Pressure Support vs. Thermal Broadening in the Lyman-alpha Forest I: Effects of the Equation of State on Longitudinal Structure

1.1 MB·English

by  Molly S. Peeples

Mon.Not.R.Astron.Soc.000,1–16(2009) Printed24January2010 (MNLATEXstylefilev2.2)
α
Pressure Support vs. Thermal Broadening in the Lyman- Forest I:
Effects of the Equation of State on Longitudinal Structure
0
Molly S. Peeples1⋆, David H. Weinberg1, Romeel Dave´2, Mark A. Fardal3, Neal Katz3
1
0 1DepartmentofAstronomyandtheCenterforCosmologyandAstro-ParticlePhysics,TheOhioStateUniversity,Columbus,OH43210
2 2UniversityofArizona,StewardObservatory,Tucson,AZ85721
3DepartmentofAstronomy,UniversityofMassachusetts,Amherst,MA01003
n
a
J

3 24January2010
2

] ABSTRACT
O
Inthelowdensityintergalacticmedium(IGM)thatgivesrisetotheLyman-αforest,gastem-
C peratureanddensityaretightlycorrelated.Thevelocityscaleofthermalbroadeningandthe
. Hubbleflow acrossthe gasJeansscale are of similar magnitude(HλJ ∼ σth). To separate
h
the effects of gas pressure support and thermal broadening on the Lyα forest, we compare
p
spectraextractedfromtwosmoothedparticlehydrodynamics(SPH)simulationsevolvedwith
-
o differentphotoionizationheatingrates(andthusdifferentJeansscales)andfromthepressure-
r lessdarkmatterdistribution,imposingdifferenttemperature-densityrelationsontheevolved
t
s particle distributions. The dark matter spectra are similar but not identical to those created
a from the full gas distributions, showing that thermal broadeningsets the longitudinal(line-
[
of-sight) scale of the Lyα forest. The turnover scales in the flux power spectrum and flux

3 autocorrelationfunctionare determinedmainlyby thermalbroadeningrather than pressure.
v However,the insensitivity to pressure arises partly from a cancellation effect with a sloped
6 temperature-density relation (T ∝ ρ0.6 in our simulations): the high density peaks in the
5 colder, lower pressure simulation are less smoothed by pressure support than in the hotter
2 simulation,anditisthishigherdensitygasthatexperiencesthestrongestthermalbroadening.
0
Changes in thermalbroadeningand pressure supporthave comparablyimportanteffects on
.
0 the flux probability distribution (PDF), which responds directly to the gas overdensity dis-
1 tributionrather thanthe scale on whichit is smooth.Tests on a lower resolutionsimulation
9 (2×1443 vs.2×2883 particlesina 12.5h−1Mpc comovingbox)showthatourstatistical
0 resultsareconvergedevenatthislowerresolution.Whilethermalbroadeninggenerallydom-
:
v inatesthelongitudinalstructureintheLyαforest,weshowinPaperIIthatpressuresupport
i determinesthetransversecoherenceoftheforestobservedtowardsclosequasarpairs.
X
Key words: cosmology: miscellaneous — cosmology: theory — intergalactic medium —
r
a methods:numerical
1 INTRODUCTION tions, most of the absorbing gas is at low density (ρ/ρ¯ 0.1–
∼
10)andnaturallydescribedasacontinuouslyfluctuatingmedium
TheLyman-αforest,causedbyLyαabsorptionofneutral hydro-
rather than a series of discrete structures (Hernquistetal. 1996;
gen atoms along the line of sight to some distant source (usu-
Bi&Davidsen 1997; Rauchetal. 1997; Croftetal. 1997, 1998).
ally a quasar), was originally described in terms of discrete in-
ThevelocitywidthofindividualfeaturesissetlargelybytheHub-
tervening gas “clouds” (Lynds 1971; Sargentetal. 1980), analo-
bleflowacrossthem,reflectingthephysicalextentoftheabsorbing
goustocloudsintheGalacticinterstellarmedium.Inthispicture,
gas along the line of sight (Hernquistetal. 1996; Weinbergetal.
thevelocitywidthsoftheobservedabsorptionstructureswerepri-
1997).Thetemperatureoftheintergalacticmedium(IGM),there-
marilyaconsequence ofthermalmotionsoftheabsorbingatoms.
fore,affectsthestructureoftheLyαforestintwoways:bysmooth-
Inthemid-1990s,three-dimensionalhydrodynamicsimulationsof
ingabsorptionalongthelineofsightthroughthethermalmotions
cold dark matter (CDM) cosmological models achieved remark-
ofatoms,andbysmoothingthephysicaldistributionofthegasin
able success in reproducing the observed properties of the Lyα
three dimensions through pressure support. In this study, we use
forest (Cenetal. 1994; Zhangetal. 1995; Hernquistetal. 1996;
twosmoothedparticlehydrodynamics(SPH)simulationswithdif-
Miralda-Escude´etal. 1996; Theunsetal. 1998). In these simula-
ferentphotoionizationheatingrates—andthusdifferentIGMtem-
peraturesandamountsofgaspressuresupport—todisentanglethe
⋆ E-mail:[email protected]
2 Peeples et al.
relativeeffectsofpressuresupport andthermal broadening inthe tering scale” below which fluctuations growth is suppressed de-
Lyαforest. pendsonthethermalhistoryofthegasratherthantheinstantaneous
Both hydrodynamical simulations (Katzetal. 1996; temperature-density relation (Gnedin&Hui 1998). We therefore
Miralda-Escude´etal. 1996; Theunsetal. 1998) and analytic expect equation (1) to describe the scale of gas pressure support
arguments (Hui&Gnedin 1997) suggest that low-density inter- onlyatanorder-of-magnitudelevel.Equation(3)showsthatJeans
galacticgasshouldhaveapower-lawtemperature-densityrelation, velocitiesandthermalvelocitiesshouldbecomparableattheover-
i.e., T = T (1 + δ)α, where 1 + δ ρ/ρ¯ is the local gas densities of typical Lyαforest features, but thecalculation isnot
0
≡
overdensity.Reasonableassumptionsforheatingandcoolingrates definitiveenoughtoshowwhetheronewilldominateinpractice.
