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Infinite statistics condensate as a model of dark matter

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JCAP11(2013)057

ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

Infinite statistics condensate as amodel of dark matter

Zahra Ebadi,a,b Behrouz Mirzaa and Hosein Mohammadzadehb

aDepartment of Physics, Isfahan University of Technology,Isfahan, 84156–83111 IranbDepartment of Physics, University of Mohaghegh Ardabili,P.O. Box 179, Ardabil, Iran

E-mail: z.ebadi@ph.iut.ac.ir, b.mirza@cc.iut.ac.ir, mohammadzadeh@uma.ac.ir

Received September 20, 2013Revised November 18, 2013Accepted November 19, 2013Published November 28, 2013

Abstract. In some models, dark matter is considered as a condensate bosonic system. In thispaper, we prove that condensation is also possible for particles that obey infinite statisticsand derive the critical condensation temperature. We argue that a condensed state of agas of very weakly interacting particles obeying infinite statistics could be considered as aconsistent model of dark matter.

Keywords: dark matter theory, rotation curves of galaxies

c© 2013 IOP Publishing Ltd and Sissa Medialab srl doi:10.1088/1475-7516/2013/11/057

JCAP11(2013)057

Contents

1 Introduction 1

2 Quons and infinite statistics 2

3 Infinite statics and holographic foam model of spacetime 4

4 Infinite statistics and dark matter 4

5 Infinite statics and condensation 5

6 Conclusion and discussion 8

1 Introduction

Investigation of cosmological nucleosynthesis leads to the conclusion that most of the mass inthe universe is not in the form of ordinary baryonic matter, i.e. atomic nuclei and electrons.This conclusion is strongly supported by the anisotropies observed in the cosmic microwavebackground [1, 2]. These observations reveal a universe composed of 4% baryons, 22% non-baryonic dark matter, and 74% dark energy [2, 3].

Dark matter plays a key role in the current models of galaxy formation [1], but oneof the most important questions in modern cosmology is what the dark matter is made upof. Observation of rotation curves [4], gravitational lensing [5], and X-ray spectra [6] areuseful for determining the mass distribution of dark matter. The data thus obtained haveled to the conclusion that galaxies are composed of a luminous galactic disk surrounded bya spherical galactic halo of dark matter which comprises roughly 95% of the total mass ofthe galaxy [7]. We expect the constituent particles of dark matter to have no charge, andto be cold and long-lived [1]. Various models have been proposed to explain dark matter.Some maintain that dark matter is made up of Bose-Einstein condensate [8] particles with avery small mass [9–13]. The condensation process is interpreted as a phase transition takingplace sometime during the cosmic history of the universe. As the normal bosonic dark mattercools below the critical temperature Tcr of condensation, it becomes energetically favorableto forming a condensate in which all the particles are in the same quantum state. Differentproperties of the Bose-Einstein condensed dark matter have been investigated [14–16].

The standard approach to treating a nonrelativistic Bose-Einstein condensation is touse the Gross-Pitaevkii equation [17], also known as the nonlinear Schrondinger equationwhich is in the form of the Schrondinger equation with an interaction term and trap poten-tial introduced. There is a constant coupling between the interacting term and the Newto-nian gravitational potential which is proportional to the scattering length of two interactingbosons and the trap potential, so that the Gross-Pitaevskii equation is coupled with the Pois-son equation [12, 15, 18, 19]. Using this equation, one can achieve the basic astrophysicalproperties of the condensate, i.e. density profile, rotational velocity, and mass profile [13, 20].Dwarf galaxies have been already studied [20], these are dark matter dominated astrophysicalobjects with a very small contribution of baryons to the total matter content, which makethem suitable for testing models of dark matter. Recently, the intermediate statistics is usedin various aspects of gravitation and cosmology [21, 22].

– 1 –

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In models based on the Bose-Einstein condensation, constituent particles of dark matterare considered to be bosons. However, theories based on the holographic model of spacetimefoam do not regard them as bosons [23]. Some physicists believe that space is composedof an ever-changing arrangement of bubbles called spacetime foam, also known as quantumfoam [23, 24]. This model necessitates that the dark energy particles obey not the ordinaryBose or Fermi statistics but the exotic infinite statistics [23]. Infinite statistics [25–27] isdefined by generalizing commutation and anticommutation relations between the creationand annihilation operators.

The present paper is organized as follows. In section 2, we review infinite statistics.In section 3, we summarize the reasons why infinite statistics must be employed in dealingwith dark matter. We will present some arguments about the infinit statistics and cold darkmatter in section 4. In section 5, we investigate the condensation and critical temperature ofa system with particles obeying infinite statistics and compare this approach with the well-known Bose-Einstein condensation. We also consider the condensation of particles obeyinginfinite statistics as a model for dark matter. Hence, we will summarize below the statisticsof quons in order to work out the critical temperature of condensation.

