Global view of QCD axion stars

Taking a comprehensive view, including a full range of boundary conditions, we reexamine QCD axion star solutions based on the relativistic Klein-Gordon equation (using the Ruffini-Bonazzola approach) and its nonrelativistic limit, the Gross-Pitaevskii equation. A single free parameter, conveniently chosen as the central value of the wave function of the axion star, or alternatively the chemical potential with range $-m<\mu <0$ (where $m$ is the axion mass), uniquely determines a spherically symmetric ground state solution, the axion condensate. We clarify how the interplay of various terms of the Klein-Gordon equation determines the properties of solutions in three separate regions: the structurally stable (corresponding to a local energy minimum) dilute and dense regions, and the intermediate, structurally unstable transition region. From the Klein-Gordon equation, one can derive alternative equations of motion including the Gross-Pitaevskii and Sine-Gordon equations, which have been used previously to describe axion stars in the dense region. In this work, we clarify precisely how and why such methods break down as the binding energy increases, emphasizing the necessity of using the full relativistic Klein-Gordon approach. Finally, we point out that, even after including perturbative axion number violating corrections, solutions to the equations of motion, which assume approximate conservation of axion number, break down completely in the regime with strong binding energy, where the magnitude of the chemical potential approaches the axion mass.

Figure 1

Top: Mass $M$ and radius $R_{99}$ of axion stars in the dilute region for different choices of axion decay constant $f$ using the EKG method [Eq. (2.6)]. The maximum mass at $M_{\max }=10M_{P}f/m$ marks the crossover from the dilute (upper, downward-sloping) to transition (lower, upward-sloping) branches of solutions. Bottom: The masses and radii of axion stars in the vicinity of the dense crossover for the different methods analyzed in this study; the inset shows the crossover from transition to dense branches of solutions. The blue, black, and red dots mark the end point for solutions of the EKG, GRB, and CEKG methods (respectively), as described in the text.

Figure 2

Total axion star mass as a function of $\mathrm{\Delta }$ (top), as well as the binding energy parameter $\mathrm{\Delta }$ (middle) and chemical potential per unit mass $\mu /m$ (bottom) as a function of central field value $Z(0)$ for different methods. In all panels, the light purple (dark purple) shaded region represents the quasirelativistic (ultrarelativistic) region defined in Sec. 2a.

Figure 3

Plot of $K/(M{\mathrm{\Delta }}^2)$ (blue) and $M_g/(M\delta )$ (red) as a function of Z(0).