  Chain Inflation is an alternative to slow roll inflation in which the
universe undergoes a series of transitions between different vacua. The density
perturbations (studied in this paper) are seeded by the probabilistic nature of
tunneling rather than quantum fluctuations of the inflaton. We find the scalar
power spectrum of chain inflation and show that it is fully consistent with a
$\Lambda$CDM cosmology. In agreement with some of the previous literature (and
disagreement with others), we show that $10^4$ phase transitions per e-fold are
required in order to agree with the amplitude of Cosmic Microwave Background
anisotropies within the observed range of scales. Interestingly, the amplitude
of perturbations constrains chain inflation to a regime of highly unstable de
Sitter spaces, which may be favorable from a quantum gravity perspective since
the Swampland Conjecture on Trans-Planckian Censorship is automatically
satisfied. We provide new analytic estimates for the bounce action and the
tunneling rate in periodic potentials which replace the thin-wall approximation
in the regime of fast tunneling. Finally, we study model implications and
derive an upper limit of $\sim 10^{10}$ GeV on the axion decay constant in
viable chain inflation with axions.
