^2}{H_0^2} = \frac{\Omega_{m,0}}{a^3} + (1-\Omega_{m,0})$ (Eq. 5.96); with $\Omega_{\Lambda,0} > 0$ the universe expands forever (Big Chill), while $\Omega_{\Lambda,0} < 0$ acts as an attractive force and forces recollapse. [R 5 135]
Negative-Λ crunch: expansion halts at $a_{\max} = \left(\frac{\Omega_{m,0}}{\Omega_{m,0}-1}\right)^{1/3}$ (Eq. 5.97) and the universe collapses to $a=0$ at $t_{\mathrm{crunch}} = \frac{2\pi}{3H_0}\frac{1}{\sqrt{\Omega_{m,0}-1}}$ (Eq. 5.98) — an exceptionally short lifetime compared with the positively curved matter-only crunch at the same $\Omega_{m,0}$; larger $\Omega_{m,0}$ means shorter life. [R 5 136]
Analytic $a(t)$ for flat, positive-Λ universes: $$H_0 t = \frac{2}{3\sqrt{1-\Omega_{m,0}}}\ln\left[\left(\frac{a}{a_{m\Lambda}}\right)^{3/2} + \sqrt{1+\left(\frac{a}{a_{m\Lambda}}\right)^{3}}\right]$$ (Eq. 5.101), with $a_{m\Lambda} = \left(\frac{\Omega_{m,0}}{1-\Omega_{m,0}}\right)^{1/3}$ (Eq. 5.100). [R 5 137]
Limiting behaviors: for $a \ll a_{m\Lambda}$, $a \approx \left(\frac{3}{2}\sqrt{\Omega_{m,0}}H_0 t\right)^{2/3}$ (Eq. 5.102), the flat matter-dominated law; for $a \gg a_{m\Lambda}$, $a \approx a_{m\Lambda}\exp\left(\sqrt{1-\Omega_{m,0}},H_0 t\right)$ (Eq. 5.103), the exponential Λ-dominated law. [R 5 138]
Age of a flat matter+Λ universe: $t_0 = \frac{2H_0^{-1}}{3\sqrt{1-\Omega_{m,0}}}\ln\left[\frac{\sqrt{1-\Omega_{m,0}}+1}{\sqrt{\Omega_{m,0}}}\right]$ (Eq. 5.104); with $\Omega_{m,0}=0.31$, $\Omega_{\Lambda,0}=0.69$ and $H_0 = 68 \pm 2\ \mathrm{km,s^{-1},Mpc^{-1}}$, $t_0 = 0.955,H_0^{-1} = 13.74 \pm 0.40$ Gyr (Eq. 5.105). [R 5 138]
When Λ took over: $t_{m\Lambda} = \frac{2H_0^{-1}}{3\sqrt{1-\Omega_{m,0}}}\ln[1+\sqrt{2}] = 0.707,H_0^{-1} = 10.17 \pm 0.30$ Gyr (Eq. 5.106) — so Λ has been the dominant component for the last ~3.6 billion years. [R 5 139]

5.4.3 Matter + Curvature + Lambda

General matter+curvature+Λ equation: $\frac{H^2}{H_0^2} = \frac{\Omega_{m,0}}{a^3} + \frac{1-\Omega_{m,0}-\Omega_{\Lambda,0}}{a^2} + \Omega_{\Lambda,0}$ (Eq. 5.107); if $\Omega_{m,0}+\Omega_{\Lambda,0} > 1$ (positive curvature) the middle term is negative, and for some parameter choices $H^2 < 0$ at intermediate $a$ — a forbidden range of scale factors. [R 5 139]
Big Bounce: a universe contracting from a low-density Λ-dominated state stops at $a_{\min} > 0$ when the negative curvature term dominates, then re-expands — an expanding universe that never had a Big Bang. [R 5 139]
Loitering universe: with ($\Omega_{m,0}$, $\Omega_{\Lambda,0}$) chosen just right, a matter-dominated $a \propto t^{2/3}$ phase is followed by a long stage with $a$ nearly constant — almost (but not quite) Einstein's static universe — before Λ drives exponential expansion (called a "Lemaître universe" in footnote, [R 5 152]). [R 5 139]
Phase diagram in the ($\Omega_{m,0}$, $\Omega_{\Lambda,0}$) plane (Fig. 5.6): Big Crunch and Big Chill regions each admit any curvature sign; Big Bounce universes contract to $a_{\min}$ then expand forever; loitering universes lie just below the Bounce–Chill dividing line, and loiter longer the closer they lie to it. [R 5 140]
Four futures with identical $\Omega_{m,0} = 0.31$ (Fig. 5.7): $\Omega_{\Lambda,0}=0.69$ → flat Big Chill; $\Omega_{\Lambda,0}=-0.31$ → zero total energy density, negatively curved, expands to $a_{\max}\approx1.93$ then Big Crunch; $\Omega_{\Lambda,0}=1.7289$ → positively curved loiterer at $a \approx 0.45$; $\Omega_{\Lambda,0}=1.8$ → Big Bounce at $a \approx 0.552$ — indistinguishable today by measuring matter density and Hubble constant alone. [R 5 141]
Bounce signature: in the $\Omega_{\Lambda,0}=1.8$ universe the largest observable redshift is $z_{\max} = 1/a_{\mathrm{bounce}} - 1 \approx 0.81$; extremely distant sources would appear blueshifted. [R 5 141]

5.4.4 Radiation + Matter

Flat radiation+matter model: valid near $a_{rm}$, $\frac{H^2}{H_0^2} = \frac{\Omega_{r,0}}{a^4} + \frac{\Omega_{m,0}}{a^3}$ (Eq. 5.108) integrates to the closed form $$H_0 t = \frac{4a_{rm}^2}{3\sqrt{\Omega_{r,0}}}\left[1 - \left(1 - \frac{a}{2a_{rm}}\right)\left(1 + \frac{a}{a_{rm}}\right)^{1/2}\right]$$ (Eq. 5.110). [R 5 142]
Its limits: $a \ll a_{rm}$: $a \approx (2\sqrt{\Omega_{r,0}}H_0t)^{1/2}$ (Eq. 5.111), the radiation law; $a \gg a_{rm}$ (before curvature/Λ matter): $a \approx \left(\frac{3}{2}\sqrt{\Omega_{m,0}}H_0t\right)^{2/3}$ (Eq. 5.112), the matter law. [R 5 143]
Time of radiation–matter equality: setting $a = a_{rm}$, $t_{rm} \approx 0.391,\frac{\Omega_{r,0}^{3/2}}{\Omega_{m,0}^2}H_0^{-1}$ (Eq. 5.113); for the Benchmark Model $t_{rm} = 3.47\times10^{-6}H_0^{-1} \approx 50,000$ yr (Eq. 5.114). [R 5 143]
The radiation era was brief: only ~50 millennia — including radiation would change the computed age $t_0 \approx 13.7$ Gyr by just a few parts per million, dwarfed by the uncertainty in $H_0$; ignoring radiation in age calculations is justified. [R 5 144]

