CMB and Matter Power Spectra of Early f(R) Cosmology in Palatini Formalism

We calculate in this article the CMB and matter power spectra of a class of early $f(R)$ cosmologies, which takes the form of $f(R) = R + \lambda_1 H_0^2\text{exp}[R/(\lambda_2 H_0^2)]$. Unlike the late-time $f(R)$ cosmologies such as $f(R) = R + \alpha(-R)^\beta$ ($\beta<0$), the deviation from $\Lambda\text{CDM}$ of this model occurs at a higher redshift (thus the name \emph{Early $f(R)$ Cosmology}), and this important feature leads to rather different ISW effect and CMB spectrum. The matter power spectrum of this model is, at the same time, again very sensitive to the chosen parameters, and LSS observations such as SDSS should constrain the parameter space stringently. We expect that our results are applicable at least qualitatively to other models that produce $f(R)$ modification to GR at earlier times (\emph{e.g.}, redshifts $\mathcal{O}(10) \lesssim z \lesssim \mathcal{O}(1)$) than when dark energy begins to dominate -- such models are strongly disfavored by data on CMB and matter power spectra.