Fossil Ionized Bubbles Around Dead Quasars During Reionization
aa r X i v : . [ a s t r o - ph ] M a r D RAFT VERSION S EPTEMBER
6, 2018
Preprint typeset using L A TEX style emulateapj v. 9/08/03
FOSSIL IONIZED BUBBLES AROUND DEAD QUASARS DURING REIONIZATION S TEVEN
R. F
URLANETTO , Z OLT ´ AN H AIMAN , & S. P ENG O H Draft version September 6, 2018
ABSTRACTOne of the most dramatic signatures of the reionization era may be the enormous ionized bubbles around lu-minous quasars (with radii reaching ∼ comoving Mpc), which may survive as “fossil” ionized regions longafter their source shuts off. Here we study how the inhomogeneous intergalactic medium (IGM) evolves insidesuch fossils. The average recombination rate declines rapidly with time: the densest pockets recombine rapidly,leaving low–density regions that recombine much more slowly. Furthermore, the brief quasar episode signif-icantly increases the mean free path inside the fossil bubbles. As a result, even a weak ionizing backgroundgenerated by galaxies inside the fossil can maintain it in a relatively highly and uniformly ionized state. Forexample, galaxies that would ionize – of hydrogen in a random patch of the IGM can maintain – ionization inside the fossil, for a duration much longer than the average recombination time in the IGM. Quasarfossils at z ∼ < can thus retain their identity for nearly a Hubble time, and will appear “gray,” distinct fromboth the average IGM (which has a “swiss-cheese” ionization topology and a lower mean ionized fraction), andfrom bubbles around active quasars (which are fully ionized). More distant fossils, at z ∼ > have a weakergalaxy-generated ionizing background and a higher gas density. They can break up and attain a swiss-cheesetopology similar to the rest of the IGM, but still with a smaller contrast between the ionized bubbles and thepartially neutral regions separating them. Analogous HeIII-fossils should exist around the epoch of HeII/HeIIIreionization at z ∼ . Rapid recombinations inside the HeIII-fossils will be more common, because many ofthem will have no HeII-ionizing background; but the time lag before another quasar appears is typically small,so . of the gas is able to recombine, even for fossils that form well before reionization is complete.Our model of inhomogeneous recombination also applies to “double reionization” models and shows that anon-monotonic reionization history is even more unlikely than previously thought. Subject headings: cosmology: theory – intergalactic medium INTRODUCTION
The reionization of hydrogen and helium throughout the in-tergalactic medium (IGM) are landmark events in the earlyhistory of structure formation. As such, they (and particu-larly hydrogen reionization) have received a great deal of at-tention – both observationally and theoretically – in the pastseveral years. At present, the observational evidence leavessignificant ambiguities (see Fan et al. 2006a for a recent re-view). The electron scattering optical depth of τ ∼ . measured through cosmic microwave background polariza-tion anisotropies implies that hydrogen reionization began at z ∼ > , albeit with a large uncertainty (Page et al. 2007;Dunkley et al. 2008). On the other hand, Ly α absorptionspectra of quasars at z ∼ show some evidence for a rapidtransition in the globally-averaged neutral fraction, ¯ x HI (e.g.,Fan et al. 2006b). However the Ly α absorption is so saturatedin the Gunn & Peterson (1965) trough (with optical depth τ GP & ¯ x HI ) that constraints derived directly from the lackof flux in that spectral region are weak (White et al. 2003;Fan et al. 2006b). Studying the properties of the ionized zonesand the flux transmitted at the corresponding wavelengthsnear the Ly α line has yielded much tighter lower limits onthe neutral fraction along a few sight lines, using Ly α andLy β transmission statistics (Mesinger & Haiman 2004, 2007)and the inferred ionized zone sizes (Wyithe & Loeb 2004a; Department of Physics and Astronomy, University of California, LosAngeles, CA 90095, USA; [email protected] Department of Astronomy, Columbia University, 550 West 120th Street,New York, NY 10027, USA; [email protected] Department of Physics, University of California, Santa Barbara, CA93106, USA; [email protected]
Mesinger & Haiman 2004), although the interpretations ofthese results is still subject to poor statistics and possible bi-ases (e.g, Lidz et al. 2006; Becker et al. 2007; Maselli et al.2007; Bolton & Haehnelt 2007b,a). Helium reionization isequally controversial: although HeII Ly α forest spectra sug-gest rapid evolution at z ∼ . (Heap et al. 2000; Smette et al.2002), secondary signatures (such as the IGM thermal his-tory and radiation background) suggest change at slightlyhigher redshifts – or in some cases no evolution at all (e.g.,Schaye et al. 2000; Theuns et al. 2002; Faucher-Gigu`ere et al.2007; Songaila 1998; Kim et al. 2002; Aguirre et al. 2004;Songaila 2005; Bernardi et al. 2003).One of the most interesting aspect of the reionization pro-cess is its inhomogeneity: different regions of the Universecan be reionized at very different times, depending on theirrelative proximity to the ionizing sources. During reioniza-tion, the resulting patterns of ionized and neutral gas can thenbe a rich source of information about the IGM, the ionizingsources, and the ways they interact (Furlanetto et al. 2004;Furlanetto & Oh 2007b). In essence, large ionized zones pin-point either large clusters of sources or exceptionally brightones.The most dramatic examples are luminous quasars, whichcan ionize enormous regions, spanning several tens of (co-moving) Mpc, in either hydrogen or helium. These sourcesare (probably) rather short-lived, with expected lifetimes ∼ – yr (see, e.g., Martini 2004 for a review). When thequasar shuts down, the ionizing background within the regiondrops precipitously: the galaxies that surround the quasar hostmay be relatively numerous (Alvarez & Abel 2007; Lidz et al.2007; Geil & Wyithe 2007) but generically cannot competewith such a strong ionizing source. As such, the IGM willbegin recombining as soon as it shuts off.However, the recombination time of IGM gas (at the meandensity) is comparable to the Hubble time at z ∼ and z ∼ for hydrogen and helium, respectively. Thus dur-ing either recombination era one expects the ionized re-gion to survive for a relatively long time after its sourcevanishes, becoming a “fossil” bubble. As some of thelargest coherent features in the Universe at these high red-shifts, these bubbles (both during their active and fossilphases) are obvious objects of interest. In particular, dur-ing hydrogen reionization they are favored targets of first-generation 21 cm surveys, which can only make images onthe largest scales – and so are limited to search for structuresof these sizes (Wyithe & Loeb 2004b; Wyithe et al. 2005).The characteristics of these regions can be used to study theproperties of the quasar (Wyithe et al. 2005; Zaroubi & Silk2005; Thomas & Zaroubi 2008; Kramer & Haiman 2007;Sethi & Haiman 2008) as well as its relation to the sur-rounding sources (Alvarez & Abel 2007; Lidz et al. 2007;Geil & Wyithe 2007). Although active bright quasars arequite rare at high redshifts, the long lifetime of fossils willmake large features much easier to find – increasing theirabundance by approximately the ratio of the recombinationtime to the source lifetime. During helium reionization, thedistribution of these fossils will affect the thermal historiesof IGM gas elements and hence their Ly α forest properties(Gleser et al. 2005; Furlanetto & Oh 2007a).Despite their promise, there has been relatively little atten-tion paid to the physics of the fossils, and in particular to howthey actually recombine. The most naive picture, in which theIGM is assumed to be uniform at the mean density, is obvi-ously too simple because, even at these high redshifts, the fil-amentary structure of the cosmic web is growing – and it haslong been recognized that the accompanying clumping is cru-cial for understanding the reionization process. In that con-text, the most common approximation for this inhomogeneityis the clumping factor C = (cid:10) n (cid:11) / h n i , which is the ratioof the recombination rate (volume-averaged over all ionizedregions) to its naive expectation in a uniform medium. This simple approach can be adequate so long as the spa-tial averaging to derive the clumping factor excludes neutralregions, which are of course not recombining, while simulta-neously accounting for the large–scale structure in the IGM(Miralda-Escud´e et al. 2000; Furlanetto & Oh 2005). In thatcase, so long as the division of the IGM into neutral andionized regions is not changing rapidly, a constant clumpingfactor approach is useful because the densest ionized regions(which account for most of the recombinations) are relativelystable.However, with fossil bubbles this assumption is manifestlyfalse: after such a precipitous drop in the ionizing back-ground, the entire region falls out of ionization equilibrium.The dense gas will initially be highly ionized by the quasar, sothe clumping factor will be large. But these regions recombinequickly and become neutral, making the total recombinationrate (and effective clumping factor) fall, eventually flatteningout at a level appropriate to the smaller ionizing backgroundfrom the remaining galaxies. In practice, it is often approximated by averaging over the entire volume,excluding only regions bound to dark matter halos (e.g., Iliev et al. 2006;Mellema et al. 2006), because weighting by the ionized fraction introducesextra complexity (but see Gnedin & Ostriker 1997; Kohler et al. 2007).
