Bounce universe and black holes from critical Einsteinian cubic gravity

We show that there exists a critical point for the coupling constants in Einsteinian cubic gravity in which the linearized equations on the maximally symmetric vacuum vanish identically. We construct an exact isotropic bounce universe in the critical theory in four dimensions. The comoving time runs from minus infinity to plus infinity, yielding a smooth universe bouncing between two de Sitter vacua. In five dimensions, we adopt a numerical approach to construct a bounce solution, in which a singularity occurs before the bounce takes place. We then construct exact anisotropic bounces that connect two isotropic de Sitter spacetimes with flat spatial sections. We further construct exact anti-de Sitter black holes in the critical theory in four and five dimensions and obtain an exact anti-de Sitter worm brane in four dimensions.

Figure 1

The left figure plots the effective cosmological constant ${\mathrm{\Lambda }}_{\mathrm{eff}}$ of the vacua. There is at least one (A)dS vacuum for all $\tilde{\lambda }$. In the region $0<\tilde{\lambda }<1$, there are three (A)dS vacua. The dashed and dotted vacua coalesce to become one vacuum when $\tilde{\lambda }=1$. The right figure plots the effective ${\kappa }_{\mathrm{eff}}$ of the linearized graviton in each vacuum, with ${\kappa }_0=1$. We see that a ghost-free vacuum (dashed line) exists only for $\tilde{\lambda }\le 1$, where ${\kappa }_{\mathrm{eff}}\ge 0$, with ${\kappa }_{\mathrm{eff}}=0$ at $\tilde{\lambda }=1$. For ${\kappa }_0=-1$, the vacua associated with dotted and solid lines become ghost free instead.

Figure 2

The left is the scaling factor $a$ as a function of comoving time. The right is $\dot{a}$ near $t=0$. We choose the parameter $\mu \sim 1.1861$ so that the bounce occurs at $t=0$, and it reaches asymptotic de Sitter as $t\to \infty$. The solution has a naked singularity in the past at $t_{*}=-0.4531$, before the bounce takes place at $t=0$.

Figure 3

The plots for $U$ and $V$, the numerator and denominator in (3.2), respectively. $U$ and $V$ both vanish at the bounce time $t=0$, giving rise to finite $\stackrel{\dots }{a}$. At $t=t_{*}$, only $V$ vanishes but not $U$, and hence $\stackrel{\dots }{a}$ is divergent.

Figure 4

Here, we plot the numerical $a$ and approximation at three different comoving time regions. The solid line represents the numerical $a$, and the dashed lines are approximated solutions. The left plot is in the large $t$ region with $a_{\mathrm{symp}}$ given in (3.21); the middle plot is in the bounce $t=0$ region with $a_{\text{bounce}}$ given in (3.24); the right plot is in the singular $t=t_{*}$ region with $a_{\mathrm{sing}}$ given in (3.25).