Title: 2MTF VI. Measuring the velocity power spectrum,
Abstract: We present measurements of the velocity power spectrum and constraints on the
growth rate of structure $f\sigma_{8}$, at redshift zero, using the peculiar
motions of 2,062 galaxies in the completed 2MASS Tully-Fisher survey (2MTF). To
accomplish this we introduce a model for fitting the velocity power spectrum
including the effects of non-linear Redshift Space Distortions (RSD), allowing
us to recover unbiased fits down to scales $k=0.2\,h\,{\rm Mpc}^{-1}$ without
the need to smooth or grid the data. Our fitting methods are validated using a
set of simulated 2MTF surveys. Using these simulations we also identify that
the Gaussian distributed estimator for peculiar velocities of
\cite{Watkins2015} is suitable for measuring the velocity power spectrum, but
sub-optimal for the 2MTF data compared to using magnitude fluctuations $\delta
m$, and that, whilst our fits are robust to a change in fiducial cosmology,
future peculiar velocity surveys with more constraining power may have to
marginalise over this. We obtain \textit{scale-dependent} constraints on the
growth rate of structure in two bins, finding $f\sigma_{8} =
[0.55^{+0.16}_{-0.13},0.40^{+0.16}_{-0.17}]$ in the ranges $k = [0.007-0.055,
0.55-0.150]\,h\,{\rm Mpc}^{-1}$. We also find consistent results using four
bins. Assuming scale-\textit{independence} we find a value $f\sigma_{8} =
0.51^{+0.09}_{-0.08}$, a $\sim16\%$ measurement of the growth rate. Performing
a consistency check of General Relativity (GR) and combining our results with
CMB data only we find $\gamma = 0.45^{+0.10}_{-0.11}$, a remarkable constraint
considering the small number of galaxies. All of our results are completely
independent of the effects of galaxy bias, and fully consistent with the
predictions of GR (scale-independent $f\sigma_{8}$ and $\gamma\approx0.55$).