Thermal vacuum, cosmic microwave radiation, neutrino masses and fractal-like self-similar structure

The behavior of thermal vacuum condensates of scalar and fermion fields is analyzed and it is shown that the condensate of Maxwell fields reproduces the characteristics of the cosmic microwave radiation. By studying fermion thermal states with the temperature of the cosmic neutrino background, we derive a value of the sum of the active neutrino masses which is compatible with its estimated lower bound. Moreover, we reveal the fractal self-similar structure of the thermal radiation and we relate it to the coherent structure of the thermal vacuum.


Introduction
In the context of the Thermo Field Dynamic [1,2], we analyze the possible links existing among the thermal vacuum states, the cosmic microwave background (CMB) radiation and the neutrino masses [3].
The CMB radiation is a thermal radiation, left over from an early stage in the expansion of the universe [4], filling almost uniformly the observable universe. Such a radiation has a thermal black body spectrum corresponding to the temperature of T γ = 2.72548 ± 0.00057 K. The anisotropy is very small [5]- [7].
Together with the CMB, there is an indirect evidence of the universe's background particle radiation composed of neutrinos, called, cosmic neutrino background (CNB). The today estimated temperature for the CNB is roughly 1.95K [7]- [12].
Neutrinos play a significant role in the understanding of many phenomena of high energy physics and astrophysics [13]- [24]. The discovery of neutrino oscillations has shown that neutrinos have mass. Oscillations do not yield a value for the mass, but do set a lower limit of order of 0.06eV on the sum of the three neutrino masses m ν [25]. The upper limit of m ν is 2eV .
In this paper we report the results of Ref. [3], in which we have shown that the thermal condensate derived in thermal field theories, behave as a perfect fluid, gives a contribution to the radiation of the universe and reproduces the behavior of CMB and CNB. Moreover, by exploiting the CMB and CNB energy contribution, we derive an upper bound on the absolute mass of the lightest neutrino. The absolute masses of the other neutrinos and the sum of the active neutrino masses are obtained by assuming the hierarchical neutrino model and using the experimental differences between the squared masses ∆m 2 12 and ∆m 2 23 . We find a value of m ν of the order of 0.06eV , which is in agreement with the estimated lower bound on the neutrino Moreover, we study the fractal self-similar structure of the thermal states. It might be interesting to investigate about a possible relation between such fractal self-similarity properties and some aspects of holographic cosmology (see e.g. [26]). The structure of the paper is the following, in Section II, we present the Thermo Field Dynamic formalism and derive the general expressions of energy density and pressure of the vacuum condensate. In Section III, we show the explicit results for Maxwell and scalar field. The results for neutrinos are reported in Section IV. The fractal structure of the thermal vacuum is studied in Section V and the conclusions are in Section VI.

