Cosmological evolution of finite temperature Bose-Einstein Condensate dark matter
Once the temperature of a bosonic gas is smaller than the critical, density dependent, transition temperature, a Bose - Einstein Condensation process can take place during the cosmological evolution of the Universe. Bose - Einstein Condensates are very strong candidates for dark matter, since they can solve some major issues in observational astrophysics, like, for example, the galactic core/cusp problem. The presence of the dark matter condensates also drastically affects the cosmic history of the Universe. In the present paper we analyze the effects of the finite dark matter temperature on the cosmological evolution of the Bose-Einstein Condensate dark matter systems. We formulate the basic equations describing the finite temperature condensate, representing a generalized Gross-Pitaevskii equation that takes into account the presence of the thermal cloud in thermodynamic equilibrium with the condensate. The temperature dependent equations of state of the thermal cloud and of the condensate are explicitly obtained in an analytical form. By assuming a flat Friedmann-Robertson-Walker (FRW) geometry, the cosmological evolution of the finite temperature dark matter filled Universe is considered in detail in the framework of a two interacting fluid dark matter model, describing the transition from the initial thermal cloud to the low temperature condensate state. The dynamics of the cosmological parameters during the finite temperature dominated phase of the dark matter evolution are investigated in detail, and it is shown that the presence of the thermal excitations leads to an overall increase in the expansion rate of the Universe.
pacs:04.50.Kd, 04.20.Cv, 04.20.Fy
The Concordance Cosmological Model, usually referred to as the CDM (+ Cold Dark Matter) model, has proved to be very successful in explaining cosmological observations across a wide range of length scales, from the cosmic microwave background (CMB) anisotropy to the Lyman- forest PeRa03 (). In this model, nonbaryonic collisionless cold dark matter makes up to 23% of the total mass content of the Universe. In the CDM model dark matter consists of cold neutral weakly interacting massive particles, beyond those existing in the Standard Model of Particle Physics. However, up to now no dark matter candidates have been detected in particle accelerators or in direct and indirect searches. Many particles have been proposed as possible candidates for dark matter, the most popular ones being the Weakly Interacting Massive Particles (WIMP) and the axions (for a review of the particle physics aspects of dark matter see OvWe04 ()). The interaction cross section of dark matter particles with normal baryonic matter is assumed to be extremely small. However, it is expected to be non-zero, and therefore the direct experimental detection of dark matter particles may be possible by some existing or future detectors det (). Superheavy particles, with mass GeV, have also been proposed as dark matter candidates. But in this case observational results show that these particles must either interact weakly with normal matter, or they must have very heavy masses above GeV AlBa03 (). Scalar field models, or other long range coherent fields coupled to gravity have been considered as possible candidates for galactic dark matter scal (). The possibility that dark matter could be described by a fluid with non-zero effective pressure was also investigated pres (); pres1 (). In particular, it was assumed that the equation of state of the dark matter halos is polytropic Sax (). The fit with a polytropic dark halo improves significantly the velocity dispersion profiles. The possibility that dark matter is a mixture of two non-interacting perfect fluids, with different four-velocities and thermodynamic parameters, was proposed recently in HaLo11 (). It has been also suggested that galactic dynamics of massive test particles can be understood, without considering dark matter, in the context of modified theories of gravity dmmodgrav ().
The experimental observation of the Bose-Einstein condensation of dilute alkali gases exp () represented a major advance in contemporary condensed matter physics. At very low temperatures, all particles in a dilute Bose gas condense to the same quantum ground state, forming a Bose-Einstein Condensate (BEC). In order for the condensation to occur particles must become correlated with each other so that their wavelengths overlap, that is, the thermal wavelength must satisfy the general condition , where is the mean inter-particles distance. The condensation takes place at a temperature , where is the mass of the particle in the condensate, is the particle number density, and is Boltzmann’s constant Da99 (). A coherent condensed state always develops if the particle density is high enough, or the temperature is sufficiently low. Quantum degenerate gases have been created in laboratory by a combination of different laser and evaporative cooling techniques, opening several new lines of research, at the border of atomic, statistical and condensed matter physics Da99 (). Recently, the Bose-Einstein condensation of photons has been observed in an optical microcavity phot ().
