# Jeans instability and turbulent gravitational collapse of Bose–Einstein condensate dark matter halos

## Abstract

We consider the Jeans instability and the gravitational collapse of the rotating Bose–Einstein condensate dark matter halos, described by the zero temperature non-relativistic Gross–Pitaevskii equation, with repulsive interparticle interactions. In the Madelung representation of the wave function, the dynamical evolution of the galactic halos is described by the continuity and the hydrodynamic Euler equations, with the condensed dark matter satisfying a polytropic equation of state with index \(n=1\). By considering small perturbations of the quantum hydrodynamical equations we obtain the dispersion relation and the Jeans wave number, which includes the effects of the vortices (turbulence), of the quantum pressure and of the quantum potential, respectively. The critical scales above which condensate dark matter collapses (the Jeans radius and mass) are discussed in detail. We also investigate the collapse/expansion of rotating condensed dark matter halos, and we find a family of exact semi-analytical solutions of the hydrodynamic evolution equations, derived by using the method of separation of variables. An approximate first order solution of the fluid flow equations is also obtained. The radial coordinate dependent mass, density and velocity profiles of the collapsing/expanding condensate dark matter halos are obtained by using numerical methods.

## 1 Introduction

After almost one hundred years of intensive study and research the properties of dark matter remain elusive. The existence of dark matter is one of the fundamental assumptions of modern cosmology and astrophysics [1, 2, 3, 4, 5, 6, 7], and its nature is one of the most important open questions in physics. Presently, all available information on the dark sector is obtained from the study of its gravitational interactions with astrophysical systems.

Perhaps the strongest evidence for the existence of the dark matter in the Universe comes from the study of the galactic rotation curves [8]. For hydrogen clouds in stable circular orbits moving around the galactic center the rotational velocities first increase near the center of the galaxy, thus following the standard (Newtonian) gravitational theory, but then they remain approximately constant at an asymptotic value of the order of \(v_{tg\infty } \sim 200{-}300~\hbox {km/s}\). Hence the rotation curves implies the existence of a mass profile of the form \(M(r) = rv^2_{tg\infty } /G\), where *G* is the gravitational constant. This fundamental result implies that within a distance *r* from the center of the galaxy, even at large distances where very little baryonic (luminous) matter can be detected, the mass profile increases linearly with *r*. This type of behavior can be explained by assuming the presence of a new (and exotic) mass component, interacting only gravitationally with ordinary matter, and which most likely consists of new particle(s) not (yet) included in the standard model of particle physics. The behavior of the galactic rotation curves still provides the most convincing and compelling evidence for the existence of dark matter [9, 10, 11, 12].

The second important astrophysical evidence for dark matter comes from the virial mass discrepancy in galaxy clusters [13]. Galaxy clusters are giant astrophysical systems formed of thousands of galaxies each, bounded together by their own gravitational interaction. The galaxies give around 1% of the mass of the clusters, while the high temperature intracluster gas represents around 9% of the cluster mass. But the total masses obtained by measuring the velocity dispersions of the galaxies exceed the total masses of all stars in the cluster by factors of the order of 200-400 [14]. Hence, in order to explain the cluster dynamics one needs to assume the presence of dark matter, representing around 90% of the mass of the cluster. Another strong evidence for the presence of dark matter follows from the measurement of the temperature of the intracluster medium, since a supplementary mass component is required to explain the determined depth of the gravitational potential of the clusters [14].

Several other astrophysical and cosmological observations have also provided compelling evidence for the presence of dark matter. From the cosmological perspective, the recent Planck satellite measurements of the Cosmic Microwave Background Radiation [15] led to the precise determination of the cosmological parameters. These results have indicated that baryonic matter only cannot explain the cosmological dynamics, and that the standard \(\Lambda \) Cold Dark Matter (\(\Lambda \hbox {CDM}\)) cosmological paradigm requiring the existence of dark matter is strongly favored by observations. For a consistent interpretation of the gravitational lensing data the existence of dark matter is also required [16, 17, 18].

Powerful observational evidences the existence of dark matter are yielded by the observations of a galactic cluster called the Bullet Cluster, consisting of two colliding clusters of galaxies. Due to collision of its two components that occurred in the past, in the Bullet Cluster cluster the baryonic and the dark matter components are separated [19]. Determinations of the cosmological parameters from the Planck data on the cosmic microwave background radiation did show convincingly that the Universe consists of 74% dark energy, 22% non-baryonic dark matter and only 4% baryonic matter [15].

Depending on the energy of the particles composing it, dark matter models can be classified generally into three major types, cold, warm and hot dark matter models, respectively, [20]. For the dark matter particle(s) the main candidates are the WIMPs (Weakly Interacting Massive Particles) and the axions, respectively [20]. WIMPs are hypothetical (and yet undetected) heavy particles, interacting through the weak force [21, 22]. Other popular dark matter candidates, the axions, are bosons that were first proposed as a solution of the strong CP (charge + parity) problem, which requires one to explain why quantum chromodynamics does not break the CP symmetry [23, 24].

There are also other approaches that attempt to interpret the observational data without resorting to dark matter. These explanations assume that on galactic or extragalactic scales the law of gravity (Newtonian or general relativistic) is modified. The earliest attempt to explain the rotation curves by modified gravity is MOND (Modified Newtonian Dynamics) theory, proposed initially in [25]. Other modified theories of gravity have also been used widely as alternative explanations to dark matter phenomenology [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. For a recent review of the dark matter problem in some modified theories of gravity see [37].

An attractive possibility for the detection of the presence of dark matter may be provided by its possible annihilation into ordinary particles. If such physical processes really take place, then a large number of positrons and gamma-ray photons are produced, thus giving some clear observational signatures for the presence of dark matter. This possibility may be supported by the detection of excess positron emission in our galaxy [38, 39, 40, 41, 42, 43, 44, 45]. Hence the excess gamma-ray and positron emissions in our galaxy could be interpreted as coming from the annihilation of dark matter with mass in the range of \(m \sim 10\)–100 GeV [38, 39, 40]. For an in depth discussion of this problem, as well as of the alternative possibilities for the interpretation of the observational data see [41, 42, 43, 44, 45].

