# Solving Olbers' paradox with the expanding universe

I'm trying to explain the Olbers' paradox using the fact that universe is expanding. First of all if we take the suppositions of the original Olbers' 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 Olbers' 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?

• Lack of capitalization of the word "universe", in English, implies that your question's based on the assumption that reality isn't divided into the causally-separated "local universes" of an "inflationary" (expanding at a quasi-exponential rate) "multiverse". There's nothing known to be wrong with that assumption, but, for a contrary view, look at such "Black Hole Genesis" models as Nikodem Poplawski's, which is torsion-based, past- & future-eternal, and described in 2010-2019 papers whose preprints can be found by his name on Cornell University's << Arxiv >> site. Commented Oct 28, 2022 at 15:19
• Note that the 1823 formulation of Olbers's Paradox presupposes the nonexistence of non-luminous matter between ourselves and distant stars. Since 1930 we have known this formulation to be invalid owing to the discovery of interstellar extinction, which happens to follow an exponential decay law, making the flux bounded from all directions, and the contribution of light from increasingly distant stars converge towards zero. en.wikipedia.org/wiki/Extinction_(astronomy) Commented Mar 5 at 22:39
• And we've known about radiative transfer for even longer. Commented May 1 at 20:24

## 2 Answers

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.

• Okay, I understand, but in this case, the integral will be reduced to, $\int r dr = \int r_c^2 \dot{a} a dt$, so in this case the comoving distance $r_c$ is a constant? And how can i integrate $a \dot{a} dt$, exist some way to perform this integral? Commented Oct 29, 2022 at 16:46
• You don't /need/ an expanding universe and comoving coordinates at all, that's just one way out of the four out of Olbers paradox. Once you put a(t) in the integral you can't solve it anymore in general but need to plug in an explicit a(t)
– rfl
Commented Oct 29, 2022 at 22:01
• "the universe is finite in size", ya sure? Commented Jul 31, 2023 at 15:58
• @MadMax A finite universe is one way out of Olber's paradox. That's not to say that it is. From all we know, it isn't, so that's the wrong way out of the paradox. But one way out it is, nonetheless.
– rfl
Commented Aug 3, 2023 at 8:21

As it seems, the problem is in geometric progression. We apply the sum of geometric series formula.

With $$\omega_0$$ being initial parameter and r being blocking factor which already seems to be less than 1 tends towards 1 as question mentions; due to blocking $$\omega$$ goes on decreasing... $${\lim_{n\to\infty}}S=\frac{\omega_0}{1−r}$$ I tried to find the exact factor but it seems very difficult visually to solve the equation, so I directly assumed $${\lim_{n\to\infty}}S_{n}=4\pi$$ Thus if we now solve for $$r$$, $$4\pi =\frac{\omega_0}{1−r} \\ r=1−\frac{\omega_0}{4\pi}.$$