Every Wrinkle Carries A Memory: An Integro-differential Bootstrap for Features in Cosmological Correlators

TL;DR

The work develops a boundary-based integro-differential bootstrap to study scale-breaking cosmological correlators arising from heavy fields with time-dependent masses during inflation. By enforcing locality, analyticity, and microcausality, it derives and solves IDEs with a built-in memory kernel, enabling analytical results at leading order in mass-oscillation amplitude and revealing nonperturbative infrared resonances. Specializing to monochromatic masses, it extracts squeezed-limit cosmological collider signals that can be exponentially enhanced by UV and IR resonances, and provides a first numerical bootstrap mapping the solution space of these boundary equations. The framework connects to EFT of inflation, yielding perturbative and non-perturbative predictions for the power spectrum and bispectrum, including distinctive resonant oscillations in the squeezed limit that could be probed by future CMB and LSS data. Overall, the paper demonstrates that scale-breaking features during inflation imprint rich, calculable nonlocal structures in boundary correlators, motivating further analytical and numerical explorations of non-scale-invariant cosmological bootstrap.

Abstract

Motivated by cosmological observations, we push the cosmological bootstrap program beyond the de Sitter invariance lamppost by considering correlators that explicitly break scale invariance, thereby exhibiting primordial features. For exchange processes involving heavy fields with time-dependent masses and sound speeds, we demonstrate that locality in the bulk implies a set of integro-differential equations for correlators on the boundary. These scale-breaking boundary equations come with a built-in memory kernel in momentum-kinematic space encapsulating the universe's evolution during inflation. Specialising to heavy fields with sinusoidal masses such as those found in axion monodromy scenarios, we show that a powerful synthesis of microcausality and analyticity allows an analytical solution of these equations at leading order in the amplitude of mass oscillations. Meanwhile, we also unveil non-perturbative information in the integro-differential equations by resumming apparent infrared divergences as parametric resonances. In addition, we provide a first-of-its-kind example of numerical bootstrap that directly maps out the solution space of such boundary equations. Finally, we compute the bispectrum and uncover, in the squeezed limit, a scale-breaking cosmological collider signal, whose amplitude can be exponentially enhanced (with respect to the Boltzmann suppression) due to particle production triggered by high-frequency mass oscillations.

Figures (13)

• Figure 1: Right: The single-exchange diagram of our interest, characterised by four external conformally coupled fields $\varphi$ and a heavy intermediate scalar $\sigma$, endowed with a time-dependent mass $m(t)$. For constant masses, the diagram satisfies the ordinary differential equation \ref{['bootstrapODE']}. Left: For time-dependent masses, the integro-differential equation \ref{['IDEschematic']} takes over, relating the exchange diagram in one momentum configuration to its values at more squeezed configurations with $u>u'$.
• Figure 2: The seed four-point function vs. the single-exchange diagrams for the bispectrum and the power spectrum of $\zeta$, related by the action of weight-shifting operators, see Section \ref{['sec: bispectrum']}
• Figure 3: Four-point and three-point function with multiple mass insertions.
• Figure 4: The plots depict two components of the Bogolyubov coefficient $\Delta\beta$, defined by \ref{['homocorrect']}--\ref{['bogoldeco']}, as functions of $\mu$. See the discussions around \ref{['alphabetalink']}. We observe that the negative frequency component $\Delta\beta_-$ (left) is enhanced within the mass window $H\ll \mu\lesssim \omega$, due to mass oscillations, while the positive frequency component $\Delta\beta_+$ (right) exponentially decays for large masses $\mu\gtrsim 1$, regardless of the frequency $\omega$. Moreover, $\Delta\beta_-$ exhibits a singularity at $\mu=\omega/2$, which is a precursor to a non-perturbative resonance effect in the infrared. See Section \ref{['IRresonanceSubSect']}.
• Figure 5: The exchange diagram corresponding to $F_{++}(k_{12},k_3,k_3)$ at linear order in $g^2$.