Improvements in the X-ray luminosity function and constraints on the Cosmological parameters from X-ray luminous clusters

0.26 MB·English

by  A. Del Popolo

Astronomy & Astrophysics manuscript no. astronastroph˙constraints1˙2colomn˙1˙u˙2col (cid:13)c ESO 2010
January 12, 2010
Improvements in the X-ray luminosity function and constraints
on the Cosmological parameters from X-ray luminous clusters
A. Del Popolo1,2, V. Costa3, G. Lanzafame4
0
1
1 Dipartimento di Fisica e Astronomia, Universit´a di Catania, Viale AndreaDoria 6, 95125 Catania, Italy
0
2 2 Argelander-Institut fu¨r Astronomie, Auf dem Hu¨gel 71, D-53121 Bonn, Germany
3 Dipartimento di Metodologie Chimiche e Fisiche per l’Ingegneria, Univesita di Catania, Viale A. Doria 6, I-95125, Catania,
n
Italy
a
4 Osservatorio Astrofisico diCatania, IstitutoNazionale di Astrofisica, Via S.Sofia 78, I-95123, Catania, Italy
J

9

]
O ABSTRACT
C
. Weshowhowtoimproveconstraintson Ωm,σ8,andthedark-energyequation-of-stateparameter, w,obtained byMantzet al.
h (2008) frommeasurementsoftheX-rayluminosityfunctionofgalaxyclusters,namelyMACS,thelocal BCSandtheREFLEX
p galaxy cluster samples with luminosities L>3×1044 erg/s in the 0.1–2.4 keV band. To this aim, we use Tinker et al. (2008)
-
o mass function instead of Jenkins et al. (2001) and the M-L relationship obtained from Del Popolo (2002) and Del Popolo et
r al. (2005). Using the same methods and priors of Mantz et al. (2008), we find, for a ΛCDM universe, Ωm = 0.28+−00..0054 and
st σ8 = 0.78+−00..0045 while the result of Mantz et al. (2008) gives less tight constraints Ωm = 0.28+−00..1017 and σ8 = 0.78+−00..1113. In the
a case of awCDM model, we findΩm =0.27+−00..0076,σ8 =0.81+−00..0056 and w=−1.3+−00..34, while in Mantzet al. (2008) they are again
[ less tight Ωm = 0.24+−00..1057, σ8 = 0.85+−00..1230 and w = −1.4+−00..47. Combining the XLF analysis with the fgas+CMB+SNIa data
1 set results in the constraint Ωm = 0.269±0.012, σ8 = 0.81±0.021 and w = −1.02±0.04, to be compared with Mantz et al.
v (2008), Ωm = 0.269±0.016, σ8 = 0.82±0.03 and w = −1.02±0.06. The tightness of the last constraints obtained by Mantz
3 et al. (2008), are fundamentally due to the tightness of the fgas+CMB+SNIa constraints and not to their XLF analysis. Our
4 findings, consistent with w=−1, lend additional support to the cosmological-constant model.
4
1 Key words.cosmology–theory–large scale structureof Universe–galaxies–formation
.
1
0
0 1. Introduction ues of σ 1 (Henry & Arnaud 1991; Reiprich & B¨oringer
8
1
2002; Schuecker et al. 2003) –a result since then con-
v: Clusterofgalaxiesarethelargestgravitationally-collapsed firmed by cosmic microwave background (CMB) studies,
i structuresintheUniverse.Evenatthepresentepochthey cosmic shear, and other experiments (Spergel et al. 2007;
X
arerelatively rare,with only a few percent ofgalaxiesbe-
Komatsu et al. 2008; Dunkley et al. 2008; Benjamin et
r ing in clusters. In the hierarchical collapse scenario for
a al. 2007; Fu et al. 2008). For precision’s sake, cluster sur-
structure formation in the universe, the number density
veys in the local universe are particularly useful for con-
of collapsed objects as a function of mass and cosmic
straining a combination of the matter density parame-
timeisasensitiveprobeofcosmology.Thegalaxyclusters
ter Ω and the normalization of the power spectrum of
m
that occupy the high-mass tail of this population provide
density fluctuations. Following the evolution of the clus-
a powerful and relatively clean tool for cosmology, since
ter space density over a large redshift baseline, one can
their growth is predominantly determined by linear grav-
break the degeneracy between σ and Ω (Rosati et al.
8 m
itational processes.
2002). Recently, X-ray studies (Vikhlinin et al. 2009b) of
Startinginthe1990’s,analysisofmassiveclustershave the evolution of the cluster mass function at z = 0-0.8
consistently indicated low values of Ωm (both from the have convincingly demonstrated that the growth of cos-
baryonicfractionarguments(White etal.1993)andmea- mic structure has sloweddown at z <1 due to the effects
surements of the evolution in the cluster number den- of dark energy, and these measurements have been used
sity (Eke et al. 1998; Borgani et al. 2001) and low val-
1 σ8 is the amplitude of the mass density fluctuation power
Send offprint requests to: A. Del Popolo, E- spectrumoverspheresofradius8h−1Mpc,andM8 isthemean
mail:[email protected] mass within thesespheres
2 A. Del Popolo: Cosmological Constraints
to improve the determination of the equation of state pa- (e.g., the total X-ray flux) or inaccurate measurements
rameter. Although the quoted cosmological test is very (e.g. temperatures with large uncertainties)
powerful, there are two main problems in practical appli- Until some yearsagothe obtainedresults for Ω were
m
cations: first, theoretical predictions provide the number several times in disagreement. Study by different authors
density of clusters of a given mass, while the mass itself (Bahcall,Fan&Cen(1997),Bahcall&Fan(1998),Sadat,
is never the directly observed quantity. Second, a cluster Blanchard& Oukbir (1998),Blanchard,Bartlett& Sadat
sampleisneededthatspansalarge-zbaselineandisbased (1998), Blanchard & Bartlett (1998), Eke et al. (1998),
on model-independent selection criteria.2 Viana&Liddle(1999),Reichartetal.(1999),Donahue&
Determiningtheevolutionofthespacedensityofclus- Voit 1999,Borganiet al.2001)found values for Ωm span-
ters requires counting the number of clusters of a given ning the entire range of acceptable values: 0.2 ≤ Ωm ≤ 1
mass per unit volume at different redshifts. Therefore, (see Reichart et al. 1999). It is interesting to note that
three essential tools are required for its application as a Viana & Liddle (1999) using the same data set as Eke et
cosmological test: (a) an efficient method to find clusters al. (1998) showed that uncertainties both in fitting local
over a wide redshift range,(b) an observable estimator of data and in the theoretical modeling could significantly
the cluster mass, and (c) a method to compute the se- change the final results: they found Ωm ≃ 0.75 as a pre-
lection function or equivalently the survey volume within ferred value with a critical density model acceptable at
which clusters are found. Observations of clusters in the <90% c.l. while Eke et al. (1998)found Ωm =0.45±0.2.
