The Optical Depth to Reionization As a Probe of Cosmological and Astrophysical Parameters Aparna Venkatesan University of San Francisco, Avenkatesan@Usfca.Edu

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2000 The Optical Depth to Reionization as a Probe of Cosmological and Astrophysical Parameters Aparna Venkatesan University of San Francisco, [email protected]

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Recommended Citation Aparna Venkatesan. The Optical Depth to Reionization as a Probe of Cosmological and Astrophysical Parameters. The Astrophysical Journal, 537:55-64, 2000 July 1. http://dx.doi.org/10.1086/309033

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THE OPTICAL DEPTH TO REIONIZATION AS A PROBE OF COSMOLOGICAL AND ASTROPHYSICAL PARAMETERS APARNA VENKATESAN Department of Astronomy and Astrophysics, 5640 South Ellis Avenue, University of Chicago, Chicago, IL 60637; aparna=oddjob.uchicago.edu Received 1999 December 17; accepted 2000 February 8

ABSTRACT Current data of high-redshift absorption-line systems imply that hydrogen reionization occurred before redshifts of about 5. Previous works on reionization by the Ðrst stars or quasars have shown that such scenarios are described by a large number of cosmological and astrophysical parameters. Here we adopt a semianalytic model of stellar reionization in order to quantify how the optical depth to reionization depends on such parameters, and combine this with constraints from the cosmic microwave background (CMB). We Ðnd this approach to be particularly useful in alleviating the well-known degeneracy in CMB parameter extraction between the optical depth to reionization and the amplitude of the primordial power spectrum, due to the complementary information from the reionization model. We also examine translating independent limits on astrophysical parameters into those on cosmological parameters, or, conversely, how improved determinations of cosmological parameters can constrain astrophysical unknowns. Subject headings: cosmic microwave background È early universe È intergalactic medium È stars: formation INTRODUCTION 1. formation in collapsing halos from the outset, and to postu- Observations of the spectra of distant quasars and gal- late a large population of faint QSOs atz Z 5, with the axies have revealed the absence of a Gunn-Peterson trough, observed turnover being true for bright QSOs only. implying that the intergalactic medium (IGM) was highly Stellar reionization is attractive for several reasons. The ionized by redshifts of about 5. Since the universe recom- Ðrst stars are expected to form atz Z 10, and are capable of bined at redshifts z D 1100, the IGM is expected to remain ionizing hydrogen. Furthermore, they create heavy ele- neutral until it is reionized through the activity of the Ðrst ments, and may account for the low (0.01Z_) but persistent luminous sources. At present, it appears that the reioniza- metallicity seen in the Lya forest clouds out to z D 4. tion of hydrogen occurred before z D 5 (Schneider, Schmidt, Finally, the detections of a He II absorption trough in high-z & Gunn 1991; Hu, McMahon, & Cowie 1999; Spinrad et quasar spectra appear to indicate a soft component to the al. 1998), while that of doubly ionized helium is thought to UV background (Haardt & Madau 1996), consistent with have occurred before z D 3 (Hogan, Anderson, & Rugers the ionizing spectrum of stars. 1997; Reimers et al. 1997). Early work on hydrogen reionization (Arons & Wingert The ionizing sources responsible for reionization can be a 1972) described the appearance of the Ðrst luminous variety of astrophysical objects, and much work has been sources, about which ionized bubbles gradually expand into done on reionization by the Ðrst stars (Haiman & Loeb a neutral IGM; eventually, such H II regions overlap and 1997; Fukugita & Kawasaki 1994), the Ðrst quasars the universe becomes transparent to Lyman-continuum (Haiman & Loeb 1998a; Valageas & Silk 1999), protoga- photons. In principle, only one ionizing photon per neutral laxies (Cen & Ostriker 1993; Gnedin 2000; Giroux & atom in the IGM is required, but the e†ects of recombi- Shapiro 1996; Ciardi et al. 2000; Miralda-Escude, Haeh- nations ensure that for an atom to stay ionized, the rate of nelt, & Rees 2000; Madau, Haardt, & Rees 1999), and ionizing photons generated by sources must be greater than related phenomena such as supernova-driven winds the rate of recombination at that epoch. This is of particular (Tegmark, Silk, & Evrard 1993) and cosmic rays (Nath & importance at high z, when the IGM had greater density. Biermann 1993). Quasars are natural candidates for both Just how much more than 1 photon per baryon is needed is H I and He II reionization, since they are more luminous a function of the cosmology and the assumptions of the and provide harder ionizing radiation than stars, and they reionization treatment; some evolution with redshift is also are seen up to the highest redshifts of current observations. expected. A qualitative assessment can be made, however; However, there are indications of a turnover in the space for example, from arguments of producing the average IGM IV ~2 density of the QSO population, which apparently decreases metallicity in C of about 10 Z_ atz [ 5, Miralda- beyond z D 3. Since this is based on optical surveys, this Escude & Rees (1997) arrived at a requirement of 20 ion- observed decline is subject to the e†ects of any dust obscur- izing photons per baryon. For the stellar reionization model ation along the line of sight; however, recent radio obser- that will be considered in this work, we will see below that vations also appear to indicate a declining QSO population about 5 ionizing photons per baryon are available for H I beyond z D 3, so that the comoving emission rate of reionization and prove sufficient. Not more than a few Lyman-continuum photons from QSOs is deÐcient by a percent of all baryons need to participate in early star for- factor of D4 relative to that required for reionization (see, mation for reionization to occur by z D 5, although this e.g., Madau 1998 and references therein). If this is real, then number may reach values of up to about 15% (° 2). QSOs may be less plausible as sources for H I reionization Although reionization by early stars would occur at red- (see also Rauch et al. 1997). The