Cosmology §

The Friedmann equation governs the expansion rate:

where $H = \dot{a}/a$ is the Hubble parameter, $a(t)$ is the scale factor, ρ is the energy density, k is the curvature, and Λ is the cosmological constant.

Key epochs:

• Inflation (t ~ 10⁻³⁶ s): exponential expansion smoothing the universe.
• Quark–hadron transition (t ~ 10⁻⁵ s): quarks confine into hadrons.
• Nucleosynthesis (t ~ 3 s to 20 min): light elements form.
• Recombination (t ~ 380,000 yr): electrons bind to nuclei; CMB is emitted.
• Structure formation: dark matter halos grow; galaxies form.

Example. The observed Hubble constant $H_0 \approx 70$ km/s/Mpc gives the Hubble time $1/H_0 \approx 14$ Gyr, consistent with the measured age of the universe.

As the universe expands and cools:

• At T ≳ 10⁹ K: neutrons and protons are in thermal equilibrium.
• At T ~ 10⁹ K: weak interactions freeze out; the neutron-to-proton ratio fixes at ≈ 1/7.
• At T ~ 10⁸ K: deuterium forms (the “deuterium bottleneck” is overcome).
• Rapid chain: D → He-3 → He-4. Almost all neutrons end up in He-4.

Predicted abundances (by mass):

• Hydrogen: ≈ 75%
• Helium-4: ≈ 25%
• Deuterium: ≈ 2 × 10⁻⁵
• Helium-3: ≈ 1 × 10⁻⁵
• Lithium-7: ≈ 1 × 10⁻¹⁰

These predictions match observations to high precision, providing strong evidence for the Big Bang model.

Example. The observed deuterium abundance in quasar absorption systems is a sensitive probe of the baryon density, since deuterium is easily destroyed in stars.

At recombination, the universe became transparent: photons decoupled from matter and have been free-streaming ever since, redshifted from visible/IR to microwave.

The CMB power spectrum shows:

• Acoustic peaks: oscillations in the photon-baryon fluid before decoupling. The peak positions and heights encode the baryon density, dark matter density, H₀, and other cosmological parameters.
• Polarization: E-modes (scalar perturbations) and B-modes (potentially from gravitational waves from inflation).

The CMB is the most precise probe of the early universe and has measured the flatness of the universe to within ≈ 0.4%.

Example. The first acoustic peak at ℓ ≈ 220 corresponds to the sound horizon at decoupling. Its position tells us the universe is spatially flat (Ωₖ ≈ 0).

The energy budget of the universe (Planck 2018):

• Dark energy: ≈ 68%
• Dark matter: ≈ 27%
• Baryonic matter: ≈ 5%

Dark matter: does not emit or absorb light; inferred from galaxy rotation curves, gravitational lensing, and the CMB. Leading candidates: a stable, weakly-interacting massive particle (WIMP) or axion.

Dark energy: causes accelerated expansion. The simplest model is a cosmological constant (w = −1), but dynamical models (quintessence) are also considered.

The acceleration was discovered in 1998 via Type Ia supernovae: distant supernovae are dimmer than expected in a decelerating universe.

Example. The rotation curve of the Andromeda galaxy shows flat velocity profiles at large radii, indicating a dark matter halo extending far beyond the visible disk.

The CMB has a peak temperature of 2.725 K. If the universe has expanded by a factor of a = 1100 since recombination, what was the temperature at the time of last scattering? What was the corresponding photon energy (in eV)?