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# Self-gravitating system made of axions 

J. Barranco 

Max-Planck-Institut f¨ ur Gravitationsphysik (Albert-Einstein-Institut), Am M¨ uhlenberg 1, D-14476 Golm, Germany ∗ 

Max-Planck-Institut f¨ ur Gravitationsphysik (Albert-Einstein-Institut), Am M¨ uhlenberg 1, D-14476 Golm, Germany † 

(Dated: October 26, 2018) 

A. Bernal 

We show that the inclusion of an axion-like effective potential in the construction of a self- gravitating system of scalar fields decreases its compactness when the value of the self-interaction coupling constant is increased. By including the current values for the axion mass m and decay constant f a , we have computed the mass and the radius for self-gravitating systems made of axion particles. It is found that such objects will have asteroid-size masses and radii of a few meters, thus a self-gravitating system made of axions could play the role of scalar mini-MACHOs and mimic a cold dark matter model for the galactic halo. 

The necessity of introducing dark matter (DM) as the main component of galactic matter has become a solid fact due to its observational support [1]. Nevertheless, the nature of DM is one of the most intriguing mysteries in physics. A large variety of particles have been consid- ered as the main component of DM in the universe and only a few of them are still considered as good prospects since they must fulfill several requirements [2]. Among the survivors, the neutralino and the axion are leading candidates [3]. The question we address in the present work is the following: If DM is mainly composed of ax- ions, what type of astrophysical objects will the axions form? 

In order to answer this question, we have solved the Einstein-Klein-Gordon (EKG) equations in the semiclas- sical limit. The source for the Einstein equations is the mean value of the energy-momentum tensor oper- ator 〈 ˆ T $^{µν}$〉 of a real, quantized scalar field constructed with potential energy density given by [4] 

It is found that the resulting self-gravitating system, the axion star, will have an asteroid-size mass ( M ∼ 10 − $^{16}$M ⊙ ) and radius of a few meters. This work im- proves our previous attempts [5, 6] by solving numerically the EKG system for current allowed values for the axion mass m and decay constant f$_{a}$ without the necessity of any interpolation. Our findings differ from previous es- timates where the effect of the potential energy density was either neglected [7] or was taken into account with a wrong sign in the self-interacting term of the poten- tial [8, 10]. In the first case, it is known that there is 

PACS numbers: 95.35.+d,14.80.Va,04.40.-b,98.35.Gi 

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a maximum mass for such self-gravitating system given by M$_{max}$ = 0 . 633 m 2 $_{p}$/m , where m is the mass associ- ated with the scalar field and m$_{p}$ is the Planck mass. For the allowed values of the axion mass, 10 − 5 eV < m < 10 − 3 eV [4, 9], the maximum mass for a self-gravitating system with the potential energy density (1) neglected lies in the range 10 − 8 M ⊙ < M axion star max < 10 − 5 M ⊙ . On the other hand, when the axion is considered to have a repulsive self-interacting term, instead of the attrac- tive one given by (1), the maximum mass will be as big as M ∼ 10 4 M ⊙ . 

Here we solve the EKG system including a Taylor ex- pansion of the potential energy density (1) and we ob- serve that its inclusion tends to decrease the mass and consequently the compactness of the self-gravitating sys- tem made of axions. Due to the smallness of the axion star’s masses they could play the role of scalar field mini- MACHOs [11] and they will be the final state of axion miniclusters [12] originated in the early universe at the QCD epoch [13]. Assuming that the axion is the main component of DM, the galactic halo will be a collisionless ensemble of axion stars and will be indistinguishable to the standard CDM scenario since N -body simulations of CDM with ultra-high resolution are insensitive to particle mass granularity smaller than 10 $^{5}$M ⊙ − 10 $^{3}$M ⊙ [14, 15]. 

The paper is organized as follows: in section I, the EKG equations for a real, quantized scalar field with a Taylor expansion of the potential energy density (1) are obtained and are solved for arbitrary values of the axion mass m and the decay constant f$_{a}$ . In Sec. II we include the current values of m and f$_{a}$ and we obtain the mass and radius of the axion-stars. We finish section II by commenting on some consequences derived in the case that the axion-star has the properties calculated here. 

