Determining the equation of state of dark energy from angular size of compact radio sources and X-ray gas mass fraction of galaxy clusters

0.19 MB·English

by  Zong-Hong Zhu

Astronomy&Astrophysicsmanuscriptno.ms˙AA0327 February2,2008
(DOI:willbeinsertedbyhandlater)
Determining the equation of state of dark energy
from angular size of compact radio sources and
X-ray gas mass fraction of galaxy clusters
4
0 Zong-HongZhu1,Masa-KatsuFujimoto1,andXiang-TaoHe2
0
2

n 1 NationalAstronomicalObservatory,2-21-1,Osawa,Mitaka,Tokyo181-8588,Japan
a
J e-mail:[email protected] [email protected]

8 2 DepartmentofAstronomy,BeijingNormalUniversity,Beijing100875,China

1 e-mail:[email protected]
v
5
Received0000,0000;accepted0000,0000
9
0
1
Abstract. Usingrecent measurements of angular sizeof high-z milliarcsecond compact
0
4 radiosourcescompiledbyGurvits,Kellermann&Frey(1999)andX-raygasmassfraction
0
/ of galaxy clusterspublished by Allen et al. (2002,2003), we explore their bounds on the
h
p equationofstate,ωx ≡ px/ρx,ofthedarkenergy,whoseexistencehasbeencongruously
-
o suggestedbyvariouscosmologicalobservations.Werelaxetheusualconstraintωx ≥ −1,
r
t andfindthatcombiningthetwodatabasesyieldsanontriviallowerboundonωx.Underthe
s
a assumptionofaflatuniverse,weobtainabound−2.22<ω <−0.62at95.4%confidence
x
:
v level.The95.4%confidenceboundgoesto−1≤ω <−0.60whentheconstraintω ≥−1
i x x
X
isimposed.
r
a
Key words. cosmological parameters — cosmology: theory — distance scale — radio
galaxies:general—X-ray:galaxies:clusters
1. Introduction
One of the most remarkablecosmologicalfindingsof recent years is, in additionalto the cold
dark matter (CDM), the existence of a componentof dark energy(DE) with negativepressure
inouruniverse.Itismotivatedtoexplaintheaccelerationoftheuniversediscoveredbydistant
typeIasupernova(SNeIa)observations(Perlmutteretal. 1998,1999;Riess etal. 1998,2001),
and to offset the deficiencyof a flat universe,favouredby the measurementsof the anisotropy
of CMB (de Bernardis et al. 2000; Balbi et al. 2000, Durrer et al. 2003; Bennett et al. 2003;
Spergel et al. 2003), but with a subcritical matter density parameter Ω ∼ 0.3, obtained from
m
Sendoffprintrequeststo:Zong-HongZhu
2 Zhu,Z.-H.,Fujimoto,M.-K.&He,X.-T.:Determiningω fromΘ-zand f data
x gas
dynamicalestimates or X-ray and gravitationallensing observationsof clustersof galaxies(for
arecentsummary,seeTurner2002).Whileacosmologicalconstantwith p = −ρ isthesim-
Λ Λ
plest candidate for DE, it suffers from the difficulties in understanding of the observed value
intheframeworkofmodernquantumfieldtheory(Weinberg1989;Carrolletal.1992)andthe
“coincidenceproblem”,theissueofexplainingtheinitialconditionsnecessarytoyieldthenear-
coincidenceof the densities of matter and the cosmologicalconstant componenttoday. In this
case,quintessence(adynamicalformofDEwithgenerallynegativepressure)hasbeeninvoked
(Ratra and Peebles 1988; Wetterich 1988; Caldwell, Dave and Steinhardt 1998; Zlatev, Wang
and Steinhardt1998).One of the importantcharacteristicsof quintessencemodelsis that their
equation of state, ω ≡ p /ρ , vary with cosmic time whilst a cosmological constant remains
x x x
a constant ω = −1. Determination of values of ω and its possible cosmic evolution plays a
Λ x
central role to distinguish various DE models. Such a challenging has triggered off a wave of
interestaimingtoconstrainω usingvariouscosmologicaldatabases,suchasSNeIa(Garnavich
x
etal.1998;Tonryetal.2003;Barrisetal.2003;Knopetal.2003;ZhuandFujimoto2003);old
high redshift objects (Lima and Alcaniz 2000a); angular size of compact radio sources (Lima
andAlcaniz2002);gravitationallensing(Chaeetal.2002;Sereno2002;Dev,JainandMahajan
2003; Huterer and Ma 2003); SNeIa plus Large Scale Structure (LSS) (Perlmutter, Turner &
White1999);SNeIaplusgravitationallensing(Wagaandmiceli1999);SNeIaplusX-raygalaxy
clusters(Schueckeretal.2003);CMBplusSNeIa(Efstathiou1999;BeanandMelchiorri2002;
Hannestad and Mo¨rtsell 2002; Melchiorri et al. 2003); CMB plus stellar ages (Jimenez et al.
