Modelling Dissipative Dynamics of r-mode Instability in Hybrid Stars

Compact star cores reach extreme densities and may contain exotic dense-matter phases. Information about the exotic interiors of rapidly rotating pulsars can be inferred from r-mode oscillations, whose stability is governed by viscous dissipation. In this work, we model a compact star containing a possible mixed phase of hadronic and quark matter and employ a hybrid statistical framework based on Bayesian inference to infer the dissipation time scales associated with the hybrid phase. Using low-mass X-ray binaries (LMXB) timing observations together with mass-radius constraints from the Neutron Star Interior Composition Explorer (NICER) mission, we estimate the shear and bulk viscosity contributions to r-mode damping for a hybrid star of two layers. Our inference yields shear and bulk viscous dissipation time scales of tau s=(4.99+0.49-0.52) times 10 8 T 5 over 3 s and tau B=(2.150+1.23-0.60) times 10 19 (T 4 10-12+T squared 10-6)-1-2 s respectively. The timescales thus obtained can be implemented to obtain the minima of the star's rotation frequency at =451.87 Hz at temperature T=0.259 MeV for a hybrid star of mass 1.5 M and =517.47 Hz at T=0.234 MeV for M=1.75 M. We find that the instability window obtained through the inference framework effectively explains the observed stability of millisecond pulsars in both the radio and LMXB populations, particularly for XTE J0929-314 and XTE J1807-294, J0437-4715, J2124-3358, respectively. These results demonstrate that Bayesian inference combined with r-mode phenomenology provides a powerful and observationally consistent framework for constraining the transport properties of dense hybrid matter.