We discuss stiffening of matter in quark-hadron continuity. We introduce a model that relates quark wavefunctions in a baryon and the occupation probability of states for baryons and quarks in dense matter. In dilute regime, the confined quarks contribute to the energy density through the masses of baryons, but do not directly contribute to the pressure, hence the equations of state are very soft. This dilute regime continues until the low momentum states for quarks get saturated; this may happen even before baryons fully overlap, possibly at density slightly above the nuclear saturation density. After the saturation the pressure grows rapidly while changes in energy density are modest, producing a peak in the speed of sound. If we use baryonic descriptions for quark distributions near the Fermi surface, we reach a description similar to the quarkyonic matter model of McLerran and Reddy. With a simple adjustment of quark interactions to get the nucleon mass, our model becomes consistent with the constraints from 1.4-solar mass neutron stars, but the high density part is too soft to account for two-solar mass neutron stars. We suggest interactions that leave low density equations of state unchanged but stiffen the high density part.
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