Cutoff in the Lyman-a forest power spectrum: Warm IGM or warm dark matter?

(1)

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Cutoff in the Lyman- α forest power spectrum: Warm IGM or warm dark matter?

Antonella Garzilli

^a^,^∗

, Alexey Boyarsky

^a

, Oleg Ruchayskiy

^b

aLorentzInstitute,LeidenUniversity,NielsBohrweg2,Leiden,NL-2333CA,TheNetherlands bDiscoveryCenter,NielsBohrInstitute,Blegdamsvej17,DK-2100Copenhagen,Denmark

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received8July2016

Receivedinrevisedform19January2017 Accepted14August2017

Availableonline23August2017 Editor:S.Dodelson

Keywords:

Cosmology,Warmdarkmatter,Largescale structureofUniverse

Methods,Numerical,Observational Quasars,Absorptionlines

We re-analyse high redshiftand highresolution Lyman-α forest spectraconsidered in[1],seekingto constrain theproperties ofwarmdarkmatterparticles.Comparedto thispreviouswork, weconsider awiderrangeofthermalhistoriesoftheintergalacticmedium.Weﬁndthatbothwarmandcolddark mattermodelscanexplainthecut-offobservedintheﬂuxpowerspectraofhigh-resolutionobservations equallywell.Thisimplies,however,verydifferentthermalhistoriesandunderlyingreionization models.

Wediscusshowtoremovethisdegeneracy.

©²⁰¹⁷^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Darkmatterisacentralingredientofthecurrentstandardcos- mologicalmodel.Itdrivestheformationofstructures,andexplains themassesofgalaxiesandgalaxyclusters. Ifdarkmatterismade ofparticles, these yet-unseen particles should have beencreated intheearlyUniverselongbeforetherecombinationepoch.Ifsuch particles were relativistic at early times, they would stream out fromoverdenseregions, smoothingout primordialdensityﬂuctu- ations. The signature of such warmdarkmatter (WDM) scenario wouldbethesuppressionofthematterpowerspectrumatscales below their free-streaming horizon. From cosmological data at largescales (CMB andgalaxysurveys) we knowthatsuch a sup- pressionshouldbesoughtatcomovingscaleswellbelowaMpc.

The Lyman-

α

forest has been used for measuring the matter powerspectrumatsuchscales[2–4].Inpreviousworksonlyupper boundshadbeenreportedonthemassofthethermalrelic[5–10].

However, while in the SDSS spectra there is no cut-off in the transmittedﬂuxpowerspectrum,thereisacut-offinthehighres- olutionspectra,forexample[4,11,7].Recently[1]hasobservedthe cut-off of the ﬂux power spectrum atscales k∼⁰.03 s/km and redshiftsz=⁴.2–5.4.

*

Correspondingauthor.

E-mailaddress:garzilli@lorentz.leidenuniv.nl(A. Garzilli).

However, theLyman-

α

forest methodmeasuresnot thedistri- butionofdarkmatteritself,butonlytheneutralhydrogendensity asaproxyfortheoverallmatter density.Theprocess ofreionization heats the hydrogen andpreventsit fromclustering atsmall scalesattheredshiftsinquestion[12].Therefore,theobservedhy- drogendistributioneventuallystopstofollowtheDMdistribution.

Indeed, it was demonstrated in [1] that within CDM cosmol- ogythereexistsasuitablethermalhistoryofintergalacticmedium (IGM) that is consistent withthe observed cutoff. This doesnot mean,however,thatthisscenarioisrealized innature.

InthisLetterweinvestigatethisissueindepth.Weaskwhether thecutoffintheﬂuxpowerspectrumcanbeattributedtothesuppression ofsmallscaleswithwarmdarkmatter andwhat thismeans forthe thermal history of IGM. Tothis end we reanalyze the data used in[1].Weusethesame suiteofhydrodynamicalsimulationsofthe IGM evolution withcoldandwarm darkmattermodels asin[1]

and demonstrate that the data is described equally well by the model, whereﬂux power spectrumsuppression is mainlydue to WDM.

