2.2. Intergalactic Continuum Opacity

Even if the bulk of the baryons in the Universe are fairly well ionized at all redshifts z 5, the residual neutral hydrogen still present in the Ly 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 _eff along the line-of-sight to redshift z,

(21)

where the average is taken over all lines-of-sight. Negleting absorption due to helium, if we characterize the Ly 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 _o for an observer at redshift z_o is

(22)

where = N_{HI H}() is the hydrogen Lyman continuum optical depth through an individual cloud at frequency = _o (1 + z) / (1 + z_o). This formula can be easily understood if we consider a situation in which all absorbers have the same optical depth ₀ independent of redshift, and the mean number of systems along the path is N = dzdN / dz. In this case the Poissonian probability of encountering a total optical depth k₀ along the line-of-sight (with k integer) is p(k₀) = e^N N^k / (₀ k!), and < e^- > = e^-k₀ p(k₀) = exp[-N(1 - e^-₀)].

If we extrapolate the N_HI^-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,

(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 _eff (_L) 0.36 (1 + z)² z. This means that at z = 3, for example, the mean free path for a photon near threshold is only z = 0.18, and sources of ionizing radiation at higher redshifts are severely attenuated.