Big Bang Cosmology

7.1 The Friedmann Equations

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):

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

7.2 Critical Density and Cosmological Parameters

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 ($\Omega_{\mathrm{total} \approx 1.0007}$).

7.3 Solutions for Different Components

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).

7.4 The Cosmic Microwave Background

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 $T \propto a^{-1} \propto (1+z)$.

7.5 CMB Anisotropy

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.

7.6 Big Bang Nucleosynthesis

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:

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

BBN predictions agree with observations of primordial abundances. The deuterium abundance is Particularly sensitive to the baryon-to-photon ratio $\eta = n_b/n_\gamma$:

$\eta \approx 6.1 \times 10^{-10}$

This value is consistent with that derived from the CMB, providing strong support for the Big Bang Model.

7.7 Dark Matter

Evidence for dark matter:

1. Galaxy rotation curves: Stars orbit as if there is much more mass than is visible. The velocity does not decrease with distance from the galactic centre (as expected from Keplerian orbits), indicating a dark matter halo.
2. Galaxy clusters: The virial theorem gives masses much larger than the luminous mass (Zwicky, 1933).
3. Gravitational lensing: The deflection of light by galaxy clusters indicates more mass than visible matter.
4. CMB anisotropy: The angular power spectrum requires $\Omega_{\mathrm{DM} \approx 0.266}$.
5. Large-scale structure: N-body simulations reproduce the observed cosmic web only with dark matter.

Properties:

• Does not emit, absorb, or scatter light (non-luminous).
• Is cold (non-relativistic at structure formation): CDM model.
• Accounts for $\sim 27\%$ of the total energy density.
• Interacts primarily via gravity (and possibly the weak force).

Candidates:

• WIMPs (Weakly Interacting Massive Particles): $m \sim 10$ GeV—1 TeV. Predicted by supersymmetry. Searched for in direct detection experiments (XENON, LUX) and at the LHC. No confirmed detection.
• Axions: Light particles ($m \sim 10^{-5}$ eV/$c^2$) proposed to solve the strong CP problem.
• Sterile neutrinos: Right-handed neutrinos that do not participate in weak interactions.
• Primordial black holes: Black holes formed in the early universe.

7.8 Dark Energy

In 1998, two teams (Riess et al., Perlmutter et al.) observed that Type Ia supernovae are fainter Than expected for a decelerating universe. This implies the expansion is accelerating: $\ddot{a} \gt 0$.

From the acceleration equation, this requires $\rho + 3P/c^2 \lt 0$Which is satisfied by a Component with $w \lt -1/3$.

The simplest explanation is Einstein’s cosmological constant $\Lambda$With equation of state $w = -1$:

$\rho_\Lambda = \frac{\Lambda c^2}{8\pi G}, \quad P_\Lambda = -\rho_\Lambda c^2$

The cosmological constant problem: Quantum field theory predicts a vacuum energy of order $M_{\mathrm{Pl}^4}$ ($\sim 10^{120}$ times larger than the observed value). The origin of the tiny Observed value is one of the deepest unsolved problems in physics.

Alternative models include quintessence (a dynamic scalar field with $-1 \lt w \lt -1/3$) And modified gravity (e.g., $f(R)$ gravity). Current data are consistent with $w = -1$ (cosmological constant), but precision is limited.

7.9 Inflation

The standard Big Bang model has several unresolved issues:

1. Horizon problem: The CMB is uniform across the sky, but regions separated by more than $\sim 1^\circ$ were causally disconnected at recombination.
2. Flatness problem: $\Omega$ must have been extraordinarily close to 1 in the early universe: $|\Omega - 1| \lt 10^{-60}$ at $t \sim 10^{-36}$ s.
3. Monopole problem: Grand unified theories predict abundant magnetic monopoles, none of which are observed.
4. Origin of structure: What seeded the density fluctuations that grew into galaxies?

Inflation (Guth, 1981) proposes a period of exponential expansion in the early universe:

$a(t) \propto e^{Ht}, \quad H \approx \mathrm{const}$

Lasting from $t \sim 10^{-36}$ s to $t \sim 10^{-33}$ s, during which the universe expanded by a Factor of $\sim e^{60} \sim 10^{26}$.

Resolutions:

• Horizon: Inflation pushes all pre-existing causal contacts outside the observable horizon, making the CMB uniform.
• Flatness: Inflation drives $\Omega$ exponentially close to 1, regardless of its initial value.
• Monopoles: Inflation dilutes their density to negligible levels.
• Structure: Quantum fluctuations during inflation are stretched to macroscopic scales, providing the primordial perturbation spectrum (nearly scale-invariant, $n_s \approx 0.965$).

Inflation is driven by a scalar field $\phi$ (the inflaton) with potential $V(\phi)$. The Slow-roll parameters are:

\epsilon = \frac{M_{\mathrm{Pl}^2}{2}\left(\frac{V'}{V}\right)^2 \ll 1, \quad \eta = M_{\mathrm{Pl}^2 \frac{V''}{V} \ll 1}}

The number of e-folds: $N = \int_{t_i}^{t_f} H\,dt$.

The spectral index: $n_s = 1 - 6\epsilon + 2\eta \approx 0.965$ (Planck 2018), consistent with Slow-roll inflation.

The tensor-to-scalar ratio: $r = 16\epsilon$. Current upper bound: $r \lt 0.036$ (Planck + BICEP/Keck).

7.10 The Hubble Tension

The Hubble constant $H_0$ measured from the CMB ($\sim 67.4$ km/s/Mpc, Planck 2018, assuming $\Lambda$CDM) disagrees with local distance-ladder measurements ($\sim 73.0$ km/s/Mpc, SH0ES).

This discrepancy is now at the $\sim 5\sigma$ level and is one of the most significant open problems In cosmology. Possible resolutions include:

1. Systematic errors in one or both measurement methods.
2. New physics prior to recombination (e.g., additional radiation, early dark energy).
3. Extensions to $\Lambda$CDM (e.g., time-varying dark energy equation of state $w(z)$).

:::caution Common Pitfall The Hubble constant $H_0$ measured from the CMB ($\sim 67.4$ km/s/Mpc, Planck) disagrees with Local distance-ladder measurements ($\sim 73.0$ km/s/Mpc, SH0ES). This “Hubble tension” is one of The most significant open problems in cosmology. When using $H_0$ in calculations, be aware of Which measurement you are referencing and the systematic uncertainties involved.

7.11 Timeline of the Universe

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