Derivation of the core mass -- halo mass relation of fermionic and bosonic dark matter halos from an effective thermodynamical model
Pierre-Henri Chavanis
TL;DR
This paper develops a thermodynamical model to derive the core mass-halo mass relation in quantum dark matter halos, explaining core-halo structures for fermions and bosons with predictions matching numerical results.
Contribution
It introduces a thermodynamical framework to analytically derive core-halo mass relations for fermionic and bosonic dark matter halos, including effects of self-interactions.
Findings
Fermionic halos follow M_c ∝ M_v^{1/2} relation.
Noninteracting bosonic halos follow M_c ∝ M_v^{1/3} relation.
Self-interacting bosonic halos follow M_c ∝ M_v^{2/3} relation.
Abstract
We consider the possibility that dark matter halos are made of quantum particles such as fermions or bosons in the form of Bose-Einstein condensates. In that case, they generically have a "core-halo" structure with a quantum core that depends on the type of particle considered and a halo that is relatively independent of the dark matter particle and that is similar to the NFW profile of cold dark matter. We model the halo by an isothermal gas with an effective temperature . We then derive the core mass -- halo mass relation of dark matter halos from an effective thermodynamical model by extremizing the free energy with respect to the core mass . We obtain a general relation that is equivalent to the "velocity dispersion tracing" relation according to which the velocity dispersion in the core is of the same order as the velocity dispersion…
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Derivation of the core mass – halo mass relation of fermionic and
bosonic dark matter halos from an effective thermodynamical model
Pierre-Henri Chavanis
Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France
Abstract
We consider the possibility that dark matter halos are made of quantum particles such as fermions or bosons in the form of Bose-Einstein condensates. In that case, they generically have a “core-halo” structure with a quantum core that depends on the type of particle considered and a halo that is relatively independent of the dark matter particle and that is similar to the NFW profile of cold dark matter. The quantum core is equivalent to a polytrope of index for fermions, for noninteracting bosons, and for bosons with a repulsive self-interaction in the Thomas-Fermi limit. We model the halo by an isothermal gas with an effective temperature . We then derive the core mass – halo mass relation of dark matter halos from an effective thermodynamical model by maximizing the entropy with respect to the core mass at fixed total mass and total energy. We obtain a general relation, valid for an arbitrary polytropic core, that is equivalent to the “velocity dispersion tracing” relation according to which the velocity dispersion in the core is of the same order as the velocity dispersion in the halo . We provide therefore a justification of this relation from thermodynamical arguments. In the case of fermions, we obtain a relation that agrees with the relation found numerically by Ruffini et al. [Mon. Not. R. Astron. Soc. 451, 622 (2015)]. In the case of noninteracting bosons, we obtain a relation that agrees with the relation found numerically by Schive et al. [Phys. Rev. Lett 113, 261302 (2014)]. In the case of bosons with a repulsive self-interaction in the Thomas-Fermi limit, we predict a relation that still has to be confirmed numerically. Using a Gaussian ansatz, we obtain a general approximate core mass – halo mass relation that is valid for bosons with arbitrary repulsive or attractive self-interaction. For an attractive self-interaction, we determine the maximum halo mass that can harbor a stable quantum core (dilute axion “star”). Above that mass, the quantum core collapses. Finally, we argue that the fundamental mass scale of the bosonic dark matter particle is and that the fundamental mass scale of the fermionic dark matter particle is , where is the cosmological constant and is the Planck mass. Their ratio is which explains the difference of mass between fermions and bosons in dark matter models. The actual value of the dark matter particle mass is equal to these mass scales multiplied by a large factor that we obtain from our model.
pacs:
95.30.Sf, 95.35.+d, 98.62.Gq
I Introduction
The nature of dark matter (DM) is still unknown and remains one of the greatest mysteries of modern cosmology. The standard cold dark matter (CDM) model works remarkably well at large (cosmological) scales and is consistent with ever improving measurements of the cosmic microwave background (CMB) from WMAP and Planck missions planck2013 ; planck2016 . However, it encounters serious problems at small (galactic) scales. In particular, it predicts that DM halos should be cuspy nfw , with a density diverging as for , while observations reveal that they have a flat core density observations . On the other hand, the CDM model predicts an over-abundance of small-scale structures (subhalos/satellites), much more than what is observed around the Milky Way satellites . These problems are referred to as the “cusp problem” and “missing satellite problem”. The expression “small-scale crisis of CDM” has been coined.
In order to solve these problems, some authors have proposed to take the quantum nature of the DM particle into account.111See our previous papers prd1 ; modeldm for an exhaustive list of references (more than ) on the subject. See also the reviews srm ; rds ; chavanisbook ; marshrevue ; leerevue ; braatenrevue . Indeed, quantum mechanics creates an effective pressure even at zero thermodynamic temperature () that may balance the gravitational attraction at small scales and lead to cores instead of cusps. The DM particle could be a fermion, like a massive neutrino, with a mass (see Appendix D of suarezchavanis3 ). It could also be a boson in the form of a Bose-Einstein condensate (BEC), like an ultralight axion, with a mass in the range depending whether the bosons are noninteracting or self-interacting (see Appendix D of suarezchavanis3 ).
In these quantum models, DM halos have a “core-halo” structure which results from a process of violent collisionless relaxation lb and gravitational cooling seidel94 ; gul0 ; gul . The core stems from the equilibrium between quantum pressure and gravitational attraction. For fermions, the quantum pressure arises from the Pauli exclusion principle like in the case of white dwarfs and neutron stars. For bosons, the quantum pressure arises from the Heisenberg uncertainty principle or from the repulsive self-interaction of the bosons like in the case of boson stars. Quantum mechanics stabilizes the halo against gravitational collapse,222This is true for the nonrelativistic systems that we consider here. For general relativistic systems, there is a maximum mass ov ; kaup ; rb ; colpi ; tkachev ; chavharko above which the system collapses towards a black hole. In our case, we will find that so that a Newtonian approach is sufficient. On the other hand, if the bosons have an attractive self-interaction, like in the case of the axion marshrevue , there exists a maximum mass prd1 for the quantum core even in the Newtonian regime. Above that limit, the quantum core (dilute axion star) undergoes gravitational collapse. leading to a flat core instead of a cusp. The quantum core is equivalent to a polytrope of index for fermions, for noninteracting bosons, and for bosons with a repulsive self-interaction in the Thomas-Fermi (TF) limit. It is responsible for the finite density of the DM halos at the center. The core mass-radius relation is for fermions, for noninteracting bosons, and for self-interacting bosons in the TF limit. On the other hand, the halo is relatively independent of quantum effects and is similar to the Navarro-Frenk-White (NFW) profile nfw produced in CDM simulations or to the empirical Burkert profile observations deduced from the observations. It is responsible for the flat rotation curves of the galaxies at large distances. We shall approximate this halo by an isothermal atmosphere with an effective temperature .333We approximate the atmosphere by an isothermal sphere but we stress that the temperature is effective and does not correspond to the true thermodynamic temperature (which is almost equal to zero). In particular, the atmosphere does not correspond to a statistical equilibrium state resulting from a “collisional” evolution of the quantum particles of mass which would be much too long (much larger than the age of the Universe) modeldm . It may rather correspond to an out-of-equilibrium thermodynamical state - or quasistationary state - resulting from a collisionless free fall evolution (independent of ) like in Lynden-Bell’s theory of violent relaxation lb . In the case of fuzzy DM, the approximately isothermal atmosphere (due to quantum interferences of excited states) may result either from a collisionless relaxation or from the “collisional” evolution of quasiparticles (granules) of the size of the solitonic core and of mass as argued in ch2 ; ch3 ; hui ; bft . In that case, the density decreases at large distance as bt , instead of for the NFW and Burkert profiles, leading exactly to flat rotation curves for . For sufficiently large halos, the halo mass-radius relation is modeldm where
[TABLE]
is the universal surface density of DM halos deduced from the observations kormendy ; spano ; donato . Ultracompact halos like dSphs ( and ) are dominated by the quantum core and have almost no atmosphere. Large halos like the Medium Spiral ( and ) are dominated by the isothermal atmosphere.
In a recent paper modeldm , we have developed of model of DM halos made of bosons with a repulsive self-interaction in the TF limit. We have obtained a generic phase diagram (see Fig. 49 of modeldm ) determining the structure of the DM halos (measured by the core mass ) as a function of their mass . There is a minimum halo mass corresponding to the ground state of the boson gas () at which the DM halo is a purely quantum object without isothermal atmosphere (). We will call it the “minimum halo”. Larger halos have a “core-halo” structure with a quantum core, representing a “nucleus” or a “bulge”, and an isothermal atmosphere. We found a branch along which the core mass decreases as the halo mass increases. Rapidly, the core mass becomes negligible and the halos behave as purely isothermal halos without quantum core. However, we found a critical point , that we interpreted as a canonical critical point, at which a bifurcation occurs. On the new branch, the core mass increases as the halo mass increases. On that branch, we found another critical point at a higher mass , that we interpreted as a microcanonical critical point, above which the quantum core becomes unstable and is replaced by a supermassive black hole resulting from a gravothermal catastrophe lbw followed by a dynamical instability of general relativistic origin balberg . Considering the bifurcated branch with the “core-halo” structure, we developed an effective thermodynamical model to analytically predict the core mass – halo mass relation .444This thermodynamical model was originally introduced in pt to analytically obtain the caloric curves of self-gravitating fermions. We showed that this relation is equivalent to the “velocity dispersion tracing” relation according to which the velocity dispersion in the core is of the same order as the velocity dispersion in the halo mocz ; bbbs ; modeldm . We could provide therefore a justification of this relation from thermodynamical arguments.
In the present paper, we extend this thermodynamical model to the case of DM halos made of fermions and to the case of DM halos made of noninteracting bosons. To unify the formalism, we model the quantum core as a polytrope of arbitrary index (with for fermions, for noninteracting bosons, and for self-interacting bosons in the TF limit) and we model the atmosphere as an isothermal gas with a uniform density confined within a “box” of radius . The radius of the box is identified with the halo radius and the total mass of the system contained within the box (core halo) is identified with the halo mass . They are related by modeldm . We analytically compute the entropy of the system. By maximizing as a function of for a given value of the mass , radius and energy , we obtain the core mass as a function of the halo mass . We find this relation to be always (for any value of ) equivalent to the velocity dispersion tracing relation, thereby generalizing our previous result modeldm . Using a Gaussian ansatz, we obtain a general approximate relation that is valid for bosons with arbitrary repulsive or attractive self-interaction. For an attractive self-interaction, we determine the maximum halo mass that can harbor a stable quantum core (dilute axion “star”). Finally, we use our results to predict the fundamental mass scale of the bosonic or fermionic DM particle in terms of the fundamental constants of physics.
The paper is organized as follows. In Sec. II we consider models of DM halos with a quantum (fermionic or bosonic) core and an isothermal atmosphere. In Sec. III we show that these models can be obtained in a unified manner from a generalized wave equation introduced in ggp ; modeldm ; nottalechaos . In Sec. IV we obtain the core mass – halo mass relation of DM halos from an analytical thermodynamical model. This relation is valid for a general polytropic core. In Sec. V we show that this relation is equivalent to the velocity dispersion tracing relation. In Secs. VI and VII we specifically apply these results to DM halos made of fermions, noninteracting bosons and self-interacting bosons.
II Quantum models of DM halos
In this section, we review quantum models of DM halos made of fermions or bosons. If DM halos are quantum objects, there must be a minimum halo radius and a minimum halo mass in the Universe corresponding to the ground state () of the self-gravitating quantum gas. This result is in agreement with the observations. Indeed, there are apparently no DM halos with a radius smaller than and a mass smaller than , the typical values of the radius and mass of dSphs like Fornax. This observational result cannot be explained by the CDM model which predicts the existence of DM halos at all scales.
