Phantom Menace in general Palatini $f(R,φ)$ theories

TL;DR

The paper investigates Palatini $f(R,\phi)$ gravity as a unified framework for early-time inflation and late-time cosmic acceleration. It maps the theory to the Einstein frame, derives a tractable dynamical-system formulation, and identifies stable late-time attractors governed by parameters $\rho$ and $\sigma$, with a Starobinsky-like curvature term playing a central role. Fixed points $P_1$ (de Sitter, stable for $\sigma<0$) and $P_3$ (phantom-like, stable for $\sigma>0$) illustrate viable late-time behavior, while other points act as transients or saddles. Confronting the model with DESI, Cosmic Chronometers, and SNeIa data constrains $\rho$ and $\sigma$, yielding a consistent evolution from a matter-dominated era to acceleration and a present $H_0$ around $71$–$72$ km/s/Mpc; the work provides a framework for dynamical dark energy with potential phantom regimes without ghosts and outlines avenues for further perturbative and model-parameter exploration.

Abstract

We study general $f(R,φ)$ theories in Palatini formalism and attempt to constrain the behavior of ones that could support both inflationary and late-time expansion era in a unified model. In particular, we find conditions for which the theories remain consistent in weak gravity regimes as well as cosmic expansion eras in both early and late universe. Assuming that the curvature part of the $f(R,φ)$ behaves as Starobinsky gravity, we assess post-inflation dynamical stability of the theory in Einstein frame and proceed to isolate two distinct fixed points that provide a stable late-time accelerating universe. Comparison with DESI, Cosmic Chronometers, and SNeIa datasets adds more stringent constraints to the behavior of the theory near the present epoch, giving us one stable fixed point where expansion is driven by a phantom scalar field. However, time scales of the two fixed points suggest that this fixed point may be transient and may eventually evolve toward a stable expansion stage driven potential domination in the distant future of the universe.

Figures (9)

• Figure 1: Point $P_4$ is stable in the given parameter space of $(\sigma,\rho)$. The dashed lines represent forbidden values of $\rho$ where the eigenvalues can be seen to diverge in Table \ref{['tab:xyzlambda_eigen']}.
• Figure 2: Evolution of $\Omega_\phi$, $\Omega_m$, $\omega_{\rm eff}$, and the dynamical variables $x$, $y$, $z$, and $\lambda$ for the initial conditions $\Omega_{\phi}(0)=0.68$, $\omega_{\phi}(0)=-0.99$, $z(0)=10^{-4}$, and $\lambda(0)=0.01$, with model parameters $\sigma=-1$ and $\rho=-1$. As the e-folding number $N$ increases, the scalar-field energy density $\Omega_\phi$ steadily grows and asymptotically approaches unity, while the matter component $\Omega_m$ diminishes to zero. Consequently, the effective equation of state $\omega_{\rm eff}$ evolves toward the cosmological-constant value $\omega_{\rm eff}\simeq -1$, signaling late-time accelerated expansion. The plots of the dynamical variables show a consistent evolution: $x$ decreases and stabilizes near $-2$, $y$ increases and settles around $3$, the potential-related variable $z$ approaches zero, and $\lambda$ remains extremely small throughout the evolution. For $\sigma=-1$, Tables \ref{['tab:xyzlambda_points']} and \ref{['tab:xyzlambda_eigen']} indicate that both critical points $P_{1}$ and $P_{7}$ can correspond to late-time acceleration. However, the trajectories of the dynamical variables clearly converge to the critical point $P_{1}$, confirming it as the late-time attractor of the model.
• Figure 3: Evolution of $\Omega_\phi$, $\Omega_m$, $\omega_{\rm eff}$, $x$, $y$, $z$, and $\lambda$ with the number of e-folds $N$, for the initial conditions $\Omega_{\phi}(0)=0.68$, $\omega_{\phi}(0)=-0.99$, $z(0)=10^{-4}$, $\lambda(0)=0.01$, and the parameters $\sigma = 0.01$, and $\rho = -0.1$. From panel (a), the scalar-field energy density parameter $\Omega_\phi$ approaches unity, while the matter density parameter $\Omega_m$ decays to zero, and the effective equation of state parameter converges to $\omega_{\rm eff}=-1.00333$, signaling a late-time phantom-like accelerated expansion. Panel (b) indicates that the dynamical variables $x$ and $y$ asymptotically decay towards vanishing values at late times. Panel (c) shows that the variable $\lambda$ decreases and tends to zero, whereas panel (d) illustrates that the variable $z$ grows and approaches to $z\simeq 1$ as $N$ increases. By comparing this asymptotic behavior with the critical points summarized in Table \ref{['tab:xyzlambda_points']} and Table \ref{['tab:xyzlambda_eigen']}, the system is seen to approach the fixed point $P_3$. For $\sigma>0$, this point is stable and therefore represents the late time dark energy dominated attractor of the system.
• Figure 4: Panels (a) and (b) respectively show the evolution of the field energy density and the effective equation of state. Panels (c) and (d) display the evolution of kinetic and hyper-kinetic terms of the field. The vertical dashed line denotes the present epoch. These plots are generated in accordance with the constraints listed in Table \ref{['tab:constraints']}.
• Figure 5: Corner plots of 1D and 2D marginalized posterior distributions of the model parameters in the presence of dataset combinations based on BAO from DESI DR2, Cosmic chronometers, and supernovae datasets. Contours at 68% ($1-\boldsymbol{\sigma}$) and 95% ($2-\boldsymbol{\sigma}$) levels showing parameter constraints and correlations within the framework.