Mon.Not.R.Astron.Soc.000,000–000(0000) Printed5February2008 (MNLATEXstylefilev2.2) Galaxy surveys, inhomogeneous reionization, and dark energy 1⋆ 2 1 Jonathan R. Pritchard , Steven R. Furlanetto †, Marc Kamionkowski ‡ 1CaliforniaInstituteofTechnology,MailCode130-33,Pasadena,CA91125,USA 2YaleCenterforAstronomyandAstrophysics,YaleUniversity,NewHaven,CT06520-8121,USA 5February2008 7 ABSTRACT 0 Weexaminetheeffectofinhomogeneousreionizationonthegalaxypowerspectrumandthe 0 consequencesforprobingdarkenergy.To modelfeedbackduringreionization,we applyan 2 ansatzsettingthegalaxyoverdensityproportionaltotheunderlyingionizationfield.Thus,in- n homogeneousreionizationmayleaveanimprintinthegalaxypowerspectrum.Weevolvethis a imprinttolowredshiftandusetheFisher-matrixformalismtoassesstheeffectonparameter J estimation.Weshowthatacombinationoflow-(z = 0.3)andhigh-(z = 3)redshiftgalaxy 7 surveyscanconstrainthesizeofcosmologicalHIIregionsduringreionization.Thisimprint 1 canalsocauseconfusionwhenusingbaryonoscillationsorotherfeaturesofthegalaxypower spectrumtoprobethedarkenergy.Weshowthatwhenbubblesarelarge,andhencedetectable, 2 ourabilitytoconstrainwcanbedegradedbyupto50%.Whenbubblesaresmall,theimprint v 8 haslittleornoeffectonmeasuringdark-energyparameters. 5 Key words: cosmology: theory – cosmological parameters – galaxies: formation – inter- 3 galacticmedium 4 0 6 0 / h 1 INTRODUCTION the IGM, makes it difficult for existing telescope facilities to de- p tect large numbers of early galaxies directly. However, a num- During the epoch of reionization, groups of star-forming - ber of large galaxy surveys now exist, which probe the distri- o regions generate significant numbers of ionizing pho- bution of galaxies in the lower-redshift regime (Efstathiouetal. r tons, which may lead to HII regions many Mpc in size t 2002; Seljaketal. 2005). These surveys focus on high-mass lu- s (Furlanetto,Zaldarriaga&Hernquist2004).Theseionizedbubbles a minous objects – e.g., luminous red galaxies (LRGs) in SDSS grow as further structure forms, eventually merging and causing : (Eisensteinetal.2005)–whichareeasilydetected.Theabundance v full reionization of the intergalactic medium (IGM). Conditions ofsuchobjectswilldependinanon-trivialwayuponthenumber i withintheseHIIregionsmaybesignificantlydifferentthaninthe X oflow-massprogenitors,especiallyupontheamountofcondensed surrounding neutral IGM. For example, the temperature in these r gas available for mergers. Thus, these late-forming galaxies will, HII regions will be raised by photoionization heating, which is a indirectly,beaffectedbytheefficiencyofgalaxyformationduring knowntosuppressstarformationinlow-masshaloes(Rees1986; reionization.Motivatedbythesearguments,weconsiderthepossi- Efstathiou 1992; Thoul&Weinberg 1996; Kitayama&Ikeuchi bilitythatlargegalaxysurveysinthelow-redshiftUniversemaybe 2000; Dijkstraetal. 2004). Also, the ionizing flux generates usedtoprobeinhomogeneousreionizationthroughitsfeedbackon more free electrons, which affects the abundance of molecular early galaxy formation. The many uncertainties remaining in our hydrogen(Oh&Haiman2002),animportantcoolant.These,and understanding of galaxy formation make it difficult to develop a other feedback mechanisms, will affect the fraction of baryons rigourousformalismforthisimprint,andmotivateasimpler,hope- that condense in haloes, and in turn modify the number density fullymorerobust,approach. of directly observable galaxies (Barkana&Loeb 2001). This Besides the possibility of detecting reionization, its imprint suppression will be inherently inhomogeneous, as highly biased inthegalaxypower spectrum mayact asasource of noisewhen regions will ionize first (Babich&Loeb 2006). Understanding probingcosmology.Modernobservationsofthecosmicmicrowave the detailed effects of feedback is one of the major remaining background (CMB) (deBernardisetal. 2000; Halversonetal. challengesinunderstandinggalaxyformation. 2002;Masonetal.2003;Benoˆıtetal.2003;Goldsteinetal.2003; Galaxiesformedduringreionizationwillbelowinmassand Spergeletal. 2003) have greatlyextended our knowledge of cos- faintbycomparisontothosefromlatergenerationsofgalaxyfor- mological parameters. One result has been the realisation that mation. This, along with absorption along the line of sight by ∼ 70% of the Universeiscomposed of anunknown formof energy that generates the accelerated expansion seen in SN Ia observations (Riessetal. 1998; Perlmutteretal. 1999). This is one ⋆ Email:jp@tapir.caltech.edu † Email:steven.furlanetto@yale.edu of the most puzzling discoveries of our times, and it is hoped ‡ Email:kamion@tapir.caltech.edu that future observations in the fields of SN Ia (Riessetal. 1998, (cid:13)c 0000RAS 2 Pritchard,Furlanetto,and Kamionkowski 2004;Perlmutteretal.1999),weaklensing(Hoekstraetal.2005), duetoanionizationfieldx (r)byδ (r) = −ǫ x (r),whereǫ i bub b i b and galaxy surveys (Seo&Eisenstein 2003; Blake&Glazebrook parametrizesthestrengthoftheeffect. 2003) willconstrain thetimeevolution of thedark energygiving Writingthenumberdensityintheformofequation(1)leads cluesastoitsnature.Forthisreason,inthispaper,wewillfocuson toagalaxypowerspectrum howreionizationmayaffectestimatesofdark-energyparameters. 