Derive the Gunn-Peterson optical depth expression...

1. May 18, 2015

dirkdiiggler

1. The problem statement, all variables and given/known data

Derive an expression for the Gunn-Peterson optical depth Tau(λ) for Lyman limit absorption from smoothly distributed intergalactic neutral Hydrogen of comoving density n0 toward a QSO of emission redshift zem. Take the absorption cross section for Lyman-limit absorption to be:

σ=5x10-18(λ/λLL)3 cm2

at wavelengths λ < λLL (and zero otherwise), where, λLL = 912 Angstroms, is the wavelength of the Lyman limit. Assume that the opitcal depth satisfies Tau<<1 so that e-Tau can be approximated as e-Tau=1-Tau, and assume an Einstein-de Sitter cosmological model.

What limit on n0 could be set from the observation Tau<0.1 at redshift z = 3?

2. Relevant equations

(1). Tau=nσ

3. The attempt at a solution

What I am trying to do is to apply equation (1) and modifying the equation given in the problem to account for redshift. That is, σ=5x10-18(λ(z+1)/λLL)3 cm2.

To end here seems a little too simple: Tau = n0 * 5x10-18(λ(z+1)/λLL)3 cm2

What else should I be looking for?