# A Soliton Solution for the Central Dark Masses in Globular Clusters and Implications for the Axiverse

###### Abstract

Compact dark masses of have been derived at the centers of well studied globular clusters orbiting our galaxy, representing of the mass of these compact stellar systems. While it is tempting to conclude these dark masses are long sought examples of “intermediate mass” black holes (IMBH), no confirming radio/X-ray emission is detected and extended sizes of are permitted by the observations that are much larger than the Schwartzchild radius (). Here we offer a standing wave explanation for the observed properties of these dark objects, as a soliton composed of light bosons, , that should develop in the deep gravitational potentials of globular clusters orbiting within the dark halo of our galaxy, from the presence of only a small fraction, , of the dark matter in this form. This would add to the dominant Universal dark matter that is increasingly interpreted as a lighter boson of , implied by the large dark cores of dwarf spheroidal galaxies. Identification of two such light bosonic mass scales of and , favors a generic string theory prediction as dimensional compactification generates a wide, discrete mass spectrum of axionic scalar fields. Observations with improved resolution can test this important theory for the dark matter by resolving our predicted soliton scale, below .

## I Introduction

Light scalar fields are a compelling choice for extending the standard model of particle physics, naturally generating axion-like dark matter with symmetry broken by the simple misalignment mechanism Preskill:1982cy ; Abbott:1982af ; Dine:1982ah ; Khlopov:1985jw . Such fields are generic to string theory from the dynamical compactification to 4 space-time dimensions describing our Universe Svrcek:2006yi . These axionic modes are expected to start out massless for symmetry reasons, subsequently picking up a relatively small mass by non-perturbative tunneling that is typically exponentially suppressedArvanitaki:2009fg ; Acharya:2010zx ; Cicoli:2012sz ; Hui:2016ltb , resulting in a discrete mass spectrum of independent axions spanning many orders of magnitude.

Each axionic field can develop rich structure on the de-Broglie scale Schive2014 ; Schive:2014hza ; Veltmaat:2018dfz ; Ringwald under gravity, summed over the ensemble of these independent axion fields, which has been shown to account for the observed coldness of dark matter and the puzzling properties of dwarf galaxies for a dominant scalar field of Schive2014 ; Schive:2014hza ; Veltmaat:2018dfz ; Ringwald . Most conspicuously, a prominent soliton develops quickly at the center of every bound halo, as identified in the first simulations in this context Schive2014 ; Schive:2014hza ; Mocz:2017wlg . These solitons represent the ground state where self-gravity is balanced by an effective pressure arising from the Uncertainty Principle, yielding a static, centrally located and highly nonlinear density peak, or soliton. The soliton scale depends on the gravitational potential depth and for the favored dominant dark matter this is predicted to be for the Milky Way Schive2014 ; Schive:2014hza ; Chen:2016unw , much smaller than the size of the galaxy. This field may be detected directly by its inherent Compton scale pressure oscillation, at frequency Khmelnitsky:2013lxt . This is feasible using pulsars near the Galactic center for which a sizable 200ns timing residual is predicted on a convenient months timescale that is boosted in amplitude within the relatively high dark matter density within the central soliton DeMartino:2017qsa .

In addition to this axion for the dominant Dark Matter, a lighter axion of may be considered to provide the dynamical dark energy from the associated quantum pressure Kamionkowski:2014zda ; Emami:2016mrt ; Poulin:2018dzj , or as a related, probabilistic consequence of the string “landscape” Tye:2016jzi . Axions that are heavier than may also be anticipated, with sub-dominant but possibly significant contributions to the total dark matter density. Here we focus on the increasingly clear detections of dark masses in the cores of several well studied globular clusters, for which a clear conclusion has yet to emerge regarding their origin as these objects do not show the expected accretion in deep radio/Xray observations that would support a black hole interpretation. Several well studied globular clusters each show evidence of a central dark compact mass with high resolution analyses by Avi-Loeb-2017 ; Mark den Brok ; Lutzgendorf:2012sz ; Lutzgendorf:2013csa for the globular clusters 47Tuc, M15, M79, M62 and M54, respectively.