X-ray band provide an efficient and physically motivated The reasons leading to the quoted discrepancies have
method of identification, which fulfills the three require- been studied in several papers (Eke et al. 1998; Reichart
ments above. The X-ray luminosity, provides a very ef- etal.1999;Donahue&Voit1999;Borganietal.2001)and
ficient method for identifying clusters down to a given can be summarized as due to: 1) The inadequate approx-
X-ray flux limit and hence within a known survey vol- imation given by the mass function used (e.g., Bryan &
ume for each luminosity L , which uniquely specifies the Norman 1998). 2) Inadequacy in the structure formation
x
cluster selection, is also a good probe of the depth of the as described by the spherical model leading to changes in
cluster gravitational potential. For these reasons most of thethresholdparameterδc(e.g.,Governatoetal.1999).3)
the cosmological studies based on clusters have used X– Inadequacy in the M-T relation obtained from the virial
ray-selected samples. theorem (see Voit & Donahue 1998; Del Popolo 2002).
According to the three points quoted above, the 4) Effects of cooling flows. 5) Determination of the X-
recipe for constraining cosmological parameters by ray cluster catalog’s selection function. 5) Missing high
means of clusters is composed of three ingredients: 1) redshift clusters in the data used (e.g., the EMSS). 6)
The predicted mass function of clusters, n(M,z), as a Evolution of the L-T relation (Voit & Donahue 1998).
function of cosmological parameters (σ , Ω , w, etc.). 7) The use of different best fitting procedures to get the
8 m
2) Sky surveys with well understood selection functions constraints (Eke et al. 1998). 8) Other effects described
to find clusters, as well as a relation linking cluster mass in more recent papers (e.g., Mantz et al. 2008 (hereafter
with an observable. A successful solution to the former M08); Vikhlinin et al 2009b).
requirement has been to identify clusters by the X-ray The situation with the cluster mass function data has
emission produced by hot intracluster gas, notably using beendramaticallyimprovedinthepastyears.Alargesam-
data from ROSAT3. 3) A tight, well-determined scaling ple of sufficiently massive clusters extending to z ≃ 0.9
relation between survey observable (e.g. L ) and mass, has been derivedfrom ROSAT PSPC pointed data cover-
x
with minimal intrinsic scatter. ing 400 deg2 (Burenin et al. 2007). Distant clusters from
the 400d sample were then observed with Chandra, pro-
Early attempts to use evolution of the cluster mass viding high-quality X-ray data and much more accurate
function as a cosmological probe were limited by small totalmassindicators(seeVikhlininetal.2009b).Chandra
sample sizes and either poor proxies for the cluster mass coveragehas also become available for a complete sample
of low-z clusters originally derived from the ROSAT All-
2 This is so that the search volume and the numberdensity SkySurvey(seeVikhlininetal.2009b).Resultsfromdeep
associated with each cluster are uniquelyidentified. Chandrapointings to a numberof low-zclustershavesig-
3 The ROSAT Brightest Cluster Sample (BCS; Ebeling et nificantly improvedourknowledgeof the outer cluster re-
al. 1998, 2000) and ROSAT-ESO Flux Limited X-ray sample gionsandprovidedamuchmorereliablecalibrationofthe
(REFLEX;B¨ohringeretal.2004)togethercoverapproximately M vs. proxy relations than what was possible before.
two-thirdsoftheskyout toredshift z≃0.3 and contain more tot
On the theoretical side, improved numerical simulations
than750clusters.TheMassiveClusterSurvey(MACS;Ebeling
resulted in better understanding of measurement biases
et al. 2001, 2007) extends these data to z ≃0.7. The ROSAT
in the X-ray data analysis (Nagai et al. 2007;Rasia et al.
160sq.degreesurvey,describedforthefirsttimebyVikhlinin
2006; Jeltema et al. 2007).
et al. (1998) is a serendipitous cluster catalogue containing
In the present paper, we want to show how tighter
201 groups/clusters, while the ROSAT 400 sq. degree survey
is based on 1610 high Galactic latitude ROSAT PSPC point- constraints can be obtained in M08 model improving the
ings(Bureninetal.2007)andincludes266opticallyconfirmed massfunctionadoptedbythem,andthescalinglawsused
galaxy clusters, groups and individual elliptical galaxies. (e.g., the M-T and M-L relationships). In this paper, we
A. Del Popolo: Cosmological Constraints 3
use the observed X-ray luminosity function to investigate for cosmological -constant models, with A = 0.316, B =
two cosmologicalscenarios,assuming a spatially flat met- 0.67, and ǫ=3.82.
ric in both cases: the first includes dark energy in the As shown in Del Popolo (2006a) and Del Popolo
form of a cosmologicalconstant (ΛCDM); the second has (2006b), the theoretical mass function obtained in the
dark energy with a constant equation-of-state parameter, quoted papers is in better agreement with high resolu-
w (wCDM). The theoretical background for this work is tion N-body simulations, namely Reed et al. 2003 (R03),
reviewed in Section 2. Section 3 presents the results and Yahagi et al. (2004) (YNY), Warren et al. 2006 (W06),
Section 4 the conclusions. and Tinker et al. (2008) (see the following and Fig. 1b)
(T08).
The mass function was calculated according to the
2. Theory
modelofDelPopolo(2006a,b).Themultiplicityfunction,
In the introduction, we discussed the ingredients needed in the quoted model, is given by:
in the recipe used to constrain cosmological parameters
β g(α ) β g(α ) β g(α )
from X-ray observations. In this section, I derive an ex- νf(ν) = A 1+ 1 1 + 2 2 + 3 3
1(cid:18) (aν)α1 (aν)α2 (aν)α3 (cid:19)
pression for the X-Ray luminosity function (XLF) (using
now the mass function obtained in Del Popolo (2006a, aνe{−2aν 1+(aνβ)1α1+(aνβ)2α2+(aνβ)3α3 2} (3)
b)) andM–T,L–TrelationsobtainedinDelPopolo2002, r2π (cid:2) (cid:3)
and Del Popolo et al. 2005, respectively) and then I set
where
someconstraintstoΩ ,σ andthedark–energyequation–
m 8
of–state parameter w, by using the data (clusters) used αi(αi−1)
g(α ) = |1−α + −...−
in M08, namely MACS (Massive Cluster Survey), BCS i i 2!
