- #1

wabbit

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## Homework Statement

In a flat FRW universe, for a source moving radially at velocity v (at emission time) relative to the local comoving frame, what is the redshift observed by a comoving observer?

## Homework Equations

##c=1##

Proper time to cosmological time ratio ##\frac{d\tau}{dt}=\sqrt{1-v^2}##

Redshift of comoving source ##\frac{\lambda_{obs}}{\lambda_{em}}=\frac{a(t_{obs})}{a(t_{em})}##

Note : Abusing terminology here, the "redshifts" here are quoted as ##S=z+1## intead of ##z##.

## The Attempt at a Solution

Based of the infinitesimal motion of the source and on light paths, I get $$\frac{\lambda_{obs}}{\lambda_{em}^{com}}=(1+v)\frac{a(t_{obs})}{a(t_{em})}$$

This is the redshift between ##\lambda_{em}^{com}## measured at emission time in a comoving frame, and ##\lambda_{obs}## measured by the comoving receiver.

And combining this with ##\frac{\lambda_{em}^{com}}{\lambda_{em}}=\frac{1}{\sqrt{1-v^2}}##, the total redshift is

$$\frac{\lambda_{obs}}{\lambda_{em}}=\sqrt{\frac{1+v}{1-v}}\frac{a(t_{obs})}{a(t_{em})}$$

Can someone confirm if this correct, and if not point to the error? This is not exactly homework, rather a calculation I did since I wondered about that case, and I didn't find the relevant formula online to check against.

Thanks