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 3-7 Notes

Chapter 8 — The Cosmic Microwave Background
Chapter 9 — Nucleosynthesis and the Early Universe
Chapter 10 — Inflation and the Very Early Universe
Chapter 11 — Structure Formation: Gravitational Instability
Chapter 12 — Structure Formation: Baryons and Photons
Epilogue

Chapter 8 — The Cosmic Microwave Background

Chapter introduction

CMB dominance at mm wavelengths: at wavelengths of a few millimetres most light in the universe comes from the CMB, not stars — the night sky really is uniformly bright, just at $T_0 = 2.7255\ \mathrm{K}$ rather than $\sim 6000\ \mathrm{K}$ (so a mm-wave Olbers would never have posed his paradox). [R 8 207]
CMB energy density today: $\varepsilon_{\gamma,0} = \alpha T_0^4 = 0.2606\ \mathrm{MeV,m^{-3}}$ (Eq. 8.1), only one part in 19 000 of the current critical density; mean photon energy $hf_{\mathrm{mean}} = 6.34\times10^{-4}\ \mathrm{eV}$. [R 8 207]
CMB number density: because each photon carries so little energy, $n_{\gamma,0} = 4.107\times10^{8}\ \mathrm{m^{-3}}$ (Eq. 8.2) is huge; baryons today have ~900× the CMB's energy density ($\varepsilon_{\mathrm{bary},0} = \Omega_{\mathrm{bary},0}\varepsilon_{c,0} \approx 234\ \mathrm{MeV,m^{-3}}$, Eq. 8.3) but each baryon carries $E_{\mathrm{bary}} \approx 939\ \mathrm{MeV}$, over a trillion times the mean photon energy. [R 8 207]
Baryon number density: $n_{\mathrm{bary},0} = \varepsilon_{\mathrm{bary},0}/E_{\mathrm{bary}} \approx 0.25\ \mathrm{m^{-3}}$ (Eq. 8.4). [R 8 208]
Baryon-to-photon ratio: $\eta = n_{\mathrm{bary},0}/n_{\gamma,0} \approx 6.1\times10^{-10}$ (Eq. 8.5) — baryons outnumbered by photons 1.6 billion to one. [R 8 208]

8.1 Observing the CMB

Serendipitous discovery: Penzias & Wilson (Bell Labs radio astronomers) found an excess signal at $\lambda = 7.35\ \mathrm{cm}$ in a horn-reflector antenna built for satellite communications; it was isotropic and constant in time, so it could not be an isolated celestial source (famously, evicting roosting pigeons didn't remove it). [R 8 208]
Interpretation: Dicke's Princeton group had deduced that a universe starting hot and dense should now be filled with microwave radiation (and were building an antenna to detect it); Penzias & Wilson's 1965 ApJ paper reported ~3.5 K of excess zenith noise temperature at 4080 Mc/s, "isotropic, unpolarized, and free from seasonal variations," with the companion Dicke–Peebles–Roll–Wilkinson paper supplying the hot-early-universe explanation. [R 8 208]
Earlier prediction: Alpher & Herman had already predicted a cosmic background at "about 5 K" in 1948, but the prediction fell into obscurity (chapter footnote). [R 8 238]
Atmospheric obstacle: the mean CMB photon energy is comparable to rotational energies of small molecules such as H₂O, so microwaves with $\lambda \lesssim 3\ \mathrm{cm}$ are strongly absorbed by atmospheric water; measurements need balloons, the cold dry South Pole, or space. Penzias & Wilson's 7.35 cm photons ($E = 1.7\times10^{-5}\ \mathrm{eV}$, 0.027× the mean) sat far down the low-energy tail. [R 8 209]
Satellite programme: COBE (1989, 900 km orbit) measured the spectrum; WMAP (2001) and Planck (2009) mapped the anisotropy at higher angular resolution from orbits librating about the Sun–Earth L2 point, 1.5 million km away. [R 8 209]
Result 1 — blackbody spectrum: at every angular position the CMB spectrum is very close to an ideal blackbody; COBE could have detected spectral deviations of one part in $10^4$ and found none in its wavelength range. [R 8 210]
Result 2 — dipole distortion: half the sky is slightly blueshifted, half redshifted — a simple Doppler shift from the observer's net motion relative to the frame in which the CMB is isotropic. [R 8 210]
Local Group velocity: subtracting WMAP's solar orbit (~30 km s⁻¹), the Sun's galactic orbit (~235 km s⁻¹), and the Galaxy's motion within the Local Group (~80 km s⁻¹) leaves the Local Group moving toward Hydra at $v_{\mathrm{LG}} = 630 \pm 20\ \mathrm{km,s^{-1}} = 0.0021c$ — as expected from gravitational acceleration toward the Virgo cluster, itself accelerated toward the Hydra–Centaurus supercluster. [R 8 211]
Result 3 — tiny residual fluctuations: with the dipole removed, the dimensionless fluctuation $\frac{\delta T}{T}(\theta,\phi) \equiv \frac{T(\theta,\phi) - \langle T \rangle}{\langle T \rangle}$ (Eq. 8.7) about the mean $\langle T \rangle = 2.7255\ \mathrm{K}$ (Eq. 8.6) has COBE rms $\langle(\delta T/T)^2\rangle^{1/2} = 1.1\times10^{-5}$ (Eq. 8.8) — only ~30 μK variation across the sky (COBE's resolution excluded scales < 10°). [R 8 212]
Case for the Hot Big Bang: a nearly perfect blackbody + near-isotropy is natural if the universe was once hot, dense, opaque, and nearly homogeneous; any alternative explanation must be far more contrived. [R 8 213]

8.2 Recombination and Decoupling

Three distinct epochs: recombination = baryonic matter goes from ionized to neutral (defined as the instant $n_{\mathrm{ions}} = n_{\mathrm{neutrals}}$); photon decoupling = photon–electron scattering rate $\Gamma$ drops below the expansion rate $H$, making the universe transparent; last scattering = when a typical CMB photon last scatters off an electron. Last scattering is very close in time to decoupling, and every observer sits inside a spherical last scattering surface from which CMB photons stream freely, continuously redshifting. [R 8 213]
Hydrogen-only simplification: helium existed at recombination but is only a complicating factor; with hydrogen alone, charge neutrality gives $n_e = n_p$, and the fractional ionization is $X \equiv \frac{n_p}{n_p + n_{\mathrm{H}}} = \frac{n_p}{n_{\mathrm{bary}}} = \frac{n_e}{n_{\mathrm{bary}}}$ (Eq. 8.9), running from 1 (fully ionized) to 0 (fully neutral). [R 8 214]
Competing reactions: photoionization $\mathrm{H} + \gamma \rightarrow p + e^-$ (Eq. 8.10, requires $hf > Q = 13.6\ \mathrm{eV}$) versus radiative recombination $p + e^- \rightarrow \mathrm{H} + \gamma$ (Eq. 8.11); their balance sets $X$. [R 8 215]
Early universe fully ionized: at $a = 10^{-5}$ ($z = 10^5$, age ≈ 70 yr), $T \approx 3\times10^5\ \mathrm{K}$ and mean photon energy $\approx 2.7kT \approx 60\ \mathrm{eV}$ (extreme UV); with 1.6 billion photons per baryon, any H atom formed was blasted apart almost immediately, so $X \approx 1$. [R 8 215]
Thomson scattering: in the ionized plasma, photons interact mainly with electrons via $\gamma + e^- \rightarrow \gamma + e^-$ (Eq. 8.12), cross-section $\sigma_e = 6.65\times10^{-29}\ \mathrm{m^2}$; mean free path $\lambda = 1/(n_e\sigma_e)$ (Eq. 8.13) and scattering rate $\Gamma = c/\lambda = n_e\sigma_e c$ (Eq. 8.14). [R 8 215]
Scattering rate when fully ionized: with $n_e = n_{\mathrm{bary},0}/a^3$ (Eq. 8.15), $\Gamma = \frac{n_{\mathrm{bary},0}\sigma_e c}{a^3} = \frac{5.0\times10^{-21}\ \mathrm{s^{-1}}}{a^3}$ (Eq. 8.16); at $a = 10^{-5}$, $\Gamma = 5.0\times10^{-6}\ \mathrm{s^{-1}}$ — a photon scattered about three times a week. [R 8 216]
Coupling criterion: photons remain coupled while $\Gamma > H$ (equivalently mean free path $\lambda < c/H$); frequent scattering keeps photons, electrons, and (via e–p interactions) protons at one common temperature. Once $\Gamma < H$, expansion dilutes electrons faster than photons can find them: photons decouple, the universe turns transparent, and baryons are no longer forced to share the CMB temperature. [R 8 216]
Radiation-era expansion rate: for $a < a_{rm} \approx 2.9\times10^{-4}$, $H = \frac{H_0\Omega_{r,0}^{1/2}}{a^2} = \frac{2.1\times10^{-20}\ \mathrm{s^{-1}}}{a^2}$ (Eq. 8.18, from $H^2/H_0^2 = \Omega_{r,0}/a^4$, Eq. 8.17); at $a = 10^{-5}$, $H = 2.1\times10^{-10}\ \mathrm{s^{-1}} \ll \Gamma$, so photons were then very well coupled. [R 8 217]
Why decoupling is sudden: if hydrogen had stayed ionized, decoupling would not have happened until $a \approx 0.0254$ ($z \approx 38$, $T \approx 110\ \mathrm{K}$); but the CMB is far too cold at that temperature to keep hydrogen ionized. Decoupling is therefore not a gradual dilution effect but a relatively sudden event driven by the plummeting free-electron density during recombination. [R 8 217]

8.3 The Physics of Recombination

Crude estimate fails: demanding mean photon energy $2.7kT = Q$ gives $T_{\mathrm{rec}} \sim Q/(2.7k) \sim 60,000\ \mathrm{K}$ (Eq. 8.19) — much too high, because it ignores the blackbody spectrum's exponential high-energy tail. [R 8 218]
Tail-plus-swarm argument: about 1 photon in 500 has $E > 10kT$, 1 in 3 million $> 20kT$, 1 in 30 billion $> 30kT$; with $1.6\times10^9$ photons per baryon, even these rare tail photons keep hydrogen ionized far below 60 000 K — so $T_{\mathrm{rec}}$ depends on $\eta$ as well as $Q$. [R 8 218]
Equilibrium framework: near recombination, photons, electrons, protons, and H atoms are in thermal equilibrium (one common $T$), and each species is in kinetic equilibrium with momentum distribution $n_x(p),dp = g_x \frac{4\pi}{h^3}\frac{p^2,dp}{\exp([E-\mu_x]/kT) \pm 1}$ (Eq. 8.21; + Fermi–Dirac, − Bose–Einstein), where $g_x$ is the statistical weight (2 for γ, e, p, n) and $\mu_x$ the chemical potential. [R 8 219]
Photon gas: with $E = pc = hf$, $g_\gamma = 2$, $\mu_\gamma = 0$, Eq. 8.21 reduces to the blackbody formula (Eq. 8.22), integrating to $n_\gamma = 0.2436,(kT/\hbar c)^3$ (Eq. 8.23). [R 8 220]
Nonrelativistic species: for $m_xc^2 \gg kT$ the distribution becomes Maxwell–Boltzmann (Eq. 8.25; note the formulas sheet corrects a spurious $c^3$ in the printed equation), with total $n_x = g_x\left(\frac{m_x kT}{2\pi\hbar^2}\right)^{3/2}\exp\left(\frac{-m_xc^2 + \mu_x}{kT}\right)$ (Eq. 8.26), boson or fermion alike. [R 8 220]
Chemical equilibrium: with $\mathrm{H} + \gamma \rightleftharpoons p + e^-$ (Eq. 8.27) balanced (one recombination per photoionization on average), chemical potentials satisfy $\mu_{\mathrm{H}} = \mu_p + \mu_e$ (since $\mu_\gamma = 0$), giving the density relation of Eq. 8.28. [R 8 221]
Saha equation: using $m_{\mathrm{H}}/m_p \approx 1$, $Q = (m_p + m_e - m_{\mathrm{H}})c^2$, and $g_{\mathrm{H}}/(g_pg_e) = 1$ (with $g_{\mathrm{H}} = 4$, $g_p = g_e = 2$): $\frac{n_{\mathrm{H}}}{n_p n_e} = \left(\frac{m_e kT}{2\pi\hbar^2}\right)^{-3/2}\exp\left(\frac{Q}{kT}\right)$ (Eq. 8.29) — first derived by Meghnad Saha for ionization in stellar atmospheres. [R 8 222]
From Saha to X(T, η): substituting $n_{\mathrm{H}} = \frac{1-X}{X}n_p$ (Eq. 8.30), $n_e = n_p$, and $n_p = 0.2436,X\eta(kT/\hbar c)^3$ (Eq. 8.33, via $\eta = n_p/(Xn_\gamma)$, Eq. 8.32) gives the quadratic $\frac{1-X}{X^2} = S(T,\eta) = 3.84,\eta\left(\frac{kT}{m_ec^2}\right)^{3/2}\exp\left(\frac{Q}{kT}\right)$ (Eqs. 8.34, 8.36), with positive root $X = \frac{-1+\sqrt{1+4S}}{2S}$ (Eq. 8.35). [R 8 222]
Recombination temperature: defining recombination as $X = \tfrac12$ and taking $\eta = 6.1\times10^{-10}$ gives $kT_{\mathrm{rec}} = 0.324\ \mathrm{eV} = Q/42$ (Eq. 8.37), i.e. $T_{\mathrm{rec}} = 3760\ \mathrm{K}$ (just above tungsten's melting point) at $z_{\mathrm{rec}} = 1380$, $t_{\mathrm{rec}} = 250,000$ yr in the Benchmark Model; the exponential dependence of $S$ on $T$ makes $T_{\mathrm{rec}}$ insensitive to the exact value of $\eta$. [R 8 223]
Fast but not instantaneous: recombination is gradual enough that equilibrium holds at each instant, yet fast by cosmological standards — $X$ drops from 0.9 ($z = 1480$) to 0.1 ($z = 1260$) in $\Delta t \approx 70,000\ \mathrm{yr} \approx 0.28,t_{\mathrm{rec}}$. [R 8 223]
Decoupling condition: with partial ionization, $\Gamma(z) = X(z)(1+z)^3 n_{\mathrm{bary},0}\sigma_e c = 5.0\times10^{-21}\ \mathrm{s^{-1}},X(z)(1+z)^3$ (Eqs. 8.38–8.39, $\Omega_{\mathrm{bary},0} = 0.048$); the matter-era Hubble parameter is $H(z) = 1.23\times10^{-18}\ \mathrm{s^{-1}}(1+z)^{3/2}$ (Eq. 8.41, $\Omega_{m,0} = 0.31$); setting $\Gamma = H$ gives $1 + z_{\mathrm{dec}} = \frac{39.3}{X(z_{\mathrm{dec}})^{2/3}}$ (Eq. 8.42). [R 8 224]
Saha vs non-equilibrium decoupling: the Saha $X(z)$ yields $z_{\mathrm{dec}} = 1120$, but once $\Gamma$ drops below $H$ photoionization falls out of equilibrium, so below $z \sim 1200$ the true $X$ exceeds the Saha prediction and decoupling is delayed; the quoted non-equilibrium result is $z_{\mathrm{dec}} = 1090$, $T_{\mathrm{dec}} = 2970\ \mathrm{K}$, $t_{\mathrm{dec}} = 371,000$ yr. [R 8 225]
Optical depth: the expected number of scatterings for a photon detected at $t_0$ since time $t$ is $\tau(t) = \int_t^{t_0}\Gamma(t),dt$ (Eq. 8.43); the time when $\tau = 1$ defines last scattering. In redshift form, $\tau(z) = 0.0041\int_0^z X(z)(1+z)^{1/2}dz$ (Eq. 8.45). [R 8 225]
Last scattering layer: $z_{\mathrm{ls}} \approx z_{\mathrm{dec}} \approx 1090$, but photons last scatter over a range of redshifts (one at $z = 1140$, another at $z = 1040$): the "surface" is really a layer of finite thickness — we see a little way into the "electron fog," as into a fog bank. [R 8 226]
Perspective: the CMB shows the universe at only one part in 37 000 of its present age. [R 8 226]
Timeline (Table 8.1): radiation–matter equality $z = 3440$, $T = 9390\ \mathrm{K}$, $t = 0.050$ Myr; recombination $z = 1380$, 3760 K, 0.25 Myr; photon decoupling $z = 1090$, 2970 K, 0.37 Myr; last scattering $z = 1090$, 2970 K, 0.37 Myr — recombination, decoupling, and last scattering all occur in the matter-dominated era. [R 8 227]
Naming quibble: cosmologists grumble that "recombination" should really be "combination" — it was the first time electrons and ions formed stable atoms (chapter footnote). [R 8 238]

