Focusing of geodesic congruences in an accelerated expanding Universe

© 2012 IOP Publishing Ltd and Sissa Medialab srl. We would like to thank prof. G. F. R. Ellis for drawing our attention to reference [10]. This work has been supported by MICINN (Spain) projects numbers FIS2011-23000, FPA2011-27853-C02-01 and Consolider-Ingenio MULTIDARK CSD2009-00064. FDA would like to thank Theoretical Physics I department, Complutense University of Madrid for providing all the required academic resources. AdlCD acknowledges financial support from NRF and URC research fellowships (South Africa). JARC acknowledges the kind hospitality of ACGC/University of Cape Town while elaborating the manuscript.
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.
Ministerio de Ciencia e Innovación (MICINN), España
Programa Consolider-Ingenio (Proyecto MULTIDARK)
National Research Foundation (NRF), Sudáfrica
URC, Sudáfrica
Depto. de Física Teórica
Fac. de Ciencias Físicas
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