Below we first relate the axion mass to the radius and mass of the associated soliton under self gravity. We obtain a convenient analytical approximation that can be modified to include the central stellar mass contribution within the soliton radius. We determine the level of the stellar mass inside the predicted dark solitonic core by comparing our final results with the numerical based mass-radius relation in Schive2014 ; Schive:2014hza .

## Ii Analytical expression for the soliton mass-radius relation.

In the following, we find the mass-radius relation for the solitonic phase of a Bose Einstein Condensate (BEC) of a bosonic system. Appealing to simplicity we neglect any self interactions between the boson Namjoo:2017nia and only focus on the self gravity as advocated by Hu:2000ke ; Schive2014 ; Schive:2014hza ; Hui:2016ltb .

### ii.1 Hydrodynamical mapping of the BEC

As it is well studied, Hui:2016ltb ; Chavanis:2011zi , one of the best ways to describe a BEC system is mapping this to a hydrodynamical system. In this picture, we describe the BEC phase with a single coherent wave function, , which is the solution of the Gross-Pitaevskii-Poisson (GPP) equation,

(1) |

The hydrodynamical map is done with the following transformation, . where both of the and are real. Using the following decomposition we define the hydrodynamical variables as, and Substituting these definitions inside Eq. (1), we obtain differential equations analogous to hydrodynamical continuity and Euler equations. While the first equation (Continuity) remain the same, there would be an extra term inside Euler equation,

(2) |

here denotes the quantum potential and it is given by,

(3) |

### ii.2 Effective classical presentation of BEC

The high number density of bosons means that the condensation can be described classically, with total energy summed over the kinetic and potential terms,

(4) |

where denotes the “classical” kinetic energy,

(5) |

here . Moreover, denotes the “quantum” kinetic energy and is given by,

(6) |

And finally refers to the gravitational energy,

(7) |

where the last equality is valid for a spherically symmetric mass profile. For this maximally symmetric configuration we have . Notice that the extra 1/2 coefficient disappears after we restrict the mass to be interior to a sphere of radius .

### ii.3 Gaussian ansatz for the density profile

Here we present the density profile for the BEC system. As it turns out, a Gaussian profile is well matched with the numerical calculation, Hu:2000ke ; Schive2014 ; Schive:2014hza ; Hui:2016ltb . Therefore we select as our density. Using this profile, total energy of the system would become,

(8) | |||||

here is the effective potential of the system and is given by . Adopting a similar technique that determines the Chandrasekhar mass, the stable, time independent “solitonic core” of the system can be found by looking at the critical point of the effective potential. This can be done by neglecting and by computing the critical point of the effective potential, which turns out to be the virial condition. This gives us,

(9) |

which can be confidentially identified as the stable minimum as the second derivative of the effective potential is positive, . Eq. (9) can be rewritten to give us the inferred axion mass,

(10) |

It is worth to compare Eq. (10) with the so-called half mass radius of a self gravitating system, Hui:2016ltb . Quite interestingly they are matched very well. Indeed rewriting the above equation in terms of the half mass radius we get anything the same with a prefactor 0.26 which is in great agreement with 0.33 from Hui:2016ltb . We could also compare our results with what was given in Schive2014 ; Schive:2014hza by Schive-Chiueh-Broadhurst (SCB). Rewriting Eq. (3) at SCB in terms of our mass scale we end up with the following equation,

(11) |

Indeed their numerically derived relation is in close agreement with our analytical relation, up to a factor of order unity, reinforcing our approach. We can now take one more step and make use of our relation to estimate the contribution of luminous stellar mass within our predicted soliton core at the center of the GCs where high star densities are typical.

