Causality and perturbative-QCD constraints on first-order phase transitions in neutron stars

Causality bounds the compactness and maximum mass of neutron stars. We generalise this bound to equations of state (EOSs) with a first-order phase transition between maximally stiff phases. In units of the transition energy density, the problem depends only on q= / and j=Δ /. The maximum compactness equals the smooth value C=0.354 for q to0 or j to0, is unaffected for q 0.67, and is otherwise reduced by typically 0.04 when the maximum-mass star is a hybrid star and by up to 0.13 when it is the last hadronic star. Matching to a Baym--Pethick--Sutherland crust and a chiral-EFT polytrope at n m=1.1,n 0 and inverting the bound, no causal EOS supports 2.01, if the energy-density jump exceeds 460 -- 495 MeV,fm-3. Additionally, imposing radius constraint of R 11.6 km it shrinks to 235 -- 395 MeV,fm-3. These limits hold whatever the matter above the transition, for onsets within 0.2,n 0 of n m. Relaxing maximal stiffness, we build EOSs with a transition at μ tr=1000 -- 1300 MeV, matched to perturbative QCD (pQCD) at 2.6 GeV. pQCD limits the jump; EOSs compatible with PSR J0740+6620, J1614 - 2230, J0437 - 4715, J0614 - 3329, GW170817, and XTE J1814-338 appear only at μ tr=1200 and 1250 MeV; and inside the causal window, the largest jump agrees with the bound within the grid resolution. Finally, inserting a transition at 2.01, into hadronic EOSs that satisfy five pulsar observations, the EOS with the lowest break density gives a narrow stable hybrid window (ΔM 0.005,), compatible with XTE J1814-338 for jumps of 6 -- 8,n 0 and with pQCD; three EOSs with higher break densities give none.