Axion condensates in neutron stars and radial oscillation modes

Technical Summary: Axion Condensates in Neutron Stars and Radial Oscillation Modes

Problem Statement
The paper investigates the physical consequences of light QCD axions forming condensates within neutron stars (NSs). While axions were originally proposed to solve the strong CP problem and are candidates for dark matter, their potential to condense inside dense nuclear matter creates a novel ground state (NGS). Previous work established that such condensation modifies the equilibrium structure of NSs, specifically thinning the stellar envelope and accelerating cooling. This study addresses a gap in the literature by exploring how these axion condensates affect the radial oscillation spectrum of neutron stars. Although radial modes do not emit gravitational waves (GWs) and are thus difficult to observe, they serve as a foundational theoretical step to understand the coupling between the fluid and the axion field, with the expectation that these mechanisms extend to nonradial modes.

Methodology
The authors employ a realistic neutron star model using the BSk26 equation of state (EOS). The analysis proceeds in three main stages:

1. Equilibrium Structure: The authors solve the coupled Tolman–Oppenheimer–Volkoff (TOV) and Klein–Gordon (KG) equations. The system includes the axion field $a$ coupled to baryonic matter via a modified nucleon mass $m^*(a)$. The potential includes a suppression parameter $\epsilon$ (set to 0.1), which lowers the axion mass and allows the baryon density to source a nontrivial axion profile in the core.
2. Linear Perturbation Analysis: The system is perturbed around the equilibrium background. The authors derive a set of linearized equations governing the perturbations of the metric ($\delta\Phi, \delta\lambda$), fluid variables ($\delta p, \delta \varepsilon, \delta n_b$), and the axion field ($\delta a$). These equations account for the mutual interactions and the coupling between the fluid and the axion field in curved spacetime.
3. Eigenvalue Problem: By imposing harmonic time dependence ($e^{-i\omega t}$) and specific boundary conditions (regularity at the center, vanishing Lagrangian pressure perturbation at the surface, and purely outgoing axion radiation at infinity), the problem is formulated as an eigenvalue problem to determine the complex quasinormal mode (QNM) frequencies $\omega = \text{Re}(\omega) + i\text{Im}(\omega)$.

To aid interpretation, a simplified model is also presented, neglecting metric perturbations and assuming planar symmetry with a step-function density profile. This model captures the essential physics of the fluid-axion coupling and the emergence of distinct mode families.

Key Contributions and Results

• Two Distinct Families of Modes: The analysis reveals two distinct families of radial quasinormal modes:
  1. Fluid-Dominated Modes: These correspond to the standard radial oscillations of the star. In the presence of the axion condensate, their real frequencies shift slightly due to changes in the background stellar structure (compactness).
  2. Axion Modes: A new family of modes emerges, driven by the oscillations of the axion field itself.
• Axion-Induced Damping: The coupling between the fluid and the axion field introduces a damping mechanism. Unlike gravitational wave emission which requires quadrupole moments ($l \ge 2$), the scalar nature of the axion allows for radiation even from purely radial pulsations ($l=0$).
  • Damping Threshold: Modes with frequencies below the axion mass ($f < m_a$) remain essentially undamped (decoupled from the axion field).
  • Strong Damping: Modes with frequencies above the axion mass ($f > m_a$) experience strong damping due to axion emission. The decay timescales for kHz-mass axions are found to be on the order of seconds.
  • Resonances: The strongest damping occurs near resonances where the fluid mode frequency matches the natural frequencies of the emitted axion modes (e.g., near $f \approx m_a$ and $f \approx 2m_a$).
• Structural Modifications: The presence of the condensate modifies the mass-radius relation, generally shrinking the stellar radius, particularly for higher axion masses. This structural change contributes to the shift in fluid mode frequencies even before considering the dynamical coupling.
• Spectral Landscape: Using the simplified model and full numerical solutions, the authors map the QNM spectrum in the complex frequency plane. The spectrum clearly separates into weakly damped fluid modes (near the real axis) and highly damped axion modes (deep in the complex plane).

Significance and Claims
The paper claims that while direct observation of radial oscillations in neutron stars is currently difficult, this work provides a necessary theoretical foundation for neutron star seismology as a probe of axion properties.

• Diagnostic Potential: The presence or absence of damping in the oscillation spectrum offers a potential observational handle on the axion mass. Specifically:
  • If undamped oscillations are detected across a range of frequencies, it implies the axion mass is higher than the maximum observed frequency.
  • Conversely, the detection of rapidly damped modes at specific frequencies would suggest the axion mass is lower than those frequencies.
• Future Outlook: The authors emphasize that the physical mechanisms identified here (coupling and damping) are expected to apply to nonradial oscillations, which do emit gravitational waves. They suggest that future extensions of this work to nonradial modes, combined with next-generation GW detectors (e.g., Einstein Telescope, Cosmic Explorer), could turn neutron star oscillations into a unique probe for constraining axion properties.

The study does not propose new experimental setups but rather interprets existing theoretical frameworks to show how future GW or electromagnetic observations of NS oscillations could constrain the axion parameter space ($\epsilon, f_a, m_a$).