# Thermal Condensate Structure and Cosmological Energy Density of the Universe

Authors: Antonio Capolupo; Gaetano Lambiase; Giuseppe Vitiello
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

The aim of this paper is to study 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, nontrivial 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.


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## Body

## 1. 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 structure. 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 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 1.95 K [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 ∑mν, which has been estimated from the neutrinos oscillations to be of the order of 0.06 eV [10].We finally discuss the fractal self-similar structure of the thermal vacuum.In Section2, 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 presented in Section 3 and, in Section 4, the fractal structure of the thermal states is analyzed. Section 5 is devoted to the conclusions.


## 2. Thermal Vacuum and Particle Condensate

The thermal vacuum state0θ, with θ=θβ, β≡1/(kBT), and kB, the Boltzmann constant, is introduced in the TFD formalism [11, 12] in such a way that the thermal statistical average Nak(θ) is given by Nak(θ)=0θ|Nak|0θ, with Nak=ak†ak, the number operator. The bosonic operators ak and ak† have usual canonical commutation relations (CCR).The explicit form of0θ is(1)0θ=∏k1cosh⁡θkexp⁡tanh⁡θkak†bk†0,and it is recognized to be a two-mode time dependent generalized SU(1,1) coherent state [13, 14], condensate of pairs of ak and bk quanta. 0 is the vacuum annihilated by ak and bk. The auxiliary boson operator bk commutes with ak and is introduced in order to produce the trace operation in computing thermal averages. The thermal vacuum |0(θ)〉 is normalized to one, 〈0(θ)|0(θ)〉=1, ∀θ, and in the infinite volume limit 0θβ|0→0 as V→∞, ∀β (for ∫d3κθκ finite and positive).One also has0θβ|0θβ′→0 as V→∞, ∀β and β′, β′≠β. Thus, {|0θβ〉} provides a representation of the CCR defined at each β and unitarily inequivalent ∀β′≠β to any other representation {|0β′〉} in the infinite volume limit.Note thatak and bk do not annihilate the state 0θ. The annihilation operators, say Ak(θk) and Bk(θk), for |0(θ)〉, Akθk0θ=0=Bkθk0θ, are obtained through the Bogoliubov transformation(2)Akθk=eiθkGake-iθkG=akcosh⁡θk-bk†sinh⁡θk,Bkθk=eiθkGbke-iθkG=bkcosh⁡θk-ak†sinh⁡θk,whose generator G is given by G=-i∑kak†bk†-akbk. The thermal vacuum expectation value of the number operator Nak=ak†ak is given by(3)Nakθ=0θak†ak0θ=sinh2θk.Minimization of the free energy (see below) then leads to the thermal statistical average ofNak(4)NakBθ=sinh2⁡θk=1eβΩk-1,which is indeed the Bose-Einstein distribution function for ak.Summing up, the “thermal background” atT is described by the quantum coherent condensate vacuum 0θ, which is the thermal physical vacuum.We now are ready to compute the contributions of the energy momentum tensorTμν to the thermal vacuum for Maxwell, scalar, and fermion fields. We observe that the off-diagonal terms of Tμν on the vacuum state are zero for these fields; that is, 0θTijx0θ=0, for i≠j. Therefore, the vacuum condensate is homogenous and isotropic and behaves as a perfect fluid (similar result holds 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 0θ,z:(5)ρz=g000θ,z:T00x:0θ,z,ρz=gjj0θ,z:Tjjx:0θ,z.Here,:⋯: denotes the normal ordering with respect to 0 and no summation on the index j is intended.