1 Large galaxy surveys contribute information on the dark P(k)= (1−ǫ Q¯)2 [Pgal(k)+2Pgal,bub(k)+Pbub(k)], (2) energy in two main ways. First the form of the matter power b spectrum, probed by galaxy surveys via the proxy of the where Q¯ is the filling fraction of the bubbles. For simplicity, we galaxy power spectrum, depends on different parameter com- choose to neglect the cross-correlation, which will be smaller or binations than the CMB, breaking many of the parameter de- comparableinsizetotheothertermsandrepresentsanunnecessary generacies (Eisenstein,Hu&Tegmark 1999). Secondly, the pre- refinementgiventhesimplicityofourtoymodel.Notetheoverall recombination oscillation of the photon-baryon fluid leaves an rescalingofthepowerspectrumbecausethemeangalaxydensity imprint in the matter power spectrum, which may be used as hni = n¯(1−ǫ Q¯).Typically,ǫ Q¯ ≪ 1andwecanneglectthis b b a standard ruler to determine the angular diameter distance correctionandtake D (z) as a function of redshift z (Seo&Eisenstein 2003; A P(k)=P (k)+P (k). (3) Blake&Glazebrook 2003). These baryon oscillations have now gal bub beendetected(Coleetal.2005;Eisensteinetal.2005)byboth2dF We now need to calculate the bubble power spectrum andSDSS.Iftheimprintinthegalaxypowerspectrumfrompatchy P (k). In order to phrase the problem as broadly as possible, bub reionizationcanmimicorconcealanyfeatureofthegalaxypower weeschewdetailedassumptionsabout reionizationinfavourofa spectrum from density fluctuations, then our ability to constrain more general approach. In this paper, we choose to associate re- darkenergyusinggalaxysurveyswillbedegraded. gionsofionizationwith“bubbles,”inanalogousfashiontothehalo In this paper, we explore the possible consequences of this model’sassociationofmasswithhaloes.Followingthehalo-model environmental dependence on the galaxy power spectrum. The formalism(Cooray&Sheth2002),P (k)isgivenbythesumof bub process of galaxy formation is still only poorly understood and two terms, P (k) = P1b(k)+P2b(k), which describe corre- bub so a detailed analysis of feedback is inappropriate. Instead we lationswithinthesamebubbleandbetweentwodifferentbubbles choose to link galaxy formation to the neutral fraction by a sim- respectively.Thesetermsaregivenby ple ansatz, by which we hope to bring out the underlying be- m 2 haviour, leaving the details for a later age. In keeping with this P1b(k)=ǫ2 dmn(m) |u(k|m)|2, (4) “simpleisbest”ideology,wechoosetomodelthevariationinneu- bZ „ρ¯« tralfractionusingananalogueofthehalomodel(Cooray&Sheth 2002). With this approach we hope to phrase the problem in a m P2b(k)=ǫ2 dm n(m ) 1 u(k|m ) generalfashion,avoidingdetailedassumptionsaboutthereioniza- bZ 1 1 „ ρ¯ « 1 tion history. To address these questions in a quantitative fashion, m we employ the Fisher-matrix formalism (Jungmanetal. 1996a,b; ×Z dm2n(m2)„ ρ¯2«u(k|m2)Pbb(k|m1,m2), (5) Tegmark,Taylor&Heavens1997).Thisallowsustoconvertathe- oreticalmodelintopredictionsfortheparameterconstraintsattain- wheren(m) isthecomoving number density of bubbles of mass ablebyimaginedexperiments. m,P (k|m ,m )isthepowerspectrumofbubblesofmassm bb 1 2 1 Theoutlineofthispaperisasfollows.In§2wedetailtheform and m2, and u(k|m) is the Fourier transform of the bubble ion- oftheionizationpowerspectrumanddescribeoursimpleansatzre- izationprofileu(r|m). Withthisnotation, wemay writethevol- latingittogalaxyformation.Then,in§3,webringthetwotogether umefillingfactorof thebubblesasQ¯ = dmn(m)(m/ρ¯),and describing thecomplete model galaxypower spectrum, including thebubble volumeasVbub = m/ρ¯ = 4πRrb3ub/3, where rbub is theeffectsofredshiftdistortionsandtheAlcock-Paczynskieffect. the comoving bubble radius. Throughout this paper, we will as- In§4weoutlinetheFisher-matrixformalism.Havingset out our sumeatop-hatprofileu(r|m)=Θ(|r|−rbub))/Vbub,forwhich model,in§5wediscussthepossibilityofdetectingreionizationus- u(k|m)=3j1(krbub)/(krbub),wherejℓ(x)isasphericalBessel inggalaxysurveys.Thisisthenexpandedtoconsidertheimplica- functionoforderℓ. tionsfordark-energyconstraintsin§6.Finally,in§7wesummarise If we assume a delta-function size distribution and treat the ourconclusions. power spectrum of the bubbles as tracing the dark-matter power spectrumP (k) ≈ P (k,z = z ),wherez istheredshiftat bb δδ ri ri whichtheimprintisformed,thisreducesto P1b(k)=ǫ2Q¯V |u(k|m)|2, (6) b bub 2 BUBBLEMODEL Wewishtorelatetheoverdensityofgalaxiestotheionizationfrac- P2b(k)=ǫ2bQ¯2|u(k|m)|2Pδδ(k|m). (7) tionwithinagivenregion.Todothis,wemakethesimpleansatz In order to keep our model simple, we ignore evolution in the thatthereisacomponenttothegalaxypowerspectrumwhichlin- bubble-size distribution. In reality, the relevant bubble sizes will earlytracestheionizedfraction.Hence,wemaywritethenumber be determined by the period when most baryons condense, an densityofgalaxiesn(r)atpositionras extended process that will average over the evolution of bubble n(r)=n¯[1+δ (r)+δ (r)], (1) growth.Wealsoignoretheeffectsofbubbleoverlap,whichisex- gal bub pected to occur for large Q¯ and undermines the halo-model ap- wheren¯ isthemeannumberdensityofgalaxies,δ (r) = bδ(r) proach. Once bubbles begin to