(Brightest Cluster Sample), and REFLEX (ROSAT ESO α (α −1)···(α −4)
i i i
| (4)
FLUX LIMITED X-Ray SAMPLE). Following M08, the 5!
constraints are obtained from measurements of the X-
where i = 1 or 2, α = 0.585, β = 0.46, α = 0.5 and
1 1 2
Ray luminosity function of the quotedsamples.The most
β = 0.35, α = 0.4 and β = 0.02, a = 0.707, and A =
2 3 3 1
straightforward mass-observable relation to complement
1.2 is the normalization constant.
these X–ray flux–limited surveys is the mass-X–ray lumi-
The “multiplicity function” is correlated with the
nosity relation. For sufficiently massive (hot) objects at
usual, more straightforwardly used, “mass function” as
the relevant redshifts, the conversion from X-ray flux to
follows. Following Sheth & Tormen (2002) (hereafter ST)
luminosity is approximately independent of temperature,
notation,iff(M,δ)dM denotesthefractionofmassthatis
in which case the luminosities can be estimated directly
containedin collapsedhaloes that have mass in the range
from the survey flux and the selection function is identi-
M-M + dM, at redshift z, and δ(z) is the redshift de-
caltotherequirementofdetection.Adisadvantageisthat
pendent overdensity,the associated“unconditional”mass
thereisalargescatterinclusterluminositiesatfixedmass;
function is:
however, sufficient data allow this scatter to be quanti-
ρ
fied empirically. More recently, a dramatic reduction in n(M,δ)dM = bf(M,δ)dM (5)
M
luminosity-mass scatter has been demonstrated when lu-
minosities are measured excluding cluster centers (typi- In Fig. 1a, we plot the multiplicity function obtained
cally r < 0.15r ; Maughan 2007; Zhang et al. 2007). inthispaper(simbolsaredescribedinthefigurecaption).
500
Alternative approaches use cluster temperature (Henry There are some differences between the quoted sim-
2000; Seljak 2002; Pierpaoli et al. 2003; Henry 2004), gas ulations and the J01 simulations. First, the multiplicity
fraction (Voevodkin & Vikhlinin 2004) or Y parameter function of the present paper, similar to that of YNY,
X
(Kravtsov et al. 2006) to achieve tighter mass-observable in the low-ν region of ν ≤ 1 systematically falls below
relationsattheexpenseofreducingthesizeofthesamples the J01 functions. In this region the multiplicity func-
available for analysis. The need to quantify the selection tion of the present paper is very close to that of YNY.
function in terms ofboth X-rayflux and a secondobserv- Additionally, the numerical multiplicity functions (and
able additionally complicates these efforts. that in Del Popolo 2006a,b) have an apparent peak at
The first ingredient of the quoted recipe (i.e., mass ν ≃insteadoftheplateauthatisseenintheJ01function.
function),usedinM08wastheJenkinsetal.(2001)(here- Similardifferencesareseeninthehigh-ν region.Thesedif-
after J01) mass function. J01 wrote the mass function of ferences between numerical multiplicity functions (R03;
galaxy clusters of mass M at redshift z as a ”universal YNY; W05; Del Popolo 2006a,b) and J01, are however
function” of σ−1(M,z) within 1–σ error–bars, and so they are overall in agree-
ment. The multiplicity function obtained in the present
M n(M,z)
f(σ−1)= (1) paperhasapeakatν ≃1asinYNYnumericalmultiplic-
ρmdlnσ−1 ity function, instead of a plateau as in the J01 function.
which was fitted by Differences are observed also in the redshift evolution of
the J01 mass function (Del Popolo 2006b). Summarizing
f(σ−1)=Ae(−|lnσ−1+B|ǫ) (2) the fitting formulas presented by J01 are accurate to
4 A. Del Popolo: Cosmological Constraints
≃10−20% (Tinker et al. 2008 (T08)). In our model, the similarity. This picture is confirmed by X-ray observa-
mass function that we used is given by Eq. (3), Eq.(4), tions, see for instance the deviation of the L-T relation
and Eq(5) which is in perfect agreement with the T08 in clusters, which is steeper than the theoretical value
mass function, as shown in Fig. (1b). So, the accuracy of predicted by the previous scenario. More precisely, until
the mass function is, as in T08, of the order of ≃ 5% for some years ago, the cluster structure was considered to
ΛCDM models for the mass and redshift rangeof interest be scale-free, which means that the global properties of
in this study. As a consequence, in this way the theoret- clusters, such as halo mass, luminosity-temperature, and
ical uncertainties in the mass function do not contribute X-ray luminosity would scale self-similarly (Kaiser 1986;
significantly to the systematic error budget. Evrard&Henry1991).Inparticular,thegastemperature
In Fig. 1b, I plot the mass function for all of our out- would scale with cluster mass as T ∝M2/3 and the bolo-
puts in the f(σ)−ln(σ−1) plane. Large values of lnσ−1 metric X-ray luminosity would scale with temperature as
correspond to rare haloes of high redshift and/or high L∝T2,inthe bremsstrahlung-dominatedregimeabove2
mass, while small values of lnσ−1 describe haloes of low keV. Studies following that of Kaiser (1986) showed that
mass and redshift combinations. Fig. 1b shows the func- the observed luminosity-temperature relation is closer to
tion f(σ) measured for all simulations in Table 1 of T08, L∝T3 (e.g.,Edge& Stewart1991),indicating that non–
the solid line the fit to the data (namely T08 eq. 3) and gravitationalprocessesshouldinfluence the density struc-
the dashed line the model of the present paper. ture of a clusters core, where most of the luminosity is
Aspreviouslyreported,oneofthemainproblemsofus- generated (Kaiser 1991; Evrard & Henry 1991; Navarro
ingthemassfunctiontoconstraincosmologicalparameter et al. 1995; Bryan& Norman 1998). One way to obtain a
is thattheoreticalpredictionsprovidethe number density scalinglawcloserto the observationaloneis tohavenon–
of clusters of a given mass, while the mass itself is never gravitational energy injected into the ICM before or dur-
the directly observed quantity. One then needs relations ing cluster formation, the so-called pre-heating (Ponman
connectingmasswithotherquantitiesmoreeasilyobtain- et al. 1999;Bower et al. 1997;Cavaliere et al. 1997,1999;
able which can be used as a surrogate for cluster mass. Tozzi & Norman 2001; Borgani et al. 2001; Voit & Brian
Over the past decade, observations of clusters of galax- 2001),feedbackprocessesthatalterthegascharacteristics
ies (e.g. ROSAT, ASCA) have shown the existence of a during the evolution of the cluster (Voit & Bryan 2001),
correlationbetweenthe totalgravitatingmassofclusters, cooling flows (Allen & Fabian 1998). A similar situation
M 4, their X-ray luminosity (L ) and the temperature is valid for the M-T relationship, namely that the self-
tot X
(T )oftheintraclustermedium(ICM)(Davidetal.1993; similarity in the M-T relation seems to break at a few
X
Markevitch 1998;Horner, Mushotzky & Scharf 1999). By keV (Nevalanien et al. 2000; Xu, Finoguenov et al. 2001;
meansofthequotedscalingrelationsonecanobtaindiffer- Muanwong et al. 2001; Bialek, Evrard & Mohr 2001).
