Suppose, as was once thought to be the case, that the universe today is flat and dominated by cold (pressure-free) matter.

(a) Show that a ∝ t ^2/3 and evaluate the age of the universe assuming the Hubble constant given in Sec. 28.2.3: H₀ = 67.7 km s⁻¹ Mpc⁻¹.

(b) Evaluate an expression for the angular diameter distance as a function of a and find its maximum value.

(c) Calculate the comoving volume within the universe back to very early times.

Data from Sec. 28.2.3. Hubble constant

Hubble constant¹⁸ H₀ ≡ H(t₀) is the contemporary value of H and is one of the key parameters of observational cosmology. It measures the age and size of the universe. We now know that the universe is accelerating q₀ ≡ q(t₀) < 0 and j₀ ≡ j (t₀) > 0.