We prove a quantitative version of the non-uniform hyperbolicity of the Teichmuller geodesic flow. Namely, at each point of any Teichmuller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the flow line in terms of an easily estimated geometric quantity on the flat surface, which is greater than or equal to the flat systole. As applications, we strengthen results of Trevi~no and Smith regarding unique ergodicity of measured foliations, and give an estimate for the spectral gaps of pseudo-Anosov homeomorphisms based on the location of their axes in the moduli space of quadratic differentials.
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