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Suppose we have an FLRW Universe with metric

$$ds^2 = -dt^2 + a(t)^2\left(\frac{dr^2}{1-Kr^2}+r^2d\Omega^2\right) = -dt^2 + a(t)^2\left(d\chi^2 +S_{K}^2(\chi)d\Omega^2\right) $$ that is filled with point sources of light, each of luminosity $L$, and that these sources are dispersed evenly with number density $n(t)$. If these sources are emitting in the time interval $[t-dt, t]$, then how many of them will be observable at present time $t_0$?

My attempt at a solution: Suppose light is emitted for a conformal interval $[\eta-d\eta,\eta]$. Let $d$ be the fixed comoving distance between the observer and one of these sources. Then the conformal time it takes light to first reach the observer (taking $c = 1$) is given by $\eta + d - d\eta$. Similarly, the last light from the source reaches the observer at a conformal time $\eta + d$. Let $a(t_0) = 1$ be the scale factor governing the expansion of the Universe at the present time, $t_0$. Then the light is detected for a proper duration of $d\eta$. We see that $d$ satisfies $$\frac{t_0 - t}{a(t_0)} \leq d \leq \frac{t_0 - t}{a(t_0)} + d\eta$$ Let $r(t) = S_K(\frac{t_0 - t}{a(t_0)}) = S_K(t_0-t)$. Therefore, the number of observable sources occupies a volume sandwiched between spheres of surface area $4\pi a^2(t)r^2(t)$ and $4\pi a^2(t) r^2(t-d\eta)$. Using the volume element implied by the second expression for our metric, this is $N = 4\pi a(t)^3 n(t)\int_{\frac{t_0 - t}{a(t_0)}}^{\frac{t_0 - t}{a(t_0)} + d\eta} S_{K}^2(\chi) \ d\chi.$ But since $d\eta$ is small we have $$N = 4\pi a(t)^3 n(t)r^2(t)d\eta = 4\pi a^2(t)n(t)r^2(t)dt.$$

Is this result correct, and, more importantly (since I am new to cosmology), is my method correct? If anyone can suggest a more mathematically rigorous approach, then I would very much appreciate it.

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