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after the big bang. Does that means the observable universe was the size of a baseball, or does it mean the entire universe? I'm guessing it means the observable universe - as we really don't know much about the size of the entire universe.

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    $\begingroup$ It means the whole universe. This is important cosmological point because inflation solves the horizon problem (the near uniform temperature distribution of the night sky). $\endgroup$ – boyfarrell Apr 23 '15 at 6:27
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    $\begingroup$ It means the observable universe. Assuming the universe is currently infinite in size it has always been infinite in size $\endgroup$ – John Rennie Apr 23 '15 at 6:50
  • $\begingroup$ @JohnRennie most scientific research and speculation points towards a finite universe. $\endgroup$ – Jaywalker Apr 23 '15 at 7:29
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    $\begingroup$ @Jaywalker: citations please. Your claim does not match my experience. $\endgroup$ – John Rennie Apr 23 '15 at 7:53
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    $\begingroup$ The point about infinite/finite is that an infinite universe cannot ever have been the size of a baseball because a baseball isn't infinite. I don't understand your point about the density fluctuating. According to the FLRW model the density has been decreasing smoothly since the end of inflation. During inflation the density is determined by the inflaton field, and we don't don't know how this behaved though it probably remained roughly constant during the inflationary era. $\endgroup$ – John Rennie Apr 23 '15 at 9:15
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When they say the universe was the size of a baseball about a billion billion billion billionth of a second after the big bang, does that means the observable universe was the size of a baseball, or does it mean the entire universe?

In the past, the answer would have been the entire universe. Big bang cosmology is all about the expanding universe starting off small circa 13.8 billion years ago. You even hear about a point singularity. However nowadays the answer tends to be the the observable universe. The change started in 2013 or thereabouts when WMAP discovered that the universe appeared to be "flat". That's when a non-sequitur crept in wherein a flat universe had to be an infinite universe. I don't know where it came from, but you can see it in respectable sources, like this NASA article:

"WMAP has confirmed this result with very high accuracy and precision. We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in extent; however, since the Universe has a finite age, we can only observe a finite volume of the Universe. All we can truly conclude is that the Universe is much larger than the volume we can directly observe."

The trouble is that it doesn't suggest that the universe is infinite in extent. That's only true if you have a flat-Earth mentality. The story goes that in the old days, people could not conceive of a world that was curved. They could only conceive of a world with an edge. Nowadays we have cosmologists who cannot conceive of a world that is not curved, they cannot conceive of a world with an edge. See this related question and note answer 3 by Danu, where he says we assume that the universe is spatially homogeneous and isotropic. The infinite universe is built on this assumption. But it's only an assumption. For all we know some guy 46 billion light years away might be looking up at the night sky wondering why half of it is black, or a mirror-image of the other. And if you've read Einstein's Leyden Address where he describes a gravitational field as a place where space is "neither homogeneous nor isotropic" you appreciate that the assumption is not a good one anyway, particularly with respect to dark matter. And remember that of the three solution as to the shape of the universe, two were always going to be wrong.

enter image description here Public domain NASA image

And none of them are actually telling you the shape of the universe. Is it shaped like a ball? Like a torus? The Planck mission said no. So how about a chocolate teapot? And how can an infinite universe expand when the energy-pressure is a counterbalanced at all locations? IMHO there's a lot of confusion here, it needs fixing, and questions like yours should help things along a little.

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  • $\begingroup$ I'd like to add that I think big bang cosmology is essentially correct. In fact I struggle to understand why Einstein didn't predict an expanding universe. IMHO it's as if his usual confidence in his own theory deserted him. $\endgroup$ – John Duffield Apr 23 '15 at 10:29
  • $\begingroup$ Dear John I'm not a historian but my understanding was that Einstein understood something like an expanding universe was possible but was influenced by (1) the steady state theory - that the universe had always been there and always will be - held considerable sway in science in 1915 and (2) in Einstein's personal case he was disturbed at how a universe expanding from something small was like many creation myths in religions (cosmic eggs and the like). His attempt to "stabilize" with the cosmological constant was misguided as the equilibrium would be an unstable one - the tiniest ... $\endgroup$ – WetSavannaAnimal Apr 23 '15 at 10:39
  • $\begingroup$ ... would lead to a runaway expansion or contraction. $\endgroup$ – WetSavannaAnimal Apr 23 '15 at 10:39
  • $\begingroup$ Rod: I've read various books etc, but I still don't understand why Einstein didn't predict the expansion. Look at the stress-energy-momentum tensor which "describes the density and flux of energy and momentum in spacetime". It's like a gravitational field is a spatial pressure gradient. And without that gradient, space is a thing with an innate pressure, and surely has to expand. Einstein was usually cocky about predictions, and yet he got bogged down with a steady state dusty hypersphere. Maybe I need to read more. Or ask. $\endgroup$ – John Duffield Apr 23 '15 at 10:52
  • $\begingroup$ Have you thought about History of Science and MAth SE? $\endgroup$ – WetSavannaAnimal Apr 23 '15 at 10:57

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