Self-gravitating Bose-Einstein Condensates Research Articles

Collective excitations of self-gravitating bose-einstein condensates: Breathing mode and appearance of anisotropy under self-gravity

Abstract We investigate the collective mode of a self-gravitating Bose-Einstein condensate (BEC) described by the Gross-Pitaevskii-Poisson (GPP) equations. The self-gravitating BEC has garnered considerable attention in cosmology and astrophysics, being proposed as a plausible candidate for dark matter. Our inquiry delves into the breathing and anisotropic collective modes by numerically solving the GPP equations and using the variational method. The breathing mode demonstrates a reduction in period with increasing total mass due to the density dependence of the self-gravitating BEC, attributed to the density-dependent nature of self-gravitating BECs, aligning quantitatively with our analytical findings. Additionally, we investigate an anisotropic collective mode in which the quadrupole mode intertwines with the breathing mode. The period of the quadrupole mode exhibits similar total mass dependence to that of the breathing mode. The characteristics of these periods differ from those of a conventional BEC confined by an external potential. Despite the differences in density dependence, the ratio of their periods equals that of the BEC confined by an isotropic harmonic potential. Furthermore, an extension of the variational method to a spheroidal configuration enables the isolation of solely the quadrupole mode from the anisotropic collective mode.

Stable vortex structures in colliding self-gravitating Bose-Einstein condensates

Stable vortex structures in colliding self-gravitating Bose-Einstein condensates

Maximum mass of relativistic self-gravitating Bose-Einstein condensates with repulsive or attractive |φ|4 self-interaction

We derive an approximate analytical expression of the maximum mass of relativistic self-gravitating Bose-Einstein condensates with repulsive or attractive $|\ensuremath{\varphi}{|}^{4}$ self-interaction. This expression interpolates between the general relativistic maximum mass of noninteracting bosons stars, the general relativistic maximum mass of bosons stars with a repulsive self-interaction in the Thomas-Fermi limit, and the Newtonian maximum mass of dilute axion stars with an attractive self-interaction [P. H. Chavanis, Phys. Rev. D 84, 043531 (2011)]. We obtain the general structure of our formula from simple considerations and determine the numerical coefficients in order to recover the exact asymptotic expressions of the maximum mass in particular limits. As a result, our formula should provide a relevant approximation of the maximum mass of relativistic boson stars for any value (positive and negative) of the self-interaction parameter. We discuss the evolution of the system above the maximum mass and consider application of our results to dark matter halos and inflaton clusters. We also make a short review of boson stars and Bose-Einstein condensate dark matter halos and point out analogies with models of extended elementary particles.

Self-interacting superfluid dark matter droplets

ABSTRACT We assume dark matter to be a cosmological self-gravitating Bose–Einstein condensate of non-relativistic ultralight scalar particles with competing gravitational and repulsive contact interactions and investigate the observational implications of such model. The system is unstable to the formation of stationary self-bound structures that minimize the energy functional. These cosmological superfluid droplets, which are the smallest possible gravitationally bound dark matter structures, exhibit a universal mass profile and a corresponding universal rotation curve. Assuming a hierarchical structure formation scenario where granular dark matter haloes grow around these primordial stationary droplets, the model predicts cored haloes with rotation curves that obey a single universal equation in the inner region ($r\, \lesssim \, 1$ kpc). A simultaneous fit to a selection of galaxies from the SPARC data base chosen with the sole criterion of being strongly dark matter dominated even within the innermost region, indicates that the observational data are consistent with the presence of a Bose–Einstein condensate of ultralight scalar particles of mass m ≃ 2.2 × 10−22 eV c−2 and repulsive self-interactions characterized by a scattering length as ≃ 7.8 × 10−77 m. Such small self-interactions have profound consequences on cosmological scales. They induce a natural minimum scale length for the size of dark matter structures that makes all cores similar in length (∼1 kpc) and contributes to lower their central densities.

