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9.1. Spectral Methods of Inferring the Reionization Redshift

9.1.1. Cosmology with Lyalpha Photons

The scattering cross-section of the Lyalpha resonance line by neutral hydrogen is given by (Section 23 of Peebles 1993)

Equation 107   (107)

where Lambdaalpha = (8pi2 e2 falpha / 3me clambdaalpha2) = 6.25 × 108 s-1 is the Lyalpha (2p -> 1s) decay rate, falpha = 0.4162 is the oscillator strength, and lambdaalpha = 1216Å and nualpha = (c / lambdaalpha) = 2.47 × 1015 Hz are the wavelength and frequency of the Lyalpha line. The term in the numerator is responsible for the classical Rayleigh scattering.

We consider a source at a redshift zs beyond the redshift of reionization (10) zreion, and the corresponding scattering optical depth of a uniform, neutral IGM of hydrogen density nH, 0(1 + z)3 between the source and the reionization redshift. The optical depth is a function of the observed wavelength lambdaobs,

Equation 108   (108)

where nuobs = c / lambdaobs and

Equation 109   (109)

At wavelengths longer than Lyalpha at the source, the optical depth obtains a small value; these photons redshift away from the line center along its red wing and never resonate with the line core on their way to the observer. Considering only the regime in which |nu - nualpha| >> Lambdaalpha, we may ignore the second term in the denominator of equation (107). This leads to an analytical result for the red damping wing of the Gunn-Peterson trough (Miralda-Escudé 1998)

Equation 110   (110)

where taus is given in equation (1), and we also define

Equation 111   (111)


Equation 112   (112)

At wavelengths corresponding to the Lyalpha resonance between the source redshift and the reionization redshift, (1 + zreion) lambdaalpha leq lambdaobs leq (1 + zs) lambdaalpha, the optical depth is given by equation (1). Since taus ~ 105, the flux from the source is entirely suppressed in this regime. Similarly, the Lybeta resonance produces another trough at wavelengths (1 + zreion) lambdabeta leq lambda leq (1 + zs) lambdabeta, where lambdabeta = (27 / 32) lambdaalpha = 1026 Å, and the same applies to the higher Lyman series lines. If (1 + zs) geq 1.18(1 + zreion) then the Lyalpha and the Lybeta resonances overlap and no flux is transmitted in-between the two troughs (see Figure 40). The same holds for the higher Lyman-series resonances down to the Lyman limit wavelength of lambdac = 912Å.

Figure 40

Figure 40. Sketch of the expected spectrum of a source at a redshift zs slightly above the reionization redshift zreion. The transmitted flux due to H II bubbles in the pre-reionization era and the Lyalpha forest in the post-reionization era is exaggerated for illustration.

At wavelengths shorter than lambdac, the photons are absorbed when they photoionize atoms of hydrogen or helium. The bound-free absorption cross-section from the ground state of a hydrogenic ion with nuclear charge Z and an ionization threshold hnu0, is given by (Osterbrock 1974),

Equation 113   (113)


Equation 114   (114)

For neutral hydrogen, Z = 1 and nuH, 0 = (c / lambdac) = 3.29 × 1015 Hz (hnuH, 0 = 13.60 eV); for singly-ionized helium, Z = 2 and nuHeII, 0 = 1.31 × 1016 Hz (hnuHeII, 0 = 54.42 eV). The cross-section for neutral helium is more complicated; when averaged over its narrow resonances it can be fitted to an accuracy of a few percent up to hnu = 50 keV by the fitting function (Verner et al. 1996)

Equation 115   (115)

where x ident [(nu / 3.286 × 1015 Hz) - 0.4434], y ident x2 + 4.563, and the threshold for ionization is nuHeI, 0 = 5.938 × 1015 Hz (hnuHeI, 0 = 24.59 eV).

For rough estimates, the average photoionization cross-section for a mixture of hydrogen and helium with cosmic abundances can be approximated in the range of 54 < hnu ltapprox 103 eV as sigmabf approx sigma0(nu / nuH, 0)-3, where sigma0 approx 6 × 10-17 cm2 (Miralda-Escudé 2000). The redshift factor in the cross-section then cancels exactly the redshift evolution of the gas density and the resulting optical depth depends only on the elapsed cosmic time, t(zreion) - t(zs). At high redshifts (equations (9) and (10) in Section 2.1) this yields,

Equation 116   (116)

The bound-free optical depth only becomes of order unity in the extreme UV to soft X-rays, around hnu ~ 0.1 keV, a regime which is unfortunately difficult to observe due to Galactic absorption (Miralda-Escudé 2000).

