Parametrized modified gravity constraints after Planck

Abstract

We constrain $f(R)$ and chameleon-type modified gravity in the framework of the Berstchinger-Zukin parametrization using the recently released Planck data, including both the cosmic mircowave background radiation (CMB) temperature power spectrum and the lensing potential power spectrum. Some other external data sets are included, such as baryon acoustic oscillation (BAO) measurements from the 6dFGS, SDSS DR7 and BOSS DR9 surveys; Hubble Space Telescope (HST) ${H}_{0}$ measurements, and supernovae from the Union2.1 compilation. We also use WMAP9 data for a consistency check and comparison. For $f(R)$ gravity, WMAP9 results can only give a quite loose constraint on the modified gravity parameter ${B}_{0}$, which is related to the present value of the Compton wavelength of the extra scalar degree of freedom, ${B}_{0}<3.37$ at 95% C.L. We demonstrate that this constraint mainly comes from the late integrated Sachs-Wolfe effect. With only Planck CMB temperature power-spectrum data, we can improve the WMAP9 result by a factor 3.7 (${B}_{0}<0.91$ at 95% C.L.). If the Planck lensing potential power-spectrum data are also taken into account, the constraint can be further strengthened by a factor 5.1 (${B}_{0}<0.18$ at 95% C.L.). This major improvement mainly comes from the small-scale lensing signal. Furthermore, BAO, HST and supernovae data could slightly improve the ${B}_{0}$ bound (${B}_{0}<0.12$ at 95% C.L.). For the chameleon-type model, we find that the data set that we used cannot constrain the Compton wavelength ${B}_{0}$ or the potential index $s$ of the chameleon field, but it can give a tight constraint on the parameter ${\ensuremath{\beta}}_{1}={1.043}_{\ensuremath{-}0.104}^{+0.163}$ at 95% C.L. (${\ensuremath{\beta}}_{1}=1$ in general relativity), which accounts for the nonminimal coupling between the chameleon field and the matter component. In addition, we find that both modified gravity models we consider favor a relatively higher Hubble parameter than the concordance $\ensuremath{\Lambda}\mathrm{CDM}$ model in general relativity.