Joint analysis of 6dFGS and SDSS peculiar velocities for the growth rate
  of cosmic structure and tests of gravity
  Measurement of peculiar velocities by combining redshifts and distance
indicators is a powerful way to measure the growth rate of cosmic structure and
test theories of gravity at low redshift. Here we constrain the growth rate of
structure by comparing observed Fundamental Plane peculiar velocities for 15894
galaxies from the 6dF Galaxy Survey (6dFGS) and Sloan Digital Sky Survey (SDSS)
with predicted velocities and densities from the 2M$++$ redshift survey. We
measure the velocity scale parameter $\beta \equiv {\Omega_m^\gamma}/b =
0.372^{+0.034}_{-0.050}$ and $0.314^{+0.031}_{-0.047}$ for 6dFGS and SDSS
respectively, where $\Omega_m$ is the mass density parameter, $\gamma$ is the
growth index, and $b$ is the bias parameter normalized to the characteristic
luminosity of galaxies, $L^*$. Combining 6dFGS and SDSS we obtain $\beta=
0.341\pm0.024$, implying that the amplitude of the product of the growth rate
and the mass fluctuation amplitude is $f\sigma_8 = 0.338\pm0.027$ at an
effective redshift $z=0.035$. Adopting $\Omega_m = 0.315\pm0.007$ as favoured
by Planck and using $\gamma=6/11$ for General Relativity and $\gamma=11/16$ for
DGP gravity, we get $S_8(z=0) = \sigma_8 \sqrt{\Omega_m/0.3} =0.637 \pm 0.054$
and $0.741\pm0.062$ for GR and DGP respectively. This measurement agrees with
other low-redshift probes of large scale structure but deviates by more than
$3\sigma$ from the latest Planck CMB measurement. Our results favour values of
the growth index $\gamma > 6/11$ or a Hubble constant $H_0 >
70$\,km\,s$^{-1}$\,Mpc$^{-1}$ or a fluctuation amplitude $\sigma_8 < 0.8$ or
some combination of these. Imminent redshift surveys such as Taipan, DESI,
WALLABY, and SKA1-MID will help to resolve this tension by measuring the growth
rate of cosmic structure to 1\% in the redshift range $0 < z < 1$.
