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So, I came recently to watch this video, which explains Olber's paradox and how it was solved with the big bang theory, doppler shift etc.

My question can be split into a more physical question and a practical observation:

Physical question: is it still a paradox even if we do not count with the more hardcore cosmology (so, universe expansion, doppler shift) and simply go about showing that the observable universe is not that uniform? What would the light intensity integral be for an infinite universe with an infinite age while assuming the rest of the universe has the same galaxy/star density as we currently know?

If the answer to the above question actually shows that the ratio is still 1 as in the Olber's paradox, what is the difference, for a universe with the same age as ours, between the light intensity for the visible spectrum considering it static (no red-shifts, so no acceleration/expansion, just like it would be frozen but just has the same age) and the currently seen one?

practical observation: I mean, even in practical terms there are only a handful of galaxies which can be seen with the naked eye. If you ever tried to see Andromeda without aid, you know you need to use your rods and not your cones and its damn hard to see. In our own galactic cluster and supercluster we can but see a few out of the hundreds of thousands of them with the naked eye. Galactic clusters are indiscernible. What I am getting at, is that I do not see the compounding effect mentioned in the Olber's paradox. Things become too small with too low flux (and stars compacted together instead of uniformly scattered) to see with the naked eye. In other words, it seems to me that the ratio between the stars in a solid-angle volume at different distances and the decay in light intensity is still lower than 1 with our current knowledge of the galactic positioning.

EDIT: TL;DR: I think that stars are more spaced out than light intensity fall-off as they conglomerate into galaxies, and also, any star will block background stars in its solid angle and there's a limit to the distance a star would be visible for the human eye, so, altogether dark sky would still be dark even for an infinite one (my opinion). I think Olber's paradox is solely a paradox due to its over simplification: not counting with solid-angles and actual star distribution and limits to human vision.

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  • $\begingroup$ "any star will block background stars in its solid angle". Not quite, since the incoming energy isn't obliterated. It's mostly absorbed and re-radiated, and some of the energy may be reflected. $\endgroup$
    – PM 2Ring
    Feb 26, 2021 at 16:34
  • $\begingroup$ yes, but for a practical you could say it shifts the peak of emission of the star's black body radiation by warming it up (and of course increases its output) but if instead of 3 photons/s on earth's surface from that star, we get 3.3 photons/s, I'm not sure that makes a huge difference. $\endgroup$
    – José Andrade
    Feb 26, 2021 at 16:40

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Let's suppose the universe is infinitely old, the galaxy distribution is homogeneous (the same everywhere) and isotropic (the same in every direction) and stars are constantly being replenished as they die out.

Then you are right that seeing an individual star becomes increasingly difficult as you try to see stars further away, since the intensity of an individual star decreases as $1/r^2$, where $r$ is the distance from the star to your eyes. You are also right that if you see one star along the line of sight, you won't see any additional stars at larger distances along the same line of sight.

However, there are also more stars in a fixed solid angle at a given distance. The number of stars in a solid angle scales as $r^2.$

The intensity in a given solid angle on the sky is then the intensity of one star, $\sim r^{-2}$, times the number of stars in that solid angle, $\sim r^2$. Since these two effects exactly cancel, you would expect the intensity of light in each solid angle to be approximately the same, given our starting assumptions -- that is Olber's paradox.

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    $\begingroup$ @JoséAndrade Sorry but you evidently don't understand Olber's paradox :) Nothing in it requires that stars be transparent. For the solid angle containing the sun, the sun provides the intensity we see, and we don't see anything behind the sun. It also is not correct to say that the faraway stars are "too far away and only a few photons reach us." At a large distance, any individual star is dim, but there are many more stars at a large distance in a given solid angle, and the collective effect of all the faraway stars is the same as one (hypothetical) nearby star taking up the same solid angle. $\endgroup$
    – Andrew
    Feb 26, 2021 at 16:40
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    $\begingroup$ "Limits to human vision" is not relevant, because the point is that the faraway stars should collectively be just as visible as the sun. "Uniformity of the universe" is relevant, and if having a non-uniform universe is a solution to the paradox since you would break one of the starting assumptions that led to the paradox to begin with. However, this is not the correct solution in our universe. $\endgroup$
    – Andrew
    Feb 26, 2021 at 16:44
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    $\begingroup$ @JoséAndrade Of course in reality the night sky is dark -- so any experimental evidence will contradict the conclusion of Olber's paradox. But the conclusion to draw from this is that at least one of the assumptions that went into Olber's argument must be wrong. All of the assumptions seem very reasonable when you first encounter them, so then the question is, which assumption is wrong? It turns out that the finite age of the Unverse, plus the idea that "stars are replenished constantly", are both not right -- if you go back far enough in the Universe's history, there are no stars. $\endgroup$
    – Andrew
    Feb 26, 2021 at 16:49
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    $\begingroup$ @JoséAndrade the Milky Way is very finite in number of stars and physical extent. It's nothing at all like an infinitely old, infinitely large universe. $\endgroup$
    – Christopher James Huff
    Feb 26, 2021 at 16:58
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    $\begingroup$ @JoséAndrade In Fractals: Form, Chance and Dimension, Mandelbrot shows another resolution of Olber's paradox which even works in an eternal universe. If the distribution of stars has a low enough fractal dimension, then the sky will be dark. I guess not everyone would agree that such a distribution is strictly uniform, but neither is the hierarchical distribution of stars in galaxies & clusters of galaxies in the real universe. ;) $\endgroup$
    – PM 2Ring
    Feb 26, 2021 at 22:02
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One should keep in mind what Olbers' paradox says. (From https://en.wikipedia.org/wiki/Olbers'_paradox#The_paradox .) "The paradox is that a static, infinitely old universe with an infinite number of stars distributed in an infinitely large space would be bright rather than dark." These assumptions were not countered with astronomical evidence until the Hubble Law phenomenon was discovered in 1912 and later understood in 1922 (Friedmann) and 1927 (Lemaitre). Hubble published the "Law" named for him in 1929. (See https://en.wikipedia.org/wiki/Hubble%27s_law#Discovery .)

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