Dark matter models can generally give good explanations of the phenomenological (and unexpected) behaviors of particle dynamics at the galactic and extragalactic level, including the constancy of the rotation curves, and the virial mass discrepancy. However, crucial conflicts do appear when one compares the results of the numerical simulations of the theoretical models with the observations. The observational data on nearly all observed galactic rotation curves indicate that in the presence of a single pressureless dark matter component they increase less sharply as compared to the predictions of the cosmological simulations of structure formation in the standard \(\Lambda \)CDM model. Moreover, the numerical simulations display dark matter density profiles that behave as \(\rho \sim 1/r\) (a cusp) at the galactic center [46]. On the other hand, observations of the galactic rotation curves show the existence of constant density cores [47, 48]. In dark matter physics this contradiction between theory and observations represents the so-called core-cusp problem. Dark matter models must also face the “too big to fail” question [49, 50]. The Aquarius simulations did show that in the dark matter halos predicted in the standard \(\Lambda \)CDM model the most massive subhalos are incompatible with the observations of the dynamics of the brightest dwarf spheroidal galaxies of the Milky Way [50]. For the dwarf spheroidal galaxies the best-fitting hosts have maximum velocities in the range \(12\; \mathrm{km/s}< V_{max} < 25~\hbox {km/s}\), while all the \(\Lambda \)CDM simulations give at least ten subhalos with velocities \(V_{max} > 25~~\hbox {km/s}\). In the framework of the \(\Lambda \)CDM-based models of the satellite population of the Milky Way these observational results cannot be interpreted. The main contradiction between theory and observations is related to the predictions of the densities of the satellites, with the predicted dwarf spheroidals having dark matter halos more massive by a factor of \(\sim \)5 than shown by the observations.

The problems mentioned above, related to the physical properties of the dark matter, may be explained if one extends the standard \(\Lambda \)CDM model by assuming that dark matter particles may have some kind of self-interaction. This model, called the self-interacting dark matter (SIDM) paradigm assumes the existence of supplementary interactions in the dark sector [51, 52, 53, 54] that may allow momentum and energy exchange between particles that compose the dark matter halos. In these models the basic quantity describing the dark matter halo properties is the self-interaction cross section \(\sigma _{DM}\) divided by the dark matter particle mass *m*, \(\sigma _{DM}/m\). If the dark matter self-interactions have cross sections per mass of the same order of magnitude as the strong nuclear force, \(\sigma _{DM}/m\sim 1 \mathrm {g/cm}^{-2}\), this would thermalize the inner regions of dark matter halos where the baryonic matter is concentrated. On the other hand for \(\sigma _{DM}/m\ge 1 \mathrm {g/cm}^{-2}\) on galactic scales Self-Interacting Dark Matter models can explain the uniformity as well as the diversity of galaxy rotation curves [55, 56, 57]. Galaxy clusters also show the same diversity of properties of their Self-Interacting Dark Matter halos [58].

The possibility of a complex self-interaction of dark matter particles received some observational backing from the study of the data obtained from the study of 72 galaxy cluster collisions. The observations did include both ‘major’ and ‘minor’ mergers, and they were done by using the Hubble and Chandra Space Telescopes [59]. Important constraints on the non-gravitational forces acting on dark matter can be obtained from the study of the collisions between galaxy. The analysis presented in [59] gave an upper limit of \(\sigma _{DM}/m\) as \(\sigma _{DM}/m < 0.47~\hbox {cm}^{2} /\hbox {g}\) (at 95% Confidence Level). In [60] an upper limit on the self-interaction cross section of dark matter of \(\sigma _{DM}/m<1.28~\hbox {cm}^{2}/\hbox {g}\) (68% Confidence Level), was obtained. Different self-interacting dark matter models were investigated in [61, 62, 63, 64]. In [65] the implications of the self-interaction of dark matter for the tidal stripping and evaporation of satellite galaxies in a Milky Way type galaxy were considered. The response of self-interacting dark matter halos to the growth of galaxy potentials using numerical simulations was investigated in [66], and a greater diversity of dark matter halo profiles was found. A self-interacting dark matter halo with \(\sigma _{DM}/m=0.1~\hbox {cm}^{2}/\hbox {g}\) gives a good fit to the measured dark matter density profile of A2667. The same halo simulated with \( \sigma _{DM}/m=0.5~\hbox {cm}^{2}/\hbox {g}\) does not produce a core profile dense enough to fit the observational data of A2667. Together with the previous findings in [59], these limits point towards the result that the constraint \(\sigma _{DM}/m\ge 0.1~\hbox {cm}^{2}/\hbox {g}\) is strongly disfavored for dark matter collision velocities greater than 1500 km/s.

Therefore, as suggested by the above observational results, we cannot reject a priori the possibility that dark matter is a self-interacting constituent of the Universe. Physical models of dark matter in which the fundamental particles are self-interacting may provide a better theoretical explanation of the observed phenomenology at galactic and extragalactic scales. But from both a phenomenological perspective, and from a fundamental theoretical and physical point of view, *the best motivated* self-interacting dark matter models can be constructed by assuming that presently dark matter is in the form of a *Bose–Einstein condensate*.

*T*is smaller than the critical one, \(T_{cr}\), which is given by the expression [70, 71, 72, 73]

*m*is the mass of the particle in the boson gas, \(k_B\) is Boltzmann’s constant, and \(\zeta \) denotes the Riemann zeta function, respectively.

The experimental creation of Bose–Einstein condensates was first realized in dilute alkali gases in 1995 by cooling a dilute vapor of approximately two thousand rubidium-87 atoms to below 170 nK, using a combination of laser cooling and magnetic evaporative cooling [74, 75, 76]. The presence of a Bose–Einstein condensate in a bosonic system is indicated, from a physical and experimental point of view, by the appearance in both coordinate and momentum space distributions of the particles of sharp peaks.

