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−3Gpc3. 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−1Mpc at z ∼0.4, 81 ± 5 h−1Mpc at z ∼0.6, and 75 ± 4 h−1Mpc at z ∼0.8. We demonstrate the robustness of our results against selection function effects, using a 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−1Mpc up to the largest scales probed by our measurement, ∼300 h−1Mpc, at 99.99 per cent confidence.
The standard theory of cosmology, ΛCDM, assumes the Universe is homogeneous and isotropic on large scales, and hence can be described by the Friedmann-Robertson-Walker (FRW) metric. However, this is merely an assumption, and it is important for it to be accurately verified by observation. Over the last decade there has been a debate in the literature as to whether the Universe really is homogeneous, or whether it has a fractal-like structure extending to large scales. It is essential to resolve this contention if we are to be justified in assuming the FRW metric.
Inflation, which ΛCDM incorporates, predicts a certain level of density fluctuations on all scales. These density fluctuations induce fluctuations in the metric, δΦ, which are virtually independent of scale, and are on the order of δΦ/c2 ∼10−5. Since these perturbations are small, the FRW metric is still valid, but it means that we expect the Universe to have a gradual approach to large-scale homogeneity rather than a sudden transition.
The most important implication of inhomogeneity is the so-called ‘averaging problem’ in General Relativity (GR). This arises when we measure ‘average’ quantities (such as the correlation function and power spectrum, and parameters such as the Hubble constant) over a spatial volume. In doing so we assume the volume is homogeneous and smooth, when it may not be. Since the Einstein equations are nonlinear, density fluctuations can affect the evolution of the average properties of the volume – this is known as the ‘backreaction mechanism’. If we observe a quantity within such a volume, we need to take averaging into account to compare it with theory. It is therefore essential to know how much inhomogeneity is present, in order to obtain meaningful results from averaged measurements (and so most, if not all, cosmological measurements).
Backreaction has also been proposed as an explanation of dark energy, which is believed to be a negative-pressure component of the Universe that drives the accelerated expansion. Some authors have suggested that instead of introducing exotic new forms of dark energy, or modifications to GR, we should revisit the fundamental assumptions of the ΛCDM model, such as homogeneity. If we assume that GR holds, but take inhomogeneities into account, it can be shown that backreaction can cause a global cosmic acceleration, without any additional dark energy component. This effect appears to be too small to explain the observed acceleration, but highlights the importance of understanding the amount of inhomogeneity in the Universe.
Another, related, consequence of inhomogeneity is that it can affect the path travelled by light rays, and thus the observed redshift of distant galaxies. This can lead to a systematic error in the measurement of the Hubble constant, and other cosmological parameters. Therefore, it is essential to understand the amount of inhomogeneity in the Universe, in order to obtain accurate measurements of these parameters.
