On the Halo Occupation of Dark Baryons

0.4 MB·English

by  Jeremy L. Tinker

DRAFTVERSIONFEBRUARY1,2008
PreprinttypesetusingLATEXstyleemulateapjv.4/12/04
ONTHEHALOOCCUPATIONOFDARKBARYONS
JEREMYL.TINKER&HSIAO-WENCHEN
KavliInstituteforCosmologicalPhysics&DepartmentofAstronomyandAstrophysics,UniversityofChicago
DraftversionFebruary1,2008
ABSTRACT
We introduce a new technique that adopts the halo occupation framework for understandingthe origin of
QSOabsorption-linesystems. OurinitialstudyfocusesspecificallyonMgIIabsorbers. Weconstructamodel
ofthegaseouscontentinwhichtheabsorptionequivalentwidthW isdeterminedbythetheamountofcoldgas,
r
intheformofdiscreteclouds,alongasightlinethroughahalo. Thetwoquantitiesthatwespecifyperhaloin
8
0 themodelare(1)themeanabsorptionstrengthperunitsurfacemassdensityAW(M),and(2)themeancovering
0 factor κg(M) of the gaseous clouds. These parameters determine the conditional probability distribution of
2 Wr as a function of halo mass, P(Wr M). Two empirical measurements are applied to constrain the model:
|
(i) the absorberfrequencydistributionfunctionand (ii) theW-dependentclusteringamplitude. We find that
n thedatademandarapidtransitioninthegascontentofhalosrat 1011.5 h- 1M , belowwhichhaloscontain
a ∼ ⊙
predominantlycoldgasandbeyondwhichgasbecomespredominantlyhot. Inordertoreproducetheobserved
J
overallstrongclusteringoftheabsorbersandtheanti-correlationbetweenW andhalomassM,roughly5%of
2 gasinhalosupto1014 h- 1M isrequiredtobecold. Thegascoveringfactrorisnearunityoverawiderange
⊙
of halo mass, supporting that MgII systems probe an unbiased sample of typical galaxies. We discuss the
]
h implicationsofourstudyinthecontextsofmassassemblyofdistantgalaxiesandtheoriginofQSOabsorption
p linesystems.
-
o Subjectheadings:Cosmology:theory—darkmatterhalos—quasars:absorptionlines—galaxies:evolution
r
t
s
a 1. INTRODUCTION MgII absorbers along common lines of sight also indicate
[ theThspeecfotrraesotfobfacLkygαrouabnsdorQpStiOons ilsinaesseynsstietmives porbosbeervoefdinin- nthoautsMggalIaIxaiebsso(rBbeerrgsearotnhz1i9=860;.6S5teairdeelaestsoacl.ia1te9d94w)iathndlutmhai-t
2 field galaxies possess extended gaseous halos to projected
tervening baryonic matter that is otherwise invisible (see
v distancesofr 50h- 1kpc(Lanzetta&Bowen1990;Steidel
0 Rauch 1998 for a comprehensive review). The gaseous ≈
1993).
7 clouds uncovered by these absorption features span a wide
Extended halos of cool gas around galaxies were pre-
4 rangeofionizationstateandneutralhydrogencolumndensity
1 over N(HI)=1012- 1022cm- 2 and are understood to origi- dicted by Spitzer (1956) and discussed in more detail in
Bahcall&Spitzer (1969). Theoretical models that explain
. nate in a range of environmentsin the dark matter distribu-
9 theoriginofextendedgaseoushalosaroundgalaxiesinclude
tion, from moderate overdensities to fully collapsed struc-
0
(i) the stripping of gas from the accretion of small, gas-
7 tures (e.g. Davéetal. 1999). Strong Lyα absorbers with
0 N(HI) & 2 1020cm- 2 are thought to be the high-redshift rich satellites into a larger system (Wang 1993); (ii) cold
× gas gravitationally bound to small-scale substructure within
: analogue to present-day galaxies (Wolfeetal. 2005). Stud-
v the dark matter halo (e.g., Sternbergetal. 2002); and (iii)
iesofthesehigh-densityneutralgascloudsshowthattheyall
Xi containlowionssuchasSi+,Fe+,andMg+ (e.g.Lu&Wolfe a two-phase medium, cold clouds condense out of a hotter
halothroughthermalinstability(Mo&Miralda-Escude1996;
1994; Prochaskaetal. 2003). Conversely, studies indicate
r Maller&Bullock2004;Cheloucheetal.2007). Thesemod-
a thatMgIIabsorbersofrest-frameabsorptionequivalentwidth
els imply thatW reflects the potentialwell of the dark mat-
W(2796) 0.3ÅariseingaseouscloudsofN(HI)=1018- r
10r22cm- 2≥(Churchilletal. 2000; Raoetal. 2006). Over the terhalo—moremassivesystemssustainbiggergaseoushalos
withahighervelocitydispersion.
redshift range z = 0.35- 2.5, the MgII λλ2796,2803 dou-
However,recentstudieshaveyieldedconflictingresultsthat
blettransitionsareshiftedtotheopticalspectralrangeλ =
obs challengethis classical picture. While some authorspresent
3800- 10,000 Å and therefore provide a convenientoptical
observations that suggest strong MgII absorbers originat-
probe of dark baryons in extended gaseous halos of typical
ing primarily in galactic superwinds from low-mass galax-
galaxiesalongrandomlinesofsight.
ies (Prochteretal. 2006; Bouchéetal. 2006; Murphyetal.
Extensive studies of the physical properties of MgII ab-
2007), others show that MgII absorbing galaxies resem-
sorbers have been carried out by various authors. Early
ble typical field galaxies (Zibettietal. 2005; Kacprzaketal.
studies that compared abundance ratios between different 2007; Nestoretal. 2007) and the covering fraction of Mg+
ions associated with MgII absorbers showed that these ab-
ions is roughly 50% (Tripp&Bowen 2005). A particularly
sorbers arise in photo-ionized gas of temperature T 104
puzzlingresultistheanti-correlationbetweenW(2796)(here-
K (e.g. Bergeron&Stasinska 1986; Hamann 1997)∼. In r
after W) and clustering amplitude by Bouchéetal. (2006)
addition, high-resolution spectra of strong absorbers also r
(hereafterB06).Theseauthorsmeasuredthecross-correlation
show that these systems are multi-component, with W
r functionofMgIIabsorbersandluminousredgalaxies(LRGs)
roughly proportional to the number of components in the
in data release three of the Sloan DigitalSky Survey(SDSS
system (Petitjean&Bergeron 1990; Churchill&Vogt 2001;
DR3; Abazajian 2005). From the relative bias with respect
Prochteretal. 2006). Finally, comparisons of galaxies and
toLRGs, B06inferredthemeanhalomassofabsorbersasa
2
functionofW,findingthatstrongabsorbersofW =2- 2.85Å strainedbyempiricaldata. Theavailableobservationsare(1)
r r
onaverageariseinlessmassivedarkmatterhalosof logM = the frequency distribution function d2N/(dW dz), the num-
r
11.11,whileabsorbersofW =0.3- 1.15Å onaverhageariise berdensityofabsorbersperunitabsorptionequivalentwidth
r
indarkmatterhalosof logM =12.49.Theauthorsconsider intervalperunitredshiftpath,and(2)thelarge-scalebiasrel-
thisanti-correlationinshtrongifavorofasuperwindoriginfor ativetoLRGs. Thehalooccupationanalysisallowsustogo
beyondsimply estimating the mean mass of dark matter ha-
W >1 Å absorbers; the large equivalentwidth in low-mass
r
losthathostabsorbersandconstrainthedistributionofthese
systems is attributed to the outflow velocity of the systems
halosasafunctionofW.