8.4 Temperature Fluctuations

Dipole vs smaller scales: the dipole reflects present-day ($z = 0$) inhomogeneity accelerating us; the smaller-angular-scale fluctuations show the universe was not perfectly homogeneous at last scattering ($z \approx 1090$). [R 8 227]
Angle-to-size conversion: a feature of angular size δθ has physical size $\ell = d_A,\delta\theta$ (Eq. 8.46) at last scattering, where $d_A \approx d_{\mathrm{hor}}(t_0)/z_{\mathrm{ls}}$ (Eq. 8.47, valid for $z \gg 1$) $\approx 14,000\ \mathrm{Mpc}/1090 \approx 12.8\ \mathrm{Mpc}$ (Eq. 8.48); numerically $\ell = 3.7\ \mathrm{kpc},(\delta\theta/1\ \mathrm{arcmin})$ (Eq. 8.49). [R 8 227]
Planck's resolution scale: the smallest fluctuations Planck resolves, $\delta\theta \approx 5$ arcmin, correspond to $\ell \approx 18\ \mathrm{kpc}$ at last scattering ≈ 20 Mpc today; a sphere of that radius contains baryon mass $M_{\mathrm{bary}} \approx 5\times10^{13}\ M_\odot$ — about a cluster of galaxies (Coma, a very rich cluster, has $\sim 2\times10^{14}\ M_\odot$). [R 8 228]
Spherical-harmonic expansion: since $\delta T/T$ lives on a sphere, expand $\frac{\delta T}{T}(\theta,\phi) = \sum_{l=0}^{\infty}\sum_{m=-l}^{l} a_{lm}Y_{lm}(\theta,\phi)$ (Eq. 8.50); what matters is not the exact hot/cold-spot pattern but its statistical properties. [R 8 228]
Correlation function: $C(\theta) = \left\langle \frac{\delta T}{T}(\hat n)\frac{\delta T}{T}(\hat n')\right\rangle_{\hat n\cdot\hat n' = \cos\theta}$ (Eq. 8.51), decomposed into multipole moments as $C(\theta) = \frac{1}{4\pi}\sum_l (2l+1),C_l,P_l(\cos\theta)$ (Eq. 8.52) with Legendre polynomials $P_l$. [R 8 228]
Multipole meaning: $l = 0$ (monopole) vanishes if $\langle T\rangle$ is defined correctly; $l = 1$ (dipole) is primarily our Doppler motion; moments $l \geq 2$ probe fluctuations at last scattering, with $l$ interchangeable with angular scale $\theta \sim 180°/l$. [R 8 229]
Plotted power spectrum: results are shown as $\Delta_T \equiv \left(\frac{l(l+1)}{2\pi}C_l\right)^{1/2}\langle T\rangle$ (Eq. 8.54) — the contribution per logarithmic interval in $l$ to the total temperature fluctuation. [R 8 229]

8.5 What Causes the Fluctuations?

Horizon at last scattering: $d_{\mathrm{hor}}(t_{\mathrm{ls}}) = a(t_{\mathrm{ls}})c\int_0^{t_{\mathrm{ls}}}dt/a$ (Eq. 8.55) $= 2.24,c,t_{\mathrm{ls}} = 0.251\ \mathrm{Mpc}$ (Eq. 8.56, using the radiation+matter scale factor since Λ is negligible then); it subtends $\theta_{\mathrm{hor}} = 0.251/12.8 \approx 0.020\ \mathrm{rad} \approx 1.1°$ (Eq. 8.57), i.e. multipole $l_{\mathrm{hor}} \approx 160$. [R 8 231]
Two regimes in the ΔT curve: on large scales $\Delta_T$ levels off at ≈ 30 μK (for $\theta > 4\theta_{\mathrm{hor}}$, $l < 40$); below $\theta_{\mathrm{hor}}$ there is a landscape of peaks and valleys — first (highest) peak at $l \approx 220$, second at $l \approx 520$, third at $l \approx 800$; the two regimes have physically different causes. [R 8 231]
Energy census at last scattering: $\varepsilon_{\mathrm{dm}} \approx 1.7\times10^{12}$ (Eq. 8.58; $\rho \sim 3\times10^{-18}\ \mathrm{kg,m^{-3}}$), $\varepsilon_\gamma \approx 3.9\times10^{11}$ (Eq. 8.60, scaling as $(1+z)^4$), $\varepsilon_{\mathrm{bary}} \approx 3.1\times10^{11}\ \mathrm{MeV,m^{-3}}$ (Eq. 8.59) — ratio dm : γ : baryons = 5.5 : 1.24 : 1, so nonbaryonic dark matter dominated the energy density and hence the gravitational potential. [R 8 232]
Potential fluctuations: writing $\varepsilon(\vec r) = \bar\varepsilon + \delta\varepsilon(\vec r)$ (Eq. 8.61), dark-matter density fluctuations source potential fluctuations via Poisson's equation $\nabla^2(\delta\Phi) = \frac{4\pi G}{c^2}\delta\varepsilon$ (Eq. 8.62). [R 8 232]
Sachs–Wolfe effect (large scales, θ > θ_hor): a photon in a potential well at last scattering loses energy climbing out (redshift → cool spot); one at a potential maximum gains energy falling down (blueshift → hot spot); the general-relativistic result (Sachs & Wolfe 1967) is $\frac{\delta T}{T} = \frac{1}{3}\frac{\delta\Phi}{c^2}$ (Eq. 8.63) — so large-scale maps of $\delta T/T$ are maps of $\delta\Phi$ at last scattering. [R 8 232]
Sachs–Wolfe plateau: $\Delta_T$ nearly constant from $l \sim 2$ to $l \sim 40$ implies the potential fluctuations $\delta\Phi$ were constant over a wide range of physical scales. [R 8 233]
Photon-baryon fluid: before decoupling, photons + electrons + protons form a single fluid with only ~40% of the dark matter's energy density, so it moves primarily under the dark matter's gravity rather than its own self-gravity. [R 8 233]
Acoustic oscillations: fluid falling into a dark-matter potential well is compressed until rising pressure drives it back out, then gravity pulls it in again — a standing sound wave in the photon-baryon fluid. [R 8 233]
Phase at decoupling sets hot/cold: fluid at maximum compression in a well when photons decouple releases hotter-than-average photons; at maximum expansion, cooler; fluid still in motion adds a Doppler shift (cooler or hotter depending on motion away from or toward us). [R 8 234]
First peak = maximum compression: the first peak ($l \approx 220$, $\theta \approx 0.8°$) marks potential wells whose fluid had just reached maximum compression at last scattering; their size is comparable to the sound horizon $d_s(t_{\mathrm{ls}}) = a(t_{\mathrm{ls}})\int_0^{t_{\mathrm{ls}}}\frac{c_s(t),dt}{a(t)}$ (Eq. 8.64); with baryons an insignificant contaminant, $c_s \approx c/\sqrt{3}$ (pure photon gas), so $d_s \approx d_{\mathrm{hor}}/\sqrt{3} \approx 0.145\ \mathrm{Mpc}$ (Eq. 8.65), subtending $\theta_s \approx 0.011\ \mathrm{rad} \approx 0.7°$ (Eq. 8.66). [R 8 234]
Peak position ⇒ flatness: the estimate $\theta_s \approx 0.7°$ assumes a flat universe; negative curvature ($\kappa = -1$) would shift the first peak to smaller angles, positive curvature ($\kappa = +1$) to larger angles. The observed position is consistent with $\kappa = 0$, i.e. $\Omega_0 = 1$. [R 8 235]
CMB + SNe complementarity: in the ($\Omega_{m,0}$, $\Omega_{\Lambda,0}$) plane the CMB best-fit ellipse is roughly perpendicular to the type Ia supernova ellipse; their small overlap selects a spatially flat universe with $\Omega_{m,0} \sim 0.3$, $\Omega_{\Lambda,0} \sim 0.7$. [R 8 235]
Peak amplitude ⇒ baryon density: the first peak's amplitude is sensitive to the sound speed $c_s$ (lower $c_s$ → higher amplitude); since $c_s$ depends on the equation-of-state parameter $w_{pb}$, which depends on the baryon-to-photon ratio, the amplitude diagnoses baryon density — detailed fits give $\eta = (6.10 \pm 0.06)\times10^{-10}$ (Eq. 8.67). [R 8 236]
Baryon budget from the CMB: $n_{\mathrm{bary},0} = \eta,n_{\gamma,0} = 0.251 \pm 0.003\ \mathrm{m^{-3}}$ (Eq. 8.68); since baryons are mostly protons, $\varepsilon_{\mathrm{bary},0} = (m_pc^2)n_{\mathrm{bary},0} = 235 \pm 3\ \mathrm{MeV,m^{-3}}$ (Eq. 8.69); hence $\Omega_{\mathrm{bary},0} = \varepsilon_{\mathrm{bary},0}/\varepsilon_{c,0} = 0.048 \pm 0.003$ (Eq. 8.70), most of the uncertainty coming from $\varepsilon_{c,0}$. [R 8 237]

Chapter 9 — Nucleosynthesis and the Early Universe

Chapter opening

Opacity barrier: the CMB shows the universe at last scattering ($t_{ls} \approx 0.37$ Myr), but the early universe's opacity hides everything at $t < t_{ls}$ — like looking at the surface of a cloud or of the Sun. [R 9 239]
Radiation-dominated simplicity: for $a \ll a_{rm} \approx 2.9\times10^{-4}$ ($t \ll t_{rm} \approx 50,000$ yr), $a(t) \propto t^{1/2}$ and blackbody $T \propto a^{-1}$, giving $T(t) \approx 10^{10},\mathrm{K},(t/1,\mathrm{s})^{-1/2}$ (Eq. 9.1), equivalently $kT(t) \approx 1,\mathrm{MeV},(t/1,\mathrm{s})^{-1/2}$ (Eq. 9.2). [R 9 239]
Mean photon energy in the early universe: $E_{\mathrm{mean}}(t) \approx 2.7,kT(t) \approx 3,\mathrm{MeV},(t/1,\mathrm{s})^{-1/2}$ (Eq. 9.3). [R 9 239]
"Poor man's particle accelerator": at $t \approx 2\times10^{-13}$ s the average particle energy equaled the LHC's beam energy of $7$ TeV ($\approx 7500\times$ the proton rest energy) — high-energy physics for free. [R 9 240]

9.1 Nuclear Physics and Cosmology

31 orders of magnitude: mean photon energy falls from $E_{\mathrm{mean}}(t_P) \sim E_P \sim 10^{28}$ eV at the Planck time to $\sim 10^{-3}$ eV today; the atomic-physics scale within this range is $\sim 10$ eV (ionization energies: H 13.6 eV, He 24.6 eV, down to ~4 eV for heavy alkalis). [R 9 240]
Nuclear vocabulary: a nucleus has $Z$ protons and $N$ neutrons (collectively nucleons); mass number $A = Z + N$; $Z$ fixes the element, isotopes differ in $N$. [R 9 240]
Binding energy $B$: energy needed to pull a nucleus apart into its nucleons, equivalently energy released on fusing it; for deuterium (²H = D, the deuteron) $p + n \rightleftharpoons \mathrm{D} + 2.22,\mathrm{MeV}$ (Eq. 9.4) — the deuteron is weakly bound; ⁴He has $B = 28.30$ MeV, $B/A = 7.07$ MeV, tightly bound for a light nucleus; the peak is $B/A \approx 8.79$ MeV at ⁵⁶Fe, ⁵⁸Fe, ⁶²Ni, so heavier nuclei release energy by fission, lighter ones by fusion. [R 9 241]
Why BBN is expected: nuclear physics has energy scale ~8 MeV (typical $B/A$) vs the atomic ~10 eV; just as recombination occurred when $T$ fell well below $Q = 13.6$ eV, protons and neutrons must fuse when $T$ falls well below $B_D = 2.22$ MeV; since $B_D/Q = 1.6\times10^5$, a first estimate scales the recombination temperature: $T_{nuc} \approx 1.6\times10^5 \times 3760,\mathrm{K}$. [R 9 242]
First estimate of the BBN epoch: $T_{nuc} \approx 6\times10^8$ K, $t_{nuc} \approx 300$ s — slightly too low in $T$ but the right order of magnitude; hence Weinberg's title The First Three Minutes. [R 9 243]
BBN was shockingly inefficient: the energetically preferred universe would be an iron–nickel alloy, yet ~3/4 of baryonic mass is still unbound ¹H, and Fe+Ni make up only 0.15% of the Galaxy's baryonic mass. [R 9 243]
Primordial helium fraction: $Y_p \equiv \frac{\rho(^4\mathrm{He})}{\rho_{\mathrm{bary}}}$ (Eq. 9.5); the Sun's outer layers have $Y = 0.27$ (contaminated recycled gas), but the minimum observed across baryonic objects is $Y = 0.24$ — everything is at least 24% helium by mass. [R 9 243]

