Stability of an isotropic cosmological singularity in higher-order gravity

We study the stability of the isotropic vacuum Friedmann universe in gravity theories with higher-order curvature terms of the form $(R_{ab}{R}^{ab}{)}^n$ added to the Einstein-Hilbert Lagrangian of general relativity on approach to an initial cosmological singularity. Earlier, we had shown that, when $n=1$, a special isotropic vacuum solution exists which behaves like the radiation-dominated Friedmann universe and is stable to anisotropic and small inhomogeneous perturbations of scalar, vector, and tensor type. This is completely different to the situation that holds in general relativity, where an isotropic initial cosmological singularity is unstable in vacuum and under a wide range of nonvacuum conditions. We show that when $n\ne 1$, although a special isotropic vacuum solution found by Clifton and Barrow always exists, it is no longer stable when the initial singularity is approached. We find the particular stability conditions under the influence of tensor, vector, and scalar perturbations for general $n$ for both solution branches. On approach to the initial singularity, the isotropic vacuum solution with scale factor $a(t)=t^{P_{-}/3}$ is found to be stable to tensor perturbations for $0.5<n<1.1309$ and stable to vector perturbations for $0.861425<n\le 1$, but is unstable as $t\to 0$ otherwise. The solution with scale factor $a(t)=t^{P_{+}/3}$ is not relevant to the case of an initial singularity for $n>1$ and is unstable as $t\to 0$ for all $n$ for each type of perturbation.

Figure 1

The variation of $P_{+}$ with $n$.

Figure 2

The variation of $P_{-}$ with $n$.

Figure 3

Power-law exponents, ${\lambda }_1$, ${\lambda }_{\pm }$, versus $n$ for tensor perturbations with $P=P_{+}$.

Figure 4

Power-law exponents, ${\lambda }_1$, ${\lambda }_{\pm }$, versus $n$ for tensor perturbations with $P=P_{-}$.

Figure 5

Power-law exponents, ${\xi }_1$, ${\xi }_{\pm }$, versus $n$ for vector perturbations with $P=P_{+}$.

Figure 6

Power-law exponents, ${\xi }_1$, ${\xi }_{\pm }$, versus $n$ for vector perturbations with $P=P_{-}$.

Figure 7

Power-law exponents, ${\rho }_1$, ${\rho }_{\pm }$, versus $n$ for scalar perturbations with $P=P_{+}$.

Figure 8

Power-law exponents, ${\rho }_1$, ${\rho }_{\pm }$, versus $n$ for scalar perturbations with $P=P_{-}$.