Presently, the only evidence for the existence of Bose Einstein condensates on a microphysical scale appeared in laboratory experiments, which involve a very small scale. On the other hand, the possibility of the existence of some forms of bosonic condensates in the cosmic environment cannot be excluded a priori. Due to their superfluid properties, in high density general relativistic objects, like, for example, neutron or quark stars, the neutrons or the quarks could form Cooper pairs, which, once the temperature or density reach their critical values, would eventually condense. Bose–Einstein condensate stars may have maximum central densities of the order of \( 0.1-0.3\times 10^{16}~\hbox {g/cm}^3\), minimum radii in the range of 10-20 km, and maximum masses of the order of \(2M_{\odot }\), respectively. The study of their interesting physical and astrophysical properties is presently an active field of research [77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87].

*R*of the zero temperature Bose–Einstein condensate dark matter halo is given by \(R=\pi \sqrt{\hbar ^{2}a/Gm^{3}}\), where

*a*is the scattering length [102]. The total mass

*M*of the condensate dark matter halo is obtained: \(M=4\pi ^2\left( \hbar ^2a/Gm^3\right) ^{3/2}\rho _c=4R^3\rho _{c}/\pi \), where \( \rho _{c}\) is the central density of the galactic halo. The mass of the dark matter particle satisfies a particle mass-galactic radius relation given by [102]

The condensate dark matter halo may not necessarily be at zero temperature. In [104] the thermal correction to the dark matter density profile where obtained. An important result on the Bose–Einstein condensate dark matter halos is that their density profiles generally indicate the presence of an enlarged core, whose presence is due to the strong interaction between dark matter particles [105]. The investigations of the properties of the Bose–Einstein condensate dark matter on cosmological and astrophysical scales is a very interesting and active field of research [106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152]. Properties of the Fuzzy Dark Matter, a particular theoretical form of dark matter, assumed to be formed of an extremely light boson (\(m\sim 10^{-22}~\hbox {eV}\)), with a de Broglie wavelength of the order of \(\lambda \sim 1~\hbox {kpc}\), were investigated in [153].

While the static properties of the Bose–Einstein condensate dark matter halos have been extensively investigated, their rotational characteristics have received less attention. The presence of vortices in a self-gravitating Bose–Einstein condensate dark matter halo, consisting of ultra-low mass scalar bosons, was investigated in [96]. Rotation of the dark matter may induce a harmonic trap potential for vortices. In [109] a detailed study of the vortices in rotating Bose–Einstein condensate dark matter halos was performed, and strong bounds for the shape and quantity of vortices in the halo, for interaction strength, for the critical rotational velocity for the nucleation of vortices, and for the boson mass were found. In [110], by assuming that a vortex lattice forms, the effects of rotation on a superfluid Bose–Einstein condensate dark matter halo were investigated. On the rotation curves sub-structures similar to some observations in spiral galaxies may form. The equilibrium properties of self-gravitating, rotating Bose–Einstein condensate haloes, which satisfy the Gross–Pitaevskii–Poisson equations were studied in [121]. For a wide range of the Bose–Einstein condensate dark matter physical parameters vortices are generated. On the other hand, vortices cannot appear for a vanishing self-interaction, and they form when the self-interaction between dark matter particles is strong enough.

One of the fundamental concepts in modern astrophysics and cosmology is that of gravitational instability, initially discussed by Jeans [154]. Its importance is related to the prospect of estimating the scale of the condensations that may occur in an extended gaseous medium under the influence of small perturbations. The possibility of such an estimation will provide at least qualitative information regarding the formation of stars and of galaxies from an original cosmic medium. The main result of the original analysis by Jeans was that a self-gravitating infinite uniform gas at rest should be unstable against small perturbations proportional to \(\exp \left[ i\left( \vec {k}\cdot \vec {r} -\omega t\right) \right] \). The lineariazation of the equations of the ideal hydrodynamics and the Poisson equation results in the well-known dispersion relation \(\omega ^{2}=c_{s}^{2}k^{2}-4\pi G\rho \), where \( c_{s}=\left( \gamma k_{B}T/m\right) ^{1/2}\) is the adiabatic sound velocity, \(\rho \) is the density, *T* is the gas temperature, and \( \gamma =5/3\) is the ratio of specifics heats, respectively. When \(\omega ^{2}\) becomes negative, an instability arises once the perturbation wavelength \(\lambda =2\pi /k\) exceeds the critical value \(\lambda _{J}=c_{s} \sqrt{\pi /G\rho }\), \(\lambda >\lambda _{J}\). Thus, an originally uniform gas, due to the instability, should break into massive components with characteristic size of the order of \(\lambda _{J}\) [13]. The effects of the rotation and of other physical effects on the stability of a self-gravitating medium were considered in [155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165]. The kinetic theory of the Jeans instability was developed in [166, 167, 168, 169, 170], mainly using methods from plasma physics. For a detailed discussion of the Jeans instability and its role in astrophysics see [13].

An interesting property of the Bose–Einstein condensation processes is represented by the collapse and the ensuing explosion of the condensates [172]. Near a Feshbach resonance the atomic scattering length *a* can be changed, by adjusting an external magnetic field, over a large range. Once the sign of the scattering length is changed, a repulsive condensate of \(^{85}\) Rb atoms is transformed into an attractive one that subsequently reaches a collapsing and an exploding phase. The collapse of a Bose–Einstein condensate was investigated by using the semi-classical Fokker–Planck equation for a gas of free bosons in [114, 173, 174], and, for a \(1/r^b\) type potential, in [175].

The mechanisms of the gravitational collapse of the Bose–Einstein condensate dark matter halos was studied in [129], by using a variational approach, and by choosing an appropriate trial wave function. This approach allows the reformulation of the Gross–Pitaevskii equation with spherical symmetry as Newton’s equation of motion for a particle in an effective potential, which is determined by the zero point kinetic energy, the gravitational energy, and the particles interaction energy, respectively. The velocity of the condensate is proportional to the radial distance, with a time-dependent proportionality function. The collapse of the condensate ends with the formation of a stable configuration, corresponding to the minimum of the effective potential. The obtained results did show that the gravitational collapse of the condensed dark matter halos can lead to the formation of stable astrophysical systems with both galactic and stellar sizes.