overlap, using isolated spheres to gal assumes galaxies trace the underlying dark-matter fluctuations δ model the HII regions will not correctly represent the true size withbiasb,andwecalculatethefractionaloverdensityofgalaxies andshapeoftheionizedregions.Toafirstapproximationthough, (cid:13)c 0000RAS,MNRAS000,000–000 Galaxysurveys, inhomogeneousreionization,anddarkenergy 3 bubble power spectrum becomes constant. If, for a given galaxy survey,r issufficientlysmall,thenthecurvatureofthebubble bub spectrumwilllieatk>k ,andthebubblespectrumwillbein- max distinguishablefromshot-noise.Includinggalaxysurveysathigher z,wherethenon-linearscaleissmaller,helpsbreakthisdegener- acy. Wenotethat,forarandomvariablewithzeromean,wewould expectthepowerspectrumtovanishonlargescales.Thatthisdoes not occur relates to a generic problem of the halo model 1-halo term, which isconstant on large scales. In most applications this is masked by a dominant 2-halo term, which does decrease on largescales.However,inourmodelthe2-bubbletermisnegligible making this issue obvious. Given that our model predicts a bub- blepower spectrum thatlookslikeshot-noiseonlargescales,we must worry both about how the removal of shot-noise will affect ourresultsandhowtodistinguishtheeffectofbubblesfromshot- noise. As mentioned above, the existence of a cutoff in the bub- blepowerspectrumdistinguishesitfromshot-noise(althoughwe mustobservethiscutoffforthistowork).Whenwecometoanal- ysetheeffectofbubblesoncosmologicalparameterestimation,we willincludeatermrepresentingwhiteshot-noisetoaccountforthis possibleconfusion. Figure1. Comparisonofthegalaxyandbubblepowerspectra. Weplot Having generated an imprint from patchy reionization, we Pgal(k) at two redshifts z = 0.3 (solid curve) and z = 3.0 (long dashedcurve).Foreachredshift,weplotthenon-linearscale:kmax(z = must evolve it to lower redshift. Our knowledge of how mergers 0.3) = 0.11hMpc−1 andkmax(z = 3.0) = 0.53hMpc−1 (dashed recyclematterfrommanysmallerhaloesintofewermoremassive vertical lines, from left to right). For comparison, we plot Pbub(k) for haloes is not sufficient tohandle this rigorously. Instead, we will the parameters (rbub = 80Mpc,ǫb = 0.6) (short dashed curve) and considerthreecasesthatoughttobracketthetruth.Wetakerbubto (rbub = 20Mpc,ǫb = 0.6)(dottedcurve).Noticehowthelattercurve beaconstant,fixingtheshapeofPbub(k),andthenconsiderhow resemblesconstantwhitenoiseintheregionk < 0.1hMpc−1.Thefor- itsamplitudevarieswithtime.Wewillconsider threemodelsfor mercurvedisplaysacutoffinpowerclosetothegalaxy-power-spectrum thistimeevolution peak. Finally, we plot a bubble power spectrum PFZH(k) (dot-dashed curve)thathasbeencalculatedusingabubblesizedistributiontakenfrom Pbub(k,z =zri), (ModelA), Furlanettoetal.(2004),withhrbubi≈20Mpc. Pbub(k,z)=8>>< Pbub(k,z =0)hGG(z(=z)0)i2, −2 (ModelB), P (k,z =z ) G(z) , (ModelC). >>: bub ri hG(z=zri)i thiseffectwillgiveaneffectivedistributionofbubblesizes,andso InModelA,weassumethat,onceproduced, thepowerspectrum shouldnotaffectourqualitativeconclusions.WewilltakeQ¯ =0.5 P (k) from bubbles remains constant in time. As the density bub andzri =6inwhatfollows,anduse(rbub,ǫb)toparametrizethe fluctuations continue to grow this means that Pbub becomes less bubble power spectrum. Note that the 2-bubble term issubdomi- significant atlater times.InModel B, weallow P (k) togrow bub nantintheregimethatweconsider,makingthedetailsofz ,and as thesquare of the linear growth function G(z). Thus in Model ri anybiasingofP (k)withrespecttoP (k),unimportant. B,P (k) remainsaconstant fractionof thetotal galaxypower bb δδ bub Figure 1 shows the form of the bubble power spectrum in spectrum.Itseemsunlikelythatthebubbleimprintwouldgrowin thismodel. Notethat the power isfairlyconstant on smallk and thisfashion,butweincludethismodelinordertoconsiderthecase cuts off sharply on linear scales smaller than the bubble radius. wherethebubbleimprintisequallyimportantatallredshifts.Note As a simple example of including a smooth bubble size distri- thatinthismodel,wechoosetonormalisethebubblespectrumto bution, Figure 1 shows P calculated using the bubble distri- thepresentday.ThisprovidesasimplewayofrestrictingP (k) bub bub butionofFurlanettoetal.(2004),assuminganionizingefficiency toamplitudescomparabletothedensitypowerspectrum. Finally, ζ =40andQ¯ =0.83,whichgivesavolumeaveragedbubblesize withModelC,weconsiderthecasewhereP (k)decreaseswith bub hr i ≈ 20Mpc.Themaineffectsof thedistributioninbubble time.Thiswillprovideanestimateoftheworst-casescenariofor bub sizesaretosmoothouttheoscillationsseeninthesinglesizemodel detectingthebubbles.ThetimeevolutionofP (k)ismostim- bub andtodecreasetherateatwhichpowerdecreasesonscalesbelow portantwhenwecancomparesurveysatdifferentredshifts.Inthe the characteristic bubble size. Having shown that the high-k cut- caseofasingleredshiftsurvey,anygrowthcanbeabsorbedintoan offoccursevenwithasmoothdistributionofbubblesizes,wewill effectiveǫ forthatsurvey. b henceforthrestrictourselvestothesimpler,singlebubblesizecase. What range of values may our two free parameters ǫ b and r reasonably take? The characteristic size of the bub- bub Theonsetofnon-linearitylimitsthescalesthatagalaxysur- bles