Cosmology: Hubble's law and Redshift

[AHSAN MUJTABA]

Homework Statement

For small z (z<<<1), the expansion of the universe can be thought of as a set of galaxies
moving apart. Therefore, the redshift (which is due to the expansion of spacetime itself)
can be interpreted as a Doppler shift arising from relative motion. Starting from the
definition of z, show that in this limit,
z=c

Relevant Equations

The equations that I intend to use are:
z=$(\lambda_o - \lambda_e)/\lambda_e$

I tried to attempt this by taking z=dv/c but I saw that relation Liddle's book on Cosmology and I really don't know where this came from.
Secondly, I will need some series ( maybe Bionomial) in terms of z from which I can show small z.
I think it's simple but here it's not clicking to me.

 

[jbsteele]

Can someone help me?A:This is the Doppler effect. The idea is that as an object moves away from you, the wavelength of light you observe is stretched compared to the wavelength of light emitted by the object. This is why galaxies that are moving away from us appear redder than they would if they were at rest with respect to us. If we know the velocity of the galaxy relative to us, we can calculate the fractional change in wavelength, $\frac{\Delta \lambda}{\lambda_0}$, where $\lambda_0$ is the rest wavelength that we would observe from the galaxy if it were at rest with respect to us.Using special relativity and assuming the velocity is small compared to the speed of light, we can show that$$ \frac{\Delta \lambda}{\lambda_0} = \frac{v}{c} $$where $v$ is the velocity of the galaxy relative to us and $c$ is the speed of light. This expression is valid for both recession (moving away from us) and blueshift (moving towards us).Z is just a convenient way of expressing this fractional change. It is defined as$$ z \equiv \frac{\Delta \lambda}{\lambda_0} - 1 $$so that$$ z = \frac{v}{c} - 1 $$