Galaxy Clusters

BACKGROUND

What are they? Why are they interesting?
Galaxy clusters are the most massive virialized objects in the Universe. They lie at the nodes of the cosmic web (as marked by circles in the left figure), corresponding to rare peaks of the initial matter density fluctuations. These most dense regions naturally form large gravitational potential wells, and thus most matter and halos accreted or merged into them are likely to be retained.

Due to their large mass, they are rare and their number density drops quickly as their mass increases. As a result, statistical measures of cluster populations depend upon the growth rate of structure and the expansion history of the universe, which in turn are sensitive to cosmological parameters. Thus, clusters are used as probes of cosmology. Additionally, as clusters are recently formed objects, details of their internal structure and surrounding regions can elucidate how halos grow and filamentary structures form.  For such studies, determining their masses is essential. For example, poorly determined or biased cluster masses can cause large errors in cluster abundance measurements, because of the sharp drop in the mass function at the cluster mass scale. These measurements would then result in mistaken estimates of cosmological parameters.

Images of a relaxed cluster, Abell 1835 at three different wavelengths: (a) X-ray, (b) optical, (c) millimeter. Figure credits: X-ray: Chandra X-ray Observatory/A. Mantz; optical: Canada-France-Hawaii Telescope/A. von der Linden et al.; millimeter: Sunyaev Zel’dovich Array/D. Marrone. From Allen et al. (2011).

Cluster mass measurements
However, determining cluster mass is not straightforward. More importantly, direct mass measurement in observations is impossible. In simulations, cluster (halos above a certain mass) masses are directly measured, from the matter density or the sum of the halo dark matter particle masses. These dark matter halos are not directly observable through telescopes. Other components of or near clusters that emit or scatter radiation are observed, instead. Many of the observables are correlated with the underlying dark matter halos, and so are used to compute the total mass of clusters. Understanding how these observables are connected with the mass is one of the key ingredients in cluster studies as discussed above.

Different components of a cluster are revealed depending on the observational wavelength used (an example is shown in the above figure). Thus, multi-wavelength observations can provide a variety of observables, many of which can be used to estimate cluster masses. In order to alleviate systematics and reduce errors in observationally obtained cluster masses, it is becoming common to combine measurements from different (often multi-wavelength) observational techniques. The advantages of complementary information and crosschecks are unfortunately mitigated by the fact that scatters from different observational methods are often correlated. Essentially, as physical properties of the clusters themselves and their environments are often the causes of mass measurement scatters, more than one measurement technique can be affected. It is important include these correlations in order to properly estimate the errors in mass measurements of any individual cluster and to avoid a bias when stacking clusters on one property and measuring another. (For discussion see Rykoff et al. (2008); Cohn & White (2009); Stanek et al. (2010); White et al. (2010))

DISENTANGLING CORRELATED SCATTER IN CLUSTER MASS MEASUREMENTS

We consider five cluster mass measurement methods with an N-body simulation. For the simulation, TreePM code was run with 2043^3 particles in a periodic box with side length 250 Mpc/h (White 2002). Halos are found via Friends of Friends (FoF) (Davis et al. 1985). Galaxies are taken to be resolved subhalos, which are found via Fof6d. The five cluster mass measurement techniques are:
1. Red galaxy richness (Nred) (Richness means the number of galaxies in a cluster),
2. Richness based upon spectroscopy (Nph),
3. Sunyaeve-Zeldovich (SZ) flux, emission by Compton scattering,
4. Velocity dispersion measurements (Vel),
5. Weak lensing (WL) measurements.
The five observables listed above are found along approximately 96 lines of sight for each cluster. We estimate cluster mass based on each measurement along each line of sight. Then we calculate the fractional mass scatters, ‘(estimated mass – true halo mass)/true halo mass’ for our further analysis. As each method doesn’t perfectly measure the true mass and measurements depend on lines of sight, the fraction of mass scatters show a broad distribution (shown in the right figure: Solid lines are the mass estimated via the methods described above ad the dashed line is a least squares Gaussian fit).

Correlated scatter

We find these mass scatters are correlated with each other. An example of cluster mass scatters for one cluster along 95 lines of sight is shown in the left figure. The correlation and covariance for each mass measurement method pair is shown at the top of each panel. Large correlations are present for many pairs of mass measurement methods. This trend persists for many clusters.

