Cosmology and Astrophysics from Relaxed Galaxy Clusters II: Cosmological Constraints
This is the second in a series of papers studying the astrophysics and cosmology of massive, dynamically relaxed galaxy clusters. The data set employed here consists of Chandra observations of 40 such clusters, identified in a comprehensive search of the Chandra archive for hot (), massive, morphologically relaxed systems, as well as high-quality weak gravitational lensing data for a subset of these clusters. Here we present cosmological constraints from measurements of the gas mass fraction, , for this cluster sample. By incorporating a robust gravitational lensing calibration of the X-ray mass estimates, and restricting our measurements to the most self-similar and accurately measured regions of clusters, we significantly reduce systematic uncertainties compared to previous work. Our data for the first time constrain the intrinsic scatter in , per cent in a spherical shell at radii 0.8–1.2 ( of the virial radius), consistent with the expected level of variation in gas depletion and non-thermal pressure for relaxed clusters. From the lowest-redshift data in our sample, five clusters at , we obtain a constraint on a combination of the Hubble parameter and cosmic baryon fraction, , that is insensitive to the nature of dark energy. Combining this with standard priors on and provides a tight constraint on the cosmic matter density, , which is similarly insensitive to dark energy. Using the entire cluster sample, extending to , we obtain consistent results for and interesting constraints on dark energy: for non-flat CDM (cosmological constant) models, and for flat models with a constant dark energy equation of state. Our results are both competitive and consistent with those from recent cosmic microwave background, type Ia supernova and baryon acoustic oscillation data. We present constraints on more complex models of evolving dark energy from the combination of data with these external data sets, and comment on the possibilities for improved constraints using current and next-generation X-ray observatories and lensing data.
keywords:cosmological parameters – cosmology: observations – dark matter – distance scale – galaxies: clusters: general – X-rays: galaxies: clusters
The matter budget of massive clusters of galaxies, and specifically the ratio of gas mass to total mass, provides a powerful probe of cosmology (White et al. 1993; Sasaki 1996; Pen 1997; Allen et al. 2002, 2004, 2008; Allen, Evrard, & Mantz 2011, and references therein). In these systems, the mass of hot, X-ray emitting gas far exceeds that in colder gas and stars (e.g. Lin & Mohr 2004; Gonzalez, Zaritsky, & Zabludoff 2007; Giodini et al. 2009; Dai et al. 2010; Leauthaud et al. 2012; Behroozi et al. 2013), and the gas mass fraction, , is expected to approximately match the cosmic baryon fraction, (Borgani & Kravtsov 2011, and references therein). Hydrodynamic simulations of cluster formation indicate that the gas mass fraction at intermediate to large cluster radii should have a small cluster-to-cluster scatter and evolve little or not at all with redshift (Eke et al. 1998; Kay et al. 2004; Crain et al. 2007; Nagai, Vikhlinin, & Kravtsov 2007; Young et al. 2011; Battaglia et al. 2013; Planelles et al. 2013). Increasingly, as simulations have incorporated more accurate models of baryonic physics in clusters, in particular modeling the effects of feedback from active galactic nuclei (AGN) in cluster cores (e.g. McNamara & Nulsen 2007), they have become able to more reliably predict the baryonic depletion of clusters relative to the Universe as a whole. Combining such predictions with measurements of cluster and constraints on , for example from cosmic microwave background (CMB) or Big Bang Nucleosynthesis (BBN) data and direct estimates of the Hubble parameter, provides a uniquely simple and robust method to constrain the cosmic matter density, . The pioneering work of White et al. (1993) was among the first to show a clear preference for a low-density universe with , a result which cluster data continue to support with ever greater precision (Allen et al. 2002, 2004, 2008; Ettori et al. 2003, 2009; Rapetti, Allen, & Weller 2005), and which has been corroborated by a variety of independent cosmological data (e.g. Percival et al. 2007, 2010; Spergel et al. 2007; Kowalski et al. 2008; Mantz et al. 2008, 2010b; Dunkley et al. 2009; Vikhlinin et al. 2009; Rozo et al. 2010; Blake et al. 2011; Komatsu et al. 2011; Hinshaw et al. 2013; Suzuki et al. 2012; Anderson et al. 2014; Benson et al. 2013; Hasselfield et al. 2013; Planck Collaboration 2013a).
Given a bound on the evolution of from theory or simulations, the apparent evolution in values measured from X-ray data can also provide important constraints on the cosmic expansion history and dark energy (Sasaki, 1996; Pen, 1997). This sensitivity follows from the fact that derived values depend on a combination of luminosity and angular diameter distances to the observed clusters, analogously to the way that type Ia supernova probes of cosmology exploit the distance dependence of the luminosity inferred from an observed flux. Allen et al. (2004) provided the first detection of the acceleration of the cosmic expansion from data, and more recently expanded and improved their analysis (Allen et al. 2008, hereafter A08; see also LaRoque et al. 2006; Ettori et al. 2009).
A key requirement for this work is that systematic biases and unnecessary scatter in the measurements be avoided. This can be achieved by limiting the analysis to the most massive, dynamically relaxed clusters available. The restriction to relaxed systems minimizes systematic biases due to departures from hydrostatic equilibrium and substructure, as well as scatter due to these effects, asphericity, and projection (Rasia et al., 2006; Nagai et al., 2007; Battaglia et al., 2013). Similarly, using the most massive clusters minimizes residual systematic uncertainties associated with details of the hydrodynamic simulations, and simplifies the analysis by restricting it to those clusters for which is expected to have the smallest variation with mass or redshift, and the smallest intrinsic scatter (Eke et al., 1998; Kay et al., 2004; Crain et al., 2007; Nagai et al., 2007; Stanek et al., 2010; Young et al., 2011; Borgani & Kravtsov, 2011; Battaglia et al., 2013; Planelles et al., 2013; Sembolini et al., 2013). Moreover, the most massive clusters at a given redshift will be the brightest at X-ray wavelengths and require the shortest observing times.
This paper is the second of a series in which we study the astrophysics and cosmology of the most massive, relaxed galaxy clusters. The first installment (Mantz et al. 2014, in preparation, hereafter Paper I) presents a procedure for identifying relaxed clusters from X-ray data based on their morphological characteristics, and identifies a suitable sample from a comprehensive search of archival Chandra data. In future work (Paper III), we will investigate the astrophysical implications of our analysis of these clusters. This paper presents the cosmological constraints that follow from measurements of for the cluster sample.
