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Early Preheating and Galaxy Formation

Winds from pregalactic starbursts and 'miniquasars' may pollute the IGM with metals and raise its temperature to a high adiabat, and so inhibit the formation of early galaxies. We compute the thermal history of the IGM when it experiences a period of rapid, homogeneous"preheating"at high redshifts. Measurements of the temperature of the Lyamn-alpha forest at z~3 constrain the redshift and energy of preheating, and rule out models that preheat too late or to too high a temperature. We predict galaxy luminosity functions in preheated universes. The results depend crucially on whether the baryonic smoothing scale in the IGM is computed globally, or in a local, density-dependent fashion. Using a globally averaged smoothing scale, we find that models with excessive preheating produce too few L_* and fainter galaxies, and are therefore inconsistent with observational data. A density-dependent smoothing scale requires more energetic preheating to achieve the same degree of suppression in the faint-end slope. All models, however, appear unable to explain the sharp cut-off in the luminosity function at bright magnitudes. Supernova-driven preheating scenarios tend to raise the mean metallicity of the universe well above the minimum levels observed in the Lyman-alpha clouds. We find that ionizing photon escape fractions must be significantly higher than 10% in order to explain the low inferred HI fraction at z~6. While early preheating causes strong suppression of dwarf galaxy formation we show that it is not able to reproduce the observed abundance of satellite galaxies in the Local Group in detail.


INTRODUCTION
In cold dark matter (CDM) cosmological scenarios, structure formation is a hierarchical process in which non-linear, massive structures grow through merging of smaller initial units. Large numbers of low-mass dark halos are predicted to be present at early times in these popular theories, and galaxies are thought to form by a two-stage collapse process: the gas first infalls along with the dark matter perturbation, gets shock-heated to the virial temperature, condenses rapidly due to atomic or molecular line cooling, and then becomes self-gravitating (but see Katz et al. 2003 andBirnboim &Dekel 2003 for alternative views of how gas reaches the galaxy phase). Massive stars subsequently form with some initial mass function (IMF), synthesize heavy-elements, and explode as supernovae (SNe) after ∼ 10 7 yr, enriching the surrounding medium. The very first zero-metallicity stars ('Population III') may in fact have been so massive to give origin to a numerous population of massive 'seed' black holes .
It is an early generation of subgalactic stellar systems around a redshift of 10-15, aided by a population of accreting black holes in their nuclei, which likely generated the ultraviolet radiation and mechanical energy that ended the cosmic "dark ages" and reheated and reionized most of the hydrogen in the intergalactic medium (IGM). The recent analysis of the first year data from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite suggests the universe was reionized at redshift zion = 20 +10 −9 (Kogut et al. 2003). The detailed history of the universe during and soon after these crucial formative stages depends on the powerspectrum of density fluctuations on small scales and on a complex network of poorly understood 'feedback' mechanisms. Yet, it is a simple expectation of the above scenario that the energy deposition by SN explosions and winds from accreting black holes (termed 'miniquasars' in Haiman, Madau & Loeb 1999) in the shallow potential wells of subgalactic systems may, depending on the efficiency with which halo gas can cool and fragment into clouds and then into massive stars and black holes, cause the blow-away of metalenriched baryons from the host galaxy and the pollution of the IGM at early times (e.g. Dekel & Silk 1986;Tegmark, Silk & Evrard 1993;Cen & Ostriker 1999;Madau, Ferrara & Rees 2001;Aguirre et al. 2001).
It has long been argued that, besides being a mechanism for spreading metals around, pregalactic outflows must also efficiently quench high redshift star formation. This is because the cooling time of collisionally ionized high density gas in subgalactic systems is much shorter than the then Hubble time, virtually all baryons are predicted to sink to the centres of these small halos in the absence of any countervailing effect (White & Rees 1978). Strong feedback is then necessary in hierarchical clustering scenarios to avoid this 'cooling catastrophe', i.e. to prevent too many baryons from turning into stars as soon as the first levels of the hierarchy collapse. The required reduction of the stellar birthrate in halos with low circular velocities may naturally result from the heating and expulsion of material due to quasar winds and repeated SN explosions from an early burst of star formation.
It has also been recognized that the radiative and mechanical energy deposited by massive stars and accreting black holes into the interstellar medium of protogalaxies may have a more global negative feedback on galaxy formation. The photoionizing background responsible for reionizing the IGM will both increase gas pressure support preventing it from collapsing into low-mass halos along with the dark matter, and reduce the rate of radiative cooling of gas inside halos (Efstathiou 1992; Thoul & Weinberg 1996;Navarro & Steinmetz 1997). Furthermore, as the blast-waves produced by miniquasars and protogalaxies propagate into intergalactic space, they may drive vast portions of the IGM to a much higher adiabat than expected from photoionization (e.g. Voit 1996;Madau 2000;Theuns, Mo & Schaye 2001;Cen & Bryan 2001), so as to 'choke off' the collapse of further galaxy-scale systems by raising the cosmological Jeans mass. The Press-Schechter theory for the evolving mass function of dark matter halos predicts a power-law dependence, dN/d ln m ∝ m (n eff −3)/6 , where n eff is the effective slope of the CDM power spectrum, n eff ≈ −2.5 on subgalactic scales. As hot outflowing gas escapes its host halo, shocks the IGM, and eventually forms a blast wave, it sweeps a region of intergalactic space the volume of which increases with the 3/5 power of the injected energy E0 (in the adiabatic Sedov-Taylor phase). The total fractional volume or porosity, Q, filled by these hot bubbles is then Q ∝ E 3/5 0 dN/d ln m. The dependence of E0 on halo mass is unknown and depends upon the complex physics of star formation occurring with each halo. For illustrative purposes we will assume that the energy per logarithmic mass interval is constant, E0dN/d ln m =constant (which, for the scales of interest, results in E0 ∝ ∼ m). In this case we find, Q ∝ (dN/d ln m) 2/5 ∝ m −11/30 . Within this simple picture it is the halos with the smallest masses which will arguably be the most efficient at heating the IGM on large scales (to avoid this would require E0 ∝ m α with α > ∼ 1.5). Note that this type of global feedback is fundamentally different from the 'in situ' heat deposition commonly adopted in galaxy formation models, in which hot gas is produced by supernovae within the parent galaxy. In the following we will refer to this global early energy input as "preheating". A large scale feedback mechanism may also be operating in the intracluster medium: studies of X-ray emitting gas in clusters show evidence for some form of non-gravitational entropy input (Ponman, Cannon & Navarro 1999). The energy required is at a level ∼ 1 keV per particle, and must be injected either in a more localized fashion or at late epochs in order not to violate observational constraints on the temperature of the Lyα forest at z ∼ 3 (see below). Of course, since this is sufficient to substantially alter the distribution of gas in cluster-sized potentials, it will have a much larger effect on gas in galaxy-sized potentials.