2 Quons and infinite statistics

The creation and annihilation operators of bosons and fermions satisfy the well known com-mutation and anticommutation algebra, aia

†j ∓ a

†jai = δij . This relation requires that all

particles be in either a symmetric or an antisymmetric state. Without such a symmetriza-tion postulate, there are many generalizations that may have intermediate properties of Boseand Fermi statistics, namely para-Bose, para-Fermi, q-deform bosons and fermions, q-mutatorand quons [25, 26, 28, 29]. Quons are characterized by a q-deformation of the commutationrelations:

aia†j − qa

†jai = δij , (2.1)

q = ±1 corresponds to bosons or fermions, while other cases (−1 < q < 1) correspond toquons. The special case q = 0 is called infinite statistics. Therefore, it is characterized by

aia†j = δij (2.2)

and a unique vacuum state annihilated by all the annihilators ai:

ai|0〉 = 0. (2.3)

This relation determines a Fock-state representation in a linear vector space. States are builtby acting on a vacuum. According to eq. (2), there is no commutation relation between twoannihilation or creation operators for infinite statistics. The quantum states are orthogonalunder any permutation of the identical particles and, therefore, all representations of thesymmetric group will occur. Hence, infinite statistics can be interpreted as the statistics ofidentical particles with an infinite number of internal degrees of freedom, which is equivalentto the statistics of nonidentical particles since they are distinguishable by their internalstates [23, 27]. The loss of local commutativity implies violation of locality, which is animportant character of quantum gravity. By virtue of these properties, infinite statisticshas been applied in a variety of areas, such as black hole statistics [30, 31], dark energyquanta [23, 32], large N matrix theory [33], and holography principle [34]. The statistical

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JCAP11(2013)057

mechanics of particles obeying infinite statistics can be derived in a way similar to the usualderivation of Boltzmann statistics. The partition function ZN is given by

ZN =∑

quantum states

e−βH (2.4)

for a given set of occupation numbers n(p), where p is a momentum, with N =∑

p n(p),there are

g[n(p)] = N !

/∏p

n(p)! (2.5)

orthogonal quantum states [26]. This is just Boltzmann counting without the Gibbs fac-tor 1/N !.

M.V. Medvedev [35] introduced an ambiguous statistics which may exhibit both Boseand Fermi statistics with the respective probabilities pb and pf . One can consider a realizationof a system with Nj particles where in each realization, the system has k bosons and Nj − kfermions while the probability of this realization is pkbp

Nj−kf . Therefore, the total number of

states of the system will be

W =∏j

Nj∑k=0

Nj !

k!(Nj − k)!wbj(k)wfj(Nj − k)pkbp

Nj−kf , (2.6)

where, wb(m) and wf (m) are the number of quantum states of n identical particles occupyinga group of G states for bosons and fermions, respectively:

wb(m) =(G+m− 1)!

m!(G− 1)!, wf (m) =

G!

m!(G−m)!

Using the maximum entropy for the most probable distribution and following the tech-nique of Lagrange’s multipliers, the occupation number is given by

nj =σρ

(ρ+ pf )((ρ− pb)

[1 +

√1−

δ2(ρ+ pf )((ρ− pb)σ2ρ(ρ+ δ)

], (2.7)

where, δ = pf − pb, σ = pf + pb, ρ = exp (εj − µ)/kBT and µ is the chemical potential [35].

Also, it has been shown that there is a conjectured relation between the particles of theambiguous statistical type and the quons [35]. It was argued that q in eq. (1) is related tothe parameters of ambiguous statistics as follows

q =pb − pfpb + pf

(2.8)

One can find the distribution function of infinite statistics (q = 0) as follows,

n(ε) =4pρ

ρ2 − p2(2.9)

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where, p = pf = pb. Therefore, the thermodynamic quantities such as the internal energyand the total number of particles for an ideal gas with infinite statistic gas will be

U =

∫ ∞0

ε n(ε)Ω(ε)dε =A

βd/2+1

∫ ∞0

4pzexxd/2

e2x − p2z2dx

N =

∫ ∞0

n(ε)Ω(ε)dε =A

βd/2

∫ ∞0

4pzexxd/2−1

e2x − p2z2dx (2.10)

where, Ω(ε) = Aεd/2−1 = VΓ( d

2 )(2mπ)d/2

hdεd/2−1 is the density of the single particle state, V is

the volume of the system, z = exp(βµ) is the fugacity, and x = βε.