Our work builds directly on that of Allen et al. (2002, 2004, 2008). Among our methodological improvements, three stand out as particularly important. First, the selection of target clusters has been automated (Paper I), enabling straightforward application to large samples. Second, the cosmological analysis uses gas mass fractions measured in spherical shells at radii near ,111Defined as the radius within which the mean cluster density is 2500 times the critical density of the Universe at the cluster’s redshift. rather than integrated at all radii . The exclusion of cluster centers from this measurement significantly reduces the corresponding theoretical uncertainty in gas depletion from hydrodynamic simulations.222Improvements in the simulated physics, particularly the inclusion of feedback processes, have also been important in reducing this uncertainty (e.g. Battaglia et al. 2013; Planelles et al. 2013). Third, the availability of robust mass estimates for the target clusters from weak gravitational lensing (von der Linden et al., 2014; Kelly et al., 2014; Applegate et al., 2014, hereafter collectively Weighing the Giants, or WtG) allows us to directly calibrate any bias in the mass measurements from X-ray data, for example due to departures from hydrostatic equilibrium (e.g. Rasia et al. 2006; Nagai et al. 2007; Battaglia et al. 2013) or instrument calibration (Applegate et al., in preparation). In addition, our procedure employs blind analysis techniques (deliberate safeguards against observer bias) including hiding measured gas mass and total mass values until all analysis of the individual clusters was complete.
Section 2 reviews the selection of our cluster sample and basic X-ray data reduction (more fully described in Paper I), as well as the additional analysis steps required to derive . The resulting measurements are presented in Section 3. The cosmology and cluster models we fit to the data are described in Section 4, and Section 5 presents the cosmological results. Section 6 summarizes the differences between our work and A08 (also discussed throughout Sections 2 and 4) and compares their cosmological constraints. In Section 7, we discuss the potential for further improvements in constraints from future observing programs targeting clusters discovered in upcoming surveys. We conclude in Section 8.
For the cosmology-dependent quantities presented in figures and tables, we adopt a reference flat CDM model with Hubble parameter , matter density with respect to the critical density , and dark energy (cosmological constant) density . However, our cosmological constraints are independent of the particular choice of reference (A08 and Section 4).
2 X-ray Data and Analysis
2.1 Cluster Sample
The data set employed here is limited to the most dynamically relaxed, massive clusters known. This restriction is critical for minimizing systematic scatter in the degree of non-thermal pressure in clusters, scatter due to global asymmetry and projection effects, and theoretical uncertainty in the implementation of relevant hydrodynamical simulations, any of which would weaken the final cosmological constraints.
Our selection of massive, relaxed clusters is described in detail in Paper I, and we provide only a short summary here. In Paper I, we introduce a set of morphological quantities which can be measured automatically from X-ray imaging data, as well as criteria for identifying relaxed clusters based on these measurements. In brief, the morphological test is based on (1) the sharpness of the peak in a cluster’s surface brightness profile, (2) the summed distances between centers of neighboring isophotes (similar in spirit to centroid variance), and (3) the average distance between the centers of these isophotes and a global measure of the cluster center (a measure of global asymmetry). The isophotes referred to in (2) and (3) typically cover the radii , a range where the signal to noise ratio is generally adequate, but which deliberately excludes the innermost regions, where complex structure (e.g. associated with sloshing or AGN-induced cavities) is ubiquitous, even in the most relaxed clusters (McNamara & Nulsen, 2007; Markevitch & Vikhlinin, 2007).
This algorithm was run over a large sample of clusters () for which archival Chandra data were available as of February 1, 2013 to generate an initial candidate list. Two additional cuts were then applied. First, to identify the most massive systems, clusters for which the global temperature (either as measured previously in the literature or estimated from X-ray luminosity–temperature scaling relations, e.g. Mantz et al. 2010a) were eliminated. Note that our final temperature requirement, in the relatively isothermal part of the temperature profile, was enforced later, using our own measurements (specifically, the projected, global temperature measured in Section 2.2) and the most recent Chandra calibration information. Second, we identified for each cluster a central region vulnerable to the aforementioned morphological complexities, which was excluded from the mass measurement procedure (Section 2.2). This circular region has a minimum radius of 50 (in our reference cosmology), but can be larger if there are visible disturbances in the cluster gas (e.g. clear cold fronts, which the morphology algorithm may not recognize if they are sufficiently symmetric in appearance or if they occupy sufficiently small cluster radii; see Table 1).333Using a fixed metric radius for the minimum exclusion region is arguably unnecessarily conservative at high redshift, given that the region will extend out to much lower densities relative to the critical density. For the two clusters in our sample, 3C186 and CL J1415.2+3612, we have therefore reduced the minimum exclusion radius to 25. In addition, there are a small number of cases where we excluded particular position angle ranges at all radii from our analysis, as in A08. These are listed in Table 1. Clusters for which this excluded region encompassed the brightest isophote identified in the morphology analysis (i.e. radii ) were removed from the sample.
Beyond the considerations described above and in Paper I, we eliminated three additional clusters from the final sample:
Abell 383: Our surface brightness profile for this cluster (Section 2.2) displays an unusual flattening between and 400 arcsec, before again decreasing at large radii. We can identify no discrete sources in the X-ray data responsible for this. There is a concentration of red galaxies at approximately these radii northeast of the cluster (Zitrin et al., 2012). However, an azimuthally resolved analysis of the X-ray surface brightness (in sectors) appears to show the excess extending over of azimuths, albeit at lower significance. Lacking a good explanation for the source of this apparent excess emission, we have removed the cluster from our sample.
MACS J0326.80043: This cluster satisfies all of our criteria for selection, but the existing data are too shallow to constrain the temperature profile at , a requirement for our measurement.
MACS J1311.00311: The spectral background in the available data does not appear to be well described by the associated Chandra blank-sky field. In particular, an excess of hard emission persists after background subtraction. Rather than attempting to model and subtract this excess, we have opted to remove the cluster from our sample.
The final sample of 40 hot, relaxed clusters used in this work appears in Table 1, along with the exclusion radii used for each, and other relevant information.
2.2 Data reduction, Spectral Analysis and Non-parametric Deprojection
The raw Chandra data were cleaned and reduced, and point source masks were created, as described in Paper I. Blank-field event lists were tailored to each observation, and cleaned in an identical manner. These blank-sky data were renormalized to match the count rates in the science observations in the 9.5–12 band on a per-CCD basis. Variations in foreground Galactic emission with respect to the blank fields were accounted for, as discussed below.
