Lambda-CDM model

Remarks

The full model also prescribes nearly scale-invariant, adiabatic, Gaussian initial perturbations, which are evolved by cosmological perturbation theory. Example: neglecting radiation, $H^2 = H_0^2(\Omega_m a^{-3} + \Omega_\Lambda)$ has the explicit solution $a(t) = \left( \Omega_m / \Omega_\Lambda \right)^{1/3} \sinh^{2/3}\left( \tfrac{3}{2}\sqrt{\Omega_\Lambda}\, H_0 t \right)$. At early times this behaves like the Einstein–de Sitter law $a \propto t^{2/3}$, and at late times like the de Sitter law $a \propto e^{\sqrt{\Omega_\Lambda} H_0 t}$. Non-example: the Einstein–de Sitter universe has $\Omega_\Lambda = 0$, so it fails clause 3. The deceleration parameter today is $q_0 = \Omega_r + \tfrac{1}{2}\Omega_m - \Omega_\Lambda$, which is negative for the observed values.