Recovering the topology of the IGM at z~2

1.7 MB·English

by  S. Caucci

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed3February2008 (MNLATEXstylefilev2.2)
Recovering the topology of the IGM at z∼2
S. Caucci1, S. Colombi1, C. Pichon1,2, E. Rollinde1, P. Petitjean1, T. Sousbie1,2
1Institutd’AstrophysiquedeParis&UPMC,98bisboulevardArago,75014Paris,France
2CentredeRechercheAstrophysiquedeLyon,9avenueCharlesAndre,69561SaintGenisLaval,France
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3February2008

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ABSTRACT

8
2 We investigate how well the 3D density field of neutral hydrogen in the Intergalactic

Medium(IGM)canbereconstructedusingtheLyman-αabsorptionsobservedalonglinesof
]
h sighttoquasarsseparatedbyarcmindistancesinprojectiononthesky.Weusecosmological
p hydrodynamicalsimulations to compare the topologies of different fields: dark matter, gas
- andneutralhydrogenopticaldepthandtoinvestigatehowwellthetopologyoftheIGMcan
o
berecoveredfromtheWienerinterpolationmethodimplementedbyPichonetal.(2001).The
r
t globalstatisticalandtopologicalpropertiesoftherecoveredfieldareanalyzedquantitatively
s
through the power-spectrum, the probability distribution function (PDF), the Euler charac-
a
[ teristics,itsassociatedcriticalpointcountsandthefillingfactorofunderdenseregions.The
localgeometricalpropertiesofthefieldareanalysedusingthelocalskeletonbydefiningthe
1
conceptofinter-skeletondistance.
v
Asaconsequenceofthenearlylognormalnatureofthedensitydistributionatthescales
5
underconsideration,thetomographyisbestcarriedoutonthelogarithmofthedensityrather
3
3 thanthedensityitself.Atscaleslargerthan∼1.4hdLOSi,wherehdLOSiisthemeanseparation
4 betweenlinesofsight,the reconstructionaccuratelyrecoversthe topologicalfeaturesofthe
. large scale density distribution of the gas, in particular the filamentary structures: the inter-
1
skeleton distance between the reconstructionand the exact solution is smaller than hdLOSi.
0
At scales larger than the intrinsic smoothing length of the inversion procedure, the power
8
0 spectrumoftherecoveredHI densityfieldmatcheswellthatoftheoriginaloneandthelow
: ordermomentsofthePDFarewellrecoveredaswellastheshapeoftheEulercharacteristic.
v TheintegralerrorsonthePDFandthecriticalpointcountsareindeedsmall,lessthan20%for
i
X ameanlineofsightseparationsmallerthan∼2.5arcmin.Thesmalldeviationsbetweenthe
reconstructionandtheexactsolutionmainlyreflectdeparturesfromthelog-normalbehaviour
r
a thatareascribedtohighlynon-linearobjectsinoverdenseregions.
Keywords: methods:statistical,hydrodynamicalsimulations–cosmology:large-scalestruc-
turesofuniverse,intergalacticmedium–quasars:absorptionlines
1 INTRODUCTION Cenetal.1994,Petitjeanetal.1995,Miralda-Escude´ etal.1996,
Theunsetal.1998,Viel,Haehnelt&Springel2004).
Aswe willshow inthe firstpart of this work, the statistical
Thestructureandcompositionoftheintergalacticmedium(IGM) andtopologicalpropertiesoftheIGMandofthedarkmatterdis-
has long been studied using the Ly-α forest in QSO absorp- tributions are the same, so that recovering the three-dimensional
tion spectra (Rauch 1998). The progress made in high resolution distributionandinferringthetopologicalpropertiesoftheIGMal-
Echelle-spectrographshasledtoaconsistentpictureinwhichthe lowsustoconstrainthepropertiesofthedarkmatterdistributionas
absorption features are related to the distribution of neutral hy- well.
drogenthroughtheLymantransitionlinesofHI.Hydrogeninthe Although topological tools have been introduced only rela-
IGMishighlyionized(Gunn&Peterson,1965). Itsphotoioniza- tively recently incosmological analysis, they have been used ex-
tion equilibrium in the expanding IGM establishes a tight corre- tensivelytocharacterizethetopologyoflargescalesstructuresas
lation between neutral and total hydrogen density and numerical revealed by the three-dimensional distribution of galaxies in the
simulationshaveconfirmedtheexistenceofthiscorrelation.They local universe (see for exemple Gott et al. (1986), Vogeley et al.
havealsoshownthatthegasdensitytracesthefluctuationsofthe (1994),Protogeros&Weinberg(1997),Tracetal.(2002),Parket
DMdensityonscaleslargerthantheJeanslength(seeforexample al.(2005)andSousbieetal.(2006)forthetopological analysisof
2 SaraCaucci et al.
galaxy surveys). The outcome of such an analysis is a quantita- improvements of the method as well as observational constrains
tive description of the complex appearance of the distribution of fromfuturesurveys.
thematter intheuniverse, with itsnetwork of clump, voids, fila-
ments and sheet-like structures. The study of the topology using
galaxysurveysisattractivebecauseoftheirlargevolumeandthe
2 THEEULERCHARACTERISTIC:ANALTERNATE
huge number of objects they contain. However the clustering of
CRITICALPOINTCOUNT
highlynonlinearobjects(galaxies,clustersofgalaxiesorQSOs)is
biasedcompared totheunderlying clusteringofdarkmatterfluc- Thispapermakesuseofvariousstatisticaltools,namelythePDF,
tuationsthatwewishtoconstrain(seeKaiser1984).Thisbiasing the Euler characteristic, the skeleton and related estimators such
resultsfromacomplicatedanddelicatecompetitionbetweenava- as the first cumulants of the PDF (connected moments), critical
rietyofprocesseswhichareoftentoocomplicatedtobetractable pointcountsandthefillingfactor, tocharacterizethetopology of
analytically. Besides, the maximum redshift in surveys is low (in the large scale density distribution. These tools will also be used
theanalysisoftheSDSSdatamadebyParketal.2005,themax- totesttheefficiencyofreconstructingthedensityfieldfromagrid
imumredshiftisz = 0.1654),sothatthiskindofanalysiscanbe of QSO sight-lines and in particular the ability to reproduce the
doneonlyinthelocalUniverse,wherethefluctuationshavealready connectivityofthelargescalestructures.