Clusters centers were determined using 0.6–7.0, background-subtracted, flat-fielded images. Initial rough centers were first determined by eye, then centroids were calculated within a radius of 300 kpc about the initial center (or the largest radius possible without including any of the gaps between CCDs). This centroiding process was iterated a further three times to ensure convergence. Individual exposures for a given object were checked for consistency, and generally the results from the longest exposure with good spatial coverage were adopted. The final centers were reviewed by eye and slightly adjusted in some cases, e.g. due to the presence of asymmetry at small cluster radii, the overall strategy being to choose a center appropriate for the large-scale cluster emission.
The spectral analyses described below were all carried out using xspec444http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/ (version 12.8.0). Thermal emission from hot, optically thin gas in the clusters, and the local Galactic halo, was modeled as a sum of Bremsstrahlung continuum and line emission components, evaluated using the apec plasma model (ATOMDB version 2.0.1). Relative metal abundances were fixed to the solar ratios of Asplund et al. (2009), with the overall metallicity allowed to vary. Photoelectric absorption by Galactic gas was accounted for using the phabs model, employing the cross sections of Balucinska-Church & McCammon (1992). For each cluster field, the equivalent absorbing hydrogen column densities, , were fixed to the values from the HI survey of Kalberla et al. (2005), except for cases where the published values are (PKS 0745 and Abell 478; in these cases, was included as a free parameter in our fits, and Table 1 lists the constraints555As discussed in Allen et al. (1993), the absorption towards Abell 478 varies significantly with projected radius within arcmin of the cluster center. We address this by fitting the data two ways. First, we perform our usual analysis, allowing to vary as a function of radius, finding a declining profile consistent with the results of Allen et al. (1993). Second, we perform a fit with a single free value of , but we exclude data at energies for radii arcsec (at radii arcsec, our measured profile is approximately constant and in agreement with Kalberla et al. 2005). The exclusion of low energies makes the modeled spectra insensitive to the column density. The data are of sufficient quality, and the cluster is hot enough, that the temperature profile can be constrained even excluding this soft band from the analysis over much of the cluster. The two approaches yield consistent mass, temperature and gas density profiles; our reported results are those of the second method.). The likelihood of spectral models was evaluated using the Cash (1979) statistic, as modified by Arnaud (1996, the -statistic). Confidence regions were determined by Markov Chain Monte Carlo (MCMC) explorations of the relevant parameter spaces.
We tested for the possibility of contamination by soft Galactic emission components, over and above that modeled by the blank-sky fields, by analyzing cluster-free regions of the data. This test utilized all regions of the detectors at distances from the cluster center (as estimated from the literature, e.g. A08; Mantz et al. 2010a; Andersson et al. 2011), provided that at least half of the relevant CCD was included. Spectra in the 0.5–7.0 band were extracted from each such region, together with appropriate response matrices. Two models were fitted to these spectra: an absorbed power law (photon index ), accounting for unresolved AGN, and the same model plus an absorbed, local, solar-metallicity thermal component. The normalizations of both components were permitted to take both positive and negative values. We compared the best-fitting -statistics for the two models using the distribution, and included a foreground thermal component in subsequent modeling only if the majority of regions tested show evidence for thermal emission at the 95 per cent confidence level. Where regions on different CCDs provided different conclusions, extra weight was given to the CCDs that are better calibrated (e.g. chips 0–3 rather than 6) or are more intrinsically sensitive to soft emission (back- rather than front-illuminated). Whenever a foreground model is required, we always fit it simultaneously with other parameters in all subsequent analysis, using the cluster-free data in addition to cluster spectra.
For the minority of nearby clusters where no appropriate cluster-free regions exist in the data, we performed the analysis described below both with and without a foreground component in the model, and discarded the foreground component if its measured normalization was consistent with zero. Table 1 lists whether a foreground model was required for each cluster.666Failing to account for an excess foreground component will typically enhance the surface brightness attributed to a cluster and reduce its inferred temperature, with the biases becoming more significant with increasing radius as the true cluster signal falls off. The impact on the values that we ultimately use in this work (measured in a shell spanning 0.8–1.2 ; see Section 3.1) depends on a number of factors, including the redshift, temperature and angular extent of each cluster, and the depth of the corresponding observations. Empirically comparing the values derived including and excluding the foreground model for the 22 clusters where our tests find it necessary, we find a bias towards higher values of typically , and as large as in the most extreme case. Here refers to the statistical measurement uncertainty on when the foreground model is erroneously not included. This tends to be slightly smaller than the correct measurement uncertainty.
Next, we constructed background-subtracted, flat-fielded surface brightness profiles for the clusters in two energy bands: 0.6–2.0 and 4.0–7.0. The soft-band profiles were used to identify radial ranges for the subsequent extraction of spectra in concentric annuli. These annuli were chosen to provide a good sampling of the shape of the brightness profile without the signal being dominated by Poisson fluctuations, with the outermost annulus still containing a clear cluster signal above the background.777The outermost radius is refined at a later stage, described below. This extra step is particularly necessary when the blank-sky fields do not account for all the non-cluster emission, e.g. when a strong Galactic foreground is present. The hard-band surface brightness profiles were similarly used to define outermost radii where there was clear cluster signal at energies , a requirement for robustly measuring the temperatures of hot clusters, such as those in our sample. Each cluster thus has three radial ranges defined for it: a central region to be excluded from the mass analysis due to expected dynamical complexity (Section 2.1), a shell at intermediate radii where temperatures can be measured robustly, and a shell at large radii where only surface brightness information is used. For each cluster, we generated source spectra and response matrices, and corresponding blank-field background spectra, for the chosen set of annuli. Source spectra were binned to have at least one count in each channel.
We next carried out an initial “projected” analysis of the cluster spectra. The cluster emission in each annulus was modeled as an absorbed, redshifted thermal component, with independent normalizations in each annulus but linked temperatures and metallicities. Metal abundances were allowed to vary by a fixed ratio relative to the solar values. For this initial analysis, the temperatures and metallicities were fitted only in the intermediate radial ranges identified for each cluster (i.e. excluding the central region and the low signal-to-noise outskirts; see above888In practice, this was accomplished by creating a duplicate spectrum for each annulus in which the energy range 0.6–2.0 was binned to a single channel, with other energies ignored. These “brightness-only” spectra were used in the central and outer radial ranges, whereas full spectra in the 0.6–7.0 band were used in the intermediate radial range.). From these fits, we obtained additional estimates of the foreground model parameters (where applicable) and accurate measurements of the (possibly foreground-subtracted) surface brightness profiles from the normalizations of the cluster components in each of the annuli. Based on these new profiles, we identified and excluded from further analysis any annuli at large radii where the brightness was consistent with zero at 95 per cent confidence, since their inclusion would be problematic for the subsequent spherical deprojection.