5.5 Benchmark Model

Component inventory (Table 5.2): $H_0 = 68\ \mathrm{km,s^{-1},Mpc^{-1}}$; photons $\Omega_{\gamma,0} = 5.35\times10^{-5}$ (CMB at $T_0 = 2.7255$ K); neutrinos $\Omega_{\nu,0} = 3.65\times10^{-5}$ (68.1% of the CMB energy density while relativistic); total radiation $\Omega_{r,0} = 9.0\times10^{-5}$; baryonic matter $\Omega_{\mathrm{bary},0} \approx 0.048$; nonbaryonic dark matter $\Omega_{\mathrm{dm},0} \approx 0.262$ (over five times the baryons); total matter $\Omega_{m,0} = 0.31$; cosmological constant $\Omega_{\Lambda,0} \approx 0.69$; spatially flat. [R 5 144]
Massive-neutrino defection: a neutrino of mass $m_\nu$ moves from the "radiation" column to the "matter" column when $a \sim 5\times10^{-4}\ \mathrm{eV}/(m_\nu c^2)$ (from Eq. 5.17). [R 5 144]
Key epochs (Table 5.2): radiation–matter equality $a_{rm} = 2.9\times10^{-4}$, $t_{rm} = 0.050$ Myr; matter–lambda equality $a_{m\Lambda} = 0.77$, $t_{m\Lambda} = 10.2$ Gyr; now $a_0 = 1$, $t_0 = 13.7$ Gyr. [R 5 145]
Smooth transitions, curious coincidence: the numerically computed $a(t)$ shows gradual (not abrupt) transitions $t^{1/2} \to t^{2/3} \to$ exponential; strikingly, we live very close to the time of matter–lambda equality (at least on a logarithmic scale). [R 5 145]
Benchmark horizon distance: as $z \to \infty$, $d_p(t_0) \to 3.20,c/H_0$; hence $d_{\mathrm{hor}}(t_0) = 3.20,c/H_0 = 3.35,ct_0 = 14,000$ Mpc (Eq. 5.115) — objects beyond ~14 Gpc are invisible because their light hasn't had time to reach us. [R 5 146]
Benchmark emission-distance maximum: $d_p(t_e)$ peaks for galaxies at $z = 1.6$, where $d_p(t_e) = 0.405,c/H_0$. [R 5 147]
Lookback time: distinct from distance — the answer to "how long has the light been traveling?" is $t_0 - t_e$; for $z \ll 1$, $t_0 - t_e \approx z/H_0$, but the relation becomes nonlinear and model-dependent at larger $z$. [R 5 147]
Lookback comparison at $z = 2$: Benchmark Model 10.5 Gyr; flat lambda-only 15.8 Gyr; flat matter-only just 7.7 Gyr (same $H_0$) — the lookback–redshift relation is a probe of the cosmological model. [R 5 148]
A $z = 10$ galaxy in the Benchmark Model: current proper distance $d_p(t_0) = 2.18,c/H_0 = 9500$ Mpc (about two-thirds of the horizon distance); at emission $d_p(t_e) = d_p(t_0)/(1+z) = 0.20,c/H_0 = 870$ Mpc; the light left when the universe was less than 4% of its current age — under half a billion years old ("a telescope is a time machine"). [R 5 148]

Chapter 6 — Measuring Cosmological Parameters

6.1 "A Search for Two Numbers"