yieldT0 104Kandα 0.6(Hui&Gnedin1997;Theunsetal. Hence, to predict the statistical properties of the Lyα forest
∼ ∼
1998). Observations, however, imply that the normalization T0 in a given cosmological model, one must calculate the predicted
could be nearly twice as high, and the slope could be much gas distribution. The most reliable way to do this uses full N-
shalloweroreveninverted(Schayeetal.1999;Ricottietal.2000; bodyplushydrodynamicsimulations,whichincludethegravityof
McDonaldetal. 2001; Boltonetal. 2008). Regardless of the dark matter and gas and the additional effects of pressure, adia-
parameter values, because the optical depth to Lyα absorption is baticheatingandcooling,heatingbyphotoionizationandshocks,
related to both the temperature and the density by a power law, andradiativecooling.However,largevolumehydrodynamicsim-
the existence of the temperature-density relation (also called the ulations with the necessary resolution are computationally inten-
“equation ofstate”)impliesatightrelationbetweentheobserved sive. Weinbergetal.(1997) and Croftetal. (1998) show that one
Lyα absorption and the IGM gas density. Since the universe has can achieve reasonable accuracy in the Lyα forest regime from
5 times as much mass in dark matter as in baryons, we can puredarkmattersimulations,applyingthetemperature-densityre-
∼
expect in general for intergalactic gas to trace the underlying lationto theevolved dark matter distribution tocompute spectra.
dark matter on scales above the gas Jeans length (Schaye 2001). Indeed,thelog-normalmodel(Bi&Davidsen1997),inwhichthe
TheLyαforest therefore provides apowerful tool for tracingthe dark matter distributioniscomputed from thelinear density field
dark matter power spectrum in the quasi-linear regime—modulo byanexponentialtransformation(Coles&Jones1991),providesa
the effects of peculiar velocities, thermal broadening, and gas qualitativelyaccuratephysicalmodel oftheLyαforest,sufficient
pressure (Croftetal. 1998, 1999, 2002; McDonaldetal. 2000, forcreatingartificialspectrawithreasonablestatisticalproperties.
2006;Vieletal.2004;Viel&Haehnelt2006). A fast way to include the effects of gas pressure in an ap-
PressureshouldbeimportantonscalesatorbelowtheJeans proximatewayisthehydrodynamicparticle-mesh(HPM)method
length, (Gnedin&Hui1998;Ricottietal.2000;Meiksin&White2001).
HPMassumesthatallthegasfollowsthepower-lawIGMequation
π 5π
λ =c =σ , (1) of state,and ituses amodificationof the standard (fast)particle-
J srGρ thr3Gρ
meshN-body methodtocomputethesumofgravitationalforces
andpressure gradient forces giventhisequation of state.Inade-
wherec = [5kT]/[3m] = σ 5/3isthespeedofsoundin
s th
tailedcomparisonoffullyhydrodynamicSPHsimulationsandthe
anidealgasepxpressedasamultiplepofthe1-Dthermalvelocityσ
th
(Miralda-Escude´etal. 1996; Schaye 2001; Desjacques&Nusser approximated HPM simulations, both evolved using GADGET-2,
Vieletal. (2004) found that while the HPM approach does con-
2005). Herem istheaveragemassof thegasparticle;inanion-
ized primordial mixture of hydrogen and helium, m = 0.59m , vergetotheSPHresults, forsomeLyαforest properties, suchas
p
thefluxprobability distributionorsmall-scalepower spectrum, it
wherem istheprotonmass. Thedensityρ isthedensityof the
p
gravitatingmedium,whichisdominatedbydarkmatter,sowetake candifferfromtheSPHcalculationbyasmuchas50%atz 2.
∼
ρ=Ω ρ (1+z)3(1+δ).Incomovingcoordinates, A simpler alternative to HPM is to use a pure N-body simula-
m,0 c,0
tionbut smooththeevolveddark matterdistributionontheJeans
5π 3 −1/2 scalebeforeextractingLyαforestspectra(Zaldarriagaetal.2001;
λ = (1+z)σ H−1 Ω (1+z)3(1+δ)
J,comv th 0 r 3 (cid:20)8π m,0 (cid:21) Desjacques&Nusser2005).BecausetheJeanssmoothingisthree-
= 782h−1kpc (2) dimensional,itisnotdegeneratewithline-of-sightthermalbroad-
ening. Zaldarriagaetal. (2001) found that, even though the ther-
σth Ωm,0(1+δ) 1+z −1/2, mal broadening dominates the pressure correction, the value of
× (cid:16)11.8kms−1(cid:17)(cid:20)(cid:18)0.25 (1+0)(cid:19)(cid:18)1+3(cid:19)(cid:21) theJeanslengthbecomesalargesourceofuncertaintyincosmo-
×
logical inferences from the Lyα forest if one tries to estimate it
wherewehavenormalizedthethermalbroadeningvelocityσ to
th
correspondtoafiducialtemperatureof104K.Bydefininga“Jeans fromthedataratherthanpredictitfromtheory.BecausetheJeans
lengthisexpectedtovarywiththegastemperatureanddensity,a
velocity”asv Hλ wecanfindtherelativeimportanceofthe
J J
≡ Zel’dovich-likeschemecanbeusedwhensmoothingthedarkmat-
Jeansscaleandσ ,
th
ter distribution in order to model these effects (Vieletal. 2002).
