Differentiating between sharp and smoother phase transitions in neutron stars

The internal composition of neutron stars is still an open issue in astrophysics. Their innermost regions are impervious to light propagation and gravitational waves mostly carry global aspects of stars, meaning that only indirect inferences of their interiors could be obtained. Here we assume a hypothetical future scenario in which an equation of state softening due to a phase transition is identified and estimate the observational accuracy to differentiate a sharp phase transition from a smoother one (associated with a mixed phase/state due to the unknown value of the surface tension of dense matter) in a region of a hybrid star by means of some electromagnetic and gravitational wave observables. We show that different transition constructions lead to similar sequences of stellar configurations due to their shared thermodynamic properties. In the most optimistic case - a strong quark-hadron density jump phase transition - radius observations require fractional uncertainties smaller than $1\%-2\%$ to differentiate mixed states from sharp phase transitions. For tidal deformabilities, relative uncertainties should be smaller than $5\%-10\%$. However, for masses around the onset of stable quark cores, relative tidal deformability differences associated with strong sharp phase transitions and mixed states could be much larger (up to around $20\%-30\%$). All the above suggests that 2.5- and 3rd generation gravitational wave detectors and near-term electromagnetic missions may be able to start assessing some particular aspects of phase transitions in neutron stars. In addition, it points to some limitations on the equation of state recovery using typical neutron star observables and the impact of systematic uncertainties on modellings of the equation of state of hybrid stars. Finally, we briefly discuss other observables that may also be relevant for the probe of mixed states in stars.