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arXiv:astro-ph/0205007v1 1 May 2002

Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed February 1, 2008 (MN LATEX style file v1.4)

Cosmological constraints from the X-ray gas mass fraction

in relaxed lensing clusters observed with Chandra

S.W. Allen, R.W. Schmidt and A.C. Fabian

Institute of Astronomy, Madingley Road, Cambridge CB3 0HA

February 1, 2008

ABSTRACT

We present precise measurements of the X-ray gas mass fraction for a sample of lumi-

nous, relatively relaxed clusters of galaxies observed with the Chandra Observatory,

for which independent confirmation of the mass results is available from gravitational

lensing studies. Parameterizing the total (luminous plus dark matter) mass profiles

using the model of Navarro, Frenk & White (1997), we show that the X-ray gas mass

fractions in the clusters asymptote towards an approximately constant value at a ra-

dius r2500, where the mean interior density is 2500 times the critical density of the

Universe at the redshifts of the clusters. Combining the Chandra results on the X-ray

gas mass fraction and its apparent redshift dependence with recent measurements of

the mean baryonic matter density in the Universe and the Hubble Constant deter-

mined from the Hubble Key Project, we obtain a tight constraint on the mean total

matter density of the Universe, Ωm= 0.30+0.04

−0.03, and measure a positive cosmological

constant, ΩΛ= 0.95+0.48

−0.72. Our results are in good agreement with recent, independent

findings based on analyses of anisotropies in the cosmic microwave background radia-

tion, the properties of distant supernovae, and the large-scale distribution of galaxies.

Key words:

lensing – cosmological parameters

X-rays: galaxies: clusters – galaxies: clusters: general – gravitational

1INTRODUCTION

The matter content of rich clusters of galaxies is thought to

provide a fair sample of the matter content of the Universe as

a whole (White et al. 1993). The observed ratio of the bary-

onic to total mass in clusters should therefore closely match

the ratio of the cosmological parameters Ωb/Ωm, where Ωb

and Ωm are the mean baryon and total mass densities of the

Universe, in units of the critical density. The combination of

robust measurements of the baryonic mass fraction in clus-

ters with accurate determinations of Ωb from cosmic nucle-

osynthesis calculations (constrained by the observed abun-

dances of light elements at high redshifts) can therefore be

used to determine Ωm.

This method for measuring Ωm, which is particularly

simple in terms of its underlying assumptions, was first high-

lighted by White & Frenk (1991) and subsequently employed

by a number of groups (e.g. Fabian 1991, White et al. 1993,

David, Jones & Forman 1995; White & Fabian 1995; Evrard

1997; Fukugita, Hogan & Peebles 1998; Ettori & Fabian

1999; Bahcall et al. 2000). In general, these studies have

found Ωm < 1 at high significance, with preferred values

lying in the range Ωm ∼ (0.1 − 0.3)h−0.5.

Sasaki (1996) and Pen (1997) described how measure-

ments of the mean baryonic mass fraction in clusters as a

function of redshift can, in principle, be used to place more

detailed constraints on cosmological parameters, since the

observed baryonic mass fraction values are sensitive to the

angular diameter distances to the clusters assumed in the

analyses. Until now, however, systematic uncertainties in the

observed quantities have seriously complicated the applica-

tion of such methods.

The baryonic mass content of rich clusters of galaxies is

dominated by the X-ray emitting intracluster gas, the mass

of which exceeds the mass of the optically luminous material

by a factor ∼ 6 (e.g. White et al. 1993; David et al. 1995;

Fukugita, Hogan & Peebles 1998). Since the X-ray emissivity

of the X-ray gas is proportional to the square of its density,

the gas mass profile can be precisely determined from the

X-ray data. With the advent of accurate measurements of

Ωb (e.g. O’Meara et al. 2001 and references therein) and

a precise determination of the Hubble Constant (Freedman

et al. 2001), the dominant uncertainty in determining Ωm

from the baryonic mass fraction in clusters has lain in the

measurements of the total (luminous plus dark) matter dis-

tributions in the individual clusters.

In this letter we report precise measurements of the X-

ray gas mass fraction for a sample of luminous, relatively

relaxed clusters spanning the redshift range 0.1 < z < 0.5,

for which precise, consistent mass models have recently

been determined from Chandra X-ray data and independent

c ? 0000 RAS

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2S.W. Allen et al.