9.2 Neutrons and Protons

Neutron–proton rest-energy difference: $Q_n = m_n c^2 - m_p c^2 = 1.29,\mathrm{MeV}$ (Eq. 9.6). [R 9 244]
Free-neutron decay: $n \rightarrow p + e^- + \bar{\nu}_e$ (Eq. 9.7) with decay time $\tau_n = 880$ s (fraction surviving after time $t$: $f = e^{-t/\tau_n}$); since $Q_n > m_ec^2 = 0.51$ MeV, the excess goes to the electron's kinetic energy and the antineutrino; a neutron bound in a stable nucleus is preserved — that is why neutrons still exist today. [R 9 244]
Universe at t = 0.1 s: $T \approx 3\times10^{10}$ K, $E_{\mathrm{mean}} \approx 10$ MeV $\gg m_ec^2$, so positrons were abundant via pair production $\gamma + \gamma \rightleftharpoons e^- + e^+$ (Eq. 9.8), and neutrinos were still coupled to baryonic matter. [R 9 244]
n↔p interconversion: maintained by the weak reactions $n + \nu_e \rightleftharpoons p + e^-$ (Eq. 9.9) and $n + e^+ \rightleftharpoons p + \bar{\nu}_e$ (Eq. 9.10). [R 9 244]
Equilibrium n/p ratio: with nucleons nonrelativistic ($kT \approx 3$ MeV $\ll m_pc^2$), Maxwell–Boltzmann number densities (Eqs. 9.11–9.12, from Eq. 8.26) with $g_p = g_n = 2$ and $(m_n/m_p)^{3/2} = 1.002 \approx 1$ give $\frac{n_n}{n_p} = \exp\left(-\frac{Q_n}{kT}\right)$ (Eq. 9.14). [R 9 245]
Ratio behavior: $n_n \approx n_p$ while $kT \gg Q_n$ ($T \gg 1.5\times10^{10}$ K, $t \ll 1$ s); below that the ratio plummets exponentially — equilibrium maintained to $t = 6$ min would leave only one neutron per million protons. [R 9 246]
Weak cross-section: neutrino–nucleon interactions have $\sigma_w \sim 10^{-47},\mathrm{m}^2,(kT/1,\mathrm{MeV})^2$ (Eq. 9.15) — compare the Thomson cross-section $\sigma_e = 6.65\times10^{-29},\mathrm{m}^2$. [R 9 247]
Freeze-out mechanism: since $\sigma_w \propto t^{-1}$ and $n_\nu \propto t^{-3/2}$, the weak interaction rate falls steeply, $\Gamma = n_\nu c \sigma_w \propto t^{-5/2}$ (Eq. 9.16), while $H \propto t^{-1}$; when $\Gamma \approx H$ the neutrinos decouple from neutrons and protons and the n/p ratio freezes (until neutron decay at $t \sim \tau_n$). [R 9 247]
Freeze-out numbers: laboratory cross-sections give $kT_{\mathrm{freeze}} = 0.8$ MeV ($T_{\mathrm{freeze}} = 9\times10^9$ K, $t_{\mathrm{freeze}} \sim 1$ s); the frozen ratio is $\frac{n_n}{n_p} = \exp\left(-\frac{1.29,\mathrm{MeV}}{0.8,\mathrm{MeV}}\right) \approx 0.2$ (Eq. 9.17) — one neutron per five protons for $t_{\mathrm{freeze}} < t \ll \tau_n$; this neutron scarcity is why BBN left 3/4 of baryons as unfused protons. [R 9 247]
n–p fusion is easy, p–p fusion nearly impossible: $p + n \rightleftharpoons \mathrm{D} + \gamma$ (Eq. 9.18) has no Coulomb barrier and is strong-force mediated (large cross-section, fast); p–p fusion must first form a diproton (²He, Eq. 9.19), which splits back with $\tau_{\mathrm{split}} \sim 10^{-23}$ s, while its weak decay ${}^2\mathrm{He} \rightarrow \mathrm{D} + e^+ + \nu_e$ (Eq. 9.20) has $\tau_{\mathrm{weak}} > 10^{-2}$ s — success probability $P \approx \tau_{\mathrm{split}}/\tau_{\mathrm{weak}} < 10^{-21}$. [R 9 248]
Sun comparison: p–p fusion does happen in the Sun, but a given core proton has only ~1 chance in $10^{10}$ per year of fusing, despite forming diprotons millions of times per second; the Sun, in hydrostatic equilibrium, can wait. [R 9 248]
BBN is a race against time: in the expanding universe $T \propto t^{-1/2}$ and $n_{\mathrm{bary}} \propto t^{-3/2}$; after less than an hour, temperature and density are too low for fusion — so only fast neutron-capture channels matter, and to lowest order BBN runs until every free neutron is bound, leftover protons remaining solitary. [R 9 249]
Maximum helium fraction: if all neutrons end in ⁴He with $n_n/n_p = 1/5$, a representative 2 n + 10 p gives one ⁴He plus 8 free protons: $Y_{\max} = \frac{4}{12} = \frac{1}{3}$ (Eq. 9.21); generally $Y_{\max} = \frac{2f}{1+f}$ with $f \equiv n_n/n_p \le 1$ (Eq. 9.48, stated inline here). [R 9 249]
Why observed $Y_p = 0.24 < Y_{\max}$ is unworrying: (i) neutron decay during any delay before nucleosynthesis, (ii) neutrons escaping fusion or ending in nuclei lighter than ⁴He (D, ³He), (iii) any synthesis beyond ⁴He — all reduce $Y_p$; only $Y_p > Y_{\max}$ would signal a misunderstanding of freeze-out. [R 9 249]
Dineutron footnote: neutron–neutron fusion is also negligible — the dineutron, like the diproton, overwhelmingly decays back to two free neutrons rather than weakly to a deuteron. [R 9 263]

9.3 Deuterium Synthesis

Setting at t ≈ 2 s: $n_n/n_p = 0.2$; neutrinos (which ceased interacting with electrons at about the same time as with nucleons) are decoupled; photons remain strongly coupled to nucleons; BBN proceeds through a series of two-body reactions building nuclei step by step. [R 9 250]
Essential first step: $p + n \rightleftharpoons \mathrm{D} + \gamma$ (Eq. 9.22), releasing the deuteron binding energy $B_{\mathrm{D}} = (m_n + m_p - m_{\mathrm{D}})c^2 = 2.22$ MeV (Eq. 9.23) as a gamma ray; conversely any photon with $E \ge B_D$ photodissociates a deuteron. [R 9 250]
Recombination analogy: Eq. 9.22 has the same structural form as $p + e^- \rightleftharpoons \mathrm{H} + \gamma$ (Eq. 9.24) — two particles bind, a photon carries off the excess — but with the strong force instead of electromagnetism and gamma-ray instead of ultraviolet energy scales. [R 9 251]
Nucleosynthetic Saha equation: chemical equilibrium between fusion and photodissociation gives $\frac{n_{\mathrm{D}}}{n_p n_n} = \frac{g_{\mathrm{D}}}{g_p g_n} \left(\frac{m_{\mathrm{D}}}{m_p m_n}\right)^{3/2} \left(\frac{kT}{2\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{[m_p + m_n - m_{\mathrm{D}}]c^2}{kT}\right)$ (Eq. 9.26), the direct analog of the recombination Saha equation (Eqs. 9.25/8.28). [R 9 251]
Simplified form: with $g_D = 3$, $g_p = g_n = 2$, $m_p = m_n = m_D/2$, and $B_D$: $\frac{n_{\mathrm{D}}}{n_p n_n} = 6 \left(\frac{m_n kT}{\pi\hbar^2}\right)^{-3/2} \exp\left(\frac{B_{\mathrm{D}}}{kT}\right)$ (Eq. 9.27) — deuterium favored as $kT \to 0$, free nucleons favored as $kT \to \infty$. [R 9 252]
Definition of $T_{nuc}$: the temperature at which $n_D/n_n = 1$, i.e. half the free neutrons have fused into deuterons. [R 9 252]
Enter η: before deuterium synthesis ~5/6 (~83%) of baryons are unbound protons, so $n_p \approx 0.8,n_{\mathrm{bary}} = 0.8,\eta,n_\gamma = 0.8,\eta\left[0.2436\left(\frac{kT}{\hbar c}\right)^3\right]$ (Eq. 9.29, using Eq. 8.23 for $n_\gamma$). [R 9 252]
Deuteron-to-neutron ratio vs T: $\frac{n_{\mathrm{D}}}{n_n} \approx 6.5,\eta \left(\frac{kT}{m_n c^2}\right)^{3/2} \exp\left(\frac{B_{\mathrm{D}}}{kT}\right)$ (Eq. 9.30), plotted for $\eta = 6.1\times10^{-10}$. [R 9 253]
The deuterium delay — why BBN waits despite $B_D \gg kT$: solving $1 \approx 6.5,\eta,(kT_{nuc}/m_nc^2)^{3/2}\exp(B_D/kT_{nuc})$ (Eq. 9.31) with $m_nc^2 = 939.6$ MeV and $\eta = 6.1\times10^{-10}$ gives $kT_{nuc} \approx 0.066$ MeV $\approx B_D/34$ ($T_{nuc} \approx 7.6\times10^8$ K): the tiny $\eta$ makes the prefactor minuscule, so the exponential must be enormous before deuterium wins over photodissociation. [R 9 253]
Neutron decay during the delay: $T_{nuc}$ is reached only at $t_{nuc} \approx 200$ s, not negligible against $\tau_n = 880$ s; decay lowers the ratio from 1/5 to $\frac{n_n}{n_p} \approx \frac{\exp(-200/880)}{5 + [1 - \exp(-200/880)]} \approx \frac{0.80}{5.20} \approx 0.15$ (Eq. 9.32), reducing $Y_{\max}$ from ≈0.33 to ≈0.27. [R 9 254]

9.4 Beyond Deuterium

Network opens after D: deuterons fuse onward via $\mathrm{D} + p \rightleftharpoons {}^3\mathrm{He} + \gamma$ (Eq. 9.33) and $\mathrm{D} + n \rightleftharpoons {}^3\mathrm{H} + \gamma$ (Eq. 9.34); tritium β-decays to ³He with a ~18-year decay time, so it is effectively stable during BBN. [R 9 254]
D–D channels: direct $\mathrm{D} + \mathrm{D} \rightleftharpoons {}^4\mathrm{He} + \gamma$ (Eq. 9.35) is less likely than $\mathrm{D} + \mathrm{D} \rightleftharpoons {}^3\mathrm{H} + p$ (Eq. 9.36) or $\mathrm{D} + \mathrm{D} \rightleftharpoons {}^3\mathrm{He} + n$ (Eq. 9.37). [R 9 255]
Fast funneling to ⁴He: ³H and ³He never accumulate — they are converted by ${}^3\mathrm{H} + p \rightleftharpoons {}^4\mathrm{He} + \gamma$, ${}^3\mathrm{He} + n \rightleftharpoons {}^4\mathrm{He} + \gamma$, ${}^3\mathrm{H} + \mathrm{D} \rightleftharpoons {}^4\mathrm{He} + n$, ${}^3\mathrm{He} + \mathrm{D} \rightleftharpoons {}^4\mathrm{He} + p$ (Eq. 9.38); all post-deuterium reactions are strong-force, neutrino-free, with large cross-sections and fast rates. [R 9 255]
Mass-5 roadblock: ⁴He is exceptionally tightly bound for a light nucleus and there are no stable $A = 5$ nuclei (⁵He, ⁵Li unstable), so ⁴He resists fusion with protons and neutrons — the orderly march to heavier nuclei stalls. [R 9 255]
Trace elements and the A = 8 gap: small amounts of ⁶Li and ⁷Li form via ${}^4\mathrm{He} + \mathrm{D} \rightleftharpoons {}^6\mathrm{Li} + \gamma$ (Eq. 9.39) and ${}^4\mathrm{He} + {}^3\mathrm{H} \rightleftharpoons {}^7\mathrm{Li} + \gamma$ (Eq. 9.40), and ⁷Be via ${}^4\mathrm{He} + {}^3\mathrm{He} \rightleftharpoons {}^7\mathrm{Be} + \gamma$ (Eq. 9.41); synthesis beyond $A = 7$ is blocked because ⁸Be (Eq. 9.42) falls back into two ⁴He with $\tau = 10^{-16}$ s. [R 9 256]
BBN codes: precise yields require computer codes tracking many reactions with temperature-dependent cross-sections; typical run (Fig. 9.4): at $t < 10$ s ($T > 3\times10^9$ K) nearly all baryons are free p and n; by $t \sim 1000$ s ($T \sim 3\times10^8$ K) BBN is essentially over, with baryons as free protons or ⁴He. [R 9 256]
Leftovers: residual free neutrons decay to protons; small D, ³H, ³He survive as relics of BBN's incompleteness (³H later decays to ³He); very small ⁶Li, ⁷Li, ⁷Be (⁷Be later converted by electron capture, ${}^7\mathrm{Be} + e^- \rightarrow {}^7\mathrm{Li} + \nu_e$). [R 9 257]
η controls the yields: a higher baryon-to-photon ratio raises $T_{nuc}$, giving BBN an earlier start in its race against falling density and temperature — more efficient ⁴He production and less leftover D and ³He. [R 9 257]
⁷Li's non-monotonic dependence: direct ⁷Li production (⁴He + ³H) decreases with η while indirect production via ⁷Be electron capture increases with η, giving a minimum in predicted ⁷Li at $\eta \approx 3\times10^{-10}$. [R 9 258]
Deuterium as the baryometer: D/H depends strongly on η in the range of interest, so a modest-accuracy D measurement determines η well; $Y_p$ depends only weakly on η, so pinning η that way would require measuring $Y_p$ with "fanatic precision." [R 9 258]
Measuring primordial D/H: local interstellar D/H ≈ 1.6×10⁻⁵ is only a lower bound (D is easily destroyed in stars); the best method uses quasars as flashlights shining through low-metallicity intergalactic clouds, where the isotopic shift of Lyman-α ($\lambda_D = 121.534$ nm vs $\lambda_H = 121.567$ nm) produces a weak D absorption line beside the strong H line at $(1+z_{cl})\lambda$. [R 9 259]
Result: quasar absorption studies give $\mathrm{D/H} = (2.53 \pm 0.04)\times10^{-5}$, translating via BBN calculations to $\eta = (6.0 \pm 0.1)\times10^{-10}$ — consistent with the value from CMB temperature fluctuations (Eq. 8.67). [R 9 259]

9.5 Baryon–Antibaryon Asymmetry

Conditions at BBN were unexotic: radiation energy density $\varepsilon_{nuc} \approx \alpha T_{nuc}^4 \approx 1.6\times10^{33},\mathrm{MeV,m^{-3}}$ (Eq. 9.43) — equivalent mass density ~2800 kg m⁻³ (≈3× water), almost all in radiation; baryon mass density $\rho_{\mathrm{bary}}(t_{nuc}) = \Omega_{\mathrm{bary},0},\rho_{c,0}(T_{nuc}/T_0)^3 \approx 0.009,\mathrm{kg,m^{-3}}$ (Eq. 9.44), about the density of Earth's stratosphere; mean photon energy $2.7kT_{nuc} \approx 0.18$ MeV (~3× a dental X-ray photon) — the physics of BBN is well understood. [R 9 260]
Two puzzling initial conditions: η ≈ 6×10⁻¹⁰ is remarkably small (strong preference for photons over baryons), and the universe strongly prefers baryons over antibaryons — at BBN, $n_{\mathrm{antibary}} \ll n_{\mathrm{bary}} \ll n_\gamma$, an imbalance rooted in the very early universe. [R 9 260]
Quark soup: for $kT \gtrsim 150$ MeV (the first few microseconds) quarks were unconfined; pairs were continually created and annihilated, $\gamma + \gamma \rightleftharpoons q + \bar{q}$ (Eq. 9.45), with numbers of up, anti-up, down, anti-down quarks and photons all nearly equal. [R 9 261]
Tiny quark asymmetry: suppose $\delta_q \equiv \frac{n_q - n_{\bar{q}}}{n_q + n_{\bar{q}}} \ll 1$ (Eq. 9.46); once cooling ended pair production, antiquarks annihilated with quarks, leaving a residue $\frac{n_q}{n_\gamma} \sim \delta_q$ (Eq. 9.47). [R 9 261]
Worked example: 800 000 003 quarks per 800 000 000 antiquarks leaves 3 surviving quarks surrounded by ~1.6 billion annihilation photons; bound into a baryon at $kT \approx 150$ MeV, this yields $\eta \sim 6\times10^{-10}$. [R 9 261]
Baryogenesis is unsolved: today's baryon–antibaryon asymmetry and the huge photon-per-baryon count are both products of the few-parts-per-billion quark excess; Sakharov (1967) outlined the necessary physical conditions, but the precise mechanism remains unknown. [R 9 262]

