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## 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

*n*toward a QSO of emission redshift

_{0}*z*. Take the absorption cross section for Lyman-limit absorption to be:

_{em}σ=5x10

^{-18}(λ/λ

_{LL})

^{3}cm

^{2}

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

*n*could be set from the observation Tau<0.1 at redshift

_{0}*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}cm^{2}To end here seems a little too simple:

**Tau =***

*n*_{0}**5x10**

^{-18}(λ(z+1)/λ_{LL})^{3}cm^{2}What else should I be looking for?

Thanks in advance!