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Axion Stars in the Infrared Limit

Following Ruffini and Bonazzola, we use a quantized boson field to describe condensates of axions forming compact objects. Without substantial modifications, the method can only be applied to axions with decay constant, $f_a$, satisfying $δ=(f_a\,/\,M_P)^2\ll 1$, where $M_P$ is the Planck mass. Similarly, the applicability of the Ruffini-Bonazzola method to axion stars also requires that the relative binding energy of axions satisfies $Δ=\sqrt{1-(E_a\,/\,m_a)^2}\ll1$, where $E_a$ and $m_a$ are the energy and mass of the axion. The simultaneous expansion of the equations of motion in $δ$ and $Δ$ leads to a simplified set of equations, depending only on the parameter, $λ=\sqrtδ\,/\,Δ$ in leading order of the expansions. Keeping leading order in $Δ$ is equivalent to the infrared limit, in which only relevant and marginal terms contribute to the equations of motion. The number of axions in the star is uniquely determined by $λ$. Numerical solutions are found in a wide range of $λ$. At small $λ$ the mass and radius of the axion star rise linearly with $λ$. While at larger $λ$ the radius of the star continues to rise, the mass of the star, $M$, attains a maximum at $λ_{\rm max}\simeq 0.58$. All stars are unstable for $λ>λ_{\rm max}$ . We discuss the relationship of our results to current observational constraints on dark matter and the phenomenology of Fast Radio Bursts.