2. Dataandmodel

The dataset isconstituted by25 high-resolutionquasar spectra, in the redshift interval 4.48≤^zQSO≤⁶.42. The spectrawere takenwiththeKeckHighResolutionEchelleSpectrometer(HIRES) andtheMagellanInamoryKyoceraEchelle(MIKE)spectrographon theMagellanclaytelescope.TheQSOspectraaredividedintofour http://dx.doi.org/10.1016/j.physletb.2017.08.022

0370-2693/©²⁰¹⁷^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

(2)

redshiftbinscenteredon:z=⁴.2,4.6,5.0,5.4.Theresultingrange ofwave-numbersprobedbythisdatasetisk=⁰.005–0.08 s/km.

Attheseredshifts,theIGMisthoughttobeinahighlyionized state,beingphoto-ionizedandphoto-heatedbyearlysources.Both theWDMcosmologyandtheIGM temperatureaffecttheamount ofﬂuxpowerspectrumatsmallscalesthroughthreedistinctphys- ical mechanisms: (1) a suppression in the initial matter power spectrum;(2) Jeansbroadening;and(3) Dopplerbroadeningofthe absorptionlines[12–17].Theﬁrstmechanismiscosmological,the lattertwo areastrophysical. TheDoppler broadeningis a onedi- mensionalsmoothingeffectthatoriginatesfromobservingthehot IGMalongalineofsight.TheMaxwelliandistributionofvelocities inthe gas then leads to the broadeningeffect. The Jeansbroad- eningsmooths the three-dimensional underlying gas distribution relativetothedarkmatter.

Thelevelofionization iscapturedbytheeffectiveopticaldepth,

τ

_eff,that iscomputedfromthe meanﬂux, ^F^,^through ^the^rela- tion^F(z)=^exp(−

τ

eff(z)).Because the IGM spans awide range of density, describing the IGM temperature may be complicated in principle. But, assuming that the IGM is heated by photo- heating, thetemperature ofthe IGM follows a simplepower-law temperature-densityrelation[18]:

T

(δ) =

^T0

(

z

)

1

+ δ

γ(z)−1

,

(1)

whereδ= δ

ρ

m/

ρ

¯m isthematteroverdensity and T₀(z),

γ

(z)are unknownfunctionsofredshift.TheresultsofRef.[1]arebasedon single power-law parametrizations, T0(z) and

γ

(z). In thisletter welettheparametersoftheIGMthermalstatevaryindependently ineach redshiftbin, witha total of 8 parameters describing the IGM thermal state (T0(zi) and

γ

(zi) in 4 distinct redshift intervals).¹

We wantto point out that T0 and

γ

are not varied in post- processing.The originalwork of[1]considered9simulationruns withdistinctthermalhistoriesforeachcosmologyconsidered.The different thermal histories are realized by changing the photo- heatingfunctioninthesimulations.TheresultingvaluesofT0 and

γ

are approximately distributed on a regular grid. In [1]the ef- fectofJeanssmoothingisaccountedbyconsideringtwoadditional simulation runs, where the time at which the ultraviolet back- groundisswitchedon,zreion,isvaried.Wecautionthereaderthat theresultingconstraintsonz_reionmustnotbeintendedasamea- surementofthetimeofreionization,becausethisdependsonthe details of the implementation of the ultraviolet background. In- stead,varying z_reion mustbe considered asa way toaccount for theunknown levelofJeanssmoothing.Finally,asin[1],weallow theeffectiveopticaldepthvaryindependentlyineachredshiftbin,

τ

eff[^zi]^.

Itshouldbenotedthatthisinterpolationschemebetweensim- ulations with different temperatures may also vary the amount of Jeans broadening (also known as the “ﬁltering scale”). While the degeneracybetween the WDM cosmologiesand the Doppler smoothinghas been extensively considered in the literature, the degeneracy between Jeans smoothing and WDM cosmology has notbeenconsideredindepthsofar.Inparticularthishasnotbeen doneforthesuiteofsimulationsintheoriginalwork[1]onwhich we base ouranalysis. We leave the studyof the degeneracy be- tweentheJeanssmoothingandWDMforfuturework.

