The Observed Growth of Massive Galaxy Clusters I: Statistical Methods and Cosmological Constraints

TL;DR

This work introduces a fully self-consistent statistical framework to derive simultaneous constraints on cosmology and X-ray scaling relations from a flux-limited, X-ray selected sample of massive galaxy clusters, using 238 clusters from the ROSAT All-Sky Survey with follow-up Chandra data for 94 of them. The authors model the cluster population through a redshift- and mass-dependent mass function (adopting the Tinker08 form with a redshift evolution parameter) and linking mass to observables via L–M and T–M scaling relations with intrinsic scatter, while rigorously accounting for selection effects and measurement covariances. Applying this framework to the X-ray luminosity function, combined with external CMB, SNIa, f_gas, and BAO data, yields competitive constraints: in flat ΛCDM, Ω_m ≈ 0.23–0.27 and σ_8 ≈ 0.79–0.82; allowing a constant w gives w ≈ -1 with uncertainties around 0.1–0.2, and evolving-w models show no strong evidence for evolution with a DETF FoM ≈ 15.5. The analysis demonstrates that partial follow-up can robustly calibrate scaling relations across a wide mass range, improving constraints beyond previous work, and offers a general approach applicable to cluster studies at other wavelengths and future large surveys.

Abstract

(Abridged) 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 clusters drawn from the ROSAT All-Sky Survey, and incorporates extensive follow-up observations using the Chandra X-ray Observatory. 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 Omega_m=0.23 +- 0.04, sigma_8=0.82 +- 0.05, and w=-1.01 +- 0.20, marginalizing over conservative allowances for systematic uncertainties. These constraints agree well and are competitive with independent data in the form of cosmic microwave background (CMB) anisotropies, type Ia supernovae (SNIa), cluster gas mass fractions (fgas), baryon acoustic oscillations (BAO), galaxy redshift surveys, and cosmic shear. The combination of our data with current CMB, SNIa, fgas, and BAO data yields Omega_m=0.27 +- 0.02, sigma_8=0.79 +- 0.03, and w=-0.96 +- 0.06 for flat, constant w models. For 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 w_0=-0.88 +- 0.21 and w_et=-1.05 +0.20 -0.36. The combined data provide constraints equivalent to a DETF FoM of 15.5. Our results highlight the power of X-ray studies to constrain cosmology. However, the new statistical framework we apply to this task is equally applicable to cluster studies at other wavelengths.

Figures (7)

• Figure 1: Left: Joint 68.3 and 95.4 per cent confidence regions for parameters of the $\Lambda$CDM cosmology from the BCS (blue), REFLEX (green) and Bright MACS (red) cluster samples individually, including all systematic allowances in Table \ref{['tab:paramlist']}. Note that only the 95.4 per cent confidence regions are visible for BCS and REFLEX. Right: Constraints from the full XLF data set (purple) and 5-year WMAP data Dunkley09. Results from the combination of the XLF and WMAP5 data are shown in gray.
• Figure 2: Marginalized posterior distributions for $\Omega_{\mathrm{m}}$ in the $\Lambda$CDM model from analysis of the XLF data (purple, short dashed line; including systematic allowances in Table \ref{['tab:paramlist']}), 5-year WMAP data Dunkley09, cluster $f_{\mathrm{gas}}$ [red, long dashed line;][including conservative systematic allowances]Allen08, SNIa Kowalski08, and BAO Percival07. Note that the XLF and $f_{\mathrm{gas}}$ results are not independent (Sections \ref{['sec:data']} and \ref{['sec:improvements']}).
• Figure 3: Joint 68.3 and 95.4 per cent confidence regions for parameters of the constant $w$ model. Left: constraints on $\Omega_{\mathrm{m}}$ and $w$ from the XLF (purple, including all systematic allowances in Table \ref{['tab:paramlist']}) are compared with those from cluster $f_{\mathrm{gas}}$ data [red;][including conservative systematic allowances]Allen08, 5-year WMAP data Dunkley09, SNIa data Kowalski08, and BAO observations Percival07. Results from combining these 5 data sets are shown in gold. Right: Constraints on $\sigma_8$ and $w$ from XLF and WMAP5 data. The combination of the XLF and WMAP5 yields the gray contours; adding the other data listed above produces the gold contours.
• Figure 4: Joint 68.3 and 95.4 per cent confidence regions for parameters of the constant $w$ model, including conservative systematic allowances. Purple contours indicate constraints from the XLF (which includes six $z<0.15$$f_{\mathrm{gas}}$ clusters; see Sections \ref{['sec:data']} and \ref{['sec:improvements']}), while constraints from all 42 $f_{\mathrm{gas}}$ clusters alone are shown in red. Results combining the XLF data with all 42 $f_{\mathrm{gas}}$ clusters appear in green. The combination of these two types of cluster data with standard priors on $h$ and $\Omega_{\mathrm{b}} h^2$ yields a competitive constraint on dark energy, $w=-1.06 \pm 0.15$.
• Figure 5: Joint 68.3 and 95.4 per cent confidence regions, including conservative systematic allowances, for parameters of the evolving $w$ models. Left: constraints on the $(w_0,w_a)$ model (Equation \ref{['eq:lindermod']}) from the combination of XLF and 5-year WMAP data are shown in gray, as well as the combination of XLF, WMAP5 Dunkley09, cluster $f_{\mathrm{gas}}$Allen08, SNIa Kowalski08 and BAO Percival07 data (gold). Right: Constraints on the $(w_0,w_{\mathrm{et}})$ model (Equation \ref{['eq:wevolmod']}). The transition scale factor is marginalized over the range $0.5<a_{\mathrm{t}}<0.95$. Crosses in each panel indicate the $\Lambda$CDM model, with constant $w=-1$.