Preheating by definition causes a large increase in the temperature of the IGM at high redshift. This consequently increases the Jeans mass, thereby preventing gas accreting efficiently into small dark matter halos. If the Jeans mass is sufficiently high bright galaxies will not be able to form, resulting in an inconsistency with the observed galaxy luminosity function. For typical preheating energies (see §3) the IGM is expected to be driven to temperatures just below the virial temperatures of halos hosting L * galaxies. Thus we may expect preheating to have a strong effect on the galaxy luminosity function at z = 0. Recently Mo & Mao (2002) and Oh & Benson (2002) have studied the effects of 'late preheating' on the formation of galaxies at lower redshifts, finding that this may have a strong impact on both the abundances and morphologies of galaxies. Here, we perform a detailed calculation of the effect of a global energy input in the IGM at the end of the cosmic dark ages. By computing the thermal history of a preheated universe we are able to constrain both the amount and epoch of energy deposition. While explosion-driven winds may also inhibit the formation of nearby low-mass galaxies through other processes, such as 'baryonic stripping' (e.g. Scannapieco, Ferrara & Madau 2002), in this work we assess the effect of the increased gas pressure after preheating on subsequent galaxy formation, and use the techniques of Benson et al. (2002a) to compute the resulting luminosity functions of galaxies. ⋆ Specifically, we investigate what constraints the observed galaxy luminosity function (LF) and inferred Hi fractions at z ≈ 6 place on preheating scenarios and ask whether an early homogeneous heat deposition in the IGM may provide sufficient suppression of galaxy formation to explain the very flat faint end slope of the LF.
The remainder of this paper is arranged as follows. In §2 we briefly describe our model while in §3 we present out results. Finally, in §4 we give our conclusions.

MODEL
We use the methods described by Benson et al. (2002a) to evolve the thermal and ionization properties of gas in the IGM and refer the reader to that paper for a detailed discussion of the calculations. Briefly, we solve the equations governing the evolution of the ionization states and temperature of gas at a representative range of density contrasts, beginning from shortly after the epoch of recombination. The distribution of gas densities is drawn from the distribution described by Benson et al. (2002a) which is chosen to reproduce a given, time-dependent clumping factor (where clumping factor is defined as f clump = ρ 2 / ρ 2 where ρ is gas density in the IGM). We solve the ionization and thermal evolution for densities spanning the range from very underdense voids to densities comparable to those found in dark matter halos. We can integrate over the suitably weighted gas properties as a function of density to compute volume or mass weighted quantities (such as the mean temperature of the IGM for example).
Both collisional and photoionization are considered in computing ionization rates. Heating of the gas occurs through photoheating, while cooling occurs due to atomic processes and Compton cooling off CMB photons. Contributions to photoheating and photoionization from both galaxies and quasars are included. We use the semi-analytic model of Benson et al. (2002a) to compute the ionizing emissivity of galaxies as a function of time, while for quasars we use the fitting function of Madau, Haardt & Rees (1999). Note that the emissivity of galaxies will be affected by preheating as we will describe below, but the quasar contribution is fixed, since it is determined from observational measurements. For galaxies, we assume that a fraction fesc of all ionizing photons produced are able to escape into the IGM ⋆ Note that similar techniques were developed by Shaprio, Giroux & Babul (1994), although they were not employed to compute the galaxy luminosity function. and so contribute to ionization and heating. Unless otherwise noted we will adopt fesc = 0.1 throughout this work (Leitherer et al. 1995;Steidel, Pettini & Adelberger 2001).
We adopt cosmological parameters (Ω0, Λ0, Ω b , σ8, h) = (0.3, 0.7, 0.045, 0.93, 0.7) consistent with current observational constraints (e.g. Netterfield et al. 2002;Freedman et al. 2001;Smith et al. 2002;Burles 2002). Benson et al. (2002a) considered photoionization by stars and quasars as the only energy input into the IGM. Madau, Ferrara & Rees (2001) showed that the IGM could be heated to a higher adiabat by pregalactic outflows at high redshift. To explore the effects of this preheating, we include a rapid deposition of energy into the IGM at early times, in addition to the photoionization. We characterize the energy input due to preheating by the energy per baryon, E preheat . This energy is deposited in the IGM at redshift z preheat . To be precise, the energy is in fact added gradually over a short time centred on this redshift. This allows for an easier numerical solution of the equations governing the thermal and ionization state of the IGM. Such rapid preheating may be relevant if the sources of the energy are Pop III stars which experience a strong negative feedback and so form a short-lived population. We find that typically we can in fact increase the length of the period over which energy is added significantly without affecting our results. For example, in a model with E preheat = 0.1keV and z preheat = 9 adding the energy over a redshift interval of ∆z = 3 has no significant effect on our results for galaxy luminosity functions (increasing ∆z to approximately z however results in the effects of preheating being almost entirely removed). Our results are therefore equally valid for both very rapid energy deposition and deposition occurring over a significant fraction of a Hubble time.
