Is the sky dark at night?

The Cool Sky Conjecture

There is a conjecture, called "Olbers' Question", or "Olbers' Paradox".  As a question, it goes "Why is the sky dark at night?"   As a statement or paradox, it goes "If the universe is infinite, then wherever you look in the sky, there should be a star.  Stars should crowd together from any point of view, 'shoulder to shoulder', and the sky should be bright with stars."

The argument also goes that, over time, the stars in an infinite, stable universe should heat the universe steadily until everything is as hot as a star.

There is a wonderful web site at Arachnoid which, along with many other neat things, has a very nice explanation of the Question/Paradox.

Enter the big bang theory.  The big bang theory is nice, because it explains the observed night sky quite well.  In an expanding, finite universe, the stars don't crowd together because the universe is merely very big...not infinite.  Also in a big bang universe, things are not required to heat up universally, and may even cool off over time.

But I don't believe it.  Entirely aside from theological arguments that make the big bang unpalatable to many, I just don't see it.

I look out at night into an infinite, OLD universe.  I see no reason why time needs a beginning (when would that be, anyway...but never mind), nor any need for the universe to have a physical bound (where would that be, anyway...but I digress again.)

The argument that the stars should crowd together does not impress me because they do, in galaxies...but even when I look at a galaxy so dense I can't see through it, the galaxy still looks dim.  I have read (and it seems reasonable...no proof there) that even relatively nearby galaxies have "surface brightness" less than the brightness of a very dark night atmosphere here on earth.  So.  In an infinite, very old (millions of millions of years old), universe, how "hot" or bright, would the night sky be, sitting here in TN?

Sadly, I lack the arithmetic skills and other needed knowledge to know.  But.  I can simplify things a bit.  Conjecture with me.

Consider me, sitting in my backyard (I don't have one, being a vile apartment-cave-dweller, so I'll borrow a friend's field).  I look up into the cool, dark winter sky.  In my simplified universe where I have borrowed the farmer's field, there are only 2 stars.  Ours (whew! thought we were going to freeze), and Theirs.   Theirs is a long way off.  Say, oh, 40 billion (thousand million) light-years away.  The light that left the surface of the star 40 billion years ago has formed a sphere whose radius is 40 billion light-years.  It has a surface area of (4*pi*r^2) 4*pi*40**2 light-years.  The surface of this sphere of photons is about 500 billion square light years, or 4.8 x 10^27 square meters.  My pupil opens to about 8mm.   This give my pupil a surface area of about 50mm, 50 MILLIONTHS of a square meter.

For me to be able to see the star at all, it must send out enough photons that a detectable number will fall into my pupil, continuously.  One every few seconds won't cut it.  Each photon has a 1/(9.5^31) chance of hitting my eye.  Not even a BRIGHT start is going to make it, and most stars aren't.  Most stars look small and dim.  In this thought excersize, Theirs is like Ours.  Small, yellow-white, pleasant.  Dim.

Imagine that there are an infinite number of stars ALL IN A ROW, behind the first star.   The sky is still completely dark and cold.

Imagine that there are stars in every single 10-square-light-year patch of a 40 billion light-year sphere around Ours.  5,000,000,000 stars, and the sky is still dark.

It actually gets much worse than this.  At a mere 100 light-years, Ours is not a naked-eye object.  A CLUSTER of stars like Ours is not a naked-eye object.   Scatter clusters of sun-size stars every 100 lightyears, all the way to the edge of forever, and the sky will still be dark: the sky will still be cool.  The photons that hit anything, hit the biggest things:  the stars.  A few more hit the planet, and practically none at all will hit the pupil.

And it still gets worse.  If you think of a star as the gases having the mass of a star, then most of the time, it seems that the gas would be cool:  the star would be "off", having blown up or burned out.  A new star has to form, then start to glow.

I look up into the cool, dark sky, and I see an infinite, ancient universe.

I realize that this is not a compelling argument, it is weak even as conjecture.   But I like it.