Growth Histories in Bimetric Massive Gravity

TL;DR

We formulate linear cosmological perturbations in Hassan–Rosen bimetric gravity for general FLRW backgrounds and show that in a pure de Sitter limit the scalar sector splits into massless and massive modes, with matter perturbations matching GR. Introducing quasi-de Sitter evolution by including matter content induces mixing between these sectors, yielding potential deviations in the growth of structure from Einstein gravity. The authors derive both analytical (Bessel/Lommel functions) and numerical solutions for the massive sector in the quasi-dS regime, demonstrating regimes where $\Lambda$CDM-like growth can be approached and highlighting observable implications for growth history. Overall, the paper clarifies how the additional spin-2 field alters late-time cosmology and provides a framework for confronting bimetric predictions with large-scale structure data.

Abstract

We perform cosmological perturbation theory in Hassan-Rosen bimetric gravity for general homogeneous and isotropic backgrounds. In the de Sitter approximation, we obtain decoupled sets of massless and massive scalar gravitational fluctuations. Matter perturbations then evolve like in Einstein gravity. We perturb the future de Sitter regime by the ratio of matter to dark energy, producing quasi-de Sitter space. In this more general setting the massive and massless fluctuations mix. We argue that in the quasi-de Sitter regime, the growth of structure in bimetric gravity differs from that of Einstein gravity.

Figures (11)

• Figure 1: The homogeneous massive perturbation $\Xi(t)=c_J J_{\nu}(x)/\sqrt{x}$, with $x$ from (\ref{['xdef']}) and $c_J$ a normalization constant, for various $\nu$. We note that for $\nu <1/2$ or $\nu$ imaginary, this homogeneous mode diverges for late time. However, for real but small $\nu$ (the second plot) it stays within reasonable values for a several times the age of the universe. We will only consider $\nu>1/2$ (as in the first plot) in this paper.
• Figure 2: Summary of some interesting values of the mass parameter $M$ and the corresponding $\nu$ values from (\ref{['nueq']}). We indicate when only one homogeneous mode goes to infinity at late time, or when both modes do, and when $\nu$ becomes imaginary (which is not a problem in itself, but see fig \ref{['fig:bessel']}) and when $Y$ and $M_P^2$ become negative (which are big problems for the background in the current model).
• Figure 3: Background solutions for the Hubble functions $H=\dot{a}/a$ and $K=\dot{Y}/Y$, with the latter for $c=6$ and $c=1/6$, cf. (\ref{['constprop']}). (Here the $c=1/6$ curves are included for illustration only, we never use values for $c$ this low). Linear (in $\epsilon$) qdS in black, quadratic qdS in dashed red, exact solution in dotted blue, and $H_0t \sim 1$ is roughly present. We see that for times $H_0 t \gtrsim 0.7$, qdS remains a good approximation for the $g$ background, and also for the $f$ background for $c=6$.
• Figure 4: Numerical qdS plots for the gravitational potential $\Psi_+$ for $k=(5/2) H_{\rm dS}$ (left panel) and $k=10 H_{\rm dS}$ (right panel). and $M^2/H_{\rm dS}^2 = \{ 1/50, 4/5, 5/4, 3/2 \}$. The shaded area is our estimate for when the qdS approximation breaks down.
• Figure 5: Numerical qdS plots of the density contrast $\delta$ for various wavenumbers $k$. Left panel: small $k$ values, $k/H_{\rm dS}= \{ 5/2, 3, 7/2, 4\}$. Right panel: intermediate $k$ values, $k/H_{\rm dS}= \{ 19/2, 10, 21/2, 11\}$. Each curve is plotted for two $M$ values, $M^2/H_{\rm dS}^2 = \{ 1/50, 3/2 \}$, but the curves for the two $M$ values are nearly coincident in some cases. The shaded area is our estimate for when the qdS approximation breaks down.