Comoving Distance

Comoving Distance

When we say a galaxy is "3 billion light-years away", we need to be careful — in an expanding universe, distance is not a single number. The comoving distance is the coordinate distance that factors out the expansion of the universe, remaining constant as the universe grows.

Definition

The comoving distance to a source at redshift $z$ is:

$\chi(z) = \frac{c}{H_0} \int_0^z \frac{dz'}{E(z')}$

where the dimensionless Hubble parameter is:

$E(z) = \frac{H(z)}{H_0} = \sqrt{\Omega_m(1+z)^3 + \Omega_\Lambda}$

for a flat universe with negligible radiation, and $D_H = c/H_0$ is the Hubble distance.

Why Comoving Distance?

Comoving coordinates expand with the universe. Two galaxies at fixed comoving positions are moving apart only because space itself is expanding — not because they have any peculiar velocity. This makes comoving distance the natural coordinate for large-scale structure.

Numerical Integration

The integral has no closed-form solution for general $\Omega_m$ and $\Omega_\Lambda$. We use the trapezoidal rule with $N = 1000$ steps:

$\chi(z) \approx D_H \cdot \Delta z \sum_{i} \frac{1}{E(z_i)}$

For standard flat $\Lambda$CDM ($H_0 = 70$, $\Omega_m = 0.3$, $\Omega_\Lambda = 0.7$):

• At $z = 0.1$: $\chi \approx 418.5$ Mpc
• At $z = 1.0$: $\chi \approx 3303.8$ Mpc

Your Task

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

• E_z(z, Omega_m, Omega_Lambda) — returns $E(z) = \sqrt{\Omega_m(1+z)^3 + \Omega_\Lambda}$
• comoving_distance_Mpc(z, H0_km_s_Mpc, Omega_m, Omega_Lambda) — returns $\chi(z)$ in Mpc using trapezoidal integration with $N = 1000$ steps and $c = 299792.458$ km/s