Non-linear bigravity and cosmic acceleration

TL;DR

This paper examines nonlinear bigravity theories with two coupled metrics, recasting their cosmological evolution as a mechanical system of two relativistic particles linked by a nonlinear spring. By analyzing Pauli-Fierz-type and brane-motivated potentials, it shows that a broad class of models naturally produce periods of cosmic acceleration and can realize a bi-de-Sitter locked state under suitable confinement of the relative lapse $\,\gamma$, yielding a tensor-quintessence-like dark energy. The work highlights two distinct dynamical regimes: spacelike worldline separations generally lead to anisotropic inflation followed by oscillations, while timelike separations can lock into isotropic de Sitter-like expansion; however, long-term stability crucially depends on the confining properties of $V(\,\gamma)$, with simple PF or brane potentials often requiring modifications. Overall, the study identifies a rich array of cosmological solutions in bigravity, emphasizes potential observational signatures (anisotropic expansion, transitions to vacuum domination), and suggests how tensor degrees of freedom in gravity could account for dark energy and possibly primordial inflation.

Abstract

We explore the cosmological solutions of classes of non-linear bigravity theories. These theories are defined by effective four-dimensional Lagrangians describing the coupled dynamics of two metric tensors, and containing, in the linearized limit, both a massless graviton and an ultralight one. We focus on two paradigmatic cases: the case where the coupling between the two metrics is given by a Pauli-Fierz-type mass potential, and the case where this coupling derives from five-dimensional brane constructions. We find that cosmological evolutions in bigravity theories can be described in terms of the dynamics of two ``relativistic particles'', moving in a curved Lorenzian space, and connected by some type of nonlinear ``spring''. Classes of bigravity cosmological evolutions exhibit a ``locking'' mechanism under which the two metrics ultimately stabilize in a bi-de-Sitter configuration, with relative (constant) expansion rates. In the absence of matter, we find that a generic feature of bigravity cosmologies is to exhibit a period of cosmic acceleration. This leads us to propose bigravity as a source of a new type of dark energy (``tensor quintessence''), exhibiting specific anisotropic features. Bigravity could also have been the source of primordial inflation.

Figures (4)

• Figure 1: General motion of the two "particle" worldlines $\alpha^{\mu}$ and $\beta^{\mu}$ in the internal Lorentzian field space. The middle worldline is the "center of mass" of the system $\sigma^{\mu}/2$. The decomposition of the "center of mass" velocity $\dot{\sigma}^{\mu}/2$ and of the worldline separation $\delta^{\mu}$ in timelike and spacelike components is shown. These projections are performed with respect to the special timelike vector $n^{\mu}$.
• Figure 2: Motion of the two "particle" worldlines $\alpha^{\mu}$ and $\beta^{\mu}$ for the exact spacelike worldline separation limit. The middle worldline is the "center of mass" of the system $\sigma^{\mu}/2$ and is directed along the special vector $n^{\mu}$. The motion of the two "particles" is symmetric with respect to the motion of their "center of mass".
• Figure 3: Numerical simulation of the motion of the two "particle" worldlines $\alpha^{\mu}$ and $\beta^{\mu}$ for the exact spacelike worldline separation limit and for the Pauli-Fierz potential. The horizontal axis is the distance $r/2$ from the "center of mass", while the vertical one is $\alpha$ and $\beta$ respectively for the two "particles". Initially the worldvolume of both metrics is inflating until the worldline separation becomes ${\mathcal{O}}(1)$. Then the two "particles" start oscillating with respect of their "center of mass". The oscillatory regime cannot be reached for the Pauli-Fierz potential if the timelike separation component $\delta$ is excited.
• Figure 4: Motion of the two "particle" worldlines $\alpha^{\mu}$ and $\beta^{\mu}$ for the exact timelike worldline separation limit. The two worldlines are collinear and the evolution of the two metrics isotropic.