general relativity - Relation between $Lambda$ and $Omega_Lambda$ in $Lambdamathrm{CDM}$

In minimal $\Lambda\mathrm{CDM}$, there is a parameter labeled $\Omega_\Lambda$, and current fits place it at around $\left( \Omega_\Lambda \sim 0.73\right)$. Meanwhile, $\Lambda$ enters the Einstein field equations as an offset to the Ricci scalar. The smallness of $\Lambda$ is a much-decried problem in physics today. My question is, what is the relation between $\Lambda$ and $\Omega_\Lambda$ in $\Lambda\mathrm{CDM}$? Are they independent?

We are currently in a cosmological constant-dominated era (since $\Omega_\Lambda$ is large). In the FRLW cosmology, we passed through epochs of matter domination, radiation domination, etc. so clearly $\Omega_\Lambda$ was changing through those epochs. How did $\Lambda$ change during that time?

Answer

They are proportional so essentially the same, but $\Omega_\Lambda$ is a convenient dimensionless number. Straight out of Weinberg's newer cosmology book:

$$\Lambda = 8 \pi G \rho_V,$$

where $\rho_V$ is the vacuum energy density, and

$$\rho_{V0} = \frac{3 H_0^2 \Omega_\Lambda}{8\pi G}.$$

Putting them together $\Lambda = 3 H_0^2 \Omega_{\Lambda0}$. Note the subscript $0$ represents the present day value. $H_0$ is the present day Hubble rate. Since $\Lambda$ is a constant you can infer $\Omega_\Lambda \sim H^{-2}$. $\Omega_\Lambda$ asymptotically approaches one as the vacuum dominates ever more and $H$ goes to the de Sitter value $H_{dS} = \sqrt{\Lambda / 3}$. In an earlier stage you have to work out $H$ using the Friedmann equation.