Table 1. Summary of the Chandra observations.

zDate Exposure (ks)

PKS0745-191

Abell 2390

Abell 1835

MS2137-2353

RXJ1347-1145(1)

RXJ1347-1145(2)

3C295

0.103

0.230

0.252

0.313

0.451

0.451

0.461

2001 Jun 16

1999 Nov 07

1999 Dec 12

1999 Nov 18

2000 Mar 05

2000 Apr 29

1999 Aug 30

17.9

9.1

19.6

20.6

8.9

10.0

17.0

gravitational lensing constraints (Allen et al. 2001a, 2002;

Schmidt, Allen & Fabian 2001). The agreement between the

mass measurements determined from the two independent

methods firmly limits the systematic uncertainties in the

baryonic mass fraction measurements to∼

accuracy comparable to the current Ωb results. With the

reduced systematic uncertainties, we show that a method

similar to those proposed by Sasaki (1996) and Pen (1997)

can be successfully applied to the data, resulting in a tight

constraint on Ωm and an interesting constraint on ΩΛ. We

show that the results obtained are in good agreement with

those from recent studies of anisotropies in the cosmic mi-

crowave background radiation, the large-scale distribution

of galaxies, and the properties of distant supernovae (e.g.

Jaffe et al. 2001; Efstathiou et al. 2001).

Resultson theX-ray

quoted for two default cosmologies: SCDM with h =

H0/100kms−1Mpc−1= 0.5, Ωm = 1 and ΩΛ = 0, and

ΛCDM with h = 0.7, Ωm = 0.3 and ΩΛ = 0.7.

<10 per cent, an

gasmass fractionsare

2OBSERVATIONS AND DATA ANALYSIS

The Chandra observations were carried out using the back-

illuminated S3 detector on the Advanced CCD Imaging

Spectrometer (ACIS) between 1999 August 30 and 2001

June 16. For our analysis we have used the the level-2 event

lists provided by the standard Chandra pipeline processing.

These lists were cleaned for periods of background flaring us-

ing the CIAO software package resulting in the net exposure

times summarized in Table 1.

The Chandra data have been analysed using the meth-

ods described by Allen et al. (2001a, 2002) and Schmidt

et al. (2001). In brief, concentric annular spectra were ex-

tracted from the cleaned event lists, centred on the peaks

of the X-ray emission from the clusters.⋆The spectra were

analysed using XSPEC (version 11.0: Arnaud 1996), the

MEKAL plasma emission code (Kaastra & Mewe 1993; in-

corporating the Fe-L calculations of Liedhal, Osterheld &

Goldstein 1995), and the photoelectric absorption models of

Balucinska-Church & McCammon (1992). Only data in the

0.5 − 7.0 keV energy range were used. The spectra for all

annuli were modelled simultaneously, in order to determine

⋆For RXJ1347-1145, the data from the southeast quadrant of

the cluster were excluded due to ongoing merger activity in that

region; Allen et al. (2002).

the deprojected X-ray gas temperature profiles under the

assumption of spherical symmetry.

For the mass modelling, azimuthally-averaged surface

brightness profiles were constructed from background sub-

tracted, flat-fielded images with a 0.984×0.984 arcsec2pixel

scale (2×2 raw detector pixels). When combined with the de-

projected spectral temperature profiles, the surface bright-

ness profiles can be used to determine the X-ray gas mass

profiles (to high precision) and total mass profiles in the clus-

ters.†For this analysis, we have used an enhanced version

of the image deprojection code described by White, Jones

& Forman (1997) with distances calculated using the code

of Kayser, Helbig & Schramm (1997).

We have parameterized the cluster mass (luminous plus

dark matter) profiles using a Navarro, Frenk & White (1997;

hereafter NFW) model with

ρ(r) =

ρc(z)δc

(r/rs)(1 + r/rs)2, (1)

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 ra-

dius, c is the concentration parameter (with c = r200/rs)

and δc = 200c3/3[ln(1 + c) − c/(1 + c)]. The normalizations

of the mass profiles may also be expressed in terms of an

equivalent velocity dispersion, σ =√50rscH(z) (with rs in

units of Mpc and H(z) in kms−1Mpc−1).

In determining the results on the X-ray gas mass frac-

tion, fgas, we have adopted a canonical radius r2500, within

which the mean mass density is 2500 times the critical den-

sity of the Universe at the redshift of the cluster. (The r2500

values are determined directly from the Chandra data, with

confidence limits calculated from the χ2grids.) The r2500

values are well-matched to the outermost radii at which

reliable temperature measurements can be made from the

Chandra data. Note that the data for PKS0745-191 do not

quite reach to r2500 and for this cluster we quote fgas at

the outermost radius at which reliable measurements can

be made ∼ 0.8r2500. Although independent confirmation of

the X-ray mass results for 3C295 is not available, we in-

clude this cluster in our simple since in most other ways

it appears similar to the other objects in the sample. The

r2500 values for the clusters are listed in Table 2. The best-fit

NFW model parameters and 68 per cent confidence limits

are summarized by Allen et al. (2001b).