v 2π√10 Ω (1+z)3+Ω 1/2 Thoughwedonotcarryoutacomprehensivecomparisonofthese
J = (1+δ)−1/2 m,0 Λ (3)
σ 3 (cid:20) Ω (1+z)3 (cid:21) methodshere,wedoinvestigatetheimpactofpressureindetail,as
th m,0
6.62(1+δ)−1/2, wellascompare full SPHsimulationstoresultsderived fromthe
≈ darkmatterdistributionalone.
which is essentially redshift-independent for the redshifts rele- Theprimarypurposeofthispaperanditscompanion(Peeples
vant to the Lyα forest. The Jeans length of equation (1) di- etal.2009,hereafterPaperII)istodisentangletherolesofpressure
vides stable from unstable modes in a static, homogeneous, self- supportandthermalbroadeningintheLyαforestbystudyingtwo
gravitatingmedium.TheIGMisexpanding,inhomogeneous,non– SPHsimulationswithdifferentthermalhistoriesandtemperature-
self-gravitating(because dark matter dominates), and evolving in densityrelations.InadditiontoexaminingthephysicsoftheLyα
density and temperature on the same timescale that fluctuations forest,thisstudyismotivatedbyobservationalevidence(discussed
grow.Evenforlinearperturbationsinabaryonicuniverse,the“fil- in 3.1)thatthetemperatureoftheIGMatz 2–4ishigherthan
§ ∼
Scaleof theLongitudinalLyman-α Forest 3
expected fromsimple photoionization modelsby afactor of 1.5– ratesstandardheatingandatomiccoolingprocessesandthestan-
2. This evidence comes from analyses of data along single lines dardGADGET-2treatmentofstarformationandmetalenrichment.
of sight. Because the higher pressure associated with hotter gas These simulations do not incorporate galactic winds, but these
would smooth theIGMinthreedimensions, usingclosely paired should have very little effect on the Lyα forest (Kollmeieretal.
linesof sight toprobe thiscoherence scale has been proposed as 2006;Bertone&White2006;Marbleetal.2008).
an alternative route for inferring the temperature-density relation We extract spectra from the SPH gas distribution using
(J. Hennawi, private communication, 2007). However, before we TIPSY1,as described byHernquistetal. (1996). Followingcom-
can understand the relative roles of temperature and pressure on mon practice, we rescale the intensity of the UV background so
thetransversestructureoftheLyman-αforest,wemustfirstunder- thatthemeanfluxdecrementoftheextractedspectramatchesob-
standtheirlongitudinaleffectsalongindependentsightlines,which servations (Table 1), as the mean decrement itself is much bet-
isthegoalofthispaper.Whileherewefindthatthermalbroaden- terknownthanthebackgroundintensity(seediscussionsby,e.g.,
ingdominatespressuresupportinsettingtheleveloflongitudinal Croftetal. 2002; Marbleetal. 2008). Our extracted spectra have
structureintheLyαforest,inPaperIIweshowthatthegasJeans 1250pixelsacrossthe12.5h−1Mpcvolume,makingthepixelsize
length dominates the level of coherence transverse to the line of 1kms−1,whichiswellbelowthesmoothingscaleimposedby
≈
sight. thermalbroadening.
This paper is organized as follows. In 2, we describe the
§
SPH simulations we have evolved to investigate these effects. In
3,wedescribethephysicsoftheLyαforestinthesesimulations 3 PHYSICSOFTHELYMAN-αFOREST
§
andexaminetheimpactofthethermalhistoryonobservablespec-
tra;thissectionalsoservestoreviewthephysicalunderstandingof 3.1 TheTemperature-DensityRelation
thehigh-redshiftLyαforestthathasemergedfromsimulationsand
Intheabsenceofshockheating,theevolutionofthetemperatureof
associatedanalyticworksincethemid-1990s.Wethenstudythese
theIGMisdescribedby
effectsonseveraltypicalstatisticalmeasuresforlearningaboutthe
IGMfromtheLyαforestin 4,withourconclusionsin 5. dT 2T 2T dδ T dm 2 dQ
§ § = + + , (4)
dz 1+z (cid:20)3(1+δ)(cid:21)dz −(cid:20)m(cid:21) dz 3kBnb dz
asshownindetailbyHui&Gnedin(1997).Here,dδ/dzandH(z)
2 SIMULATIONS dependonthecosmology,theoverdensity1+δ (ρgas/ρ¯b),m
≡
isthemeanparticlemass,anddQ/dzisthenetpowerperunitvol-
WeanalyzetwoSPHsimulationswithidenticalinitialconditions, ume owing to the ambient radiation field. The first two terms in
one with a fiducial photoionization heating rate and one with a equation(4)describeheatingandcoolingowingtoadiabaticpro-
heating rate from photoionization that is four times higher than cesses.Afterreionization,thechangeintemperatureowingtothe
the fiducial; hereafter, we refer to these simulations as the “fidu- changeintheionizationfraction(thethirdterminequation[4])is
cial” and “H4” simulations, respectively. In the terminology of effectivelyzeroatallredshiftsandrelevantdensities.