It is the goal of the present paper to investigate the Jeans stability and the collapsing properties of the Bose–Einstein condensate dark matter halos in the presence of vortices, induced by the rotation of the halo. Rotation is a general feature of galactic dynamics, probably generated by some physical instability processes in the early Universe. To describe the Bose Einstein condensate galactic dark matter halos as a multi-particle bosonic system we adopt an effective approach based on the Gross–Pitaevskii equation [102]. The Gross–Pitaevskii equation gives an effective mean-field description of the gravitationally confined dark matter halo. The mathematical analysis of the condensed dark matter halos is greatly simplified by the introduction of the Madelung representation of the wave function, which allows the description of the dark matter in terms of the equations of the classical fluid dynamics, the continuity and the Euler equation, respectively. In this approach dark matter halos can de described as fluid structures obeying a polytropic equation of state, with polytropic index \(n=1\). The Euler evolution equation also contains the quantum force, derived from the quantum potential, and which represents a purely quantum effect. The assumptions of the rotation for the condensate dark matter halo leads to the necessity of taking into account the presence of quantized vortices. A vortex is an excitation of the bosonic system, and hence it is a state whose energy is higher than that of the ground state.

In order to investigate the Jeans stability of Bose–Einstein condensate dark matter halos we consider the perturbation of the hydrodynamic flow equations with respect to an appropriately chosen ground state, and we perform a local linear stability analysis, which includes the effects of the quantum pressure, of the quantum potential and of the rotation, as well as the self-gravity of the system. The dispersion relation for the propagation of the instabilities in the system is obtained, and the Jeans wave number for the rotating Bose–Einstein condensate dark matter halo is obtained. Once the Jeans wave number is known, we can immediately obtain the Jeans radius and mass, which give the critical length and mass scales above which condensate dark matter collapses. In the approximation when rotation and quantum force are ignored, the Jeans radius and mass coincide with the radius and mass of the static condensate. The effects of the rotation are also considered, and the Jeans radius and mass are also obtained for rotation dominated dark matter halos. Similar results can be obtained in the framework of the Thomas–Fermi approximation, by requiring that the total energy of the system is minimal.

Once the radius and mass of the Bose–Einstein condensate dark matter halo exceeds the Jeans limits for the critical stability, gravitational collapse or expansion follows. In order to study the collapse or the expansion of the condensate we derive first a semi-analytical solution to the spherical hydrodynamic equations governing the time and space evolution of a Bose–Einstein condensate dark matter halo. First we reduce the system of the coupled nonlinear partial equations describing the evolution of the galactic halo to a single, strongly nonlinear partial differential equation for the mass \(M=M(r,t)\) of the condensate. Then we factorize the mass function into products of a time-dependent factor and another factor depending only on the spatial variable *r*. This factorization means that both spatial and temporal profiles of the gravitational, mechanical and thermodynamic variables of the halo have a universal behavior, once they are factorized appropriately. To study the spatial profiles of the collapsing/expanding rotating dark matter halos we use approximate and numerical methods. A first order approximative solution of the mass function is explicitly obtained. The general mass, density and velocity spatial profiles are obtained by numerically integrating the mass equation. In relation with the general properties of the gravitational collapse we also show that homologous solutions to the hydrodynamic equations describing the time evolution of a Bose–Einstein condensate dark matter, wherein thermodynamic variables factorize into products of a time-dependent factor and another factor depending only on the scaled spatial variable \(\xi \equiv r/R(t)\), *do not exist*. This factorization means that spatial profiles of the thermodynamic variables remain time invariant. Since Bose–Einstein condensate dark matter halos do not have this property, it follows that the dynamics of the collapse of a condensed galactic halo is very different from other types of gravitational collapse, like, for example, the pre-supernova stellar core collapse.

The present paper is organized as follows. The basic properties of the Bose–Einstein condensate dark matter halos are presented in Sect. 2, where, with the use of the Madelung representation, the evolution equations of the condensed galactic halos are formulated in terms of the continuity and Euler equations of classical hydrodynamics. The Jeans instability of condensed dark matter halos confined by their own gravitational field is discussed in Sect. 3, by fully taking into account the effects of the rotation and the quantum effects. The limiting cases of the general dispersion relation are considered in detail. The time evolution of the Bose–Einstein condensate dark matter halos is investigated in Sect. 4, and it is shown that the hydrodynamic equations describing the dynamics of a rotating polytropic gas with polytropic index \(n=1\) do admit a separable solution, in which all physical quantities can be expressed as products of two functions, one depending on time only, and the second a function of the radial coordinate only. An approximate first order solution of the evolution equations is also obtained. The spatial dependence of the physical quantities is investigated by using numerical methods. We discuss and conclude our results in Sect. 5. In Appendix A we explicitly show that the hydrodynamic evolution equations describing a polytropic fluid with polytropic index \(n=1\) do not admit self-similar homologous solutions.

## 2 Gravitationally confined Bose–Einstein dark matter halos

Generally, Bose–Einstein condensation processes take place in a Bose gas with particle number density *n* when the thermal de Broglie wave length \(\lambda _{dB}=\sqrt{2\pi \hbar ^2/mk_BT}\), exceeds the mean interparticle distance \(n^{1/3}\). Then, as a result, the wave packets percolate in space. The critical condensation temperature can then be obtained qualitatively as \(T\le 2\pi \hbar ^2n^{2/3}/mk_B\) [102]. Under the assumption of an adiabatic cosmological expansion of the Universe, the temperature dependence of the number density of the particle is \(T\propto n^{2/3}\). Hence Bose–Einstein condensation occurs if the mass of the particle satisfies the condition \(m<1.87~\hbox {eV}\) [176], and therefore particles satisfying this mass limit could Bose–Einstein condense, and form large scale cosmological or astrophysical structures.

