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In the causal set approach to quantum gravity, Poincar\'{e} symmetry ismodified by swerving in spacetime, induced by the random lattice discretizationof the space-time structure. The broken translational symmetry at shortdistances is argued to lead to a residual diffusion in momentum space, wherebya particle can acquire energy and momentum by drift along its mass shell and asystem in equilibrium can spontaneously heat up. We consider bounds on the rateof momentum space diffusion coming from astrophysical molecular clouds, nuclearstability and cosmological neutrino background. We find that the strongestlimits come from relic neutrinos, which we estimate to constrain the momentumspace diffusion constant by $k < 10^{-61} {\rm GeV}^3$ for neutrinos withmasses $m_\nu > 0.01 {\rm eV}$, improving the previously quoted bounds byroughly 17 orders of magnitude.Comment: Additional discussion about behavior of alpha particles in nuclei  added. Version matches that accepted in PR

Low energy bounds on Poincaré violation in causal set theory