Similar physics should describe any global “recombinationera”, during which the global average ionized fraction is de-creasing with cosmic time. This may occur in “double reion-ization” scenarios, in which there is a sharp transition in themode of star formation so that the IGM is reionized twice,with substantial recombinations in between (Cen 2003b,a;Wyithe & Loeb 2003a,b), although in practice such historiesare difficult to arrange self-consistently (Haiman & Holder2003; Furlanetto & Loeb 2005; Iliev et al. 2007). In this case,the entire IGM would be “fossilized”, with the second genera-tion of stars unable to maintain the same level of ionization asthe previous one. Such models have only been studied usinga simple clumping prescription, and they rely on a relativelylarge value for C to speed up recombinations during the inter-mediate phase (Furlanetto & Loeb 2005). But the same argu-ments apply as for fossils: a large clumping factor will onlybe appropriate until the dense gas recombines. Afterward, theeffective clumping factor will decrease, and the global recom-bination rate will slow down.The goal of this paper is to describe the physics of this re-combination process in some detail and thus to evaluate bet-ter the structure of fossil ionized regions. We will aim toanswer several basic questions. Does the clumpiness of theIGM allow most of the gas to recombine, or only the densestregions? Can the residual ionizing background from galax-ies maintain high ionization, even if they could not providethe initial ionization of the gas on their own? How does theionization topology evolve inside the fossils? Does the in-homogeneous IGM allow regions far from galaxies to shieldthemselves and recombine faster than those near the galaxies– so that the fossil resembles the swiss–cheese topology of therest of the IGM? Or, does the ionization remain more uniforminside fossils than outside? And, finally, can quasar fossils bedistinguished, either individually or statistically, from large,galaxy-generated ionized bubbles, and from active quasars,through the topology of their ionized gas and their abundance?This paper is organized as follows. In § § §
7, we apply these models to fossils dur-ing helium reionization. Finally, in §
8, we discuss some ofthe observational prospects to detect these fossils, and we of-fer our conclusions in § Ω m = 0 . , Ω Λ = 0 . , Ω b = 0 . , H =100 h km s − Mpc − (with h = 0 . ), n = 0 . , and σ = 0 . , consistent with the most recent measurements(Dunkley et al. 2008; Komatsu et al. 2008). Unless otherwisespecified, we use comoving units for all distances. THE RECOMBINATION HISTORY
Our first task is to compute the recombination history ofan individual gas parcel. We will use a simplified version ofHui & Gnedin (1997); see Furlanetto & Oh (2007a) for moredetails. The full solution requires us to follow the abundancesof three independent species (including hydrogen, helium,and electrons), the gas temperature (which affects the recom-bination and collisional ionization rates), and the density ofthe parcel.Consider a gas element of fractional overdensity δ and tem-perature T illuminated by an ionizing background of the form J ν = J HI , − (cid:18) νν HI (cid:19) − α × (cid:26) ν HI < ν < ν HeII f ν
HeII < ν , (1)where J HI , − is the angle-averaged specific intensity of thebackground, in units of − cm − s − Hz − sr − , ν HI and ν HeII are the frequencies corresponding to the HI and HeIIionization edges, and f is a constant. We will assume forsimplicity that f = 0 before helium is reionized and f = 1 afterward. In our fiducial models, we adopt α = 1 . (typi-cal of quasar spectra). After the quasar that initially ionizeda region turns off, the ionizing background will be providedby stellar sources, and the spectrum will soften considerably.However, the shape only changes the photoheating rate, whichdoes not substantially affect our calculations. (It does not af-fect the HeIII fraction because we set f = 0 before heliumreionization anyway.)We define the number density of species i to be n i ≡ (1 + δ ) ˜ X i ¯ ρ b /m p , where ¯ ρ b is the mean proper mass densityof baryons. Note that the ˜ X i are the species number fractions, not the neutral fractions.The thermal evolution of the element is determined by d T d t = − HT + 2 T δ ) d δ d t − T P i ˜ X i d( P i ˜ X i )d t + 23 k B n b d Q d t , (2)where d / d t is the Lagrangian derivative. The first term on theright hand side describes the Hubble expansion, the seconddescribes adiabatic cooling or heating from structure forma-tion, the third accounts for the change of internal energy perparticle from changing the total particle density, and in the lastterm d Q/ d t is the net heat gain or loss per unit volume fromradiation processes (see below).The second term requires an expression for the growth ofthe nonlinear density field. Hui & Gnedin (1997) used theZel’dovich approximation, along with an analytic estimate forthe distribution of the strain tensor, which has been shown toprovide an accurate analytic approximation to the density evo-lution in the quasilinear regime. However, to maintain com-putational simplicity, we will assume that the gas elementsevolve following the spherical collapse (or expansion) model.We map the linear densities to nonlinear overdensities via thefollowing fitting formula (Mo & White 1996), δ L = D ( z ) δ L = δ c − . δ ) / − . √ δ + 0 . δ ) . , (3)where D ( z ) is the linear growth function and δ c ≈ . isthe threshold for virialization in the spherical collapse model.Under this assumption, we can write d δ d t = δ L d D d t d δ d δ L . (4)In the present context, this approximation affects the temper-ature distribution (and hence recombination coefficient) sub-stantially only at densities far from the mean, so it is not asignificant concern for us.The fourth term includes a number of radiative heating andcooling processes. The most important heating mechanism isphotoionization itself; each species i contributes a term d Q d t (cid:12)(cid:12)(cid:12)(cid:12) i = n i Z ∞ ν i d ν (4 πJ ν ) σ i h ν − h ν i h ν , (5) where σ i is the photoionization cross section for species i , ν i is its ionization threshold, and h denotes Planck’s constant(to differentiate it from the Hubble constant). We use thefits of Verner et al. (1996) for the photoionization cross sec-tions. The other relevant mechanisms cool the gas, and in-clude Compton cooling off the CMB (which dominates dur-ing the era of hydrogen reionization), recombinations (radia-tive and, for He II , dielectronic), collisional ionization, col-lisional line excitation, and free-free emission. We use thefits of Hui & Gnedin (1997) for all of these processes ex-cept Compton cooling (for which we use the exact form inSeager et al. 1999) and free-free emission (for which we usethe fit in Theuns et al. 1998). We have verified that our resultsare unchanged if we use the fits presented in Theuns et al.(1998) (an updated form of those in Cen 1992) for all thesemechanisms.Finally, we have an ionization balance equation for eachspecies; e.g., for hydrogen, d ˜ X HI d t = − d ˜ X HII d t = − ˜ X HI Γ HI + α HI B ( T ) ˜ X e ˜ X HII [¯ n b (1+ δ )] , (6)where α B is the case-B recombination coefficient and Γ i = Z ∞ ν i d ν πJ ν h ν σ i (7)is the ionization rate of species i . We will use Γ =Γ / (10 − s − ) for convenience. Note that, unlike inFurlanetto & Oh (2007a), we do not assume photoionizationequilibrium for these calculations. THE DENSITY DISTRIBUTION
Equations (2), (4), and (6) allow us to follow the evolutionof a single gas parcel. We wish, however, to consider the ag-gregate evolution of large regions of the IGM; we thereforeneed the distribution of gas density inside the IGM. Unfortu-nately, there is no well-tested model at high redshifts. In thefollowing we will use the density distribution P V (∆) (wherethe distribution is over volume and ∆ = 1 + δ ) recommendedby Miralda-Escud´e et al. (2000, henceforth MHR00), whichfits cosmological simulations at z = 2 – quite well: P V (∆) d∆ = A ∆ − β exp (cid:20) − (∆ − / − C ) δ / (cid:21) d∆ . (8)Intuitively, the underlying Gaussian density fluctuations aremodified through nonlinear void growth and a power law tailat large ∆ . MHR00 argued that the form could be extrapo-lated to higher redshifts in the following way. First, δ essen-tially represents the variance of density fluctuations smoothedon the Jeans scale for an ionized medium (appropriate for ourpurposes, where we will only consider gas that has been pre-ionized); thus δ ∝ (1 + z ) − . The power-law exponent β determines the behavior at large densities; we set β = 2 . for z > . The remaining constants ( A and C ) can beset by demanding proper mass and volume normalization.Note, however, that this distribution comes from a simula-tion with somewhat different cosmological parameters thanthe currently preferred values and so is probably not accurateenough for detailed comparisons.MHR00 also offer a prescription for determining λ i , themean free path of ionizing photons, in a highly-ionized uni-verse. They assume that the gas density field has two phases:low-density gas is highly ionized and transparent to ionizingphotons, while high density regions (with ∆ > ∆ i ) are neu-tral because of self-shielding. We will examine the accuracyof this assumption later, but it will be quite useful for mostof our work (see also Furlanetto & Oh 2005). In this picture,the mean free path equals the mean distance between the self-shielded clumps along a random line of sight, which is ap-proximately λ i = λ [1 − F V (∆ i )] − / . (9)Here F V (∆ i ) is the fraction of volume with ∆ < ∆ i and λ isa (redshift-dependent) normalization factor. Formally, this ex-pression is valid only if the number density and shape (thoughnot total cross section) of absorbers is independent of ∆ i . Thisis obviously not true in detail for the cosmic web. However,MHR00 found that equation (9) provides a good fit to numer-ical simulations at z = 2 – with λ H ( z ) = 60 km s − (inphysical units). We will extrapolate the same prescription tohigher redshifts. Equation (9) was derived in the highly ion-ized limit, ignoring photoelectric absorption by low-columndensity systems, and therefore generically overestimates λ i at the ionization edge. The correction can be up to a fac-tor of ∼ even in highly ionized cases (Furlanetto & Oh2005). Some of this decrease can be compensated by thelonger mean free paths of higher-energy photons, but to beconservative, we set λ H ( z ) = 30 km s − whenever we re-fer to the MHR00 mean free path in our calculations. We willdiscuss the accuracy of this mean free path estimate in moredetail below.The MHR00 prescription was developed for the post-overlap universe, when dense regions are self-shielded and themean-free path is controlled by Lyman-limit systems. In thepre-overlap universe, assuming that all regions with ∆ < ∆ i are ionized implicitly assumes an “outside-in” reionizationtopology, in apparent contradiction with expectations fromsemi-analytic models and simulations (e.g., Furlanetto et al.2004; McQuinn et al. 2007) that reionization is an “inside-out” process due to source bias. The MHR00 prescription islikely applicable during reionization once the mean free pathis sufficiently large that a typical photon samples a fair (andrelatively unbiased) fraction of the density distribution dur-ing its flight (Zhang et al. 2007). The MHR00 prescription isparticularly well suited to quasar fossil bubbles. For one, the initial mean free path within the quasar bubble (tens of Mpc)is comparable to that of the post-overlap universe, when F V isclose to unity; one is considering a sufficiently large volumethat it is a fair sample of the density distribution. Lidz et al.