Inflationary Bispectra and IR Physics from Quantum Simulators

Abstract: Bose-Einstein condensates have been used successfully as quantum simulators for early-universe particle production. Here we extend this analogy to the interacting theory. To this end, we compute the leading interaction terms expected in a general experimental setup that determine the 3-point correlations. We apply them to cosmologically motivated expanding model universes, but also periodically oscillating ones. While expanding scale factors do not necessarily lead to easily observable signatures due to experimental constraints on the realizable scale factor, periodically driven ones produce resonant correlators that grow linearly over time and lend themselves well to experimental detection. One direct application of this analysis is the experimental study of late-time infrared divergencies often discussed in the context of massless scalars in de Sitter space.

• The paper derives an exact interacting phase-fluctuation action for a two-dimensional BEC, whose derivative vertices reproduce key operators from the effective field theory of inflation and predict an experimentally relevant coupling of approximately 0.133.
• The paper finds secularly growing density and phase bispectra across exponential and polynomial scale factors, with density signals reaching amplitudes of roughly 10⁻⁴–10⁻² while highlighting important observable and effective-theory limitations.
• The paper shows that periodic driving can produce bispectra that grow linearly through interaction resonances while suppressing particle-production resonances, offering a laboratory route to probing infrared physics and primordial non-Gaussianity.

Overview

This paper extends the program of Bose–Einstein condensate (BEC) quantum simulators for QFT in curved spacetime beyond the free theory, deriving and analyzing the leading self-interactions of the analog scalar field in an effective FLRW background (2608.19327). The central results are threefold: the interacting action of a 2D BEC simulator is computed exactly in terms of phase fluctuations; the resulting momentum-space bispectra of density and phase observables are evaluated numerically for expanding scale factors, exhibiting secular growth characteristic of perturbative IR divergences in de Sitter-like backgrounds; and periodically driven scale factors are shown to produce linearly growing, resonant bispectra under conditions where resonant particle production can be suppressed.

The theoretical framework models the cold-atom gas as a relativistic complex scalar $\Phi$ with charge $q$ coupled to an external gauge field, with a Feshbach-tunable quartic coupling $\lambda(t,\vec x)$. Expanding around a classical condensate via a radial split $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$ — a parametrization distinct from earlier work on these simulators — the authors obtain the Madelung/Euler equations for the background and, after integrating out the density fluctuation $\delta\rho = \rho - \rho_0$, arrive at the scale-free action

$$S = \hbar \int dt\, d^2x \left[ \frac{a^2(t)}{2}\left(\dot\phi + \frac{\kappa}{2}(\vec\nabla\phi)^2\right)^2 - (\vec\nabla\phi)^2 \right],$$

where $\phi$ is the canonically normalized phase fluctuation, $a^2(t) \propto 1/\lambda(t)$ plays the role of the spatial scale factor, and $\kappa$ is a single dimensionless interaction parameter. Using experimental parameters for potassium-39 (scattering length $a_s^{\mathrm i} = 50\,a_\mathrm B$), they estimate $q$0: small enough to justify perturbation theory, yet large enough that its effects should be experimentally resolvable.

Relation to the EFT of inflation

A notable structural result is that the two leading interaction vertices of the simulator,

$q$1

carry only shift-symmetric derivative interactions with the correct $q$2 prefactor, matching the operator structure obtained from the EFT of inflation after the Stückelberg replacement in $q$3 spacetime dimensions (2608.19327). This means the BEC does not merely simulate free particle production but realizes a subset of the phenomenologically relevant interactions governing primordial non-Gaussianity, opening the possibility of laboratory rehearsal of bispectrum analysis pipelines relevant to cosmic microwave background and large-scale-structure data.

Observables

The authors identify two complementary measurement channels. Direct imaging of the density contrast yields correlators of $q$4: a careful Heisenberg-versus-interaction-picture analysis shows that the composite-operator contribution from $q$5 inside $q$6 cancels against the commutator term arising from time derivatives of the evolution operator, so that $q$7 exactly. Homodyne techniques, which access the phase directly, instead measure correlators of $q$8 itself. All correlators are computed within the Schwinger-Keldysh formalism using Wightman propagators built from glued mode functions across a three-phase experimental protocol (constant–driven–constant), with vacuum initial conditions imposed via an $q$9 prescription; finite-temperature generalizations are given but not pursued.

