The Atacama Cosmology Telescope inside its ground screen on Cerro Toco in the Chilean Andes

Unit 14 · Advanced + Cosmology · 75 min

The Hot Big Bang and the Cosmic Microwave Background

The thermal history of the universe and its oldest light

Image: M. Devlin, University of Pennsylvania (CC BY-SA 4.0)

This unit follows the universe as it cools: the first second, the three minutes in which helium formed, and the moment 372,000 years later when atoms appeared and space became transparent. You learn to read the cosmic microwave background, whose acoustic peaks give the geometry and the contents of the universe to percent accuracy, and you meet the puzzles that led to the idea of inflation.

Preview: the beginning of the unit

1 A universe that cools

Run the expansion of Unit 13 backward and the universe becomes denser and hotter without limit. The hot Big Bang model claims that this extrapolation can be trusted back to the first fraction of a second. Unit 13 dealt with the first piece of evidence, the expansion itself. This unit covers the other two: the abundances of the lightest elements, and a bath of microwave photons that fills all of space.

1.1 Temperature and redshift

The photon bath is the cosmic microwave background (CMB). Its spectrum today is that of a black body (Unit 3) with temperature $T_0 = 2.7255\unit{K}$ . Expansion stretches every wavelength in proportion to the scale factor $$a = 1/(1+z)$$ . A black body spectrum depends on wavelength and temperature only through the product $\lambda T$ , so stretching all wavelengths by one common factor turns it into another black body spectrum whose temperature is lower by that factor.

The relation has been tested. Carbon monoxide molecules in a gas cloud at $$z = 2.418$$ are excited by the microwave bath of their own epoch, and their level populations give $9.15 \pm 0.72\unit{K}$ . The prediction is $2.7255\unit{K}\times 3.418 = 9.32\unit{K}$ .

The energy density of black body radiation is $u = a_{\mathrm{rad}}T^4$ with the radiation constant $a_{\mathrm{rad}} = 4\sigma/c = 7.566\times10^{-16}\unit{J\,m^{-3}\,K^{-4}}$ . With $T \propto 1/a$ this gives $u \propto a^{-4}$ as in Unit 13, while matter dilutes as $a^{-3}$ . Today radiation (photons and neutrinos) makes up $\Omega_r = 9.2\times10^{-5}$ of the critical density and matter $\Omega_m = 0.315$ . They were equal at $1+z_{\mathrm{eq}} = \Omega_m/\Omega_r \approx 3400$ , 51,000 years after the Big Bang. Everything earlier is the radiation era.

1.2 The radiation era

In the radiation era the density follows from the temperature alone, and curvature and the cosmological constant are negligible. The Friedmann equation then works as a clock.

DerivationThe clock of the radiation era

Without curvature and cosmological constant the Friedmann equation of Unit 13 reads $H^2 = (\dot a/a)^2 = 8\pi G\rho/3$ , where $\dot a$ is the time derivative of the scale factor and $\rho$ the mass density. Radiation has $\rho \propto a^{-4}$ , so $\dot a \propto 1/a$ , or $a\,\dd a \propto \dd t$ . Integrating both sides gives $a^2 \propto t$ , hence $a \propto t^{1/2}, \qquad H = \frac{\dot a}{a} = \frac{1}{2t}, \qquad t = \sqrt{\frac{3}{32\pi G\rho}} .$ The age depends on the density only. The density is $\rho c^2 = \tfrac{1}{2}g_*\,a_{\mathrm{rad}}T^4$ , where $$g_*$$ counts the kinds of relativistic particles: 2 for the two polarization states of the photon and $$7/8$$ for each state of a fermion. Near $kT = 1\unit{MeV}$ photons, electrons, positrons and three kinds of neutrinos and antineutrinos give $g_* = 2 + \tfrac{7}{8}(4+6) = 10.75$ and $t = \sqrt{\frac{3c^2}{16\pi G\,g_*\,a_{\mathrm{rad}}}}\;\frac{1}{T^2} = \frac{0.99\times10^{20}\unit{s\,K^2}}{T^2}.$

A temperature is also an energy scale: particles in a gas at temperature $$T$$ have energies of order $$kT$$ , and $kT = 1\unit{MeV}$ corresponds to $1.16\times10^{10}\unit{K}$ . Each time $$kT$$ drops below a rest energy or a binding energy, the contents of the universe change (Table 1).

Table 1. Thermal history of the universe. Values before one second are rounded to one significant figure.

Event	Time	$$T$$ (K)	$$kT$$
Quarks bind into protons and neutrons	$2\times10^{-5}\unit{s}$	$2\times10^{12}$	155 MeV
Neutrinos decouple	1 s	$10^{10}$	0.9 MeV
Electrons and positrons annihilate	10 s	$3\times10^{9}$	0.3 MeV
Light nuclei form	4 min	$9\times10^{8}$	0.075 MeV
Matter-radiation equality	51,000 yr	9270	0.80 eV
Recombination and last scattering	372,000 yr	2970	0.26 eV
Reionization (midpoint)	670 Myr	24	2.0 meV
Today	13.8 Gyr	2.7255	0.23 meV

One rule explains most rows of the table. A reaction keeps particles in equilibrium as long as each particle reacts many times per expansion time $$1/H$$ . When the reaction rate per particle falls below the expansion rate $$H$$ , the reaction freezes out and the abundances stay where they were. Neutrinos are the first case: at about one second the weak interaction becomes too slow, and they have traveled freely ever since. Soon afterwards electrons and positrons annihilate into photons. This heats the photons and leaves the neutrinos alone, so the neutrino background should today have $(4/11)^{1/3}\,T_0 = 1.95\unit{K}$ . No experiment has detected it directly so far.

A black body at temperature $$T$$ contains $n_\gamma = 2.03\times10^{7}\,T^3$ photons per cubic meter ( $$T$$ in kelvin), which for $$T_0$$ gives 411 photons per cubic centimeter. The baryons add up to 4.9% of the critical density (Section 5), or 0.25 protons and neutrons per cubic meter. Photons therefore outnumber baryons by $n_\gamma/n_b = 1.6\times10^{9}$ . Both densities dilute as $a^{-3}$ , so this ratio has stayed the same since the first minutes, and it decides when nuclei and atoms can form.

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