Probing gravity at sub-femtometer scales through the pressure distribution inside the proton

Recently, a measurement of the pressure distribution experienced by the quarks inside the proton has found a strong repulsive (positive) pressure at distances up to 0.6 femtometers from its center and a (negative) confining pressure at larger distances. In this paper we show that this measurement puts significant constraints on modified theories of gravity in which the strength of the gravitational interaction on microscopic scales is enhanced with respect to general relativity. We consider the particular case of Eddington-inspired Born-Infeld gravity, showing that strong limits on $\kappa$, the only additional parameter of the theory with respect to general relativity, may be derived from the quark pressure measurement ($|\kappa| \lsim 10^{-1} \, {\rm m^5 \, kg^{-1} \, s^{-2}}$). Furthermore, we show how these limits may be significantly improved with precise measurements of the first and second moments of the pressure distribution inside the proton.