alternative is to allow QSO shifts well beyond current observations, it has many distinct 55 56 VENKATESAN Vol. 537 consequences that can be feasibly constrained by current dark matter (SCDM) model are considered to be relatively and future experiments. Some of these include, as men- successful at describing the observed universe. This picture tioned above, the evolving IGM metallicity and the cosmic postulates a critical-density universe, with cold dark matter ionizing background as derived from spectral features in dominating the matter content; structures, made up of high-z absorption-line systems. High-z reionization will baryons and CDM, originated in primordial adiabatic Ñuc- also leave a signature in the CMB through the Thomson tuations and evolved subsequently through gravitational scattering of CMB photons from free electrons (Sugiyama, instability. Current modiÐcations to this paradigm include, Silk, & Vittorio 1993; Dodelson & Jubas 1995; Tegmark, e.g., the addition of a cosmological constant. Silk, & Blanchard 1994; Tegmark & Silk 1995; Hu 2000; The SCDM model, and its variants, are described by a set Zaldarriaga 1997). Depending on the epoch and degree of of parameters that characterize the primordial power spec- reionization, we expect an overall (somewhat scale- trum of Ñuctuations, the cosmology of the universe, and dependent) damping of primary temperature anisotropies in quantities related to primordial nucleosynthesis. At present, the CMB, the generation of new temperature anisotropies the extraction of such parameters from observations has on the appropriate scales through the e†ects of second- been made feasible by the quality of data from large-scale order processes and the degree of inhomogeneity in the structure surveys, from cosmic velocity Ñows (Zehavi & reionization process (Gruzinov & Hu 1998; Knox, Scocci- Dekel 1999), from Type Ia SNe (Tegmark 1999), and from marro, & Dodelson 1998), and Ðnally, the creation of a new current and projected future data from the CMB polarization signal, as the process of Thomson scattering (Zaldarriaga, Spergel, & Seljak 1997; Eisenstein, Hu, & introduces some degree of polarization even for incident Tegmark 1999). Typically, a 9È13 parameter set describes radiation that is unpolarized. Scattering from the ionized the adiabatic CDM model, and can be solved for given the IGM, or the reprocessing of starlight into far-infrared wave- data. One of these parameters isqe, which is by nature lengths by dust formed from early supernovae (SNe), will somewhat unique, in that it is the only quantity that is not also cause the CMB to undergo some spectral distortion set purely by the physics prior to the Ðrst few minutes after (Loeb & Haiman 1997); this can be measured experimen- the Big Bang. Thus, it can potentially provide clues as to tally through the Compton y-parameter. These and other postrecombination astrophysics, assuming that the other observational signatures that have the potential to con- (cosmological) parameters that also a†ectqe are compara- strain the epoch, and hence possibly the source, of reioniza- tively well constrained. tion have been examined in the literature (see Haiman & Several of the semianalytic works on reionization men- Knox 1999 for a summary). tioned above have explored the e†ects onqe of varying A model of reionization is therefore, in principle, emin- model parameters. Other authors have performed CMB ently testable. Current detections of the Ðrst Doppler peak analyses that have revealed inherent degeneracies in con- in the CMBÏs temperature anisotropies limit the total straining speciÐc combinations of parameters, e.g.,qe and optical depth to electron scattering,qe, such as may arise the amplitude of the primordial power spectrum, A (see, e.g., from reionization, to beqe [ 1 (Scott, Silk, & White 1995). Zaldarriaga et al. 1997; Eisenstein et al. 1999). In this paper, Future experiments such as the Next Generation Space we examine the results of cross-constraining the range in a Telescope (NGST ) or the Space Infrared Telescope Facility cosmological parameter, given the allowed band inqe from (SIRT F) may detect the high-z sites of reionizing sources a reionization model due to astrophysical parameter uncer- (see, e.g., Haiman & Loeb 1998b), or at least exclude cur- tainty, with the permitted range from CMB observations. rently viable candidates, while upcoming CMB experiments SpeciÐcally, we Ðnd that for the combinationqe-A, the well- such as MAP or the Planck surveyor can measureqe to very known degeneracy in their e†ects on the CMB can be high accuracies by combining information from tem- broken when used in combination with the constraints from perature anisotropies and polarization in the CMB. a reionization model. Sinceqe depends on both cosmo- The optimistic prospects for testing reionization and the logical and astrophysical parameters, however, such an increasing multiwavelength view of the high-z universe have analysis can be extended to mutual constraints involving generated a large body of work on reionization models in these two independent classes of parameters, by eliminating the last few years, whose techniques fall broadly into qe. The advantage of this is that, given a model of structure numerical (Gnedin 2000; Chiu & Ostriker 2000; Gnedin & formation and a reasonable framework describing reioniza- Ostriker 1997; Cen & Ostriker 1993) or semianalytic tion, as well as the data from the CMB, we can use known methods (Haiman & Loeb 1997, 1998a; Tegmark et al. astrophysics to further constrain cosmology and place 1994; Valageas & Silk 1999). The former have the advan- tighter limits on cosmological parameters, even those that tage of being able to track the details of radiative transfer, will be determined to high accuracies by future experiments. incorporating the clumpiness of the IGM and the essen- Conversely, a well-determined cosmological parameter can tially nonuniform development of ionizing sources, and, be used to constrain the astrophysics of ill-known details of perhaps most importantly, describing the process of reioni- early star formation. This will, at the least, be a powerful zation in a quantitative fashion. The advantage of semi- test of the cosmology, if our understanding of reionization is analytic approaches is their inherent Ñexibility and ability reasonably