Katzetal.(1996),wecomputethephotoionizationratesΓandpho- Hydrogen reionization produces one energeticphotoelectron
toionization heating rates ǫ for the fiducial simulation assuming perhydrogenatom,anditisexpectedtoheattheIGMtoatempera-
the Haardt&Madau (2001) quasar+galaxy photoionizing back- tureT 2–5 104K,dependingonthespectralshapeoftheioniz-
ground,andfortheH4simulationweincreaseǫHI,ǫHeI,andǫHeII ingsou∼rcesan×dradiativetransfereffects(Miralda-Escude´&Rees
byafactoroffour.Inprinciple,theseheatingratescouldarisefrom 1994). Thereafter, adiabatic cooling reduces the overall tempera-
a much harder UV background spectrum that yields more resid- ture, but denser regions remain hotter because they have higher
ual energy per photo-electron. However, we do not propose any neutralfractionsandthushigherphotoionizationheatingrates.Ata
specificmodelfortheseheatingrates—theyareacomputationally givenredshift,thesimulatedtemperature-density(T-ρ)relationof
simplewaytoobtainIGMtemperaturesthatarehigherthanthose thephotoionizedmediumcanbewellapproximatedbyapowerlaw,
inthefiducialmodel and closertothoseestimatedfromobserva-
tions(see 3.1forfurtherdiscussion). T(z)=T0(z)(1+δ)α(z). (5)
§
WeevolvetheseSPHsimulationsusingtheparallelGADGET- Theslope α(z) approaches 0.6 well after reionization. Evolution
2code(Springel2005)totracetheevolutionof2883 darkmatter
with the Haardt&Madau (2001) UV background spectrum and
and2883 gasparticlesina12.5h−1Mpccomovingcubicvolume Ω h2 0.022 yields T 104K at z = 2–4, for reionization
b 0
fromz = 15toz = 2.Wealsoevolvedanothersimulationusing ≈ ≈
atz &7(e.g.,Hui&Gnedin1997;Theunsetal.1998;Dave´etal.
thefiducial heating ratesand thesame initialconditions but only
1999; Table 1). Figure 1 shows the distribution of SPH particles
2 1443particlestoserveasatestforresolutionconvergence.We
× forourfiducialsimulationinthetemperature-overdensityplaneat
adoptastandardΛCDMcosmologicalmodelwiththeparameters
z=3.Atlowoverdensities(1+δ .10),mostofthegasfallsalong
(Ω ,Ω ,Ω ,h,σ ,n ) = (0.25,0.75,0.044,0.7,0.8,0.95), all
M Λ b 8 s atightlocus,asexpectedfromtheabovediscussion.Athigherden-
of which are in good agreement with the Wilkinson Microwave
sities(1+δ & 100),thegashasbeguntocooltoformgalaxies.
AnisotropyProbe(WMAP)five-yearresults(Hinshawetal.2009);
Thehighertemperaturegashasbeenshockheated.(Theapparent
our choice of σ8 = 0.8, however, is somewhat lower than what increaseintemperatureat1+δ 104owestothewayGADGET-2
is typically considered for Lyα studies. These parameters give a ∼
treatsthemultiphaseinterstellarmedium.)Formanyofoursubse-
massperSPHparticleof1.426×106M⊙,whichismuchlessthan quentanalyses,wewillisolatephysicaleffectsbyimposingoneof
thetypicalJeansmassofMJ ≡ ρλ3J ∼ 7×109M⊙.Thespline thethreeT-ρrelations,denotedbydashedlinesinFigure1.Specif-
gravitationalforcesofteninghasanequivalentPlummerlengthof
ically,weassigneachgas(ordarkmatter)particlethetemperature
0.875h−1kpccomoving( 1/50oftheinitialparticlegridspac-
∼
ing). The SPH smoothing lengths are chosen to enclose 33 2
±
neighbors within the smoothing kernel. The simulation incorpo- 1 UniversityofWashingtonversion
4 Peeples et al.
impliedbyitsoverdensityandagivenT-ρrelationbeforeextract-
ingspectra.TheyellowlineisaneyeballfittotheT-ρrelationin
thefiducialsimulation,whilethepinklineisthecorrespondingfit
totheH4relation;theH4T-ρnormalizationis 2.3timeshigher
∼
than thefiducial normalization. In both of these cases weset gas
with1+δ > 10toa“shocked”temperatureofT = 5 105K,
×
sothat thishigh densitygaswillbeentirelyionizedand thusnot
contribute to the Lyα forest. The mass- and volume-fraction of
gas in these high density regions is relatively tiny; for all of the
statisticspresented here, thefiducial and H4gasdistributionsus-
ingtheimposedfiducialandH4T-ρrelations,respectively,yield
nearlyidenticalresultstothoseobtainedusingtheactualtempera-
turescalculatedbyGADGET-2.Parametersforthesefitsarelisted
inTable1.Forsomecomparisons, weimposeaflatT-ρrelation,
withallparticlessettoT =2 104K,asshownbythebluedashed
×
line.
Although most Lyα forest features are broadened by Hub-
ble flow, the narrowest features arise at velocity caustics and
have widths set by thermal broadening. Analyses of observed
line width distributions imply T 1.5–2 104K at 2 6
0
≈ ×
z 6 4 (Gnedin&Hui 1998; Ricottietal. 2000; Schayeetal.
2000; McDonaldetal. 2001; we adopt McDonaldetal.’s con-
straintsinTable 1 and Figure2). Thisissignificantly hotter than
the value T 104K expected for a Haardt&Madau (2001) Figure1.Distributionof1%ofthegasparticlesforthefiducialsimulation
0
≈ inthetemperature-densityplaneatz =3.Insubsequentanalyses,weuse
ionizing background. Theunsetal. (2000) and Zaldarriagaetal.
either thesimulation temperatures themselves oroneofthethree plotted
(2001)findasimilarresultbyfittingthesmall-scalecutoffofthe
T-ρrelations(dashedlines).
one-dimensionalfluxpowerspectrum.Morerecently,Boltonetal.