Chapter strategy: $a(t)$ is not directly observable; the Friedmann-equation argument runs both ways — knowing $\varepsilon$ for each component gives $a(t)$, and conversely determining $a(t)$ from observations of distant objects constrains $\varepsilon$ for each component. [R 6 153]
Taylor-expansion approach: since the exact form of $a(t)$ is hard to determine, expand it about $t_0$ (Eq. 6.1); truncation at a few terms is justified because $a$ doesn't fluctuate wildly — model universes all have smoothly varying scale factors and there's no evidence of wild oscillation in the real one. [R 6 153]
Two-number expansion: keeping three terms and setting $a(t_0)=1$, $a(t) \approx 1 + H_0(t-t_0) - \frac{1}{2}q_0 H_0^2 (t-t_0)^2$ (Eq. 6.4) — the recent expansion described entirely by $H_0$ and $q_0$. [R 6 154]
Hubble constant: $H_0 \equiv \frac{\dot a}{a}\big|_{t=t_0}$ (Eq. 6.5). [R 6 154]
Deceleration parameter: dimensionless, $q_0 \equiv -\left(\frac{\ddot a, a}{\dot a^2}\right){t=t_0} = -\left(\frac{\ddot a}{aH^2}\right){t=t_0}$ (Eq. 6.6). [R 6 154]
Sign convention of $q_0$: $q_0 > 0$ means decelerating expansion (relative velocity of any two points decreasing), $q_0 < 0$ accelerating; the name and sign date from the mid-1950s when the limited data favored a decelerating matter-dominated universe — a large enough cosmological constant makes $q_0$ negative. [R 6 155]
The Taylor expansion (6.4) is physics-free — pure kinematics of the expansion near $t_0$, saying nothing about the forces acting; Allan Sandage's famous 1970 review described all of cosmology as "a search for two numbers," $H_0$ and $q_0$. [R 6 155]
$q_0$ from the acceleration equation: for $N$ components, $\frac{\ddot a}{a} = -\frac{4\pi G}{3c^2}\sum_i \varepsilon_i(1+3w_i)$ (Eq. 6.7); dividing by $H^2$ (the bracketed factor being $1/\varepsilon_c$) and evaluating at $t_0$ gives $q_0 = \frac{1}{2}\sum_{i=1}^{N}\Omega_{i,0}(1+3w_i)$ (Eq. 6.10). [R 6 155]
Component form: for radiation + matter + $\Lambda$, $q_0 = \Omega_{r,0} + \frac{1}{2}\Omega_{m,0} - \Omega_{\Lambda,0}$ (Eq. 6.11); acceleration ($q_0<0$) requires $\Omega_{\Lambda,0} > \Omega_{r,0} + \Omega_{m,0}/2$; the Benchmark Model has $q_0 \approx -0.53$. [R 6 156]
Measuring $H_0$ in principle: at small $z$ Hubble's law $cz = H_0 d$ (Eq. 6.12) is linear, so the slope of $cz$ vs. $d$ gives $H_0$; in practice distance is not only difficult to measure but also somewhat difficult to define. [R 6 156]
Proper distance–scale factor link: light emitted at $t_e$, observed at $t_0$, gives $d_p(t_0) = c\int_{t_e}^{t_0}\frac{dt}{a(t)}$ (Eq. 6.13); with the Taylor expansion this yields $d_p(t_0) \approx c(t_0-t_e) + \frac{cH_0}{2}(t_0-t_e)^2$ (Eq. 6.15) — first term is the static-universe distance (lookback time times $c$), second the correction for expansion during flight. [R 6 157]
Photons stamp $a(t_e)$, not lookback time: what we observe is redshift, $z = \frac{1}{a(t_e)} - 1$ (Eq. 6.16), so invert the expansion to get lookback time from redshift: $t_0 - t_e \approx H_0^{-1}\left[z - \left(1+\frac{q_0}{2}\right)z^2\right]$ (Eq. 6.18). [R 6 157]
Proper distance vs. redshift (low $z$): $d_p(t_0) \approx \frac{c}{H_0}z\left[1 - \frac{1+q_0}{2}z\right]$ (Eq. 6.19); the linear Hubble relation $d_p \propto z$ holds only for $z \ll 2/(1+q_0)$, and if $q_0 > -1$ the proper distance at moderate redshift ($z \sim 0.1$) is less than the linear relation predicts. [R 6 158]

6.2 Luminosity Distance

Proper distance is unmeasurable: a tape measure would keep lengthening as you unreeled it; measuring $d_p(t_0)$ would require an infinitely fast tape measure or halting the expansion — neither physically possible, so distance must be computed from observed properties. [R 6 158]
Radar ranging works within the solar system ($d = c,\delta t/2$) and fixes $1\ \mathrm{AU} = 149,597,870.7$ km, but beyond $\sim 10$ AU reflected signals are too faint. [R 6 159]
Trigonometric parallax: $d_\pi = 1,\mathrm{pc}\left(\frac{b}{1,\mathrm{AU}}\right)\left(\frac{\theta}{1,\mathrm{arcsec}}\right)^{-1}$ (Eq. 6.20); Gaia (launched 2013) reaches errors $\sim 10\ \mu\mathrm{as}$, but a galaxy at 100 Mpc would need $< 0.01\ \mu\mathrm{as}$ with an Earth-orbit baseline — cosmological parallaxes are unmeasurably small. [R 6 159]
Observables at cosmological distances: bolometric flux $f$ (W m$^{-2}$, integrated over all wavelengths), redshift $z$ (from spectral lines), and angular diameter $\delta\theta$ (for extended sources); in practice flux over a limited wavelength range is measured. [R 6 160]
Standard candle: an object of known luminosity $L$; its luminosity distance is $d_L \equiv \left(\frac{L}{4\pi f}\right)^{1/2}$ (Eq. 6.21) — a "distance" because it has distance dimensions and equals the proper distance the candle would have in a static Euclidean universe, where $f = L/(4\pi d^2)$. [R 6 160]
Curvature effect on flux: in the RW metric, photons emitted at comoving coordinate $r$ are now spread over a sphere of proper area $A_p(t_0) = 4\pi S_\kappa(r)^2$ (Eq. 6.24); positive curvature gives $A_p < 4\pi r^2$ (photons concentrated), negative curvature $A_p > 4\pi r^2$ — a geometric effect present even in a static universe. [R 6 161]
Expansion dims flux by $(1+z)^{-2}$, via two effects: each photon's energy drops, $E_0 = \frac{E_e}{1+z}$ (Eq. 6.26), and the interval between photon detections is stretched, $\delta t_0 = \delta t_e(1+z)$. [R 6 162]
Master flux relation: $f = \frac{L}{4\pi S_\kappa(r)^2(1+z)^2}$ (Eq. 6.27), hence $d_L = S_\kappa(r)(1+z)$ (Eq. 6.28). [R 6 162]
Near-flat simplification: evidence indicates $R_0 \gg d_{\mathrm{hor}}(t_0)$, and any finite-$z$ object is well inside the horizon, so $r \ll R_0$ and $S_\kappa(r) \approx r$; then $d_L = d_p(t_0)(1+z)$ (Eq. 6.29) — a naive inverse-square estimate overestimates the true proper distance by the factor $(1+z)$ even in perfectly flat space. [R 6 163]
Low-$z$ luminosity distance: $d_L \approx \frac{c}{H_0}z\left(1 + \frac{1-q_0}{2}z\right)$ (Eq. 6.31), from multiplying the low-$z$ proper distance (Eq. 6.30) by $(1+z)$. [R 6 163]