The results also depend on the cosmological parameters ns,

M,

σ

8, H₀. However the small scale Lyman-

α

data by itself doesnotsuﬃcientlyconstrainthecosmologicalparameters.There- fore,intheﬁnallikelihoodfunctionfortheseparametersweused

1 Ref.[1]alsoperformedsucha“binnedanalysis”,seethedetailedcomparison below.

Table 1

ParameterestimationfromBayesiananalysis.Weshowthe1-σand2-σconﬁdence intervals.Weonlyshowtheparametersthatareconstrainedat1or2-σlevel.

parameter mean 1-σ 2-σ

H0[km/s/Mpc] 63 <67 –

mWDM[keV] 3.9 [143,2.3] >2.1

T0(z=4.2)[10³K] 10.6 [9.4,11.8] [8.3,12.9] T0(z=⁴.6)[10³K] 9.8 [⁸.6,11.1] [⁷.5,12.2] T0(z=⁵.0)[10³K] 4.0 [².0,5.6] <6.9

T0(z=⁵.4)[10³K] 3.8 <4.5 <8.2

τeff(z=⁴.2) 1.12 [¹.05,1.19] [¹.00,1.25] τeff(z=⁴.6) 1.30 [¹.21,1.39] [¹.15,1.47] τeff(z=5.0) 1.88 [1.74,2.00] [1.64,2.13] τeff(z=5.4) 2.91 [2.69,3.10] [2.54,3.31]

γ(z=⁴.2) 1.3 >1.1 –

γ(z=⁵.4) 1.3 >1.1 –

Fig. 1. Measuredfluxpowerspectrumindimensionlessunits,²_F(k)=^PF(k)×^k/π, comparedwiththetheoreticalmodelwiththebest-fittingvaluesoftheastrophys- icalandcosmologicalparametersforWDMandCDMcosmologies.Thesolidrefer thebest-fittingvaluesforWDMcosmology.Thedottedlinesrefertothebest-fitting caseforCDMcosmology.Thesebest-fittingmodelslargelyoverlap,exceptatthe highestredshift andonthesmallest scales.Theblue,grayand greenpoints are SDSS-III/BOSSDR9dataforz=⁴.0,z=⁴.2 andz=⁴.4 from[20].(Forinterpreta- tionofthereferencestocolorinthisfigurelegend,thereaderisreferredtothe webversionofthisarticle.)

bestﬁtPlanck values[19]withGaussian priors(asin[1]), M= 0.315±⁰.017,

σ

8=⁰.829±⁰.013,ns=⁰.9603±⁰.0073.

3. Results

InTable 1wegivetheresultoftheparameterestimation.Fig. 1 shows the theoretical ﬂux power spectrum for the mean values oftheparameters,comparedwiththeMIKE andHIRESdataused inthis analysis. Inorder toclarify the effectofdifferent thermal histories on our constraints, we show the effect ofchanging the thermal parameters (T0 and

γ

) and ionization parameters (

τ

eff) andthemassofthethermalrelic(1/mwdm)inFig. 2,analogousto Figs. 5and6of[1].

In Fig. 3we show the 2D confidence regions between m_wdm, and T0≡^T(δ=⁰)(marginalizing over theother parameters).We see that at redshifts z=⁴.2, 4.6 there is no degeneracy andan IGM temperature T₀∼¹⁰⁴ ^K îs ^needed^to êxplain ^the ôbserved fluxpowerspectrumindependentlyofm_wdm.Ifdarkmatteris“too warm”(m_wdm<1.5 keV)itproducestoosharpofacut-offinthe powerspectrumandisinconsistentwiththedata.

At the z=⁵.0 bin the situation is different. For the masses m_wdm∼².2–3.3 keVevenverylowtemperaturesT₀^{2500 K}^are consistentwiththedata.Inthiscasethecutoffintheﬂuxpower spectrum is explained by WDM rather than by the temperature.