Principal Component Analysis for individual clusters

To understand correlated scatters, we perform Principal Component Analysis (PCA). PCA can be used when there are several correlated or covariant quantities. It is essentially a rotation of axes to find linearly independent bases (i.e. quantities which are not covariant or correlated), and is based on a model where some underlying average linear relation is present.  Our starting application will be for individual clusters. For each individual cluster and line of sight, we have several different methods to estimate the true cluster mass. Each line of sight can thus be associated with five numbers, where each number is the mass measured in one method. These numbers can then be thought of as coordinates in some five dimensional abstract space, with each axis in this space corresponding to a different measurement method.

For illustration, we take a set hypothetical measurements for two methods, as shown in the left figure. Each pair of measurements, by the two methods, is a position, i.e. a dot, in this plane labeled by two coordinates. The origin is at the average values. For example, the point marked as a star is M* = 10^{14} e_{red} + 10^{14} e_{WL} = \sqrt{2} \times 10^{14} PC0 + 0PC1. The generalization to more measurement methods and thus a higher dimensional space is immediate. Amongst the orthonormal PCi, we choose PC0 to be along the direction of largest scatter (i.e. to correspond to the largest variance), PC1 to be along the direction of second largest variance, etc.

We apply PCA to the covariances for M_{obs}/M_{true} for each individual cluster, to get a new basis. We find that many clusters have similar PC0 (i.e. the combination of mass scatters that dominates is similar for many of the clusters).

Also, PC0 accounts for 70% of total variances on average (left figure in below). The relatively large contribution from PC0 means that the variance is strongly dominated by the single combination of mass scatters in the direction of PC0.  PC0, PC1, PC2, together comprise almost all the variance for most clusters (right figure in below). The presence of some mass measurement methods with small scatter suggests that there are some directions of the combined measurement methods which would also have small scatter.

Correlations of mass scatter with cluster properties depending on line-of-sight for individual clusters
To get more understanding of the PCA decomposition, we compare values of PCA quantities along lines of sight to cluster properties along those lines of sight. We calculate the correlations between the following two: 1. |cosΘ| between line-of-sight direction and each of directional properties 2. cos Φ of the mass scatter along each line-of-sight to PC0 direction. An example illustration of the two is shown on right. Six specific physical cluster directions that we consider for “1.” are listed in below figure. Note that many of these axes are close to each other in direction. (For filament related properties, e.g. filament plane, halo mass plane, see Filaments.)

Then the full distribution of correlation coefficients between “1.” and “2.” of all clusters in the sample is shown in the left figure. x and y axes represent correlation coefficients and the number of clusters, respectively. Vertical dotted lines denote medians. From top to bottom, the correlation coefficients with the physical cluster directions corresponding to the properties shown from left to right in the figure above. The largest contributions to PC0, for most clusters, seem to come when observing along the long axis of the cluster, but there is a wide scatter. The next largest signals are with the direction orthogonal to the mass plane normal, the orthogonal to the filament plane normal and the direction of the largest substructure.

Principal Component Analysis for physical properties of cluster ensembles
Now we turn to line-of-sight independent properties. We take some characteristic mass scatter scalars for each cluster and compare them with other cluster properties, intrinsic to the cluster or due to its environment, 24 in total. We use correlations rather than covariances, to take out the dimensional dependence. Then we perform PCA. The result of correlation coefficients between many cluster properties and their projection on the first four principal components is shown in the figure on right. Horizontal dotted lines at ±0.4 are to guide the eye to larger positive or negative correlations. The top box is for PC0, and the second is for PC1 and so on. Then we can classify scalar properties of clusters into groups if they show strong and similar tendencies with correlations with PC’s. For example, higher triaxiality clusters tend to have higher richness fractions in their largest subgroup.

It is interesting to think about applying these methods directly to observation. One would need more information, in particular estimates of M_{true}(M_{est}), rather than the oppo-site which we have here. This requires calibrations from simulations which better sample an expected observational sample at the high mass end, and which also include estimated masses from lower mass halos (as well as faithfully reproducing the observational systematics and selection function). Another technical issue is to improve estimates of the correlations and covariances, so that sets of inconsistent measurements can be more easily recognized. It would be very interesting to do such analyses on a larger box, and/or with other measurement methods.