enteredthehighlynon-linearregime. FollowingColombi,Pogosyan&Souradeep(2000,hereafter
Given the strong correlation existing between dark matter CPS),thissectionintroducestheEulercharacteristic,χ˜+,asanal-
distribution and the low-density intergalactic medium, one could ternatecriticalpointcountinanoverdenseexcursionwithdensity
probe the underlying distributionof matter via thesignature pro- contrastslargerthanathresholdδTH.Itisshownhowthebehavior
duced by diffuse hydrogen in quasar spectra, namely absorption ofχ˜+ isrelatedtoconnectivityinthefield.Thenumericalimple-
features observed in the Ly-α forest. Indeed, absorption spectra mentationusedtomeasureitisdescribedandtestedonGaussian
provideapicturecomplementarytothosedrawnbygalaxysurveys randomrealizations.
to infer the large scale distribution of the matter in the Universe,
sincetheabsorptionfeaturesproducedbytheIGMinLy-αforest
2.1 DefinitionoftheEulercharacteristic
canbedetectedalsoatlargeredshiftandsincetheIGMprobesthe
lowdensityrange,whereasthegalaxydistributiondoesnot.Even- Letδ(x)beascalarfunctiondefinedina3DvolumeV.Givena
tually, higher density contrasts can be recovered from the analy- thresholdvalueδTH,considertheexcursionsetE+formedbythe
sisoftheLy-αforestifhigherordertransitionsareincludedinthe pointsxwithδ(x) δTH,asexpressedbythefollowingequation:
analysis;forexample,theLy-βtransitionsshouldallowustoprobe ≥
densitycontrastsuptoδ 15. E+ xδ(x)>δTH . (1)
≈ ≡{ | }
The flux along a single line-of-sight towards a quasar only Theanalysisofthegeometricalpropertiesofpointsthatbelongto
providesonedimensionalinformation,whichcanbeusedtocon- theexcursionsetE+asafunctionofδTHgivesinformationabout
strainthefluctuationamplitude andthematterdensity(Nusser & theglobaltopologyofthescalarfieldδ(x)andallowsforthechar-
Haehnelt1999,Rollindeetal.2001,Zaroubietal.2006).Thetrans- acterizationoflargescalestructures.
verseinformation,foundinpairsofquasars,hasbeenusedtostudy Asimplequalitativelinkcanbeestablishedbetweenthedis-
the extension of the absorbing regions (e.g. Petitjeanet al. 1998; tributionofcriticalpoints(definedby δ=0),aontheonehand,
∇
Crotts & Fang 1998; Young, Impey & Foltz 2001; Aracil et al. andconnectivityontheother,whicharerelatedtolocalandglobal
2002) and thegeometryof theUniverse atz 2(Hui,Stebbins propertiesoftheexcursionsetrespectively.Ifoneconsidersover-
∼
&Burles1999;McDonald&Miralda-Escude´1999;Rollindeetal. denseregions,connectivityhappensalongridges(filaments)pass-
2003,Coppolanietal.2006). ingthroughsaddlepointsandconnectinglocalmaxima.Thesame
Givenasetoflinesofsight(LOSs)towardagroupofQSOs reasoningcanbeappliedtounder-denseregionswhereminimaare
with small angular separation, inversion methods can be used to connected through tunnels (pancakes) via another kind of saddle
recover the three-dimensional distribution of low density gas, as point.ThisideaisinfactsupportedonrigorousgroundsbyMorse
demonstratedinPichonetal.(2001).Theyshowedthatthevisual theory(seeMilnor1963).TheMorsetheoremestablishesthelink
characterizationofthedensityfield(withitsnetworkoffilaments, betweenthedistributionofcriticalpointsandtheglobalconnectiv-
clumps, voids and pancakes) is correctly reproduced if the mean ityoftheexcursionset,viatheEulercharacteristic.Thisquantity
separationbetweentheLOSsislessthan dLOS 5Mpc. representstheintegraloftheGaussiancurvatureoveraniso-density
h i≤ surfacethatmarkstheboundaryoftheexcursionset(seeforexem-
Inthispaperwetestquantitativelywhethersuchaninversion
ple Gott, Melott & Dickinson, 1986). It is usually defined as the
can recover the global properties of connectivity of the density
followingcount(seeforexampleMecke,Buchert&Wagner,1994,
field, using topological tools such as the Euler characteristic and
fordetails):
theprobabilitydistributionfunction.
Thepaperisorganizedasfollows.InSection2,theEulerchar- χ˜+ =connectedcomponents tunnels+cavities. (2)
−
acteristicisdefinedasanalternatecriticalpointscountandimple-
According toMorse theorem, itcanalsobeexpressed asalinear
mentedforaGaussianfield.Thedifferencebetweenthetopolog-
combinationofthenumberofcriticalpointsofdifferenttypesthat
ical properties of the dark matter, of the gas and of the observed
optical depth is then discussed using outputs of a hydrodynami- arefoundintheexcursionsetasafunctionofδTH.