The data for this refined set of annuli were then fitted with a non-parametric model for the deprojected, spherically symmetric intracluster medium (ICM) density and temperature profiles (the projct model in xspec). In this model, the cluster atmosphere is described as a set of concentric, spherical shells, with radii corresponding to the set of annuli from which spectra were extracted. Within each shell, the gas is assumed to be isothermal. Given the temperature, metallicity and emissivity (directly related to the density) of the gas in each shell, the spectrum projected onto each annulus can be calculated straightforwardly (e.g. Kriss et al. 1983). For more details and results based on these fits, including the non-parametric thermodynamic profiles for the clusters, see Paper III. For the present work, these profiles provide a way to assess the goodness of fit for the Navarro et al. (1997, hereafter NFW) mass model used in determining cluster masses (below). Specifically, the good agreement between the temperature and density profiles obtained under the assumption of an NFW mass profile in hydrostatic equilibrium with the ICM and the non-parametric temperature and density profiles described above (which make no such assumptions) verifies that clusters in our sample are well described by the NFW-hydrostatic equilibrium model (Figure 1; similar profiles for all clusters in the sample will be presented in Paper III).
2.3 Mass and Profile Constraints
To determine the mass of each cluster, we fit a model that simultaneously describes its three-dimensional mass profile and thermodynamic structure, under the assumptions of spherical symmetry and hydrostatic equilibrium. In this step, we exclude completely data from the central region of each cluster, due to concerns about the validity of these assumptions.999To be precise, we include annuli from the central region in the xspec model, but ignore the corresponding data. Gas temperatures associated with these regions were fixed to broadly reasonable values based on the earlier, non-parametric fits; gas densities are then inferred from these temperatures and the mass profile model. In this way, integrated quantities such as gas masses will provide for the presence of some non-zero amount of gas in the central region, consistent with the remaining model parameters. The gas mass associated with the central region may not be accurate; however, the influence of this exact value on volume-integrated quantities drops rapidly with the outer radius of integration. In particular, the contribution to quantities integrated to is negligible. Note that, in any case, our cosmological analysis uses measurements in a spherical shell that excludes this central region, making these considerations moot for the cosmological results. Otherwise, the data are used similarly to the projected case, with full spectral information at intermediate radii and only surface brightness at large cluster radii.
The model itself is an adaptation of the nfwmass code of Nulsen et al. (2010, distributed as part of the clmass package for xspec; see also Appendix A).101010The nfwmass code contains an option to account for projected emission from spherical radii larger than those otherwise included in the model (i.e. beyond the spatial extent of the employed data) by assuming a -model continuation of the surface brightness profile. Our sole modification is to set the parameter of this model dynamically, by requiring that the slope of the density (or surface brightness) profile be continuous across this boundary. That is, the value of is set based on the predicted density profile in the outermost shell of the model, itself determined by the mass profile model and the temperature in that shell. The ICM is again described as a series of concentric, isothermal shells. The mass profile of the cluster is modeled by the NFW form, with two free parameters. Under the assumption of hydrostatic equilibrium, this piecewise-constant temperature profile and NFW mass profile determine the gas density profile up to an overall normalization. (In contrast, the non-parametric model fit in Section 2.2 allows the temperature and density profiles to be independent, but without additional assumptions it provides no information about the mass.) We have argued elsewhere (Mantz & Allen, 2011) that “semi-parametric” models of the kind used here, combining a non-parametric description of the ICM with a theoretically well motivated, parametrized model for the mass profile, presently provide the least biased approach to X-ray mass determination, given that current data cannot meaningfully constrain non-parametric mass profiles. In contrast, the common assumption of parametrized forms for both the ICM density and temperature profiles represents a complex and non-intuitive prior on the mass profile, and is more constraining than the data require.
A convenient feature of nfwmass is that the model itself is completely independent of cosmological assumptions. That is, the fitting procedure described above requires no assumptions about cosmology. The parameter constraints translate to profiles of mass, gas density and temperature (hence also pressure and entropy) of a cluster in physically unmeaningful units, which can be related to physical quantities through a cosmology-dependent factor (see details in Appendix A). By keeping the results in this cosmology-independent form, and by furthermore multiplying the gas density and total mass profiles of each cluster by different, random values when evaluating the results of individual cluster fits, we effectively blinded ourselves to the value of each cluster, the level of agreement among clusters, and any trends with redshift, until the analysis of all clusters was complete and final.
The analysis in Section 2.3 produces temperature, gas density and mass profiles for each cluster, from which gas mass fraction profiles can be derived (see also Appendix A). This section presents those results; for ease of interpretation, these are displayed for a reference CDM cosmology with , and . Uncertainties for each cluster are based on the distribution of MCMC samples from the spectral analysis, and incorporate the statistical uncertainty in the science observations themselves, the modeling of the astrophysical and instrumental background using the blank-field data, and the constraints on foreground contamination (where applicable).
3.1 Profiles and Cosmological Measurements
Figure 2 shows the differential profiles (i.e. the ratio of gas mass density to total mass density) for the relaxed cluster sample as a function of overdensity, , where is the critical density. The left panel of the figure contains the 13 lowest-redshift clusters (), while the right panel shows the entire sample. For each cluster, we show results only in the radial range where temperature measurements were performed. While there is greater dispersion at small radii, the profiles largely converge and have small scatter at (). Outside the cluster centers, the profiles rise with a regular, power-law shape, for , or equivalently for . At larger radii, fewer than the full sample of 40 clusters provide data; nevertheless the measured profiles remain consistent with this power law, with no indication of flattening.
To investigate the intrinsic scatter as a function of radius, we extracted gas mass fractions for each cluster in a series of spherical shells, spanning radial ranges of width . The data for each shell were fitted with a linear function of redshift, to approximately marginalize any cosmological signal (Section 4), with the fractional intrinsic scatter as a free parameter. The results of this exercise are shown in the left panel of Figure 3, with the scatter minimized at radii and significantly increasing at smaller radii.