### ii.4 The effect of interior luminous matter on the mass-radius relation

So far we have neglected the back-reaction of the luminous matter on the above solitonic mass-radius relation. Here we model this effect. This arises from the contribution of the luminous matter on the gravitational potential. The interaction gravitational energy is equal to,

(12) |

where refers to the gravitational potential from the luminous matter. Hereafter, sub index refers to the luminous matter. For a spherical distribution of the matter, it is given as, . Finally . Here has the following ansatz in our solitonic system, James HH Chan ,

(13) |

where and denote the globular cluster mass and radius, respectively which are related through the velocity dispersion as, . In addition, refers to the solitonic radius. We put the sub-index temporary to avoid doing the variation with respect to this parameter. We will identify this with after the variation. Plugging the above profile back inside Eq. (12), we have the following expression for the gravitational energy as,Bar:2018acw ,

(14) |

This would be added up into the effective potential of the system. Therefore the “modified” potential would be,

This modifies the critical point of the system as,

therefore the new mass-radius relation at would be,

(17) |

where we have dropped the sub-index from . This is the most important expression that we have found in this work. In Fig. 1 we present the behavior of the mass-radius relation for different reasonable values of the boson mass as well as the velocity dispersion. Hereafter we define

GCs | soliton Mass () | soliton Radius (pc) | Axion Mass ( eV) | Ref |
---|---|---|---|---|

47 Tuc | Avi-Loeb-2017 | |||

M15 | Mark den Brok | |||

M79 | Lutzgendorf:2012sz | |||

M62 | Lutzgendorf:2012sz | |||

M54 | Lutzgendorf:2013csa |

Next we could draw the density profile for the soliton, stellar and their combination (soliton + stellar). This gives us an intuitive picture of how they behave. We present this behavior in Fig. 2. We consider two different examples of the GCs. Here (1) means , and pc while (2) refers to , and pc.

Having presented the generalized mass-radius relation, we could finally try to estimate the axion mass for some very well studied populations of the Globular Clusters with a proposed dense core as listed in Tab. 1. In Fig. 3 we present the inferred axion mass in terms of the soliton radius for these populations. We end up with the following mass range for the axion .

## Iii Conclusions

We have shown that the IMBH interpretation for dark central masses claimed for several well studied GCs, is not the only physically viable explanation. A light scalar field can also generate a sufficiently compact dark mass corresponding to an axion of given the limited resolution of the dynamical data to sub parsec. This has the advantage over an IMBH interpretation as it does not then conflict with the stringent lack of gas accretion affecting he credibility of the IMBH interpretation. Hence, it a major priority is to obtain better constrain better the radius of this central mass which if resolved into a distinctive top-hat density profile of a soliton would be of great importance for our understanding of the nature of dark matter.

It may also be tempting to associate the origin of super-massive black holes with the formation of such compact solitons of , which can represent the long sought ”seed” that is understood to promote the early formation of SMBH, for which massive high redshift examples of up to are a challenge to physical models of gas and dark matter. The physically complex link between such black holes and axionic scalar fields is being explored Helfer:2016ljl .

The greatest physical importance of our soliton interpretation is in relation to String Theory. Such a axion together with the lighter , as a viable candidate for the dark matter, and much lighter axion, , to be responsible for the current expansion of the universe, Kamionkowski:2014zda , could greatly support the idea of an ”Axiverse” Arvanitaki:2009fg ; Acharya:2010zx ; Cicoli:2012sz , of a discrete mass spectrum of several light axions spanning a wide range of axion mass, generically resulting from higher dimensional compactification.

## Acknowledgments

We are grateful to Henry Tye for the fruitful discussions. The work of R.E. was supported by Hong Kong University through the CRF Grants of the Government of the Hong Kong SAR under HKUST4/CRF/13. GFS acknowledges the IAS at HKUST and the Laboratoire APC-PCCP, Université Paris Diderot and Sorbonne Paris Cité (DXCACHEXGS) and also the financial support of the UnivEarthS Labex program at Sorbonne Paris Cité (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02). TJB thanks IAS hospitality, SDSS/BOSS data etc. TC acknowledges the grant MOST 103-2112-M-002-020-MY3 of Ministry of Sciences and Technologies, Taiwan.

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