Probing the inflaton: Small-scale power spectrum constraints from measurements of the CMB energy spectrum

TL;DR

The paper investigates how CMB spectral distortions from Silk-damped small-scale perturbations constrain the primordial power spectrum up to $k\sim 10^4\,\mathrm{Mpc}^{-1}$. It develops $k$-space distortion window-functions and a tight-coupling heating formalism to compute mu and y distortions for arbitrary $P_\zeta(k)$, and compares current COBE/FIRAS bounds with PIXIE forecasts. By applying the framework to steps, bends, particle production, running-mass, and small-field GW-generating models, it shows that PIXIE could exclude or tightly constrain many inflationary scenarios, sometimes ruling them out in the absence of distortions. The method provides a robust, largely model-independent constraint on small-scale power that complements PBH/UCMH limits, while highlighting caveats from other energy-release processes and recombination physics.

Abstract

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.

Figures (10)

• Figure 1: Effective heating rate for the standard power spectrum, $P^{\rm st}_\zeta(k)$, with one sharp feature at $k_\delta$. For illustration we chose $(n_{\rm S},n_{\rm run})=(0.96,0)$. Furthermore, we set $A^\delta_\zeta=2.4 \times 10^{-9}$ in both shown cases.
• Figure 2: The bounds on $A_\zeta \equiv {\mathcal{P}}_\zeta$ derived from COBE/FIRAS with the same assumptions used to derive corresponding limits from PBHs and UCMHs.
• Figure 3: Effective heating rate for the standard power spectrum, $P^{\rm st}_\zeta(k)$, with a step at $k_{\rm s}$. For illustration we chose $(n_{\rm S},n_{\rm run})=(0.96,0)$. Furthermore, we set $A^{\rm s}_\zeta=3.5 \times 10^{-9}$ in both shown cases.
• Figure 4: Limits on the total amplitude of one step in the small-scale power spectrum occurring at $k_{\rm s}$. The energy release of modes with $k<k_{\rm s}$ was included, however, this only matters for the case of PIXIE. For illustration, the power law index was also varied.
• Figure 5: Effective heating rate for the standard power spectrum with a kink at $k_{\rm b}$. For illustration we chose $(n_{\rm S},n_{\rm run})=(0.96,0)$. Furthermore, we set $n_{\rm S}^\ast=1.2$ in both examples.