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When an isolated point source of ionizing radiation turns on, the ionized volume initially grows in size at a rate fixed by the emission of UV photons, and an ionization front separating the H II and H I regions propagates into the neutral gas. Most photons travel freely in the ionized bubble, and are absorbed in a transition layer. The evolution of an expanding H II region is governed by the equation

Equation 8 (8)

where VI is the proper volume of the ionized zone, Ndotion is the number of ionizing photons emitted by the central source per unit time, nbarH is the mean hydrogen density of the expanding IGM, H is the Hubble constant, and tbarrec is the hydrogen mean recombination timescale,

Equation 9a
Equation 9b (9)

One should point out that the use of a volume-averaged clumping factor, C, in the recombination timescale is only justified when the size of the H II region is large compared to the scale of the clumping, so that the effect of many clumps (filaments) within the ionized volume can be averaged over (see Figure 5). Across the I-front the degree of ionization changes sharply on a distance of the order of the mean free path of an ionizing photon. When tbarrec >> t, the growth of the H II region is slowed down by recombinations in the highly inhomogeneous medium, and its evolution can be decoupled from the expansion of the universe. Just like in the static case, the ionized bubble will fill its time-varying Strömgren sphere after a few recombination timescales,

Equation 10 (10)

In analogy with the individual H II region case, it can be shown that hydrogen component in a highly inhomogeneous universe is completely reionized when the number of photons emitted above 1 ryd in one recombination time equals the mean number of hydrogen atoms [39]. At any given epoch there is a critical value for the photon emission rate per unit cosmological comoving volume,

Equation 11a (11)
Equation 11b

independently of the (unknown) previous emission history of the universe: only rates above this value will provide enough UV photons to ionize the IGM by that epoch. One can then compare our estimate of Ndotion to the estimated contribution from QSOs and star-forming galaxies. The uncertainty on this critical rate is difficult to estimate, as it depends on the clumpiness of the IGM (scaled in the expression above to the value inferred at z = 5 from numerical simulations [22]) and the nucleosynthesis constrained baryon density. The evolution of the critical rate as a function of redshift is plotted in Figure 4 (right). While Ndotion is comparable to the quasar contribution at z gtapprox 3, there is some indication of a deficit of Lyman continuum photons at z = 5. For bright, massive galaxies to produce enough UV radiation at z = 5, their space density would have to be comparable to the one observed at z approx 3, with most ionizing photons being able to escape freely from the regions of star formation into the IGM. This scenario may be in conflict with direct observations of local starbursts below the Lyman limit showing that at most a few percent of the stellar ionizing radiation produced by these luminous sources actually escapes into the IGM [34]. (4)

Figure 5

Figure 5. Propagation of an ionization front in a 1283 cosmological density field produced by a mini-quasar with Ndot = 5 x 1053 s-1. The box length is 2.4 comoving Mpc. The quasar is turned on at the densest cell, which is found in a virialized halo of total mass 1.3 x 1011 Msmsun. The solid contours give the position of the I-front at 0.15, 0.25, 0.38, and 0.57 Myr after the quasar has switched on at z = 7. The underlying greyscale image indicates the initial H I density field. (From [1].)

It is interesting to convert the derived value of Ndotion into a ``minimum'' SFR per unit (comoving) volume, rhodot* (hereafter we assume Omegabh2 = 0.02 and C = 30):

Equation 12a
Equation 12b (12)

The star formation density given in the equation above is comparable with the value directly ``observed'' (i.e., uncorrected for dust reddening) at z approx 3 [41]. The conversion factor assumes a Salpeter IMF with solar metallicity, and has been computed using a population synthesis code [6]. It can be understood by noting that, for each 1 Msmsun of stars formed, 8% goes into massive stars with M > 20 Msmsun that dominate the Lyman continuum luminosity of a stellar population. At the end of the C-burning phase, roughly half of the initial mass is converted into helium and carbon, with a mass fraction released as radiation of 0.007. About 25% of the energy radiated away goes into ionizing photons of mean energy 20 eV. For each 1 Msmsun of stars formed every year, we then expect

Equation 13 (13)

4 Note that, at z = 3, Lyman-break galaxies would radiate more ionizing photons than QSOs for fesc gtapprox 30%. Back.

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