Angular Correlations of the X-Ray Background and Clustering of Extragalactic X-Ray Sources

The information content of the autocorrelation function (ACF) of intensity fluctuations of the X-ray background (XRB) is analyzed. The tight upper limits set by ROSAT deep survey data on the ACF at arcmin scales imply strong constraints on clustering properties of X-ray sources at cosmological distances and on their contribution to the soft XRB. If quasars have a clustering radius r_0=12-20 Mpc (H_0=50), and their two point correlation function, is constant in comoving coordinates as indicated by optical data, they cannot make up more 40-50% of the soft XRB (the maximum contribution may reach 80% in the case of stable clustering, epsilon=0). Active Star-forming (ASF) galaxies clustered like normal galaxies, with r_0=10-12 Mpc can yield up to 20% or up to 40% of the soft XRB for epsilon=-1.2 or epsilon=0, respectively. The ACF on degree scales essentially reflects the clustering properties of local sources and is proportional to their volume emissivity. The upper limits on scales of a few degrees imply that hard X-ray selected AGNs have r_0<25 Mpc if epsilon=0 or r_0<20 Mpc if epsilon=-1.2. No significant constraints are set on clustering of ASF galaxies, due to their low local volume emissivity. The possible signal on scales >6 deg, if real, may be due to AGNs with r_0=20 Mpc; the contribution from clusters of galaxies with r_0~50 Mpc is a factor 2 lower.Comment: ApJ, in press (20 July 1993); 28 pages, TeX, ASTRPD-93-2-0

Impact of Systematic Errors in Sunyaev-Zel'dovich Surveys of Galaxy Clusters

Future high-resolution microwave background measurements hold the promise of detecting galaxy clusters throughout our Hubble volume through their Sunyaev-Zel'dovich (SZ) signature, down to a given limiting flux. The number density of galaxy clusters is highly sensitive to cluster mass through fluctuations in the matter power spectrum, as well as redshift through the comoving volume and the growth factor. This sensitivity in principle allows tight constraints on such quantities as the equation of state of dark energy and the neutrino mass. We evaluate the ability of future cluster surveys to measure these quantities simultaneously when combined with PLANCK-like CMB data. Using a simple effective model for uncertainties in the cluster mass-SZ flux relation, we evaluate systematic shifts in cosmological constraints from cluster SZ surveys. We find that a systematic bias of 10% in cluster mass measurements can give rise to shifts in cosmological parameter estimates at levels larger than the $1\sigma$ statistical errors. Systematic errors are unlikely to be detected from the mass and redshift dependence of cluster number counts alone; increasing survey size has only a marginal effect. Implications for upcoming experiments are discussed.Comment: 12 pages, 6 figures; accepted to JCAP; revised to match submitted versio

Cauchy's formulas for random walks in bounded domains

Cauchy's formula was originally established for random straight paths crossing a body $B \subset \mathbb{R}^{n}$ and basically relates the average chord length through $B$ to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length travelled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in $B$. Similar results are obtained also for the average number of collisions performed by the walker in $B$, and an extension to absorbing media is discussed.Comment: 12 pages, 6 figure