Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models

TL;DR

This work derives general ghost- and Laplacian-instability conditions for the most general Horndeski scalar-tensor theories on a flat FLRW background with two perfect fluids and applies these criteria to extended Galileon dark energy models. A tracker solution with $H\dot{\phi}^{2q}=$ constant is generalized, enabling $w_{DE}$ to approach -1 during matter domination, and the de Sitter fixed point is shown to be stable under a set of positivity constraints on perturbation parameters. The analysis yields a theoretically viable parameter space where scalar and tensor perturbations remain ghost- and gradient-stable, and numerical simulations confirm viable cosmological evolution from radiation to de Sitter epochs. The results offer a framework to construct and test dark energy models within Horndeski theories that are compatible with current observations while avoiding fundamental instabilities. The work also highlights the importance of the transition-era tensor speed, $c_T^2$, in constraining model parameters. Overall, the paper provides both analytic and numerical tools to assess the cosmological viability of broad classes of scalar-tensor theories for dark energy.

Abstract

In the Horndeski's most general scalar-tensor theories with second-order field equations, we derive the conditions for the avoidance of ghosts and Laplacian instabilities associated with scalar, tensor, and vector perturbations in the presence of two perfect fluids on the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background. Our general results are useful for the construction of theoretically consistent models of dark energy. We apply our formulas to extended Galileon models in which a tracker solution with an equation of state smaller than -1 is present. We clarify the allowed parameter space in which the ghosts and Laplacian instabilities are absent and we numerically confirm that such models are indeed cosmologically viable.

Figures (6)

• Figure 1: Allowed parameter space in the ($\alpha,\beta$) plane (the area in black) for $p=1$ and $q=5/2$. We have used all the available conditions considered so far, i.e. (\ref{['eq:allANDoth']}) and (\ref{['eq:r2mincond']}).
• Figure 2: Evolution of the density parameters $\Omega_{{\rm DE}}$, $\Omega_{m}$, $\Omega_{r}$, $w_{{\rm eff}}$, and $w_{{\rm DE}}$ versus the redshift $z$ for $p=1$, $q=5/2$, $\alpha=1$, $\beta=0.45$, and $x_{{\rm dS}}=1$. The initial conditions are chosen to be $r_{1}=1$, $r_{2}=10^{-30}$, and $\Omega_{r}=0.9998$ at $z=1.76\times10^{7}$. In this case the solution is on the tracker from the beginning.
• Figure 3: Variation of $w_{{\rm DE}}$ versus $z$ for $p=1$, $q=5/2$, $\alpha=1$, $\beta=0.45$, and $x_{{\rm dS}}=1$ with several different initial conditions. The solid line corresponds to the tracker with the initial conditions same as those given in Fig. \ref{['omega']}. The initial conditions for the cases (a)-(c) are (a) $r_{1}=4.0\times10^{-2}$, $r_{2}=5.0\times10^{-26}$, $\Omega_{r}=0.9998$ at $z=1.82\times10^{7}$, (b) $r_{1}=1.0\times10^{-5}$, $r_{2}=1.0\times10^{-13}$, $\Omega_{r}=0.9998$ at $z=1.76\times10^{7}$, and (c) $r_{1}=1.0\times10^{-7}$, $r_{2}=1.0\times10^{-9}$, $\Omega_{r}=0.99995$ at $z=6.64\times10^{7}$, respectively.
• Figure 4: Variation of $Q_{S}/M_{{\rm pl}}^{2}$ versus $z$ for the same model parameters and initial conditions as those given in Fig. \ref{['wde']}. The solid line represents the tracker solution, whereas the cases (a), (b), and (c) correspond to the evolution for the initial conditions as those given in Fig. \ref{['wde']}.
• Figure 5: Evolution of $c_{S}^{2}$ versus $z$ for the same model parameters and initial conditions as those given in Fig. \ref{['wde']}. The solid line represents the tracker solution, whereas the cases (a), (b), and (c) correspond to the evolution for the initial conditions as those given in Fig. \ref{['wde']}.