The right panel of Figure 3 compares our cumulative profiles (i.e. integrated within a sphere) to the simulations of Battaglia et al. (2013). These simulations include the effects of cooling and star formation, as well as heating from AGN feedback, on the ICM, and we specifically plot their results for relatively massive () and relaxed clusters, where relaxation is defined in terms of the ratio of kinetic to thermal energy. Our measurements are displayed as a dark (light) shaded blue regions, corresponding to 68 (95) per cent confidence at each radius, and representing the combined effect of measurement uncertainties and intrinsic scatter; the thick, blue line is the median profile across the cluster sample, again accounting for the measurement uncertainties for each cluster. For context, the horizontal, dot-dashed line shows the cosmic baryon fraction measured by Planck (Planck Collaboration, 2013a). We note very good agreement between the shapes of the simulated and measured profiles over a wide range in radius, encompassing the radii of interest for the cosmological measurements, and extending to (where our data become increasingly noisy and other astrophysical effects, such as gas clumping, may become important; e.g. Simionescu et al. 2011; Urban et al. 2014; Walker et al. 2013).111111Note that the agreement in the normalization of the profiles, while also good, is irrelevant, since the simulations only directly address the depletion parameter, . In Figure 3, we have scaled the predicted depletion profile by the cosmic baryon fraction adopted in the simulations. Note that incompleteness (in the sense that fewer than 40 clusters contribute to the results; see Figure 2) increases rapidly beyond ; while it is not clear that selection effects should introduce any particular bias in this case, the combined profile should be treated with caution at large radii.
Our cosmological analysis uses the gas mass fraction integrated within a shell spanning , which is shown as a shaded, vertical band in Figures 2 and 3 (for a typical NFW concentration parameter). The exclusion of smaller radii is intended to minimize both uncertainties in the prediction of the gas depletion factor from hydrodynamic cluster simulations (see Section 4.2) and the intrinsic scatter seen at small radii in the figures, which should result in tighter cosmological constraints. At the same time, temperature profiles (and thus ) cannot be reliably measured at radii much larger than for most clusters, as can be seen in Figure 2. In practice, the 0.8–1.2 shell represents a good compromise between these considerations and the need to maintain good statistical precision of the measurements. Table 2 contains our measurements in this shell, along with masses within and redshifts for each cluster. Note that the tabulated values are marginalized over the uncertainty in (or , equivalently).
The behavior with redshift of measured in the 0.8–1.2 shell (for the adopted reference cosmology with and ) is shown in the left panel of Figure 4. Qualitatively, it is clear that there is little or no evolution with redshift for this cosmological model. The right panel of the figure shows the values derived from the same data, but assuming a cosmology with no dark energy and ; for this model, there is an evident redshift dependence. As described more fully in Section 4, this dependence of the apparent evolution of on the cosmic expansion is the basis of dark energy constraints using these data.
3.2 Mass Dependence
Hydrodynamic simulations of cluster formation generally predict a mild increasing trend of the cumulative gas mass fraction, e.g. , with mass when fit over a wide mass range extending from group to cluster scales (e.g. Young et al. 2011; Battaglia et al. 2013; Planelles et al. 2013, and other references in Section 1). Comparison of values measured for groups and intermediate-mass clusters supports this picture (Sun et al., 2009). It is less clear whether an increasing trend persists at the high masses relevant for this work; measurements by Vikhlinin et al. (2006) and A08 are both consistent with being constant with temperature (hence with mass) for clusters. We address this question with the current data in Paper III. Here, we are concerned only with a possible mass trend of integrated in a shell about , which has not been studied previously in either simulations or real data.
In Figure 5, we show our measurements in the shell versus . Also shown is the best-fitting power-law – relation (and 95.4 per cent limits), derived using the Bayesian regression code of Kelly (2007). Critically, this method accounts for both intrinsic scatter in and the significant anti-correlation between measured values of and (typical correlation coefficients ). The best-fitting slope is slightly negative and consistent with zero (; 68.3 per cent confidence limits).121212This question can be investigated in a less cosmology-dependent way by incorporating a power-law mass dependence into the model given below in Section 4 and fitting for the slope of this relation simultaneously with the full set of model parameters, again accounting for the anti-correlation between and mass measurements. In this way, uncertainty in the cosmic expansion history can be straightforwardly marginalized over. We obtain consistent results from this analysis, with no evidence for a trend in the shell value. As there is no theoretical motivation for a decreasing trend with mass at radii , and since marginalizing over an – slope has a negligible effect on our cosmological constraints, we fix the mass dependence to zero in the subsequent sections.
3.3 Intrinsic Scatter
Thanks to new X-ray observations obtained since A08, our data are now precise enough to detect the presence of intrinsic scatter in the measurements. This scatter reflects cluster-to-cluster variations in gas depletion, non-thermal pressure, asphericity, and departures from the NFW mass model. A log-normal scatter in , , is included in the complete model described in Section 4 and constrained simultaneously with the rest of the parameters in all our subsequent results. However, constraints on the scatter itself are independent of the cosmological model employed; we find . This 7.4 per cent intrinsic scatter in corresponds to only per cent intrinsic scatter in the cosmic distance inferred from a single cluster (Section 4.4).
Qualitatively, Figure 4 appears to show an increase in scatter from to , although the highest redshift points again appear to have little dispersion. Although a trend of scatter with redshift is certainly astrophysically plausible for the cluster population at large, it is not clear that we should expect one for a sample which is restricted to the hottest, most dynamically relaxed clusters at all redshifts. To test for such a trend, we break the data into the redshift ranges 0.0–0.2, 0.2–0.3, 0.3–0.4, 0.4–0.5 and 0.5–1.1, respectively containing 5, 8, 12, 8 and 7 clusters, and fit each subset individually.131313Specifically, we marginalize over non-flat CDM models with , and (see Section 4.1). The cosmological parameters are not well constrained by these sub-samples of the data (though see Section 5.1), but this procedure effectively marginalizes over a wide range of plausible cosmic expansion histories within each redshift bin. The constraints on the intrinsic scatter in each bin agree at confidence. Consistently, a weighted linear regression on using these measurements finds no evidence for a non-zero slope with redshift. We henceforth adopt a constant-scatter model throughout this work, while noting that the possibility of evolution will be an interesting question to return to as the number of known high-redshift relaxed clusters continues to grow.