ent methods for tracing the evolution of the cluster num- Consequently, if, as in M08, one starts with self-similar
ber density: (1) The X-ray temperature function (XTF), scaling laws in order to have consistent scaling relations
whichhas beenpresentedfor local(e.g.,Henry & Arnaud one has to compare the self-similar scaling relations to
1991) and distant clusters (Eke et al. 1998; Henry 2000). observations (Morandi et al. 2007).
2) The evolution of the X-ray luminosity function (XLF). Different from the M08 approach,in the following, we
In this case, we need a relation between the observed L use models for the L–T, T–M, relationships taking into
X
and the cluster virial mass. account the non-self similarity: namely, the M −Tx rela-
In the following, following M08, we shall use the XLF tion obtained analytically using the model of Del Popolo
to constraincosmologicalparameters.Then the next cru- (2002), while the Lx−Tx relation is that obtained in Del
cial step, after having a mass function, is to convert it Popolo, Hiotelis & Pen´arrubia (2005) based on an im-
in a Luminosity function (XLF). This can be done by provement of the Punctuated Equilibrium Model (PEM)
first converting mass into intra-cluster gas temperature, of Cavaliere et al. (1997, 1998, 1999). The drawbacks of
by means of the M − T relation, and then converting usingself-similarrelationshipsfittedtothedata(clusters)
x
the temperature into X-ray luminosity, by means of the and the reasons to use a different approach, were already
L −T relation. M08 used a self-similar relationship be- discussed in Del Popolo (2003) (their sect. 3), and in the
x x
tween mass and X-ray luminosity for massive clusters remainder of this section.
(e.g., Bryan & Norman 1998) modified by an additional Similarly, to the present study, in Del Popolo (2003)
redshift–dependent factor (see Morandi et al. 2007). At we used the models for the L–T, T–M, relationships
this point, we must stress an important point. Numerical instead of the scaling relations obtained from simulations
simulations confirm that the DM component in clusters of Chandra data (see, e.g., Pierpaoli et al. 2001, 2003
of galaxies,which represents the dominant fraction of the for references).5 Eq. (5) in M08 similar to that Eq.
mass,has a remarkablyself-similar behavior;howeverthe (13) of Pierpaoli et al. (2001) or Eq. (4) of Pierpaoli
baryonic component does not show the same level of self- et al. (2003) comes from rather simplistic arguments
(dimensionalanalysisandanassumptionthatclustersare
4 Since Mtot compares with the ICM temperature measure-
ments that can be obtained through X-ray spectroscopy, this 5 Notice that Eq. 22 in Del Popolo 2003, and Eq. 5 in M08,
explainstheimportanceofamass-temperature(MT)relation. are very similar.
A. Del Popolo: Cosmological Constraints 5
self-similar) not taking into account important physical The luminosity function likelihood is the same as in
effects that gives rise to a non-self-similar behavior of M08 (Sect. 4.2).
the quoted relation, as previously discussed. The fitting The likelihood that N clusters with inferred luminosi-
procedure used by M08, trying to take account of the ties in a range dLˆ exist in a volume dV can in general
previous physics and the non-self-similar behavior of the be written as a Poisson probability plus a correction due
relationship, is complicated by several effects. In fact, to the clustering of halos with one another. If the plane
in the fit one uses data that may contain small groups of redshift and inferred luminosity is divided into non-
which can be influential in the estimation of the slope overlappingcells,thenthelikelihoodofourdataissimply
of the model, and one has then to choose accurately
the data to be used in the fit. This choice mitigates N˜Nje−N˜j
P (N ,N ,...)= j , (6)
the possibility of obtaining biased results if slope of the 1 2 N !
Yj j
mass-luminosity relation is different for massive clusters
compared with smaller groups. In M08 they fitted only where N andN˜ arethe number ofclusters detectedand
the data (clusters) with L > 3 × 1044 erg/s in the j j
predicted in the jth cell, respectively.
0.1–2.4 keV band. Moreover, the process of fitting the
If the cells are taken to be rectangular, with the jth
model in Eq. 7 of M08 is complicated by the presence cell given by z(1) ≤z <z(2) and Lˆ(1) ≤Lˆ <Lˆ(2), then
of Malmquist bias. Close to the flux limit for selection, j j j j
any X-ray selected sample will preferentially include the
most luminous sources for a given mass. This results N˜ = zj(2)dzdV(z) Lˆ(j2)dLˆdn˜(z,Lˆ), (7)
in a steepening of the derived mass-luminosity relation j Zz(1) dz ZLˆ(1) dLˆ
j j
and a bias in the inferred intrinsic scatter in luminosity
where V(z) is the comoving volume within redshift z. In
for a given mass. The use of the extended sample of
the absenceof intrinsicscatter inthe mass–luminosityre-
Reiprich & B¨oringer (2002) (RB02), rather than only
lation and measurement errors in the observed luminosi-
their flux–limited HIFLUGCS sample, partially mitigates
ties,thederivativeofthecomovingnumberdensitywould
this effect by softening the flux limit. A further problem
be simply
is that as a consequence of Malmquist bias there is a
strong apparent, but not necessarily physical, correlation
dn˜(z,L) dM(L)dn(z,M)
between luminosity and redshift due to the fact that the =f (z,L) (8)
sky
dL dL dM
flux limit corresponds to higher luminosities at higher
redshifts. Here fsky is the sky coverage fraction of the surveys as
a function of redshift and inferred luminosity, dn/dM is
nolongertheJenkinsmassfunctionbuttheonediscussed
Forwhatconcernsthedata(clusters)usedintheanal-
in the present paper and M(L) is the mass–luminosity
ysis, they are the same of those used by M08: the follow-
relation discussed in the present paper.
