I When was the matter density equal to the vacuum energy density?

[LeoChan]

TL;DR Summary

In ΛCDM, to find t and z when the matter density equal to the vacuum energy density.

In ΛCDM, H(t0) = 70km/s/Mpc,
Ωd(t0) = 0.3, Ωr(t0) = 0 and ΩΛ(t0) =0.7,
so that Ω(t0) = Ωd(t0) + Ωr(t0) + ΩΛ(t0) = 1and the universe is spatially flat.

I want to know the t and z when the matter density equal to the vacuum energy density. By total energy density equation, I think Ωd(t) + ΩΛ(t) = 1, so they are both equal to 0.5 .

Maybe 0.5 = Λ / ( 3 * H(t) ^ 2 ). As for the matter, I am not sure since I only know it is proportional to a^-3. Is it related to the redshift dependent Hubble parameter, H(z)?

Thank you for your attention.

 

[Arman777]

Just use

$$\Omega_{m,0}(1+z_{eq})^3 = \Omega_{\Lambda,0}$$
to find $z_{eq}$ and then type it into some cosmological calculator to obtain $t$

 

[LeoChan]

Just use

$$\Omega_{m,0}(1+z_{eq})^3 = \Omega_{\Lambda,0}$$
to find $z_{eq}$ and then type it into some cosmological calculator to obtain $t$

Thank you.That is quite straight forward.