Tobemorespecific,letusconsider thecriticalpointsofthe
calsimulation(Section3)andrelyingondifferentstatisticaltools.
field.Forthesepoints,theHessianmatrix,whosecomponentsare
In Section4, the abilityto reconstruct the global topology of the
givenby:
threedimensionaldistributionfromasimpleWienerinterpolation
ofadiscretegroupoflinesofsightisconsidered.Finally,Section5 ∂2δ
= , (3)
summarizes theresultsof thispaper and discusses some possible Hi,j ∂xi∂xj
The topologyof IGM athighredshift 3
0.04 1.0
Minima
Pancakes
0.02 old 0.8 Filaments
h
s
e Maxima
0.00 hr
e t 0.6
h
c +−0.02 e t
v
o
b 0.4
a
−0.04 n
o
cti
−0.06 Fra 0.2
−0.08 0.0
−4 −2 0 2 4 −4 −2 0 2 4
d /s d /s
Figure1.MeanEulercharacteristics, χ+ (seeEqs.4and6),foraGaus- Figure2.Evolutionofthenumberofcriticalpointsenteringthecomputa-
sianrandomfield(GRF,pointswithsmallerrorbars)asafunctionofthe tionoftheEulercharacteristicfortheGRFconsideredinFig.1.Thefrac-
densitythreshold,δ/σ,comparedtothetheoreticalprediction(Doroshke- tionofdifferenttypesofcriticalpointsabovethethresholdisplottedasa
vich,1970,smoothcurve).Themeaniscarriedover5realizationsofaGRF functionofδ/σandeachdistributioniscomparedtotheanalyticalpredic-
whosepowerspectrumisgivenbyapower-lawwithspectralindexn= 1 tion.Again,symbolswitherrorbarsrepresentthemeanover5realizations
ona2563 grid,whileadditionalsmoothingisperformedwithaGauss−ian ofthesameGRF,while thesmoothcurves give theanalytical prediction
windowofsize5pixels. (whichcanbeeasilyderivedfromBardeenetal.1986).
2.2 InterpretationoftheEulercharacteristics
is calculated and its eigenvalues are estimated. According to the
Forclarity,letusrecallheretheinterpretationoftheshapeofthe
number of negativeeigenvalues, I,of the Hessianmatrix, thelo-
Eulercharacteristicasafunctionofdensitythreshold(CPS).Letus
calstructuresofthefieldcanbeclassifiedinthefollowingway:a
firststudythesimplecaseofaGaussianrandomfield(GRF).The
clump, a filament, a pancake and a void corresponding to I =3,
analyticpredictionsforaGRFaregiven,forexample,inDoroshke-
2, 1 and 0 respectively. The Morse theorem states that the Euler
vich(1970) (see also, Schmalzing& Buchert 1997). Inwhat fol-
characteristiccanbeexpressedasacountofthenumberofcritical
lows, a slightly different normalization from Eq. (4) is used: the
pointsbelongingtoeachofthesefourclasses:
volumeindependentquantity
χ˜+ =NI=3−NI=2+NI=1−NI=0, (4) χ+ =χ˜+/Ntot, (6)
whereNI=iisthenumberofcriticalpointswithinegativeeigen- whereNtot = iNI=iisthetotalcriticalpointcountinthevol-
values.Withthisapproach,itissufficienttodeterminethenumber umeconsideredP,inthelimitδTH .
→−∞
distribution of the four kinds of critical point. However this dif- InFigure1,thenumericalestimatesoftheEulercharacteristic
ferentialmethodrequiresthefieldunderconsiderationtobesuffi- aregivenasafunctionofthedensitythresholdδTH =δ/σwhere
cientlysmoothandnondegenerate.Tothisend,inthesubsequent δ = (ρ ρ¯)/ρ¯isthedensitycontrastandσ = rms(δ)fromfive
−
analysesofthispaper,thefieldwillbesmoothedwithaGaussian Gaussianrandomfield(GRF)realizations(pointswitherrorbars)
window(usingstandardFFTtechnique), whosepowerspectrumisgivenbyapower-lawwithspectralindex
n = 1, i.e. P(k) kn. The result is compared withthe ana-
1 r2 lyticp−rediction(solid∝line).Theshapeofthecurveasafunctionof
W(r)= exp , (5)
(2π)3/2L3s (cid:18)−2L2s(cid:19) thedensitythresholdcanbeunderstoodfromEquation(4)andFig-
ure2whichdisplaysthecriticalpointcounts. Atverylowvalues
of sufficiently large size L compared to the sampling grid pixel ofthethreshold(δ/σ< 4)theexcursionsetincludesalmostall
s
−
size in order to minimize the impact of numerical artefacts com- pointsand,duetothe∼symmetrybetweenhighandlow-densityre-
ing from the discretization of the field on a grid (see, e.g., CPS gions,thenumberofminima(pancakes)compensatesthenumber
for athorough analysis of measurement issues). In what follows, of maxima (filaments)so that theEuler characteristicapproaches
thesmoothingscaleusedtomeasuretheEulercharacteristicisal- zero. When the value of the threshold is increased, local minima
wayslargerthan0.01 NpixgridpixelswhereNpix=256pixels firstdropoutofE+,creatingcavitiesandthusincreasingthevalue
istheboxresolutiono×f thesimulations. Inprinciplethissmooth- ofχ+.Atδ/σ∼> 2,pancakesstarttodropouttooandcavities
ingscaleislargeenoughtohaveanunbiasedmeasurementofthe connecttogether,th−usthevalueofχ+decreases,reachingitsmini-
Eulercharacteristic.TheprescriptionofCPSisusedtodetectand mumatδ/σ 0.Intherange0∼<δ/σ∼<2filamentsalsodropout,
≈
classifycriticalpoints.Thismethodinvolveslocallyfittingasec- breaking up the ridges to create isolated clusters, thus increasing
ondorderhypersurfaceonthesmootheddensityfield,whiletaking χ+ again.Finallyintheregionδ/σ∼>2onlyclumpsarefoundto
intoaccounteachpointonthegridunderconsiderationandits26 lieintheexcursionset,buttheyareprogressivelylostasthethresh-
neighbors. oldincreases,explainingthefinaldecreaseofthecurve.
