Evolution of cosmological gravitational waves in $f(R)$ gravity

We give a rigorous and mathematically clear presentation of the covariant and gauge-invariant theory of gravitational waves in a perturbed Friedmann-Lemaître-Robertson-Walker universe for fourth order gravity, where the matter is described by a perfect fluid with a barotropic equation of state. As an example of a consistent analysis of tensor perturbations in fourth order gravity, we apply the formalism to a simple background solution of $R^n$ gravity. We obtain the exact solutions of the perturbation equations for scales much bigger than and smaller than the Hubble radius. It is shown that the evolution of tensor modes is highly sensitive to the choice of $n$ and an interesting new feature arises. During the radiation dominated era, their exists a growing tensor perturbation for nearly all choices of $n$. This occurs even when the background model is undergoing accelerated expansion as opposed to the case of general relativity. Consequently, cosmological gravitational wave modes can in principle provide a strong constraint on the theory of gravity independent of other cosmological data sets.

Figure 1

The exponents of each mode of the solution [Eq. (83)] against $n$ in the vacuum case. The black (gray) line represents the growing (decaying) mode. The points represent the value of the exponents in the case of GR ($n=1$). As $n$ is increased the background expansion rate increases, resulting in a weaker growth rate for tensor perturbations. When the critical value of $n\approx 1.366$ is reached, no growing modes can be supported in this model.

Figure 2

The exponents of each mode of the solution for the normalized shear against $n$ in the dust and radiation dominated eras. (a) The left panel represents the exponents of the mode in the dust dominated era. (b) The right panel represents the exponents of the mode in the radiation dominated era. The black (grey) line represents the growing (decaying) mode. The points represent the value of the exponents in the case of GR ($n=1$).

Figure 3

The range of parameters for which the modes grow or decay as a function of $n$ and $w_m$ The black thick line represents the change from growth to decay for the ${\tilde{\Sigma }}_1$ mode. The grey thick line represents the change from growth to decay for the ${\tilde{\Sigma }}_2$ mode. The thin black line represents the GR case ($n=1$). In region I, ${\tilde{\Sigma }}_1$ decays and ${\tilde{\Sigma }}_2$ grows. In region II, both modes grow. In region III, ${\tilde{\Sigma }}_1$ grows and ${\tilde{\Sigma }}_2$ decays. Finally in region IV, both modes decay.