correct; the additional hope is that this will to probe the parameter space of a reionization model at prove to be an alternative way of constraining the activity will, which is of value given the many input cosmological of the Ðrst stars. and astrophysical parameters involved. The plan of the paper is as follows. In ° 2, we outline the For astrophysical sources, the process of reionization is stellar reionization scenario that we consider, and set up a strongly related to the evolution of structure in the universe, parameter set that describes reionization. In ° 3, we review and could result in feedback e†ects for subsequent object the standard methodology related to parameter extraction formation (see, e.g., Ciardi et al. 2000). Of the current theo- from the CMB, and incorporate the parameter set from ° 2 ries of structure formation, variants of the standard cold into this formalism. In ° 4, we present our results, and show No. 1, 2000 REIONIZATION AND PARAMETER ESTIMATION 57 the combined constraints from a reionization model and the (Tegmark et al. 1997; Haiman, Rees, & Loeb 1996) have CMB. We present our conclusions in ° 5. argued for a higher value ofMC, based on the requirement THE STELLAR REIONIZATION MODEL of an e†ective coolant for baryons in a halo to fragment into 2. stars. The picture is as follows: the very Ðrst stars form from We assume a SCDM primordial power spectrum of metal-free gas and cool through primordial molecular density Ñuctuations, given by P(k) \ AknT 2(k), where A and hydrogen. The universe at that epoch is transparent to n are, respectively, the amplitude and index of the power photons in the energy range 11.2È13.6 eV, but is opaque to spectrum, and where the matter transfer function, T (k), is more energetic photons. This initial trace level of stellar taken to be of the form given in Bardeen et al. (1986), with activity easily photodissociates the remainingH2 in the uni- the baryon correction as given by Peacock & Dodds (1994). verse (whose abundance relative to H I is very low), well Here we evaluate the power-spectrum normalization A before the H I ionizing Ñux has built to values sufficient for through the value of the rms density contrast over spheres reionization (see HL97 and references therein). Subsequent ~1 of radius 8 h Mpc today,p8. The cosmology of our model halo formation continues, but star formation halts since is also described by) , the cosmological density param- there is no coolant available, and can only resume when 0 8 ] ~1.5 eter;) , the density parameter of baryons; and h, the halos more massive than 10 M_[(1 z)/10] collapse, b ~1 ~1 Hubble constant in units of 100 km s Mpc . We assume which can utilize line cooling by atomic H. We setMC to be \ that)0 1 throughout this work, and thus do not include this higher value, with the understanding thatFB now rep- it in the parameter set to be varied in what follows. We resents the fraction of all baryons that are in star-forming assume that there are no tensor contributions to P(k), and halos. Finally,z is the earliest redshift at which the Ðrst * set the cosmological constant to be zero. Thus, our cosmo- stars can form, and here we set it to be 100. Prior to this, the logical parameter set is (A, n, )b, h). temperature of the IGM is coupled to that of the CMB, and The reionization of the universe by the Ðrst generations of the CMB photons are still energetic enough to photo- ~ stars is described by the model developed in Haiman & destroy H , thus preventing the formation ofH2 through Loeb (1997, hereafter HL97), with the minor modiÐcations the H~ channel, an important cooling mechanism for struc- described below. BrieÑy, the fraction of baryons in col- tures at these high redshifts to fragment into stars. lapsed dark matter halos,FB, is followed using the Press- The evolution of an individual ionization front is charac- Schechter formulation; of these baryons, a fractionf cool terized by the ionization radiusri, and, for a time-dependent and form stars in a Scalo initial mass function (IMF).* A source luminosity, can be solved for through a di†erential fractionfesc of the generated ionizing photons is assumed to equation as in HL97, where the rate of emission of ionizing escape from the host object and propagate isotropically into photons from a stellar population of metallicity Z \ 10~4 the IGM. One can then solve for the size of the ionized Z_ was constant for about 2 Myr before declining with the regions associated with each such star-forming cloud, death of the massive stars in the IMF. In this work, we use which, when integrated over all haloes, yields at each z the the analytic solution from Shapiro & Giroux (1987; here- average ionization fraction of the universe, given by the after SG87) for the evolution ofri in an expanding universe Ðlling factor of ionized hydrogen by volume(FH II). We in units of theStro mgren radius,rS. The Stro mgren radius assume that the IGM is homogeneous, in which case the represents the equilibrium reached in the IGM between a ionized region created by each source can be taken to be sourceÏs ionizing photon rate and the IGMÏs recombination spheres of radiusr . Reionization is deÐned to occur when rate;rS increases with decreasing z, or equivalently, with \ i FH II 1. The total optical depth for electron scattering, decreasing average IGM density. The maximum value that qreion, to the reionization redshift,zreion, is given by inte- ri can have isrS ; only sources at very high redshifts (D100) grating the product of the electron density, the ionization have ionized regions that Ðll theirStro mgren spheres fraction, and the Thomson cross section along the line of (SG87). Note that since the SG87 solution does not account sight from the present tozreion. The cosmology of the uni- for time-varying sources, we expectqreion to be overesti- verse entersqreion through the Ðrst two terms of the inte- mated (reionization occurs earlier) compared to HL97, but grand, and also through the path length of the photons last as we show in the next section, this is a very slight e†ect. scattered at z . Thus, in this work,r \ (r /r ) r , where r3\3S(0)/ reion 2 \ ]i ~13i S SG873 S~1 S Our adopted model is summarized by the following equa- [4naB nH(z)], aB 2.6 10 cm s , and the initial \ tions for an)0 1 universe: emission rate of ionizing photons leaving the host object is