Cosmological perturbations of self-accelerating universe in nonlinear massive gravity

TL;DR

Perturbations of self-accelerating FRW solutions in a ghost-free nonlinear massive gravity are analyzed with a gauge-invariant formalism. By integrating out the nondynamical scalar and vector Stückelberg sectors, the gravity sector behaves like GR for the scalar and vector perturbations, while tensor perturbations acquire a time-dependent mass $M_{GW}^2(t)$ that modifies gravitational-wave dispersion. The tensor sector is the only dynamical gravity-d.o.f., with the mass term entering through the fiducial metric; background evolution is governed by an effective cosmological constant $\,\\Lambda_{\\pm}$. These results imply potential observational signatures in the stochastic gravitational-wave spectrum and underscore the need for nonlinear or UV-complete analyses to address possible strong coupling on the nontrivial cosmological branches.

Abstract

We study cosmological perturbations of self-accelerating universe solutions in the recently proposed nonlinear theory of massive gravity, with general matter content. While the broken diffeomorphism invariance implies that there generically are 2 tensor, 2 vector and 2 scalar degrees of freedom in the gravity sector, we find that the scalar and vector degrees have vanishing kinetic terms and nonzero mass terms. Depending on their nonlinear behavior, this indicates either nondynamical nature of these degrees or strong couplings. Assuming the former, we integrate out the 2 vector and 2 scalar degrees of freedom. We then find that in the scalar and vector sectors, gauge-invariant variables constructed from metric and matter perturbations have exactly the same quadratic action as in general relativity. The difference from general relativity arises only in the tensor sector, where the graviton mass modifies the dispersion relation of gravitational waves, with a time-dependent effective mass. This may lead to modification of stochastic gravitational wave spectrum.

Figures (1)

• Figure 1: Sign of the effective cosmological constant $\Lambda_\pm$ in the positive (left panel) and negative (right panel) branches. In the red (green) region with $+45^\circ$ ($-45^\circ$) lines, $\Lambda_{\pm}$ is positive (negative). The white region and the dotted squared region correspond to $1+\alpha_3+\alpha_3^2-\alpha_4<0$ and $X_{\pm}<0$, respectively, and are excluded since the cosmological solutions (\ref{['eq:fsolcosmo']}) do not exist there. Along the dotted black line (defining the boundary between the red and green regions), $\Lambda_\pm =0$ and the background solution reduces to the GR one. The solid line corresponds to $X_{\pm}=0$ and thus defines one of the boundaries between the allowed (red or green) and excluded (dotted squared) regions. Along the dashed line, both $X_{\pm}$ and $\Lambda_{\pm}$ diverge, and it defines another boundary between the allowed (red or green) and excluded (dotted squared) regions.