2003);and combinationsof various databases (Kujat et al. 2002).Other potential methodsfor
the determination of ω have also widely discussed in literatures, such as the proposed SNAP
x
satellite1 (HutererandTurner1999;WellerandAlbrecht2001;WellerandAlbrecht2002);ad-
vanced gravitationalwave detectors (Zhu, Fujimoto and Tatsumi 2001; Biesiada 2001); future
SZ galaxy cluster surveys(Haiman,Mohr and Holder2001);and gammaray bursts (Choubey
andKing2003;Takahashietal.2003).
In this work, we shall consider the observational constraints on the DE equation of state
parameterizedbyaredshiftindependentpressure-to-densityratioω arisingfromthelatestob-
x
servationsofangularsizeofhigh-zmilliarcsecondcompactradiosourcescompiledbyGurvits,
Kellermann&Frey(1999)andtheX-raygasmassfractiondataofclustersofgalaxiespublished
byAllenetal. (2002,2003).Thebasicsofa constantω assumptionare twofolds:ontheone
x
hand,theangulardiameterdistanceDAusedinthisworkisnotsensitivetovariationsofω with
x
redshiftbecauseitdependsonω throughmultipleintegrals(Maoretal.2001;Maoretal.2002;
x
Wasserman2002);ontheotherhand,forawideclassofquintessencemodels(particularly,those
withtrackingsolutions),bothofΩ andω varyveryslowly(Zlatevetal.1999;Steinhardtetal.
x x
1999;Efstathiou1999),andaneffectiveequationofstate,ω ∼ ω (z)Ω (z)dz/ Ω (z)dzisa
eff x x x
R R
goodapproximationforanalysis(Wangetal.2000).Werelaxetheusualconstraintω ≥−1,be-
x
1 SNAPhomepage,http://snap.lbl.gov
Zhu,Z.-H.,Fujimoto,M.-K.&He,X.-T.:Determiningω fromΘ-zand f data 3
x gas
causerecentyearstherehavebeenseveralmodelswhichpredictaDEcomponentwithω <−1
x
(Parker and Raval 1999; Schulz and White 2001; Caldwell 2002;Maor et al. 2002; Frampton
2003) and also we hope to explore its effects on the ω determination. The confidence region
x
on the (ω , Ω ) plane obtained through a combined analysis of the two databases suggests
x m
−2.22 < ω < 0.62 at 95.4% confidence level, which goes to −1 ≤ ω < 0.60 when the
x x
constraintω ≥−1isimposed.
x
The planofthepaperisas follows.In thenextsection,we providethe boundsonω from
x
the angular size-redshift data. Constraints from the X-ray gas mass fraction of galaxy clusters
arediscussedinsection3.Finally,wepresentacombinedanalysis,ourconcludingremarksand
discussionin section4.Throughoutofthepaper,weassumea flatuniversewhichissuggested
bythemeasurementsoftheanisotropyofCMBandfavouredbyinflationscenario.
2. Constraintsfromtheangularsize-redshiftdata
WebeginbyevaluatingtheangulardiameterdistanceDAasafunctionofredshiftz.Theredshift
dependentHubbleparametercanbewrittenasH(z)= H E(z),whereH =100hkms−1Mpc−1is
0 0
theHubbleconstantatthepresenttime.Foraflatuniversethatcontains(baryonicandcolddark)
matteranddarkenergywithaconstantω (weignoretheradiationcomponentsintheuniverse
x
that are not important for the cosmological tests considered in this work), we get (Turner and
White1997;Chibaetal.1997;Zhu1998)
c 1 z dz′
DA(z;Ωm,ωx)= H01+zZ0 E(z′;Ωm,ωx) , E2(z;Ωm,ωx)=Ωm(1+z)3+(1−Ωm)(1+z)3(1+ωx).(1)
Wefirstanalyzetheangularsize-redshitdataformilliarcsecondradiosourcesrecentlycom-
piled by Gurvits, Kellermannand Frey (1999)to constrain ω . The basics of the angularsize-
x
redshittestinthecontextofdarkenergywasfirstdiscussedinatheoreticalviewpointbyLima
and Alcaniz (2000b) without using any database. They also provide an analytical closed form
whichdetermineshowtheredshiftz ,atwhichtheangularsizetakesitsminimalvalue,depends
m
on ω . Later on, using the same database compiled by Gurvits, Kellermann and Frey (1999),
x
LimaandAlcaniz(2002)obtainedΩ ∼ 0.2andω ∼ −1.. A distinguishingcharacteristicof
m x
ouranalysisisthattheusualconstraintsω ≥ −1isrelaxed.ThisdatabaseshowninFigure1is
x
145sourcesdistributedintotwelveredshiftbinswithaboutthesamenumberofsourcesperbin.