Thesituationisanalogousatz=⁵.4.Table 1summarizesthepa- rameterestimation.

(3)

Fig. 2. EffectoftheIGMparametersandmwdmonthefluxpowerspectrumindimensionlessunits,²_F(k)=^PF(k)×^k/π.Inthetop-left(top-right,bottom-left,bottom-right) panelweshowtheeffectofvaryingT0(γ,τeff,1/mwdm)by±^{10% with}^respect^to^thebest-fittingvaluesforWDMcosmology.Thesolidlinecorrespondstothebest-fitting caseforWDMcosmology,thedashed(dotted)linecorrespondstotherelevantparameterincreased(decreased)by10%.(Forinterpretationofthereferencestocolorinthis figurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 3. Confidenceregionsbetweenmwdm,andT0andγ atallredshift,andzreion.Weshow1/mwdminsteadofmwdmforvisualizationpurposes.mwdmisdegeneratewith zreion,thatistheredshiftatwhichtheultravioletbackgroundhasbeenswitchedoninthesimulations,and T0attheredshiftz=5.0.mwdm isnotdegeneratewiththe T0fortheotherredshiftintervals.Thereisnoobviousdegeneracywithγ.RegardingmwdmandT0,attheredshiftsz=4.2,4.6 thereisnodegeneracyandT0∼10⁴Kis neededtoexplaintheobservedfluxpowerspectrum,independentlyofmwdm.Atz=⁵.0 evenverylowtemperaturesT0^{2500 K}âre^consistent^with^the^data,ând^the cutoffinthefluxpowerspectrumisexplainedbyWDMratherthanbythetemperature.Atz=⁵.4 theanalysispreferslowvaluesofT0∼⁵×¹⁰³^K,independentlyofmwdm.

(4)

Fig. 4. TheevolutionoftheIGMmeantemperature,T0,inredshift.Blackvertical barsare1-σ confidencelimits;redverticalbarsare2-σ confidencelimits.Filled dotsaretheparametermean;thearrowsmarkthe upperlimits.Thehorizontal barsspantheredshiftintervalofLymanαabsorbersconsideredforeachmeasure- mentofthefluxpowerspectrum.Thesolid(dotted)linesrefertotheconstraintson temperatureforWDM(CDM)cosmology(theconstraintsinCDMhavebeenshifted byz=0.05 forimprovingthereadabilityofthefigure).Atz=5.0 thereisa1-σ leveldetectionandonlyanupperlimitat2-σ levelinWDMcosmology,instead thereareboth1and2-σdetectionsforCDMcosmology.Atz=⁵.4,thereareonly upperlimitsat1and2-σ levelsforWDMcosmologyand1-σ detectionand2-σ upperlimitforCDMcosmology.Hence,theconstraintsonthetemperaturearesub- stantiallyequivalentinthetwocosmologies.Thebluedashedlineistheasymptotic IGMmeantemperatureinthe caseofearlyhydrogen andfirstheliumreioniza- tionfromastellarionizingspectrumwithslopeα=2,beingtheionizingspectrum Jν∝ν^−α.(Forinterpretationofthereferencestocolorinthisfigurelegend,the readerisreferredtothewebversionofthisarticle.)

Another important property of Fig. 3 is that even assuming CDMcosmology,thetemperatureT₀ isanon-monotonic function ofredshiftandshouldbecolderthan∼^{8000 K}^at^z=⁵.0–5.4,see Fig. 4.²

Theresulting

χ

² for theBayesian analysisis∼^25,^for ³⁰^de- grees of freedom (49 data points − ^{19 free} parameters).This is inagreementwiththefactthatthecovariancematrixisuncertain andthat hasbeen multiplied by a factor that boosts the result- ing error bars by 30%, with respect to the error bars computed by bootstrapping. This is done in the original analysis in order toaccount forpresumed sample variance effect that affect other statisticslikethetransmittedﬂuxPDF. Thesamplevarianceeffect mayaffectthetransmittedﬂuxpowerspectrum,evenifadetailed computationhasnotbeenperformed.

