  Chain inflation is an alternative to slow-roll inflation in which the
inflaton tunnels along a large number of consecutive minima in its potential.
In this work we perform the first comprehensive calculation of the
gravitational wave spectrum of chain inflation. In contrast to slow-roll
inflation the latter does not stem from quantum fluctuations of the
gravitational field during inflation, but rather from the bubble collisions
during the first-order phase transitions associated with vacuum tunneling. Our
calculation is performed within an effective theory of chain inflation which
builds on an expansion of the tunneling rate capturing most of the available
model space. The effective theory can be seen as chain inflation's analogue of
the slow-roll expansion in rolling models of inflation. We show that chain
inflation produces a very characteristic double-peak spectrum: a faint
high-frequency peak associated with the gravitational radiation emitted during
inflation, and a strong low-frequency peak associated with the graceful exit
from chain inflation (marking the transition to the radiation-dominated epoch).
There exist very exciting prospects to test the gravitational wave signal from
chain inflation at the aLIGO-aVIRGO-KAGRA network, at LISA and /or at pulsar
timing array experiments. A particularly intriguing possibility we point out is
that chain inflation could be the source of the stochastic gravitational wave
background recently detected by NANOGrav, PPTA, EPTA and CPTA. We also show
that the gravitational wave signal of chain inflation is often accompanied by
running/ higher running of the scalar spectral index to be tested at future
Cosmic Microwave Background experiments.
