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We study the stochastic gravitational waves from string gas cosmology. Withthe help of the Lambert W function, we derive the exact energy density spectrumof the stochastic gravitational waves in term of tensor-to-scalar. New featherswith the spectrum are found. First, the non-Hagedorn phase can be ruled out bythe current B-mode polarization in the cosmic microwave background. Second, theexact spectrum from the Hagedorn phase with a logarithmic term is shown to beunique in the measurable frequency range. Third, which is the most important,we find the string length can be constrained to be lower than 7 $\sim$ ordersof that Planck scale.

String length constraining from the stochastic gravitational waves background