A Note on Magnetized and Self-Gravitating Dark Matter Halos with Bose-Einstein Condensation

In the present study, we assume a rotating, self-gravitating dark matter which is magnetized under the influence of Bose-Einstein condensation. For the condensate dark matter halos, the non-relativistic Gross-Pitaevskii equation is used and for the hydrodynamical evolution, continuity and modified Euler equation are considered. The modified Maxwell equation is also considered to show the influences of the magnetic field and finite electrical resistivity. We determine the dispersion relation and the Jeans wave number by assuming slight perturbations of the quantum hydrodynamical equations. From the findings, we found that the Jeans wave number depends on quantum potential and rotation. In the absence of resistivity, we observe that the Jeans wave number modifies due to the magnetic field also.


Introduction:
In the study of extragalactic astronomy and cosmology, dark matter is a major aspect.Over one-fourth of the total cosmic energy comes from dark matter, whereas nearly 70 % comes from dark energy.This amount makes up around 96% of the overall contribution to the current cosmic energy budget.The theory of dark matter was first suggested by Zwicky [1,2].He used the virial theorem to study the Coma Cluster and discovered proof of a sizable amount of invisible mass, whose gravitational influence must tie the cluster together.Numerous cosmological measurements have since proven the existence of this puzzling element in the cosmos.One of the fundamental assumptions of the current prevailing cosmological hypothesis also called the Cold Dark Matter (CDM) concept, is the possibility of the existence in the Universe of many types of yet unseen and undiscovered matter.Its success is largely dependent on the presumption that the universe contains dark energy and dark matter, individually.The analysis of galaxy rotation curves provides the best evidence for the presence of dark matter in the cosmos [3].According to the classic gravitational theory, rotational velocities for hydrogen clouds in stable circular orbits traveling about the galactic center first grow there, but after that, they roughly remain stable at an intermediate value of the order of  ∞ ~200 − 300 /.The strongest and most persuasive evidence for the presence of dark matter is still found in the galaxy rotation curves [4][5][6][7].
The measurements of a galaxy cluster labeled as the Bullet Cluster offer significant yet gravitational observational proof for the existence of dark matter [8].The idea of gravitational instability was first introduced by Jeans [9] and is one of the core ideas in contemporary astrophysics and cosmology.Many authors found the Jeans instability in Bose-Einstein condensate dark matter [10][11] and it is the central problem of current astrophysics and cosmology.
In the present investigation, we introduced magnetic field and finite electrical resistivity in self-gravitating Bose-Einstein condensate dark matter.The gravitationally constrained dark matter halo is effectively described by a mean-field using the Gross-Pitaevskii equation.The mathematical analysis of the condensed dark matter halos has been made significantly easier by the development of the Madelung demonstration of the wave function, which enables the dark matter to be described according to the continuity equation, the classical fluid dynamics equations, and the Euler equation, particularly.This hypothesis states that dark matter halos are fluid structures that correspond to a polytropic equation of state with a polytropic index n = 1.The Euler evolution equation also includes the quantum force, which represents a wholly quantum phenomenon.For the magnetic field and the impact of limited electrical resistivity, we employ the Maxwell equation.We obtain the dispersion relation by applying perturbations to the hydrodynamical equations.From this, we further find out the Jeans wave number of the proposed study.

Magnetized Bose-Einstein condensate dark matter:
For the gravitationally confined magnetized Bose-Einstein condensate dark matter, the Gross-Pitaevskii equation, Poisson equation, and modified Maxwell equation can be written as where
H (0,0, H) is the magnetic field, which has been taken in the z-direction,  is the electrical resistivity,  is the mass density, h is the perturbed magnetic field, m is the mass of the condensate particle, and u is the velocity of condensation.
We now take into consideration the following set of equations as the dynamical behavior of the condensate dark matter halos to discover the Jean instability of the turbulent Bose-Einstein condensate dark matter in the presence of a magnetic field and finite electron resistivity: The continuity equation is written as The altered hydrodynamical Euler equation is given as The Poisson equation is given as The maxwell equation is written as As a result of the instability in the dark matter halo, the system experiences gravitational interaction and small perturbations of the hydrodynamical quantities.In the first-order linear approximation, these phenomena may be characterized as  =  0 + ,  =  0 + ,  =   + ,  =  0 + , Φ = Φ 0 + Φ,  =   + ,  =   + .
Here we consider that −1 We consider that all the perturbed quantities vary as exp (   ̂+ ).byusing this, we can find the components of the magnetic field from equation (7) as Therefore the linear equations ( 4)-( 7) take the form after introducing perturbations as Now, take the partial derivative of the equation ( 8) with regard to time, and apply the ∇ operator of eq. ( 9), we determine the density perturbation's propagation equation and then substitute δρ = exp (   ̂+ ), we acquire the magnetized Bose-Einstein condensate dark matter fluid's dispersion relation as follows: where is the Alfven velocity.This is the required dispersion relation of our considered problem.We can say that this dispersion relation depends on the speed of sound, finite electrical resistivity, Alfven velocity, uniform rotation, and quantum force of the magnetized BEC dark matter fluid.
For the Jeans wave number   , vanishing the  in the eq.( 12) and the algebraic equation has the following solution: Equation ( 13) represents the Jeans wave number in the proposed system.We can predict that this condition depends on quantum force, rotation, and adiabatic speed.It is independent of magnetic field and finite electrical resistivity.This relation is the same as Harko [10] eliminating the influence of the magnetic field as well as electrical resistivity.Now, in the absence of finite electrical resistivity or the infinitely conducting medium, eq. ( 12) takes the form From equation ( 14), we observed that the in the absence of finite electrical resistivity, the dispersion relation is affected by adiabatic speed, quantum force, rotation, and magnetic field.
The Jean wave number is given by The Jeans wave number presented in equation (15) shows the effect of adiabatic speed, quantum force, rotation, and magnetic field.
The dispersion relation of equation ( 14) can be rewritten as For || < | 2 |, we get the angular frequency  imaginary.This circumstance, therefore, corresponds with the instability of magnetized Bose-Einstein Condensate dark matter halos.

Summary:
In the current work, we have investigated some possibilities of astrophysical consequences of the Jeans instability in the self-gravitating magnetized Bose-Einstein condensate dark matter halos.We have considered the condensate dark matter in the magnetic field with the consequence of finite electrical resistivity.We know that the Gross-Pitaevskii equation, which provides the ground state of a quantum system, defines the Bose-Einstein condensation.In the current approach, we include the effect of the magnetic field in the same equation and the impact of finite electrical resistivity by the modified maxwell's equation.For simplification, we suppose that the temperature of dark matter is zero.We have also considered the rotation of condensate dark matter halos.
From the linearization of the hydrodynamic equations and some assumptions, we get the dispersion relation, which displays the results of rotation, resistivity, magnetic field, and quantum force.The Jeans wave number given by this dispersion relation depends on quantum force and rotation and is not affected by magnetic field and resistivity.In the infinitely conducting medium, we observed that the Jeans wave number depends on the magnetic field also.So, we can conclude that the finite electrical resistivity minimizes the impact of the magnetic field.Thus, we can say that the magnetic field plays a significant component in the dark matter halo of the Bose-Einstein condensate.There are a wide variety of future possibilities regarding magnetized BEC dark matter.We hope that this will play a great impact on the new era of research in astrophysics.
magnetic dark matter fluid's adiabatic sound's speed.