Analytic dependence of the tachyonic GW spectral peak on the trilinear coupling q3

Derive an analytical relation for the dependence of the comoving wavenumber at the gravitational-wave spectral peak generated by tachyonic resonance during preheating in the α-attractor model with trilinear interaction h φ χ^2 and χ self-interaction λχ χ^4 on the dimensionless trilinear resonance parameter q3 ≡ h φ*/ω*^2; specifically, determine k̃_peak(q3) from the field dynamics rather than empirical lattice fits.

Background

The paper studies preheating and gravitational-wave production in an α-attractor inflationary model with a trilinear coupling h φ χ^2 between the inflaton φ and a light scalar χ, supplemented by a χ^4 self-interaction to bound the potential. In dimensionless variables, the dynamics are governed by parameters including q3 ≡ h φ/ω^2 and qχ, and preheating proceeds through a combination of parametric resonance in φ and tachyonic bursts in χ.

The resulting GW spectrum exhibits two peaks: a dominant low-frequency peak from the parametric channel and a subdominant high-frequency peak from the tachyonic channel. Lattice simulations show that the tachyonic GW peak shifts to higher k̃ as q3 increases, and an empirical fit yields k̃_peak ∝ q3^0.25. However, an analytic derivation of the functional dependence of the tachyonic GW peak on q3 is not provided, motivating the explicit open problem.