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Radio pulsars provide some of the most important constraints for ourunderstanding of matter at supranuclear densities. So far, these constraintsare mostly given by precision mass measurements of neutron stars (NS). Bycombining single measurements of the two most massive pulsars, J0348$+$0432 andJ0740$+$6620, the resulting lower limit of 1.98 $M_\odot$ (99% confidence) ofthe maximum NS mass, excludes a large number of equations of state (EOSs).Further EOS constraints, complementary to other methods, are likely to comefrom the measurement of the moment of inertia (MOI) of binary pulsars inrelativistic orbits. The Double Pulsar, PSR J0737$-$3039A/B, is the mostpromising system for the first measurement of the MOI via pulsar timing.Reviewing this method, based in particular on the first MeerKAT observations ofthe Double Pulsar, we provide well-founded projections into the future bysimulating timing observations with MeerKAT and the SKA. For the first time, weaccount for the spin-down mass loss in the analysis. Our results suggest thatan MOI measurement with 11% accuracy (68% confidence) is possible by 2030. Ifby 2030 the EOS is sufficiently well known, however, we find that the DoublePulsar will allow for a 7% test of Lense-Thirring precession, or alternativelyprovide a $\sim3\sigma$-measurement of the next-to-leading order gravitationalwave damping in GR. Finally, we demonstrate that potential new discoveries ofdouble NS systems with orbital periods shorter than that of the Double Pulsarpromise significant improvements in these measurements and the constraints onNS matter.

Constraining the dense matter equation-of-state with radio pulsars