4 SaraCaucci et al.
ThissimpleanalysisshowshowthefeaturesseenintheEuler Bouchet&Schaeffer1994)andtheEulernumber(see,e.g.,CPS).
characteristic are closely related to the network of filaments and Forthereconstruction,thetypicalseparation dLOS betweenlines
h i
pancakesthatconnectclumpsandvoids. ofsightdefinesanaturalsmoothingscaleLs dLOS .Notethat,
≃h i
unfortunately, the upper bound of Ls 4 Mpc corresponds to
∼
a lower bound on dLOS in current state of the art observations
h i
(Rollindeetal.2003),butonecanexpecttolowerthislimitinfu-
3 FROMDARKMATTERTOOPTICALDEPTH
turesurveys(Theuns&Srianand2006).Hencethefollowinganal-
TheLyman-αabsorptionlinesobservedinQSOsspectraandpro- ysesareperformedintherange2Mpc Ls 4Mpc.
duced by the HI structures intercepted by the line-of-sight can ≤ ≤
be used to study the topology of the Universe at high redshift
(z∼>1 2). However, theinformation derived fromobservations 3.2 PDFandEulercharacteristicofphysicaldensityfields
−
of QSOsspectra is more directly related to the HI optical depth,
ItthisSectionwecomparethelarge-scaledistributionandthetopo-
whereas here the aim is to constrain the underlying dark matter
logicalpropertiesofthedifferentdensityfields(darkmatter,total
densityfieldforwhichtheorymakesdirectpredictions.Hence,one
amountofgasandneutralgas)bylookingattheirprobabilitydis-
hastorelyonsimulationsinordertocalibratetherelationbetween
tribution function (PDF) and their Euler characteristic (χ+). Our
thedensityfieldoftheneutralhydrogenandthatofthedarkmatter.
knowledge of the physics of the intergalactic medium is used to
Inthissectionwefirstpresentthehydrodynamicalsimulations
usedinthepresentworkandwethenanalysetheshapeofthePDF performtheanalysisandtolinkthedistributionsofHIandH.In-
andtheEulercharacteristicofthethreedensityfields(darkmatter, deed,theobservationsgiveaccesstotheHIopticaldepththrough
absorptionspectra.Wealsoconsiderthermalbroadeningandred-
gas and HI) and of the optical depth field, explaining how these
shiftdistortioneffects.
curvesarerelated.
3.1 Numericalsimulations 3.2.1 FromdarkmattertoHI:IGMequationofstate
Weanalyseacosmologicalhydrodynamicalsimulationthatevolves ItiswellknownthatonscaleslargerthantheJeanslengththedis-
both dark matter particles and a gaseous component to study the tributionofthegasfollowsthedistributionofdarkmatter,sothat
globaltopologyoftheintergalacticmediumatredshiftz =2.The their statistical and topological properties are expected to be the
dynamicalevolutionandthephysicalpropertiesofthegasandthe sameat thesescales. Thisischecked by comparing thePDFand
of the HI component are calculated taking into account the heat- the Euler characteristic of the two density fields smoothed using
ingandcoolingprocessesandtheeffectoftheionizingUVback- differentvaluesofLs,asshowninFigs.3and4.Notetheagree-
groundinastandardway.ThecorrespondingParticle-Mesh(PM) mentofthePDFsandoftheEulercharacteristicsofthetwofields
codeusedtoperformthesimulationisdescribedindetailinCop- forallvaluesofthesmoothingscaleconsidered,aresultwhichcan
polanietal.(2006). be expected since the scaling regime probed is largely above the
Inthisrun,thestandardΛCDMmodelisassumedwithaset Jeanslengthofthegas.
ofcosmological parametersgivenby: Ωm = 0.3andΩΛ = 0.7, Thecomparisonofthedistributionoftheneutralgas(HI)with
whiletheassumedbaryondensityisΩ =0.04.TheHubblecon- thatofthetotalamountofgasandthedarkmattercallsforaslightly
b
stantisH0 = 70kms−1Mpc−1 andtheamplitudeofthefluctu- more elaborate approach, given the non-linearity involved in the
ationsofthematterdensityfieldinasphereofradius8h−1 Mpc expressionthatrelatesthedistributionofthegastothedistribution
isσ8 =1.Whiletheothercosmologicalparametersareroughlyin of HI.In fact, numerical simulationssupport theidea that atight
agreementwithrecentobservationalconstraints,thevalueofσ8is correlationexistsbetweenneutralandtotalhydrogendensity(Cen
somewhat largecompared tothevaluesuggested byWMAP(see etal.1994, Miralda-Escude´ et al.1996, Theunset al.1998, Viel,
Spergeletal.,2007).However,thisshouldnothaveanyincidence Haehnelt&Springel2004).Thiscorrelationisexpectedtofollow
ontheresultsderivedinthispaper. apower-lawoftheform
withTphereiosdimicublaotuionndairnyvoclovneddit5io1n2s3odfarckommoavttienrgpsairzteiclLesin=ab4o0x ρgas ≈A·(ρHi)α. (7)
box
Mpc. The gaseous component was also followed on a 5123 grid Wethusintroducehereanewdensityfieldρ˜Hidefinedastheright-
whichwasusedtocomputegravitationalforces.Althoughthissim- hand-sideofEquation(7),sothatρ˜Hi A (ρHi)α.Inwhatfollows
≡ ·
ulationmarginallyresolvestheJeanslengthofthegas,Coppolani thisnewdensityfieldwillbeusedinordertoapproximatetheden-
et al. (2006) checked with higher resolution runs that numerical sityofthegas.However,Equation(7)isnotfulfilledinthewhole
convergencewasachievedatsmallscales. rangeofρHivalues.
Although5123gridpointswereavailable,thisresolutionwas Toillustratethis,Fig.5(a)displaysthegasdensitydistribution
degraded to a 2563 resolution (using standard donner cell pro- (top),withitsnetworkoffilamentsoutlinedintheleftpanelwith
cedure), in order to make the calculations more tractable. Obvi- acontourcorrespondingtoδ=1,andthetemperaturedistribution
ouslythisadditionalsmoothingmakestheeffectsofsubclustering inunitsof 104 K (bottom) for which wehavedrawn thecontour
within the Jeans length irrelevant. Therefore the gaseous compo- corresponding to (T/104) = 2 intheleft panel. Notethat along
nentshouldbenearlyindistinguishablefromthedarkmattercom- filamentsandattheirintersectionthegasishot.Thisindicatesthat
ponent. shockwavespropagatealongfilaments,risingthetemperatureand
Themainlimitintheseanalysesremainstheboxsize,which ionizing thegas. Thisisconfirmed by Fig. 5(b) which shows the
isstillsmallandonlyallowsforafairstatisticalmeasureatscales ratioR = ρ˜HI/ρgas measureddirectlyinthe2563 grid(top),and
LslargerthanLmax Lbox/10,i.e.4Mpc.Indeedfinitevolume afteraGaussiansmoothingwithawindowwhosesizeisLs =2.2
effects are known to∼become significant for Ls>Lmax for stan- Mpc(bottom).Inbothcases,thepanelsontheleftshowthecon-
dardstatisticssuchastheprobabilityfunction(se∼e,e.g.,Colombi, toursrelativetoR = 0.7.Tocompletethepicture,letusconsider
The topologyof IGM athighredshift 5
0.05 0.1 0.2 0.5 1.0 0.1 0.2 0.5 1.0 2.
