Abstract

We present constraints on the mean matter density, Ωm, dark energy density, ΩDE, and the dark energy equation of state parameter, w, using Chandra measurements of the X-ray gas mass fraction (fgas) in 42 hot (kT > 5 keV), X-ray luminous, dynamically relaxed galaxy clusters spanning the redshift range 0.05 < z < 1.1. Using only the fgas data for the six lowest redshift clusters at z < 0.15, for which dark energy has a negligible effect on the measurements, we measure Ωm= 0.28 ± 0.06 (68 per cent confidence limits, using standard priors on the Hubble constant, H0, and mean baryon density, Ωbh2). Analysing the data for all 42 clusters, employing only weak priors on H0 and Ωbh2, we obtain a similar result on Ωm and a detection of the effects of dark energy on the distances to the clusters at ∼99.99 per cent confidence, with ΩDE= 0.86 ± 0.21 for a non-flat ΛCDM model. The detection of dark energy is comparable in significance to recent type Ia supernovae (SNIa) studies and represents strong, independent evidence for cosmic acceleration. Systematic scatter remains undetected in the fgas data, despite a weighted mean statistical scatter in the distance measurements of only ∼5 per cent. For a flat cosmology with a constant dark energy equation of state, we measure Ωm= 0.28 ± 0.06 and w=−1.14 ± 0.31. Combining the fgas data with independent constraints from cosmic microwave background and SNIa studies removes the need for priors on Ωbh2 and H0 and leads to tighter constraints: Ωm= 0.253 ± 0.021 and w=−0.98 ± 0.07 for the same constant-w model. Our most general analysis allows the equation of state to evolve with redshift. Marginalizing over possible transition redshifts 0.05 < zt < 1, the combined fgas+ CMB + SNIa data set constrains the dark energy equation of state at late and early times to be w0=−1.05 ± 0.29 and wet=−0.83 ± 0.46, respectively, in agreement with the cosmological constant paradigm. Relaxing the assumption of flatness weakens the constraints on the equation of state by only a factor of ∼2. Our analysis includes conservative allowances for systematic uncertainties associated with instrument calibration, cluster physics and data modelling. The measured small systematic scatter, tight constraint on Ωm and powerful constraints on dark energy from the fgas data bode well for future dark energy studies using the next generation of powerful X-ray observatories, such as Constellation-X.

1 INTRODUCTION

The matter content of the largest clusters of galaxies is expected to provide an almost fair sample of the matter content of the Universe (e.g. White et al. 1993; Eke, Navarro & Frenk 1998; Frenk et al. 1999). The ratio of baryonic-to-total mass in clusters should, therefore, closely match the ratio of the cosmological parameters Ωbm. The baryonic mass content of clusters is dominated by the X-ray emitting gas, the mass of which exceeds the mass of optically luminous material by a factor of ∼6, with other sources of baryonic matter being negligible (Fukugita, Hogan & Peebles 1998; Lin & Mohr 2004). The combination of robust measurements of the baryonic mass fraction in clusters from X-ray observations together with a determination of Ωb from cosmic microwave background (CMB) data or big bang nucleosynthesis calculations and a constraint on the Hubble constant, can therefore be used to measure Ωm (e.g. Fabian 1991; White & Frenk 1991; Briel, Henry & Böhringer 1992; White et al. 1993; David, Jones & Forman 1995; White & Fabian 1995; Evrard 1997; Ettori & Fabian 1999; Mohr, Mathiesen & Evrard 1999; Roussel, Sadat & Blanchard 2000; Grego et al. 2001; Allen, Schmidt & Fabian 2002a; Allen et al. 2003; Ettori, Tozzi & Rosati 2003; Lin, Mohr & Stanford 2003; Sanderson & Ponman 2003; Allen et al. 2004; LaRoque et al. 2006). This method currently provides one of our best constraints on Ωm and is remarkably simple and robust in terms of its underlying assumptions.

Measurements of the apparent evolution of the cluster X-ray gas mass fraction, hereafter fgas, can also be used to probe the acceleration of the Universe (Allen et al. 2004; see also Sasaki 1996; Pen 1997; Allen et al. 2002a, 2003; Ettori et al. 2003; LaRoque et al. 2006). This constraint originates from the dependence of the fgas measurements, which derive from the observed X-ray gas temperature and density profiles, on the assumed distances to the clusters, fgasd1.5.1 The expectation from non-radiative hydrodynamical simulations is that for the largest (kT≳ 5 keV), dynamically relaxed clusters and for measurement radii beyond the innermost core (rr2500), fgas should be approximately constant with redshift (Eke et al. 1998; Crain et al. 2007). However, possible systematic variation of fgas with redshift can be accounted for in a straightforward manner, so long as the allowed range of such variation is constrained by numerical simulations or other, complementary data (Eke et al. 1998; Bialek, Evrard & Mohr 2001; Muanwong et al. 2002; Borgani et al. 2004; Ettori et al. 2004; Kay et al. 2004; Kravtsov, Nagai & Vikhlinin 2005; Ettori et al. 2006; Nagai, Vikhlinin & Kravtsov 2007a).

The first clear detection of cosmic acceleration using the fgas technique was made by Allen et al. (2004) using Chandra observations of 26 hot (kT≳ 5 keV), X-ray luminous (Lbol≳ 1045h−270 erg s−1), dynamically relaxed clusters spanning the redshift range 0.07–0.9. The total Chandra exposure used in that work, after all screening procedures were applied, was ∼830 ks. That study led to a ∼3σ detection of the acceleration of the Universe and a tight constraint on the mean mass density Ωm= 0.25 ± 0.04 (see also Allen et al. 2002a, 2003; Ettori et al. 2003; LaRoque et al. 2006) in excellent agreement with independent findings from CMB studies (e.g. Spergel et al. 2003, 2007), type Ia supernovae (SNIa) data (e.g. Riess et al. 2004; Astier et al. 2006), galaxy redshift surveys (e.g. Cole et al. 2005; Eisenstein et al. 2005; Percival et al. 2007) and X-ray cluster number counts (e.g. Mantz et al. 2007).

Here we present a significant extension of the Allen et al. (2004) work. Our expanded sample contains 42 clusters spanning the redshift range 0.05 < z < 1.1. We incorporate new, deeper exposures for some of the original clusters, as well as new targets, approximately doubling the total exposure time used. Our analysis method incorporates conservative allowances for systematic uncertainties associated with instrument calibration, cluster physics and data modelling. As before, we employ rigorous selection criteria, restricting the analysis to the hottest, most dynamically relaxed clusters. We show that this leads to remarkably small intrinsic scatter in the fgas measurements, with no apparent systematic dependence of fgas on temperature for clusters with kT > 5 keV. Our method imposes a minimum of prior constraints and does not require that the density and temperature profiles of the X-ray emitting gas follow simple parametrized forms. We make our fgas measurements for each cluster at the radius r2500 in the reference Lambda cold dark matter (ΛCDM) cosmology, corresponding to an angle θΛCDM2500, for which the mean enclosed mass density is 2500 times the critical density of the Universe at the redshift of the cluster. This corresponds to about one quarter of the virial radius2 and represents a near-optimal choice for Chandra studies, being sufficiently large to provide small systematic scatter but not so large as to be hampered by systematic uncertainties in the background modelling. We compare our fgas measurements to results from other, independent studies and to the predictions from current hydrodynamical simulations.

Our analysis of cosmological parameters employs a Markov Chain Monte Carlo (MCMC) approach, which is efficient and allows for the simple inclusion of priors and a comprehensive study of the effects of systematic uncertainties. We present results based on studies of the fgas data alone (adopting simple priors on Ωbh2 and h) and for the fgas data used in combination with current CMB constraints (in which case the priors on Ωbh2 and h can be dropped) and SNIa data (Astier et al. 2006; Jha, Riess & Kirshner 2007; Riess et al. 2007; Wood-Vasey et al. 2007). We highlight the power of the data combinations for cosmological work, particularly in constraining the mean matter and dark energy densities of the Universe and the dark energy equation of state.

The fgas measurements are quoted for a flat ΛCDM reference cosmology with h=H0/100 km s−1Mpc−1= 0.7 and Ωm= 0.3.

2 X-RAY OBSERVATIONS AND ANALYSIS

2.1 Sample selection

Our sample consists of 42 hot, X-ray luminous, dynamically relaxed galaxy clusters spanning the redshift range 0.05 < z < 1.1. The systems have mass-weighted X-ray temperatures measured within r2500, kT2500≳ 5 keV and exhibit a high degree of dynamical relaxation in their Chandra images (Million et al., in preparation), with sharp central X-ray surface brightness peaks, short central cooling times (tcool≤ a few ×109yr) minimal isophote centroid variations (e.g. Mohr et al. 1995) and low X-ray power ratios (Buote & Tsai 1995, 1996; Jeltema et al. 2005). Although target selection is based only on these morphological X-ray characteristics, the clusters also exhibit other signatures of dynamical relaxation including minimal evidence for departures from hydrostatic equilibrium in X-ray pressure maps (Million et al., in preparation). The notable exceptions are Abell 2390, RXJ1347.5−1145, MACS1427.3+4408 and MACSJ0744.9+3927, for which clear substructure is observed between position angles of 255°–15°, 90°–190°, 160°–280° and 210°–330°, respectively (Allen, Schmidt & Fabian 2002b; Morris et al., in preparation; Million et al., in preparation). The regions associated with obvious substructure in these clusters have been excluded from the analysis. The bulk of the clusters at z > 0.3 were identified in the MACS survey (Ebeling, Edge & Henry 2001; Ebeling et al. 2007). Of the 70 MACS clusters with sufficient data on the Chandra archive at the time of observation to enable detailed spatially resolved spectroscopy, 22/70 are identified as being sufficiently relaxed to be included in the present study.

The restriction to clusters with the highest possible degree of dynamical relaxation, for which the assumption of hydrostatic equilibrium should be most valid, minimizes systematic scatter in the fgas data (Section 5.3) and allows for the most precise and robust determination of cosmological parameters. The restriction to the hottest (kT > 5 keV), relaxed systems further simplifies the analysis: for galaxies, groups and clusters with kT≲ 4 keV, the baryonic mass fraction is both expected and observed to rise systematically with increasing temperature, with the systematic scatter being largest in the coolest systems (e.g. Bialek et al. 2001; Muanwong et al. 2002; Ettori et al. 2004; Kravtsov et al. 2005; Vikhlinin et al. 2006). As shown in Sections 3.1 and 5.3, for the hot, relaxed clusters studied here, fgas exhibits no dependence on temperature and the intrinsic scatter is small.

2.2 Data reduction

The Chandra observations were carried out using the Advanced CCD Imaging Spectrometer (ACIS) between 1999 August 30 and 2005 June 28. The standard level-1 event lists produced by the Chandra pipeline processing were reprocessed using the ciao (version 3.2.2) software package, including the appropriate gain maps and calibration products. Bad pixels were removed and standard grade selections applied. Where possible, the extra information available in VFAINT mode was used to improve the rejection of cosmic ray events. The data were cleaned to remove periods of anomalously high background using the standard energy ranges and time bins recommended by the Chandra X-ray Centre. The net exposure times after cleaning are summarized in Table 1. The total good exposure is 1.63 Ms, approximately twice that of the Allen et al. (2004) study.

Table 1

Summary of the Chandra observations. Columns list the target name, observation date, detector used, observation mode, net exposure after all cleaning and screening processes were applied and right ascension (RA) and declination (Dec.) for the X-ray centres. Where multiple observations of a single cluster have been used, these are listed separately.