ing three flux-limited surveys are included in our analy-
Similar to M08, the presence of scatter requires us to
sis: the BCS (Ebeling et al. 1998) and REFLEX sample
take into accountthat a cluster detected with inferredlu-
(Bo¨hringeret al. 2004)atlow redshifts (z <0.3), andthe minosity Lˆ could potentially have any true luminosity L
MACS(Ebelingetal.2001)at0.3<z <0.5(seeM08).In
and mass M, with some associated probability. To cal-
the analysis, the sample was chosen to cover the redshift
culate the predicted number density correctly, we must
range z <0.5,since at higher redshifts the number of un-
therefore convolve with these probability distributions:
relaxedclustersdecrease,andthe L–TandT–Mrelations
are appropriate for relaxed clusters. The purpose of this dn˜(z,Lˆ) ∞
paperistopresentananalysisbasedonlyontheX-raylu- dLˆ = fsky(z,Lˆ)Z0 dL P(Lˆ|L) (9)
minosityfunction(XLF)datadescribedabove,alongwith
∞ dn(z,M)
the priors described in Sect. 4 of M08. × dM P(L|M) .
Z dM
0
Following M08, we parametrize the full model fit-
ted to the X-ray luminosity function data as h, Ω h2, P(L|M)isalog-normaldistributionwhosewidthislikein
b
Ω h2, σ , n , w where Ω and Ω are the baryon and M08 the intrinsic scatter in the mass–luminosity relation,
c 8 s b c
cold dark matter densities (Ω = Ω + Ω ). In addi- η(z), andP(Lˆ|L)is a normaldistribution whose width as
m b c
tion to the assumption of spatial flatness, we adopt the a function of flux is modeled as a power law, as described
Gaussian priors h = 0.72±0.08 (Freedman et al. 2001) in Sect. 3.2 of M08.
andΩ h2 =0.0214±0.002(Kirkmanetal.2003)fromthe
b
Hubble Key Project and Big Bang nucleosynthesis stud-
3. Results
ies, respectively. Since the results are insensitive to the
spectralindexwithinareasonablerange(seeM08),wefix In Fig. 2, we compare, for a ΛCDM the joint Ω -σ con-
m 8
n =0.95 in accordance with (Spergel et al. 2007)for the straints obtained from the BCS, REFLEX and MACS
s
standard analysis. The dark-energy equation of state was data sets combination The marginalized constraints from
bounded by a uniform prior, −5<w <0. the combination of the three cluster samples are Ω =
m
6 A. Del Popolo: Cosmological Constraints
0.28+0.05 andσ =0.78+0.04 while theresultofM08gives the L-M relationship gives rise to tight constraints even
−0.04 8 −0.05
less tight constraints Ω =0.28+0.11 and σ =0.78+0.11. when using only the XLF function.
m −0.07 8 −0.13
In order to understand why the results of our analysis
Our previous constraints are in good agreement with
are different from those of M08, we have to stress a key
recent, independent results from the CMB (Spergel et al.
point. The M08 paper, as well as several others papers
2007) and cosmic shear, as measured in the 100 Square
in the literature, used two different data sets: REFLEX,
DegreeSurvey(Benjaminetal.2007)andCFHTLSWide
BCS, and MACS to constrain cosmological parameters
field (Fu et al. 2008). Our results are also in good overall
andanotherexternaldatasettoconstraintheluminosity–
agreementwithpreviousfindingsbasedontheobservedX-
mass relation (RB02 data set), which in M08 is a power-
rayluminosityandtemperaturefunctionsofclusters(Eke
lawwiththreefreeparameters,withoutexplicitlyaccount-
et al 1998, Donahue & Voit 1999, Henry 2000, Borgani
ingforselectionbias.Consequently,itwasnecessarytore-
et al. 2001, Seljak 2002, Allen et al. 2003, Pierpaoli et
strict that external data set (RB02) to low redshifts and
al. 2003, Schuecker et al. 2003, Henry 2004). Our result
high fluxes in order to minimize the effects of selection
on Ω is in excellent agreement with current constraints
m
bias,makingitimpossibletotestfordeparturesfromself-
basedonclusterf data(Allenetal.2008andreferences
gas
similarevolutioninthescalingrelation.Inordertohavea
therein) and the power spectrum of galaxies in the 2dF
“self-consistent”analysis,itisnecessarythatasinglelike-
galaxyredshiftsurvey(Coleetal.2005)andSloanDigital
lihood function be applied to the full data set which en-
SkySurvey(SDSS)(Eisensteinetal.2005,Tegmarketal.
compassestheentiretheoreticalmodel(cosmology+scal-
2006, Percival et al. 2007), as well as the combination of
ing relations) so as to ensure that the covariance among
CMB data with a variety of external constraints (Spergel
all the model parameters is fully captured and that the
2007).
effects of the mass function and selection biases are prop-
InFig.3a,wesetconstraintsforthewCDMmodel,and erly accounted for throughout. This kind of analysis was
weplotthejointconstraintsonΩm andσ8 fromthe lumi- performed for the first time by Vikhlinin et al. (2009a,b),
nosity function data using our standardpriors,while Fig. whousedthesameclustersampletoconstrainthescaling
3b displays constraints on Ωm and w obtained indepen- relations, thus obtaining tighter constraints.