undergoingstarburstepisodes. r
WewillconsidertwoscenariosforP(W M).Inthefirstsce-
Thecross-correlationfunctionofabsorbersandgalaxieson r
|
large scales (&1h- 1Mpc) providesa quantitativecharacteri- nario,thefractionofcoldgasrelativetodarkmattermassisa
smoothlyvaryingfunctionofhalomass. Thisscenerioismo-
zationoftheoriginoftheseabsorbers. Theclusteringofab-
tivatedbythecanonicalmodelforhalogasinwhichaccreting
sorbersisaconsequenceofthehalosinwhichtheyarefound.
gasis always shock-heatedto the virial temperaturewhen it
Low-masshalosforminawiderangeofenvironmentswhile
crossesthevirialradius(e.g.,White&Frenk1991). Thegas
high-masshalos can only form in the most dense regionsof
thencoolsfromtheinsideout, creatinga coolingradiusthat
the darkmatterdistribution, thushigh-masshalosare highly
varies with the properties of the halo. We will demonstrate
clustered. The clustering of halos depends only on simple
that this model cannot simultaneously fit the frequency and
gravitational physics, and the bias of dark matter halos as a
bias data; models of this type that reproduce the frequency
functionofmassisawell-establishedrelationinthestandard
distributionpredicta positive correlationbetweenW and M
CDM cosmology. Thusit is straightforwardto infer a mean r
becausemoremassivehalosareexpectedtocontainmoreab-
halomassfromlarge-scaleclustering,aspresentedinB06.
sorbinggas. TakingintoaccountthefactthattheMgIItran-
However, when interpreting the clustering of gaseous
sitionsprobeprimarilywarm, photo-ionizedgas, we include
clouds based on the presence of a specific ion, it is neces-
inthesecondmodelatransitionalmassscale,atwhichshock
sarytoconsiderthestate ofthegasinthedarkmatterhalos.
For example, if the observed Mg+ ions originate in photo- heatedgasprogressivelytakesoverhighermasshalosandthe
ionized gas of temperature T 104 K, then the observed absorptionefficiencyissignificantlyreduced.Inthetransition
∼ region between entirely ‘cold-mode’ and ‘hot-mode’ halos,
anti-correlationbetweenW and M maybeproducedwhen
r
h i heatingoccursasaninside-outprocess.Namely,ahighlyion-
the halo gas in massive dark matter halos becomes too hot
for abundant Mg+ to survive. To investigate how the con- izedcoreexpandsuntilitenvelopestheentiregashalo(e.g.,
Dekel&Birnboim2006). Weallowthemassscaleandwidth
tentofwarm/coldgasdependsonhalomass, weintroducea
of this transition to be free parameters. In addition, we al-
newtechniquethatadoptsthehalooccupationframeworkfor
low some fraction of cold gas in the hot halos that is also
studyingthe originofMgII absorbers. Thissame technique
constrainedby empiricaldata. We demonstratethat the sec-
canalsobeappliedforstudyingthenatureofQSOabsorption-
linesystems. ondscenarioreproducesboththeobservedd2N/(dWrdz)and
biasdatawell. Weobtainabest-fittransitionalmassscaleof
Our halo occupation approach is similar to the Halo Oc-
1011.5h- 1M andacoldgasfraction(asrepresentedbythe
cupation Distribution (HOD) method that was developed to ⊙
establish an empirical mapping between galaxies and dark M∼g+ions)of<10%inmassivehalos.
Giventheagreementbetweenobservationsandourhalooc-
matter halos (e.g., Seljak 2000; Scoccimarroetal. 2001;
cupationmodel,wearguethatitisnotnecessarytoinvokead-
Berlind&Weinberg 2002; see Zheng&Weinberg 2007 and
ditional, complicated star formationfeedback for explaining
referencestherein). This mappingis purelystatistical in na-
theinversecorrelationbetweenW and M . Wedemonstrate
ture; it interpretsgalaxy bias based on the probabilitythat a r
h i
thatthe halooccupationapproachallowsusto gainphysical
haloofmassMcontainsN galaxiesofagivenclass,P(N M).
| insights for (1) understanding the origin of intervening ab-
Theprobability,P(N M), isconstrainedby measurementsof
| sorbersobservedinthespectraofbackgroundQSOs,and(2)
the space density of the class of objects and their two-point
constrainingthedistributionofbaryonsindarkmatterhalos.
auto-correlation function. HOD analysis of galaxy cluster-
Wediscusstheseresultsinthecontextofgasaccretioninre-
ingdatacanprovideinsightintogalaxyformationandevolu-
centtheoreticaldevelopments(i.e., Birnboim&Dekel2003;
tion (vandenBoschetal. 2003b; Cooray 2006; Zhengetal.
Dekel&Birnboim 2006; Kerešetal. 2005; Birnboimetal.
2007; Whiteetal. 2007; Tinkeretal. 2007) and to constrain
2007).
cosmological parameters (vandenBoschetal. 2003a, 2007;
This paper is organizedas the following. In § 2, we out-
Tinkeretal.2005;Tinker2007;Zheng&Weinberg2007). In
linethetheoreticalframeworkforconstructingtheconditional
thispaper,weextendthisframeworktocharacterizethecold
probability distribution of equivalent widths P(W M). In §
gasdetectedthroughQSOabsorptionlinesandfocusspecifi- r
|
3, we derive the observables that characterize the statistical
callyonMgIIabsorbers.
properties of MgII absorbers, based on known dark matter
TostudythedarkmatterhalopopulationofMgIIabsorbers,
halo statistics and P(W M). In § 4, we compare the model
we adopta conditionalprobabilitydistribution of equivalent r
|
predictionswithempiricalmeasurementsforMgIIabsorbers.
widthsP(W M) to describe the halo occupationofcold gas.
r
| In § 5, we apply the best-fit model to predictadditionalsta-
Our technique is fundamentally the same as the conditional
tisticalquantitiesonthegalaxy–absorbercorrelationthatcan
luminosity function for interpreting galaxy bias (Yangetal.
