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And to Demosthenes Kazanas’ 1980 paper Dynamics of the universe and spontaneous symmetry breaking.
There’s more on Philip Cole’s blog entry the early stages of cosmic inflation where you can also see John Peacock saying the first inflation paper was by Erast Gliner in 1965.
Whilst only four pages long it’s rather technical and it doesn’t use the word inflation.
So it’s not surprising that it didn’t get much attention.
It’s by Stuart Clark, who says Alexei Starobinsky published his theory of inflation in December 1979.
Starobinsky’s paper was Spectrum of relict gravitational radiation and the early state of the universe.
He had bubbles forming in a supercooled environment which is the opposite of boiling, he was treating space like a liquid instead of a ghostly elastic solid, and he had nothing to curtail the exponential expansion. It talks about false vacuum decaying into empty bubbles of true vacuum expanding at the speed of light, but in a universe expanding faster than light.He gave an overview of the horizon and flatness problems and talked about the standard model of cosmology, saying the universe is assumed to be homogeneous and isotropic and described by the Robertson-Walker metric.He also said the evolution of the scale factor R(t) is governed by the Einstein equations, he referred to conservation of energy, and he said He said the above equations must be supplemented by an equation of state for matter which can be approximated to an ideal quantum gas of massless particles.The theory of cosmic or cosmological inflation is related to Big Bang theory.It’s been around now for the thick end of forty years. By 1980 a lot of cosmologists were happy that the universe was expanding, and that it wasn’t a steady-state universe.
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See the 2005 Symmetry magazine article the growth of inflation by Davide Castelvecchi. Hence the bubbles couldn’t coalesce to reheat the universe.