Hidden Twin Star Solutions from an Agnostic Speed-of-Sound Model: Confronting XTE J1814--338's Extreme Compactness

The twin star configuration, where two neutron stars share the same mass but exhibit different radii, arises from a strong first-order phase transition within the stellar interior. In widely used equation of state (EoS) meta-models, such as the Polytrope (PP) and Speed-of-Sound (CS) models, this first-order phase transition behavior can be naturally mimicked by tuning some model parameters. Here, we systematically explore the under-explored parameter space within one of a widely adopted CS model that leads to twin stars via a strong first-order phase transition. Within this twin-star subspace, we perform a comprehensive Bayesian analysis that integrates mass--radius (MR) constraints from X-ray observations of rotation-powered millisecond pulsars. The resultant twin star branch, situated within the 1--1.2 $M_{\odot}$ mass range and approximately 7 km in radius, surprisingly coincides with the MR ranges proposed for the recent anomaly in the Accreting Millisecond X-ray Pulsars XTE J1814--338 (J1814), suggesting a hybrid twin star configuration. Moreover, incorporating the J1814 observation as an additional constraint yields an extreme phase transition pressure $P_{\text{trans}} = 108.9_{-4.85}^{+6.46}$ MeV/fm$^3$, a transition density of $\varepsilon_{\text{trans}}/\varepsilon_0 = 4.847_{-0.134}^{+0.271}$(where $\varepsilon_0$ is the nuclear saturation energy density) and an energy density jump $\Delta \varepsilon = 558.7_{-278.7}^{+303.6}$ MeV/fm$^3$, corresponding to $\Delta \varepsilon/\varepsilon_0 = 3.716_{-1.854}^{+2.020}$. Notably, to satisfy all astrophysical constraints, the speed of sound inside of the hybrid twin star core is driven toward the speed of light ($c_s^2/c^2>0.9$), indicating the potential presence of strongly interacting, exotic matter in this core region.