(2008) havefitthefluxprobabilitydistributionfunction(PDF)in
high-resolutionspectrainferringasimilar,highT andashallow, ventionally assumed) or a physical process that has not yet been
0
possibly inverted (α < 0) slope. Lidzetal. (2009) find T identified.Preciselybecauseofthistension,itisimportanttoun-
0
≈
2 104Kovertheredshiftrangez =2–4,usingawaveletanaly- derstand the physics and observational consequences of pressure
×
sisof40high-resolutionspectra.Inallcases,theobservationsare supportintheLyαforest,aswestudyinthispaperandinPaperII.
interpretedbycomparingthemtoasuiteofcosmologicalsimula- In Figure 2, the dashed lines show the approximate T-ρ
tions, which incorporate many parameters in addition to the T-ρ relationships found in our two SPH simulations; the z = 3
relationitself. dashed lines are the same as the low-density yellow and pink
Accordingtoequation(4),therearetwobasicwaystoincrease linesinFigure1.Thetemperature-density parameters for theob-
the gas temperature: either have hotter “initial conditions,” or a servations and our simulations at each redshift are given in Ta-
higherheatingratedQ/dz.Evenwiththehighestplausiblereion- ble 1. The fiducial simulation clearly disagrees with the obser-
izationtemperatures, itisdifficulttoreproduce theinferredT at vations at z = 3 and 2.4, while the H4 simulation is in closer
0
z =3,giventheobservationalevidencethatz >7(Fanetal. agreementbutsomewhattoohot,i.e.,thetwosimulationsbracket
reion
2006;Hinshawetal.2009;seealsoHui&Haiman2003).Energy the central observation estimates. The points in Figure 2 are nu-
injectionfromHe IIreionizationoveranextended periodoftime merical integrations of equation (4), where we approximate the
could alsokeep theIGMhot downtoz 3(Boltonetal.2008; overdensity evolutionusing amodifiedZel’dovichapproximation
∼
Furlanetto&Oh 2008), though even this mechanism appears un- (Reisenegger&Miralda-Escude 1995). H I reionization is mod-
able to produce an inverted T-ρ relation (McQuinnetal. 2009). eledbyinitializingthetemperatureata“reionizationtemperature”
Aharderphotoionizingspectrumproducesmoreenergyinputper Trforalloverdensitiesatareionizationredshiftzreion =9.45;the
photoelectronandthusahigherIGMtemperature.2 Itisthissolu- temperature-densityrelationisfairlyindependentofTr byz = 4,
tionthatweadoptforourH4simulation,thoughtheheatingrates assumingreionizationoccursatz&6.
weadopt correspondtoanimplausiblyhardspectrum.Thestruc-
ture of the Lyα forest should depend on the T-ρ relation but be
3.2 TheFluctuatingGunn-PetersonApproximation
fairlyinsensitivetothedetailedmechanismthatproducesit,sowe
expectourconclusionsabouttheimpactofpressuresupporttoap- The H I optical depth τ isproportional tothe neutral hydrogen
HI
ply to a broad range of such mechanisms. However, the tension density,with
betweentheT-ρrelationspredictedfromtheoryandthoseinferred
πe2
fromobservationsremainpuzzling,anditisnotclearwhetherthe τ = fλ H−1(z)n , (6)
HI m c 0 HI
resolutionliesinmodestchanges(e.g.,temperaturesatthelowend e
ofobservationalestimatesandaspectrumthatisharderthancon- wherefλ istheoscillatorstrengthoftheLyαtransitionandλ0 is
the center of the Lyα transition, 1216A˚ (Gunn&Peterson 1965;
Miralda-Escude1993).Inthelimitofhighionizationfraction,the
neutralhydrogendensity,
2 Note that in photoionization equilibrium, higher intensity background
radiation—with thesamespectrum—affects onlythephotoionization and n = αHII n n , (7)
recombinationrates,nottheelectrontemperature. HI (cid:18)Γ (cid:19) e H
UV
Scaleof theLongitudinalLyman-α Forest 5
Figure2.Acomparisonofthetemperature-densityrelationevolutionfromsimulations(dashedlines),calculationsusingequation(4)(points),andobservations
(shadedregions).Theshadedregioncorrespondstothe1-σerrorregionfromMcDonaldetal.(2001),withtheorangelinesindicatingthebest-fitrelationsat
eachredshift.(Therearetwosuchlinesforz=3and2.4becauseMcDonaldetal.usedsimulationsattwodifferentredshiftstocomparetotheseobservations.)
Atz=3,thedashedpinkandblacklinesarethesameasthedashedpinkandgoldlines,respectively,inFig.1.
Table1.ObservedmeanfluxdecrementshDi≡h1−e−τiarefromMcDonaldetal.(2000)andobservedtemperature-densityrelations(T =T0[1+δ]α)
arefromMcDonaldetal.(2001).
z hDi observedT0[K] observedα fiducialT0[K] fiducialα H4T0[K] H4α
4.0 0.525±0.012 17400±3900 0.43±0.45 11700 0.54 28200 0.55
3.0 0.316±0.023 18300±1800 0.33±0.26 11000 0.57 25000 0.57
or18400±2100 0.29±0.30
2.4 0.182±0.021 17400±1900 0.52±0.14 10000 0.56 23000 0.57
or19200±2000 0.51±0.14
isproportionaltothegasdensitysquared;Γ isthephotoioniza- agnosticoftheIGMtemperatureT ,butinpracticeitisnot well
UV 0
tionrateowingtotheambientUVbackground.Attherelevanttem- enough known. Hence, werelyonstructureintheLyαforestfor
peratures(T 106),therecombinationcoefficientα ispropor- IGMdiagnosticsandcosmologicaltests.
tional to T−0≪.7 (Katzetal. 1996). Following RauchHeItIal. (1997) Our results below rely on our full hydrodynamical simula-
andCroftetal.(1998),wecancombineequations(6)and(7)with tions, but equation (8) is useful to understand our results, and it
apower-lawT-ρrelationtoobtaintheopticaldepth canbeausefulbasisforsimpleranalyticornumericaltreatments.
