k $k$ -essence in Horndeski models

Abstract

In this paper, we investigate a simple class of Horndeski models where the scalar field plays the role of a k-essence fluid. We present several solutions for early-time universe, namely inflation and cosmological bounce, by making use of some reconstruction technique. Moreover, we furnish the formalism to calculate perturbations in FRW space-time and we compute the spectral index and the tensor-to-scalar ratio during inflation.

Reconstruction of k-essence inflation in Horndeski gravity

“…where the prime denotes the derivative with respect to N . The first Friedmann-like equation is derived as [29],…”

“…and the spectral index, after the introduction of the function Q [22] is derived as, in terms of the e-folds number [29],…”

“…Generally speaking, the field equations of modified gravity appear at the fourth order, but in 1974 Horndeski found the most general class of scalar-tensor models which lead to second order differential equations like in General Relativity [17]. The Horndeski Lagrangian is quite involved and contains the coupling of a scalar field with several curvature invariants (the Ricci scalar, the Einstein tensor...), and in the last years has been investigated in many works in the context of the early-time inflation [18,19,20,21,22,23,24,25,26,27,28,29,30] (see also Refs. [31,32,33,34,35,36]).…”

Axionic black branes in the k -essence sector of the Horndeski model

“…In k-essence, the acceleration of the Universe (both at early and late times) can be driven by the kinetic energy instead of the potential energy of the scalar field [25]. The model was first introduced in [26] and then specifically used as dark energy models in [27][28][29][30][31][32]. These models are characterized by a nonlinear kinetic term for the scalar field and are expressed in (1) by the arbitrary function K(X) (together with with G 4 = 1 and G 3 = G 5 = 0).…”

k-Essence Non-Minimally Coupled with Gauss–Bonnet Invariant for Inflation

“…Moreover, k-essence is one of the possible descriptions for inflation (see also Refs. [43,44]). The advantage for dealing with a Horndeski model is that-despite the involved form of the Lagrangian-the equations of motion are at the second order, like in General Relativity.…”