Chapter 10 — Inflation and the Very Early Universe

Overview

Three problems of the Hot Big Bang: despite successes reaching back to neutron–proton freezeout ($t \approx 1$ s), the standard radiation-dominated Big Bang has three flaws recognized after the CMB's discovery: the flatness, horizon, and monopole problems. [R 10 264]
One-line versions: flatness = "the universe is nearly flat today, and was even flatter in the past"; horizon = "the universe is nearly isotropic and homogeneous today, and was even more so in the past"; monopole = "the universe is apparently free of magnetic monopoles." [R 10 264]

10.1 The Flatness Problem

Curvature–density link: the Friedmann equation (form of Eq 4.34) ties $\Omega$ to spatial curvature, $1 - \Omega(t) = -\kappa \left( \frac{c/H(t)}{a(t)R_0} \right)^{2}$ (Eq. 10.1); today $1 - \Omega_0 = -\kappa \left( \frac{c/H_0}{R_0} \right)^{2}$ (Eq. 10.2). [R 10 265]
Present flatness constraint: CMB + type Ia supernovae + other observations give $|1 - \Omega_0| \leq 0.005$ (Eq. 10.3), implying $R_0 \geq 14,c/H_0$; a priori $\Omega_0$ could have been $10^{-6}$ or $10^{6}$ without violating physics. [R 10 265]
Backward extrapolation: combining 10.1 and 10.2, $1 - \Omega(t) = \frac{H_0^{2}(1 - \Omega_0)}{H(t)^{2}a(t)^{2}}$ (Eq. 10.4). [R 10 265]
Deviation grows with time: in the radiation+matter era ($t \ll t_{m\Lambda} \approx 10$ Gyr, using Eq 10.5 for $H$), $1 - \Omega(t) = \frac{(1 - \Omega_0)a^{2}}{\Omega_{r,0} + a\Omega_{m,0}}$ (Eq. 10.6), so $|1-\Omega|_r \propto a^{2} \propto t$ (Eq. 10.7) when radiation-dominated and $|1-\Omega|_m \propto a \propto t^{2/3}$ (Eq. 10.8) when matter-dominated — flatness is unstable. [R 10 266]
At radiation–matter equality ($a_{rm} \approx 2.9\times10^{-4}$, Benchmark values $\Omega_{r,0} \approx 9.0\times10^{-5}$, $\Omega_{m,0} \approx 0.31$, $\Omega_{\Lambda,0} \approx 0.69$): $|1 - \Omega_{rm}| \leq 2 \times 10^{-6}$ (Eq. 10.9). [R 10 266]
At Big Bang nucleosynthesis ($a_{\mathrm{nuc}} \approx 3.6\times10^{-9}$): $|1 - \Omega_{\mathrm{nuc}}| \leq 7 \times 10^{-16}$ (Eq. 10.10) — flat to better than a part in a quadrillion when deuterium formed. [R 10 266]
At the Planck time ($t_P \sim 5\times10^{-44}$ s): $|1 - \Omega_P| \leq 2 \times 10^{-62}$ (Eq. 10.11); analogy — changing the Sun's mass by 2 parts in $10^{62}$ means adding or subtracting a twentieth of an electron. [R 10 267]
Why it matters: had $|1-\Omega|$ at nucleosynthesis been a part per million instead, the universe would have hit a Big Crunch or low-density Big Chill within a few decades — no galaxies, stars, or cosmologists; a physical flattening mechanism is preferable to fanatically fine-tuned initial conditions. [R 10 267]

10.2 The Horizon Problem

The problem: large-scale homogeneity and isotropy — normally a computational blessing (Robertson–Walker metric, Friedmann equation) — is unexplained in the standard Hot Big Bang, since widely separated regions were never in causal contact. [R 10 267]
Antipodal points are causally disconnected: the current proper distance to the last scattering surface is $d_p(t_0) = c \int_{t_{\mathrm{ls}}}^{t_0} \frac{dt}{a(t)}$ (Eq. 10.12) $= 0.98,d_{\mathrm{hor}}(t_0)$ for the Benchmark Model, so antipodal points on that surface are separated by $1.96,d_{\mathrm{hor}}(t_0)$ — they have never exchanged signals or thermalized with each other. [R 10 268]
Yet antipodal CMB patches have the same temperature to one part in $10^5$ (after subtracting the dipole distortion) — properties identical without any chance to swap information. [R 10 269]
Horizon scale at last scattering: in the standard scenario $d_{\mathrm{hor}}(t_{\mathrm{ls}}) = 0.251$ Mpc (Eq 8.56); with angular-diameter distance $d_A \approx 12.8$ Mpc, a horizon-sized patch subtends $\theta_{\mathrm{hor}} = \frac{d_{\mathrm{hor}}(t_{\mathrm{ls}})}{d_A} \approx \frac{0.251,\mathrm{Mpc}}{12.8,\mathrm{Mpc}} \approx 0.020,\mathrm{rad} \approx 1.1^{\circ}$ (Eq. 10.13); yet patches with $\theta > 1.1^\circ$ ($l < 160$) agree in temperature to within 30 μK, one part in $10^5$ of $T_0 = 2.7255$ K. [R 10 269]
Scale of the coincidence: the last scattering surface divides into ~40 000 causally disconnected patches $1.1^\circ$ across; Ryden's potluck analogy — 40 000 guests all bringing potato salad implies they communicated. [R 10 269]

10.3 The Monopole Problem

Origin: not purely cosmological — it arises from combining the Hot Big Bang with Grand Unified Theories (GUTs), field theories unifying the electromagnetic, weak, and strong forces. [R 10 270]
Electroweak unification (Weinberg, Salam, Glashow, experimentally confirmed): above $E_{ew} \sim 1$ TeV ($T_{ew} \sim 10^{16}$ K, $t_{ew} \sim 10^{-12}$ s) the electromagnetic and weak forces merge, leaving three fundamental forces before a picosecond. [R 10 270]
GUT scale: extrapolation suggests strong–electroweak unification at $E_{\mathrm{GUT}} \sim 10^{12}$ TeV, i.e. $T_{\mathrm{GUT}} \sim 10^{28}$ K and $t_{\mathrm{GUT}} \sim 10^{-36}$ s — about four orders of magnitude below the Planck energy $E_P \sim 10^{16}$ TeV, where a Theory of Everything would add gravity. [R 10 271]
GUT phase transition: as $T$ drops below $T_{\mathrm{GUT}}$ the strong–electroweak symmetry is spontaneously lost (analogy: liquid water's isotropy lost when it freezes into an anisotropic crystal). [R 10 271]
Topological defects form where causally disconnected domains' fields mismatch: domain walls (2D), cosmic strings (1D), and point-like (0D) defects; GUTs predict the point-like defects act as magnetic monopoles — isolated north or south poles — with rest energy $m_M c^2 \sim E_{\mathrm{GUT}} \sim 10^{12}$ TeV, over a nanogram (a bacterium's mass) per particle. [R 10 272]
Expected abundance: roughly one defect per horizon volume at formation, $n_{\mathrm{M}}(t_{\mathrm{GUT}}) \sim \frac{1}{(2ct_{\mathrm{GUT}})^{3}} \sim 10^{82},\mathrm{m^{-3}}$ (Eq. 10.14). [R 10 272]
Monopole energy density at creation: $\varepsilon_{\mathrm{M}}(t_{\mathrm{GUT}}) \sim (m_{\mathrm{M}}c^{2}),n_{\mathrm{M}} \sim 10^{94},\mathrm{TeV,m^{-3}}$ (Eq. 10.15) — ten orders of magnitude below the radiation density $\varepsilon_{\gamma}(t_{\mathrm{GUT}}) \approx \alpha T_{\mathrm{GUT}}^{4} \sim 10^{104},\mathrm{TeV,m^{-3}}$ (Eq. 10.16), so radiation still dominated then. [R 10 273]
But monopoles win eventually: massive monopoles go non-relativistic, $\varepsilon_M \propto a^{-3}$ vs. $\varepsilon_\gamma \propto a^{-4}$, so they would dominate once $a$ grew by $\sim 10^{10}$ — at $T \sim 10^{18}$ K, age $t \sim 10^{-16}$ s. [R 10 273]
Observation: monopole searches give $n_{M,0} < 10^{-29} n_{\mathrm{bary},0}$, hence $\Omega_{M,0} < 5\times10^{-16}$ even at $m_M c^2 \sim 10^{15} m_p c^2$ — no monopole domination, indeed no evidence they exist. Some GUT variants make no monopoles, but the flatness and horizon problems aren't so easily dismissed; Alan Guth (1981) proposed inflation to solve all three at once. [R 10 273]

10.4 The Inflation Solution

Definition of inflation: an epoch in the early universe when expansion accelerated, $\ddot a > 0$; the acceleration equation $\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^{2}}(\varepsilon + 3P)$ (Eq. 10.17) requires $P < -\varepsilon/3$, i.e. temporary domination by a component with $w < -1/3$. [R 10 274]
Simplest implementation: temporary domination by a positive cosmological constant $\Lambda_i$ ($w = -1$): $\frac{\ddot{a}}{a} = \frac{\Lambda_i}{3} > 0$ (Eq. 10.18) and $\left(\frac{\dot{a}}{a}\right)^{2} = \frac{\Lambda_i}{3}$ (Eq. 10.19) give constant $H_i = (\Lambda_i/3)^{1/2}$ and exponential growth $a(t) \propto e^{H_i t}$ (Eq. 10.20). [R 10 274]
Toy model: exponential expansion switched on at $t_i$ and off at $t_f$ inside the radiation era, $a(t) = \begin{cases} a_i (t/t_i)^{1/2} & t < t_i \ a_i, e^{H_i(t - t_i)} & t_i < t < t_f \ a_i, e^{H_i(t_f - t_i)}(t/t_f)^{1/2} & t > t_f \end{cases}$ (Eq. 10.21); growth factor $\frac{a(t_f)}{a(t_i)} = e^{N}$ (Eq. 10.22) with the number of e-foldings $N \equiv H_i(t_f - t_i)$ (Eq. 10.23). [R 10 275]
Fiducial numbers: inflation starting at $t_i \approx t_{\mathrm{GUT}} \approx 10^{-36}$ s with $H_i \approx 10^{36},\mathrm{s^{-1}}$, ending at $t_f \approx (N+1)t_{\mathrm{GUT}}$; this needs $\varepsilon_{\Lambda_i} = \frac{c^{2}}{8\pi G}\Lambda_i = \frac{3c^{2}}{8\pi G}H_i^{2} \sim 10^{105},\mathrm{TeV,m^{-3}}$ (Eq. 10.24), over 107 orders of magnitude larger than today's $\varepsilon_{\Lambda,0} \approx 0.0034,\mathrm{TeV,m^{-3}}$. [R 10 275]
Flatness fix: writing $|1 - \Omega(t)| = \frac{c^{2}}{R_0^{2},a(t)^{2},H(t)^{2}}$ (Eq. 10.25), exponential expansion drives $|1 - \Omega(t)| \propto e^{-2H_i t}$ (Eq. 10.26), so $|1 - \Omega(t_f)| = e^{-2N}|1 - \Omega(t_i)|$ (Eq. 10.27); even a strongly curved pre-inflation universe with $|1 - \Omega(t_i)| \sim 1$ (Eq. 10.28) ends at $|1 - \Omega(t_f)| \sim e^{-2N}$ (Eq. 10.29). [R 10 276]
E-foldings required: extrapolating $|1-\Omega_0| \leq 0.005$ back to $t_f$, where $a(t_f) \approx 2 \times 10^{-28}\sqrt{N + 1}$ (Eq. 10.30), demands $|1 - \Omega(t_f)| \leq 2 \times 10^{-54}(N + 1)$ (Eq. 10.31); comparing with $e^{-2N}$ gives a minimum $N \approx 60$, i.e. growth $a(t_f)/a(t_i) \sim e^{60} \sim 10^{26}$ ($N$ may be far larger since data allow $|1-\Omega_0| \ll 1$). [R 10 277]
Horizon fix: with $d_{\mathrm{hor}}(t) = a(t),c \int_0^{t} \frac{dt}{a(t)}$ (Eq. 10.32), the pre-inflation (radiation-dominated) horizon is $d_{\mathrm{hor}}(t_i) = 2ct_i$ (Eq. 10.33) $\approx 6\times10^{-28}$ m (Eq. 10.36); for large $N$ the post-inflation horizon is $d_{\mathrm{hor}}(t_f) = e^{N} c,(2t_i + H_i^{-1})$ (Eq. 10.35); taking $N = 65$ and $H_i \approx t_i^{-1}$, $d_{\mathrm{hor}}(t_f) \approx e^{N},3ct_i \sim 15$ m (Eq. 10.37) — in $\sim 10^{-34}$ s the horizon jumps from submicroscopic to whale-sized. [R 10 277]
Consequence at last scattering: the horizon then grows as $d_{\mathrm{hor}} \propto a \propto t^{1/2}$ (with $a(t_f) \sim 2\times10^{-27}$, $a(t_{\mathrm{ls}}) \approx 9.1\times10^{-4}$), reaching $d_{\mathrm{hor}}(t_{\mathrm{ls}}) \sim 200$ Mpc — 800 times the no-inflation value of 0.25 Mpc, large enough that antipodal points on the last scattering surface are causally connected. [R 10 278]
Visible universe repackaged: the currently visible universe ($d_p(t_0) \approx 14,000$ Mpc) had, immediately after inflation, proper radius $d_p(t_f) = a_f,d_p(t_0) \sim 0.9,\mathrm{m}$ (Eq. 10.38) — all future galaxies inside a sphere six feet across. [R 10 279]
Pre-inflation causal contact: just before inflation the currently visible universe had radius $d_p(t_i) = e^{-N}d_p(t_f) \sim 4 \times 10^{-29},\mathrm{m}$ (Eq. 10.39), smaller than the pre-inflation horizon $\sim 6\times10^{-28}$ m — so it had time to exchange information and become uniform before inflation began. [R 10 280]
Monopole fix: during inflation $n_M \propto e^{-3H_i t}$; after 65 e-foldings $n_M(t_f) = e^{-195} n_M(t_{\mathrm{GUT}}) \approx 0.002,\mathrm{m^{-3}}$, and subsequent expansion to $a_0 = 1$ leaves $n_M(t_0) \approx 2\times10^{-83},\mathrm{m^{-3}} \approx 5\times10^{-16},\mathrm{Mpc^{-3}}$ — probably not a single monopole within the last scattering surface. [R 10 280]
Redefined horizons (footnote): after inflation, the "horizon distances" of earlier chapters must be read as post-inflationary horizon distances — light travel distance since the end of inflation at $t_f \sim 10^{-34}$ s, not since $t = 0$. [R 10 289]

