Solving Olber's paradox with the expanding universe I'm trying to explain the Olber's paradox using the fact that universe is expanding. First of all if we take the suppositions of the original Olber's paradox:

*

*Infinite universe ($r=0\rightarrow r=\infty$)

*Static in time

*Constant density and luminosity of the stars

We can calculate the flux that arrive to us from all these stars, that from the flux definition, $f=2\pi L r^2$, and so $df=4 \pi L r dr$, so integrating with some star density that is constant $\rho_s$ we have,
$$f=\int_0^{\infty} \rho_s 4 \pi L rdr= 4 \pi \rho_s L \int _0^{\infty} r dr \rightarrow \infty$$
So this is the Olber's paradox, that tells the flux i.e. the light received should be infinite. In order to solve this we can introduce cosmology concepts, as the space expansion, and that the universe and stars has born at some point. So I want to perform the before integral but now adding this restrictions, but  I don't know how to implement it because,
$$r=a(t)r_{c}$$
With $r_{c}$ the comoving distance, and so,
$$dr=\dot{a} r_c dt \quad ; \quad dr=adr_c$$
And as I have understood I will need to integrate over space ($r_c$) and over time, from the born of the universe to now. How can I convert the flux integral in a integal that depends on time and space?
 A: You've done the substitution already, so now your integral is over time, not over space anymore. You simply need to fix the limits of integration too and you're done. There's thus a number of ways out of Olber's paradox:

*

*the universe is finite in age, which just means that you integrate out to just that time instead of out to infinity

*the universe is finite in size, so the integral doesn't go out to infinity... of course for that you don't need to substitute anything and can keep your original integral

*the universe isn't static, in which case you only integrate to that time where star formation sets in

*the universe expands, which is just to say that you find a $a(t)$ such that your integral is finite despite integrating out to infinite time, e.g. some exponential.

I think the issue you expressed in your question (if I understood it right) is a particular nice example where one may be misled by one's intuition in cosmology. Strictly focusing on the math can help in those cases - you have formulated your problem as an integral, and the substitution to go from radius to time is just math. Spacetime magic doesn't enter that picture.
