  In the early Universe, energy stored in small-scale density perturbations is
quickly dissipated by Silk-damping, a process that inevitably generates mu- and
y-type spectral distortions of the cosmic microwave background (CMB). These
spectral distortions depend on the shape and amplitude of the primordial power
spectrum at wavenumbers k < 10^4 Mpc^{-1}. Here we study constraints on the
primordial power spectrum derived from COBE/FIRAS and forecasted for PIXIE. We
show that measurements of mu and y impose strong bounds on the integrated
small-scale power, and we demonstrate how to compute these constraints using
k-space window functions that account for the effects of thermalization and
dissipation physics. We show that COBE/FIRAS places a robust upper limit on the
amplitude of the small-scale power spectrum. This limit is about three orders
of magnitude stronger than the one derived from primordial black holes in the
same scale range. Furthermore, this limit could be improved by another three
orders of magnitude with PIXIE, potentially opening up a new window to early
Universe physics. To illustrate the power of these constraints, we consider
several generic models for the small-scale power spectrum predicted by
different inflation scenarios, including running-mass inflation models and
inflation scenarios with episodes of particle production. PIXIE could place
very tight constraints on these scenarios, potentially even ruling out
running-mass inflation models if no distortion is detected. We also show that
inflation models with sub-Planckian field excursion that generate detectable
tensor perturbations should simultaneously produce a large CMB spectral
distortion, a link that could potentially be established by PIXIE.
