The observed growth of massive galaxy clusters – I. Statistical methods and cosmological constraints

A. Mantz 1 2 S. W. Allen 1 2 D. Rapetti 1 2 H. Ebeling 0 0 Institute for Astronomy , 2680 Woodlawn Drive, Honolulu, HI 96822, USA 1 SLAC National Accelerator Laboratory , 2575 Sand Hill Road, Menlo Park, CA 94025, USA 2 Kavli Institute for Particle Astrophysics and Cosmology at Stanford University , 452 Lomita Mall, Stanford, CA 94305-4085, USA A B S T R A C T This is the first of a series of papers in which we derive simultaneous constraints on cosmological parameters and X-ray scaling relations using observations of the growth of massive, X-ray flux-selected galaxy clusters. Our data set consists of 238 cluster detections from the ROSAT All-Sky Survey, and incorporates follow-up observations of 94 of those clusters using the Chandra X-ray Observatory or ROSAT . Here we describe and implement a new statistical framework required to self-consistently produce simultaneous constraints on cosmology and scaling relations from such data, and present results on models of dark energy. In spatially flat models with a constant dark energy equation of state, w, the cluster data yield m = 0.23 0.04, 8 = 0.82 0.05 and w = 1.01 0.20, incorporating standard priors on the Hubble parameter and mean baryon density of the Universe, and marginalizing over conservative allowances for systematic uncertainties. These constraints agree well and are competitive with independent data in the form of cosmic microwave background anisotropies, type Ia supernovae, cluster gas mass fractions, baryon acoustic oscillations, galaxy redshift surveys and cosmic shear. The combination of our data with current microwave background, supernova, gas mass fraction and baryon acoustic oscillation data yields m = 0.27 0.02, 8 = 0.79 0.03 and w = 0.96 0.06 for flat, constant w models. The combined data also allow us to investigate evolving w models. Marginalizing over transition redshifts in the range 0.05-1, we constrain the equation of state at late and early times to be respectively w0 = 0.88 0.21 and wet = 1.05+00..2306, again including conservative systematic allowances. The combined data provide constraints equivalent to a Dark Energy Task Force figure of merit of 15.5. Our results highlight the power of X-ray studies, which enable the straightforward production of large, complete and pure cluster samples and admit tight scaling relations, to constrain cosmology. However, the new statistical framework we apply to this task is equally applicable to cluster studies at other wavelengths. - Clusters of galaxies have a long history as cosmological laboratories, beginning with the discovery of dark matter (Zwicky 1937) through studies of cluster galaxy orbits, and providing early evidence for a low matter density universe (White et al. 1993) through studies of their baryonic and dark matter content. These breakthroughs rested on observations of the properties of individual systems, but the population of clusters as a whole also contains a great deal of cosmological information. Clusters represent the most massive gravitationally bound systems in the Universe, and as such their abundance probes the amount of structure in the Universe and its growth over cosmic time. The local cluster population has been used to jointly constrain the average density of matter, m, and the amplitude of density perturbations, 8 (recently by Henry et al. 2009; Rozo et al. 2010, but see also Reiprich & B ohringer 2002; Allen et al. 2003; Schuecker et al. 2003 and additional references in Mantz et al. 2008, hereafter M08). Most recently, the construction of X-ray flux-limited cluster samples out to redshift z = 0.5 and beyond has enabled studies of the growth of structure (M08; Vikhlinin et al. 2009b). Such studies provide an important new window on cosmology, as various dark energy and modified gravity models designed to explain the acceleration of the cosmic expansion are potentially distinguishable by their effects on structure formation (e.g. Rapetti et al. 2009, 2010; Schmidt, Vikhlinin & Hu 2009). The key ingredients for investigations of the cluster population are (1) a cluster survey with a well-understood selection function,1 and (2) scaling relations that link cluster mass to observable quantities. Currently, the most successful approach to finding massive clusters over a range of redshifts is through the X-ray emission of the hot intracluster gas, although SunyaevZeldovich and optical surveys are also making significant progress. Because Malmquist and Eddington biases are ubiquitous in current X-ray flux-limited samples, great care is warranted in their analysis. In particular, precise, robust cosmological constraints can only be obtained by simultaneously fitting the X-ray luminositymass relation, as we do here. Conversely, rigorous analysis of the scaling relations must take into account the cluster mass function and the selection function of the data set, as discussed by Mantz et al. (2010a, hereafter Paper II; see also Stanek et al. 2006; Pacaud et al. 2007). While a flux-limited sample, combined with more detailed follow-up observations of a subset of clusters in that sample, may contain the necessary information to provide simultaneous constraints on cosmology and scaling relations, a fully self-consistent statistical framework for this analysis has been lacking to this point. By self-consistent, we mean that a single likelihood function applying to the full data set (survey + follow-up observations) and encompassing the entire theoretical model (cosmology + scaling relations) should be derived from first principles, ensuring that the covariance among all the model parameters is fully captured and that the effects of the mass function and selection biases are properly accounted for throughout. This is the first of a series of papers in which we address these issues. Paper II contains details of the follow-up X-ray observations and their reduction, presents the constraints on scaling relations from our simultaneous analysis, and discusses their astrophysical implications. In this paper, we present the statistical methods applied to the problems described above and the resulting cosmological constraints. The dark energy models we address here include the simple cosmological constant model; models with a constant dark energy equation of state, w; and simple evolving w models. As we will show, our analysis produces some of the tightest constraints on dark energy parameters of any experiment to date. In Papers III (Rapetti et al. 2010) and IV (Mantz, Allen & Rapetti 2010b), we respectively apply our analysis to investigations of modified gravity and neutrino properties. In Section 2, we briefly review the cluster data set, which is more fully described in Paper II. Section 3 contains a full description of the theoretical model fit to the data, including the cosmology, cluster mass function and scaling relations. The new, self-consistent analysis method is presented in Section 4.1. Sections 5 and 6 respectivel (...truncated)