We examine a homogeneous energy deposition since the filling factor of pregalactic outflows is expected to be large (Madau, Ferrara & Rees 2001). Recent numerical simulations have shown that outflows from starbursting dwarf galaxies can enrich ∼ 20% of the simulation volume at the end of the cosmic dark ages (Thacker, Scannapieco & Davis 2002), while semi-analytical models that include H2 cooling in minihalos and the formation of 'Population III' very massive stars can yield filling factor of unity (Furlanetto & Loeb 2003).
As described by Benson et al. (2002a) we use the resulting thermal history of the IGM to compute the filtering mass (Gnedin 2000), which in turn allows us to determine the effects of the IGM temperature on the accretion of gas into dark matter halos. This is input into the galform semianalytic model of galaxy formation (Cole et al. 2000) in order to compute the luminosity function of galaxies. The galform model follows the formation of galaxies in a merging hierarchy of dark matter halos. By calculating the rate at which gas is able to cool into a star forming phase (and adopting simple rules for the rate of star formation in that phase) galform is able to estimate the luminosity of galaxies as a function of time. The most massive galaxies are typically built up through merging of smaller systems (a process driven by dynamical friction). By simulating galaxy formation in dark matter halos spanning a broad range of masses we are able to construct the expected luminosity function of galaxies at the present day.
The filtering mass is conventionally computed using the volume averaged temperature of the IGM. However, unlike purely photo-heated models the temperature distribution in a preheated IGM at late times can be highly inhomogeneous (as will be discussed in §3.2). As such, we consider a possibly more reasonable approach in our preheated models, and compute filtering masses using the density-dependent temperature predicted by our IGM model, for several representative densities. Using the same probability distribution function (PDF) for the density distribution as used in our IGM model (see Benson et al. 2002a) we also compute the fraction of the IGM's mass which exists in each of these density bins. We then apply our galaxy formation model to compute the properties of galaxies existing in dark matter halos. For each such halo modeled we select one of the density-dependent filtering masses. This selection is done at random, weighting by the mass fraction present in each density bin such that the probability for a halo to exist in each density range is proportional to the IGM mass in that density range. † Unlike Benson et al. (2002a) we do not allow the ionizing background to heat gas already in halos (due to the high computational cost of this calculation). As shown by Benson et al. this causes only a minor additional suppression of galaxy formation. Its effect will be even more negligible in this work, where we consider filtering masses which are much higher than those in Benson et al. We adopt the same parameters for the semi-analytic model as did Benson et al. (2002a), with the exception of using the more realistic value Ω b = 0.045 for the baryon density parameter. As we are interested in whether preheating can produce a galaxy luminosity function with a flat faint-end slope (as is observed), we switch off the effects of supernova feedback in galform. This local heating mechanism is normally required to produce a flat luminosity function at the present epoch.

RESULTS
Theoretical modeling of the first stars and galaxies provides a valuable guide for the range of preheating energies and redshifts which should be considered. Lowenstein (2001) suggests that Pop III stars may preheat the intracluster medium at a level of ∼ 0.1keV per baryon at z > ∼ 10. Madau, Ferrara & Rees (2001) find preheating energies < 0.1keV † Here, we are treating the density in our IGM model as describing the large scale density environment within which a dark matter halo forms (note that the density PDF used is equally applicable to dark matter and gas when we are considering large scales). While this seems to be the most reasonable approach (since the filtering mass prescription is based upon a linear theory calculation), it has not been tested in numerical simulations and so must be viewed with some degree of caution. It should also be noted that weighting the selection of filtering masses by the mass fraction in each density bin assumes that dark matter halo formation is unaffected by the large scale density environment. at z ≈ 9 from pregalactic winds. In order to determine what constraints galaxy formation can place on preheating scenarios we choose to study models spanning the range E preheat = 0.05-0.3keV and z preheat = 6-12. This incorporates the theoretical estimates described above, and also covers more extreme models, both higher in energy (motivated by observations of X-ray clusters) and with more recent preheating (when preheating occurs recently there is less time for the IGM to cool and return to its previous thermal state).
Note that all our models satisfy the limit on the thermal pressure of intergalactic gas imposed by the lack of a Compton y-distortion to the spectrum of the CMB observed by the Cosmic Background Explorer (COBE) satellite. The mean thermal energy density introduced into the IGM at z preheat is (1) where H is the Hubble constant and tcomp is the Compton cooling time of hot electrons off CMB photons, inverse Compton scattering will transfer all the energy released to the CMB for z > zcomp = 7h 2/5 − 1 ≈ 5. Here me is the electron mass, σT the Thomson cross-section, UCMB is the energy density of the CMB, and we have assumed a pure hydrogen plasma such that the total number density of particles, ntot is twice the number density of electrons, ne.
The amount of y-distortion expected to the spectrum of the CMB is (Sunyaev & Zel'dovich 1980). The COBE satellite measured y < 1.5 × 10 −5 at the 2σ level (Fixsen et al. 1996), implying E preheat < 0.07 (1 + z preheat ) keV. This limit holds for z preheat > zcomp; at redshifts z preheat < zcomp the cooling time for Comptonization exceeds the expansion timescale, and only a small fraction of the thermal energy released is transferred to the CMB.

