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Assuming that the early universe had (i) a description using perturbativestring theory and its field theory limit (ii) an epoch of slow-roll inflationwithin 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 theexpansion parameters. For example, allowing couplings and mass-squaredhierarchies $\lesssim 0.2$, implies $r \lesssim 0.05$, but asking more safelyfor hierarchies $\lesssim 0.1$, the bound becomes $r \lesssim 10^{-6}$.Moreover, large volumes -- typically used in string models to keep backreactionand moduli stabilisation under control -- drive $r$ down. Consequently, anydetection of inflationary gravitational waves would present an interesting butdifficult challenge for string theory.

Prospects for Primordial Gravitational Waves in String Inflation