Recent cosmological observations indicate that the reionized universe may have started at around z = 6, where a significant suppression around Lyα has been observed from the neutral intergalactic medium. The associated neutral hydrogen column density is expected to exceed 10 21 cm −2 , where it is very important to use the accurate scattering cross section known as the Kramers-Heisenberg formula that is obtained from the fully quantum mechanical time-dependent second order perturbation theory.

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... e present the Kramers-Heisenberg formula and compare it with the formula introduced in a heuristic way by Peebles (1993) considering the hydrogen atom as a two-level atom, from which we find a deviation by a factor of two in the red wing region far from the line center. Adopting a representative set of cosmological parameters, we compute the Gunn-Peterson optical depths and absorption profiles. Our results are quantitatively compared with previous work by Madau & Rees (2000) , who adopted the Peebles approximation in their radiative transfer problems. We find deviations up to 5 per cent in the Gunn-Peterson transmission coefficient for an accelerated expanding universe in the red off-resonance wing part with the rest wavelength ∆λ ∼ 10Å.