$${\sigma }_{8}$$ Tension. Is Gravity Getting Weaker at Low z? Observational Evidence and Theoretical Implications

Abstract

Dynamical observational probes of the growth of density perturbations indicate that gravity may be getting weaker at low redshifts $z$. This evidence is at about $2-3\sigma$ level and comes mainly from weak lensing data that measure the parameter $S_8=\sigma_8 \sqrt{\Omega_{0m}/0.3}$ and redshift space distortion data that measure the growth rate times the amplitude of the linear power spectrum parameter $f\sigma_8 (z)$. The measured $f\sigma_8$ appears to be lower than the prediction of General Relativity (GR) in the context of the standard $\Lambda$CDM model as defined by the Planck best fit parameter values. This is the well known $f\sigma_8$ tension of $\Lambda$CDM, which constitutes one of the two main large scale challenges of the model along with the $H_0$ tension. We review the observational evidence that leads to the $f\sigma_8$ tension and discuss some theoretical implications. If this tension is not a systematic effect it may be an early hint of modified gravity with an evolving effective Newton's constant $G_{eff}$ and gravitational slip parameter $\eta$. We discuss such best fit parametrizations of $G_{eff}(z)$ and point out that they can not be reproduced by simple scalar-tensor and $f(R)$ modified gravity theories because these theories generically predict stronger gravity than General Relativity (GR) at low $z$ in the context of a $\Lambda$CDM background $H(z)$. Finally, we show weak evidence for an evolving reduced absolute magnitude of the SnIa of the Pantheon dataset at low redshifts ($z<0.1$) which may also be explained by a reduced strength of gravity and may help resolve the $H_0$ tension.