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The hypergeometric amplitude is a one-parameter deformation of the Veneziano amplitude for four-point tachyon scattering in bosonic string theory that is consistent with $S$-matrix bootstrap constraints. In this article we construct a similar hypergeometric generalization of the Veneziano amplitude for type-I superstring theory. We then rule out a large region of the $(r,m^2,D)$ parameter space as non-unitary, and establish another large subset of the $(r, m^2, D)$ parameter space where all partial wave coefficients are positive. We also analyze positivity in various limits and special cases. As a corollary to our analysis, we are able to directly demonstrate positivity of a wider set of Veneziano amplitude partial wave coefficients than what has been presented elsewhere., Comment: Based on a Brown University undergraduate thesis, 37 pages, 6 figures

In the canonical tachyonic resonance preheating scenario, only an order one fraction of energy density in the inflaton is transferred to radiation, due to backreaction effects. One possible way to improve the energy transfer efficiency is to allow for the perturbative decays of the resonantly produced daughter particles, which serve as the "spillway" to drain the direct decay products from inflaton and to reduce the backreaction. In this article, we study two observational consequences of spillway preheating. The first is on the inflationary observables: the scalar spectrum tilt $n_s$ and tensor-to-scalar ratio $r$. The spillway scenario modifies the evolution of the equation of state between the end of inflation and the thermal big bang. As a result, it affects the time elapsed from inflation to the Cosmic Microwave Background (CMB), as well as the fits of inflationary models and their corresponding prediction for $n_s$ and $r$. We map out the equation of state by systematically scanning the parameter space of the spillway scenario, and show that the most efficient spillway scenario predicts a bluer spectrum, compared to the tachyonic preheating scenario. Another consequence is the production of high-frequency gravitational waves (GWs). Comparing the simulation results with those of tachyonic preheating, we find that the existence of spillways leads to sharper-peaked GW spectra with a mildly damped amplitude., Comment: 28 pages, 13 figures