A sketch of the overall spectrum of a source slightly above the reionization redshift, i.e., with 1 < [(1 + zs) / (1 + zreion)] < 1.18, is shown in Figure 40. The transmitted flux between the Gunn-Peterson troughs due to Lyalpha and Lybeta absorption is suppressed by the Lyalpha forest in the post-reionization epoch. Transmission of flux due to H II bubbles in the pre-reionization epoch is expected to be negligible (Miralda-Escudé 1998). The redshift of reionization can be inferred in principle from the spectral shape of the red damping wing (Miralda-Escudé & Rees 1998; Miralda-Escudé 1998) or from the transmitted flux between the Lyman series lines (Haiman & Loeb 1999a). However, these signatures are complicated in reality by damped Lyalpha systems along the line of sight or by the inhomogeneity or peculiar velocity field of the IGM in the vicinity of the source. Moreover, bright sources, such as quasars, tend to ionize their surrounding environment (Wood & Loeb 2000) and the resulting H II region in the IGM could shift the Lyalpha trough substantially (Cen & Haiman 2000; Madau & Rees 2000).

The inference of the Lyalpha transmission properties of the IGM from the observed spectrum of high-redshift sources suffers from uncertainties about the precise emission spectrum of these sources, and in particular the shape of their Lyalpha emission line. The first galaxies and quasars are expected to have pronounced recombination lines of hydrogen and helium due to the lack of dust in their interstellar medium (see Section 4.1.3 for more details). Lines such as Halpha or the He II 1640Å line should reach the observer unaffected by the intervening IGM, since their wavelength is longer than that of the Lyalpha transition which dominates the IGM opacity (Oh 1999). However, as described above, the situation is different for the Lyalpha line photons from the source. As long as zs > zreion, the intervening neutral IGM acts like a fog and obscures the view of the Lyalpha line itself [in contrast to the situation with sources at zs < zreion, where most of the intervening IGM is ionized and only photons more energetic than Lyalpha are suppressed by the Lyalpha forest (see Figure 3)]. Photons which are emitted at the Lyalpha line center have an initial scattering optical depth of ~ 105 in the surrounding medium.

The Lyalpha line photons are not destroyed but instead are absorbed and re-emitted (11) . Due to the Hubble expansion of the IGM around the source, the frequency of the photons is slightly shifted by the Doppler effect in each scattering event. As a result, the photons diffuse in frequency to the red side of the Lyalpha resonance. Eventually, when their net frequency redshift is sufficiently large, they escape and travel freely towards the observer (see Figure 41). As a result, the source creates a faint Lyalpha halo on the sky (12) . The well-defined radiative transfer problem of a point source of Lyalpha photons embedded in a uniform, expanding neutral IGM was solved by Loeb & Rybicki (1999). The Lyalpha halo can be simply characterized by the frequency redshift relative to the line center, (nu - nualpha), which is required in order to make the optical depth from the source [equation (110] equal to unity. At high redshifts, the leading term in equation (110) yields

Equation 117   (117)

This is the frequency interval over which the damping wing affects the source spectrum. A frequency shift of nu* = 8.85 × 1012 Hz relative to the line center corresponds to a fractional shift of (nu* / nualpha) = (v/c) = 3.6 × 10-3 or a Doppler velocity of v ~ 103 km s-1. The halo size is then defined by the corresponding proper distance from the source at which the Hubble velocity provides a Doppler shift of this magnitude,

Equation 118   (118)

Typically, the Lyalpha halo of a source at zs ~ 10 occupies an angular radius of ~ 15" on the sky and yields an asymmetric line profile as shown in Figures 41 and 42. The scattered photons are highly polarized and so the shape of the halo would be different if viewed through a polarization filter (Rybicki & Loeb 1999).

Figure 41

Figure 41. Loeb-Rybicki halos: Scattering of Lyalpha line photons from a galaxy embedded in the neutral intergalactic medium prior to reionization. The line photons diffuse in frequency due to the Hubble expansion of the surrounding medium and eventually redshift out of resonance and escape to infinity. A distant observer sees a Lyalpha halo surrounding the source, along with a characteristically asymmetric line profile. The observed line should be broadened and redshifted by about one thousand km s-1 relative to other lines (such as Halpha) emitted by the galaxy.