(2007) find that the bias is modest when averaged over thequasar bubble volume. Moreover, dense regions recombinefirst, leaving underdense regions ionized—precisely the as-sumptions made in the MHR00 model, provided the ionizingbackground is relatively uniform (see § λ i as a function of mass-averaged ion-ized fraction ¯ x i ≡ F M (∆ i ) for several different redshifts( z = 2 , , , , , , and , from top to bottom). Note that λ i decreases as redshift increases because the IGM gets lessand less clumpy, so the spacing between gas above a givendensity threshold decreases. At higher redshifts, ¯ x i ≈ . corresponds to ∆ ≈ , so these regions are typically sepa-rated by a fraction of a Mpc.It is important to note that the MHR00 mean free path is much larger than one would expect in a uniformly ionized Note that the details of the small-scale density distribution within andaround halos are generally absorbed into the escape fraction. F IG . 1.— Comoving mean free path of photons at the hydrogen ionizationedge as a function of the ionized fraction ¯ x i in the MHR00 model. The curvesassume z = 2 , , , , , , and , from top to bottom. medium. The optical depth in a uniform medium, measuredacross one fiducial MHR00 mean free path, can be expressedas τ λ ∼ x HI ¯ n H λ i σ HI (10) ∼ x HI (cid:18) λ i Mpc (cid:19) (cid:18) z (cid:19) where x HI is the mean neutral fraction of the gas outside ofself-shielded regions, ¯ n H is the mean density of hydrogen nu-clei, and σ HI is the photoionization cross section. For thelatter, we used σ HI = ¯ σ = 2 × − cm , the value averagedover a typical stellar spectrum (Miralda-Escud´e 2003). Fig-ure 1 shows that under the MHR00 geometry, the mean freepath reaches ∼ Mpc scales when x HI ∼ . − . . In com-parison, in a uniform IGM, reaching this large mean free pathwould require x HI . − . The reason is that the MHR00procedure calculates the mean free path in an entirely dif-ferent way: it assumes that the bulk of the IGM is highly-ionized (with negligible absorption) except for regions abovethe self-shielding threshold (which are entirely neutral). Inother words, in the MHR00 geometry, ¯ x i (which is a mass av-erage) refers to the fraction of gas outside these self-shieldedregions, not to the volume-averaged ionized fraction. We willcheck for the consistency of this picture later and see someexamples in which it becomes untenable. FOSSIL BUBBLES WITH ZERO IONIZING BACKGROUND
We begin by considering the simple case of an ionized bub-ble whose generating source turns off instantaneously, withno sources remaining or turning on. Thus the gas simply re-combines, without any further influence from a radiation field.In the context of hydrogen reionization, pre–existing galaxiesin the fossil bubble will always maintain an ionizing back-ground. Considering the zero–background case is neverthe-less a useful pedagogical exercise; furthermore, in the con-text of helium reionization, where the death of the quasar mayleave no ionizing sources, this case is also physically relevant.To produce our ionization histories, we initialize our cal-culation at an assumed redshift z i where the quasar has fully F IG . 2.— Ionization histories for fossil HII regions that are produced whenthe ionizing source turns off at some initial redshift z i . The fossils are as-sumed to see no ionizing flux, so the gas is allowed to recombine. The threesets of curves assume x i = 1 at z i = 8 , , and , from left to right.Within each set, the solid and dashed curves show the mass– and volume–averaged ionized fractions, ¯ x i,m and ¯ x i,v , respectively, using the MHR00 fitfor the density distribution. The dotted curves show the histories for elementsat the mean density (with ∆ = 1 ). ionized a region. We ignore the residual neutral gas at thisstage, including any self-shielded, high–density clumps thatmay remain neutral even in the presence of a quasar. We thencompute x i (∆ , z ) at a final redshift z . The mass-averagedionized fraction within the bubble, ¯ x i,m , is ¯ x i,m = f − Z ∆ max d∆ x i (∆ , z )∆ P V (∆) . (11)We set ∆ max = 50 so as to exclude virialized gas from thecalculation (since its ionization properties are irrelevant tothe IGM properties). The prefactor f − is the fraction ofmass in gas with ∆ < ∆ max and normalizes the result togive the ionized fraction of just the IGM gas. The volume-averaged ionized fraction, ¯ x i,v is computed similarly, exceptwith ∆ P V (∆) → P V (∆) inside the integrand and f IGM thevolume fraction of gas with ∆ < ∆ max .Figure 2 shows the evolution of the ionized fraction in afossil bubble with Γ = 0 . We consider three initializationredshifts, z i = 8 , , and . For each case, we show ¯ x i,m , ¯ x i,v , and x i (∆ = 1) by the solid, dashed, and dotted curves(respectively). As one would expect from the short recom-bination time at these high redshifts, all of these cases areable to recombine significantly. Also as one would expect, ¯ x i,m < ¯ x i,v , because underdense regions (which recombineslowly) of course fill more than their share of the volume.However, the two are not so far apart, and in particular themass-averaged value is always quite close to the expectationfor gas elements at the mean density.Many models (including both semi-analytic and numeri-cal varieties) compute reionization histories using a so-calledclumping factor, C ≡ (cid:10) n e (cid:11) / h n e i to estimate the total re-combination rate. The dotted curve in Figure 3 shows thisfactor in the MHR00 model as a function of redshift, assum-ing that all gas with ∆ < is ionized. The effective value isalways larger than unity, but not by an extremely large value; F IG . 3.— The effective clumping factors for the recombining fossils inFig. 2. For reference, the dotted curve shows C for the MHR00 IGM model,assuming that all gas at ∆ < is fully ionized (with gas above this thresholdresiding inside collapsed halos and excised from the IGM). it is significantly smaller than often assumed because we haveexcluded virialized systems and also assumed that the gas isJeans-smoothed on the scale appropriate for ionized gas. Itincreases toward lower redshifts as structure formation con-tinues and amplifies density contrasts.Unfortunately, blindly applying this clumping simplifica-tion only works when all the gas is in ionization equilib-rium. In reality, this is never a particularly good assump-tion. For example, during reionization, the dense interiors ofself-shielded clumps remain neutral, hardly ever see ionizingphotons, and do not contribute to the ionized volume. Ion-izing photons need not battle recombinations in this gas, butonly the slower recombinations in the more rarefied outskirtsof such regions. The dotted curve essentially deals with thisproblem by excluding dense gas from the clumping calcula-tion (as is typical in numerical simulations; e.g., Iliev et al.2006; McQuinn et al. 2007). In reality, one should allow themaximum density threshold ∆ i to evolve to reflect the actualionizing background (and neutral fraction) within the bubble.The problem is somewhat more insidious for fossil bubbles.The solid, long-dashed, and short-dashed curves in Figure 3show the effective clumping factors for fossil bubbles, againwith initial ionization at z i = 8 , , and . Here we definethe clumping factor C eff by comparing the overall recombina-tion rate to that for a gas element at the mean density. Afteran initial spike caused by rapid cooling of the dense gas (re-member α ∝ T − . ), C eff quickly decreases to a value closeto unity.This behavior occurs because there is (by definition) zeroionizing background inside fossils, so no gas elements can bein ionization equilibrium. Although dense elements recom-bine quickly, they cannot help their less-dense neighbors todo so; once a region is mostly neutral, it breaks off from the This spike is probably not physical, because the gas is actually heatedover the lifetime of the quasar and expands and cools continuously, ratherthan being set to an arbitrary initial temperature; see Shapiro et al. (2004);Mesinger et al. (2006) for discussions of the thermal behavior of photoheatedgas in simulations. F IG . 4.— Neutral fraction as a function of density inside fossil HII regions. (a): z i = 15 . The curves show z = 14 – , from bottom to top, with ∆ z = 2 ;these have ¯ x HI ,m = 0 . , . , . , . , and . . (b): z i = 10 .The curves show z = 9 – , from bottom to top, with ∆ z = 1 ; these have ¯ x HI ,m = 0 . , . , . , and . . The filled hexagons mark the nominal ∆ i values, which, in the MHR00 model, produce the same average neutralfraction. recombining gas. Thus the density of an “average” recom-bining parcel decreases with cosmic time, and with it the ef-fective clumping factor. It eventually levels off at C eff . ,because the recombination time for gas at the mean density iscomparable to the Hubble time. This behavior is vital for an-alytic models of fossil bubbles – and also for any model witha “recombination era”, including so-called “double reioniza-tion” – and a proper treatment requires a full accounting of theIGM density distribution. This also implies that cosmologicalsimulations cannot use simple subgrid clumping in any cellswhere the local ionizing background is zero; instead the in-stantaneous recombination rate depends on the integrated re-combination history of the cell.Another way to see this is through the neutral fraction asa function of density. Figure 4 shows sequences of x HI (∆) at several different times, with z i = 15 and z i = 10 inthe top and bottom panels, respectively. For the most part,low-density gas remains highly-ionized, while dense regionsquickly recombine. However, the transition between the twois rather extended (because the recombination time is propor-tional to ∆ − ). Indeed, even deep voids can have a substantialneutral fraction long after the quasar turns off. The Mean Free Path
