Scaling laws in the distribution of galaxies

Past surveys have revealed that the large-scale distribution of galaxies in the universe is far from random: it is highly structured over a vast range of scales. Surveys being currently undertaken and being planned for the next decades will provide a wealth of information about this structure. The ultimate goal must be not only to describe galaxy clustering as it is now, but also to explain how this arose as a consequence of evolutionary processes acting on the initial conditions that we see in the cosmic microwave background anisotropy data. In order to achieve this we need to build mathematically quantifiable descriptions of cosmic structure. Identifying where scaling laws apply and the nature of those scaling laws is an important part of understanding which physical mechanisms have been responsible for the organization of clusters of galaxies, superclusters, and the voids between them. Finding where these scaling laws are broken is equally important since this indicates the transition to different underlying physics. In describing scaling laws it is helpful to make analogies with fractals, mathematical constructs that can possess a wide variety of scaling properties. We must beware, however, of saying that the universe is a fractal on some range of scales: it merely exhibits a specific kind of fractal-like behavior on those scales. The richness of fractal scaling behavior is an important supplement to the usual battery of statistical descriptors. This article reviews the history of how we have learned about the structure of the universe and presents the data and methodologies that are relevant to an understanding of any scaling properties that structure may have. The ultimate goal is to have a complete understanding of how that structure emerged. We are getting close! 

Figure 1

(Color in online edition) Scaling relations in one-dimensional Brownian motion $x\left(t\right)$. In successive zooms the vertical coordinate $\left(x\right)$ is multiplied by 2, while the horizontal coordinate (the time $t$) is multiplied by 4 to properly rescale the curve.

Figure 2

Hierarchical universe. This was very popular at the end of the 19th century and the first half of the 20th century. Courtesy of Edward Harrison, reproduced with permission.

Figure 3

Idealized diagram by de Vaucouleurs, showing two hierarchical frequency distributions of the number of clumps per unit volume. In the top panel there are no characteristic scales in the distribution. This is the model proposed by 216. The bottom panel shows a more sophisticated alternative in which the overall decrease in the number of clumps per unit volume does not behave monotonically with the scale, but it displays a series of local maxima corresponding to the characteristic scales of different cosmic structures: galaxies, groups, clusters, superclusters, etc. Reproduced from 100, with permission of the Editors of the Publications of the Astronomical Society of the Pacific.

Figure 4

The cluster of galaxies Abell 1689 at redshift $z=0.18$ seen by the Hubble Space Telescope with its recently installed Advanced Camera for Surveys (ACS). The arcs observed amongst hundreds of galaxies comprising the cluster are multiple images of far-away individual galaxies whose light has been amplified and distorted by the total cluster mass (visible and dark) acting as a huge gravitational lens. Image courtesy of NASA, N. Benitez (JHU), T. Broadhurst (The Hebrew University), H. Ford (JHU), M. Clampin (STScI), G. Hartig (STScI), G. Illingworth (UCO/Lick Observatory), the ACS Science Team, and the European Space Agency.

Figure 5

A view of the three-dimensional distribution of galaxies in which the members of the Coma cluster have been highlighted to show the characteristic “finger-of-god” pattern. From 66.

Figure 6

Redshift slices: (a) two slices of 4° width and depth $z=0.25$ from the 2dF galaxy redshift survey. From 296; (b) the first CfA-II slice from 96, shown to scale; (c) circular diagram with radius corresponding to redshift $z=0.2$, showing 53 965 galaxies from the Sloan Digital Sky Survey (SDSS). Courtesy of Jon Loveday.

Figure 7

The near-infrared view of the local universe provided by the 2MASS survey. Beyond the Milky Way lying at the Galactic equator, more than 1.5 million galaxies are depicted using a gray-scale code based on their photometrically deduced redshift. From 193, reproduced with permission from the Astronomical Society of Australia and CSIRO Publishing.

Figure 8

(Color in online edition) The agreement between the estimated power spectrum of the cosmic microwave background anisotropies from four different experiments with similar sensitivity. Reproduced from 101, with permission from Blackwell Science Ltd.

Figure 9

(Color in online edition) The angular correlation function from the SDSS as a function of magnitude. The correlation function is determined for the magnitude intervals $18<r*<19$, $19<r*<20$, $20<r*<21$, and $21<r*<22$. The fits to these data, over angular scales of $1^{\prime }–{30}^{\prime }$, are shown by the solid lines. From 77.

Figure 10

The correlation function $1+\xi \left(r\right)$ for different samples calculated with different estimators. We can see that the small-scale fractal regime is followed by a gradual transition to homogeneity.

Figure 11

The (projected) real-space two-point correlation function of the SDSS data. The solid and the dashed lines show different fits corresponding to using the full covariance matrix or just the diagonal, respectively. From 418.

Figure 12

The correlation length as a function of the sample depth for the CfA-II catalog. The observed plateau argues against the fractal interpretation of the galaxy distribution. From 258.

Figure 13

(Color in online edition) The two-point correlation function for ●, the x-ray selected clusters from the ROSAT-ESO Flux-Limited X-ray (REFLEX) survey and ∎, the Las Campanas galaxy redshift survey. The solid and dashed lines are the expected results for a similar x-ray survey in a $\Lambda$ cold-dark-matter model with different values for the cosmological parameters. From 48.

Figure 14

(Color in online edition) The two-point correlation functions for the Abell clusters and two subsamples of the APM survey. The best power-law fits are shown in the plot. From 318. Abell data from 319. APM data from 86.

Figure 15

The correlation length of different cluster samples as a function of the intercluster distance. The solid line shows the relation $r_c=2.6\sqrt{d_c}$ that fits well the observations and the $\Lambda$ cold-dark-matter (LCDM) model. From 16.

Figure 16

(Color in online edition) A multifractal mass distribution over a square of side $L_0$. The distribution was generated following a multiplicative cascade process. The gray scale represents the quantity of mass $\left(X\right)$ in each pixel. Successive enlargements of two different regions of the original plot illustrate the inhomogeneity of the mass distribution.

Figure 17

(Color in online edition) A deep simulated wedge of the universe. Figure by Gus Evrard and Andrzej Kudlicki, courtesy of the Virgo Consortium; details in text.

Figure 18

(Color in online edition) The scaling of the two-point correlation function, for different subsamples of a Voronoi vertices model. The subsamples have been selected according to given “richness” criteria that mimic that of real galaxy clusters. From 390.

Figure 19

(Color in online edition) The halo model. The simulated dark-matter distribution (left panel) and its halo model (right panel). Reproduced from 78, with permission from Elsevier.