Focusing of geodesic congruences in an accelerated expanding Universe

TL;DR

This paper investigates how accelerated cosmic expansion affects geodesic focusing using the Raychaudhuri equation, revealing conditions under which geodesics may still converge despite acceleration.

Contribution

It demonstrates that geodesic focusing can occur in an accelerated universe and clarifies the role of space-time geometry contributions in this process.

Findings

Space-time contribution to Raychaudhuri equation is positive in accelerated expansion.

Focusing of geodesics remains possible despite cosmic acceleration.

High-speed geodesics can exhibit attractive gravity effects in an accelerated universe.

Abstract

We study the accelerated expansion of the Universe through its consequences on a congruence of geodesics. We make use of the Raychaudhuri equation which describes the evolution of the expansion rate for a congruence of timelike or null geodesics. In particular, we focus on the space-time geometry contribution to this equation. By straightforward calculation from the metric of a Robertson-Walker cosmological model, it follows that in an accelerated expanding Universe the space-time contribution to the Raychaudhuri equation is positive for the fundamental congruence, favoring a non-focusing of the congruence of geodesics. However, the accelerated expansion of the present Universe does not imply a tendency of the fundamental congruence to diverge. It is shown that this is in fact the case for certain congruences of timelike geodesics without vorticity. Therefore, the focusing of geodesics remains feasible in an accelerated expanding Universe. Furthermore, a negative contribution to the Raychaudhuri equation from space-time geometry which is usually interpreted as the manifestation of the attractive character of gravity is restored in an accelerated expanding Robertson-Walker space-time at high speeds.