If a CMB photon traveled for 13.7 billion years to reach me, how far away was the source of that CMB photon when it first emitted it?

Question

If a CMB photon traveled for 13.7 billion years (- 374,000 years) to reach me. How far away was the source of that CMB photon when it first emitted it?

My attempt to solve this question was to use the following assumptions:

1. Temperature of CMB photon today is 2.725 K (will use value of 3 K here)

2. Temperature of CMB photon when it was first emitted is 3000 K

3. A factor of x1000 in temperature decrease results in a factor of x1000 in wavelength increase. (According to Wien's displacement law)

Does this mean that the source of the CMB photon that just reached me today, was actually 13.7 billion light years / 1000 = 13.7 million light years away from me when it first emitted the photon?

Answer 1

The comoving distance traveled by light in vacuum between cosmological times $t_i$ and $t_f$ is $\displaystyle \int_{t_i}^{t_f} \frac{c\,dt}{a(t)}$. The metric distance at cosmological time $t$ is $a(t)$ times the comoving distance. The distance you're looking for is therefore $\displaystyle \int_{t_i}^{t_f} \frac{a(t_i)}{a(t)} c\,dt$.

This value depends on $a(t)$ over the whole time interval from $t_i$ to $t_f$, not just at the endpoints. The redshift factor is equal to $\displaystyle\frac{a(t_f)}{a(t_i)}$, but you can't just divide $cΔt$ by that factor to get the distance.

You can, however, divide the usually quoted comoving distance to the CMBR (around $46\text{ Gly}$ or $14\text{ Gpc}$) by the CMBR redshift (around $1100$) to get the correct distance (around $42\text{ Mly}$ or $13\text{ Mpc}$). This is because the $46\text{ Gly}$ distance is calculated using that first integral, without multiplying by $a(t_i)$, and $a(t_f)=a(t_0)=1$ by convention.

Answer 2

"13.7 million light years away from me when it first emitted the photon?" Based on your assumptions, this is sort-of correct. There are two not-quite-correct details. (1) The temperature you assune for the production of the CMBR is corrected in the comment by benrg. (2) The "me" was not present in the univese when the observed CMBR was emitted. A better dscription is the location of the comoving point where you are now at the location it was when the CMRB was emitted.

Answer 3

The light was emitted 367.000 years after the big bang from the edge of the universe. So The distance to where you now are at that epoch was 367.000 lightyears.