be tested with future observations. Finally, we discuss in §
2003;vandenBoschetal.2007).Inourapproach,wechoose
6theinterpretationsofouranalysis. We adopta flatΛCDM
to parameterize P(W M)—the distribution of W within a
halo of given mass—ra|ccordingto expectationsofra uniform cosmologywithΩM=0.25andσ8=0.8andadimensionless
gaseoushalo,inwhichtheequivalentwidthisproportionalto Hubbleconstanth=H0/(100km s- 1 Mpc- 1) throughoutthe
the density-weighted path-length along a sight-line through paper.
the halo. The parameters of P(W M) are left free and con-
r| 2. THEORETICALFRAMEWORK
3
In this section, we describe the approachto populate dark found.Wenotethatanisothermaldistributionofdiscretegas
matter halos with cold baryons that are represented by the clumps is expected to have a mean absorption strength that
presence of Mg+ ions. An implicit assumption of our halo declineswithradius. Atacertainimpactparameterthemean
occupation approach is that all absorbers originate in halos, numberofcloudsperlineofsightwillrapidlydeclineandbe-
rather than in metal enriched intergalactic medium (IGM) comenegligible(see,e.g.,Figure1inCheloucheetal.2007).
outside of individual halos. This assumption is justified by Weapproximatethiseffectbycreatinganeffectivegasradius,
the strong correlation between absorbers and galaxies; the R ,beyondwhichnogasclumpsarepresent.
g
LRG–MgII cross-correlation function exhibits a strong sig- Considering the total absorption equivalent widthW as a
r
nal at separations less than halo radii (Bouchéetal. 2004). sumoverallclumpsencounteredalongasightlineleadsto
In addition, for each absorber a galaxy is nearly always
found at a separation significantly less than the typical halo W(sM)=W 2σcl √R2g- s2ρ (√s2+l2)dl (4)
rmaduilutis-ceoxmppeocnteedntgnivaetunrethoef gMaglaIxIysylusmteminsosditeym.onFsitnraaltleys,tthhaet r | 0×"Mcl Z0 g #
the absorbing medium is highly clumped, in contrast to the W σ f 2 R2- s2
smoothergasintheIGM. = 0 cl g G0 tan- 1 g , (5)
For each dark matter halo, we will specify two quantities: (cid:18) Mcl (cid:19) s2+a2 ss2+a2h
h
(1)themeanabsorptionstrengthperunitsurfacemassdensity
q
wheresistheimpactparameterofthesightlinewith respect
A (M),and(2)themeancoveringfactorκ (M)ofthegaseous
W g
clouds. These parameters determine the conditional proba- tothecenterofthehalo,σcl andMclarethecrosssectionand
mean mass of individual gas clumps, andW is the absorp-
bilitydistributionofW asa functionofhalomass, P(W M) 0
r r
| tionperclump. Itisclearthattheparametersthatcontrolthe
which, when combined with known statistics of dark matter
relation between impact parameter and W, such as the gas
halos, allows us to derive theoretical predictions of various r
fraction f , and the mean absorption strengthW and cross-
statisticalpropertiesofMgIIabsorbersfordirectcomparisons g 0
withobservations. sectionσcl ofindividualclumps,aredegenerate. We express
theirproductasasinglefreeparameter
2.1. ConstructingtheGaseousHalo
W σ f
0 cl g
A , (6)
Adarkmatterhaloisdefinedasa collapsedandvirialized W ≡ M
cl
object with a mean interior density equal to 200 times the
which represents the mean absorption equivalent width per
backgrounddensity.Thustheradiusofahalois
unitsurfacemassdensityofthecoldgas. Inthelimitthatall
3M 1/3 the componentsof the absorptionline are optically thin, AW
Rh=R200= 4π(200ρ¯ ) . (1) wouldbeameasureofthemeanparticledensityofMg+ions.
(cid:20) m (cid:21) Some fraction of the lines will be optically thick, however,
Thechoiceof200ρ¯mismotivatedbyestimatesofthevirialra- so we are unable to constrainthe density fromthe values of
diusfromthesphericalcollapsemodel,whichpredictavalue A . Therelationbetweenthepredictedabsorptionequivalent
W
of 180.Whiletheseestimatesareidealized,theyalsomatch widthW atimpactparametersofagivenhaloisdescribedby
∼ r
the halos identified in N-body simulations, from which the
globalpropertiesofhalosarecalibrated. AccordingtoEqua- 2 R2- s2
hti-o1nkp(1c),,wah1i0le12Rh- 1Mfo⊙r aha1lo01h4ahs-a1Mco-mhoavloingisra1d.1iuhs-o1fM2p4c3. Wr(s|M)=AW(M) s2G+0a2tan- 1ss2g+a2h. (7)
200 ⊙ h
With this definition, all halos of mass M have the same co- q
Equation (7) shows that if the fraction of cold gas and its
movingradius at any redshift. We adopt the halo mass pro-
properties do not change across all halo masses, then A is
fileofNavarroetal.(1997)andtheconcentration-massrela- W
expectedtobeaconstantandthetotalmeanequivalentwidth
tion of Bullocketal. (2001) (with updated parameters from
is expected to increase with halo mass due to the larger gas
Wechsleretal.2006)tocalculatethehalomassasafunction
massinmoremassivehalos.
ofradius.
We choose a value of a =0.2R , noting that our results
Webeginwiththeempiricalresultthattheequivalentwidth h g
areinsensitiveto theexactchoiceforthisparameter. We set
W of an absorptionsystem is proportionalto the numberof
r R = 80 comoving h- 1kpc (50 h- 1kpc physical at z = 0.6)
absorptioncomponents, which we interpretas gas clouds or g
clumpsencounteredalongthesightlinethroughthehalo1.We for halos of mass 1012 h- 1M⊙. The gas radius scales as
adoptan isothermalprofilewith a coreradiusah to approxi- Rg=80(M/1012)1/3 h- 1kpc, the same scaling asthe halo ra-
matethemeangasdensitythatfollows diusinequation(1). Withthisnormalization,thegasradiusis
roughlyonethirdofthehaloradiusforallhalomasses. The
ρg(r)= fgG0 r2+a2h - 1, (2) choice of 50 h- 1kpc for our fiducial radius is motivated by
observationsoftheradialextenttowhichMgIIabsorptionof
where f isthegasfraction,
g (cid:0) (cid:1)
Wr 0.3ÅisdetectedaroundL∗galaxies(e.g.,Steidel1993).