is the easiest question to address. Furlanettoetal. (2004) veyisabletoprobe.Wechoosetodefinethiscutoffscalebyrequir- present a model for bubbles forming around highly biased re- ingthattheaveragefluctuationonascaleRsatisfiesσ(R) ≤ 0.5 gions leading to typical sizes of ∼ 5Mpc, when Q¯ = 0.5 forR=π/(2k )(Seo&Eisenstein2003).Thiscutoffcanlead (see also Furlanetto,McQuinn&Hernquist 2006). In contrast, max toadegeneracybetweenthebubblespectrumandtheconstantshot- Wyithe&Loeb (2004) use arguments based on light-travel times noiseexpectedonlargescalesfromnon-Gaussianclusteringofthe andcosmicvariancetoobtainbubblesizesof∼60Mpcattheend galaxies(Seljak2000).WeseeinFigure1thatonlargescalesthe ofreionization.Thislattervaluecanbetakenasanupperlimiton (cid:13)c 0000RAS,MNRAS000,000–000 4 Pritchard,Furlanetto,and Kamionkowski reasonablebubblesizes,whiletheformergivesamorereasonable andwheretheenergydensityindarkenergyisgivenby estimateofwhatwemightexpect.Thesevaluesareinbroadagree- z 1+w(z′) ment with the results of computer simulation (Ilievetal. 2005; Ω (z)=Ω exp 3 dz′ . (13) Zahnetal.2006),whichyieldsizes∼10Mpc. X X „ Z0 1+z′ « The range of ǫb begs the question of how exactly to inter- In the special case of a cosmological constant, the growth factor pretthisparameter.Wehaveassumedalinearrelationbetweenthe maybeexpressedas ionizationfractionofaregionanditsgalaxyoverdensity. Wecan readily see that n(r) ≥ 0, which implies a solid upper limit of G(z)= 5Ω H(z) z 1+z′ dz′, (14) ǫb ≤ 1.Analternativeapproachistoconsider thesuppressionof 2 m H0 Z∞ [H(z′)/H0]3 galaxy formation in haloes of low mass. Simulations at low red- but for a general dark-energy model where w(z) 6= −1, the shift(z<3)(Thoul&Weinberg1996;Kitayama&Ikeuchi2000) full numerical integration is necessary (Wang&Steinhardt 1998; indicatesignificantsuppressionofgalaxyformationinhaloeswith Weinberg&Kamionkowski2003). circular velocities V ≤ 50kms−1. At higher z, photoioniza- circ Figure 1 shows P (k,µ,z) averaged over angle and eval- tionislesseffectiveduetothedecreasedcoolingtime,decreased gal uated at z = 0.3 and z = 3. It displays a clear peak at UV flux, increased self-shielding from the higher densities, and k ≈ 0.02hMpc−1,correspondingtothescaleofmatter-radiation collapse beginning before any UV background can be generated equality, and visible baryon oscillations on smaller scales. These (Dijkstraetal. 2004). In this case, Dijkstraetal. (2004) find that features arise from the acoustic oscillation of the baryon-photon only haloes with V ≤ 20kms−1 suffer significantly reduced circ fluidduringtheperiodoftightcouplingbeforerecombination.The condensationofbaryons.Toestimatethemassfractioningalaxies sound speed, which governs thepeak positions, iswellmeasured affectedbyphotoionizationfeedback,wetakethislattervalueand fromtheCMB.Consequently,thebaryonoscillationsmaybeused applyittothePress-Schecterdistribution(Press&Schechter1974) asastandardruler,allowingadirectmeasurementoftheangular- asalow-masscutoffbelowwhichnogalaxiesform.Thisgivesan diameterdistance.Thesefeatureshavenowbeendetectedingalaxy estimateofthedecrementingalaxiesduetophotoionizationfeed- surveys (Coleetal. 2005; Eisensteinetal. 2005), and their usein back, probing the dark energy is well known (Seo&Eisenstein 2003; ǫ ≈∆¯ ≡ F(M >Mfeedback)−F(M >Mcool) , (8) Blake&Glazebrook2003). b g » F(M >M ) – When converting the observed redshift and angular position cool of galaxies into linear space, we must assume a particular cos- whereF(M)isthefractionofmassinhaloesofmassgreaterthan mology.Ifthisreferencecosmologyisdifferentfromthetruecos- M, M is the mass corresponding to V = 20kms−1, feedback circ mology, then we will introduce distortions into the inferred dis- and M is the mass corresponding to the virial temperature cool tribution of galaxies. This is the Alcock-Paczynski (AP) effect T ≈ 104K needed for effective cooling by atomic hydrogen. vir (Alcock&Paczynski1979)andisessentiallyacosmologicalred- Evaluatingequation(8)givesǫ ≈0.18atz=10andǫ ≈0.10at b b shift distortion. We may express the power spectrum inferred by z = 6,whichgiveanindicationofsensiblevalues.Onceagalaxy ourobservationsintermsofthetruepowerspectrumPtr(ktr,µtr) growslargeenough, gravitywillovercome feedback of thisform by and damp this effect. Thus, these numbers represent an effective upperlimitinthemostplausiblemodel. D2(z)Htr(z) P (k,µ)= A Ptr(ktr,µtr), (15) obs Dtr2(z)H(z) A 3 GALAXYPOWERSPECTRUM whereHandDAarecalculatedusingthereferencecosmology,and HtrandDtrwiththetruecosmology.Wewritethecomponentsof A In constructing our galaxy power spectrum, we follow aFourierwavevectorparallelandperpendiculartothelineofsight Seo&Eisenstein (2003). Incorporating the effects of bias, asktr = (Htr/H)k andktr = (D /Dtr)k .Theinformation || || ⊥ A A ⊥ linear redshift distortions (Kaiser 1987), and linear growth, the containedintheAPeffectcanbeusefulinprobingtheevolutionof galaxypowerspectrumtakestheform thedarkenergy,soweincludeitinthisanalysis. G(z) 2 Thefinalobservedgalaxypowerspectrumincorporatesallof P (k,µ,z)= b2(1+βµ2)2P (k,z=0), (9) theeffectsthatwehavediscussedbeforeandtakestheform gal »G(z=0)– δ where Pδ(k,z = 0) isthe power spectrum