 Solution

ID: past-exam-of-the-relativistic-astrophysics-and-cosmology-course-of-the-university-of-cambridge/2012/paper-1/2/a/solution

Density parameters at the same epoch share a common critical-density denominator. Thus compare the densities directly:

ρ_{m} = ρ_{m 0} (1 + z)^{3}

and

ρ_{γ} = ρ_{γ 0} (1 + z)^{4}

. Equality gives

1 + z_{mγ} = \frac{Ω _{m 0}}{Ω _{γ 0}} = 6245.89, z_{mγ} ≃ 6244.89 ≃ 6.24 \times 1 0^{3} .

This is the photon-only matter-radiation equality appropriate to the specified radiation density, not the equality including all relativistic species.

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