Name Date Detector Mode Exposure (ks) RA (J2000) Dec. (J2000) 
Abell 1795(1) 2002 June 10 ACIS-S VFAINT 13.2 13 48 52.4 26 35 38 
Abell 1795(2) 2004 January 14 ACIS-S VFAINT 14.3 13 48 52.4 26 35 38 
Abell 1795(3) 2004 January 18 ACIS-I VFAINT 9.6 13 48 52.4 26 35 38 
Abell 2029(1) 2000 April 12 ACIS-S FAINT 19.2 15 10 56.2 05 44 41 
Abell 2029(2) 2004 January 08 ACIS-S FAINT 74.8 15 10 56.2 05 44 41 
Abell 2029(3) 2004 December 17 ACIS-I VFAINT 9.4 15 10 56.2 05 44 41 
Abell 478(1) 2001 January 27 ACIS-S FAINT 39.9 04 13 25.2 10 27 55 
Abell 478(2) 2004 September 13 ACIS-I VFAINT 7.4 04 13 25.2 10 27 55 
PKS0745−191(1) 2001 June 16 ACIS-S VFAINT 17.4 07 47 31.7 −19 17 45 
PKS0745−191(2) 2004 September 24 ACIS-I VFAINT 9.2 07 47 31.7 −19 17 45 
Abell 1413 2001 May 16 ACIS-I VFAINT 64.5 11 55 18.1 23 24 17 
Abell 2204(1) 2000 July 29 ACIS-S FAINT 10.1 16 32 47.2 05 34 32 
Abell 2204(2) 2004 September 20 ACIS-I VFAINT 8.5 16 32 47.2 05 34 32 
Abell 383(1) 2000 November 16 ACIS-S FAINT 18.0 02 48 03.5 −03 31 45 
Abell 383(2) 2000 November 16 ACIS-I VFAINT 17.2 02 48 03.5 −03 31 45 
Abell 963 2000 October 11 ACIS-S FAINT 35.8 10 17 03.8 39 02 49 
RXJ0439.0+0520 2000 August 29 ACIS-I VFAINT 7.6 04 39 02.3 05 20 44 
RXJ1504.1−0248 2005 March 20 ACIS-I VFAINT 29.4 15 04 07.9 −02 48 16 
Abell 2390 2003 September 11 ACIS-S VFAINT 79.2 21 53 36.8 17 41 44 
RXJ2129.6+0005 2000 October 21 ACIS-I VFAINT 7.6 21 29 39.9 00 05 20 
Abell 1835(1) 1999 December 11 ACIS-S FAINT 18.0 14 01 01.9 02 52 43 
Abell 1835(2) 2000 April 29 ACIS-S FAINT 10.3 14 01 01.9 02 52 43 
Abell 611 2001 November 03 ACIS-S VFAINT 34.5 08 00 56.8 36 03 24 
Zwicky 3146 2000 May 10 ACIS-I FAINT 41.4 10 23 39.4 04 11 14 
Abell 2537 2004 September 09 ACIS-S VFAINT 36.0 23 08 22.1 −02 11 29 
MS2137.3−2353(1) 1999 November 18 ACIS-S VFAINT 20.5 21 40 15.2 −23 39 40 
MS2137.3−2353(2) 2003 November 18 ACIS-S VFAINT 26.6 21 40 15.2 −23 39 40 
MACSJ0242.6−2132 2002 February 07 ACIS-I VFAINT 10.2 02 42 35.9 −21 32 26 
MACSJ1427.6−2521 2002 June 29 ACIS-I VFAINT 14.7 14 27 39.4 −25 21 02 
MACSJ2229.8−2756 2002 November 13 ACIS-I VFAINT 11.8 22 29 45.3 −27 55 37 
MACSJ0947.2+7623 2000 October 20 ACIS-I VFAINT 9.6 09 47 13.1 76 23 14 
MACSJ1931.8−2635 2002 October 20 ACIS-I VFAINT 12.2 19 31 49.6 −26 34 34 
MACSJ1115.8+0129 2003 January 23 ACIS-I VFAINT 10.2 11 15 52.1 01 29 53 
MACSJ1532.9+3021(1) 2001 August 26 ACIS-S VFAINT 9.4 15 32 53.9 30 20 59 
MACSJ1532.9+3021(2) 2001 September 06 ACIS-I VFAINT 9.2 15 32 53.9 30 20 59 
MACSJ0011.7−1523(1) 2002 November 20 ACIS-I VFAINT 18.2 00 11 42.9 −15 23 22 
MACSJ0011.7−1523(2) 2005 June 28 ACIS-I VFAINT 32.1 00 11 42.9 −15 23 22 
MACSJ1720.3+3536(1) 2002 November 03 ACIS-I VFAINT 16.6 17 20 16.8 35 36 27 
MACSJ1720.3+3536(2) 2005 November 22 ACIS-I VFAINT 24.8 17 20 16.8 35 36 27 
MACSJ0429.6−0253 2002 February 07 ACIS-I VFAINT 19.1 04 29 36.1 −02 53 08 
MACSJ0159.8−0849(1) 2002 October 02 ACIS-I VFAINT 14.1 01 59 49.4 −08 49 58 
MACSJ0159.8−0849(2) 2004 December 04 ACIS-I VFAINT 28.9 01 59 49.4 −08 49 58 
MACSJ2046.0−3430 2005 June 28 ACIS-I VFAINT 8.9 20 46 00.5 −34 30 17 
MACSJ1359.2−1929 2005 March 17 ACIS-I VFAINT 9.2 13 59 10.3 −19 29 24 
MACSJ0329.7−0212(1) 2002 December 24 ACIS-I VFAINT 16.8 03 29 41.7 −02 11 48 
MACSJ0329.7−0212(2) 2004 December 06 ACIS-I VFAINT 31.1 03 29 41.7 −02 11 48 
RXJ1347.5−1145(1) 2000 March 03 ACIS-S VFAINT 8.6 13 47 30.6 −11 45 10 
RXJ1347.5−1145(2) 2000 April 29 ACIS-S FAINT 10.0 13 47 30.6 −11 45 10 
RXJ1347.5−1145(3) 2003 September 03 ACIS-I VFAINT 49.3 13 47 30.6 −11 45 10 
3C295(1) 1999 August 30 ACIS-S FAINT 15.4 14 11 20.5 52 12 10 
3C295(2) 2001 May 18 ACIS-I FAINT 72.4 14 11 20.5 52 12 10 
MACSJ1621.6+3810(1) 2002 October 18 ACIS-I VFAINT 7.9 16 21 24.8 38 10 09 
MACSJ1621.6+3810(2) 2004 December 11 ACIS-I VFAINT 32.2 16 21 24.8 38 10 09 
MACSJ1621.6+3810(3) 2004 December 25 ACIS-I VFAINT 26.1 16 21 24.8 38 10 09 
MACS1427.3+4408 2005 February 12 ACIS-I VFAINT 8.70 14 27 16.2 44 07 31 
MACSJ1311.0−0311 2005 April 20 ACIS-I VFAINT 56.2 13 11 01.6 −03 10 40 
MACSJ1423.8+2404 2003 August 18 ACIS-S VFAINT 113.5 14 23 47.9 24 04 43 
MACSJ0744.9+3927(1) 2001 November 12 ACIS-I VFAINT 17.1 07 44 52.9 39 27 27 
MACSJ0744.9+3927(2) 2003 January 04 ACIS-I VFAINT 15.6 07 44 52.9 39 27 27 
MACSJ0744.9+3927(3) 2004 December 03 ACIS-I VFAINT 41.3 07 44 52.9 39 27 27 
MS1137.5+6625 1999 September 30 ACIS-I VFAINT 103.8 11 40 22.4 66 08 15 
ClJ1226.9+3332(1) 2003 January 27 ACIS-I VFAINT 25.7 12 26 58.1 33 32 47 
ClJ1226.9+3332(2) 2004 August 07 ACIS-I VFAINT 26.3 12 26 58.1 33 32 47 
CL1415.2+3612 2003 September 16 ACIS-I VFAINT 75.1 14 15 11.2 36 12 02 
3C186 2002 May 16 ACIS-S VFAINT 15.4 07 44 17.5 37 53 17 
Name Date Detector Mode Exposure (ks) RA (J2000) Dec. (J2000) 
Abell 1795(1) 2002 June 10 ACIS-S VFAINT 13.2 13 48 52.4 26 35 38 
Abell 1795(2) 2004 January 14 ACIS-S VFAINT 14.3 13 48 52.4 26 35 38 
Abell 1795(3) 2004 January 18 ACIS-I VFAINT 9.6 13 48 52.4 26 35 38 
Abell 2029(1) 2000 April 12 ACIS-S FAINT 19.2 15 10 56.2 05 44 41 
Abell 2029(2) 2004 January 08 ACIS-S FAINT 74.8 15 10 56.2 05 44 41 
Abell 2029(3) 2004 December 17 ACIS-I VFAINT 9.4 15 10 56.2 05 44 41 
Abell 478(1) 2001 January 27 ACIS-S FAINT 39.9 04 13 25.2 10 27 55 
Abell 478(2) 2004 September 13 ACIS-I VFAINT 7.4 04 13 25.2 10 27 55 
PKS0745−191(1) 2001 June 16 ACIS-S VFAINT 17.4 07 47 31.7 −19 17 45 
PKS0745−191(2) 2004 September 24 ACIS-I VFAINT 9.2 07 47 31.7 −19 17 45 
Abell 1413 2001 May 16 ACIS-I VFAINT 64.5 11 55 18.1 23 24 17 
Abell 2204(1) 2000 July 29 ACIS-S FAINT 10.1 16 32 47.2 05 34 32 
Abell 2204(2) 2004 September 20 ACIS-I VFAINT 8.5 16 32 47.2 05 34 32 
Abell 383(1) 2000 November 16 ACIS-S FAINT 18.0 02 48 03.5 −03 31 45 
Abell 383(2) 2000 November 16 ACIS-I VFAINT 17.2 02 48 03.5 −03 31 45 
Abell 963 2000 October 11 ACIS-S FAINT 35.8 10 17 03.8 39 02 49 
RXJ0439.0+0520 2000 August 29 ACIS-I VFAINT 7.6 04 39 02.3 05 20 44 
RXJ1504.1−0248 2005 March 20 ACIS-I VFAINT 29.4 15 04 07.9 −02 48 16 
Abell 2390 2003 September 11 ACIS-S VFAINT 79.2 21 53 36.8 17 41 44 
RXJ2129.6+0005 2000 October 21 ACIS-I VFAINT 7.6 21 29 39.9 00 05 20 
Abell 1835(1) 1999 December 11 ACIS-S FAINT 18.0 14 01 01.9 02 52 43 
Abell 1835(2) 2000 April 29 ACIS-S FAINT 10.3 14 01 01.9 02 52 43 
Abell 611 2001 November 03 ACIS-S VFAINT 34.5 08 00 56.8 36 03 24 
Zwicky 3146 2000 May 10 ACIS-I FAINT 41.4 10 23 39.4 04 11 14 
Abell 2537 2004 September 09 ACIS-S VFAINT 36.0 23 08 22.1 −02 11 29 
MS2137.3−2353(1) 1999 November 18 ACIS-S VFAINT 20.5 21 40 15.2 −23 39 40 
MS2137.3−2353(2) 2003 November 18 ACIS-S VFAINT 26.6 21 40 15.2 −23 39 40 
MACSJ0242.6−2132 2002 February 07 ACIS-I VFAINT 10.2 02 42 35.9 −21 32 26 
MACSJ1427.6−2521 2002 June 29 ACIS-I VFAINT 14.7 14 27 39.4 −25 21 02 
MACSJ2229.8−2756 2002 November 13 ACIS-I VFAINT 11.8 22 29 45.3 −27 55 37 
MACSJ0947.2+7623 2000 October 20 ACIS-I VFAINT 9.6 09 47 13.1 76 23 14 
MACSJ1931.8−2635 2002 October 20 ACIS-I VFAINT 12.2 19 31 49.6 −26 34 34 
MACSJ1115.8+0129 2003 January 23 ACIS-I VFAINT 10.2 11 15 52.1 01 29 53 
MACSJ1532.9+3021(1) 2001 August 26 ACIS-S VFAINT 9.4 15 32 53.9 30 20 59 
MACSJ1532.9+3021(2) 2001 September 06 ACIS-I VFAINT 9.2 15 32 53.9 30 20 59 
MACSJ0011.7−1523(1) 2002 November 20 ACIS-I VFAINT 18.2 00 11 42.9 −15 23 22 
MACSJ0011.7−1523(2) 2005 June 28 ACIS-I VFAINT 32.1 00 11 42.9 −15 23 22 
MACSJ1720.3+3536(1) 2002 November 03 ACIS-I VFAINT 16.6 17 20 16.8 35 36 27 
MACSJ1720.3+3536(2) 2005 November 22 ACIS-I VFAINT 24.8 17 20 16.8 35 36 27 
MACSJ0429.6−0253 2002 February 07 ACIS-I VFAINT 19.1 04 29 36.1 −02 53 08 
MACSJ0159.8−0849(1) 2002 October 02 ACIS-I VFAINT 14.1 01 59 49.4 −08 49 58 
MACSJ0159.8−0849(2) 2004 December 04 ACIS-I VFAINT 28.9 01 59 49.4 −08 49 58 
MACSJ2046.0−3430 2005 June 28 ACIS-I VFAINT 8.9 20 46 00.5 −34 30 17 
MACSJ1359.2−1929 2005 March 17 ACIS-I VFAINT 9.2 13 59 10.3 −19 29 24 
MACSJ0329.7−0212(1) 2002 December 24 ACIS-I VFAINT 16.8 03 29 41.7 −02 11 48 
MACSJ0329.7−0212(2) 2004 December 06 ACIS-I VFAINT 31.1 03 29 41.7 −02 11 48 
RXJ1347.5−1145(1) 2000 March 03 ACIS-S VFAINT 8.6 13 47 30.6 −11 45 10 
RXJ1347.5−1145(2) 2000 April 29 ACIS-S FAINT 10.0 13 47 30.6 −11 45 10 
RXJ1347.5−1145(3) 2003 September 03 ACIS-I VFAINT 49.3 13 47 30.6 −11 45 10 
3C295(1) 1999 August 30 ACIS-S FAINT 15.4 14 11 20.5 52 12 10 
3C295(2) 2001 May 18 ACIS-I FAINT 72.4 14 11 20.5 52 12 10 
MACSJ1621.6+3810(1) 2002 October 18 ACIS-I VFAINT 7.9 16 21 24.8 38 10 09 
MACSJ1621.6+3810(2) 2004 December 11 ACIS-I VFAINT 32.2 16 21 24.8 38 10 09 
MACSJ1621.6+3810(3) 2004 December 25 ACIS-I VFAINT 26.1 16 21 24.8 38 10 09 
MACS1427.3+4408 2005 February 12 ACIS-I VFAINT 8.70 14 27 16.2 44 07 31 
MACSJ1311.0−0311 2005 April 20 ACIS-I VFAINT 56.2 13 11 01.6 −03 10 40 
MACSJ1423.8+2404 2003 August 18 ACIS-S VFAINT 113.5 14 23 47.9 24 04 43 
MACSJ0744.9+3927(1) 2001 November 12 ACIS-I VFAINT 17.1 07 44 52.9 39 27 27 
MACSJ0744.9+3927(2) 2003 January 04 ACIS-I VFAINT 15.6 07 44 52.9 39 27 27 
MACSJ0744.9+3927(3) 2004 December 03 ACIS-I VFAINT 41.3 07 44 52.9 39 27 27 
MS1137.5+6625 1999 September 30 ACIS-I VFAINT 103.8 11 40 22.4 66 08 15 
ClJ1226.9+3332(1) 2003 January 27 ACIS-I VFAINT 25.7 12 26 58.1 33 32 47 
ClJ1226.9+3332(2) 2004 August 07 ACIS-I VFAINT 26.3 12 26 58.1 33 32 47 
CL1415.2+3612 2003 September 16 ACIS-I VFAINT 75.1 14 15 11.2 36 12 02 
3C186 2002 May 16 ACIS-S VFAINT 15.4 07 44 17.5 37 53 17 