6.3 Angular-diameter Distance

Standard yardstick: an object of known proper length $\ell$; conveniently one tightly bound (by gravity, duct tape, etc.) so it does not expand with the universe. [R 6 164]
Angular-diameter distance: from the small-angle formula, $d_A \equiv \frac{\ell}{\delta\theta}$ (Eq. 6.32); equals the proper distance only in a static Euclidean universe. [R 6 165]
Derivation from the RW metric: the yardstick's ends at $(r,\theta_1,\phi)$, $(r,\theta_2,\phi)$ emit light traveling on geodesics of constant $\theta,\phi$; the metric gives $\ell = a(t_e)S_\kappa(r),\delta\theta$ (Eq. 6.34), hence $d_A = \frac{S_\kappa(r)}{1+z}$ (Eq. 6.35). [R 6 165]
Relation to $d_L$: $d_A = \frac{d_L}{(1+z)^2}$ (Eq. 6.36) — for an object that is both a standard candle and a standard yardstick, the angular-diameter distance is always smaller than the luminosity distance. [R 6 166]
Flat-universe meaning of $d_A$: $d_A(1+z) = d_p(t_0) = \frac{d_L}{1+z}$ (Eq. 6.37); so in a flat universe $d_A$ equals the proper distance at the time of emission, $d_p(t_e)$, not the current proper distance. [R 6 166]
Low-$z$ expansion: $d_A \approx \frac{c}{H_0}z\left(1 - \frac{3+q_0}{2}z\right)$ (Eq. 6.38); as $z \to 0$ all measures converge: $d_A \approx d_L \approx d_p(t_0) \approx (c/H_0)z$. [R 6 167]
Opposite high-$z$ behavior: in models with a finite horizon, $d_p(t_0) \to d_{\mathrm{hor}}(t_0)$ as $z\to\infty$, so $d_L \approx z, d_{\mathrm{hor}}(t_0)$ diverges (Eq. 6.39) while $d_A \approx \frac{d_{\mathrm{hor}}(t_0)}{z} \to 0$ (Eq. 6.40). [R 6 167]
Critical redshift: in models other than lambda-only, $d_A$ has a maximum at some $z_c$; for the Benchmark Model $z_c = 1.6$ with $d_A(\mathrm{max}) = 0.405,c/H_0 = 1770$ Mpc — identical glow-in-the-dark yardsticks would shrink in angular size out to $z_c$, then grow: a sky full of big, faint, redshifted yardsticks. [R 6 168]
Minimum angular size: a yardstick's $\delta\theta$ is minimized at $z_c$; in the Benchmark Model $\delta\theta(\mathrm{min}) = \frac{\ell}{1770,\mathrm{Mpc}} \approx 0.1,\mathrm{arcsec}\left(\frac{\ell}{1,\mathrm{kpc}}\right)$ (Eq. 6.41). [R 6 168]
Practical trouble with yardsticks: galaxies and clusters are large enough to resolve but lack sharply defined edges, and both grow with time (mergers; infall of galaxies into clusters) — correcting for evolution is difficult, which long plagued the method. [R 6 168]

6.4 Standard Candles and H₀

Recipe for $H_0$ (Hubble's own method): identify a population of standard candles of luminosity $L$; measure $z$ and $f$ for each; compute $d_L = (L/4\pi f)^{1/2}$; plot $cz$ vs. $d_L$; the slope at $z \ll 1$ is $H_0$ — but as in the rabbit-stew recipe ("first catch your rabbit"), the hardest step is finding a good standard candle. [R 6 169]
Cepheid variables: highly luminous supergiants, $L = 400 \to 40,000\ L_\odot$, pulsationally unstable with periods $P = 1.5 \to 60$ days; luminosity varies with the pulsation partly through surface area, partly through surface temperature. [R 6 169]
Period–luminosity relation: discovered by Henrietta Leavitt (Harvard) from Cepheids in the SMC — longer period means larger mean flux; since the SMC's depth is small compared to its distance, flux differences reflect luminosity differences, not distance; if universal, the relation makes Cepheids standard candles. [R 6 170]
Relative distances from equal-period Cepheids: measured $\bar f_{\mathrm{LMC}}/\bar f_{\mathrm{M31}} = 230$ (Eq. 6.42) gives $\frac{d_L(\mathrm{M31})}{d_L(\mathrm{LMC})} = \sqrt{230} = 15.2$ (Eq. 6.43); in practice astronomers use many Cepheids per galaxy and shift the whole P–L relations into coincidence. [R 6 170]
Absolute calibration problem: relative fluxes give only relative distances; anchoring needs $L$ for a given $P$, ideally from parallax to a Galactic Cepheid — but Cepheids are rare (nearest: Polaris at $d_\pi = 130 \pm 10$ pc, then $\delta$ Cephei at $270 \pm 10$ pc), so historically the P–L relation was normalized via a secondary distance to the LMC. [R 6 171]
Key numbers of the ladder: consensus $d_L(\mathrm{LMC}) = 50 \pm 2$ kpc, implying $d_L(\mathrm{M31}) = 760 \pm 30$ kpc; Cepheids are measurable out to $d_L \sim 30$ Mpc; the Virgo cluster is at $d_L = 300, d_L(\mathrm{LMC}) = 15$ Mpc. [R 6 171]
Hubble Key Project result: using Cepheids (a motivating purpose for building HST), the data are best fitted with $H_0 = 75 \pm 8\ \mathrm{km,s^{-1},Mpc^{-1}}$. [R 6 171]
Virgocentric flow: within the $\sim 30$ Mpc Cepheid range the universe is not homogeneous and isotropic; the Local Group's infall toward Virgo makes the measured $cz$ of the Virgo cluster $250\ \mathrm{km,s^{-1}}$ less than in a perfectly homogeneous universe, so recession velocities must be corrected for this peculiar motion. [R 6 171]
Cautionary footnote: in his 1929 paper Hubble underestimated luminosity distances by a factor $\sim 7$ because he underestimated his standard candles' luminosity by a factor $\sim 49$; also note that within our (non-expanding) galaxy, parallax, luminosity, and proper distances are identical. [R 6 180]

