Cosmological perturbations in extended massive gravity

We study cosmological perturbations around self-accelerating solutions to two extensions of nonlinear massive gravity: the quasi-dilaton theory and the mass-varying theory. We examine stability of the cosmological solutions, and the extent to which the vanishing of the kinetic terms for scalar and vector perturbations of self-accelerating solutions in massive gravity is generic when the theory is extended. We find that these kinetic terms are in general nonvanishing in both extensions, though there are constraints on the parameters and background evolution from demanding that they have the correct sign. In particular, the self-accelerating solutions of the quasi-dilaton theory are always unstable to scalar perturbations with wavelength shorter than the Hubble length.

Figure 1

The stability of the scalar sector implied by the determinant of the kinetic matrix (49). For modes with $k/(aH)$ below the solid line, the determinant is positive, so there are no ghost degrees of freedom [see Eq. (a1)] for the field basis in which this is manifest). On the other hand, above the solid line, one degree of freedom has a positive kinetic term while the other is a ghost.

Figure 2

Plot of the dispersion relation of scalar modes versus the rescaled momenta $\tilde{k}$, for the example (a27), with $\Lambda =0$, $\omega =1$, ${\alpha }_3=-10$, ${\alpha }_4=6$, $\xi =0$, $m_g^2<0$, in the positive branch defined in Eq. (16). The dashed line corresponds to $M_{\mathrm{GW}}^2/H^2$ which is the mass term for mode 2 (blue) in the low momentum regime. At the critical point $\tilde{k}=1$, ${\omega }_1^2$ (red) diverges to $-\infty$, and then becomes positive and finite after the transition.

Figure 3

Order of magnitude plots of the squared-frequency in Hubble units for the case $\Lambda =0$, $\omega =1$ and $\xi =0$, and for a benchmark momentum value $\tilde{k}=2$. The left (right) column shows the positive (negative) branch of solutions given in Eq. (16), while the upper (lower) panel shows ${\omega }_1^2$ (${\omega }_2^2$). From dark to light, the colors represent the following values of the squared-frequency: (0,1]; (1,10]; $(10,{10}^2]$; $({10}^2,{10}^3]$; $({10}^3,{10}^4]$; $({10}^4,\infty )$. In the shaded parameter region, there are no background solutions. The gridded region corresponds to ${\omega }^2<0$, while the thick black line corresponds to ${\alpha }_4=-3(1+{\alpha }_3)$ [for the positive (negative) branch, ${\alpha }_3>-2$ (${\alpha }_3<-2$) only], where $X\sim 0$.