Ultracompact dwarf DM halos just have a quantum core without atmosphere (ground state). Larger DM halos have a core-halo structure with a quantum core corresponding to the ground state () of the quantum gas and an “atmosphere” resulting from violent relaxation and gravitational cooling. The atmosphere has a density profile that can be fitted by the empirical Burkert profile or by NFW profile. In this paper, we shall approximate this density profile by an isothermal profile of effective temperature . It is the atmosphere that fixes the size of large halos and explains why their radius increases with their mass as (since is constant). By contrast, the radius of the quantum core usually decreases or remains constant as its mass increases (see below).
To determine the parameters of the DM particle, we proceed as follows (see Appendix D of suarezchavanis3 ).555Our aim here is not to make an accurate model of DM halos. Therefore, an order of magnitude of the DM particle parameters is sufficient. We assume that the smallest halo that has been observed, with a typical radius and a typical mass
[TABLE]
corresponds to the ground state of a self-gravitating quantum gas. Using the mass-radius relation of the self-gravitating quantum gas at , we can obtain the parameters of the DM particle. We can then check that the nonrelativistic treatment used in this paper is valid by showing that .
II.1 Fermionic DM
The equation of state of a nonrelativistic Fermi gas at is chandrabook
[TABLE]
This is a polytropic equation of state of index (i.e. ) and polytropic constant
[TABLE]
In the TF approximation, which amounts to neglecting the quantum potential, the fundamental differential equation of hydrostatic equilibrium determining the density profile of a nonrelativistic fermion star666In this paper, the name “fermion star” refers to ultracompact DM halos made of fermions or to the fermionic core of larger DM halos (the same comment applies to the names “boson stars”, “BEC stars” and “axion stars” used below). at with the equation of state (3) writes (see Appendix A)
[TABLE]
It can be reduced to the Lane-Emden equation (248) of index . This profile has a compact support (see Fig. 1) and the fermion star is stable. The mass-radius relation is
[TABLE]
where . On the other hand, from Eqs. (270) and (272) which reduce to777In the main text, we denote by , and what are called , and in the Appendices.
[TABLE]
[TABLE]
and from the Betti-Ritter formula (278), we obtain
[TABLE]
Combined with Eq. (6), we find that the energy of a nonrelativistic fermion star at (ground state) is
[TABLE]
Let us assume that the smallest DM halo that we know, with mass and radius , corresponds to the ground state of a nonrelativistic self-gravitating Fermi gas. From the mass-radius relation (6), we get
[TABLE]
Using the reference values of and corresponding to Fornax [see Eq. (2)], we find a fermion mass (see Appendix D of suarezchavanis3 ):
[TABLE]
Alternatively, using the results of Appendix G and taking in the numerical applications, we find that the minimum halo radius, the minimum halo mass and the maximum central density are
[TABLE]
[TABLE]
[TABLE]
where is the universal density of DM halos given by Eq. (1). These values can be improved if we have a more reliable expression of the fermion mass , but they are sufficient for our purposes (the same comment applies to the bosonic models considered below).
The maximum mass of a fermion star at set by general relativity is and its minimum radius is ov . They can be written as
[TABLE]
For a fermion of mass , we obtain and . The maximum mass is much larger than the typical core mass of a DM halo. Assuming that a fermion star at describes the quantum core of a DM halo, we conclude that such cores are nonrelativistic since in general. Since the maximum mass is much larger than the core mass, gravity can be treated within a Newtonian framework.
II.2 Noninteracting bosonic DM
We consider a gas of noninteracting bosons at forming a BEC. The wavefunction of a self-gravitating BEC without self-interaction is governed by the Schrödinger-Poisson equation prd1 . Using Madelung’s hydrodynamic representation of the Schrödinger equation madelung , we find that the fundamental differential equation of quantum hydrostatic equilibrium determining the density profile of the BEC is prd1
[TABLE]
This equation can be solved numerically rb ; membrado ; gul0 ; gul ; prd2 ; ch2 ; ch3 ; pop ; hui . The density profile of a noninteracting BEC star at (ground state) extends to infinity (see Fig. 2) and the BEC star is stable. The mass-radius relation is
[TABLE]
where is the radius enclosing of the mass. From Eqs. (264) and (269) which reduce to
[TABLE]
[TABLE]
we obtain
[TABLE]
From numerical computations rb ; membrado ; prd2 , we find that the total energy of a noninteracting BEC star at (ground state) is
[TABLE]
Actually, we find in Appendix E that there exists another solution of Eq. (17). It has a compact support (see Fig. 2) and its profile corresponds to a polytrope of index (i.e. ). Its energy is lower than the energy of the solution considered here, suggesting that it is more stable, even if comparing the energies of stable states may not be decisive in view of the very long lifetime of metastable states in systems with long-range interactions. In the following, in order to develop a unified description of fermions and bosons based on polytropic equations of state, we will use the solution from Appendix E. However, as far as scalings and orders of magnitude are concerned, we would get similar results by using the more conventional (but maybe less stable) solution of this section.
Let us assume that the smallest DM halo that we know, with mass and radius , corresponds to the ground state of a noninteracting BEC star. From the mass-radius relation (18), we get
[TABLE]
Using the reference values of and corresponding to Fornax [see Eq. (2)], we find a boson mass (see Appendix D of suarezchavanis3 ):
[TABLE]
Alternatively, using the results of Appendix G and taking in the numerical applications, we find that the minimum halo radius, the minimum halo mass and the maximum central density are (computed from the polytropic solution of Appendix E)
[TABLE]
[TABLE]
[TABLE]
where is the universal density of DM halos given by Eq. (1).
The maximum mass of a noninteracting boson star at set by general relativity is and its minimum radius is kaup ; rb . They can be written as
[TABLE]
For a boson of mass , we obtain and . The maximum mass is much larger than the typical core mass of a DM halo. Assuming that a BEC star at (soliton) describes the quantum core of a DM halo, we conclude that such cores are nonrelativistic since in general. Since the maximum mass is much larger than the core mass, gravity can be treated within a Newtonian framework.
II.3 Bosonic DM with a repulsive self-interaction in the TF limit
We consider a gas of self-interacting bosons at forming a BEC. The wavefunction of a self-gravitating BEC with a quartic self-interaction is governed by the GPP equations prd1 . The equation of state of a self-interacting BEC is prd1
[TABLE]
where is the scattering length of the bosons. This is a polytropic equation of state of index (i.e. ) and polytropic constant
[TABLE]
We assume that the self-interaction is repulsive (). Using Madelung’s hydrodynamic representation of the GPP equations, and taking the quantum potential into account, we find that the fundamental differential equation of quantum hydrostatic equilibrium determining the density profile of the BEC is prd1
[TABLE]
This equation can be solved numerically prd2 . The density profile of a noninteracting BEC star at (ground state) extends to infinity. The mass-radius relation has been obtained in prd1 ; prd2 .
In the TF approximation, which amounts to neglecting the quantum potential, the fundamental differential equation of hydrostatic equilibrium determining the density profile of a self-interacting BEC star at with the equation of state (29) writes (see Appendix A)
[TABLE]
It can be reduced to the Lane-Emden equation (248) of index which has a simple analytical solution chandrabook . This profile has a compact support (see Fig. 3) and the self-interacting BEC star is stable. A self-gravitating BEC with a repulsive self-interaction in the TF approximation has a unique radius tkachev ; maps ; leekoh ; goodman ; arbey ; bohmer ; prd1 ,
[TABLE]
that is independent of its mass. From Eqs. (270) and (272) which reduce to
[TABLE]
[TABLE]
and from the Betti-Ritter formula (278), we obtain
[TABLE]
Combined with Eq. (33), we find that the energy of a self-interacting BEC star at (ground state) in the TF limit is
[TABLE]
Let us assume that the smallest DM halo that we know, with mass and radius , corresponds to the ground state of a self-interacting BEC star. From Eq. (33), we get
[TABLE]
This formula depends only on . Using the reference value of corresponding to Fornax [see Eq. (2)], we find that the ratio of the boson parameters and is given by (see Appendix D of suarezchavanis3 ):
[TABLE]
In order to determine the mass of the boson, we need another relation. In this respect, we note that and must satisfy the constraint set by the Bullet Cluster bullet , where is the self-interaction cross section. In Appendix D of suarezchavanis3 we have considered two extreme cases corresponding to an upper boson mass
[TABLE]
and a lower boson mass
[TABLE]
The upper boson mass (40) corresponds to the bound . The lower boson mass (41) corresponds to the transition between the TF regime and the noninteracting regime. We note that when a self-interaction between the bosons is allowed, a large mass window is open. In particular, a repulsive self-interaction allows one to have a larger boson mass than in the noninteracting case. As discussed in Appendix D.4 of suarezchavanis3 this may be interesting in view of the fact that the mass of a noninteracting boson () is in tension with observations of the Lyman- forest hui . This tension could reflect the fact that bosons have a repulsive self-interaction.
Alternatively, using the results of Appendix G and taking in the numerical applications, we find that the minimum halo radius, the minimum halo mass and the maximum central density are
[TABLE]
[TABLE]
[TABLE]
where is the universal density of DM halos given by Eq. (1).
The maximum mass of a self-interacting boson star set by general relativity is and its minimum radius is colpi ; tkachev ; chavharko . They can be written as
[TABLE]
[TABLE]
We note that these results do not depend on the specific mass and scattering length of the bosons, but only on the ratio . For a ratio , we obtain and . The maximum mass is much larger than the typical core mass of a DM halo. Assuming that a self-interacting BEC star at describes the quantum core of a DM halo, we conclude that such cores are nonrelativistic since in general. Since the maximum mass is much larger than the core mass, gravity can be treated within a Newtonian framework.
II.4 Bosonic DM with an attractive self-interaction
We consider a gas of self-interacting bosons at forming a BEC. The wavefunction of a self-gravitating BEC with a quartic self-interaction is governed by the GPP equations prd1 . The equation of state of a self-interacting BEC is given by Eq. (29). We assume that the self-interaction is attractive (). This is the case for the axion marshrevue . Using Madelung’s hydrodynamic representation of the GPP equations, and taking the quantum potential into account, we find that the fundamental differential equation of quantum hydrostatic equilibrium determining the density profile of the BEC is given by Eq. (31) prd1 . This equation can be solved numerically prd2 . The density profile of a noninteracting BEC star at (ground state) extends to infinity. The mass-radius relation has been obtained in prd1 ; prd2 . There is a maximum mass prd1
[TABLE]
corresponding to a minimum stable radius
[TABLE]
When the axion star is expected to collapse and form a dense axion star, a black hole or a bosenova as discussed in braaten ; cotner ; bectcoll ; ebycollapse ; tkachevprl ; helfer ; phi6 ; visinelli ; moss .
II.5 Mass-radius relation of isothermal DM halos
In the previous subsections, we have focused on the quantum core of DM halos. Ultracompact dwarf DM halos just have a quantum core (ground state). Larger DM halos have a “core-halo” structure with a quantum core surrounded by an atmosphere. We have seen that the structure of the quantum core strongly depends on the nature of the DM particle. By contrast, the structure of the atmosphere is relatively independent of the DM particle. We assume that it has an isothermal equation of state
[TABLE]
with an effective temperature . For sufficiently large DM halos, the isothermal atmosphere dominates the core. Indeed, the DM halo mass is much larger than the core mass and it is a good approximation to assume that the DM halo is purely isothermal.888We have in mind, for example, the Medium Spiral ( and ). Therefore, from the “outside”, large DM halos behave as classical isothermal spheres. If represents the halo mass and the halo radius as defined in Appendix G, then the mass-radius and temperature-radius relations of isothermal DM halos are modeldm
[TABLE]
where is the universal surface density of DM halos given by Eq. (1). On the other hand, the circular velocity at the halo radius is
[TABLE]
III A generalized wave equation
III.1 Coarse-grained dynamics
The previous results can be obtained in a unified manner from the generalized GPP equations ggp ; modeldm ; nottalechaos
[TABLE]
[TABLE]
As discussed in more detail in ggp ; modeldm , the thermal () and dissipative () terms present in the generalized GPP equations (III.1) and (53) parametrize the complicated processes of violent relaxation lb and gravitational cooling seidel94 experienced by a collisionless system of self-gravitating fermions or bosons. As a result, the generalized GPP equations (III.1) and (53) describe the evolution of the system on a “coarse-grained” scale.