Next we will consider how the MHR00 model for the ion-ized gas distribution and mean free path fares in this scenario.Qualitatively, one might expect it to perform adequately: af-ter all, the distribution goes from relatively highly-ionized gaswhen ∆ ≪ to nearly neutral gas at high densities. Infact, blindly using the ionized fraction to fix ∆ i (employ-ing the physical scenario of MHR00, and assuming that gaswith ∆ < ∆ i is actually highly ionized) even roughly repro-duces the regime for which x i ∼ . (see Fig. 4). However , ∆ i clearly does not mark a sharp separation between highly-ionized and nearly neutral gas; rather, the low-density gas isalso marginally neutral. As a result, the mean free path for these fossils will be far smaller than shown in Figure 1. Sim-ply assuming a uniform medium (eq. 10) yields τ ∼ acrossa region of size λ i ∼ kpc at x HI ∼ . – no larger than atypical halo!We can do slightly better than this uniform IGM approx-imation by dividing the gas into discrete systems in a simi-lar way to the Ly α forest at lower redshifts (Schaye 2001).The typical length scale of such a system is the Jeans length, L J ∼ G/ √ c s ρ , where c s is the local sound speed. Then thecolumn density of neutral hydrogen (which determines theoptical depth of each system to ionizing photons) is N HI ∼ n H L J ∝ n / H T / . At the Lyman edge, τ ∼ (cid:18) ∆0 . x HI . (cid:19) / (cid:18) z (cid:19) / . (12)Comparison to Figure 4 shows that the neutral fractions re-quired to produce τ > are typically exceeded even in low-density gas. Thus, effectively, every discrete absorber – evena deep void – is a Lyman-limit system, and the mean free pathof an ionizing photon is extremely small. The MHR00 modelis obviously not applicable in this scenario. A NONZERO IONIZING BACKGROUND
We now include a small, but nonzero, ionizing backgroundin our calculations. This background could represent pre–existing galaxies, as well as any new galaxies that continueto form within the fossil bubble, and/or any low-level resid-ual emission from the quasar after it has turned off (see § d f coll / d t . Here f coll is the collapsed fraction in halosabove the minimum mass m min able to form stars. We willparameterize this threshold mass in terms of the mass m of halos with virial temperatures larger than K, so thatatomic cooling is efficient (Barkana & Loeb 2001). We as-sume an ionizing efficiency ζ , such that the globally-averagedionized fraction ¯ x gi at the initial redshift z i is ¯ x gi ( z i ) = ζf coll ( z i ) . (13) Gas at the Mean Density
We can now take two possible approaches. In this sec-tion, we will discuss an approach similar to §
4, i.e. track-ing the evolution of each gas element independently, simplyadding a redshift-dependent ionizing background to the evolu-tion equations. This approach is imperfect, as we will discussbelow, but it is useful to provide a feel for the effectiveness ofan ionizing background in suppressing recombinations. In thenext section, we will follow an alternative (more global) ap-proach, based directly on the emissivities (rather than fluxes).We begin by examining the evolution of gas with ∆ = 1 .The ionization rate can be written as
Γ = Z dν ǫ ν λ ν σ HI ( ν ) (14) = ¯ n b ¯ σ ¯ λ d( ζf coll )d t , (15) Note that it is possible for the effective m min to differ between theuniverse at large and the highly-ionized fossil bubble, because (for exam-ple) photoheating may increase the Jeans mass (Rees 1986; Efstathiou 1992;Thoul & Weinberg 1996; Dijkstra et al. 2004). Note that this expression ignores recombinations, but in our calculationsthey can be simply swept into the free parameters described below, so longas they are roughly uniform. F IG . 5.— Ionization histories for gas with ∆ = 1 under a uniform ion-izing background. The two sets of curves assume x i = 1 at z i = 8 and . The solid curves assume Γ = 0 as in Fig. 2. The long-dashed, short-dashed, short dot-dashed, dotted, and long dash-dotted curves within each set(i.e., from top to bottom) take ¯ x gi ( z i ) λ i = 0 . , , , and kpc,respectively. Note that this calculation breaks down when self-shielding be-comes important; see text. where ǫ ν is the emissivity per unit frequency and λ ν is the(proper) mean free path of a photon with frequency ν . Substi-tuting values appropriate for our cosmology, we find Γ = 0 . s − ¯ x gi ( z i ) f coll ( z i ) (cid:18) λ i Mpc (cid:19) (cid:18) z (cid:19) / (cid:12)(cid:12)(cid:12)(cid:12) d f coll d z (cid:12)(cid:12)(cid:12)(cid:12) , (16)where λ i is the comoving mean free path (averaged over fre-quency) and we have used equation (13) to eliminate the effi-ciency ζ in terms of the more physically transparent globally-averaged ionized fraction when the bubble is first ionized. Figure 5 shows ionization histories for gas at the mean den-sity, initially ionized at z i = 8 and . The solid curvesassume Γ = 0 . The long-dashed, short-dashed, short dot-dashed, dotted, and long dash-dotted curves within each set(from top to bottom) take ¯ x gi ( z i ) λ i = 0 . , , , and kpc, respectively. This combination of parameters setsthe overall amplitude of the ionizing background in equa-tion (16); the background can be large either because one be-gins late in reionization ( ¯ x gi is large) or because the mean freepath in the fossil bubble is large. For comparison to resultsbelow, it is useful to think of these as having ¯ x gi ( z i ) = 0 . , sothat the bubble is created early in reionization (but not tooearly) and λ i = 0 . – Mpc. Comparison to Figure 1shows that the MHR00 model could apply to the IGM inthe latter two cases (3-30 Mpc); the other cases imply verysmall mean free paths and correspond to a regime where theMHR00 assumption breaks down.
The Mean Free Path
We will next explicitly consider how the MHR00 model forthe mean free path fares in this picture. We must immediately In fact, bright quasars form in the most massive halos and hence in biasedregions of the IGM, inside of which f coll should be larger than its globalmean. Lidz et al. (2007) have shown, however, that although the bias is largewithin a few Mpc of the quasar, it is relatively modest when averaged overthe entire volume of a large fossil bubble, so we ignore it for simplicity. note that Figure 5 shows the evolution only for gas at the meandensity. However, in the interesting limit in which this gas re-mains highly ionized, the fossils approach an asymptotic ion-ized fraction relatively quickly: this is the value appropriatefor photoionization equilibrium, which states x HI (1 − x HI ) = 1 . × − ∆ (cid:20) ¯ x gi ( z i ) λ i Mpc (cid:21) − (cid:18) z (cid:19) − / × (cid:20) f coll ( z i )d f coll / d z (cid:21) − . (17)Here we have assumed T = 10 K; in fact, the temperatureevolves as the universe expands, so this expression is onlyapproximately correct. Except for the brief initial adjust-ment phase, the equilibrium value works extremely well for ¯ x gi ( z i ) λ i & kpc (so long as the appropriate temperatureis chosen). Thus we expect x HI ∝ ∆ in other gas parcels,so long as the ionizing background illuminates every regionequally.Now let us suppose that a quasar appears at z i , ionizes alarge bubble (with comoving radius & Mpc), and turnsoff. For a concrete example, let us assume that ¯ x gi ( z i ) = 0 . .The fossil will immediately begin recombining. At this ini-tial stage, the mean free path must be at least as large as thequasar bubble (otherwise a single quasar could not have ion-ized the entire region). Thus the bottom curve in Figure 5 willbe most appropriate, and the neutral fraction will initially be ∼ × − . Substituting into equation (10), the optical depthacross one mean free path is ∼ . . Thus the mean free pathwould remain large, and the entire fossil would clearly remainhighly ionized.Such long mean free paths are quite reasonable at high red-shifts, even outside of fossils. For example, Lidz et al. (2007)show that extrapolations of the Ly α forest to z & yield meanfree paths at the ionization edge of ∼ – Mpc. Moreover,as described above, the MHR00 model predicts λ i & Mpcfor the highly ionized gas immediately after the quasar turnsoff.With the help of equation (17), we can also see that densergas will also remain quite ionized: in the example above, ∆ = 50 gas would initially have x HI ∼ . . However, suchan approach ignores radiative transfer, which is crucial for ourpurposes (it is instead similar to the way many cosmologicalsimulations mimic a uniform radiation field by illuminatingeach and every gas particle identically). The basic point isthat dense gas elements are not isolated: rather they are partof discrete regions, surrounded by other dense elements. Theaccumulated opacity of the outskirts of such objects reducesthe effective ionizing background in the densest elements – orin other words they become self-shielded. When this processis included, it is possible to approximate the IGM as two-phase, with gas at ∆ < ∆ i highly ionized (and eventuallysettling into ionization equilibrium) and gas at ∆ > ∆ i self-shielded and fully neutral (MHR00, Furlanetto & Oh 2005).It is these self-shielded elements that set the mean free pathin the MHR00 picture, and it is also these regions that thesimple-minded approach of this section ignores. FOSSIL BUBBLES AND INHOMOGENEOUS REIONIZATION