dently fromthe XLF data. The marginalizedresults from In the analysis of the present paper, the L–M relation
the X-ray luminosity function data are Ωm = 0.27+−00..0076, is a physically motivated relation (not a power-law with
σ8 = 0.81+−00..0056 and w = −1.3+−00..34, while in M08 they free parameters) which does not require fits to data, as
are again less tight Ω = 0.24+0.15, σ = 0.85+0.13 and in M08. Since we do not need the double analysis of M08
m −0.07 8 −0.20
w = −1.4+0.4. Our new XLF results are consistent with and previous papers, the first to get the L–M fitting pa-
−0.7
the cosmological-constantmodel (w =−1). rameters from RB02 data, and the second to obtain the
cosmologicalconstraintsusingBCS,REFLEXandMACS,
An improvement on the previous results can
we bypass the quoted drawback in the M08 analysis.
be obtained by adding to the XLF analysis the
Itis interesting to note thata monthafter the present
f +CMB+SNIadataset.Thef +CMB+SNIacombi-
gas gas
paper was submitted, two papers, Mantz et al. (2009a,b),
nation already provides tight constraints on Ω , h, Ω h2
m b
appeared in arXiv showing that the key point that I pre-
and n (hence no priors on these parameters are used in
s
viously stressed, namely generalizing M08 to allow the
either combined analysis), but the degeneracy between w
quoted simultaneous andself-consistentfit and using T08
and σ (right panel of Fig. 4) limits the precision of the
8
mass function (instead of that in Jenkins et al. 2001) re-
darkenergy results.The additionof the XLF data breaks
sultincosmologicalconstraintsthatareafactor2-3better
the degeneracy in the Ω -σ plane (left panel), resulting
m 8
thanthoseinM08,basedonthesameflux-limitedsample
in tighter constraints on Ω , σ and w. The degeneracy
m 8
ofclusters.Inthepresentpaper,wehavealsocheckedthat
breaking power of other combinations of data with the
usingthesameL–MrelationusedinM08,wereobtainthe
CMB is discussed by Spergel et al. (2007). The result-
same set of constraints derived by M086 .
ing constraint are Ω = 0.269±0.012, σ = 0.81±0.021
m 8
Another point to stress concerns the use of our non-
and w = −1.02±0.04, to be compared with M08 Ω =
m
self similar L–M relation for clusters of luminosity L >
0.269±0.016, σ = 0.82±0.03 and w = −1.02±0.06. The
8 3×1044 erg/s in the 0.1–2.4keV band. Since the clusters
previous constraints are in agreement within the errors
included in the M08 sample are high X-ray luminosity
with Vikhlinin et al. (2009b) constraints, namely Ω =
m (above 3×1044 erg/s), one could think that the changes
0.255±0.043, σ =0.786±0.011and w=−0.991±0.045.
8
in the L–M relation of the present paper, with respect to
It is important to note that the tight constraints ob-
the classical self-similar model, will not produce signifi-
tained by M08 when combining XLF analysis with the
cant changes in constraints on the cosmological parame-
f +CMB+SNIadata setareprimarilydue to the tight-
gas ters. Even if major differences between the L–M model
ness of the constraints obtained from f +CMB+SNIa
gas of the present paper and the self–similar model are ob-
data itself and not to the precision of the XLF analysis
served at gas temperatures below 3 keV, we stress that
of M08,as shown by comparing our results for Ω , σ , w
m 8
obtained by the XLF analysis with those of M08. In our 6 Mantz et al. (2009a,b) obtain Ωm = 0.27 ± 0.02, σ8 =
model the improvement in the mass function model and 0.79±0.03 and w=−0.96±0.06.
A. Del Popolo: Cosmological Constraints 7
the present L–M relation depends on the M–T and L–T 4. Conclusions
relationships,andespecially the secondone(basedonthe
In the present paper, we showedhow to improve the con-
ModifiedPunctuatedEquilibriumModel(MPEM))never
straints onΩ , σ , andthe dark-energyequation-of-state
behaves in a self-similar way as shown in Del Popolo et m 8
parameter, w, from measurements of the X-ray luminos-
al.(2005)(evenatgastemperatureshigher than10 keV).
ity function of galaxy clusters, as performed by M08.
Moreover,aspreviouslyreported,the improvementinthe
Improving the mass function by means of Del Popolo
constraintsisstrictlyconnectedtothefactthatwebypass
(2006a, b) model, which was shown to be in good agree-
the quoteddrawbackinthe M08analysisbymeansofour
mentwithT08andusingtheL–Mrelationshipobtainedin
L–M relation not depending on parameters that must be
DelPopolo(2002)andDelPopoloetal.(2005),weshowed
fixed using external data.
that the XLF alone can give tight constraints on the cos-
In order to obtain tighter and tighter constraints one mologicalparameters.Usingthesamemethodsandpriors
needstotrytoreducetotheminimumthesystematicun- ofM08,wefind,foraΛCDMuniverse,Ω =0.28+0.05and
m −0.04
certaintiesintheanalysis.Muchprogressisexpectedover σ =0.78+0.04andsimilarlyinthecaseofawCDMmodel,
thecomingyearsinrefiningtherangesoftheseallowances, 8 −0.05
wefindΩ =0.27+0.07,σ =0.81+0.05andw =−1.3+0.3,
bothobservationallyandthroughimprovedsimulations.A m −0.06 8 −0.06 −0.4
both tighter than M08 results. Combining the XLF anal-
reductioninthesizeoftherequiredsystematicallowances
ysis with the f +CMB+SNIa data set results in the
will tighten the cosmological constraints. Improved nu- gas
constraint Ω = 0.269± 0.012, σ = 0.81± 0.021 and
merical simulations of large samples of massive clusters, m 8
w =−1.02±0.04, in agreement with the most recent de-
includingamorecompletetreatmentofstarformationand
termination of the quoted parameters (Allen et al. 2008;
feedback physics that reproduces both the observed opti-
Vikhlinin et al.2009b;Percivalet al. 2009).Our findings,
cal galaxy luminosity function and cluster X-ray proper-
consistent with w = −1 lends additional support to the
ties,willbe ofmajorimportance.Further deepX-rayand
cosmological-constantmodel.
optical observations of nearby clusters will provide better
constraints on the viscosity of the cluster gas. Improved
Acknowledgements. The author acknowledges the financial
optical/near infrared observations of clusters should pin support from the German Research Foundation (DFG) under
down the stellar mass fraction in galaxy clusters and its grant NO KR 1635/16-1.