of the dark matter at P (k,µ)= DA2(z)Htr(z) P (ktr,µtr)+P (ktr) the present day, b = Ω (z)0.6/β is the bias, and µ2 = k2/k2 obs Dtr2(z)H(z) gal bub m || A ˆ ˜ isthedirectioncosinebetweentheFourier-modewavenumberand +P , (16) shot thelineof sight. Wedefinethe redshift-distortionparameter β in termsofσ8,g andσ8,thefluctuationsingalaxiesanddarkmatter, wherePshotisresidualshot-noisefromnon-Gaussianclusteringof respectively,smoothedonscalesof8h−1Mpc,bytherelation, galaxies (Seljak 2000), which we treat as a constant white-noise term. σ =σ b 1+2β/3+β2/5. (10) 8,g 8 p InordertocalculatethelineargrowthfactorG(z),weintegratethe perturbationequation, 4 FISHERMATRIX G¨(t)+2HG˙(t)−4πGρ G(t)=0, (11) m To quantitatively constrain the effect of bubbles on the galaxy with powerspectrum, weturntotheFishermatrix.Thisformalismal- H2 lowsustoestimatetheuncertaintiesonasetofmodelparameters =Ω (1+z)3+(1−Ω −Ω )(1+z)2+Ω (z), (12) H02 m m X X Θ = (θ1,θ2,...,θN) given some data set. We define the Fisher (cid:13)c 0000RAS,MNRAS000,000–000 Galaxysurveys, inhomogeneousreionization,anddarkenergy 5 matrix(Tegmarketal.1997) Table1.SpecificationforCMBexperiments ∂2logL Fij ≡−fi ∂θ ∂θ fl˛ , (17) Experiment Frequency θbeam σT σP i j ˛Θ0 ˛ WMAP 40 28.2 17.2 24.4 whereListhelikelihood function describ˛ing theprobabilitydis- 60 21.0 30.0 42.6 tributionoftheparametersandΘ0 istheplaceinparameterspace 90 12.6 49.9 70.7 wheretheFishermatrixisevaluated, typicallythepoint ofmaxi- Planck 143 8.0 5.2 10.8 mumlikelihood.GiventheFishermatrix,theCramer-Raoinequal- 217 5.5 11.7 24.3 itystatesthattheminimumuncertaintyonaparameterθ isgiven i by∆θ ≥(F−1)1/2.Thisestimateoftheuncertaintywillbereli- ablepriovidedthatiiΘ0isneartothetruevaluesoftheparameters. SNeontseist:ivFitrieeqsuσenTciaensdarσePinaGreHinz.µBKeapmersFizWeθHbMeambeaismt,hweF=WH(θMbeainmaσrc)s−e2c.. ToevaluateFij,weneedtospecifyamodel,whichdetermines TakenfromEisensteinetal.(1999). thedependenceofthelikelihoodfunctiononΘ,andapointinpa- rameterspacewherewewishtodetermineparameteruncertainties. aretheinversesquareofthedetectornoisefortemperatureandpo- InthecasethatthemodelparametersareGaussiandistributed,the larization,respectively.Formultiplefrequencychannelswereplace Fishermatrixtakestheform w B2withthesumofthisquantityforeachofthechannels. T ℓ 1 ∂µ ∂µ Movingnowtogalaxysurveys,wemaywritetheappropriate F = tr(C−1C, C−1C, )+ C−1 , (18) αβ 2 α β ∂θ ∂θ Fishermatrixas(Tegmark1997) α β whereC isthecovariancematrixforthedata,andµisthedata’s FGAL = kmax ∂lnP(k)∂lnP(k)V (k) d3k , (21) mean.ThiswillbeagoodapproximationinthecaseofbothCMB αβ Z0 ∂θα ∂θβ eff 2(2π)3 observations and galaxy survey. Note that, for our purposes, we wherethederivativesareevaluatedusingthecosmologicalparam- willneedtocombineinformationfromboththeCMBandgalaxy etersofthefiducial modelandV istheeffectivevolumeofthe eff surveys.Whenusedtogetherthesedatasetsbreakmanydegenera- survey,givenby ciesthatarepresentwhentheyareusedalone.Letusconsiderthe Fishermatrixfromeachoftheseinturn. V (k,µ) = d3r n(r)P(k,µ) 2 A CMB experiment may be characterised by a beam eff Z »n(r)P(k,µ)+1– size θbeam and sensitivities to temperature σT and polariza- n¯P(k,µ) 2 tion σP. Given these quantities, the Fisher matrix is given = »n¯P(k,µ)+1– Vsurvey. (22) by(Jungmanetal.1996a,b;Kamionkowski,Kosowsky&Stebbins 1997;Zaldarriaga&Seljak1997) Here the galaxy survey is parametrized by the survey volume V and the galaxy density n(r), which in the last equality survey ∂CX ∂CY we assume to be uniform n¯. In addition, we must specify k , FCMB = ℓ (Cov )−1 ℓ , (19) max αβ ∂θ ℓ XY ∂θ a cutoff on small scales to avoid the effects of non-linearity. We Xℓ XX,Y α α choosetodefinethiscutoffscalebythecriterionσ(R) ≤ 0.5for where CℓX is the power in the ℓth multipole for X = T,E,B, R=π/(2kmax)(Seo&Eisenstein2003). and C, the temperature, E-mode polarization, B-mode polariza- Toapplytheaboveframework,weneedatheoryrelatingthe tion, and TE cross-correlation respectively. The elements of the observablesClX andP(k,µ)totheparameters.FortheCMBthis covariance matrix Cov between the various power spectra are isstandard,whileinthecaseofthegalaxysurveysweuseequation ℓ (Kamionkowskietal.1997;Zaldarriaga&Seljak1997) (16), which arose from our discussion in §2 and §3. Using these models,wecalculatetheFishermatricesforindividualgalaxysur- 2 (Cov ) = (C +w−1B−2)2, veysandourCMBexperiment,andthencombinethem ℓ TT (2ℓ+1)f Tℓ T ℓ sky (Cov ) = 2 (C +w−1B−2)2, FαTβOT =FαCβMB+ FαGβAL,i, (23) ℓ EE (2ℓ+1)f Eℓ P ℓ Xi sky 2 whereilabelsthedifferentgalaxysurveys.ThistotalFishermatrix (Cov ) = (C +w−1B−2)2, ℓ BB (2ℓ+1)f Bℓ P ℓ istheninvertedtogetparametererrorpredictions. sky Forthiscalculation,weneedthespecificationsofourexperi- 1 (Covℓ)CC = (2ℓ+1)f [CC2ℓ+(CTℓ+wT−1Bℓ−2) ments.ThesearegiveninTables1and2.Weconsidertwogalaxy sky surveys.ThefirstusesparameterscorrespondingtotheSDSSLRG ×(C +w−1B−2)], Tℓ T ℓ survey,whichiscurrentlyunderway.Thesecondisahypothetical (Cov ) = 2 C2 , survey at z = 3, based upon a survey of Lyman break galaxies ℓ TE (2ℓ+1)fsky Cℓ (Seo&Eisenstein2003). 