In general description of the Raychaudhuri equation it is found that this first order non-linear differential equation can be written as a second order linear differential equation in the form of Harmonic Oscillator with varying frequency. Further, the integrability of the Raychaudhuri equation has been studied and also the expansion scalar is obtained in an explicit form. Subsequently, $f(R)$ gravity theory has been studied in the background of inhomogeneous FLRW spacetime with an aim to formulate the Raychaudhuri Equation. A congruence of time-like geodesics has been investigated using the Raychaudhuri Equation to examine whether the geodesics converge or not and some possible conditions are determined to avoid singularity. Finally, a brief quantum description has been presented.

The present work deals with the classical and quantum aspects of the Raychaudhuri equation in the framework of f(T)-gravity theory. In the background of homogeneous and isotropic Friedmann–Lemaître–Robertson-Walker space-time, the Raychaudhuri equation has been formulated and used to examine the focusing theorem and convergence condition for different choices of f(T). Finally in quantum cosmology, the wave function of the universe has been shown to be the energy eigen function of the time-independent Schrödinger equation of a particle. Also probability measure on the mini-superspace has been examined at zero volume for singularity analysis in the quantum regime. Lastly, the Bohmian trajectory for the present quantum system has been explicitly determined for some particular choices.

No abstract