Olbers’ Paradox says that in an infinite universe every line of sight will end on a star. Surface brightness is independent of distance (moving a star further away makes it smaller and reduces its flux but not its surface brightness), so why is the night-sky dark rather than uniformly painted at the brightness of an average star?

Now the explanation I have been given is that the universe is $1.4\cdot10^9$ years old so the furthest we can see is $1.4\cdot10^9$ light years away, but the average distance to a star is $2\cdot10^{24}$ light years away, hence the result.

Now suppose hypothetically that the earth (and sun) are still here the way they are now when the universe is $2\cdot10^{24}$ years old, then will an observer on earth looking at the night sky see uniform brightness?

• I'm not sure, but I think that the average distance at that time will have grown too due to inflation. Commented May 8, 2015 at 20:27
• In an infinite universe, how could you possibly know the average distance to a star? Commented May 8, 2015 at 20:32
• More on Olbers’ paradox. Commented May 8, 2015 at 20:54

Olbers’ Paradox says that in an infinite universe every line of sight will end on a star.

That statement is incomplete. The paradox requires not only an infinite universe, but also one that is both static and infinitely old. Neither of the second two statements are true for our universe.

Your question considers the effect of aging. As our position in the universe gets older, light has had more time to reach us and we can "see" a greater volume of space.

But the static condition is not there, so you still don't get a bright sky in the future. Stars die, and the material to make them is not infinitely abundant. After a long-enough period of time, there may be no stars to see. Besides that fact, the universe is expanding faster over time. The light from distant galaxies may be unable to reach us in the future, limiting the light present. Our current model of the universe suggests that we will always have dark skies.

• It also requires a transparent universe, and this is not (perfectly) true in the visible. Interstellar dust will intercept a certain fraction of the light from far away, and the greater the distance (assuming statistically "uniform" dust distribution) the greater the absorption. I've no idea at what point this would become percepitable. Commented May 9, 2015 at 20:39
• Transparency is not required. If every spot "seen" is of a star, then the combined energy will raise the temperature of the dust to that of the stars. When that happens, it becomes incandescent and cannot contribute to a dark sky. Commented May 10, 2015 at 3:56
• What about blackholes? Image there are uniform distribution of blackholes which will only light to go in but not getting out (infinite time) Commented Jan 2, 2022 at 7:22
• @Splash Black holes are very messy eaters, and their accretion disks are some of the brightest objects we can see Commented Apr 9, 2023 at 5:52

I agree with what BowlOfRed said, but I'm going to give an answer with a different nuance.

So why is the night-sky dark rather than uniformly painted at the brightness of an average star?

Because the universe isn't infinite. Big bang cosmology describes a universe that started small some 13.8 billion years ago and has been expanding ever since. It's been expanding for a finite time, it can't be infinite. Unfortunately, in recent years a non-sequitur has crept in wherein a "flat" universe is assumed to be an infinite universe. Articles which used to say the universe was the size of a grapefruit now say the observable universe was the size of a grapefruit. IMHO this is a temporary situation which will be sorted out in a few years once cosmologists appreciate a subtlety of Einstein's greatest blunder wherein a flat universe doesn't have to be infinite, and an infinite universe can't expand.

By the by, see Expanding Confusion by Davis and Lineweaver: "We show that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light". And note that the night sky isn't actually dark, it's painted at the brightness of the CMBR which predates all stars.

• A flat universe that isn't infinite would, literally, be a box, and, as no astronomical object I've ever heard of has that shape, this answer doesn't seem to represent a viable conjecture, except maybe in an extremely creationistic context. Commented Feb 22, 2023 at 16:39
• It's not the universe that's expanding, it's space itself. That can expand without being finite in amount, as each local bit of space does not care about the global structure of the universe. Whether the universe is finite or not is independent of the fact that space expands, or even whether or not it has an edge. This is where the "flat universe can't be finite" issue comes from, which, to be fair, is a non-sequitur. Even if the universe has no edge, the flat torus is a thing, it's just hard to conceptualize. Just like an infinite expanding universe. Commented Apr 9, 2023 at 6:09

Olbers's Paradox actually fails to demonstrate a finite or nonuniform universe. The hypothetical is moot because universal expansion isn't the most natural explanation for why the night sky is dark. The supposed paradox ignores the existence of non-luminous interstellar matter, which we have since learned about (and which modern astronomical theories explicitly rely on). Such matter is known to attenuate light according to an exponential decay law (see the Beer-Lambert attenuation law). Olbers himself disregarded this law as it was not documented with respect to starlight until 1930, more than a century after he proposed this supposed paradox as a thought experiment. Those arguing against the existence and prevalence of such matter are asking us to presuppose the non-existence of planets, moons, asteroids, dust and gas (from which the stars themselves are said to form), nebulae, dark matter, etc.

