Evolution of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:msub><mml:mi>σ</mml:mi><mml:mn>8</mml:mn></mml:msub></mml:math> tension with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mtext>Planck</mml:mtext><mml:mn>15</mml:mn><mml:mo>/</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>CDM</mml:mi></mml:mrow></mml:math> determination and implications for modified gravity theories

We construct an updated extended compilation of distinct (but possibly correlated) $f{\ensuremath{\sigma}}_{8}(z)$ redshift space distortion (RSD) data published between 2006 and 2018. It consists of 63 datapoints and is significantly larger than previously used similar data sets. After fiducial model correction we obtain the best fit ${\mathrm{\ensuremath{\Omega}}}_{0m}\ensuremath{-}{\ensuremath{\sigma}}_{8}$ $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ parameters and show that they are at a $5\ensuremath{\sigma}$ tension with the corresponding $\text{Planck}15/\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ values. Introducing a nontrivial covariance matrix correlating randomly 20% of the RSD datapoints has no significant effect on the above tension level. We show that the tension disappears (becomes less than $1\ensuremath{\sigma}$) when a subsample of the 20 most recently published data is used. A partial cause for this reduced tension is the fact that more recent data tend to probe higher redshifts (with higher errorbars) where there is degeneracy among different models due to matter domination. Allowing for a nontrivial evolution of the effective Newton's constant as ${G}_{\mathrm{eff}}(z)/{G}_{\mathrm{N}}=1+{g}_{a}{(\frac{z}{1+z})}^{2}\ensuremath{-}{g}_{a}{(\frac{z}{1+z})}^{4}$ (${g}_{a}$ is a parameter) and fixing a $\text{Planck}15/\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ background we find ${g}_{a}=\ensuremath{-}0.91\ifmmode\pm\else\textpm\fi{}0.17$ from the full $f{\ensuremath{\sigma}}_{8}$ data set while the 20 earliest and 20 latest datapoints imply ${g}_{a}=\ensuremath{-}{1.28}_{\ensuremath{-}0.26}^{+0.28}$ and ${g}_{a}=\ensuremath{-}0.4{3}_{\ensuremath{-}0.41}^{+0.46}$ respectively. Thus, the more recent $f{\ensuremath{\sigma}}_{8}$ data appear to favor GR in contrast to earlier data. Finally, we show that the parametrization $f{\ensuremath{\sigma}}_{8}(z)=\ensuremath{\lambda}{\ensuremath{\sigma}}_{8}\mathrm{\ensuremath{\Omega}}(z{)}^{\ensuremath{\gamma}}/(1+z{)}^{\ensuremath{\beta}}$ provides an excellent fit to the solution of the growth equation for both GR (${g}_{a}=0$) and modified gravity (${g}_{a}\ensuremath{\ne}0$).

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