Sources of Preheating
It is interesting at this stage to set some general constraints on the early star-formation episode and stellar populations that may be responsible for preheating the IGM at the levels envisaged here. Let Ω * be the mass density of stars formed at z preheat in units of the critical density, ESN the mechanical energy injected per SN event, and fw the fraction of that energy that is eventually deposited into the IGM. Denoting with η the number of SN explosions per mass of stars formed, one can write where mp is the proton mass. For a Salpeter initial mass function (IMF) between 0.1 and 100 M⊙, the number of Type II SN explosions per mass of stars formed is η = 0.0074 M −1 ⊙ , assuming all stars above 8 M⊙ result in SNe II. Numerical simulations of the dynamics of SN-driven bubbles from subgalactic halos have shown that up to 40% of the available SN mechanical luminosity can be converted into kinetic energy of the blown away material, fw ≈ 0.4, the remainder being radiated away (Mori et al. 2002). With ESN = 1.2 × 10 51 ergs, equation (4) implies SN-driven pregalactic outflows efficiently carry metals into intergalactic space . For a normal IMF, the total amount of metals expelled in winds and final ejecta (in SNe or planetary nebulae) is about 1% of the input mass. Assuming a large fraction, fZ = 0.5, of the metal-rich SN ejecta escape the shallow potential wells of subgalactic systems, the star-formation episode responsible for early preheating will enrich the IGM to a mean level where we take Z⊙ = 0.02. The weak C IV absorption lines observed in the Lyα forest at z = 3 − 3.5 imply a minimum universal metallicity relative to solar in the range Songaila 1997). The metal abundances of the Lyα clouds may underestimate the average metallicity of the IGM if there existed a significant warmhot gas phase component with a higher level of enrichment, as detected for example in O VI (Simcoe, Sargent & Rauch 2002). Today, the metallicity of the IGM may be closer to ∼ 1/3 of Solar if the metal productivity of galaxies within clusters is to be taken as representative of the universe as a whole (e.g. Renzini 1997). ‡ Preheating energies in excess of 0.1 keV appear to require values of Ω * and Z that are comparable to the total mass fraction in stars seen today (e.g. Glazebrook et al. 2003) and well in excess of the minimum enrichment of the IGM inferred at intermediate redshifts, respectively. Uncertainties in the early IMF make other preheating scenarios possible and perhaps even more likely. The very first generation of metal-free Population III stars may have formed with an IMF biased towards very massive members (i.e. stars a few hundred times more massive than the Sun), quite different from the present-day Galactic case (Bromm, Coppi & Larson 1999;Abel, Bryan & Norman 2000). Population III stars with main-sequence masses of approximately ‡ Note that a metallicity ∼ 0.5Z ⊙ at z = 10 increases the gas radiative cooling rate to a level comparable to inverse Compton cooling. Our calculations assume a cooling function for a primordial plasma. At the low metallicities typical of Lyα forest clouds, the thermal behaviour can be modeled to a good approximation by a gas with primordial abundances (e.g. Sutherland & Dopita 1993).
140 − 260 M⊙ will encounter the electron-positron pair instability and be completely disrupted by a giant nuclearpowered explosion (Heger & Woosley 2002). A fiducial 200 M⊙ Population III star will explode with a kinetic energy at infinity of ESN = 4 × 10 52 ergs, injecting about 90 M⊙ of metals (Heger & Woosley 2002). For a very 'top-heavy' IMF with η = 0.005 M −1 ⊙ , equation (4) now yields (assuming fw = 1) and a mean IGM metallicity (assuming fZ = 1) This scenario can yield large preheating energies by converting only a small fraction of the comic baryons into Population III stars. This is even more true for preheating from winds produced by an early, numerous population of faint miniquasars. § Thin disk accretion onto a Schwarzchild black hole releases about 50 MeV per baryon. If a fraction fw of this energy is used to drive an outflow and is ultimately deposited into the IGM, the accretion of a trace amount of the total baryonic mass onto early black holes, may then suffice to preheat the whole universe. Note that this value is about 50 fw times smaller than the density parameter of the supermassive variety found today in the nuclei of most nearby galaxies, ΩSMBH ≈ 2×10 −6 h −1 (Merritt & Ferrarese 2001).

Thermal Evolution
In Figure 1 we show the thermal history of IGM gas at the mean density of the Universe for a variety of E preheat and z preheat . In each case, the gas is initially heated by photoionization from the first stars (beginning at z ≈ 30). The preheating energy causes a rapid increase in the temperature at z preheat . Note that the temperature never becomes as high as 2E preheat /3kB since the heating ionizes the gas, freeing electrons and thereby increasing the number density of particles (recall that E preheat specifies the energy per baryon). We include the effects of inverse Compton cooling, adiabatic expansion, and atomic cooling. For gas close to the mean density of the Universe, Compton cooling and adiabatic expansion dominate the cooling of gas after preheating. However, for high densities (i.e. densities typical of regions forming galaxies) cooling is dominated by atomic processes. Consequently, at high densities the gas is typically § Because the number density of bright quasi-stellar objects (QSOs) at z > 3 is low (Fan et al. 2001), the thermal and kinetic energy they expel into intergalactic space must be very large to have a global effect, i.e. for their blastwaves to fill and preheat the universe as a whole. The energy density needed for rare, luminous QSOs to shock-heat the entire IGM would in this case violate the COBE limit on y-distortion (Voit 1994(Voit , 1996. cooler than the results shown in Fig. 1 (which are for gas at mean density). This will have important consequences for the filtering mass and galaxy luminosity function as will be discussed in §3.5 and §3.6, where we will compute filtering masses as a function of density, and follow galaxy formation for each different filtering mass. A model with no preheating is also shown, whose only heat source is therefore photoheating.