III.2 Madelung transformation
Making the Madelung transformation
[TABLE]
[TABLE]
we can show that the generalized GPP equations (III.1) and (53) are equivalent to the fluid equations
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
is the quantum potential and is the pressure determined by the equation of state
[TABLE]
This equation of state has a linear part and a polytropic part. The linear (isothermal) equation of state accounts for effective thermal effects. The polytropic equation of state takes into account the self-interaction of the bosons or the quantum pressure arising from the Pauli exclusion principle for fermions. The equation of state (60) defines a composite model of DM halos with a core-halo structure. The polytropic equation of state dominates in the core where the density is high and the isothermal equation of state dominates in the halo where the density is low (we assume that ). As a result, the corresponding DM halos present a quantum (fermionic/bosonic) core surrounded by an isothermal envelope. The quantum core solves the cusp problem and the isothermal envelope leads to flat rotation curves. This model has been studied in detail in modeldm for self-interacting BECs. Its extension to noninteracting BECs and fermions will be presented in a forthcoming paper.
III.3 Condition of quantum hydrostatic
equilibrium
The equilibrium state of the hydrodynamic equations (56) and (57) satisfies the condition of quantum hydrostatic equilibrium ggp
[TABLE]
It describes the balance between the quantum potential arising from the Heisenberg uncertainty principle, the quantum pressure (due to the Pauli exclusion principle for fermions or due to the self-interaction of the bosons), the pressure due to the effective temperature, and the gravitational attraction. Combining Eq. (61) with the Poisson equation (58), we obtain the fundamental differential equation of quantum hydrostatic equilibrium ggp
[TABLE]
For the equation of state (60), it takes the form
[TABLE]
This differential equation determines the general equilibrium density profile of a quantum DM halo in our model ggp ; modeldm . This profile generically has a core-halo structure with a polytropic core and an isothermal halo.
In the core, the differential equation (63) reduces to
[TABLE]
It determines the structure of the quantum core as described in Secs. II.1-II.4.
In the halo, the differential equation reduces to
[TABLE]
It is equivalent to the Emden equation chandrabook . It determines the structure of the isothermal atmosphere of large DM halos as described in Sec. II.5.
III.4 Free energy
The free energy associated with the generalized GPP equations (III.1) and (53) or equivalently with the hydrodynamic equations (56) and (57) is
[TABLE]
The energy is the sum of the classical kinetic energy
[TABLE]
the quantum kinetic energy
[TABLE]
the internal energy associated with the polytropic equation of state
[TABLE]
and the gravitational energy
[TABLE]
On the other hand,
[TABLE]
is the Boltzmann entropy associated with the isothermal equation of state.
The generalized GPP equations (III.1) and (53) or equivalently the hydrodynamic equations (56) and (57) satisfy an -theorem ggp
[TABLE]
The free energy decreases monotonically when (or is constant when ). At equilibrium, we have implying . Then, Eq. (57) leads to the condition of quantum hydrostatic equilibrium (61). When , using Lyapunov’s direct method, one can show that the system relaxes, for , towards a stable equilibrium state which is a (local) minimum of free energy at fixed mass.
The extremization of the free energy at fixed mass, corresponding to the variational principle where is a Lagrange multiplier, returns the condition of quantum hydrostatic equilibrium (61). Furthermore, (local) minima of free energy are stable while maxima or saddle points are unstable.
The generalized GPP equations (III.1) and (53) are associated with a canonical description in which the temperature is fixed. It is possible to modify these equations so that the temperature evolves in time in order to conserve the energy (see Appendix I of ggp ). This corresponds to a microcanonical description. As is well-known ijmpb , the equilibrium states are the same in the microcanonical and canonical ensembles. However, their stability may be different in case of ensembles inequivalence. In particular, equilibrium states that are unstable in the canonical ensemble may be stable in the microcanonical (this is because the microcanonical ensemble is more constrained than the canonical ensemble). For example, equilibrium states with a negative specific heat are always unstable in the canonical ensemble while they may be stable in the microcanonical ensemble. This property will play a fundamental role in the following analysis.
IV Analytical model of DM
halos with a polytropic core and an isothermal atmosphere
In this section, we develop an approximate analytical model of DM halos with a quantum core surrounded by an isothermal atmosphere. For simplicity, we assume that the density of the isothermal atmosphere is uniform. In all the DM models discussed in Sec. II the quantum core can be described by a polytropic equation of state. Therefore, by considering a polytropic core with an arbitrary index , we can account for a wide diversity of situations and, in particular, unify the treatment of fermionic and bosonic DM halos. We shall enclose the system within a box of radius . The box is necessary to have a finite mass . In order to connect this model with real DM halos, we shall identify the box radius with the halo radius and the mass with the halo mass . Therefore, we set
[TABLE]
The mass and the radius of sufficiently large DM halos are related to each other by the first relation of Eq. (50).
IV.1 Polytropic core
We modelize the core of a DM halo by a pure polytrope of index . Its mass and its radius satisfy the mass-radius relation (see Appendix A) chandrabook
[TABLE]
The internal energy and the gravitational energy of the polytropic core are given by (see Appendices B and C)
[TABLE]
Therefore, its total energy is
[TABLE]
Combined with Eq. (74), we obtain
[TABLE]
IV.2 Isothermal atmosphere of uniform density
We modelize the halo by an isothermal atmosphere of mass contained between the spheres of radius and . The internal energy of a gas with the isothermal equation of state (49) is ggp
[TABLE]
It can be rewritten as
[TABLE]
where is the de Broglie wavelength. Treating the atmosphere as a gas with a uniform density, we obtain
[TABLE]
where is the total volume of the system. On the other hand, the gravitational energy of the uniform atmosphere in the presence of the “external” polytropic core is given by (see Appendix F)
[TABLE]
To obtain these results, we have assumed that which is a very good approximation in all cases of physical interest.
IV.3 Free energy
Using the foregoing results, the total free energy of the system (core halo) is
[TABLE]
For a given value of , and , the free energy is a function of the core mass. The extrema of this function determine the possible equilibrium states of the system. More precisely, they determine the possible equilibrium core masses as a function of , and . We recall that the equilibrium states are the same in the canonical and in the microcanonical ensembles. Indeed, the extrema of at fixed mass coincide with the extrema of at fixed mass and energy. However, their stability may be different in the canonical and in the microcanonical ensembles. This is the important notion of ensembles inequivalence for systems with long-range interactions ijmpb . In the canonical ensemble, a minimum of at fixed mass corresponds to a stable equilibrium state (most probable state) while a maximum of at fixed mass corresponds to an unstable equilibrium state (less probable state). In the microcanonical ensemble, a maximum of at fixed mass and energy corresponds to a stable equilibrium state (most probable state) while a minimum of at fixed mass and energy corresponds to an unstable equilibrium state (less probable state). We first consider the canonical ensemble. The microcanonical ensemble is treated in Sec. IV.8.
It is convenient to introduce the dimensionless variables
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and
[TABLE]
so that Eq. (82) can be rewritten as
[TABLE]
with . On the other hand, introducing
[TABLE]
the core mass-radius relation (74) can be written as
[TABLE]
The condition that () when () implies .
IV.4 Connection to DM halos
Before going further, let us connect the dimensionless variables introduced previously to the parameters of the DM halos. The variable represents the normalized core mass. Using Eq. (73) it can be written as
[TABLE]
The variable represents the normalized inverse temperature. For DM halos, using Eqs. (50) and (73), we get
[TABLE]
We see that the normalized inverse temperature is of order . This is essentially a consequence of the virial theorem. Since our approach is approximate, we will allow to vary slightly around this value. To fix the ideas we take . Finally, the variable characterizes the mass of the DM halos. For fermionic DM halos, we have999This parameter is related to the “degeneracy parameter” introduced in ijmpb and given by Eq. (86). We have
(93)
[TABLE]
Using Eqs. (50) and (73), we get
[TABLE]
Normalizing the halo mass by the minimum halo mass from Eq. (14) we obtain
[TABLE]
For noninteracting BECDM halos, we have
[TABLE]
Using Eqs. (50) and (73), we get
[TABLE]
Normalizing the halo mass by the minimum halo mass from Eq. (26) we obtain
[TABLE]
For self-interacting BECDM halos in the TF limit, we have101010This parameter is related to the parameter introduced in modeldm . We have .
[TABLE]
Using Eqs. (50) and (73), we get
[TABLE]
Normalizing the halo mass by the minimum halo mass from Eq. (43) we obtain
[TABLE]
IV.5 Equilibrium states
The equilibrium states of DM halos, corresponding to , are the solutions of the equation
[TABLE]
This equation determines the normalized core mass as a function of , and . It is convenient to introduce the notation
[TABLE]
so we can rewrite Eq. (103) as
[TABLE]
We note that depends weakly (logarithmically) on and so, in a first approximation, it can be treated as a constant. The solutions of Eq. (103) can be easily found by studying the inverse function
[TABLE]
for a given value of and (see Fig. 4). Our analytical model is valid for sufficiently large values of and (corresponding to large DM halos). On the other hand, the results depend on the value of the polytropic index . In the following, we assume .111111The index is special and has been treated in modeldm . The condition is required in order to have a stable core chandrabook . For , we get
[TABLE]
Close to , the curve is always increasing. For , we get
[TABLE]
where we have assumed in order to avoid unphysical results due to the invalidity of our model for small values of .
We note that the inverse temperature becomes infinite at some when the denominator in Eq. (106) vanishes, i.e., when
[TABLE]
Instead of solving Eq. (109) for as a function of , it is simpler to study the inverse function
[TABLE]
This function (not represented) has the following properties: (i) is positive provided that ; (ii) ; (iii) there is a maximum
[TABLE]
at . In order to avoid unphysical results related to the divergence of the inverse temperature at some , we assume that . We find for and for . In a sense, this critical value is the counterpart of the canonical critical point that appears in the exact caloric curve of self-gravitating fermions and bosons (see Fig. 32 of ijmpb for fermions) although it manifests itself in a singular manner in our simple analytical model. Its exact value for fermions is corresponding to ijmpb . When several equilibrium states exist for the same value of the temperature leading to canonical phase transitions (see below).
When the curve presents a maximum at and a minimum at . They are determined by Eq. (103) and by the equation
[TABLE]
obtained by differentiating Eq. (103) with respect to and writing .
For , we find that
[TABLE]
and
[TABLE]
We know that the maximum inverse temperature of a classical isothermal self-gravitating gas confined within a box is emden . For self-gravitating fermions and bosons, the maximum inverse temperature of the gaseous phase is close to and tends to this value when (see Fig. 14 of ijmpb for fermions). Therefore, the above results indicate that our simple analytical model is not valid for and . In particular, the maximum at is an artifact of our model. The true maximum is at .
On the other hand, for , we find that
[TABLE]
and
[TABLE]
For fermions (), using Eq. (93), we obtain at leading order and pt . We note that decreases as increases. This is consistent with the properties of the exact caloric curve of self-gravitating fermions (see Fig. 34 in ijmpb ).
IV.6 Stability of the equilibrium states
Let us now consider more specifically the function giving the free energy of the system as a function of the core mass for a given value of , and . Using Eq. (104), we can rewrite Eq. (88) as
[TABLE]
Its first derivative is
[TABLE]
The condition determines the possible equilibrium states of the system as we have just seen. The stability of these equilibrium states in the canonical ensemble is then determined by the sign of the second derivative of the free energy:
[TABLE]
In the canonical ensemble an equilibrium state is stable when , corresponding to a minimum of free energy, and unstable when , corresponding to a maximum of free energy. In Fig. 5 we have plotted the curve in the case and where the system has three equilibrium states as detailed in the following section.
