Relation between comoving distance and conformal time?

Question

In cosmology, we have two quantities and I want to understand the physical relation between these two :

$\chi = \int_{t_e}^{t_0}c\frac{dt'}{a(t')}$ : the comoving distance with $t_e$ the time at emission and $t_0$ the current cosmic time

$\eta_0 = \int_{0}^{t_0}\frac{dt'}{a(t')}$ : the conformal time with $0$ the time at Big-Bang and $t_0$ the current cosmic time

My question are :

1) Is it possible to know the conformal time of a galaxy only knowing its comoving distance from us

2) What are the physical meaning difference between $\eta$ and $\frac{\chi}{c}$ ?

3) What are the physical meaning difference between $\chi$ and $c \eta$ ?

Than you very much !

Answer 1

These are essentially the same, as indicated by the formulas, with one having units of length and the other of time. So, (1) yes, the conformal time of emission of the light we see is simply today's conformal time minus ($1/c$ times) the comoving distance to the galaxy (though this is not directly measurable but rather must be found with e.g. the luminosity distance or the angular diameter distance).

(2/3) $\eta$ is a rescaling of "normal" time $t$ (proper time for an observer moving with the Hubble flow in an FRW universe), and so it still measures a "time." If you foliate spacetime with spacelike slices of constant time, then $\eta$ can measure the distance between the slices containing two different events. $\chi$ you can take to be the proper distance between two events, if both events are projected onto our current time slice by following the Hubble flow.

Note that the purpose of conformal time and comoving distance is to make the Robertson-Walker metric conformally equivalent to Minkowski (or something very similar in the non-flat case): $ds^2 \propto -c^2d\eta^2 + d\chi^2 + S_k(\chi)^2 d\Omega^2$. Thus photons ($ds^2 = 0$) traveling in the radial ($d\Omega = 0$) direction simply move a comoving distance $\Delta\chi = c \Delta\eta$ in a conformal time $\Delta\eta$, giving the simple relation between comoving distance and conformal lookback time applicable to (1).