# Redshift of non-comoving galaxy

1. May 15, 2015

### wabbit

1. The problem statement, all variables and given/known data
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?

2. Relevant 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 $z$.

3. 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

2. May 15, 2015

### Staff: Mentor

I think you can split the problem in two parts. Find the frequency or wavelength observed by a comoving observer at (nearly) the same place, then apply the usual scale factor for distant observers. And the result is exactly what you got if the object is moving away.

3. May 15, 2015

### wabbit

Thanks - Indeed, I didn't see it this way, the result is just a Doppler shift times an expansion shift, that's a much better way to get it than what I did !