S(0). HereFH II(z) (see eq. [2]) is determined at each redshift \ C dc D z by integrating over the product of the rate of new halos FB(z) erfc , (1) J2p(R, z) that formed stars at a turn-on redshiftzon (where z \ zon \ z ) and the ionized volume per unit mass associated with z dF 4n * F (z) \ o (z) P dz B (z )C r3(z , z)D , (2) such objects, for a source mass M. A detailed treatment of H II B on dz on 3M i on the evolution ofF andF with z, given various choices of z* B H II zreion input parameters, can be found in HL97. For their standard \ P J ] [ ~3 qreion 0.053)b h dz 1 z[1 f FB(z)]FH II(z) . (3) model,FB rises rapidly, from values of D10 at z D 20 to 0 * about 0.1 at z D 10; during this period,FH II rises steeply The critical overdensity, d 4 1.686, p(R, z) is evaluated from about 10~4 (z D 32) to unity at z D 18, so that reioni- c P with a spherical top-hat window function over a scale R zation occurs relatively quickly with the growth of FB. MC, whereMC is the minimum halo mass that collapses at a We now see thatqreion is a function of several cosmo- given redshift. While a natural choice forMC is the baryonic logical and astrophysical parameters. The astrophysical Jeans mass, given byD106 M [(1 ] z)/100]1.5, this parameter set is( f , M , S(0)). The parameter S(0) is itself a _ * C assumes that collapsing halos at a given mass scale are function of several variables: the choice of the IMF, the equivalent to star-forming halos. However, several authors metallicity Z of the progenitor stars,f , f , and the haloÏs esc * 58 VENKATESAN Vol. 537 mass. The last two factors account for the fraction of star- intermediate-mass stars (2È8M_). Thus, if the IMF was forming baryons in each halo, whilefesc represents the loss di†erent in the past, the carbon abundance in the Lya forest of ionizing photons to the host cloud before reaching the does not constrain the massive or reionizing stellar activity IGM. Let us now consider the Ðrst two variables. Although in early halos. The second point to note is that the pause in there have been some arguments for the IMF to be biased star formation caused by the initial dissociation ofH2 led to toward high-mass stars in the early universe (Larson 1998), the choice ofMC, as in HL97. This minimum halo mass the details of the nature of star formation under those con- corresponds to objects with virial temperatures of D104 K. ditions are still not well understood. In the absence of a At this temperature, the host object is immune to photoion- convincing theory of primordial star formation, the most ization heating (as pointed out in HL97), and so if outÑows reasonable assumption is to take a present-day IMF and are desired to expel the generated metals into the IGM (i.e., calculate the luminosity expected from metal-poor stars. the Lya forest), the mechanical energy of SNe must be Since reionization is a†ected primarily by the massive stars invoked. Again, the massive reionizing stars end their lives in any IMF, an IMF in the past biased toward high-mass as Type II SNe an order of magnitude in time before their stars would still have the same emission spectrum of ion- intermediate-mass carbon-producing compatriots. Since all izing photons, while one dominated by low-mass stars is the mechanical input lies with Type II SNe, the question unlikely to reionize the universe by z D 5. We therefore take arises of how the carbon, produced signiÐcantly later, leaves \ ~4 S(0) to be as given in HL97 for Z 10 Z_ stars from the host halo to mix with the IGM. Furthermore, Type II standard stellar evolutionary models; it includes the ion- SNe occur on much more predictable timescales, i.e., imme- izing radiation from stars only, and is steady at f f ] diately following the progenitorÏs death, than do Type Ia 46 ~1 ~1 * esc 10 photons s M_ for about 2 Myr before declining SNe (3È10 Gyr), and there is not much more than a wheeze rapidly, which is consistent with the value in, e.g., Ciardi et to be had from the deaths of intermediate-mass stars as al. (2000). As a rough estimate, this translates to D5 ion- planetary nebulae. izing photons per baryon in the universe, for f \ 0.05, Having voiced these objections, we point out that while f \ 0.2. * the 12C connection as made above betweenM andf is esc C * The ionizing photon contribution from SNe is relatively not ideal, postulating a general relation between these two small (HL97), but most of the mass assigned to forming variables is not ad hoc. The value ofMC does intrinsically stars is eventually returned to the IGM. Therefore, the determine the stellar history, metallicity, and luminosity second factor on the right-hand side of equation (3) for qreion evolution of the universe; the high value ofMC in HL97 and should have an extra contribution,fSN f FB, to account for other works is physically well-motivated by the necessary the extra baryons that are available for* new star formation, step of having an available coolant to aid star formation. \ 8 ] ~1.5 wherefSN is an IMF-averaged fraction of the progenitor We proceed to setMC 10 M_[(1 z)/10] for the mass that is expelled into the IGM at the end of the starÏs semianalytic treatment here, and now narrow our astro- life. We expect between D50% and 95% of the parent starÏs physical parameter set to ( f , f ). esc * mass as ejecta, until a point is reached (for stellar masses We end here by addressing some of the issues that are not ranging from 50 to 100M_) where the entire star collapses accounted for in this work. The IGM is assumed to be into a black hole. For the low values off that will be homogeneous, but clearly some clumpiness will develop in * considered here, the productf fSN will be a small correction the IGM from the growth of initial density inhomogeneities, and can be ignored. Moreover,* the mass of the ejecta from a and the assumption of the average ionized fraction at a dying star depends sensitively on the stellar metallicity, with given redshift being equal to the H II Ðlling factor will even- low-Z stars having higher remnant masses and less ejected tually break down. However, this appears to be a relevant material relative to solar-Z stars (Woosley & Weaver 1995). e†ect only at ““ late ÏÏ times(z [ 10), when the fraction of Thus,fSN is likely to be highly variable, both spatially and baryons in collapsed structures becomes signiÐcant (Gnedin with z, due