Constraints on a $f(R)$ gravity dark energy model with early scaling evolution

The modified gravity with $f(R)=R^{1+ϵ}$ ($ϵ>0$) allows a scaling solution where the density of gravity sector follows the density of the dominant fluid. We present initial conditions of background and perturbation variables during the scaling evolution regime in the modified gravity. As a possible dark energy model we consider a gravity with a form $f(R)=R^{1+ϵ}+q{R}^{-n}$ ($-1<n\le 0$) where the second term drives the late-time acceleration. We show that our $f(R)$ gravity parameters are very sensitive to the baryon perturbation growth and baryon density power spectrum, and present observational constraints on the model parameters. Our analysis suggests that only the parameter space extremely close to the $\mathrm{\Lambda }\text{CDM}$ model is allowed.