Comment on a quintessence particle mass in the Kaluza–Klein Theory and properties of its field, arXiv: astro-ph/0310300v6

We calculate a mass of a quintessence particle of order 10−5 eV and we find several solutions for quintessence field equation. We consider also a quintessence speed of sound in several schemes. In this paper we consider several consequences of the Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory. We consider a value of a mass of quintessence particle, several interesting relations among energy scales, radiation density in the second de Sitter phase. We find a spatial dependence of a quintessence field (and an effective gravitational constant Geff) in a case of spherical-statical symmetry, cylindrical-statical symmetry, flat-static symmetry. We find a time dependences of a quintessence field (with no spatial dependence). We get a solution for a quintessence field (a travelling wave) and two-dimensional wave solution applying those solutions for Geff. We propose some kind of statistical approach to our results. We calculate a speed of sound for a quintessence field. Let us consider a value of mass of a quintessence particle (a scalar particle) (see Ref. [1]) obtained from Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory (see Ref. [2]): m0 = √ n 2 ( n n+ 2 )n/2 |γ| ( |γ| β )n/2 . (1) (see Eq. (79) from Ref. [1]). The value of this mass has been obtained by this particle during the second de Sitter phase. Moreover during our contemporary epoch it is the same. The parameters n, γ and β are defined in Refs [1], [2]. During the second de Sitter phase a cosmological constant has been calculated in Ref. [2] and one finds λc0 = 6H 2 1 = 2|γ|(n+2)/2nn/2 βn/2(n+ 2)(n+2)/2 . (2) The value of a cosmological constant remains the same during our contemporary epoch. H1 is a Hubble constant during the second de Sitter phase (see Eq. (8) of Ref. [1]). Thus for our contemporary epoch one gets m0 = 1 2 √ n(n+ 2) √ λc0 . (3) 1 In this way we can connect the value of a cosmological constant of our contemporary epoch to the value of a mass of a quintessence particle. From recent observational data we get λc0 = Λ = 10 −52 1 m2 . (4) Moreover in order to get a correct dimension for a mass we should add a factor with h̄ and c (now we abandon the system of units with h̄ = c = 1). One gets m0 = 1 2 h̄ c √ n(n+ 2) √ λc0 . (3a) Using the value of λc0 one finally gets m0 ≃ √ n(n+ 2) · 0.17 · 10−39 g (5) or m0 ≃ √ n(n+ 2) · 0.95 · 10−5 eV. (5a) For example, if we take n = 14(= dimG2), one finally gets m0 ≃ 14.2 · 10−5 eV. (6) This value is bigger than that considered by different authors. Moreover, still sufficiently small. The particle interacts only gravitationally and because of this it is undetectable by using known experimental methods. Taking a density of dark energy as 0.7 of a critical density, ρc = 1.88h 2 · 10−29 g cm3 , (7) one gets a number of quintessence particles per unit volume n = h √ n(n+ 2) · 1.31 · 10 1 cm3 (8) where h is a dimensionless Hubble constant 0.7 < h < 1. Taking n = 14 and h = 0.7 one finally gets n = 4 · 10 1 cm3 (8a) which is many orders of magnitude smaller than Loschmidt number. Thus a gas of quintessence particles is not so dense from the point of view of our earth conditions. However, if this number of particles per unit volume is considered in a container of size 200Mpc, the gas can be considered as extremely dense. 2 In order to settle—is this gas dense or not—we should calculate a mean scattering length. The scattering cross-section for a quintessence particle σ = 1 λc0 = 10 m. (∗) A mean scattering length l = 1 σn (∗∗) where n is a number of quintessence particles per unit volume (Eq. (8a)). One gets l = 10−60 m. (∗∗∗) It means that a gas of quintessence particles is extremely dense (if we apply the Knudsen criterion—a gas is dense if l ≪ L, where L is the size of the container) even in the Solar System. Let us consider the Eq. (111) from Ref. [3] which connects several scales of energy and gives an account of the smallness of gravitational interactions in our theory. We rewrite this equation in the form ( mpl mEW )( mà mEW )n1 = ( n|γ| (n+ 2)β )(n+2)(n1+2)/2 (9) where mEW is an electro-weak interactions energy scale. Taking mà = mEW n1 = 2 (M = S ) n = 14 (H = G2) one gets mpl mEW = ( 7 8 κ )24 (10) where κ = |γ| β (11) and eventually one gets κ = ( mpl mEW )1/24 · 8 7 . (12) Taking mEW ≃ 80GeV and mpl ≃ 2.4 · 10 GeV one gets κ ≃ 6.19 (13) 3 which is very reasonable for it establishes a relation between two cosmological terms γ and β as of the same order. Simultaneously this is a consistency condition for our model with energy scales (the 6-dimensional Planck’s mass is equal to mEW). In this way we can calculate a mass of a quintessence particle for our contemporary epoch from Eq. (1) m0 = 2 · 10 eV · |P̃ |. (14) This gives us an estimation for |P̃ | and R̃(Γ̃ ): R̃(Γ̃ ) = 1 κ ( mEW mplαem )2 |P̃ | (15) where αem = 1 137 is a fine coupling constant. Using Eq. (13) and values of mpl and mEW one gets R̃(Γ̃ ) = 24.75 · 10−31|P̃ |. (16) From Eq. (14) and Eq. (6) one gets |P̃ | = 5 · 10−39 (17) and R̃(Γ̃ ) ∼= 1.2 · 10−68. (18) Moreover in our simplified theory we have R̃(Γ̃ ) = 2(2μ + 7μ + 5μ+ 20) (μ2 + 4)2 (19) and μ should be very close to the root of the polynomial W (μ) = 2μ + 7μ + 5μ+ 20. (20) From (18) and (19) one gets 2μ + 7μ + 5μ+ 20 ∼= 1.3 · 10−66. (21) Thus μ is very close to the 70-digit approximation of the root of the polynomial (20) (W (μ) = 0.1 · 10−67). Due to this we can control the cosmological terms. In the case of |P̃ | we have the formula (28) from Ref. [1] and one gets P̃ (ζ0 + ε) ∼= P̃ (ζ0) + dP̃ dζ (ζ0)ε (22) P̃ (ζ0 = ±1.38 . . . ) = 0 (23) ∣∣ dP̃ dζ (|ζ0| = 1.38 . . . ) ∣∣ = 25. (24) 4 Thus one gets from (17) and (23–24) ε ≃ 2 · 10−40. (25) Thus we need an approximation of ζ0 up 40-digit arithmetics. We see that cosmological terms coming from the Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory are very small, but not zero, and that they are easily controllable by μ and ζ parameters. Let us consider a self-interaction potential for a quintessence field for our contemporary epoch (which is the same as for the second de Sitter phase). One gets from cosmological terms λc0(Ψ) λc0(Ψ) = − 1 2 ( βe − |γ| ) e (26) U(q0) = − |γ| 2(n+ 2) ( n|γ| (n+ 2)β )n 2 exp   nmpl 2 √ 2π|M | q0   ×  n exp   mpl √ 2π|M | q0  − (n+ 2)   (27)