Dynamical system analysis of the cosmological phases in Palatini $k$-essence gravity

TL;DR

This work develops a generalized Palatini $f(\mathcal{R})$+$k$-essence framework, recasting it as a biscalar-tensor theory with an auxiliary scalaron $\phi$ and a coupled kinetic sector $U(\phi,\xi,X)$. It derives stability and hyperbolicity conditions, linking to DHOST classifications in vacuum, and specializes to a tractable choice $U(\phi,\xi,X)=\lambda^2\phi X+W(\phi,\xi)$ to study cosmology. Through a flat FLRW dynamical-systems analysis, the authors identify fixed points corresponding to de Sitter, scaling, and quintessence-like phases, including heteroclinic trajectories that connect early- and late-time acceleration eras. The two-fluids and single-fluid cases reveal a robust richness of cosmological histories, with inflationary and late-time de Sitter attractors, together with matter-/radiation-dominated phases that can be realized as transient dynamics. Overall, the Palatini formulation of $k$-essence gravity provides a versatile mechanism to reproduce diverse cosmological epochs and to explore connections between early- and late-time acceleration within a well-posed, ghost-free framework.

Abstract

We formulate a generalized $k$-essence model in the presence of a Palatini $f(\mathcal{R})$ gravitational sector. In the corresponding biscalar-tensor theory, we discuss the distinguished dynamical properties of the two scalar fields, elucidating how the Palatini scalaron can be still algebraically solved in terms of matter, the $k$-essence field and its kinetic term. We derive the conditions ensuring the absence of Ostrogradsky modes and the well-posedness of the initial data problem, also providing an intriguing analogy with a specific class of DHOST theories. Then, we investigate the cosmology of a flat Friedmann-Lemaître-Robertson-Walker spacetime according a dynamical system approach, with the aim of determining the set of fixed points in the phase space, representing specific periods of the Universe evolution and characterized by different effective barotropic index $w_{\text{eff}}$. The analysis reveals the presence of a range of possible configurations, with the existence of (quasi) de-Sitter epochs connected by heteroclinic orbits, scaling solutions and quintessence phases.

Figures (3)

• Figure 6: Local dynamics in the proximity of the point $P_1$ (red dot) for $k=1$ (left) and $k=-2$ (right). The remaining parameters are fixed as $p=1$, $\phi_0=1$, $\lambda=1$, $W_0=1$. The colors of the arrows indicate different magnitudes of the derivatives in each point, growing from blue to red.
• Figure 7: Transition of the point $P_4$ from saddle (left) to repeller (right). The change of character is observed by varying the parameter $s$ from the value $s=1$ (left) to $s=2$ (right). The remaining parameters are set as $a=1$, $\lambda=1$, $W_0=1$, $w=0$.
• Figure 8: Real part of the Jacobian eigenvalues calculated for the points $P_{2,3}$ (left) and $P_{4,5}$ (right).