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 $\Lambda_1, \Lambda_2$, the effective, or observable, tidal deformability that enters the lowest-order GW phasing correction is

$$\tilde{\Lambda} = \frac{16}{13}\,\frac{(m_1 + 12 m_2)\,m_1^4\,\Lambda_1 + (m_2 + 12 m_1)\,m_2^4\,\Lambda_2}{(m_1 + m_2)^5}.$$

For equal-mass, equal-radius binaries, this reduces to $\tilde{\Lambda} = \Lambda_1 = \Lambda_2$.

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

$$\mathcal{M}_c = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}.$$

2. Calculation of $\Lambda$ and Its Microphysical Dependence

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

$$r\,\frac{dy}{dr} + y^2 + y F(r) + r^2 Q(r) = 0, \quad y(0) = 2,$$

extracting the surface value $y_R = y(R)$, and inserting it into the full expression for the Love number $k_2$. The EoS determines the radius–mass relation $R(M)$ and thus the compactness and $\Lambda(M)$.

For tidal phase transitions or non-nucleonic matter, discontinuities in the EoS lead to abrupt changes in $k_2$ and $\Lambda$, which propagate into the structure of $\tilde{\Lambda}(q)$ for fixed $\mathcal{M}_c$ (Han et al., 2018). The effective deformability is highly sensitive to the EoS stiffness at densities of order $2 n_0$ (nuclear saturation), with stiff EoS (large radii) producing larger $\Lambda$ and soft EoS producing smaller values.

Quantitatively, for realistic EoS, canonical neutron-star radii $R_{1.4} \sim 11.5\!-\!12$ km correspond to $\Lambda_{1.4} \sim 250\!-\!400$ and typical $\tilde{\Lambda}$ for GW170817-like systems in the range $197 \leq \tilde{\Lambda} \leq 720$ (Sammarruca et al., 29 Nov 2025).

3. Thermal and Compositional Effects on $\tilde{\Lambda}$

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 $T$ to values as high as $1$ MeV leads to a $20\%$ decrease in $k_2$ and a $5\%$ increase in $R$ for a 1.4 $M_\odot$ star, yet the product $k_2 R^5$ (hence $\lambda$) remains nearly constant. For adiabatic (isentropic) configurations with entropy per baryon $S \le 0.2\,k_B$, both $k_2$ and $R$ are stable to within $\sim 1\%$.

The cancellation of the opposing effects of $R$ and $k_2$ leads to negligible change in $\Lambda$ and $\tilde{\Lambda}$ for $T \lesssim 1$ MeV or $S \lesssim 0.2\,k_B$. Thus, the use of cold EoS in GW inference of $\tilde{\Lambda}$ is robust:

• For GW170817-like mass ranges, increasing $T$ from 0.01 to 1 MeV shifts $\tilde{\Lambda}$ curves by only a few percent.
• For realistic entropies, curves of $\tilde{\Lambda}(q)$ for different $S$ are virtually coincident for $S \le 0.2\,k_B$ (Kanakis-Pegios et al., 2022).

For composition effects, the difference between frozen and beta-equilibrium configurations shifts $\tilde{\Lambda}$ 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: $$\Psi_{\rm tidal}(f) = -\frac{39}{2}\,\tilde{\Lambda}\,(\pi M f)^{5/3}.$$ Higher-order tidal terms (6PN and beyond) contribute additional corrections, but for $T \lesssim 1$ MeV or $S \lesssim 0.2\,k_B$, the principal effect is through $\tilde{\Lambda}$ 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 $\tilde{\Lambda}$ improves with higher SNR and stiffer EoS, but systematic thermal and compositional corrections are sub-dominant at present sensitivity $[2202.01820]$.

5. Microphysical and Astrophysical Consequences

The robust connection between $\tilde{\Lambda}$, the EoS stiffness, and stellar radius enables tight constraints on dense-matter physics:

• Upper bounds on $\tilde{\Lambda}$ from GW170817 exclude very stiff EoS with $R_{1.4} \gtrsim 13$ km.
• The insensitivity of $\tilde{\Lambda}$ to pre-merger temperature up to $T \lesssim 1$ MeV justifies EoS inference assuming cold stars.
• Detection of anomalies in $\tilde{\Lambda}(q)$, such as kinks or gaps, could indicate phase transitions or exotic constituents in the core (Han et al., 2018).

Combination of GW measurements of $\tilde{\Lambda}$, 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 $M_\odot$ neutron star as a function of temperature for the Lattimer–Swesty EoS (Kanakis-Pegios et al., 2022):

$T$ (MeV)	$k_2$	$R$ (km)	$\lambda$ ($10^{36}$ g cm$^2$ s$^2$)
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 $T = 1$ MeV, the product $k_2 R^5$ and $\lambda$ remains effectively unchanged; in the adiabatic sequence up to $S = 0.2 k_B$, variations are sub-percent.

7. Implications for Data Analysis and Future Measurements

Given the thermal invariance of $\tilde{\Lambda}$ for inspiral temperatures relevant to current binary neutron-star mergers, current and next-generation GW analyses can safely interpret observed $\tilde{\Lambda}$ using cold EoS. However, independent radius measurements in conjunction with $\tilde{\Lambda}$ could, in principle, reveal nonzero pre-merger temperatures if anomalously large radii are measured at fixed $\tilde{\Lambda}$.

As statistical errors in $\tilde{\Lambda}$ 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 $\tilde{\Lambda}$ 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 $\tilde{\Lambda}$ as a key parameter in GW astrophysics for constraining the microphysics of dense matter (Kanakis-Pegios et al., 2022).