Thelowestandhighestredshiftbinsarecenteredatredshiftsz = 0.52andz = 3.6respectively.
Wedeterminethemodelparametersω andΩ throughaχ2minimizationmethod.Therangeof
x m
ω spanstheinterval[-3,0]instepsof0.01,whiletherangeofΩ spanstheinterval[0,1]also
x m
instepsof0.01.
[θ(z;l;Ω ,ω )−θ ]2
χ2(l;Ω ,ω )= i m x oi , (2)
m x σ2
Xi i
whereθ(z;Ω ,ω ) = l/DA istheanglesubtendedbyanobjectofproperlengthltransverseto
i m x
thelineofsightandθ istheobservedvaluesoftheangularsizewitherrorsσ oftheithbinin
oi i
thesample.Thesummationisoverall12observationaldatapoints.
4 Zhu,Z.-H.,Fujimoto,M.-K.&He,X.-T.:Determiningω fromΘ-zand f data
x gas
Fig.1. Diagram of angular size vs redshift data for 145 compact radio sources (binned into
12 bins) of Gurvits, Kellermann and Frey (1999). We assume the charateristic linear size
l=22.64h−1pcfortheoreticalcurves.Thesolidcurvecorrespondstoourbestfitwithω =−1.19
x
andΩ = 0.23,whilethedashedanddot-dashedcurvescorrespondtoaΛ-dominateduniverse
m
andthestandardcolddarkmatter(SCDM)modelrespectively.
Aspointedoutbytheauthorsofpreviousanalysesonthisdatabase(Gurvits,Kellermannand
Frey 1999;Vishwakarma 2001;Alcaniz 2002;Zhu and Fujimoto 2002;Jain, Dev and Alcaniz
2003;ChenandRatra2003),whenoneuse theangularsize datatoconstrainthecosmological
parameters, the results will be strongly dependenton the characteristic length l. Therefore,in-
stead of assuming a specific value for l, we have workedon the intervall = 15h−1−30h−1pc.
In order to make the analysis independentof the choice of the characteristic length l, we also
minimizeequation(2)forl,ω andΩ simultaneously,whichgivesl=22.64h−1pc,ω =−1.19
x m x
andΩ = 0.23asthebestfit.Figure2displaysthe68.3%and95.4%confidencelevelcontours
m
inthe(Ω ,ω )planeusingthelowershadedandthelowerplusdarkershadedareasrespectively.
m x
Itisclearfromthefigurehatω ispoorlyconstrainedfromtheangularsize-redshiftdataalone,
x
which only gives ω < −0.32 at 95.4% confidence level. However, as we shall see in Sec.4,
x
when we combinethis test with the X-ray gas mass fraction test, we could get fairly stringent
constraintsonbothω andΩ .
x m
3. ConstraintsfromthegalaxyclustersX-raydata
Clusters of galaxies are the largest virialized systems in the universe, and their masses can be
estimated by X-ray and opticalobservations,as well as gravitationallensing measurements.A
comparisonof the gasmass fraction, f = M /M , as inferred fromX-ray observationsof
gas gas tot
clustersofgalaxiestothecosmicbaryonfractioncanprovideadirectconstraintonthedensity
Zhu,Z.-H.,Fujimoto,M.-K.&He,X.-T.:Determiningω fromΘ-zand f data 5
x gas
Fig.2. Confidence region plot of the best fit to the database of the angular size-redshift data
compiled by Gurvits, Keller and Frey (1999) – see the text for a detailed description of the
method.The68%and95%confidencelevelsinthe (Ω , ω ) planeare shownin lowershaded
m x
andlower+darkershadedareasrespectively.
parameteroftheuniverseΩ (Whiteet.al.1993).Moreover,assumingthegasmassfractionis
m
constantincosmictime,Sasaki(1996)showthatthe f dataofclustersofgalaxiesatdifferent
gas
redshiftsalsoprovideanefficientwaytoconstrainothercosmologicalparametersdecribingthe
geometryoftheuniverse.Thisisbasedonthefactthatthemeasured f valuesforeachcluster
gas
ofgalaxiesdependontheassumedangulardiameterdistancestothesourcesas f ∝ [DA]3/2.
gas
Theture,underlyingcosmologyshouldbetheonewhichmakethesemeasured f valuestobe
gas
invariantwithredshift(Sasaki1996;Allenatal.2003).
