Evolution of cosmological perturbations in Bose–Einstein condensate dark matter

Abstract

We consider the global cosmological evolution and the evolution of the density contrast in the Bose–Einstein condensate dark matter model, in the framework of a post-Newtonian cosmological approach. In the Bose–Einstein model, dark matter can be described as a non-relativistic, Newtonian gravitational condensate, whose density and pressure are related by a barotropic equation of state. For a condensate with quartic non-linearity, the equation of state is polytropic with index n= 1.The basic equation describing the evolution of the perturbations of the Bose–Einstein condensate is obtained, and its solution is studied by using both analytical and numerical methods. The global cosmological evolution as well as the evolution of the perturbations of the condensate dark matter shows 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.

equation of state, instabilities, cosmology: theory, dark matter, large-scale structure of Universe

1 INTRODUCTION

Cosmological observations provide compelling evidence that about 95 per cent of the content of the Universe resides in two unknown forms of energy that we call dark matter and dark energy: the first residing in bound objects as non-luminous matter (Persic, Salucci & Stel 1996; Boriello & Salucci 2001; Binney & Tremaine 2008) and the latter in the form of a zero-point energy that pervades the whole Universe (Padmanabhan 2003; Peebles & Ratra 2003). The dark matter is thought to be composed of cold neutral weakly interacting massive particles (WIMPs), beyond those existing in the standard model of particle physics, and not yet detected in accelerators or in dedicated direct and indirect searches. There are many possible candidates for dark matter, the most popular ones being the axions and the weakly interacting massive particles [for a review of the particle physics aspects of dark matter, see Overduin & Wesson (2004)]. Their interaction cross-section with normal baryonic matter, while extremely small, is expected to be non-zero and we may expect to detect them directly. It has also been suggested that the dark matter in the Universe might be composed of superheavy particles, with a mass of ≥10¹⁰ GeV (Chung, Kolb & Riotto 1998; Chung et al. 2005; Kolb, Starobinsky & Tkachev 2007; Chuzhoy & Kolb 2009). But observational results show that the dark matter can be composed of superheavy particles only if these interact weakly with normal matter or if their mass is above 10¹⁵ GeV (Albuquerque & Baudis 2003). Scalar fields or other long-range coherent fields coupled to gravity have also been intensively used to model galactic dark matter (Lee & Koh 1996; Nucamendi, Salgado & Sudarsky 2000; Matos & Guzman 2001; Mielke & Schunk 2002; Arbey, Lesgourgues & Salati 2003; Fuchs & Mielke 2004; Hernández et al. 2004; Giannios 2005; Khlopov, Rubin & Sakharov 2005; Arbey 2006, 2008; Bernal & Guzman 2006; Briscese 2011). Alternative theoretical models to explain the galactic rotation curves have also been elaborated recently (Milgrom 1983; Mannheim 1993; Bekenstein 2004; Mak & Harko 2004; Brownstein & Moffat 2006; Harko & Cheng 2006; Bertolami et al. 2007; Boehmer & Harko 2007a; Boehmer, Harko & Lobo 2008a,b; Capozziello et al. 2009).

In order to explain the recent observational data, the Λ cold dark matter (ΛCDM) model was developed (Padmanabhan 2003; Peebles & Ratra 2003). The ΛCDM model successfully describes the accelerated expansion of the Universe, the observed temperature fluctuations in the cosmic microwave background radiation, the large-scale matter distribution, and the main aspects of the formation and the evolution of virialized cosmological objects.

Despite these important achievements, at galactic scales of ∼10 kpc, the ΛCDM model meets with severe difficulties in explaining the observed distribution of the invisible matter around the luminous one. In fact, N -body simulations, performed in this scenario, predict that bound haloes surrounding galaxies must have very characteristic density profiles that feature a well-pronounced central cusp, ρ_NFW(r) =ρ_s/(r/r_s)(1 +r/r_s)² (Navarro et al. 1997), where r_s is the scale radius and ρ_s is the characteristic density. On the observational side, high-resolution rotation curves show, instead, that the actual distribution of dark matter is much shallower than the above, and it presents a nearly constant density core: ρ_B(r) =ρ₀r³₀/(r+r₀)(r²+r²₀) (Burkert 1995), where r₀ is the core radius and ρ₀ is the central density.

At very low temperatures, all particles in a dilute Bose gas condense to the same quantum ground state, forming a Bose–Einstein condensate (BEC), i.e. a sharp peak over a broader distribution in both coordinates and momentum space. A coherent state develops when the particle density is sufficiently high or the temperature is sufficiently low. The Bose–Einstein condensation process was first observed experimentally in 1995 in dilute alkali gases, such as vapours of rubidium and sodium, confined in a magnetic trap and cooled to very low temperatures. A sharp peak in the velocity distribution was observed below a critical temperature, indicating that condensation has occurred, with the alkali atoms condensed in the same ground state and showing a narrow peak in the momentum space and in the coordinate space (Anderson et al. 1995; Bradley et al 1995; Davis et al. 1995). Quantum degenerate gases have been created by a combination of laser and evaporative cooling techniques, opening several new lines of research, at the border of atomic, statistical and condensed matter physics (Dalfovo et al. 1999; Cornell & Wieman 2002; Ketterle 2002; Pitaevskii & Stringari 2003; Duine & Stoof 2004; Chen et al. 2005; Pethick & Smith 2008).

