$f(\mathcal{R})$ gravity as a dark energy fluid

We study the equations for the evolution of cosmological perturbations in $f(\mathcal{R})$ and conclude that this modified gravity model can be expressed as a dark energy fluid at background and linearized perturbation order. By eliminating the extra scalar degree of freedom known to be present in such theories, we are able to characterize the evolution of the perturbations in the scalar sector in terms of equations of state for the entropy perturbation and anisotropic stress which are written in terms of the density and velocity perturbations of the dark energy fluid and those in the matter, or the metric perturbations. We also do the same in the much simpler vector and tensor sectors. In order to illustrate the simplicity of this formulation, we numerically evolve perturbations in a small number of cases.

Figure 1

Evolution of some relevant $f(\mathcal{R})$ quantities that appear as coefficients in the EOS for the dark sector anisotropic stress (4.9) and entropy perturbation (4.10), for a designer $f(\mathcal{R})$ that mimics $\mathrm{\Lambda }\mathrm{CDM}$ ($w_{\mathrm{de}}=-1$) with $B_0=1$. When these are negative valued, their absolute value is plotted in order to accommodate the logarithmic scale.

Figure 2

Spectrum of the ratio $Z/Y$ (or $-\mathrm{\Phi }/\mathrm{\Psi }$ in the notation of [22]) for different values of the equation-of-state parameter when $B_0=1$ (left) and different designer $f(\mathcal{R})$ scenarios parametrized by $B_0$ and with $w_{\mathrm{de}}=-1$ (right). On the $x$ axis, the wave number is written in units “$h/\mathrm{Mpc}$,” where $h=0.73$ is the reduced Hubble constant.

Figure 3

Evolution of the metric perturbations $\mathrm{\Phi }=-Z$ (left) and ${\mathrm{\Phi }}_{-}=-\frac{Y+Z}{2}$ (right) for the $\mathrm{\Lambda }\mathrm{CDM}$ expansion history ($w_{\mathrm{de}}=-1$, ${\mathrm{\Omega }}_{\mathrm{de}}^0=0.76$). The dotted lines correspond to the $\mathrm{\Lambda }\mathrm{CDM}$ scenario without $f(\mathcal{R})$. In this case the amplitudes of the perturbed potentials $\mathrm{\Phi }$ and ${\mathrm{\Phi }}_{-}$ do not depend on the wave number. The black lines correspond to an $f(\mathcal{R})$ scenario that mimics the cosmological constant ($w_{\mathrm{de}}=-1$, $B_0=1$). The amplitudes of the metric perturbations are now sensitive to the wave number ${\mathrm{K}}_0=\frac{k}{H_0}$. Even in the infrared limit $K_0=0$ represented by the dashed curves closest to the dotted lines, there is a disagreement with the $\mathrm{\Lambda }\mathrm{CDM}$ predictions.