3
Ls = 1.6 Mpc 1.5
2 1.0
1 0.5
F 0 0.0
D
P 1.5 Ls = 2.2 Mpc Ls = 3.6 Mpc 1.5
1.0 1.0
0.5 0.5
0.0 0.0
0.1 0.2 0.5 1.0 2. 0.1 0.2 0.5 1.0 2.
1+d
Figure3.Probabilitydistributionfunctionofdensityfieldsatdifferentsmoothingscales(fromlefttoright,toptobottom,nosmoothing,Ls=1.6,2.2and3.6
Mpc).Thethicksolid,thickdashedandthinsolidcurvescorrespondtodarkmatter,gasandHI(rescaledaccordingtoEq.7),respectively.Thedottedcurveis
abestfitofalognormaldistributiontothethinsolidcurve,showingthatallthesePDFsarereasonablyclosetolognormal,apropertythatwillbeusefulfor
thereconstruction.Intheunsmoothedcase,thegasandHIPDFsmatchverywellfor1+δ∼<1butdepartfromeachotherathigherdensity.Theapparent
verygoodmatchintheunsmoothedcasecomesfromthefactthattheun-shockedpartoftheintergalacticmediumtotallydominatesthepartofthePDFwhich
isvisibleinthispanel.ThematchbetweenHIandgasPDFsdecreaseswithincreasingsmoothingscale,dueto“mixingeffects”,asexplainedinthemaintext.
Notefinallythatthedarkmatterisnotdisplayedintheunsmoothedpanelbecausetheresultwouldbecontaminatedbythecloud-in-cellinterpolationusedto
computethedensityonthegrid.
Figure4.SameasinFig.3butforthemeasuredEulercharacteristicasafunctionofdensitythresholdatdifferentsmoothingscales.Again,thethicksolid,
thickdashedandthinsolidcurvescorrespondrespectivelytodarkmatter,gasandHI(rescaledaccordingtoEq.7withthevalues(A,α)giveninTable1).
Whilethecurvesforthedarkmatterandforthegassuperposeexactlyatallsmoothingscalesandallvaluesofthethreshold,theHI,evenafterthescalingis
applied,behavesinadifferentwayinthehighdensityregion.Asexplainedinthetext,thisisaconsequenceofthepresenceofshocksandcondensedobjects,
whoseeffectisachangeinconnectivityproperties.
6 SaraCaucci et al.
18.95
y
nsit 12.63
e
d
s
a
G 6.32
0.00
9.47
e
r
u
at 6.32
r
e
p
m
e 3.16
T
0.00
(a) Gasdensityandtemperature
0.95
d
e
h
ot
o 0.63
m
s
n
U
− 0.32
R
0.00
0.95
c
p
M
2 0.63
2.
=
s
L 0.32
−
R
0.00
(b)RatioR=ρ˜HI/ρgas
Figure5.Top:Gasdensityandtemperature(inunitsof104K)spatialdistributionsinaonepixel( 15.6kpc)slice.Theintensityofthefieldsiscolor-coded
withthescalegivenontheright.Thepanelsontheleftgivethecontourscorrespondingtoδ =≈1forthedensityandtoT/104 = 2forthetemperature.
Bottom:ForthesamesliceasaboveweshowontherightthespatialdistributionoftheratioR=ρ˜HI/ρgasfortheunsmoothedfield(up)andforthefield
smoothedwithaGaussianwindowofsize(FWHM)Ls=2.2Mpc(down).ThecolorscaleissuchthatdarkerregionscorrespondtolowvaluesofR.Onthe
leftthecontourscorrespondtoR=0.7.
Fig.6whichshowsthescatterbetweenρgas andρHi fordifferent inaqualitativeway,bytheslicesshowninFig.5(b).Morequantita-
smoothingscales,asindicatedineachpanel.Asexpected,thetight- tively,forthefieldsRshowninFigure5(b)wehavecalculatedthe
nessofthecorrelationisveryhighinunderdenseandmoderately fractionofthevolumeoccupiedbytheregionswithR < 0.7(i.e.
dense regions, but shock heating on theone hand and theforma- thevolumeoftheregionsenclosedbythecontoursintheleftpan-
tionofcondensedobjectsontheotherproduceasignificantscatter els).Forthesliceshown,thisfractionisf(R < 0.7) = 0.07and
(whereR<1)alongdensestfilamentsandattheirintersection(in f(R<0.7)=0.19fortheun-smoothedandthesmoothedcasere-
clusters).Forthepurposeofthereconstruction,somesmoothingis spectively,whilewhenthewholethreedimensionalboxesarecon-
required. Unfortunately, smoothing also mixestheseregions with sidered, the fraction of volume occupied by shocked regions are
theun-shockedpartoftheintergalacticmedium.Thisisconfirmed, f(R < 0.7) = 0.05andf(R < 0.7) = 0.11.Asaresultofsuch
The topologyof IGM athighredshift 7
Figure6.ScatterplotsdisplayingtherelationbetweenthegasdensityandtheHIdensityatdifferentsmoothingscales.Thedashedblacklinesineachpanel
representthebestfit,followingEq.(7),withtheparameters(A,α)giveninTable1.Notethatthedispersionincreaseswhenthesmoothingscalesincreases,
duetothemixingeffectdiscussedinFigure5.
