Enhanced Stochastic Gravitational Waves signals from Wess-Zumino chiral superfield

Abstract: In this work, we investigate the possibility that supersymmetric structures may leave observable imprints in the stochastic gravitational-wave (GW) background generated during the reheating era. To this end, we construct a phenomenological interaction vertex describing the coupling between a single inflaton and the D-term sectors of a pair of chiral and anti-chiral superfields. In contrast to the conventional Yukawa coupling between the inflaton and structureless matter fields, we find that the supersymmetry-preserving chiral multiplet structure leads to a substantial enhancement, by at least one order of magnitude, in the amplitude of the resulting GWs spectrum. Our results therefore suggest that the interplay between reheating-era stochastic GWs and supersymmetric phenomenology merits further exploration and development.

• The paper discovers that using a supersymmetric D-term coupling between the inflaton and a Wess-Zumino chiral superfield increases the stochastic gravitational wave amplitude during reheating by about an order of magnitude (compared to Yukawa couplings).
• The enhancement in gravitational wave signal strength is attributed to derivative terms introduced by the supersymmetry preserving D-term coupling, reaching amplitudes within the sensitivity range of future detectors such as BBO, DECIGO, and Einstein Telescope.
• The cubic derivative interactions from the D-term lead to a more sensitive gravitational wave spectrum with a one order of magnitude gain, suggesting graivitational bremsstrahlung during reheating as a potential cosmological probe of supersymmetric dynamics ensuring benchmark consistency.

Motivation and model construction

The paper examines whether supersymmetric field structure can leave a measurable imprint on the stochastic gravitational-wave (GW) background produced by graviton bremsstrahlung during reheating. The starting point is the observation, drawn from earlier work on inflaton decay into pairs of particles of different spin (Barman et al., 2023), that the GW amplitude depends only weakly on the spin of the final-state pair — differences remain below an order of magnitude. The authors ask whether organizing the final states into chiral supermultiplets changes this near-universality.

The model couples the inflaton $\varphi$ to the D-term sector of a Wess–Zumino chiral superfield $\Phi$ via

$$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$

with $M_P \simeq 2.44\times 10^{18}$ GeV. After component expansion, this yields two cubic interactions: a derivative coupling of the inflaton to two Majorana fermions, and a coupling containing kinetic and mass terms for a complex scalar pair. Crucially, unlike the standard Yukawa interaction $y_\phi\,\varphi\phi_i\phi_i$, these vertices contain momentum-dependent (derivative) structures inherited from the supersymmetric D-term construction. The appendices provide a self-contained derivation of the free Majorana propagators in two-component notation, all cubic Feynman rules via path-integral methods, and the graviton–Majorana–Majorana vertex derived from the four-component minimal-gravity interaction decomposed into two-component spinors.

Two- and three-body decay rates

The two-body widths are

$$\Gamma_{\varphi\to\psi\psi}^{(0)} = \frac{\kappa^2 \mathsf{y}_{\text{D}}^2 M^3}{8\pi} y^2(1-4y^2)^{3/2}, \qquad \Gamma_{\varphi\to\phi\phi^\star}^{(0)} = \frac{\kappa^2 \mathsf{y}_{\text{D}}^2 M^3}{64\pi}\left(4y^2-\tfrac12\right)^2(1-4y^2)^{1/2},$$

where $y = m/M$. Both expressions are derived perturbatively for massive final states and become ill-defined as $m\to 0$; consequently the analysis excludes light final states, and the case $y=0.01$ is explicitly dropped from the GW phenomenology despite being kinematically allowed.

For the three-body decays $\varphi \to \psi\psi h$ and $\Phi$0, the full set of diagrams includes graviton emission from both the inflaton line and the matter lines. A notable technical result is that the cross terms between the inflaton-emission and fermion-emission contributions vanish after helicity and polarization sums, so the squared amplitude splits cleanly. The closed-form differential widths $\Phi$1 are given analytically in terms of $\Phi$2 and $\Phi$3, involving logarithms $\Phi$4 with $\Phi$5. The scalar-channel result is independent of $\Phi$6 at leading structure, whereas the Majorana channel retains non-trivial $\Phi$7-dependence throughout.

Gravitational-wave spectrum and enhancement

The spectrum follows the standard graviton-bremsstrahlung treatment: Boltzmann equations for $\Phi$8, $\Phi$9, and $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$0 are solved in the regime $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$1, with the soft divergence regularized by an infrared cutoff $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$2 and kinematic upper bound $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$3. The present-day spectrum takes the form

$$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$4

with the frequency range bounded by $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$5 Hz, using the BICEP/Keck bound on $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$6.

The central numerical result concerns the coefficient $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$7: the scalar channel from the D-term interaction gives $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$8, compared with $$S_{\text{int}} = \mathsf{y}_{\text{D}}\,\kappa \int d^8 z\; \varphi\,[\bar{\Phi}\Phi]_D, \qquad \kappa = \sqrt{16\pi}/M_P,$$9 and $M_P \simeq 2.44\times 10^{18}$0 for the conventional Yukawa channels studied previously — an enhancement of roughly one order of magnitude. For the Majorana channel no single fixed coefficient exists because of the $M_P \simeq 2.44\times 10^{18}$1-dependence; at $M_P \simeq 2.44\times 10^{18}$2 the value approaches the scalar result. For benchmarks $M_P \simeq 2.44\times 10^{18}$3 GeV$M_P \simeq 2.44\times 10^{18}$4 and $M_P \simeq 2.44\times 10^{18}$5 GeV$M_P \simeq 2.44\times 10^{18}$6, peak amplitudes reach $M_P \simeq 2.44\times 10^{18}$7 and $M_P \simeq 2.44\times 10^{18}$8 respectively, within projected sensitivity of BBO, DECIGO, Cosmic Explorer, Einstein Telescope, resonant cavity detectors, and above the BBN bound $M_P \simeq 2.44\times 10^{18}$9 is respected. The authors attribute the enhancement primarily to the derivative terms generated by the D-term component expansion, which motivates studying higher-derivative superfield couplings such as $y_\phi\,\varphi\phi_i\phi_i$0 or vector-inflaton analogues.

Limitations and open questions

The framework rests on several assumptions stated by the authors themselves. The interaction is a phenomenological toy construction — a single Wess–Zumino chiral superfield with $y_\phi\,\varphi\phi_i\phi_i$1 and only quadratic D-term contributions retained — not embedded in a UV-complete supersymmetric model; whether realistic Kähler potentials, gauge multiplets, or multiple superfields preserve the order-of-magnitude gain is left open. All results are tree-level. The IR cutoff $y_\phi\,\varphi\phi_i\phi_i$2 is introduced ad hoc to regulate the soft-graviton divergence, and the sensitivity of the spectrum to this choice is not quantified. The analytic decay-width expressions fail in the massless limit, restricting the viable parameter space to moderately heavy final states. Finally, the component-expansion approach generates a large number of diagrams even in this minimal setup; the authors note that extending to realistic models would require the supergraph formalism in superspace, which they defer to future work.

Conclusion

The paper demonstrates that replacing a structureless Yukawa coupling with a SUSY-preserving D-term coupling between the inflaton and a chiral supermultiplet enhances the reheating-era stochastic GW amplitude by roughly one order of magnitude, driven largely by derivative interactions in the component expansion. Within the tested benchmarks the signal lies in the reach of proposed high-frequency detectors, suggesting that graviton bremsstrahlung during reheating could serve as an indirect probe of supersymmetric dynamics. The robustness of this conclusion beyond the minimal Wess–Zumino toy model, and the quantitative role of the IR regularization, remain open questions.