A heuristic wave equation parameterizing BEC dark matter halos with a quantum core and an isothermal atmosphere

The Gross–Pitaevskii–Poisson equations that govern the evolution of self-gravitating Bose–Einstein condensates, possibly representing dark matter halos, experience a process of gravitational cooling and violent relaxation. We propose a heuristic parametrization of this complicated process in the spirit of Lynden-Bell’s theory of violent relaxation for collisionless stellar systems. We derive a generalized wave equation that was introduced phenomenologically in Chavanis (Eur Phys J Plus 132:248, 2017) involving a logarithmic nonlinearity associated with an effective temperature $$T_\mathrm{eff}$$ and a damping term associated with a friction $$\xi $$ . These terms can be obtained from a maximum entropy production principle and are linked by a form of Einstein relation expressing the fluctuation-dissipation theorem. The wave equation satisfies an H-theorem for the Lynden-Bell entropy and relaxes towards a stable equilibrium state which is a maximum of entropy at fixed mass and energy. This equilibrium state represents the most probable state of a Bose–Einstein condensate dark matter halo. It generically has a core-halo structure. The quantum core prevents gravitational collapse and may solve the core-cusp problem. The isothermal halo leads to flat rotation curves in agreement with the observations. These results are consistent with the phenomenology of dark matter halos. Furthermore, as shown in a previous paper (Chavanis in Phys Rev D 100:123506, 2019), the maximization of entropy with respect to the core mass at fixed total mass and total energy determines a core mass–halo mass relation which agrees with the relation obtained in direct numerical simulations. We stress the importance of using a microcanonical description instead of a canonical one. We also explain how our formalism can be applied to the case of fermionic dark matter halos.

Rotating self-gravitating Bose-Einstein condensates with a crust: A model for pulsar glitches

We develop a minimal self-gravitating model for pulsar glitches by introducing a solid-crust potential in the three-dimensional Gross-Pitaevskii-Poisson equation, which we have used earlier to study gravitationally bound Bose-Einstein condensates, i.e., bosonic stars. In the absence of the crust potential, we show that, if we rotate such a bosonic star, it is threaded by vortices. We then show, via extensive direct numerical simulations, that the interaction of these vortices with the crust potential yields (a) stick-slip dynamics and (b) dynamical glitches. We demonstrate that, if enough momentum is transferred to the crust from the bosonic star, then the vortices are expelled from the star, and the crust's angular momentum ${J}_{c}$ exhibits features that may be interpreted as glitches. From the time series of ${J}_{c}$, we compute the cumulative probability distribution functions (CPDFs) of event sizes, event durations, and waiting times, which are consistent with the previous work. We show that these CPDFs have signatures of self-organized criticality, which are similar to those seen in observations of pulsar glitches and are consistent with previous work.

Jeans mass-radius relation of self-gravitating Bose-Einstein condensates and typical parameters of the dark matter particle

We study the Jeans mass-radius relation of Bose-Einstein condensate dark matter in Newtonian gravity. We show at a general level that it is similar to the mass-radius relation of Bose-Einstein condensate dark matter halos [P.H. Chavanis, Phys. Rev. D {\bf 84}, 043531 (2011)]. Bosons with a repulsive self-interaction generically evolve from the Thomas-Fermi regime to the noninteracting regime as the Universe expands. In the Thomas-Fermi regime, the Jeans radius remains approximately constant while the Jeans mass decreases. In the noninteracting regime, the Jeans radius increases while the Jeans mass decreases. Bosons with an attractive self-interaction generically evolve from the nongravitational regime to the noninteracting regime as the Universe expands. In the nongravitational regime, the Jeans radius and the Jeans mass increase. In the noninteracting regime, the Jeans radius increases while the Jeans mass decreases. The transition occurs at a maximum Jeans mass which is similar to the maximum mass of Bose-Einstein condensate dark matter halos with an attractive self-interaction. We use the mass-radius relation of dark matter halos and the observational evidence of a ``minimum halo'' (with typical radius $R\sim 1\, {\rm kpc}$ and typical mass $M\sim 10^8\, M_{\odot}$) to constrain the mass $m$ and the scattering length $a_s$ of the dark matter particle.