2.3 Spectral analysis

The spectral analysis was carried out using an updated version of the techniques described by Allen et al. (2004) and Schmidt & Allen (2007). In brief, concentric annular spectra were extracted from the cleaned event lists, centred on the coordinates listed in Table 1. Emission associated with X-ray point sources or obvious substructure (Table 2) was excluded. The spectra were analysed using xspec (version 11.3; Arnaud 1996), the mekal plasma emission code (Kaastra & Mewe 1993; incorporating the Fe-L calculations of Liedhal, Osterheld & Goldstein 1995) and the photoelectric absorption models of Balucinska-Church & McCammon (1992). The emission from each spherical shell was modelled as a single-phase plasma. The abundances of the elements in each shell were assumed to vary with a common ratio, Z, with respect to solar values. The absorbing column densities were fixed to the Galactic values determined from H i studies (Dickey & Lockman 1990), with the exception of Abell 478 and PKS0745−191 where the value was allowed to fit freely. (For Abell 478, the absorbing column density was allowed to vary as a function of radius, as was shown to be required by Allen et al. 1993). We have included standard correction factors to account for time-dependent contamination along the instrument light path. In addition, we have incorporated a small correction to the high-resolution mirror assembly model in ciao 3.2.2, which takes the form of an ‘inverse’ edge with an energy, E= 2.08 keV and optical depth τ=−0.1 (H. Marshall, private communication) and also boosted the overall effective area by 6 per cent, to better match later calibration data (A. Vikhlinin, private communication). These corrections lead to an excellent match with results based on later calibration data, available in ciao 3.4. Only data in the 0.8–7.0 keV energy range were used in the analysis (with the exceptions of the earliest observations of 3C 295, Abell 1835 and Abell 2029, where a wider 0.6–7.0 keV band was used to enable better modelling of the soft X-ray background).

Table 2

Clusters with regions of localized substructure that have been excluded or down-weighted in the analysis. Column 2 lists the position angles (PA) that have been excluded in the case of Abell 2390, RXJ1347.5−1145, MACS1427.3+4408 and MACSJ0744.9+3927. Column 3 lists the radii (in h−170 kpc) within which the spectral data have been down-weighted by including a systematic uncertainty of ±30 per cent in quadrature with the statistical errors on the temperature measurements.

Cluster Excluded PA Down-weighted r 
Abell 1795 – 75 
Abell 2029 – 30 
Abell 478 – 15 
PKS0745−191 – 55 
Abell 1413 – 40 
Abell 2204 – 75 
Abell 383 – 40 
RXJ1504.1−0248 – 80 
Abell 2390 255–15 50 
RXJ2129.6+0005 – 40 
Zwicky 3146 – 240 
Abell 2537 – 40 
MACSJ2229.8−2756 – 40 
MACSJ0947.2+7623 – 40 
MACSJ1931.8−2635 – 40 
MACSJ1115.8+0129 – 85 
MACSJ1532.9+3021 – 40 
RXJ1347.5−1145 90–190 – 
MACSJ1621.6+3810 – 45 
MACSJ1427.3+4408 160–280 – 
MACSJ0744.9+3927 210–330 – 
Cluster Excluded PA Down-weighted r 
Abell 1795 – 75 
Abell 2029 – 30 
Abell 478 – 15 
PKS0745−191 – 55 
Abell 1413 – 40 
Abell 2204 – 75 
Abell 383 – 40 
RXJ1504.1−0248 – 80 
Abell 2390 255–15 50 
RXJ2129.6+0005 – 40 
Zwicky 3146 – 240 
Abell 2537 – 40 
MACSJ2229.8−2756 – 40 
MACSJ0947.2+7623 – 40 
MACSJ1931.8−2635 – 40 
MACSJ1115.8+0129 – 85 
MACSJ1532.9+3021 – 40 
RXJ1347.5−1145 90–190 – 
MACSJ1621.6+3810 – 45 
MACSJ1427.3+4408 160–280 – 
MACSJ0744.9+3927 210–330 – 

For the nearer clusters (z < 0.3), background spectra were extracted from the blank-field data sets available from the Chandra X-ray centre. These were cleaned in an identical manner to the target observations. In each case, the normalizations of the background files were scaled to match the count rates in the target observations measured in the 9.5–12 keV band. Where required, e.g. due to the presence of strong excess soft emission in the field, a spectral model for additional soft background emission was included in the analysis. For the more distant systems (as well as for the first observation of Abell 1835, the ACIS-I observation of Abell 383, and the observations of Abell 2537, RXJ 2129.6+0005 and Zwicky 3146) background spectra were extracted from appropriate, source free regions of the target data sets. (We have confirmed that similar results are obtained using the blank-field background data sets.) In order to minimize systematic errors, we have restricted our spectral analysis to radii within which systematic uncertainties in the background subtraction (established by the comparison of different background subtraction methods) are smaller than the statistical uncertainties in the results. All results are drawn from ACIS chips 0, 1, 2, 3 and 7 which have the most accurate calibration, although ACIS chip 5 was also used to study the soft X-ray background in ACIS-S observations.

Separate photon-weighted response matrices and effective area files were constructed for each region using calibration files appropriate for the period of observations. The spectra for all annuli for a given cluster were modelled simultaneously in order to determine the deprojected X-ray gas temperature and metallicity profiles, under the assumption of spherical symmetry. The extended C-statistic, available in xspec, was used for all spectral fitting.

2.4 Measuring the mass profiles

The details of the mass analysis and results on the total mass and dark matter profiles are presented by Schmidt & Allen (2007). In brief, X-ray surface brightness profiles in the 0.8–7.0 keV band were extracted from background subtracted, flat-fielded Chandra images with 0.984 × 0.984 arcsec2 pixel. The profiles were centred on the coordinates listed in Table 1. Under the assumptions of hydrostatic equilibrium and spherical symmetry, the observed X-ray surface brightness profiles and deprojected X-ray gas temperature profiles may together be used to determine the X-ray emitting gas mass and total mass profiles in the clusters. For this analysis, we have used an enhanced version of the Cambridge X-ray deprojection code described by e.g. White, Jones & Forman (1997). This method is particularly well suited to the present task in that it does not use parametric fitting functions for the X-ray temperature, gas density or surface brightness in measuring the mass; the use of such functions introduces strong priors that complicate the interpretation of results and, in particular, can lead to an underestimation of uncertainties. The only additional important assumption in the analysis is the choice of a Navarro, Frenk & White (1995, 1997; hereafter NFW) model to parametrize the total (luminous-plus-dark) mass distributions:  

1
formula
where ρ(r) is the mass density, ρc(z) = 3H(z)2/8πG is the critical density for closure at redshift z, rs is the scale radius, c is the concentration parameter (with c=r200/rs) and δc= 200 c3/3[ln(1 +c) −c/(1 +c)].3Schmidt & Allen (2007) show that the NFW model provides a good description of the mass distributions in the clusters studied here.

Given the observed surface brightness profile and a particular choice of parameters for the total mass profile, the deprojection code is used to predict the temperature profile of the X-ray gas. (In detail, the median model temperature profile determined from 100 Monte Carlo simulations for each mass model is used.) This model temperature profile is then compared with the observed spectral, deprojected temperature profile and the goodness of fit is calculated using the sum over all temperature bins:  

2
formula
where Tobs is the observed, spectral deprojected temperature profile and Tmodel is the model, rebinned to the same spatial scale. For each cluster, the mass parameters are stepped over a grid of values and the best-fitting values and uncertainties determined via χ2 minimization techniques. The X-ray emitting gas density, pressure, entropy, cooling time and mass, and the integrated X-ray gas mass fraction, fgas, are then determined in a straightforward manner from the Monte Carlo simulations and χ2 values at each grid point.

A number of systematic issues affect the accuracy of the fgas measurements and their interpretation; these are discussed in detail in Section 4.2. In particular, our analysis incorporates allowances for effects associated with calibration and modelling uncertainties and non-thermal pressure support in the X-ray emitting gas, employing priors that span conservative ranges for the likely magnitudes of these effects.

Finally, for a number of the clusters, small but noticeable substructure is present at small radii. This is likely to result from interactions between the central radio sources and surrounding gas (e.g. Böhringer et al. 1993; Fabian et al. 2000, 2003a, 2005, 2006; Birzan et al. 2004; Dunn & Fabian 2004; Dunn, Fabian & Taylor 2005; Forman et al. 2005; Allen et al. 2006; Rafferty et al. 2006) and/or ‘sloshing’ of the X-ray emitting gas within the central potentials (e.g. Churazov et al. 2003; Markevitch et al. 2003; Ascasibar & Markevitch 2006). The regions affected by such substructure are listed in Table 2. A systematic uncertainty of ±30 per cent has been added in quadrature to all spectral results determined from these regions, leading to them having little weight in the mass analysis.

2.5 The stellar baryonic mass fraction

Observations of nearby and intermediate redshift clusters show that for clusters in the mass/temperature range studied here, the average mass fraction in stars (in galaxies and intracluster light combined) fstar∼ 0.16 h0.570fgas (Lin & Mohr 2004; see also White et al. 1993; Fukugita et al. 1998; Balogh et al. 2001).

For the present analysis, we ideally require the ratio s=fstar/fgas measured within r2500 for each cluster. However, such measurements are not yet available for the bulk of the clusters studied here. For hot, massive clusters, the relative contribution of the central dominant galaxy to the overall cluster light is less significant than for cooler, less massive systems (e.g. Lin & Mohr 2004). We have therefore assumed that the stellar mass fraction within r2500 is similar to that measured within the virial radius, i.e. s= 0.16 h0.570, but have both included a conservative 30 per cent Gaussian uncertainty in this value and allowed for evolution at the ±20 per cent level, per unit redshift interval. Since the stellar mass accounts for only ∼14 per cent of the overall baryon budget within r2500 and less than 2 per cent of the total mass, these systematic uncertainties do not have a large effect on the overall error budget. A program to measure the evolution of the optical baryonic mass content of the largest relaxed clusters is underway.

3 THE X-RAY GAS MASS FRACTION MEASUREMENTS

3.1 New fgas measurements

As mentioned above, in compiling the results on the X-ray gas mass fraction, fgas, we have adopted a canonical measurement radius of r2500. The r2500 value for each cluster is determined directly from the Chandra data, with confidence limits calculated from the χ2 grids. In general, the values are well matched to the outermost radii at which reliable temperature measurements can be made from the Chandra data, given systematic uncertainties associated with the background modelling.

Fig. 1(a) shows the observed fgas(r) profiles for the six lowest redshift clusters in the sample, for the reference ΛCDM cosmology. Although some dispersion in the profiles is present, particularly at small radii, the profiles tend towards a common value at r2500. Fitting the fgas measurements at r2500 for the six lowest redshift systems with a constant value we obtain fgas= 0.113 ± 0.003, with χ2= 4.3 for five degrees of freedom. Fitting the results for all 42 clusters gives fgas= 0.1104 ± 0.0016, with χ2= 43.5 for 41 degrees of freedom.

Figure 1

The X-ray gas mass fraction profiles for the ΛCDM reference cosmology (Ωm= 0.3, ΩΛ= 0.7, h= 0.7) with the radial axes scaled in units of r2500. Left-hand panel: Results for the six lowest redshift clusters with z≲ 0.15. Right-hand panel: Results for the entire sample. Note fgas(r) is an integrated quantity and so error bars on neighbouring points in a profile are correlated.

Figure 1

The X-ray gas mass fraction profiles for the ΛCDM reference cosmology (Ωm= 0.3, ΩΛ= 0.7, h= 0.7) with the radial axes scaled in units of r2500. Left-hand panel: Results for the six lowest redshift clusters with z≲ 0.15. Right-hand panel: Results for the entire sample. Note fgas(r) is an integrated quantity and so error bars on neighbouring points in a profile are correlated.

Fig. 1(b) shows the fgas(r/r2500) profiles for all 42 clusters in the sample. Fitting the data in the range 0.7–1.2r2500 with a power-law model, we measure fgas= 0.1105 ± 0.0005(r/r2500)0.214±0.022. Note that the error bars on the mean fgas measurements quoted above reflect only the statistical uncertainties in these values. A systematic uncertainty of ∼10–15 per cent in the global, absolute fgas normalization is also present due to uncertainties in e.g. instrument calibration, X-ray modelling and non-thermal pressure support; this must be accounted for in the determination of cosmological constraints (Section 4.2).

