Disordered Heterogeneous Universe: Galaxy Distribution and Clustering across Length Scales

The studies of disordered heterogeneous media and galaxy cosmology share a common goal: analyzing the disordered distribution of particles and/or building blocks at microscales to predict physical properties of the medium at macroscales, whether it be a liquid, colloidal suspension, composite material, galaxy cluster, or entire Universe. The theory of disordered heterogeneous media provides an array of theoretical and computational techniques to characterize a wide class of complex material microstructures. In this work, we apply them to describe the disordered distributions of galaxies obtained from recent suites of dark matter simulations. We focus on the determination of lower-order correlation functions, void and particle nearest-neighbor functions, certain cluster statistics, pair-connectedness functions, percolation properties, and a scalar order metric to quantify the degree of order. Compared to analogous homogeneous Poisson and typical disordered systems, the cosmological simulations exhibit enhanced large-scale clustering and longer tails in the void and particle nearest-neighbor functions, due to the presence of quasi-long-range correlations imprinted by early Universe physics, with a minimum particle separation far below the mean nearest-neighbor distance. On large scales, the system appears hyperuniform, as a result of primordial density fluctuations, while on the smallest scales, the system becomes almost antihyperuniform, as evidenced by its number variance. Additionally, via a finite-scaling analysis, we compute the percolation threshold of the galaxy catalogs, finding this to be significantly lower than for Poisson realizations (at reduced density ${\eta }_c=0.25$ in our fiducial analysis compared to ${\eta }_c=0.34$), with strong dependence on the mean density; this is consistent with the observation that the galaxy distribution contains voids of up to 50% larger radius. However, the two sets of simulations appear to share the same fractal dimension on scales much larger than the average intergalaxy separation, implying that they lie in the same universality class. We also show that the distribution of galaxies is a highly correlated disordered system (relative to the uncorrelated Poisson distribution), as measured by the $\tau$ order metric. Finally, we consider the ability of large-scale clustering statistics to constrain cosmological parameters, such as the Universe’s expansion rate, using simulation-based inference. Both the nearest-neighbor distribution and pair-connectedness function (which includes contributions from correlation functions of all order) are found to considerably tighten bounds on the amplitude of quantum-mechanical fluctuations from inflation at a level equivalent to observing 25 times more galaxies. The pair-connectedness function in particular provides a useful alternative to the standard three-particle correlation, since it contains similar large-scale information to the three-point function, can be computed highly efficiently, and can be straightforwardly extended to small scales (though likely requires simulation-based modeling). This work provides the first application of such techniques to cosmology, providing both a novel system to test heterogeneous media descriptors and a tranche of new tools for cosmological analyses. A range of extensions are possible, including implementation on observational data; this will require further study on various observational effects, necessitating high-resolution simulations.

Popular Summary

The distribution of galaxies traces the distribution of matter in the early Universe. As such, it encodes information on a wealth of cosmological parameters, such as the density of matter. An open question is how best to analyze the data. Most works focus on measuring the correlation functions of the galaxy distribution and comparing them to physical models, though this is known to be suboptimal. While many alternatives have been proposed, there is little consensus on which have practical utility, and few are natural from a theoretical standpoint. Here, we show that tools from the theory of disordered heterogeneous media are well suited to characterizing the distribution of galaxies and yield new information on the Universe’s structure and evolution.

The galaxy distribution is a unique 3D set of irregularly arranged points. As such, heterogeneous-media and statistical-mechanical techniques designed to quantify the spatial structure and topology of many-particle material microstructures can be used to understand the galaxy distribution. We use several statistical descriptors from the theory of disordered heterogeneous media to characterize galaxy distributions seen in recent dark matter simulations. We find that the new descriptors can constrain cosmological parameters such as the amplitude of early-Universe quantum fluctuations, yielding bounds that are 5 times tighter than those of conventional statistics, with negligible additional computational requirements.

Employing these tools enables us to extract new information from galaxy distributions and offers new challenges to devise even better statistical descriptors. A key next step will be the application to observational data, which will require updated modeling and consideration of all relevant observational effects.

Figure 1

Schematic depicting the various statistical descriptors used in this work. The black points indicate the positions of random particles (here visualized in ${\mathbb{R}}^2$), with the groups of colored circles (each of diameter $D$) demonstrating the clusters. The pair correlation function $g_2$ counts pairs of particles belonging both to the same cluster (pink lines) and to different clusters (gray lines), whereas the pair-connectedness function $P_2$ contains only particles within the same cluster (pink lines). We show also the nearest-neighbor functions: $H_V$ encodes the distance between a randomly positioned point and the nearest galaxy (red arrows), while $H_P$ gives the separation between a galaxy and its closest neighbor (blue lines). Further details on these statistics are given in Sec. 2.

