We construct an updated and extended compilation of growth-rate data based on recent redshift-space distortion measurements. The data set consists of 34 data points and includes corrections for model dependence. In order to minimize overlap and maximize the independence of the data points, we also construct a subsample of this compilation (a “gold” growth data set) which consists of 18 data points. We test the consistency of this data set with the best-fit $\mathrm{Planck}15/\mathrm{\Lambda }\mathrm{CDM}$ parameters in the context of General Relativity using the evolution equation for the growth factor $\delta (a)$ with a $w\mathrm{CDM}$ background. We find tension at the $\sim 3\sigma$ level between the best-fit parameters $w$ (the dark energy equation of state), ${\mathrm{\Omega }}_{0m}$ (the matter density parameter), and ${\sigma }_8$ (the matter power spectrum normalization on scales $8h^{-1}\mathrm{Mpc}$) and the corresponding $\mathrm{Planck}15/\mathrm{\Lambda }\mathrm{CDM}$ parameters ($w=-1$, ${\mathrm{\Omega }}_{0m}=0.315$, and ${\sigma }_8=0.831$). We show that the tension disappears if we allow for evolution of the effective Newton constant, parametrized as $G_{\mathrm{eff}}(a)/G_{\mathrm{N}}=1+g_a(1-a{)}^n-g_a(1-a{)}^{2n}$ with $n\ge 2$ where $g_a$ and $n$ are parameters of the model, $a$ is the scale factor, and $z=1/a-1$ is the redshift. This parametrization satisfies three important criteria: a) positive energy of the graviton ($G_{\mathrm{eff}}>0$), b) consistency with big bang nucleosynthesis constraints ($G_{\mathrm{eff}}(a\ll 1)/G_{\mathrm{N}}=1$), and c) consistency with Solar System tests ($G_{\mathrm{eff}}(a=1)/G_{\mathrm{N}}=1$ and $G_{\mathrm{eff}}^{\prime }(a=1)/G_{\mathrm{N}}=0$). We show that the best-fit form of $G_{\mathrm{eff}}(z)$ obtained from the growth data corresponds to weakening gravity at recent redshifts (decreasing function of $z$), and we demonstrate that this behavior is not consistent with any scalar-tensor Lagrangian with a real scalar field. Finally, we use MGCAMB to find the best-fit $G_{\mathrm{eff}}(z)$ obtained from the Planck cosmic microwave background power spectrum on large angular scales and show that it is a mildly increasing function of $z$, in $3\sigma$ tension with the corresponding decreasing best-fit $G_{\mathrm{eff}}(z)$ obtained from the growth data.

We construct an updated extended compilation of distinct (but possibly correlated) $f{\sigma }_8(z)$ 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 ${\mathrm{\Omega }}_{0m}-{\sigma }_8$ $\mathrm{\Lambda }\mathrm{CDM}$ parameters and show that they are at a $5\sigma$ tension with the corresponding $\text{Planck}15/\mathrm{\Lambda }\mathrm{CDM}$ 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 $1\sigma$) 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 $G_{\mathrm{eff}}(z)/G_{\mathrm{N}}=1+g_a{(\frac{z}{1+z})}^2-g_a{(\frac{z}{1+z})}^4$ ($g_a$ is a parameter) and fixing a $\text{Planck}15/\mathrm{\Lambda }\mathrm{CDM}$ background we find $g_a=-0.91\pm 0.17$ from the full $f{\sigma }_8$ data set while the 20 earliest and 20 latest datapoints imply $g_a=-{1.28}_{-0.26}^{+0.28}$ and $g_a=-0.43_{-0.41}^{+0.46}$ respectively. Thus, the more recent $f{\sigma }_8$ data appear to favor GR in contrast to earlier data. Finally, we show that the parametrization $f{\sigma }_8(z)=\lambda {\sigma }_8\mathrm{\Omega }(z{)}^{\gamma }/(1+z{)}^{\beta }$ provides an excellent fit to the solution of the growth equation for both GR ($g_a=0$) and modified gravity ($g_a\ne 0$).

