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This is a follow up to How do quantum fluctuations lead to overdense regions in the CMB? (which still hasn't been answered). The general logic of Inflation (if anything about the theory can be described as 'logical') is that some field popped into existence just long enough to fix a major problem with GR, then disappeared again when it was no longer needed. This field had 'fluctuations' in it which later led to overdense regions of space.

Setting aside the fact that we still don't have an explanation for what is fluctuating or why the momentary fluctuations lead to coherent pools, why is inflation needed for these fluctuations to grow to cosmic scales? That is, why can't fluctuations in a plain-old quantum field grow into overdensities with GR expansion?

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You misunderstood quite a few things, here. For what is knows a universe doesn't need inflation right after its beginning. However, we'd expect to see a different universe today if inflation never happened.

Do be more precise: We see that the cosmic microwave is the same in every direction, except for really, really tiny variations. Furthermore we don't assume that physics changes as you wrote in your answer. We assume that physics doesn't change at all. Since right after the universe started, everything was really close together (a heck of a lot of mass per volume), we'd expect gravity to have pulled mass together with a lot of force. Therefore, we should see cluster. But we don't.

This is why inflation was proposed: If during some time at the very beginning of the universe space expended really, really rapidly, gravitation wouldn't stand a chance in pulling everything together fast enough to make these huge lumps we'd expect to see otherwise but don't.

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