Table 3 summarizes the results on the X-ray gas mass fraction for each cluster measured at r2500, together with the r2500 values, for the reference ΛCDM cosmology. Fig. 2 shows a comparison of the fgas results, plotted as a function of redshift, for the reference ΛCDM cosmology and a flat, standard cold dark matter (SCDM) cosmology with Ωm= 1.0, h= 0.5. Whereas the results for the ΛCDM cosmology appear consistent with the expectation of a constant fgas(z) value from non-radiative simulations (e.g. Eke et al. 1998; Crain et al. 2007), as evidenced by the acceptable χ2 value quoted above, the results for the reference SCDM cosmology indicate a clear, apparent drop in fgas as the redshift increases. The χ2 value obtained from a fit to the SCDM data with a constant model, χ2= 144 for 41 degrees of freedom, shows that the SCDM cosmology is clearly inconsistent with a prediction that fgas(z) should be constant.

Table 3

The redshifts, r2500 values, mean mass-weighted temperatures within r2500 and the X-ray gas mass fractions within r2500 for the reference ΛCDM cosmology. Error bars are statistical uncertainties and are quoted at the 68 per cent confidence level. A systematic uncertainty of ∼10–15 per cent is associated with the global, absolute normalization of the fgas values due to uncertainties in instrument calibration, X-ray modelling and non-thermal pressure support (Section 4.2). The redshifts for the MACS clusters are from Ebeling et al. (2007, in preparation).

 z r2500(h−170 kpc) kT2500 fgash1.570 
Abell 1795 0.063 570+18−24 6.51 ± 0.23 0.1074 ± 0.0075 
Abell 2029 0.078 611+10−13 8.58 ± 0.44 0.1117 ± 0.0042 
Abell 478 0.088 643+16−15 7.99 ± 0.43 0.1211 ± 0.0053 
PKS0745−191 0.103 682+42−41 9.50 ± 1.13 0.1079 ± 0.0124 
Abell 1413 0.143 599+17−19 7.80 ± 0.35 0.1082 ± 0.0058 
Abell 2204 0.152 628+38−24 10.51 ± 2.54 0.1213 ± 0.0116 
Abell 383 0.188 502+25−23 5.36 ± 0.23 0.0903 ± 0.0080 
Abell 963 0.206 540+24−27 7.26 ± 0.28 0.1144 ± 0.0102 
RXJ0439.0+0521 0.208 454+37−25 4.86 ± 0.45 0.0917 ± 0.0127 
RXJ1504.1−0248 0.215 671+44−33 9.32 ± 0.59 0.1079 ± 0.0111 
Abell 2390 0.230 662+42−30 11.72 ± 1.43 0.1257 ± 0.0110 
RXJ2129.6+0005 0.235 507+65−57 7.38 ± 0.88 0.1299 ± 0.0299 
Abell 1835 0.252 684+27−26 10.57 ± 0.62 0.1197 ± 0.0082 
Abell 611 0.288 518+43−30 7.39 ± 0.48 0.1020 ± 0.0133 
Zwicky 3146 0.291 679+66−66 8.27 ± 1.08 0.0943 ± 0.0163 
Abell 2537 0.295 518+57−33 8.12 ± 0.78 0.0949 ± 0.0147 
MS2137.3−2353 0.313 479+18−10 5.65 ± 0.30 0.1106 ± 0.0061 
MACSJ0242.6−2132 0.314 478+29−20 5.51 ± 0.47 0.1268 ± 0.0131 
MACSJ1427.6−2521 0.318 412+42−37 5.24 ± 0.77 0.1052 ± 0.0220 
MACSJ2229.8−2756 0.324 414+41−29 5.42 ± 0.68 0.1452 ± 0.0265 
MACSJ0947.2+7623 0.345 594+65−49 7.80 ± 0.69 0.1048 ± 0.0196 
MACSJ1931.8−2635 0.352 581+131−46 7.49 ± 0.77 0.1193 ± 0.0266 
MACSJ1115.8+0129 0.355 664+118−108 8.92 ± 1.31 0.0925 ± 0.0283 
MACSJ1532.9+3021 0.363 543+45−33 7.69 ± 1.34 0.1280 ± 0.0162 
MACSJ0011.7−1523 0.378 497+40−27 6.56 ± 0.37 0.1067 ± 0.0125 
MACSJ1720.3+3536 0.391 520+39−32 8.11 ± 0.55 0.1153 ± 0.0151 
MACSJ0429.6−0253 0.399 439+19−24 6.10 ± 0.58 0.1375 ± 0.0154 
MACSJ0159.8−0849 0.404 597+33−48 10.62 ± 0.69 0.1097 ± 0.0160 
MACSJ2046.0−3430 0.423 413+62−50 5.81 ± 1.02 0.1253 ± 0.0398 
MACSJ1359.2−1929 0.447 458+91−56 6.73 ± 0.96 0.0845 ± 0.0290 
MACSJ0329.7−0212 0.450 481+26−23 6.85 ± 0.45 0.1262 ± 0.0129 
RXJ1347.5−1144 0.451 776+43−31 14.54 ± 1.08 0.0923 ± 0.0078 
3C295 0.461 419+20−15 5.09 ± 0.42 0.1067 ± 0.0096 
MACSJ1621.6+3810 0.461 496+53−39 9.15 ± 1.01 0.0954 ± 0.0172 
MACS1427.3+4408 0.487 428+67−36 6.65 ± 1.40 0.1201 ± 0.0294 
MACSJ1311.0−0311 0.494 461+30−26 6.07 ± 0.71 0.1066 ± 0.0168 
MACSJ1423.8+2404 0.539 467+18−14 7.80 ± 0.44 0.1141 ± 0.0086 
MACSJ0744.9+3927 0.686 466+40−23 8.67 ± 0.98 0.1151 ± 0.0140 
MS1137.5+6625 0.782 435+84−44 6.89 ± 0.78 0.0716 ± 0.0235 
ClJ1226.9+3332 0.892 521+123−54 11.95 ± 1.97 0.0769 ± 0.0198 
CL1415.2+3612 1.028 278+33−25 5.59 ± 0.84 0.1086 ± 0.0262 
3C186 1.063 292+54−57 5.62 ± 1.00 0.1340 ± 0.0777 
 z r2500(h−170 kpc) kT2500 fgash1.570 
Abell 1795 0.063 570+18−24 6.51 ± 0.23 0.1074 ± 0.0075 
Abell 2029 0.078 611+10−13 8.58 ± 0.44 0.1117 ± 0.0042 
Abell 478 0.088 643+16−15 7.99 ± 0.43 0.1211 ± 0.0053 
PKS0745−191 0.103 682+42−41 9.50 ± 1.13 0.1079 ± 0.0124 
Abell 1413 0.143 599+17−19 7.80 ± 0.35 0.1082 ± 0.0058 
Abell 2204 0.152 628+38−24 10.51 ± 2.54 0.1213 ± 0.0116 
Abell 383 0.188 502+25−23 5.36 ± 0.23 0.0903 ± 0.0080 
Abell 963 0.206 540+24−27 7.26 ± 0.28 0.1144 ± 0.0102 
RXJ0439.0+0521 0.208 454+37−25 4.86 ± 0.45 0.0917 ± 0.0127 
RXJ1504.1−0248 0.215 671+44−33 9.32 ± 0.59 0.1079 ± 0.0111 
Abell 2390 0.230 662+42−30 11.72 ± 1.43 0.1257 ± 0.0110 
RXJ2129.6+0005 0.235 507+65−57 7.38 ± 0.88 0.1299 ± 0.0299 
Abell 1835 0.252 684+27−26 10.57 ± 0.62 0.1197 ± 0.0082 
Abell 611 0.288 518+43−30 7.39 ± 0.48 0.1020 ± 0.0133 
Zwicky 3146 0.291 679+66−66 8.27 ± 1.08 0.0943 ± 0.0163 
Abell 2537 0.295 518+57−33 8.12 ± 0.78 0.0949 ± 0.0147 
MS2137.3−2353 0.313 479+18−10 5.65 ± 0.30 0.1106 ± 0.0061 
MACSJ0242.6−2132 0.314 478+29−20 5.51 ± 0.47 0.1268 ± 0.0131 
MACSJ1427.6−2521 0.318 412+42−37 5.24 ± 0.77 0.1052 ± 0.0220 
MACSJ2229.8−2756 0.324 414+41−29 5.42 ± 0.68 0.1452 ± 0.0265 
MACSJ0947.2+7623 0.345 594+65−49 7.80 ± 0.69 0.1048 ± 0.0196 
MACSJ1931.8−2635 0.352 581+131−46 7.49 ± 0.77 0.1193 ± 0.0266 
MACSJ1115.8+0129 0.355 664+118−108 8.92 ± 1.31 0.0925 ± 0.0283 
MACSJ1532.9+3021 0.363 543+45−33 7.69 ± 1.34 0.1280 ± 0.0162 
MACSJ0011.7−1523 0.378 497+40−27 6.56 ± 0.37 0.1067 ± 0.0125 
MACSJ1720.3+3536 0.391 520+39−32 8.11 ± 0.55 0.1153 ± 0.0151 
MACSJ0429.6−0253 0.399 439+19−24 6.10 ± 0.58 0.1375 ± 0.0154 
MACSJ0159.8−0849 0.404 597+33−48 10.62 ± 0.69 0.1097 ± 0.0160 
MACSJ2046.0−3430 0.423 413+62−50 5.81 ± 1.02 0.1253 ± 0.0398 
MACSJ1359.2−1929 0.447 458+91−56 6.73 ± 0.96 0.0845 ± 0.0290 
MACSJ0329.7−0212 0.450 481+26−23 6.85 ± 0.45 0.1262 ± 0.0129 
RXJ1347.5−1144 0.451 776+43−31 14.54 ± 1.08 0.0923 ± 0.0078 
3C295 0.461 419+20−15 5.09 ± 0.42 0.1067 ± 0.0096 
MACSJ1621.6+3810 0.461 496+53−39 9.15 ± 1.01 0.0954 ± 0.0172 
MACS1427.3+4408 0.487 428+67−36 6.65 ± 1.40 0.1201 ± 0.0294 
MACSJ1311.0−0311 0.494 461+30−26 6.07 ± 0.71 0.1066 ± 0.0168 
MACSJ1423.8+2404 0.539 467+18−14 7.80 ± 0.44 0.1141 ± 0.0086 
MACSJ0744.9+3927 0.686 466+40−23 8.67 ± 0.98 0.1151 ± 0.0140 
MS1137.5+6625 0.782 435+84−44 6.89 ± 0.78 0.0716 ± 0.0235 
ClJ1226.9+3332 0.892 521+123−54 11.95 ± 1.97 0.0769 ± 0.0198 
CL1415.2+3612 1.028 278+33−25 5.59 ± 0.84 0.1086 ± 0.0262 
3C186 1.063 292+54−57 5.62 ± 1.00 0.1340 ± 0.0777 
Figure 2

The apparent variation of the X-ray gas mass fraction measured within r2500 as a function of redshift for the (left-hand panel) reference ΛCDM and (right-hand panel) reference SCDM (Ωm= 1.0, ΩΛ= 0.0, h= 0.5) cosmologies. The plotted error bars are statistical rms 1σ uncertainties. The global, absolute normalization of the fgas value should be regarded as uncertain at the ∼10–15 per cent level due to systematic uncertainties in instrument calibration, modelling and the level of non-thermal pressure support (Section 4.2).

Figure 2

The apparent variation of the X-ray gas mass fraction measured within r2500 as a function of redshift for the (left-hand panel) reference ΛCDM and (right-hand panel) reference SCDM (Ωm= 1.0, ΩΛ= 0.0, h= 0.5) cosmologies. The plotted error bars are statistical rms 1σ uncertainties. The global, absolute normalization of the fgas value should be regarded as uncertain at the ∼10–15 per cent level due to systematic uncertainties in instrument calibration, modelling and the level of non-thermal pressure support (Section 4.2).

Table 3 also lists the mass-weighted temperatures measured within r2500 for each cluster. Fig. 3 shows fgas as a function of kT2500 for the reference ΛCDM cosmology. The dotted line in the figure shows the best-fitting power-law model, fgas(r2500) ∝kTα2500, which provides a good description of the data (χ2= 43.5 for 40 degrees of freedom) and is consistent with a constant value (α= 0.005 ± 0.058). The solid lines shows the 2σ limits on the steepest and shallowest allowed power-law models. It is clear from the figure that fgas is independent of temperature for the clusters in the present sample.

Figure 3

The X-ray gas mass fraction as a function of mass-weighted temperature measured within r2500 for the reference ΛCDM cosmology. The dotted line shows the best-fitting power-law model which provides a good description of the data (χ2= 43.5 for 40 degrees of freedom) and is consistent with a constant value (slope α= 0.005 ± 0.058). The solid lines shows the 2σ limits on the slopes allowed by the data. The figure demonstrates that fgas is essentially independent of temperature for the massive, dynamically relaxed clusters in the present sample.

Figure 3

The X-ray gas mass fraction as a function of mass-weighted temperature measured within r2500 for the reference ΛCDM cosmology. The dotted line shows the best-fitting power-law model which provides a good description of the data (χ2= 43.5 for 40 degrees of freedom) and is consistent with a constant value (slope α= 0.005 ± 0.058). The solid lines shows the 2σ limits on the slopes allowed by the data. The figure demonstrates that fgas is essentially independent of temperature for the massive, dynamically relaxed clusters in the present sample.