3 Infinite statics and holographic foam model of spacetime

In this section, we summarize the reason one has to use infinite statistics for dark energy.Quantum fluctuations of spacetime give rise to quantum foam, and black hole physics dictatesthat the foam is of holographic type [23]. Applied to cosmology, the holographic modelrequires the existence of a specific dark energy that is composed of an enormous number ofinert particles of extremely long wavelengths [23]. The entropy of an ideal gas is given by

S = NkB

[ln

(V

Nλ3

)+ 5/2

]. (3.1)

where, N is the total number of particles in a volume V and λ = (2π~2/mkBT )1/2 is thermalwavelength [8]. Y. Ng has employed this equation for a gas of dark energy [37]. In the aboveequation, if V ∼ λ3 (long wavelength), the entropy will be negative unless N ∼ 1 or if theGibbs factor 1/N ! in the partition function is removed from the calculation of the entropy.The first condition is not possible because the holographic foam model of spacetime has beenused to show that dark energy is a cosmological manifestation of quantum foam and that itconsists of a large number of very low energy particles [23]. According to the holographicfoam model, N should not be too different from (RH/lP )2 1 [23, 36], where RH and lP arethe Hubble radius and the planck length, respectively. Therefore, the alternative conditionwill be feasible which will lead to the following equation:

S = NkB

[ln

(V

λ3

)+ 3/2

]. (3.2)

Removing the Gibbs factor shows that these particles are neither fermions nor bosonsand that they obey some other statistics [37]. The only known consistent statistics in D > 2without the Gibbs factor is infinite statistics or quantum Boltzmann statistics [26, 38].

4 Infinite statistics and dark matter

If we use the eq. (3.1) for cold dark matter at very low temperature, the negative entropyproblem will be occurred. In other words, in very low temperature limit Nλ3/V 1 andthe first term in eq. (3.1) will be negative. We argue that in this limit to avoid the negativeentropy, we need a statistics with removed Gibbs factor. There are some models based on thecondensation of the boson gas for cold dark matter [14–16] and therefore, we could proposethe condensate of infinite statistics as a model of dark matter. In comparison with the Y. Ng

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JCAP11(2013)057

model for dark energy, we notice that V is not in order of λ3 but the temperature is lowenough. Therefore, we could employ the infinite statistic for dark matter in low temperaturelimit. We expect that the condensation in low temperature for ideal boson gas and we explorethe possibility of condensation for infinite statistics in next section. Also, we will show thatinfinite statistics condensate (ISC) model for dark matter may be lead to less interactingdark matter particles with respect to the BEC models. Recently, it has been shown that theunconventional quanta of MONDian dark matter must obey infinite statistics which containssome similar ideas as this paper [39].

5 Infinite statics and condensation

Based on the observations in the preceding section, we explore the condensation of particlesobeying infinite statistics as a model for dark matter and work out the critical condensa-tion temperature for this statistics. Investigation of the thermodynamic geometry of somethermodynamic systems has led to the introduction of a qualitative tool, namely, the ther-modynamic curvature that is singular at the phase transition point and its sign specifies thestatistical interaction of the system [40–44]. We construct the thermodynamic geometry of anideal gas with particles obeying infinite statistics. The metric elements of the thermodynamicparameter space using the Fisher-Rao representation are given by

Gββ =∂2 lnZ∂β2

= −(∂U

∂β

)γ

,

Gβγ = Gγβ =∂2 lnZ∂γ∂β

= −(∂N

∂β

)γ

, (5.1)

Gγγ =∂2 lnZ∂γ2

= −(∂N

∂γ

)β

,

where, β = 1/kBT and γ = −µ/kBT . The curvature of two dimensional thermodynamicparameter space can be evaluated by the Christoffel symbol, Γijk = 1

2 (Gij,k +Gik,j −Gjk,i),using the derivative of the metric elements with respect to the thermodynamic parameters.In the following, one can evaluate the well-known Riemann tensor, Ricci tensor and, finally,Ricci scalar which will be called the thermodynamic curvature because of the identity of theconstructed geometry. The thermodynamic curvature of an ideal gas with particles obeyinginfinite statistics in a three dimensional box as a function of fugacity for isothermal processeshas been depicted in figure 1. Clearly, the thermodynamic curvature is positive and thestatistical interaction is, therefore, attractive. Also, there is a critical point (z = 1/p) wherethe thermodynamic curvature is singular. This is similar to the ideal boson gas wherein thestatistical interaction is positive and the thermodynamic curvature is singular at the Bose-Einstein condensation point (z = 1). Bose-Einstein condensation occurs at the singular pointof the occupation number at ε = 0 and µ = 0. According to eq. (10), within the infinitestatistics limit, condensation occurs at z = 1/p or equivalently at,

µ = −kBT ln p. (5.2)

Now, we can evaluate the phase transition temperature using the total number of particlesobeying infinite statistics. For non-relativistic particles with the dispersion relation ε(k) =~2k2/2m and at z = 1/p, according to eq. (2.10) we will have

n =N

V=

2(π)d/2

(2π)dΓ(d/2)

