Rotational effects in quark stars: comparing different models

Technical Summary: Rotational Effects in Quark Stars: Comparing Different Models

Problem and Motivation
The theoretical proposition that strange quark matter (QM) may constitute the true ground state of nuclear matter motivates the study of strange stars (QSs), compact objects composed of up, down, and strange quarks. A definitive identification of such objects requires confronting theoretical predictions with astrophysical observations, particularly those probing rapidly rotating compact stars. While early investigations employed simplified equations of state (EOS), modern descriptions incorporating quark masses and interactions yield substantially higher maximum masses and rotational frequencies. However, distinguishing between competing QM descriptions and differentiating QSs from hadronic neutron stars (NSs) remains a challenge. This work investigates the rotational properties of bare, self-bound strange stars to determine how rotational observables can break the degeneracy between different QM models and provide multimessenger signatures for their identification.

Methodology
The authors perform a fully relativistic comparison of two representative QM EOS: the Density-Dependent Quark Mass (DDQM) model and the Vector MIT Bag model. These models were selected because they provide physically distinct descriptions of QM: the DDQM model yields a relatively soft EOS with larger radii and lower maximum masses, while the Vector MIT Bag model, incorporating repulsive vector interactions, produces a stiff, self-bound EOS capable of supporting more massive configurations.

The study utilizes the rns code to solve Einstein's field equations for stationary, axisymmetric spacetimes, generating sequences of uniformly rotating configurations. The free parameters for both models were constrained using the same set of astrophysical data (including PSR J0952−0607, PSR J0740+6620, PSR J0030+0451, HESS J1731−347, and GW170817) via Bayesian inference, as detailed in previous work [5]. Four parameter sets were derived for each model to satisfy different subsets of observational constraints.

Key observables computed and analyzed include:

• Mass–radius relations ($M$–$R$)
• Moments of inertia ($I$)
• Quadrupole moments ($Q$)
• Surface redshifts ($Z_p$)
• Keplerian frequencies ($\nu_K$)
• A full decomposition of the stellar energy budget (gravitational $|W|$, internal $U$, rotational kinetic $T$, and binding energy $E_{bind}$).

Key Results
The analysis reveals that rotation amplifies the intrinsic differences between the two EOS models:

1. Mass and Compactness: The stiffer Vector MIT Bag model supports significantly more massive stars, reaching maximum gravitational masses ($M_{max}$) of $\sim 3.3 M_\odot$ under rapid rotation, compared to $\sim 2.8 M_\odot$ for the DDQM model. Consequently, MIT configurations are more compact, exhibiting smaller radii for a given mass.
2. Rotational Limits: MIT models sustain higher Keplerian frequencies ($\nu_K \gtrsim 1450$ Hz) compared to DDQM models ($\nu_K < 1300$ Hz). While both models show a mass enhancement of $\sim 29\text{--}30\%$ near the Kepler limit (exceeding the typical $\sim 20\%$ for hadronic stars), the frequency required to achieve this enhancement differs, offering a potential discriminator.
3. Moment of Inertia ($I$) and Quadrupole Moment ($Q$):
   • At the canonical mass ($1.4 M_\odot$), the softer DDQM model produces stars with larger radii and consequently higher moments of inertia and quadrupole moments compared to the more compact MIT configurations.
   • At maximum mass, the MIT model dominates in $I$ due to its ability to support higher total masses, despite being more compact.
   • The dimensionless quadrupole moment $\bar{Q}$ decreases monotonically with compactness ($M/R_e$), but DDQM configurations systematically yield larger $\bar{Q}$ values than MIT ones at fixed compactness, reflecting the higher deformability of the softer EOS.
4. Surface Redshift ($Z_p$): The Vector MIT Bag model predicts significantly higher polar redshifts, reaching $Z_p \gtrsim 0.85$ near maximum mass, indicative of extreme compactness. In contrast, DDQM models exhibit more moderate redshifts ($Z_p \lesssim 0.6$).
5. Energy Decomposition: A central result is the full energy decomposition. The Vector MIT Bag model exhibits negative internal energy ($U < 0$), confirming the self-bound nature of the strange quark matter (energetically favorable like atomic nuclei). Conversely, the DDQM model shows positive internal energy ($U > 0$), indicating that the resulting stars are bound purely by gravity. In both models, gravitational binding energy ($|W|$) dominates the total energy budget.

Significance and Claims
The paper claims that combined measurements of mass, radius, moment of inertia, surface redshift, and spin frequency can effectively break the EOS degeneracy between these two QM scenarios.

• Differentiating QM Models: Massive, rapidly rotating pulsars ($M \gtrsim 3 M_\odot$, $\nu \gtrsim 1400$ Hz) would favor the stiff Vector MIT Bag model, whereas canonical-mass pulsars with larger radii and moments of inertia would point toward the softer DDQM model.
• Differentiating QSs from NSs: The study identifies distinctive multimessenger signatures. Specifically, the detection of a large polar surface redshift ($Z_p \gtrsim 0.8$) or a very large moment of inertia at high mass would strongly favor QSs with stiff, self-bound EOS over standard hadronic NSs.
• Energy Budget as a Diagnostic: The sign of the internal energy ($U$) serves as a critical physical criterion: negative $U$ implies intrinsic self-binding (MIT), while positive $U$ implies gravitational binding (DDQM).

The authors conclude that while current observations do not yet provide strong candidates for strange stars, the distinct rotational signatures identified—such as the magnitude of mass enhancement, characteristic behaviors of $I$, $Q$, and $Z_p$, and the energy decomposition—establish specific benchmarks. These observables, accessible through X-ray timing (e.g., NICER), binary pulsar campaigns, and future gravitational-wave detectors, offer a robust framework for testing the existence and composition of quark matter in compact stars.