Thermal condensate structure and cosmological energy density of the Universe

The aim of this paper is the study of thermal vacuum condensate for scalar and fermion fields. We analyze the thermal states at the temperature of the cosmic microwave background (CMB) and we show that the vacuum expectation value of the energy momentum tensor density of photon fields reproduces the energy density and pressure of the CMB. We perform the computations in the formal framework of the thermo field dynamics. We also consider the case of neutrinos and thermal states at the temperature of the neutrino cosmic background. Consistency with the estimated lower bound of the sum of the active neutrino masses is verified. In the boson sector, non trivial contribution to the energy of the universe is given by particles of masses of the order of $10^{-4}eV$ compatible with the ones of the axion-like particles. The fractal self-similar structure of the thermal radiation is also discussed and related to the coherent structure of the thermal vacuum.


INTRODUCTION
The task of this paper is the analysis of the thermal vacuum condensate for scalar and fermion fields, with specific reference to temperatures characteristic of cosmic microwave background (CMB). The interest in considering the vacuum condensate in relation with CMB resides in the fact that it is a thermal radiation filling almost uniformly the observable universe and one expects that it plays a relevant role in the universe thermal vacuum stracture. CMB appears as a radiation left over from an early stage in the expansion of the universe [1] and has a thermal black body spectrum corresponding to the temperature of 2.72548 ± 0.00057 K [2]. The anisotropies contained in the spatial variation in the spectral density are attributed to small thermal variations, presumably generated by quantum fluctuations of matter [1], [3], [4].
In our analysis, we compute the expectation value of the energy momentum tensor density of photon fields on the thermal vacuum. As a result, we obtain the energy density and pressure of the CMB.
Together with the CMB, there is an indirect evidence of the existence of the cosmic neutrino background (CNB) which represents the universe's background particle radiation composed of neutrinos (relic neutrinos) [5]- [9]. The CNB estimated temperature is roughly 1.95K [5]. It is therefore interesting to extend our study of thermal vacuum condensate also to the CNB case. Thus, we assume the hierarchical neutrino model and, by computing the energy density of the neutrino thermal vacuum, we check the lower bound of the sum of the active neutrino masses m ν , which has been estimated from the neutrinos oscillations to be of the order of 0.06eV [10]. We finally discuss the fractal self-similar structure of the thermal vacuum.
In Section II, the Thermo Field Dynamics (TFD) formalism is introduced and the general expressions of its energy density and pressure are shown. Explicit computations for Maxwell, scalar and fermion fields are pre-sented in Section III and, in Section IV, the fractal structure of the thermal states is analyzed. Section V is devoted to the conclusions.

THERMAL VACUUM AND PARTICLE CONDENSATE
The thermal vacuum state |0(θ) , with θ = θ(β), β ≡ 1/(k B T ) and k B the Boltzmann constant, is introduced in the TFD formalism [11,12] in such a way that the thermal statistical average N a k (θ) is given by N a k (θ) = 0(θ)|N a k |0(θ) , with N a k = a † k a k , the number operator. The bosonic operators a k and a † k have usual canonical commutation relations (CCR).
Note that a k and b k do not annihilate the state |0(θ) . The annihilation operators, say A k (θ k ) and B k (θ k ), for |0(θ) , A k (θ k )|0(θ) = 0 = B k (θ k )|0(θ) , are obtained through the Bogoliubov transformation . The thermal vacuum expectation value of the number operator N a k = a † k a k is given by Minimization of the free energy (see below) then leads to the thermal statistical average of N a k which is indeed the Bose-Einstein distribution function for a k . Summing up, the "thermal background" at T is described by the quantum coherent condensate vacuum |0(θ) , which is the thermal physical vacuum.
We now are ready to compute the contributions of the energy momentum tensor T µν to the thermal vacuum for Maxwell, scalar and fermion fields. We observe that the off-diagonal terms of T µν on the vacuum state are zero for these fields, i.e. 0(θ)|T ij (x)|0(θ) = 0, for i = j. Therefore, the vacuum condensate is homogenous and isotropic and behaves as a perfect fluid (similar result hold for mixed particles [15]- [19] and for curved space [20]). Then the energy density and pressure induced by the condensate (3), at a given time (we consider the red shift z of the universe), can be defined by computing the expectation value of the (0, 0) and (j, j) components of the energy-momentum tensor of a field on |0(θ, z) , Here : ... : denotes the normal ordering with respect to |0 and no summation on the index j is intended.