To rectify the shortcomings of the previous section, we notethat the principal problem is that a uniform ionizing back-ground allows too much ionization. Because each gas parcelis treated independently, and because the ionizing backgroundis not re-adjusted to account for losses due to recombinations, F IG . 6.— Minimum global ionized fraction required for the backgroundionizing sources to counteract recombinations in fossil bubbles. The thickcurves use the MHR00 clumping factor, while the thin curves assume C eff =1 . The solid curves assume m min = m , while the dashed curves assume m min = 10 m . The figure suggests that fossils can be kept highly ionized,at a level well above the ionized fraction in the rest of the IGM. photons are not conserved in such a scheme. In more concreteterms, the scheme above not only puts photons in the outskirtsof dense regions but also in their centers; self-shielding de-mands that the outskirts consume all the photons, so addingmore to the center must “over-ionize” the universe.We will therefore follow a more global approach to the evo-lution of the ionized fraction by enforcing photon conserva-tion as our starting point. We first ask what emissivity isrequired to overcome the recombinations within an ionizedbubble. As in equation (13), we parameterize the emissivityin terms of the equivalent global ionized fraction outside thefossil.Figure 6 shows ¯ x gi,min , the minimum ¯ x gi required to over-come recombinations in a fully-ionized bubble (in this con-text, “fully-ionized” implies that all gas elements below ∆ i =50 are ionized). Note that if the emissivity was constant atall times, then ¯ x gi,min would simply equal t gal /t rec , the ratioof the time elapsed since the onset of galaxy formation to theinstantaneous recombination time. In practice, the emissiv-ity increases steeply with cosmic time, which decreases thevalue of ¯ x gi,min . Note further that the recombination rateis proportional to the ionized fraction, so the minimum re-quirement for a particular bubble should actually be multi-plied by its mean ionized fraction at the appropriate redshift.We show results for two clumping factors: the upper thickset of curves take the MHR00 value and are appropriate forthe period shortly after the fossil forms, while the thin lowercurves assume C = 1 . The latter is appropriate for evaluat-ing whether relatively old fossil bubbles that have partially re-combined will continue to do so once the dense gas is alreadyneutral (see Fig. 3). The solid curves assume m min = m ,while the dashed curves assume m min = 10 m . The lat- Unlike in the previous section, here we take the global ionized fractionat each redshift, not at the moment of the bubble’s creation. Also note that ourconversion from emissivity to ¯ x gi ignores recombinations in the backgroundmedium, so in reality the required value will be even smaller. ter, higher mass threshold presents an even weaker require-ment on ¯ x gi , because more massive halos evolve more quicklyover cosmic time, so that the emissivity that can keep the fos-sils ionized corresponds to an even lower background ionizedfraction.We see that, even taking a conservative clumping factor,fossil bubbles can only recombine when ¯ x gi . . – . . Withthe MHR clumping value, the minimum increases slowly asredshift decreases as the IGM becomes clumpier. If, on theother hand, we look at the later stages when C ∼ (or roughly ¯ x i . . ), the requirement decreases to ¯ x gi & . – . . In thelatter case, the minimum decreases as redshift decreases be-cause the mean IGM density (and hence recombination rate)falls as well. This simple calculation suggests that quasar bub-bles will only recombine significantly if they are created wellbefore reionization is underway, at ¯ x gi . . – . .To give some concrete scenarios, we track the ionized frac-tion within the fossil via d¯ x i d z = ζ d f coll d z − ¯ x i Cα A ¯ n e d t d z , (18)where ¯ n e and ¯ n p are the the mean electron and proton den-sities at redshift z . We assume (unless stated otherwise) that m min = m to calculate f coll . For the effective clumping fac-tor, we will generally use the MHR00 value C MHR but alsocompare to a case with C = 1 (where the recombination rateis slower). This assumes that the same stellar sources that set ¯ x gi ( z i ) continue to illuminate the fossil. Of course, if the fos-sil persists for long enough, and if quasars are relatively com-mon, it is possible that another quasar will form in the sameregion; in that case, the fossil will quickly become completelyionized again.Figure 7 shows the resulting ionization histories of the fos-sil bubbles, in comparison to those of the background, ¯ x gi ( z ) ,for a series of assumed emissivities (for the latter, we can alsouse eq. 18, but with different initial conditions; we use C MHR for the background, but it makes little difference). In eachpanel, the solid and dotted curves are for fossil bubbles with C = C MHR and C = 1 , respectively. The dashed curves show ¯ x gi . In the left panel, z i = 10 , with the three curves in each settaking ¯ x gi ( z i ) = 0 . , . , and . , from top to bottom. In theright panel, z i = 15 , with the three curves in each set taking ¯ x gi ( z i ) = 0 . , . , and . , from top to bottom. We nowdiscuss each of these cases in turn. A Typical Quasar
As seen in Figure 7, the requirement that reionization com-pletes by z = 6 forces ¯ x gi ( z i ) & . for z i ≤ . So,for quasars at z i = 10 , the contrast between the backgroundand the fossils (or, indeed, between the background and activequasar bubbles) is never more than a factor of a few. Unlesssupermassive black holes form long before reionization, thiscase is therefore typical of most fossils. Figure 7 shows that,as expected, the ionizing background inside the bubble effi-ciently suppresses recombinations – in no case does ¯ x i fallbelow ∼ . for these fossils. This is true regardless of theclumping prescription – while the MHR00 model allows amore rapid initial decline, the two prescriptions converge oncethe dense gas recombines and the clumping factor approachesunity.Our description is still, however, incomplete, because wehave not answered one important question: as they recom-bine, do the fossils re-develop the “swiss cheese” topologytypical of reionization, with islands of ionized gas separated F IG . 7.— Top:
Comparison of the ionization histories for fossil bubbles and the universe as a whole. In each panel, the solid and dotted curves are for fossilbubbles with C = C MHR and C = 1 , respectively. The dashed curves show ¯ x gi (with C = C MHR ). Left: z i = 10 , with the three curves in each set taking ¯ x gi ( z i ) = 0 . , . , and . , from top to bottom. Right: z i = 15 , with the three curves in each set taking ¯ x gi ( z i ) = 0 . , . , and . , from top to bottom. Bottom:
Mean free paths of ionizing photons in the fossil and in the background universe. The solid curves show λ i from the MHR00 model; the dashed curvesshow a fiducial clustering length R c for galaxies according to ¯ x gi ( z ) (see text for discussion). The parameters are the same as in the top panel, except that we donot show the cases with C = 1 . by a sea of neutral (or in this case partially neutral) hydro-gen? Up to this point, we have assumed that “shadowing”and other radiative transfer effects are unimportant, so thatphotons are delivered to any region that is not self-shieldedas needed. Of course, in reality the ionizing sources are clus-tered, which modulates the local emissivity. During reioniza-tion of the background universe, this strongly affects the dis-tribution of bubble sizes (Furlanetto et al. 2004). Is the sametrue for recombining fossils?To answer this, we will need to compare the mean free pathwithin the fossil to the typical scale of fluctuations in the ion-izing background generated by the clustering of the ionizingsources. First we consider the fossils. Our primary tool forcomputing λ i is the MHR00 model. Given an ionized frac-tion, this sets a threshold density ∆ i above which gas is as-sumed to be self-shielded, and below which it is highly ion-ized. In our case, ∆ i can be determined by demanding thatthe fraction of mass with ∆ < ∆ i is equal to ¯ x i ; with that, λ i follows from equation (9).The solid lines in the bottom panels of Figure 7 show thisprescription; they correspond to the same models as in the up-per panels, all using C MHR for the clumping. Obviously,the mean free path falls immediately after the bright quasarturns off; it reaches a minimum simultaneously with ¯ x i andthen increases again. In our models, we assume that all gaswith ∆ > remains neutral, so the mean free paths even-tually match onto λ i corresponding to this density threshold(the turnover in the solid curves when ¯ x i ≈ ).The mean free path typically falls to λ i ∼ – Mpc be-fore increasing again. If we consider the uniform radiation Note that we use λ = 30 km s − here, as described in §
3. This isa conservative lower bound on the mean free path, because of high-energyphotons, so if anything we overestimate the probability that regions break offfrom the uniform background. In actuality, the initial mean free path is probably no larger than the fos-sil, so the decline may not be quite so severe except for the brightest quasars. field models shown in Figure 5, this is a rather large valueand [for the ¯ x gi ( z i ) assumed in this panel] the resulting ion-izing background should easily maintain a high level of ion-ization in the gas. This is important, because the gas between the self-shielded regions must remain optically thin for theMHR00 model to apply. To examine whether this is the case,we show in Figure 8 b the neutral fraction as a function ofdensity at several stages during this recombination process, assuming a uniform ionizing background and neglecting self-shielding . The solid, dashed, and dotted curves show x HI (∆) at z = 7 , , and 9, respectively (note that the entire universe isreionized by z = 6 ). We take ¯ x gi ( z i ) = 0 . for all the curves,so they correspond to the middle case in Figure 7. We havealso used the mean free paths and emissivity from this modelas input in the calculation, so it provides the actual x HI for allthe gas that lies below the self-shielding threshold. The filledhexagons show this threshold ∆ i according to the MHR00model (and again taking the recombination history in Fig. 7).Obviously, even though the mean free path has declinedsignificantly, the low-density gas remains quite highly ion-ized throughout the history. At z = 9 , , and , the volume-averaged neutral fractions are ≈ (1 . , . , . × − (in-cluding only gas with ∆ < ∆ i ), respectively. Using equa-tion (10), these correspond to optical depths across one meanfree path of τ λ ≈ (0 . , . , . , from high to low redshift.Because these optical depths are somewhat below unity, theMHR00 model appears to be a reasonable description of theIGM inside the fossil: the accumulated opacity of gas between self-shielded regions is small. However, the absorption is notcompletely negligible, so there may be some pockets wherethe emissivity is smaller than normal (i.e., voids that are farfrom nearby galaxies), particularly in the earliest phases orwhen ¯ x gi ( z i ) . . , where the optical depths are somewhatlarger. To gauge whether this will occur, we must consider thescales over which the emissivity itself fluctuates significantly.Before proceeding to do so, we note that Figure 8 also0 F IG . 8.— Neutral fractions as a function of density at a sequence of red-shifts, assuming that the ionizing background and mean free paths are set bythe middle of the set of three curves in Fig. 7 . Top: (a)