= M(<Rg)/4π , (3) We≥willdemonstratein§5thatthevalueofRgmakesdistinct
G0 R - a tan- 1(R /a ) predictionsforthedistributionofimpactparameters,andthat
g h g h
ourchoiceforR matchesavailableobservationswell.
g
and M(<R ) is the mass of dark matter within the effective
g
radius Rg of the gaseous halo. Beyond Rg, no absorption is 2.2. TheConditionalProbabilityDistributionofWr—A
SmoothGaseousHaloModel
1WenotethatanabsorptionlineofwidthWrimpliesamaximumvelocity
differencebetweencomponentsof∼100kms- 1 perAngstrom. Themaxi- ToconstructtheconditionalprobabilitydistributionofWr,
mumcircularvelocityofa1012h- 1M⊙NFWhalois∼150kms- 1. P(Wr M),weconsidertheprobabilitydistributionofarandom
|
4
lineofsightinterceptingahaloatimpactparametersandthe 2.3. IncorporatingaCold-Mode/Hot-ModeTransition
mean absorption expected at the impact parameter s. For a
As we summarized in § 1, the MgII absorbersare known
random line of sight, the probability that a sightline passes
tooriginateinprimarilycool,photo-ionizedgas. Recentnu-
throughahaloofmassMwithimpactparametersis
merical simulations show that stable shocks develop as gas
2s/R2 if s R accretes onto a dark matter halo, and above a critical mass
P(sM)= g ≤ g (8)
| 0 if s>Rg. thresholdall gasis shock heatedto the virialtemperatureor
(cid:26) above. Atlowmassscales,however,gasaccretedontohalos
ThedistributionofW isrelatedtothedistributionofimpact
r isnotshockheated,butremainscoldasitsinkstothecenter
parametersaccordingto
ofthehalo(e.g.,Kerešetal.2005).Dekel&Birnboim(2006)
P(Wr M)dWr=κg(M)P(sM)ds, (9) demonstratethatthisisduetothegascoolingtimebeingtoo
| |
whereκ isthetotalprobabilityofdetectinganabsorberina shortwithrespecttothecompressiontimeinlowmasshalos,
g
haloofmassM,whichmaybelessthanunity.Thisintegrated thusshocksarenotstableatanyradiuswithinthehalo.
probabilitythatasightlinethroughahaloyieldsanabsorber Itisthereforeexpectedthathalosaboveacertainmassscale
is governed by two physical quantities: the mean covering may not contribute significantly to the observed MgII ab-
fractionofcoldgasperhaloandthefractionofhalosthathost sorbersduetoshockheatingoftheirgas. Inthissecondsce-
coldgasclouds. Thesetwoquantitiesaredegenerate,andwe nario, we include a transitionalmass scale, abovewhich the
parameterizetheirproductwiththequantityκ . gasbecomespredominantlyshock-heatedandunabletocon-
g
From equations (8) and (9), the probability distribution tributeto MgII absorption. We denotethe shockradiusR .
sh
functionofequivalentwidthsis The absorption efficiency at r<R is reduced to a fraction
sh
2s(Wr M) ds fcold, which representsthe fractionof gasthat is cold within
P(Wr|M)=κg(M) R2| dW , (10) the shock radius relative to gas outside the shock. For im-
g r pactparameterssmallerthantheshockradius,equation(7)is
wheres(W M)istheinversionofequation(7),whichisper- modifiedasthefollowing,
r
|
formed numerically. The derivative of s with respect to W
r
is 2 R2 - s2
ds s tan- 1x 1+x2 (a2+R2)(s x) - 1 Wr(s|M)=Wr(s|M)(Rsh=0)- (1- fcold)AW s2G+0a2tan- 1s s2sh+a2h ,
=(2A )- 1 × + h g × , h
dWr WG0 (s2+a2h)3/2 s2+a2(R2g- s2)(a2h+s2) q (13)
h where the first term on the left hand side is the result from
 q (11) equation(7),inwhichR =0.Equation(11)iscalculatedfor
sh
wherex (R2- s2)/(a2+s2). s<Rshinthesamemanner.
≡ g h Figure 1 shows P(W) (normalized to κ =1) for halos of
Insteadoqfparameterizingthemassdependenceofκgusing 1011.5h- 1M (leftpanerl)and1013h- 1M g(rightpanel). The
power-laws or polynomials, we adopt a non-parametric ap- ⊙ ⊙
valueofA issetsuchthatasightlineatanimpactparameter
proach and specify the values of κ at four differentmasses W
andsplineinterpolatebetweenthemg. Thus,theformofκ is ofRg/2througha1012h- 1M⊙haloyieldsanequivalentwidth
g
fully non-parametric. The four mass values, M though M of 1.5 Å forcomparisonpurpose. In each panel, we present
1 4
arelogM =10.0,11.33,12.66,and14.0. Wedenotethefour threedifferentprobabilitydistributionfunctions(PDFs). The
i
parametersoftheκg functionasκ1, κ2, κ3, andκ4. We per- dotted curve shows equation (10), which represents the Wr
formtheinterpolationin logM and logκ and extrapolateto distributionforacontinuousgasdensityprofilewithnoshock.
g
higherandlowermasses. Astheimpactparameterapproachestheedgeofthehalo,Wr
Inthissmoothgaseoushalomodel,whichwewillreferto becomessmall. Itismorelikelytogothroughathaloatlarge
asthe“classical”model,weadoptadoublepowerlawtoac- impactparameters,buttheprojecteddensitydropssteeplyas
count for the mass dependence of the mean absorption per sapproachedRg, thusP(Wr)=0 atWr =0. Themodeofthe
uniAtWsu(rMfa)c=emaAAss0MMde1αβn2sityAW,i.e., iiff MM12≤>11. (12) Wdfoirslltoriwb3us.t6WioÅrn- 3ao.tcsTcu=hre0s.daitstsri∼bu0ti.o9nRrge,aacnhdesatalmaragxeirmWurmthvealcuuervoef
(cid:26) 0 12 12 T∼he dashed curve shows the distribution of equivalent
Asa result, thisclassical modelcontainssevenfreeparame-
widthsfor a halowith R =0.3R , and f =5%. The pri-
sh g cold
terstobeconstrainedbythedata.Threegoverntheamountof
mary effect of shock heated gas in the inner sphere is to re-
absorptionasafunctionofhalomass,A0,α,β,andfourchar- duce the maximumWr. In the case of a 1011.5 h- 1M⊙ halo,
acterizethemassdependenceofgascoveringfractionκ -κ .
