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

• dirkdiiggler
In summary: This is related to the distance d by the equation:d = (1+z) * D(z)where D(z) is the comoving distance at redshift z. Substituting this into our expression for the optical depth, we get:Tau = n0 * 5x10^-18 * (λ0/λLL)^3 * cm^2 * (1+z) * D(z)Finally, we can consider the fact that the optical depth satisfies Tau << 1, so we can approximate e^-Tau as 1-Tau. This allows us to simplify our expression for the optical depth to:Tau = n0 * 5x10^-18 * (λ0/λLL)^3
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 zem. Take the absorption cross section for Lyman-limit absorption to be:σ=5x10-18(λ/λLL)3 cm2at 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?

(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)/λLL)3 cm2.To end here seems a little too simple: Tau = n0 * 5x10-18(λ(z+1)/λLL)3 cm2What else should I be looking for?Thanks in advance!

To derive an expression for the Gunn-Peterson optical depth, we need to consider the following factors:

1. The number density of intergalactic neutral hydrogen, n0, which is the total number of neutral hydrogen atoms per unit volume.
2. The absorption cross section, σ, which is the probability that a photon will be absorbed by a neutral hydrogen atom.
3. The distance between the QSO and the observer, which is related to the redshift, z.
4. The wavelength of the photon, λ, which is also related to the redshift, z.
5. The Lyman limit, λLL, which is the wavelength at which the absorption cross section becomes significant.

To start, let's rewrite equation (1) as:

Tau = n0 * σ * d

where d is the distance between the QSO and the observer. We can rewrite the absorption cross section as:

σ = 5x10^-18 * (λ/λLL)^3 * cm^2

Since we are dealing with a cosmological model, we need to take into account the expansion of the universe, which is described by the scale factor, a. The scale factor is related to the redshift by the equation:

a = 1/(1+z)

Therefore, we can rewrite the absorption cross section as:

σ = 5x10^-18 * (λ/λLL)^3 * a^3 * cm^2

Now, we need to consider the fact that the photon's wavelength, λ, is also affected by the expansion of the universe. This is described by the equation:

λ = λ0 * a

where λ0 is the rest wavelength of the photon. Substituting this into our expression for the absorption cross section, we get:

σ = 5x10^-18 * (λ0/a * a/λLL)^3 * a^3 * cm^2

Simplifying, we get:

σ = 5x10^-18 * (λ0/λLL)^3 * cm^2

Now, we can substitute this expression for σ into our original equation for the optical depth:

Tau = n0 * 5x10^-18 * (λ0/λLL)^3 * cm^2 * d

Since we are dealing with a cosmological model, we need to take into account the comoving distance, which is the distance between the QSO and the observer at the time the

## 1. What is the Gunn-Peterson optical depth expression?

The Gunn-Peterson optical depth expression is a mathematical equation that describes the absorption of light by neutral hydrogen gas in the early universe. It relates the optical depth, or the likelihood of absorption, to the density of neutral hydrogen gas along the line of sight.

## 2. Why is the Gunn-Peterson optical depth expression important?

The Gunn-Peterson optical depth expression is important because it allows scientists to study the early universe and the formation of galaxies. The absorption of light by neutral hydrogen gas can tell us about the density and distribution of matter in the early universe.

## 3. How is the Gunn-Peterson optical depth expression derived?

The Gunn-Peterson optical depth expression is derived from a combination of physical principles and mathematical equations, including the equations of radiative transfer and the equations of cosmological structure formation. It also takes into account the properties of neutral hydrogen gas and the expansion of the universe.

## 4. What are the assumptions made in deriving the Gunn-Peterson optical depth expression?

Some of the assumptions made in deriving the Gunn-Peterson optical depth expression include a uniform distribution of neutral hydrogen gas, a homogeneous and isotropic universe, and a constant rate of expansion. It also assumes that the gas is optically thin, meaning that light can easily pass through it without being significantly absorbed.

## 5. How is the Gunn-Peterson optical depth expression used in research?

The Gunn-Peterson optical depth expression is used in research to study the early universe and the formation of galaxies. It is also used to measure the density of neutral hydrogen gas in the intergalactic medium, which can help us understand the evolution of cosmic structures. Additionally, it is used to test and refine our understanding of cosmological models.

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