# Cosmological Redshift

• Joella Kait
In summary: Since part (d) is asking for a specific situation in the same universe described in part (b), we can still use the same equations and assumptions.
Joella Kait

## Homework Statement

Consider a point in the intergalactic medium at some cosmic time ## t_{obs}##, the time of arrival of a photon of wavelength ##λ_{obs}## as seen by a hydrogen atom at that location. The source of this photon a comoving distance ##r## away emitted it at wavelength ##λ_{em}## at time ##t_{em}##. Assume the scale factor at present ##a_0= 1##.

(a) Express ##r## as a function of ##t_{obs}##, ##t_{em}## and the scale factor ##a(t)##.

(b) Solve for the dependence of ##a(t)## on ##t## for a universe in which the Hubble constant varies with time according to
$$\ H(t)=\frac{2}{3t}.$$

(c) What is the ratio of ##λ_{em}## to ##λ_{obs}## in terms of ##t_{em}## and ##t_{obs}##, in a universe described in (b)?

(d) The same photon will later reach a telescope on Earth today at ##λ_0=3645 Angstroms##. Suppose ##λ_{obs}=1215 Angstroms##, the H atom Lyman-alpha line transition. What is ##λ_{em}## if the source is located a comoving distance ##r=500 Mpc## (in present-day units) away from the H atom? Assume ##H_0=70 km s^{-1} Mpc^{-1}##.

I mainly just want help with part (d).

## Homework Equations

[/B]
## ds^2=-c^2dt^2+a(t)^2[dr^2+s_k(r)^2d\Omega^2## but ##ds=0## and ##d\Omega=0##
##H(t)=\frac{1}{a}*\frac{da}{dt}##
##\frac{\lambda_{em}}{a(t_{em})}=\frac{\lambda_{obs}}{a(t_{obs})}##

## The Attempt at a Solution

[/B]
For part (a) I intergrated and got ## r=c\int\limits_{t_{em}}^{t_{obs}} \frac{1}{a(t)} \ dt ##.
For part (b) I used the second equation to get ##a(t)=t^{2/3}##
For part (c) I used the third equation to get ##\frac{\lambda_{em}}{\lambda_{obs}}=\frac{t_{em}^{2/3}}{t_{obs}^{2/3}}##
I'm really lost on part (d) though. I was told that I'm supposed to integrate my answer from (a), but I am not quite sure how to go about that.
##r=c\int\limits_{t_{em}}^{t_{obs}} \frac{1}{t^{2/3}} \ dt = 3c(t_{obs}^{1/3}-t_{em}^{1/3})##
I'm not sure how to get ##\lambda's## from this, and I'm also not sure how to include ##t_0/\lambda_0##

Last edited:
##\lambda_{obs}## is not the wavelength observed on Earth at the present time. It is the wavelength observed by a comoving hydrogen atom at some intermediate time ##t_{obs}##.

Joella Kait
Orodruin said:
##\lambda_{obs}## is not the wavelength observed on Earth at the present time. It is the wavelength observed by a comoving hydrogen atom at some intermediate time ##t_{obs}##.
Thanks! I somehow over looked that it said hydrogen atom. For part (d) do you think it's still assuming the same universe as in part (b)?

Joella Kait said:
Thanks! I somehow over looked that it said hydrogen atom. For part (d) do you think it's still assuming the same universe as in part (b)?
Yes.

## 1. What is cosmological redshift?

Cosmological redshift is a phenomenon in which light from distant galaxies appears to be shifted towards the red end of the electromagnetic spectrum. This is due to the expansion of the universe, which causes the wavelengths of light to stretch as they travel through space.

## 2. How does cosmological redshift relate to the expansion of the universe?

As the universe expands, the space between objects also expands. This means that the wavelengths of light traveling through this expanding space will also stretch, resulting in a redshift. The amount of redshift is directly proportional to the distance the light has traveled through the expanding universe.

## 3. What causes cosmological redshift?

Cosmological redshift is caused by the Doppler effect, which is the change in wavelength of light due to the relative motion between the source of light and the observer. In the case of cosmological redshift, the source of light (distant galaxies) is moving away from the observer due to the expansion of the universe.

## 4. How is cosmological redshift measured?

Cosmological redshift is measured using a unit called redshift (z), which is a measure of the amount by which the wavelength of light has been stretched. This can be calculated by comparing the observed wavelength of light to the expected wavelength of that light based on its known properties.

## 5. What implications does cosmological redshift have for our understanding of the universe?

The observation of cosmological redshift has led to the discovery of the expanding universe and the theory of the Big Bang. It also provides evidence for the existence of dark energy, a mysterious force that is thought to be responsible for the acceleration of the expansion of the universe. Cosmological redshift also allows us to measure the distances to faraway galaxies and study the history of the universe.

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