1 4 thelargestequivalentwidthisreducedbyafactoroftwo. For
Insummary,thegasdensityprofiledeterminestherelative
s<R , P(W) has a very different form. As s falls just in-
sh r
absorption between different impact parameters in a halo—
side the shock radius, W decreases rapidly. At s R /2,
sh
the density profile sets the shape of P(W M). The relative ∼
r W(s)reachesa minimumvalueasthe smallfractionof cold
| r
absorption between halos of different masses is determined
gaswithin R accumulatesenoughto contribute to the total
sh
by A ; at s=0, the line of sight directly through the center
W absorption. The minimumvalue ofW dependson f , but
r cold
ofahalo,theabsorptionscalesasWr(0|M)≃M112/3(AW/10.9) aroundthe minimum, the derivative dWr/ds becomessmall,
where AW is in units of (hÅcm2/gm). The incidence of the producinga sharp featurein the total P(Wr) curve atWr ∼1
absorptionis set by κ . Some sightlines that pass througha Å. This featureis even more pronouncedin the PDF for the
g
halo will encounter no absorbing gas, either due to the fact highermasshalo(rightpanel),alsowithR =0.3R .
sh g
thattheabsorbinggasishighlyclumpyandthecoveringfrac- Thesharpfeatureis dueto theidealizednatureof thecal-
tion is low or due to the fact that the halo does not contain culation.Inpractice,thepropagationoftheshocksisunlikely
coldgas. tobesphericallysymmetric,andahaloofPoisson-distributed
5
FIG. 1.— Thedistributionofequivalent widthsP(Wr|M)forhalosof1011.5h- 1M⊙ (leftpanel)and1013 h- 1M⊙ (rightpanel). Inbothpanels,thedotted
curveisP(Wr|M)foramodelinwhichthecoldgasfollowsanisothermaldensityprofilethroughoutthehalo.Thedashedcurvesrepresentatmodelinwhichthe
coldgaswithinR=0.3Rgisreducedbyafactorof1- fcold=0.95. ThesharpfeatureinthePDFisdueslowchangeinWrassapproaches0. Thesolidcurve
representsamodelwherethisfeatureissmoothedout,asexpressedinequation(14).
clouds will also smear out this feature. To facilitate the andrestrictR /R tobenosmallerthen0andnolargerthan
sh g
calculations, however, we approximate the distribution of 1. Wewillrefertothe“transitionscale”asthemassatwhich
W(s<R ) as a Gaussian centered on µ =W(s= R /2) R /R =0.5.Atlowmassestheshockradiusiszero,whilein
r sh W r sh sh g
withwidthσ =[W(R )/2- W(0)]/2.Theamplitudeofthe thetransitionregionthefractionofthegasradiusthatiswithin
W r sh r
Gaussian function is set by the fraction of the cross section theshockedcoreincreaseslinearlywithlogM untilR =R .
sh g
within the shock radius, which for R =0.3R is 9%. The ForhaloswithR =R ,theequivalentwidthiscalculatedas
sh g sh g
Gaussian function is then added to the PDF for s > R .
sh
For this shocked halo model, the probability distribution is 2 R2- s2
W(sM)= f A G0 tan- 1 g . (16)
r | cold W s2+a2 ss2+a2h
P(Wr|M)=κg(1- fσ)s(WRr2g|M)ddWsr + Nominally, this “transitioqn” modhel has eight free parame-
ˆ
κ2gπfσσ2 exp - (W2rσ- 2µW)2 , Wr≤Wmax,(14) wteersl:eaAvWe,AκW1 ttohrboeuginhvaκr4i,anRt0shw,iγthshh,aalnodmfacoslsd.anIndathttirsibsucteenraerlieo-,
W (cid:20) W (cid:21) vantmass-dependentgasfractiontoR and f ,inorderto
sh cold
where fσ =R2sh/pR2g is the probabilitythat a sight line has an simplifythemodelanalysis. RemovingfreedomfromAW in
impactparameterwithinRsh,andWmax=max[Wr(Rsh),Wr(0)] favor of Rsh is beneficial in that it highlights the differences
fromequation(4).WenotethatP(W)=0forW >W . The between the two scenarios and allows us to isolate features
r r max
results are shown in the solid lines in both panels of Figure fromeachmodel.Aswewillshowin§4,theclassicalmodel
1. Instead of a sharp peak in P(W), the smooth model cre- andthetransitionmodelhavedistinctpredictionsforvarious
r
atesa‘shoulder’inthePDF,smoothingoutthediscontinuity statistical quantities of the MgII absorbers. These distinc-
inthedashedcurve. TheparametersoftheGuassianarecho- tionscanbeattributedtothedifferentspatialdistributionsof
sensuchthatthisshouldertracksthepeakinthePDFasR the cold gas in halos of different mass, and serve as strong
sh
changes. constraintsofthemodels.
In the one-dimensional simulations of Birnboim&Dekel
(2003)andDekel&Birnboim(2007),shockheatingfirstbe- 3. DERIVATIONOFTHEOBSERVEDSTATISTICAL
comestableatthe innerregionsofthehalo, thenpropagates PROPERTIESOFMGIIABSORBERS
outwardasthehaloevolvesandbecomesmoremassive.Thus Both the frequency distribution of MgII absorbers as a
at a given epoch, the width of the transition region in their functionofW andtheirclusteringamplitudewithrespectto
r
simulations is roughly a factor of 2- 5 in halo mass. In LRG are known from previous work. Here we summarize
morerealisticthree-dimensionalsimulations,butinwhichthe these known statistical propertiesfrom observationsand de-
shockfrontsarenotfullyresolved,thefractionofcoldgasde- rivetheoreticalformulaebasedonthetwoscenariosdescribed
creaseslinearlywithlogMover 1dexinmass(Kerešetal. in§2.
∼
2005; Birnboimetal. 2007). We parameterize the mass de-
pendenceofRshas 3.1. TheFrequencyDistributionFunction
Rsh =Rˆ0 +γ log M , (15) Thefrequencydistributionfunctionisdefinedasthe num-
R sh sh 10 12 ber of absorbers per absorption equivalent width interval
g
6
per unit redshift path length. Here we combine the re- ourempiricaldataforcomparisonswithmodels2.
sults from Prochteretal. (2006) for W > 1Å with the re-
r
sultsfromSteidel&Sargent(1992)for0.3Å W 1Å.The 3.2. TheMgII–LRGCross-CorrelationFunction
r
≤ ≤
Prochteretal. (2006) data is nearly complete (>95%) over The cross-correlation function between absorbers and
the redshift range 0.4 z 2.2 for absorbers ofW >1 Å. galaxiesalongcommonlinesofsightdeterminesthecluster-
r
≤ ≤
TheSteidel&Sargent(1992)dataiscompletetoW =0.3Å ingstrength(andthereforetherelativebias)oftheabsorbers
r
overtheredshiftrange0.2 z 2.2. with respect to the galaxies. Given the mean mass scale of
For empirical measurem≤ent≤s, we take the raw number thehalosthathostthegalaxies,onecanderivethemeanmass
counts from Prochteretal. (2006) and divide the number scaleofthehaloshostingtheabsorbers. Bouchéetal.(2006)
counts by the total co-movingpath length of all the lines of presentedtheprojectedcross-correlationfunctionofMgIIab-
sight,ltot.Thetotalco-movingpathlengthiscalculatedbyin- sorbersandLRGsonscalesofrp=1- 10h- 1Mpc. TheMgII
tegratingthedistance-redshiftrelation,weightedbythenum- -LRGsamplecoversaredshiftrangeof0.35<z<0.8.These
beroflinesofsightg(z)ateachredshiftintervaldzasfollows, authors calculate the relative bias of absorbers to LRGs by
takingtheratioofabsorber-LRGcross-correlationfunctionto
ltot= dzg(z) Ωm(1+z)3+ΩΛ - 1/2. (17) theLRGautocorrelationfunction,
Equation(17)yieZldstheto(cid:2)talco-movingra(cid:3)dialdistancecov- ˆbW bW = ξWG. (19)
≡ b ξ
ered by the lines of sight to 45,000 quasars available in G GG
∼
SDSS DR3. The total number of absorbers is N =4835, Equation(19)isonlyexactatlargescalesbecauseitassumes
abs
with l = 4.55 107 h- 1Mpc. There is no systematic or linear bias, but the scale dependence of halo bias is largely
tot
×
cosmic variance error estimate for the frequency data, so independentofhalomass(Tinkeretal.2005)andwillcancel
we apply Poisson fluctuations for estimating random errors, inthe ratio. We takethe absolutebiasof LRGswith respect
including a 5% ‘systematic’ uncertainty for each of the 38 todarkmattertobeb =1.85atz=0.6(Whiteetal.2007,
LRG
bins that have at least one absorber. This additional sys- whoadoptthesamecosmologyaswhatisassumedhere),and
tematic error only alters the total uncertainty for the lowest estimatetheabsolutebiasoftheabsorbersfollowingEquation
few W bins from the Prochter et al. sample (W = 1- 1.5 (19).