to the evolving metallicity of subsequent gener- & Ostriker 1997), or for baryon-dominated universes ations of stars, and is more appropriately modeled in a (SG87). Therefore, we assume that the clumping factor is simulation rather than in a semianalytic model. The calcu- unity (homogeneous IGM) for the rest of this work. We lated values ofqreion here can be taken as a lower limit. have also neglected corrections from doubly ionized helium, This leaves the astrophysical parameters,f , fesc, and MC. which is not problematic, since the spectrum of photons We note that in most semianalytic models,f* is set by the produced by stars is softer than that from quasars, and is * 12 I II choice ofMC, since the stellar Z output, particularly in C, more relevant for H than for He reionization (see, is combined with the evolution ofF to produce the obser- however, Tumlinson & Shull 2000 on the helium-ionizing B \ vationally detected average carbon abundance of 0.01 Z_ spectrum from zero-metallicity stars). We have set FHe II in the Lya forest clouds at z D 3 (Songaila & Cowie 1996). FH II, but this introduces an error of not more than a few Thus,f andMC are not independent of each other if we percent (Tegmark & Silk 1995). Finally, we have not choose* the above normalization; for the HL97 choice of included the e†ects of bias in the normalization of the \ MC, f 0.13. This is the maximum value thatf can have matter power spectrum, i.e., we assume that light traces the from* arguments of avoiding IGM overenrichment;* given underlying mass distribution. the approximately order-of-magnitude scatter in the CONSTRAINTS FROM THE MICROWAVE BACKGROUND average metal abundance of the Lya clouds, and that one 3. need not require the reionizing stars to solely account for As discussed in the introduction, signatures from reioni- Z , f may be smaller than D0.15. zation are expected in the CMB; an accurate measurement IGM * As an aside for the interested reader, we note here two ofqreion or the detection of postrecombination features in drawbacks of normalizingf via 12C. One is that the the CMB anisotropies have the power to constrain the * massive stars in the IMF(Z10 M_) are the ones relevant reionization epoch and the nature of the sources through for reionization, while 12C is produced dominantly by the the angular scale h(Pl~1) on which they a†ected the CMB. No. 1, 2000 REIONIZATION AND PARAMETER ESTIMATION 59 \ \ Here l is deÐned from expanding the angular power spec- We will consider two cases here: lmax 400, fsky 0.01, trum of the CMB in terms of its multipole momentsC and which is roughly representative of data from current CMB l \ \ Legendre polynomials: experiments; andlmax 3000, fsky 0.8, for the data expected from Planck. Since we neglect any experimental or = (2l ] 1) systematic e†ects, the power of theC Ïs to constrainP , as C(h) 4 ; Cl Pl(cos h) . (4) l N l/2 4n presented here, is the ““ best possible ÏÏ case. Note also that the above formulae are valid when only the temperature The e†ect ofqreion is to introduce an overall damping of information from the CMB is used. More general expres- the temperatureClÏs (HL97, and references therein), except at the largest scales. As discussed by, e.g., Zaldarriaga et al. sions for the case of including polarization data can be (1997), this is practically indistinguishable from the gener- found in, e.g., Zaldarriaga et al. (1997). The derivatives of theC Ïs with respect toP were com- ally reduced values ofCl expected from simply having a l N lower amplitude, A, of the primordial power spectrum; the puted for each parameter using two-sided derivatives with di†erence between these two e†ects at the smallest lÏs is step sizes chosen so that the value of this derivative obscured by cosmic variance. While the amount of the remained stable (see, e.g., Eisenstein et al. 1999, Appendix B.1). The values of theC Ïs themselves for a given damping due toqreion is l-dependent and can potentially l parameter set were found using the publicly available break this degeneracy, the accuracy to whichqreion can be estimated from temperature anisotropy maps alone is not CMBFAST code (version 2.4.1).1 Note that the parameter set describing the reionization model is (A, ) , h, n, f , f ), sufficient to distinguish between these two di†erent e†ects b esc * (Zaldarriaga 1997). which yieldsqreion, whereas the CMB data can determine When combined with the polarization data from the the cosmological parameters andqreion, or equivalently, [P , q (P , P )]. Therefore, when we specify q CMB however,qreion can be constrained with far greater cosmo reion cosmo astro precision. Linear polarization is generated by the primary (e.g., to CMBFAST), the cosmological and the astro- temperature quadrupole anisotropy photons scattering o† physical parameter sets(Pcosmo, Pastro) are no longer inde- the free electrons in the reionized IGM, and is a relatively pendent, but are related through the reionization model, clean probe of the epoch of reionization (Zaldarriaga 1997). and theCl derivatives become The polarization signal is expected at low levels compared LC LC LC Lq to that from temperature anisotropies, and may prove diffi- l \ l K ] l K , (7) LP LP Lq LP cult to measure, especially for low optical depths. Neverthe- cosmo cosmo q Pcosmo cosmo less,q should be able to be detected, in principle, by LC LC Lq reion l \ l . (8) future experiments to within 1 p errors of, e.g., 0.69 without LP Lq LP polarization (0.022 with polarization) information for MAP, astro astro and 0.59 (0.004) correspondingly for Planck (Eisenstein et al. Once the Fisher matrix,Fij, has been constructed, it can 1999). be inverted to give the covariance matrix C between the parametersP ; C represents the minimum variance in the We wish to combine the constraints on cosmological or N ii ] astrophysical parameters from a reionization scenario with estimate ofPi. Any 2 2 submatrix of C can then be those from the CMB; in order to do this for the latter, we extracted, giving the ellipse equation for the joint con- follow the standard prescription as outlined in, e.g., Ðdence region in the two-parameter subspace of interest, [ Æ ~1 Æ [ \ Jungman et al. (1996) and Knox (1995). We assume Gauss- [P PN] (C2C2) [P PN] * , (9) C ian initial perturbations, and that the multipole moments l where * is set throughout this work to be at the 68% con- are determined by a ““ true ÏÏ set of N theoretical parameters, Ðdence level. 