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The rate of expansion of the universe after the Big Bang is described as Inflation. It is stated as being initially very fast then slowed. But the expansion is still continuing. Some say expansion is currently powered by Dark Energy. The equations unlike gravity say that the rate appears to be linearly proportional to distance, like Hooke's Law for springs. Initially the universe was very small, so if the spring constant is dependent on the size of the universe, could this explain the inflation? What experiment could be devised to test this?

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The expansion caused by a cosmological constant (a particularly simple version of dark energy) is not exactly Hooke's law, it is an expansion that maintains a fixed Hubble constant. So this means $\mathrm{d}a/\mathrm{d}t = Ha$, for scale parameter $a$ and constant $H$. The solution is exponentially growing in time. Hooke's law (without the minus sign) would have the acceleration $\mathrm{d}^2 a/\mathrm{d}t^2$ being proportional to $a$, which permits more complicated solutions, but you're right it certainly has similarities if one picks out the exponentially growing solution.

What's more, you are right that some treatments of inflation look like that of a cosmological constant, in that they are also exponential in time, but with a much much larger H constant (like 50 orders of magnitude larger). So if it's the same thing, then for some reason the cosmological constant must have dropped vastly in value. It's thought to be for a rather different reason, though this is all speculation. The common speculation is that the "false vacuum" went through a phase change during inflation, leading to the vacuum we have now, and the vacuum we have now has a very different cosmological constant. Maybe you can think of it as a phase change in the cosmological constant itself, I don't know. So there could be some connection, but they can't be just the same.

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The big similarity between inflation and dark energy is that the preferred models for both can be described by scalar fields, however there is a huge difference between the energy scale, with the energy scale of inflation being 27 orders of magnitude bigger than the energy scale of dark energy!

Could they be related? Possibly, but the difference between the energy scales is not easy to explain. There are a number of papers proposing a relationship, but nothing that has really gone beyond speculation. Any test of a connection between the two would depend on the model used.

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