Observationally, we cannot distinguish between the various possible causes of scatter at this point (though a larger weak lensing/X-ray calibration sample, coupled with ASTRO-H or other X-ray measurements of gas motions, may eventually directly constrain the scatter in non-thermal support), but note that the observed per cent scatter places an upper limit on the individual contributions of the sources mentioned above. This limit is consistent with expectations; for example, the simulations of Battaglia et al. (2013) indicate a fractional scatter of per cent in the integrated gas depletion for massive, relaxed clusters. A similar level of dispersion is expected due to non-thermal pressure (Nagai et al., 2007; Rasia et al., 2012; Nelson et al., 2014).
This section describes the complete model fitted to the data, including descriptions of both the cosmological expansion and the internal structure of clusters. Table 3 summarizes the parameters of the cluster model and associated priors, as well as the parametrization of the cosmological background used when analyzing cluster or supernova data alone (discussed in more detail below). For completeness, Table 4 provides the equivalent information for the alternative cosmological parametrization used when analyzing CMB or baryon acoustic oscillation (BAO) data, either alone or in combination with other data sets (this is the standard parametrization in cosmomc).
4.1 Cosmological Model
In this paper, we consider cosmological models with a Friedmann-Robertson-Walker metric, containing radiation, baryons, neutrinos, cold dark matter, and dark energy. We adopt an evolving parametrization of the dark energy equation of state (Rapetti et al., 2005),
where is the scale factor. In this model, takes the value at the present day and in the high-redshift limit, with the timing of the transition between the two determined by . Equation 1 contains as special cases the cosmological constant model (CDM; and ), constant- models (), and the simpler evolving- model adopted by Chevallier & Polarski (2001) and Linder (2003) (). A08 provide details on the calculation of cosmic distances using this model.
Beyond the dark energy equation of state, the relevant cosmological parameters for the analysis of cluster data are the Hubble parameter and the present-day densities of baryons, matter, and dark energy. As noted in Appendix A, the interpretation of our X-ray data also depends (extremely weakly) on the primordial mass fraction of helium, . This we derive self-consistently from the baryon density, , assuming the standard effective number of neutrino species, , using the BBN calculations of Pisanti et al. (2008, see also ). We note, however, that simply taking results in identical cosmological constraints from the data.
4.2 Gas Depletion
Following A08, we describe the depletion of X-ray emitting gas in the 0.8–1.2 shell relative to the cosmic baryon fraction as , where and parametrize the normalization and evolution of this quantity. Key differences from previous work are the use of in a shell rather than the cumulative quantity , and the fact that we model directly the hot gas depletion rather than both the baryonic depletion and the ratio of mass in stars and cold gas to hot gas. The latter development is due to improvements in hydrodynamical simulations of cluster formation, which now account for a realistic amount of energy feedback from AGN in addition to radiative cooling and star formation. The decision to make our measurements in spherical shells excluding the clusters’ centers makes the predictions from simulations yet more reliable.
Specifically, we consider the recent simulations of Battaglia et al. (2013) and Planelles et al. (2013), which implement both cooling and AGN feedback in the smoothed particle hydrodynamics (SPH) framework. The gas depletion from these simulations is shown in Figure 6, evaluated both in a sphere of radius and in a spherical shell encompassing . The figure shows that the results of the two independent simulations are in much closer agreement for the spherical shell, excluding the cluster center, than for the full volume. Agreement between the two is at the per cent level, similar to the level of agreement between these entropy-conserving SPH codes and simulations using adaptive mesh refinement (e.g. Kravtsov et al. 2005). On this basis, we adopt a uniform prior on centered on (the average of the two cooling+feedback simulation results) and with a full width of 20 per cent, shown by a shaded band in the figure. Note that this conservative prior also encompasses the depletion values derived from the adiabatic and cooling-only simulations of Planelles et al. (2013) for the 0.8–1.2 shell.
The available information from the published simulations is insufficient to repeat this exercise at to obtain a prior on for the shell. However, both works do consider the evolution of the cumulative depletion factor for cooling+feedback models. Neither set of simulations shows evidence for evolution in the gas depletion in massive clusters at the radii of interest (see Figure 10 of Battaglia et al. 2013 and Figure 7 of Planelles et al. 2013). We therefore adopt a conservative uniform prior .141414Planelles et al. (2013) suggest a prior for the baryonic (not gas) depletion. This range encompasses cumulative results at both and for adiabatic and cooling-only simulations in addition to cooling+feedback. Given that the only results in that work that display a trend with redshift apply to the baryonic depletion within in simulations without feedback (in particular, the gas depletion is always consistent with zero evolution), we have chosen to adopt a prior whose width is similar to the Planelles et al. (2013) recommendation, but which is centered at zero.
|Cosmic baryon fraction,|
|Total matter density normalized to|
|Dark energy density normalized to|
|Present-day dark energy equation of state|
|Evolution parameter for|
|Transition scale factor for|
|Primordial helium mass fraction|
|Early-time dark energy equation of state|
|Clusters||Gas depletion normalization|
|Gas depletion evolution|
|Power-law slope of shell|
|Intrinsic scatter of shell measurements|
|Mass calibration at|
|Mass calibration evolution|
|Intrinsic scatter of lensing/X-ray mass ratio|
|Cold dark matter density|
|Angular size of the sound horizon at last scattering|
|Effective density from spatial curvature|
|Present-day dark energy equation of state|
|Evolution parameter for|
|Transition scale factor for|
|Optical depth to reionization|
|Scalar power spectrum amplitude|
|Scalar spectral index|
|Species-summed (degenerate) neutrino mass in eV|
|Effective number of neutrino species|
|Primordial helium mass fraction|
|Early-time dark energy equation of state|
4.3 Measurement, Calibration and Scatter
Any inaccuracies in instrument calibration, as well as any bias in measured masses due to substructure, bulk motions and/or non-thermal pressure in the cluster gas, will cause the measured values of to depart from the true values. With the advent of robust gravitational lensing measurements (WtG), these effects can now be directly constrained from data.151515Strictly speaking, the lensing data can only calibrate bias in the X-ray mass determinations, not any bias in gas masses. However, the current level of uncertainty in total mass, per cent, is significantly greater than the systematic uncertainty in the flux calibration of Chandra (for example, taking the level of disagreement between the ACIS and XMM-Newton detectors as the scale of the uncertainty). The lensing mass measurements themselves are expected to be unbiased (see Becker & Kravtsov 2011 and WtG). From the 12 clusters in common between this work and the WtG sample, we (Applegate et al., in preparation) find a mean weak lensing to Chandra X-ray mass ratio of for our reference cosmology.161616This analysis incorporates allowances for systematic uncertainties, as detailed in WtG. In particular, systematics associated with galaxy shear measurements, photometry and projection are individually controlled at the few per cent level.171717Note that an underestimate of the total mass by the X-ray analysis, as one might expect due to non-thermal support (e.g. Nagai et al. 2007), would correspond to values . The measurement of (albeit at a relatively low confidence level) implies that temperature measurements based on fitting the Bremsstrahlung continuum to Chandra observations (with the current calibration) may be overestimated by per cent at the typical temperatures of our cluster sample (5–12). This estimate would place the “correct” temperatures roughly midway between Chandra ACIS and XMM-Newton MOS results from continuum fitting, and in broad agreement with results from fitting the Fe emission line with either instrument (8th IACHEC meeting; http://web.mit.edu/iachec/meetings/2013/index.html). See Applegate et al. (in preparation) for more details.
