The motto of this work is to generate a general formalism of $f(\bar{R}, L(X))-$gravity in the context of dark energy under the framework of the {\bf K-}essence emergent geometry with the Dirac-Born-Infeld (DBI) variety of action, where $\bar{R}$ is the familiar Ricci scalar, $L(X)$ is the DBI type non-canonical Lagrangian with $X={1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ and $\phi$ is the {\bf K-}essence scalar field. The emergent gravity metric $\G_{\mu\nu}$ and the well known gravitational metric $g_{\mu\nu}$ are not conformally equivalent. We have constructed a modified field equation using the metric formalism in $f(\bar{R}, L(X))$-gravity incorporating the corresponding Friedmann equations in the framework of the background gravitational metric which is of Friedmann-Lema{\^i}tre-Robertson-Walker (FLRW) type. The solution of modified Friedmann equations have been deduced for the specific choice of $f(\bar{R}, L(X))$, which is of Starobinsky-type, using power law expansion method. The consistency of the model with the accelerating phase of the Universe has been shown, when we restrict ourselves to consider the value of the dark energy density, as $\dot\phi^{2}=\frac{8}{9}=0.888 <1$, which indicates that the present Universe is dark energy dominated. Graphical plots for the energy density ($\rho$), pressure ($p$) and equation of state parameter ($\o$) w.r.t. time ($t$) based on parametric values are interestingly consistent with the dark energy domination and hence accelerating features. We also put some light on the corresponding energy conditions and constraints of the $f(\bar{R}, L(X))$ theory with one basic example.

In this paper, we study the collapsing scenario for the {\bf k-}essence emergent Vaidya spacetime in the context of massive gravity's rainbow. For this study, we consider that the background metric is Vaidya spacetime in massive gravity's rainbow. We show that the {\bf k-}essence emergent gravity metric resembles closely to the new type of generalized Vaidya massive gravity metric with the rainbow deformations for null fluid collapse where we consider the {\bf k-}essence scalar field as a function solely of the advanced or the retarded time. The {\bf k-}essence emergent Vaidya massive gravity rainbow mass function is also different. This new type {\bf k-}essence emergent Vaidya massive gravity rainbow metric has satisfied the required energy conditions. The existence of the locally naked central singularity, the strength and the strongness of the singularities for the rainbow deformations of the {\bf k-}essence emergent Vaidya massive gravity metric are the interesting outcomes of the present work.

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.