10.5 The Physics of Inflation

Open questions motivating the mechanism: what turns inflation on at $t_i$ and off at $t_f$; why photons aren't diluted to undetectability like monopoles (answer: reheating); why local density fluctuations aren't flattened away entirely (the CMB is not perfectly isotropic); no consensus exists on the exact mechanism — the chapter treats one plausible one. [R 10 281]
Inflaton field: a scalar field $\phi(\vec r, t)$ with potential energy density $V(\phi)$; early implementations identified it with the Higgs field, but generically it is just called the inflaton. [R 10 281]
Scalar-field thermodynamics: for homogeneous $\phi$ (units of energy, $V$ of energy density), $\varepsilon_{\phi} = \frac{1}{2}\frac{1}{\hbar c^{3}}\dot{\phi}^{2} + V(\phi)$ (Eq. 10.40) and $P_{\phi} = \frac{1}{2}\frac{1}{\hbar c^{3}}\dot{\phi}^{2} - V(\phi)$ (Eq. 10.41). [R 10 281]
Slow-roll makes a cosmological constant: if $\dot{\phi}^{2} \ll \hbar c^{3} V(\phi)$ (Eq. 10.42) then $\varepsilon_{\phi} \approx -P_{\phi} \approx V(\phi)$ (Eq. 10.43), i.e. $w \approx -1$; the inflaton drives exponential inflation when it also dominates the universe's energy density. [R 10 282]
Equation of motion: from the fluid equation $\dot{\varepsilon}{\phi} + 3H(\varepsilon{\phi} + P_{\phi}) = 0$ (Eq. 10.44) one gets $\ddot{\phi} + 3H(t)\dot{\phi} = -\hbar c^{3}\frac{dV}{d\phi}$ (Eq. 10.45) — a particle pushed by $-dV/d\phi$ and impeded by "Hubble friction" $3H\dot\phi$, which slows the field's descent toward the potential minimum. [R 10 282]
Terminal velocity: like a skydiver, the field can reach steady speed ($\ddot\phi = 0$) when $3H\dot{\phi} = -\hbar c^{3}\frac{dV}{d\phi}$ (Eq. 10.46), i.e. $\dot{\phi} = -\frac{\hbar c^{3}}{3H}\frac{dV}{d\phi}$ (Eq. 10.47). [R 10 282]
Slow-roll condition on the potential: at terminal velocity, $\varepsilon_\phi \approx -P_\phi$ requires $\left(\frac{dV}{d\phi}\right)^{2} \ll \frac{9H^{2}V}{\hbar c^{3}}$ (Eq. 10.48); substituting $H = \left(\frac{8\pi G V}{3c^{2}}\right)^{1/2}$ (Eq. 10.49) yields the compact form $\left(\frac{E_P}{V}\frac{dV}{d\phi}\right)^{2} \ll 1$ (Eq. 10.51), with $E_P$ the Planck energy — a sufficiently shallow slope plus a dominant amplitude gives exponential expansion. [R 10 283]
False vacuum: for the plateau potential of Figure 10.4 — global minimum ("true vacuum") at $\phi = \phi_0$, $V = 0$; plateau at $\phi \approx 0$ with $V \approx V_0$ — the field near $\phi = 0$ sits in a metastable false vacuum: nudged off, it slowly rolls toward the true vacuum, taking many Hubble times if the plateau is broad and shallow, satisfying $\left(\frac{dV}{d\phi}\right)^{2} \ll \frac{V_0^{2}}{E_P^{2}}$ (Eq. 10.52), and contributing $\varepsilon_\phi \approx V_0 \approx$ const meanwhile. [R 10 284]
Onset of inflation: the inflaton is dynamically negligible while $\varepsilon_r \sim \alpha T^4 \gg V_0$; inflation begins once radiation cools below the false-vacuum density, at $T_i \approx \left(\frac{V_0}{\alpha}\right)^{1/4} \approx 2 \times 10^{28},\mathrm{K}\left(\frac{V_0}{10^{105},\mathrm{TeV,m^{-3}}}\right)^{1/4}$ (Eq. 10.53), equivalently $kT_i \approx (\hbar^3 c^3 V_0)^{1/4} \approx 2\times10^{12}$ TeV (Eq. 10.54), at time $t_i \approx \left(\frac{c^{2}}{GV_0}\right)^{1/2} \approx 3 \times 10^{-36},\mathrm{s}\left(\frac{V_0}{10^{105},\mathrm{TeV,m^{-3}}}\right)^{-1/2}$ (Eq. 10.55). [R 10 285]
Duration and e-foldings: while rolling, $H_i \approx \left(\frac{8\pi G V_0}{3c^{2}}\right)^{1/2} \approx t_i^{-1}$ (Eq. 10.56); inflation ends when $\phi$ reaches the true vacuum, and the e-folding count for the plateau potential is $N \sim H_i \frac{\phi_0}{\dot{\phi}} \sim \left(\frac{E_P}{V_0}\frac{dV}{d\phi}\right)^{-1}\left(\frac{\phi_0}{E_P}\right)$ (Eq. 10.57) — a broad, high, shallowly sloped plateau gives many e-foldings. [R 10 285]
Reheating: after rolling off the plateau, $\phi$ oscillates about $\phi_0$; Hubble friction damps the oscillations, but coupling to other fields damps them faster, dumping the inflaton's energy into photons and other relativistic particles that reheat the universe; the released energy $V_0$ is the latent heat of the false→true vacuum transition (analogy: freezing water releases $3\times10^8,\mathrm{J,m^{-3}} \sim 2\times10^{15},\mathrm{TeV,m^{-3}}$). [R 10 286]
Supercooling and recovery: the temperature drops by $e^{-N}$ during inflation — for $N = 65$ starting at the GUT time, from $T(t_i) \sim 10^{28}$ K to $T(t_f) \sim 0.6$ K — so inflation dilutes every particle species including photons; reheating then restores the temperature to roughly its pre-inflationary value $T_i$. [R 10 286]
Inflation is "too successful": it flattens local as well as global curvature; if pre-inflation fluctuations were $\delta\varepsilon/\varepsilon \sim 1$, a naive calculation gives post-inflation $\frac{\delta\varepsilon}{\bar{\varepsilon}} \sim e^{-65} \sim 10^{-28}$ (Eq. 10.59) — even with later growth, far smoother than the observed CMB. [R 10 287]
Quantum seeds of structure: inflation blows submicroscopic scales up to macroscopic ones (recall $4\times10^{-29},\mathrm{m} \to 0.9,\mathrm{m}$); on quantum scales the vacuum — true or false — is intrinsically inhomogeneous, with virtual particles popping in and out; inflation stretches these quantum fluctuations of the inflaton field to macroscopic scale, and the resulting energy fluctuations are the origin of today's inhomogeneities: "Great superclusters from tiny quantum fluctuations grow." [R 10 287]

Chapter 11 — Structure Formation: Gravitational Instability

Chapter intro — the observed large-scale structure

The universe is homogeneous/isotropic only when smoothed with a filter ~100 Mpc across; on smaller scales it holds density fluctuations from subatomic quantum fluctuations up to superclusters and voids ~50 Mpc across. Large scale structure = all structure bigger than individual galaxies, mapped by measuring galaxy redshifts and estimating distances via the Hubble relation $d = (c/H_0)z$. [R 11 290]
2dF Galaxy Redshift Survey: two narrow stripes near the north (Coma Berenices) and south (Sculptor) galactic poles, ~100 000 galaxy redshifts each, giving two 2-D slices reaching z ≈ 0.22, i.e. proper distance $d_p(t_0) \sim 1000$ Mpc. [R 11 290]
Galaxies are emphatically not Poisson-distributed: they lie in long superclusters — flattened (planar) or elongated (linear) structures just now collapsing under self-gravity — separated by roughly spherical underdense voids; a cluster is a fully collapsed, more-or-less virialized object embedded in a supercluster. [R 11 291]
Morphology nicknames: "bubbly" or "spongy" (voids as the bubbles/holes) if you focus on underdensities; a cosmic web of filaments and walls if you focus on the overdense superclusters. [R 11 292]

11.1 The Matthew Effect

Gravitational instability is the basic mechanism growing voids and superclusters: density fluctuations existed at last scattering (stamped on the CMB); in a matter-dominated universe overdense regions expand less rapidly than the whole, can collapse into bound objects, and then draw in matter from underdense surroundings — the Matthew effect ("the rich get richer, the poor get poorer"). [R 11 292]
Density fluctuation (density contrast): $\delta(\vec r,t) \equiv \frac{\varepsilon(\vec r,t) - \bar\varepsilon(t)}{\bar\varepsilon(t)}$ (Eq. 11.2), with $\bar\varepsilon$ averaged over a volume V large compared to the biggest structure (Eq. 11.1); for matter/radiation (ε ≥ 0) the minimum is δ = −1, but δ has no upper bound. [R 11 293]
Static-universe toy model: uniform overdense sphere (δ ≪ 1) in a static, homogeneous, pressureless medium; the excess mass gives $\frac{\ddot R}{R} = -\frac{4\pi G\bar\rho}{3},\delta(t)$ (Eq. 11.4), so a mass excess drives collapse. (Caveat: no truly static homogeneous matter-only universe exists — the awkward fact behind Einstein's Λ; "static" holds only locally, like room air stabilized by a large-scale pressure gradient.) [R 11 294]
Mass conservation closes the system: for δ ≪ 1, $R(t) \approx R_0[1 - \tfrac13\delta(t)]$ (Eq. 11.8), so $\ddot R/R \approx -\tfrac13\ddot\delta$ (Eq. 11.10). [R 11 295]
Combining gives $\ddot\delta = 4\pi G\bar\rho,\delta$ (Eq. 11.11), with solution $\delta(t) = A_1 e^{t/t_{\rm dyn}} + A_2 e^{-t/t_{\rm dyn}}$ (Eq. 11.12): small fluctuations in a static pressureless medium grow exponentially. If the sphere starts at rest, $A_1 = A_2 = \delta(0)/2$; after a few dynamical times only the growing mode matters. [R 11 296]
Dynamical time: $t_{\rm dyn} = \frac{1}{(4\pi G\bar\rho)^{1/2}} \approx 9.6,\text{hours}\left(\frac{\bar\rho}{1,\mathrm{kg,m^{-3}}}\right)^{-1/2}$ (Eq. 11.13) — depends only on $\bar\rho$, not on R. [R 11 296]

11.2 The Jeans Length

Pressure is what keeps air (ρ ≈ 1 kg m⁻³, $t_{\rm dyn}$ ~ 10 hours) from runaway collapse: a nonrelativistic gas has $w \approx kT/(\mu c^2)$ (Eq. 11.14), so its pressure only vanishes in the limit T → 0. [R 11 296]
Pressure gradients build up at the sound speed, in time $t_{\rm pre} \sim R/c_s$ (Eq. 11.16) with $c_s = c,(dP/d\varepsilon)^{1/2} = \sqrt{w},c$ (Eq. 11.17); hydrostatic equilibrium is attainable only if $t_{\rm pre} < t_{\rm dyn}$ (Eq. 11.18). [R 11 297]
Jeans length (precise version): $\lambda_J = c_s\left(\frac{\pi c^2}{G\bar\varepsilon}\right)^{1/2} = 2\pi c_s t_{\rm dyn}$ (Eq. 11.20). Overdense regions larger than $\lambda_J$ collapse under their own gravity; smaller regions merely oscillate as stable standing sound waves. [R 11 298]
Worked example — Earth's atmosphere: $c_s \approx \tfrac13$ km/s and $t_{\rm dyn} \approx 10$ hours give $\lambda_J \sim 10^5$ km, far exceeding the atmospheric scale height — no catastrophic collapse of air. [R 11 298]
In an expanding flat universe the Hubble time is comparable to the collapse time, $H^{-1} = (3/2)^{1/2},t_{\rm dyn} \approx 1.22,t_{\rm dyn}$ (Eq. 11.22), so a component with EoS parameter w has $\lambda_J = 2\pi\left(\frac{2}{3}\right)^{1/2}\sqrt{w},\frac{c}{H}$ (Eq. 11.24). [R 11 299]
Radiation (w = 1/3, $c_s = c/\sqrt3$): $\lambda_J \approx 3.0,c/H$ (Eq. 11.25) — radiation perturbations smaller than ~3 Hubble distances are stable sound waves; a pure-radiation universe can't form collapsed structures ≪ c/H, so gravitationally collapsed sub-Hubble structure requires a nonrelativistic component. [R 11 299]
Before decoupling ($z_{\rm dec} \approx z_{\rm ls} \approx 1090$) photons, electrons and baryons formed a single photon–baryon fluid; since $\varepsilon_\gamma > \varepsilon_{\rm bary}$, baryons were (with mild exaggeration) a dynamically insignificant contaminant, and $\lambda_J(\text{before}) \approx 3c/H(z_{\rm dec}) \approx 0.66$ Mpc (Eq. 11.26). [R 11 300]
Baryonic Jeans mass: $M_J \equiv \rho_{\rm bary}\left(\frac{4\pi}{3}\lambda_J^3\right)$ (Eq. 11.27); just before decoupling ($\rho_{\rm bary} \approx 5.6\times10^{-19}$ kg m⁻³), $M_J(\text{before}) \approx 2\times10^{49},\mathrm{kg} \approx 10^{19},M_\odot$ (Eq. 11.28) — huge even compared with the baryonic mass of the Coma cluster (~2×10¹⁴ M⊙) or a supercluster (~10¹⁶ M⊙). [R 11 300]
After decoupling the two gases separate: photons keep $c_s = c/\sqrt3 \approx 0.58c$ (Eq. 11.29), while the baryonic gas drops to $c_s = (kT/mc^2)^{1/2}c \approx 1.5\times10^{-5}c$ (Eqs. 11.30–11.31) ≈ 5 km/s, using $kT_{\rm dec} \approx 0.26$ eV and mean atomic rest energy $mc^2 = 1.22,m_pc^2 \approx 1140$ MeV (helium fraction $Y_p = 0.24$). [R 11 300]
So at decoupling the baryonic Jeans length fell by $F \approx 2.6\times10^{-5}$ (Eq. 11.32) and the Jeans mass by $F^3 \approx 1.8\times10^{-14}$: $M_J(\text{after}) \approx 2\times10^5,M_\odot$ (Eq. 11.33) — small even next to the Small Magellanic Cloud (~10⁹ M⊙), let alone our galaxy (~10¹¹ M⊙). [R 11 301]
This abrupt drop marks a key epoch: baryonic perturbations from supercluster scales down to the smallest dwarf galaxies could not grow until photon decoupling at $t_{\rm dec} \approx 0.37$ Myr; afterwards baryonic collapse was "off and running". [R 11 301]

