[zbhfw (OP)]

When cosmologists say things like "Approximately 10 to the −37 seconds into the expansion, a phase transition caused a cosmic inflation, during which the Universe grew exponentially"  what time frame of what hypothetical observer are they using? 
Since time varies according to gravity, and the universe shortly after the big bang was very small and dense, does that mean that time right after the big bang was moving slower (compared to our own time under 1 g)?
Are cosmologists using the 1 g time frame when describing the early universe?
Maybe I'm asking a stupid question, but I'm just wondering what the relativistic implications are considering how dense and massive the early universe was.

[yor_on]

You hit the cheese on the head there. There are two interpretations of relativity, one that discuss it as 'real time', also assuming a lack of simultaneity to explain it. Or maybe three, as you have those assuming that it is 'real time pockets' existing in the universe, although that's not relativity as defined from Einstein. And then those, as me, that don't really care about comparisons, assuming everyone to use a locally same arrow, treating your life span the (locally) same, not caring where you go, or, how fast you go. The first and the last one define a time from local measurements, meaning that we all should be able to agree on it, as we don't measure anything other that locally. So you, and me, both locally (Earth) being 'at rest' with it, reading the CBR (Cosmic Background radiation) should be able to agree on a time scale.
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The first one isn't 'real time' in the aspect that all share it though, although real enough for you, and me, also called 'proper time', the one defined by your wristwatch. Einstein discussed the arrow of time as a 'illusion' if I remember right. The last one do not define that arrow as a illusion though, instead, possibly, defining a whole universe as one

[HellsMascot]

zbhfw this is definitely not a stupid question, and the assertions you make in your post are something I wonder about myself. I don't find the single reply to satisfactorily answer your question, but I am also not well-versed enough in early universe cosmology to provide answers to a question that should have concrete answers. The definition of these occurrences in the nascent development of the universe by these cosmologists is of course confounded by relativity. However, it is even further complicated by the idea that the physical laws of the universe evolve over time. In the very early universe, did relativistic 'conditions' exist, and therefore does the time frame matter? Were the fundamental forces even established? Something like the cosmic background radiation cannot tell us these things.

[yor_on]

Sure it does. How can one expect to define anything without a arrow? Without it one may have a probability, but a probability is not what we measure. We measure outcomes, and those need a arrow. You can assume probability to co-exist with a arrow if you like, but the measurements defining ones logic, and theory, need to be done inside it.

[zbhfw (OP)]

Thanks for those replies.  It's been a long time since I got an undergraduate degree in physics (it was 1976), but I find myself suddenly getting interested in hitting the books again and relearning what I have forgotten.  I recently got the latest edition of Halliday, Resnick et. al. on Kindle and am going through it from the beginning, like someone having a second childhood.
But I read somewhere that if you approach a black hole, your time would slow so much relative to the rest of the universe, that you could watch billions of years pass by on Earth before falling in.  And of course, that's only a miniscule fraction of the mass of the whole universe.

[yor_on]

You have to differ between 'proper time' and 'time' as described relative other frames of reference. Your proper time is your wrist watch, and that one will give you a same life span wherever you are. But the proportions between that proper time and the way a universe will behave is indeed out of proportion thought of it in terms of 'energy' (and mass) expended by you hovering close to a event horizon. And that one is interesting as it speaks about 'energy' spent relative what a universe might be seen to spent. But there is a way to make it fit possibly, and that is looking at relative motion from a local definition. Locally there is no way I know of to measure any energy expended by Earth following a geodesic. And that goes for all heavenly orbits, none spends energy locally. So looking at it as 'energy', 'compressed' over time by the universe relative you, from this point of view, it's only you spending any energy (constantly uniformly accelerating) as you hover over that event horizon. But this is my way of looking at it, and I agree, it's been confounding me too when you think of it in terms of energy needed for compressing and defining a energy spent relative a motion, or gravity.

[yor_on]

Where it falls as a explanation is when you imagine yourself to accelerate close to light speed, to then coast in a uniform motion. If this reasoning is correct you should now fit perfectly with all other heavenly bodies, all in uniform motion, expending no energy. But you will still have created a same effect as when hovering, the universe 'speeding up' relative your wristwatch as well as you should observe a Lorentz contraction of the universe, in your direction of motion, shrinking it. And if we discuss that in form of energy then what you spent is nothing, compared to shrinking a universe.
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Which to me mean that we can't use 'energy' for describing it. We can use 'symmetry' though, and we can use 'four dimensions' as parameters, having a sliding scale relative each other, but not think of it in form of energy.