Effective Deformability of Binary Systems

• The paper shows that effective deformability (tilde Λ) is derived from a mass-weighted average of individual tidal responses, providing key constraints on dense matter EoS.
• It details a methodology using the TOV and tidal perturbation equations to calculate individual deformabilities, ensuring accurate gravitational-wave phase predictions.
• The study demonstrates that finite-temperature and compositional effects cause minimal changes, validating cold EoS approximations in current GW data analyses.

The effective deformability of a binary system, often denoted as $\tilde{\Lambda}$, is the primary tidal polarizability parameter that enters the gravitational-wave (GW) phasing of an inspiraling neutron-star binary. It encapsulates the combined tidal response of both compact objects in a mass-weighted average, allowing constraints to be placed on the dense-matter equation of state (EoS) from GW observations. The following sections present the precise formalism, EoS dependence, finite-temperature and composition effects, observational implications, and key constraints on neutron-star microphysics, based on current literature and especially on (Kanakis-Pegios et al., 2022).

1. Definition and Formalism of Effective Tidal Deformability

For each star of mass $M$, radius $R$, and quadrupolar Love number $k_2$, the induced quadrupole moment in response to an external tidal field $E_{ij}$ is

$$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$

with $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$ the dimensional tidal deformability. The dimensionless tidal deformability, widely used in GW analyses, is

$$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$

where $\beta = GM/(R c^2)$ is the compactness.

For a binary system of masses $m_1 \ge m_2$ and individual deformabilities $M$0, the effective, or observable, tidal deformability that enters the lowest-order GW phasing correction is

$M$1

For equal-mass, equal-radius binaries, this reduces to $M$2.

The chirp mass, which is measured with high accuracy in GW observations, is

$M$3

2. Calculation of $M$4 and Its Microphysical Dependence

The dimensionless polarizability $M$5 for a given EoS is computed by integrating the Tolman–Oppenheimer–Volkoff (TOV) equations in parallel with the linearized tidal perturbation equations,

$M$6

extracting the surface value $M$7, and inserting it into the full expression for the Love number $M$8. The EoS determines the radius–mass relation $M$9 and thus the compactness and $R$0.

For tidal phase transitions or non-nucleonic matter, discontinuities in the EoS lead to abrupt changes in $R$1 and $R$2, which propagate into the structure of $R$3 for fixed $R$4 (Han et al., 2018). The effective deformability is highly sensitive to the EoS stiffness at densities of order $R$5 (nuclear saturation), with stiff EoS (large radii) producing larger $R$6 and soft EoS producing smaller values.

Quantitatively, for realistic EoS, canonical neutron-star radii $R$7 km correspond to $R$8 and typical $R$9 for GW170817-like systems in the range $k_2$0 (Sammarruca et al., 29 Nov 2025).

3. Thermal and Compositional Effects on $k_2$1

The impact of finite temperature and non-barotropic stellar structure on the effective tidal deformability during the late inspiral has been systematically investigated (Kanakis-Pegios et al., 2022, Kanakis-Pegios et al., 2021, Andersson et al., 2019).

For isothermal models, increasing the temperature $k_2$2 to values as high as $k_2$3 MeV leads to a $k_2$4 decrease in $k_2$5 and a $k_2$6 increase in $k_2$7 for a 1.4 $k_2$8 star, yet the product $k_2$9 (hence $E_{ij}$0) remains nearly constant. For adiabatic (isentropic) configurations with entropy per baryon $E_{ij}$1, both $E_{ij}$2 and $E_{ij}$3 are stable to within $E_{ij}$4.

The cancellation of the opposing effects of $E_{ij}$5 and $E_{ij}$6 leads to negligible change in $E_{ij}$7 and $E_{ij}$8 for $E_{ij}$9 MeV or $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$0. Thus, the use of cold EoS in GW inference of $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$1 is robust:

• For GW170817-like mass ranges, increasing $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$2 from 0.01 to 1 MeV shifts $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$3 curves by only a few percent.
• For realistic entropies, curves of $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$4 for different $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$5 are virtually coincident for $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$6 (Kanakis-Pegios et al., 2022).