ENERGY DENSITY OF THERMAL VACUUM AND CMB TEMPERATURE
In the photon fields case, the explicit expressions of the energy momentum tensor density T µν [21,22]. As usual F αβ = ∂ β A α − ∂ α A β , (g µν = (1, −1, −1, −1), µ = 0, 1, 2, 3 ; = 1 = c will be used throughout the paper). The thermal vacuum condensate energy density is then where Ω k = k for photons. The result we obtain is In a similar way, the contribution given to the pressure by the thermal vacuum condensate of photons field is The equation of state is then w γ (z) = p γ (z)/ρ γ (z) = 1/3, which is the equation of state of the radiation. Eqs. (7) and (8) reproduce of course the results obtained by solving the Boltzmann equation for the distribution function of photons in thermal equilibrium [5]. The advantage of the present computation is that the role of the boson condensate in obtaining such a result is underlined. Taking the present CMB temperature, T γ = 2.72548 ± 0.00057 K, and the present red shift of the universe, z = 0, one obtains the value of the thermal vacuum energy density, ρ γ = 2 × 10 −51 GeV 4 , which of course coincides with the energy density of the CMB [5].
Leaving apart the photon case, we consider now massive boson and fermion fields. The energy momentunm tensor density is given by T µν At any epoch, the thermal vacuum energy and thermal pressure are given by Eqs. (4) and (5), which in the case of the field φ give In the case of the isotropy of the momenta k 1 = k 2 = k 3 , these can be written as Explicitly they become Notice that the vacuum energy density at thermal equilibrium ρ B (z), Eqs. (13), coincides with the result obtained by solving the Boltzmann equation for the particle Bose distribution function [5]. The difference to the pressure p B (z) between the contributions coming from the vacuum condensate and the ones coming solely from the Bose distribution function appears in Eq. (14). The second term on the R.H.S. of Eq.(14) appears due to the condensate of the physical vacuum contributing with non-vanishing values of 0(θ, z)|a k a −k |0(θ, z) and 0(θ, z)|a † k a † −k |0(θ, z) . Would the vacuum be the trivial one |0 , these contributions would be identically zero.
By considering the present epoch, z = 0, T = T γ , and by solving numerically the integral in Eq.(13), one has the contribution to the vacuum energy given by ρ B ≃ 9 × 10 −52 GeV 4 for masses less or equal than the CMB temperature m ≤ T γ , i.e. m ≤ 2.3 × 10 −4 eV (for example, possible candidates are axion-like with m a ∈ (10 −3 − 10 −6 )eV ). The maximum value of ρ B is obtained for m ≪ 10 −4 eV . In this case, one has ρ B ≃ 10 −51 GeV 4 . Negligible values of ρ B are obtained for boson masses m ≫ 10 −3 eV .
In the fermion case, the Fermi-Dirac distribution function is obtained The thermal vacuum contribution to the energy density and to the pressure, are respectively. In Eq.(16), the relation i 2ψ γ 0 ← → ∂ 0 ψ = iψ γ 0 ∂ 0 ψ = −iψ γ j ∂ j ψ + mψψ is used. For Majorana fields, Eqs. (16) and (17) give where α r k , r = 1, 2, is the annihilator of fermion field. The explicit expressions of the energy density and pressure are respectively. These equations coincide with the energy density and pressure obtained by solving the Boltzmann equation for the fermion distribution function [5]. For z = 0 and masses m ≤ T γ , we find at T = T γ the maximum value of ρ F , i.e. ρ F ∼ 1.6 × 10 −51 GeV 4 which is of the same order of CMB energy. The state equation is w F ∼ 1/3. Condensates of heavier fermions give negligible contributions to the universe energy. Only particles with masses less or equal to 10 −4 eV , e.g. neutrinos, may give relevant contributions.