I have written a basic simulation to validate the effects of non-luminous matter quantitatively and qualitatively. Code here. It directly applies the exponential decay law by assuming a uniform random density of stars and a uniform density of non-luminous matter throughout space. The simulation even cheats massively in favor of Olbers's Paradox by modeling the infinite backdrop of stars at varying distances as a purely luminous wall of light at a fixed, finite and even shortest possible distance beyond the foreground model of direct sampling. This means we have a strict theoretical upper bound on the total light emanating from distant stars. The density of non-luminous matter is controllable as a parameter p. The simulation shows that for even very small values of p, a purely luminous wall would be practically invisible, or be indistinguishable from cosmic microwave background radiation (interstellar attenuation not only decreases apparent intensity, it also induces reddening. The observation that microwaves predominate at such distances is therefore a natural consequence of known laws of optics, and is therefore not evidence for a different nature of the universe so far back in time or at such a distance).

If one sets p=0, the entire field of view is indeed saturated by pure, contiguous starlight under this simulation. But for any non-infinitesimal density p of attenuating matter, the visualization -- and the mathematical solution for expected light intensity over all points in space--converges sharply towards a model that closely resembles what we see in our night sky, regardless of all other parameter settings.

In the following two images, the absorption value is increased only slightly, starting as close as the simulation allows to zero at first:

In both images, there is a perfectly bright wall of simulated stars a short, fixed distance away from the observer. Radiation from that wall leaks and is visible as a uniform grey background when the simulated wall is very close and the density of attenuating matter approaches zero, but if the wall of light is slightly more distant or the density of attenuating matter is only slightly increased, the background approaches pitch black, despite there still being an infinite number of perfectly bright stars intersecting every possible line of sight.

You can bump up the stellar density, radii and luminosity as high as you want, and even a tiny percentage of non-luminous matter still overwhelms the resultant apparent brightness calculations with the exponential decay induced by material attenuation.

Of course arguments about thermodynamics are just begging the question. How nonluminous matter is able to be there and coexist with stars is a separate matter of discussion. (How stars got there is too, come to think of it). The fact that it is there and that we know it is there, even at great distances, disproves the idea that Olbers's paradox is a paradox at all, as it completely fails to falsify an infinite and uniform universe given what we know.

In case someone thinks this idea is flamingly absurd, that even a small amount of matter between stars can block nearly all of the light from them, remember that despite how brightly the Sun shines at noonday and how close it is to the Earth, you can block almost all of that light using a thin cardboard cereal box or a parasol consisting of cloth a couple millimeters thick. This gives you an idea of how effective exponential attenuation is. Can there not be the equivalent matter of one millimeter of cardboard between you and a star 3 billion light years away? If not then what are stars made of?

• I think this answer offers some useful information but it needs to address the point that if conditions were static then eventually the matter absorbing the light would heat up and come to thermal equilibrium with the stars. So the fact that this has not happened indicates that conditions have not been static. Also it would be better to remove the first sentence because inflation is largely irrelevant here. (I have not voted). Commented Mar 5 at 22:19
• @AndrewSteane Thermal equilibrium is a subject that should be treated separately. The very existence of planets, cosmic dust, or even stars violates many a priori assumptions about thermal equilibrium. The observation that there is cosmic dust and other interstellar matter suffices to prove that, if stars and the material from which they are organized are allowed to exist or coexist at all (and they do), then the above model applies. Perfect thermal equilibrium presupposes there would be no stars at all, which is counter to the very premises of the paradox (as well as our own observations). Commented Mar 5 at 22:29

And still a different answer: Olber's Paradox fails to take the inverse square law into account. Light from a star is attenuated by distance. It is possible for the viewer to be so far away from a star that the star is invisible to the naked eye. We see that even within our "visible" universe as we record images viewed telescopically that are otherwise invisible to the unaided viewer. That aspect remains regardless of age or whether the Universe is static or expanding so an infinite Universe would not result in a bright night sky.

• Welcome Thorkil-2 to to Physic Stack Exchange. Unfortunately, this answer is wrong. Olber's paradox does not fail to take the inverse square law into account, and actually follows from applying the inverse square law in an infinite universe with a uniform density of stars. See wikipedia. Commented Jan 22, 2023 at 16:59
• Sorry, but the reasoning in the Wikipedia argument makes no sense. Consider a single star at a sufficient distance that it is invisible from the Earth. Its light is so greatly attenuated by distance that it contributes no light to our night sky, so it doesn't matter how many such stars are in each shell. If one of a given surface brightness beyond a critical distance is invisible, all like-stars are invisible, even an infinite number, unless the wave fronts from the greater number of those stars are in phase in some visible wavelength. Commented Jan 24, 2023 at 14:13
• "greatly attenuated by distance" does not mean a star "contributes no light to our night sky". The amount of light from a star falls off as an inverse square law, but $1/r^2$ is never zero except at infinity. In an infinite, uniform density, flat universe, the number of stars in a shell at distance $r$ increases as $r^2$. Olber's paradox follows from $\int_0^\infty 1/r^2 \times r^2 dr = \infty$. Physics SE comments are not for long discussions, so I suggest reading more carefully the many explanations of Olber's paradox that can be found online. Commented Jan 25, 2023 at 23:28
• I should have been more specific and said "contributes no visually detectable light, but I'll drop it here for now. Commented Jan 27, 2023 at 1:02