This constraint has a mild dependence on the cosmological background, due to the dependence of the lensing signal on angular diameter distances. Rather than taking the above result as a prior, therefore, we directly incorporate the data and analysis used by Applegate et al. into our model (see that work for details of the gravitational lensing likelihood calculation). Specifically, we model the mean ratio of lensing to X-ray mass as , with a log-normal intrinsic scatter, and constrain these parameters simultaneously with the rest of the model. The evolution parameter, , cannot be constrained by the 12 clusters in the calibration sub-sample; while there is no particular theoretical expectation for evolution in, e.g., the amount of non-thermal pressure in the most relaxed clusters, we nevertheless marginalize over a uniform prior .
Additionally, we must account for the fact that our X-ray measurements are made under the assumption of a particular reference cosmological model. The tabulated values are thus proportional to , where is the true cosmic distance to the cluster, and is the distance evaluated assuming the reference model.181818We do not distinguish between angular diameter and luminosity distances in this section, but see Appendix A. Another, smaller dependence arises through the dependence of the reference value of (actually the equivalent angular radius, ) on the critical density, . For a given trial cosmology, we need to predict the gas mass fraction in the reference measurement shell rather than the true 0.8–1.2 shell (according to the trial cosmology’s ). As in A08, we take advantage of the fact that the profiles of our clusters are consistent with a simple power law at the relevant radii (Figure 2). For each cluster, we fit a power-law model to the function , as varies from 0.7 to 1.3; averaging over the cluster sample, we find a power-law slope of .
Including these terms, the complete model which we fit to the data is
Equation 4.3 represents the predicted mean for each of our cluster measurements. In addition, we model and fit for a log-normal intrinsic scatter in the measured value, , as described in Section 3.3. The measurement errors of , after marginalizing over , are approximately log-normal as well, and we model them as such. The likelihood associated with each cluster thus has a simple, Gaussian form, with mean given by the logarithm of Equation 4.3 and variance equal to the sum of and the square of the associated fractional measurement error.191919The log-normal form was chosen for computational convenience. However, we have explicitly verified that our cosmological constraints are unchanged if the intrinsic scatter and measurement errors are instead modeled as Gaussian. The residuals from the best fit are consistent with either hypothesis, reflecting the fact that the two distributions are similar for small values of the total fractional scatter.
4.4 Summary of the Model and Priors
Along with the intrinsic scatter in , Equation 4.3 constitutes a complete model for the X-ray measurements. The normalization of this function depends on the product , and is systematically limited by the nuisance parameters and . In practice, the calibration parameter, , dominates the error budget, with the statistical uncertainty on the mean value, especially at low redshift, being small. Section 5.1 outlines the constraints on cosmological parameters obtained from the low-redshift clusters, for which uncertainties related to the model of dark energy and the evolution of the depletion factor () are negligible. In particular, combining the low-redshift cluster data with priors on and produces a tight constraint on which is independent of the cosmic expansion.
The redshift dependence of provides constraints on dark energy, through the dependence. This is illustrated in Figure 7, which shows our data along with the predictions (from Equation 4.3) of three dark energy models. The normalizations of the model curves have been fitted to the cluster data to demonstrate the difference between models that might be acceptable to those low-redshift data alone.202020Note that this normalization effectively measures , as described above. Hence, it is instructive to compare the curves for various models of dark energy but with the same value of , as in the figure. Our sensitivity to the redshift-dependent signal is limited by the systematic uncertainty represented by and , and the sparsity of data at redshifts ; in practice, the latter dominates the uncertainty on dark energy parameters from current data (see also Section 7).
4.5 Fitting the Models
The cluster model described in the preceding sections, and the associated likelihood evaluation, have been coded into a stand-alone library that can straightforwardly be linked to cosmomc212121http://cosmologist.info/cosmomc/ or other software.222222http://www.slac.stanford.edu/~amantz/work/fgas14/ The results presented here were produced using cosmomc (Lewis & Bridle 2002; October 2013 version). Cosmological calculations were evaluated using the camb package of Lewis, Challinor, & Lasenby (2000), suitably modified to implement the evolving- model of Rapetti et al. (2005), including the corresponding dark energy density perturbations.232323To calculate the dark energy perturbations in evolving- models, we do not use the standard Parametrized Post-Friedmann (PPF) framework in cosmomc, but rather an extension of the fluid description used for constant- models. Especially for cases far from CDM, this gives us more accurate results by construction. We have verified that the prescription we use to avoid the divergence at the crossing of the phantom divide () allows us to appropriately match the PPF results designed to overcome that theoretical problem (Fang, Hu, & Lewis, 2008).
In Section 5, we compare and combine our cosmological constraints with those of other cosmological probes. Specifically, we include all-sky CMB data from the Wilkinson Microwave Anisotropy Probe (WMAP 9-year release; Bennett et al. 2013; Hinshaw et al. 2013) and the Planck satellite (1-year release, including WMAP polarization data; Planck Collaboration 2013b), as well as high-multipole data from the Atacama Cosmology Telescope (ACT; Das et al. 2013) and the South Pole Telescope (SPT; Keisler et al. 2011; Reichardt et al. 2012; Story et al. 2013). For these data, we use the likelihood codes provided by the WMAP242424http://lambda.gsfc.nasa.gov (December 2012 version) and Planck252525http://pla.esac.esa.int/pla/aio/planckProducts.html teams, where the latter also evaluates the ACT and SPT likelihoods. When using CMB data, we marginalize over the default set of nuisance parameters associated with each data set in cosmomc (e.g. accounting for the thermal Sunyaev-Zel’dovich effect and various astrophysical foregrounds). In addition, we include the Union 2.1 compilation of type Ia supernovae (Suzuki et al., 2012) and BAO data from the combination of results from the 6-degree Field Galaxy Survey (6dFGS; ; Beutler et al. 2011) and the Sloan Digital Sky Survey ( and ; Padmanabhan et al. 2012; Anderson et al. 2014). For these data sets, likelihood functions are included as part of cosmomc.