The possibility that dark matter, representing a significant amount of the total matter content of our Universe, is in the form of a Bose-Einstein Condensate was analyzed in detail in BoHa07 (). By introducing the Madelung representation of the wave function, it follows that dark matter can be described as a non-relativistic, Newtonian Bose-Einstein gravitational condensate gas, whose density and pressure are related by a barotropic equation of state, which, for a condensate with quartic non-linearity, has the polytropic index one. The validity of the model was tested by fitting the predicted galactic rotation curves with a sample of rotation curves of low surface brightness and dwarf galaxies, respectively BoHa07 (); jcap (); robles (). In all cases a very good agreement was found between the theoretical rotation curves and the observational data. Therefore dark matter halos can be described as an assembly of light individual bosons, occupying the same ground energy state, and acquiring a repulsive interaction. This interaction prevents gravity from forming the central density cusps. The condensate particle is light enough to naturally form condensates of very small masses that later may coalesce. Different astrophysical properties of condensed dark matter halos have been extensively investigated BEC ().
The cosmological study of the Bose-Einstein condensate dark matter was initiated recently in mnr () and chav (), respectively, with the study of the global cosmological evolution of the cold Bose-Einstein Condensate dark matter, and with the analysis of the perturbations of the condensate dark matter. The obtained results show significant differences with respect to the pressureless dark matter model, considered in the framework of standard cosmology. Therefore the presence of condensate dark matter could have modified drastically the cosmological evolution of the early Universe, as well as the large scale structure formation process. The cosmological details of the Bose-Einstein Condensation process have been analyzed for the case of the cold dark matter in Har (). The evolution of the cosmological inhomogeneities in the condensed dark matter and the observational implications on the Cosmic Microwave Background (CMB) spectra have been considered in CMB ().
All the mentioned previous researches have been done by assuming a zero temperature condensed dark matter. This assumptions is certainly a very good approximation for the description of dark matter in thermodynamic equilibrium with the cosmic microwave background, and for the analysis of the galactic rotation curves. It is already well established in condensed matter physics that the zero-temperature Gross-Pitaevskii equation gives an excellent quantitative descriptions of the Bose-Einstein condensates for , where is the Bose-Einstein transition temperature Gr (). This condition is obviously satisfied by the dark matter halos of the low-redshift galaxies. However, in the early Universe, immediately after the condensation, finite temperature effects could have played an important role in the dark matter dynamics, and significantly influence the cosmological evolution. The study of the finite temperature Bose-Einstein Condensate dark matter was initiated in HaM (), where the first order temperature corrections to the density profile of the galactic halos and the static properties of the condensates interacting with a thermal cloud have been obtained.
It is the purpose of the present paper to study the cosmological evolution of the finite temperature gravitationally self-bound Bose-Einstein dark matter condensates. As a first step in our study, by using the Hartree-Fock-Bogoliubov and Thomas-Fermi approximations, respectively, we obtain the equations of state of the arbitrary finite temperature dark matter condensate interacting with a thermal cloud. A finite temperature condensate can be described in terms of two fluids, the condensate proper, and the thermal excitations. Once the temperature of the system decreases, the number of the thermal excitations drops, and at zero temperature the system consists of the condensate only. Therefore the dynamics of a finite temperature Bose - Einstein -Condensate can be described as a system of two interacting fluids. By using the specific equations of state of the thermal excitations and of the condensate, the cosmological dynamics of the Universe is considered within a two-fluid dark matter model for a flat Friedmann-Robertson-Walker (FRW) type cosmological model. The physical parameters of the dark matter are obtained by numerically solving the evolution equations of the system.
The present paper is organized as follows. The basic equations describing finite temperature Bose-Einstein condensates, as well as the properties of the condensate dark matter particles, are reviewed in Section II. The equations of state of the finite temperature dark matter, trapped by a gravitational potential, are obtained in Section III. The cosmological evolution of the finite temperature condensed dark matter is considered in Section IV. We discuss and conclude our results in Section V.