3.2 Comparison with previous fgas results

Approximately 0.75 Ms of the ∼1.6 Ms of Chandra data used here were also included in the Allen et al. (2004) study. The current work includes a reanalysis of those data using improved calibration information, where available. The fgas results from the two studies show excellent overall agreement: the new fgas values are, on average, ∼6 per cent lower than those reported by Allen et al. (2004), a difference consistent with expectations given the modification to the effective area calibration described in Section 2.3.

LaRoque et al. (2006) present fgas measurements for 38 X-ray luminous clusters, including 10 of the large, dynamically relaxed systems studied here. Their best-fitting results at r2500 are in good overall agreement with the present work, with their fgas values being, on average, ∼6 per cent higher than those reported here, for the systems in common.

Pointecouteau et al. (2004) present an analysis of XMM–Newton data for Abell 478, for which they measure an fgas value at r2500 of 0.13 ± 0.02, in good agreement with this work. These authors also report a value of 0.11 for Abell 1413, based on the data of Pratt & Arnaud (2002), which is consistent with the results reported here.

Vikhlinin et al. (2006) present fgas measurements for 13 clusters of which six are in common with this study. On average, the Vikhlinin et al. (2006)fgas results are ∼10 per cent lower than those reported here after correcting their values to the same reference ΛCDM cosmology.

We note that the statistical uncertainties on the fgas measurements listed in Table 3 are, typically, larger than those reported by other authors. Two contributing factors to this difference are: (1) that the present analysis does not impose strong priors on the shapes of the temperature and density profiles in the clusters through the use of parametric models (the use of such parameterizations can lead to spuriously tight constraints in cases where they do not provide an adequate description of the data); and (2) the fgas measurement errors reported here are marginalized over the uncertainties in all other parameters, including the uncertainties in r2500.

4 COSMOLOGICAL ANALYSIS

4.1 Markov Chain Monte Carlo method

Our determination of cosmological parameters uses an MCMC method. We employ a modified version of the cosmomc code4 of Lewis & Bridle (2002; see Rapetti, Allen & Weller 2005; Rapetti et al. 2007 for details of the enhancements), which uses a Metropolis–Hastings MCMC algorithm to explore parameter space. We run the code on four to 16 processors simultaneously, creating multiple chains and using the message passing interface to dynamically update the proposal matrix based on the covariance of post-burn-in samples. This leads to a much faster convergence than would be obtained from a single chain run on a single compute node.

Convergence is assessed using the Gelman–Rubin criterion (Gelman & Rubin 1992). Convergence is deemed acceptable when the ratio of between-chain to mean-chain variances, R, satisfies R− 1 < 0.1. (We have also visually compared individual chains to ensure that consistent final results were obtained.) In general, our combined chains typically have lengths of at least 105 samples and have R− 1 ≪ 0.1. (For the evolving-w models, R− 1 ∼ 0.1.) Conservative burn-in periods of at least 10 000 samples were allowed for each chain.

4.2 Analysis of the fgas data: modelling and systematic allowances

The differences between the shapes of the fgas(z) curves in Figs 2(a) and (b) reflect the dependence of the measured fgas values on the assumed angular diameter distances to the clusters. Under the assumption (Section 1) that fgas should, in reality, be approximately constant with redshift, as suggested by non-radiative simulations of large clusters (Eke et al. 1998; Crain et al. 2007; uncertainties in the predictions from simulations are discussed below) inspection of Fig. 2 would clearly favour the ΛCDM over the SCDM cosmology.

To determine constraints on cosmological parameters, it is not necessary to generate fgas(z) data sets for every cosmology of interest and compare them to the expected behaviour. Rather, one can fit a single, reference fgas(z) data set with a model that accounts for the expected apparent variation in fgas(z) as the underlying cosmology is varied. We choose to work with the ΛCDM reference cosmology, although similar results can in principle be derived for other reference cosmologies.

The model fitted to the reference ΛCDM data is  

3
formula
where dA(z) and dΛCDMA (z) are the angular diameter distances to the clusters in the current test model and reference cosmologies,  
4
formula
with E(z) defined as in Section 4.4. The factor A in equation (3) accounts for the change in angle subtended by r2500 as the underlying cosmology is varied5:  
5
formula
Here, η is the slope of the fgas(r/r2500) data in the region of r2500, as measured for the reference ΛCDM cosmology. For simplicity, we use the best-fitting average slope of η= 0.214 ± 0.022 determined from a fit to the whole sample over the range 0.7 < r/r2500 < 1.2 (Section 3) and marginalize over the slope uncertainty. This angular correction factor, which is close to unity for all cosmologies and redshifts of interest, has not been employed in previous studies and, indeed, can be neglected without significant loss of accuracy for most work. Nevertheless, we include it here for completeness and note that its inclusion leads to slightly tighter constraints on dark energy than would otherwise be obtained.

The parameter γ in equation (3) models non-thermal pressure support in the clusters. Based on hydrodynamical simulations, Nagai et al. (2007a) estimate a bias of ∼9 per cent in fgas measurements at r2500 for relaxed clusters. This bias originates primarily from subsonic motions in the intracluster gas and, as discussed by those authors (see also Section 5.3), can be regarded as an upper limit, given observational indications that the gas viscosity in real clusters appears likely to exceed that modelled in the simulations. For the large, relaxed clusters and measurement radii of interest here, non-thermal pressure support due to cosmic rays (Pfrommer et al. 2007) and magnetic fields (Dolag & Schindler 2000) is expected to be small. Based on these considerations, our default analysis assumes a uniform prior of 1.0 < γ < 1.1, although we also consider the case where the non-thermal pressure support may be up to twice as large, i.e. 1.0 < γ < 1.2.

The parameter s(z) =s0(1 +αsz) in equation (3) models the baryonic mass fraction in stars. As discussed in Section 2.5, we include a 30 per cent Gaussian uncertainty on s0, such that s0= (0.16 ± 0.05) h0.570, and a 20 per cent uniform prior on αs, such that −0.2 < αs < 0.2, allowing for evolution in the stellar baryonic mass fraction of ± 20 per cent per unit redshift interval.

The factor b(z) =b0(1 +αbz) is the ‘depletion’ or ‘bias’ factor, i.e. the ratio by which the baryon fraction measured at r2500 is depleted with respect to the universal mean; such depletion is a natural consequence of the thermodynamic history of the gas. The non-radiative simulations of hot, massive clusters published by Eke et al. (1998; see also Crain et al. 2007) give b0= 0.83 ± 0.04 at r2500, and are consistent with no redshift evolution in b for z < 1. We use these simulations as a benchmark because other simulations that include cooling currently tend to significantly overproduce young stars in the largest galaxies (see e.g. Balogh et al. 2001), which is problematic for the prediction of b(z). We note also the good agreement between the observed, scaled fgas(r) profiles determined from the Chandra data and the b(r) profiles for the three most relaxed clusters in the simulations of Eke et al. (1998; the red curves in Fig 4); this suggests that the non-radiative simulations provide a useful approximation for the purpose of predicting b(z). (The profiles for the less relaxed simulated clusters are shown as dashed green curves in the figure.) Nevertheless, to account for systematic uncertainties in the predictions of b(z), we include a conservative 20 per cent uniform prior on b0, such that 0.65 < b0 < 1.0, and allow for moderate, systematic evolution in b(z) over the observed redshift range, setting −0.1 < αb < 0.1. This encompasses a range of evolution allowed by recent simulations including various approximations to the detailed baryonic physics (e.g. Kay et al. 2004; Ettori et al. 2006; Crain et al. 2007; Nagai et al. 2007a).

Figure 4

The X-ray depletion or bias factor, b (i.e. the enclosed baryon fraction relative to the universal value) as a function of radius, in units of the virial radius rvir, from the simulations of Eke et al. (1998). The simulated clusters have similar masses to the systems studied here. The results (at zero redshift) for the three most dynamically relaxed clusters in the simulations are shown as bold red curves. Less relaxed simulated clusters are shown as dashed green curves. The Chandra observations for the six lowest redshift clusters in the fgas sample are plotted as blue circles, with error bars. (The Chandra profiles are identical to those shown in Fig. 1, but are scaled assuming Ωm= 0.27, Ωb= 0.0413 and r2500= 0.25 rvir.) The agreement between the observed and predicted profiles argues that the non-radiative simulations provide a reasonable approximation for the purpose of predicting the baryonic mass distributions.

Figure 4

The X-ray depletion or bias factor, b (i.e. the enclosed baryon fraction relative to the universal value) as a function of radius, in units of the virial radius rvir, from the simulations of Eke et al. (1998). The simulated clusters have similar masses to the systems studied here. The results (at zero redshift) for the three most dynamically relaxed clusters in the simulations are shown as bold red curves. Less relaxed simulated clusters are shown as dashed green curves. The Chandra observations for the six lowest redshift clusters in the fgas sample are plotted as blue circles, with error bars. (The Chandra profiles are identical to those shown in Fig. 1, but are scaled assuming Ωm= 0.27, Ωb= 0.0413 and r2500= 0.25 rvir.) The agreement between the observed and predicted profiles argues that the non-radiative simulations provide a reasonable approximation for the purpose of predicting the baryonic mass distributions.

The factor K in equation (3) is a ‘calibration’ constant that parametrizes residual uncertainty in the accuracy of the instrument calibration and X-ray modelling. Contributing factors include uncertainty in the instrument effective area, variations in element abundance ratios, modelling the effects of gas clumping and asphericity (the latter effects are expected to be small for large, relaxed clusters; Nagai et al. 2007a. See also Piffaretti, Jetzer & Schindler 2003; Gavazzi 2005). We conservatively include a 10 per cent Gaussian uncertainty in K to model the combined effect of these factors, such that K= 1.0 ± 0.1. The small intrinsic dispersion in fgas values (Section 5.3) means that Malmquist bias is expected to have a negligible effect on the derived cosmological parameters. Uncertainties associated with other systematic factors are expected to be negligible in comparison to the allowances listed above.

In cases where the Chandrafgas data are not combined with CMB data, we include simple Gaussian priors on Ωbh2 and h. Two separate sets of priors were used: ‘standard’ priors with Ωbh2= 0.0214 ± 0.0020 (Kirkman et al. 2003) and h= 0.72 ± 0.08 (Freedman et al. 2001), and ‘weak’ priors in which the nominal uncertainties were tripled to give Ωbh2= 0.0214 ± 0.0060 and h= 0.72 ± 0.24. In cases where the CMB data are included, no priors on Ωbh2 or h are needed or used. The complete set of standard priors and allowances included in the fgas analysis are summarized in Table 4.

Table 4

Summary of the standard systematic allowances and priors included in the Chandrafgas analysis. The priors on Ωbh2 and h (Freedman et al. 2001; Kirkman et al. 2003) are used when the CMB data are not included. We have also examined the case where the allowance for non-thermal pressure support has been doubled i.e. 1.0 < γ < 1.2 (see text for details).

Cluster Parameter Allowance 
Calibration/modelling K 1.0 ± 0.1 (Gaussian) 
Non-thermal pressure γ 1.0 < γ < 1.1 
Gas depletion: normalization b0 0.65 < b0 < 1.0 
Gas depletion: evolution αb −0.1 < αb < 0.1 
Stellar mass: normalization s0 0.16 ± 0.048 (Gaussian) 
Stellar mass: evolution αs −0.2 < αs < 0.2 
fgas(rr2500) slope η 0.214 ± 0.022 (Gaussian) 
Standard prior Ωbh2 Ωbh2 0.0214 ± 0.0020 
Standard prior h h 0.72 ± 0.08 
Weak prior Ωbh2 Ωbh2 0.0214 ± 0.0060 
Weak prior h h 0.72 ± 0.24 
Cluster Parameter Allowance 
Calibration/modelling K 1.0 ± 0.1 (Gaussian) 
Non-thermal pressure γ 1.0 < γ < 1.1 
Gas depletion: normalization b0 0.65 < b0 < 1.0 
Gas depletion: evolution αb −0.1 < αb < 0.1 
Stellar mass: normalization s0 0.16 ± 0.048 (Gaussian) 
Stellar mass: evolution αs −0.2 < αs < 0.2 
fgas(rr2500) slope η 0.214 ± 0.022 (Gaussian) 
Standard prior Ωbh2 Ωbh2 0.0214 ± 0.0020 
Standard prior h h 0.72 ± 0.08 
Weak prior Ωbh2 Ωbh2 0.0214 ± 0.0060 
Weak prior h h 0.72 ± 0.24 

Finally, we note how inspection of equation (3) can provide useful insight into the strength of the fgas experiment. The pre-factors before the square brackets shows how the normalization of the fgas(z) curve is used to constrain Ωm, given prior information on Ωb, h, K, γ, b and s. The ratio of distances inside the square brackets (and to a small extent the angular correction factor) shows how the shape of the fgas(z) curve constrains the geometry of the Universe and therefore dark energy. The combination of information from both the normalization and shape breaks the degeneracy between Ωm and the dark energy parameters in the distance equations.

4.3 Other data used in the analysis

In addition to the analysis of the Chandrafgas data alone, we have examined the improved constraints on cosmological parameters that can be obtained through combination of the fgas data with CMB and SNIa studies.