The possibility that dark matter could be in the form of a BEC was considered in Sin (1994) and Ji & Sin (1994). The condensate was described by the non-relativistic Gross–Pitaevskii equation, and its solution was obtained numerically. An alternative approach was developed in Boehmer & Harko (2007b). By introducing the Madelung representation of the wavefunction, the dynamics of the system can be formulated in terms of the continuity equation and of the hydrodynamic Euler equations. Hence, 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. In the case of a condensate with quartic non-linearity, the equation of state is polytropic with index n = 1. To test the validity of the model, the Newtonian tangential velocity equation of the model was fitted with a sample of rotation curves of low surface brightness and dwarf galaxies, respectively. A very good agreement was found between the theoretical rotation curves and the observational data for the low surface brightness galaxies. Therefore, dark matter haloes can be described as an assembly of light individual bosons that acquire a repulsive interaction by occupying the same ground energy state. This prevents gravity from forming the central density cusps. The condensate particle is light enough to naturally form condensates of very small masses that may later coalesce, forming the structures of the Universe in a similar way to the hierarchical clustering of the bottom-up CDM picture. Then, at large scales, BEC perfectly mimics an ensemble of cold particles, while at small scales quantum mechanics drives the mass distribution. Different properties of the BEC dark matter haloes, such as the effects of the rotation and of the vortices, as well as the cosmological effects of the condensation were also investigated (Ferrer & Grifols 2004; Fukuyama, Morikawa & Tatekawa 2008; Brook & Coles 2009; Fukuyama & Morikawa 2009; Lee 2009; Rindler-Daller & Shapiro 2009; Sikivie & Yang 2009; Kain & Ling 2010; Lee & Lim 2010).

It is the purpose of this paper to study the global cosmological dynamics of gravitationally self-bound Bose–Einstein dark matter condensates and the evolution of the small cosmological perturbations in the condensate. The equations of motion of the condensate dark matter are obtained in a post-Newtonian approximation by using the conservation of the general relativistic energy–momentum tensor and considering the small velocity limit. The cosmological dynamics of the BEC is also studied. The exact solution of the Friedmann equations is obtained, and it is compared with the standard Einstein–de Sitter cosmological model. In order to study the evolution of the small cosmological perturbations the equation describing the Newtonian perturbations with pressure is obtained in a general form, by also taking into account a term which was neglected in the previous studies of this problem (McCrea 1951; Harrison 1965; Lima, Zanchin & Brandenberger 1997; Reis 2003; Abramo et al. 2007; Pace, Waizmann & Bartelmann 2010). The equation of the density contrast for the BEC is investigated by using both analytical and numerical methods. The presence of the condensate dark matter significantly modifies the cosmological dynamics of the Universe as well as the large-scale structure formation.

This paper is organized as follows. The basic properties of the BEC dark matter haloes are reviewed in Section 2. The post-Newtonian hydrodynamical equations of motion for a perfect fluid with pressure are derived in Section 3. The cosmological dynamics of the BEC dark matter is considered in Section 4. The equation describing the small cosmological perturbations of a fluid with pressure is derived in Section 5. The evolution of the small cosmological perturbations in a BEC is considered in Section 6. We discuss and conclude our results in Section 7.

2 DARK MATTER AS A BOSE–EINSTEIN CONDENSATE

In a quantum system of N interacting condensed bosons, most of the bosons lie in the same single-particle quantum state. For a system consisting of a large number of particles, the calculation of the ground state of the system with the direct use of the Hamiltonian is impracticable, due to the high computational cost. However, the use of some approximate methods can lead to a significant simplification of the formalism. One such approach is the mean field description of the condensate, which is based on the idea of separating out the condensate contribution to the bosonic field operator. We also assume that in a medium composed of scalar particles with non-zero mass, when the medium makes a transition to a Bose–Einstein condensed phase, the range of Van der Waals-type scalar-mediated interactions among particles becomes infinite.

2.1 Gross–Pitaevskii equation

The many-body Hamiltonian describing the interacting bosons confined by an external potential V_ext is given, in the second quantization, by

1

where and are the boson field operators that annihilate and create a particle at the position r, respectively, and is the two-body interatomic potential (Dalfovo et al. 1999; Chen et al. 2005). is the potential associated with the rotation of the condensate.

The use of some approximate methods can lead to a significant simplification of the formalism. One such approach is the mean field description of the condensate, which is based on the idea of separating out the condensate contribution to the bosonic field operator. For a uniform gas in a volume V, BEC occurs in the single particle state ⁠, having zero momentum. The field operator can then be decomposed in the form ⁠. By treating the operator as a small perturbation, one can develop the first-order theory for the excitations of the interacting Bose gases (Dalfovo et al. 1999; Barcelo, Liberati & Visser 2001).