Ii Finite temperature Bose-Einstein condensate dark matter
In the present Section we briefly review the basic properties of the high temperature Bose-Einstein condensate, and of the dark matter condensed particles. For a detailed discussion of the considered issues and of the derivation of the main results we refer the reader to the papers and books in Gr ()-HFB ().
ii.1 The generalized Gross-Pitaevskii equation and the hydrodynamic representation
where is the mass of the condensed particle, is the gravitational trapping potential, and , with the -wave scattering length. Eq. (1) is obtained under the assumption that the interaction potential can be represented as a zero-range pseudo-potential of strength . By taking the average of Eq. (1) with respect to a broken symmetry non-equilibrium ensemble, in which the quantum field operator takes a non-zero expectation value, we obtain the evolution equation for the condensate wave-function . Hence for the exact equation of motion of we find
By introducing the non-condensate field operator so that , where the average value of is zero, , we can separate out the condensate component of the quantum field operator to obtain the equation of motion for as follows Za99 (); HaM (); Prouk (); Gr ()
where we have denoted
and we have introduced the local condensate mass density
the non-condensate mass density
the off-diagonal (anomalous) mass density
and the three-field correlation function density
In the following we will restrict our analysis to the range of finite temperatures where the dominant thermal excitations can be approximated as high energy non-condensed particles moving in a self-consistent Hartree-Fock mean field, with local energy Za99 (); HaM (); Prouk (); Gr ()
where . Therefore in the present approximation we neglect the mean field effects associated with the anomalous density and and with the three-field correlation function . In the case of dark matter halos with a large number of particles this represents a very good approximation, the contribution of the anomalous density and of the three-field correlation function to the total density being of the order of a few percents HaM ().
In the thermal cloud the collision between particles forces a non-equilibrium distribution to evolve to the static Bose-Einstein distribution Gr (). Hence the particles in the thermal cloud are in thermodynamic equilibrium among themselves. By using a single-particle representation spectrum the equilibrium distribution of the thermal cloud can be written as
where , with Boltzmann’s constant, and is the chemical potential of the thermal cloud. In order to determine we assume that the condensate and the thermal cloud components are in local diffusive equilibrium with respect to each other. The requirement of a diffusive equilibrium between the cloud and the condensate imposes the condition Za99 (); HaM (); Gr ()
where is the chemical potential of the condensate. Therefore also determines the static equilibrium distribution of the particles in the thermal cloud.
where is the de Broglie thermal wavelength, is a Bose-Einstein function, and the fugacity is defined as
The generalized Gross-Pitaevskii equation can be transformed to a hydrodynamic form by introducing the Madelung representation of the wave function as . Then by taking into account that in the present approach we neglect the effects of the mean field associated with the anomalous density and the three-field correlation function, it follows that Eq. (3) is equivalent to the following hydrodynamic type system Za99 (); HaM (); Gr ()
where the local velocity of the condensate is given by . The chemical potential of the condensate is defined as
Eq. (17) can be reformulated as the Euler equation of fluid dynamics for the condensate,
ii.2 Physical properties of the dark matter condensate particle
The zero-temperature approximation, which gives a very good description of the galactic dark matter halos BoHa07 (); jcap (), allows us to make an estimate of the physical properties of the dark matter particle. From the analysis of the static Bose-Einstein condensate dark matter halos it follows that the radius of the condensate dark matter halo is given by BoHa07 (). The total mass of the condensate dark matter halo can be obtained as , where is the central density of the galactic halo, giving for the mean value of the condensate density the expression . The dark matter particle mass in the condensate is given by BoHa07 ()
For fm and kpc, the typical mass of the condensate particle is of the order of meV. For fm, corresponding to the values of observed in terrestrial laboratory experiments, eV.
An important method of observationally obtaining the properties of dark matter is the study of the collisions between clusters of galaxies, like the bullet cluster (1E 0657-56) and the baby bullet (MACSJ0025-12). From these studies one can obtain constraints on the physical properties of dark matter, such as its interaction cross-section with baryonic matter, and the dark matter-dark matter self-interaction cross section. If the ratio of the self-interaction cross section and of the dark matter particle mass is known from observations, then the mass of the dark matter particle in the Bose-Einstein condensate can be obtained from Eq. (20) as HaM ()
By comparing results from X-ray, strong lensing, weak lensing, and optical observations with numerical simulations of the merging galaxy cluster 1E 0657-56 (the Bullet cluster), an upper limit (68 % confidence) for of the order of was obtained in Bul (). By adopting for a value of , we obtain for the mass of the dark matter particle an upper limit of the order
By using this value of the particle mass we can estimate the scattering length as
This value of the scattering length , obtained from the observations of the Bullet cluster 1E 0657-56, is much smaller than the value of fm corresponding to the BEC’s obtained in laboratory terrestrial experiments exp ().