Figure 42

Figure 42. Monochromatic photon luminosity of a Lyalpha halo as a function of frequency redshift, nutilde ident (nualpha - nu) / nu*. The observed spectral flux of photons F(nu) (in photons cm-2 s-1 Hz-1) from the entire Lyalpha halo is F(nu) = (Ltilde(nutilde) / 4pi dL2)(Ndotalpha / nu*)(1 + zs)2 where Ndotalpha is the production rate of Lyalpha photons by the source (in photons s-1), nu = nutilde nu* / (1 + zs), and dL is the standard luminosity distance to the source (from Loeb & Rybicki 1999).

Detection of the diffuse Lyalpha halos around bright high-redshift sources (which are sufficiently rare so that their halos do not overlap) would provide a unique tool for probing the distribution and the velocity field of the neutral intergalactic medium before the epoch of reionization. The Lyalpha sources serve as lamp posts which illuminate the surrounding H1 fog. On sufficiently large scales where the Hubble flow is smooth and the gas is neutral, the Lyalpha brightness distribution can be used to determine the cosmological mass densities of baryons and matter. Due to their low surface brightness, the detection of Lyalpha halos through a narrow-band filter is much more challenging than direct observation of their sources at somewhat longer wavelengths. However, NGST might be able to detect the Lyalpha halos around sources as bright as the quasar discovered by Fan et al. (2000) at z = 5.8 or the galaxy discovered by Hu et al. (1999) at z = 5.74, even if these sources were moved out to z ~ 10 (see Section 4 in Loeb & Rybicki 1999). The disappearance of Lyalpha halos below a certain redshift can be used to determine zreion.

9.1.2. 21 cm Tomography of the Reionization Epoch

The ground state of hydrogen exhibits hyperfine splitting involving the spins of the proton and the electron. The state with parallel spins (the triplet state) has a slightly higher energy than the state with anti-parallel spins (the singlet state). The 21 cm line associated with the spin-flip transition from the triplet to the singlet state is often used to detect neutral hydrogen in the local universe. At high redshift, the occurrence of a neutral pre-reionization IGM offers the prospect of detecting the first sources of radiation and probing the reionization era by mapping the 21 cm emission from neutral regions. While its energy density is estimated to be only a 1% correction to that of the CMB, the redshifted 21 cm emission should display angular structure as well as frequency structure due to inhomogeneities in the gas density field (Hogan & Rees 1979; Scott & Rees 1990), hydrogen ionized fraction, and spin temperature (Madau, Meiksin, & Rees 1997). Some of the resulting signatures during the pre-overlap phase of reionization (Section 6.3.1) and during the overlap phase are discussed by Tozzi et al. (2000) and Shaver et al. (1999), respectively. Also, the 21 cm signatures have been explored in a numerical simulation by Gnedin & Ostriker (1997). Indeed, a full mapping of the distribution of H1 as a function of redshift is possible in principle. Although detecting the presence of the largest H II regions may be within the reach of proposed instruments such as the Square Kilometer Array (hereafter SKA; see Taylor & Braun 1999), these instruments may not have sufficient sensitivity at the sub-arcminute resolution that would be necessary for a detailed mapping. Moreover, serious technical challenges and problems due to foreground contamination must be overcome even for an initial detection of the reionization signal.

The basic physics of the hydrogen spin transition is determined as follows (for a more detailed treatment, see Madau et al. 1997). The ground-state hyperfine levels of hydrogen tend to thermalize with the CMB background, making the IGM unobservable. If other processes shift the hyperfine level populations away from thermal equilibrium, then the gas becomes observable against the CMB in emission or in absorption. The relative occupancy of the spin levels is usually described in terms of the hydrogen spin temperature TS, defined by

Equation 119   (119)

where n0 and n1 are the singlet and triplet hyperfine levels in the atomic ground state (n = 1), and T* = 0.07 K is defined by kB T* = E21, where the energy of the 21 cm transition is E21 = 5.9 × 10-6 eV, corresponding to a frequency of 1420 MHz. In the presence of the CMB alone, the spin states reach thermal equilibrium with TS = TCMB = 2.73(1 + z) K on a time-scale of T* / (TCMBA10) appeq 3 × 105(1 + z)-1 yr, where A10 = 2.9 × 10-15 s-1 is the spontaneous decay rate of the hyperfine transition. This time-scale is much shorter than the age of the universe at all redshifts after cosmological recombination.