It is often referred to as the “fluctuating Gunn-Peterson approxi-
τ =1.54 T0 −0.7 10−12s−1 1+z 6 0.7 mation” (FGPA; Weinbergetal. 1997; Croftetal. 1998) because
HI ×(cid:18)104K(cid:19) (cid:18) ΓUV (cid:19)(cid:18)1+3(cid:19) (cid:18) h (cid:19) it describes Lyα absorption as a continuous phenomenon analo-
(8)
Ω h2 2 4.0927 1 dV −1 goustotheGunn-Peterson(1965)effect,butarisinginafluctuating
b,0 (1+δ)2−0.7α 1+ los . medium.Aswritten,itignoresthermalbroadening(atemperature-
×(cid:18)0.02156(cid:19) (cid:20)H(z)/H0(cid:21) (cid:20) H(z) dx (cid:21) and therefore density-dependent convolution) and shock heating
The last factor represents the impact of line-of-sight peculiar ve- (i.e., gas not falling on the temperature-density relation). Also,
locity gradients dV /dx, which change the density of atoms in while1+δ ρ /ρ¯ ,thegasoverdensityisoftenapproximatedas
los ≡ gas gas
frequencyspacerelativetorealspace. thedarkmatteroverdensityρ /ρ¯ inN-body simulations,per-
DM DM
IndependentlyoftheLyαforest,thephotoionizationrateΓ hapssmoothedbyaneffectiveJeanslength.
UV
is difficult to constrain; the other quantities entering the normal-
ization pre-factor of equation (8) are also uncertain. We there-
fore follow standard practice (e.g., Miralda-Escude´etal. 1996;
3.3 TheLyαforestinthefiducialcase
Marbleetal. 2008) and choose Γ for each model so that it re-
UV
produces the observed mean flux decrement D 1 F = Figure3shows,ontheleft,aslicethroughourfiducialsimulation
1 exp( τ ) at the redshift under investihgatiio≡n. Shpe−cificailly, atz =3,withadepthof125h−1kpc.Thesmall-scalefilamentary
h − − HI i
weadopt D valuesfromMcDonaldetal.(2000),whicharelisted structure of the high redshift universe is evident. While the dark
h i
inTable1;attheoverlappingredshifts(z =2.4, 3)these D are matterandbaryonshavesimilarlarge-scalestructure,thezoom-in
h i
consistentwiththemorerecentmeasurementsofKimetal.(2007). panel ontherightshowsthatthegasdistributionismorediffuse.
IfΓ wereperfectlyknown,then D coulditselfbeusedasadi- Inparticular,thedensestfilamentsofdarkmatterliewithinthicker
UV h i
6 Peeples et al.
Figure3.A125h−1kpcthickslice(left)anda125×125h−1kpccomovingregion(right)ofthefiducialsimulationatz=3withgasparticles(red),dark
matterparticles(black),andstarparticles(blue)allshown.Thefivegreenlines(left)denotesightlinesreferredtoinsubsequentfigures.
filamentsofgas.Thisdifferencereflectstheimpactofgaspressure velocityspace.3 Featurescontractingalongthelineofsightcause
support. physicallydistinctgasregionstoconvergetothesameregionofthe
observedspectrum.However,iftheconnectinglinesdonotcross,
thentheregionstillhasnetexpansion. Boththethermallybroad-
enedandnon-thermallybroadenedspectrahavethesamecharacter-
isticbroadfeatures,implyingthatresidualHubbleflowdominates
Figure4showshow density, temperature,velocity, andther- thevelocitywidthofthesefeatures.However,thermalbroadening
malbroadeningcombinetoproduceLyαforestspectra,alongthe smoothsthesmallscaleroughness. Atvelocitycaustics(converg-
sightlinesmarked1–4inFigure3.Ineachpanel,thebottommost inglinesinFigure4), peculiar velocities cancel the Hubbleflow,
plotshowsthegastemperatureinblackandtheneutralhydrogen andthefeaturesdobecomenarrowerwhenoneremovesthether-
fraction in blue. Because nHI/nH ∝ nHT−0.7 and T ∝ n0H.6 for malbroadening.
gas on the T-ρ relation, the neutral fraction and temperature are
positivelycorrelatedatmosttemperatures.However,hightemper-
aturegashasusuallybeenshock heatedoff theT-ρrelation, and 3.4 Impactsoftemperatureandpressureongasevolution
the recombination coefficient α falls more steeply than T−0.7
HII Theobviousconsequenceofhavingahigherphotoionizationheat-
athightemperature(Katzetal.1996),causingtheneutralfraction
ing rate is that gas temperatures in the H4 simulation are higher
todecreasedramatically(see,e.g.,thefeaturenear100kms−1 in
thaninthefiducialsimulation,asshownatz = 3inthetoppan-
sightline #2). In the middle graphs, weplot the neutral hydrogen
elsofFigure5.Amoresubtleeffect,isthatthelargerJeanslength
numberdensity[cm−3]inpurple,usingthescaleontheleft-hand
oftheH4simulationsmoothsthegasdistribution,asshowninthe
axis.Thegasanddarkmatteroverdensitiesareplottedinblackand
density-codedbottompanelsofFigure5.Thereisarelativepaucity
cyan, respectively, using thescaleon theright-hand axis. Ingen-
ofverydenseclumpsintheH4simulation;thisisespeciallynotice-
eral,thedarkmatterandgasoverdensitiesaresimilar,butthedark
ableinthelowerdensityfilaments.