The values of the function at and are
[TABLE]
and
[TABLE]
For , we find that
[TABLE]
In practice so the term in parenthesis is positive. Since the function is defined for , and since the slope of the function at is positive, the solution (gaseous phase) is a local minimum of even though . We shall therefore consider that the solution is a stable equilibrium state.
IV.7 The different equilibrium states
After these mathematical preliminaries, we are now ready to perform the complete analysis of the equilibrium states of our simple analytical model. As explained previously we assume .
The curve is made of a vertical branch at up to , then it decreases, reaches a minimum at , and finally increases up to infinity when (see Fig. 4). When , there is a unique equilibrium state (). It corresponds to the gaseous phase (G). It is stable (minimum of free energy). When , there is a unique equilibrium state (). It corresponds to the condensed phase (C). It is stable (minimum of free energy). When there are three equilibrium states (see Figs. 4 and 5): (i) a gaseous phase (G); (ii) a core-halo phase (CH); (iii) a condensed phase (C). Let us analyze these solutions in more detail in the limit :
(i) The gaseous solution (G) corresponds to a purely isothermal halo without core. The core mass is equal to zero: . This solution is stable, being a minimum of free energy, although the derivative of does not vanish at as explained above.
(ii) The core-halo solution (CH) corresponds to an isothermal halo harboring a core with a small mass (). From Eq. (105), we find that the normalized core mass scales as
[TABLE]
For fermions, using Eq. (93), we obtain at leading order pt . This asymptotic formula is compared with the exact value of in Fig. 6. Substituting Eq. (123) into Eq. (119) we find that
[TABLE]
Therefore, the core-halo solution is unstable in the canonical ensemble being a maximum of free energy.
(iii) The condensed solution (C) corresponds to a quantum core surrounded by a tenuous atmosphere (). From Eq. (105), we find that the normalized core mass scales as
[TABLE]
showing that the quantum core contains almost all the mass. Substituting Eq. (125) into Eq. (119) we find that
[TABLE]
Therefore, the condensed solution is stable, being a minimum of free energy.
The occurence of three equilibrium states at the same temperature reveals the existence of a canonical phase transition associated with an isothermal collapse aaiso . Such gravitational phase transitions are analyzed in detail in ijmpb . The gaseous (G) and condensed (C) solutions are stable while the core-halo (CH) solution is unstable. It has a negative specific heat (see Sec. IV.8) which is forbidden in the canonical ensemble. It plays the role of a “germ” or a “critical droplet” in the langage of phase transitions and nucleation (the quantum core is the analogue of the droplet). It creates a barrier of free energy (see Fig. 5) that the system must cross to pass from the gaseous phase to the condensed phase (or the converse). Depending on the value of the temperature with respect to a temperature of transition ijmpb , the gasous and the condensed states may be either fully stable (global minimum of free energy) or metastable (local minimum of free energy). However, for systems with long-range interactions, metastable states have very long lifetimes scaling as where is the number of particles in the system lifetime , so they are strongly stable in practice.121212The system initially in the metastable gaseous phase (G) must spontaneously form a quantum core of mass to overcome the barrier of free energy and collapse in the condensed phase (C). The probability to spontaneously form such a core, thanks to energy fluctuations, is extremely weak, scaling as . This is a such a rare event that the metastable gaseous phase is stable in practice. By contrast, the core-halo state (CH) is unstable in the canonical ensemble, being a maximum (not a minimum) of free energy at fixed mass. This looks like a bad news since this core-halo structure is the most interesting structure from a physical point of view. Fortunately, we show in the next section that this core-halo structure is stable in the microcanonical ensemble (if the halo mass is not too large), being a maximum of entropy at fixed mass and energy. This is a manifestation of ensembles inequivalence for systems with long-range interactions ijmpb . Since the core-halo structure seems to appear in observations and numerical simulations we suggest, following our previous paper modeldm , that the microcanonical ensemble is more relevant than the canonical ensemble in the physics of DM halos.
IV.8 Microcanonical ensemble
In the microcanonical ensemble, a stable equilibrium state is obtained by maximizing the entropy at fixed mass and energy . Let us first compute the energy and the entropy of the system (quantum core isothermal atmosphere) by using the same analytical model as in the previous sections.
The energy of the quantum core is given by Eq. (76). On the other hand, the energy of an isothermal self-gravitating gas is
[TABLE]
where the first term is the kinetic (thermal) energy and the second term is the gravitational energy. Treating the atmosphere as a gas with a uniform density, we get
[TABLE]
where is the gravitational energy of the atmosphere in the presence of the core. It is given by Eq. (81). Therefore, the total energy of the system (core atmosphere) is
[TABLE]
In the microcanonical ensemble, the total energy is fixed. Therefore, Eq. (129) determines the temperature of the halo as a function of the core mass .
The entropy of an isothermal self-gravitating gas is
[TABLE]
Treating the atmosphere as a gas with a uniform density, we get
[TABLE]
Since the entropy of the core is equal to zero, Eq. (IV.8) represents the total entropy of the system.
From the above expressions, we note that the free energy of an isothermal self-gravitating gas (which is the relevant thermodynamic potential in the canonical ensemble) is
[TABLE]
This justifies the expression of the internal energy given by Eq. (78). On the other hand, the free energy calculated with Eqs. (129) and (IV.8) returns Eq. (82) of our analytical model.
Introducing the dimensionless variables
[TABLE]
we get
[TABLE]
and
[TABLE]
The first equation determines the inverse temperature as a function of the core mass for a fixed value of the energy . Then, the maximization of the entropy (at fixed energy ) determines the equilibrium state(s) in the microcanonical ensemble. From the above expressions, we note that the dimensionless free energy (relevant in the canonical ensemble) is given by
[TABLE]
where now the inverse temperature is fixed and the energy depends on the core mass . This returns Eq. (88).
One can easily check that the condition for a fixed value of yields Eq. (103) which was previously obtained from the condition for a fixed value of . As a result, the equilibrium states (extrema of entropy at fixed mass and energy and extrema of free energy at fixed mass) are the same in the microcanonical and canonical ensembles. However, their stability may be different in the microcanonical and canonical ensembles. This is the notion of ensembles inequivalence for systems with long-range interactions ijmpb .
Let us first consider all the possible equilibrium states determined by Eq. (103). For each of them, we can compute the inverse temperature and the energy and plot the caloric curve , relating the temperature to the energy, parametrized by the core mass . The minimum energy, corresponding to , is . We can also add the caloric curve of the gaseous phase () which is simply given by . The complete caloric curve for (fermions) and (corresponding to ) is represented in Fig. 7. For , we recover the three solutions (G), (CH) and (C) studied in Sec. IV.7. We note for future reference that the core-halo solution has an energy and a core mass . We clearly see that the core-halo phase has a negative specific heat implying that it is unstable in the canonical ensemble as we have found. However, it is known that negative specific heats are allowed in the microcanonical ensemble for systems with long-range interactions ijmpb . In the present case, the core-halo solution (CH) turns out to be stable in the microcanonical ensemble. This is confirmed in Fig. 8 by plotting the entropy versus the core mass at the energy . We see that the core-halo solution at is a maximum of entropy at fixed energy. Therefore, the core-halo solution (CH), with a negative specific heat, is unstable in the canonical ensemble while it is stable in the microcanonical ensemble. This makes the core-halo solution extremely important since it is now justified as being the “most probable” configuration of the system (maximum entropy state).
This result is valid only when , where is the microcanonical critical point (its exact value for fermions is corresponding to ijmpb ). When the caloric curve becomes multivalued (see Fig. 9) leading to microcanonical phase transitions associated with the gravothermal catastrophe lbw ; ijmpb . In that case, we can have a microcanonical phase transition from the gaseous phase (G’) to the condensed phase (C’) that we do not analyze in detail here (see ijmpb for a detailed discussion). We just point out that the core-halo solution (CH) is now both canonically and microcanonically unstable (for we have and ). This is confirmed in Fig. 10 by plotting the entropy versus the core mass at the energy . We see that the core-halo solution is a minimum of entropy at fixed energy. Therefore, the core-halo solution (CH) is unstable in the canonical and in the microcanonical ensembles.
IV.9 General scenario of DM halos
Recalling that is a measure of the size of a DM halo (see Sec. IV.4), and that the virial theorem fixes the value of the normalized temperature to [see Eq. (92)], the preceding results confirm and refine the scenario developed in modeldm :
(i) There is a “minimum halo” of mass (value obtained from the observations) corresponding to a purely quantum core without isothermal atmosphere (ground state).
(ii) For , the caloric curve is monotonic. As a result, the virial condition determines a unique solution: a quantum phase (Q). It corresponds to a DM halo with a quantum core and a tenuous isothermal atmosphere.
(iii) For , the caloric curve has an -shape structure (see Fig. 7 of this paper and Fig. 10 of pt ). As a result, the virial condition determines three solutions: a gaseous phase (G), a core-halo phase (CH) and a condensed phase (C). Among these three solutions, the core-halo state (CH) corresponding to a DM halo with a quantum core and an isothermal atmosphere is the most relevant. This solution is unstable in the canonical ensemble but it is stable in the microcanonical ensemble. It has a negative specific heat. The quantum core may mimic a large bulge (not a black hole) as argued in modeldm . The core mass – halo mass relation , which is of prime interest, is studied in the following sections.
(iv) For , the caloric curve has a -shape structure (see Fig. 9 of this paper and Fig. 7 of pt ). As before, the virial condition determines three solutions: (G), (CH) and (C). However, the core-halo solution (CH) is now unstable in both canonical and microcanonical ensembles. In that case, the system collapses and undergoes a gravothermal catastrophe lbw . A possibility is that it collapses from the gaseous phase (G’) to the condensed phase (C’) by forming a quantum core of mass acf and expulsing a hot atmosphere at large distances.131313As argued in clm2 ; modeldm , this scenario may not be relevant for DM halos. However, in another context, it may be relevant to explain the supernova phenomenon of massive stars pomeau2 ; acf . Alternatively, the core of the system may become relativistic during the gravothermal catastrophe and finally undergo a dynamical instability of general relativistic origin leading ultimately to a supermassive black hole (this is the scenario of Balberg et al. balberg advocated in clm2 ; modeldm ). We stress that the process of gravothermal catastrophe is generally very long (secular) but it may be relevant in galactic nuclei or if the DM particle has a self-interaction balberg .
In conclusion, we predict that DM halos with a mass harbor a quantum core (bosonic soliton or fermion ball) while DM halos with a mass harbor a supermassive black hole. This result is connected to the fundamental existence of a microcanonical critical point in the statistical mechanics of self-gravitating systems ijmpb . If our scenario turns out to be correct, it would provide a beautiful illustration of the curious thermodynamics of self-gravitating systems in a cosmological context. The value of depends on the type of particles that compose DM. For self-interacting bosons in TF limit we found (corresponding to and ) and for fermions we found (corresponding to and ). The case of noninteracting bosons is under investigation modeldm .141414In comparison, the canonical critical point is (corresponding to and ) for self-interacting bosons in TF limit and (corresponding to and ) for fermions Presently, the obtained value of is a rough estimate which should be improved in the future if our scenario is relevant. On the other hand, we do not the preclude the presence of supermassive black holes in DM halos of mass if they are formed by a mechanism different from the one balberg that we have advocated.
Remark: The gaseous solutions (G) constitute the lower branch of the generic phase diagram reported in Fig. 49 of modeldm . The core-halo solutions (CH) constitute the upper branch of this phase diagram that appears above the canonical critical point (bifurcation point) . This is the branch of most physical interest in the physics of DM halos. The core mass – halo mass relation on the core-halo branch (CH) is studied in detail in the following sections. This branch becomes unstable above where the quantum core (bulge) is replaced by a supermassive black hole.
