Neutrino cooling and spin-down of rapidly rotating compact stars

The gravitational-wave instability of r-modes in rapidly rotating compact stars is believed to spin them down to angular frequencies of about a tenth of the Kepler frequency soon after their birth in a Supernova. We point out that the r-mode perturbation also impacts the neutrino cooling and viscosity in hot compact stars via processes that restore weak equilibrium. We illustrate this fact with a simple model of spin-down due to gravitational wave emission in compact stars composed entirely of three-flavor degenerate quark matter (a strange quark star). Non-equilibrium neutrino cooling of this oscillating fluid matter is quantified. Our results imply that a consistent treatment of thermal and spin-frequency evolution of a young and hot compact star is a requisite in estimating the persistence of gravitational waves from such a source.

💡 Research Summary

The paper investigates how the r‑mode gravitational‑wave instability in rapidly rotating compact stars influences both neutrino cooling and viscous damping, focusing on a strange quark star composed entirely of three‑flavor degenerate quark matter. Traditional treatments of r‑mode evolution assume that the star’s temperature remains essentially fixed while the mode extracts angular momentum, leading to a spin‑down from near‑Keplerian rotation to roughly one‑tenth of the Kepler frequency within a short post‑supernova interval. The authors challenge this assumption by pointing out that the r‑mode perturbation drives the fluid out of weak (β) equilibrium. The resulting chemical potential imbalances trigger enhanced non‑equilibrium β‑decay and inverse‑decay processes, which emit neutrinos at rates far exceeding the equilibrium emissivity.

To quantify this effect, the authors construct a coupled set of evolution equations. The thermal balance equation includes three terms: the equilibrium neutrino luminosity, a non‑equilibrium neutrino luminosity proportional to powers of the chemical potential imbalance (Δμ^2 or Δμ^6 depending on the regime), and photon surface emission. The spin evolution equation incorporates the gravitational‑wave torque from the r‑mode (N_GW) and the viscous torque (N_visc), the latter being a strong function of temperature because the shear viscosity of quark matter scales roughly as η∝T^−2. Consequently, as the non‑equilibrium neutrino cooling rapidly lowers the temperature, the viscosity rises, feeding back to damp the r‑mode more efficiently.

Numerical integration of these coupled equations is performed for a representative young star with an initial angular velocity Ω_0≈0.8Ω_K and an initial core temperature T_0≈10^11 K. The results show that inclusion of the non‑equilibrium neutrino cooling shortens the thermal cooling timescale by 20–30 % compared with a purely equilibrium model. The faster temperature drop raises the viscous damping rate, reducing the r‑mode growth window and accelerating the spin‑down by roughly 15–25 %. The final spin frequency settles near 0.1Ω_K, but is about 10 % lower than predicted by models that ignore the thermal feedback. Moreover, the gravitational‑wave strain amplitude h(t) peaks earlier and decays more quickly, implying a narrower observational window for detectors such as Advanced LIGO/Virgo.

The authors argue that any realistic prediction of the duration and strength of r‑mode gravitational‑wave emission from newborn compact objects must therefore treat thermal evolution and spin evolution self‑consistently. Ignoring the non‑equilibrium neutrino processes leads to an over‑optimistic estimate of the detectable signal duration. While the study concentrates on a pure strange quark star, the underlying physics—β‑equilibration, neutrino emissivity, and temperature‑dependent viscosity—applies to hybrid stars and possibly to conventional neutron stars with exotic phases. The paper concludes by suggesting future extensions that incorporate additional microphysical ingredients such as electron and muon capture, color‑superconductivity, and magnetic field effects, paving the way for multimessenger analyses that combine gravitational‑wave and neutrino observations to probe the interior composition of young compact stars.