Borde–Guth–Vilenkin theorem

Remarks

The key estimate shows that $\int_{\lambda_i}^{\lambda_f} H\, d\lambda$ is bounded above by a quantity that depends only on the endpoint $\lambda_f$, so $(\lambda_f - \lambda_i) H_{\min}$ is bounded and the past affine length is finite. In a flat Friedmann–Lemaître–Robertson–Walker model with scale factor $a$, a null geodesic has affine parameter with $d\lambda = a(t)\, dt / a(t_f)$, so $\int H\, d\lambda = \int \dot a\, dt / a(t_f) = (a(t_f) - a(t_i))/a(t_f) \le 1$, and the past affine length is at most $1/H_{\min}$. The theorem uses no energy condition and does not use the Einstein equations, only the averaged expansion condition. Example: the flat slicing of de Sitter space, $a = e^{Ht}$, satisfies the hypothesis, and indeed it is past incomplete. The full de Sitter space extends it, and in the extension the averaged expansion condition fails. Arvind Borde, Alan Guth and Alexander Vilenkin published it in 2003.

Going deeper

For a student

Classical singularity theorems of Penrose and Hawking derive geodesic incompleteness from energy conditions, which inflationary matter can violate. The Borde–Guth–Vilenkin theorem avoids energy conditions entirely and replaces them by a purely kinematic hypothesis on the average expansion. Its conclusion is incompleteness, not a curvature singularity: it says the classical description must break down or be extended into a region where the hypothesis fails. It is often discussed together with the Big Bang and with cosmic inflation.