The IGM is observable when the kinetic temperature TK of the gas differs from TCMB and an effective mechanism couples TS to TK. Although collisional de-excitation of the triplet level (Purcell & Field 1956) is a possible mechanism, in the low-density IGM the dominant mechanism is scattering by Lyalpha photons (Wouthuysen 1952; Field 1958). Continuum UV photons produced by early radiation sources redshift by the Hubble expansion into the local Lyalpha line at a lower redshift. These photons mix the spin states via the Wouthuysen-Field process whereby an atom initially in the n = 1 state absorbs a Lyalpha photon, and the spontaneous decay which returns it from n = 2 to n = 1 can result in a final spin state which is different from the initial one. Since the neutral IGM is highly opaque to resonant scattering, the shape of the radiation spectrum near Lyalpha is determined by TK (Field 1959), and the spin temperature is then a weighed mean of TK and TCMB:

Equation 120   (120)

where (if TS >> T*) the Lyalpha pumping efficiency is

Equation 121   (121)

Here P10 is the indirect de-excitation rate of the triplet n = 1 state via the Wouthuysen-Field process, related to the total scattering rate Palpha of Lyalpha photons by P10 = 4Palpha / 27 (Field 1958). Thus the critical value of Palpha is given by the thermalization rate (Madau et al. 1997)

Equation 122   (122)

A patch of neutral hydrogen at the mean density and with a uniform TS produces an optical depth at 21(1 + z) cm of

Equation 123   (123)

assuming a high redshift z. Since the brightness temperature through the IGM is Tb = TCMB e-tau + TS (1 - e-tau), the observed differential antenna temperature of this region relative to the CMB is (Madau et al. 1997, with the Omegam dependence added)

Equation 124   (124)

where tau << 1 is assumed and deltaTb has been redshifted to redshift zero. In overdense regions, the observed deltaTb is proportional to the overdensity, and in partially ionized regions deltaTb is proportional to the neutral fraction. Thus, if TS >> TCMB then the IGM is observed in emission at a level that is independent of TS. On the other hand, if TS << TCMB then the IGM is observed in absorption at a level that is a factor ~ TCMB / TS larger than in emission. As a result, a number of cosmic events are expected to leave observable signatures in the redshifted 21 cm line.

Since the CMB temperature is only 2.73(1 + z) K, even relatively inefficient heating mechanisms are expected to heat the IGM above TCMB well before reionization. Possible preheating sources include soft X-rays from early quasars or star-forming regions, as well as thermal bremsstrahlung from ionized gas in collapsing halos. However, even the radiation from the first stars may suffice for an early preheating. Only ~ 10% of the present-day global star formation rate is required (Madau et al. 1997) for a sufficiently strong Lyalpha background which produces a scattering rate above the thermalization rate Pth. Such a background drives TS to the kinetic gas temperature, which is initially lower than TCMB because of adiabatic expansion. Thus, the entire IGM can be seen in absorption, but the IGM is then heated above TCMB in ~ 108 yr (Madau et al. 1997) by the atomic recoil in the repeated resonant Lyalpha scattering. According to Section 8.1 (also compare Gnedin 2000a), the required level of star formation is expected to be reached already at z ~ 20, with the entire IGM heated well above the CMB by the time overlap begins at z ~ 10. Thus, although the initial absorption signal is in principle detectable with the SKA (Tozzi et al. 2000), it likely occurs at ltapprox 100 MHz where Earth-based radio interference is highly problematic.

As individual ionizing sources turn on in the pre-overlap stage of reionization, the resulting H II bubbles may be individually detectable if they are produced by rare and luminous sources such as quasars. If the H II region expands into an otherwise unperturbed IGM, then the expanding shell can be mapped as follows (Tozzi et al. 2000). The H II region itself, of course, shows neither emission nor absorption. Outside the ionized bubble, a thin shell of neutral gas is heated above the CMB temperature and shows up in emission. A much thicker outer shell is cooler than the CMB due to adiabatic expansion, but satisfies TS = TK and produces absorption. Finally, at large distances from the quasar, TS approaches TCMB as the quasar radiation weakens. For a quasar with an ionizing intensity of 1057 photons s-1 observed after ~ 107 yr with 2' resolution and 1 MHz bandwidth, the signal ranges from -3 to 3 µJy per beam (Tozzi et al. 2000). Mapping such regions would convey information on the quasar number density, ionizing intensity, opening angle, and on the density distribution in the surrounding IGM. Note, however, that an H II region which forms at a redshift approaching overlap expands into a preheated IGM. In this case, the H II region itself still appears as a hole in an otherwise emitting medium, but the quasar-induced heating is not probed, and there is no surrounding region of absorption to supply an enhanced contrast.