In the general case of a non-uniform and time-dependent configuration, the field operator in the Heisenberg representation is given by ⁠, where ⁠, also called the condensate wavefunction, is the expectation value of the field operator, ⁠. It is a classical field, and its absolute value fixes the number density of the condensate through ⁠. The normalization condition is ⁠, where N is the total number of particles in the condensate.

The equation of motion for the condensate wavefunction is given by the Heisenberg equation corresponding to the many-body Hamiltonian given by equation (1):

2

The zeroth-order approximation to the Heisenberg equation is obtained by replacing with the condensate wavefunction ψ. In the integral containing the particle–particle interaction ⁠, this replacement is in general a poor approximation for short distances. However, in a dilute and cold gas, only binary collisions at low energy are relevant, and these collisions are characterized by a single parameter, the s-wave scattering length l_a, independent of the details of the two-body potential. Therefore, one can replace with an effective interaction ⁠, where the coupling constant λ is related to the scattering length l_a through λ = 4πℎ²l_a/m, where m is the mass of the condensed particles. With the use of the effective potential the integral in the bracket of equation (2) gives ⁠, and the resulting equation is the Schrödinger equation with a quartic non-linear term (Chen et al. 2005). However, in order to obtain a more general description of the BEC stars, we shall assume an arbitrary non-linear term g (| ψ (r, t) |²) =g (ρ) (Barcelo et al. 2001).

Therefore, the generalized Gross–Pitaevskii equation describing a gravitationally trapped rotating BEC is given by

3

where we denoted g′ = d g/d ρ. As for ⁠, we assume that it is the gravitational potential V, V_ext=V, and it satisfies the Poisson equation

4

where is the mass density inside the BEC.

2.2 Hydrodynamical representation

The physical properties of a BEC described by the generalized Gross–Pitaevskii equation given by equation (3) can be understood much easily by using the so-called Madelung representation of the wavefunction (Dalfovo et al. 1999), which consists in writing ψ in the form

5

where the function has the dimensions of an action. By substituting the above expression of into equation (3), it decouples into a system of two differential equations for the real functions ρ_m and v, given by

6

7

where we have introduced the quantum potential,

8

and the velocity of the quantum fluid,

9

respectively, and we have denoted

10

From its definition it follows that the velocity field is irrotational, satisfying the condition ∇×v = 0. Therefore, the equations of motion of the gravitational ideal BEC take the form of the equation of continuity and of the hydrodynamic Euler equations. The Bose–Einstein gravitational condensate can be described as a gas whose density and pressure are related by a barotropic equation of state (Pethick & Smith 2008). The explicit form of this equation depends on the form of the non-linearity term g.

When the number of particles in the gravitationally bounded BEC becomes large enough, the quantum pressure term makes a significant contribution only near the boundary of the condensate. Hence, it is much smaller than the non-linear interaction term. Thus, the quantum stress term in the equation of motion of the condensate can be neglected. This is the Thomas–Fermi approximation, which has been extensively used for the study of the BECs (Dalfovo et al. 1999; Chen et al. 2005). As the number of particles in the condensate becomes infinite, the Thomas–Fermi approximation becomes exact. This approximation also corresponds to the classical limit of the theory (it corresponds to neglecting all terms with powers of ℎ or, equivalently, to the regime of strong repulsive interactions among particles). From a mathematical point of view, the Thomas–Fermi approximation corresponds to neglecting in the equation of motion all terms containing ∇ρ and ∇S.

In the standard approach to the BECs, the non-linearity term g is given by

11

where u₀ = 4πℎ²l_a/m (Dalfovo et al. 1999; Chen et al. 2005). The corresponding equation of state of the condensate is

12

with

13

Therefore, the equation of state of the BEC with quartic non-linearity is a polytrope with index n = 1. However, in the case of low dimensional systems Kolomeisky et al. (2000) have shown that in many experimentally interesting cases the non-linearity will be cubic, or even logarithmic, in ρ_m. The strong interaction assumption is valid only if the interaction energy per particle is much bigger than the ground-state energy (due to the zero-point motion) per particle. This is the case for condensates in the dilute limit below two dimensions. But as space dimensionality decreases, it becomes increasingly harder for the repulsive particles to avoid collisions. Thus the correlations between particles dominate, and the quartic non-linearity should be replaced by a more general, power-law term (Kolomeisky et al. 2000). Hence more general models, with the non-linearity term of the form g(ρ_m) =αρ^Γ_m, where α = constant and Γ = constant, can also be considered. In this case, the equation of state of the gravitational BEC is the standard polytropic equation of state, P(ρ_m) =α (Γ− 1)ρ^Γ_m, and the structure of the static gravitationally bounded BEC is described by the Lane–Emden equation, (1/ξ²) d (ξ² d θ/d ξ)/d ξ+θⁿ = 0, where n = 1/(Γ− 1) and θ is a dimensionless variable defined by ρ=ρ_cmθⁿ, where ρ_cm is the central density of the condensate. The dimensionless radial coordinate ξ is defined by the relation r = [(n+ 1)Kρ^1/n−1_cm/4πG]^1/2ξ. Hence, the BEC dark matter can generally be described as fluid satisfying a polytropic equation of state of index n.

