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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.