We compare our model results to the observational determinations of Schaye et al. (2000) and Ricotti, Gnedin & Shull (2000). This comparison will be used to discard models which are strongly inconsistent with the data. It is clear that the measurements of the IGM temperature at z ∼ 3 rule out models in which E preheat is too high, or z preheat is too low. For sufficiently low E preheat or high z preheat the IGM is able to recover to close to the thermal state of the no preheating case, which lies close to the data, by z = 3. The result is that models which heat to TIGM > ∼ 10 6 K at z < ∼ 10 are inconsistent with the z ≈ 3 temperature data. The models which adequately fit the temperature data are indicated in Table 1, where we also indicate which models are consistent with the measured Compton y-distortion in the CMB. We consider only those models consistent with both constraints for the remainder of the paper. (Note that we will typically not plot lines for the E preheat = 0.05keV models to avoid overcrowding the figures.) Table 1 also lists the optical depth to Thomson scattering for CMB photons, τe. There is very little variation between the models since the bulk of hydrogen reionization occurs through photoionization prior to preheating. Our models are consistent with the constraints on the optical depth from the WMAP experiment (Kogut et al. 2003), τ = 0.17 ± 0.04, only at the 1.5σ level. This discrepancy can be resolved somewhat by exploring models with higher values of fesc. In Table 1 we show results for models with E preheat = 0.1keV and z preheat = 9 and 12, for fesc = 20% and 50%. All are consistent with the measured IGM temperature and the Compton-y constraint, but produce higher optical depths due to partial photoionization of hydrogen at high redshifts. The models with fesc = 50% achieve τ = 0.15, very close to the WMAP value.
Before considering the effect of preheating on the galaxy luminosity function we examine two other predictions from our model-the ionization state of the IGM and the entropy of IGM gas.

Ionization States
The large amount of energy deposited into the IGM during preheating will necessarily affect the ionization state of gas in the IGM. The fractional densities of Hi and Heiii in our models are shown in Figure 2. Hi and Heii are collisionally ionized at z preheat in all of our models.
In some cases, the neutral fraction is of order 0.001 or higher. As such, these models would still produce a Gunn-Peterson trough after z preheat . Hi is replenished after z preheat by radiative recombinations (the features in the curves are due to changes in recombination rates as the IGM cools), and is finally almost fully ionized between z ≈ 6 and z ≈ 2 through photoionizations. The neutral fraction at these redshifts is often larger than in the no preheating case. This occurs because preheating exerts a strong negative feedback on galaxy formation, resulting in fewer ionizing photons be- Table 1. Properties of the nine fesc = 10% models and additional higher fesc models are considered. Columns 1 and 2 list the preheating energy and redshift respectively. Column 4 notes whether the model produces a reasonable match to the measured IGM temperature at z ∼ 3, Column 5 indicates whether the model is consistent with the measured limit on the Compton y-distortion of the CMB, while Column 6 lists the optical depth to Thomson scattering for CMB photons. ing available at these redshifts and consequently a higher neutral fraction. This may allow these models to explain the Gunn-Peterson troughs seen in the spectra of the most distant Sloan Digital Sky Survey quasars (Becker et al. 2001) at z ≈ 6. However, the observed lack of a Gunn-Peterson trough at z < ∼ 6 clearly rules out these models. A possible solution to this problem lies in increasing the escape fraction of ionizing photons as discussed below.
For helium (lower panel), we see that preheating typically causes ionization to Heiii, which then remains at an almost constant level until z = 0. Note that this is at variance with more conventional scenarios in which the double reionization of helium occurred later, at a redshift of 3 or so (see Kriss et al. 2001, and references therein), due to the integrated radiation emitted above 4 Ryd by QSOs (but see Oh et al. 2001).
As noted above, the presence of a significant fraction of neutral hydrogen at z < ∼ 6 in our preheated models would conflict with the lack of an observed Gunn-Peterson effect at these redshifts. A possible solution to this problem is to increase the rate of photoionization by increasing the escape fraction, fesc for galaxies. Figure 3 shows the ionization fractions for models with E preheat = 0.1keV, z preheat = 9 and 12 and with increased escape fractions of fesc = 20% and 50%. While there is no observational evidence for such high escape fractions at low-redshifts our ignorance of the nature of very high redshift galaxies makes it interesting to consider the consequences of such high escape fractions.
Increasing fesc to 20% is sufficient to reduce the neutral fraction to negligible levels in the z preheat = 9 model, while a higher escape fraction still is required for the z preheat = 12 model. We conclude that an increased escape fraction will remove the residual neutral hydrogen which is problematic for our fesc = 10% models. Furthermore, this higher escape fraction has only a small impact on the thermal evolution of the IGM, and the filtering mass and luminosity functions remain largely unchanged. Our conclusions regarding these quantities in the remainder of the paper are therefore equally  valid for these higher escape fractions. It is also interesting to note that these higher escape fractions result in somewhat better agreement with the WMAP optical depth measurements. An escape fraction of 20% results in τ = 0.13 while fesc = 50% results in τ = 0.15.
As described in §2 we have also performed calculations using a density-dependent filtering mass in order to approxi-mately account for the significant inhomogeneity in the IGM temperature in preheated models. Figure 4 shows the ionization fractions for two such models. Galaxy formation in high density regions is significantly less suppressed in these models since, as we will see in §3.5, the temperature and filtering mass are lower. As such, the neutral hydrogen fractions in these models drop to very low values, albeit somewhat later than a model with no preheating. This helps reconcile these models with the SDSS quasar observations, although clearly some additional increase in fesc is still required.