At redshifts approaching overlap, the IGM should be almost entirely neutral but with TS >> TCMB. In this redshift range there should still be an interesting signal due to density fluctuations. The same cosmic network of sheets and filaments that gives rise to the Lyalpha forest observed at z ltapprox 5 should lead to fluctuations in the 21 cm brightness temperature at higher redshifts. At 150 MHz (z = 8.5), for observations with a bandwidth of 1 MHz, the root mean square fluctuation should be ~ 10 mK at 1', decreasing with scale (Tozzi et al. 2000).

A further signature, observable over the entire sky, should mark the overlap stage of reionization. During overlap, the IGM is transformed from being a neutral, preheated and thus emitting gas, to being almost completely ionized. This disappearance of the emission over a relatively narrow redshift range can be observed as a drop in the brightness temperature at the frequencies corresponding to the latter stages of overlap (Shaver et al. 1999). This exciting possibility, along with those mentioned above, face serious challenges in terms of signal contamination and calibration. The noise sources include galactic and extragalactic emission sources, as well as terrestrial interference, and all of these foregrounds must be modeled and accurately removed in order to observe the fainter cosmological signal (see Shaver et al. 1999 for a detailed discussion). For the overlap stage in particular, the sharpness of the spectral feature is the key to its detectability, but it may be significantly smoothed by inhomogeneities in the IGM.

9.2. Effect of Reionization on CMB Anisotropies

In standard cosmological models, the early universe was hot and permeated by a nearly uniform radiation bath. At z ~ 1200 the free protons and electrons recombined to form hydrogen atoms, and most of the photons last scattered as the scattering cross-section dropped precipitously. These photons, observed today as the Cosmic Microwave Background (CMB), thus yield a snapshot of the state of the universe at that early time. Small fluctuations in the density, velocity, and gravitational potential lead to observed anisotropies (e.g., Bennett et al. 1996) that can be analyzed to yield a great wealth of information on the matter content of the universe and on the values of the cosmological parameters (e.g., Hu 1995; Jungman et al. 1996).

Reionization can alter the anisotropy spectrum, by erasing some of the primary anisotropy imprinted at recombination, and by generating additional secondary fluctuations that could be used to probe the era of reionization itself (see Haiman & Knox 1999 for a review). The primary anisotropy is damped since the rescattering leads to a blending of photons from initially different lines of sight. Furthermore, not all the photons scatter at the same time, rather the last scattering surface has a finite thickness. Perturbations on scales smaller than this thickness are damped since photons scattering across many wavelengths give canceling redshifts and blueshifts. If reionization occurs very early, the high electron density produces efficient scattering, and perturbations are damped on all angular scales except for the very largest.

The optical depth to scattering over a proper length dl is dtau = sigmaT ne dl, where sigmaT is the Thomson cross-section and ne the density of free electrons. If reionization occurs instantaneously at redshift z, then the total scattering optical depth in LambdaCDM is given by (e.g., Section 7.1.1 of Hu 1995)

Equation 125   (125)

With our standard parameters (end of Section 1) this implies tau = 0.037 at the current lower limit on reionization of z = 5.8 (Fan et al. 2000), with tau = 0.10 if z = 11.6 and tau = 0.15 if z = 15.3. Recent observations of small-scale anisotropies (Lange et al. 2000; Balbi et al. 2000) revealed a peak in the power spectrum on a ~ 1° scale, as expected from the primary anisotropies in standard cosmological models. This indicates that the reionization damping, if present, is not very large, and the observations set a limit of tau < 0.33 at 95% confidence (Tegmark & Zaldarriaga 2000) and, therefore, imply that reionization must have occurred at z ltapprox 30.

