Symmetry-Driven $k$-Essence Cosmological Dynamics

Abstract

We investigate a family of $k$-essence cosmological models selected by the requirement that the field equations admit a nontrivial variational symmetry. For the $k$-essence theory with Lagrangian function $F(R,X,ϕ)=f_{1}R+f_{2}(ϕ,X)$, where $f_{2}(ϕ,X)=αX+β\exp\left((n-1)ϕ\right)X^{n}$, the existence of a scaling symmetry introduces a conservation law which we use to obtain an exact vacuum solution. We generalize our analysis and investigate the structure of the phase-space of the cosmological field equations by performing a dynamical systems analysis, both in the absence and the presence of a pressureless matter source. We employ Hubble-normalized variables and the Poincaré compactification, so as to determine the asymptotic behavior of the model and identify all admissible cosmological epochs, from early-time to late-time behavior. We discuss in detail the existence and stability conditions of each fixed point as functions of the free parameters, and characterize the full asymptotic structure of the cosmological history as provided by this symmetry-driven $k$-essence model. We explore the conditions under which the model can describe cosmic acceleration. We find that the symmetry driven $k$-essence model can describe the inflationary epoch with a natural exit to the matter dominated era, where the $k$-essence mimics the pressureless matter component.