The solid, longdashed, short dashed, dash-dotted, and dotted curves take z = 8 , , , ,and , respectively. (b) The solid, dashed, and dotted curves take z = 7 , ,and 9, respectively. Note that all the x HI curves ignore self-shielding. Thefilled hexagons indicate where the MHR00 model predicts that self-shieldingbecomes important. lets us check another necessary condition for the MHR00model: the self-shielded gas must have had sufficient time torecombine and become neutral. This is a particular worry forthese fossils, because the typical densities of the self-shielded“Lyman-limit systems” are not too far above the mean (see thefilled hexagons in Fig. 8). Fortunately, the gas is indeed stillable to recombine quickly: the recombination time of (fully-ionized) gas with ∆ = ∆ i is, at worst, roughly on-tenth of theHubble time. Thus the self-shielded systems should be ableto reach x i ∼ . without any trouble.Now we return to the question of whether isolated, low den-sity regions will be able to break off and recombine morequickly. A necessary, but not sufficient, condition to avoidreturning to a swiss–cheese morphology and to maintain auniform ionizing background is λ i > ¯ d gal ≡ ¯ n − / , where n gal is the mean number density of galaxies. If we include allhalos with T vir > K, ¯ d gal ≈ . , . , and . Mpc and z = 6 , , and , respectively. In this example, λ i remainswell above this level, and ionizing photons will, on average,be able to reach neighboring galaxies. Thus we would notexpect the most extreme swiss-cheese topology.However, inhomogeneity can actually begin before λ i reaches the mean galaxy spacing, because galaxies are highlyclustered at high redshifts. A convenient way to includeclustering is to compare λ i to the characteristic size R c of ionized bubbles around clumps of galaxies in the back-ground universe. We compute R c as a function of ionizedfraction using the analytic model of Furlanetto et al. (2004),which is accurate to within a factor of ∼ during mostof reionization, and better in the later stages (Zahn et al.2007; Mesinger & Furlanetto 2007). We show R c by thedashed curves in the bottom panels of Figure 7 (see alsoFurlanetto et al. 2006a). We cap R c at the mean free path when ∆ i = 50 for consistency with We see that λ i > R c throughout the initial decline andplateau in ¯ x i . This means that source clustering will be rela-tively unimportant while the bubble is recombining; the typ-ical scale of variation is smaller than the residual mean freepath in the bubble, so fluctuations in the ionizing sources willbe washed out by the long mean free path (as in the post-reionization universe). On the other hand, stochastic fluctu-ations in the galaxy distribution could still cause some inho-mogeneity through a “runaway” effect, in which a region thathappens to see a lower flux will become more neutral, whichwill help to shield neighboring gas parcels, leading to lowerfluxes, etc. Such a process would require τ λ > , which iscertainly possible in the earliest stages, or if ¯ x gi ( z i ) . . . Soit seems likely that some rare regions far from galaxies willbegin to recombine. In practice, this runaway effect will prob-ably be reduced by the fact that under-illuminated regions arealready the farthest from the galaxies and so are most likelyto remain outside the ionized bubbles anyway.Once ¯ x i begins to increase, the left panel in Figure 7 showsthat λ i ≈ R c . (This near equality is coincidental; see the rightpanel for a counter-example.) At this stage, source cluster-ing within the bubble will become important. In fact, eventhough the characteristic bubble size in the background uni-verse is comparable to or only slightly larger than λ i , the“sub–bubbles” within the fossil will actually be somewhatlarger than in the surrounding universe.To understand this, note that ¯ x i reaches a plateau when theemissivity is barely enough to overcome the modest recombi-nation rate inside the bubbles (we find C ≈ at these times).The fossil is then in a “photon-starved” regime where a largeremissivity is required in order to ionize additional hydrogenatoms. Once R c ≈ λ i , overdense clumps of sources are ableto do just that over the scale of the mean free path in the back-ground universe. But in the fossil, only . of the hydro-gen atoms are actually neutral – so a clump of sources in thefossil can ionize a volume ∼ times larger than an equivalentclump in the background universe. The actual sub–clump sizemay even be somewhat larger than this naive prediction, be-cause the new, larger sub–bubble can merge with its neighborsas well. On the other hand, the quasar is also likely to be in-side an overdense region, which would decrease the expectedsize, at least in the central, biased parts of the fossil (Lidz et al.2007). We also note that the actual size of the galaxy sub–bubbles can be rather large (several Mpc at z ∼ < ), so that atypical fossil may contain only a few dozen such sub–bubbles.Meanwhile, if these highly-ionized regions around clumpsof sources are able to grow because their emissivity is largerthan average, we also know that underdense voids withinthe fossil will continue recombining even during the phaseswhere ¯ x i is either flat or increasing. Thus a contrast be-tween the fully ionized regions and their recombining neigh-bors will develop, and the fossil will look much like the back-ground universe but with larger bubbles (by a factor ∼ / ¯ x i in volume) and a significantly smaller contrast between thefully ionized regions and the surrounding partially neutral gas.Even these shielded regions would not provide much contrastbecause much of the dense gas has already recombined, sothe effective clumping is modest. We emphasize that, even inthe absence of any ionizing sources, fossils recombine quiteslowly in their late stages (see Fig. 2), and only a modest the fossil treatment. This roughly simulates the appearance of self-shieldedregions in the background universe; see Furlanetto & Oh (2005) for a morecomplete model. ¯ x i occurs at a quite largevalue.) An Early Quasar
Quasars at z i = 15 will produce fossils with more dra-matic contrasts, because reionization at z ∼ – implies ¯ x gi ( z i ) . . . Thus there is a long time for the bubble toremain visible against the mostly neutral background (and,eventually, to recombine). However, even in this extremecase, the right panels of Figure 7 show that the bubbles neverbecome more than about one-half neutral before the back-ground catches up to them. One interesting side note is that inthis high–redshift case, the MHR00 clumping factor implies more residual neutral gas; as seen in Figure 3, C . exceptin the initial stages, because only underdense gas (where therecombination time is long) remains ionized after the initialphases. (At z & , the MHR00 clumping factor does lead toslightly faster recombinations, although that is difficult to seein the figure.)The degree of recombination in these bubbles also suggeststhat the MHR00 prescription may break down. Figure 8 con-firms this explicitly for the ¯ x gi ( z i ) = 0 . case (the middlecurves in the right panels of Figure 7). Here we find that ∆ i ≈ for most of the evolution, which leads to mean freepaths ∼ Mpc. (At this extremely high redshift, the IGMis still fairly uniform, so there is little variation of the meanfree path with the ionized fraction; see Fig. 1. Thus λ i fallsquickly to ∼ Mpc and then levels off for a long period, in-cluding the plateau and initial increase in ¯ x i .) Until the endstages of reionization, the ionized fraction in the ∆ < ∆ i gas (nominally optically thin in the MHR00 picture) is a fewpercent at z & , so τ λ ≈ , . , . , . , and . at z = 14 , , , , and , respectively. Thus, even assuminguniform illumination , the IGM recombines sufficiently to ren-der it opaque to ionizing photons on Mpc scales. The MHR00picture breaks down, and the mean free path will become afraction of an Mpc.Of course, luminous sources will be able to maintain highionization in their own environs, while more distant gas willcontinue to recombine. The mean spacing between galaxiesat z = 15 (10) is . . Mpc, respectively. Comparing toFigure 7, λ i falls below this level, at least until ¯ x gi . . , evenusing the overestimate from the MHR00 picture. Thus theearly stages of the recombination process must be highly in-homogeneous: regions surrounding rare galaxies will remainhighly ionized, but other regions will recombine much likethe cases shown in Figure 2. As a result, the ionized fractioninside of these fossils will be smaller than shown in Figure 7,because much of the gas will be recombining without inter-ference while the ionizing sources waste photons repeatedlyionizing dense gas in their environs. This could lead to muchhigher contrast between the ionized bubbles and partially ion-ized surroundings than is ever possible for a fossil surround-ing a z i = 10 quasar.Once the clustering of the ionizing sources becomes signif-icant, R c quickly becomes larger than λ i (again, even withthe overestimate from the MHR00 picture). This means thatsource clustering will induce stronger variations in the fos-sil’s ionization structure that begin at a smaller ¯ x gi (comparedto the z i = 10 fossil). Again, the ionization pattern insidethe fossil will have a larger characteristic scale than that in the background universe, because the surrounding gas has lessneutral hydrogen, but the contrast between the fully ionizedbubbles and the rest of the IGM will be somewhat smallerthan in the background universe.Overall, we find that only in extreme cases like this one –where the quasar shuts off while the background universe isonly a few percent ionized – do we expect a well-developedswiss-cheese topology inside the fossil. Attaining such a stateis difficult for two reasons. First, the low-density gas requiressignificant time to recombine; because & of the masshas ∆ . , the quasar must appear early: at least severalrecombination times before the completion of reionization.Furthermore, a swiss-cheese topology also requires that themean free path within the fossil falls below R c (or, even bet-ter, the mean galaxy spacing) so that the clustering patternof the galaxies can induce fluctuations in the emissivity andhence ionized bubble pattern. From Fig. 7, we see that λ i grows more steeply with ¯ x i than does R c ; only if significantrecombination takes place does λ i fall below R c . Residual Emission from the Quasar
Thus far, we have assumed a “lightbulb” model of quasarevolution, in which quasars only turn on for some short pe-riod of time, and then shut off, leaving only the remaininggalaxies to supply an ionizing background to suppress recom-binations. However, recent models motivated by hydrody-namical simulations of galaxy mergers – in which black holesspend the majority of the time well below the peak luminos-ity of the associated quasar, but nevertheless radiating – pro-vide better fits to multi-wavelength quasar luminosity func-tions and a host of other empirical constraints (Hopkins et al.2005a,b). If the quasar continues shining at some lower lumi-nosity, it may suffice to prevent the bubble from recombining,even without the aid of neighboring galaxies. If the quasarshines at L hi for time t hi and then reverts to L lo for an ex-tended period, the requirement that the bubble does not re-combine during the latter period is ( L lo /L hi ) ∼ > ( t hi /t rec ) ∼ .