6.5 Standard Candles and Acceleration

Why go deep: to escape Virgocentric flow and other peculiar velocities one needs candles at $d_L > 100$ Mpc ($z > 0.02$); to measure acceleration one needs redshifts where $d_L(z)$ deviates measurably from linear, $d_L \approx \frac{c}{H_0}z\left[1 + \frac{1-q_0}{2}z\right]$ (Eq. 6.44) — at $z = 0.2$ the Benchmark Model ($q_0 = -0.53$) gives $d_L$ 5 percent larger than an empty universe ($q_0 = 0$). [R 6 172]
Supernova taxonomy: type II spectra have strong hydrogen absorption lines, type I none; type II are core-collapse deaths of $M > 8,M_\odot$ stars; type Ib are the same mechanism after winds strip the hydrogen envelope — their difference from type II is literally superficial; type Ia are a completely different species. [R 6 172]
Type Ia mechanism: a white dwarf (supported by electron degeneracy pressure) is pushed toward the Chandrasekhar mass $M \approx 1.4,M_\odot$ by merger with another white dwarf or accretion from a companion; approaching/exceeding the limit triggers collapse, then runaway nuclear fusion that blows the star apart, leaving no condensed remnant. [R 6 173]
SN Ia as candles: roughly one per century in our galaxy, but peak luminosity $L = 4\times 10^9\ L_\odot$ — 100 000 times the brightest Cepheid — briefly rivaling all other stars of a moderately bright galaxy combined, so visible at $z \sim 1$. [R 6 173]
Standardization complication: peak luminosities actually span $L \approx (3\to 5)\times 10^9\ L_\odot$, but peak luminosity is tightly correlated with light-curve shape — fast rise-and-fall means dimmer than average, leisurely rise-and-fall means brighter — so the rise/fall time gives the peak luminosity, just as period does for a Cepheid. [R 6 173]
Two teams: the Supernova Cosmology Project and the High-z Supernova Search Team searched for distant supernovae at the end of the 20th century to constrain the acceleration. [R 6 173]
Apparent magnitude: $m \equiv -2.5\log_{10}(f/f_x)$ (Eq. 6.45) with reference flux $f_x = 2.53\times 10^{-8}\ \mathrm{W,m^{-2}}$ (a nod to Hipparchus: naked-eye stars get $0 < m < 6$); smaller $m$ = larger flux; the Sun ($f = 1361\ \mathrm{W,m^{-2}}$) has $m = -26.8$. [R 6 174]
Absolute magnitude: the apparent magnitude an object would have at $d_L = 10$ pc; $M \equiv -2.5\log_{10}(L/L_x)$ (Eq. 6.46) with $L_x = 78.7\ L_\odot$; the Sun has $M = 4.74$; $m$ is a logarithmic flux measure, $M$ a logarithmic luminosity measure. [R 6 174]
Distance modulus: $m - M = 5\log_{10}\left(\frac{d_L}{1,\mathrm{Mpc}}\right) + 25$ (Eq. 6.49); e.g. LMC ($d_L = 0.050$ Mpc) has $m-M = 18.5$; Virgo (15 Mpc) has $m-M = 30.9$. [R 6 175]
Distance modulus vs. redshift (low $z$): $m - M \approx 43.23 - 5\log_{10}\left(\frac{H_0}{68,\mathrm{km,s^{-1},Mpc^{-1}}}\right) + 5\log_{10} z + 1.086(1-q_0)z$ (Eq. 6.51). [R 6 176]
Reading the Hubble diagram: as $z\to 0$, $m-M$ vs. $\log z$ is a straight line whose amplitude at fixed $z$ gives $H_0$; at slightly larger $z$ the deviation from the line reveals acceleration — at a given $z$, standard candles are fainter in an accelerating ($q_0 < 0$) universe than in a decelerating ($q_0 > 0$) one. [R 6 176]
The acceleration result: at $z \approx 1$, Benchmark-Model supernovae are ~0.6 mag fainter than in a flat matter-only universe — the observed faintness of type Ia supernovae at $z > 0.3$ led to the conclusion that the expansion is accelerating; but they are also ~0.6 mag brighter than in a flat lambda-only universe, so the same data set useful upper limits on the acceleration. [R 6 176]
The data: Figure 6.5 compiles the distance modulus vs. redshift for 580 type Ia supernovae (Suzuki et al. 2012), well fitted by the Benchmark Model. [R 6 177]
Parameter constraints from SNe alone: fitting matter + $\Lambda$ models (not required flat) gives a 95% confidence ellipse in the $(\Omega_{m,0}, \Omega_{\Lambda,0})$ plane; decelerating ($q_0 > 0$), Big Crunch, and Big Bounce universes are strongly excluded, but positive, negative, and flat curvature are all still allowed — only when combined with CMB observations (Ch. 8) do the data point to a flat accelerating universe with $\Omega_{m,0} \sim 0.3$, $\Omega_{\Lambda,0} \sim 0.7$. [R 6 177]

Chapter 7 — Dark Matter

Introduction

Why measure Ωm,0: the matter density parameter helps determine the spatial curvature and expansion rate of the universe; even with nonzero Λ, matter is non-negligible today and was the dominant component in the fairly recent past. The census also asks what matter is: stars, other baryons, or dark matter. [R 7 181]
SN Ia degeneracy: type Ia supernova fluxes vs. redshift are consistent with a flat universe (Ωm,0 ≈ 0.3, ΩΛ,0 ≈ 0.7), but neither parameter is individually well constrained — the data also allow (Ωm,0 = 0, ΩΛ,0 ≈ 0.3) or (Ωm,0 = 0.45, ΩΛ,0 ≈ 0.9); independent matter censuses are needed. [R 7 181]