Entropy
Preheating has been suggested as the origin of the entropy floor seen in clusters of galaxies (Ponman, Cannon & Navarro 1999). These observations imply an "entropy" of S(= kBT /n 2/3 e ) ∼ 100keV cm 2 for gas at z = 0. We show, in Figure 5, the entropy of IGM gas in our models as a function of redshift. The entropy is never constant (as would be expected for gas cooling by adiabatic expansion) due to the other cooling and heating processes included in our calculation. Note that none of our models ever reach S = 100keV cm 2 . The requirement that the IGM temperature match that which is measured at z < ∼ 4 limits the amount of entropy which can be deposited into the IGM (the entropy produced by preheating is increased by increasing E preheat and/or decreasing z preheat , both of which tend to result in temperatures which are too high at z ≈ 3), while Oh & Benson 2002 note that preheating must occur prior to z ≈ 2 in order to affect the cores of clusters. Alternatively, entropy generation spatially localized to regions which are destined to become clusters could circumvent these constraints. Finally, we note that the use of a density-dependent filtering mass has little effect on the volume-averaged entropies shown in Fig. 5. For very dense regions, little entropy is generated by the preheating models considered here, although considerable entropy is produced through photo-heating at late times reaching S = 50keV cm 2 for regions with densities comparable to dark matter halos at z = 0.

Filtering Mass
The filtering mass is central to our calculation of the effects of preheating on the luminosity function of galaxies. If the IGM has a non-zero temperature, then pressure forces will prevent gravitational collapse of the gas on small scales. For gas at constant temperature, and ignoring the expansion of the Universe, the effects of pressure on the growth of density fluctuations in the gas are described by a simple Jeans criterion, such that density fluctuations on mass scales below the Jeans mass MJ are stable against collapse. However, this simple criterion needs to be modified in the case of an expanding Universe in which the gas temperature is a function of time. Gnedin & Hui (1998) have obtained an analytical description of the effects of gas pressure in this case. From linear perturbation analysis, the growth of density fluctuations in the gas is suppressed for comoving wavenumbers k > kF, where the critical wavenumber kF is related to the Jeans wavenumber kJ by  (Table 1).
and kJ is defined as In the above,ρtot is the mean total mass density including dark matter, D(t) and H(t) are the linear growth factor and Hubble constant respectively as functions of cosmic time t, and a dot over a variable represents a derivative with respect to t. Gnedin & Hui (1998) define T IGM to be the volume-weighted mean temperature of the IGM. We therefore compute the volume weighted temperature of IGM gas from our IGM model by averaging over the temperatures of gas at each density considered in the calculations. Such a global approach seems reasonable if the IGM temperature is reasonably homogeneous, such as happens in the case of a purely photoionized IGM (i.e. with no preheating). With preheating however, there can be considerable inhomogeneity in the IGM temperature since, after preheating, at the mean density cooling is dominated by Compton cooling and adiabatic expansion (both of which cool at a rate proportional to the gas density), while at high densities atomic cooling processes dominate (which are proportional to gas density squared). In this case it may be more realistic to compute the filtering mass as a function of density, using the density-dependent temperature in eqn. (11) to do so. We will use the volume-weighted temperature of the IGM to compute filtering masses unless stated otherwise, but consider the alternative approach also. The above expression for kF accounts for arbitrary thermal evolution of the IGM, through kJ(t). Corresponding to the critical wavenumber kF there is a critical mass MF which we will hereafter call the filtering mass, defined as The Jeans mass MJ is defined analogously in terms of kJ.
In the absence of pressure in the IGM, a halo of mass Mtot would be expected to accrete a mass (Ω b /Ω0)Mtot in gas when it collapsed. Gnedin (2000) found that in cosmological gas-dynamical simulations with a photoionized IGM, the average mass of gas Mgas which falls into halos of mass Mtot can be fit with the formula with the same value of MF as given by equations (11) and (12). The denominator in the above expression thus gives the factor by which the accreted gas mass is reduced because of the IGM pressure. Specifically, MF gives the halo mass for which the amount of gas accreted is reduced by a factor 2 compared to the case of no IGM pressure. The resulting filtering mass for each of our models is shown in Figure 6. The filtering mass begins to rise at z ≈ 20 due to the initial photoheating of the IGM by early star formation. At the epoch of preheating the filtering mass begins to rise sharply. For models with large E preheat , the filtering mass subsequently remains almost constant to z = 0. It is important to note that the filtering mass prescription results in a much more aggressive suppression of galaxy formation than the simpler prescription in which halos with virial temperature Tvir < TIGM are assumed to be unable to form galaxies. By z = 0, the hottest model we consider has TIGM ≈ 4000K, corresponding to the virial temperature of a 10 8 h −1 M⊙ halo, while the filtering mass for this model is a few times 10 11 h −1 M⊙. The "thermal memory" of the IGM as encapsulated in the filtering mass is therefore of crucial importance in determining the extent to which galaxy formation is suppressed. Consequently, it would be extremely valuable to conduct tests of the filtering mass prescription in preheated N-body simulations of galaxy formation to validate its use in this regime.
For the no preheating case the filtering mass is approximately 10 11 h −1 M⊙ at z = 0. Three of our models produce a filtering mass at z = 0 which is within a factor of three of this value. The remaining two predict filtering masses roughly an order of magnitude larger. In these latter two models, the filtering mass is comparable to the mass of halos thought to host L * galaxies. As such, we may expect these models to produce a dearth of L * and fainter galaxies. Note that the filtering mass resulting from a given E preheat depends strongly on z preheat .