Jeans Instability of Dissipative Self-Gravitating Bose–Einstein Condensates with Repulsive or Attractive Self-Interaction: Application to Dark Matter

We study the Jeans instability of an infinite homogeneous dissipative self-gravitating Bose–Einstein condensate described by generalized Gross–Pitaevskii–Poisson equations [Chavanis, P.H. Eur. Phys. J. Plus2017, 132, 248]. This problem has applications in relation to the formation of dark matter halos in cosmology. We consider the case of a static and an expanding universe. We take into account an arbitrary form of repulsive or attractive self-interaction between the bosons (an attractive self-interaction being particularly relevant for the axion). We consider both gravitational and hydrodynamical (tachyonic) instabilities and determine the maximum growth rate of the instability and the corresponding wave number. We study how they depend on the scattering length of the bosons (or more generally on the squared speed of sound) and on the friction coefficient. Previously obtained results (notably in the dissipationless case) are recovered in particular limits of our study.

Light trapping by the dark matter

Considering the dark matter as self-gravitating Bose-Einstein condensate in a gravito-chemical equilibrium state, we have studied the light trajectory through the halo. Found that depending on the initial direction cosine of the light ray, there are three types of trajectories: (a) the refracted light i.e. light which can refracts away from the halo to the out side, (b) the trapped light i.e. light that is trapped into the halo i.e. an observer, situated outside of the halo, can not observe it and (c) the reflected light: in this case the halo behaves as a perfect reflector to the outside light. The light trapping can introduce asymmetry in the observed spectrum and modify the apparent number of stars in a dark matter dominated galaxy.

Core mass-halo mass relation of bosonic and fermionic dark matter halos harboring a supermassive black hole

We study the core mass-halo mass relation of bosonic dark matter halos, in the form of self-gravitating Bose-Einstein condensates, harboring a supermassive black hole. We use the ``velocity dispersion tracing'' relation according to which the velocity dispersion in the core ${v}_{c}^{2}\ensuremath{\sim}G{M}_{c}/{R}_{c}$ is of the same order as the velocity dispersion in the halo ${v}_{h}^{2}\ensuremath{\sim}G{M}_{h}/{r}_{h}$ (this relation can be justified from thermodynamical arguments) and the approximate analytical mass-radius relation of the quantum core in the presence of a central black hole obtained in our previous paper [P.H. Chavanis, Eur. Phys. J. Plus 134, 352 (2019)]. For a given minimum halo mass $({M}_{h}{)}_{\mathrm{min}}\ensuremath{\sim}{10}^{8}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ determined by the observations, the only free parameter of our model is the scattering length ${a}_{s}$ of the bosons (their mass $m$ is then determined in order to match the characteristics of the minimum halo). For noninteracting bosons and for bosons with a repulsive self-interaction, we find that the core mass ${M}_{c}$ increases with the halo mass ${M}_{h}$ and achieves a maximum value $({M}_{c}{)}_{\mathrm{max}}$ at some halo mass $({M}_{h}{)}_{*}$ before decreasing. The whole series of equilibria is stable. For bosons with an attractive self-interaction, we find that the core mass achieves a maximum value $({M}_{c}{)}_{\mathrm{max}}$ at some halo mass $({M}_{h}{)}_{*}$ before decreasing. The series of equilibria becomes unstable above a maximum halo mass $({M}_{h}{)}_{\mathrm{max}}\ensuremath{\ge}({M}_{h}{)}_{*}$. In the absence of black hole $({M}_{h}{)}_{\mathrm{max}}=({M}_{h}{)}_{*}$. At that point, the quantum core (similar to a dilute axion star) collapses. We perform a similar study for fermionic dark matter halos. We find that they behave similarly to bosonic dark matter halos with a repulsive self-interaction, the Pauli exclusion principle for fermions playing the role of the repulsive self-interaction for bosons.

Bose–Einstein condensate of ultra-light axions as a candidate for the dark matter galaxy halos

We suggest that the dark matter halo in some of the spiral galaxies can be described as the ground state of the Bose–Einstein condensate of ultra-light self-gravitating axions. We have also developed an effective “dissipative” algorithm for the solution of nonlinear integro-differential Schrödinger equation describing self-gravitating Bose–Einstein condensate. The mass of an ultra-light axion is estimated.