11.3 Instability in an Expanding Universe

In an expanding universe, losing pressure support does not mean exponential growth: the self-gravity timescale $t_{\rm dyn} \sim (c^2/G\varepsilon)^{1/2}$ is comparable to the expansion timescale $H^{-1}$, so gravity (densifying overdense regions) and expansion (diluting them) compete on equal footing. [R 11 302]
Newtonian analysis of a sphere with $\rho = \bar\rho(t)[1+\delta(t)]$ (Eq. 11.34) in a matter-only expanding universe: mass conservation gives $R(t) \propto a(t)[1+\delta(t)]^{-1/3}$ (Eq. 11.39) — a slightly overdense sphere expands slightly slower than a(t), an underdense one slightly faster. [R 11 302]
Subtracting the unperturbed acceleration equation $\ddot a/a = -\frac{4\pi}{3}G\bar\rho$ (Eq. 11.42, matching Eq. 4.49) and keeping terms linear in δ yields the growth equation $\ddot\delta + 2H\dot\delta = 4\pi G\bar\rho,\delta$ (Eq. 11.44); the extra $2H\dot\delta$ term is Hubble friction, which slows perturbation growth (H = 0 recovers the static exponential case, Eq. 11.45). [R 11 304]
The fully relativistic version is $\ddot\delta + 2H\dot\delta = \frac{4\pi G}{c^2}\bar\varepsilon_m,\delta$ (Eq. 11.46), applicable when radiation (w = 1/3) or Λ (w = −1) are present — but δ is the fluctuation in the matter density alone, $\delta = (\varepsilon_m - \bar\varepsilon_m)/\bar\varepsilon_m$ (Eq. 11.47). [R 11 304]
In terms of the matter density parameter: $\ddot\delta + 2H\dot\delta - \frac{3}{2}\Omega_m H^2\delta = 0$ (Eq. 11.49) — matter perturbations grow rapidly only in epochs when matter dominates. [R 11 305]
Radiation era ($\Omega_m \ll 1$, H = 1/(2t)): $\ddot\delta + \dot\delta/t \approx 0$ (Eq. 11.50) gives $\delta(t) \approx B_1 + B_2\ln t$ (Eq. 11.51) — dark matter fluctuations grow only logarithmically. [R 11 305]
Λ era ($H = H_\Lambda$ = const): $\delta(t) \approx C_1 + C_2 e^{-2H_\Lambda t}$ (Eq. 11.53) — fluctuations freeze at constant fractional amplitude while the average matter density itself keeps plummeting. [R 11 306]
Flat matter era ($\Omega_m = 1$, H = 2/(3t)): power-law ansatz $\delta = Dt^n$ gives $n(n-1) + \frac43 n - \frac23 = 0$ (Eq. 11.56), i.e. n = 2/3 or −1, so $\delta(t) \approx D_1 t^{2/3} + D_2 t^{-1}$ (Eq. 11.57); once the decaying mode dies, $\delta \propto t^{2/3} \propto a(t) \propto \frac{1}{1+z}$ (Eq. 11.58), valid while |δ| ≪ 1. [R 11 306]
Dark matter's head start: baryon-only perturbations could grow only from $z_{\rm dec} = 1090$ (factor ~1090 by today), but dark matter perturbations started growing effectively at $z_{rm} = 3440$; at decoupling the baryons fell into pre-existing dark-matter gravitational wells (Figure 11.4). [R 11 307]

11.4 The Power Spectrum

Equations 11.44/11.46 apply to low-amplitude perturbations of any shape, not just spheres — so today's bubbly large-scale structure (the "tree") lets us deduce the early low-amplitude fluctuation field (the "nut"). [R 11 308]
While |δ| ≪ 1 the Robertson–Walker metric (Eq. 11.59) still describes the geometry and each bit of matter keeps a nearly constant comoving coordinate; one studies $\delta(\vec r)$ at a time $t_i$ — the exact choice (after perturbations are in place, before |δ| ~ 1) changes only the amplitude, not the shape, of the field. Cosmologists care about its statistical properties, not exact peak locations. [R 11 309]
Fourier decomposition in a comoving box of volume V: $\delta(\vec r) = \frac{V}{(2\pi)^3}\int \delta_{\vec k},e^{-i\vec k\cdot\vec r},d^3k$ (Eq. 11.61), with each component a complex number $\delta_{\vec k} = |\delta_{\vec k}|e^{i\phi_{\vec k}}$ (Eq. 11.63) of comoving wavelength λ = 2π/k. [R 11 309]
Each Fourier mode independently obeys the growth equation $\ddot\delta_{\vec k} + 2H\dot\delta_{\vec k} - \frac32\Omega_m H^2\delta_{\vec k} = 0$ (Eq. 11.64) as long as its proper wavelength a(t)2π/k lies between the Jeans length and the Hubble distance; phases stay constant while amplitudes are small — so linear theory still describes very large scales even after galaxies and clusters have already collapsed. [R 11 310]
Power spectrum: $P(k) = \langle|\delta_{\vec k}|^2\rangle$ (Eq. 11.65), averaged over all orientations of $\vec k$. [R 11 310]
Gaussian field: phases $\phi_{\vec k}$ uncorrelated; if homogeneous and isotropic, all statistical properties are contained in P(k), and δ at a random point follows a Gaussian pdf with variance $\sigma_\delta^2 = \frac{V}{2\pi^2}\int_0^\infty P(k),k^2,dk$ (Eq. 11.67). Most inflationary scenarios predict exactly such a field. [R 11 311]
Inflation also predicts a power law $P(k) \propto k^n$ (Eq. 11.68) with n ≈ 1; n = 1 exactly is the Harrison–Zel'dovich spectrum, proposed independently by Harrison and Zel'dovich as early as 1970, before inflation existed as a concept. [R 11 311]
Observed tilt: CMB temperature fluctuations on large scales (θ > 1.1°) measured by Planck give n = 0.97 ± 0.01 — only slightly "tilted" from Harrison–Zel'dovich. [R 11 312]
Mean nonrelativistic-matter mass in a randomly placed comoving sphere of radius r: $\langle M\rangle = \frac{4\pi}{3}r^3\rho_{m,0} = 1.67\times10^{11},M_\odot\left(\frac{r}{1,\mathrm{Mpc}}\right)^3$ (Eq. 11.69); the mean square mass fluctuation follows from P(k) through a top-hat window involving the spherical Bessel function $j_1(x) = (\sin x - x\cos x)/x^2$ (Eq. 11.70). [R 11 312]
RMS mass fluctuation for $P \propto k^n$: $\frac{\delta M}{M} \propto r^{-(3+n)/2}$ (Eq. 11.72), equivalently $\propto M^{-(3+n)/6}$. Randomly scattered point masses (Poisson) give $\delta M/M \propto M^{-1/2}$, i.e. n = 0 — so the n = 1 spectrum has more power on small scales than Poisson. [R 11 313]
n = 1 is the only power law for which the potential fluctuations diverge neither at large nor at small scales — hence Harrison–Zel'dovich is also called the scale-invariant spectrum. [R 11 313]

11.5 Hot versus Cold

The primordial $P(k) \propto k^n$ spectrum is modified between the end of inflation $t_f$ and radiation–matter equality $t_{rm} \approx 5.0\times10^4$ yr; the shape at $t_{rm}$ — when perturbations start growing significantly — depends on whether the dark matter is cold or hot. [R 11 313]
"Hot" vs "cold" refers to particle thermal velocity at the moment the mass inside a Hubble volume equaled a large galaxy's mass, $M_{\rm gal} \approx 10^{12},M_\odot$: with radiation-era $H(a) = H_0\sqrt{\Omega_{r,0}}/a^2$ (Eq. 11.73), $M_H = 1.6\times10^{28},M_\odot,a^3$ (Eq. 11.74), which hits $10^{12},M_\odot$ at a ≈ 4×10⁻⁶, kT ≈ 60 eV, cosmic time t ≈ 12 yr. [R 11 314]
Cold dark matter: nonrelativistic at t ≈ 12 yr. Examples: WIMPs — decoupled at t ~ 1 s when 3kT ~ 3 MeV, but with mc² ~ 100 GeV they were already nonrelativistic at decoupling; axions — formed out of equilibrium with very low thermal velocities. [R 11 314]
Hot dark matter: relativistic when it decoupled and still relativistic at t ≫ 12 yr. Neutrinos with $5\times10^{-4},\mathrm{eV} \ll m_\nu c^2 \ll 180,\mathrm{eV}$ qualify: relativistic at t ≈ 12 yr, nonrelativistic today. [R 11 315]
HDM of particle mass $m_h$ becomes nonrelativistic when 3kT ≈ $m_hc^2$, at $T_h \approx \frac{m_hc^2}{3k} \approx 12,000,\mathrm{K}\left(\frac{m_hc^2}{3,\mathrm{eV}}\right)$ (Eq. 11.75), i.e. at $t_h \approx 42,000,\mathrm{yr}\left(\frac{m_hc^2}{3,\mathrm{eV}}\right)^{-2}$ (Eq. 11.76) (still radiation-dominated if $m_hc^2 > 2.4$ eV). [R 11 315]
Free streaming: before $t_h$ the HDM particles fly in random directions at ~c, wiping out density fluctuations with wavelength ≲ $ct_h$: proper scale $d_{\min} \approx ct_h \approx 13,\mathrm{kpc},(m_hc^2/3,\mathrm{eV})^{-2}$ (Eq. 11.77), comoving scale $r_{\min} \approx 55,\mathrm{Mpc},(m_hc^2/3,\mathrm{eV})^{-1}$ (Eq. 11.78), mass scale $M_{\min} \approx 2.7\times10^{16},M_\odot,(m_hc^2/3,\mathrm{eV})^{-3}$ (Eq. 11.79). Rough guide: $m_hc^2$ ~ 50 eV erases below Local-Group scale, ~8 eV below Coma-cluster scale, ~3 eV below supercluster scale. [R 11 315]
HDM power spectrum (for $m_hc^2 = 3$ eV, $r_{\min} \approx 55$ Mpc): close to the primordial P ∝ k for $k \ll 2\pi/r_{\min} \approx 0.1,\mathrm{Mpc^{-1}}$, but severely suppressed for $k \gg 2\pi/r_{\min}$; correspondingly δM/M levels off below $M_{\min} \sim 3\times10^{16},M_\odot$. [R 11 316]
Top-down scenario (HDM): the first structures to collapse are supercluster-mass regions (M ~ M_min) of nearly uniform overdensity; clusters and galaxies form later by fragmentation. Observation contradicts this: superclusters are only now collapsing while galaxies have existed since z ~ 10 (universe ~0.5 Gyr old) — so most dark matter must be cold, with negligible free streaming. [R 11 317]
CDM growth rule: during radiation domination, sub-Hubble modes (proper wavelength < c/H) stagnate, but modes with proper wavelength larger than the Hubble distance keep growing regardless of what dominates. [R 11 317]
Two key comoving scales: WIMP decoupling ($t_d$ ~ 1 s, $a_d$ ~ 3×10⁻¹⁰): $r_d = 2ct_d/a_d \sim 60$ pc (Eq. 11.80), $M_d \sim 0.05,M_\odot$ (Eq. 11.81), $k_d \sim 10^5,\mathrm{Mpc^{-1}}$; radiation–matter equality ($t_{rm}$ = 0.050 Myr, $a_{rm}$ = 2.9×10⁻⁴, c/H ≈ 1.8c$t_{rm}$ ≈ 0.027 Mpc): $r_{rm} = 1.8ct_{rm}/a_{rm} \approx 90$ Mpc (Eq. 11.82), $M_{rm} \approx 1.3\times10^{17},M_\odot$, $k_{rm} \approx 0.07,\mathrm{Mpc^{-1}}$. [R 11 318]
Resulting CDM spectrum at $t_{rm}$: modes with $k < k_{rm}$ stayed super-Hubble through the whole radiation era and retain the primordial P ∝ k; modes with $k_d > k > k_{rm}$ grew only until their proper wavelength $a(t)r \propto t^{1/2}$ fell below $c/H \propto t$, then froze until $t_{rm}$ — suppression is greatest at the largest k (earliest horizon entry); the spectrum has a broad maximum at $k \sim k_{rm} \sim 0.1,\mathrm{Mpc^{-1}}$. [R 11 319]
Normalization: δM/M = 1 at $M = 10^{14},M_\odot$ today, matching observed fluctuations on very large (still linear, δM/M < 1) scales; at radiation–matter equality the amplitude was smaller by a factor $\sim a_{rm} \sim 3\times10^{-4}$. [R 11 319]
Bottom-up scenario (CDM): δM/M is largest at the smallest masses, so the smallest objects collapse first — galaxies, then clusters, then superclusters — consistent with the observed relative ages of galaxies and superclusters. [R 11 319]
Mixed hot+cold fits to observed large-scale structure require $\Omega_{\rm hdm,0} \le 0.007$; if the HDM is entirely the three standard neutrino flavors of the cosmic neutrino background, then $[m(\nu_e)+m(\nu_\mu)+m(\nu_\tau)]c^2 \le 0.3,\mathrm{eV}$ (Eq. 11.83) — more HDM would free-stream the universe too smooth on small scales. [R 11 320]
ΛCDM model: the Hot Big Bang variant whose energy density today is dominated by dark energy as a cosmological constant plus cold dark matter; basic version is spatially flat; the name dates from the mid-1990s and it is now the standard model of cosmology. [R 11 320]

11.6 Baryon Acoustic Oscillations

Baryons (15% of the matter) gravitate exactly like CDM but also couple electromagnetically to photons: radiative cooling allows dense galaxies ($M < 10^{12},M_\odot$) and ultra-dense stars ($M < 100,M_\odot$) to form; and on scales beyond the largest superclusters ($M > 10^{17},M_\odot$) the photon coupling marks the power spectrum via baryon acoustic oscillations. [R 11 320]
Origin: at last scattering the densest regions of the photon–baryon fluid were those that had just compressed to maximum density before rebounding under their own pressure; their size ≈ the sound horizon, $d_s(t_{\rm ls}) = 0.145$ Mpc (Eq. 11.84); the photon overdensities show up as CMB hot spots of angular size θ ≈ θ_s ≈ 0.7°. [R 11 321]
The baryon overdensities survive as the acoustic scale: comoving $r_s = d_s(t_{\rm ls})(1+z_{\rm ls}) \approx 160$ Mpc (Eq. 11.85), corresponding to mass scale $M_s = \frac{4\pi}{3}r_s^3\rho_{m,0} \approx 7\times10^{17},M_\odot$ (Eq. 11.86). [R 11 321]
The baryon overdensity at $r_s$ was initially small, but Matthew-effect growth has raised it to detectable levels today — while still leaving the universe homogeneous and isotropic above ~100 Mpc. [R 11 322]
Two-point correlation function ξ(r): pick a random galaxy; the expected number in volume dV at comoving distance r is $dN = n_{\rm gal}[1+\xi(r)],dV$ (Eq. 11.87). Poisson distribution ⇒ ξ = 0; real galaxies cluster, giving ξ > 0 at small separations. [R 11 322]
ξ(r) is the Fourier transform of the power spectrum: for isotropic P(k), $\xi(r) = \frac{V}{2\pi^2}\int P(k),j_0(kr),k^2,dk$ (Eq. 11.89, $j_0 = \sin x/x$) — the two contain identical information; ξ(r) is the more convenient choice for galaxy redshift surveys. [R 11 323]
BAO bump: since $r_s \approx 160,\mathrm{Mpc} \approx 0.04,c/H_0$, surveys reaching well beyond z ≈ 0.04 with enough galaxies can detect an extra bump in ξ(r) at r ~ r_s; it is seen in a sample of nearly a million luminous galaxies at 0.4 < z < 0.7 — a modern relic of the photon–baryon fluid fluctuations at $t_{\rm ls} \approx 0.37$ Myr. [R 11 323]
BAO as standard yardstick: the acoustic scale's length rests on well-understood physics, so tracking the angular size of the BAO bump with redshift probes dark energy (is w = −1? does w evolve?); adding BAO to supernova + Planck CMB constraints shrinks the 95% confidence region to a tiny ellipse at $\Omega_{m,0} = 0.305$, $\Omega_{\Lambda,0} = 0.696$ — the basis of the Benchmark Model — hugging the κ = 0 line, the source of the $|\Omega_0 - 1| \le 0.005$ flatness limit. [R 11 324]

