2MTF VI. Measuring the velocity power spectrum

We present measurements of the velocity power spectrum and constraints on the growth rate of structure ƒσ₈, 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 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 Watkins & Feldman (2015) is suitable for measuring the velocity power spectrum, but sub-optimal for the 2MTF data compared to using magnitude fluctuations δm, 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 scale-dependent constraints on the growth rate of structure in two bins, finding ƒσ₈ = [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 Mpc^-1. We also find consistent results using four bins. Assuming scale-independence we find a value ƒσ₈ = 0.51 +0.09 -0.08, a ∼16% measurement of the growth rate. Performing a consistency check of General Relativity (GR) and combining our results with CMB data only we find γ = 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 ƒσ8 and γ ≈ 0.55).