Co-moving distance and universal time

I am reading about Cosmology, in particular, the concept of co-moving distance. I am trying to derive the equation for calculating the co-moving distance with the following argument.

If the source and the observer were in a static universe then the distance traveled by a photon between the source and observer would just have been, $$d\chi = c dt$$ Here, $$d\chi$$ is the co-moving distance, $$dt$$ is the time interval and c is the speed of light. However, in a dynamic universe the scale factor changes with time hence, the distance actually traveled by the photon would be, $$a(t)d\chi=cdt$$ which would give you the standard equation for calculating the co-moving distance, $$\chi=\int_{t_e} ^{t} c\frac{dt'}{a(t')}$$ Here, is my question: Although it gives the right equation (I am not sure if I have reasoned it right though) I am concerned if I have, mistakenly, treated time as measured by a universal clock. I mean, computationally, how do I know in the observer frame when the photon was emitted from the source? Have I assumed that a universal clock is giving the time stamps for the source as well as the observer?

• Wikipedia says The comoving time coordinate is the elapsed time since the Big Bang according to a clock of a comoving observer – PM 2Ring Jul 3 at 8:46
• I think that is called the conformal time. Of course, both are the same when the photon in question is coming from the Big Bang. But the comoving time can be just the time taken by a photon to travel from a source to the observer in an expanding universe. – Shaz Jul 3 at 8:53
• Just as you have implicitly chosen the that the comoving coordinate of the observer is 0 in your derivation, we can choose the comoving time coordinate to be that of the observer in this formulation. There is no need for a universal clock. – alex1stef2 Jul 3 at 9:11
• Yes but, computationally, how do I know when the photon was emitted in the observer frame? All I can do is take the observer time for photons arriving at two different times. How do I calculate the co-moving distance between the source and the observer? – Shaz Jul 3 at 9:21
• The conformal time is to the proper distance as the comoving time is to the comoving distance. So the conformal time is given by $\eta=\int_0^{t}\frac{dt'}{a(t')}$ – PM 2Ring Jul 3 at 9:55