Probing the inflaton: Small-scale power spectrum constraints from measurements of the cosmic microwave background energy spectrum

In the early universe, energy stored in small-scale density perturbations is quickly dissipated by Silk damping, a process that inevitably generates μ- 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 ≲ 10⁴ Mpc^-1. Here, we study constraints on the primordial power spectrum derived from COBE/FIRAS and forecasted for PIXIE. We show that measurements of μ 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 with PIXIE.