As discussed above, the IGM temperature can be highly inhomogeneous in a preheated universe. As such, the use of a volume averaged IGM temperature in eqn. (11) may be inappropriate. In Fig. 7 we show the filtering mass computed for a model with E preheat = 0.1keV and z preheat = 9, using the temperature history of gas at several different densities, and compare this to the result obtained using the volume averaged temperature. Not surprisingly, the filtering mass of gas at mean density is very similar to that obtained using a volume averaged temperature. Furthermore, very low density Figure 6. The filtering mass as a function of redshift in our models. Results are shown for all models which adequately fit the z ≈ 3 temperature data and the Compton y-distortion constraint (Table 1). Figure 7. The filtering mass as a function of redshift for E preheat = 0.1keV and z preheat = 9. The thin solid line indicates the result when the volume average IGM temperature is used in eqn. (11), while the dotted line shows the result for gas at mean density. Short and long-dashed lines indicate the filtering mass for gas with density comparable to that of a void and of a dark matter halo at redshift zero respectively. For reference, the heavy, solid line shows the result for no preheating. gas (which will form part of a void at z = 0) has a filtering mass very similar to that of gas at mean density, since at low densities the dominant cooling mechanisms (Compton cooling and adiabatic expansion) are proportional to density. Fig. 7 also shows the filtering mass for gas which is at a density similar to that of a virialized halo at z = 0. Prior to z ≈ 2, the filtering mass for this gas is much lower than that for gas at mean density, lying close to the filtering mass for a model with no preheating. Here, the high density of the gas has allowed nearly all of the preheating energy to be rapidly radiated away, and so it has little effect on the filtering mass. After z ≈ 2 the filtering mass for this high density gas begins to rise due to photoheating (as this high density gas cools abundance of neutral species increases, thereby raising the photoheating rate), resulting in the filtering mass at z = 0 being somewhat higher than for gas at mean density. Nevertheless, over a wide range of redshift the high density gas has a significantly lower filtering mass than gas at mean density (and than that calculated using a volume averaged IGM temperature). Since galaxies are expected to form in high density regions this may have important consequences for the galaxy luminosity function. We will explore this possibility in §3.6. Figure 8 shows the luminosity functions predicted by the semi-analytic model of Benson et al. (2002a) when the filtering masses from Figure 6 are used. The model with no preheating actually succeeds in matching the bright end of the luminosity function rather well, but at the expense of over predicting the number of faint galaxies by almost an order of magnitude. As was noted by Benson et al. (2002a), photoheating alone is not sufficient to explain the paucity of faint galaxies.

Luminosity Functions
As anticipated in §3.5 those models which produced a filtering mass of several times 10 11 h −1 M⊙ at z = 0 result in too few galaxies faintwards of L * . As such, these models are clearly inconsistent with the observational data (dashed line in Figure 8). Of the remaining models we see that all perform better at matching the faint end of the luminosity function than the model with no preheating. In fact, the model with E preheat = 0.1keV and z preheat = 9 produces a very good match the the faint end of the luminosity function. However, this model fails to produce a sufficiently sharp cut off at the bright end and so over predicts the abundance of bright galaxies. This excess of bright galaxies occurs because, when the faint end of the luminosity function is sufficiently suppressed, too much gas remains available for cooling at late times. This gas is then able to cool in massive halos, producing an overabundance of bright galaxies. Note that the model with E preheat = 0.1keV and z preheat = 12 does do reasonably well at matching the bright end cut-off, but over predicts at the faint end. Similarly, a model with E preheat = 0.05keV and z preheat = 9 gets reasonably close to the bright end, but again fails to suppress the faint end sufficiently. Cole et al. (2000) were able to obtain a good match to the bright end of the luminosity function in a model using SNe feedback to suppress the faint end. Our failure to match the bright end in a preheated model is due to our use of a higher Ω b than Cole et al. (2000) (0.045 instead of 0.02). With the higher Ω b used here (and which is now preferred observationally) both preheated models and models with SNe feedback suffer the same problems in trying to match the bright end of the luminosity function (see Benson et al. 2003b for a detailed study of the problem of the correctly matching the bright end of the luminosity function).
As discussed in §3.5, if we compute the filtering mass using the temperature history as a function of density in the IGM (as opposed to using a volume averaged temperature history), we find that high density regions have a much lower filtering mass than low or average density regions. This will of course impact the luminosity function of galaxies. In Fig. 9 we show luminosity functions computed using these alternative filtering masses.
We compare these luminosity functions to their counterparts computed using the filtering mass for the volume averaged IGM temperature. The luminosity functions are intermediate between that for the volume averaged temperature calculation (since regions at average or lower density produce luminosity functions of this type) and the no preheating case (since high density regions produce luminosity functions of this type). The z preheat = 9, E preheat = 0.1keV model, which fits the observational data well using the volume averaged filtering mass, is a poor match in the new calculation. However, the z preheat = 12, E preheat = 0.3keV model which previously caused too much suppression is now a reasonable match to the faint-end of the luminosity function. Clearly, stronger preheating is required to produce a good match to the luminosity function when we account for the variation of filtering mass with the environment. Figure 9. B-band luminosity functions of galaxies at z = 0, as predicted by the semi-analytic model of Benson et al. (2002a) and using the filtering masses shown in Fig. 7 (using the IGM temperature as a function of density, T IGM (ρ)), are shown as lines, and are compared to those computed using the filtering mass appropriate for a volume averaged IGM temperature, T IGM . The observational determination of Norberg et al. (2002) is shown as circles.

Local Group Satellites
Finally, we examine the results of our models for very faint galaxies, namely the satellite galaxies found in the Local Group. As first shown by Kauffmann, White & Guiderdoni (1993), CDM models typically over predict the number of faint satellite galaxies in the Local Group. Recently, Benson et al. (2002b) examined the effect of photoheating on this abundance and concluded that while photoheating produced a large (almost an order of magnitude) reduction in the abundance of satellites, it was unable to fully reconcile the theory and observations.
