Big Bang

Remarks

If the Einstein equations hold, then at a Big Bang the curvature invariant $R_{ab}R^{ab}$ blows up together with the density. So the metric has no $C^2$ extension through $t = t_b$, and $t_b$ is an initial singularity. Example: the Einstein–de Sitter model, $a(t) = t^{2/3}$ and $\rho = 1/(6\pi G t^2)$ on $(0, \infty)$, has a Big Bang at $t_b = 0$. Non-example: the de Sitter model $a(t) = e^{Ht}$ is defined for all $t \in \R$ and $a$ never tends to $0$ at a finite time, so it has no Big Bang. A Big Bang is forced under mild conditions: if $\rho + 3p > 0$, $\Lambda \le 0$ and $H(t_0) > 0$, then the second Friedmann equation gives $\ddot a < 0$. So $a$ is concave and lies below its tangent line at $t_0$, which reaches zero at $t_0 - 1/H(t_0)$; hence the solution breaks down at some time $t_b \ge t_0 - 1/H(t_0)$.