4. RESULTS (PN). If we deÐne the likelihood function L of observing a set ofClÏs, givenPN, then the behavior of L near its We present our results here from combining the reioniza- maximum can be quantiÐed in terms of the Fisher information model (° 2) and the constraints from the CMB (° 3). tion matrix, whose elements are given by the second deriv- This analysis assumes that the density perturbation spec- ative of the logarithm of L with respect to pairs of trum at the CMB and structure formation scales is parameters inPN. The Fisher matrix then represents the described by the same power law. For the choice of param- accuracy with whichP can be estimated from a given data eters in HL97, wheref \ 0.13 andf \ f (z), we obtain N * esc esc set; here, the CMBÏs experimentally measuredC Ïs. Further q \ 0.0734, orz D 18.4, which only slightly exceeds l reion reion \ assuming that L has a Gaussian form near its maximum, the HL97 value ofqreion 0.07. Henceforth, we will refer to the Fisher matrix is given by qreion as q for convenience, noting that q is always evaluated to the reionization epoch in this work. We deÐne our stan- = 1 LC (P ) LC (P ) F \ ; C l N l N D ,1¹ i, j ¹ N , (5) dard model (SM), Ðxing) \ 1, as given by A(p \ 0.7) \ ij p2 LP LP ] 6 \ \0 \ \ 8 \ l/2 l i j 1.55 10 , )b 0.05, h 0.5, n 1.0, fesc 0.2, f 0.05, and q \ 0.0573, with reionization occurring at z D 15.5.* wherepl is a measure of how the observedClÏs are distrib- As a simple example, we begin with the q-A plane, shown uted about the mean value of the trueCl(PN)Ïs. We assume in Figure 1, where we isolate the dependence of q on A, thatpl is cosmic variance limited, and we ignore terms arising from the instrumental noise associated with an keeping all the other parameters in the SM Ðxed. The range experiment and from any foregrounds. For a sky fraction ofp8 is D0.5È0.8, from the large-scale distribution of clus- ters of galaxies (see, e.g., Bunn & White 1997, and references fsky that has been mapped,pl can be approximated by therein), but is 1.2 when normalized to COBE for the SM 4 2 choice of cosmological parameters. As an illustrative range S C (P ) l ¹ l p \ 50 (2l ] 1) f l N max (6) l sky 1 O l [ l . CMBFAST is available at: http://www.sns.ias.edu/Dmatiasz/ 06 max CMBFAST/cmbfast.html. 60 VENKATESAN Vol. 537

FIG. 1.ÈCombined constraints in the q-A plane from the reionization FIG. 2.ÈMagniÐed version of Fig. 1; the additional light nested band model and current CMB data. The standard model (SM) is shown by the represents the uncertainty in the value offesc alone. solid line; the shaded band represents the astrophysical uncertainty in the reionization model, given the allowed ranges for( f , f ); and the ellipse esc * shows the 1 p joint conÐdence region from current CMB data. cosmology and the astrophysics of reionization to constrain each other, given a reionization model. The ellipse here, as \ in Figure 1, reveals the inherent degeneracy between for the plots, we normalize A forp8 0.5È1.2. The solid line in Figure 1 represents the SM, while the light-shaded region q-related quantities and A through the long narrow ellipse. represents the uncertainty due to the astrophysics of the We now overplot the DSF permitted band forfesc (0.1È0.2); clearly, even this approximate range inf considerably reionization model (or AS, for astrophysical slop), with ( fesc, esc f ) \ (1.0, 0.15) for the upper and (0.1, 0.01) for the lower narrows the allowed range in A. We note again that our envelope.* The range forf (0.01È0.15) is set as follows: the choice of the DSF range forf was motivated by the * esc lower limit comes from numerical simulations of star forma- reasons outlined earlier, and that other ranges forfesc are tion (see, e.g., Ciardi et al. 2000, and references therein), possible; the main point demonstrated by Figure 3 is the while the upper limit corresponds approximately to that in power of using any such band of independently known astrophysics to constrain a cosmological parameter. Note HL97 and Haiman & Loeb (1998a). The value offesc has been estimated through a number of theoretical and obser- also that A would have to be known to great accuracy to vational methods (see Wood & Loeb 2000, and references place any limits onfesc that are stronger than the DSF band. So far, we have been using the Fisher matrix formalism therein). Here, we take the range forfesc to be 0.1È1.0, the lower limit coming from Dove, Shull, & Ferrara (2000, here- for speciÐc pairs of parameters, while Ðxing the values of the after DSF), who modeled the escape fraction of ionizing other parameters in the SM. The more general and proper photons from OB stellar associations in the H I disk of the way to do this is to construct a 6 ] 6 matrix for the parameter set (A, ) , h, n,f , f ), which yields q. However, the Milky Way, and found that for a coeval star formation b esc * \ ^ analysis described in ° 3 implies that only any Ðve of these history,fesc 0.15 0.05. Our choice of this limit from DSF is motivated by the similarity of their modelÏs lumi- six parameters will be independent, since the CMB data will nosity history to that in HL97; as noted above, there are determine the cosmological parameters and q. Indeed, the 6 ] 6 matrix, when constructed, proves to be singular. We alternate values forfesc in the literature for a variety of astrophysical environments. We will use both the full range forfesc and the more narrow DSF band in later plots. To combine this with the CMB constraint, a shaded ellipse representing the two-parameter 68% joint con- Ðdence region is overplotted, assuming that the true model describing the universe is given by the SM and [q, A] \ \ [0.057,A(p8 0.7)]. This ellipse is narrow enough that we show a magniÐed version of Figure 1 in Figure 2, with an additional lighter band, nested within the AS band, showing the e†ect of varying onlyfesc while Ðxingf to its SM value. We see that the CMB does not really constrain* q or A separately at all, a near degeneracy that was expected from the discussion in the previous section. However, the combination of the CMB conÐdence region and the AS band is much more constraining: this translates to a 1 p error of about 0.02 forp8, which is noticeably better than the corresponding value of D0.2 from the CMB ellipse alone. We now extend Figure 2 to connect, through q, two a IG F . 