However, measuring a small tau from the temperature anisotropies alone is expected to be very difficult since the anisotropy spectrum depends on a large number of other parameters, creating a near-degeneracy which limits our ability to measure each parameter separately; the degeneracy of tau with other cosmological parameters is due primarily to a degeneracy with the gravitational-wave background. However, Thomson scattering also creates net polarization for incident radiation which has a quadrupole anisotropy. This anisotropy was significant at reionization due to large-scale structure which had already affected the gas distribution. The result is a peak in the polarization power spectrum on large angular scales of order tens of degrees (Zaldarriaga 1997). Although experiments must overcome systematic errors from the detector itself and from polarized foregrounds (such as galactic dust emission and synchrotron radiation), parameter estimation models (Eisenstein, Hu, & Tegmark 1999; Zaldarriaga, Spergel, & Seljak 1997) suggest that the peak can be used to measure even very small values of tau: 2% for the upcoming MAP satellite, and 0.5% for the Planck satellite which will reach smaller angular scales with higher accuracy.

Reionization should also produce additional temperature anisotropies on small scales. These result from the Doppler effect. By the time of reionization, the baryons have begun to follow dark matter potentials and have acquired a bulk velocity. Since the electrons move with respect to the radiation background, photons are given a Doppler kick when they scatter off the electrons. Sunyaev (1978) and Kaiser (1984) showed, however, that a severe cancellation occurs if the electron density is homogeneous. Opposite Doppler shifts on crests and troughs of a velocity perturbation combine to suppress the anisotropy induced by small-scale velocity perturbations. The cancellation is made more severe by the irrotational nature of gravitationally-induced flows. However, if the electron density varies spatially, then the scattering probability is not equal on the crest and on the trough, and the two do not completely cancel. Since a non-zero effect requires variation in both electron density and velocity, it is referred to as a second-order anisotropy.

The electron density can vary due to a spatial variation in either the baryon density or the ionized fraction. The former is referred to as the Ostriker-Vishniac effect (Ostriker & Vishniac 1986; Vishniac 1987). The latter depends on the inhomogeneous topology of reionization, in particular on the size of H II regions due to individual sources (Section 6.2) and on spatial correlations among different regions. Simple models have been used to investigate the character of anisotropies generated during reionization (Gruzinov & Hu 1998; Knox et al. 1998; Aghanim et al. 1996). The Ostriker-Vishniac effect is expected to dominate all anisotropies at small angular scales (e.g., Jaffe & Kamionkowski 1998), below a tenth of a degree, because the primary anisotropies are damped on such small scales by diffusion (Silk damping) and by the finite thickness of the last scattering surface. Anisotropies generated by inhomogeneous reionization may be comparable to the Ostriker-Vishniac effect, and could be detected by MAP and Planck, if reionization is caused by bright quasars with 10 Mpc-size ionized bubbles. However, the smaller bubbles expected for mini-quasars or for star-forming dwarf galaxies would produce an anisotropy signal which is weaker and at smaller angular scales, likely outside the range of the upcoming satellites (see, e.g., Haiman & Knox 1999 for discussion). Gnedin & Jaffe (2000) used a numerical simulation to show that, in the case of stellar reionization, the effect on the CMB of patchy reionization is indeed sub-dominant compared to the contribution of non-linear density and velocity fluctuations. Nevertheless, a signature of reionization could still be detected in future measurements of CMB angular fluctuations on the scale of a few arcseconds (see also Bruscoli et al. 2000, who find a somewhat higher power spectrum due to patchy reionization).

9.3. Remnants of High-Redshift Systems in the Local Universe

At the end of the reionization epoch, the heating of the IGM resulted in the photo-evaporation of gas out of halos of circular velocity Vc above ~ 10-15 km s-1 (Section 6.4). The pressure of the hot gas subsequently shut off gas infall into even more massive halos, those with Vc ~ 30 km s-1 (Section 6.5). Thus, the gas reservoir of photo-evaporating halos could not be immediately replenished. Some dwarf galaxies which were prevented from forming after reionization could have eventually collected gas at z = 1-2, when the UV background flux declined sufficiently (Babul & Rees 1992; Kepner, Babul, & Spergel 1997). However, Kepner et al. (1997) found that even if the ionizing intensity J21 declines as (1 + z)4 below z = 3, only halos with Vc gtapprox 20 km s-1 can form atomic hydrogen by z = 1, and Vc gtapprox 25 km s-1 is required to form molecular hydrogen. While the fact that the IGM was reionized has almost certainly influenced the abundance and properties of dwarf galaxies observed today, the exact manifestations of this influence and ways to prove that they occurred have not been well determined. In this section we summarize recent work on this topic, which should remain an active research area.