matterhassharper,higheroverdensitypeaksaswasseenvisually
Figure6plotsthedistributionofgasoverdensityinthefidu-
inFigure 3. As expected from theT-ρ relation, the gasoverden-
cialand H4simulations. Thevisual differencesintherightpanel
sityandthegastemperaturefollowoneanotherexceptatveryhigh
ofFigure5manifestthemselvesasahigherfrequencyofparticles
gasoverdensity.Thehighestgasoverdensitiescorrespondtocon-
withρ /ρ¯ 10–100 inthefiducialsimulationandahigher
densedhalos—i.e.,galaxies—andthusthegashascooledtolower gas gas ∼
frequency of ρ /ρ¯ 1 particles in the H4 simulation. This
temperaturesintheseregions.Theneutralhydrogendensityshows gas gas ∼
differenceleadstoacancellationinmanyofthestatisticalcompar-
more variation than the gas density because it is proportional n2
H isonsin 4;though theH4gasishotter(andthushasmorether-
(actuallyρ1.6oncetemperatureeffectsareincluded).Thetopmost §
malbroadening),itshigherpressureimpliesthattherethereisrela-
plothasthetransmittedfluxF exp( τ )inblackandthetrans-
≡ − HI tivelylesshigh-densityandthereforerelativelyhigher-temperature
mittedfluxthatwouldbeobservedintheabsenceofthermalbroad-
eningingrey.Thelinesbetweenthemiddleplot andthetopplot
showtheeffectsofpeculiarvelocitieswhenconvertingfromneu- 3 Thebar-likeappearanceofthelinesshowingtheeffectsofpeculiarve-
tral hydrogen density in physical space to an observable flux in locityalsolendsthesecomplicatedplotsthenameof“zooplots.”
Scaleof theLongitudinalLyman-α Forest 7
Figure4.PhysicalquantitiesandLyαforestspectraalongfourofthesightlineslabelledinFigure3;velocityv=0correspondstothetopofFigure3.Shown
inrealspacearethetemperature(black,bottom),theneutralhydrogenfraction(blue,bottom),theH Inumberdensity(purple,middle),thegasoverdensity
(black,middle),andthedarkmatteroverdensity(cyan,middle).Thebarsseparatingthetopandmiddlepanelsshowtheeffectsofpeculiarvelocitieswhen
transitioningfromrealspacetotheobservedfluxtransmissionwith(top,black)andwithout(top,grey)thermalbroadening.See§3.3formoredetails.
8 Peeples et al.
Figure6.Distributionsofgasoverdensitiesatz=3inthefiducial(black)
andH4(pink)simulations,aswellasthedissipationlessdarkmatter(cyan)
inthefiducialsimulation.
gaswithlargethermalbroadening,andthedarkmatterdistribution
would have more. We, therefore, consider an additional, flat T-ρ
relation,withT =2 104Katalldensities,sothatwecanexam-
Figure5.Temperature(top)anddensity(bottom)evolutionforgasparticles
×
ina2.5×2.5×0.05h−1Mpccomovingsliceatz=3inthefiducial(left) inetheeffectsofpressureinthepresenceofpressure-independent
andH4(right)simulations.Thetemperaturescalerunslogarithmicallyfrom thermal-broadening.
fromlogT = 3.5(black/darkblue)tologT = 5.5(yellow/white). The We now turn to the effects of pressure support and the T-ρ
densityscalerunsfromlog(ρgas/ρ¯gas) ≡ log(1+δ) = −1.3(black)to relationontheLyαforestspectra( 4.1)andonfluxstatistics(the
log(1+δ)=1.3(green). fluxpowerspectrum, 4.2;theauto§correlationfunction, 4.3;and
§ §
the probability distribution function, 4.4. For all these analyses
§
weusethesamesightlinesineachsimulationtoensuretheeffects
gas.Forcomparison, wealsoshow inFigure6thedensitydistri-
ofsamplearevariancethesame.
butionofthedarkmatter,withdensitiescomputedusingtheSPH
smoothingkernel.Becausethedarkmatterispressureless,ittypi-
callyreaches higher overdensities, though itdoes not achieve the 4.1 Spectra
highest overdensities seen in the gas distributions because these
arisefromdissipation,i.e.cooling.Extrapolatingfromthedensity Figure7showshow pressureandtheT-ρrelationaffecttheLyα
distributionsinourlow-resolutionsimulation,thedarkmatterden- spectra along the five sightlines labelled in Figure 3, with sight-
line #1 corresponding to the topmost spectrum in each panel; as
sity distribution would probably be somewhat less skewed if we
increasedthemassresolutionofthesimulation,butatlowoverden- in Figure 4, v = 0 corresponds to the top of Figure 3. Any dif-
sitythegasdistributionsareessentiallyconvergedsinceweresolve ferencesbetweenthemodelsareusuallymostnoticeableinunsat-
urated lines. The top-left panel shows spectra computed directly
theJeansmass.
from the simulated gas distributions, with differences in pressure
support,differencesinthermalbroadeningbecauseofthedifferent
T-ρnormalization,anddifferencesinthermalbroadeningbecause
4 EFFECTSOFPRESSUREANDTHERMAL
ofthesamplingoftheT-ρrelation.Lightgreylinesshowspectra
BROADENINGONTHELyαFOREST
fromour lowerresolutionsimulation.Theimpactof resolutionis
To isolate the effects of gas pressure and thermal broadening in generallyverysmall,buttherearesomeslightdifferences.
our subsequent analyses, we examine both the original gas parti- TherearetwonotabledifferencesbetweenthefiducialandH4
cle distributions and distributions with one of the three imposed spectraexemplifiedbythev 450kms−1 featureinsightline#5
T-ρrelationsillustratedinFigure1.ForanimposedT-ρrelation, andthev 750kms−1 feat∼ureinsightline#4. First,featuresin
∼
wereplaceeachgasparticle’stemperaturebythetemperaturethat theH4spectraarebroaderthaninthefiducialspectra,whichcould
correspondstoitsoverdensity.Weusethesameproceduretocre- arisebecauseofgreaterthermalbroadeningand/orbecauseofthe
ateLyα forest spectra fromthe dark matter distribution(“gasify- larger Jeans scale (and thus thelarger Hλ ).Second, features in
J
ing” the dark matter). The fiducial T-ρ relation (yellow in Fig- theH4spectraarenot asdeepasinthefiducialcasebecause the
ure1) matchesthat found fortheIGMinthefiducial simulation; hottergasdoesnotreachashighoverdensityand/orbecausehigher
above1+δ = 10,thegasis“shockheated”toatemperatureof thermalbroadeningsmearsoutinherentlysharpfeatures.