In the following, we will consider only the case of the condensate with quartic non-linearity. In this case the physical properties of the condensate are relatively well known from laboratory experiments, and its properties can be described in terms of only two free parameters, the mass m of the condensate particle and the scattering length l_a, respectively.

2.3 Dark matter as a Bose–Einstein condensate

In the case of a static BEC, all physical quantities are independent of time. Moreover, in the first approximation we can also neglect the rotation of the condensate, by taking V_rot = 0. Therefore, the equations describing the static BEC in a gravitational field with potential V take the form

14

15

These equations must be integrated together with the equation of state and some appropriately chosen boundary conditions. The density distribution ρ_BE of the static gravitationally bounded single component dark matter BEC is given by (Boehmer & Harko 2007b)

16

where and ρ^(c)_BE is the central density of the condensate, ρ^(c)_BE=ρ_BE(0). The mass profile of the BEC galactic halo is

17

with a boundary radius R_BE. At the boundary of the dark matter distribution ρ_BE(R_BE) = 0, giving the condition kR_BE=π, which fixes the radius of the condensate dark matter halo as ⁠. The tangential velocity of a test particle moving in the condensed dark halo can be represented as (Boehmer & Harko 2007b)

18

The mass of the particle in the condensate can be obtained from the radius of the dark matter halo in the form (Boehmer & Harko 2007b)

19

From this equation, it follows that m is of the order of eV. For l_a≈ 1 fm and R_BE≈ 10 kpc, the mass is of the order of m≈ 14 meV. For values of l_a of the order of l_a≈ 10⁶ fm, corresponding to the values of l_a observed in terrestrial laboratory experiments, m≈ 1.44 eV. These values are perfectly consistent with the limit m < 1.87 eV obtained for the mass of the condensate particle from cosmological considerations (Fukuyama et al. 2008).

3 POST-NEWTONIAN HYDRODYNAMICS OF THE BOSE–EINSTEIN CONDENSATES

In order to study gravitational effects on the evolution of BEC dark haloes, a full general relativistic treatment is needed. The equations of motion of the condensate are obtained from the conservation of the energy–momentum tensor, T^μ_ν;μ = 0, with; denoting the covariant derivative with respect to the metric g_μν and

20

where u^μ is the 4-velocity of the fluid, satisfying the conditions u_μu^μ = 1 and u^μ_;νu_μ = 0. By taking the covariant divergence of T^μ_ν, we obtain the equation

21

where the comma denotes the ordinary derivative with respect to the coordinate x^μ. By contracting equation (21) with u^ν, we obtain

22

In the Newtonian limit of small condensate velocities, the 4-velocity is given by u^μ = (1, v/c), where v is the 3-velocity of the condensate. The 4-divergence of the 4-velocity is given by ⁠, where (−g) is the determinant of the metric tensor. In the Newtonian limit we assume that ⁠, that is, the deviations from the Minkowski-type geometry are small. Under these assumptions, from equation (22) we obtain the equation of continuity of the BEC as

23

where all differential operations are considered with respect to the physical coordinate r. By contracting equation (21) with the projection operator h^ν_α=δ^ν_α−u_αu^ν, with the property h^ν_αu_ν≡ 0, we obtain the relativistic Euler equation of motion as

24

In the Newtonian approximation, the generalized Euler equation of motion becomes

25

The gravitational potential V satisfies the generalized Poisson equation,

26

Equations (23), (25) and (26) represent the basic equations describing the dynamics of a gravitationally bounded BEC in the first post-Newtonian approximation (McCrea 1951; Harrison 1965; Lima et al. 1997; Reis 2003; Abramo et al. 2007; Pace et al. 2010).