Mass-radius relation of self-gravitating Bose-Einstein condensates with a central black hole

We determine the mass-radius relation of self-gravitating Bose-Einstein condensates with an attractive -1/r external potential created by a central mass. Following our previous work (P.H. Chavanis, Phys. Rev. D 84, 043531 (2011)), we use an analytical approach based on a Gaussian ansatz. We consider the case of noninteracting bosons as well as the case of self-interacting bosons with a repulsive or an attractive self-interaction. These results may find application in the context of dark matter halos made of self-gravitating Bose-Einstein condensates. In that case, the central mass may mimic a supermassive black hole. We apply our results to ultralight axions with an attractive self-interaction. We determine how the central black hole affects the mass-radius relation and the maximum mass of axionic halos found in our previous papers. Our approximate analytical results based on the Gaussian ansatz are compared with exact analytical results obtained in particular limits.

Head-on collisions and orbital mergers of Proca stars

Proca stars are self-gravitating Bose-Einstein condensates obtained as numerical stationary solutions of the Einstein-(complex)-Proca system. These solitonic can be both stable and form dynamically from generic initial data by the mechanism of gravitational cooling. In this paper we further explore the dynamical properties of these solitonic objects by performing both head-on collisions and orbital mergers of equal mass Proca stars, using fully non-linear numerical evolutions. For the head-on collisions, we show that the end point and the gravitational waveform from these collisions depends on the compactness of the Proca star. Proca stars with sufficiently small compactness collide leaving a stable Proca star remnant. But more compact Proca stars collide to form a transient ${\it hypermassive}$ Proca star, which ends up decaying into a black hole, albeit temporarily surrounded by Proca quasi-bound states. The unstable intermediate stage can leave an imprint in the waveform, making it distinct from that of a head-on collision of black holes. The final quasi-normal ringing matches that of Schwarzschild black hole, even though small deviations may occur, as a signature of sufficiently non-linear and long-lived Proca quasi-bound states. For the orbital mergers, the outcome also depends on the compactness of the stars. For most compact stars, the binary merger forms a Kerr black hole which retains part of the initial orbital angular momentum, being surrounded by a transient Proca field remnant; in cases with lower compactness, the binary merger forms a massive Proca star with angular momentum, but out of equilibrium. As in previous studies of (scalar) boson stars, the angular momentum of such objects appears to converge to zero as a final equilibrium state is approached.

Jeans-type instability of a complex self-interacting scalar field in general relativity

We study the gravitational instability of a general relativistic complex scalar field with a quartic self-interaction in an infinite homogeneous static background. This quantum relativistic Jeans problem provides a simplified framework to study the formation of the large-scale structures of the Universe in the case where dark matter is made of a scalar field. The scalar field may represent the wave function of a relativistic self-gravitating Bose-Einstein condensate. Exact analytical expressions for the dispersion relation and Jeans length $\lambda_J$ are obtained from a hydrodynamical representation of the Klein-Gordon-Einstein equations. When relativistic effects are fully accounted for, we find that the perturbations are stabilized at very large scales of the order of the Hubble length $\lambda_H$. Numerical applications are made for ultralight bosons without self-interaction (fuzzy dark matter), for bosons with a repulsive self-interaction, and for bosons with an attractive self-interaction (QCD axions and ultralight axions). We show that the Jeans instability is inhibited in the ultrarelativistic regime (early Universe and radiationlike era) because the Jeans length is of the order of the Hubble length ($\lambda_J\sim\lambda_H$), except when the self-interaction of the bosons is attractive. By contrast, structure formation can take place in the nonrelativistic regime (matterlike era) for $\lambda_J\le \lambda\le \lambda_H$. Since the scalar field has a nonzero Jeans length due to its quantum nature (Heisenberg uncertainty principle or quantum pressure due to the self-interaction of the bosons), it appears that the wave properties of the scalar field can stabilize gravitational collapse at small scales, providing halo cores and suppressing small-scale linear power. This may solve the CDM crisis such as the cusp problem and the missing satellite problem.

