In the previous calculations, (13) and (21), where we described the source terms for the radiation field within the IGM we introduced opacity factors to model the absorption of the radiation emitted by quasars and stars by the IGM and Lyman- clouds. We shall deal with these terms in this section.
where K is the recombination rate (within the ionized bubbles), the mean number density of hydrogen obtained from (26), the clumping factor from (27), the emission rate of ionizing photons from the source and t its age.
We have so we neglected here the influence of the expansion of the universe and the time-dependence of the source luminosity over its age. We take for galaxies and for quasars. Although this procedure is consistent with our prescription for the quasar luminosity function (we assumed quasars to shine at the Eddington luminosity on the time-scale and then to fade) we somewhat overestimate the radiative output of galaxies since the galaxy luminosity function decreases with z over the time-scale . However, the relation (33) should still provide a correct estimate of the magnitude of this effect. Note that is smaller than the usual "Stromgren" radius which corresponds to the limit in (33). Indeed, the exponential term in (33) can also be written as which shows that at redshifts where the recombination time is larger than the age of the source (which is smaller than ) the ionization front is smaller than the Stromgren radius. This effect was also described by Shapiro & Giroux (1987). In addition, these authors took into account the expansion of the universe but assumed a fixed number of sources. Here since we consider sources with a life-time we neglect the influence of the expansion of the universe but we take into account the increasing number of galaxies and quasars.
Next, we obtain the volume fraction (i.e. the filling factor) occupied by such ionized bubbles around galaxies or quasars as:
where is the recombination time within the ionized bubbles. This differs from the quasar mass function (18) through the term because a region remains ionized over a time-scale which may be longer than the quasar life-time . Note that our general procedure only provides an upper bound to the actual efficiency of radiative processes since we did not include absorption within the host galactic halo itself. We do not integrate over time since this is already done in (33) and the sharp rise with time of the luminosity functions (before reionization) ensures that the radiative output is dominated by recent epochs. Moreover, since we have , as shown below in Fig. 8 (curve ), ionized bubbles do not survive more than a Hubble time unless new sources (galaxies or quasars) appear. The filling factor within the IGM is written as the sum of the contributions from galaxies and quasars:
Of course the previous considerations only apply to high redshifts prior to reionization when the universe is almost completely neutral. At reionization these bubbles overlap and the background UV flux gets suddenly very large as absorption drops. At later times the whole universe is ionized so there are no more discrete bubbles (and formally ). We also define in a similar fashion the filling factors and which describe regions around quasars where helium is singly or doubly ionized. Since the stellar radiation shows an exponential decrease at high frequencies quasars are the only relevant source for this process. The filling factor obtained above will be used to obtain the IGM opacity.
However, as we explained previously we also consider the universe to contain numerous clouds which contribute to the opacity seen by the radiation field. We shall first consider that these clouds are ionized (or more exactly that their number density drops significantly) within the radius defined in (33) from the quasar. In other words, most of the opacity comes from clouds located deeply within the IGM where the background radiation is very small (before reionization) since close to quasars (within ) the local radiation suddenly gets much higher. Note that the distribution of Lyman- clouds we calculate is indeed obtained from the IGM background radiation, which calls for the cutoff . However, at low z after reionization the "sphere of influence" of a quasar is no longer given by (since the whole medium is ionized). Instead, we define the radius by:
Thus, the "sphere of influence" of a quasar is defined by the region of space around the source where the radiation emitted by this quasar is significantly larger than the background radiation (at high z when the medium is neutral this corresponds to , while at low z when the IGM is ionized this is given by ). Thus, in practice we shall simply use to obtain the volume fraction where the number density of Lyman- clouds is significantly lower than within the IGM:
As we shall see from the numerical results, at low redshifts when the universe is reionized the opacity is very low (the UV flux is large) so that absorption plays no role for the evolution of . Thus, in practice the radius is irrelevant. It only provides some information on the properties of the universe but it does not influence the behaviour of the latter. Thus, while increases with time until it reaches unity as the universe gets reionized, will first grow before reionization as the volume occupied by the ionized bubbles increases and then decrease at because the quasar luminosity function drops at low z. Since a fraction of volume Q translates into the same fraction Q along a random line of sight (neglecting correlations in the distributions of sources) we write the opacity seen from a point in the IGM to a source located at the distance r as:
where is the neutral hydrogen filling factor () and corresponds to discrete clouds while describes the IGM contribution. The typical distance between galactic sources characterized by their parameter x, density contrast and radius R, is given by their number density, see (2),
as in (35). Here is the recombination time within the ionized bubbles, see Fig. 8. Since at low redshift we usually have . Next, we define an effective opacity over the region of size l and volume V by:
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999