Hubble's Law

Hubble's Law

In 1929, Edwin Hubble discovered that galaxies are receding from us at velocities proportional to their distance — the observational cornerstone of the expanding universe.

The Law

$v = H_0 \cdot d$

where $H_0 \approx 70$ km/s/Mpc is the Hubble constant today, $d$ is the distance in megaparsecs (Mpc), and $v$ is the recession velocity in km/s.

One parsec is 3.0857 × 10¹⁶ m (about 3.26 light-years). A megaparsec is 10⁶ pc.

The Hubble Time

A rough estimate of the age of the universe is the Hubble time — the time it would take for a galaxy to reach its current distance if it had always moved at today's velocity:

$t_H = \frac{1}{H_0}$

To convert $H_0$ from km/s/Mpc to SI units (s⁻¹):

$H_0^{\text{SI}} = H_0 \times \frac{1000}{3.0857 \times 10^{22}}$

Then $t_H = 1/H_0^{\text{SI}}$ in seconds. Dividing by $3.156 \times 10^{16}$ s/Gyr gives gigayears.

The Hubble Distance

The Hubble distance $D_H = c / H_0$ sets the characteristic length scale of the observable universe:

$D_H = \frac{c}{H_0} \approx 4283 \text{ Mpc}$

using $c = 299792.458$ km/s and $H_0 = 70$ km/s/Mpc.

Your Task

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

• hubble_velocity(H0_km_s_Mpc, d_Mpc) — returns recession velocity in km/s
• hubble_time_Gyr(H0_km_s_Mpc) — returns Hubble time in Gyr
• hubble_distance_Mpc(H0_km_s_Mpc) — returns Hubble distance in Mpc using $c = 299792.458$ km/s