4 COSMOLOGICAL DYNAMICS OF BOSE–EINSTEIN CONDENSATES

The Bose–Einstein condensation takes place when particles (bosons) become correlated with each other. This happens when their wavelengths overlap, that is, the thermal wavelength is greater than the mean interparticles’ distance a, λ_T > a. The critical temperature for the condensation to take place is T_cr < 2πℎ²n^2/3/mk_B (Dalfovo et al. 1999). On the other hand, cosmic evolution has the same temperature dependence, since in an adiabatic expansion process the density of a matter-dominated Universe evolves as ρ∝T^3/2 (Fukuyama & Morikawa 2009). Therefore, if the boson temperature is equal, for example, to the radiation temperature at z = 1000, the critical temperature for the Bose–Einstein condensation is at present T_cr = 0.0027 K (Fukuyama & Morikawa 2009). Since the matter temperature T_m varies as T_m∝a⁻², where a is the scalefactor of the Universe, it follows that during an adiabatic evolution the ratio of the photon temperature T_γ to the matter temperature evolves as T_γ/T_m∝a. Using for the present-day energy density of the Universe the value ρ_cr = 9.44 × 10⁻³⁰ g cm⁻³, the BEC takes place provided the fact that the boson mass satisfies the restriction m < 1.87 eV (Fukuyama et al. 2008). Thus, once the temperature T_cr of the boson is less than the critical temperature, the BEC can always take place at some moment during the cosmological evolution of the Universe. On the other hand, we expect that the Universe is always under critical temperature, if it is at the present time (Fukuyama & Morikawa 2009). Another cosmological bound on the mass of the condensate particle can be obtained as m < 2.696(g_d/g)(T_d/T_cr)³ eV (Boyanovsky, de Vega & Sanchez 2008), where g is the number of internal degrees of freedom of the particle before decoupling, g_d is the number of internal degrees of freedom of the particle at decoupling and T_d is the decoupling temperature. In the Bose condensed case T_d/T_c < 1, and it follows that the BEC particle should be light, unless it decouples very early on, at high temperature and with a large g_d. Therefore, depending on the relation between the critical and the decoupling temperatures, in order for a BEC light relic to act as cold dark matter, the decoupling scale must be higher than the electro-weak scale (Boyanovski et al. 2008).

The set of equations (23), (25) and (26) admits a homogeneous and isotropic cosmological background solution with ρ_m=ρ_b(t) and P=P_b(t). In this case, the fluid’s velocity is given by

27

and the evolution of the scalefactor a is determined by the Friedmann equations:

28

and

29

respectively, where we have denoted ⁠. The continuity equation (23) reduces to

30

In the case of the BECs the equation of state is given by P_b=U₀ρ²_b, and equation (30) can be integrated immediately to obtain

31

where C is an arbitrary constant of integration. By assuming that the present-day density of the BEC, ρ_{m, 0}, is obtained for a value a=a₀ of the scalefactor, we obtain C=ρ_m,0a³₀/(1 +ρ_m,0U₀/c²), and the background cosmological density of the condensate can be written as

32

where we have denoted

33

The energy density of the BEC diverges as a→ρ^1/3₀. The equation determining the time evolution of the scalefactor is given by

34

where we have denoted

35

where Ω_{BE, 0}=ρ_{m, 0}/ρ_{cr, 0} is the present-day density parameter of the BEC, ρ_cr,0 = 3H²₀/8πG is the present-day critical density of the Universe and H₀ is the present-day value of the Hubble parameter.

Equation (34) can be integrated immediately to give the time evolution of the scalefactor of the BEC as

36

where C₁ is an arbitrary constant of integration. The constant C₁ can be determined from the condition t = 0 when (a/a₀)³=ρ₀, thus obtaining C₁ = 0. Therefore, the time evolution of the scalefactor is described by the equation

37

where we denoted t_H = 1/H₀.

In the case of the standard dark matter models, dark matter is assumed to be a pressureless fluid, and the background cosmological evolution is described by the Einstein–de Sitter model, with the scalefactor given by a/a₀ = (9Ω_{DM, 0}/4)^1/3(t/t_H)^2/3, where Ω_{DM, 0} is the present-day density parameter of the dark matter, and we have assumed that a(0) = 0. In the following for the Hubble constant we adopt the value H₀ = 70 km s⁻¹ Mpc⁻¹ = 2.273 × 10⁻¹⁸ s⁻¹ (Hinshaw et al. 2009), giving for the critical density a value of ρ_{c, 0} = 9.248 × 10⁻³⁰ g cm⁻³. The constant ρ₀ can be represented as

38

while for Ω_BE we obtain

39

The two parameters ρ₀ and Ω_BE, describing the global properties of the condensate, are related by the relation Ω_BE=Ω_{DM, 0}× (1 −ρ₀).

By assuming that the entire existing dark matter is in the form of a BEC, it follows that Ω_{BE, 0}≈Ω_{DM, 0}≈ 0.228 (Hinshaw et al. 2009). By assuming that m = 1 meV and l_a = 10¹⁰ fm, we obtain ρ₀ = 0.7426, while Ω_BE = 0.058 66. For these values of the physical parameters of the condensate, the energy density of the dark matter diverges for a/a₀→ 0.9056. The time evolution of the scalefactor a for the BEC dark matter, for different values of the parameters m and l_a, and for the pressureless dark matter, are represented in Fig. 1. The cosmological dynamics of the condensate dark matter shows significant differences as compared to the standard pressureless dark matter model, with the condensate expanding much faster than the cosmological fluid of the standard ΛCDM model, with the speed of expansion increasing with increasing ρ₀.