5 Cosmological Results
This section presents the cosmological constraints obtained from our analysis of the cluster data. Section 5.1 discusses the constraints available from the lowest redshift clusters, with minimal external priors. The subsequent sections explore progressively more complex cosmological models using the cluster data, as well as independent cosmological probes. When combining data sets, we consider separately combinations which include WMAP or Planck CMB data. For simplicity, the figures and discussion in this section refer to the WMAP version of these results. The combined results using Planck data are quantitatively similar; for completeness we include the corresponding figures in Appendix B. Our results are summarized in Tables 5 and 6.
5.1 Dark Energy-Independent Constraints from Low-Redshift Data
The amount and nature of dark energy have a very small effect on cosmic expansion at the lowest redshifts in our data set, in particular for the 5 clusters with .262626We have explicitly verified that our cluster results in this section are identical whether we marginalize over CDM or flat, constant- models. This insensitivity is not absolute; for example, it breaks down if the dark energy equation of state is allowed to evolve rapidly at redshifts , as in our most general dark energy model. To the extent that the cosmology-dependent curvature of and the variation of are negligible over this redshift range, Equation 4.3 reduces to
As our data are very precise for these nearby clusters, constraints on the product will be systematically limited, specifically by the calibration parameter, (Table 3). We obtain .272727Note that this result marginalizes over the complete model; the simplified form in Equation 4 is for illustration only.
Figure 8 shows this constraint from cluster in the – plane, along with measurements of the local Hubble expansion (Riess et al., 2011) and the tight constraints for flat CDM models from WMAP and Planck. The data are consistent with all of these data individually, although the figure shows clearly the tension in the value of derived from Planck compared with that from the local distance ladder (). Combining the WMAP and data for flat CDM models, we obtain a constraint on the Hubble parameter, , consistent with Planck.
The cluster constraint on can be combined with direct Hubble parameter measurements of Riess et al. (2011) to obtain a CMB-free constraint on the cosmic baryon fraction. Applying their constraint of , we find , consistent with the best-fitting WMAP-only and Planck-only values at the and levels, respectively.282828Note that using instead the results of the Carnegie Hubble Project, (Freedman et al., 2012), shifts this constraint by per cent. When we additionally use a prior on , below and in subsequent sections, the influence of is even smaller (the residual dependence being ; see Equation 4). The effect on dark energy constraints in later sections is completely negligible. Alternatively, using a prior on from BBN data allows the low- clusters to constrain the combination . We employ a prior based on the deuterium abundance measurements of Cooke et al. (2014), which yields . Combining priors on both and with our measurement of provides a direct constraint on (e.g. White et al. 1993). We find from the clusters, in good agreement with the full data set (below), as well as the combination of CMB data with other probes of cosmic distance (e.g. Hinshaw et al. 2013; Planck Collaboration 2013a).
The above priors on and constitute the “standard” priors that we use together with the cluster data in subsequent sections (Table 3). In models where the equation of state of dark energy is a free parameter, CMB data provide a relatively weak upper bound on . However, because the CMB still tightly constrains in this case, the combination of CMB and data provides tight constraints on both and (see also A08). Consequently, we do not require or use the priors on and in later sections where the data are used in combination with CMB measurements.
5.2 Constraints on CDM Models
For non-flat CDM models, the constraints obtained from the full data set (plus standard priors) are shown as red contours in Figure 9. We obtain and , with relatively little correlation between the two parameters, as can be seen in the figure. Also shown in Figure 9 are independent constraints from WMAP+ACT+SPT (hereafter CMB; Keisler et al. 2011; Hinshaw et al. 2013; Reichardt et al. 2012; Story et al. 2013; Das et al. 2013), type Ia supernovae (Suzuki et al., 2012) and BAO (Beutler et al. 2011; Padmanabhan et al. 2012; Anderson et al. 2014), where the latter constraints also incorporate our standard priors on and . The four independent data sets are in good agreement. Combining them (without additional priors), we obtain tight constraints strongly preferring a flat universe: and individually, with .
5.3 Constraints on Constant- Models
We next consider spatially flat models with a constant dark energy equation of state, . The constraint on is , identical to the CDM case. Our constraint on the equation of state is . The constraints appear in the left panel of Figure 10 along with independent constraints from CMB, supernova and BAO data, and the combination of all four. Again, the different cosmological probes are in good agreement; from the combination we obtain and .
Allowing global spatial curvature in the model, the combination of , CMB, supernova and BAO data yields and (right panel of Figure 10), again consistent with the flat CDM model.
5.4 Constraints on Evolving- Models
Allowing the parameter in Equation 1 governing the evolution dark energy equation of state, , to be free, we investigate the constraints available from the combination of , CMB, supernova and BAO data in two cases: fixing the transition scale factor at (i.e. the model is that of Chevallier & Polarski 2001 and Linder 2003) and marginalizing over the range , as in Rapetti et al. (2005) and A08. The resulting constraints on and are shown in the left panel of Figure 11 as gray and gold shaded contours, respectively. Curvature is allowed to vary, remaining tightly constrained and consistent with zero, in both cases. For completeness, Table 6 shows results for models with both free and fixed curvature. In every case, the data are consistent with the CDM model (, ). The right panel of the figure shows the constraints on and versus for models with curvature and free. Even for this general model, the combination of data provides a tight constraint on , .
5.5 Impact of the Data
As a simple measure of the influence of the data on our combined constraints, we compare the areas of the plotted 95.4 per cent confidence regions from the full combination of data to those obtained from combining only CMB, supernova and BAO data (i.e. excluding ). For the CDM and flat, constant- models [respectively the and confidence regions] we find 11 per cent reductions in uncertainty when including the data in the combination. For the evolving- models (with free curvature), the allowed areas in the