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.