r r
Å) and prevents those data from driving the total χ2 when Tocomparethemodelsto thedata, we calculatetheabso-
evaluating a model. For Wr 1Å, we use the results from lutebiasofabsorbersanddividethecalculationbybLRG. The
Steidel&Sargent(1992). Thi≤sisamuchsmallersample,but bias data are in four bins ofWr: [0.3,1.15] Å, [1.15,2.0] Å,
is valuable for constraining the shape of d2N/(dWdl). The [2.0,2.85] Å, and [2.85,4.0] Å. The highest-Wr bin has lit-
Steidel & Sargent measurements are scaled to those of the tle influence on the parameter constraints due to the small
Prochteretal.(2006)datafor1Å<W 1.3Åtoaccountfor number of objects in that bin. When comparing models to
r
thedifferenceinthetotalredshiftsurvey≤pathsofthetwosam- biasdata,wetakethenumber-weightedaveragebiasoverthe
ples. givenrangeinWr. TheB06sampleisincompleteforWr<1
To obtain theoretical predictions, we calculate the condi- Å. The true frequency distribution monotonically increases
tionalprobabilityofahalocontaininganabsorberofagiven with smaller W, but the [0.3,1.15] Å bin is dominated by
r
tsetrres.ngTthhePi(nWterg|Mral)oafstshpeecpirfioedducinto§f2thfeorcarogsisv-esnecsteiotnofwpeairgahmteed- ∼tro1duÅceatbhseosrabmeres.incWomhepnlecteanlceuslsaitnintgheˆbWmoidnetlhbiys mbiunl,tiwpleyiinng-
halo mass function and the probability of detecting an ab- P(W M) by 21.4(1- W) for W < 1 Å. With this factor, a
r r r
sorberofW inahaloofagivenmassleadstothefrequency |
r modelthataccuratelyre-producesthetrued2N/(dW dl)data
oftheabsorbersversusW,i.e. r
r willproducea frequencydistributionthatmatchesthe shape
oftheB06sample.
d2N dn Inthecontextofthehalomodel,thebiasofabsorberswith
f(Wr) = dM σg(M)P(Wr M), (18) respect to the dark matter is equal to the mean halo bias,
≡ dW dl dM |
r
Z weightedbytheprobabilityoffindinganabsorberW atmass
r
where σ (M)=πR2 is the cross section of the gas halo and M;
g g
dn/dMisthehalomassfunction. Thefrequencydistribution
in equation (18) has units of Å- 1(h- 1Mpc)- 1 in co-moving 1 dn
b = dM σ (M)b (M)P(W M), (20)
coordinates. W f(W) dM g h r|
r
We note that two scaling factors need to be accountedfor Z
beforeadirectcomparisonbetweenempiricalmeasurements wherebh isthehalobias. Forthehalomassfunctionweuse
andmodelpredictionscanbedone. First, theredshiftdistri- thefittingfunctionofWarrenetal.(2006),whileforthehalo
butionsof the frequencydata andthe biasdata are different. bias we use the fitting function of Tinkeretal. (2005). All
Asdiscussed in § 3.2below,the biasdata are limitedby the halo properties are calculated at the effective redshift of the
redshiftrangeforLRGsat0.35<z<0.8.Wethereforemust B06absorbersample. Theredshiftrangeofthedataislarge
re-normalize the frequency function to the effective redshift enoughsuchthatintegratingequations(18)and(20)overred-
oftheB06sampleatz=0.6. Theamplitudeofthefrequency shiftwillyieldslightlydifferentresults,butforthepurposeof
functionrisesbetweenz=0.5andz=1,andtheratioofdN/dz thisstudyitissufficienttochooseaneffectiveredshiftforall
atz=0.6tothefullsampleis0.55. Second,theamplitudeof
f(W)basedonaSDSSDR5MgIIabsorbersampleisfound 2 We note that the slope of the full Prochteretal. (2006) sample is
r dlogN/dW =0.72, a value consistent with the results from Nestoretal.
to increase by a factor of 1.1 (J. X. Prochaska, privatecom-
(2005),whocalculate dlogN/dW =0.75over0.4<z<0.8forasmaller
munication). We have therefore included this correction in sampleculledfromtheSDSSearlydatarelease.
7
calculations. In our subsequentanalyses, the redshift evolu- massrange. Giventheconstraints,thereislittle contribution
tion of f(W) will be used as an additionalconstrainton the toMgIIabsorptionfromhalosmoremassivethan1013h- 1M
r ⊙
models. andlessmassivethan3 1010h- 1M .
⊙
×
Figure 3b shows constraints for the absorption column
4. COMPARISONSBETWEENOBSERVATIONSAND
A (M). Forlow-masshalos,theconstraintsontheslopeand
BEST-FITMODELS W
normalization are strong; 1012h- 1M halos produce 1Å
⊙
Thescenariosdiscussedin§2havedistinctpredictionsfor ∼
absorbers. The best-fit model has a single power-law, with
theempiricaldatadiscussedin§3. Weconstrainthemodels α β - 0.17. Thenegativevaluesofαandβ increasethe
usingaMonteCarloMarkovChaintechniquethatdetermines ≈ ≈
absorptionefficiencyinlowermasshalos,butnotsufficiently
thebest-fitparametersbasedoncomparisonsofthedatawith
to produce an anti-correlation between W and M. A slope
r
randomrealizationsfromvariousmodelpredictions.Foreach more negative than - 1/3 in A (M) versus M is required to
realizationinthechain,thetotalχ2isthesumoftheχ2values makethemeanW constantasaWfunctionofM,buttheseneg-
r
fromboththefrequencyandbiasdata. Wewillrefertothese
ative values would violate the constraints on the frequency
quantities as χ2f and χ2b, respectively. Because of the larger distributionbyoverproducinghigh-Wrabsorbers.