Thermal vacuum condensate
We describe the observed black body spectrum of the CMB at T = T γ , by using the Thermo Field Dynamics (TFD) formalism [1,2] in quantum field theory (QFT). Thus we introduce the thermal vacuum state |0(θ) , with θ = θ(β), β ≡ 1/(k B T ) and k B the Boltzmann constat. The thermal vacuum |0(θ) is defined in such a way that the thermal statistical average N a k (θ) is given by the expectation value of the number operator N a k = a † k a k , with a k denoting the bosonic CMB modes, on |0(θ) , i.e. N a k (θ) = 0(θ)|N a k |0(θ) .
The boson modes a k have usual canonical commutation relations (CCR) and |0(θ) is given by |0(θ) is a two-mode generalized SU (1, 1) squeezed coherent state [27,28] at finite temperature, condensate of pairs of a k and b k quanta, with b k auxiliary boson mode, commuting with a k . The quantum b k is introduced in order to produce the trace operation in the computing thermal averages of the observable quantities. In Eq.(1), |0 is the vacuum for a k and b k . One has that 0(θ)|0(θ) = 1, ∀ θ , and in the infinite volume limit (for d 3 κ θ κ finite and positive) 0(θ(β))|0 → 0 as V → ∞, ∀ β . Moreover, the thermal vacuum |0(θ) provides a representation of the CCR defined at each β and unitarily inequivalent to any other representation {|0(β ) , ∀ β = β} in the infinite volume limit. Indeed one has 0(θ(β))|0(θ(β )) → 0 as V → ∞, ∀ β and β , β = β. This means that in thermal equilibrium (β constant in time t) at a given temperature T , the system sits in the representation of the CCR corresponding to such a T . In thermal non-equilibrium conditions, with temperature changing in time, i.e. β = β(t), the system evolves in time through unitarily inequivalent representations of the CCR and it is known that such a time evolution is controlled by the entropy operator [1,2]. It is crucial to note that the annihilation operators for the state |0(θ) , denoted with A k (θ k ) and B k (θ k ), are different from a k and b k which annihilate the state |0 . The relation among the two sets of annihilators is provided by the Bogoliubov transformation The condensate density is given by the expectation value of the number operator a † k a k on the thermal vacuum, N a k (θ) = 0(θ)|a † k a k |0(θ) = sinh 2 θ k . In the boson case, minimization of the free energy leads to the Bose-Einstein distribution function for a k Thus, we can reproduce the observed black body spectrum of the CMB by computing the expectation value N B a k (θ), at T = T γ . For fermion fields we can obtain in a similar way the corresponding thermal SU (2) vacuum and Bogoliubov transformations [1,2]. We then obtain the Fermi-Dirac distribution function for fermions: In the following, we compute the energy momentum tensor T µν of the thermal vacuum for Maxwell, for massive boson and fermion fields and we show that particle with masses much higher than the CMB temperature, i.e. m 2 × 10 −4 eV , give negligible contribution to the energy of the universe. Next we derive the absolute mass of the neutrinos [3].
Notice that the vacuum condensate is homogenous and isotropic and behaves as a perfect fluid, indeed the off-diagonal terms of T µν on the vacuum state are zero for any field, i.e. 0(θ)|T ij (x)|0(θ) = 0, for i = j. Then, the energy density and pressure of the vacuum condensates (2), (3) at a given time (we consider the red shift z), are represented by the (0, 0) and (j, j) components of T µν of the vacuum |0(θ, z) , with : ... :, normal ordering with respect to |0 .

Thermal vacuum, CMB radiation and boson fields
In the photon fields case, Eqs. (4) and (5) become and the state equation is the one of the radiation w γ (z) = p γ (z)/ρ γ (z) = 1/3. These results coincides with the ones obtained by solving the Boltzmann equation for the distribution function of photons in thermal equilibrium [12]. By considering the present CMB temperature and the present red shift, z = 0, one obtains a value of the thermal vacuum energy density ρ γ = 2 × 10 −51 GeV 4 , coinciding with the energy density of the CMB [12].
Let us now study the energy momentum tensor of the thermal vacuum |0(θ, z) for massive boson and fermion fields. For bosons, at any epoch, one has [3]  In particular, at the present epoch, z = 0 one obtains 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 . The maximum value of ρ B is achieved for masses m 10 −4 eV . In this case, the energy density is ρ B 10 −51 GeV 4 . Negligible values of ρ B are obtained for boson masses m 10 −3 eV . Therefore, the thermal vacuum contribution of bosons to the energy of the universe is completely negligible with respect to ρ γ and with respect to the critical density of the present universe ρ cr 4.5 × 10 −47 GeV 4 . Only hypothetical particles as axions with masses m a ∈ (10 −3 − 10 −6 )eV , could generate non trivial contributions [3].

Thermal vacuum and neutrino masses
The thermal vacuum contribution of fermion fields to the vacuum energy density and to the pressure, are At the present epoch, z = 0, and for masses m ≤ T γ , one has the maximum value of ρ F at T = T γ , which is ρ F ∼ 1.6 × 10 −51 GeV 4 . In this case, the state equation is w F ∼ 1/3. We note that only the thermal vacuum condensate of sub-eV massive particles, like the neutrinos, is relevant, since the condensate of more heavy fermions give negligible contributions to the energy of universe.
We consider now the relic neutrino temperature T ν = 1.95K and we study the energy density of the thermal vacuum for cosmic neutrino background (CNB). We use massive Dirac neutrinos. Let |0(θ, z) denote the vacuum condensate of the neutrino modes with temperature T ν . We derive the absolute neutrino masses from the plausible value of the CNB energy.
In Eqs. (10) and (11), if one considers T = T ν and the sum on the three neutrino fields with masses m i , one has only small variations from the above results. Indeed, for z = 0 and m i ∼ 10 −4 eV , we find the maximum value of the energy density ρ ν ∼ 10 −51 GeV 4 , and the state equation is w ν ∼ 1/3. We assume ρ ν ≤ ρ γ , and the mass of the lighter neutrino m ν,1 ∼ 10 −3 eV [3]. Considering the hierarchical neutrino model, according to which ∆m 2 12 = 8 × 10 −5 eV 2 and ∆m 2 23 = 2.7 × 10 −3 eV 2 , one can derive the following values of the other neutrino masses: m ν,2 = 9 × 10 −3 eV and m ν,3 = 5.3 × 10 −2 eV . Then, the sum of the three neutrino masses is m ν = 6 × 10 −2 eV , which is in agreement with its estimated lower bound. Notice that the contributions to the energy density ρ ν given by ν 2 and ν 3 are negligible.