Due to the low temperature of the Bose–Einstein condensates, their physical properties can be understood within the so-called mean-field approximation. The success of the mean-field approach is determined by the dilute nature of the Bose gases. We can always write the Bose field operator as a sum of the condensate wave function and an operator describing the non-condensed bosons, so that \(\hat{\Psi }\left( \vec {r},t\right) =\Psi \left( \vec {r} ,t\right) +\hat{\Psi }^{\prime }\left( \vec {r},t\right) \), where \(\Psi \left( \vec {r},t\right) \equiv \left<\hat{\Psi }\left( \vec {r},t\right) \right>\) is the average value of \(\hat{\Psi }\left( \vec {r},t\right) \), and \(\hat{\Psi }^{\prime }\left( \vec {r},t\right) \) represents the fluctuations in the system. The single-particle density matrix \(\rho \) is given by \(\rho \left( \vec {r}, \vec {r}\;^{\prime }\right) =\left<\hat{\Psi }^{+}\left( \vec {r}\right) \hat{ \Psi }\left( \vec {r}\;^{\prime }\right) \right>\), where \(\hat{\Psi }^{+}\left( \vec {r}\right) \) is the field operator creating a particle at a point \(\vec {r} \), and \(\hat{\Psi }\left( \vec {r}\;^{\prime }\right) \) is the field operator annihilating a particle at \(\vec {r}\;^{\prime }\). In a dilute Bose gas close to \(T = 0\), one can neglect with a very good approximation the non-condensed bosons \(\hat{\Psi }^{\prime }\left( \vec {r},t\right) \). In this case the mean-field order parameter is given exactly by the quantum mechanical wave function \(\Psi \left( \vec {r},t\right) \), with well defined phase. Hence, the zero temperature dynamics of a Bose gas consisting of confined weakly interacting particles is described by a mean-field macroscopic wave function \(\Psi \). The wave function of the condensate behaves like a complex order parameter whose absolute value and phase contains all relevant information as regards the Bose–Einstein condensate system, and satisfies a nonlinear Schrödinger equation, called the Gross–Pitaevskii equation [100, 101].

*a*is the coherent scattering length (defined as the zero-energy limit of the scattering amplitude \(f_{scat}\)),

*m*is the mass of the condensate particle, and \(V_{ext}\) is the external potential. In the following we assume that the exterior potential \(V_{ext}( \vec {r},t)\) is the gravitational potential \(V_{grav}\), \(V_{ext}(\vec {r} ,t)=V_{grav}\left( \vec {r},t\right) \equiv \phi \left( \vec {r},t\right) \). For a single component condensate dark matter halo \(\phi \left( \vec {r} ,t\right) \) satisfies the Poisson equation

*G*is the gravitational constant, respectively. The probability density \(\left| \Psi \left( \vec {r},t\right) \right| ^{2}\) is normalized according to \( \int _{V}{n\left( \vec {r},t\right) d^{3}{\vec {r}}}=\int _{V}{\left| \Psi \left( \vec {r},t\right) \right| ^{2}d^{3}\vec {r}}=N\), where

*N*is the total particle number in the dark matter halo, which can be obtained by integrating the norm of the wave function over the entire volume

*V*of the Bose–Einstein condensate.

*L*is given by [177]

*phase*of the wave function, we obtain

*V*of the condensate, using the Gauss theorem, leads immediately to the equation of the particle number conservation, which can be formulated as

*S*that encompasses the total volume

*V*. For a static condensate, \(\vec {v}\equiv 0\), and \(\rho (R)\equiv 0 \), where

*R*is the radius of the condensate, a condition which implies that the particle flux \(\vec {j}(R)=\rho (R)\vec {v}\equiv 0\). Hence, from Eq. (18), it follows that the total particle number in the Bose–Einstein condensate dark matter halo at zero temperature is a constant, \(N=\mathrm {constant}\).

For superfluid for which the phase of the wave function is * nonsingular*, the continuity and Euler Eqs. (8) and (17) represent a *potential (irrotational) flow*, since the condition \( \nabla \times \vec {v}=0\) is always satisfied, due to the definition (13) of the fluid velocity.

*the phase singularity lines*[178, 179, 180]. Since the phase of the wave function is defined within a factor of \(2\pi \) only, Eq. (13), defining the condensate velocity, implies that the circulation of the velocity field along a closed contour must be quantized in units of \(\hbar /m\), according to the rule [178, 179, 180]

*n*, called the topological charge of the flow, is an integer number. It also represents the winding number of the phase of the wave function along the contour. To obtain a non-zero circulation, the contour must winds around a line of zero density, along which the phase, as well as the velocity field, are no longer defined. The nucleation of the quantized vortices is a powerful experimental evidence for the existence of the macroscopic wave function describing the Bose–Einstein condensate. In laboratory experiments the observation of the quantized vortices is quite difficult due to the smallness of the size of the vortex core. This size is of the order of the so-called

*healing length*[72]

*the macroscopic angular velocity of the condensate*, given by

*the areal vortex number density*, defined as \(n_V = N_V /A_{\perp }\), and obtained by assuming that the singular velocity fields of the vortices are distributed uniformly in the plane of rotation with area \(A_{\perp }\). As we move away from the vortex, the velocity slowly decreases. If we move towards the vortices then the superfluid density tends to zero.

*distributed vorticity*, and it has been used very successfully to describe the dynamics of Bose–Einstein condensates containing vortex lattices. In the distributed vorticity approximation we introduce a rotational component in the velocity field of the condensate, so that we can write [178, 179, 180]

## 3 The Jeans instability for turbulent Bose–Einstein condensate dark matter halos

### 3.1 Jeans stability of the nonrotating Bose–Einstein condensate matter halos

On the other hand, one could also ask the question if Bose–Einstein condensate dark matter could form astrophysical objects and structures with the limiting mass given by the Chandrasekhar mass as given by Eq. (52). For the adopted mass of the dark matter particle \(m=1 \;\mathrm { meV}=1.78\times 10^{-36}~\hbox {g}\), the Chandrasekhar limit gives a mass of the order of \(M_{Ch}=1.62\times 10^{34}M_{\odot }=3.25\times 10^{57}~\hbox {g}\), which exceeds by three orders of magnitude the total mass of the ordinary matter in the Universe, \(4.5\times 10^{54}~\hbox {g}\) [?]. Therefore, it follows that the Chandrasekhar mass limit cannot lead to realistic descriptions of the physical properties of condensate dark matter clouds composed of particles having masses of the order of the mass values assumed for condensate dark matter particles.