number of frequency data and smaller errors on those data, At M >1012 h- 1M⊙, the constraints on AW are poor be-
χ2f dominatesthetotalχ2. However,theχ2f valuesshouldbe cause κg falls off rapidlyat these masses. There is a degen-
taken with caution, because we have no true estimate of the eracybetweenκ4 andβ; if κg fallsoffforM>1012h- 1M⊙,
cosmicvariance. high-W systemscanonlybeproducedifβislarge.However,
r
ˆ
4.1. TheClassicalGaseousHaloModel thissteepenstheslopeofbW andthesemodelsfareevenworse
incomparisontothebiasdata.Themodelswhichproducethe
As described in § 2.2, the classical model contains seven
lowestχ2havenegativevaluesofbothαandβ.
freeparameters,includingthreethatcharacterizetheabsorb- b
Figure3cshowstherelativecontributiontoabsorbercounts
ing gas column, A , for halos of different mass scales and
W fromhalosofdifferentmassbasedonthebest-fitmodel. We
fourthatcharacterizegascoveringfractionκ -κ ofindivid-
1 4 refer to this mass distribution as the “occupation function”.
ual halos. The mass dependencein A allows the predicted
W ThecurvesshowthreevaluesofW:0.5Å,1Å,and2Å,and
W versus impact parameter s relation (Equation 7) to vary r
r therelativeareasunderthecurvesrepresenttherelativenum-
smoothly with halo mass. The best-fit parameters are listed
ber of absorbers. The y-axis, in arbitrary units, is the prob-
inTable1.Figure2comparesobservationaldatawithpredic-
ability of detecting an absorber ofW in a halo of mass M,
tionsbasedonthe“classical”model(dottedcurves). r
i.e.,P(W M)P(M),whereP(M)istheprobabilitythatagiven
Wepresentinpanel(a)thefrequencydataandinpanel(b) r|
line of sight intersects a halo of mass M based on a cross-
the residuals normalized by the correspondingerror of each
section weighted halo mass function, σ (M)dn/dM. These
bin. The frequency data are adequately fit by this model, g
curveshavetheGaussian-typeshapeoftheκ function,with
yielding χ2 =43.3 for 41 data points and seven free param- g
f the sharp edgesnear the peak resultingfrom the mass range
eters. The classical model matches the abundance of strong
over which κ is truncated at unity. The mass dispersion is
g
absorbers,W >2Å,butatlowerequivalentwidthstheresid-
r 1 dex in logM for all three curves. The modes of each
ualsbecomelarge. Specifically,thismodeloverproducesthe ∼
curvemovetohighermass,producingtheincreasingtrendof
number of absorbers by nearly 60% atW 0.3 Å due to a W withMshowninFigure2d.
∼ r
largecontributionfromlow-masshalos. In summary, the “classical” modelwith a smoothgaseous
Wepresentthecomparisonforthehalobiasoftheabsorbers halo cannot fit both the frequencydata and the bias data si-
in panel (c) and the inferred mean halo mass in panel (d). multaneously. Under this scenario, more massive halos are
Thebiasdata(pointswitherrorbars)arederivedfromMgII– expected to contain more absorbing gas, resulting in a pos-
ˆ
LRG cross-correlation measurements. The relative bias b itive correlation betweenW and halo mass M. Forcing the
W r
derivedfromobservationsdecreasesfrom0.8at W 0.8Å meanabsorptionA perhalotodeclinesteeplywithincreas-
r W
h i≈
tonearly0.4at W 2.4Å.ThemeasuredbiasforW 2.8 ing halo mass would improvethe model fit to the bias data,
r r
h i≈ ≥ butatthesametimeitwouldover-producetheabundanceof
Å increasesslightly to 0.5, butthe numberof objectsin this
binissmallandithaslittleinfluenceonourfittingresults.Us- strong(Wr>2Å)absorbers.
ingasimplemean-massapproximation,B06inferadecrease
in mean halo mass of nearly 1.5 dex over this range in Wr. 4.2. TheSignificanceofShock-HeatedGasinMassive
Thebest-fitclassicalmodelisunabletoreproducetheinverse Halos
ˆ
correlationbetweenb andW. Asexpected,thebiasmono-
W r The best-fit parameters of the transition model are also
tonically increases with W, placing the weakest absorbers
r listed in Table 1. This transition model has eight free pa-
in M =1011.5 h- 1M⊙ halos while locating the strongest ab- rameters: A , κ through κ , Rˆ0 , γ , and f . As dis-
sorbersinthenearlysamehalosasLRGs. W 1 4 sh sh cold
cussed in § 2.3, we have left A to be invariant with halo
Figures 3a-3b present the parameter constraints for this W
mass and attributed relevant mass-dependent gas fraction to
model. In both panels, the inner and outer shaded regions
R and f . Comparisons between observations and pre-
show the allowed parameter range, constrained respectively sh cold
by ∆χ2 < 1 and < 4 from the best-fit model. Figure 3a dictions based on the transition model are also presented in
Figure2(solidcurves).
presentstheconstraintsonthemeangascoveringfractionκ
g The addition of the cold-hot transition produces a best-fit
asafunctionofhalomass. ThefunctionpeaksatlogM 12;
nearly all halos of logM 11.5- 12.5 are expected to≈pro- modelthatpreciselyfitsthefrequencydataovertheentireWr
duce an absorber if the im≈pact parameter is less than Rg to range,includingthosepointsatW <1Å,withχ2f =19.8.The
abackgroundquasar. Atlowerandhighermasses,κg rapidly mˆodel also reproduces the anti-correlation between Wr and
fallsoff,buttheconstraintsbecomeweakattheextremaofthe b forW <2.5 Å, shownin Figure 2c. Althoughthe slope
W r
8
FIG.2.— Panel(a)ThefrequencydistributionofMgIIabsorbersasafunctionofequivalentwidthWr.Thepointswitherrorbarsrepresentacombinationof
thedatasetsofProchteretal.(2006)(Wr>1Å)andSteidel&Sargent(1992)(Wr<1Å).Thetransitionbetweenthetwodatasetsisseeneasilyintherelative
errorbarsbetweenlowandhighWr. Thesolidanddottedcurvesrepresentthebest-fitmodelslistedinTable1. Thesolidcurveisthemodelwithacold-hot
transitioninhalogas,whilethedottedcurvehasasmoothlyvaryingcoldgasfractionasafunctionofhalomass. Theχ2valuesarebrokenintotheχ2forthe
frequencyandbiasdata,respectively. Panel(b):Theresidualsofthemodel f(Wr)curves,normalizedbytheerrorsonthedata.Panel(c):Thebiasofabsorbers
ˆ
relativetoLRGs,bW,asafunctionofWr.ThepointswitherrorsarethemeasurementsofBouchéetal.(2006).Thesolidanddottedcurvesarethetwomodels
frompanel(a).Panel(d):ThemeanlogarithmichalomassasafunctionofWr.PointswitherrorbarsarethevaluesinferredfromthebiasdatabyB06(notused
inthefit).Solidanddottedcurvesarethetwobest-fitmodels.
