Age of the Universe

Age of the Universe

How old is the universe? The answer depends on the cosmological model — specifically the Hubble constant H₀ and the density parameters.

The Hubble Time

A simple first estimate is the Hubble time — the time it would take for the universe to reach its current size if it had always expanded at today's rate:

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

With H₀ = 70 km/s/Mpc ≈ 2.27 × 10⁻¹⁸ s⁻¹:

$t_H \approx 13.97 \text{ Gyr}$

Matter-Only Universe

For a flat, matter-dominated universe the exact age is:

$t_0 = \frac{2}{3} t_H \approx 9.31 \text{ Gyr}$

This is younger than the oldest stars, which ruled out a pure matter universe.

Flat ΛCDM (the real universe)

The current standard model includes matter (Ω_m ≈ 0.3) and dark energy (Ω_Λ = 1 − Ω_m ≈ 0.7). The exact age is:

$t_0 = \frac{2}{3H_0\sqrt{1-\Omega_m}} \ln\left(\frac{1 + \sqrt{1-\Omega_m}}{\sqrt{\Omega_m}}\right)$

With H₀ = 70 km/s/Mpc and Ω_m = 0.3: t₀ ≈ 13.47 Gyr

With the Planck values (H₀ = 67.4, Ω_m = 0.315): t₀ ≈ 13.80 Gyr

Converting to Gyr

1 Gyr = 10⁹ years. The conversion factor: 1 Gyr ≈ 3.156 × 10¹⁶ s.

Your Task

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

• hubble_time_Gyr(H0_km_s_Mpc) — returns 1/H₀ in Gyr
• age_matter_only_Gyr(H0_km_s_Mpc) — returns (2/3) × t_H in Gyr
• age_LCDM_Gyr(H0_km_s_Mpc, Omega_m) — returns flat ΛCDM age in Gyr