Chapter 12 — Structure Formation: Baryons and Photons

Overview

Baryons complicate structure formation: before decoupling ($z_{\rm dec} = 1090$) photon–baryon coupling prevented dense baryonic lumps; at lower redshift the interaction helps, since radiating away excess thermal energy is what permits objects of high mass density. [R 12 328]
Mean baryon density today ($t_0 \approx 13.7$ Gyr): $\rho_{\mathrm{bary},0} = 4.2 \times 10^{-28},\mathrm{kg,m^{-3}} = 6.2 \times 10^{9},\mathrm{M_\odot,Mpc^{-3}}$ (Eq. 12.1). [R 12 328]
Baryonic overdensities span huge range: solar neighborhood ($\rho_{\rm sn} \approx 0.095,\mathrm{M_\odot,pc^{-3}} \approx 6.4\times10^{-21},\mathrm{kg,m^{-3}}$) has $\delta_{\rm sn} \sim 2\times10^{7}$ (Eq. 12.2); the Sun's interior ($\rho_\odot \approx 1400,\mathrm{kg,m^{-3}}$) has $\delta_\odot \sim 3\times10^{30}$. Yet most baryons remain low-density intergalactic gas. [R 12 328]

12.1 Baryonic Matter Today

Stellar mass density: $\rho_{\star,0} \approx 4\times10^{8},\mathrm{M_\odot,Mpc^{-3}} \approx 3\times10^{-29},\mathrm{kg,m^{-3}}$ (Eq. 12.3) — only ~7% of the baryonic mass; even this easiest-to-see component is a rough estimate. [R 12 329]
Baryon census in bound systems: stars/remnants/brown dwarfs/planets ~7%, interstellar gas ~1%, circumgalactic gas ~3% (together ~4% "gas associated with galaxies"), intracluster gas ~4% — so gravitationally bound systems hold only ~15% of all baryons. [R 12 329]
"Missing baryon problem" (1990s): baryons in galaxies + clusters fall far short of $\rho_{\mathrm{bary},0}$; resolution — they are not missing, just inconspicuous: ~40% is diffuse intergalactic gas ($T < 10^5$ K, smoothly distributed) and ~45% is warm-hot intergalactic gas ($10^5,\mathrm{K} < T < 10^7,\mathrm{K}$, in long filaments between clusters). [R 12 330]
Intergalactic gas properties: very low density (most of the universe's volume has $\delta \le 0$; warm-hot gas typically $3 < \delta < 300$) and almost fully ionized ($X \approx 1$) — yet recombination drove $X$ from 0.9 to 0.1 between $z = 1480$ and $z = 1260$, so something later reionized the hydrogen. [R 12 331]

12.2 Reionization of Hydrogen

CMB as reionization probe: reionized intergalactic gas is an "obstacle course" of free electrons; a CMB photon scatters at rate $\Gamma = n_e \sigma_e c$ (Eq. 12.4), with $\sigma_e = 6.65\times10^{-29},\mathrm{m^2}$. [R 12 331]
Optical depth to reionized gas: $\tau_* = \int_{t_}^{t_0} \Gamma(t),dt = c\sigma_e \int_{t_}^{t_0} n_e(t),dt$ (Eq. 12.5); the fact that we see small-angular-scale CMB fluctuations at all requires $\tau_* < 1$ (otherwise the CMB would be smeared as through a translucent screen). [R 12 332]
Measured value: Planck sees slight suppression of temperature fluctuations at high multipole $l$, consistent with $\tau_* = 0.066 \pm 0.016$ — about one CMB photon in 15 scatters from a free electron at low redshift. [R 12 332]
Model assumptions: baryons treated as pure hydrogen, $n_{\mathrm{H}} + n_p = n_{\mathrm{bary},0}/a^3$ (Eq. 12.6), undergoing complete instantaneous reionization at $t_$, so $n_e = n_p = n_{\mathrm{bary},0}/a^3$ for $t > t_$ (Eq. 12.7). (Helium ignored — complicates the maths without changing conclusions.) [R 12 332]
Present-day scattering rate scale: $\Gamma_0 = c\sigma_e n_{\mathrm{bary},0} = 1.58\times10^{-4},\mathrm{Gyr^{-1}} \approx 0.0023,H_0$ (Eq. 12.9) — the rate if baryons were uniform, fully ionized hydrogen today. [R 12 333]
Optical depth as redshift integral: $\tau_* = \Gamma_0 \int_0^{z_} \frac{(1+z)^2,dz}{H(z)}$ (Eq. 12.11); inserting $H(z) = H_0[\Omega_{m,0}(1+z)^3 + \Omega_{\Lambda,0}]^{1/2}$ (Eq. 12.12) gives the analytic solution $$\tau_ = \frac{2}{3\Omega_{m,0}} \frac{\Gamma_0}{H_0} \left( \left[ \Omega_{m,0}(1+z_)^3 + \Omega_{\Lambda,0} \right]^{1/2} - 1 \right) \quad \text{(Eq. 12.13)},$$ which for Benchmark parameters is $\tau_ = 0.00485\left([0.31(1+z_*)^3 + 0.69]^{1/2} - 1\right)$ (Eq. 12.14). [R 12 333]
Redshift of reionization: matching $\tau_* = 0.066 \pm 0.016$ gives $z_* = 7.8 \pm 1.3$, i.e. $t_* \sim 650$ Myr; the "era of neutrality" was a brief interlude, $t_* - t_{\rm rec} \sim 0.05,t_0$. [R 12 334]
Ionizing sources — O stars: photons with $hf > Q = 13.6$ eV ionize H; galaxies exist by $z \sim 10$, so they were present at reionization. Only the most massive O stars ($M \ge 30,\mathrm{M_\odot}$) are hot enough to contribute significantly; a $30,\mathrm{M_\odot}$ O star lives $t \approx 6,\mathrm{Myr} \approx 2\times10^{14}$ s. [R 12 334]
Ionizing sources — AGN: compact galactic nuclei powered by accretion onto a supermassive black hole (radiated energy up to $0.1mc^2$ per accreted mass $m$), much of it as $hf > 13.6$ eV UV; ionizing-photon production rate $\dot N_* \approx 3\times10^{56},\mathrm{s^{-1}},(L_{\rm AGN}/10^{13},\mathrm{L_\odot})$ (Eq. 12.15). AGN with $L > 10^{13},\mathrm{L_\odot}$ are quasars; one such quasar emits as many ionizing photons in a month as an O star in its whole 6 Myr lifetime. [R 12 334]
Supermassive black hole examples: Milky Way, $M_{\rm bh} \approx 4\times10^{6},\mathrm{M_\odot}$ (Schwarzschild radius $\approx 0.08$ AU); NGC 4889 in Coma, $M_{\rm bh} \approx 2\times10^{10},\mathrm{M_\odot}$ ($\approx 400$ AU). [R 12 335]

12.3 The First Stars and Quasars

Dark Ages: the era between recombination and the first stars/AGN — "dark" only in starlight; the CMB was there, and at $t_{\rm ls} \approx 0.37$ Myr ($T \approx 2970$ K) the entire sky was as bright as an M-star's surface (no Olbers' paradox then). [R 12 335]
End of the Dark Ages: by $z \approx 50$ ($t \approx 50$ Myr) the background had cooled to $T \approx 140$ K — a far-infrared background useless for ionizing H — and this is roughly when the first stars formed (estimates span $z = 50 \to 20$, $t = 50 \to 180$ Myr). Ionized bubbles grew around isolated galaxies, then merged into a single ionized intergalactic medium at $z \sim 8$. [R 12 336]
Escape fraction: much ionizing radiation is absorbed by gas and dust inside the source galaxy; $f_{\rm esc}$ = fraction leaking into intergalactic space; poorly determined, optimistic fiducial $f_{\rm esc} = 0.2$ for both hot stars and AGN. [R 12 336]
Photon budget for reionization: comoving baryon density in intergalactic space $n_{\rm bary} = 0.25,\mathrm{m^{-3}} = 7.3\times10^{66},\mathrm{Mpc^{-3}}$ (Eq. 12.16); photons that must be produced is $n_* = \frac{n_{\rm bary}}{f} = 3.7\times10^{67},\mathrm{Mpc^{-3}},\left(\frac{0.2}{f}\right)$ (Eq. 12.17). [R 12 337]
Scale of the budget: with $f_{\rm esc} \approx 0.2$, reionizing a comoving Mpc$^3$ takes $N_* \sim 4\times10^{67}$ ionizing photons — the lifetime output of ~40 000 O stars (i.e. $\sim 10^{63}$ photons per O star), or a $10^{13},\mathrm{L_\odot}$ quasar shining for 4000 yr. [R 12 337]
Quasars didn't do it: quasars are breathtakingly rare; their comoving number density peaked at $z \approx 2.5$ ($t \approx 2.6$ Gyr), long after reionization; at $z_* \approx 8$ even optimistic extrapolation gives ~1 luminous quasar per $10^{10}$ comoving Mpc$^3$, requiring ~40 000 Gyr to reionize — vs. $t_* \sim 0.65$ Gyr available. [R 12 337]
Low-luminosity AGN don't rescue it: if the AGN luminosity function at $z \sim 8$ resembles today's, adding fainter AGN is still inadequate — most reionizing photons came from hot, luminous stars. [R 12 338]
Star formation history: today ~3 parts per million of the universe's baryons are converted into stars per Myr; the comoving star formation rate peaked in the range $z = 4 \to 1$ ($t = 1.5 \to 6$ Gyr) at ~10× today's value — the "intense star-forming era." [R 12 338]
SFR at reionization: at $z_* \approx 8$ the comoving rate was about the same as today, $\dot\rho_\star(z{=}8) \approx 20,000,\mathrm{M_\odot,Myr^{-1},Mpc^{-3}}$ (Eq. 12.18); only ~10% of that mass goes into O stars with $M \ge 30,\mathrm{M_\odot}$ — ~67 new O stars per comoving Mpc$^3$ per Myr, hence ~400 O stars present at any time (6 Myr lifetime). [R 12 339]
Photon production rate: 400 O stars each emitting $\dot N_* \approx 5\times10^{48},\mathrm{s^{-1}}$ give $\dot n_* \approx 2\times10^{51},\mathrm{s^{-1},Mpc^{-3}} \approx 6\times10^{64},\mathrm{Myr^{-1},Mpc^{-3}}$ (Eq. 12.19). [R 12 340]
Timescale check: star formation must persist for $t = n_/\dot n_ \approx 600,\mathrm{Myr},(0.2/f_{\rm esc})$ (Eq. 12.20) — not impossibly long compared to $t_* \approx 650$ Myr provided $f_{\rm esc} \gtrsim 0.2$; so massive stars can supply the reionizing photons. [R 12 340]
Cheap side effect: converting just ~1 part in 500 (~0.2%) of the baryonic mass into stars reionizes the remaining gas; ongoing star formation then drives $\rho_\star$ up to today's $4\times10^{8},\mathrm{M_\odot,Mpc^{-3}}$ and keeps the IGM ionized despite radiative recombination. [R 12 340]