Phase transitions between dilute and dense axion stars

We study the nature of phase transitions between dilute and dense axion stars interpreted as self-gravitating Bose-Einstein condensates. We develop a Newtonian model based on the Gross-Pitaevskii-Poisson equations for a complex scalar field with a self-interaction potential $V(|\psi|^2)$ involving an attractive $|\psi|^4$ term and a repulsive $|\psi|^6$ term. Using a Gaussian ansatz for the wave function, we analytically obtain the mass-radius relation of dilute and dense axion stars for arbitrary values of the self-interaction parameter $\lambda\le 0$. We show the existence of a critical point $|\lambda|_c\sim (m/M_P)^2$ above which a first order phase transition takes place. We qualitatively estimate general relativistic corrections on the mass-radius relation of axion stars. For weak self-interactions $|\lambda|<|\lambda|_c$, a system of self-gravitating axions forms a stable dilute axion star below a general relativistic maximum mass $M_{\rm max,GR}^{\rm dilute}\sim M_P^2/m$ and collapses into a black hole above that mass. For strong self-interactions $|\lambda|>|\lambda|_c$, a system of self-gravitating axions forms a stable dilute axion star below a Newtonian maximum mass $M_{\rm max,N}^{\rm dilute}=5.073 M_P/\sqrt{|\lambda|}$, collapses into a dense axion star above that mass, and collapses into a black hole above a general relativistic maximum mass $M_{\rm max,GR}^{\rm dense}\sim \sqrt{|\lambda|}M_P^3/m^2$. Dense axion stars explode below a Newtonian minimum mass $M_{\rm min,N}^{\rm dense}\sim m/\sqrt{|\lambda|}$ and form dilute axion stars of large size or disperse away. We determine the phase diagram of self-gravitating axions and show the existence of a triple point $(|\lambda|_*,M_*/(M_P^2/m))$ separating dilute axion stars, dense axion stars, and black holes. We make numerical applications for QCD axions and ultralight axions.

Dissipative self-gravitating Bose-Einstein condensates with arbitrary nonlinearity as a model of dark matter halos

We develop a general formalism applying to Newtonian self-gravitating Bose-Einstein condensates. This formalism may find application in the context of dark matter halos. We introduce a generalized Gross-Pitaevskii equation including a source of dissipation (damping) and an arbitrary nonlinearity. Using the Madelung transformation, we derive the hydrodynamic representation of this generalized Gross-Pitaevskii equation and obtain a damped quantum Euler equation involving a friction force proportional and opposite to the velocity and a pressure force associated with an equation of state determined by the nonlinearity present in the generalized Gross-Pitaevskii equation. In the strong friction limit, we obtain a quantum Smoluchowski equation. These equations satisfy an H-theorem for a free energy functional constructed with a generalized entropy. We specifically consider the Boltzmann and Tsallis entropies associated with isothermal and polytropic equations of state. We also consider the entropy associated with the logotropic equation of state. We derive the virial theorem corresponding to the generalized Gross-Pitaevskii equation, damped quantum Euler equation, and quantum Smoluchowski equation. Using a Gaussian ansatz, we obtain a simple equation governing the dynamical evolution of the size of the condensate. We develop a mechanical analogy associated with this gross dynamics. We highlight a specific model of dark matter halos corresponding to a generalized Gross-Pitaevskii equation with a logarithmic nonlinearity and a cubic nonlinearity. It corresponds to a damped quantum Euler equation associated with a mixed entropy combining the Boltzmann and Tsallis entropies. It leads to dark matter halos with an equation of state $P=\rho k_{B} T_{\rm eff}/m+2\pi a_{s}\hbar^{2}\rho^{2}/m^{3}$ presenting a condensed core (BEC/soliton) and an isothermal halo with an effective temperature $T_{\rm eff}$ . We propose that this model provides an effective coarse-grained parametrization of dark matter halos experiencing gravitational cooling. Specific applications of our formalism to dark matter halos will be developed in future papers.