03 ( t hi / yr) C ([1 + z ] / , where t rec = 1 / [ Cα A n e ] isthe recombination time within the bubble. This accords wellwith expectations from the (Hopkins et al. 2005a) model, inwhich quasars shine on cosmological timescales with Edding-ton ratios between l ∼ . and l ∼ at peak. A fit to theirsimulations which also agrees with fits to luminosity functionsis: d t dlog L = t ∗ Q exp (cid:0) − L/L ∗ Q (cid:1) , (19)where the fitted parameters L ∗ Q = α L L peak , t ∗ Q = t (10) ∗ ( L peak / [10 L ⊙ ]) α T , and α L = 0 . , t (10) ∗ = 1 . × yr and α T = − . . This light curve allows us to cal-culate the radiative output as a function of luminosity. Forinstance, although most of the energy is radiated (and hencemost of the bubble growth occurs) while the quasar is at peakluminosity, a L peak ∼ L ⊙ quasar spends ∼ Gyr with
L > . L peak , sufficient to prevent the bubble from recom-bining until the universe is reionized. In this model, lowermass black holes radiate an even large fraction of energyat low Eddington ratios. Of course, the applicability of theHopkins et al. (2005a) model to high redshifts is uncertain, Strictly speaking, in the simulations motivating the Hopkins et al.(2005a) fitting formula, a somewhat larger fraction of the low-Eddington ra-tio emission is produced before , rather than after the bright, near-Eddingtonphase. F IG . 9.— Left panel:
Ionization histories for fossil HeIII regions with zero ionizing background. The four sets of curves that decrease from right to left assume ¯ x HeIII = 1 at z i = 4 , , , and . Within each set, the solid and dashed curves show ¯ x HeIII ,m and ¯ x HeIII ,v , respectively. The dotted curves show the historiesfor elements with ∆ = 1 . The dot-dashed curve (increasing from the right) shows a sample global reionization history. Right panel:
Same as Figure 3, but forhelium reionization. but it is clear that low-luminosity quasar remnants could con-tribute very significantly to the ionizing background in ’fossil’bubbles. Hence, our arguments considering only the galacticcontribution provide a strict lower bound on the ionizationstate. HELIUM REIONIZATION
Zero Ionizing Background
Figure 9 shows corresponding recombination histories forfossil HeIII bubbles produced during quasar reionization. Weconsider four initialization redshifts here, z i = 4 , , , and , from left to right, and assume zero ionizing background.Qualitatively, the results are similar to those for HI: the mass-averaged ionized fraction is typically smaller than the ionizedfraction for gas at the mean density, but not by a large amount.However, once most of the gas has recombined, ¯ x HeIII ,m be-comes slightly larger than the ionized fraction of gas with ∆ = 1 : beyond this point, most of the neutral gas lies in un-derdense voids, and the effective clumping factor is substan-tially less than unity. The curves are also somewhat steeperthan for hydrogen because the recombination time for heliumis several times smaller and because the Universe is clumpierat lower redshifts.Figure 9 also shows that the clumping factor behaves sim-ilarly to fossils during hydrogen reionization (although herethere is no initial spike, because the temperatures are assumedto be hotter at the initial instant of reionization). The asymp-totic C eff are actually smaller here, mostly because the re-combination rate is slightly faster relative to the Hubble timeand so only gas below the mean density remains substantiallyionized.As before, it is useful to compare x HeIII within bubbles tothe background universe. The dot-dashed curve in the leftpanel of Figure 9 shows such a history (see Furlanetto & Oh2007b for details). It uses the quasar luminosity function fromHopkins et al. (2007), assumes that quasars have spectra typi-cal of observed quasars at lower redshifts (Vanden Berk et al.2001; Telfer et al. 2002) and ignores IGM recombinations (so is no more than a rough guide). We have reduced the to-tal luminosity of each quasar by a factor of ∼ . to forcereionization to complete at z = 3 , consistent with a num-ber of observational probes (see Furlanetto & Oh 2007b for asummary). Interestingly, the background ionized fraction in-creases extremely rapidly, with most of the ionizations occur-ring at z . . On the one hand, this means that most bubblesform not long before reionization is complete and have rela-tively little time to recombine. On the other hand, the fossilsthat do form around the rare luminous quasars at z & havean extremely high contrast with the background universe andhave ample time to recombine nearly fully. A Non-zero Ionizing Background
We have so far assumed that fossil helium bubbles evolvein isolation, without being exposed to any HeII–ionizing ra-diation. In contrast to hydrogen reionization, that is not a badassumption – at least for a time – because quasars are so rare,and galaxies (probably) do not contribute significantly to he-lium reionization. Thus, once a bubble appears, it will remainfree of any ionizing sources until another quasar forms in thesame region. The key question is how long this empty phasewill last.To address this question, we must first determine whatsphere of influence a new quasar will have in the partiallyionized medium. L ⋆ quasars (with lifetimes of yr) canionize all the helium inside regions within ∼ Mpc oftheir host galaxy. Once regions are mostly ionized, so thatbattling recombinations presents the main challenge (ratherthan ionizing new material), Furlanetto & Oh (2007b) showedusing the MHR00 model that their light is attenuated onlyover ∼ Mpc distances. Thus the relevant question is thetime lag before a quasar appears in this maximum volume V max ∼ Mpc. To estimate this, we write the quasar num-ber density as n QSO = 10 − n − Mpc − and the lifetime as t QSO = 10 t yrs. For the purposes of a simple estimate, wethen assume that the quasar population over the time lag ∆ t can be divided into N discrete generations. The number of3 F IG . 10.— Ionization histories inside fossils and in the background uni-verse, as in Fig. 7 with z i = 10 . The solid and dashed curves are identicalto the ¯ x gi ( z i ) = 0 . and 0.4 curves in Fig. 7 with C = C MHR . The dottedcurves also take ¯ x gi ( z i ) = 0 . and 0.4 with C = C MHR , but they assumethat photoheating raises the Jeans mass by a factor of ten within the fossils. generations required to have a probability of one-half to finda second source in the region is then ∼ (2 n QSO V max ) − , or H ( z )∆ t ∼ . t n − (cid:18) z (cid:19) / . (20)According to the Hopkins et al. (2007) model, the numberdensity of quasars with L & . L ⊙ (roughly L ⋆ ) is n − ∼ (2 . , . , . at z = (3 , , , implying H ( z )∆ t ∼ (0 . , . , . t at these redshifts. Thus once reionizationgets underway, the time lag between successive generationsof ionizing sources is rather small – only if a region is ionizedearly in reionization will it have sufficient time to recombinesignificantly. For example, a z = 4 (5) bubble will reach ¯ x HeIII ,m ∼ . ( . ) before it is reionized by another quasar.We therefore expect roughly one-half of all photons producedat z & to be “wasted” through fossils. But by z = 3 . , thefossil lifetimes will be so short that few photons are wasted. DISCUSSION
Photoionization Feedback
As noted above, it is possible that photoheating increasesthe Jeans mass in the fossil bubbles (Rees 1986; Efstathiou1992; Thoul & Weinberg 1996), and with it the effective min-imum galaxy mass m min over that in the universe at large.The efficiency of this feedback may be reduced at redshiftsmuch larger than z ∼ > (Dijkstra et al. 2004); nevertheless, itcan decrease the emissivity in fossils. It is therefore natural toask whether such a reduction may allow fossils to recombinemore rapidly than we found above.Figure 10 shows a set of ionization histories that addressthis issue. The solid curves are identical to the ¯ x gi ( z i ) = 0 . and . curves in Figure 7. In the dotted curves, we assumethat photoheating inside the fossil bubble has increased theminimum mass to m min = 10 m (Rees 1986; Efstathiou1992; Thoul & Weinberg 1996). Although photoheating de-creases the effective collapsed fraction within the bubble bynearly an order of magnitude, the effect on ¯ x i is relatively modest, and the region still remains highly ionized. In part,this is because these massive galaxies evolve more rapidly, sothe emissivity (proportional to d f coll / d z ) does not fall by aslarge a factor as f coll itself. Note also that we still compareto a background universe with m min = m ; this is why thefossil can have a smaller ionized fraction than the universe atlarge. Of course, in reality photoheating will also slow downthe background evolution once ¯ x gi & . , and the two willmap more smoothly onto each other (Haiman & Holder 2003;Furlanetto & Loeb 2005). But this simple treatment sufficesto show that photoheating feedback will not allow fossils torecombine significantly more than in our fiducial cases; in-stead, they are roughly equivalent to a normal history withone-half the emissivity.However, because massive galaxies are much rarer, inho-mogeneity will be easier to re-establish. In particular, λ i can be comparable to or smaller than the mean galaxy spac-ing if photoheating feedback suppresses galaxy formation in-side of fossil HII regions. If we only include galaxies with T vir > × K (i.e., maximal photoheating feedback), ¯ d gal = 4 . , . , . , and Mpc at z = 6 , , , and .Comparing to Figure 10, the mean free path inside these bub-bles can fall below ¯ d gal in moderately ionized volumes, evenat relatively low redshifts. However, the effect of photoheat-ing feedback is probably smaller during or just after a regionis reionized, because objects already in the midst of collapseare not very susceptible to it (Dijkstra et al. 2004). Thus it isunlikely that suppression at this level can be achieved in therelatively short time interval between z = 10 and the end ofreionization. We would nevertheless generically expect moreinhomogeneity inside the bubbles if photoheating feedbackoccurs. Detecting Fossil Bubbles
We have now established that most fossil bubbles formed inthe midst of hydrogen reionization will remain highly ionizedthroughout the entire process. We will now consider strate-gies to find them. To do so, we must rely on two features offossils: their large size, and possibly a different topology ofionized gas. We have already seen that the latter effect is notso important; most bubbles will remain highly ionized, andthose that do begin to recombine significantly will resemblethe background universe but with larger bubbles and less con-trast – which will probably only be distinguished with 21 cmsurvey instruments well beyond those currently being built,such as the Square Kilometer Array. Instead, it is their largesizes that are often taken to be the distinguishing characteris-tic of fossils.The first potential problem is distinguishing quasar-drivenfossils from large ionized regions generated by “normal”star formation during reionization. As illustrated in Fig-ure 7, Furlanetto et al. (2004) established that HII regions in-evitably reach extremely large sizes during reionization (seealso Furlanetto et al. 2006a). During most of the process, thecharacteristic size is relatively small in comparison to the ion-ized regions generated by luminous quasars (which are ∼ – Mpc across, assuming typical quasar parameters). But thisis only the characteristic size – in reality, the bubble distri-bution has a tail of rare, large objects. Because the quasarsthemselves are also extremely rare, it is not obvious whichtype of rare large bubble will outnumber the other.Figure 11 illustrates the problem quantitatively. The threethick curves give estimates for the number density of large4 F IG . 11.— Comparison of the number densities of massive halos (assumedto have hosted luminous quasars) and large bubbles from “normal” reioniza-tion (dominated by stellar sources). The thick solid, long-dashed, and short-dashed curves show the abundance of regions with R > , , and Mpc,respectively, from stellar reionization, based on the model of Furlanetto et al.