String length constraining from the stochastic gravitational waves background

We study the stochastic gravitational waves from string gas cosmology. With the help of the Lambert W function, we derive the exact energy density spectrum of the stochastic gravitational waves in term of tensor-to-scalar. New feathers with the spectrum are found. First, the non-Hagedorn phase can be ruled out by the current B-mode polarization in the cosmic microwave background. Second, the exact spectrum from the Hagedorn phase with a logarithmic term is shown to be unique in the measurable frequency range. Third, which is the most important, we find the string length can be constrained to be lower than 7 $\sim$ orders of that Planck scale.
Comment: 9 pages, 4 figures