This paper studies how our Universe could pass from a period of “higher‑dimensional inflation” to the familiar hot Big Bang without flooding space with unwanted bulk gravitons. The idea behind the larger framework is that one or two extra spatial dimensions could have expanded to roughly a micrometer during inflation. That possibility can help explain two big puzzles at once: why gravity is so weak and why the observable Universe is so large. But it also creates a technical problem: the extra dimensions come with a dense tower of Kaluza‑Klein (KK) gravitons — copies of the graviton with different masses — that can be thermally produced and, if abundant, spoil cosmology.

The authors assume the extra dimensions are stabilized at the end of inflation and then work out a detailed post‑inflation history up to the reheating temperature that avoids overproducing these bulk gravitons. They focus mainly on the case with one extra dimension (d = 1) and also comment on two (d = 2). Two key constraints motivate their scenario. First, measurements of the cosmic microwave background require an almost scale‑invariant spectrum of primordial fluctuations down to very large scales, which limits how inflation and the later expansion can behave. Second, thermal production in the primordial plasma can generate many KK gravitons; keeping their relic abundance acceptable imposes an upper bound on the reheating temperature, called the “normalcy temperature.” For d = 1 this bound is around 1 GeV, while for d = 2 it is about 4 MeV.

A separate theoretical constraint, the Higuchi bound, requires that during a de Sitter‑like inflationary phase a spin‑2 particle (like a graviton) have mass squared m2 ≥ 2H2, where H is the Hubble rate. This bound would force the Hubble scale to be extremely small for a static large extra dimension. To avoid this and other fine tunings, the authors adopt a period of higher‑dimensional inflation in which the extra dimension itself grows exponentially from a microscopic size up to the micron scale. That expansion can produce the required large non‑compact volume (many e‑folds) while remaining compatible with the observed near scale invariance of cosmic fluctuations for the d = 1 case.