I-C-Q relations for rapidly rotating stable hybrid stars

A number of hadronic equations of state for neutron stars have been investigated for the purpose of the present paper, considering the fact that at sufficiently high density, heavy baryons and quark phases may appear. The observational limits from NICER, GW170817, etc., are obeyed by our choice of equations of state. The universal relations are investigated for both slowly and rapidly rotating neutron stars with heavy baryons present inside the core. For slowly rotating stars, the universality of the I-Love-Q relations is verified, and the I-C-Q relations are inferred to be universal for rapidly rotating stars. Further, we extend the investigation to obtain the universal relations for compact stars containing the quark core, where the connected stable branch of such hybrid stars is considered. The parameters of the I-Love-Q and I-C-Q universal relations are obtained for slowly rotating and rapidly rotating hybrid stars, respectively. These relations would enable extracting information, within the context of general relativity, from astrophysical systems involving rapidly rotating neutron stars.

💡 Research Summary

The paper investigates the existence and robustness of universal relations among the moment of inertia (I), compactness (C = M/R), and quadrupole moment (Q) for neutron stars that contain heavy baryons (hyperons and Δ resonances) and for hybrid stars that also host a quark core. A suite of equations of state (EoS) is selected from the CompOSE database, including the purely nucleonic DD2 model, DD2Λ (with Λ hyperons), DD2Λφ (adding repulsive Λ–Λ interaction via the φ meson), DD2Y (full baryon octet), and three variants of DD2YΔ with different Δ potential depths. Hybrid EoS are constructed by coupling these hadronic models to a quark phase (DD2Q, DD2ΛQ, etc.) using a first‑order phase transition. All chosen EoS satisfy current astrophysical constraints from NICER mass‑radius measurements of PSR J0030+0451 and PSR J0740+6620, as well as the tidal‑deformability limits from GW170817.

Static stellar configurations are obtained by solving the Tolman‑Oppenheimer‑Volkoff equations, while rapidly rotating equilibria are computed with the RNS code, varying the dimensionless spin parameter χ = J/M² from 10⁻³ (effectively non‑rotating) up to the Keplerian limit (χ≈0.6). For each model and spin, the authors calculate the mass‑radius curve, the tidal Love number k₂ (and the dimensionless Λ), the moment of inertia I, the quadrupole moment Q, and the compactness C. They present a series of plots: pressure versus energy density, speed of sound, M‑R for static and Keplerian sequences, Λ(C), I(Λ), Q(Λ), I(Q), and I(C) for several spin values.

The first major result is a verification that the classic I‑Love‑Q relation holds for slowly rotating stars (χ≈10⁻³) across all EoS, with fractional deviations below 0.1 %. This confirms previous findings that the relation is essentially EoS‑independent even when hyperons or Δ resonances are present.

The second, novel contribution is the extension to rapidly rotating stars via an I‑C‑Q relation. The authors fit the three‑dimensional data with a logarithmic polynomial of the form
 log I = a₀ + a₁ log C + a₂ log Q + a₃ (log C)² + a₄ (log Q)² + a₅ log C log Q,
determining the coefficients a_i for each EoS family. The fitted surfaces reproduce the numerical data with typical fractional errors of 5–8 % for both pure hadronic and hybrid configurations, indicating that the universality survives the inclusion of heavy baryons and a deconfined quark core. Two‑dimensional projections (I‑C, I‑Q, C‑Q) also display a tight clustering, reinforcing the quasi‑universal character.

Additional analysis links the maximum attainable spin χ_max to the compactness and gravitational mass, showing that χ_max decreases with increasing C, while hybrid stars with a stiff hadronic segment can sustain slightly larger χ_max (≈5 % higher) than softer counterparts. This behavior is consistent with the expectation that a stiffer EoS provides greater centrifugal support.

The paper concludes that universal I‑Love‑Q and I‑C‑Q relations are robust tools for extracting neutron‑star interior properties from observations, even in the most general scenario considered: a fully populated baryon octet plus a possible quark phase, rotating up to the Kepler limit. The authors suggest future work to incorporate strong magnetic fields, anisotropic pressures, and more sophisticated quark‑matter models to test the limits of these relations.