Raychaudhuri Equation in K-essence Geometry: Conditional Singular and Non-Singular Cosmological Models

We investigate how the Raychaudhuri equation behaves in the k-essence geometry. As far as we are concerned, both the early and current epochs of the universe are relevant to the k-essence theory. Here, we have studied the k-essence geometry using the Dirac-Born-Infeld (DBI) variety of non-standard action. The corresponding k-essence emergent spacetime is not conformally equivalent to the usual gravitational metric. We assume that the background gravitational metric is of the Friedmann-Lemaitre-Robertson-Walker (FLRW) type in this case. We have found that both the conditional singular and non-singular cosmological models of the universe through the modified Raychaudhuri equation are possible where we have used the spacetime as the flat k-essence emergent FLRW-type. We have also addressed the Focusing theorem and conditional caustic universe construction. These conditional effects are caused by the additional interactions that arise as a result of the coupling that exists between gravity and the k-essence scalar field.