Derive Redshift Parameter Expression using Cosmological Principle

[Verdict]

Homework Statement

The cosmological principle can also be used to derive an expression for the redshift of light
(from far away galaxies) in terms of the scale factors at emission and observation. Use the fact
that the wavelength in terms of the co-moving coordinates Δx is independent of time to derive an expression for

in terms of the scale factors a(tobs) and a(tem). The redshift parameter z is the typical parameter used to indicate distance in cosmology.

(I suppose it is obvious, but obs stands for observer and em for emitter)

Homework Equations

The Attempt at a Solution

To be honest, I do not even know where to begin with this. I have looked over http://en.wikipedia.org/wiki/Redshift#Mathematical_derivation but that seems far more complicated than anything we have been doing so far. I suppose I have to utilize one of the Friedmann equations, but which one in what way I don't know. I understand that this forum isn't a solve my homework for me forum, so what I'm asking for is a push in the right direction, so I can start working on this.

 

[mark726]

Hello there,

Thank you for bringing up this interesting topic about the cosmological principle and its application in deriving an expression for the redshift of light. I can understand that this may seem overwhelming at first, but let me provide you with some guidance to help you get started.

Firstly, it is important to understand the cosmological principle and its implications. The cosmological principle states that the universe is homogeneous and isotropic on large scales, meaning that it looks the same in all directions and at all points in space. This principle allows us to make certain assumptions and simplifications in our calculations.

Now, let's take a look at the equation for the redshift parameter z:

z = (λobs - λem)/λem

where λobs is the observed wavelength and λem is the emitted wavelength. Using the fact that the wavelength in terms of co-moving coordinates Δx is independent of time, we can rewrite this equation as:

z = (a(tobs)Δxobs - a(tem)Δxem)/a(tem)Δxem

where a(tobs) and a(tem) are the scale factors at the time of observation and emission, respectively, and Δxobs and Δxem are the co-moving coordinates of the observer and emitter, respectively.

Now, we can use the definition of the scale factor, a(t) = R(t)/R0, where R(t) is the scale of the universe at time t and R0 is the present scale of the universe. This allows us to rewrite the equation as:

z = (R(tobs)/R0Δxobs - R(tem)/R0Δxem)/(R(tem)/R0Δxem)

Simplifying this further, we get:

z = (R(tobs)Δxobs/R0Δxem - R(tem)/R0)/R(tem)/R0

Now, we can use the fact that the wavelength in terms of co-moving coordinates is independent of time to rewrite this equation as:

z = (Δxobs/Δxem - 1)/(a(tem)/a(tobs))

This is the desired expression for the redshift parameter z in terms of the scale factors at emission and observation. I hope this helps to guide you in the right direction. If you have any further questions, please do not hesitate to ask. Good luck with your work