The suppression of gas infall mentioned above suggests that the abundance of luminous halos as a function of circular velocity should show a break, with a significant drop in the abundance below Vc = 30I>km s-1. Such a drop may in fact be required in order to reconcile the LambdaCDM model with observations. Klypin et al. (1999) and Moore et al. (1999) found that the abundance of halos with Vc ~ 10-30 km s-1 in numerical simulations of the Local Group environment is higher by an order of magnitude than the observed dwarf galaxy abundance. The predicted and observed abundances matched well at Vc > 50 km s-1. Bullock et al. (2000a) considered whether photoionization can explain the discrepancy at the low-mass end by preventing dark matter halos from forming stars. They assumed that a sub-halo in the Local Group can host an observable galaxy only if already at reionization its main progenitor contained a fraction f of the final sub-halo mass. Using semi-analytic modeling, they found a close match to the observed circular velocity distribution for zreion = 8 and f = 0.3.

These results neglect several complications. As mentioned above, halos with Vc gtapprox 20 km s-1 may be able to accrete gas and form stars once again at z ltapprox 1. Any accreted gas at low redshift could have been previously enriched with metals and molecules, thus enabling more efficient cooling. On the other hand, if the progenitors had a Vc ltapprox 17 km s-1 at zreion then they were not able to cool and form stars, unless molecular hydrogen had not been dissociated (Section 3.3). In order to reconcile the photoionization scenario with the recent episodes of star formation deduced to have occurred in most dwarf galaxies (e.g., Mateo 1998; Grebel 1998), a continuous recycling of gas over many generations of stars must be assumed. This would mean that supernova feedback (Section 7.1) was unable to shut off star formation even in these smallest known galaxies. In addition, the existence of a large abundance of sub-halos may be problematic even if the sub-halos have no gas, since they would interact with the disk dynamically and tend to thicken it (Toth & Ostriker 1992; Moore et al. 1999; Velazquez & White 1999).

However, the photoionization scenario is useful because it may be testable through other implications. For example, Bullock et al. (2000b) used semi-analytic modeling to show that many subhalos which did form stars before reionization were tidally disrupted in the Milky Way's gravitational field, and the resulting stellar streams may be observable. The formation of the Milky Way's stellar component has also been investigated by White & Springel (2000). They combined a scaled-down dark matter cluster simulation with semi-analytic prescriptions for star formation in halos, and showed that the oldest stars in the Milky Way should be located mostly in the inner halo or bulge, but they cannot be easily identified because the populations of old stars and of low metallicity stars are only weakly correlated.

Gnedin (2000c) pointed out several observed features of dwarf galaxies that may be related to their high-redshift histories. First, almost all Local Group dwarf galaxies with measured star formation histories show a sharp decline in the star formation rate around 10 Gyr ago. Gnedin noted that this drop could correspond to the suppression of star formation due to reionization (Section 6.5) if the measured ages of the old stellar populations are somewhat underestimated; or that it could instead correspond to the additional suppression caused by helium reionization (Section 6.3.2) at z gtapprox 3. He showed that if only the old stellar population is considered, then the Schmidt Law (Section 5.2) implies that the luminosity of each dwarf galaxy, divided by a characteristic volume containing a fixed fraction of all the old stars, should be proportional to some power of the central luminosity density. This assumes that the present central luminosity density of old stars is a good measure of the original gas density, i.e., that in the core almost all the gas was transformed into stars, and feedback did not play a role. It also assumes that the gas density distribution was self-similar in all the dwarf galaxies during the period when they formed stars, and that the length of this period was also the same in all galaxies. These assumptions are required in order for the total stellar content of each galaxy to be simply related to the central density via the Schmidt Law evaluated at the center. Taking ten galaxies which have well-measured star formation histories, and which formed most of their stars more than 10 Gyr ago, Gnedin found a correlation with a power law of 3/2, as expected from the Schmidt Law (Section 5.2). Clearly, the theoretical derivation of this correlation combines many simplistic assumptions. However, explaining the observed correlation is a challenge for any competing models, e.g., models where feedback plays a dominant role in regulating star formation.