†The observed surface brightness profile and a particular pa-

rameterized mass model are together used to predict the temper-

ature profile of the X-ray gas. (We use the median temperature

profile determined from 100 Monte-Carlo simulations. The outer-

most pressure is fixed using an iterative technique which ensures

a smooth pressure gradient in these regions.) The predicted tem-

perature profile is rebinned to the same binning as the spectral

results and the χ2difference between the observed and predicted,

deprojected temperature profiles is calculated. The parameters for

the mass model are stepped through a regular grid of values in

the rs-σ plane (see text) to determine the best-fit values and 68

per cent confidence limits. (The best-fit models generally provide

good descriptions of the data). The gas mass profile is determined

to high precision at each grid point directly from the observed sur-

face brightness profile and model temperature profile. Spherical

symmetry and hydrostatic equilibrium are assumed throughout.

c ? 0000 RAS, MNRAS 000, 000–000

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Cosmological constraints from the X-ray gas mass fraction in relaxed clusters3

Figure 1. The observed X-ray gas mass fraction profiles with

the radial axis scaled in units of r2500. Symbols are as follows:

PKS0745-191 (light circles), Abell 2390 (light triangles), Abell

1835 (dark triangles), MS2137-2353 (light squares), RXJ1347-

1145 (dark circles), 3C295 (dark squares). The default ΛCDM

cosmology is assumed. Note that fgas(r) is an integrated quan-

tity and so the error bars on neighbouring points in a profile are

correlated.

3 RESULTS

3.1The X-ray gas mass fraction measurements

Fig. 1 shows the observed fgas(r) profiles for the six clus-

ters assuming the standard ΛCDM cosmology. We see that

although some variation is present from cluster to cluster,

the profiles tend towards a similar value at r2500. Table 2

lists the results on the X-ray gas mass fractions measured

at r2500 for both the SCDM and ΛCDM cosmologies. Tak-

ing the weighted mean of the fgas results for all six clusters

studied, we obtain¯fgas = 0.160 ± 0.007 for SCDM (h=0.5)

and¯fgas = 0.113 ± 0.005 for ΛCDM (h=0.7).

In calculating the total baryonic mass in the clusters, we

assume that the optically luminous baryonic mass in galaxies

is 0.19h0.5times the X-ray gas mass (White et al. 1993;

Fukugita, Hogan & Peebles 1998). Other sources of baryonic

matter are expected to make only very small contributions

to the total mass and are ignored.

Given the baryonic masses, and assuming that the re-

gions of the clusters within r2500 provide a fair sample of the

matter content of the Universe, we can write

Ωm =

Ωb

fgas(1 + 0.19h0.5). (2)

For Ωbh2= 0.0205 ± 0.0018 (O’Meara et al. 2001) and us-

ing the ΛCDM (h = 0.7) fgas values, we obtain the (self-

consistent) result Ωm = 0.319 ± 0.032. Using the SCDM

(h = 0.5) fgas values, we obtain Ωm = 0.452 ± 0.044.

3.2Cosmological constraints from the fgas(z) data

In addition to the simple calculation of Ωm based on the

weighted-mean fgas values, described above, the data for

the present sample can be used to obtain more rigorous

constraints on cosmological parameters from the apparent

variation of fgas with redshift.

Fig. 2 shows the fgas values as a function of redshift for

the SCDM and ΛCDM cosmologies. We see that whereas

the results for the ΛCDM cosmology are consistent with a

constant fgas value, the results for SCDM indicate an ap-

parent drop in fgas as the redshift increases. The differences

in the fgas(z) behaviour for the SCDM and ΛCDM cos-

mologies reflect the dependence of the fgas(z) measurements

on the assumed angular diameter distances to the clusters

(fgas ∝ D1.5

should be invariant with redshift, as would be expected if

rich, relaxed clusters provide a fair sample of the matter

content of the Universe, we can see from inspection of Fig. 2

that the data for the present sample favour the ΛCDM over

the SCDM cosmology.

In order to quantify more precisely the degree to which

our data can constrain the relevant cosmological param-

eters, we have fitted the data in Fig. 2(a) with a model

which accounts for the expected apparent variation in the

fgas(z) values, which are measured assuming an SCDM cos-

mology, for different underlying cosmologies. The ‘true’ cos-

mology should be the cosmology that provides the best fit

to the measurements. (We work with the SCDM data. Note

that the fgas(r) profiles exhibit only small variations around

r2500, and so the effects of changes in r2500 as the cosmology

is varied can be ignored.)