3.ÈConstraint from current CMB data in thefesc-A plane, as priori independent parameters,fesc and A, shown in Figure extended from Fig. 2; the horizontal band shows the permitted range of 3. The purpose of this plot is to probe the potential of 0.1È0.2 forfesc from DSF. No. 1, 2000 REIONIZATION AND PARAMETER ESTIMATION 61 brieÑy note here some informative aspects of performing singular value decomposition (SVD; Press et al. 1992) of \ T ] F6C6, so that F UW V , where W is a diagonal 6 6 matrix containing the singular values w. An element that has an anomalously low value (close to zero) in W implies that the corresponding column in V is a linear combination of parameters that will not be well constrained. Table 1 shows the matrix V with the corresponding column weights w resulting from this decomposition; also shown are the parameters associated with each row in V , 2 where A is expressed asp8. We see that the sixth weight is very close to zero, so that the sixth column of V contains combinations of(Pcosmo, Pastro) that are poorly constrained by this analysis. This column has terms corresponding to essentially onlyPastro, the dominant contribution coming fromfesc. This implies thatfesc will not be well determined from the CMB (via q), given the reionization model con- IG F . 4.ÈConstraints from the CMB in thefesc-h plane; larger ellipse sidered here. The Ðrst Ðve columns of V also convey what represents current data, and smaller nested ellipse (line) is from Planck. combinations of these six parameters will be constrained; we note thatfesc has very small contributions in these, i.e., to the information that can be extracted from the CMB. In These plots are intended to be examples of the constraints comparison,f can be better determined from the CMB, as in variousP ÈP subspaces. * astro cosmo seen from columns 1È5, particularly the Ðfth, where the Figure 4 shows the case offesc versus h; the larger ellipse dominant term is fromf . This insensitivity of the CMB corresponds to current CMB data, and the nested one * data tofesc can be traced back to the stellar reionization (appearing as a tiny line) is from Planck. For all subsequent model we adopted here; variations inf a†ect q more sig- cases, we show these ellipses separately; the astrophysical * niÐcantly than do those infesc (see Figs. 12 and 15 of HL97). range forfesc is omitted from this plot for visual clarity. The covariance matrix for F can be found by Figure 5 shows the results expected from Planck alone for ~1 \ ~1 T C 4 F VW U . Because the ratio of the minimum to fesc versus h, with the light-shaded horizontal band rep- ] ~20 the maximum value of W , D3 10 , is very small com- resenting the full range offesc (0.1È1.0), and the dark band pared to machine precision, we follow the usual technique representing the DSF values forfesc (0.1È0.2). The case of fesc of adjusting the anomalously low singular value in W , here versus)b is shown in Figures 6 and 7, for current CMB w6, to zero (Press et al. 1992); despite this, the SVD inver- data and Planck, respectively, with overplotted bands the sion of F still produces an inaccurate covariance matrix, i.e., same as in Figure 5. C Æ F D I. Figures 8, 9, 10, and 11 display the respective cases of f * We now proceed to work with the independent 5 ] 5 versusp2 (wherep2 P A) andf versus n. For these four 8 8 * subportions of the full 6 ] 6 matrix, which translates to plots, the horizontal dark band represents the maximum P and any one ofP . These 5 ] 5 matrices are astrophysical range of 0.01È0.15 forf ; values below this cosmo astro * inverted, and the 2 ] 2 submatrix of interest is projected range are unlikely to be sufficient for reionization, and into the two-parameter plane as the appropriate error values above this range must invoke IMFs other than that ellipse, which displays the conÐdence region after margin- of present-day galaxies in order to not violate metal pro- alizing over the other parameters. The results of this general duction or background light constraints. Fisher matrix analysis are presented below for the idealized We note here some generic features of Figures 5È11. In all speciÐcations of current data and for those expected from cases, the inclusion of known constraints on the astro- Planck (° 3). Only the temperature anisotropies from the physical parameters strengthens the CMBÏs limits on CMB are used for Figures 4È11; the polarization informa- cosmological parameters, even for the data expected from tion expected from Planck is included for Figures 12È13. Planck. This is particularly the case withf , due to the * TABLE 1 V AND DIAGONAL ELEMENTS OF W FROM F \ UW V T

Column

Row 1 2 3 4 5 6 2 [ [ [ p8 ...... 0.3922 0.1470 0.5116 0.7498 0.0277 0.000 [ [ [ [ [ )b ...... 0.5024 0.8592 0.0916 0.0320 0.0028 8.42E 16 h ...... 0.5012 [0.3673 [0.7611 [0.1856 [0.0111 3.613E[15 n ...... [0.5840 0.3242 [0.3857 [0.6342 [0.0545 1.599E[14 [ [ fesc ...... 0.0098 0.0039 0.0111 0.0041 0.2539 0.9671 f ...... 0.0373 [0.0148 0.0420 0.0155 [0.9652 [0.2544 * Weight: w ...... 18110.877 9251.715 330.074 12.455 0.0191 6.366E[16 OTE \ T N .ÈResults of the singular value decomposition ofF6C6 UW V ; shown are V , weights w for each column in V , and parameters associated with each row in V . 62 VENKATESAN Vol. 537

FIG. 5.ÈConstraint from Planck in thef -h plane; light shaded band IG 2 esc F . 8.ÈConstraint from current CMB data in thef -p8 plane, where represents the entire allowed astrophysical range of 0.1È1.0 forf , and the 2 P * esc p8 A; shaded band represents the permitted astrophysical range of 0.01È dark shaded band represents the permitted values of 0.1È0.2 forfesc from 0.15 for f . DSF. *

straint for cosmological parameters extracted from the greater sensitivity of q tof relative tof . Thus, the 1 p esc CMB. As some illustrative examples involving Planck data, error forf from the CMB is* signiÐcantly smaller than that the entire astrophysical permitted band forf reduces the forf for* all the cases shown here, making independent esc 1 p error forp from about 0.02 to less than* 0.01 (Fig. 9), limits on the former a more powerful complementary con- 8

IG FIG. 9.ÈSame as Fig. 8, but for data from Planck F . 6.ÈConstraint from current CMB data in thefesc-)b plane; shaded bands are the same as in Fig. 5.