Our analysis of CMB observations uses the three-year Wilkinson Microwave anisotropy Probe (WMAP) temperature (TT) data for multipoles l < 1000 (Hinshaw et al. 2007; Spergel et al. 2007) and temperature-polarization (TE) data for l < 450 (Page et al. 2007). We use the October 2006 version of the WMAP likelihood code available from http://lambda.gsfc.nasa.gov/product/map/current/m_sw.cfm. Like most authors, we have ignored the small contribution to the TT data expected to arise from the Sunyaev–Z'eldovich (SZ) effect in clusters and groups (e.g. Komatsu & Seljak 2002) and do not account for gravitational lensing of the CMB (Lewis & Challinor 2006), which has a negligible effect on the derived cosmological parameters. To extend the analysis to higher multipoles (smaller scales), we also include data from the Cosmic Background Imager (Mason et al. 2003; Pearson et al. 2003), the Arcminute Cosmology Bolometer Array Receiver (Kuo et al. 2003) and BOOMERanG (Piacentini et al. 2005; Jones et al. 2006; Montroy et al. 2006), as incorporated into the current version of the cosmomc code (Lewis & Bridle 2002). We use a modified version of camb (Lewis, Challinor & Lasenby 2000) to calculate CMB power spectra, which includes a consistent treatment of the effects of dark energy perturbations for evolving-w models (Rapetti et al. 2005; we assume that the sound speed in the dark energy fluid is equal to the speed of light).

Our analysis of SNIa data uses two separate supernova samples. In the first case, we use the compilation of Davis et al. (2007) which includes results from the ESSENCE survey (60 targets; Miknaitis et al. 2007; Wood-Vasey et al. 2007), the SNLS first-year data (57 targets; Astier et al. 2006), 45 nearby supernovae (Jha et al. 2007) and the 30 high-redshift supernovae discovered by Hubble Space Telescope (HST) and reported by Riess et al. (2007) for which a ‘gold’ rating was awarded. This sample includes 192 SNIa in total. The second supernova sample is the full ‘gold’ sample of Riess et al. (2007) which totals 182 SNIa, including the HST-discovered objects. For both samples we marginalize analytically over the absolute normalization of the distance moduli.

4.4 Dark energy models

We have considered three separate dark energy models in the analysis: (1) standard ΛCDM, for which the dark energy equation of state w=−1; (2) a model that allows any constant dark energy equation of state, including ‘phantom’ models with w < −1; (3) a model in which the dark energy equation of state is allowed to evolve as  

6
formula
where a= 1/(1 +z) is the scalefactor, w0 and wet are the equation of state at late (present day) and early times, and zt and at are the redshift and scalefactor at the transition between the two, respectively (Rapetti et al. 2005; see also Chevallier & Polarski 2001; Linder 2003, 2007; Corasaniti et al. 2004). We employ a uniform prior on the transition scalefactor such that 0.5 < at < 0.95. As discussed by Rapetti et al. (2005), this model is both more general and more applicable to current data, which primarily constrain the properties of dark energy at redshifts z < 1, than models which impose a transition redshift z= 1, e.g. w(a) =w0+wa(1 −a).

Energy conservation of the dark energy fluid leads to an evolution of the energy density with scalefactor  

7
formula
where ρde,0 is the energy density of the dark energy fluid today. Using the parameterization of equation (6) we obtain  
8
formula
with  
9
formula
The Friedmann equation, which relates the first time derivative of the scalefactor of the Universe to the total density, can be conveniently expressed as forumla, with  
10
formula
Here Ωk is the curvature, ΩDE is the dark energy density and f(a) is its redshift dependence. (Note that we have ignored the density contributions from radiation and relativistic matter in this expression, although they are included in the analysis.) For our most general dark energy parameterization (equation 6)  
11
formula
For ΛCDM cosmologies, the dark energy density is constant and f(a) = 1. For w < −1 the dark energy density increases with time. For constant w models with w < −1/3, dark energy accelerates the expansion of the Universe. The results from a purely kinematic modelling of the data, which does not rely on the Friedmann equation and is independent of the assumptions of general relativity, are discussed by Rapetti et al. (2007).

Our combined analysis of Chandrafgas, SNIa and CMB data therefore has up to 10 interesting parameters: the physical dark matter and baryon densities in units of the critical density, the curvature Ωk, the ratio of the sound horizon to the angular diameter distance for the CMB (Kosowsky, Milosavljevic & Jimenez 2002), the amplitude of the scalar power spectrum, the scalar spectral index, the optical depth to reionization, and up to three parameters associated with the dark energy equation of state: w0, wet and at. In all cases, we assume an absence of both tensor components and massive neutrinos and, for the analysis of the CMB data alone, include a wide uniform prior on the Hubble parameter, 0.2 < h < 2.0. (Tests in which tensor components are included with ΛCDM models lead to similar results on dark energy, but take much longer to compute.)

5 CONSTRAINTS ON COSMOLOGICAL PARAMETERS

5.1 Constraints on Ωm from the low-zfgas data

In the first case, we have used the Chandrafgas data for only the six, lowest redshift clusters in the sample, with z≲ 0.15, to constrain the mean matter density of the Universe. The restriction to low-z clusters minimizes correlated uncertainties associated with the nature of the dark energy component (dark energy has only a very small effect on the space–time metric over this redshift range; we employ a broad uniform prior such that 0.0 < ΩΛ < 2.0) and renders negligible uncertainties associated with the evolution of the depletion factor and stellar baryonic mass fraction (αb and αs). Fig. 5 shows the marginalized constraints on Ωm for a ΛCDM model with free curvature, using the standard priors on Ωbh2 and h, for which we obtain a result of Ωm= 0.28 ± 0.06. The full set of conservative systematic allowances, as described in Table 4, were included.

Figure 5

The marginalized constraints on Ωm from the Chandrafgas data for the six lowest redshift clusters, using the non-flat ΛCDM model and standard priors on Ωbh2 and h. Uncertainties due to the evolution in b and s and the nature of the dark energy component are negligible in the analysis (although allowances for these uncertainties are included). We obtain a marginalized result Ωm= 0.28 ± 0.06 (68 per cent confidence limits).

Figure 5

The marginalized constraints on Ωm from the Chandrafgas data for the six lowest redshift clusters, using the non-flat ΛCDM model and standard priors on Ωbh2 and h. Uncertainties due to the evolution in b and s and the nature of the dark energy component are negligible in the analysis (although allowances for these uncertainties are included). We obtain a marginalized result Ωm= 0.28 ± 0.06 (68 per cent confidence limits).

The result on Ωm from the six lowest redshift clusters is in good agreement with that obtained for the whole sample, as discussed below. It is also consistent with the result on Ωm found from an analysis of all clusters except the six lowest redshift systems, Ωm= 0.29 ± 0.06, i.e. the six lowest redshift clusters do not dominate the Ωm constraints. Note that the error bars on Ωm are dominated by the widths of the priors on Ωbh2 and h and the magnitudes of the systematic allowances on K, b and γ, which are all at the ∼10–20 per cent level. In contrast, the statistical uncertainty in the normalization of the fgas(z) curve is small (Section 3.1) and has a negligible impact on the Ωm results.

The result on Ωm is consistent with previous findings based on fgas data (see references in Section 1) and independent constraints from the CMB (e.g. Spergel et al. 2007), galaxy redshift surveys (e.g. Eisenstein et al. 2005) and other leading cosmological data. Note that the agreement in cosmological parameters determined from the fgas and CMB data argues against any unusual depletion of baryons within r2500 in hot, relaxed clusters (see e.g. the discussions in Ettori 2003; Afshordi et al. 2007; McCarthy, Bower & Balogh 2007).

5.2 Constraints on the ΛCDM model using the fgas (+ CMB + SNIa) data

We next extended our analysis to measure Ωm and ΩΛ for a non-flat ΛCDM model using the Chandrafgas data for the full sample of 42 clusters. The results are shown as the red contours in Fig. 6. Using the systematic allowances summarized in Table 4 and the standard priors on Ωbh2 and h, we measure Ωm= 0.27 ± 0.06 and ΩΛ= 0.86 ± 0.19 (68 per cent confidence limits) with χ2= 41.5 for 40 degrees of freedom. The low χ2 value obtained is important and indicates that the model provides an acceptable description of the data (see Section 5.3 below). The result on Ωm is in excellent agreement with that determined from the six lowest redshift clusters only (Section 5.1). The result is also consistent with the value reported by Allen et al. (2004) using the previous release of fgas data, although the more conservative systematic allowances included here lead to the quoted uncertainties in Ωm being larger by ∼50 per cent.

Figure 6

The 68.3 and 95.4 per cent (1 and 2 σ) confidence constraints in the (Ωm, ΩΛ) plane for the Chandrafgas data (red contours; standard priors on Ωbh2 and h are used). Also shown are the independent results obtained from CMB data (blue contours) using a weak, uniform prior on h (0.2 < h < 2), and SNIa data (green contours; the results for the Davis et al. 2007 compilation are shown). The inner, orange contours show the constraint obtained from all three data sets combined (no external priors on Ωbh2 and h are used). A ΛCDM model is assumed, with the curvature included as a free parameter.

Figure 6

The 68.3 and 95.4 per cent (1 and 2 σ) confidence constraints in the (Ωm, ΩΛ) plane for the Chandrafgas data (red contours; standard priors on Ωbh2 and h are used). Also shown are the independent results obtained from CMB data (blue contours) using a weak, uniform prior on h (0.2 < h < 2), and SNIa data (green contours; the results for the Davis et al. 2007 compilation are shown). The inner, orange contours show the constraint obtained from all three data sets combined (no external priors on Ωbh2 and h are used). A ΛCDM model is assumed, with the curvature included as a free parameter.

Fig. 7 shows the marginalized constraints on ΩΛ obtained using both the standard and weak priors on Ωbh2 and h. We see that using only the weak priors (Ωbh2= 0.0214 ± 0.0060, h= 0.72 ± 0.24), the fgas data provide a clear detection of the effects of dark energy on the expansion of the Universe, with ΩΛ= 0.86 ± 0.21: a model with ΩΛ≤ 0 is ruled out at ∼99.98 per cent confidence. (Using the standard priors on Ωbh2 and h, a model with ΩΛ≤ 0 is ruled out at 99.99 per cent confidence; Table 5.) The significance of the detection of dark energy in the fgas data is comparable to that of current SNIa studies (e.g. Riess et al. 2007; Wood-Vasey et al. 2007). The fgas data provide strong, independent evidence for cosmic acceleration.

Figure 7

The marginalized constraints on ΩΛ determined from the Chandrafgas data using the non-flat ΛCDM model and standard (solid curve) and weak (dashed curve) priors on Ωbh2 and h. The fgas data provide a detection of the effects of dark energy at the ∼99.99 per cent confidence level.

Figure 7

The marginalized constraints on ΩΛ determined from the Chandrafgas data using the non-flat ΛCDM model and standard (solid curve) and weak (dashed curve) priors on Ωbh2 and h. The fgas data provide a detection of the effects of dark energy at the ∼99.99 per cent confidence level.

Table 5

Summary of the constraints on cosmological parameters determined from the Chandrafgas data and complementary data sets. Error bars reflect the combined statistical and systematic uncertainties, incorporating the allowances and priors described in Section 4.2. For the low-z fgas data (z < 0.15), the constraint on Ωm is almost independent of the details of the dark energy component (Section 5.1). The SNIa(1) and SNIa(2) labels denote the supernovae samples of Davis et al. (2007) and Riess et al. (2007), respectively (Section 4.3).

Data Model Ωbh2, h priors Cosmological constraints 
Ωm ΩDE w0 wet 
low-z fgas ΛCDM (0 < ΩΛ < 2.0) Standard 0.28 ± 0.06 – – – 
fgas ΛCDM Standard 0.27 ± 0.06 0.86 ± 0.19 – – 
fgas ΛCDM Weak 0.27 ± 0.09 0.86 ± 0.21 – – 
fgas+ CMB ΛCDM None 0.28 ± 0.06 0.73 ± 0.04 – – 
fgas+ CMB + SNIa(1) ΛCDM None 0.275 ± 0.033 0.735 ± 0.023 – – 
fgas Constant w (flat) Standard 0.28 ± 0.06 – −1.14+0.27−0.35 – 
fgas Constant w (flat) Weak 0.29 ± 0.09 – −1.11+0.31−0.45 – 
fgas+CMB Constant w (flat) None 0.243 ± 0.033 – −1.00 ± 0.14 – 
fgas+ CMB + SNIa(1) Constant w (flat) None 0.253 ± 0.021 – −0.98 ± 0.07 – 
fgas+ CMB + SNIa(1) Constant w None 0.310 ± 0.052 – −1.08+0.13−0.19 – 
fgas+ CMB + SNIa(1) Evolving w (flat) None 0.254 ± 0.022 – −1.05+0.31−0.26 −0.83+0.48−0.43 
fgas+ CMB + SNIa(1) Evolving w None 0.29+0.09−0.04 – −1.15+0.50−0.38 −0.80+0.70−1.30 
fgas+ CMB + SNIa(2) Evolving w (flat) None 0.287 ± 0.026 – −1.19+0.29−0.35 −0.33+0.18−0.34 
Data Model Ωbh2, h priors Cosmological constraints 
Ωm ΩDE w0 wet 
low-z fgas ΛCDM (0 < ΩΛ < 2.0) Standard 0.28 ± 0.06 – – – 
fgas ΛCDM Standard 0.27 ± 0.06 0.86 ± 0.19 – – 
fgas ΛCDM Weak 0.27 ± 0.09 0.86 ± 0.21 – – 
fgas+ CMB ΛCDM None 0.28 ± 0.06 0.73 ± 0.04 – – 
fgas+ CMB + SNIa(1) ΛCDM None 0.275 ± 0.033 0.735 ± 0.023 – – 
fgas Constant w (flat) Standard 0.28 ± 0.06 – −1.14+0.27−0.35 – 
fgas Constant w (flat) Weak 0.29 ± 0.09 – −1.11+0.31−0.45 – 
fgas+CMB Constant w (flat) None 0.243 ± 0.033 – −1.00 ± 0.14 – 
fgas+ CMB + SNIa(1) Constant w (flat) None 0.253 ± 0.021 – −0.98 ± 0.07 – 
fgas+ CMB + SNIa(1) Constant w None 0.310 ± 0.052 – −1.08+0.13−0.19 – 
fgas+ CMB + SNIa(1) Evolving w (flat) None 0.254 ± 0.022 – −1.05+0.31−0.26 −0.83+0.48−0.43 
fgas+ CMB + SNIa(1) Evolving w None 0.29+0.09−0.04 – −1.15+0.50−0.38 −0.80+0.70−1.30 
fgas+ CMB + SNIa(2) Evolving w (flat) None 0.287 ± 0.026 – −1.19+0.29−0.35 −0.33+0.18−0.34 

In contrast to the Ωm constraints, the error budget for ΩΛ includes significant contributions from both statistical and systematic sources. From the analysis of the full sample of 42 clusters using the standard priors on Ωbh2 and h, we find ΩΛ= 0.86 ± 0.19; the error bar comprises ±0.15 statistical error and ±0.12 systematic uncertainty. Thus, whereas improved measurements of Ωm from the fgas method will require additional information leading to tighter priors and systematic allowances, significant improvements in the precision of the dark energy constraints should be possible simply by gathering more data (e.g. doubling the present fgas data set).

