  Inflation correlators with massive exchanges are central observables of
cosmological collider physics, and are also important theoretical data for us
to better understand quantum field theories in dS. However, they are difficult
to compute directly due to many technical complications of the
Schwinger-Keldysh integral. In this work, we initiate a new bootstrap program
for massive inflation correlators with dispersion relations on complex momentum
planes. We classify kinematic variables of a correlator into vertex energies
and line energies, and develop two distinct types of dispersion relations for
both of them, respectively called vertex dispersion and line dispersion
relations. These dispersion methods allow us to obtain full analytical results
of massive correlators from a knowledge of their oscillatory signals alone,
while the oscillatory signal at the tree level can be related to simpler
subgraphs via the cutting rule. We further apply this method to massive loop
correlators, and obtain new analytical expressions for loop diagrams much
simpler than existing results from spectral decomposition. In particular, we
show that the analyticity demands the existence of an "irreducible background"
in the loop correlator, which is unambiguously defined, free of UV divergence,
and independent of renormalization schemes.
