Reconstruction of $f(Q)$ Gravity from Geodesic Congruence Dynamics: Raychaudhuri Equation

Abstract

In this study, we use the Raychaudhuri equation as the essential geometrical tool to reconstruct the functional form of $f(Q)$ gravity within the context of symmetric teleparallel gravity. The expansion scalar and the congruence dynamics recorded in the Raychaudhuri equation are used to relate the non-metricity scalar $Q$ with the kinematical evolution of the Universe in order to establish the reconstruction method. We obtain explicit forms of $f(Q)$ that can explain the observed accelerated expansion of the Universe without the need for an additional dark energy component by taking into account appropriate cosmological assumptions. Moreover, the reconstructed model can also be extended to the infinite past to model the early universe for certain scenarios. In an FLRW spacetime we have considered a constant deceleration parameter and a constant jerk parameter to facilitate our reconstruction analysis. We further discuss the, general relativity limit, Friedmann consistency and background physical viability of the reconstructed models. The analysis indicates that Raychaudhuri equation, interpreted as an equation governing the focusing/defocusing of cosmological geodesic congruences, can be used as the central equation in an inverse reconstruction of $f(Q)$ gravity.