FRACTAL STRUCTURE OF THE THERMAL STATES
Finally, we show that the thermal vacuum |0(θ) has a fractal self-similar structure. Let us consider the time dependent case θ = θ(t). We will use the notation |0(θ(t)) ≡ |0(t) . The boson vacuum |0(t) provides the quantum representation of the system of couples of damped/amplified oscillators [23] mẍ + γẋ + kx = 0, mÿ − γẏ + ky = 0, where "dot" denotes time derivative, m, γ and κ are positive real constants and L is the Lagrangian from which Eqs. (22) and (23) It is found [23] that the time evolution of the system ground state (the vacuum) leads out of the Hilbert space of the states, and thus the proper quantization setting is the one of the quantum field theory (QFT). One has therefore to consider operators a k , b k and their hermitian conjugates, so to perform, as customary in QFT, the continuum momentum limit (or the infinite volume limit) by use of the relation κ → (V /(2π) 3 ) d 3 κ at the end of the computations. The Hamiltonian H of the system is found to be [23] H = H 0 + H I , with where it has been used θ κ (t) = Γ k t ≡ (γ k /2m) t for each κ-mode. The group structure is the one of the SU (1, 1), [ H 0 , H I ] = 0 and the Casimir operator C is given by The initial condition of positiveness for the eigenvalues of H 0 are thus protected against transitions to negative energy states. One then finds that the time evolution of the vacuum |0 for a k and b k is controlled by H I and given by |0(θ(t)) = e −it H |0 = e −it H I |0 which gives in fact Eq. (1).
One also finds that |0(t) turns out to be a squeezed coherent state characterized by the q-deformation of Lie-Hopf algebra and provides a representation of the CCR at finite temperature which is equivalent [23] to the Thermo Field Dynamics representation {|0(β) } [11,12]. In the limit of quasi-stationary case with β(t) slowly changing in time, minimization of the free energy gives again the Bose-Einstein distribution function Eq. (3).
Indeed, let us now introduce the functional F a for the a−modes where H a is the free Hamiltonian relative to the a−modes, H a = k Ω k a † k a k , and S a is given by Inspection of Eqs.(29) and (30) suggests that F a and S a can be considered as free energy and the entropy, respectively. Minimization of the functional F a , ∂Fa ∂θ k (t) = 0, ∀k [11,12] (we consider = c = 1) then leads to Eq.(3) which is the Bose-Einstein distribution function for a k . The first principle of thermodynamics at constant temperature can be then expressed as where, the change in time of the particle condensed in the vacuum turns out into heat dissipation dQ = 1 β dS whereṄ k a (t) denotes the time derivative of N k a (t). We now remark that the system of Eqs. (22) and (23)) posses self-similarity properties. To see this, let us put with z 1 (t) = r 0 e − i Ω t e −Γt and z 2 (t) = r 0 e + i Ω t e +Γ t , Γ ≡ γ/2m and Ω 2 = (1/m)(κ − γ 2 /4m), κ > γ 2 /4m.
Solutions of Eqs. (35) and (36) are in fact z 1 (t) = r 0 e − i Ω t e −Γt and z 2 (t) = r 0 e + i Ω t e +Γ t and they describe the parametric time evolution of clockwise and the anti-clockwise logarithmic spirals, r = r 0 e −dα and r = r 0 e dα , with α(t) = Γ t/d and Ω t = Γ t/d = α(t) [24]. Thus, Eqs. (22) and (23) (or equivalently Eqs. (35) and (36)), whose quantum representation is provided by |0(t) , are found to describe the self-similar fractal structure of their logarithmic spiral solutions [25,26]. This establish the link between the SU (1, 1) coherent states and fractal-like self-similarity [24]. The relation of the photon energy-momentum tensor T µν γ with Eqs. (22) and (23) can also be shown. For details see [24]. Similar discussions can be done for the fermion vacuum.

CONCLUSIONS
We have studied the thermal vacuum structure at the temperature of the CMB. In the framework of TFD, the the energy momentum tensor density of photon has expectation value on the vacuum which agrees with the energy density and pressure of the CMB. In the case of neutrinos and thermal states at the temperature of the CNB consistency has been verified with the estimated lower bound of the sum of the active neutrino masses. The fractal self-similar structure of the thermal vacuum has been also discussed.