Figure 2

Example of the cosmological simulations used in this work. This shows a $(250\times 250\times 50)h^{-3}{\mathrm{Mpc}}^3$ slice of the dark matter distribution from a single quijote simulation, with the color bar indicating the fractional density of simulation particles in each pixel. The colored points show the positions of dark matter halos; these are used as a proxy for galaxies. To facilitate analysis of percolation and connectedness, galaxies are assigned to clusters (indicated by the various colors), via a burning (or “friends-of-friends”) algorithm with a separation corresponding to reduced density $\eta$, here set to 0.2. Throughout this work, length is given in cosmologists’ units of megaparsec divided by the reduced expansion rate $h\approx 0.7$), in which the mean pairwise particle separation is $\approx 8h^{-1}\mathrm{Mpc}$.

Figure 3

Measurements of the pair correlation function (a) and structure factor (b) from 1000 quijote cosmological simulations (blue) alongside 1000 Poisson realizations (red). Both sets of simulations have the same number density ($\approx 1\times {10}^{-4}{h}^3{\mathrm{Mpc}}^{-3}$) and a volume of $1h^{-3}{\mathrm{Gpc}}^3$, with a mean pairwise particle separation of $\approx 8h^{-1}\mathrm{Mpc}$. The shaded regions show the statistical variance between realizations (which grows large on large scales), and we note that the two statistics are related by a Fourier transform. In (a), the inset shows the pair correlation function in the typical cosmologists’ normalization, plotting $r^2{h}_2(r)\equiv r^2[g_2(r)-1]$; this clearly brings out the structure imprinted by early Universe physics. At large $r$, we find $h_2(r)\sim r^{-(3+n_s)}$, where $n_s\approx 0.96$: this indicates the presence of quasi-long-range correlations. Similarly, the inset of (b) shows $k\tilde{h}(k)$, equal to the cosmologists’ $kP(k)$ (with a slope of $k^{n_s}$ on large scales). The oscillatory features at $k\sim 0.1h{\mathrm{Mpc}}^{-1}$ arise from acoustic waves in the early Universe.

Figure 4

Local number variance for the cosmological (blue) and Poisson (red) simulations as a function of scale $R$. This is defined as the variance of the number of particles found in spheres of radius $R$ and computed directly from the correlation function shown in Fig. 3. The dashed lines show an extrapolation to large $R$, using the theoretical asymptotic structure factor form [with $\mathcal{S}(k)\sim k^{0.96}$]. The vertical lines show two characteristic scales: the sound horizon at recombination (dashed), which sources acoustic wave in the early Universe, giving the bump in $h(r)$, and the size of the Universe as it transitioned from radiation to matter dominated (dotted).

Figure 5

Comparison of the void and particle nearest-neighbor probability density functions for the cosmological and Poissonian dataset, as defined in Eqs. (14) and (15). Panel (a) shows the probability distribution of finding a void of radius $r$ in the cosmological (blue) and Poisson (red) datasets, averaged over 100 realizations, while panel (b) gives the distribution function of the distance of a given particle from its nearest neighbor. For the Poisson case, the dashed lines show theory curves (defined in Sec. 2c), which are in excellent agreement with the simulations. The void distribution in the quijote simulations follows the (mean-density-matched) Poisson distribution at small $r$, but has an excess of large voids (shown by much broader tails), due to the quasi-long-range correlations. In contrast, the particle distribution $H_P(r)$ differs between the cosmological and Poissonian simulations on all scales, notably with an absence of small separations (due to halo exclusion effects) and an enhancement on large scales.

Figure 6

Pair-connectedness function $P_2(r,\eta )$ for the Poisson (left) and cosmological (right) simulations. Results are shown for a variety of reduced densities $\eta$, corresponding to clustering distances $D(\eta )$ in the range $[10,20]h^{-1}\mathrm{Mpc}$, and shaded regions show the $1\sigma$ deviations expected from statistical fluctuations. The quijote simulations show significantly enhanced correlations on large scales, due to the underlying correlations of matter imprinted in the early Universe. This additionally suggests that the galaxy sample will percolate at lower $\eta$: this will be explored in Sec. 5.

Figure 7

Direct-connectedness function $C_2(r,\eta )$ for the Poisson (left) and cosmological (right) simulations. This follows Fig. 6, but focuses on smaller scales, and additionally includes analytic predictions from the Percus-Yevick model. We additionally normalize all quantities by $g_2(r)$, and exclude values of $\eta$ for which $C_2(r,\eta )$ is not well-behaved (beyond the percolation threshold).

