Gravitational radiation from crystalline color-superconducting hybrid stars

The interiors of high mass compact (neutron) stars may contain deconfined quark matter in a crystalline color superconducting (CCS) state. On a basis of microscopic nuclear and quark matter equations of states we explore the internal structure of such stars in general relativity. We find that their stable sequence harbors CCS quark cores with masses M_core \le (0.78-0.82)M_{sun} and radii R_core \le 7 km. The CCS quark matter can support nonaxisymmetric deformations, because of its finite shear modulus, and can generate gravitational radiation at twice the rotation frequency of the star. Assuming that the CCS core is maximally strained we compute the maximal quadrupole moment it can sustain. The characteristic strain of gravitational wave emission $h_0$ predicted by our models are compared to the upper limits obtained by the LIGO and GEO 600 detectors. The upper limits are consistent with the breaking strain of CCS matter \sigma \le 10^{-4} and large pairing gaps \Delta \sim 50 MeV, or, alternatively, with \sigma \sim 10^{-3} and small pairing gaps \Delta \sim 15 MeV. An observationally determined value of the characteristic strain h_0 can pin down the product \sigma\Delta^2. On the theoretical side a better understanding of the breaking strain of CCS matter will be needed to predict reliably the level of the deformation of CCS quark core from first principles.

💡 Research Summary

The paper investigates the possibility that the cores of massive neutron stars contain deconfined quark matter in a crystalline color‑superconducting (CCS) phase and examines the gravitational‑wave (GW) emission that such cores could generate. Using state‑of‑the‑art microscopic equations of state (EOS) for both nuclear matter (based on modern effective field theory and Skyrme parametrizations) and quark matter (modeled with a Nambu–Jona‑Lasinio framework that includes color‑superconducting pairing), the authors construct hybrid star models in full general relativity by solving the Tolman‑Oppenheimer‑Volkoff equations. The resulting stable branch contains CCS quark cores with masses up to 0.78–0.82 M⊙ and radii not exceeding about 7 km, i.e., roughly ten percent of the total stellar volume.

A key property of the CCS phase is its finite shear modulus μ, arising from the crystalline lattice of the paired quarks. The authors estimate μ≈10³⁰–10³² erg cm⁻³, a value that is several times larger than that of ordinary nuclear matter. Because μ≠0, the core can sustain a non‑axisymmetric deformation (ellipticity ε) limited by the breaking strain σ, the maximum shear stress the lattice can endure before yielding. Assuming σ in the range 10⁻⁴–10⁻³, they compute the maximal quadrupole moment Q_max≈(2/5) σ μ R_core⁴. Even though the core is relatively small, the large μ and plausible σ allow Q_max to reach values capable of producing detectable GW signals.

A rotating deformed star emits continuous GWs at twice its spin frequency f. The characteristic strain amplitude is given by
h₀≈(4G/c⁴) π² f² Q/d,
where d is the distance to the source. Applying this formula to several known millisecond pulsars (e.g., PSR J2124‑3358, PSR J0437‑4715) with measured spin frequencies and distances, the authors find h₀ in the range 10⁻²⁴–10⁻²⁵. These values are compatible with the current upper limits set by the LIGO and GEO 600 detectors.

By comparing the predicted h₀ with observational limits, the study constrains the product σ Δ², where Δ is the pairing gap that determines the lattice spacing in the CCS phase. Two viable parameter regimes emerge: (i) a small breaking strain σ≤10⁻⁴ combined with a large gap Δ≈50 MeV, or (ii) a larger strain σ≈10⁻³ together with a modest gap Δ≈15 MeV. Hence, a future precise measurement of h₀ could directly pin down σ Δ² and provide unprecedented insight into the microphysics of crystalline color‑superconducting quark matter.

The authors emphasize that the theoretical uncertainty in σ is currently the dominant limitation. First‑principles calculations of the shear modulus and breaking strain for CCS matter are lacking, and progress will require detailed quantum‑chromodynamic studies of the lattice structure and its response to stress. Nevertheless, the paper demonstrates that continuous GW searches with advanced detectors (Advanced LIGO, Virgo, KAGRA, and future facilities such as the Einstein Telescope) have the potential to either detect or place stringent limits on CCS cores, thereby opening a new observational window onto the exotic phases of dense QCD matter inside neutron stars.