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2.2. Intergalactic Continuum Opacity

Even if the bulk of the baryons in the Universe are fairly well ionized at all redshifts z ltapprox 5, the residual neutral hydrogen still present in the Lyalpha forest clouds and Lyman-limit systems significantly attenuates the ionizing flux from cosmological distant sources. To quantify the degree of attenuation we have to introduce the concept of an effective continuum optical depth taueff along the line-of-sight to redshift z,

Equation 21 (21)

where the average is taken over all lines-of-sight. Negleting absorption due to helium, if we characterize the Lyalpha forest clouds and LLS as a random distribution of absorbers in column density and redshift space, then the effective continuum optical depth of a clumpy IGM at the observed frequency nuo for an observer at redshift zo is

Equation 22 (22)

where tau = NHI sigmaH(nu) is the hydrogen Lyman continuum optical depth through an individual cloud at frequency nu = nuo (1 + z) / (1 + zo). This formula can be easily understood if we consider a situation in which all absorbers have the same optical depth tau0 independent of redshift, and the mean number of systems along the path is DeltaN = integ dzdN / dz. In this case the Poissonian probability of encountering a total optical depth ktau0 along the line-of-sight (with k integer) is p(ktau0) = eDeltaN DeltaNk / (tau0 k!), and < e-tau > = e-ktau0 p(ktau0) = exp[-DeltaN(1 - e-tau0)].

If we extrapolate the NHI-1.5 power-law in equation (16) to very small and large columns, the effective optical depth becomes an analytical function of redshift and wavelength,

Equation 23 (23)

Due to the rapid increase with lookback time of the number of absorbers, the mean free path of photons at 912 Å becomes so small beyond a redshift of 2 that the radiation field is largely `local'. Expanding equation (23) around z, one gets taueff (nuL) approx 0.36 (1 + z)2 Deltaz. This means that at z = 3, for example, the mean free path for a photon near threshold is only Deltaz = 0.18, and sources of ionizing radiation at higher redshifts are severely attenuated.

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