Forcompleteness we havealsoperformed frequentistanalysis forthesame

χ

² consideredintheBayesiananalysis. Asshownin Fig. 5thetwoanalysesareinbroadagreementwitheachother.

Wewouldliketostressthatourresultsdependcruciallyonal- lowingforanon-monotonicredshiftdependenceofT0(z).In[1]it wasshownthatassumingapower-law(monotonic)redshiftdepen- denceforT₀(z)and

γ

(z),onepredictshighertemperaturesofIGM forthe same data.In this case the CDM cosmology is preferred overWDM, leading tothe 2

σ

lower bound mwdm≥³.3 keV [1].

The“binnedanalysis”of[1]gaveresultssimilartothose,reported here.Theauthorsof[1]howeverrejectedtheseresults,considering atemperature jumpat z=^5–5.4 tobe “unphysical” andarguing thatthelow

χ

² isasignofoverﬁtting.

Inour opinion the present analysisimpliesthat moredata is neededto study such a scenario, as it currently does not allow to make any deﬁnitive conclusion andin particular does not allow torule it out. Moreover, asmentioned above, theerror bars in [1] were inﬂated by 30% and therefore we consider the re- duced

χ

²=²⁵/30≈⁰.83 tobeconsistentwith1.Weseethat2

σ

2 Thetemperaturevaluesthatwehaveestimatedathighredshiftcouldbeinac- curate,becausethelowesttemperatureinthesimulationgridwas5400 K.

Fig. 5. Theresultsofthefrequentistanalysis:theχ²−χ_min² versustheWDMmass, mWDM.Therearetwominimaoftheχ²curve,CDMandmwdm=2.7 keV.

lowerbound ontheWDMmassrelaxesdowntom_wdm≥².1 keV (consistently with the results of binned analysis of [1]). More- over, the non-monotonic thermal history makes the WDM with mwdm=^{2–3 keV} ân êqually^good ^fitâs^CDM. ^The ^best^fit ^values of T₀ can be inaccurate as they lie belowthe lowest simulation point inT0 grid.Thereforemoresimulations areneededtosettle thisquestion.Thisiscurrentlyworkinprogress.Intheabsenceof such additionalstudiestheproposed non-monotonicthermalhis- torycannotberuledoutbasedontheexistingLyman-

α

data.

Fortheinterpretation oftheseresultsitis crucialtooverview what is known about the thermal state of the IGM both theoretically and observationally.We arguebelow that the measured thermalhistoryisinagreementwithcurrentmodelsofgalaxyfor- mationandreionization.

4. StateoftheIGMatz∼⁵

The IGM temperature can be determined from the broadening oftheLyman-

α

absorptionlinesinQSOspectra[21–31,16].Alter- natively,it hasbeenproposedto determinetheIGM temperature bymeasuringthelevelofthetransmittedﬂux[32–34,30],however thereisnoagreementbetweenthetwomethodsyet,see[35].

Allthe measurementofthe IGM temperatureintheliterature assumedCDMcosmology. Becauseofthe existingdegeneracybe- tween the IGM temperature and WDM, the assumption of the WDM cosmology could change the deduced values of the IGM temperature. Nevertheless,inthe absenceofsuch measurements, wecompareourestimatesfortheIGMtemperaturewiththemea- surementsbasedontheCDMassumption.

The IGM temperature at z<5 is constrained relatively well to be atthe level T₀ (^8–10)×¹⁰³ ^K [25,22,27,28]. At z=⁶.0 thereisasingle measurement,[29],thatrestrictsT0 totherange 5000<T0<10000 K (68% CL) (see e.g. [1] for discussion). The simplestinterpretation ofthesedata (alsoadopted in[1]) isthat the temperature is growing monotonically with redshift. Instead, given the large error bars of the measurements, and taking into accountadiabatingcoolingonemayexpectadropoftemperature atz∼^{5 with}^a^subsequent^rise^to∼¹⁰⁴^K^at^z∼⁴.6 inagreement withothermeasurementsfrom[25,22,27,28].ThisincreaseinIGM temperature can be explained with an early start of HeII reion- izationpredictedbysomemodels ofreionizationbyquasars,[36]

(seerecentdiscussionofsuch“two-component”reionizationmod- elsin[37]).

