Recently while I was thinking about answers I was getting in another thread I realized that I have problem of conceptually grasping something. Problem is not mathematical in nature, it arises from what math is telling us. Average density of universe determines its spatial curvature. In order to preserve assumption of homogeneity and isotropy spatially flat universe is required to be infinite in extent. As demonstrated in flatness-oldness problem, incredibly tiny variation in density would quickly lead to departure from flatness. At the age of 1 ns, 1 part in 10^{59}, and flatness is gone. Here I make one assumption that I think is the correct. If the universe is infinite now, it was infinite to begin with. How the universe "knew" that if it was infinite in extent that it must have very precisely determined density? I mean, why you can't have universe that is infinite, but slightly sparser or denser? If it was infinite and slightly denser it couldn't later close into itself, like math is telling us. It is sort of the case where future determines past. Maybe inflation solves this, but I don't see how. Anyway, is the presumption of spatially flat infinite universe made only as an attempt to preserve cosmological principle, or it has some other foundation?
You are right, bad wording on my part. Here is the assumption that universe begun with infinite extent (I am from some reason skipping singularity issues, and take that we are talking about very hot and dense state), and very precisely determined density, because the two are linked together. So, let me rephrase my dilemma. Somehow universe is born. From something, or from nothing, it doesn't matter. Quantum fluctuation blown out of proportion, or whatever. It has density slightly higher then critical, thus it is closed, and finite in extent, mass, etc. Now, if the same thing happened with slightly lower density it would be infinite. That is what I can't properly understand.
The density of the universe has no bearing on whether it is infinite in size or not. It simply determines if the universe will become more or less dense in the future.
Inflation theory was proposed principally in order to resolve the flatness problem, although the inflation process is not yet completely understood. It is also not entirely certain that inflation theory, as currently presented, will stand the test of future experiments.
Just a note for the historical record, according to Guth's book "The Inflationary Universe" inflation came about to solve the monopole problem, not the flatness problem, nor the horizon problem. It seems everyone gets this wrong, not suprisng I guess as the original paper by Guth had horizon and flatness problem in the title, but readign Guths book , its clear that he was not even thinking of either the flatness or the horizon problems when he came up with inflation.
To elaborate on the comment of inflation surviving future experimental tests - ESA's PLanck is in space right now and should deliver data on the CMB in maybe 12/18 months from now. So far WMAP team gave a ringing endorsement to inflation. Inflation also predicts gravitational waves from the big bang. Planck maybe able to indirectly detect these by measuring the B mode polarisation. http://cosmology.berkeley.edu/~yuki/CMBpol/CMBpol.htm But if it fails then people will probably say PLanck wasnt senstive enough and look to the next proposed mission, for example CORE: http://arxiv.org/abs/1102.2181 Alternatively gravity waves might be measured directly by LISA (whcih may or may not get funded) , but I think a lot of people suspect even LISA wont be good enough to detect gravity waves from inflation and it would take something like the BBO to do that: http://en.wikipedia.org/wiki/Big_Bang_Observer I think if all these missions fly and still detect no primoridal gravity waves people will give up on inflation but this is likely decades away, unless of course Planck may throw a spanner in the works, we shall see.
You seem to be thinking that the space of initial conditions is continuous, that one can simply dial the initial energy density from subcritical to supercritical continuously (and this seems problematic given that the size of the universe jumps discontinuously from finite to infinite in the process. I would suggest not to think of the space of initial conditions as continuous: you instead have a discrete set of initial states, and the universe simply starts out in one of them.
To add to this, I don't think it's fair to say that if primordial gravity waves aren't discovered that people will "give up on inflation." That would be incredibly shortsighted. Not saying I endorse them, but a vast majority of inflation models developed within string theory predict an unobservable tensor amplitude. Gravity waves aside, inflation remains the only theory that I know of that explains the non-zero correlation of polarization anisotropies on superhorizon scales.
I sort of knew that that would be an answer. But what does it mean? There are uncountable number of failed universes for few of them that are close to critical density? I will try to pose my OP question in a different manner. If, at any point of time, universe has exactly critical density, and thus is flat and infinite, does it mean that it will always has critical density, and that departure isn't possible because you can't make infinity finite?
If the universe began perfectly flat, [itex]k=0[/itex], it would remain so forever. But this is seen at the level of the Friedmann equation, from which we derive the density parameter, [tex]\Omega -1 = \frac{k}{a^2}{H^2}[/tex] It does not follow from the apparent mathematical impossibility of continuously moving from a finite to an infinite manifold, because a flat universe need not be infinite. By considering manifolds with nontrivial topologies, we find that finite, flat manifolds exist -- like the torus.