Decoupling of large-scale, adiabatic inflationary perturbations from enhanced small-scale modes at one-loop

TL;DR

This work analyzes the back-reaction of amplified short-scale inflationary perturbations on a long-wavelength adiabatic mode using the δN formalism within the separate-universe framework. It shows that, at 1-loop, back-reaction arises from either non-linear δN evolution or from initial-condition corrections, and that for a broad band of enhanced short modes the loop corrections decouple from the detailed peak properties and become scale-invariant. By recasting the 1-loop contributions as total derivatives and boundary terms—under adiabaticity and soft-theorem constraints—the authors argue that the net effect on the large-scale power spectrum is largely unobservable in typical single-field scenarios, though the precise structure is model-dependent and can involve renormalized δN coefficients via multi-point propagators. The results clarify the interplay between long- and short-scale physics in inflationary loops, connect separate-universe methods with in-in formalisms in a controlled regime, and set practical boundaries for when back-reaction constraints on enhanced small-scale power (e.g., PBH-related scenarios) are meaningful.

Abstract

We reconsider back-reaction from large amplitude, short-scale perturbations onto a long wavelength adiabatic mode. In a loop expansion of the long-mode power spectrum, this back-reaction appears first at 1-loop. Due to the separation between the long and short scales, the separate universe method provides a simple and efficient framework for this computation. In this paper, building on our earlier work, we employ a $δN$ formula for the long mode, which captures the effect of short scales. We show that back-reaction at 1-loop is due to either (i) non-linearity of the $δN$ formula, or (ii) 1-loop corrections to the initial conditions. We argue that contributions of type (ii) cannot themselves be described within the separate universe framework, but their properties can be constrained using soft theorems and a ''multi-point propagator'' expansion. When applied to a band of enhanced short-scale perturbations that crossed the horizon during inflation, our result shows that the loop correction decouples from their detailed properties. Furthermore, the back-reaction we obtain is scale-invariant. Its magnitude is model-dependent, but is degenerate with effects from modes that were still sub-horizon at the end of inflation. In this scenario (but not necessarily in all scenarios), we conclude that the effect is not observable.

Figures (5)

• Figure 1: Time-evolution of $\epsilon_1$ and $\epsilon_2$ for two models leading to amplified fluctuations on small scales; see main text for details. In both panels the horizontal axis is $\Delta N \equiv N-N_\text{end}$, where $N_\text{end}$ labels the end of inflation. We highlight times during the non-attractor phase $(\epsilon_2<-3)$ in orange, and the following dual attractor phase in green.
• Figure 2: Tree-level scalar power spectrum ${\mathcal{P}_\zeta(k)}_\text{tree}$, for the models of Fig. \ref{['fig: background examples']}; see main text for details. In both panels, the vertical dotted line identifies the long adiabatic mode $p$, for which we wish to estimate the back-reaction. Modes highlighted in blue cross the horizon during the non-attractor CR phase, defined by the condition $\epsilon_2<-3$, as in Fig. \ref{['fig: background examples']}. We label the first and last scale of the blue band as $k_s$ and $k_e$, respectively. We highlight the broad peak in red.
• Figure 3: Schematic illustration of tree-level power spectra displaying a peak on short scales. These are not computed from realistic inflationary models (such as those in Fig. \ref{['fig:integral boundaries']}), but rather are toy examples. The vertical, black arrow indicates the (model-dependent) quantity $\mathcal{P}_\zeta(q_{\text{max}};t)_\text{tree} - \mathcal{P}_\zeta(q_{\text{min}};t)_\text{tree}$. In each panel, a reference spectrum is plotted in black. The red lines represent examples of other spectra with the same infrared and ultraviolet plateaus as the reference one, but with different amplitude of the peak.
• Figure 4: Evolution of the Mukhanov--Sasaki mass, $\nu^2$, and slow-roll parameter $\epsilon_1$, computed for the toy USR model analysed in Ref. Iacconi:2023ggt. On the horizontal axis we display the number of e-folds to the end of inflation, $\Delta N \equiv N_\text{end}-N$. The two vertical dashed lines mark the two choices of initialization time for the separate universe computations, $\eta_i=-1/k_\text{peak}$ (brown) and $\eta_i = -1/k_e$ (blue). The yellow region highlights USR evolution, defined by the condition $\epsilon_2<-3$.
• Figure 5: Linear power spectrum, $\mathcal{P}_\zeta(k; t)_\text{tree}$, computed by applying the separate universe approach (pink, dashed line), initialized at $\eta_i = -1/k_\text{peak}$ (left) and $\eta_i = -1/k_e$ (right). In both panels, the black line represents results obtained by numerically solving the Mukhanov--Sasaki equation.