Figure 8

Mean number of particles per cluster $S(\eta ,L)$, as a function of the reduced density $\eta$ and (cubic) simulation box size $L$. We show results for both cosmological data and Poisson realizations, plotting the reciprocal $1/S(\eta ,L)$ averaged over 1000 realizations. Note that $S(\eta )$ is bounded by the number of particles in the dataset ($N$), shown by dotted lines. We additionally show the Percus-Yevick prediction in black, which is inaccurate for all but the smallest $\eta$.

Figure 9

Comparison of the cluster size distribution in the cosmological and Poisson simulations. We plot the mean number of $s$-mers per unit particle (defined as clusters containing $s$ member) as a function of $s$, normalizing to the Poisson prediction. Results are shown for various values of the reduced density $\eta$ and we assume an aperiodic simulation volume of size $L=800h^{-1}\mathrm{Mpc}$, averaging over 100 realizations. At large $\eta$, the two sets of simulations have a similar $n_s$ distribution (at least for small $s$), while at low $\eta$, the enhanced clustering in the cosmological simulations leads to significantly more $s$-mers, with the mean number of particles per cluster increasing with $\eta$ (cf. Fig. 8).

Figure 10

Percolation probability $\mathrm{\Pi }(\varphi ,L)$ obtained from 1000 quijote and Poisson simulations, for various nonperiodic box sizes $L$ and volume filling fractions $\varphi$. This is defined as the fraction of realizations containing a sample spanning cluster, and asymptotes to a Heaviside function centered at the percolation threshold for $L\to \infty$ (shown as a dotted line). Points represent the values computed from simulations, while the solid lines show a fit using the sigmoid function of Eq. (34). Using a finite-scaling analysis, we find the percolation thresholds of ${\eta }_c=0.252$ (0.343) for the quijote (Poisson) simulations, though we caution that the cosmological result depends on the sample density.

Figure 11

Correlation matrix of the pair correlation function ($g_2$), the pair-connectedness function ($P_2$, with reduced density $\eta =0.2$), the three-particle correlation function ($g_3$), and the void nearest-neighbor function ($H_V$) obtained from 512 cosmological simulations run with a fiducial set of cosmological parameters. The correlation matrix is defined as the covariance matrix normalized by its leading diagonal, i.e., $C_{ij}/\sqrt{C_{ii}{C}_{jj}}$, with all values lying in the range $[-1,1]$. The labels show the statistic of interest, which are demarcated by the dotted lines. In each statistic, the bins are ordered from small scale (bottom left) to large scale (top right). For $g_3$, we use four Legendre multipoles $g_{3,\ell }$ with $\ell \in \{0,1,2,3\}$. Notably, the individual bins of $g_{3,0}$ and $P_2$ are highly correlated, and there are a number of nontrivial correlations between observables, though few with $H_V$.

Figure 12

Forecasted constraints on key cosmological parameters using the pair correlation function $g_2$ and the pair-connectedness function $P_2$, using a Fisher forecast (left) and simulation-based inference (right). The dark (light) ellipses represent our 68% (95%) confidence intervals on the various parameters possible from a single galaxy simulation (as in Fig. 2), with the gray regions showing the prior used in the simulation-based approach (from 8192 simulations). We show results for four parameters: $H_0$, giving the Universe’s expansion rate, the matter density ${\mathrm{\Omega }}_m$, the clustering amplitude ${\sigma }_8$, and the total neutrino mass $\sum m_{\nu }$. The diagonal figures give the marginalized constraints on single parameters, while the off diagonals show the correlations; e.g., the bottom left-hand panel shows the correlation between $h$ and ${\sigma }_8$. We find that $P_2$ is a poor predictor of $H_0$ and ${\mathrm{\Omega }}_m$ (shown by the broad contours), but can tightly constrain ${\sigma }_8$. The combination $g_2+P_2$ significantly increases the precision by which this parameter can be measured, and we find similar results from the Fisher forecasts (which are, in general, overoptimistic), and the simulation-based inference (which is usually conservative).

Figure 13

As Fig. 12 but comparing the constraining power from the combination of the pair correlation function with the pair-connectedness function (green), the three-particle correlation function (red), and the void nearest-neighbor function (yellow). The pair-connectedness function and void nearest-neighbor functions are found to perform almost as well as the three-particle function in this scenario (in terms of constraining ${\sigma }_8$) and are much faster to compute and analyze.