∫ ∞0

dkkd−1 4 exp(ε/kBTcr)

exp(2ε/kBTcr)− 1(5.3)

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JCAP11(2013)057

Figure 1. Thermodynamic curvature of an ideal gas with particles obeying infinite statistics withrespect to the fugacity of gas for isotherm processes (β = 1).

where, k is the wave number. The finite critical temperature for particles obeying infinitestatistics can be compared to Bose-Einstein condensation temperature. Solving eq. (5.3) forthe critical temperature yields,

kBTcr =2π~2

mq=0

(n

ζ(d/2)

)2/d 21−4/d

(2d/2 − 1)2/d, (5.4)

where, mq=0 is the quon mass that constitutes the dark matter and ζ(x) is the Riemannzeta function analytically continued to all x 6= 1. Also, eq. (5.4) indicates that there is afinite critical temperature for d > 2. We can obtain some properties of quon condensation bycomparison with a boson gas condensation. For a bosonic system, the critical temperatureof Bose-Einstein condensation is given by [8]

kBTcr =2π~2

mb

(n

ζ(d/2)

)2/d

. (5.5)

– 6 –

JCAP11(2013)057

where, mb is the boson mass that constitutes the dark matter. One can compare the criticaltemperature in eq. (5.4) with that of Bose-Einstein condensation given by eq. (5.5) to findthat the critical temperature of the infinite statistics model has a correction factor unlike theBose-Einstein condensation temperature

21−4/d

(2d/2 − 1)2/d. (5.6)

The critical temperatures of quon and boson particles, eqs. (5.4), (5.5), with a specifiedvalue of density number will be the same if the relation between mb and mq=0 is of thefollowing form

mq=0

mb=

21−4/d

(2d/2 − 1)2/d. (5.7)

We obtain mq=0 ' 0.53mb for d = 3. It has been shown that the current energy densityof the universe is ρcr = 9.44 × 10−30g/cm3 and so Bose-Einstein condensation takes placeprovided that the boson mass satisfies the restriction mb < 1.87 eV [14]. It is easy to find asimilar limit for the mass of quons. So we propose that a condense state of quon particlesthat obey infinite statistics is a new good candidate for investigating the dark matter.

One can also consider mb = mq=0 and according to eqs. (5.4) and (5.5), the criticaltemperature of infinite statistics model will be smaller than Bose-Einstein condensation tem-perature. It is known that the self interaction of dark matter particles must be very weak.This is because of the observations of elongated shapes of clusters of galaxies [45]. In thefollowing we will show that our model represents a condensed state of very weakly interact-ing particles which is almost similar to non-interacting particles of an ideal gas. When thecondensation occurs, we can work out the equation of state. It has been shown that theequation of state for an ideal gas of particles obeying the infinite statistics is given by [35]

PV =2

3

∫ ∞0

ε n(ε)Ω(ε)dε

= V(2mπ)3/2

h3

25/2 − 1

21/2ζ(5/2)(kBT )5/2. (5.8)

Using eqs. (2.10) and (5.4) at the critical point, we obtain

PV = αNkBTcr (5.9)

where, α = ζ(5/2)2ζ(3/2)

25/2−123/2−1

' 0.65 is a constant coefficient. Roughly the equation of state is

PV ' NkBT ; which means that the particles effectively behave as an ideal gas (very weaklyinteracting) in the condensate regime. The related condensation temperature is also lowerthan the Bose-Einstein condensation temperature, which means a lower pressure for the gas.At low temperature in the condensate regime, the thermal wavelength is very long, whichindicate that the particles or the physical degrees of freedom are highly correlated. Although,the system is correlated, but the internal interaction of particles is very weak.

Therefore, condensed state of a gas of very weakly interacting particles obeying infinitestatistics presents a model which indicates that the level of self interaction is small enoughto produce effectively collisionless dark matter particles.

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JCAP11(2013)057

6 Conclusion and discussion

Different models have been proposed to describe dark matter. Some of these models arebased on the Bose-Einstein condensate and are consistent with such observational data asdensity profile of dwarf galaxies [13]. We argued that constituent particles of dark mattercould obey infinite statics rather than Bose or Fermi statistics. Therefore, we investigatedthe possibility of quon condensation using infinite statistics. We introduced the first modelof a condensed state of a gas of very weakly interacting particles obeying infinite statistics.

According to the thermodynamic curvature of the thermodynamic parameter space, theISC has an attractive internal intrinsic interaction and at the critical point the equation ofstate is PV ' NkBT , which describes an ideal gas with non-interacting particles. Therefore,it is obvious that the pressure will go to zero at the condensate regime where the temperatureis very low. Dark matter is thought to be cold and nearly pressureless [45]. Therefore, thepresented model again is in a good consistency with the standard cold dark matter.

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