Fig. 6 also shows the constraints on Ωm and ΩΛ obtained from the CMB (blue contours) and SNIa (green contours) data (Section 4.3). The agreement between the results for the independent data sets is excellent and motivates a combined analysis. The inner, orange contours in Fig. 6 show the constraints on Ωm and ΩΛ obtained from the combined fgas+ CMB + SNIa data set. We obtain marginalized 68 per cent confidence limits of Ωm= 0.275 ± 0.033 and ΩΛ= 0.735 ± 0.023. Together, the fgas+ CMB + SNIa data also constrain the Universe to be close to geometrically flat: Ωk=−0.010 ± 0.011. No external priors on Ωbh2 and h are used in the analysis of the combined fgas+ CMB + SNIa data (see also Section 5.6).

Finally, we have examined the effects of doubling the allowance for non-thermal pressure support in the clusters, i.e. setting 1.0 < γ < 1.2. For the analysis of the fgas data alone, this boosts the best-fitting value of Ωm by ∼5 per cent but leaves the results on dark energy unchanged. This can be understood by inspection of equation (3) and recalling that the constraint on Ωm is determined primarily from the normalization of the fgas curve, whereas the constraints on dark energy are driven by its shape (Section 4.2). For the combined fgas+ CMB + SNIa data set, doubling the width of the allowance on γ has a negligible impact on the results, since in this case the value of Ωm is tightly constrained by the combination of data sets.

5.3 Scatter in the fgas data

Hydrodynamical simulations suggest that the intrinsic dispersion in fgas measurements for the largest, dynamically relaxed galaxy clusters should be small. Nagai et al. (2007a) simulate and analyse mock X-ray observations of galaxy clusters (including cooling and feedback processes), employing standard assumptions of spherical symmetry and hydrostatic equilibrium and identifying relaxed systems based on X-ray morphology in a similar manner to that employed here. For relaxed clusters, these authors find that fgas measurements at r2500 are biased low by ∼9 per cent, with the bias primarily due to non-thermal pressure support provided by subsonic bulk motions in the intracluster gas. They measure an intrinsic dispersion in the fgas measurements of ∼6 per cent, with an indication that the scatter may be even smaller for analyses limited to the hottest, relaxed systems with kT≳ 5 keV. Nagai et al. (2007a) also suggest that the true scatter may be yet smaller if their simulations have underestimated the viscosity of the X-ray emitting gas.6 In contrast, for unrelaxed simulated clusters, Nagai et al. (2007a) find that fgas measurements are biased low by on average 27 per cent with an intrinsic dispersion of more than 50 per cent. Thus, the dispersion in fgas measurements for unrelaxed clusters is expected to be an order of magnitude larger than for relaxed systems. This is in agreement with the measurement of very low intrinsic systematic scatter in the fgas data for relaxed clusters reported here (see below) and the much larger scatter measured in previous works that included no such restriction to relaxed clusters. Earlier, non-radiative simulations by Eke et al. (1998) also argued for a small intrinsic scatter in fgas, at the few per cent level, for large, relaxed clusters (see also Crain et al. 2007). Likewise, Kay et al. (2004) measure a small intrinsic dispersion in fgas measurements from simulations including cooling and moderate star formation.

The expectation of a small intrinsic dispersion in the fgas measurements for hot, dynamically relaxed clusters is strikingly confirmed by the present data. Even without including the allowances for systematic uncertainties associated with γ, b0, αb, s and αs described in Table 4 (i.e. keeping only the 10 per cent systematic uncertainty on the overall normalization, as described by K) the best-fitting non-flat ΛCDM model gives an acceptable χ2= 41.9 for 40 degrees of freedom, when fitting the full fgas sample. (The χ2 drops only to 41.5 with the full set of systematic allowances included; this small change in χ2 illustrates the degeneracies between the systematic allowances and model parameters.) The acceptable χ2 for the best-fitting model rules out the presence of significant intrinsic, systematic scatter in the current fgas data. This absence of systematic scatter is observed despite the fact that the rms scatter in the fgas data is only 15 per cent. Moreover, the rms scatter is dominated by those measurements with large statistical uncertainties; the weighted mean scatter of the fgas data about the best-fitting ΛCDM model is only 7.2 per cent, which corresponds to only 7.2/1.5 = 4.8 per cent in distance.

5.4 Constraints on the constant w model using the fgas (+ CMB + SNIa) data

We have next examined the ability of our data to constrain the dark energy equation of state parameter, w. In the first case, we examined a geometrically flat model in which w is constant with time. Fig. 8 shows the constraints in the Ωm, w plane for this model using the Chandrafgas data and standard priors/allowances (red contours), the CMB data (blue contours) and SNIa data (green contours). The different parameter degeneracies in the data sets are clearly evident. For the fgas data alone, we measure Ωm= 0.28 ± 0.06 and w=−1.14+0.27−0.35.

Figure 8

The 68.3 and 95.4 per cent (1 and 2σ) confidence constraints in the (Ωm, w) plane obtained from the analysis of the Chandrafgas data (red contours) using standard priors on Ωbh2 and h. Also shown are the independent results obtained from CMB data (blue contours) using a weak, uniform prior on h (0.2 < h < 2.0) and SNIa data (green contours; Davis et al. 2007). The inner, orange contours show the constraint obtained from all three data sets combined: Ωm= 0.253 ± 0.021 and w=−0.98 ± 0.07 (68 per cent confidence limits). No external priors on Ωbh2 and h are used when the data sets are combined. A flat cosmology with a constant dark energy equation of state parameter w is assumed.

Figure 8

The 68.3 and 95.4 per cent (1 and 2σ) confidence constraints in the (Ωm, w) plane obtained from the analysis of the Chandrafgas data (red contours) using standard priors on Ωbh2 and h. Also shown are the independent results obtained from CMB data (blue contours) using a weak, uniform prior on h (0.2 < h < 2.0) and SNIa data (green contours; Davis et al. 2007). The inner, orange contours show the constraint obtained from all three data sets combined: Ωm= 0.253 ± 0.021 and w=−0.98 ± 0.07 (68 per cent confidence limits). No external priors on Ωbh2 and h are used when the data sets are combined. A flat cosmology with a constant dark energy equation of state parameter w is assumed.

The results for the three data sets shown in Fig. 8 are each, individually, consistent with the ΛCDM model (w=−1). The consistent nature of these constraints again motivates a combined analysis of the data, shown as the small, central (orange) contours. For the three data sets combined, we measure Ωm= 0.253 ± 0.021 and w=−0.98 ± 0.07 (68 per cent confidence limits). No priors on Ωbh2 and h are required or used in the combined fgas+ CMB + SNIa analysis. The constraints on w from the combined data set are significantly tighter than 10 per cent.

We note that our analysis accounts for the effects of dark energy perturbations, which must exist for dark energy models other than ΛCDM; neglecting the effects of such perturbations can lead to spuriously tight constraints (see Rapetti et al. 2005 for details).

5.5 Constraints on the evolution of w from the combined fgas+ CMB + SNIa data

Fig. 9 shows the constraints on w0 and wet obtained from a combined analysis of fgas+ CMB + SNIa data using the general, evolving dark energy model (equation 6) and assuming geometric flatness (Ωk= 0). The left- and right-hand panels show the results obtained for the two separate SNIa samples (Section 4.3). Using the Davis et al. (2007) SNIa compilation (left-hand panel), we find no evidence for evolution in the dark energy equation of state over the redshift range spanned by the data: the results on the dark energy equation of state at late and early times, w0=−1.05+0.31−0.26 and wet=−0.83+0.48−0.43 (68 per cent confidence limits), are both consistent with a cosmological constant model (w=−1, constant). A similar conclusion is drawn by Davis et al. (2007) using SNIa + CMB + baryon acoustic oscillation (BAO) data.

Figure 9

The 68.3 and 95.4 per cent confidence limits in the (Ωm; w0, wet) plane determined from the fgas+ CMB + SNIa data using our most general dark energy model (equation 6) with the transition scalefactor marginalized over the range 0.5 < at < 0.95. The solid, purple contours show the results on (Ωm, w0). The dashed, turquoise lines show the results on (Ωm, wet). The horizontal dotted line denotes the cosmological constant model (w0=wet=−1). The left- and right-hand panels show the results obtained for the two SNIa samples: (left-hand panel) Davis et al. (2007) and (right-hand panel) Riess et al. (2007). A flat geometry (Ωk= 0) is assumed. The data provide no significant evidence for evolution in w and are consistent with the cosmological constant (ΛCDM) model (w=−1; Section 5.5).

Figure 9

The 68.3 and 95.4 per cent confidence limits in the (Ωm; w0, wet) plane determined from the fgas+ CMB + SNIa data using our most general dark energy model (equation 6) with the transition scalefactor marginalized over the range 0.5 < at < 0.95. The solid, purple contours show the results on (Ωm, w0). The dashed, turquoise lines show the results on (Ωm, wet). The horizontal dotted line denotes the cosmological constant model (w0=wet=−1). The left- and right-hand panels show the results obtained for the two SNIa samples: (left-hand panel) Davis et al. (2007) and (right-hand panel) Riess et al. (2007). A flat geometry (Ωk= 0) is assumed. The data provide no significant evidence for evolution in w and are consistent with the cosmological constant (ΛCDM) model (w=−1; Section 5.5).

We note, however, a hint of evolution in the dark energy equation of state when the Riess et al. (2007)‘gold’ SNIa sample is used instead (right-hand panel of Fig. 9). In this case, the marginalized constraints on dark energy at late and early times, as defined in Section 4.4, differ at the 2–3σ level. Similar indications are also apparent in the analysis of the same SNIa (+ CMB + BAO) data by Riess et al. (2007). However, the analysis using the Davis et al. (2007) SNIa compilation (left-hand panel), which includes the high-quality, high-redshift HST supernovae from Riess et al. (2007) and which shows no suggestion of a departure from the ΛCDM model, argues that the hint of evolution in the right-hand panel of Fig. 9 may be systematic in origin (see also Conley et al. 2007; Riess et al. 2007 for discussions).

5.6 The degeneracy breaking power of the combined fgas+ CMB (+ SNIa) data

The degeneracy breaking power of the combined fgas+ CMB data set is evidenced in the left-hand panel of Fig. 10, which shows the constraints on Ωm versus ΩDE for a ΛCDM model with free curvature for the CMB data alone (blue contours) and the combined fgas+ CMB data set (orange contours). For the fgas+ CMB data, we measure Ωm= 0.278+0.064−0.050 and ΩΛ= 0.732+0.040−0.046 (68 per cent confidence limits), with the curvature Ωk=−0.011+0.015−0.017. As mentioned above, no external priors on Ωbh2 and H0 are required when the fgas and CMB data are combined. The degeneracy breaking power of other combinations of data with the CMB is discussed by Spergel et al. (2007).

Figure 10

The degeneracy-breaking power of the fgas+ CMB data. Contours show the 68.3 and 95.4 per cent confidence limits determined from the CMB data alone (larger, blue contours) and combined fgas+ CMB data (smaller, orange contours). Left-hand panel: The constraints on Ωm and ΩDE for the ΛCDM model with the curvature included as a free parameter. Right-hand panel: The tight constraints on H0 and Ωbh2 for the flat, constant w model, demonstrating why external priors on these two parameters are not required when the fgas and CMB data are combined.

Figure 10

The degeneracy-breaking power of the fgas+ CMB data. Contours show the 68.3 and 95.4 per cent confidence limits determined from the CMB data alone (larger, blue contours) and combined fgas+ CMB data (smaller, orange contours). Left-hand panel: The constraints on Ωm and ΩDE for the ΛCDM model with the curvature included as a free parameter. Right-hand panel: The tight constraints on H0 and Ωbh2 for the flat, constant w model, demonstrating why external priors on these two parameters are not required when the fgas and CMB data are combined.

