Some inequalities bridging stringy parameters and cosmological observables

By demanding the validity of an effective field theory description during inflation, in this note we derive some peculiar inequalities among the three interesting stringy and cosmological parameters, namely the tensor-to-scalar ratio ($r$), the string coupling ($g_s$) and the compactification volume (${\cal V}$). In deriving these inequalities, we explicitly demand that the inflationary scale and the Hubble parameter during inflation are well below the Kaluza-Klein (KK) mass scale, string scale, and the four dimensional Planck mass. For the inflationary models developed within the framework of type IIB orientifold comapctification, we investigate the regions of parameters space spanned by the three parameters $(r, g_s, {\cal V})$ by satisfying our inequalities, and we find that the same can reduce the size of available parameter space quite significantly. Moreover, we comment on obtaining further constraints on the parameters by comparing gravitino mass ($m_{3/2}$) with the Hubble scale ($H$), which also provides a lower bound on tensor-to-scalar ratio ($r$), for the cases when $m_{3/2} <H$. We also illustrate the outcome of our bounds in some specific class of string(-inspired) models.