We have made the largest volume measurement to date of the transition to large‐scale homogeneity in the distribution of galaxies. We use the WiggleZ survey, a spectroscopic survey of over 200 000 blue galaxies in a cosmic volume of ∼1 h−3 Gpc3. A new method of defining the ‘homogeneity scale' is presented, which is more robust than methods previously used in the literature, and which can be easily compared between different surveys. Due to the large cosmic depth of WiggleZ (up to z = 1), we are able to make the first measurement of the transition to homogeneity over a range of cosmic epochs. The mean number of galaxies N(< r) in spheres of comoving radius r is proportional to r3 within 1 per cent, or equivalently the fractal dimension of the sample is within 1 per cent of D2 = 3, at radii larger than 71 ± 8 h−1Mpc at z ∼ 0.2, 70 ± 5 h−1 Mpc at z ∼ 0.4, 81 ± 5 h−1 Mpc at z ∼ 0.6 and 75 ± 4 h−1 Mpc at z ∼ 0.8. We demonstrate the robustness of our results against selection function effects, using a Λ cold dark matter (ΛCDM) N‐body simulation and a suite of inhomogeneous fractal distributions. The results are in excellent agreement with both the ΛCDM N‐body simulation and an analytical ΛCDM prediction. We can exclude a fractal distribution with fractal dimension below D2 = 2.97 on scales from ∼80 h−1 Mpc up to the largest scales probed by our measurement, ∼300 h−1 Mpc, at 99.99 per cent confidence.

We present scale-dependent measurements of the normalised growth rate of structure f σ 8 (k, z = 0) using only the peculiar motions of galaxies. We use data from the 6degree Field Galaxy Survey velocity sample (6dFGSv) together with a newly-compiled sample of low-redshift (z < 0.07) type Ia supernovae. We constrain the growth rate in a series of ∆k ∼ 0.03hMpc −1 bins to ∼ 35% precision, including a measurement on scales > 300h −1 Mpc, which represents one of the largest-scale growth rate measurement to date. We find no evidence for a scale dependence in the growth rate, or any statistically significant variation from the growth rate as predicted by the Planck cosmology. Bringing all the scales together, we determine the normalised growth rate at z = 0 to ∼ 15% in a manner independent of galaxy bias and in excellent agreement with the constraint from the measurements of redshift-space distortions from 6dFGS. We pay particular attention to systematic errors. We point out that the intrinsic scatter present in Fundamental-Plane and Tully-Fisher relations is only Gaussian in logarithmic distance units; wrongly assuming it is Gaussian in linear (velocity) units can bias cosmological constraints. We also analytically marginalise over zero-point errors in distance indicators, validate the accuracy of all our constraints using numerical simulations, and demonstrate how to combine different (correlated) velocity surveys using a matrix 'hyper-parameter' analysis. Current and forthcoming peculiar velocity surveys will allow us to understand in detail the growth of structure in the low-redshift universe, providing strong constraints on the nature of dark energy.