smoothing length (i.e. going from leftto right inFig. 4) worsens
Ls A α thematch, asexpected, but thisisinpart lost inthenoisedue to
finitevolumeeffects.Notethatχ+measuredinHIis,intheδ>0
Unsmoothed 1.275 0.63
regime,morepeakedthanforthetotalgas.Thisagreeswithintu-
1.6Mpc 0.915 0.56879
ition,sincegalaxies forminfilaments:inthesehighly condensed
2.2Mpc 0.85 0.55209
objects,gasconcentratesandcoolsdown.HencetheHIdensitybe-
3.6Mpc 0.795 0.5389
comessignificantagaininsidetheseclumps,butisdepletedintheir
surroundingsduetoshockheatingascanbeseenfromFig.5.The
Table1.ValuesoftheparametersAandαenteringinthescalingrelation resultingdistributionofHIinfilamentsisthereforeexpectedtobe
betweengasandHI(seeEq.7)asafunctionofthesmoothingscaleLs. more clumpy than thetotal gasi.e. less efficientlyconnected, re-
sultinginalargerincreaseofχ+ forδ > 0.Theestimatesmade
insidefilamentsare however certainly not freeof numerical arte-
mixing, the tightness of the correlation is weakened, but remains facts since they are limited by the simulation’s spatial resolution
good as shown in Fig. 6. However, the best fit values of the pa- (followingaccuratelytheformationofcondensedobjectsrequires
rametersAandαchangesslightlywhenthefieldissmoothed(see muchhigherspatialresolutionthanoursimulation).Thereforeal-
Table1).WefitthesevalueswiththelowdensitytailofthePDF thoughonecandefinitelytrusttheδ<0measurements,theresults
(seeFigure3).Asexpected,thehigherdensitytailmatchworsens derivedforδ > 0arelikelytoyield∼therightqualitativebehavior,
withsmoothing. butarecertainlyquantitativelybiased.
Given that the scaling relation (7) is monotonous, should it
apply exactly, the topology of the neutral gas should be exactly
the same as of the total gas/matter distribution. However, given
thedispersionofthisrelation,oneexpectstheEulercharacteristic InthenextSections,theδ > 0disagreementwillbeignored
of the ρ˜Hi fieldtodepart fromthat of ρgas for largedensity con- anditwillbeassumedthatthescalingrelation(7)isalwaysvalid,
trasts. This is confirmed by Fig. 4: a nearly perfect agreement is keeping inmindthelimitationof suchanassumption. Hence,re-
foundbetweenthegasandHIforδ<0,whiledifferencesbecome constructions will be performed on the optical depth without at-
significant at larger values of the d∼ensity contrast. Increasing the temptingtodirectlyrecoverthegasdistribution.
8 SaraCaucci et al.
Figure7.TheeffectofredshiftdistortionontheHIdensity.Thesameslice(whosewidthis6pixels,correspondingto0.94Mpc)oftheHIdensitycontrastis
shownwithoutdistortion(leftpanels)andwithdistortion(rightpanels,usingtheinfinitelydistantobserverapproximationwithadistortionalongthexaxis)
inthecasewherethefieldsarenotsmoothed(toppanels)andwhenthefieldsaresmoothedatLs=1.6Mpc(bottompanels).
Figure8.EffectofredshiftdistortionontheEulercharacteristicofHIatdifferentsmoothingscales:Ls =1.6Mpc,2.2Mpcand3.6Mpc.Thesolidblack
lineisforHIwithoutanydistortions,thedashedyellowlinehasbeenobtainedbyincludingonlytheeffectofpeculiarvelocities,whilefortheredthinline
bothredshiftdistortionandthermalbroadeningaretakenintoaccount.
3.2.2 FromHItoopticaldepth:redshiftspacedistortionsand wheredistortionsinducedbythepeculiarmotionsoperate.More-
thermalbroadening over,theprofilesofabsorptionlinesarebroadenedatsmallscales
by the effect of the temperature. Sincethe thermal broadening is
Inthe above discussion we argued that the mainfeatures of dark importantonlyatscalesofthesameorderorsmallerthantheJeans
mattertopology,astracedthroughtheEulercharacteristicandthe length,thissecondeffectshouldbenegligeableinthescalingrange
probability density function, can be recovered through the topol- consideredinthispaper,sinceitwillbesweptoutbythesmooth-
ogyoftheHIforsmalldensitycontrasts,δ<0.However,alonga
line-of-sight,theopticaldepthisinfactobse∼rvedinredshiftspace,
The topologyof IGM athighredshift 9
ing.Onthecontrary,redshiftdistortionshouldapriorinotbene- (LOS)towardstheQSO.However,ifasetofLOSstowardsagroup
glected. ofquasarsisavailable,theinformationalongeachLOScanbein-
Intheory,itispossibletopartiallycorrectforredshiftdistor- terpolatedtoconstructa3-dimensionalopticaldepthfield.
tioneffects(seeforinstancePVRCP).However,thecorresponding InthisSectionwefirstbrieflyoutlinetheinversiontechnique
treatmentofthepeculiarvelocitiesinvolvesasimultaneousdecon- implementedtorecovertheopticaldepthanddescribeshowtoset
volutionoftheHIdensityfieldwiththevelocityfieldontopofthe theparameters that enter the inversion procedure. Wethen check
inversiondiscussedinnextsection.Thisrequiresnotonlyaprior how the reconstruction performs by measuring various statistical
forthedensityfieldbutalsoforitscorrelationwiththepeculiarve- quantities,inparticularthePDFofthedensityfieldanditsEuler
locityfieldandmakestheinversionquiteconvolvedandthiswould characteristic. As argued in the previous Section, the focus is on
gobeyondthescopeofthispaper.Inwhatfollows,itisshownthat the optical depth: no attempt is made to recover the gas or dark
infactredshiftdistortionshaveasmalleffectonthetopologyofthe matterdistributiondirectly.Thermalbroadening,redshiftdistortion
overalldensitydistributionfortheprobedscales;theyshallthusbe andeffectsofsaturationorinstrumentalnoiseareneglected.Given
neglected in the reconstruction part of this work. Moreover, one these assumptions, studying the optical depth distribution is then
of theinteresting outcomes of thereconstruction istopredict the equivalenttostudyingtheHIdensitydistribution,ρHi.
positionsoffilamentsinthethreedimensionalmatterdistribution.
