## The Importance of Being Homogeneous

A recent article in New Scientist reminded me that I never completed the story I started with a couple of earlier posts (here and there), so while I wait for the rain to stop I thought I’d make myself useful by posting something now. It’s all about a paper available on the arXiv by Scrimgeour et al. concerning the transition to homogeneity of galaxy clustering in the WiggleZ galaxy survey, the abstract of which reads:

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 (Gpc/h)^3. 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 r^3 within 1%, or equivalently the fractal dimension of the sample is within 1% of D_2=3, at radii larger than 71 \pm 8 Mpc/h at z~0.2, 70 \pm 5 Mpc/h at z~0.4, 81 \pm 5 Mpc/h at z~0.6, and 75 \pm 4 Mpc/h at z~0.8. We demonstrate the robustness of our results against selection function effects, using a LCDM N-body simulation and a suite of inhomogeneous fractal distributions. The results are in excellent agreement with both the LCDM N-body simulation and an analytical LCDM prediction. We can exclude a fractal distribution with fractal dimension below D_2=2.97 on scales from ~80 Mpc/h up to the largest scales probed by our measurement, ~300 Mpc/h, at 99.99% confidence.

To paraphrase, the conclusion of this study is that while galaxies are strongly clustered on small scales – in a complex `cosmic web’ of clumps, knots, sheets and filaments –  on sufficiently large scales, the Universe appears to be smooth. This is much like a bowl of porridge which contains many lumps, but (usually) none as large as the bowl it’s put in.

Our standard cosmological model is based on the Cosmological Principle, which asserts that the Universe is, in a broad-brush sense, homogeneous (is the same in every place) and isotropic (looks the same in all directions). But the question that has troubled cosmologists for many years is what is meant by large scales? How broad does the broad brush have to be?

I blogged some time ago about that the idea that the  Universe might have structure on all scales, as would be the case if it were described in terms of a fractal set characterized by a fractal dimension $D$. In a fractal set, the mean number of neighbours of a given galaxy within a spherical volume of radius $R$ is proportional to $R^D$. If galaxies are distributed uniformly (homogeneously) then $D = 3$, as the number of neighbours simply depends on the volume of the sphere, i.e. as $R^3$, and the average number-density of galaxies. A value of $D < 3$ indicates that the galaxies do not fill space in a homogeneous fashion: $D = 1$, for example, would indicate that galaxies were distributed in roughly linear structures (filaments); the mass of material distributed along a filament enclosed within a sphere grows linear with the radius of the sphere, i.e. as $R^1$, not as its volume; galaxies distributed in sheets would have $D=2$, and so on.

We know that $D \simeq 1.2$ on small scales (in cosmological terms, still several Megaparsecs), but the evidence for a turnover to $D=3$ has not been so strong, at least not until recently. It’s just just that measuring $D$ from a survey is actually rather tricky, but also that when we cosmologists adopt the Cosmological Principle we apply it not to the distribution of galaxies in space, but to space itself. We assume that space is homogeneous so that its geometry can be described by the Friedmann-Lemaitre-Robertson-Walker metric.

According to Einstein’s  theory of general relativity, clumps in the matter distribution would cause distortions in the metric which are roughly related to fluctuations in the Newtonian gravitational potential $\delta\Phi$ by $\delta\Phi/c^2 \sim \left(\lambda/ct \right)^{2} \left(\delta \rho/\rho\right)$, give or take a factor of a few, so that a large fluctuation in the density of matter wouldn’t necessarily cause a large fluctuation of the metric unless it were on a scale $\lambda$ reasonably large relative to the cosmological horizon $\sim ct$. Galaxies correspond to a large $\delta \rho/\rho \sim 10^6$ but don’t violate the Cosmological Principle because they are too small in scale $\lambda$ to perturb the background metric significantly.

The discussion of a fractal universe is one I’m overdue to return to. In my previous post  I left the story as it stood about 15 years ago, and there have been numerous developments since then, not all of them consistent with each other. I will do a full “Part 2” to that post eventually, but in the mean time I’ll just comment that this particularly one does seem to be consistent with a Universe that possesses the property of large-scale homogeneity. If that conclusion survives the next generation of even larger galaxy redshift surveys then it will come as an immense relief to cosmologists.

The reason for that is that the equations of general relativity are very hard to solve in cases where there isn’t a lot of symmetry; there are just too many equations to solve for a general solution to be obtained.  If the cosmological principle applies, however, the equations simplify enormously (both in number and form) and we can get results we can work with on the back of an envelope. Small fluctuations about the smooth background solution can be handled (approximately but robustly) using a technique called perturbation theory. If the fluctuations are large, however, these methods don’t work. What we need to do instead is construct exact inhomogeneous model, and that is very very hard. It’s of course a different question as to why the Universe is so smooth on large scales, but as a working cosmologist the real importance of it being that way is that it makes our job so much easier than it would otherwise be.

P.S. And I might add that the importance of the Scrimgeour et al paper to me personally is greatly amplified by the fact that it cites a number of my own articles on this theme!