We used to think that there was one planet earth, inside a universe that is very large (at least 10^26 meters in diameter) but that the reachable universe (future light-cone in the terminology of special relativity, or causal future in the terminology of GR) was finite. Anything outside the reachable universe is irrelevant, since we can't affect it. However, cosmologists went on to study the process that probably created the universe, known as inflation.

The scientific idea of a spatially infinite universe, and the recognition that this would have weird implications, is independent of and long predates inflation. Spatial infinity is Tegmark's Level I, while eternal inflation is Level II. Eternal inflation gives rise to some more variation in physical laws than a 'normal' infinite universe, but not anything qualitatively new (if an infinite universe contains everything computable, it contains simulations of every possible set of computable physical laws). As for timing, well, see this comment by Mitchell Porter.

The role of eternal inflation in scientific Eternal inflation and its implications 10 thinking,

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