(2004). These are evaluated at z = 8 but only depend on redshift slightly.The horizontal dotted and solid lines show the total abundance of halos with m > M ⊙ and M ⊙ , respectively, at the specified redshifts, repre-senting possible quasar hosts. ionized bubbles from normal star formation. We show R = 20 , , and Mpc, from top to bottom. We calculatethese with the analytic model of Furlanetto et al. (2004); nu-merical simulations show that this model underestimates thecharacteristic bubble size early in reionization (by a factor ofa few), but is quite accurate in the later stages most applicableto this argument (Mesinger & Furlanetto 2007).The thin horizontal lines show the number density of halosabove mass thresholds M ⊙ and M ⊙ (solid and dot-ted lines, respectively) calculated from the Sheth & Tormen(1999) mass function at z = 7 , , and 10 (from top tobottom within each set). If we assume that these massivehalos all host supermassive black holes, then the numberdensity of halos above that threshold will equal the maxi-mum number density of fossils as well (ignoring recombi-nations). The luminous z = 6 quasars are thought to liein halos of m ∼ – M ⊙ from both number den-sity arguments (Fan et al. 2000; Haiman & Loeb 2001) andspectral signatures in the optical (Barkana & Loeb 2003) andmolecular line spectra (Walter et al. 2004; Narayanan et al.2008). This mass range is also consistent with convert-ing the empirically-estimated black hole masses (Vestergaard2004) to a halo mass using the black hole mass-halo massand black hole mass-velocity dispersion relations calibratedat lower redshifts (Magorrian et al. 1998; Ferrarese & Merritt2000; Gebhardt et al. 2000).Comparing the bubble and halo number densities, it is clearthat fossils will not necessarily be distinguishable from morenormal (galaxy–driven) ionized zones past about the mid-way point of reionization. Evidently, the rare tail of ionized In detail, we compute x i ( > R ) / (4 πR / , where x i ( > R ) is thefraction of space filled by bubbles larger than R . This is more robust to un-certainties in the maximum bubble size set by recombinations, which breaksextremely large HII regions into smaller sections (Furlanetto & Oh 2005). bubbles may provide a comparable number of objects to thequasars themselves. That said, the uncertainties in both thetail of the bubble distribution (which depends on the overlapof smaller regions) and the conversion from quasar luminosityto halo mass (and hence number density) are highly uncertain,and it is possible that fossils will, in fact, stand out relativelywell. There are also two morphological effects that couldbe helpful (although difficult to interpret robustly). First, thelarge halos hosting quasars are also surrounded by substantialclusters of sources, so they already sit in large ionized bub-bles from which the quasar can build a somewhat larger ion-ized region (Lidz et al. 2007). Second, quasars may presentdistinctive ionization patterns (perhaps two collinear cones, ifthe emission is beamed, or a spherical region if it is isotropic).Furthermore, high-redshift galaxy surveys over the samefields may allow us to break this degeneracy. 21 cmand galaxy surveys should have a strong detectable anti-correlation (Furlanetto & Lidz 2007; Wyithe & Loeb 2006).Moreover, on the largest scales where bubbles are directly de-tectable, we could simply compare the observed comovingemissivity within the bubble to its size. This relation willhave some natural scatter due to the past star formation his-tory within the bubble, variations in gas clumping, and ob-servational error. However, since quasars constitute a strongimpulsive spike in the comoving emissivity, fossil quasar bub-bles will induce a decorrelation between the observed comov-ing emissivity and bubble size. In particular, quasar fossilswill lie far off the tail of the emissivity distribution for a givenbubble size (with the deviation being proportional to the frac-tion of ionized volume contributed by the quasar). Such ef-fects could potentially allow us to distinguish quasar-blownbubbles (the majority of which will be fossil regions), permit-ting a lower bound on the contribution of quasars to reioniza-tion.Another potential problem in identifying these bubbles isthat the underlying ionization (and density) fields are fluctu-ating on large scales from the cumulative effects of smallerregions. We need to be able to distinguish the fossil region’ssignal from this background “confusion noise”. The best wayto discover fossil bubbles will probably be through the 21 cmline (Furlanetto et al. 2006b), for which the brightness is pro-portional to the total density of neutral gas. A fully ionizedfossil would have a contrast ∼ x HI mK with the mean back-ground. To estimate the confusion limit in such an observa-tion, we use the simulations of Mesinger & Furlanetto (2007)to measure the r.m.s. fluctuation amplitude at wavenumbers k ∼ (2 π/ Mpc − . This varies from ∼ . – mK overthe range ¯ x HI ∼ . – . . Thus we would expect fossils tostand out very well – many standard deviations from the mean– except near the end of reionization, when they will beginto blend into the background fluctuations as well. The mainobstacle toward direct detection is therefore telescope noise,though sufficiently large bubbles may still be detectable, evenby the first generation of instruments, at least at low redshifts(see § Note that the bubble size is proportional to the integrated star formationhistory ∝ f coll , rather than the instantaneous comoving emissivity ∝ ˙ f coll .However, the two are strongly correlated in the sense that the rarest high den-sity regions with the highest f coll will also have the highest rate of collapse. Fossil and Active Bubbles
As we have seen above, although the fossils can be kepthighly ionized, they are not fully ionized, and will thus ap-pear different from active quasar bubbles (for example, in 21cm imaging, they will have a somewhat reduced contrast).Since active quasars, with the luminosities required to pro-duce large ( ∼ > Mpc) fossil bubbles, should be relativelyeasily detectable even at z > , active bubbles can probablybe distinguished routinely from fossil bubbles, simply by thepresence or absence of a luminous ionizing source.We now turn to the question of how many fossil bubbleswe might expect relative to those surrounding active quasars.Fossils forming relatively close to the end of reionization (at z . for reionization at z ∼ ) do not significantly re-combine so would appear as mostly empty fossils long aftertheir source quasar fades. This will boost the number of large(mostly) ionized regions well above that expected from thequasar luminosity function alone, and can potentially allowone to constrain the quasar lifetime (Wyithe & Loeb 2004b;Wyithe et al. 2005).However, because fossils persist for so long, the quasarabundance itself probably evolves significantly over the re-combination time. As an example, let us suppose that thenumber density of bright quasars follows n QSO ∝ (1 + z ) − β ,with β > , and that fossils remain ionized so long as theyform at z < z max . In that case, assuming that the quasar life-time t QSO is much smaller than the elapsed time from z max to the observed time z obs , we expect n fossil ( z obs ) ∼ n QSO ( z obs )( β + 3 / H ( z obs ) t QSO . (21)The number density of bright quasars has β ∼ over therange z ∼ – (Fan et al. 2001, 2004). If this continues tohigher redshifts, it will provide a factor ∼ suppression fromthe naive value. Unless it can be measured independently,this evolutionary factor will pose a significant (factor of sev-eral) uncertainty in the measurement of t QSO from the fos-sil to active ratio. One way to break the degeneracy is bymeasuring the luminosity function over the entire time inter-val – but doing so would require a search for active quasarsthat extends to much higher redshifts than the fossil searchitself. Another potential method would be a series of mea-surements of the number density of bubbles over a redshiftinterval comparable to the recombination time of a bubble.This could probe the redshift evolution of the quasar lumi-nosity function, particularly if sensible priors for the quasarlifetime are adopted. Finally, we note that the shapes of ac-tive quasar bubbles could, in principle, contain information onthe host quasars and the IGM, through finite light–travel timeeffects (Wyithe & Loeb 2004b; Yu 2005). The results we ob-tained here, i.e. that most of the fossil gas takes significantlylonger to recombine than one would estimate using a fidu-cial IGM clumping factor, will hamper statistical versions ofthis measurement (Sethi & Haiman 2008), since fossils willoutnumber active quasar bubbles and not show the apparentdistortion due to finite light travel time.
Double Reionization
Up to this point, we have studied recombinations in the con-text of fossil bubbles, but much of our discussion also appliesto so-called “double reionization” scenarios (Cen 2003b,a;Wyithe & Loeb 2003a,b) in which a first generation of mas-sive, metal free stars reionizes most of the universe but alsoenriches it – inducing a transformation to much less efficient,but more normal, stars. In the scenarios originally proposed,photons from the second generation did not suffice to bat-tle recombinations and so the universe became substantially(up to 80%) neutral, until structure formation had progressedenough for the more normal stars to complete reionization.However, these models rely on relatively large clumpingfactors to speed up recombinations so that a recombinationera can occur during the relatively short interval between thetwo ionizing source generations. Reionization requires that(at least) one ionizing photon per hydrogen atom be produced;this is relatively difficult at high redshifts. But maintaining ahigh ionization fraction is less stringent: it requires only thatone ionizing photon be produced per recombination. For gasat the mean density, where the recombination time is roughlythe Hubble time, this is relatively easy. We have seen that theeffective clumping factor declines as more gas recombines,suggesting that the “recombination era” will probably be con-fined to dense pockets of gas.Our models strengthen this conclusion. Figure 7 can beused to analyze double reionization scenarios in addition tofossil bubbles: we can simply think of the ¯ x gi ( z ) curves asdescribing the second generation, and we suppose that thefossil’s initialization redshift describes the moment at whichthe first generation shuts off. The left and right panels thenmimic moderate and more extreme double reionization sce-narios. Our most important conclusion is that, even in themost extreme case – when the second generation is able toachieve only ¯ x i ∼ . at the initial point – less than halfof the gas is actually able to recombine. In more moderatescenarios, only ∼ can recombine, and that is generallydense gas surrounding galaxies and halos.Note finally that these scenarios assume a sudden tran-sition between the two generations – in reality, that ismuch too simplistic, because all of the feedback mech-anisms that can mediate such a transition (metal enrich-ment, photoheating, etc.) act locally and on relatively longtime scales (Haiman & Holder 2003; Dijkstra et al. 2004;Furlanetto & Loeb 2005; Iliev et al. 2007; McQuinn et al.2007). Both of these effects (slow recombinations and slowtransitions) will combine to make double reionization excep-tionally unlikely. SUMMARY
We have examined recombinations in inhomogeneous fos-sil ionized regions, adapting the simple MHR00 model of thedensity distribution of the IGM, and tracking the ionizationhistories of individual gas elements. Without a residual ion-izing background, the process occurs quickly at first, becausethe dense IGM gas can recombine rapidly (during both hy-drogen and helium reionization). But once these relativelyrare pockets become neutral, recombinations slow dramati-cally, and eventually C eff . . Thus it is quite difficult for aregion to recombine entirely, even in the absence of any ion-izing flux.The ionizing background from galaxies inside the fossil, to-gether with any residual low–level emission from the blackhole past its bright quasar phase, efficiently further suppresses6recombinations during hydrogen reionization. So long asgalaxies are able to ionize & of the universe on average,they will be able to halt recombinations in all but the dens-est gas inside the fossil regions. In most cases, the mean freepath of ionizing photons remains large enough to wash out anyfluctuations in the ionizing source population, and the fossilremains uniformly ionized (outside of self-shielded regions).In more extreme scenarios, when the fossil is created long be-fore reionization, the gas can recombine significantly (downto roughly the mean density), which will probably allow largeregions far from galaxies to shield themselves and begin re-combining faster. In such a case, the fossil IGM will lookqualitatively similar to the swiss–cheese ionization topologyof the IGM elsewhere, but with larger ionized regions and asmaller contrast between the ionized and “gray” partially neu-tral regions.These fossil regions may therefore be somewhat harder todistinguish from the normal IGM than often assumed – moreso because “normal” ionized regions can reach comparablesizes to those of quasar zones near the middle of reionization,and may be as or even more abundant (albeit both of these estimates rely on the rare tails of theoretical distributions, ingalaxies and ionized bubbles, and have very large uncertain-ties).During helium reionization, fossil bubbles evolve some-what differently. Because the ionizing sources are rare, oncea region’s source quasar shuts down, it is unlikely to be illu-minated by any other source for a time. 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