1.3. Gunn-Peterson Effect

Consider radiation emitted at some frequency _e that lies blueward of Ly by a source at redshift z_e, and observed at Earth at frequency _o = _e (1 + z_e)^-1. At a redshift z such that (1 + z) = (1 + z_e) / _e, the emitted photons pass through the local Ly resonance as they propagates towards us through a smoothly distributed sea of neutral hydrogen atoms, and are scattered off the line-of-sight with a cross-section (neglecting stimulated emission) of

(11)

where f = 0.4162 is the upward oscillator strength for the transition, is the line profile function [with normalization () d = 1], c is the speed of light, and e and m_e are the electron charge and mass, respectively. The total optical depth for resonant scattering at the observed frequency is given by the line integral of this cross-section times the neutral hydrogen proper density n_HI(z),

(12)

where dl /dz = c H₀^-1 (1 + z)^-1 [_M (1 + z)³ + _K (1 + z)² + ]^-1/2 is the proper line element in a Friedmann-Robertson-Walker metric, and _M, , and _K = 1 - _M - are the matter, vacuum, and curvature contribution to the present density parameter. As the scattering cross-section is sharply peaked around , we can write

(13)

In an Einstein-de Sitter (_M = 1, = 0) Universe, this becomes

(14)

The same expression for the opacity is also valid in the case of optically thin (to Ly scattering) discrete clouds as long as n_HI is replaced with the average neutral density of individual clouds times their volume filling factor.

In an expanding Universe homogeneously filled with neutral hydrogen, the above equations apply to all parts of the source spectrum to the blue of Ly. An absorption trough should then be detected in the level of the rest-frame UV continuum of the quasar; this is the so-called ``Gunn-Peterson effect''. Between the discrete absorption lines of the Ly forest clouds, quasar spectra do not show a pronounced Gunn-Peterson absorption trough. The current upper limit at z_e 5 is _GP < 0.1 in the region of minimum opacity, implying from equation (14) a neutral fraction of n_HI / _H < 10^-6 h. Even if 99% of all the cosmic baryons fragment at these epochs into structures that can be identified with quasar absorption systems, with only 1% remaining in a smoothly distributed component, the implication is a diffuse IGM which is ionized to better than 1 part in 10⁴.

In modern interpretations of the IGM, it is difficult to use the Gunn-Peterson effect to quantify the amount of ionizing radiation that is necessary to keep the neutral hydrogen absorption below the detection limits. This is because, in hierarchical clustering scenarios for the formation of cosmic structures (the Cold Dark Matter model being the most studied example), the accumulation of matter in overdense regions under the influence of gravity reduces the optical depth for Ly scattering considerably below the average in most of the volume of the Universe, and regions of minimum opacity occur in the most underdense areas (expanding `cosmic minivoids').