From Geometry to Observation: Gravitational Waves and the Raychaudhuri Equation

Gravitational waves (GWs) are independent of any particular theory of gravity. The universality of this notion is highlighted by the Raychaudhuri equation (RE), which is independent of any theory of gravity and contains the Ricci tensor $R_{\mu\nu}$ as a key ingredient, thereby connecting spacetime geometry with matter-energy content. Under small metric perturbations, $R_{\mu\nu} \propto \Box h_{\mu\nu}$, where $h_{\mu\nu}$ is the perturbation, indicating that various gravity theories, via their corresponding $R_{\mu\nu}$, produce different gravitational wave equations. In the framework of Einstein's gravity, this leads to the standard wave equation. This study analyzes a modified form, {\it GW-inspired RE}, within the homogeneous and isotropic FLRW background to investigate late-time cosmic acceleration and structure formation. We employ {\it Pantheon+ SNe Ia, Hubble, and BAO} datasets to constrain model parameters through Bayesian inference utilizing NUTS in {\it NumPyro}. A nuisance parameter $\mu_0$ is introduced to address residual systematics. This facilitates a robust estimation of $H_0$, $\Omega_{DE,0}$, and $r_d$, which addresses the resolution of the Hubble tension. We analyze the redshift evolution of the deceleration parameter, $q(z)$, both with and without $\mu_0$, emphasizing its influence on cosmic dynamics. The GW-inspired RE is reformulated as a harmonic oscillator, providing insight into expansion and geodesic focusing. A graphical comparison demonstrates the relationship $d^{GW}_L(z) = d^{EM}_L(z)$ utilizing GWOSC data. Thus, the RE in the context of small perturbation of the metric opens up whole new vistas of {\it observational astronomy.}