V Justification of the velocity
dispersion tracing relation and determination of the relation
V.1 The velocity
dispersion tracing relation
We consider the core-halo solution (CH) of Sec. IV.7. The normalized core mass is given by Eq. (123). We first show that this result is equivalent to the “velocity dispersion tracing” relation mocz ; bbbs ; modeldm
[TABLE]
stating that the velocity dispersion in the core is of the same order as the velocity dispersion in the halo . Using Eq. (74), this relation can be rewritten as
[TABLE]
On the other hand, using Eqs. (83) and (IV.3), we find that Eq. (123) is equivalent to
[TABLE]
The formulae (138) and (V.1) are consistent with each other up to a multiplicative factor of order unity containing a logarithmic correction. As a result, Eq. (123) is equivalent to the “velocity dispersion tracing” relation from Eq. (137). Our study provides therefore a justification of this relation from (effective) thermodynamical arguments.
V.2 The relation
If we now substitute the relation from Eq. (50) into Eq. (138) we obtain
[TABLE]
This equation gives the relation between the core mass and the halo mass . It displays the fundamental scaling
[TABLE]
Combining Eq. (140) with the minimum halo mass from Eq. (G.1), we get
[TABLE]
where
[TABLE]
is a constant that depends on the polytropic index . We find below that the prefactor in Eq. (142) is of order unity. Therefore, the ratio between the core mass and the halo mass is . Accordingly, the core mass becomes negligible in front of the halo mass when with .
V.3 The relation
Following our previous work modeldm , we have defined the halo mass and the halo radius such that represents the distance at which the central density is divided by (see Appendix G). However, Schive et al. ch3 use another definition of the halo mass and halo radius . They are connected by
[TABLE]
where is the present background matter density in the Universe and is a prefactor of order . For the numerical applications, we shall take and giving planck2016 . Using
[TABLE]
in consistency with Eq. (137), and Eqs. (50) and (144), we obtain
[TABLE]
The scaling was previously noted in modeldm . Normalizing the halo mass by the minimum halo mass, we get
[TABLE]
with
[TABLE]
Combining Eqs. (142) and (147), we finally obtain the core mass – halo mass relation
[TABLE]
It exhibits the fundamental scaling .
Remark: Since is a dimensioness constant of order (see below), Eq. (148) provides a relation between the DM particle parameters [via ], the universal DM surface density and the present density of matter in the Universe . Expressing and in terms of the cosmological constant , we will be able (see Sec. VI) to obtain the DM particle parameters in terms of the fundamental constants of physics.
VI Application to quantum models of DM halos
We now apply these results to quantum models of DM halos made of fermions, noninteracting bosons, and self-interacting bosons as described in Sec. II. For reasons that will become clear below, we treat the case of noninteracting bosons first.
VI.1 Noninteracting bosons
A noninteracting self-gravitating BEC () is equivalent to a polytrope of index with a polytropic constant given by Eq. (286) (see Sec. II.2 and Appendix E). Using Eqs. (142), (143) and the results of Appendix G, we get
[TABLE]
On the other hand, using Eqs. (26) and (148), we find that
[TABLE]
Therefore, Eq. (147) takes the form
[TABLE]
Combining Eqs. (150) and (152), we obtain the core mass – halo mass relation
[TABLE]
It exhibits the fundamental scaling . This theoretical scaling is consistent with the scaling found numerically by Schive et al. ch3 (see also veltmaat ). These authors also presented an heuristic argument to justify this relation. As discussed in Refs. mocz ; bbbs ; modeldm , their argument is equivalent to assuming the velocity dispersion tracing relation (137). We stress, however, that this relation is not obvious a priori and that other relations, such as the energy tracing relation corresponding to , could be contemplated as well mocz . They would lead to different results. The fact that the velocity dispersion tracing relation (137) can be justified from a maximum entropy principle (most probable state), as shown in the present paper, may provide a physical basis for it.
On the other hand, reversing Eq. (151) following the remark at the end of Sec. V.3, we get
[TABLE]
The present matter density in the Universe is given by and the density of dark energy is given by where is the present fraction of dark energy and is the cosmological constant. Therefore, we can write the present DM density in terms of the cosmological constant as
[TABLE]
On the other hand, in the framework of the logotropic model epjp ; lettre ; ouf , we have theoretically predicted that the surface density of the DM halos is constant and that its universal value is given in terms of an effective cosmological constant (whose value is the same as Einstein’s cosmological constant) by151515We emphasize that there is no free parameter in the logotropic model epjp ; lettre ; ouf . In particular, the prefactor in Eq. (156) is predicted by our model.
[TABLE]
Now that this formula has been isolated, we can use it independently from the theory developed in epjp ; lettre ; ouf . Combining Eqs. (154), (155) and (156) we find that the mass of the noninteracting bosonic particle is given by
[TABLE]
where
[TABLE]
This mass scale is often interpreted as the smallest mass of the bosons predicted by string theory axiverse or as the upper bound on the mass of the graviton graviton .161616It is simply obtained by equating the Compton wavelength of the particle with the Hubble radius (the typical size of the visible Universe) giving . By comparison, if we identify the Compton wavelength with the Schwarzschild radius we get the Planck mass . This is also the mass of an hypothetical particle called the cosmon ouf . We see that it fixes the mass scale of the DM particle in the case where it is a noninteracting boson. Nevertheless, there is a huge proportionality factor between them, of the order of . We have also found this result in Appendix F of ouf from considerations based on the Jeans instability. These considerations are further developed in Appendix I and generalized to the case of self-interacting bosons and fermions.
Remark: Returning to original variables, and using Eqs. (50), (137) and (288), the core mass – halo mass relation of DM halos made of noninteracting bosons can be written as
[TABLE]
leading to .
VI.2 Self-interacting bosons
A self-gravitating BEC with a repulsive self-interaction in the TF limit () is equivalent to a polytrope of index with a polytropic constant given by Eq. (30) (see Sec. II.3). Using Eqs. (142), (143) and the results of Appendix G, we get
[TABLE]
On the other hand, using Eqs. (43) and (148), we find that
[TABLE]
Therefore, Eq. (147) takes the form
[TABLE]
Combining Eqs. (160) and (162), we obtain the core mass – halo mass relation
[TABLE]
It exhibits the fundamental scaling . This is a new theoretical prediction modeldm that still has to be tested with direct numerical simulations of self-interacting bosons.
On the other hand, reversing Eq. (161) following the remark at the end of Sec. V.3, we get
[TABLE]
Using Eqs. (155) and (156) we obtain
[TABLE]
where
[TABLE]
In this expression, is the mass of the cosmon given by Eq. (158) and
[TABLE]
is the gravitational radius of the cosmon ouf .
Remark: Returning to original variables, and using Eqs. (33), (50) and (137), the core mass – halo mass relation of DM halos made of self-interacting bosons in the TF limit can be written as
[TABLE]
leading to .
VI.3 Fermions
A fermionic core is equivalent to a polytrope of index with a polytropic constant given by Eq. (4) (see Sec. II.1). Using Eqs. (142), (143) and the results of Appendix G, we get
[TABLE]
On the other hand, using Eqs. (14) and (148), we find that
[TABLE]
Therefore, Eq. (147) takes the form
[TABLE]
Combining Eqs. (169) and (171), we obtain the core mass – halo mass relation
[TABLE]
It exhibits the fundamental scaling . This theoretical scaling, previously given in the form of Eq. (169) in Appendix H of clm2 , is consistent with the scaling found numerically by Ruffini et al. rar (they find an exponent equal to instead of ).
On the other hand, reversing Eq. (170) following the remark at the end of Sec. V.3, we get
[TABLE]
Using Eqs. (155) and (156) we obtain
[TABLE]
where
[TABLE]
To our knowledge, this mass scale has not been introduced before. It is the geometric mean of the cosmon mass given by Eq. (158) and the Planck mass .171717In comparison, using the results of ouf , the mass of the electron may be written in terms of the fundamental constants of physics as where is the fine structure constant.
Remark: Comparing Eqs. (151), (161) and (170), we note that the value of defined by Eq. (148) is relatively insensitive on the value of the polytropic index for the cases contemplated. This is because the parameters have been chosen so that typically has a fixed value ().
Remark: Returning to original variables, and using Eqs. (6), (50) and (137), the core mass – halo mass relation of DM halos made of fermions may be written as
[TABLE]
leading to .
VI.4 Semiclassical limit
It is interesting to study how the mass , the radius , the velocity dispersion and the energy in the core behave in the semiclassical limit . For noninteracting bosons, using Eq. (159), we find and . For self-interacting bosons, using Eq. (168), we find and . For fermions, using Eq. (176), we find and . Therefore, in the semiclassical limit , the mass , the size and the energy of the quantum core go to zero while the velocity dispersion remains finite.
VII Gaussian ansatz
In this section, we obtain the core mass – halo mass relation of DM halos by combining the velocity dispersion tracing relation (137) with the approximate core mass-radius relation of a self-gravitating BEC at obtained in prd1 from a Gaussian ansatz. This allows us to recover the preceding results and to generalize them to the case of a repulsive self-interaction (), without making the TF approximation, and to the case of an attractive self-interaction (). Throughout this section, we introduce appropriate normalizations in order to clearly see the physical origin of the parameters and be able to refine the numerical applications when more precise data will be available.
VII.1 Core mass-radius relation
Using a Gaussian ansatz, it is found in prd1 that the approximate mass-radius relation of a self-gravitating BEC at (ground state) is given by
[TABLE]
with the coefficients , and . Inversely, the radius can be expressed in terms of the mass as
[TABLE]
with when and with when . The results of prd1 describe the “minimum halo” (ground state) or the quantum core of larger halos. For noninteracting BECs (), the mass-radius relation (177) reduces to
[TABLE]
In the repulsive case () the mass-radius relation is monotonic (see Fig. 2 of prd1 ). There is a minimum radius
[TABLE]
corresponding to (TF limit). Measuring the DM particle mass in units of and the scattering length in units of , we get . For we are in the noninteracting limit (179). In the attractive case () the mass-radius relation is nonmonotonic (see Fig. 3 of prd1 ). There is a maximum mass
[TABLE]
corresponding to the radius
[TABLE]
Measuring the DM particle mass in units of and the scattering length in units of , we get and .
No equilibrium state exists with a mass . For the branch (corresponding to the solutions (178) with the sign ) is stable and the branch (corresponding to the solutions (178) with the sign ) is unstable. For we are in the noninteracting limit (179) and for we are in the (unstable) nongravitational limit where
[TABLE]
VII.2 The minimum halo mass
We first determine the minimum halo mass . As explained previously, the minimum halo corresponds to the ground state () of the self-gravitating BEC. In our approximate approach we write the surface density as
[TABLE]
where is a constant of order unity (in the numerical applications we take for the reason explained in footnote 17). Eliminating between Eqs. (177) and (184), and treating as a universal constant, we get the minimum halo mass as a function of and (we recall that for the ground state since there is no isothermal atmosphere by definition). For noninteracting BECs () we find that the minimum halo mass is
[TABLE]
The prefactor is . This result can be compared with Eq. (26). Measuring the DM particle mass in units of , we get . In the general case, valid for an arbitrary value of , we find that the minimum halo mass is determined by the equation
[TABLE]
where we have introduced the appropriate scattering length scale
[TABLE]
The prefactor is . Measuring the DM particle mass in units of , we get . For we recover . More generally, the noninteracting limit is valid for . The relation vs is plotted in Fig. 11. For a given mass , we see that is larger than when and smaller when .
In the repulsive case, for , we have
[TABLE]
This corresponds to the TF limit. Returning to the original variables, we obtain
[TABLE]
The prefactor is . This result can be compared with Eq. (43). Measuring the DM particle mass in units of and the scattering length in units of , we get .
In the attractive case, for , we have
[TABLE]
This corresponds to the nongravitational limit in which the configurations are unstable. Returning to the original variables, we obtain
[TABLE]
The prefactor is . On the other hand, the normalized maximum mass (181) can be written as
[TABLE]
We find that the minimum halo is critical (i.e. ) for
[TABLE]
These relations determine the minimum scattering length of the DM particle and the minimum mass of the minimum halo. Returning to the original variables, we obtain
[TABLE]
[TABLE]
The prefactors are and . Measuring the DM particle mass in units of , we get and . These relations can be directly obtained by writing and using Eqs. (181) and (182). The minimum halo is stable only for . It has a mass . When the minimum halo is unstable (it lies on the branch of the mass-radius relation). We note that is relatively close to . Therefore, when , the minimum halo mass is always of the order of (see the stripe in Fig. 11).