In such a scenario,the temperature at 5<z<6 depends on howlong thefirststageofreionization lastedandwhatthetem- peratureofIGM wasat z^6.Âs ^mentionedâbove,^the ^measurement[29]atz∼^{6 has}^largeuncertainties.Theoretically,T0(z=⁶) dependsonhowearlythefirststageofhydrogen(andHeI)reion-

(5)

Fig. 6. Comparisonbetween thelinear transfer functions, T(k), ofthermal relic (WDM)and sterile neutrinos (SN). The dashed (dotted)black lineisthe linear transferfunctionformWDM=2.1 keV (mWDM=3.3 keV)ascomputedin[10].The colored(green,red,cyan)linesarerealisticlineartransferfunctionsfor someof thesterileneutrinomodelswithm^NRP_SN =7 keV.Thelineartransferfunctionswith L6=^{10 and} ¹²^(redând^cyan^lines) âre^partially^warmer^that^the ^lower^bound of[1](thedottedblackline),butstillsatisfytheconstraintsfromthisletter(the dashedblackline)untilthemaximumk-modeusedinthereferencenumericalsim- ulations.Thelinear transferfunctionwithL6=^{8 (green}^line)îs^colder^than^the boundof[1].ThelineartransferfunctionwithL6=0 (blueline)violatesthecon- straintfromthisletter.Thesolidverticallineisthemaximumk-modeusedinthe referencesimulations.(Forinterpretationofthe referencestocolorinthisfigure legend,thereaderisreferredtothewebversionofthisarticle.)

izationshasended, andwhat sources droveit(cf. [38,39]).It has beenspeculatedthathydrogenisreionizedbythemetal-free(Pop- ulationIII)stars, whosespectral hardness predicts highvaluesof thetemperature.However,thepropertiesofPopulation IIIstarsare purely speculative – we do not know how long they lasted and whetherthey wereindeed thesources ofreionization.Forexam- ple,reionizationcould beduetoa moremetalrich populationof starswithsofterstellar spectra[40],leading toa lowervalues of IGMtemperatureatz∼^6.^To^settle^this^question,ânindependent constraint on the ultraviolet background at high redshift would beneeded,howevertherearenosuchmeasurementsto-date.The lower limit of[29] is T0(z=⁶)≈⁵×¹⁰³ ^K ôrêven ^slightly^below,fullyconsistent withthe low valuesatz=⁵.0–5.4 (Table 1) reachedviaadiabaticcooling.

Wenote that an indirectargumentinfavour oftheIGM tem- peraturesathighredshiftsbeing∼¹⁰⁴^K,^is^the^“missing^satellite problem” –high temperaturewould prevent gas from collapsing into dark matter halos with a mass below ∼¹⁰⁷^M, thus sup- pressingtheformationofsmallgalaxies(seee.g.[41–44]),explain- inginparticularthesmallnumberofsatellitesoftheMilkyWay.

However,inWDMcosmologiesthematterpowerspectrumissup- pressed at the smallest scales, thus solving the missing satellite problemevenifthegaswassuﬃcientlycooler.

Finally,we useour results to explore the constraintson ster- ileneutrinodarkmatter[45],resonantlyproducedinthepresence of lepton asymmetry [46–48]. This is a non-thermal warm dark matter,whoseprimordialphase-spacedensitydistributionresem- bles amixture ofcold + ^warm ^dark^matter ^components ^[49,50], demonstratinga shallower cut-off.In Fig. 6 we comparethe lin- eartransferfunction (thesquare rootoftheratioofthemodiﬁed linearmatterpowerspectrumtothat ofcolddarkmatter, T(k)=

