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We review the Equation of State (EoS) approach to dark sector perturbationsand apply it to $f(\mathcal{R})$ gravity models of dark energy. We show thatthe EoS approach is numerically stable and use it to set observationalconstraints on designer models. Within the EoS approach we build an analyticalunderstanding of the dynamics of cosmological perturbations for the designerclass of $f(\mathcal{R})$ gravity models, characterised by the parameter $B_0$and the background equation of state of dark energy $w$. When we use the PlanckCosmic Microwave Background (CMB) temperature anisotropy, polarisation andlensing data as well as the Baryonic Acoustic Oscillation (BAO) data from SDSSand WiggleZ, we find $B_0<0.006$ (95\%CL) for the designer models with $w=-1$.Furthermore, we find $B_0<0.0045$ and $|w+1|<0.002$ (95\%CL) for the designermodels with $w\neq -1$. Previous analyses found similar results for designerand Hu-Sawicki $f(\mathcal{R})$ gravity models using the Effective Field Theory(EFT) approach (Raveri et al. 2014, Hu et al. 2016), therefore this hints forthe fact that generic $f(\mathcal{R})$ models with $w\neq-1$ can be tightlyconstrained by current cosmological data, complementary to solar system tests(Brax et al. 2008, Faulkner et al. 2007). When compared to a $w$CDM fluid withthe 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 $w$.

Do cosmological data rule out $f(\mathcal{R})$ with $w\neq-1$?