UsingtheChandraobservationaldata,Allenetal.(2002;2003)havegotthe f profilesfor
gas
the 10 relaxedclusters. Exceptfor Abell963,the f profilesof the other9 clustersappearto
gas
haveconvergedorbecloseto convergingwithacanonicalradiusr ,whichisdefinedasthe
2500
radiuswithinwhichthe meanmassdensityis 2500timesthecriticaldensityofthe universeat
the redshiftofthe cluster (Allenet al. 2002,2003).Thegasmassfractionvaluesof these nine
clustersatr (orattheoutermostradiistudiedforPKS0745-191andAbell478)wereshown
2500
in Figure 5 of Allen et al. (2003).We will use this database to constrain the equation of state
of the dark energy component, ω . Our analysis of the present data is very similar to the one
x
performedbyLimaetal.(2003).However,inadditionaltoincludingnewdatafromAllenetal.
(2003),wealsotakeintoaccountthebiasbetweenthebaryonfractionsingalaxyclustersandin
theuniverseasawhole.FollowingAllenetal.(2002),wehavethemodelfunctionas
fmod(z;ω ,Ω )= bΩb h DSACDM(zi) 3/2 (3)
gas i x m 1+0.19h1/2 Ω 0.5DA(z;ω ,Ω )
m i x m
(cid:0) (cid:1)  
6 Zhu,Z.-H.,Fujimoto,M.-K.&He,X.-T.:Determiningω fromΘ-zand f data
x gas
Fig.3. Confidence region plot of the best fit to the f of 9 clusters published by Allen et al.
gas
(2002,2003)– see the text for a detailed description of the method. The 68% and 95% confi-
dence levels in the ω –Ω plane are shown in lower shaded and lower + darker shaded areas
x m
respectively.
wherethebiasfactorb≃0.93(Bialeketal.2001;Allenetal.2003)isaparametermotivatedby
gasdynamicalsimulations,whichsuggestthatthebaryonfractioninclustersisslightlydepressed
withrespecttotheUniverseasawhole(CenandOstriker1994;Eke,NavarroandFrenk1998;
Frenketal. 1999;Bialek etal. 2001).Theterm (h/0.5)3/2 representsthe changein theHubble
parameterfromthedefautvalueofH =50kms−1Mpc−1andtheratioDA (z)/DA(z;ω ,Ω )
0 SCDM i i x m
accountsforthedeviationsofthemodelconsideringfromthedefaultstandardcolddarkmatter
(SCDM)cosmology.
Again,wedetermineω andΩ throughaχ2minimizationmethodwiththesameparameter
x m
ranges and steps as last section. We constrain Ω h2 = 0.0205± 0.0018, the bound from the
m
primodialnucleosynthesis(O’Mearaetal.2001),andh = 0.72±0.08,thefinalresultfromthe
Hubble Key Project by Freedmanet al. (2001).The χ2 difference between the modelfunction
andSCDMdataisthen(Allenetal.2003)
2
χ2(ωx,Ωm)= 9 hfgmasod(zi;ωxσ,2Ωm)− fgas,oii +"Ωbh02.−0001.80205#2+"h−0.008.72#2, (4)
Xi=1 fgas,i
where fmod(z;ω ,Ω )referstoequation(3), f isthemeasured f withthedefautSCDM
gas i x m gas,oi gas
cosmology,andσ isthesymmetricroot-mean-squareerrors(ireferstotheithdatapoint,with
fgas,i
totally9data).Thesummationisoveralloftheobservationaldatapoints.
Figure 3 displaysthe 68.3%and 95.4%confidencelevelcontoursin the (ω , Ω ) plane of
x m
ouranalysisusingthelowershadedandthelowerplusdarkershadedareasrespectively.Thebest
fithappansatω = −0.86andΩ = 0.30.Asshowninthefigure,althoughtheX-raygasmass
x m
Zhu,Z.-H.,Fujimoto,M.-K.&He,X.-T.:Determiningω fromΘ-zand f data 7
x gas
Fig.4. Confidence region plot of the best fit from a combined analysis for the angular size-
redshift data (Gurvits et al. 1999) and the X-ray gas mass fractions of 9 clusters (Allen et al.
2002, 2003). The 68% and 95.4% confidence levels in the ω –Ω plane are shown in lower
x m
shaded and lower + darker shaded areas respectively.The best fit happansat ω = −1.16 and
x
Ω =0.29.
m
fractiondataconstrainsthedensityparameterΩ verystringently,itstillpoorlylimitsthedark
m
energyequationofstateω .Thesituationcanbedramaticallyimprovedwhenthetwodatabases
x
are combinedto analysis, in particularly,a nontriviallower boundon ω will be obtained(see
x
below).