### 3.2 The effects of the rotation

*G*, and they are fully determined by the fundamental quantum parameters

*m*and

*a*, characterizing the condensate, as well as by the angular velocity of the halo \(\left| \vec {\Omega }_{0}\right| \).

### 3.3 The Thomas–Fermi approximation

*N*bosons in a volume

*V*, extended over a radius

*R*, in the presence of a confining gravitational potential. The total mass of the system is denoted by

*M*. Moreover, we assume that all bosons are in the same quantum state. The total energy \(E_{tot}\) of the system is given by \(E=E_{kin}+E_{int}+E_{grav}\). For a single particle the kinetic energy can be approximated as \(\hbar ^2/2mR^2\) [72], and hence the total kinetic energy of the system can be obtained: \(E_{kin}=N\hbar ^2/2mR^2\). The interaction energy is given by \( E_{int}=(1/2) \left( N^2/V\right) mg\) [72], while the gravitational potential energy can be approximated as \(E_{grav}=-\alpha GM^2/R\), where \(\alpha \) is a constant. For the \(n=1\) polytropic equation of state \(\alpha =3/4\) [103]. Therefore for the total energy of the bosonic system in a gravitational field we obtain the expression

*exact*.

On the other hand, the physical length scales of the order of \(R\approx \sqrt{m/4\pi \rho a}\), where \(\rho \) is the mean dark matter density, the Thomas–Fermi approximation is not valid anymore. For \(\rho =10^{-24}\; \mathrm {g/cm}^3\), we obtain \(R\approx 377.65~\hbox {cm}\), a length scale that is insignificant from the point of view of the galactic dark matter distribution. For the gravitational energy of the Bose–Einstein condensate dark matter halo we obtain the value \(E_{grav}=8.89\times 10^{56}\) ergs, why for the considered numerical values of the parameters of the dark matter halos the interaction energy has the value \(E_{int}=1.35\times 10^{56}\) ergs. Hence in gravitationally trapped galactic size condensates the interaction energy is of the same order of magnitude as the gravitational energy of the bosons. However, it is important to point out that \( E_{grav}>E_{int}\), a relation that shows that the dark matter halo is trapped by its gravitational energy.

## 4 Time evolution of Bose–Einstein condensate dark matter halos

*r*. Consequently we obtain for the centrifugal acceleration the expression \(2\Omega ^2r/3\) [183].

*M*(

*r*,

*t*), the total mass of the dark matter halo within radius

*r*, is defined as

*r*, we obtain

*v*and

*M*fully determine the dynamical evolution of the collapsing Bose–Einstein condensate dark matter halo. This system of equations can be reduced to a single equation describing the dynamical evolution of the collapse. From Eq. (80) we obtain

### 4.1 The stationary solution

### 4.2 Collapsing Bose–Einstein condensate dark matter halos

Obtaining some analytical or semi-analytical solutions of the equations describing the time evolution of Bose–Einstein condensate dark matter halos would allow one to construct a clearer picture of the relationship between the input physics and the behavior of the system, also making possible to understand astrophysical effects that are more difficult to understand with the extensive use of numerical methods for solving the hydrodynamical evolution equations. One of the powerful mathematical approaches for the study of the collapse processes in astrophysical phenomena is based on the idea of self-similarity, which consists in the rescaling of the radial coordinate *r* as \(r\rightarrow \xi =r/\alpha (t)\), where \(\alpha (t)\) is a function only dependent on time, and to assume that all the other physical parameters can be expressed as functions of \(\xi \) and of some function of time, so that an arbitrary physical quantity \(\Phi (r,t)\) can be written as \( \Phi (r,t)=\beta (t)\phi (\xi )\). An interesting consequence of the self-similar (homologous) evolution is that the initial density and mass profiles of the collapsing systems do not change. From a mathematical point of view after introducing the appropriately chosen self-similarity transformations, the system of nonlinear partial differential equations (80) and (81) can be reduced in many cases to a system of ordinary nonlinear differential equations. Some astrophysically relevant self-similar solutions describe the gravitational collapse of isothermal spheres or of polytropic spheres. It would then be interesting to consider self-similar solutions of the hydrodynamic equations describing the time evolution of Bose–Einstein condensate Dark matter halos. Unfortunately, the hydrodynamic equations describing the evolution of a polytropic gas with equation of state \(p\propto \rho ^2\) do not admit self-similar or homologous solutions (for a proof of this results see “Appendix A”. The main reason is that after introducing the self-similar variables in the standard way, the explicit time dependence of the equations cannot be eliminated for any choice of the time similarity functions.

#### 4.2.1 The evolution equations

*f*(

*t*) and

*m*(

*r*) are functions depending on the time and the radial coordinate

*r*only. Then the velocity of the dark matter halo can be immediately obtained from Eq. (82) as

*f*(

*t*) and the angular velocity \( \Omega (t)\) satisfy the conditions

*dimensional*constants. Eq. (111) can be integrated immediately to give

*s*. The function

*f*(

*t*) is dimensionless. Hence once the solution of Eq. (115) is known, the evolution of the physical parameters of the collapsing/expanding Bose–Einstein dark matter halos are given by