FIG. 10.ÈConstraint from current CMB data in thef -n plane; shaded FIG. 7.ÈSame as Fig. 6, but for data from Planck band is the same as in Fig. 8. * No. 1, 2000 REIONIZATION AND PARAMETER ESTIMATION 63

FIG. 13.ÈConstraint from Planck in thef -n plane using temperature FIG. 11.ÈSame as Fig. 10, but for data from Planck and polarization; the allowed astrophysical range* forf is the entire y-axis. * and for n, from 0.006 to D0.004. The power to increase such Ðdence regions in all the Ðgures. While this only strengthens constraints will only become better asf (orf ) become * esc the argument for the power of astrophysical limits in con- better constrained themselves, but it may not matter much straining cosmology, the converse situation, which appears for most cosmological parameters in the post-Planck era hopeful from Figures 12È13, is realistically tentative at best. (since they will already be determined with great precision), In short, it is possible in principle that the astrophysics of a with the exception of A orp8. For this case, the method here stellar reionization model can be constrained by limits on may prove to be a valuable cross-check. cosmological parameters from PlanckÏs temperature and Figures 5È11 also reveal that even the most promising polarization data, although this may prove difficult to cases of cosmological parameter determination from the achieve. We may simply have to await the view from CMBÏs temperature information will not help to constrain SIRT F and the NGST to determine the reionizing activity astrophysical parameters such asf orfesc, whose currently of the Ðrst stars. known ranges as shown through* the horizontal bands in each Ðgure are typically much smaller than what would be 5. CONCLUSIONS deduced from the joint conÐdence region. This is partly due We have examined the power of a reionization model, to the low value of q itself, D0.06 in our SM, which hinders given its many cosmological and astrophysical parameters, its accurate determination from the CMB data. to constrain these input quantities when combined with When polarization is included for the projected data from parameter extraction from the CMB. In the case of the Planck, we see, from the two examples shown in Figures 12 well-known degeneracy between q and A in their e†ects on and 13, thatfesc can be determined to about the same accu- the CMB, we have found that this can be alleviated by the racy as the DSF allowed band, but that the 1 p error for f * complementary information from a reionization model, and is signiÐcantly smaller than its astrophysical uncertainty. that this remains a useful cross-constraint even when allow- Thus, future CMB data may be able to constrain the astro- ing for the astrophysical uncertainty in q. physical aspects of reionization. We recall, however, that we When we eliminate q and perform a more general Fisher have neglected e†ects from experimental noise or from fore- matrix analysis, we Ðnd that the astrophysical details of grounds in our analysis, which will enlarge the joint con- reionization can be useful in further constraining the CMBÏs limits on cosmological parameters, even in the case of the expected temperature data from Planck. We have shown that independent limits on the astrophysical inputs to reionization, despite the current uncertainty in their values, reduce the errors for cosmological parameters by a factor of at least D2. Given that we have considered the most optimistic parameter yield from CMB experiments (° 3), the use of known astrophysics can only become more valuable for realistic experimental results. This is of particular value for p8 (or A) and n, given their implications for structure formation and for theoretical models of the origin of the seeds of structure in the early universe. The converse situationÈusing a projected exquisite determination of a cosmological parameter to constrain astrophysical reionization parametersÈdoes not yield quite as interesting results with temperature data from current experiments or from Planck, even though we made the most FIG. 12.ÈConstraint from Planck in thef -) plane using tem- optimistic assumptions; the 1 p errors forfesc orf are esc b larger than what are already known to be reasonable. When* perature and polarization; lines represent the allowed range forfesc from DSF. the projected polarization data from Planck is included, we 64 VENKATESAN found thatf in particular can be constrained to far greater contribution, we expect larger average values of q for a Ðxed * accuracy than its current astrophysical uncertainty; in prac- reionization model, since structures freeze out earlier, tice, however, this may prove difficult to achieve, given the resulting in a longer line of sight to the last scattering e†ects of foregrounds and instrumental noise, which we surface at the reionization epoch. Increased qÏs can also have neglected here. result from higher values off orf , or from a lower value * esc In summary, the astrophysical details of reionization can ofMC (° 2), which would allow the Ðrst stars to form earlier strengthen the limits on the cosmology of our universe, and more ubiquitously. Since higher qÏs will have a better beyond even the projected parameter yield from future chance of being accurately determined from the CMB, it CMB data, and there is more potential to a measurement of would be interesting to analyze the constraints in this paper, q than the determination of a single number out of a large from both the reionization scenario and the CMB, for a parameter space describing adiabatic CDM models. These more general parameter space; we examine this in a forth- broad conclusions are naturally subject to the assumptions coming work. made in this analysis. The sizes of joint conÐdence regions derived from the CMB data for any two-parameter sub- This work was presented as part of a dissertation to the space is determined by the full covariance matrix, whose Department of Astronomy and Astrophysics at The Uni- elementsÏ values are dependent on the dimension of the versity of Chicago, in partial fulÐllment of the requirements chosen parameter space and the selected parameters. The for the Ph.D. degree. I thank Scott Dodelson for his many inclusion of more parameters has the generic result of valuable suggestions toward this project, Angela Olinto for increasing the sizes of the error ellipses; therefore, the helpful comments, and Daniel Eisenstein for useful dis- primary results of this paper can only be strengthened when cussions. Parameter estimation from the CMB was per- parameter spaces larger than that analyzed here are con- formed using CMBFAST by Uros Seljak and Matias sidered. Zaldarriaga. This research was partly supported by the In the SCDM cosmology assumed here, the values of q in NSF through the collaborative US-India project INT- our standard model were relatively low (D0.06). In an open 9605235, by NSF grant AST 94-20759, and by DOE grant universe, or one dominated by a cosmological-constant DE-FG0291 ER40606 at the University of Chicago.

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