We review the equation of state (EoS) approach to dark sector perturbations and apply it to $f(\mathcal{R})$ gravity models of dark energy. We show that the EoS approach is numerically stable and use it to set observational constraints on designer models. Within the EoS approach we build an analytical understanding of the dynamics of cosmological perturbations for the designer class of $f(\mathcal{R})$ gravity models, characterized by the parameter $B_0$ and the background equation of state of dark energy $w$. When we use the Planck cosmic microwave background temperature anisotropy, polarization, and lensing data as well as the baryonic acoustic oscillation data from SDSS and WiggleZ, we find $B_0<0.006$ (95% C.L.) for the designer models with $w=-1$. Furthermore, we find $B_0<0.0045$ and $|w+1|<0.002$ (95% C.L.) for the designer models with $w\ne -1$. Previous analyses found similar results for designer and Hu-Sawicki $f(\mathcal{R})$ gravity models using the effective field theory approach [Raveri et al., Phys. Rev. D 90, 043513 (2014); Hu et al., Mon. Not. R. Astron. Soc. 459, 3880 (2016)]; therefore this hints for the fact that generic $f(\mathcal{R})$ models with $w\ne -1$ can be tightly constrained by current cosmological data, complementary to solar system tests [Brax et al., Phys. Rev. D 78, 104021 (2008); Faulkner et al., Phys. Rev. D 76, 063505 (2007)]. When compared to a $w\mathrm{CDM}$ fluid with the same sound speed, we find that the equation of state for $f(\mathcal{R})$ models is better constrained to be close to $-1$ by about an order of magnitude, due to the strong dependence of the perturbations on $w$.

• Received 22 December 2017

DOI:https://doi.org/10.1103/PhysRevD.97.104070