We have performed a comparative analysis of three recent and reliable SnIa datasets available in the literature: the Full Gold (FG) dataset (157 data points $0<z<1.75$), a Truncated Gold (TG) dataset (140 data points $0<z<1$) and the most recent Supernova Legacy Survey (SNLS) dataset (115 data points $0<z<1$). We have analyzed and compared the likelihood of cosmological constant and dynamical dark energy parametrizations allowing for crossing of the phantom divide line (PDL). We find that, even though the constraints obtained using the three datasets are consistent with each other at the 95% confidence level, the latest (SNLS) dataset shows distinct trends which are not shared by the Gold datasets. We find that the best fit dynamical $w(z)$ obtained from the SNLS dataset does not cross the PDL $w=-1$ and remains above and close to the $w=-1$ line for the whole redshift range $0<z<1$ showing no evidence for phantom behavior. The $\Lambda +\mathrm{\text{Cold}}\mathrm{\text{Dark}}\mathrm{\text{Matter}}$ (LCDM) parameter values ($w_0=-1$, $w_1=0$) almost coincide with the best fit parameters of the dynamical $w(z)$ parametrizations. In contrast, the best fit dynamical $w(z)$ obtained from the Gold datasets (FG and TG) clearly crosses the PDL and departs significantly from the PDL $w=-1$ line while the LCDM parameter values are about $2\sigma$ away from the best fit $w(z)$. In addition, the $({\Omega }_{0m},{\Omega }_{\Lambda })$ parameters in a LCDM parametrization without a flat prior, fit by the SNLS dataset, favor the minimal flat LCDM concordance model. The corresponding fit with the Gold datasets mildly favors a closed universe and the flat LCDM parameter values are $1\sigma –2\sigma$ away from the best fit $({\Omega }_{0m},{\Omega }_{\Lambda })$.

Phantom dark energy ($w<-1$) can produce amplified cosmic acceleration at late times, thus increasing the value of $H_0$ favored by CMB data and releasing the tension with local measurements of $H_0$. We show that the best fit value of $H_0$ in the context of the CMB power spectrum is degenerate with a constant equation-of-state parameter $w$, in accordance with the approximate effective linear equation $H_0+30.93w-36.47=0$ ($H_0$ in $\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$). This equation is derived by assuming that both ${\mathrm{\Omega }}_{0m}{h}^2$ and $d_A={\int }_0^{z_{\mathrm{rec}}}\frac{dz}{H(z)}$ remain constant (for an invariant CMB spectrum) and equal to their best fit Planck/$\mathrm{\Lambda }\mathrm{CDM}$ values as $H_0$, ${\mathrm{\Omega }}_{0m}$, and $w$ vary. For $w=-1$, this linear degeneracy equation leads to the best fit $H_0=67.4\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$ as expected. For $w=-1.22$, the corresponding predicted CMB best fit Hubble constant is $H_0=74\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$, which is identical with the value obtained by local-distance ladder measurements, while the best fit matter density parameter is predicted to decrease, since ${\mathrm{\Omega }}_{0m}{h}^2$ is fixed. We verify the above $H_0-w$ degeneracy equation by fitting a $w\mathrm{CDM}$ model with fixed values of $w$ to the Planck TT spectrum, showing also that the quality of fit (${\chi }^2$) is similar to that of $\mathrm{\Lambda }\mathrm{CDM}$. However, when including SnIa, baryon acoustic oscillation, or growth data, the quality of fit becomes worse than $\mathrm{\Lambda }\mathrm{CDM}$ when $w<-1$. Finally, we generalize the $H_0-w(z)$ degeneracy equation for the parametrization $w(z)=w_0+w_{1}z/(1+z)$ and identify analytically the full $w_0-w_1$ parameter region (straight line) that leads to a best fit $H_0=74\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$ in the context of the Planck CMB spectrum. This exploitation of $H_0-w(z)$ degeneracy can lead to immediate identification of all parameter values of a given $w(z)$ parametrization that can potentially resolve the $H_0$ tension.