Big Bang Cosmology

The dynamics of a homogeneous, isotropic universe are described by the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric:

ds^2 = -c^2 dt^2 + a(t)^2\left[\frac{dr^2}{1 - kr^2} + r^2(d\theta^2 + \sin^2\theta,d\phi^2)\right]

Where $a(t)$ is the scale factor and $k \in \{-1, 0, +1\}$ is the curvature parameter.

The Friedmann equation (from Einstein’s equations with the FLRW metric):

H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}

Where $H = \dot{a}/a$ is the Hubble parameter and $\Lambda$ is the cosmological constant.

Derivation of the Friedmann equation. Starting from the Einstein field equations:

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}

The FLRW metric gives the Einstein tensor components $G_{00} = 3(\dot{a}^2 + kc^2)/(c^2 a^2)$ and $G_{ij} = -(2a\ddot{a} + \dot{a}^2 + kc^2)g_{ij}/a^2$. For a perfect fluid with $T_{\mu\nu} = \mathrm{diag}(\rho c^2, P, P, P)$ The $00$ component yields:

\frac{3(\dot{a}^2 + kc^2)}{c^2 a^2} + \Lambda = \frac{8\pi G}{c^4}\rho c^2

Multiplying through by $c^2 a^2/3$:

\dot{a}^2 + kc^2 = \frac{8\pi G}{3}\rho a^2 + \frac{\Lambda c^2 a^2}{3}

Dividing by $a^2$:

H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \quad \blacksquare

The acceleration equation:

\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3P}{c^2}\right) + \frac{\Lambda c^2}{3}

The fluid equation (from energy conservation):

\dot{\rho} + 3\frac{\dot{a}}{a}\left(\rho + \frac{P}{c^2}\right) = 0

The critical density: $\rho_c = \frac{3H_0^2}{8\pi G} \approx 9.2 \times 10^{-27}$ kg/m$^3$.

The density parameters: $\Omega_i = \rho_i/\rho_c$ With $\sum_i \Omega_i = 1$ for a flat universe.

Current best-fit values (Planck 2018):

• $\Omega_{\mathrm{matter} \approx 0.315}$ (matter: baryonic + dark)
• $\Omega_b \approx 0.049$ (baryonic matter)
• $\Omega_{\mathrm{DM} \approx 0.266}$ (dark matter)
• $\Omega_\Lambda \approx 0.685$ (dark energy)
• $\Omega_r \approx 9 \times 10^{-5}$ (radiation)
• $H_0 \approx 67.4$ km/s/Mpc

The universe is very close to flat ($\Omega_{\mathrm{total} \approx 1.0007}$).

For an equation of state $P = w\rho c^2$:

\rho \propto a^{-3(1+w)}

Derivation. From the fluid equation with $P = w\rho c^2$:

\dot{\rho} + 3\frac{\dot{a}}{a}\rho(1 + w) = 0

\frac{\dot{\rho}}{\rho} = -3(1+w)\frac{\dot{a}}{a}

Integrating: $\ln\rho = -3(1+w)\ln a + \mathrm{const}$ Giving $\rho \propto a^{-3(1+w)}$. $\blacksquare$

• Matter-dominated era: $a(t) \propto t^{2/3}$, $H \propto t^{-1}$.
• Radiation-dominated era: $a(t) \propto t^{1/2}$, $H \propto t^{-1}$.
• Dark-energy-dominated era: $a(t) \propto e^{Ht}$ (exponential expansion).

At $T \sim 3000$ K ($z \sim 1100$, $t \sim 380\,000$ years), the universe cooled enough for electrons And protons to combine into neutral hydrogen (recombination). Before this, photons were Continuously scattered by free electrons (the universe was opaque). After recombination, photons Decoupled and travelled freely.

These photons have been redshifted by the expansion of the universe and are observed today as the Cosmic Microwave Background (CMB) at $T_0 \approx 2.725$ K, with a peak wavelength $\lambda_{\mathrm{peak} \approx 1.9}$ mm (microwave).

Temperature-redshift relation. The CMB temperature at redshift $z$ is:

T(z) = T_0(1 + z)

This follows from the adiabatic expansion of a photon gas: $\rho_\gamma \propto a^{-4}$ and $\rho_\gamma \propto T^4$ So $T \propto a^{-1} \propto (1+z)$.

The CMB is nearly isotropic but has small temperature fluctuations: $\Delta T / T \sim 10^{-5}$.

The angular power spectrum $C_l$ of these fluctuations encodes cosmological parameters:

• The first peak at $l \sim 200$ confirms the universe is spatially flat.
• The relative heights of peaks determine the baryon density $\Omega_b$.
• The damping tail constrains the matter density and Hubble constant.

The CMB is also polarised at the level of $\sim 10^{-6}$. Two modes:

• E-mode: Gradient-type polarisation, produced by density fluctuations (scalar perturbations).
• B-mode: Curl-type polarisation, produced by gravitational waves (tensor perturbations).

Detection of primordial B-mode polarisation would provide evidence for inflation.

In the first few minutes after the Big Bang, the universe was hot and dense enough for nuclear Reactions to occur. The main products were:

The nuclear reaction chain starts at $T \sim 0.1$ MeV ($t \sim 1$ s):

p + n \to d + \gamma

d + p \to {^3\mathrm{He} + \gamma, \quad d + d \to {^3\mathrm{He} + n, \quad d + d \to t + p}}

t + d \to {^4\mathrm{He} + \gamma, \quad {^3\mathrm{He} + d \to {^4\mathrm{He} + p}}}

The process stops at $^4$ He because there are no stable nuclei with $A = 5$.

At $T \gg 1$ MeV, weak interactions maintain $n/p$ in thermal equilibrium:

\frac{n}{p} = e^{-\Delta m c^2/(k_B T)}

Where $\Delta m = m_n - m_p \approx 1.293$ MeV/$c^2$.

When $T \sim 0.8$ MeV ($t \sim 1$ s), the weak interaction rate falls below the expansion rate ($\Gamma_W \lt H$), and the $n/p$ ratio freezes out at $n/p \approx 1/6$. By the time Nucleosynthesis begins ($t \sim 200$ s), neutron decay has reduced this to $n/p \approx 1/7$.

The predicted helium mass fraction:

Y_p = \frac{2(n/p)}{1 + n/p} \approx 0.25