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.
Highlights
C (2019) 79:787 little baryonic 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 mass component, interacting only gravitationally with ordinary matter, and which most likely consists of new particle(s) not included in the standard model of particle physics
These results have indicated that baryonic matter only cannot explain the cosmological dynamics, and that the standard Cold Dark Matter ( CDM) cosmological paradigm requiring the existence of dark matter is strongly favored by observations
In relation with the general properties of the gravitational collapse we 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 ξ ≡ r/R(t), do not exist
Summary
The probability density | (r , t)|2 is normalized according to V n (r , t) d3r = V | (r , t)|2 d3r = 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. 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 = constant. After taking the partial derivative with respect to the time of the continuity equation (28), and applying the ∇ operator to Eq ( 29), we obtain the propagation equation of the density perturbations in the Bose–Einstein condensate dark matter fluid as.
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