Production of Gravitational Waves from Preheating and Tachyonic Instabilities

We analyze GW production during preheating for an $\alpha$-attractor potential terminating in the positive-curvature regime, with energy transfer via $\phi\chi^{2}$. Linear Floquet analysis and nonlinear simulations show that $\phi$ fluctuations grow by parametric resonance, while $\chi$ undergoes tachyonic bursts. The GW spectrum features two peaks: a dominant low-frequency peak from the parametric channel and a subdominant high-frequency peak from the tachyonic channel. Redshifted to today, the peak reaches $h^{2}\Omega_{\rm GW}^{(0)} \sim 10^{-11}$ at $f^{(0)}_{p} \sim 10^{7}$ Hz. This multi-peak structure is a characteristic imprint of trilinear preheating in $\alpha$-attractors.