Derivation of a generalized Schrödinger equation from the theory of scale relativity

Using Nottale’s theory of scale relativity relying on a fractal space-time, we derive a generalized Schrodinger equation taking into account the interaction of the system with the external environment. This equation describes the irreversible evolution of the system towards a static quantum state. We first interpret the scale-covariant equation of dynamics stemming from Nottale’s theory as a hydrodynamic viscous Burgers equation for a potential flow involving a complex velocity field and an imaginary viscosity. We show that the Schrodinger equation can be directly obtained from this equation by performing a Cole-Hopf transformation equivalent to the WKB transformation. We then introduce a friction force proportional and opposite to the complex velocity in the scale-covariant equation of dynamics in a way that preserves the local conservation of the normalization condition. We find that the resulting generalized Schrodinger equation, or the corresponding fluid equations obtained from the Madelung transformation, involve not only a damping term but also an effective thermal term. The friction coefficient and the temperature are related to the real and imaginary parts of the complex friction coefficient in the scale-covariant equation of dynamics. This may be viewed as a form of fluctuation-dissipation theorem. We show that our generalized Schrodinger equation satisfies an H-theorem for the quantum Boltzmann free energy. As a result, the probability distribution relaxes towards an equilibrium state which can be viewed as a Boltzmann distribution including a quantum potential. We propose to apply this generalized Schrodinger equation to dark matter halos in the Universe, possibly made of self-gravitating Bose-Einstein condensates.

Cosmological evolution of a complex scalar field with repulsive or attractive self-interaction

We study the cosmological evolution of a complex scalar field with a self-interaction potential $V(|\varphi|^2)$, possibly describing self-gravitating Bose-Einstein condensates, using a fully general relativistic treatment. We generalize the hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field approximation developed in our previous paper. We establish the general equations governing the evolution of a spatially homogeneous complex scalar field in an expanding background. We show how they can be simplified in the fast oscillation regime and derive the equation of state of the scalar field in parametric form for an arbitrary potential. We explicitly consider the case of a quartic potential with repulsive or attractive self-interaction and determine the phase diagram of the scalar field. We show that the transition between the weakly self-interacting regime and the strongly self-interacting regime depends on how the scattering length of the bosons compares with their effective Schwarzschild radius. We also constrain the parameters of the scalar field from astrophysical and cosmological observations. Numerical applications are made for ultralight bosons without self-interaction (fuzzy dark matter), for bosons with repulsive self-interaction, and for bosons with attractive self-interaction (QCD axions and ultralight axions).

Supermassive black holes from collapsing dark matter Bose–Einstein condensates

The discovery of active galactic nuclei at redshifts 6 suggests that supermassive black holes (SMBHs) formed early on. Growth of the remnants of population III stars by accretion of matter, both baryonic as well as collisionless dark matter (DM), leading up to formation of SMBHs is a very slow process. Therefore, such models encounter difficulties in explaining quasars detected at . Furthermore, massive particles making up collisionless DM have not only so far eluded experimental detection but they also do not satisfactorily explain gravitational structures on small scales.In recent years, there has been a surge in research activities concerning cosmological structure formation that involve coherent, ultra-light bosons in a dark fluid-like or fuzzy cold DM state. In this paper, we study collapse of such ultra-light bosonic halo DM that is in a Bose–Einstein condensate (BEC) phase to give rise to SMBHs on dynamical time scales. Time evolution of such self-gravitating BECs is examined using the Gross–Pitaevskii equation in the framework of time-dependent variational method. Comprised of identical dark bosons of mass m, BECs can collapse to form black holes of mass Meff on time scales ∼108 yrs provided . In particular, ultra-light dark bosons of mass can lead to SMBHs with mass at . Recently observed radio-galaxies in the ELAIS-N1 deep field with aligned jets can also possibly be explained if vortices of a rotating cluster size BEC collapse to form spinning SMBHs with angular momentum , where nW and M are the winding number and mass of a vortex, respectively.

Covariant theory of Bose-Einstein condensates in curved spacetimes with electromagnetic interactions: The hydrodynamic approach

We develop a hydrodynamic representation of the Klein-Gordon-Maxwell-Einstein equations. These equations combine quantum mechanics, electromagnetism, and general relativity. We consider the case of an arbitrary curved spacetime, the case of weak gravitational fields in a static or expanding background, and the nonrelativistic (Newtonian) limit. The Klein-Gordon-Maxwell-Einstein equations govern the evolution of a complex scalar field, possibly describing self-gravitating Bose-Einstein condensates, coupled to an electromagnetic field. They may find applications in the context of dark matter, boson stars, and neutron stars with a superfluid core.