Researchers derived the first fully analytic expressions for a class of primordial three-point signals (bispectra) that keep the quadratic mixing between curvature fluctuations and a second, massive field fully non-perturbative. In simpler terms: they found an exact way to include the direct coupling between the main density perturbation (the curvature mode) and an extra scalar field (an isocurvature mode) when computing a particular signature of early-universe interactions. Previous analytic work treated that mixing as a small effect; this paper lifts that restriction.

The authors work inside the effective field theory of inflation, a framework that organizes possible fluctuations and interactions by their importance. They build on an operator representation of the exact linear solutions and convert it into a single, simpler integral. Any scale-invariant tree-level one-vertex bispectrum reduces to a one-dimensional Schwinger-parameter integral over a small set of pre-computable “leg” kernels. This representation disentangles the dressed frequencies coming from mixing, so once the kernels are tabulated the three-point function for any triangle of momenta can be evaluated quickly.

They illustrate the method using the cubic, time-derivative interaction written as ˙π3 (here π is the Goldstone mode related to the curvature perturbation). When the dimensionless mixing strength λ (defined as ρ/H, where ρ parametrizes the quadratic mixing and H is the Hubble scale) is small, the resulting bispectrum matches the familiar single-field “equilateral” shape. But as λ becomes order one or larger, the shape shifts away from the equilateral template and becomes genuinely multifield, with a large amplitude. The authors show this change is substantial enough that it could motivate dedicated searches in data.

They also obtain the squeezed limit (one momentum much smaller than the others) in closed form for any mixing strength. In that limit the signal displays the so-called cosmological collider oscillations. The frequency of those oscillations is controlled by an effective parameter ν_eff, which the authors write as ν_eff = i μ_eff with μ_eff = sqrt(9/4 − m^2/H^2 − λ^2). In plain terms, the effective mass that sets the collider signal is “dressed” by the mixing λ. For a non-tachyonic bare mass (m2 ≥ 0), sufficiently strong mixing (for example λ ≳ 3/2) makes the dressed field heavy enough that oscillations are unavoidable.