Testing the effect of $H_0$ on $f\sigma_8$ tension using a Gaussian
  Process method
  Using the $f\sigma_8(z)$ redshift space distortion (RSD) data, the
$\sigma_8^0-\Omega_m^0$ tension is studied utilizing a parameterization of
growth rate $f(z) = \Omega_m(z)^\gamma$. Here, $f(z)$ is derived from the
expansion history $H(z)$ which is reconstructed from the observational Hubble
data applying the Gaussian Process method. It is found that different priors of
$H_0$ have great influences on the evolution curve of $H(z)$ and the constraint
of $\sigma_8^0-\Omega_m^0$. When using a larger $H_0$ prior, the low redshifts
$H(z)$ deviate significantly from that of the $\Lambda$CDM model, which
indicates that a dark energy model different from the cosmological constant can
help to relax the $H_0$ tension problem. The tension between our best-fit
values of $\sigma_8^0-\Omega_m^0$ and that of the \textit{Planck} 2018
$\Lambda$CDM (PLA) will disappear (less than $1\sigma$) when taking a prior for
$H_0$ obtained from PLA. Moreover, the tension exceeds $2\sigma$ level when
applying the prior $H_0 = 73.52 \pm 1.62$ km/s/Mpc resulted from the Hubble
Space Telescope photometry. By comparing the $S_8 -\Omega_m^0$ planes of our
method with the results from KV450+DES-Y1, we find that using our method and
applying the RSD data may be helpful to break the parameter degeneracies.