A stronger constraint for was proposed in Bul1 (), so that , giving a dark matter particle mass of the order
and a scattering length of the order of
Iii Equation of state of the finite temperature Bose-Einstein condensate
In the present Section we obtain the equation of state of the Bose-Einstein condensate dark matter in thermal equilibrium with the thermal excitations described as a non-condensate cloud. To obtain the equation of state of the global condensate plus thermal excitations cloud we proceed in two steps: first we explicitly construct the equation of state of the thermal excitations, and then we obtain the temperature dependent equation of state of the condensate. The thermodynamic parameters of the condensate plus thermal excitations system can be obtained by adding the densities and pressures of the thermal excitations and of the condensate, respectively.
iii.1 The equation of state of the thermal excitations
In order to obtain an easy to handle form of the matter density of the finite temperature non-condensate particle in thermodynamic equilibrium with the condensate, we power expand the Bose-Einstein function, so that
For Eq. (26) approximates the function with an error smaller than 1%. Therefore the density of the thermal cloud can be represented as
The previous results can be written in a more transparent form if we introduce the condensation temperature , given by Da99 ()
where is the Riemann zeta function, and is the density of the dark matter at the condensation moment. Then we obtain immediately
respectively. By introducing the dimensionless condensate density , related to the condensate density by the relation
we obtain the thermal cloud density as
At the initial moment of the condensation , and therefore . An order of magnitude estimate of the maximum value of can be obtained by assuming that the density of the condensate is of the same order of magnitude as the initial transition density, , thus giving . For K, we obtain g/cm, and . However, due to the incertitude in the numerical values of the physical parameters, we will consider the behavior of the condensed dark matter system for larger values of . The variation of the thermal cloud density is represented, as a function of , and for different values of , in Fig. 1.
To obtain the pressure of the non-condensate particles we start with the expansion
valid for , which gives
In terms of the transition temperature we have
and thus we obtain for the non-condensate pressure the expression
The variation of the thermal cloud pressure , where , is represented, as a function of , in Fig. 2.
The equation of state of the thermal cloud is represented in Fig. 3.
The obtained equations of state of the thermal excitations are valid only in the temperature range determined by the condition
iii.2 The equation of state of the finite temperature condensate
In order to obtain the equation of state of the condensate we introduce the Thomas-Fermi approximation for the condensate wave function Gr () - HFB (). In the Thomas-Fermi approximation, the kinetic energy term of the condensate particles is neglected. Hence in this approximation the chemical potential of the condensate is given by HaM ()
Eq. (17) can be reformulated as the Euler equation of fluid dynamics for the condensate,
where the pressure of the finite temperature Bose-Einstein condensate in thermal equilibrium with a gas of thermal excitations is given by
In terms of the dimensionless density , defined by Eq. (31), we obtain
The ratio for the condensate is determined by the numerical value of the dimensionless parameter
The variation of the condensate pressure with respect to is represented, for , Fig. 4.
Iv Cosmological evolution of the Universe with finite temperature BEC dark matter
We assume that the space-time geometry of the Universe is the flat Friedmann-Robertson-Walker (FRW) metric, given by
where is the scale factor describing the cosmological expansion. The Hubble function is defined as .
For the matter energy-momentum tensor we consider the case of the perfect fluid energy-momentum tensor, given by
where and represent the total energy density and pressure of the Universe.
As for the matter content of the Universe, we assume that it consists of radiation, with energy density and pressure , pressureless () baryonic matter, with energy density , dark matter, with energy density , and pressure , respectively, and dark energy, described by the cosmological constant . Hence the total energy density and pressure of the matter in the Universe is given by
respectively. In the following we neglect any possible interaction between these components, by assuming that the energy of each component is individually conserved. Thus, the gravitational field equations, corresponding to the line element given by Eq. (45), become
From Eq. (51) it follows that the energy density of the radiation and of the baryonic matter are given by
respectively, where and are the mass densities corresponding to .