Figure 1

Time evolution of the scalefactor of the pressureless dark matter, described by the Einstein–de Sitter metric (solid curve), and of the BEC dark matter, for different values of ρ₀: ρ₀ = 10⁻⁶ (dotted curve), ρ₀ = 10⁻³ (dashed curve), ρ₀ = 0.005 (long-dashed curve) and ρ₀ = 0.01 (ultra-long-dashed curve).

In the case of a Universe filled with dark energy, radiation, baryonic matter with negligible pressure and Bose–Einstein condensed dark matter, the time evolution of the scalefactor is given by

40

where Ω_{B, 0}, Ω_{rad, 0} and Ω_Λ are the present-day values of the density parameters of the baryonic matter, radiation and dark energy, respectively. For Ω_{B, 0}, Ω_{rad, 0} and Ω_Λ, we adopt the numerical values Ω_{B, 0} = 0.0456, Ω_{rad, 0} = 8.24 × 10⁻⁵ and Ω_Λ = 0.726 (Hinshaw et al. 2009). In the case of the standard ΛCDM cosmological model, the Friedmann equation describing the evolution of the Universe containing baryons, pressureless dark matter, radiation and dark energy is given by

41

The time evolutions of the scalefactors for Universes containing BEC dark matter and standard pressureless dark matter are represented, for different values of the BEC parameter ρ₀, in Fig. 2. The presence of the condensate dark matter changes the global cosmological dynamics of the Universe, and the magnitude of the changes increases with the increase of the BEC parameter ρ₀.

Figure 2

Time evolution of the scalefactor of a Universe filled with dark energy, baryonic matter and BEC dark matter, respectively, for different values of ρ₀: ρ₀ = 10⁻³ (dotted curve), ρ₀ = 10⁻² (dashed curve), ρ₀ = 0.05 (long-dashed curve) and ρ₀ = 0.1 (ultra-long-dashed curve). The solid curve represents the time evolution of the scalefactor in the standard ΛCDM model, in which the dark matter is pressureless.

5 COSMOLOGICAL PERTURBATIONS OF AN EXPANDING BOSE–EINSTEIN CONDENSATE

In the gravitationally bounded BEC, we assume small perturbations of the physical quantities around the homogeneous background of the form

42

43

44

45

In equations (42)–(45), the index b denotes the background quantities. Substituting these equations into the continuity equation (23), we obtain

46

The variation of the equation of motion (equation 25) gives

47

The Poisson equations for the perturbation of the gravitational potential, obtained by perturbing equation (26), can be written as

48

In order to describe the cosmological evolution we make a change to the comoving coordinate system, so that r=aq, ∇_q=∇=a∇_r and

49

To simplify the notation, we define the parameters w=P_b/ρ_bc² and c²_eff=δP/δρ, respectively, which are generally functions of the time only. We also introduce the density contrast as δ=δρ/ρ_b. The time derivative of the background pressure is related to the speed of sound of the background condensate c²_s=∂P_b/∂ρ_b by the relation

50

Therefore, the perturbation equations (43)–(45) can be written as

51

52

53

In the following, we denote

54

and

55

respectively. α_s is related to the time derivative of w by the relation

56

By taking the time derivative of equation (51), the divergence of equation (52), by eliminating ∇·u by using the perturbed equation of continuity, and with the use of equation (53), we obtain the equation giving the evolution of the density contrast as

57

Changing the independent variable from the time t to the scalefactor a using the relations ∂/∂t=aH(a)∂/∂a and ∂²⁠, we obtain the evolution equation in the form

58

Equation (57) is different from the perturbation equation obtained in the Newtonian cosmology with pressure by Lima et al. (1997), Reis (2003) and Abramo et al. (2007). The reason is that we have included in our analysis the term ⁠, which was neglected in the previous studies. This term generates the new term −(3c²_s/c²)Hu on the left-hand side of equation (52), which modifies the final perturbation equation. On the other hand, in the present approach the term was neglected.

In order to numerically integrate equation (57) or equation (58), we have to chose some physically appropriate initial conditions. In the current standard model for structure formation in the Universe, it is supposed that quantum fluctuations were generated during an initial period of inflation. These fluctuations inflated up to super-horizon scales, producing a near scale-invariant, and near Gaussian, set of primordial potential fluctuations. At the end of inflation, the Universe is reheated, and particles and radiation are produced. In this hot early phase, cold thermal relics (dark matter) are also formed (Padmanabhan 2003; Peebles & Ratra 2003). Dark matter particles interact gravitationally and possibly through the weak interaction. Therefore, in order to obtain some physically realistic initial conditions for the state of the Universe during large-scale structure formation, one needs to evolve cosmological perturbations, starting from initial conditions, deep inside the radiation epoch and far outside the Hubble radius. Initial conditions for photons, neutrinos, cold dark matter and baryons have been obtained, in the framework of the standard ΛCDM cosmological model, in both the synchronous and Newtonian gauges, by Ma & Bertschinger (1995). In the conventional method, the power spectrum of the matter fluctuations in the Universe is computed by numerically solving the Boltzmann equation. The power spectrum is usually obtained in the linear theory and then extrapolated to the present epoch. In the standard cosmology, in first-order Eulerian perturbation theory, all modes evolve independently, and the power spectrum can be scaled back to the initial epoch via the growth function (Ma & Bertschinger 1995). Moreover, the effect of the dark matter pressure is generally ignored in the conventional methods of generating initial conditions. On the other hand, the exact moment in the history of the Universe when the Bose–Einstein condensation occurred is not known. That is why obtaining the rigorous and physically well-motivated initial conditions for the density contrast δ and for its derivative for BEC dark matter requires the full investigation of the cosmological dynamics from the reheating era, by taking into account the dark matter condensation and pressure effects.

