Cosmological epoch of matter-radiation equality

[gboff21]

Homework Statement

If Ω_0m=0.25 and Ω_0R=7.4*10^-5 calculate the redshift when the two densities Ω_m and Ω_R are equal.

Relevant Equations
1+z=1/a
$\Omega = \frac{rho}{rho_{crit}}$
$\rho_{0,crit} = \frac{3 H_{0}^{2}}{8 \pi G}$

The attempt at a solution

convert matter density: $\epsilon_{0,m} = \rho_{0,m} c^{2} = \Omega_{m,0} \rho_{crit,0} c^{2}$

sub in for critical density: $\epsilon_{0,m} = \Omega_{m,0} \frac{3 H_{0}^{2}}{8 \pi G} c^{2}$

calculate ratio of matter to radiation: $\frac{\epsilon_{R}}{\epsilon{M}} = \frac{\Omega_{R,0}}{\Omega{M,0}} \frac{8 \pi G c^{2}}{3 H_{0}^{2}}$

and as $\epsilon_{R} \propto 1/a^{4}$ and $\epsilon_{M} \propto 1/a^{3}$ and ρ0/a^3 = ρ

$\frac{\epsilon_{R}}{\epsilon{M}} = \frac{\epsilon_{0,R}}{\epsilon{0,M}} 1/a$

put $\frac{\epsilon_{R}}{\epsilon{M}} = 1$ so

$1 = \frac{\Omega_{M,0}}{\Omega{R,0}} \frac{3 H_{0}^{2}}{8 \pi G c^{2}} (1+z)$

This comes out as 1+z = 3.215*10^-6
and so gives a negative redshift!

Now I have either done something terribly wrong or the Omegas given are for an arbitrary universe in which the equality epoch has yet to occur!Thanks

 

[gboff21]

Sudden realisation (maybe):

Is it because
(ϵR)(ϵM) = (ϵ_0,R) (ϵ_0,M) a
not

(ϵR)(ϵM) = (ϵ_0,R) (ϵ_0,M) 1/a??

1+z≈311000?