Growth index after the Planck results

Abstract

The growth index ${\ensuremath{\gamma}}_{L}$ was proposed to investigate the possible deviation from the standard $\ensuremath{\Lambda}\mathrm{CDM}$ model and Einstein's gravity theory in a dynamical perspective. Recently, thanks to the measurement of the cosmic growth rate via the redshift-space distortion, one can understand the evolution of the density contrast through $f{\ensuremath{\sigma}}_{8}(z)$, where $f(z)=d\mathrm{ln}\ensuremath{\delta}/d\mathrm{ln}a$ is the growth rate of matter and ${\ensuremath{\sigma}}_{8}(z)$ is the rms amplitude of the density contrast $\ensuremath{\delta}$ at the comoving $8{h}^{\ensuremath{-}1}\text{ }\text{ }\mathrm{Mpc}$ scale. In this paper, we use the redshift-space distortion data points to study the growth index on the bases of Einstein's gravity theory and a modified gravity theory under the assumption of $f={\ensuremath{\Omega}}_{m}(a{)}^{{\ensuremath{\gamma}}_{L}}$. The cosmic background evolution is fixed by the cosmic observations from the type Ia supernovae SNLS3 data, cosmic microwave background radiation data from Planck, and baryon acoustic oscillations. Via the Markov Chain Monte Carlo method, we find the following ${\ensuremath{\gamma}}_{L}$ values for Einstein's gravity with a cosmological constant ($w=\mathrm{const}$) dark energy and a modified gravity theory in the 1, 2, and $3\ensuremath{\sigma}$ regions, respectively: ${0.675}_{\ensuremath{-}0.0662\ensuremath{-}0.120\ensuremath{-}0.155}^{+0.0611+0.129+0.178}$, ${0.745}_{\ensuremath{-}0.0819\ensuremath{-}0.146\ensuremath{-}0.190}^{+0.0755+0.157+0.205}$, and ${0.555}_{\ensuremath{-}0.0167\ensuremath{-}0.0373\ensuremath{-}0.0516}^{+0.0193+0.0335+0.0436}$. In Einstein's gravity theory, the values of the growth index ${\ensuremath{\gamma}}_{L}$ show an almost $2\ensuremath{\sigma}$ deviation from the theoretical prediction of $6/11$ for the $\ensuremath{\Lambda}\mathrm{CDM}$ model. However, in the modified gravity framework, a deviation from Einstein's relativity is not detected in the $1\ensuremath{\sigma}$ region. This implies that the currently available cosmic observations do not predict an alternative modified gravity theory beyond the $\ensuremath{\Lambda}\mathrm{CDM}$ model under Einstein's gravity, but that the simple assumption of $f={\ensuremath{\Omega}}_{m}^{{\ensuremath{\gamma}}_{L}}$ should be improved.