In Figure 10 we show as circles the V-band luminosity function of Local Group satellite galaxies (from Benson et al. 2002b) per host halo (i.e. the halo in which the satellite orbits at the present time). The results of our models are overlaid as lines. As expected, the model with no preheating does not match the observational data, and predicts too many faint galaxies. ¶ Adding in preheating causes a rapid increase in the filtering mass just after z preheat . The effects of this increase can be understood quite simply-in particular the reader is referred to Benson et al. (2003a) who give a detailed discussion of the effects of the filtering mass on the luminosity function. Briefly, after z preheat , galaxy formation will be suppressed in dark matter halos with mass less ¶ Note that this luminosity function differs from that of Benson et al. (2002b) since we do not include any feedback due to supernovae here. Figure 10. The V-band luminosity function of satellite galaxies in the Local Group. We plot the differential luminosity function per host halo (n.b. we consider there to be two host halos-those of the Milky Way and M31-in the Local Group). Points show the observational result. Lines show the mean luminosity function of satellites in halos of mass 10 12 h −1 M ⊙ in our models.
than the filtering mass. Since the typical formation redshift of halos increases as the halo mass decreases we expect the filtering mass to steepen the luminosity function for faint galaxies. For halos with mass comparable to the filtering mass we expect a flattening of the luminosity function as the filtering mass gradually causes more suppression as halo mass decreases.
This is exactly what is seen in Figure 10. For example, the thin solid line (E preheat = 0.1keV, z preheat = 9) is much steeper than the no preheating model (heavy solid line) in the range MV − 5 log h = −12 to −15, and then rapidly flattens in the range MV − 5 log h = −16 to −18. Evidently, none of these models is able to satisfactorily fit the observational data. While the flattening at bright magnitudes helps match the observed luminosity function (e.g. the model with E preheat = 0.1keV, z preheat = 12 does well brightwards of MV − 5 log h = −15), the steeper slope at faint magnitudes results in an overabundance of faint satellites. Not surprisingly, no model using density-dependent filtering masses performs any better at matching the Local Group luminosity function.

CONCLUSIONS
We have calculated the thermal evolution of the IGM when it is rapidly preheated at a given redshift. Observations of the temperature of the IGM at z ≈ 3 allow us to rule out models in which this preheating occurs too late or to too high a temperature (simply speaking, the IGM must have sufficient time before z = 3 to cool down after preheat-ing). Unlike a purely photoionized IGM, the temperature after preheating can become highly inhomogeneous, since different cooling mechanisms dominate for gas of different densities. Consequently, the effects of preheating on galaxy formation depend strongly on whether we compute filtering masses using the volume averaged IGM temperature, or a local, density-dependent temperature.
Preheating causes an early reionization of the Universe, but in most cases using a globally averaged filtering mass hydrogen is able to mostly recombine before becoming highly ionized again at late times through photoionization by stars and quasars. When density-dependent filtering masses are used hydrogen does not recombine after preheating, although full reionization is delayed relative to a model with no preheating. For an escape fraction similar to current observational limits we find that after preheating there is a significant fraction of neutral hydrogen remaining, which would cause a Gunn-Peterson effect at low redshifts. The observed lack of a Gunn-Peterson effect by z ≈ 6 is therefore a strong constraint on preheating, or may imply the need for much higher escape fractions at high redshifts.
An important result from this work is that no model consistent with the z < ∼ 4 temperature data produces sufficient entropy to explain the high observed entropies in cluster cores. Furthermore, much of the entropy which is injected into the IGM is lost through cooling soon after preheating occurs.
Filtering masses computed from the volume averaged thermal history of the IGM can reach values comparable to the mass of halos hosting L * galaxies today if preheating is particularly energetic or early. In such cases we have shown that far too few L * and fainter galaxies are produced, allowing us to rule out these models. However, we find other preheated models which produce a galaxy luminosity function in excellent agreement with the data, at least for faint magnitudes, and without the need for supernovae feedback at late times. This comes at the expense of over predicting the abundance of bright galaxies however. When densitydependent filtering masses are used we find that, in the dense regions of the IGM where galaxies are most likely to form, rapid cooling after preheating keeps the filtering mass low until late times, resulting in much less suppression of galaxy formation. Consequently, more energetic preheating is required to achieve the same degree of suppression in the luminosity function compared to models with a globally averaged filtering mass. As a result, we find no preheating model consistent with the IGM temperature data that is able to fully match the observed luminosity function at the faint end.
An interesting conclusion is that we find no model which is able to adequately fit the luminosity function and is also consistent with the observed lack of a Gunn-Peterson effect at z < ∼ 6 for an escape fraction fesc = 10%. Too much neutral hydrogen remains after preheating, resulting in a large optical depth. This occurs as preheating strongly suppresses galaxy formation, reducing the number of ionizing photons produced below the number needed to fully ionize the Universe. However, this small neutral fraction does allow the models to potentially explain the Gunn-Peterson effect seen in Sloan Digital Sky Survey quasar spectra at z ≈ 6. This problem may be alleviated by adopting a higher escape fraction (e.g. 20% for z preheat = 9 and E preheat = 0.1keV) without significantly altering the thermal evolution of the IGM or the z = 0 galaxy luminosity function. Alternatively, if quasars are much more abundant at z > 6 than assumed in our calculations (which use the fitting function of Madau, Haardt & Rees 1999, which in turn is derived from observations of quasars at z < 4.5) they may provide sufficient photoionizations at high redshift to adequately reduce the neutral hydrogen fraction.
Finally, we examined the abundances of satellite galaxies in the Local Group. While preheating is able to flatten the predicted luminosity function for relatively bright satellites-bringing it into agreement with the observational data-it steepens the luminosity function at faint magnitudes and so is unable to explain the paucity of the faintest satellites.
Preheating alone can produce a galaxy luminosity function almost as flat as that observed without the need for feedback from supernovae as is commonly assumed in galaxy formation models. However, preheating acting alone is not able to fully match the observational constraints. The problem here is that when the faint end of the luminosity function is significantly suppressed too much gas remains available for cooling at late times. This gas is then able to cool in massive halos, producing an overabundance of bright galaxies (Benson et al. 2003b-see also Kauffmann et al. 1999;Cole et al. 2000;Somerville & Primack 1999). In conclusion, an early epoch of preheating has important consequences for galaxy formation at recent times, and may remove or reduce the need for more traditional forms of feedback in CDM models.