4. Combinedanalysis,conclusionanddiscussion
Now we present our combined analysis of the constraints from the angular size-redshift data
and the X-raygas mass fractionof galaxyclustersand summarize our results. In Figure 4, we
display the 68.3% and 95.4% confidencelevel contoursin the (ω , Ω ) plane using the lower
x m
shadedandthelowerplusdarkershadedareasrespectively.Thebestfithappansatω = −1.16
x
and Ω = 0.29. As it shown, fairly stringent bounds on both ω and Ω are obtained, with
m x m
−2.22 < ω < −0.62and 0.28 < Ω < 0.32at the 95.4%confidencelevel. The boundon ω
x m x
goesto−1≤ω <−0.60whentheconstraintω ≥−1isimposed.
x x
Althoughprecisedeterminationsofω anditspossibleevolutionwithcosmictimearecrucial
x
fordecipheringthemysteryofDE,currentlyω hasnotbeendeterminedquitewellevenwithan
x
assumptionofω beingconstant(HannestadandMo¨rtsell2002;Spergeletal.2003;Takahashi
x
etal.2003).Itisworthyofdeterminingω usingajointanalysis.Inthispaperwehaveshown
x
that stringent constraints on ω can be obtained from the combination analysis of the angular
x
size-redshift data and the X-ray mass fraction data of clusters, which is a complementary to
8 Zhu,Z.-H.,Fujimoto,M.-K.&He,X.-T.:Determiningω fromΘ-zand f data
x gas
other joint analyses. At this point we compare our results with other recent determinations of
ω from independentmethods. For the usual quintessence model (i.e., the constraintω ≥ −1
x x
is imposed), Garnavich et al. (1998)found ω < −0.55 using the SNeIa data from the High-z
x
SupernovaSearchTeam,whileLimaandAlcaniz(2002)obtainedω <−0.50usingtheanugular
x
size-redshiftdata fromGurvits, Kellermanand Frey(1999)(95%confidencelevel). Ourresult
of ω < −0.60 is a little bit more stringent than theirs. However Bean and Melchiorri (2002)
x
foundanevenbetterconstraint,ω <−0.85,byanalyzingSNeIadataandmeasurementsofLSS
x
andthepositionsoftheacousticpeaksintheCMBspectrum.Forthemoregeneraldarkenergy
model including either normal XCDM, as well as the extended or phantom energy (i.e., the
constraintω ≥−1isrelaxed),HannestadandMo¨rtsell(2002)combinedCMB,LSSandSNeIa
x
dataforanalyzingandobtained−2.68<ω <−0.78at95.4%confidencelevel,whoselowerand
x
upperboundsarea little bitlowerthanours(−2.22 < ω < −0.62at95.4%confidencelevel).
x
Recently, Schuecker et al. (2003)combinedREFLEX X-ray clusters and SNeIa data to obtain
−1.30 < ω < −0.65with 1σ statistical significance.From Figure 4, it is foundour 1σ result
x
is−1.72 < ω < −0.83,whichiscomparablewiththeresultsofSchueckeretal.(2003).Using
x
theX-raygasmassfractionof6galaxyclusters,Limaetal.(2003)found−2.08 < ω < −0.60
x
(1σlevel),whichislessstringentthantheresultpresentedinthiswork.Thisisbecauseweused
moreX-raygasmassfractiondataofgalaxyclustersandcombinedtheangularsize-redshiftdata
of compactradio sources for analyzing. The analysis presented here reinforcesthe interest in
precisemeasurementsofangularsizeofdistantcompactradiosourcesandstatisticalstudiesof
the intrinsic length distribution of the sources. It is also hopefully that our constraints will be
dramaticallyimprovedaftermoreacurateX-raydatafromChandraandXMM-Newtonbecome
availablenearfuture.
Acknowledgements. WewouldliketothankL.I.Gurvitsforsendingustheircompilationoftheangular
size-redshift data and helpful explanation of the data, S. Allen for providing us the X-ray mass fraction
dataandvarious helpabout dataanalysis, J. S.Alcanizand D.Tatsumi fortheir helpful discussion. Our
thanksgototheanonymouserefereeforvaluablecommentsandusefulsuggestions,whichimprovedthis
work very much. This work was supported by aGrant-in-Aid for ScientificResearch on PriorityAreas
(No.14047219)fromtheMinistryofEducation,Culture,Sports,ScienceandTechnology.
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