A New Scaling of Neutron Star Tidal Deformability for Directly Probing the Core Equation of State

Abstract: The dimensionless tidal deformability, $Λ$, of neutron stars (NSs), inferred from gravitational-wave (GW) observations, has thus far been used primarily to constrain the pressure of dense matter near twice nuclear saturation density, leaving the core equation of state (EOS) largely inaccessible to inspiral-phase GW observations. We show that the core EOS can be probed directly through $Λ$ using a perturbative analysis of the dimensionless stellar-structure and tidal-response equations formulated in terms of scaled intrinsic variables, without invoking any specific EOS model. We uncover a remarkable EOS-insensitive scaling relation between $Λ$ and the central EOS parameter $\mathrm{X}\equiv P_{\rm c}/\varepsilon_{\rm c}$, where $P_{\rm c}$ and $\varepsilon_{\rm c}$ denote the central pressure and energy density, respectively. The relation is validated against a broad ensemble of physically viable EOSs. Applying it to tidal deformabilities inferred from events such as GW170817 enables a direct determination of $\mathrm{X}$. We further derive a tight lower bound, $$Λ<em>{\rm{TOV}}\gtrsim 9.2^{+1.2}</em>{-1.2}$$, for maximum-mass NSs along stable mass-radius sequences, quantitatively demonstrating that even the most compact stable NSs remain distinctly separated from black holes, for which $Λ_{\rm{BH}}=0$. These findings reveal a previously unrecognized connection between inspiral-phase tidal deformability and the core EOS, establishing a direct link between GW observables and the microphysics of ultradense matter in the strong-gravity regime. The resulting scaling establishes inspiral-phase tidal deformability as a direct and largely model-insensitive probe of the EOS of NS cores.

• The paper introduces an analytic, EOS‐insensitive scaling that directly links tidal deformability with central core properties of neutron stars.
• It employs the IPAD‐TOV formalism to derive explicit scaling relations validated across extensive EOS ensembles, achieving high precision (R² ~ 0.96).
• The findings set robust lower bounds on tidal deformability, confirming a finite separation between neutron stars and black holes in gravitational-wave observables.

Direct Scaling of Neutron Star Tidal Deformability with Core Equation of State

Introduction

The paper "A New Scaling of Neutron Star Tidal Deformability for Directly Probing the Core Equation of State" (2606.21402) develops a fundamentally new scaling relation connecting the dimensionless tidal deformability $\Lambda$ of neutron stars (NSs) to a central equation-of-state (EOS) parameter $X \equiv P_c/\varepsilon_c$, with $P_c$ and $\varepsilon_c$ denoting the central pressure and energy density, respectively. Traditional analyses have employed tidal deformability constraints largely to probe EOS behavior at a few times nuclear saturation density, but direct information about core properties remained inaccessible without strong modeling assumptions. This work derives an EOS-insensitive, analytic scaling—validated across a comprehensive EOS ensemble—that enables direct inference of $X$ from GW-measured $\Lambda$, establishing tidal deformability as a probe of core microphysics without assuming a specific EOS. Significant implications include a quantitative lower bound on the tidal deformability of maximum-mass NSs, confirming the finite separation between the most compact NSs and black holes in the $\Lambda$ observable.

IPAD-TOV Framework and Scaling Relations

The analysis employs the IPAD-TOV formalism (Intrinsic and Perturbative Analyses of the Dimensionless Tolman–Oppenheimer–Volkoff equations), which isolates relevant dimensionless variables governing relativistic stellar structure and tidal response. The main object of study is the dimensionless tidal deformability $\Lambda = 2 k_2 / (3 \xi^5)$, where $k_2$ is the quadrupolar Love number and $\xi = M_{\mathrm{NS}}/R$ is the compactness. By recasting the structure and tidal equations purely in terms of dimensionless quantities, the paper demonstrates that $X \equiv P_c/\varepsilon_c$0 depends predominantly on the central EOS stiffness $X \equiv P_c/\varepsilon_c$1, with sub-leading influence from the log-stability slope $X \equiv P_c/\varepsilon_c$2.