Fractal structure of the thermal vacuum
In this Section we show the fractal self-similar structure of the thermal vacuum |0(θ) considered above for the CMB and CNB [3]. Consider the boson case. In full generality, we consider timedependent θ: θ = θ(t). We use the notation |0(t) ≡ |0(θ(t)) . As shown in [29], |0(t) provides the quantum representation of the system of couples of damped/amplified oscillators mẍ + γẋ + kx = 0, (12) 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. (12) and (13) satisfying the relations [ α, . As shown in [29], the proper quantization setting is the one of the quantum field theory (QFT) and the Hamiltonian H of the system is H . In this case, the parameter θ κ (t) of the thermal vacuum is θ κ (t) = Γ k t ≡ (γ k /2m) t for each κ-mode [29]. The time evolution of the vacuum |0 for a k and b k is driven by H I and given by |0(θ(t)) = e −it H |0 = e −it H I |0 , which leads to the explicit expression for |0(θ(t)) similar to the one in Eq. (1) for time dependent β. The free energy for the a−modes is defined by where H a is the free Hamiltonian for the a−modes, H a = k Ω k a † k a k , and S a is the entropy: For β(t) slowly varying in time, the Bose-Einstein distribution function Eq. (2) is derived by minimizing F a . Therefore, |0(t) provides a finite temperature representation of the CCR which is equivalent to the Thermo Field Dynamics representation {|0(β) }, with time dependent β. We now remark that Eqs. (12) and (13) describe a system which has self-similarity properties. We indeed note that solutions of Eqs. (12) and (13) are the parametric time evolution of clockwise and anti-clockwise logarithmic spirals. Indeed, by defining x(t) ≡ [z 1 (t) + z * 2 (−t)]/2 and y(t) ≡ [z * 1 (−t) + z 2 (t)]/2, Eqs. (12) and (13) can be rewritten as [30] mz 1 + γż 1 + κ z 1 = 0, whose solutions are the clockwise and anti-clockwise logarithmic spirals z 1 (t) = r 0 e − i Ω t e −Γt and z 2 (t) = r 0 e + i Ω t e +Γ t , respectively, with Γ ≡ γ/2m and Ω 2 = (1/m)(κ−γ 2 /4m), κ > γ 2 /4m [30,31]. Therefore, Eqs. (12) and (13), whose quantum representation is given by |0(t) , describe the self-similar fractal structure of their logarithmic spiral solutions [32,33]. The link between SU (1, 1) coherent states, fractal-like self-similarity and CMB is thus established [3]. A similar result may be obtained in the fermion case for CNB by considering that the su(2) algebra contracts in the infinite volume limit to the e(2) algebra (as it can be seen by considering the Holstein-Primakoff non-linear boson realization [34,35,36]), which is isomorph to the Weyl-Heisenberg algebra.

Conclusion
We have shown that the structure of CMB is characterized by the arrow of time and by coherent, thermal and fractal-like self-similar properties. The observed black body spectrum of CMB is confirmed by the results here presented. Moreover, by analyzing the thermal vacuum energy for neutrinos and considering the neutrino hierarchical model, we have derived the value of the neutrino masses which are in agreement with estimated lower bounds. An interesting question to ask is on a possible relation between the fractal self-similarity structures considered in this paper and the recently reported observations on the holographic structure of the Universe [26].