For bosons with an attractive self-interaction, like the axion marshrevue , it is more convenient to express the results in terms of the decay constant (see, e.g., phi6 )
[TABLE]
rather than the scattering length . Measuring the DM particle mass in units of and the scattering length in units of , we get . We can write
[TABLE]
with
[TABLE]
The prefactor is . Measuring the DM particle mass in units of , we get . Eq. (186) can be rewritten as
[TABLE]
It determines the minimum halo mass in terms of and . This relation is plotted in Fig. 12. The maximum mass (181) can be written as
[TABLE]
or, in normalized form, as
[TABLE]
Using Eqs. (193) and (197), the minimum decay constant corresponding to the critical minimum halo is
[TABLE]
Returning to the original variables, we find
[TABLE]
The prefactor is . Measuring the DM particle mass in units of , we get . Only the upper part of the curve starting from the point () is stable. The noninteracting limit corresponds to .
VII.3 The and relations
If we consider that the minimum halo mass is known from the observations, and take (Fornax) to fix the ideas, we can use the relation (186) to determine the mass that the DM particle must have as a function of its scattering length in order to match the value of the minimum halo mass . For we find from Eq. (185) that
[TABLE]
The prefactor is . In that case we obtain which can be compared to Eq. (24) and Eq. (326). We can then write
[TABLE]
On the other hand, we can write
[TABLE]
where we have introduced the appropriate scattering length scale
[TABLE]
The prefactor is . We find . Substituting Eqs. (205) and (206) into Eq. (186), we obtain
[TABLE]
This relation determines the mass of the DM particle as a function of its scattering length in order to yield a minimum halo of mass . It is plotted in Fig. 13. For , we recover which is the mass of a noninteracting boson. More generally, the noninteracting limit corresponds to . We see that is larger than when and smaller when . Therefore, we can increase the DM particle mass by allowing for a repulsive self-interaction between the bosons. As discussed in Appendix D.4 of suarezchavanis3 this could alleviate some tensions with observations of the Lyman- forest encountered in the noninteracting model hui . By contrast, an attractive self-interaction implies a (slightly) smaller DM particle mass and may therefore be even more in conflict with observations of the Lyman- forest. As a result, a repulsive self-interaction () is priviledged over an attractive self-interaction (). In this respect, we recall that theoretical models of particle physics usually lead to particles with an attractive self-interaction (e.g., the QCD axion). However, some authors fan ; reig have pointed out the possible existence of particles with a repulsive self-interaction (e.g. the light majoron).
In the repulsive case, for , we have
[TABLE]
This corresponds to the TF limit. Returning to the original variables, we obtain
[TABLE]
The prefactor is . We find which can be compared with Eq. (39) and Eq. (330).
In the attractive case, the curve presents a turning point at
[TABLE]
However, this turning point does not correspond to the critical minimum halo for which [see Eqs. (193), (205) and (206)]
[TABLE]
Returning to the original variables, we obtain
[TABLE]
[TABLE]
The prefactors are and . We find and which can be compared with Eq. (D19) in Appendix D of suarezchavanis3 . Only the upper part of the curve starting from the point () is stable. The existence of a stable minimum halo in the Universe implies that . In that case, the DM particle mass satisfies . We note that is relatively close to . Therefore, when , the minimum DM particle mass is always of the order of (see the stripe in Fig. 13).
For bosons with an attractive self-interaction, like the axion marshrevue , it is more convenient to express the results in terms of the decay constant (196). We can write
[TABLE]
with
[TABLE]
The prefactor is . We find . Eq. (208) can be rewritten as
[TABLE]
It determines the relation between and in order to have a minimum halo (ground state) of mass . This relation is plotted in Fig. 14 (note that ). Using Eqs. (212) and (215), the minimum decay constant corresponding to the critical minimum halo is
[TABLE]
Returning to the original variables, we obtain
[TABLE]
The prefactor is . We find . Only the upper part of the curve starting from the point () is stable. The existence of a stable minimum halo in the Universe implies that . In that case, the DM particle mass satisfies . The noninteracting limit corresponds to .
There is an interesting by-product of our analysis. Indeed, particle physics and cosmology lead to the following relation between and hui :
[TABLE]
Taking and , this relation can be rewritten as
[TABLE]
This relation is independent from Eq. (217). Equating Eqs. (217) and (221), we obtain and . Therefore, we can determine and individually. We note that has the same value as in the noninteracting case while has a finite value . It corresponds to . Interestingly, lies in the range expected in particle physics hui (we stress that the value of has been deduced from our model based on the core mass-radius relation (177)). Since , we are essentially in the noninteracting regime.
Remark: we note that for the critical minimum halo the ratio
[TABLE]
is independent of . The prefactor is . We find .
VII.4 The relation
To obtain the core mass – halo mass relation we use the velocity dispersion tracing relation (137), the core mass-radius relation (178) and the halo-mass radius relation from Eq. (50). We obtain
[TABLE]
For convenience, we have taken . In this manner, when , we recover the condition (186) determining the minimum halo mass.181818This can be understood as follows. Combining Eqs. (137) and (50) we get
(224)
This equation, that uses the relation , is valid only for sufficiently large halos (see Sec. II.5). However, if we extrapolate this equation to the minimum halo for which , we get . Comparing this equation with Eq. (184) we obtain . For , we get
[TABLE]
This relation is also valid for and . We recover the scaling from Eq. (150). Returning to the original variables, we get
[TABLE]
The prefactor is . For a DM halo of mass similar to the one that surrounds our Galaxy, we obtain a core mass (we have taken ). The corresponding core radius is [see Eq. (179)]. The quantum core represents a bulge or a nucleus (it cannot mimic a black hole).
In the repulsive case, for or , we have
[TABLE]
This corresponds to the TF limit. Using Eq. (188), we obtain
[TABLE]
We recover the scaling from Eq. (160). Returning to the original variables, we get
[TABLE]
The prefactor is . For a DM halo of mass similar to the one that surrounds our Galaxy, we obtain a core mass (we have taken ). The core radius is [see Eq. (180)]. The quantum core represents a bulge or a nucleus (it cannot mimic a black hole). The core mass – halo mass relation for is plotted in Figs. 15 and 16. These solutions are valid for .
In the attractive case, the core mass vanishes at
[TABLE]
On the other hand the core mass is maximum at
[TABLE]
with the value
[TABLE]
This corresponds to the maximum core mass given by Eq. (192). The core mass – halo mass relation for is plotted in Figs. 17 and 18. These solutions are valid for . On the other hand, the branch corresponds to unstable states so that only the branch , corresponding to stable states, is physical. In summary, when there is no halo with a stable quantum core (see Sec. VII.2). When a stable quantum core exists only in the range . It has a mass . Coming back to the original variables, the maximum halo mass is
[TABLE]
The prefactor is . Measuring the DM particle mass in units of and the scattering length in units of , we get . Note that if we determine the maximum halo mass approximately by equating Eqs. (225) and (192) [or equivalently Eqs. (226) and (181)], we obtain a value that differs from the real one [Eq. (231) or equivalently (233)] by a factor .
For bosons with an attractive self-interaction, like the axion marshrevue , it is more convenient to express the results in terms of the decay constant (196). When there is no halo with a stable quantum core. When a stable quantum core exists only in the range . The minimum halo mass is close to given by Eq. (185). The maximum halo mass and the maximum core mass can be written as
[TABLE]
[TABLE]
We note that the maximum halo mass depends only on while the maximum core mass depends on and . The prefactors are and . Measuring the DM particle mass in units of and the decay constant in units of , we get and . For and (see Sec. VII.3), we find and . Since the largest DM halos observed in the Universe have a mass , these results suggest that the effect of an attractive self-interaction is negligible: Everything happens as if the bosons were not self-interacting. This favors the consideration of a repulsive self-interaction modeldm .
In this section, we have expressed the core mass – halo mass relation in terms of . This relation could easily be expressed in terms of by using Eq. (147) with (we have taken ).
Remark: We can directly obtain the expression (233) of the maximum halo mass from the relation [see Eq. (224)]
[TABLE]
with Eqs. (181) and (182). If we consider a self-gravitating BEC with a central black hole it can be shown that the ratio is independent of the black hole mass epjpbh . This implies that the expression (233) of the maximum halo mass is unchanged for a fixed central black hole. The case where the black hole mass changes with the halo mass is treated in mcmhbh .
VII.5 Summary
In the noninteracting case () the halos with a mass [see Eq. (185)] contain a quantum core of mass given by Eq. (225). All the configurations are stable.
For a repulsive self-interaction () the halos with a mass [see Eq. (186)] contain a quantum core of mass given by Eq. (223). For and not too large (), we are in the noninteracting limit discussed previously. For or sufficiently large () we are in the TF limit. In that case, the minimum halo mass is given by Eq. (189) and the mass of the quantum core is given by Eq. (228). All the configurations are stable.
For an attractive self-interaction () the halos can contain a stable quantum core only if [see Eq. (194)] or equivalently if [see Eq. (203)]. When this condition is fulfilled the quantum core is stable for . The minimum halo mass [see Eqs. (186) and (199)] is of the order of [see Eq. (185)]. When we reach the maximum halo mass [see Eqs. (233) and (234)], the core reaches its maximum limit [see Eqs. (181) and (235)] and collapses. The result of the collapse (dense axion star, black hole, bosenova…) is discussed in braaten ; cotner ; bectcoll ; ebycollapse ; tkachevprl ; helfer ; phi6 ; visinelli ; moss .
These results are summarized in Fig. 19 representing the general core mass – halo mass relation with a normalization such that the minimum halo mass is fixed to a common value obtained from the observations (typically ). The minimum halo mass determines the relation between and as explained in Sec. VII.3. The construction of Fig. 19 is detailed below and the selected values of the DM particle parameters are given in Table I. On this representation, we clearly see that, as compared to the noninteracting case (), the core mass increases more rapidly with in the case of a repulsive self-interaction () and less rapidly in the case of an attractive self-interaction (). For , there is a stable core for any halo mass. For , there is a maximum halo mass associated with the existence of a maximum core mass . Above that mass, the quantum core collapses.
To make Fig. 19, corresponding to a fixed minimum halo mass , we have proceeded in the following manner. For a given value of we can obtain from Eq. (208). Then, we get from Eq. (206) or, equivalently, from
[TABLE]
Finally, we get from Eq. (205) or, equivalently, from Eq. (186). We can then plot as a function of by using Eq. (223). We stress that this procedure yields a “universal” curve for a given value of . The mass of the minimum halo then determines the scales and according to Eqs. (204) and (207).191919We have assumed that the surface density of the DM halos is universal [see Eq. (1)]. If this were not the case, our general model would remain valid but the problem would depend on and instead of just . For convenience, we have proceeded the other way round. We have chosen a value of , determined from Eq. (186) and plotted as a function of by using Eqs. (186) and (223). We have then used Eqs. (205) and (208) to obtain the values of and corresponding to our choice of . The selected values of and the corresponding DM particle parameters appearing in Fig. 19 are reported in Table I.
In the case of an attractive self-interaction, using Eqs. (205), (206), (215), (231) and (232), the maximum halo mass and the maximum core mass normalized by the minimum halo mass are given by
[TABLE]
and
[TABLE]
where , and are related to each other by Eqs. (208) and (217). They are plotted as a function of the decay constant in Fig. 20. For (correspondingly and ), the maximum halo mass and the maximum core mass increase monotonically with , starting from the value . For (i.e. ) they behave as and . As indicated at the end of Sec. VII.4, the largest DM halos observed in the Universe have a mass . Therefore, if we will never see the effect of an attractive self-interaction (for any DM halo). Taking , this condition corresponds to . Therefore, for (correspondingly and ) everything happens as if the bosons were noninteracting. This is particularly true for the value predicted in Sec. VII.3 and, more generally, for values of in the range expected in particle physics hui .
