If the universe is "bounded" today, could it one day become "unbounded"?
Our text talks of a universe being "bounded" and "unbounded". Would it be reasonable to assume that being bounded or unbounded is a function of changes in density as a result of expansion? I mean, as the universe expands its mass is spread more thinly decreasing density until ultimately the average density would be less than the critical density needed to maintain expansion... in other words a density-dependent boundary.
This is a tricky concept. Yes, it's true that the density of the universe decreases with time as it expands. And the textbooks say that if the density is below a certain value, then the universe will expand forever--and if it's above that same value, the universe will recollapse.
But the "critical density" is really a function of how fast the universe is expanding. Given the rate at which the universe is expanding, and given how the rate of expansion changes over time, there is a "critical density" at any given point in time. If the density of the universe is greater than that critical density, then it will *eventually* recollapse. If it's less than that critical density, then it will expand forever.
Let me put it another way. The value of the critical density is something that changes with time. It had one value 10 billion years ago; has another value today; and will have another value in another 10 billion years. The actual density of the universe is something that has been changing over the course of the universe's history. (Both numbers have been getting smaller.) But whether or not the actual density of the universe is greater than the critical density is something that doesn't change. If it's greater than the critical density today, that it always has, and always will be greater.
So if you know the current density of the universe, how fast it's expanding, and how that rate of expansion changes with time, then you can figure out whether the universe will eventually recollapse or expand forever.
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