Matter-Radiation Equality

Matter-Radiation Equality

In the very early universe, radiation dominated — photons and relativistic neutrinos made up the dominant energy content. As the universe expanded, however, the energy densities diluted at different rates:

• Radiation: ρ_r ∝ a⁻⁴ (energy density drops faster because photon wavelengths also redshift)
• Matter: ρ_m ∝ a⁻³

Eventually these two densities became equal at the matter-radiation equality epoch.

Scale Factor and Redshift of Equality

At equality, ρ_r = ρ_m. Using density parameters (normalised to critical density today):

$a_{eq} = \frac{\Omega_r}{\Omega_m}$

$z_{eq} = \frac{1}{a_{eq}} - 1 = \frac{\Omega_m}{\Omega_r} - 1$

With Ω_m ≈ 0.3 and Ω_r ≈ 9.4 × 10⁻⁵ (photons + neutrinos):

$z_{eq} \approx 3190$

Temperature at Equality

Using T ∝ (1 + z):

$T_{eq} = T_0 \cdot (1 + z_{eq}) = T_0 \cdot \frac{\Omega_m}{\Omega_r}$

Significance

Matter-radiation equality marks the transition from a radiation-dominated to a matter-dominated universe. Before equality, radiation pressure suppressed the growth of density fluctuations. After equality, matter could begin to cluster, eventually forming the large-scale structure we see today.

Your Task

Implement three functions. All constants must be defined inside each function body.

• scale_factor_equality(Omega_m, Omega_r) — returns a_eq = Ω_r / Ω_m
• redshift_equality(Omega_m, Omega_r) — returns z_eq = Ω_m / Ω_r − 1
• temperature_equality_K(Omega_m, Omega_r) — returns T_eq = T₀ × (Ω_m / Ω_r) with T₀ = 2.725 K