5 105K.Likewise,thepinklinesinFigure1show theT-ρre- Thetoprightpanelisolatestheimpactofthermalbroadening,
×
lation used to mimic the H4 simulation. Figure 6 shows that gas applying thefiducial and H4T-ρrelationstothegasdistribution
distributions with different pressure also sample the T-ρ relation ofthefiducialsimulation.Thelower-leftpanelisolatestheimpact
differently,andiftheT-ρrelationisslopedtheywillthereforeex- of pressure, applying the flat T-ρ relation to the fiducial and H4
periencedifferentthermalbroadeningForexample,usingthefidu- gasdistributionsandtothedarkmatterdistributionfromthefidu-
cialT-ρrelation,theH4simulationwouldhavelesshighdensity cialsimulation.Insomecases,suchasthev 150kms−1feature
∼
Scaleof theLongitudinalLyman-α Forest 9
Figure7.Comparisonofspectraalongthesightlines labelled inFigure3,isolatingdifferentphysicaleffects.Topleft:fulleffects ofthermalbroadening,
thermalhistoryandresolutionareshown.Topright:effectsofthermalbroadeningareisolated.Bottomleft:effectsofpressuresupportareisolated.Bottom
right:effectsofpressuresupportanddifferentsamplingsoftheunderlyingoverdensitydistributionareisolated.See§4.1fordetails.
10 Peeples et al.
insightline#1, thev 1025kms−1 featureinsightline#3, and
thev 250kms−1 fe∼atureinsightline#4,differencesinthefull
∼
simulationspectra(upperleft)arelargelyerasedintheupperright
panelbutremainsimilarinthelowerleftpanel,whichshowsthat
theymostlyarisefromdifferentpressureeffectsinthetwosimula-
tionsratherthanfromdifferencesinthethermalbroadening.There
arefewercasesofthereverse,wheredifferencespresentinthefull
spectraremainintheupperrightpanelsbutdisappearinthelower
left, though some are visible in sightline #5. The differences in
the upper-right (thermal broadening isolated) are always system-
atic, with lower thermal broadening yielding deeper and slightly
narrowerfeatures.ThedifferencesbetweentheH4andfiducialgas
distributionsinthelower-left(pressureisolated)aremorerandom,
andwewillseebelowthattheirstatisticalsignatureisweaker.
Thepressurelessdarkmatterdistributiondoesleadtospectra
withmoresmallscalestructure,asshownbythethindarklinesin
thelowerleft.However,despitethecleardifferencesbetweenthe
gasanddarkmatterdistributionsevidentinFigures3,4,and6,the
differencesinthespectracreatedfromthesedistributionsaresmall.
Intheabsenceofthermalbroadening,darkmatterspectraaremuch
morejaggedthantheirgascounterpartsshowninFigure4,butreal-
isticthermalbroadeningmasksthesedifferencestoalargeextent.
Thevelocitywidthsofmostfeaturesarenotsensitivetothelevel
of thermalbroadening (asseenintheupper right)butneither are
theysetbytheJeansscale,orelsethedarkmatterandgasspectra
wouldhavegreaterdifferences.Instead,thermalbroadeningerases
thefinestscalestructuresinthedensityfieldsothattypicalfeatures
correspondtocoherent,moderateoverdensitystructuresexpanding
withtheHubbleflow.
AdoptingthefiducialT-ρrelationinplaceoftheflatT-ρre-
lation(bottomright)makesonlyasmalldifferencetotheusualap-
pearance of the spectra (top left). Features that are flat-bottomed
in the dark matter spectra with the flat T-ρ relation often be-
come less saturated with the fiducial T-ρ relation, such as the
v 775kms−1 featureinsightline#3andthev 1000kms−1
∼ ∼
featureinsightline#5.Intheseregions,thedarkmatteroverdensity
ishigh,sowithT (1+δ)0.6 theyhavehigherthermalbroad-
∝
ening,whichspreadsthefeatureinvelocityspaceandreducesits
saturation.Insomecases,suchasthev 725kms−1 featureof
∼
sightline#1, thewings ofthe darkmatter featurebecome notice-
ably broader than those of the gas featuresbecause of the higher
temperaturesathigheroverdensities.
4.2 The1-DFluxPowerSpectrum
The Lyman-α flux power spectrum is a powerful tool for prob-
ing the dark matter mass power spectrum on the smallest scales.
Becauseofthehighredshift,thecontinuoussamplingoftheline-
of-sight density field, and the moderate overdensity of absorbing
structures, the Lyα forest provides a more direct link to the lin-
Figure8.The1-Dfluxpowerspectrumatz=4.0(top),z=3.0(middle),
ear theory power spectrum than other small-scale tracers. (Pri-
andz = 2.4(bottom),withthefiducialsimulationinblack,theH4sim-
mary cosmic microwave background anisotropies are damped on
ulationinpink,andthelower-resolutionsimulationingrey.Forreference,
these scales.) However, to infer information about the structure
wealsoshowthefluxpowerspectrumintheabsenceofthermalbroadening
of the underlying dark matter, one must understand the thermal
forthefiducialgas(green)andtheH4gas(purple),aswellastheobserved
structure of the IGM, as the amplitude and the shape of the flux fluxpowerspectrumfromMcDonaldetal.(2001,squares)andCroftetal.
power spectrum are connected to the gas temperature-density re- (2002,triangles).
lationviaequation (8).Thebias of thefluxpower spectrum, i.e.,
b2(k) = P (k)/P (k),isalsoinfluencedbynon-lineargravita-
F lin
tionalevolution,thermalbroadening,pressuresupport,peculiarve-
locities,andshockheating(Croftetal.1998,1999,2002;Vieletal.
2004,2008;McDonaldetal.2005).
Figure8showstheone-dimensional line-of-sightfluxpower