7.1 Visible Matter

V band: filter passing 500 nm < λ < 590 nm (green/yellow, where the retina is most sensitive); ~12% of the Sun's luminosity passes, so $L_{\odot,V} \approx 0.12,L_\odot \approx 4.6\times10^{25}$ W. [R 7 182]
Local luminosity density: galaxy surveys out to d ~ 0.1c/H₀ give a V-band luminosity density $\Psi_V = 1.1\times10^{8},L_{\odot,V},\mathrm{Mpc^{-3}}$ (Eq. 7.1). [R 7 182]
Mass-to-light ratio: converts Ψ_V into a stellar mass density; if all stars were solar, M/L_V = 1 M⊙/L⊙,V ≈ 43 metric tons per watt of yellow-green light. Main-sequence values span a huge range: an O star (M = 60 M⊙, L_V ≈ 20 000 L⊙,V) has M/L_V ≈ 0.003, an M star (M = 0.1 M⊙, L_V ≈ 5×10⁻⁵ L⊙,V) has M/L_V ≈ 2000 M⊙/L⊙,V. [R 7 182]
Initial mass function: χ(M)dM = number of stars formed with mass in M → M+dM; star formation favors low masses. High-mass fit: power law $\chi(M) \propto M^{-\beta}$ for M > 1 M⊙ (Eq. 7.2), typically β ≈ 2.3; low-mass fit: log-normal $\chi(M) \propto \frac{1}{M}\exp\left(-\frac{(\log M-\log M_c)^2}{2\sigma^2}\right)$ (Eq. 7.3), typically M_c ≈ 0.2 M⊙, σ ≈ 0.5. [R 7 183]
Chabrier function: the log-normal-plus-power-law-tail IMF is named after astronomer Gilles Chabrier. [R 7 206]
Brown dwarfs: gaseous spheres with M < 0.08 M⊙ never ignite core hydrogen fusion, so they are cooler and dimmer than M stars; the IMF peaks in the range 0.02–0.2 M⊙. At formation there are ~250 M stars per O star — comparable total mass, but negligible total V-band light next to the one O star. [R 7 183]
Star-forming vs. quiescent M/L: galaxies actively forming stars today reach M/L_V ≈ 0.3 M⊙/L⊙,V. [R 7 183] An O star (60 M⊙) exhausts its fuel in t ≈ 3 Myr and explodes as a type II supernova, so quiescent galaxies lack O stars and rise to M/L_V ≈ 8; the local mix averages to ⟨M/L_V⟩ ≈ 4 M⊙/L⊙,V. [R 7 184]
Stellar mass density: $\rho_{\star,0} = \langle M/L_V\rangle,\Psi_V \approx 4\times10^{8},M_\odot,\mathrm{Mpc^{-3}}$ (Eq. 7.4); the critical density as a mass density is ρ_c,0 = 1.28×10¹¹ M⊙ Mpc⁻³. [R 7 184]
Density parameter of stars: $\Omega_{\star,0} \approx 0.003$ (Eq. 7.5) — stars are only 0.3% of the critical density; even adding stellar remnants (white dwarfs, neutron stars, black holes) and brown dwarfs, Ω⋆,0 < 0.005. [R 7 185]
Interstellar gas: in our galaxy and M31, gas is ~20% of the stellar mass; the ratio is higher in irregulars such as the Magellanic Clouds, and there is significant gas between galaxies. [R 7 185]
Coma cluster inventory (stars vs. gas): total V-band luminosity L_Coma,V ≈ 5×10¹² L⊙,V (its two brightest galaxies, NGC 4889 and NGC 4874, have L_V ≈ 2.5×10¹¹ each) gives stellar mass M_Coma,⋆ ≈ 2×10¹³ M⊙; X-ray images reveal hot intracluster gas at T ≈ 10⁸ K (photon energies E ~ kT ~ 9 keV) with M_Coma,gas ≈ 2×10¹⁴ M⊙ — roughly ten times the stellar mass. [R 7 185]
Most baryons are intergalactic: ~85% of the universe's baryons are in extremely tenuous intergalactic gas, much of it too low-density to detect with current technology. [R 7 186]
Baryon density parameter: CMB temperature fluctuations (sensitive to the baryon-to-photon ratio at t ~ 250 000 yr) and primordial nucleosynthesis (sensitive to it at t ~ few minutes) both give $\Omega_{\mathrm{bary},0} = 0.048 \pm 0.003$ (Eq. 7.6) — ten to twenty times Ω⋆,0: visible stars are a minority of the baryons. [R 7 187]

7.2 Dark Matter in Galaxies

Nonbaryonic dark matter: most matter is not even baryonic — it doesn't absorb, emit, or scatter light of any wavelength; it is detected via its gravitational influence on visible matter. The Sun's orbit (R = 8.2 kpc, v = 235 km s⁻¹) exemplifies the disk stars' near-circular orbits used as tracers. [R 7 187]
Rotation-curve relation: equating centripetal acceleration $a = v^2/R$ (Eq. 7.7) with gravity $a = GM(R)/R^2$ (Eq. 7.8) gives $v = \sqrt{GM(R)/R}$ (Eq. 7.10), where M(R) is the mass inside radius R (spherical symmetry assumed). [R 7 188]
Exponential disk → Keplerian expectation: disk surface brightness falls as $I(R) = I(0),e^{-R/R_s}$ (Eq. 7.11), with R_s ≈ 4 kpc for our galaxy and ≈ 6 kpc for M31; a few scale lengths out, the enclosed stellar mass is essentially constant, so if stars dominated, v would fall as $v \propto R^{-1/2}$ ("Keplerian rotation"). [R 7 188]
Flat rotation curves: Rubin & Ford (1970) measured M31's ionized-gas emission lines out to R = 24 kpc = 4R_s with no sign of Keplerian decline; 21 cm observations of atomic hydrogen extend this to R = 35 kpc ≈ 6R_s with v(R) ≈ 230 km s⁻¹ nearly constant. (Slipher first detected M31's rotation in 1914.) [R 7 189]
Dark halo: since v at R > 3R_s exceeds what stars and gas alone provide, the visible disk must be embedded in a dark halo whose mass gravitationally anchors the high-speed outer stars and gas. Most if not all spirals have comparable halos; our own galaxy's orbital speed is roughly constant at R > 15 kpc. [R 7 189]
Spiral galaxy mass: with v ≈ constant, $M(R) = \frac{v^2 R}{G} = 1.05\times10^{11},M_\odot \left(\frac{v}{235,\mathrm{km,s^{-1}}}\right)^2 \left(\frac{R}{8.2,\mathrm{kpc}}\right)$ (Eq. 7.12); our galaxy's luminosity is L_gal,V = 2.0×10¹⁰ L⊙,V. [R 7 189]
Galactic mass-to-light ratio: $\langle M/L_V\rangle_{\mathrm{gal}} \approx 64,\frac{M_\odot}{L_{\odot,V}}\left(\frac{R_{\mathrm{halo}}}{100,\mathrm{kpc}}\right)$ (Eq. 7.13) — grows with the (poorly known) halo radius. [R 7 190]
Halo size estimates: keeping globular clusters and satellite galaxies (e.g. Magellanic Clouds) bound requires R_halo ≈ 75 kpc → M_gal ≈ 9.6×10¹¹ M⊙ and ⟨M/L_V⟩_gal ≈ 48 — an order of magnitude above the stellar value; a speculated R_halo ≈ 300 kpc (nearly halfway to M31) gives M_gal ≈ 3.8×10¹² M⊙ and ⟨M/L_V⟩_gal ≈ 190 M⊙/L⊙,V. [R 7 190]