2 Finally,wemustdecideuponourchoiceofparameterspace. (Cov ) = C (C +w−1B−2), ℓ TC (2ℓ+1)f Cℓ Tℓ T ℓ Wespecifyourcosmologyusing7parametersdescribingthetotal sky (Covℓ)EC = (2ℓ+21)fskyCCℓ(CEℓ+wP−1Bℓ−2), amryattaemrfprlaictutidoenAΩ2Sm,hs2c,aΩlamr ,sptheectbraarlyionndefxracntsio,noΩptbicha2l,dtheeptihnfltaotiloans-t scatteringτ,andthetensor-scalarratioT/S;eachgalaxysurveyis (Cov ) = (Cov ) =(Cov ) =0. (20) ℓ TB ℓ EB ℓ CB describedbyfiveparameters(logH,logD ,logG,logβ,P ); A shot Here B2 is the beam window function, assumed Gaussian with to these, we add two parameters (ǫ , r ) to describe our bub- ℓ b bub B2 =exp[−ℓ(ℓ+1)θ2 /8ln2],whereθ isthefull-width, blemodel(wechoosetosetQ¯ = 0.5andz = 6).Inchoosing ℓ beam beam ri half-maximum(FWHM)ofthebeaminradians.Also,w andw these parameters, we are following Seo&Eisenstein (2003). We T P (cid:13)c 0000RAS,MNRAS000,000–000 6 Pritchard,Furlanetto,and Kamionkowski Table2.Specificationforgalaxysurveys 0.15 Survey z Vsurvey n¯ kmax σ8,g (h−3Gpc)3 (h3Mpc−3) (hMpc−1) SDSS 0.3 1.0 10−4 0.11 1.8 b 0.1 ε S2 3.0 0.50 10−3 0.53 1.0 Notes:TakenfromSeo&Eisenstein(2003). 0.05 1 20 40 60 80 100 0.8 r (Mpc) bub Figure3.ModelB:Contourmapofdetection.AsforFigure2.Forcom- 0.6 parisonwithotherfigures,notethatG(z =6)/G(z =0)=0.18,sothat b ε [G(z=0)/G(z=6)]ǫbliesintherange[0,1]. 0.4 1 0.2 20 40 60 80 100 0.8 r (Mpc) bub Figure2.Model A:Contour mapofdetection ǫb > 2σǫb andrbub > 0.6 2σrbub inthebubbleparameterplane.Thewhiteregionisdetectable by εb SDSSalone,thegreyregionisdetectablebySDSS+S2,andtheblackre- gionisundetectabletoallsurveys.PlanckCMBdataisassumedinallcal- 0.4 culations. 0.2 treatalloftheaboveparametersasbeingindependentandthenex- tractinformationabout thedarkenergyfromouruncertaintieson 20 40 60 80 100 (logH,logD )fromeachsurvey. Wechoose toparametrizethe r (Mpc) A bub darkenergyusingthreeparameters(Ω ,w ,w ),takingthedark- X 0 1 energy equation-of-state parameter to be w(z) = w +w z. In Figure4.ModelC:Contourmapofdetection. AsforFigure2.Notethe 0 1 deciding on our fiducial values, we follow the results of WMAP greatlydecreasedabilityofSDSSalonetodetectbubbleswhencompared (Spergeletal. 2003) for the cosmological parameters. These are withFigure2. broadlyconsistentwiththeupdatedresultsofSpergeletal.(2006), except forthe decreased Ω andτ.Evaluating theFishermatrix m at these different best-fitting parameters modifies our constraints only slightly.Thebubble parametersarehighly uncertain, and so to make a detection. We shade the region of the (rbub,ǫb) plane wechoosetoexplorealargeparameterspace. whereadetectioncanbemadebySDSSalone(white),SDSSand S2combined(grey),andwherenodetectioncanbemade(black). For models A and C, we consider the region of parameter space withr ∈[5Mpc,100Mpc]andǫ ∈[0.1,1].FormodelB,we 5 POSSIBLITYOFDETECTINGBUBBLES bub b chooseaslightlydifferentnormalisationsothat[G(z =0)/G(z = Nowthatwehaveestablishedatheoreticalframework,wewishto 6)]ǫ ∈ [0.1,1]. This makes the range of amplitude of P at b bub determinewhethertheimprintcanbedetectedusingthespecified z =0identicalintheregioncoveredinFigures2and3.Notethat galaxysurveys.Ournullhypothesisisthatthereisnoimprint,and G(z=6)/G(z =0)=0.18inourfiducialcosmology. weassumethatadetectionrequiresthatwecandistinguishbothǫ First,compareFigures2and4.InmodelA,weseethatSDSS b andr fromzeroatapproximatelythe2σ level;i.e.,werequire alone isable to detect bubbles over a wide range of ǫ , provided bub b bothǫ > 2σ andr > 2σ .Throughout, weassumethe thatthebubblesarelarge(r > 40Mpc).Incontrast,whenwe b ǫb bub rbub bub inclusionofCMBinformationatthelevelofPlanck.Lessprecise allowthebubbleamplitudetodecreasewithtime,asinModelC, CMBdatawillrelaxconstraintsoncosmologicalparameters,caus- weseethatSDSSaloneisalmost unable toconstrain either bub- ingparameterdegeneraciestodecreasethesensitivityofthegalaxy bleparameter.Inbothcases,additionoftheS2surveygreatlyim- surveytothebubbleimprint. provesthesituation,allowingawiderrangeofparameterspaceto Figures2,3,and4showcontourplotsformodelsA,B,and beprobed. However, evenwithS2,thetheoreticallypreferredre- C,denotingregionsofparameterspacewhereoursurveysareable gionwithr < 10Mpcandǫ < 0.2,towardsthebottomleft bub b (cid:13)c 0000RAS,MNRAS000,000–000 Galaxysurveys, inhomogeneousreionization,anddarkenergy 7 handcorner,remainsunconstrained.Theprospectsfordetectionare 6 IMPLICATIONSFORDARK-ENERGYCONSTRAINTS clearlyenhancedbyincludinggalaxysurveysathigherredshift. Currentconstraintsondark-energyparametersarisefromthecom- Figure 3, for Model B, shows that growth improves the binationofhigh-precisionCMBdatawithinformationfromgalaxy prospectsforprobingsmallervaluesofǫ ,butmakeslittlediffer- b surveys. The combination of high-z (z > 3) information, long ence to our ability to constrain the bubble size. As in model A, before dark energy becomes dynamically important, with low- SDSSalonecanonlyprobebubbleswithr > 40Mpc,andS2 bub z (z < 3) information, deep within the dark-energy-dominated isrequiredtoprobesmaller scales.Notethatwhenwenormalize regime,servestobreakmanyofthedegeneraciesthateitherdataset tothepresentday,theinclusionofgrowthinmodelBreducesthe possesseswhenusedalone.Addinginmoregalaxysurveysatdif- amplitudeofthebubbleimprintseenbytheS2surveybyafactor of[G(z =3)/G(z =0)]2 ≈0.1overthatinmodelA.Thisisre- ferentredshiftsfurtherconstrainstheevolutionofthedarkenergy, allowingconstraintsonbothΩ anditsequation-of-stateparame- sponsiblefortheincreasedregionthatisundetectabletoSDSS+S2 X terw(z).Intheprevioussection,weconsideredthebubbleimprint in Figure 3. The amplitude of P at z = 0.3 is very similar bub asausefulsignal;inthissectionweconsideritasapotentialsource inthesetwomodels,resultinginthenearlyidenticalcontoursfor ofnoiseforgalaxysurveys.Ifthebubblepowerspectrumisableto SDSSonly(whenrescaledtoaccount forthedifferentnormalisa- mimictheeffectsofdarkenergy,thenitwilldegradeourabilityto tion)inFigures2and3.Thestrikingdifferencesbetweenthethree constraindark-energyparameters.Throughoutthissection,wewill modelsindicatestheimportanceofthetimeevolutionofthebubble consider a dark-energy model with w = −1 and w = 0. Our imprint. 0 1 numericalresultsdependuponthischoiceofmodel,buttheoverall Itisworthpausingforamomenttoconsiderwhereourlever- pictureremainsthesamewhenw andw takeothervalues. age on thebubble power spectrum originates. When wecombine 0 1 thetwosurveys, thebulkoftheimprovement iscomingfromthe Galaxy surveys provide direct constraints on the dark en- S2surveyalone.Thisisunsurprising,asthegrowthofthedensity ergy through both the baryon oscillations and from the Alcock- fluctuationsmeansthebubbleimprintisamoresignificantcontri- Paczynskieffect.Theyalsoprovideindirectconstraintsincombi- butiontothegalaxypowerspectrumatearlytimes.Further,ifwe nationwithCMBdata,astheyprobeΩmindependentlyofΩmh2, considerFigure1,weseethatforverysmallbubblesizes,thebub- theparameterdirectlyprobedbytheCMB.ThisallowstheCMB blespectrumbeginstoresemblewhitenoiseovertheregionprobed indicationofflatnessΩk ≈0toconstrainΩX.Thebubbleimprint bythegalaxysurveys.Thiswouldfurthercomplicatedetectingthe must interfere with one of these measurements to be a source of bubbleimprintasitcouldthenbeconfusedwithresidualPoisson confusion. shot-noiseinthegalaxycounts.Thisproblemisgreatestatlowz Measurementofthebaryonoscillationsallowadetermination where the non-linear scale is larger. Both of these motivate per- oftheangular-diameterdistance.Theirdistinctiveoscillatorystruc- formingthistestingalaxysurveysatincreasingredshift,ideallyat tureisverydifferentfromthesmoothstructurethatweexpectfrom theredshiftofreionization,whereanHαsurveymaybepossible. anyplausiblebubbleimprintandsowedonot expect theretobe We conclude that galaxy surveys should be sensitive to the anyconfusionbetweenthetwo.Theinferredpeakposition,ampli- imprint inthegalaxypower spectrum leftover after reionization. tude,andoverallshapeofthegalaxypowerspectrum,ontheother However,detectingthisimprintwillbedifficultunlessthecharac- hand,couldbeaffectedbythesmoothformofthebubbleimprint, teristicsizeofHIIregionsislarge(r >10Mpc)andtheeffects makingthesethemostlikelypointsofconfusion.Thus,weexpect bub offeedbacksignificant(ǫb>0.1).Thisshouldbesufficienttocon- parameterssuchasΩmandnstobesensitivetothebubbleimprint. strainthemoreextrememodelsforreionization,butisunlikelyto ThissimplepictureismodifiedbyinclusionofCMBdata,which impactmorereasonablescenarios.Thereissignificantuncertainty places tight constraints on many of these parameters making the in this prediction stemming from the difficulty in predicting the effectofthebubbleimprintmoresubtle. evolutionoftheimprinttomorerecenttimes. Thecorrelationbetweenthedifferentparametersisindicated Currently, the best hope for measuring the size of HII inFigure5,foramodelwithrbub = 10Mpcandǫb = 0.1.Note regions during the early stages of reionization lies with up- thatthereisaweakcorrelationbetweenrbub andthedark-energy coming 21cm observations (e.g., LOFAR1, MWA2, or PAST3). parameters.Aslightlylargercovarianceisseenbetweenrbuband Direct imaging of the HII regions is unlikely with the first ns. At larger values of (rbub,ǫb), the picture remains unchanged generation of detectors, but the prospects for statistical detec- exceptforabreakingofthedegeneracybetweenrbubandǫbasthe tion at z ≤ 10 are good (Zaldarriaga,Furlanetto&Hernquist cutoffinPbub(k)onsmallscalesbecomesmorepronounced. 2004;Bowman,Morales&Hewitt2006;McQuinnetal.2005).At Beforedetailingtheeffectthebubbleimprinthasonstatistical higherredshifts,z > 10,correspondingtolowerfrequencies,sky errors,let usconsider thepossibilityof systematicbiasing of our noise increases dramatically making observations more difficult. best-fittingvalues,ifanexistingbubbleimprintwasignoredinthe Animprintdirectlyuponthegalaxypowerspectrumavoidsthese analysisofdata.Thiswillberelevantonlyinthecasethatthebub- technicalissues,makingpossibleacomplementarymeasurement. ble imprint is not easily detectable, as an obvious imprint would Intheeventofveryearlyreionization,detectionoftheimprintdis- certainlybeincluded inthedataanalysis. For thecase wherethe cussedinthispapermightprovideweakconstraintsonreionization bubblesarenotdetected–i.e.,ǫ <2σ andr <2σ –we b ǫb bub rbub before21cmexperimentsreachthedesiredsensitivity. haveestimatedthissystematicoffsetbetweentheinferredandtrue parameters,usingtheFishermatrixtoapproximatethefulllikeli- hoodsurface.Wefindthattheoffsetissignificantlysmallerthanthe