TABLE1
PARAMETERSOFTHEBEST-FITMODELS
Model AW Rˆ0sh γsh logκ1 logκ2 logκ3 logκ4 α β fcold χ2f χ2b
Classical 33.2 — — -1.721 -0.012 -0.198 -1.763 -0.172 -0.176 — 43.3 11.1
Transition 136 1.02 1.03 -9.27 -0.205 -0.006 -0.168 — — 0.061 19.8 3.5
NOTE.— TheunitsofAW are[hÅcm2/gm].
ofthetrendisnotasstrongasthatseenintheB06measure- W =2Å. This declining trend in the absorber bias is due to
r
ments, χ2=3.5forfourdata points. Therelativebiasin the adecreaseinthemeanhalomasswithincreasingW (Figure
b r
ˆ ˆ
modeldecreasesfrombW =0.7atWr=0.4ÅtobW =0.55for 2d). ThemeanmassfallsbynearlyadecadefromWr=0.4Å
9
FIG.3.— Panel(a):Constraintsonthecoveringfractionκgasafunctionofhalomassfromtheclassicalmodel.Innerandoutercontoursrepresenttherangein
modelsthatproduce∆χ2<1and4withrespecttothebest-fitmodel.Panel(b):ConstraintsonthemeanabsorptionstrengthperunitgasmassdensityAW(M).
Contoursarefortherangeofmodelsdescribedinpanel(a).Panel(c):Theoccupationfunctionofabsorbersforthreedifferentequivalentwidthvalues:0.5Å,1
Å,and2Å.They-axis,inarbitraryunits,istheprobabilitythatasightlinepassesthroughahaloofmassM(P(M))multipliedbytheprobabilitythatahaloof
massMyieldsaWrabsorber(P(Wr|M)).
toW =4Å.3 sharperthantheclassicalmodel: κ =1forhalosoflogM
r g
The total χ2 for this modelis 23.3 for 37 degreesof free- 11.5- 12.5andcutsoffsharplyatlowermasses. Overnear≈ly
dom. Inapurelystatisticalsensethissuggeststhatthemodel adecadeinhalomass,allmodelswith∆χ2<4yieldκ =1.
g
hastoomanyfreeparameters. However,wenoteonceagain However, the transition from cold mode (R /R =0) to hot
sh g
thattheerrorson f(Wr)areestimatesonly,andwewishtoin- mode (Rsh/Rg = 1) begins at M 1011 h- 1M⊙ and ends
cludeasmuchfreedominthemodeltopreventthefrequency at M 1012 h- 1M . This indica≃tes that massive halos of
⊙
data from drivingthe constraintson the halo occupation. In ≃
logM 12contributetotheobservedMgIIstatisticsthrough
thissense, χ2b is the mostrelevantquantity,notingthateight thepre≥senceofcoldflowintheshock-heatedhalos. Thecon-
freeparametersisbarelysufficienttoreproducethebiasmea-
straintsonthecoldfractionwithintheshockarestrong,with
surements.
thebest-fitmodelyielding f =0.06withanallowedrange
cold
Figures 4a and 4b present the parameter constraints for of 0.03 for models with ∆χ2 < 4. Both the transitional
the transition model. The covering fraction is narrower and ±
massscale andcoldgasfractioninmassivehalosagreewell
withthesimulationsofKerešetal.(2005)andBirnboimetal.
3WenotethatthemassesderivedbyB06dependonboththemodelused
(2007).Inthepost-shockregimelogM>12.5,theconstraints
andthecosmologyadopted,andtheyarenotutilizedinthefit.Evenaperfect
fittothebiasdatainpanel(c)willnotnecessarilyyieldtheB06massesin on the coveringfactor are poor; models with a low κ4 yield
panel(d). betterfitstothefrequencydatawhilehighcoveringfactorsat
10
FIG.4.— Panel(a):Constraintsonthecoveringfractionκgasafunctionofhalomassfromthetransitionmodel.Innerandoutercontoursrepresenttherange
inmodelsthatproduce∆χ2<1and4withrespecttothebest-fitmodel.Panel(b):ConstraintsontheshockradiusRsh(relativetothegasradius).Contoursare
fortherangeofmodelsdescribedinpanel(a).ThehorizontalbandstotherightoftheRshcontoursrepresenttheconstraintson fcold.Panel(c):Theoccupation
functionofabsorbersfromthebest-fitmodel.TheabrupttransitionatM=1012h- 1M⊙isduetoRshbecominglargerthanRg.
1014h- 1M⊙producehigherbiasfactorsandlowerχ2b. For Wr ≥ 1 Å absorbers, halos in the post-shock regime
Figure 4c shows the occupation functions for the best-fit do not initially have enough gas to produce absorbers of
model. For halos of M = 1011- 1012h- 1M , the absorber this strength, and so the distribution of halo masses occu-
⊙
countsrisesasκ increaseswithhalomassasshowninpanel pied by these absorbers becomes bimodal; only at M >
g
(a). ForWr =0.5 Å, the occupation function peaks are 1012 1012.5- 13h- 1M⊙isthereenoughcoldgastoproduceabsorbers
h- 1M , thenabruptlydropswhenR =R andallhalosen- of this strength. For W =2 Å, for example, halos between
⊙ sh g r
ter the hot phase. For W =1 and 2 Å, the peak in the oc- 1011.6 and 1013.2 h- 1M⊙ halos do not produce strong ab-
cupationfunctionsoccurart1011.7 h- 1M and1011.5 h- 1M , sorbers. This bimodal distribution is what causes ˆb to in-
⊙ ⊙ W
respectively. Thistrendisduetotheinside-outnatureofthe creaseforW >3Å,while logM remainsconstantinFigure
r
h i
heating;higher-W systemsareproducedatsmallerimpactpa- 2;biasincreasesnon-linearlywithmass,weightinghigh-mass
r
ˆ
rametersand the shockradiusenvelopesthatradiusatlower halosmoreinb thanin logM .
halomasses. Above1012h- 1M⊙,theoccupationhasatailout In summary,Wthe transihtion miodel can simultaneously re-
to1014h- 1M⊙. Inthebest-fitmodel,κ4=0.67,implyingthat produce the observed frequency data and bias data of MgII
asignificantfractionofcluster-sizedhaloscontainsomecold absorbers.Theanti-correlationbetweenW andMinthetran-
r
gas with a high covering fraction. However, we note again sition modelis producedthroughtwo mechanisms. First, as
thattheconstraintonκ4 ispoordueto thelow frequencyof the shock propagates outward from the center of a massive
suchmassivehalos.