evolution. Ground and space-based gravitational lensing
studieswillprovideimportant,independentconstraintson
References
the mass distributions in clusters; a large program using
the Subaru telescope and Hubble Space Telescope is un- Allen S. W., Schmidt R. W., Fabian A. C., Ebeling H., 2003,
derway, as is similar work by other groups (e.g. Hoekstra MNRAS,342, 287
2007). Allen et al. 2008, MNRAS383, Issue 3, 879
Allen, S. W., & Fabian, A. 1998, MNRAS,297, L57
In the near future, continuing programs of Chandra
Bahcall, N.A. & Fan, X.1998, ApJ,504, 1
and XMM-Newton observations of known, X-ray lumi-
Bahcall,N. A.,Fan, X., & Cen, R.1997, ApJ,485, L53
nousclustersshouldallowimportantprogresstobemade,
BardeenJ.M.,Bond,J.R.,Kaiser,N.,Szalay,A.S.1986,ApJ
both by expanding the f sample (e.g. Chandra snap-
gas 304, 15
shot observations of the entire MACS sample; Ebeling et Benjamin, J. et al., 2007, MNRAS,381, 702
al. 2001, 2007) and through deeper observations of the Bialek, J.J., Evrard,A.E., &Mohr J.J. 2001, ApJ,555, 597
current target list. A new, large area X-ray survey such Blanchard, A.,& Bartlett, J.G. 1998,A&A, 332, L49
as that proposed by the Spectrum-RG/eROSITA project Blanchard, A., Bartlett, J. G., & Sadat, R. 1998, preprint
could make a substantial contribution, finding hundreds (astro-ph/9809182)
of suitable systems at high redshifts. B¨ohringer H.et al., 2004, A&A,425, 367
Borgani, S.et al., 2001, ApJ, 561, 13
Looking a decade ahead, the International X-ray
Bower, R. G., Castander, F. J., Couch, W., Ellis, R. S., &
Observatory (IXO), result of the merging of NASA’s
B¨ohringer, H.1997, MNRAS,291, 353
Constellation-X and ESA/JAXA’s XEUS mission con- Bryan,G. L., & Norman,M. L. 1998,ApJ, 495, 80
cepts,willofferthe possibilityto carryoutprecisestudies Burenin, R. A., Vikhlinin, A., Hornstrup, A., Ebeling, H.,
of dark energy using the fgas technique.7 Quintana,H., & Mescheryakov, A. 2007, ApJS,172, 561,
Cavaliere, A., Menci, N., & Tozzi, P. 1997, ApJ, 484, L21
(CMT97)
7 Thelargecollectingareaandcombinedspatial/spectralre- . 1998, ApJ,501, 493 (CMT98)
solvingpowerofIXOshouldpermitprecisefgas measures.An . 1999, MNRAS,308, 599 (CMT99)
investment of ≃ 10 Ms of IXO time to measure fgas to 5% Cash, A., 1979, ApJ 228, 939
(corresponding to 3.3% accuracy in distance) in each of the Cole, S., et al., 2005, MNRAS,362, 505
500 hottest, most X-ray luminous, dynamically relaxed clus- David L.P., SlyzA.,Jones C., Forman W., VrtilekS.D., 1993,
ters detected in future cluster surveys, spanning the redshift ApJ,412, 479
range 0 < z < 2 (typical redshift z ≃ 0.6), will be sufficient Del Popolo A., 2002, MNRAS 336, 81
to constrain cosmological parameters with a DETF figure of Del Popolo A., 2003, ApJ 599, 723
merit of 2040. Del Popolo A., Hiotelis S., Pen´arrubia G., 2005, ApJ628, 76
8 A. Del Popolo: Cosmological Constraints
Del Popolo, A., 2006a, ApJ637, 12 Rasia, E., et al. 2006, MNRAS,369, 2013
Del Popolo, A., 2006b, AJ 131, 2367 Reed, D., et al. 2003, MNRAS,346, 565 (R03)
Donahue,M., & Voit,G.M. 1999,ApJ, 523, L137 Reichart, D. E., Nichol, R. C., Castander, F. J., Burke, D. J.,
Dunkley,J. et al., 2008, arXiv:0811.4280 Romer, A. K., Holden, B. P., Collins, C. A., & Ulmer, M.
Ebeling H., Edge A. C., Allen S.W., Crawford C. S., Fabian P.1999b, ApJ,518, 521
A. C., HuchraJ. P., 2000, MNRAS,318, 333 Reiprich, T. H. & B¨ohringer, H., 2002, ApJ, 567, 716
Ebeling H., Edge A. C., Bohringer H., Allen S. W., Crawford Rosati, P., Borgani S., and Norman, C., Ann. Rev. Astron.
C. S., Fabian A.C., Voges W., HuchraJ. P., 1998, Astrophys2002, 40: 539-77
Ebeling H., Edge A. C., HenryJ. P., 2001, ApJ, 553, 668 Sadat,R., Blanchard, A., & Oukbir,J. 1998,A&A, 329, 21
EbelingH.,BarrettE.,DonovanD.,MaC.J.,EdgeA.C.,van Schuecker,P.et al., 2003, A&A,398, 867
Speybroeck L., 2007, ApJL, 661, L33 Seljak U., 2002, MNRAS,337, 769
Edge, A. C., & Stewart, G. C. 1991, MNRAS,252, 414 Sheth R.K., & Tormen G., 2002, MNRAS 329, 61
Eke, V.R. et al., 1998, MNRAS,298, 1145 Spergel, D.N. et al., 2007, ApJS,170, 377
Eke, V.R., Cole, S., & Frenk,C. S. 1996, MNRAS,282, 263 Tegmark M. et al., 2006, Phys.Rev.D, 74, 123507
Eisenstein D.J. et al., 2005, ApJ,633, 560 Tinker et al. 2008, ApJ688, 709
Finoguenov, A.,Reiprich, T. H.,& B¨ohringer, H.2001, A&A, Tozzi, P., & Norman, C. 2001, ApJ,546, 63
368, 749 Tru¨mper J., 1993, Science, 260, 1769
Freedman W. L. et al., 2001, ApJ, 553, 47 Viana, P. T. P., & Liddle, A. R.1999, MNRAS,303, 535
Fu,L. et al., 2008, A&A,479, 9 Voit, C.M. 2000, ApJ, 543, 113
Gao L., Navarro J. F., Cole S., Frenk C., White S. D. M., Vikhlinin, A., McNamara, B. R., Forman, W., Jones, C.,
Springel V., Jenkins A., Neto A. F., 2007, MNRAS, sub- Quintana,H., & Hornstrup, A.1998, ApJ502, 558
mitted, arXiv:astro-ph/0711.0746 Vikhlinin, A.et al., 2009a, ApJ692, 1033
Governato, F., Babul, A., Quinn, T., Tozzi, P., Baugh, C., Vikhlinin, A.et al., 2009b, ApJ692, 1060
Katz, N.,& Lake,G. 1999,MNRAS, 307, 949 Voevodkin A., Vikhlinin A.,2004, ApJ, 601, 610
Gregory P. C., & Loredo T., 1992, ApJ398, 146 Voit, C.M., & Donahue,M. 1998,ApJ, 500, L111
Henry J. P., 2000, ApJ,534, 565 Voit, G. M., & Bryan, G. 2001, Nature,414, 425
Henry J. P., 2004, ApJ,609, 603 Warren,M.S.,Abazajian,K.,Holz,D.E.,&Teodoro,L.2006,
Henry,J. P. & Arnaud,K. A.,1991, ApJ, 372, 410 ApJ,646, 881
HornerD.J.,MushotzkyR.F.,ScharfC.A.,1999,ApJ,520,78 White S.D.M., Navarro J.F., Evrard A.E., Frenk C.S., 1993,
Hoekstra H., 2007, MNRAS 379, 317 Nature,366, 429.
