Stability of the early universe in bigravity theory

We study the stability of a spherically symmetric perturbation around the flat Friedmann–Lemaître–Robertson–Walker spacetime in the ghost-free bigravity theory, retaining nonlinearities of the helicity-0 mode of the massive graviton. It has been known that, when the graviton mass is smaller than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the Higuchi-type ghost or the gradient instability against the linear perturbation in the bigravity. Hence, the bigravity theory has no healthy massless limit for cosmological solutions at linear level. In this paper we show that the instabilities can be resolved by taking into account nonlinear effects of the scalar graviton mode for an appropriate parameter space of coupling constants. The growth history in the bigravity can be restored to the result in general relativity in the early stage of the Universe, in which the Stückelberg fields are nonlinear and there is neither ghost nor gradient instability. Therefore, the bigravity theory has the healthy massless limit, and cosmology based on it is viable even when the graviton mass is smaller than the Hubble parameter.

Figure 1

The stable regions of the coupling constants for flat FLRW backgrounds for the cases of (a) $w=-1$, (b) $w=0$, (c) $w=1/3-{10}^{-5}$, (d) $w=1/3+{10}^{-5}$, (e) $w=2/3$, and (f) $w=1-{10}^{-5}$. The ${\mu }_0={\mu }_{+}$-branch is stable in the regions denoted by ${\mu }_{+}$ (red regions), the ${\mu }_0={\mu }_{-}$-branch is stable in the regions denoted by ${\mu }_{-}$ (blue regions), and both ${\mu }_0={\mu }_{\pm }$ are stable in the regions denoted by ${\mu }_{\pm }$ (purple regions). The black solid curves correspond to ${\beta }_3={\beta }_2^2$, and the black dotted lines correspond to ${\beta }_2={\beta }_3$.

Figure 2

The same figures as Fig. 1 for the cases of (a) $w=1+{10}^{-5}$ and (b) $w=2$.

Figure 3

Case A: There exists the stable branch with nonlinear ${\mu }_{\infty }(\sim O(1))$. We plot the roots of Eq. (5.8) for ${\tilde{\delta }}_g={10}^{-2},{\tilde{\delta }}_f={10}^{-3}$ (red curves) and for ${\tilde{\delta }}_g={10}^{-3},{\tilde{\delta }}_f={10}^{-2}$ (blue dashed curves). We set ${\beta }_2=-3,{\beta }_3=3,m_g^2=m_f^2$. The branch with ${\mu }_0\simeq 0.40$ for $w=0$ and ${\mu }_0\simeq -0.10$ for $w=2/3$ are stable in the early stage of the Universe ($m_{\text{eff}}/H\ll 1$). In the late stage ($m_{\text{eff}}/H\gg 1$), the branches with ${\mu }_0\simeq 0$ and 0.70 are stable. The Universe may evolve from the stable ${\mu }_0$-branch to the stable ${\mu }_{\infty }$-branch. For example, for $w=0$, $\mu$ changes from ${\mu }_0=0.4$ (GR phase) to ${\mu }_{\infty }=0.7$ (nonlinear bigravity phase), while for $w=2/3$, it does from ${\mu }_0=-0.1$ (GR phase) to ${\mu }_{\infty }=0$ (linear bigravity phase).

Figure 4

Case B: There exists only one stable branch with ${\mu }_{\infty }=0$. We plot the roots of Eq. (5.8) for ${\tilde{\delta }}_g={10}^{-2},{\tilde{\delta }}_f={10}^{-3}$ (red curves) and ${\tilde{\delta }}_g={10}^{-3},{\tilde{\delta }}_f={10}^{-2}$ (blue dashed curves). We set ${\beta }_2=-1,{\beta }_3=1/2,m_g^2=m_f^2$. The stable branches for $m_{\text{eff}}/H\ll 1$ are given by ${\mu }_0\simeq 1.1$ for $w=0$ and ${\mu }_0\simeq -0.21$ for $w=2/3$. For $m_{\text{eff}}/H\gg 1$, only ${\mu }_{\infty }\simeq 0$ is stable. The Universe with $w=2/3$ may evolve from ${\mu }_0=-0.21$ (GR phase) to ${\mu }_{\infty }=0$ (linear bigravity phase), but for $w=0$, there is no stable adiabatic solution.

Figure 5

The left figure shows asymptotically homothetic branches $(\widehat{\mu }\to 0)$ of Eq. (a13). We set ${\beta }_2=-3,{\beta }_3=3,m_g^2=m_f^2,{\mathrm{\Lambda }}_g=0$, and $K^2\mathcal{G}{\mathcal{M}}_f/G{M}_g={10}^{-3}$ (red solid curve), $K^2\mathcal{G}{\mathcal{M}}_f/G{M}_g={10}^{-1}$ (blue dashed curve), and $K^2\mathcal{G}{\mathcal{M}}_f/G{M}_g=10$ (green dotted curve). Except for $K^2\mathcal{G}{\mathcal{M}}_f/G{M}_g={10}^{-3}$, the asymptotically homothetic branches do not extend below the Vainshtein radius. The right figure shows the existence of the Vainshtein screening for $K^2\mathcal{G}{\mathcal{M}}_f/G{M}_g={10}^{-3}$, i.e., GR is recovered for $R<R_V$, where we define ${\widehat{\mathrm{\Phi }}}_{\mathrm{GR}}^{\prime }={\widehat{\mathrm{\Phi }}}_g^{\prime }{|}_{m_g=0}$ and ${\widehat{\mathrm{\Psi }}}_{\mathrm{GR}}={\widehat{\mathrm{\Psi }}}_g{|}_{m_g=0}$.

Figure 6

The same as Fig. 5 for nonhomothetic branches. The asymptotically stable nonhomothetic branch is characterized by $\widehat{\mu }\to 0.70$ as $R/R_V\to \infty$. The deviation of bigravity from GR diverges as ${\widehat{\mathrm{\Psi }}}_g^{\prime }/{\widehat{\mathrm{\Psi }}}_{\mathrm{GR}}^{\prime }\propto R^3$ beyond the Vainshtein radius and similar to the ${\widehat{\mathrm{\Phi }}}_g/{\widehat{\mathrm{\Phi }}}_{\mathrm{GR}}$.

Figure 7

The roots of Eq. (a13). We set ${\beta }_2=-3,{\beta }_3=3,m_g^2=m_f^2$, and ${\mathrm{\Lambda }}_g=10m_g^2$. An asymptotically stable branch is characterized by $\widehat{\mu }\to {\widehat{\mu }}_{\infty }=0.28$ as $R/R_V\to \infty$. The gravitational property is almost same everywhere as discussed in Sec. 5b.

Figure 8

The parameter region of the existence of a stable asymptotically nonhomothetic branch. We set $m_g^2=m_f^2$ and ${\mathrm{\Lambda }}_g=0$. In the colored region, there is a stable branch with ${\widehat{\mu }}_{\infty }>-1$ other than ${\widehat{\mu }}_{\infty }=0$. The curves for ${\beta }_3={\beta }_2^2$ (solid curve) and ${\beta }_3={\beta }_2$ (dotted line) are also shown for the comparison with the stability condition (4.62) in the regime $H\gg m_{\text{eff}}$, obtained in Sec. 4c.