6 COSMOLOGICAL EVOLUTION OF SMALL PERTURBATIONS IN A BOSE–EINSTEIN CONDENSATE

By taking into account the equation of state of the BEC, we immediately obtain w=P_b/ρ_bc² = (U₀/c²)ρ_b=ρ₀/[(a/a₀)³−ρ₀] and c²_s/c²=c²_eff/c² = 2w. The conditions c²_s/c²≤ 1 and c²_eff/c²≤ 1 impose the constraint (a/a₀)³≥ 3ρ₀, and in the following we will consider the fact that the model considered in this paper is valid only for this range of values of the scalefactor. Hence, for the time evolution of the linear density perturbations of the BEC dark matter we successively obtain

59

and

60

respectively, where we have denoted α=a/a₀. In the limit of large α, when (a/a₀)³≫ρ₀, w→ 0, and equation (60) becomes

61

with the solution

62

where C₁ and C₂ are arbitrary constants of integration. Hence, in the limit of a pressureless fluid, we recover the standard general relativistic result. In the limit of small α, so that α³→ 3ρ₀, we can approximate equation (60) as

63

with the solution

64

where and are two arbitrary constants of integration.

By introducing w=ρ₀/(α³−ρ₀) as a new independent variable, equation (60) can be written as

65

By representing the density contrast δ as δ(w) =w^−1/3(1 +w)^−5/3u(w), it follows that the new function u(w) satisfies the equation

66

Therefore, the general solution of equation (65) can be obtained as

67

where ₂F₁(a, b; c; z) is the hypergeometric function, ⁠, |z| < 1 and C′₁ and C′₂ are arbitrary constants of integration. The constants of integration can be determined from the initial conditions. When a³ = 3ρ₀, w = 1/2, and u(1/2) = (3^5/3/4)δ_i and ⁠, where we have denoted δ_i=δ(1/2) and δ′_i=δ′ (1/2), respectively. In the limit of large a, w→ 0, and the density contrast can be approximated as

68

Near the initial state w = 1/2, the density contrast can be approximated as

69

Since equation (60) is valid only in the linear regime of small perturbations, we assume that the initial value of the perturbation, δ(a_i/a₀), occurring for a value a=a_i of the scalefactor, satisfies the condition δ(a_i/a₀) ≪ 1. Since the equation describing the perturbations is a second-order differential equation, two initial values have to be given, one for the initial perturbation δ(a_i/a₀) and one for the initial rate of evolution of the perturbation, δ′(a_i/a₀). We consider two cases, namely the case of a perturbation with an initial low evolution rate, of the order of δ′(a_i/a₀) = 10⁻⁵, and the case of a perturbation with a very high initial evolution rate, δ′(a_i/a₀) = 1.5, respectively. The comparison between the evolution of the linear perturbations for pressureless dark matter in an expanding Einstein–de Sitter cosmological background and the evolution of the density perturbations in a BEC dark-matter-dominated Universe is presented in Figs 3 and 4, respectively.

Figure 3

Evolution of the density perturbations δ(a/a₀) as a function of a/a₀ of the pressureless dark matter, described by the Einstein–de Sitter metric (solid curve), and of the BEC dark matter, for different values of ρ₀: ρ₀ = 10⁻³ (dotted curve), ρ₀ = 2 × 10⁻³ (dashed curve), ρ₀ = 3 × 10⁻³ (long-dashed curve) and ρ₀ = 5 × 10⁻³ (ultra-long-dashed curve). In all cases, the initial conditions are δ(3ρ₀)^1/3 = 10⁻⁵ and δ′(3ρ₀)^1/3 = 10⁻⁵, respectively.

Figure 4

Evolution of the density perturbations δ(a/a₀) as a function of a/a₀ of the pressureless dark matter, described by the Einstein–de Sitter metric (solid curve), and of the BEC dark matter, for different values of ρ₀: ρ₀ = 10⁻³ (dotted curve), ρ₀ = 2 × 10⁻³ (dashed curve), ρ₀ = 3 × 10⁻³ (long-dashed curve) and ρ₀ = 5 × 10⁻³ (ultra-long-dashed curve). In all cases, the initial conditions are δ(3ρ₀)^1/3 = 10⁻⁵ and δ′(3ρ₀)^1/3 = 1.5, respectively.