Barkana & Loeb (1999) noted that a particularly acute puzzle is presented by the very smallest galaxies, the nine dwarf spheroidals in the Local Group with central velocity dispersions sigma ltapprox 10 km s-1, including five below 7 km s-1 (see recent reviews by Mateo 1998 & van den Bergh 2000). These galaxies contain old stars that must have formed at z gtapprox 2, before the ionizing background dropped sufficiently to allow them to form. There are several possible solutions to the puzzle of how these stars formed in such small halos or in their progenitors which likely had even smaller velocity dispersions. The solutions are that (i) molecular hydrogen allowed these stars to form at z > zreion, as noted above, (ii) the measured stellar velocity dispersions of the dwarf galaxies are well below the velocity dispersions of their dark matter halos, or (iii) the dwarf galaxies did not form via the usual hierarchical scenario.

One major uncertainty in comparing observations to hierarchical models is the possibility that the measured velocity dispersion of stars in the dwarf spheroidals underestimates the velocity dispersion of their dark halos. Assuming that the stars are in equilibrium, their velocity dispersion could be lower than that of the halo if the mass profile is shallower than isothermal beyond the stellar core. The velocity dispersion and mass-to-light ratio of a dwarf spheroidal could also appear high if the galaxy is non-spherical or the stellar orbits are anisotropic. The observed properties of dwarf spheroidals require a central mass density of order 0.1 Msun pc-3 (e.g., Mateo 1998), which is ~ 7 × 105 times the present critical density. Thus, only the very inner parts of the halos are sampled by the central velocity dispersion. Detailed observations of the velocity dispersion profiles of these galaxies could be used to determine the circular velocity of the underlying halo more reliably.

A cosmological scenario for the formation of dwarf spheroidal galaxies is favored by the fact that they are observed to be dark matter dominated, but this may not rule out all the alternatives. The dwarf dark halos may have formed at low redshift by the breakup of a much larger galaxy. Under this scenario, gas forming stars inside the large parent galaxy would have been unaffected by the photoionizing background. At low redshift, this galaxy may have collided with the Milky Way or come close enough to be torn apart, forming at least some of the dwarf spheroidal systems. Simulations of galaxy encounters (Barnes & Hernquist 1992; Elmegreen, Kaufman, & Thomasson 1993) have found that shocks in the tidal tails trigger star formation and lead to the formation of dwarf galaxies, but these galaxies contain only small amounts of dark matter. However, the initial conditions of these simulations assumed parent galaxies with a smooth dark matter distribution rather than clumpy halos with dense sub-halos inside them. As noted above, simulations (Klypin et al. 1999; Moore et al. 1999) suggest that galaxy halos may have large numbers of dark matter satellites, and further simulations are needed to test whether these subhalos can capture stars which form naturally in tidal tails.

A common origin for the Milky Way's dwarf satellites (and a number of halo globular clusters), as remnants of larger galaxies accreted by the Milky Way galaxy, has been suggested on independent grounds. These satellites appear to lie along two (e.g., Majewski 1994) or more (Lynden-Bell & Lynden-Bell 1995, Fusi-Pecci et al. 1995) polar great circles. The star formation history of the dwarf galaxies (e.g., Grebel 1998) constrains their merger history, and implies that the fragmentation responsible for their appearance must have occurred early in order to be consistent with the variation in stellar populations among the supposed fragments (Unavane, Wyse, & Gilmore 1996; Olszewski 1998). Observations of interacting galaxies (outside the Local Group) also demonstrate that ``tidal dwarf galaxies'' do indeed form (e.g., Duc & Mirabel 1997; Hunsberger, Charlton, & Zaritsky 1996).

10 We define the reionization redshift to be the redshift at which the individual H II regions overlapped and most of the IGM volume was ionized. In most realistic scenarios, this transition occurs rapidly on a time-scale much shorter than the age of the universe (see Section 6.3.1). This is mainly due to the short distances between neighboring sources. Back.

11 At the redshifts of interest, zs ~ 10, the low densities and lack of chemical enrichment of the IGM make the destruction of Lyalpha photons by two-photon decay or dust absorption unimportant. Back.

12 The photons absorbed in the Gunn-Peterson trough are also re-emitted by the IGM around the source. However, since these photons originate on the blue side of the Lyalpha resonance, they travel a longer distance from the source, compared to the Lyalpha line photons, before they escape to the observer. The Gunn-Peterson photons are therefore scattered from a larger and hence dimmer halo around the source. The Gunn-Peterson halo is made even dimmer relative to the Lyalpha line halo by the fact that the luminosity of the source per unit frequency is often much lower in the continuum than in the Lyalpha line. Back.

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