parameteruncertainty,typicallybeingoforder∼0.1%.Fromthis, weconcludethatfailingtoincludetheimprintshouldnotsystem- aticallyaffectparameterestimatesinthenearfuture.Whengalaxy 1 Seehttp://www.lofar.org/. surveysbegintoprobecosmologicalparametersbelowthepercent 2 Seehttp://web.haystack.mit.edu/arrays/MWA/. levelthiseffectwillneedtobeincluded.Wenowturntotheeffect 3 SeePen,Wu&Peterson(2005) oftheimprintonparameterconstraints. (cid:13)c 0000RAS,MNRAS000,000–000 8 Pritchard,Furlanetto,and Kamionkowski 1 0.54 0.5 0.8 0.46 ǫb 0.6 0.42 b ε rbub 0.38 0.4 0.34 A2 S 0.3 ns 0.2 0.26 w1 20 40 60 80 100 r (Mpc) w0 bub ΩX Figure6.ModelA:Contourmapoferrorsinw0inthebubbleparameter plane.Weplotcontoursspanningtherangeσw0 =0.22–0.58inintervals Ωmh2 of0.04.Thefiducialmodeltakesw0=−1andw1=0.Whennobubbles arepresent,wefindσw0 =0.39. Ωmh2 ΩX w0 w1 ns A2S rbub ǫb 0.54 Figure 5. Illustration of the reduced covariance matrix. (Ωmh2,ΩX,w0,w1,ns,A2s,rbub,ǫb). The model uses rbub = 10Mpc 0.15 0.5 andǫb = 0.1.Blackindicatesstrongcorrelationandwhiteindicateslittle correlation. 0.46 0.42 b 0.1 ε Figures 6, 7, and 8 indicate error contours for w over the 0.38 0 (r ,ǫ )plane.Thesameshadingschemeisusedinallthreefig- bub b 0.34 urestoalloweasycomparison.FirstconsiderFigure6.Weseethat the uncertainty on w0 is maximal for bubble parameters rbub ≈ 0.05 0.3 80Mpcandǫ ≈ 0.5.TheformofP (k)isplottedinFigure1 b bub whereweseethatitcutsoffclosetothemaximumofthedensity 0.26 powerspectrum.Thisisconsistentwithourabovestatements.We 20 40 60 80 100 find a maximum uncertainty of σw0 = 0.48 in contrast withthe r (Mpc) uncertaintyσ = 0.39intheabsenceofbubbles.Thisindicates bub w0 thatbubblescanbeanimportantsourceofnoiseinattemptstocon- straindarkenergy.However,thelargevaluesof(r ,ǫ )required Figure7.ModelB:Contourmapoferrorsinw0inthebubbleparameter bub b plane.AsforFigure6. forthiseffectseemtheoreticallyunlikelyandfromthediscussion in §5 would allow direct detection of the bubbles. For more rea- sonablechoicesofbubbleparameters(r <10Mpc,ǫ < 0.1), bub b theuncertaintyonw reducestoσ = 0.39.Thus,theeffectof bubbles are large and feedback strong. This is an interesting ex- 0 w0 thebubbleimprintislikelytobesomewhatimportantinfutureat- ampleoftheAPeffect.Inthisregion,thebubblepowerspectrum temptstoconstraindarkenergy. dominatesoverthedensitycontributionandtheoverallshapeofthe Now consider Figures7 and 8. Theincreased uncertainty in galaxypowerspectrumdisplaysawelldefined,sharpcutoff.Dis- w ismorepronouncedinModelB,wheretheuncertaintyrisesas tortionofthisscalebytheAPeffectplacesgoodconstraintsonthe 0 highasσ = 0.62.Evenhere,forsmallvaluesof(r ,ǫ ),the dark energy. Galaxy surveys already show that the galaxy power w0 bub b bubbleimprintbecomesunimportantandwerecoverσ =0.39, spectrumcloselytracestheunderlyingdensityfield,sothisregion w0 theno-bubbleuncertainty. Againthismaximaluncertaintyoccurs isruledout. closetor ≈80Mpcandǫ ≈0.5.Theincreasedvalueofσ Having considered dark-energy parameters, it would seem bub b w0 is a consequence of the bubble imprint growing at the same rate natural to also consider inflationary parameters; e.g., the tilt n s asthedensityfluctuations.Consequently, theoverallshapeofthe and amplitude A2. For the surveys that we have analysed, inclu- S totalgalaxypowerspectrumremainsconstantintime.Thus,com- sion of the bubble power spectrum makes little difference to the bininginformationattworedshiftsprovidesmuchlessleverageon uncertainty on these parameters. Essentially, all of the informa- separatingoutthetwocomponents,leadingtolargerparameterun- tion needed for constraining these quantities is contained within certainties. In model C,the damping of the imprint means that it the CMB. In the absence of information on the optical depth τ, hasmuchlesseffectonthedark-energyparameters. oriftherearesignificanttensormodes,galaxy-surveyinformation Finally, wenote that Figures 6, 7, and 8 display a region of becomes important in breaking degeneracies. This is not true in decreaseduncertaintyinw inthetoprighthandcorner,whenthe the cases that we consider, where CMB-polarization information 0 (cid:13)c 0000RAS,MNRAS000,000–000 Galaxysurveys, inhomogeneousreionization,anddarkenergy 9 1 constraintheshapeandamplitudeofthebubbleimprint.Withour 0.54 presentunderstandingofreionization,thisseemspremature. Future galaxy surveys will greatly add to our knowledge of 0.5 0.8 thedistributionofgalaxiesandthenatureofthedarkenergy.Ifwe 0.46 aretoextractmaximuminformationfromthesesurveys,wemust tighten our understanding of the biasing of galaxy formation and 0.6 0.42 thepossibleeffectofreionizationonearlygenerationsofgalaxies. b ε ThisworkwassupportedatCaltechinpartbyDoEDE-FG03- 0.38 92-ER40701andNASANNG05GF69G. 0.4 0.34 0.3 REFERENCES 0.2 0.26 AlcockC.,PaczynskiB.,1979,Nature,281,358 20 40 60 80 100 BabichD.,LoebA.,2006,ApJ,640,1 r (Mpc) BarkanaR.,LoebA.,2001,PhysicsReports,349,125 bub BenoˆıtA.,etal.,2003,A&A,399,L25 BlakeC.,GlazebrookK.,2003,ApJ,594,665 Figure8.ModelC:Contourmapoferrorsinw0inthebubbleparameter BowmanJ.D.,MoralesM.F.,HewittJ.N.,2006,ApJ,638,20 plane.AsforFigure6. 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Galaxy surveys, inhomogeneous reionization, and dark energy

Similar Galaxy surveys, inhomogeneous reionization, and dark energy