How Does the Lyman-Alpha Forest Inform Us About the Universe's Expansion?

[ghetom]

Homework Statement

You are observing a high redshift (z ≥ 5) quasar. Suppose that there are $$n(z) = n_{0}(1+z)^3$$ damped Lyman-alpha (DLA) clouds per Mpc3 at redshift z, each with cross-sectional area a. Explain why we expect to see through n(z)al of them along a length l of the path towards the quasar. What is the variation dl of the path l between z and z+dz? Show that in the case of a flat universe ($$\Omega_{k} = 0 , \Omega_{M} + \Omega_{\Lambda} = 1$$ and at high redshift i.e. when $$(1 + z)^3 >> \Omega_{\Lambda}/\Omega_{m}$$ you can express dN/dz as a function of $$\Omega_{m}$$ only.

Locally we find dN/dz ≈ 0.045. If the cross section a does not change, what is the value we expect at z = 5? We measure dN/dz ≈ 0.4. How does this compare to the expected value and what is the interpretation of this result?

Homework Equations

$$Hd/c = z$$ : Hubbles law

The Attempt at a Solution

I'm stuck on finding dl of the path, l, between z and z+dz; I've tried using hubble's law but I don't think this is correct because the universe is accelerating at high redshift, and it also doesn't give an answer in terms of $$\Omega_{m}$$ only. I tried using the FRW equations, but didnt get anywhere.

Any help is much appreciated.

 

[mark726]

To find the variation dl of the path l between z and z+dz, we can use the distance-redshift relation in a flat universe:

d_l = c/H_0 * ∫(1+z)^-1 * dz

where H_0 is the Hubble constant. Since we are considering a high redshift (z ≥ 5), we can assume that (1+z)^3 >> Ω_Λ/Ω_m. Therefore, we can approximate the distance-redshift relation as:

d_l ≈ c/H_0 * ∫(1+z)^-1 * dz = c/H_0 * ln(1+z)

Now, to find dN/dz as a function of Ω_m only, we need to use the relation between the number density of DLA clouds and the Hubble parameter:

n(z) = n_0 * (1+z)^3 = (H_0/c)^3 * (1+z)^3

Therefore, dN/dz can be written as:

dN/dz = n(z) * a * dl/dz = (H_0/c)^3 * (1+z)^3 * a * c/H_0 * (1+z)^-1 = n_0 * a * (1+z)^2

Since we are given that locally dN/dz ≈ 0.045, we can set this equal to the expression above and solve for n_0:

0.045 = n_0 * a * (1+z)^2

n_0 = 0.045/[a * (1+z)^2]

Now, at z = 5, we can substitute this value of n_0 into the expression for dN/dz to find the expected value:

dN/dz = n_0 * a * (1+z)^2 = (0.045/[a * (1+z)^2]) * a * (1+5)^2 = 0.045 * 36 = 1.62

Therefore, we expect to see 1.62 DLA clouds along a length l of the path towards the quasar. However, we measure dN/dz ≈ 0.4, which is significantly higher than the expected value. This could indicate that the cross-sectional area a is larger than expected, or that there are other factors affecting the number density of DLA clouds at high redshift. Further analysis