Cross-correlation of such a distribution with for instance the ob-
serveddistributionof galaxiesathigh redshiftcan infact bealso
performedinredshiftspace.
4.1 Theinversionmethod:Wienerinterpolation
Figure7displaystheHIdistributioninrealandredshiftspace
withandwithoutsmoothing:themaineffectofredshiftdistortion Thetechnique used tointerpolate the optical depth fieldbetween
on HI is an enhancement of large scale density contrasts orthog- linesofsightisdescribedanddiscussedindetailsinPVRCP.
onally to the line of sight due to large scale motions (this is the LetDbea1-dimensionalarrayrepresentingthedataset(i.e.
so-calledKaisereffect,e.g.Kaiser1987):the“voids”(underdense thevaluesofγLOS = ln(ρHi)alongtheLOSs,whichweassume
regions)aremorepronounced,andthefilamentsorthogonaltothe tobeparalleltoeachother);wecallMthe3-dimensionalarrayof
LOSaremorecontrasted.Thereisaswellasmallscale“fingerof theparametersthatneedtobeestimated(herethevaluesofγ3D =
god”effect,duetointernalmotionsinsidelargedarkmatterhaloes, ln(ρHi) inthe 3-dimensional volume) by fittingthe data. Wiener
butitisnotverypronouncedatsuchahighredshift,andisinampli- interpolationreads(seeEq.20ofPVRCP),assumingthenoiseis
tudeofthesameorderofthermalbroadening.Notehowever,that uniformanduncorrelated,
nontrivialshellcrossingscanstilloccur,e.g.twofilamentscross-
ingeachotherthankstopeculiarvelocities,butthiseffectremains M=CMD (CDD+N)−1 D, (8)
· ·
small, and is clearly damped out by smoothing; after smoothing
onlytheKaisereffectremains. where N = n2I is the diagonal noise contribution, CMD is the
These qualitative arguments are illustrated by Figure 8. The mixedparameters-datacovariancematrixandCDDisthedataco-
measuredEulercharacteristicsbeforeandafterredshiftdistortion variancematrix:
differ only slightly. When the redshift distortion istaken intoac-
count,forδ<0,ashifttowardstheleftisinduced(dashedcurve) CMD =Cγ3DγLOS, CDD =CγLOSγLOS. (9)
ascompared∼tothenon-distortedcase(solidone);whiletheoppo-
Here an ad-hoc prior is used and a Gaussian shape for the co-
siteoccursfor δ>0(althoughinthelattercase, theeffect seems variances is assumed. In this cases the matrices C and
tobenearlyinsig∼nificant).Thisshiftremainsquitesmallasargued C aregivenby γ3DγLOS
before.Noteaswellthatthermalbroadening(thincurve)istotally γLOSγLOS
negligFeinaabllley., one last point should be mentioned. When one con- C(x1,x2,x1⊥,x2⊥)=σ2×exp − (x1−L2x2)2 ×
(cid:16) x (cid:17)
siders real absorption spectra, instrumental noise has to be taken x x 2
1⊥ 2⊥
intoaccountintheanalysis.Thisnoise,combinedwithsaturation exp | − | , (10)
− L2
ofthefluxoftheLy-αabsorptionlinesarisinginhighdensityre- (cid:16) T (cid:17)
gions(withδ>10) cancomplicate theinterpretation ofthemea- where(xi,xi⊥)representsthecoordinatesofthepointsalongand
surements.In∼thiscase,someoftheinformationabouttheintensity perpendiculartotheLOSsrespectively,L andL arecorrelation
x T
of the density field cannot be recovered, unless, say, Lyman-β is lengthsalongandperpendicular totheLOSs,whileσ2 quantifies
alsoavailable.Inthiswork,however,themaininterestlaysinre- thetypicalapriorifluctuationsinavolumeofsizeL L2.The
producingthelow-densitypartoftheHIdistribution,forwhichthe meaningandchoiceoftheseparameterswillbediscuxs×sedTfurther
relation(7)holdsandforwhichthetopologyoftheunderlyingdark in 4.2.
matterdistributionistheoreticallyconstrained.Inthisregime,the § Notethattheshapeofthecovariancematrixcanbecalculated
Ly-αlinesarenotsaturated, thusacompletetreatment ofsatura- withamoresophisticatedapproach.Thiswouldinvolvetheuseof
tioneffectsinhighdensityregionsisnotrequiredfortheaimofthe theoretical priors relying on our knowledge of large scale struc-
presentwork. ture dynamics. If, for instance, the reconstruction was performed
onthepuredarkmatterdensitycontrast,onecouldbetemptedto
derivethesecorrelationmatricesfromthenon-linear power spec-
trumobtained,forexample,byPeacock&Dodds(1996),givena
4 TOPOLOGICALANDSTATISTICALPROPERTIESOF
cosmologicalmodel.Here,asimplerinterpolationschemeisused.
THERECOVEREDFIELDS
Thisschemehastheadvantageofdependingonlyonthreetuning
The absorption spectrum towards a quasar gives access to one- parameters:theassumedtypicaloverallsignal-to-noiseratio,σ/n,
dimensionalinformationi.e.theopticaldepthalongthelineofsight andtwotypicallengthsintheinterpolation,L andL .
x T
10 SaraCaucci et al.
4.2 Choiceoftheparametersintheinterpolation
EachreconstructionisperformedonanumberNLOSofLOSsex- NLOS S(eapracrmatiino)n (MLpTc) σ
tracted at random from the simulation box. Since the distant ob-
server approximation is implemented, all the LOSs are parallel. 400 1.33 2 1.12
ForagivenvalueofNLOS,themeaninter-LOSdistance, dLOS , 320 1.49 2.24 1.17
reads: h i 225 1.77 2.67 1.23
200 1.88 2.83 1.25
hdLOSi≡pL2box/NLOS. (11) 114250 22.4.22 33..6352 11..3219
Thisparameterobviouslydefinesanaturalscaleinthereconstruc- 100 2.65 4 1.34
tion:onecannot,intuitively,expecttoreconstructdetailsofthedis-
tributionatscales <hdLOSi,atleastinthedirectionorthogonalto Table 2.Parameters used inthe reconstructions performed inthis paper.