*r*, the mass

*m*(

*r*), the density \(\rho (r,t)\) and the velocity

*v*(

*r*,

*t*) according to the transformations

### 4.3 The first order approximation

### 4.4 Exact numerical profiles

The variations of the dimensionless spatial density, mass and velocity profiles of the contracting/expanding Bose–Einstein condensate dark matter halos are represented in Figs. 1, 2 and 3, respectively. To numerically integrate the system of Eqs. (136) and (137) we have used the initial conditions \(m_0\left( \eta _0\right) =\left( 4/\pi \right) \eta _0^3\) and \(m_0'\left( \eta _0\right) =\left( 12/\pi \right) \eta _0^2\), with \(\eta _0=10^{-3}\). The spatial density profile of the dark matter halo during its dynamical evolution is represented in Fig. 1. The static solution in the absence of rotation is also represented, as the lower curve in the figure. In all cases the density profiles are described by monotonically decreasing functions of the dimensionless radius \(\eta \). The effects of the rotation of the halo can be clearly seen, and they lead to a significant impact on the density profiles. Near the origin, for \(\eta \) in the range \(0<\eta <0.25\), the density profiles are basically indistinguishable, but for larger values of \(\eta \) the effects of the rotation modify the overall density distribution. The impact of the rotation on the spatial mass distribution of the collapsing/expanding dark matter halos is presented in Fig. 2. The mass profiles are characterized by linearly increasing functions of \(\eta \). For \(\eta \) in the range \(0<\eta <1.1\), the mass distribution of the rotating collapsing/expanding condensate dark matter halo essentially coincides with the static mass distribution. The effects of the rotation become important for large values of the radius, once we are approaching the lower density regions at the outer boundary of the halo. The rotation can lead to a significant increase in the total mass of the halo, the increase being of the order of 20% for \(\gamma _0=0.9\).

The spatial velocity profiles of the dynamically evolving condensate dark matter halos are presented in Fig. 3. Similarly to the mass profile, the spatial distribution of the velocity is a monotonically increasing function of \(\eta \). For \(\eta \) in the range \(0<\eta <1.1\), the velocity profiles are (almost) identical, and the effects of rotation are extremely small. For small rotation velocities the velocity can be well approximated by a linear function of \(\eta \), so that \(V(\eta )\propto \eta \). However, for larger values of the dimensionless radial coordinate, the effects of the rotation become important at large distances from the halo center, and the velocity – radial coordinate relations is not linear anymore.

## 5 Discussions and final remarks

In the present paper we have investigated some of the possible physical and astrophysical consequences of the instabilities in a Bose–Einstein condensate dark matter halo. We have considered two distinct, but interrelated topics, the Jeans stability of condensate dark matter clouds, and the dynamics of the gravitational collapse that follows once the real size of the gravitationally confined system exceeds the Jeans scales.

The Bose–Einstein condensates are generally described by the Gross–Pitaevskii equation, which gives the ground state of a quantum system of identical bosons, and which is obtained by using the Hartree–Fock approximation and the pseudopotential interaction model. In the present approach we assume that dark matter is a Bose–Einstein condensate of a gas of bosons, which are in the same quantum state, and thus can be described by the same wavefunction. Hence, we interpret the galactic dark matter halos as huge quantum systems, whose properties can be described by a single wave function. To simplify our formalism we also adopt the assumption that dark matter is at zero temperature. Of course the presence of the excitations and of non-zero temperature effects may have important effects on the properties of condensate dark matter.

An important mathematical and physical property of the Gross–Pitaevskii equation is that, similarly to the standard Schrödinger equation, it admits a hydrodynamical representation, easily obtainable after representing the wave function in the Madelung variables. It turns out that in the hydrodynamic representation the Bose–Einstein condensates can be described as a perfect fluid, and its properties are characterized by the local density and local velocity only. The fluid satisfies a continuity and an Euler type equations, which contains the contributions of the quantum pressure \(p=u_0\rho ^2\), generated by the self-interaction of the field, and of the quantum potential \(V_Q\), both of these terms being essentially quantum mechanical in their origin. The velocity \(\vec {v}\) of the quantum fluid is related to the phase *S* of the wave function by the relation \(\vec { v}=\left( \hbar /m\right) \nabla S\), and the flow of the condensate fluid is rotationless, as long as *S* contains no singularities, as, for example, in a vortex. In the present approach we have also allowed the presence of vortices in the dark matter halo. The simplest method to create vortices in a condensate is through their rotation. Hence, under the natural assumption that Bose–Einstein condensate dark matter halos are rotating, one must also take into account the existence of vortices in the system. A Bose–Einstein condensate dark matter halo can rotate only due to the existence of quantized vortex lines. When the rotation frequency \(\Omega \) of the halo exceeds a critical value \(\Omega _c\), vortex nucleation occurs. In the present paper we have adopted the simple relation (22) between the angular velocity of the rotating condensate dark matter halos, and their vortex number density, which indicates that the quicker the dark matter halo rotates, the higher the number of vortices. On the other hand in rotating gravitationally trapped Bose–Einstein condensate dark matter halos the nucleation of vortices may also be a result of the instabilities of collective excitations.

In the present manuscript we have used the term turbulence in the sense that it is common in the physics of quantum fluids, meaning the presence in the fluid of quantized vortices [184, 185]. In this nomenclature turbulence is more related to the quantization of vortices than to the lack of viscosity of superfluids. On the other hand the term turbulence may also be interpreted as describing a phase of temporally and spatially disordered fluid motion, characterized by a large number of degrees of freedom interacting essentially nonlinearly [185]. The nonlinear interaction is usually a consequence of the presence of the term \(\left( \vec {v}\cdot \nabla \right) \vec {v}\) in the Euler equation.

*T*is the temperature of the gas, \(\mu \) is the mean molecular weight, and \(k_B\) is the Boltzmann constant, we have \(v_s=\sqrt{ k_BT/m\mu }\), and \(R_J=\sqrt{\pi k_BT/G\rho _0m\mu }\). For an ideal gas the Jeans length is proportional to the temperature of the system, and inversely proportional to the density. On the other hand for a Bose–Einstein condensate in the same approximation we obtain \(R_J=\sqrt{4\pi ^2\hbar ^2a/Gm^3}\), and expression that depends only on the fundamental physical characteristics of the dark matter halo (particle mass and scattering length), and on the Planck constant. Hence, for a Bose–Einstein condensate at zero temperature the Jeans radius has an universal expression, independent on the macroscopic properties of the medium.