The model function fitted to the data is

A). Under the assumption that the fgas values

fmod

gas (z) =

Ωb

?1 + 0.19√h?Ωm

?

h

0.5

DΩm=1,ΩΛ=0

A

DΩm, ΩΛ

A

(z)

(z)

?1.5

, (3)

which depends on Ωm, ΩΛ, Ωb and h. The ratio (h/0.5)1.5

accounts for the change in the Hubble Constant between the

considered model and default SCDM cosmology, and the ra-

tio of the angular diameter distances accounts for deviations

in the geometry of the Universe from the Einstein-de Sitter

case. We constrain Ωbh2= 0.0205 ± 0.0018 (O’Meara et al.

2001) and h = 0.72 ± 0.08, the final result from the Hub-

ble Key Project reported by Freedman et al. (2001). The χ2

difference between the model and SCDM data is then

χ2

=

?

?

all clusters

?fmod

gas (zi) − fgas, i?2

σ2

fgas, i

+

Ωbh2− 0.0205

0.0018

?2

+

?h − 0.72

0.08

?2

, (4)

where fgas, iand σfgas, iare the best-fit values and symmetric

root-mean-square errors for the SCDM data from Table 2,

and zi are the redshifts of the clusters. We have examined a

grid of cosmologies covering the plane 0.0 < Ωm < 1.0 and

0.0 < ΩΛ < 1.5. The joint 1, 2 and 3 σ confidence contours

on Ωm and ΩΛ (corresponding to ∆χ2values of 2.30, 6.17

and 11.8, respectively) determined from the fits are shown

in Fig. 3.

The best-fit cosmological parameters and marginalized

1σ error bars are Ωm = 0.30+0.04

with χ2

the model provides an acceptable description of the data.

The best-fit cosmological parameters are similar to those

−0.03and ΩΛ = 0.95+0.48

−0.72,

min= 1.7 for 4 degrees of freedom, indicating that

c ? 0000 RAS, MNRAS 000, 000–000

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4S.W. Allen et al.

Table 2. The observed X-ray gas mass fractions (and 68 per cent confidence limits) measured at r2500 (in Mpc) for the default SCDM

and ΛCDM cosmologies.

SCDMΛCDM

r2500

fgas

r2500

fgas

PKS0745-191

Abell 2390

Abell 1835

MS2137-2353

RXJ1347-1145

3C295

0.85+0.04

−0.05

0.69+0.14

−0.09

0.72+0.05

−0.03

0.49+0.03

−0.01

0.72+0.10

−0.08

0.42+0.03

−0.03

0.174+0.013

−0.012

0.209+0.060

−0.046

0.164+0.016

−0.016

0.159+0.009

−0.016

0.142+0.034

−0.027

0.128+0.020

−0.016

0.68+0.03

−0.03

0.64+0.15

−0.09

0.66+0.06

−0.02

0.46+0.02

−0.03

0.73+0.08

−0.09

0.41+0.04

−0.03

0.112+0.008

−0.009

0.138+0.047

−0.033

0.114+0.006

−0.013

0.117+0.015

−0.009

0.108+0.031

−0.018

0.105+0.019

−0.016

Figure 2. The apparent variation of the observed X-ray gas mass fraction (with root-mean-square 1σ errors) as a function of redshift for

the default (a: left panel) SCDM (h=0.5) and (b: right panel) ΛCDM (h=0.7) cosmologies. The dashed curves show the results of fitting

a constant value to the data in each case. The solid line in (a) shows the predicted curve for the best-fit cosmology with Ωm= 0.30 and

ΩΛ= 0.95 (see Section 3.2).

assumed for the default ΛCDM cosmology in Fig. 2b, which

is expected given the approximately constant nature of the

fgas(z) values shown in that Figure.

4DISCUSSION

The result on the mean matter density of the Universe,

Ωm = 0.30+0.04

X-ray gas mass fraction for the present sample of relaxed,

lensing clusters, represents one of the tightest constraints

on this cosmological parameter to date. The variation of the

gas mass fraction with redshift also yields the measurement

of a positive cosmological constant with ΩΛ = 0.95+0.48

in good agreement with previous results based on studies

of the properties of distant supernovae (Riess et al. 1998;

Perlmutter et al. 1999)

In Fig. 3 we show a comparison of the joint con-

straints on Ωmand ΩΛdetermined from the Chandra fgas(z)

data, with the results of Jaffe et al. (2001) from studies

of cosmic microwave background (CMB) anisotropies (in-

corporating the COBE Differential Microwave Radiometer,

−0.03, determined from the Chandra results on the

−0.72,

BOOMERANG-98 and MAXIMA-1 data of Bennett et al.