In the case of a repulsive self-interaction, for a given value of , the transition betwen the noninteracting limit and the TF limit occurs at a typical halo mass
[TABLE]
For we are in the noninteracting limit and for we are in the TF limit. Using the same argument as above, if we will not see the effect of a repulsive self-interaction (for any DM halo). Taking , this condition corresponds to . Therefore, for (correspondingly ) everything happens as if the bosons were noninteracting. To our knowledge, there is no strong constraint on a repulsive self-interaction (see, however, the Bullet Cluster constraint mentioned in Sec. II.3) so that large positive values of (), corresponding to large values of the boson masse (), are possible in principle.
VIII Conclusion
In this paper, we have analytically derived the core mass – halo mass relation of fermionic and bosonic DM halos from an effective thermodynamical approach. We have modeled the DM halos by a quantum core of mass surrounded by an isothermal atmosphere of uniform density. We first determined an analytical expression of the free energy and entropy of the DM halos. The equilibrium core mass is then obtained by extremizing the free energy at fixed mass or by extremizing the entropy at fixed mass and energy. By representing the quantum core by a polytrope of index we have developed a unified description for fermions (), noninteracting bosons () and self-interacting bosons in the TF approximation (). This allowed us to treat fermionic and bosonic DM halos with the same formalism. In the generic case, the extremization problem determines three solutions corresponding to a gaseous phase (G), a core-halo phase (CH) and a condensed phase phase (C). The most important solution is the core-halo phase. We showed that this phase is always unstable in the canonical ensemble (maximum of free energy at fixed mass) while it is stable in the microcanonical ensemble (maximum of entropy at fixed mass and energy) when (the gaseous and the condensed phases are stable in all statistical ensembles). When the core-halo phase is unstable in all statistical ensembles. In that case, the quantum core may be replaced by a supermassive black hole modeldm resulting from a gravothermal catastrophe lbw followed by a dynamical instability of general relativistic origin balberg .
Our thermodynamical approach leads to the velocity dispersion tracing relation (137) put forward heuristically in mocz ; bbbs ; modeldm . Therefore, this relation can be justified by an effective thermodynamical approach (maximum entropy principle). For noninteracting bosons, we obtain the mass-radius relation (153) which is consistent to the one found by Schive et al. ch3 (see also veltmaat ). For fermions, we obtain the mass-radius relation (172) which is consistent to the one found by Ruffini et al. rar . For bosons with an attractive self-interaction in the TF limit, we predict the mass-radius relation (163) which still has to be confirmed numerically. Combining the velocity dispersion tracing relation mocz ; bbbs ; modeldm with the core mass – core radius relation derived in prd1 we have obtained an approximate general core mass – halo mass relation [see Eq. (223)] that is valid for bosons with arbitrary repulsive or attractive self-interaction. For an attractive self-interaction, corresponding to axions marshrevue , we have determined the maximum halo mass [see Eq. (234)] that can harbor a stable quantum core (dilute axion star).
The mass of the minimum halo (ground state) determines the parameters of the DM particle (depending on the strength of the self-interaction). Observations reveal that . For fermions, we find [see Eq. (321)]. For noninteracting bosons we find [see Eq. (326)]. For self-interacting bosons in the TF limit we find [see Eq. (330)]. We can then use the core mass – halo mass relation to determine the characteristics of the quantum core residing in a given DM halo. Let us consider a DM halo of mass similar to the one that surrounds our Galaxy. For fermions, we find and [see Eqs. (6) and (169)]. For noninteracting bosons we find and [see Eqs. (288) and (150)]. For self-interacting bosons in the TF limit we find and [see Eqs. (33) and (160)]. In our model, the quantum core represents a bulge or a nucleus. It cannot mimic a black hole as it has been sometimes suggested.
Finally, we have argued that the mass scale of noninteracting DM bosons is determined in terms of fundamental constants by while the mass scale of fermions is determined by . We found that the prefactor between the actual DM particle mass and these fundamental mass scales can be very large ( orders of magnitude for bosons and orders of magnitude for fermions). However, these fundamental mass scales can explain the intrinsic difference of mass between bosonic and fermionic DM particles. Their ratio is , corresponding to a difference of orders of magnitude. Finally, in the case of self-interacting bosons, we have found that the fundamental scale of the ratio is .
In the present paper, we have developed an analytical model in which the isothermal atmosphere has a uniform density. This approximation is sufficient to obtain the correct scaling of the core mass – halo mass relation. However, in order to develop more accurate models of fermionic and bosonic DM halos, and in particular to be able to determine their density and circular velocity profiles, we need to solve a generalized Emden equation numerically. The case of self-gravitating BECs with a repulsive self-interaction has been treated in detail in modeldm . The case of noninteracting bosons and fermions can be treated with the same method. These models are being presently investigated inpreparation .
It will be important in future works to determine if DM is made of fermions or bosons (either noninteracting, with a repulsive self-interaction, or with an attractive self-interaction). All these models are very interesting from a physical point of view, with fascinating properties, but it is possible that some of them will be ruled out by observations. Alternatively, all these models could be of interest if DM is made of several types of particles (bosons and fermions) as suggested in modeldm .
It will also be important in future works to go beyond certain approximations made in this paper. For example, we have ignored the influence of baryons. They may have several significant effects on DM halos and, in particular, they can alleviate the cusp problem romano . It would be interesting to check the validity of our results in the presence of baryons. We have also used a nonrelativistic approach. As we have shown, this nonrelativistic approach is expected to be sufficient in most applications of DM halos. However, in the case of bosons with an attractive self-interaction, if the mass of the quantum core becomes greater than the maximum mass prd1 , the core collapses. In that case, relativistic effects can be important and the GPP equations must be replaced by the Klein-Gordon-Einstein (KGE) equations tkachevprl ; helfer ; phi6 ; visinelli ; moss . The KGE equations must also be used mabc in order to study boson stars and scalar field configurations in highly relativistic environments, such as scalar field dark matter laying in the vicinity of compact objects like black holes or neutron stars.
Appendix A Polytropic spheres
In this Appendix, we recall general results pertaining to self-gravitating polytropic spheres emden ; chandrabook . We apply them to the quantum models of DM halos at (ground states) discussed in Sec. II.
For classical self-gravitating systems, or for quantum self-gravitating systems in the TF approximation, the condition of hydrostatic equilibrium
[TABLE]
combined with the Poisson equation
[TABLE]
leads to the fundamental differential equation
[TABLE]
For a polytropic equation of state of the form
[TABLE]
where is the polytropic constant and is the polytropic index, the differential equation (243) becomes
[TABLE]
In the following, we restrict ourselves to spherically symmetric distributions. We also assume and (i.e. ) for reasons explained below. With the substitution
[TABLE]
where is the central density and
[TABLE]
is the polytropic radius, we obtain the Lane-Emden equation
[TABLE]
with the boundary conditions
[TABLE]
According to the general results of Refs. emden ; chandrabook , a polytrope of index has a compact support provided that . In that case, the density falls off to zero at a finite radius. This characterizes a complete polytrope.202020Polytropes with have an infinite mass. The polytrope is unbounded but has a finite mass. For this index, the Lane-Emden equation has a simple analytical expression discovered by Schuster schuster . It was used by Plummer plummer to fit the density profile of globular clusters (Plummer’s model). It is customary to denote by the normalized radius at which the density vanishes: . The radius and the mass of a complete polytrope then are
[TABLE]
[TABLE]
Eliminating between these two relations, we find that the mass-radius relation of a complete polytrope of index is
[TABLE]
where is a constant determined by the Lane-Emden equation (248). A complete polytrope of index is dynamically stable with respect to the Euler-Poisson equations if and linearly unstable if .
For the polytrope (fermion stars), using Eq. (4), we find
[TABLE]
[TABLE]
[TABLE]
[TABLE]
For the polytrope (noninteracting boson stars), using Eq. (286), we find
[TABLE]
[TABLE]
[TABLE]
[TABLE]
For the polytrope (self-interacting boson stars), using and Eq. (30), we find
[TABLE]
[TABLE]
[TABLE]
Appendix B Total energy, eigenenergy and virial theorem of a self-gravitating BEC
We consider a self-gravitating BEC described by the GPP equations with a self-interaction corresponding to a power-law potential ggp . In the hydrodynamic representation of the GPP equations, a power-law potential of interaction gives rise to a polytropic equation of state. In that case, the total energy of the BEC is given by
[TABLE]
This is the sum of the quantum kinetic energy
[TABLE]
the internal energy
[TABLE]
and the gravitational energy
[TABLE]
The eigenenergy satisfies the relation
[TABLE]
On the other hand, the equilibrium scalar virial theorem writes
[TABLE]
For classical self-gravitating systems, or for self-gravitating BECs in the TF approximation where the quantum potential can be neglected, the foregoing equations reduce to
[TABLE]
[TABLE]
[TABLE]
From these equations, we obtain the relations
[TABLE]
Appendix C Betti-Ritter formula
For classical self-gravitating systems, or for self-gravitating BECs in the TF approximation, the condition of hydrostatic equilibrium is given by Eq. (241). For a polytropic equation of state (244) we have
[TABLE]
As a result, the condition of hydrostatic equilibrium (241) can be integrated into
[TABLE]
where is a constant of integration. For a self-gravitating BEC it represents the eigenenergy ggp . Multiplying Eq. (275) by and integrating over the whole configuration, we obtain Eq. (271). Assuming (i.e. ) so that on the boundary of the system where the density vanishes, we find from Eq. (275) that
[TABLE]
This equation determines the eigenenergy . As a result, Eq. (271) can be rewritten as
[TABLE]
Combining this relation with the equilibrium scalar virial theorem (272) we obtain the Betti-Ritter formula
[TABLE]
determining the gravitational energy of a polytropic sphere chandrabook . From Eqs. (278) and (273), we get
[TABLE]
and
[TABLE]
From the last relation, we can directly conclude that complete polytropes with index , i.e. , are stable (because ) while complete polytropes with index , i.e. , are unstable (because ) chandrabook .
Appendix D Ledoux formula
The complex pulsation of a polytrope of index is approximately given by the Ledoux formula ledouxpekeris :212121This formula can also be obtained from a variational principle based on a Gaussian ansatz ggp .
[TABLE]
where is the moment of inertia of the system. Using the results of Appendix A it can be written as
[TABLE]
Combining this equation with the Betti-Ritter formula (278), we can rewrite Eq. (281) as
[TABLE]
For , we find . For , we find . For , we find .
Appendix E Density profile of a noninteracting self-gravitating BEC with a
compact support: polytrope
The fundamental differential equation of quantum hydrostatic equilibrium determining the density profile of a noninteracting self-gravitating BEC is given by Eq. (17). On the other hand, the fundamental differential equation of classical hydrostatic equilibrium determining the density profile of a polytrope of index is given by Eq. (245). For it becomes
[TABLE]
Dividing Eq. (284) by , applying the Laplacian operator, and using Eq. (284) again, we obtain
[TABLE]
Remarkably, this equation coincides with Eq. (17) provided that we make the identification
[TABLE]
As a result, the density profile of a polytrope of index and polytropic constant given by Eq. (286) is a particular solution of Eq. (17).222222We note that Eq. (284) implies Eq. (285) but the converse is wrong. As a result Eqs. (17) and (284) are not equivalent. Using the variables defined in Appendix A, it can be written as
[TABLE]
where is the solution of the Lane-Emden equation (248) of index .
A polytrope of index has a compact support and is stable. Furthermore, according to Eq. (252), the mass-radius relation is
[TABLE]
where we have used Eq. (288) displays the same scaling as the mass-radius relation from Eq. (18) but the prefactor is different. This is because the density profile given by Eq. (287) is different from the density profile of the soliton that has been reported previously in the literature rb ; membrado ; gul0 ; gul ; prd2 ; ch2 ; ch3 ; pop ; hui (see Sec. II.2). Indeed, the authors of Refs. rb ; membrado ; gul0 ; gul ; prd2 ; ch2 ; ch3 ; pop ; hui have looked for a solution of Eq. (17) corresponding to a density profile that goes to zero at infinity. Actually, there exists another solution of Eq. (17), given by Eq. (287), corresponding to a density profile with a compact support that vanishes at a finite radius .232323In Ref. prd2 we found a solution of Eq. (17) for which “the program breaks down because the density achieves too small values ().” This solution corresponds to the density profile (287) with a compact support. The two profiles are plotted in Fig. 2. Apparently, they are both stable. It may therefore be useful to compare their respective energy in order to determine which profile has the lowest energy (ground state).