iv.1 Evolution equations of the two-component finite temperature BEC dark matter
Even if the total energy-density of the condensate-thermal excitations system is conserved, the energies of the condensate and of the thermal cloud are not independently conserved, due to the growth of the condensate through absorption of the particles from the thermal cloud. This process must naturally occur during the expansion and subsequent cooling of the Universe. Moreover, the collisions in the thermal cloud will transfer a particle to the condensate, so that the number of the condensed particles will increase as , and there is of course the reverse process, where a collision of a thermal cloud particle with one within the condensate transfers a particle from the condensate into the thermal excitation band, so that . However, in the following we will consider only a simplified model, in which the particles of the thermal cloud turn into the condensate at a rate particles per second, leading to a decrease of a number of particles in the thermal cloud and to the corresponding increase of the number of the condensate particles. Therefore the equations of the energy conservation of the two dark matter components take the form
respectively, where and describe the time variation of the density of the two components in the finite temperature dark matter system. From Eqs. (54) and (55) it follows that the total dark matter density satisfies the conservation equation
where is the total pressure of the condensate-thermal excitations dark matter system.
The time rate at which the excited components turn into the condensate may be approximated by the transition probability that can be estimated by using the quantum kinetic theory, so that W (). gives the rate at which the thermal particles above the condensate energy band enter the condensate due to inter-particle collisions. The main assumption in obtaining is that the condensate does not readily act back on the thermal component to change its temperature. This assumption is expected to be valid when one considers equilibrium or quasi-equilibrium situations. The transition rate of the thermal excitations to the condensed state can be given as W ()
where and are the chemical potentials of the thermal cloud and of the condensate, and is a modified Bessel function. Eq. (57) can be simplified to the form
where the correction factor , assumed to be a constant, incorporates the effects of the exponential and of the Bessel function, respectively. The transition rate can be written as
Generally, the source terms in the thermal cloud and continuity equations can be written as Gr ()
where is the scattering cross section, and denote the particle momentum and energy, and , are the Bose-Einstein distribution functions of the thermal particles and of the condensate particles, respectively. The out collision rate represents the scattering of an incoming thermal particle (2) from the condensate to produce two thermal atoms (3) and (4). The designation ”out” means that a particle is leaving the condensate. Thus, is the rate of increase of the number of thermal particles per unit volume and per unit time as a result of the collision with a condensate particle. The reverse process gives the ”in” collision rate of a thermal particle . In the thermodynamic equilibrium state
and in the following we assume that this condition holds for all times. The equality between the transition rates between the thermal clod and the condensate also allows us to define the equilibrium collision time , given by Eq. (58). Generally, and can be obtained by numerical integration of Eqs. (60) and (61). However, in order to obtain a semi-analytical model of the cosmological evolution of the finite temperature Bose-Einstein condensate dark matter, for the energy density transfer rate we introduce the simplifying phenomenological assumption that it is proportional to the density of the condensate , and we write it as
where is a constant. In terms of the dimensionless condensate density we obtain
iv.2 Cosmological dynamics of the finite temperature BEC dark matter
In the following we denote the ”reduced” dimensionless temperature by , . By taking into account that in the case of the finite temperature Bose-Einstein condensate dark matter the energy densities and are functions of the reduced temperature and of the condensate density , , , the energy conservation equation for the dark matter takes the form
where and , respectively.
In the present Subsection we consider the cosmological evolution of the condensate dark matter component only, by ignoring the effects of the radiation and of the baryonic matter. In order to simplify the formalism we introduce the critical density of the Universe as , where is the value of the Hubble function at , and we define the density parameter of the dark matter as . We also rescale the comoving time variable by introducing the dimensionless time variable
respectively. Accordingly, the constant in the expression of is rescaled as , and we denote the corresponding energy density transition rate by . Moreover, we normalize the scale factor so that gives the value of the scale factor corresponding to the present age of the Universe.
respectively, where we have denoted , and we have taken into account that is independent of the reduced temperature , .
The evolution of the scale factor of the Universe is described by the equation
Eqs. (68), (69) and (70) give a complete description of the cosmological evolution of the finite temperature Bose-Einstein condensed dark matter. In order to completely determine the cosmological dynamics of the dark matter, we need the initial value of and at , . Since at the initial moment of the transition there is no dark matter in the form of a condensate, it follows that at . As for the initial value of the reduced temperature, it is given by . In the following we denote
The cosmological evolution of the Bose-Einstein condensed dark matter depends on two dimensionless parameters, and . By assuming for the condensation temperature a value of the order of K Har (), for a particle mass of the order of g, we obtain <