Cosmological behavior in extended nonlinear massive gravity

TL;DR

The paper analyzes cosmology in extended (varying-mass) nonlinear massive gravity by recasting the background equations as autonomous dynamical systems for flat and open FRW geometries. It compares two input schemes—imposing the varying graviton mass $V(\psi)$ at will and imposing the Stückelberg function $b(t)$ at will—under exponential potentials $W(\psi)$ and $V(\psi)$, and identifies a wide array of late-time attractors, including quintessence, phantom, and cosmological-constant–like states, with the graviton mass often tending to zero at late times. In flat geometry, a constraint on $V(\psi)$ leads to pathologies, whereas open geometry admits a richer, more viable set of attractors, and can alleviate the coincidence problem by allowing $0\le\Omega_{DE}\le1$ at late times. The results demonstrate the extended theory’s potential to describe late-time acceleration and phantom-divide crossing using a canonical scalar, while highlighting the need for a full perturbative stability analysis to ensure ghost-free behavior. Overall, the extended varying-mass framework offers a broader and potentially more realistic cosmological repertoire than standard quintessence or constant-mass massive gravity, albeit with caveats tied to the chosen input functions.

Abstract

We perform a detailed dynamical analysis of various cosmological scenarios in extended (varying-mass) nonlinear massive gravity. Due to the enhanced freedom in choosing the involved free functions, this cosmological paradigm allows for a huge variety of solutions that can attract the universe at late times, comparing to scalar-field cosmology or usual nonlinear massive gravity. Amongst others, it accepts quintessence, phantom, or cosmological-constant-like late-time solutions, which moreover can alleviate the coincidence problem. These features seem to be general and non-sensitive to the imposed ansantzes and model parameters, and thus extended nonlinear massive gravity can be a good candidate for the description of nature.

Figures (6)

• Figure 1: Trajectories in the $Y$-$Z$ plane of the cosmological scenario \ref{['ODEs1']}, where the varying graviton mass square $V(\psi)$ is imposed at will, in a flat universe. We use $\gamma=1, \lambda_V=2,\lambda_W=-1, \alpha_3=\alpha_4=0.1$. The physical part of the phase space is marked by the shadowed region limited by the red lines. In this specific example the universe is led to the phantom stable late-time solution $P_3$.
• Figure 2: Trajectories in the $X$-$Y$ plane of the cosmological scenario \ref{['ODEs2']}, where the varying graviton mass square $V(\psi)$ is imposed at will, in an open universe. We focus on the invariant set $\Omega_k=U=Z=0$ and we choose $\gamma=1, \lambda_V=-2,\lambda_W=1, \alpha_3=\alpha_4=0.1$. In this specific example the stable late-time state of the universe is the phantom solution $Q_1$.
• Figure 3: Trajectories in the $Y$-$Z$ plane of the cosmological scenario \ref{['ODEs2']}, where the varying graviton mass square $V(\psi)$ is imposed at will, in an open universe. We focus on the invariant set $\Omega_k=X=Z=0$ and we choose $\gamma=1, \lambda_V=2,\lambda_W=1, \alpha_3=\alpha_4=0.1$ and $U_c=\frac{\lambda_V}{\sqrt{6}}$. In this specific example the stable late-time state of the universe is the quintessence-like point $Q_{14}$.
• Figure 4: Trajectories of the cosmological scenario \ref{['ODEs2']}, where the varying graviton mass square $V(\psi)$ is imposed at will, in an open universe, in the subset $X=Z=0$, which is invariant provided $3+3\alpha_3+\alpha_4=0,\alpha_3\neq -2,\alpha_4\neq 3$. We use the parameters values $\gamma=1,\lambda_W=3$. In this specific example the stable late-time solutions of the universe are the expanding, non-accelerating $Q_6^+$ (its basin of attraction is the half-subspace $\Omega_k>0$), and the contracting $Q_6^-$ (its basin of attraction is the half-subspace $\Omega_k<0$). Additionally, we can see the saddle points $Q_{15}$ (non-accelerating with $0<\Omega_{DE}<1$), $Q_{16}$ (non-accelerating, matter-dominated), $Q_{17}$ (curvature-dominated, contracting) and $Q_{18}$ (non-accelerating, curvature-dominated, expanding), as well as the unstable points $Q_{12}$ and $Q_{13}$ (non-accelerating, dark-energy dominated, with stiff $w_{DE}$).
• Figure 5: Trajectories of the cosmological scenario (\ref{['autonomous1']})-(\ref{['autonomous4']}), where the Stückelberg field function function $b(t)$ is imposed at will, in a flat universe, using $\gamma=1, \lambda_W=1, \alpha_3=\alpha_4=0.5, B=1.7$. In this specific example the stable late-time state of the universe is the expanding, dark-energy dominated, quintessence-like point $R_4^+$. Additionally, we depict the saddle points $R_1$ (non-accelerating, matter-dominated), and $R_2$,$R_3$ (non-accelerating, dark-energy dominated).