  Over 30 years ago, Barrow & Tipler proposed the principle according to which
the action integrated over the entire 4-manifold describing the universe should
be finite. Here we explore the cosmological consequences of a related
criterion, namely that semi-classical transition amplitudes from the early
universe up to current field values should be well defined. On a classical
level, our criterion is weaker than the Barrow-Tipler principle, but it has the
advantage of being sensitive to quantum effects. We find significant
consequences for early universe models, in particular: eternal inflation and
strictly cyclic universes are ruled out. Within general relativity, the first
phase of evolution cannot be inflationary, and it can be ekpyrotic only if the
scalar field potential is trustworthy over an infinite field range. Quadratic
gravity eliminates all non-accelerating backgrounds near a putative big bang
(thus imposing favourable initial conditions for inflation), while the expected
infinite series of higher-curvature quantum corrections eliminates Lorentzian
big bang spacetimes altogether. The scenarios that work best with the principle
of finite amplitudes are the no-boundary proposal, which gives finite
amplitudes in all dynamical theories that we have studied, and string-inspired
loitering phases. We also comment on the relationship of our proposal to the
swampland conjectures.