12.4 Making Galaxies

Stars live in galaxies: today's $\rho_\star$ yields luminosity density $\Psi_V = 1.1\times10^{8},\mathrm{L_{\odot,V},Mpc^{-3}}$ and stellar number density $n_\star \sim 10^{9},\mathrm{Mpc^{-3}}$; stars are concentrated in galaxies embedded in dark halos. [R 12 341]
Schechter luminosity function: $\Phi(L),dL = \Phi^* \left(\frac{L}{L^}\right)^{\alpha} \exp\left(-\frac{L}{L^}\right)\frac{dL}{L^}$ (Eq. 12.21); V-band fits give slope $\alpha \approx -1$, normalization $\Phi^ \approx 0.005,\mathrm{Mpc^{-3}}$, and characteristic luminosity $L^*$ comparable to our own galaxy's. [R 12 341]
Exponential cutoff: galaxies with $L > L^$ are exponentially rare; $\sim 10L^$ giants (NGC 4889, NGC 4874) exist only in rich clusters, grown by "cannibalism". Milky Way: $M_{\rm bary} \approx 1.2\times10^{11},\mathrm{M_\odot}$, $M_{\rm tot} \approx (1\to2)\times10^{12},\mathrm{M_\odot}$. Making galaxies with $L_V > 2\times10^{10},\mathrm{L_{\odot,V}}$, $M_{\rm bary} > 10^{11},\mathrm{M_\odot}$, $M_{\rm tot} > 10^{12},\mathrm{M_\odot}$ is evidently very hard. [R 12 342]
The limit is baryonic physics, not dark matter: dark halos far above $10^{12},\mathrm{M_\odot}$ exist (Coma's halo: $M \approx 2\times10^{15},\mathrm{M_\odot}$) but embrace many smaller galaxies rather than one gargantuan galaxy. [R 12 342]
Spherical collapse model: an overdense sphere at radiation–matter equality ($t_{rm} \approx 0.050$ Myr) has $M = \frac{4\pi}{3}\rho_m(t)[1+\delta(t)]R(t)^3$ (Eq. 12.22); it expands nearly with the Hubble flow while $\delta \ll 1$, turns around when $\delta(t_{\rm coll}) \approx 1$; since $\delta \propto a \propto t^{2/3}$ in the matter era, $1+z_{\rm coll} \approx \delta_{rm}(1+z_{rm})$ (Eq. 12.23) — higher initial overdensity ⇒ earlier collapse. [R 12 343]
Virialization density: at turnaround $\bar\rho(t_{\rm coll}) \approx 2\rho_m(t_{\rm coll})$ (Eq. 12.24); the collapsing sphere oscillates and settles ("virialization") at half its turnaround radius, so $\bar\rho_{\rm halo} \approx 8\bar\rho(t_{\rm coll}) \approx 16\rho_{m,0}(1+z_{\rm coll})^3$ (Eq. 12.25) — halo density encodes collapse epoch. [R 12 343]
Milky Way's collapse redshift: halo mass $M = 1.9\times10^{12},\mathrm{M_\odot},(R_{\rm halo}/0.15,\mathrm{Mpc})$ (Eq. 12.26) gives $\bar\rho \approx 3400,\rho_{m,0},(R_{\rm halo}/0.15,\mathrm{Mpc})^{-2}$ (Eq. 12.27), implying $1+z_{\rm coll} \approx 6,(R_{\rm halo}/0.15,\mathrm{Mpc})^{-2/3}$ (Eq. 12.28). [R 12 344]
Virial temperature: virialization shock-heats the gas; hydrostatic equilibrium (Eq. 12.29) evaluated at $R_{\rm halo}$ with an isothermal power-law profile $\rho_{\rm gas} \propto r^{-\beta}$ gives $kT_{\rm gas} = \frac{GM_{\rm tot}\mu}{\beta R_{\rm halo}}$ (Eq. 12.30); ionized primordial gas ($Y_p = 0.24$) has mean particle mass $\mu = 0.59,m_p$. [R 12 344]
Virial temperature vs mass and epoch: with $R_{\rm halo}$ from Eq. 12.31 one gets $kT_{\rm gas} = \frac{4}{\beta}\left(\frac{\pi}{3}\right)^{1/3} G\mu,\rho_{m,0}^{1/3} M_{\rm tot}^{2/3}(1+z_{\rm coll})$ (Eq. 12.32); numerically, with $\beta \approx 2$, $T_{\rm gas} \approx 1.0\times10^{6},\mathrm{K},\left(\frac{M_{\rm tot}}{10^{12},\mathrm{M_\odot}}\right)^{2/3}\left(\frac{1+z_{\rm coll}}{5}\right)$ (Eq. 12.33). Massive, early-collapsing halos are hottest. [R 12 345]
Weak mass dependence of collapse epoch: CDM is "bottom-up," but $\delta M/M \propto M^{-0.14}$ near $10^{12},\mathrm{M_\odot}$, so $z_{\rm coll}$ depends only weakly on mass. For 2σ fluctuations: $M_{\rm tot} = 10^{10},\mathrm{M_\odot}$ → $z_{\rm coll} \approx 6.7$, $T_{\rm gas} \approx 7\times10^{4}$ K; $10^{12},\mathrm{M_\odot}$ → $z_{\rm coll} \approx 3.4$, $9\times10^{5}$ K; $10^{14},\mathrm{M_\odot}$ → $z_{\rm coll} \approx 1.0$, $9\times10^{6}$ K. [R 12 346]
Cooling is the key to galaxy formation: to form a dense star-filled galaxy, halo gas must radiate away thermal energy (into intergalactic space), lose pressure support, and fall to the center. Halos with $T_{\rm gas} < 10^{6}$ K retain some bound electrons and cool quickly by line emission (He$^+$, then He and H); halos with $T_{\rm gas} > 10^{6}$ K are almost fully ionized and can cool only by bremsstrahlung (free–free emission). [R 12 346]
Bremsstrahlung cooling time: luminosity density $\Psi = 5.3\times10^{-32},\mathrm{W,m^{-3}},(\rho_{\rm gas}/10^{-24},\mathrm{kg,m^{-3}})^{2}(T/10^{6},\mathrm{K})^{1/2}$ (Eq. 12.34) and thermal energy density $\varepsilon = \frac{3}{2}nkT$ (Eq. 12.35) give $$t_{\rm cool} = \frac{\varepsilon}{\Psi} = 13,\mathrm{Gyr}\left(\frac{\rho_{\rm gas}}{10^{-24},\mathrm{kg,m^{-3}}}\right)^{-1}\left(\frac{T}{10^{6},\mathrm{K}}\right)^{1/2} \quad \text{(Eq. 12.36)};$$ hot ($T > 10^6$ K) gas cools within the age of the universe only if $\rho_{\rm gas} > 10^{-24},\mathrm{kg,m^{-3}}$. [R 12 347]
Halo gas density: with cosmic baryon fraction $f = 0.048/0.31 = 0.15$, $\bar\rho_{\rm bary} = f\bar\rho_{\rm halo} \approx 0.8\times10^{-24},\mathrm{kg,m^{-3}},(f/0.15),((1+z_{\rm coll})/5)^{3}$ (Eq. 12.37) — so a $10^{12},\mathrm{M_\odot}$ halo collapsing at $z_{\rm coll} > 4$ is both hot enough for bremsstrahlung and dense enough to cool within the age of the universe. [R 12 347]
Why no jumbo galaxies — the density lottery: total mass inside the last scattering surface ($d_p(t_0) \approx 14,000$ Mpc) is $M \approx 4.3\times10^{23},\mathrm{M_\odot}$ (Eq. 12.38), i.e. $4.3\times10^{9}$ regions of $10^{14},\mathrm{M_\odot}$; the very first such halo to collapse is a $P \approx 2.3\times10^{-10}$ (6.2σ) fluctuation, collapsing at $z_{\rm coll} \approx 5.2$ with $T_{\rm gas} \approx 2.3\times10^{7}$ K and $\rho_{\rm gas} \approx 1.5\times10^{-24},\mathrm{kg,m^{-3}}$, giving $t_{\rm cool} \approx 42$ Gyr $> t_0$. Later-collapsing $10^{14},\mathrm{M_\odot}$ halos have much lower density and even longer cooling times — so cluster-mass halos never cool into a single gargantuan galaxy. [R 12 348]
Cold flows caveat: in realistic non-spherical collapse, some baryonic gas is never shock-heated to $T_{\rm gas}$ and flows cold to the halo center on timescales shorter than $t_{\rm cool}$, enabling the first galaxies to form at higher redshift (ending the Dark Ages); the shock-heated portion still obeys the cooling argument — which is why Coma's intracluster gas fails to form a single galaxy with $M_{\rm bary} \gg 10^{11},\mathrm{M_\odot}$. [R 12 349]

12.5 Making Stars

What cooled gas becomes: infalling cool gas is not swallowed by the central black hole ($M_{\rm bh} < 1%$ of a galaxy's baryonic mass, even in NGC 4889), nor does it form one "megastar" or $10^{12}$ "microstars": $10^5,\mathrm{M_\odot}$ of gas makes about a million stars and brown dwarfs with typical mass $M_\star \sim 0.1,\mathrm{M_\odot}$ plus a power-law tail (the initial mass function). [R 12 349]
Molecular cloud cores: stars form in the dense, cold cores of molecular clouds (H as H$2$), with $\rho{\rm core} \sim 10^{-15},\mathrm{kg,m^{-3}}$ — over $10^{12}\times\rho_{\mathrm{bary},0}$ yet under $10^{-18}\times\rho_\odot$; enriched composition $Y = 0.27$, $X = 0.72$, $Z = 0.01$ (C and O partly in molecules, partly in silicate/graphite dust grains); cores opaque at visible wavelengths appear as "dark nebulae". [R 12 350]
Core temperature: $T_{\rm core} \approx 20$ K, set by the balance between heating by cosmic rays (which penetrate the opaque core) and cooling by far-infrared radiation from dust grains. [R 12 350]
Core timescales and sound speed: dynamical time $t_{\rm dyn} = (4\pi G\rho_{\rm core})^{-1/2} \approx 1.1\times10^{12},\mathrm{s},(\rho_{\rm core}/10^{-15},\mathrm{kg,m^{-3}})^{-1/2}$ (Eq. 12.39); with mean molecular mass $\mu = 2.3,m_p$ (H$2$ + He), isothermal sound speed $c_s = (kT{\rm core}/\mu)^{1/2} \approx 270,\mathrm{m,s^{-1}}$ at 20 K (Eq. 12.40) — barely slower than sound in room-temperature air. [R 12 350]
Jeans mass in a core: Jeans length $\lambda_J = 2\pi c_s t_{\rm dyn} \approx 1.9\times10^{15}$ m (Eq. 12.41) gives $$M_J = \frac{4\pi}{3}\rho_{\rm core}\lambda_J^3 \approx 15,\mathrm{M_\odot}\left(\frac{\rho_{\rm core}}{10^{-15},\mathrm{kg,m^{-3}}}\right)^{-1/2}\left(\frac{T_{\rm core}}{20,\mathrm{K}}\right)^{3/2} \quad \text{(Eq. 12.42)}.$$ Naively regions below $\sim 15,\mathrm{M_\odot}$ can't collapse — but the IMF peaks at $M_\star \sim 0.1,\mathrm{M_\odot}$ (stars above $15,\mathrm{M_\odot}$ are the rarest), so this conclusion is "nonsense": the resolution is that collapsing cores can cool. [R 12 351]
Isothermal luminosity: a 15 $\mathrm{M_\odot}$ core has thermal energy $E_{\rm core} = N(\frac{3}{2}kT_{\rm core}) = 3.2\times10^{36}$ J (Eq. 12.43); adiabatic compression would heat it (Eqs. 12.44–12.45), so holding $T_{\rm core}$ constant requires radiating $L^{\rm iso}{\rm core} = \frac{2E{\rm core}}{R_{\rm core}}\frac{dR_{\rm core}}{dt}$ (Eq. 12.46); for free collapse ($dR/dt \approx R/t_{\rm dyn}$), $L^{\rm iso}{\rm core} \approx 0.015,\mathrm{L\odot},(M_{\rm core}/15,\mathrm{M_\odot})(\rho_{\rm core}/10^{-15},\mathrm{kg,m^{-3}})^{1/2}$ (Eq. 12.47). [R 12 351]
Actual core luminosity: emission is far-infrared light from dust; cores opaque in the visible need not be opaque in the far IR (Barnard 68: $\tau \approx 23$ at 550 nm, $\tau \approx 4$ at 1650 nm, $\tau \approx 0.03$ at 0.1 mm), so write $L_{\rm core} = 4\pi R_{\rm core}^2 f_e \sigma_{\rm sb} T_{\rm core}^4$ with efficiency $f_e \le 1$ (Eq. 12.48). [R 12 352]
Cooling is easy: with $R_{\rm core} \approx 10^4$ AU (Eq. 12.49), $L_{\rm core} \approx 1100,\mathrm{L_\odot},f_e,(M_{\rm core}/15,\mathrm{M_\odot})^{2/3}(\rho_{\rm core}/10^{-15},\mathrm{kg,m^{-3}})^{-2/3}$ (Eq. 12.50) — any $f_e \gtrsim 10^{-5}$ keeps the collapse isothermal. [R 12 354]
Hierarchical fragmentation: at constant $T$, $M_J \propto \rho^{-1/2}T^{3/2}$ falls as the core contracts; when density has grown ×4 (radius ×$4^{-1/3}$), $M_J$ has halved and the 15 $\mathrm{M_\odot}$ core splits into two 7.5 $\mathrm{M_\odot}$ fragments; repeated subdivision follows. [R 12 354]
Origin of the IMF power law: if fragmentation is slightly inefficient (e.g. 1% of cores per round collapse directly to a star instead of splitting), after $n$ rounds one gets $2^n$ stars of mass $2^{-n}M_{\rm core}$, giving $\frac{dN}{d\log M} \propto \frac{1}{M}$ (Eq. 12.51), equivalent to $\chi(M) \propto M^{-2}$; the observed IMF is steeper ($\propto M^{-2.3}$), a sign the simple model misses some physics. [R 12 354]
Why fragmentation stops: after $n$ rounds $M_{\rm frag} = 2^{-n}M_{\rm core}$ and $\rho_{\rm frag} = 2^{2n}\rho_{\rm core}$; the required isothermal luminosity (Eq. 12.52) is the same for every fragment, but the actual luminosity falls as $R^2$; their ratio $$\frac{L_{\rm frag}}{L^{\rm iso}{\rm frag}} \approx 73,000,f_e,\left(\frac{M{\rm frag}}{15,\mathrm{M_\odot}}\right)^{-1/3}\left(\frac{\rho_{\rm frag}}{10^{-15},\mathrm{kg,m^{-3}}}\right)^{-7/6} \approx 73,000,f_e,2^{-2n} \quad \text{(Eqs. 12.53–12.54)}$$ drops fourfold per round; fragmentation halts when it reaches 1. [R 12 355]
Why stars have stellar masses: cooling round $n = 9$ would need an unphysical $f_e \approx 3$, so at most 9 rounds occur, giving minimum fragment mass $M_{\rm frag} \sim 2^{-9} \times 15,\mathrm{M_\odot} \sim 0.03,\mathrm{M_\odot}$ — within the broad maximum of the observed IMF. Once fragments can't stay cool, $T$ rises, $M_J$ starts increasing, and they become protostars (near hydrostatic equilibrium, contracting on timescales ≫ $t_{\rm dyn}$); a protostar with $M > 0.08,\mathrm{M_\odot}$ ignites fusion and becomes a star (below that, a brown dwarf). [R 12 356]
Population III / first stars: low dust content (low C, O, Si) lowers $f_e$ and halts fragmentation earlier, so metal-poor star formation has a top-heavy IMF; the very first stars had no dust and could cool only via inefficient molecular-hydrogen processes — likely no hierarchical fragmentation at all, so they were all extremely massive (comparable to the Jeans mass of their gas), lived fast, died young, and ejected the first C, O, and other heavy elements. [R 12 356]
Model caveats and planets: real star formation involves magnetic fields, turbulence, and angular momentum — a contracting, spinning fragment spins up and yields a protostar plus a rotationally supported protoplanetary disk, which is unstable to forming planets. Scaling from the Solar System ($M_{\rm pl} = 0.0013,\mathrm{M_\odot}$, mostly Jupiter): $\Omega_{\mathrm{pl},0} \sim 0.0013,\Omega_{\star,0} \sim 4\times10^{-6}$ (Eq. 12.55) — dynamically negligible, but vital as habitats for beings who ask where we come from. [R 12 357]

Epilogue

State of play (2016): the evidence supports a Benchmark Model that is spatially flat with accelerating expansion — 69% of the energy density from a cosmological constant (or other negative-pressure dark energy), 31% from matter, of which only 4.8% is baryonic. [R Ep 360]
Open question — parameters: the precise values of $H_0$, $\Omega_{m,0}$, $\Omega_{\Lambda,0}$ are still not pinned down absolutely. [R Ep 360]
Open question — dark matter: it cannot be entirely baryons, nor entirely neutrinos; most must be exotic stuff not yet detected in laboratories. [R Ep 360]
Open question — dark energy: vacuum energy acting as a cosmological constant, or a component with $-1 < w < -1/3$? If vacuum energy, is it a false vacuum driving a temporary inflationary stage, or the true vacuum at last? [R Ep 360]
Open question — inflation: the particle physics behind inflation is incomplete; some cosmologists even ask whether inflation happened at all, or whether the flatness, horizon, and monopole problems can be solved another way. [R Ep 361]
Open question — initial conditions: why is the universe expanding? Extrapolating back to the Planck time gives $H \sim 1/t_P$; what set these initial conditions — "What put the Bang in the Big Bang?" [R Ep 361]
The deepest questions are unposed: as with ancient Egyptian questions about the sky-dome, erroneous models generate irrelevant questions; future observations may render the Benchmark Model obsolete. [R Ep 361]

regstuff/cosmology_notes_8-12.md

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

Contents

Chapter 8 — The Cosmic Microwave Background

Chapter introduction

8.1 Observing the CMB

8.2 Recombination and Decoupling

8.3 The Physics of Recombination

8.4 Temperature Fluctuations

8.5 What Causes the Fluctuations?

Chapter 9 — Nucleosynthesis and the Early Universe

Chapter opening

9.1 Nuclear Physics and Cosmology

9.2 Neutrons and Protons

9.3 Deuterium Synthesis

9.4 Beyond Deuterium

9.5 Baryon–Antibaryon Asymmetry

Chapter 10 — Inflation and the Very Early Universe

Overview

10.1 The Flatness Problem

10.2 The Horizon Problem

10.3 The Monopole Problem

10.4 The Inflation Solution

10.5 The Physics of Inflation

Chapter 11 — Structure Formation: Gravitational Instability

Chapter intro — the observed large-scale structure

11.1 The Matthew Effect

11.2 The Jeans Length

11.3 Instability in an Expanding Universe

11.4 The Power Spectrum

11.5 Hot versus Cold

11.6 Baryon Acoustic Oscillations

Chapter 12 — Structure Formation: Baryons and Photons

Overview

12.1 Baryonic Matter Today

12.2 Reionization of Hydrogen

12.3 The First Stars and Quasars

12.4 Making Galaxies

12.5 Making Stars

Epilogue