The quantum kinetic energy of a BEC is given by Eq. (265). Integrating the second expression by parts, we obtain
[TABLE]
Since for a polytrope of index [see Eq. (287)] and since and (see Appendix A), we have . As a result, the surface term vanishes and Eq. (289) reduces to
[TABLE]
Using Eq. (284) we then find that
[TABLE]
This result can be compared to the internal energy (266) of a polytrope of index which is
[TABLE]
We have the relation
[TABLE]
Therefore, the energy of a self-gravitating noninteracting BEC with the density profile (287) is different from the energy of the corresponding polytrope.
For a self-gravitating noninteracting BEC, we have (see Sec. II.2)
[TABLE]
On the other hand, the gravitational energy of a polytrope is (see Appendix C)
[TABLE]
Combining Eqs. (294) and (295) and using the mass-radius relation from Eq. (288), we find that the energy of a self-gravitating noninteracting BEC with the density profile (287) is
[TABLE]
This energy is lower than the one given by Eq. (22). Therefore, the solution of Eq. (17) that has a compact support [see Eq. (287)] has a lower energy than the solution of Eq. (17) that extends to infinity (see Sec. II.2). The ground state of the self-gravitating noninteracting BEC corresponds therefore to the solution considered in this Appendix, not to the solution that has been considered in Refs. rb ; membrado ; gul0 ; gul ; prd2 ; ch2 ; ch3 ; pop ; hui (see Sec. II.2). However, in the case of systems with long-range interactions, we know that a metastable state (i.e. a local but not a global energy minimum) may have a very long lifetime and can be fully relevant. This may be the case of the solution considered in Refs. rb ; membrado ; gul0 ; gul ; prd2 ; ch2 ; ch3 ; pop ; hui which seems to be selected in direct numerical simulations ch2 ; ch3 .
Remark: the energy of a polytrope of index can be obtained from the relations (see Appendices B and C):
[TABLE]
[TABLE]
[TABLE]
implying
[TABLE]
It differs from Eq. (296) by a factor .
Appendix F Gravitational energy
The gravitational energy of a spherically symmetric system is given by (see, e.g., Appendix G of ggp )
[TABLE]
where
[TABLE]
is the mass contained within the sphere of radius . In our model, the system is made of a core of mass and radius and a uniform atmosphere of density and mass contained within the spheres of radius and . Therefore, we can write
[TABLE]
where the first term is the gravitational energy of the core and the second term is the gravitational energy of the atmosphere in the presence of the core. For , the cumulated mass is
[TABLE]
Substituting Eq. (304) into Eq. (303) and evaluating the integral we get
[TABLE]
with
[TABLE]
These expressions are exact. For our problem, it is a good approximation to assume that . As a result, the foregoing equation simplifies into
[TABLE]
with
[TABLE]
We finally obtain
[TABLE]
The first term is the potential energy of the core, the third term is the potential energy of the atmosphere and the second term is the interaction energy. This is as if we had a point mass at the center of a distribution of mass inpreparation .
Appendix G The minimum halo radius, the minimum halo mass and the maximum halo
central density
In this Appendix, we determine the radius, the mass and the central density of the “minimum halo” assuming that it corresponds to the ground state () of a self-gravitating quantum gas. We have seen in Sec. II that it can be represented by a polytropic sphere. For the sake of generality we treat the case of an arbitrary polytropic index , then consider particular cases corresponding to fermions, noninteracting bosons and self-interacting bosons in the TF limit.
G.1 General results
The halo radius is defined as the distance at which the central density is divided by . Using Eqs. (246) and (247), it is given by
[TABLE]
where is determined by the equation
[TABLE]
The value of can be obtained by solving the Lane-Emden equation (248). The halo mass, which is the mass contained within the sphere of radius , is given by
[TABLE]
Eliminating the central density between Eqs. (310) and (312), we obtain the minimum halo mass-radius relation
[TABLE]
which is analogous to the mass-radius relation (252). On the other hand, using Eqs. (310) and (312) and introducing the universal surface density of DM halos (1) we find that the minimum halo radius, the minimum halo mass, and the maximum halo central density are given by
[TABLE]
and
[TABLE]
[TABLE]
G.2 Fermions
For the polytrope (fermion stars) using Eq. (4), we have
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Using Eq. (1), we obtain Eqs. (13)-(15). Inversely, assuming that the mass of the minimum halo is known, we obtain the fermion mass
[TABLE]
If we take we obtain .
G.3 Noninteracting bosons
For the polytrope (noninteracting boson stars) we have
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Using Eq. (1), we obtain Eqs. (25)-(27). Inversely, assuming that the mass of the minimum halo is known, we obtain the noninteracting boson mass
[TABLE]
If we take we obtain .
G.4 Self-interacting bosons in the TF limit
For the polytrope (self-interacting boson stars) we have
[TABLE]
[TABLE]
[TABLE]
Using Eq. (1), we obtain Eqs. (42)-(44). Inversely, assuming that the mass of the minimum halo is known, we obtain the self-interacting boson parameter (in the TF limit)
[TABLE]
If we take we obtain .
Appendix H Results of the quantum Jeans instability theory
In this Appendix, we recapitulate the main results of the quantum Jeans instability theory developed in Refs. prd1 ; abriljeans , restricting ourselves to the nonrelativistic regime. We refer to these papers for details about their derivation and for some generalizations.
H.1 Fermions
If DM is made of fermions, the Jeans length and the Jeans mass are given by
[TABLE]
[TABLE]
Eliminating the density between these expressions, we get the Jeans mass-radius relation
[TABLE]
We can also define a Jeans surface density
[TABLE]
These equations display the same scalings as Eqs. (318)-(320) and (15) for DM halos.
H.2 Noninteracting bosons
If DM is made of noninteracting bosons, the Jeans length and the Jeans mass are given by
[TABLE]
[TABLE]
Eliminating the density between these expressions, we get the Jeans mass-radius relation
[TABLE]
The Jeans surface density is
[TABLE]
These equations display the same scalings as Eqs. (323)-(325) and (27) for DM halos.
H.3 Self-interacting bosons in the TF limit
If DM is made of self-interacting bosons in the TF limit, the Jeans length and the Jeans mass are given by
[TABLE]
[TABLE]
The Jeans surface density is
[TABLE]
These equations display the same scalings as Eqs. (328), (329) and (44) for DM halos.
Appendix I The mass of the DM particle
In this Appendix, we relate the mass of the DM particle to the cosmological constant and to the other fundamental constants of physics.
I.1 Fermions
We have seen in Sec. II.1 that the minimum mass of DM halos made of fermions is given by
[TABLE]
It is obtained by requiring that the smallest DM halo in the Universe corresponds to the ground state of the self-gravitating Fermi gas.
On the other hand, the minimum mass of DM halos may be obtained from a quantum Jeans instability theory (see, e.g., Refs. prd1 ; abriljeans ) leading to the Jeans mass (332). Let us compute the Jeans mass at the present epoch where . For a fermion mass , the Jeans mass is abriljeans . It is orders of magnitude smaller than the minimum mass of observed DM halos. Actually, we cannot expect to have a perfect agreement between the Jeans mass computed at the present epoch and the observed minimum mass of DM halos because the linear Jeans instability took place at an earlier epoch (see Appendix I.4) and the present DM halos result from a nonlinear evolution. Therefore, we write , where is a dimensionless factor that is difficult to predict theoretically (for fermions the previous estimate gives ). Using Eq. (332), we obtain
[TABLE]
Comparing (342) and (343) we get
[TABLE]
This relation gives the surface density of the smallest DM halo if we know the fermion mass . Inversely, since appears to have a universal value (see Eq. (1)), we can use Eq. (344) to obtain the fermion mass . More precisely, since and can be expressed in terms of the cosmological constant by Eqs. (155) and (156), we find that the mass of the fermionic particle is given by
[TABLE]
It is equal to the mass scale given by Eq. (VI.3) multiplied by a large numerical factor of order (for ). This gives which is the correct order of magnitude of the fermion mass usually advocated in DM models (see Appendix D of suarezchavanis3 ). We note that, up to the dimensionless factor , this mass scale has been predicted in terms of the fundamental constants of physics independently from the observations.
Remark: We can obtain these results (without the prefactor) directly from the Jeans scales of Appendix H.1. From Eq. (334), we have
[TABLE]
If we take and (see Eqs. (155) and (156)), we get
[TABLE]
Inversely, if we assume that and we find that . We also note that and are the cosmological scales corresponding to the size and to the mass of the Universe ouf .
I.2 Noninteracting bosons
We have seen in Sec. II.2 that the minimum mass of DM halos made of noninteracting bosons is given by
[TABLE]
On the other hand, the quantum Jeans instability theory prd1 ; abriljeans leads to the Jeans mass (336). For a boson mass , the Jeans mass computed at the present epoch where is abriljeans . It is order of magnitude smaller than the minimum mass of observed DM halos. Writing with , we obtain
[TABLE]
Comparing Eqs. (348) and (349) we get
[TABLE]
Using Eqs. (155) and (156), we find that the mass of the bosonic particle (in the noninteracting case) is given by
[TABLE]
It is equal to the mass scale given by Eq. (158) multiplied by a huge numerical factor of order (for ). This is because is raised in Eq. (351) to the power . This gives which is the correct order of magnitude of the mass of ultralight axions usually advocated in DM models (see Appendix D of suarezchavanis3 ).
Remark: We can obtain these results (without the prefactor) directly from the Jeans scales of Appendix H.2. From Eq. (338), we have
[TABLE]
If we take and (see Eqs. (155) and (156)), we get
[TABLE]
Inversely, if we assume that and we find that .
I.3 Self-interacting bosons in the TF limit
We have seen in Sec. II.3 that the minimum mass of DM halos made of self-interacting bosons in the TF limit is given by
[TABLE]
On the other hand, the quantum Jeans instability theory prd1 ; abriljeans leads to the Jeans mass (340). For a ratio , the Jeans mass computed at the present epoch where is abriljeans . It is order of magnitude smaller than the minimum mass of observed DM halos. Writing with , we obtain
[TABLE]
Comparing (354) and (355) we get
[TABLE]
Using Eqs. (155) and (156), we find that the ratio is given by
[TABLE]
It is equal to the scale given by Eq. (166) multiplied by a very small numerical factor of order (for ). This gives which is the correct order of magnitude of the parameter of self-interacting bosons usually advocated in DM models (see Appendix D of suarezchavanis3 ).
Remark: We can obtain these results (without the prefactor) directly from the Jeans scales of Appendix H.3. From Eq. (341), we have
[TABLE]
If we take and (see Eqs. (155) and (156)), we get
[TABLE]
Inversely, if we assume that and we find that .
I.4 The epoch where we can apply the Jeans instability theory
Let us consider how the minimum halo mass depends on the density and on the dimensionless constant in Eqs. (343), (349) and (355). For fermions, they appear in the combination . For noninteracting bosons, we have . For self-interacting bosons in the TF limit, we have . Remarkably, the last terms in these equalities involve the same typical number . These relations suggest that is not equal to the present Jeans length but rather to the Jeans length at the epoch where the density of the Universe is . Indeed, if we calculate the Jeans mass at the epoch where the density of the Universe is , we find in the three cases considered above that . Using and (redshift), this epoch corresponds to
[TABLE]
It is intermediate between the epoch of radiation-matter equality () and the present epoch (). It may correspond to the epoch where we can apply the linear Jeans instability theory to predict the typical mass of DM halos.
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