# Number of Stars vs Value of Omega (Crtitical Density of the Universe)

I may be badly mixing things up here. If I am, please kindly correct me.

As I understand it, if the universe was too dense at the start of the big bang, it would have collapsed back in on itself. Too sparse, and it would have expanded violently leaving no galaxies. I also read that the ratio of the actual density to the critical density (Omega) leaves little room for error, if you want to have a universe that lasts. The value seems to not be able to vary more than one part in 10^62.

Now I've also read that a recent estimate says there are about 10^23 stars (300 sextillion). I noticed that the number of stars is far fewer than the amount of variance allowed for in Omega.

So my question is: If you took away one solar mass worth of matter from the early universe, would that have thrown off the balance of the expansion of the universe? Would the universe have not been here if one star was missing?

The TL;DR: The Acccepted Answer is "Yes, one star would make a difference. It would take even less to throw off Omega."

(For the benefit of others coming across this question)

The point is that during the ordinary phase of the Big Bang expansion, the difference $|\Omega-1|$ was rather dramatically increasing with time. Today, we know that $|\Omega-1|\lt 0.01$ or so. If we use the cosmological equations to reconstruct what $|\Omega-1|$ had to be when the Universe was a second old, or very young, we find out that the following inequality had to hold $$|\Omega(t=1\,{\rm s}) -1 |\lt 10^{-62}$$ or so.

For some extended discussion what the flatness problem is and how the cosmic inflation solves it, see e.g.

http://motls.blogspot.com/2014/03/alan-guth-and-inflation.html?m=1

Now, if the Universe didn't have this fine-tuned $\Omega$ when it was one second old, the density would immediately begin to diverge away from $\Omega=1$ to either hopelessly high values (implying Big Crunch) or hopelessly low values (implying sparse space without stars) billions of years after the Big Bang. That's what the FRW and similar equations – effectively, Einstein's equations (the field equations from the general theory of relativity) specialized for a uniform Universe – imply.

Here everywhere, the cosmological principle is assumed, so the density $\Omega$ is assumed to be basically uniform across the Universe. Even if you insist that it remains a uniform, a tiny change – by "one star or much less" – would have led to consequences incompatible with life.

If you changed the average $\Omega$ by changing $\Omega$ "more dramatically" but only in a smaller region of the Universe (if you literally tried to remove a "seed" of what is a star today), the consequences for that region of the Universe would be even more dramatic – it would have crashed or diluted or became incompatible with life much earlier.

So indeed, the mass density one second after the Big Bang had to be adjusted with a much greater accuracy than "one missing star per visible Universe". I sort of vaguely understand why it may sound counterintuitive but there is really no paradox here. If the early Universe were a consequence of the present, then it would mean that someone had to adjust the number of stars etc. with the precision better than "1 star per Universe", and it would be extremely unlikely or unnatural if not a downright inconsistency.

But the point is that the events of the early Universe are not a consequence of the present. It's exactly the other way around. The present and the stars around (the "future") are a consequence of events in the past (the "past"), including the events in the early Universe. So it makes sense to ask what events or observations would be different if the early cosmology weren't this exactly flat – the answer is that (almost) no stars would exist 14 billion years after the Big Bang, It does not make sense to ask how the early Universe were different if the number of stars today were smaller by one – because the history of the early Universe in no way depends on the number of stars today (causality: future depends on the past, not the other way around). We don't have to fine-tune the number of stars in the visible Universe today with an unrealistic accuracy in order to allow the young Universe to exist. The young Universe did whatever it did long before we were here and we can no longer change that!

• Yes, I understand we're not talking about removing a star from the present. So you're saying if the early universe was missing one solar mass of mass-energy just after the singularity, you would have had a "Big Rip"?
– John
Jul 28 '14 at 16:04
• Big Crunch (or, on the other side, an exponentially diluted space), not Big Rip, but otherwise Yes. Big Rip is a physically impossible evolution that may only occur if equations allow $w=p/\rho \lt -1$. Jul 28 '14 at 16:12
• So, wow, one star really would've made a difference? I almost find it hard to believe but it's awesome.
– John
Jul 28 '14 at 17:25
• One human body-like fraction of the mass of the Universe would've made a lethal difference! The right intuition that this seems very "supernatural" is why we say that the Big Bang Theory depends on unnatural initial conditions, and it's the explanation why Alan Guth's and Andrei Linde's cosmic inflation is needed. With the inflationary prehistory before the universe was a femtosecond (much less) second old, the proximity of Omega to 1 is explained because inflation drives Omega ever closer towards one as time goes by - it reverses the otherwise "normal diverging from 1" evolution of Omega. Jul 28 '14 at 19:29
• Except that they currently don't know how to make inflation stop, right? The equations point to an ever-inflating, bubbling, branching-off bunch of universes?
– John
Jul 28 '14 at 19:35