7.3 Dark Matter in Clusters

Zwicky's argument (1930s): the radial velocity dispersion of Coma cluster galaxies (~1000 km s⁻¹) is far too large for the visible stars and gas to bind the cluster; to keep the galaxies from flying off, the cluster must contain much "dunkle Materie" — dark matter. [R 7 190]
Historical footnote: Zwicky popularized the phrase, but Henri Poincaré discussed possible "matière obscure" in our galaxy already in 1908. [R 7 206]
Model setup: treat the cluster as N point-mass galaxies — gravitationally bound, not expanding with the Hubble flow, Newtonian, and isolated: $\ddot{\vec{x}}i = G\sum{j\neq i} m_j \frac{\vec{x}_j-\vec{x}_i}{|\vec{x}_j-\vec{x}_i|^3}$ (Eq. 7.14). [R 7 191]
Potential energy: $W = -\frac{G}{2}\sum_{i,j,(j\neq i)} \frac{m_i m_j}{|\vec{x}_j-\vec{x}_i|}$ (Eq. 7.15), rewritable as $W = -\alpha,\frac{GM^2}{r_h}$ (Eq. 7.16), where M = total galaxy mass, r_h = half-mass radius (sphere about the center of mass containing M/2), and α ≈ 0.45 fits observed clusters. [R 7 191]
Kinetic energy: $K = \frac{1}{2}M\langle v^2\rangle$ (Eq. 7.18), with $\langle v^2\rangle \equiv \frac{1}{M}\sum_i m_i|\dot{\vec{x}}_i|^2$ (Eq. 7.19) the mass-weighted mean square velocity. [R 7 192]
Virial theorem derivation: define moment of inertia $I \equiv \sum_i m_i |\vec{x}_i|^2$ (Eq. 7.20); differentiating twice gives $\ddot{I} = 2\sum_i m_i(\vec{x}_i\cdot\ddot{\vec{x}}_i) + 4K$ (Eq. 7.22); symmetrizing the first term under i ↔ j exchange shows it equals 2W (Eq. 7.26), yielding the virial theorem $\ddot{I} = 2W + 4K$ (Eq. 7.27). [R 7 192]
Steady-state virial theorem: first derived in the 19th century for kinetic theory of gases; for I = constant (no expansion/contraction, center-of-mass frame), $0 = W + 2K$ (Eq. 7.28), so $M = \frac{\langle v^2\rangle, r_h}{\alpha G}$ (Eq. 7.31) — like Eq. 7.12, mass ≈ (characteristic velocity)² × characteristic radius / G. [R 7 194]
Coma observables: mean redshift ⟨z⟩ = 0.0232 (Eq. 7.32) → distance d = (c/H₀)⟨z⟩ = 102 Mpc (Eq. 7.33); line-of-sight velocity dispersion σ_r = 880 km s⁻¹ (Eq. 7.34); assuming isotropic dispersion, $\langle v^2\rangle = 3\sigma_r^2 = 2.32\times10^{12},\mathrm{m^2,s^{-2}}$ (Eq. 7.35). Only line-of-sight velocities are measurable, hence the isotropy assumption. [R 7 195]
Half-mass radius: unknown a priori (the dark matter distribution is what we seek); assuming mass-to-light ratio constant with radius (half-mass sphere = half-light sphere) and intrinsic sphericity, the galaxy distribution gives r_h ≈ 1.5 Mpc (Eq. 7.36). [R 7 195]
Coma virial mass: $M_{\mathrm{Coma}} = \frac{\langle v^2\rangle r_h}{\alpha G} \approx 2\times10^{15},M_\odot$ (Eq. 7.37) — so stars are ~1% and hot gas ~10% of the cluster mass; the implied mass-to-light ratio is ⟨M/L_V⟩_Coma ~ 400 M⊙/L⊙,V (Eq. 7.38), far above our galaxy's. [R 7 196]
Gas confinement as confirmation: without dark matter to anchor it gravitationally, the hot X-ray gas would have expanded out of the cluster on time scales much shorter than the Hubble time. [R 7 196]
Hydrostatic equilibrium: pressure-supported intracluster gas obeys $\frac{dP_{\mathrm{gas}}}{dr} = -\frac{GM(r),\rho_{\mathrm{gas}}(r)}{r^2}$ (Eq. 7.39); with the perfect gas law $P_{\mathrm{gas}} = \rho_{\mathrm{gas}} kT_{\mathrm{gas}}/\mu$ (Eq. 7.40), the total mass profile is $M(r) = \frac{kT_{\mathrm{gas}}(r),r}{G\mu}\left[-\frac{d\ln\rho_{\mathrm{gas}}}{d\ln r} - \frac{d\ln T_{\mathrm{gas}}}{d\ln r}\right]$ (Eq. 7.41), assuming uniform composition and ionization (constant μ). [R 7 196]
X-ray mass estimate: cluster X-rays combine bremsstrahlung (free electrons accelerated by protons/He nuclei) and line emission from highly ionized iron and other heavy elements; model fits to the spectrum give T_gas(r) and ρ_gas(r), yielding M ≈ 1.3×10¹⁵ M⊙ within r ≈ 4 Mpc for Coma — consistent with the virial estimate. [R 7 197]
Cluster density parameter: rich clusters generally show Coma-like mass-to-light ratios; summing all cluster masses gives $\Omega_{\mathrm{clus},0} \approx 0.2$ (Eq. 7.42) — a lower limit on Ωm,0, since smoothly distributed matter in the intercluster voids is not counted. [R 7 197]

