Prospects for Primordial Gravitational Waves in String Inflation

Assuming that the early universe had (i) a description using perturbative string theory and its field theory limit (ii) an epoch of slow-roll inflation within a four-dimensional effective field theory and a hierarchy of scales $M_{inf} < m_{kk} < m_s \lesssim M_{pl}$ that keeps the latter under control, we derive an upper bound on the amplitude of primordial gravitational waves. The bound is very sensitive to mild changes in numerical coefficients and the expansion parameters. For example, allowing couplings and mass-squared hierarchies $\lesssim 0.2$, implies $r \lesssim 0.05$, but asking more safely for hierarchies $\lesssim 0.1$, the bound becomes $r \lesssim 10^{-6}$. Moreover, large volumes -- typically used in string models to keep backreaction and moduli stabilisation under control -- drive $r$ down. Consequently, any detection of inflationary gravitational waves would present an interesting but difficult challenge for string theory.