theLOSs. ∼
Thelongitudinal correlation lengthasbeenfixedtothevalueLx = 0.4
ThemeaningoftheparametersL andL inEq.(10)isthen
T x Mpcforallthereconstructions.
quitestraightforward.ThecorrelationlengthsL=L andL=L
T x
stabilize the inversion by ensuring the smoothness of the recon-
struction.Inordertoavoidtheformationoffictitiousstructures,the tweenadjacentpatches.Thesizeofthebufferregionischosento
transversecorrelationlengthmustbeoftheorderofthemeansepa- benover 2LT (ingridpixelunits),whichimpliesatypicalresid-
≃
rationbetweentheLOSs,LT dLOS (wehavechosentotakeit ualcontaminationofedgeeffectsduetothepartitioningoflessthan
∼h i
exactlyequalto dLOS ),whilethechoiceofthelongitudinalcor- 2percent.
h i
relationlengthdependsontheproblemconsidered.Sinceredshift
distortionisnotaconcerninthepresentwork,thisparametercan
4.3 Biasinthereconstruction
bechosen tobeof theorder oftheJeanslengthinordertoavoid
informationlossforsmallscalesalongtheLOSs,hereLx = 0.4 Firstnotethattheinversionisnotdirectlyperformedonthedensity,
Mpc. butonitslogarithm,i.e.thenormalizeddensityfieldiswrittenas
Fromapracticalpointofview,thevarianceparameterσofthe ρHi = 1+δHi exp(γHi)andthefieldγHi isinterpolated by
correlationmatrixfixestherelativecontributionofsignaltonoise usingthemethod≡describedabove.Thishastwoadvantages:(i)it
in Eq. (8), σ/n. In our reconstruction, only ideal LOSs are con- ensures the positivity of the reconstructed density; (ii) since the
sidered. Thus, strictlyspeaking, there isno instrumental noise or densityturnsouttoberoughlylog-normal1 (seeforexampleBi&
saturationeffects.However,theinversionofthematrixCDD+N Davidsen,1997;Choudhury,Padmanabhan&Srianand,2001;Viel
isnumericallyunstablewhenNissettozero,giventhefinitesam- etal.,2002a;Zaroubietal.;2006,Desjacquesetal.,2007;andsee
plingandthedegeneracyofthematrix,Eq(10),closetoitsdiag- Coles&Jones,1991,forthestatisticalpropertiesofthelog-normal
onal, (x1,x1⊥) (x2,x2⊥). In practice one has to “tune” the distribution),performingthereconstructiononthelogarithmofthe
≃
signal-to-noiseratio,σ/n,toobtainthebestcompromisebetween field is expected to reproduce more realistic results as shown on
numericalstabilityand“exactness”ofthefinalreconstruction.This Fig.3.
choiceisad-hoc:(σ/n)2istheestimatedvarianceσ2(LT,Lx)of However, as a result of the reconstruction, the recovered
theunderlyingfieldinaboxofsizeLT ×LT ×Lx.Thisisequiv- field, γHi,rec, will be smooth over anisotropic volumes of size
alenttoassumingthat,asthenoisegoestozero,theinverseofthe L L L ,whichmeansthatatbest,onecanidentifystruc-
x T T
non-reduced second order correlation(intheappropriate units) is ∼turesa×tthis×levelofsmoothnessonalogarithmicspace.Although
used, I+CDD, instead of the reduced one, C−DD1, to perform a theoreticalpredictions(namelygravitationalclustering,primordial
stablereconstruction. nonGaussianities,etc.)doexistforthedensityfielditself,thatis
Inthisworkweestimatedirectlyσ2(LT,Lx)fromthesimu- forρHi = exp(γHi)anditssmoothed counterparts, theycannot
lation.Itishoweverimportanttonotethatσ2(LT,Lx)caninprin- beapplied directlyinour casebecause smoothing and taking the
ciplebederivedfromtheLOSsalonebymeasuringthe1-Dpower- exponential are operations which do not commute, except in the
spectrum of ρHi. Fromthis1-D power-spectrum, one can indeed weakly nonlinear regime, δHi 1. In particular, recovering the
infera3-Dpower-spectrumwithstandarddeconvolutionmethods resultsforδHionalinearspace≪,bytakingtheexponentialofγHi
andthenanestimateofσ2(LT,Lx)bytheappropriateintegralon andsubsequentlysmoothingit,aneffectivebias,essentiallydueto
the3-Dpower-spectrum. rarepeaksintheγHi,rec field,isintroduced. Theeffectof sucha
Themeasuredvaluesofσ(LT,Lx)arelistedinTable2.They nonlinearbiasisdifficulttocontrolandcaninsomecasesbeim-
areoftheorderunity:theassumedsignal-to-noiseratioisaboutone portantasshownbelowandstudiedinmoredetailsinAppendixA.
intheregimeconsideredhere.Hence,inpractice,thead-hocpro-
cedureusedtoperformtheinversiondoesnotchangesignificantly
byincludingthecontributionoftheactualinstrumentalnoise.How- 4.4 Testingthereconstruction:statisticalandtopological
ever,inthiscasethepresenceofthesaturatedregionsintheLyman- analysis
αspectraremainsaproblem.
Wenowtestthequalityofthereconstructionusingthesamestatis-
Finally, note that due to the large size of the matrices, re-
ticaltoolsasinSection3,namelythePDFofthefieldandtheEuler
constructionscanonlybeperformedbypartioningthesimulation
characteristic.Otherstatisticsareconsidered,suchasthevariance
box in blocks of smaller size, that contain N3 grid points with
sub and the skewness of the PDF, the power-spectrum of the density
N = 32. Thereconstruction isperformedoneachblockindi-
sub
vidually. In order to avoid edge effects, neighboring patches are
overlapped by adding a buffer region in which LOSs still con- 1 Notethatifafieldsuchasρgasislognormal,the(inverseofthe)trans-
tribute.In thisway, thea priori correlation ensure continuity be- formation(7)leavesthenewfield,e.g.ρHi,lognormalaswell.