The paper discusses the production of gravitational waves during the reheating phase following inflation. The authors compute the spectrum of gravitational waves from the inflaton condensate, focusing on the Starobinsky model and T-models, which are phenomenologically consistent with CMB anisotropy measurements. They show that the shape of the gravitational wave spectrum can distinguish between these models. For the Starobinsky model and T-models with \( k = 2 \), the spectra differ significantly compared to models with \( k = 4 \) or \( k > 4 \). The observation of multiple harmonics in the spectrum can provide a direct measurement of the inflaton mass, and the cutoff frequency can be used to determine the reheating temperature. The paper also explores the implications for future gravitational wave detectors and the cosmic axion background.The paper discusses the production of gravitational waves during the reheating phase following inflation. The authors compute the spectrum of gravitational waves from the inflaton condensate, focusing on the Starobinsky model and T-models, which are phenomenologically consistent with CMB anisotropy measurements. They show that the shape of the gravitational wave spectrum can distinguish between these models. For the Starobinsky model and T-models with \( k = 2 \), the spectra differ significantly compared to models with \( k = 4 \) or \( k > 4 \). The observation of multiple harmonics in the spectrum can provide a direct measurement of the inflaton mass, and the cutoff frequency can be used to determine the reheating temperature. The paper also explores the implications for future gravitational wave detectors and the cosmic axion background.