1996, de Bernardis et al. 2000 and Hanany et al. 2000, re-

spectively)‡and the properties of distant supernovae (in-

corporating the data of Riess et al. 1998 and Perlmutter et

al. 1999). The agreement between the results obtained from

the independent methods is striking: all three data sets are

consistent, at the 1σ confidence level, with a cosmological

model with Ωm = 0.3 and ΩΛ = 0.7 − 0.8. These results are

also consistent with the findings of Efstathiou et al. (2001)

from a combined analysis of the 2dF Galaxy Redshift Survey

and CMB data.

An important aspect of the present work is that, in

addition to the exquisite data quality provided by Chan-

dra, the clusters studied are all regular, relatively relaxed

systems for which independent confirmation of the mass re-

‡We note that the results on Ωm and ΩΛfrom the CMB data

reported by Jaffe et al. (2001) are consistent with, though less

constraining than, the later analyses of de Bernardis et al. (2002)

and Stompor et al. (2001) using the full BOOMERANG and

MAXIMA-1 data sets.

c ? 0000 RAS, MNRAS 000, 000–000

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Cosmological constraints from the X-ray gas mass fraction in relaxed clusters5

Chandra

SNIa

CMB

Figure 3. The joint 1, 2 and 3 σ confidence contours on Ωm and ΩΛdetermined from the Chandra fgas(z) data (bold contours), and

independent analyses of cosmic microwave background (CMB) anisotropies and the properties of distant supernovae (from Jaffe et al.

2001).

sults is available from gravitational lensing studies. The sys-

tematic uncertainties in the fgas measurements are therefore

greatly reduced with respect to previous X-ray studies. For

both Abell 2390 and RXJ1347-1145, the X-ray and weak

lensing mass profiles are consistent within their 68 per cent

confidence limits. For Abell 1835, 2390, MS2137-2353 and

PKS0745-191, the observed strong lensing configurations (on

scales r ∼ 20 − 80h−1kpc) can be explained by mass mod-

els within the 68 per cent Chandra confidence contours, al-

though redshift measurements for the arcs (which are re-

quired to define the lensing masses precisely) are not avail-

able in all cases.§The presence of significant non-thermal

pressure support (e.g. arising from turbulent and/or bulk

motions and/or magnetic fields) on scales ∼ r2500 can there-

fore be excluded, and the residual systematic uncertainties

in the fgas values are small (∼

ically, than the statistical uncertainties. We note that the

effects of departures from spherical symmetry on the fgas

results are expected to be∼

Canizares 1996).

The observed fgas(r) profiles are essentially flat around

r2500, which supports the assumption that the measured fgas

values represent a fair sample of the matter content of the

<10 per cent i.e. smaller, typ-

<a few per cent e.g. Buote &

§For RXJ1347-1145, a two-component mass model, consistent

with the complex X-ray structure observed in the southeast quad-

rant, is required to explain the strong lensing data.

Universe. If, however, the values were to rise by a further

∼ 10 per cent beyond r2500, the result on Ωm would drop

by a corresponding amount. The fgas values measured at

r2500 are not sensitive to the choice of using an NFW model

to parameterize the total mass distributions in the clusters.

Repeating the analysis presented here using either a non-

singular isothermal sphere or a Moore et al. (1998) model to

parameterize the total mass distributions leads to results on

the weighted-mean fgas values in good agreement with those

quoted in Section 3.1. We note, however, that the fgas(r)

profiles determined using the different mass models exhibit

some systematic variation, particularly at small radii (r∼

0.1r2500), and when extrapolated to large radii (r > r2500),

as can be expected given the different asymptotic slopes.

In a future paper, we will examine in detail the ability of

different parameterized models to describe the Chandra data

for relaxed clusters.

<

The constraints on Ωm and ΩΛ should improve as fur-

ther Chandra, XMM-Newton and high-quality gravitational

lensing data become available for more regular, relaxed clus-

ters, especially at high redshifts (although relaxed systems,

like those studied here, are expected to be very rare at high

redshifts). This work can also be extended to include less

relaxed clusters, or clusters which appear relaxed at X-ray

wavelengths but for which independent confirmation of the

mass results from gravitational lensing studies is not yet

available, although this will require careful consideration of

the additional systematic uncertainties involved.

c ? 0000 RAS, MNRAS 000, 000–000

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6S.W. Allen et al.

We thank Andrew Jaffe for providing the CMB and

supernovae results shown in Fig. 3. SWA and ACF thank

the Royal Society for support.

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