Late-time behavior and IR divergences

For exponential ($\lambda(t,\vec x)$0) and polynomial ($\lambda(t,\vec x)$1) scale factors, the leading late-time scaling of the bispectra was derived by inserting the asymptotic mode-function expansion into the Schwinger-Keldysh expressions. The key findings, summarized in the paper's classification table, are:

Scale factor	Observable	Leading late-time growth
Exponential (de Sitter)	$\lambda(t,\vec x)$2	$\lambda(t,\vec x)$3
Exponential (de Sitter)	$\lambda(t,\vec x)$4	$\lambda(t,\vec x)$5
Accelerating polynomial, $\lambda(t,\vec x)$6	$\lambda(t,\vec x)$7	up to $\lambda(t,\vec x)$8 ($\lambda(t,\vec x)$9 vertex); $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$0 (derivative vertex)
Decelerating polynomial, $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$1	$\Phi = e^{i(S_0+\sigma)}(R_0 + r)$2	oscillatory power-law growth, $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$3 or $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$4

Every interaction considered produces secularly growing contributions for some combination of scale factor and observable, including the purely derivative $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$5 vertex at polynomial scale factors. The authors explicitly note that this bears on a recent claim in the literature that fields with only derivative interactions do not exhibit de Sitter IR divergences: their analysis exhibits such divergences, albeit for polynomial rather than exponential expansion, providing a concrete counterpoint and, more importantly, an experimental avenue to probe the debated IR structure of massless minimally coupled scalars. The numerical evaluation (via adaptive Monte Carlo integration over realistic momentum windows) reproduces these scalings qualitatively, with density-bispectrum amplitudes of order $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$6 to $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$7 in the realistic parameter range.

Two caveats attach directly to this result. First, the strongest growth in $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$8 is partly driven by the explicit $\Phi = e^{i(S_0+\sigma)}(R_0 + r)$9 prefactor in the observable, which is absent for the homodyne phase bispectrum $\delta\rho = \rho - \rho_0$0. Second — and decisive experimentally — the secular growth only dominates after the scale factor has grown by a factor of order ten, whereas the accessible range before the effective theory breaks down is roughly a factor of three.

Periodically driven scale factors

To circumvent the limited dynamical range, the paper analyzes a triangle-wave scale factor, reducing the mode equation to a Hill equation amenable to exact Floquet analysis. For long-wavelength modes no resonance occurs as long as $\delta\rho = \rho - \rho_0$1, which is always satisfied experimentally. Short-wavelength modes exhibit infinitely many resonant bands determined by the condition

$\delta\rho = \rho - \rho_0$2

within which mode functions grow exponentially, producing the peaked two-point function previously observed in driven-BEC experiments (2608.19327).

The central proposal of this section is the separation of interaction resonances from particle-production resonances. By analogy with a toy model in which the bispectrum grows linearly in time whenever $\delta\rho = \rho - \rho_0$3, while mode-function resonances require much larger momenta, one can tune parameters so that the resonant two-point function lies outside the validity window of the EFT while the three-point function remains resonant. Numerical evaluation confirms sharply defined resonant bands near kinematic boundaries $\delta\rho = \rho - \rho_0$4, with envelopes growing linearly in time, while off-resonant configurations oscillate without growth. If the required fine-tuning proves viable in practice, this would provide arbitrarily large interaction signals without ever-growing scale factors.

Limitations and open questions

The paper concedes several limitations at the points where they bind. The breakdown of the simulator EFT once correlations grow large is not treated here; prior analyses cover the two-point sector only, and a stochastic-inflation-style treatment would likely be needed for higher correlators. Boundary effects, heating from the driving magnetic fields, and the finite condensation-density window (which fixes the maximum achievable $\delta\rho = \rho - \rho_0$5 since $\delta\rho = \rho - \rho_0$6) further constrain the platform; genuinely late-time de Sitter physics may require a physically expanding BEC, whose compatibility with the condensation bound remains unclear. On the formalism side, Schwinger-Keldysh methods predict ensemble averages only and cannot describe realization-level features such as the positions of localized momentum-space peaks seen in driven experiments, motivating wave-function-based or field-level inference approaches. Finally, whether the proposed fine-tuned separation of interaction and particle-production resonances survives experimental noise is left open, as is the identification of an optimal driving waveform beyond the analytically convenient triangle wave.

Conclusion

By computing the interacting action and leading bispectra of a BEC quantum simulator for QFT in curved spacetime, this work moves the analog-gravity program past the free-theory regime and connects it to the EFT of inflation. Its most consequential results are the demonstration that simulator observables exhibit the secular growth associated with IR divergences across a broad class of scale factors, and the Floquet-based mechanism by which resonant three-point signals can be isolated from resonant particle production. Both results convert standing theoretical debates about infrared physics in cosmological spacetimes into concrete, repeatable laboratory measurements.