Derive the Gunn-Peterson optical depth expression....

[dirkdiiggler]

Homework Statement

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 z_em. Take the absorption cross section for Lyman-limit absorption to be:


σ=5x10^-18(λ/λ_LL)³ cm²


at wavelengths λ < λ_LL (and zero otherwise), where, λ_LL = 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?

Homework Equations

(1). Tau=nσ

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)/λ_LL)³ cm².


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


What else should I be looking for?


Thanks in advance!