PWDM(k)/PCDM(k)) ofthermalrelicWDM withamassm_wdm= 2.1 keV (lowerboundfromthiswork)andamwdm=³.3 keV[1]

withthoseofresonantlyproducedsterileneutrinoswiththemass 7 keV (motivated by the recent reports of an unidentiﬁed spec- tral line at the energy E ∼³.5 keV in the stacked X-ray spectra of Andromeda galaxy,Perseus galaxy clusters, stacked galaxy clusters andthe Galactic Center of the Milky Way [51–53]). We showthat dependingonthevalueoftheleptonasymmetry, L₆≡

10⁶(nνe −ⁿν¯e)/s (see [50] for details) the linear power can be colder than that of thermalrelics with m_wdm=².1 keV (Fig. 6), thus being fully admissible by the data.³ Notice that the non- resonant sterile neutrino dark matter witha 7 keV mass would be excluded atmore than 3

σ

levelby previous constraintsfrom theSDSS[7,6].

5. Conclusionandfuturework

We demonstratedthatthecut-offintheﬂux powerspectrum, observed in the high resolution Lyman-

α

forest data may either be dueto free-streamingofdarkmatterparticles orbeexplained by the temperature of the intergalactic medium. Taking into ac- countmeasurementsatredshiftsz∼6 andatz<5 weseethatif darkmatteriswarm,thisrequiresnon-monotonicdependenceon theIGMtemperatureonz withthelocalminimumatz∼⁵.0–5.4.

Evencolddarkmatterslightlyprefersanon-monotonic T0(z).⁴Im- proving our knowledge of the IGM temperature at z∼^{5–6 will} thereforeeitherresultinverystrongLyman-

α

boundsonDMfree- streaming,essentially excludingitsinﬂuenceonobservablesmall- scalestructures,or(iftemperaturewillbefoundtobewellbelow 5000 K)wouldleadtothediscoveryofWDM.

AmethodthatwouldallowtomeasuretheIGMtemperatureat the redshiftsofinterest waspresented in[16]. Itisbasedon the following idea: for highresolution spectra it is not necessary to studyaveragedeviationfromtheQSOcontinuumperredshiftbins (asitisdoneinlowerresolutioncase)butitispossibletoidentify individual absorption lines andto measuretheir broadening. The thermalDopplereffectbroadensthenaturallorentzianlineproﬁle oftheLyman-

α

transitionproportionallytothesquarerootofthe temperature, andonewouldliketousethisinformationtodeter- minethetemperatureoftheIGMdirectly.However,thereareother effectsthatcontribute tothelinewidth – thephysicalextentand theclusteringoftheunderlyingfilaments.Themethodof[16]po- tentiallyallowstodisentangletheseeffects.Inview ofourresults it is important to attemptto apply this methodto observational data. This is a method that hasbeen tested with simulations at redshift∼^3,ândît^still^has^to^be^seenîfît^worksât^redshift^5.

Acknowledgements

The authors are grateful to Matteo Viel, for sharing with them the code used in [1], and making this analysis possible.

The authors thank James Bolton, Joop Schaye and Tom The- uns for useful discussions on the IGM temperature at high redshift. AG thanks Mark Lovell for sharing his knowledge about generating SN power spectra. She also thanks Bin Hu, Samuel Leach, Matteo Martinelli and Jesus Torrado for useful discussions on MCMC method. The authors thank the anonymous ref- eree for his helpful comments, that improved the manuscript by large extent. The research was supported in part by the Eu- ropean Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreements 278594-GasAroundGalaxies.

3 Ourcomputationsofthephase-spacedistributionfunctionsforsterileneutrinos arebasedon[48]andthelinear powerspectrumisobtainedwiththe modiﬁed CAMBcodedevelopedin[49].Wedonotexpectthemostrecentcomputations[54, 55]toaffectourresults.

4 Asstatedin[56],a modelofﬂuctuatingUVB,withspatiallyconstantmeanfree pathforthehydrogen-ionizingphotons(similartotheoneusedin[1]andinthis work)maynotbeadequatetoexplaintheobservedscatterintheopticaldepthat theredshifts5.1≤^z≤⁵.7.Propermodelingofpatchyreionizationmayaffectthe conclusionsabouttheIGMstate.Weleavethisforfuturework.