The right-hand panel of Fig. 10 shows the constraints on the Hubble constant, H0, and mean baryon density, Ωbh2, determined using the flat, constant w model for the CMB data alone (blue contours) and the combined fgas+ CMB data set (orange contours). The improvement in the constraints on these parameters determined from the fgas+ CMB data over the CMB data alone is substantial. The tight constraints for the fgas+ CMB data, H0= 72.5 ± 4.6 km s−1Mpc−1 and Ωbh2= 0.0223 ± 0.0007, demonstrate clearly why external priors on these two parameters are not required when the fgas and CMB data are combined. Indeed, the constraints on H0 and Ωbh2 obtained from the fgas+ CMB data are significantly tighter than the external priors on these parameters that are employed when the fgas data are used alone (Table 4). Similar constraints on H0 and Ωbh2 are presented by the WMAP team (Spergel et al. 2007) for flat ΛCDM models using various data combinations.

Fig. 11 shows the constraints on the dark energy equation of state obtained from an analysis of the combined fgas+ CMB + SNIa data set where the curvature is also included as a free parameter. The marginalized results for the constant w model (left-hand panel), w=−1.08+0.13−0.19 and Ωk=−0.024+0.022−0.018, are comparable to those of Spergel et al. (2007; see their fig. 17) from a combined analysis of CMB, SNIa and Galaxy Redshift Survey data. The constraints for the non-flat evolving w model (right-hand panel), though weaker than those for the flat model (Fig. 9), remain interesting and are also consistent with a cosmological constant. As discussed by Rapetti et al. (2005; see also Spergel et al. 2007), such results demonstrate the power of the fgas+ CMB + SNIa data to constrain the properties of dark energy without the need to assume that the Universe is flat.

Figure 11

Left-hand panel: The 68.3 and 95.4 per cent confidence limits on the dark energy equation of state and curvature from the analysis of the fgas+ CMB + SNIa data using the non-flat, constant w model. The SNIa compilation of Davis et al. (2007) has been used. The horizontal and vertical dotted lines denote the loci for cosmological constant models and geometric flatness, respectively, both of which are consistent with the data. Right-hand panel: The 68.3 and 95.4 per cent confidence limits in the (Ωm; w0, wet) plane determined from the fgas+ CMB + SNIa data for the general dark energy model (equation 6) with the curvature also included as a free parameter. Other details are as in the left-hand panel of Fig. 9.

Figure 11

Left-hand panel: The 68.3 and 95.4 per cent confidence limits on the dark energy equation of state and curvature from the analysis of the fgas+ CMB + SNIa data using the non-flat, constant w model. The SNIa compilation of Davis et al. (2007) has been used. The horizontal and vertical dotted lines denote the loci for cosmological constant models and geometric flatness, respectively, both of which are consistent with the data. Right-hand panel: The 68.3 and 95.4 per cent confidence limits in the (Ωm; w0, wet) plane determined from the fgas+ CMB + SNIa data for the general dark energy model (equation 6) with the curvature also included as a free parameter. Other details are as in the left-hand panel of Fig. 9.

Using the non-flat evolving w model but fixing the transition redshift zt= 1 in equation (6), we recover the model used by the Dark Energy Task Force (DETF) to assess the power of future dark energy experiments. The combination of current fgas+ CMB + SNIa data provides a DETF figure of merit ∼2.

6 DISCUSSION

The new Chandrafgas results and analysis presented here build upon those of Allen et al. (2004) and Rapetti et al. (2005). The present study includes 16 more objects, approximately twice as much Chandra data and extends the study beyond a redshift of 1. Our analysis includes a comprehensive and conservative treatment of systematic uncertainties (Section 4.2; see also Table 4). Allowances for such uncertainties are easily incorporated into the MCMC analysis.

As with SNIa studies, the fgas data constrain dark energy via its effects on the distance–redshift relation to a well-defined source population – in this case, the largest, dynamically relaxed galaxy clusters – using measurements of a ‘standard’ astrophysical quantity – the ratio of baryonic-to-total mass in the clusters. Our results provide a clear and independent detection of the effects of dark energy on the expansion of the Universe at ∼99.99 per cent confidence for a standard non-flat ΛCDM model, an accuracy comparable to that obtained from current SNIa work (e.g. Astier et al. 2006; Miknaitis et al. 2007; Riess et al. 2007; Wood-Vasey et al. 2007). Like SNIa studies, the fgas data trace the evolution of dark energy over the redshift range 0 < z < 1, where it grows to dominate the overall energy density of the Universe. Our results for the fgas data alone, and the combination of fgas+ CMB + SNIa data, show that this growth is consistent with that expected for models in which the dark energy is a cosmological constant (w=−1).

Despite some clear similarities, important complementary differences between the fgas and SNIa experiments exist. In the first case, the physics of the astrophysical objects – large, relaxed galaxy clusters and SNIa – are very different; the fact that such similar cosmological results are obtained from the distance–redshift information for these separate source populations is reassuring. Future studies, combining the two techniques but using larger target samples, should open the possibility for precise distance–redshift measurements and good control of systematic uncertainties, employing both kinematic and dynamical analyses (e.g. Rapetti et al. 2007; Riess et al. 2007, and references therein).

An important strength of the fgas method is the tight constraint on Ωm provided by the normalization of the fgas curve; this breaks the degeneracy between the mean matter density and dark energy density inherent in the distance measurements. Our result on Ωm is consistent with a host of previous X-ray studies (Section 1).

A further strength, which is of relevance when considering observing strategies for future dark energy work, is the small intrinsic dispersion in the fgas distance measurements. SNIa studies have established the presence of a systematic scatter of ∼7 per cent in distance measurements for individual SNIa using high-quality data (Jha et al. 2007; see also e.g. Riess et al. 2004; Astier et al. 2006; Riess et al. 2007; Wood-Vasey et al. 2007). In contrast, systematic scatter remains undetected in the present Chandrafgas data for hot, relaxed clusters, despite the fact that the weighted mean statistical scatter in fgas data corresponds to only ∼5 per cent in distance. This small systematic scatter for large, dynamically relaxed clusters (identified as relaxed on the basis of their X-ray morphologies) is consistent with the predictions from hydrodynamical simulations (e.g. Nagai et al. 2007a), although the results for both observed and simulated clusters are, at present, based on relatively small samples and more data are required. We stress that such small systematic scatter is neither expected nor observed in studies where a restriction to morphologically relaxed clusters is not employed, e.g. compare the small scatter measured here with the much larger scatter observed in the studies of LaRoque et al. (2006) and Ettori et al. (2003); see also Nagai et al. (2007a). The restriction to the hottest, relaxed clusters, for which fgas is independent of temperature (Fig. 3) also simplifies the determination of cosmological parameters.

As mentioned above, the allowances for systematic uncertainties included in the analysis are relatively conservative. Much progress is expected over the coming years in refining the ranges of these allowances, both observationally and through improved simulations. As discussed in Sections 5.1 and 5.2, a reduction in the size of the required systematic allowances will tighten the cosmological constraints. Improved numerical simulations of large samples of massive clusters, including a more complete treatment of star formation and feedback physics that reproduces both the observed optical galaxy luminosity function and cluster X-ray properties, will be of major importance. Progress in this area has been made (e.g. Bialek et al. 2001; Muanwong et al. 2002; Ettori et al. 2004; Kay et al. 2004; Kravtsov, Nagai & Vikhlinin 2005; Ettori et al. 2006; Rasia et al. 2006; Nagai et al. 2007a; Nagai, Kravtsov & Vikhlinin 2007b), though more work remains. In particular, this work should improve the predictions for b(z). Further deep X-ray and optical observations of nearby clusters will provide better constraints on the viscosity of the cluster gas. Improved optical/near infrared observations of clusters should pin down the stellar mass fraction in galaxy clusters and its evolution.

Ground and space-based gravitational lensing studies will provide important, independent constraints on the mass distributions in clusters; a large programme using the Subaru telescope and HST is underway, as is similar work by other groups (e.g. Hoekstra 2007). Follow-up observations of the SZ effect will also provide additional, independent constraining power in the measurement of cosmological parameters (the combination of direct observations of the SZ effect using radio/submillimetre data and the prediction of this effect from X-ray data provides an additional constraint on absolute distances to the clusters, e.g. Molnar, Birkinshaw & Mushotzky 2002; Schmidt, Allen & Fabian 2004; Bonamente et al. 2006 and references therein). Moreover, the independent constraints provided by the SZ observations should allow a reduction of the priors required in future work (e.g. Rapetti & Allen 2007).

In the near future, continuing programmes of Chandra and XMM–Newton observations of known, X-ray luminous clusters should allow important progress to be made, both by expanding the fgas sample (e.g. Chandra snapshot observations of the entire MACS sample; Ebeling et al. 2001, 2007) and through deeper observations of the current target list. The advent of new, large area SZ surveys (e.g. Ruhl et al. 2004) will soon provide important new target lists of hot, X-ray luminous high-redshift clusters. A new, large area X-ray survey such as that proposed by the Spectrum-RG/eROSITA project7 could make a substantial contribution, finding hundreds of suitable systems at high redshifts.

Looking a decade ahead, the Constellation-X Observatory (Con-X)8 and, later, XEUS9 offer the possibility to carry out precise studies of dark energy using the fgas technique. As discussed by Rapetti & Allen (2007; see also Rapetti et al. 2006), the large collecting area and combined spatial/spectral resolving power of Con-X should permit precise fgas measurements with ∼5 per cent accuracy for large samples (≳ 500) of hot, massive clusters (kT≳ 5 keV) spanning the redshift range 0 < z < 2 (typical redshift z∼ 0.5). The predicted constraints on dark energy from such an experiment, assuming Planck priors (Albrecht et al. 2006), have a DETF figure of merit ≳20, which is comparable to other leading proposed dark energy techniques such as SNIa, cluster number counts, weak lensing and BAO studies. The high spectral resolution offered by the Con-X calorimeters will also permit precise measurements of bulk motions and viscosity in the cluster gas, addressing directly one of the main sources of systematic uncertainty in the method.

An ASCII table containing the redshift and fgas(z) data is available at http://xoc.stanford.edu or from the authors on request. The analysis code, in the form of a patch to cosmomc, will be made available at a later date.

1
To understand the origin of the fgasd1.5 dependence, consider a spherical region of observed angular radius θ within which the mean gas mass fraction is measured. The physical size, R, is related to the angle θ as RdA. The X-ray luminosity emitted from within this region, LX, is related to the detected flux, FX, as LX= 4πd2LFX, where dL is the luminosity distance and dA=dL/(1 +z)2 is the angular diameter distance. Since the X-ray emission is primarily due to collisional processes (bremsstrahlung and line emission) and is optically thin, we may also write LXn2V, where n is the mean number density of colliding gas particles and V is the volume of the emitting region, with V= 4π(θdA)3/3. Considering the cosmological distance dependences, we see that ndL/d1.5A, and that the observed gas mass within the measurement radius MgasnVdLd1.5A. The total mass, Mtot, determined from the X-ray data under the assumption of hydrostatic equilibrium, MtotdA. Thus, the X-ray gas mass fraction measured within angle θ is fgas=Mgas/MtotdLd0.5A.
2
The virial radius is defined as the radius within which the density contrast Δc= 178 Ωm(z)0.45, with respect to the critical density (Lahav et al. 1991; Eke et al. 1998).
3
Note that the outermost pressure, at the limit of the X-ray surface brightness profile, is fixed using an iterative method that ensures a smooth, power-law pressure gradient in these regions. The model temperature profiles, for radii spanned by the spectral data, are not sensitive to any reasonable choices for the outer pressures.
5
To see the origin of the correction factor A, recall that equation (3) predicts the fgas value at the measurement radius in the reference ΛCDM cosmology. This measurement radius corresponds to a fixed angle θΛCDM2500 for each cluster, which will differ slightly from θ2500, the angle corresponding to r2500 for that cluster in the current test cosmology. The mass contained within radius r2500, M2500= 104πr32500ρcrit/3. Given that the temperature, and temperature and density gradients, in the region of θ2500 are likely to be approximately constant, the hydrostatic equation gives forumla. Thus, since ρcrit= 3 H(z)2/8πG, we have forumla, and the angle spanned by r2500 at redshift forumla. Since the fgas profiles follow a smooth power-law form in the region of θ2500, the ratio of the model fgas value at θΛCDM2500 to that at θ2500 can be described by equation (5).
6
Recent work on the morphologies of X-ray cavities and Hα filaments suggest a relatively high gas viscosity (low Reynolds number) in nearby cluster cores (Fabian et al. 2003a,b; Ruszkowski, Brüggen & Begelman 2004; Fabian et al. 2005; Reynolds et al. 2005).

We thank Sarah Church, Vince Eke, Bob Kirshner, Gary Mamon, Herman Marshall, Rich Mushotzky, Jerry Ostriker, Harvey Tananbaum, Alexey Vikhlinin, Jochen Weller and Nick White for discussions over the course of this work. We also thank Antony Lewis for help with cosmomc. The computational analysis presented here was carried out using the KIPAC XOC compute cluster at the Stanford Linear Accelerator Centre (SLAC). We acknowledge support from the National Aeronautics and Space Administration through Chandra Award Numbers DD5-6031X, GO2-3168X, GO2-3157X, GO3-4164X, GO3-4157X and G07-8125X, issued by the Chandra X-ray Observatory Centre, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics and Space Administration under contract NAS8-03060. This work was supported in part by the US Department of Energy under contract number DE-AC02-76SF00515.

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