As one can see from Figs 3 and 4, for all initial conditions, in the case of the BEC, for a given a, the amplitude of the density contrast is higher as compared to the case of the standard dark matter model. The condensate enters more rapidly in the non-linear phase (δ≫ 1) than the pressureless dark matter. Thus, the presence of the BEC dark matter can significantly accelerate the process of cosmic structure formation.

7 DISCUSSIONS AND FINAL REMARKS

In this paper, we have considered the global cosmological evolution and the evolution of the small cosmological perturbation in a Bose–Einstein dark matter condensate. The basic equation describing the evolution of the small perturbations in the post-Newtonian regime was obtained, and its solutions have been studied by using both analytical and numerical methods. The evolution of the density perturbations of the condensate has been compared to the evolution of the small cosmological perturbations in a pressureless fluid evolving in an Einstein–de Sitter cosmological background. Depending on the numerical values of the physical parameters describing the condensate (the mass of the particle and the scattering length, respectively), significant differences could appear in the evolution of the BEC dark matter haloes, as compared to the standard pressureless dark matter models. These differences appear at the level of both the global cosmological evolution and the behaviour of the small perturbations in the dark matter fluid, and they could have fundamental implications for the formation of the large-scale structure in the Universe.

One of the most important problems present-day cosmology faces is the problem of the galaxy formation. To explain galaxy formation, the evolution of the linear density and temperature perturbations in a Universe with dark matter, baryons and radiation must be computed. For pressureless dark matter the evolution of the perturbations of all cosmic components, from cosmic recombination until the epoch of the first galaxies, was obtained in Naoz & Barkana (2005). The evolution of sub-horizon linear perturbations can be described by two coupled, second-order differential equations, with the pressureless dark matter interacting gravitationally with itself and with the baryons, while the baryons experience both gravity and pressure. Starting from very low values on sub-horizon scales, the baryon density perturbations gradually approach those in the dark matter, and the temperature perturbations approach the value expected for an adiabatic gas. The presence of the baryons does not modify significantly the evolution of the dark matter perturbations. By including the effect of the BEC pressure in the perturbation equations for baryons and dark matter, a more general (and realistic) description of the galaxy formation process can be obtained. The presence of the BEC modifies the dynamical evolution of the baryons, and the growth of linear perturbations, which provide the initial conditions for the formation of galaxies. In the BEC model the dark matter perturbations grow more rapidly than in the standard cosmology, and therefore this could lead to a much faster growth rate of the baryonic perturbations, accelerating the galaxy formation process.

A recent major experimental advance in the study of the Bose–Einstein condensation processes was the observation of the collapse and subsequent explosion of the condensates (Rybin et al. 2004). A dynamical study of an attractive ⁸⁵Rb BEC in an axially symmetric trap was done, where the interatomic interaction was manipulated by changing the external magnetic field, thus exploiting a nearby Feshbach resonance. In the vicinity of a Feshbach resonance the atomic scattering length can be varied over a huge range, by adjusting an external magnetic field. Consequently, the sign of the scattering length is changed, thus transforming a repulsive condensate of ⁸⁵Rb atoms into an attractive one, which naturally evolves into a collapsing and exploding condensate. From a simple physical point of view, the collapse of the BECs can be described as follows. When the number of particles becomes sufficiently large, so that N > N_c, where N_c is a critical number, the attractive interparticle energy overcomes the quantum pressure, and the condensate implodes. In the course of the implosion stage, the density of particles increases in the small vicinity of the trap centre. When it approaches a certain critical value, a fraction of the particles gets expelled. In a time period of an order of a few milliseconds, the condensate again stabilizes. There are two observable components at the final stage of the collapse: remnant and burst particles. The remnant particles are those which remain in the condensate. The burst particles have an energy much larger than that of the condensed particles. There is also a fraction of particles, which is not observable. This fraction is usually referred to as the missing particles.

The scattering length l_a is defined as the zero-energy limit of the scattering amplitude f (Dalfovo et al. 1999). Depending on the spin dependence of the underlying particle interaction, the scattering length may in general be also spin dependent. The spin-independent part of the quantity is referred to as the coherent scattering length l_a. The scattering lengths can be obtained for some systems in the laboratory, but for dark matter they are unknown. Another essential parameter is the mass m of the condensate particle, which, due to the lack of information about the physical nature of the dark matter, is a free parameter, which must be constrained by observations. Due to the lack of any physical information about the numerical values of these two fundamental parameters, in the numerical estimations performed in this paper we have given different numerical values to a combination of these two basic quantities.

Since BECs are less stable with respect to perturbations than usual non-condensate matter, we expect that in such a condensate evolution of the perturbations and the subsequent collapse could take place much faster than in the usual non-condensed matter. This would strongly affect the formation of the large-scale structure in the early Universe. In this paper, we have provided some basic theoretical tools necessary for an in-depth comparison of the predictions of the condensate model and of the observational results.

I would like to thank to the anonymous referee for comments and suggestions that helped me to considerably improve the manuscript. The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China.