The Friedmann Equation

The Friedmann Equation

The Friedmann equation is the master equation governing how the universe expands. Derived from Einstein's field equations applied to a homogeneous, isotropic universe, it relates the Hubble parameter $H(a)$ to the energy content of the cosmos.

The Full Equation

$H^2(a) = H_0^2 \left[ \frac{\Omega_m}{a^3} + \frac{\Omega_r}{a^4} + \Omega_\Lambda + \frac{\Omega_k}{a^2} \right]$

where $a$ is the scale factor (normalised so $a_0 = 1$ today), and:

• $\Omega_m$ — matter density parameter (dark + baryonic)
• $\Omega_r$ — radiation density parameter
• $\Omega_\Lambda$ — dark energy (cosmological constant)
• $\Omega_k = 1 - \Omega_m - \Omega_r - \Omega_\Lambda$ — curvature term

For the observed flat universe ($\Omega_k = 0$) with negligible radiation today, the standard $\Lambda$CDM values are $\Omega_m \approx 0.3$, $\Omega_\Lambda \approx 0.7$.

Dimensionless Hubble Parameter

It is convenient to define $E(a) = H(a)/H_0$:

$E(a) = \sqrt{\frac{\Omega_m}{a^3} + \frac{\Omega_r}{a^4} + \Omega_\Lambda}$

At $a = 1$ (today), $E(1) = 1$ when $\Omega_m + \Omega_\Lambda = 1$ (flat).

Deceleration Parameter

The deceleration parameter $q_0$ measures whether the expansion is speeding up or slowing down today:

$q_0 = \frac{\Omega_m}{2} + \Omega_r - \Omega_\Lambda$

For $\Omega_m = 0.3$, $\Omega_\Lambda = 0.7$: $q_0 = -0.55 < 0$ — the universe is accelerating.

Your Task

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

• H_over_H0(Omega_m, Omega_r, Omega_Lambda, a) — returns $E(a) = H(a)/H_0$ for flat universe
• H_at_z(H0_km_s_Mpc, Omega_m, Omega_r, Omega_Lambda, z) — returns $H(z)$ in km/s/Mpc
• deceleration_parameter(Omega_m, Omega_r, Omega_Lambda) — returns $q_0$