Key analytic results include explicit scaling formulae for $X \equiv P_c/\varepsilon_c$3 and $X \equiv P_c/\varepsilon_c$4, with corrections derived from a systematic expansion about the stellar center. Remarkably, despite the nonlinearity of the relativistic equations, the mapping $X \equiv P_c/\varepsilon_c$5 is found to be both robust and highly insensitive to the details of the EOS at lower densities or exotic components (e.g., hyperons, deconfinement), provided basic physical requirements—causality, stability, NS mass constraints—are respected. This relation is confirmed by direct numerical integration across broad EOS ensembles.

Extraction of Core EOS Parameter from GW Data

Fitting to extensive EOS suites constructed from both meta-modeling and nonlinear Walecka frameworks, constrained by nuclear theory, laboratory, and astrophysical bounds, the correlation $X \equiv P_c/\varepsilon_c$6 is established with high precision ($X \equiv P_c/\varepsilon_c$7) for canonical NSs. For instance, for the $X \equiv P_c/\varepsilon_c$8 NS, the scaling enables direct inversion of observational $X \equiv P_c/\varepsilon_c$9 measurements to central $P_c$0, thus bypassing the conventional degeneracy with EOS model parameterizations. Adopting the LIGO/Virgo GW170817 constraint $P_c$1, the resulting allowed range for the central stiffness $P_c$2 is $P_c$3. The method generalizes to provide allowed intervals for $P_c$4 and $P_c$5 for canonical masses, with central energy density determined to the $P_c$6 nuclear saturation value and corresponding pressures extracted with quantifiable uncertainty.

Lower Bound on Tidal Deformability of TOV Neutron Stars

Extending the scaling to the maximum-mass TOV sequence ($P_c$7), a direct lower bound is derived: for all physically allowed EOSs consistent with causality ($P_c$8), the maximum-mass stable NS must satisfy $P_c$9. Even when allowing for a broad upper limit on $\varepsilon_c$0 ($\varepsilon_c$1), this minimal $\varepsilon_c$2 remains $\varepsilon_c$3. This result quantitatively demonstrates that, in the inspiral GW regime, no stable NS can approach the zero-deformability (black hole) limit. The result is robust across all realistic EOS models, including those with strong phase transitions or exotic matter.

Theoretical and Practical Implications

This scaling directly bridges global GW observables with microscopic ultradense EOS properties, substantially improving the interpretability of GW constraints without the need to marginalize over arbitrary or subjective EOS parameterizations. The practical consequence is the ability to infer central EOS properties—and thus microphysics such as sound speed, stiffness, and possible proximity to conformal limits—directly from observational data. The analysis also yields updated, tight bounds on canonical NS radii, central sound speeds, and energy densities.

At a deeper level, the emergence of this behavior suggests that the relativistic stellar structure and tidal response may possess universality features beyond those encapsulated by I-Love-Q relations, perhaps due to an underlying compression of microphysics into a small set of effective dimensionless variables. This has implications for future GW analyses, systematic EOS inference, and the theoretical understanding of universality in compact object astrophysics.

Perspective and Future Directions

The direct mapping between $\varepsilon_c$4 and $\varepsilon_c$5 established here paves the way for model-independent, high-fidelity inference of the EOS at the highest densities accessible in nature. As next-generation GW observatories improve constraints on tidal deformabilities (both inspiral and post-merger), the precision of central EOS determinations will increase, enabling discrimination of microphysical scenarios deep in the NS interior, and offering stringent tests for the appearance of exotic phases, causality bounds, and the approach to or violation of conformal sound speed limits. The analytic nature of the scaling further facilitates direct connection with theoretical calculations of the nuclear and quark matter EOS.

Conclusion

This work provides a rigorous, EOS-insensitive analytic scaling that allows direct extraction of neutron star core physics from inspiral gravitational-wave observations. The established mapping dramatically reduces the degeneracy inherent in prior EOS inference and sets robust lower bounds on observable properties of compact objects. The result has significant implications for both dense matter theory and astrophysical observation, and forms the foundation for more precise, less model-dependent EOS constraints as observational data improve.

---

Reference:

"A New Scaling of Neutron Star Tidal Deformability for Directly Probing the Core Equation of State" (2606.21402)