7.4 Gravitational Lensing

Lensing as a probe: dark matter affects not only the trajectories of matter but also of photons — it can bend and focus light, acting as a gravitational lens; used to search for dark matter both in our galaxy's halo and in distant clusters. [R 7 198]
MACHOs: MAssive Compact Halo Objects — hypothetical dim compact halo objects (cold white dwarfs, black holes, brown dwarfs, etc.) that would be nearly undetectable by emitted light. [R 7 198]
Deflection angle: a photon passing a compact mass M at impact parameter b is deflected by $\alpha = \frac{4GM}{c^2 b}$ (Eq. 7.43); grazing the Sun's surface, α = 1.7 arcsec (Eq. 7.44). [R 7 198]
1919 eclipse test: comparison of eclipse photographs of stars near the Sun with plates taken six months earlier confirmed Einstein's predicted deflection, bringing experimental support to general relativity. [R 7 199]
Einstein radius: a lens exactly on the line of sight images the source into a perfect ring of angular radius $\theta_E = \left(\frac{4GM}{c^2 d},\frac{1-x}{x}\right)^{1/2}$ (Eq. 7.45), where d = observer–source distance and xd = observer–lens distance; for a MACHO halfway to an LMC star (x ≈ 0.5), $\theta_E \approx 4\times10^{-4},\mathrm{arcsec}\left(\frac{M}{1,M_\odot}\right)^{1/2}\left(\frac{d}{50,\mathrm{kpc}}\right)^{-1/2}$ (Eq. 7.46) — far too small to resolve. [R 7 199]
Microlensing signature: an imperfectly aligned lens gives two or more arcs; the observable is flux amplification, significant when the MACHO–star angular separation ≲ θ_E. Even if the halo were entirely MACHOs, the lensing probability per LMC star is only P ~ 5×10⁻⁷ at any moment, so surveys monitored millions of LMC stars; an event brightens, then dims, as the separation shrinks and grows. [R 7 200]
Event time scale: the time to cross θ_E, $\Delta t = \frac{d,\theta_E}{2v} \approx 90,\mathrm{days}\left(\frac{M}{1,M_\odot}\right)^{1/2}\left(\frac{v}{200,\mathrm{km,s^{-1}}}\right)^{-1}$ (Eq. 7.47), with v the relative transverse velocity; more massive MACHOs give longer events. [R 7 200]
MACHO survey results: a scarcity of short-duration events implies no significant halo population of brown dwarfs or free-floating planets (M < 0.08 M⊙); the total event count caps MACHOs at ≤ 8% of the halo mass. Conclusion: the dark halo is mostly a smooth distribution of nonbaryonic dark matter, not stellar/planetary-mass compact objects. [R 7 201]
Cluster lensing: for a cluster (M ~ 10¹⁴ M⊙, ~500 Mpc away) lensing a background galaxy at d ~ 1000 Mpc, $\theta_E \approx 0.5,\mathrm{arcmin}\left(\frac{M}{10^{14},M_\odot}\right)^{1/2}\left(\frac{d}{1000,\mathrm{Mpc}}\right)^{-1/2}$ (Eq. 7.48) — large enough that the arc-shaped images are resolvable. [R 7 201]
Abell 2218: z = 0.176, proper distance d = 740 Mpc; its elongated curved arcs are lensed background galaxies at z > 0.176. Cluster masses from lensing generally agree with the virial-theorem and hydrostatic-equilibrium estimates. [R 7 201]

7.5 What's the Matter?

Candidate mass range: proposals span from axions (rest energy m_ax c² ~ 10⁻⁵ eV, m_ax ~ 2×10⁻⁴¹ kg — ~50 billion axions per electron mass) to primordial black holes (masses up to m_BH ~ 10⁵ M⊙ ~ 2×10³⁵ kg — ~30 billion Earths each; "primordial" = formed very early, not by stellar collapse) — two candidates differing by 76 orders of magnitude in mass, a sign of vast ignorance. [R 7 202]
Cosmic neutrino background: the one known nonbaryonic massive particle is the neutrino; the CνB is a relic of when the universe was opaque to neutrinos (as the CMB is for photons). Each of the three flavors has number density (3/11)n_γ, so $n_\nu = 3\left(\frac{3}{11}\right)n_\gamma = 3.36\times10^{8},\mathrm{m^{-3}}$ (Eq. 7.49) — about twenty million cosmic neutrinos pass through your body at any instant. [R 7 203]
Required neutrino mass: to supply all nonbaryonic dark matter, $m_\nu c^2 = \frac{\Omega_{\mathrm{dm},0},\varepsilon_{c,0}}{n_\nu}$ (Eq. 7.50); with Ω_dm,0 ≈ 0.262 this gives m_ν c² ≈ 3.8 eV average (Eq. 7.51). [R 7 203]
Why neutrinos fail: neutrino oscillations plus large-scale-structure studies constrain the average mass to $0.019,\mathrm{eV} < m_\nu c^2 < 0.1,\mathrm{eV}$ (Eq. 7.52) [R 7 203] — implying $0.0013 < \Omega_{\nu,0} < 0.007$ (Eq. 7.53): neutrinos account for less than 3% of the dark matter. [R 7 204]
WIMPs: supersymmetric extensions of the Standard Model predict massive nonbaryonic particles (photinos, gravitinos, axinos, sneutrinos, gluinos, ...) interacting only through gravity and the weak nuclear force; weakly interacting particles much heavier than the neutrino mass limit are generically Weakly Interacting Massive Particles (WIMPs). Since WIMPs occasionally interact with atomic nuclei, direct-detection experiments search for them — so far, no convincing detections. [R 7 204]

regstuff/cosmology_notes_3-7.md

Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — Revision Notes

Contents

Chapter 3 — Newton versus Einstein

3.4 Describing Curvature

3.5 The Robertson–Walker Metric

3.6 Proper Distance

Chapter 4 — Cosmic Dynamics

Chapter introduction — geometric limits on curvature

4.1 Einstein's Field Equation

4.2 The Friedmann Equation

4.3 The Fluid and Acceleration Equations

4.4 Equations of State

4.5 Learning to Love Lambda

Chapter 5 — Model Universes

5.1 Evolution of Energy Density

5.2 Empty Universes

5.3 Single-component Universes

5.3.1 Matter only

5.3.2 Radiation only

5.3.3 Lambda only

5.4 Multiple-component Universes

5.4.1 Matter + Curvature

5.4.2 Matter + Lambda

5.4.3 Matter + Curvature + Lambda

5.4.4 Radiation + Matter

5.5 Benchmark Model

Chapter 6 — Measuring Cosmological Parameters

6.1 "A Search for Two Numbers"

6.2 Luminosity Distance

6.3 Angular-diameter Distance

6.4 Standard Candles and H₀

6.5 Standard Candles and Acceleration

Chapter 7 — Dark Matter

Introduction

7.1 Visible Matter

7.2 Dark Matter in Galaxies

7.3 Dark Matter in Clusters

7.4 Gravitational Lensing

7.5 What's the Matter?