Instability of certain bimetric and massive-gravity theories

Stability about cosmological background solutions to the bimetric Hassan-Rosen theory is studied. The results of this analysis are presented, and it is shown that a large class of cosmological backgrounds is classically unstable. This sets serious doubts on the physical viability of the Hassan-Rosen theory—and in turn also of the de Rham-Gadabaze-Tolley model. A way to overcome this instability by means of curvature-type deformations is discussed.

Figure 1

Absolute values of the functions $\alpha (a)$ (black, solid curves) and $\beta (a)$ (red, dashed curves) as functions of the scale factor $a$ (double-logarithmic scale). The parameters are ${\beta }_2=-0.3$, $M=3$, $M_{\star }=2.5$. Note that the two lower curves are twofold degenerate.

Figure 2

Real parts of the eigenvalues of the matrix $\mathbb{A}$ [cf. Eq. (12)] as functions of the scale factor $a$ (log-axis). The two large panels represent the regular solutions to Eq. (8), and the small graph in the lower panel shows the one for which $\alpha (a)$ is nonanalytic. Dashed lines indicate the eigenvalues of the undeformed theory. The parameters are ${\beta }_2=-0.3$, $M=3$, $M_{\star }=2.5$.

Figure 3

Absolute values of the relative metric fluctuations $y_{23}^{(1)}≔\delta f_{23}/f_{22}^{(0)}$ (red curve) and $y_{23}^{(3)}≔\delta g_{23}/g_{22}^{(0)}$ (blue curve) as functions of the scale factor $a$. Dashed, colored lines correspond to the undeformed theory, solid lines to the deformed one. The dot-dashed, vertical line is at $a=a_{\star }\approx 0.3$ [for the parameter set (14)].

Figure 4

Stability parameter plot in the $\beta \mathrm{\text{−}}\gamma$ plane, exemplary for $\alpha =0$, ${\beta }_2=-0.4$ and $M=1$. Green dots represent fully stable regions, yellow ones indicate classical instability, and red points stand for unitarity violation (cf. main text).