The Wouthuysen-Field effect in a clumpy intergalactic medium

We show that the high optical depth of the intergalactic medium to Ly alpha photons before the Epoch of Re-ionization results in a negligible magnitude for the Wouthuysen-Field effect produced by a radiation source on its distant surroundings, unless (i) the scattering medium has sufficient time for the impinging resonance line photons to establish a steady-state frequency distribution or (ii) the scattering gas is undergoing internal expansion or has a peculiar motion of tens to hundreds of km s(-1) away from the source. Because of the intergalactic attenuation, discrete structures will receive only radiation from a source displaced from the resonance line frequency by typically hundreds to thousands of Doppler widths. The incident radiation must diffuse across the resonance line to produce a substantial scattering rate. We present steady-state solutions in the radiative diffusion approximation for the radiation field trapped in a clump of gas and show that this may result in an enhancement of the strength of the Wouthuysen-Field effect by as much as a factor of 10(6) over the free-streaming (single-scattering) limit. Solutions to the time-dependent diffusion equation, however, show that the time-scales required to establish a steady state will generally exceed the lifetime of the sources, resulting in a substantially reduced scattering rate. In the presence of internal expansion, a steady state may be established as photons are redshifted across the resonance line and into the red wing, and significant enhancement in the scattering rate over the free-streaming limit may again be produced. Alternatively, a substantial scattering rate may arise in systems with a peculiar motion away from the source that redshifts the received radiation into the resonance line centre. As a consequence, at epochs z less than or similar to 30, when collisional decoupling of the hyperfine structure of hydrogen from the cosmic microwave background is small except in dense regions, and prior to the establishment of any large-scale diffuse radiation field of resonance line photons, the 21-cm signature from the intergalactic medium produced by the Wouthuysen-Field effect will, in general, trace the peculiar velocity field of the gas in addition to its density structure.