Bounds on the minimum sound speed above neutron star densities

We show that the existence of massive neutron stars and asymptotic freedom of QCD place robust upper bounds on the lowest sound speed of the ultra-dense matter unattainable in neutron stars. Centered on worst-case scenarios, our limits are the most conservative among physical equations of state in the density range $\sim 2-40 n_0$. Discovery of $\gtrsim 2.6 M_\odot$ neutron stars, in combination with current multimessenger astrophysical constraints on the equation of state, would strongly support first-order phase transitions at high baryon densities.