Can someone explain in somewhat simple terms why a quantum field in a hot universe would have negative zero point energy (is this right?) and cause an inflationary epoch?
I can try and answer some of this, though this isn’t my area of expertise. “Quantum field” isn’t really a useful term; these predictions are made based on quantum field theory, in which everything of substance is considered to be a quantum field.
I think you may mean a “scalar field,” which means a field that can be described by a single number at each point in space. Scalar fields have the simplest behavior; neither the electromagnetic nor gravitational fields are scalar, for instance. Scalar fields are in some sense the generator of massive particles without spin, or perhaps the other way around.
The equations of general relativity are such that it’s possible for a substance or field to have a negative pressure, despite having a conventional positive energy density. To the best of my knowledge this means the field “pulls” instead of “pushes” on ordinary mass, but its effect on space is different, and leads somehow to inflation. Some rough information is on Wikipedia here. I don’t have a good explanation for this; I still can’t figure out why the expansion of space doesn’t violate conservation of energy if space has an inherent nonzero energy density.
Pretty technical, but at least you can be sure that Guth gets his inflationary terminology right:
Was Cosmic Inflation the ‘Bang’ of the Big Bang?
There are some questions that even the great minds here at the Dope can’t answer in a paragraph or two, because so much context and preparedness is required to get to the answer.
This is one of them.
I recommend reading a book. Specifically this one. The Fabric of the Cosmos: Space, Time, and the Texture of Reality, by Brian Greene. Greene leads you through all the precursors slowly so that by the time he get to the mind-blowing stuff, your brain holds together long enough to get to the end.