For composition effects, the difference between frozen and beta-equilibrium configurations shifts $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$7 by at most a few percent (Andersson et al., 2019).

4. Role in Gravitational-Wave Phasing and Observational Inference

The post-Newtonian expansion of the GW phase incorporates the leading-order tidal contribution at 5PN order: $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$8 Higher-order tidal terms (6PN and beyond) contribute additional corrections, but for $$Q_{ij} = -\frac{2}{3}\,k_2\,\frac{R^5}{G}\,E_{ij} \equiv -\lambda\, E_{ij},$$9 MeV or $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$0, the principal effect is through $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$1 computed using cold EoS (Park et al., 4 Feb 2025, Choi et al., 2018).

For parameter estimation, Fisher-matrix studies and Bayesian analyses demonstrate that the measurement precision on $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$2 improves with higher SNR and stiffer EoS, but systematic thermal and compositional corrections are sub-dominant at present sensitivity $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$3.

5. Microphysical and Astrophysical Consequences

The robust connection between $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$4, the EoS stiffness, and stellar radius enables tight constraints on dense-matter physics:

• Upper bounds on $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$5 from GW170817 exclude very stiff EoS with $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$6 km.
• The insensitivity of $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$7 to pre-merger temperature up to $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$8 MeV justifies EoS inference assuming cold stars.
• Detection of anomalies in $\lambda = \frac{2}{3}\,\frac{R^5}{G}\,k_2$9, such as kinks or gaps, could indicate phase transitions or exotic constituents in the core (Han et al., 2018).

Combination of GW measurements of $$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$0, independent radius constraints (e.g., from X-ray pulse profiling), and electromagnetic signatures (kilonova modeling) has the potential to disentangle finite-temperature, compositional, and phase structure effects (Kanakis-Pegios et al., 2022, Sammarruca et al., 29 Nov 2025).

6. Representative Quantitative Results

The following table summarizes the thermal stability of tidal deformability parameters for a 1.4 $$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$1 neutron star as a function of temperature for the Lattimer–Swesty EoS (Kanakis-Pegios et al., 2022):

$$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$2 (MeV)	$$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$3	$$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$4 (km)	$$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$5 ($$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$6 g cm$$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$7 s$$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$8)
0.01	0.1005	12.21	2.73
0.10	0.0984	12.26	2.73
1.00	0.0788	12.82	2.73

Even at $$\Lambda = \frac{\lambda}{M^5} = \frac{2}{3}\,k_2 \left( \frac{R c^2}{G M} \right)^5 = \frac{2}{3}\,k_2\,\beta^{-5},$$9 MeV, the product $\beta = GM/(R c^2)$0 and $\beta = GM/(R c^2)$1 remains effectively unchanged; in the adiabatic sequence up to $\beta = GM/(R c^2)$2, variations are sub-percent.

7. Implications for Data Analysis and Future Measurements

Given the thermal invariance of $\beta = GM/(R c^2)$3 for inspiral temperatures relevant to current binary neutron-star mergers, current and next-generation GW analyses can safely interpret observed $\beta = GM/(R c^2)$4 using cold EoS. However, independent radius measurements in conjunction with $\beta = GM/(R c^2)$5 could, in principle, reveal nonzero pre-merger temperatures if anomalously large radii are measured at fixed $\beta = GM/(R c^2)$6.

As statistical errors in $\beta = GM/(R c^2)$7 shrink with improved detector sensitivity, precision in the percent regime may expose the small systematic uncertainties due to temperature, composition, and EoS phase structure (Kanakis-Pegios et al., 2022).

In summary, for binary neutron stars in the late inspiral, the effective tidal deformability $\beta = GM/(R c^2)$8 is given by a precise, mass-weighted combination of the component deformabilities and, for realistic temperatures and entropies, is robustly predicted by the cold EoS. This establishes $\beta = GM/(R c^2)$9 as a key parameter in GW astrophysics for constraining the microphysics of dense matter (Kanakis-Pegios et al., 2022).