Hu W., Kravtsov A. V.,2003, ApJ, 584, 702 Xu,H., Jing, G., & Wu,X.2001, ApJ, 553, 78
Jenkins A., Frenk C. S., White S. D. M., Colberg J. M., Cole Yahagi, H., Nagashima, M., & Yoshii, Y. 2004, ApJ, 605, 709
S., Evrard A. E., Couchman H. M. P., Yoshida N., 2001, (YNY04)
MNRAS 321, 372 Zhang,Y.Y.,FinoguenovA.,B¨ohringerH.,KneibJ.P.,Smith
Jeltema, T. E., Hallman, E. J., Burns, J. O., & Motl, P. M. G. P., Czoske O., Soucail G., 2007, A&A467, 437
2007, ApJ, in press, (arXiv:0708.1518, 708
Kaiser, N.1986, MNRAS,222, 323
Kaiser, N.1991, ApJ,383, 104
Kirkman D., Tytler D., Suzuki N., OMeara J. M., Lubin D.,
2003, ApJS,149, 1
Komatsu, E. et al., 2008, arXiv:0803.0547
Kravtsov A. V.,Vikhlinin A., Nagai D., 2006, ApJ, 650, 128
Maughan B. J., 2007; ApJ668, 772
Mantz et al 2008, MNRAS,387, 1179 (M08)
Mantz et al 2009a, arXiv: 0909.3098
Mantz et al 2009b, arXiv: 0909.3099
Markevitch M., 1998, ApJ, 503, 77
MorandiA.,EttoriS.,Moscardini L.,2007, MNRAS,379,518
Muanwong, O., Thomas, P. A., Kay, S. T., Pearce, F. R., &
Couchman, H.M. P. 2001, ApJ,552, L27
Nagai,D.,Vikhlinin,A.,&Kravtsov,A.V.2007,ApJ,655,98
Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1995,MNRAS,
275, 720
Nevalainen,J.,Markevitch,M.,&Forman,W.2000,ApJ,532,
694
Percival W. J. et al., 2007, ApJ,657, 51
Percival W. J. et al., 2009, astro-ph.CO/0907.1660
Pierpaoli E., Scott D., White M., 2001, MNRAS,325, 77
Pierpaoli E., Borgani S., Scott D., White M., 2003, MNRAS,
342, 163
Ponman,T.J.,Cannon,D.B.,&Navarro,J.F.1999,Nature,
397, 135
Press W., & SchechterP., 1974, ApJ 187, 425
A. Del Popolo: Cosmological Constraints 9
0.9
0.85
0.3
0.80
0.75
0.2
0.70
0.65
0.1
0.2 0.3 0.4 0.5 0.6
0 -0.4 -0.2 0 0.2 0.4 0.6 Fig.2. Joint68.3and95.4percentconfidenceconstraints
onΩ andσ for aΛCDM modelfromMACS,BCS,and
(a) (b) m 8
REFLEXcombinationusingstandardpriors(asdescribed
in the text).
0.85
0.80
0.75
0.70
0.0 0.2 0.3 0.4 0.5 0.6
Fig.3. Panel(a).SameasFig.2butforawCDMmodel.
Panel (b). Joint 68.3 and 95.4 per cent confidence con-
straints on Ω and w for a constant-w model using the
m
X-ray luminosity function data and standard priors (as
described in the text).
(b) (b)
Fig.1. In Panel (a), the solid line represents the multi-
plicity function obtained in this paper, the short–dashed
line the fitting formula proposed by Yahagi et al. (2004)
(their Eq. 7), the dotted line the Sheth & Tormen
(2002)(ST)multiplicityfunction,thelong-dashedlinethe
Jenkins et al. (2001) multiplicity function. The errorbars
with open circles represent the run 140 of YNY, those
with filled squares the case 70b, those with open squares
the case 70a, those with filled circles the case 35b, those
withcrossesthecase35a.Panel(b).Massfunctionplotted
inredshift-independentform.Themeasuredf(σ)fromall
simulations in Table 1 of Tinker et al. (2008). The solid
line is the best fit function of equation (3) (Tinker et al.
2008). The dashed line the model in the present paper.
10 A. Del Popolo: Cosmological Constraints
0.85
0.80
0.75
0.70
0.0 0.2 0.3 0.4 0.5 0.6
Fig.4. Joint68.3and95.4percentconfidenceconstraints
on Ω and σ (left panel) and σ and w (rightpanel) ob-
m 8 8
tainedfromacombinedf +CMB+SNIaanalysis(blue)
gas
andtheimprovedconstraintsobtainedbycombiningthese
data with the XLF (gold). No priors on h, Ω h2 or n
b s
are imposed in either analysis. In the left panel, the re-
sultsfromthe XLFaloneusingstandardpriorsareshown
(purple) in order to illustrate the degeneracy breaking.
Notethatinthe leftpanelweplottedjustthe innerconfi-
dencecontoursinthef +CMB+SNIaanalysis,inorder
gas
to have a more readable plot.