*p*, and the speed of sound in the dark matter halo is zero. By assuming a zero rotational velocity, for the Jeans radius and the Jeans mass of the dark matter halo in the Schrödinger–Poisson model we obtain the relations

We have also discussed the effects of the rotation on the condensate dark matter halos. In realistic astrophysical situations the condition \(\vec { \Omega }^{2}/\pi G\rho _{0}\gg 1\) may be satisfied. In this case the Jeans wave number is proportional to the angular velocity of the halo, while the Jeans radius and Jeans mass are inversely proportional to \(\left| \vec {\Omega } _{0}\right| \), and \(\left| \vec {\Omega }_{0}\right| ^{3}\), respectively. For large angular velocities the Jeans radius is very small, of the order of \(10^{-2}\) kpc, while the corresponding Jeans mass is of the order of \(90M_{\odot }\). These radius and mass values may correspond to stellar mass black holes, or other types of small compact objects (small in an astrophysical sense). In the case of ordinary baryonic matter the condition of instability for waves propagating perpendicularly to the axis of rotation is \(k_J^2v_s^2<4\pi G\rho _0-4\Omega ^2\) [156].

*N*in the galactic halo becomes sufficiently high, so that \(N > N_c\), where \(N_c\) is a critical particle number, the attractive interparticle energy exceeds the quantum pressure, and the dark matter halo begins to collapse. During the collapsing stage, the density of dark matter particles increases in the vicinity of the galactic center.

To study the dynamical evolution of the halo one need to solve the full system of hydrodynamic equations describing the time and space evolution of the Bose–Einstein condensate. In the Thomas–Fermi approximation the basic equations describing the dynamics of the condensate coincide with the continuity and Euler equations of classical hydrodynamics. The effect of the rotation was also included by means of a term, averaged over the angle \(\theta \), proportional to the radial coordinate, and with a time-dependent angular frequency. In order to preserve spherical symmetry we assume that the rotation of the condensate is slow. The effect of the gravitational field is taken into account via the Poisson equation, which relates the gravitational potential of the confined condensate to its density, which is essentially quantum in its nature. After integrating the Poisson equation, the evolution equations can be reduced to a system of two coupled nonlinear partial differential equations, which can subsequently be reduced to a single nonlinear second order partial differential equations for the mass of the condensate *M*(*r*, *t*). For many hydrodynamic flow models in gravitational fields, with matter obeying a polytropic equation of state the equations of motion admit semi-analytical self-similar or homologously collapsing solutions [190, 191, 192, 193, 194].

When these solutions are valid scale radial profiles of thermodynamic variables remain time invariant. The homologous solutions are stable against linear nonspherical perturbations, a result which can explains the remarkable stability of the galactic dark matter halos. However, in the case of index \(n=1\) polytropes no self-similar solution does exist, and the solutions of the equations of motion cannot be written in the form \(\Phi (r,t)=f(t)g\left( r/t\right) \) [190, 191, 192, 193, 194]. This means that during their evolution the physical quantities describing Bose–Einstein condensate dark matter halos *are not scale and time invariant*. However, an exact semi-analytical solution of the evolution equations of the Bose–Einstein condensates self-confined by the gravitational field can be obtained by assuming the separability of the mass variable in the for \(M(r,t)=f(t)m(r)\). The function *f*(*t*) has the analytical form \(f(t)=\left( t_0\pm t\right) ^{-2}/4\pi ^2 G\rho _c\), where the minus sign corresponds to the collapse of the condensate, while the plus sign describes the expansion of the condensate dark matter halo. As for the mass function *m*(*r*), time evolution generates a universal condensate dark matter profile, which in the absence of rotation just slightly modifies the static mass and density profiles. However, once the rotation is taken into account, the mass and density profiles are significantly modified as functions of the angular velocity.

*f*(

*t*), so that \(f(t)=\left( t_0+ t\right) ^{-2}/4\pi ^2 G\rho _c\). The Bose–Einstein condensate disperses in the space, with the local values of the physical parameters decreasing in time. In the large time limit the density as well as the radial and angular velocities tend to zero, and the Bose–Einstein condensate reaches a vacuum state. The minus sign in the function

*f*(

*t*) describes the collapse of the condensate. Both solutions are singular, for the expanding case the singularity occurs at the initial moment \(t=0\) (by a rescaling of the time variable one can remove the constant \(t_0\) from the results, \(t\rightarrow t+t_0\)), while in the case of the gravitational collapse the singularity is reached at the finite time \(t_0\). In order to obtain a rough estimate of \(t_0\), we assume that the mass distribution

*m*(

*r*) during the collapsing phase does not differ much from the stationary one. By series expanding Eq. (94) we obtain \(m(r)\approx M_0(r)\approx \left( 4\pi /3\right) \rho _c r^3\), an expression which allows us to estimate the velocity of the collapsing Bose–Einstein condensate dark matter halos as

It is interesting to note that, at least in the first order of approximation, the velocity of the condensate dark matter halo has the mathematical form of the cosmological Hubble law, \(v(t)=H(t)r\), with the Hubble function given by \(H(t)=2/\left( t_0\pm t\right) \). Moreover, as shown by the numerical simulations presented in Fig. 3, for small angular velocities the simple linear proportionality between the dimensionless velocity *V* and the radial dimensionless coordinate \(\eta \) is valid during the entire phase of the collapsing/expanding evolution.

A large number of cosmological and astrophysical observations, including the behavior of the galactic rotation curves, or the virial mass discrepancy in galaxy clusters points towards the possibility of the existence of dark matter in the Universe. If dark matter is present in the form of bosonic particles, then the possibility that it has the form of a Bose–Einstein condensate is strongly supported by the laws of quantum physics that show that a phase transition to a condensate state must occur, once the temperature reaches its critical value. A major adjustment in our interpretation of the basic principles of astrophysics and cosmology will be required if further observations would confirm this hypothesis. In the present papers we have introduced some basic theoretical tools that could help in the analysis and interpretation of the large scale structure formation in the presence of Bose–Einstein condensate dark matter.

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