CMB and matter power spectra of earlyf(R)cosmology in the Palatini formulation

Abstract

We calculate in this article the cosmic microwave background (CMB) and matter power spectra of a class of early $f(R)$ cosmologies, which takes the form of $f(R)=R+{\ensuremath{\lambda}}_{1}{H}_{0}^{2}\mathrm{exp}[R/({\ensuremath{\lambda}}_{2}{H}_{0}^{2})]$. Unlike the late-time $f(R)$ cosmologies such as $f(R)=R+\ensuremath{\alpha}(\ensuremath{-}R{)}^{\ensuremath{\beta}}$ ($\ensuremath{\beta}l0$), the deviation from $\ensuremath{\Lambda}\mathrm{CDM}$ of this model occurs at a higher redshift (thus the name ``early $f(R)$ cosmology''), and this important feature leads to rather different integrated Sachs-Wolfe (ISW) effect and CMB spectrum. The matter power spectrum of this model is, however, again very sensitive to the chosen parameters, and large scale structure (LSS) observations such as Sloan digital sky survey (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 (e.g., redshifts $\mathcal{O}(10)\ensuremath{\lesssim}z\ensuremath{\lesssim}\mathcal{O}(1)$) than when dark energy begins to dominate---such models are strongly disfavored by data on CMB and matter power spectra.