## 3. Energy Density of Thermal Vacuum and CMB Temperature

In the photon fields case, the explicit expression of the energy momentum tensor densityTγμν is Tγμν=-FμαFαν+(1/4)gμνFαβFαβ [21, 22]. As usual, Fαβ=∂βAα-∂αAβ (gμν=(1,-1,-1,-1), μ=0,1,2,3; ħ=1=c will be used throughout the paper). The thermal vacuum condensate energy density is then(6)
ρ
γ
z
=
∫
d
3
k
Ω
k
0
θ
,
z
:
a
k
†
a
k
:
0
θ
,
z
,where Ωk=k for photons. The result we obtain is(7)ργz=π2kB41+z4Tγ415ħ3c3.In a similar way, the contribution given to the pressure by the thermal vacuum condensate of photons field is(8)pγz=π2kB41+z4Tγ445ħ3c3.The equation of state is then wγ(z)=pγ(z)/ργ(z)=1/3, which is the equation of state of the radiation. Equations (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, Tγ=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, ργ=2×10-51 GeV4, 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 momentum tensor density is given byTBμν(x)=∂μϕ(x)∂μϕ(x)-(1/2)gμν(∂ρϕ(x)∂ρϕ(x)-m2ϕ(x)2) for free real scalar fields ϕ and Tμν=(i/2)ψ¯γμ∂ν⃡ψ for free Majorana spinor fields ψ.At any epoch, the thermal vacuum energy and thermal pressure are given by (5), which in the case of the field ϕ give(9)ρB=120θ,z:π2x+∇→ϕx2+m2ϕ2x:0θ,z;pB=0θ,z:∂jϕx2+12π2x-∇→ϕx2-m2ϕ2x:0θ,z.In the case of the isotropy of the momenta k1=k2=k3, these can be written as(10)ρB=∫d3k2π3Ωk0θ,zak†ak0θ,z;(11)
p
B
=
∫
d
3
k
2
π
3
1
3
k
2
Ω
k
0
θ
,
z
a
k
†
a
k
0
θ
,
z
-
1
3
k
2
Ω
k
+
1
2
m
2
Ω
k
0
θ
,
z
a
k
a
-
k
e
-
2
i
Ω
k
t
+
a
k
†
a
-
k
†
e
2
i
Ω
k
t
0
θ
,
z
.Explicitly, they become (12)ρBz=12π2∫0∞dkk2Ωkexp⁡Ωk/kBTγ1+z-1,(13)pBz=16π2∫0∞dkk2k2Ωk1exp⁡Ωk/kBTγ1+z-1-k2Ωk+3m22Ωkexp⁡Ωk/2kBTγ1+zexp⁡Ωk/kBTγ1+z-1cos⁡2Ωkt.Notice that the vacuum energy density at thermal equilibriumρB(z), (12), coincides with the result obtained by solving the Boltzmann equation for the particle Bose distribution function [5]. The difference to the pressure pB(z) between the contributions coming from the vacuum condensate and the ones coming solely from the Bose distribution function appears in (13). The second term on the RHS of (13) appears due to the condensate of the physical vacuum contributing with nonvanishing values of 〈0(θ,z)|aka-k|0(θ,z)〉 and 〈0(θ,z)|ak†a-k†|0(θ,z)〉. Should the vacuum be the trivial one |0〉, these contributions would be identically zero. To our knowledge, the presence of the time dependent term of RHS of (13) is a new result which has not been reported till now in the literature.By considering the present epoch,z=0, T=Tγ, and by solving numerically the integral in (12), one has the contribution to the vacuum energy given by ρB≃9×10-52 GeV4 for masses less than or equal to the CMB temperature m≤Tγ; that is, m≤2.3×10-4 eV (e.g., possible candidates are axion-like with ma∈(10-3–10-6) eV). The maximum value of ρB is obtained for m≪10-4 eV. In this case, one has ρB≃10-51 GeV4. Negligible values of ρB are obtained for boson masses m≫10-3 eV.In the fermion case, the Fermi-Dirac distribution function is obtained:(14)NakFθ=sin2⁡θk=1eβΩk+1.The thermal vacuum contribution to the energy density and to the pressure is(15)ρF=120θ,z:-iψ¯γj∂jψ+mψ¯ψ:0θ,z;(16)pF=0θ,z:i2ψ¯γj∂j⃡ψ:0θ,z,respectively. In (15), the relation (i/2)ψ¯γ0∂0⃡ψ=iψ¯γ0∂0ψ=-iψ¯γj∂jψ+mψ¯ψ is used. For Majorana fields, (15) and (16) give (17)ρF=∑r∫d3k2π3Ωk0θ,zαkr†αkr0θ,z;(18)pF=13∑r∫d3k2π3k2Ωk0θ,zαkr†αkr0θ,z, where αkr,  r=1,2, is the annihilator of fermion field.The explicit expressions of the energy density and pressure are(19)ρFz=1π2∫0∞dkk2Ωkexp⁡Ωk/kBTγ1+z+1,(20)pFz=13π2∫0∞dkk4Ωk1exp⁡Ωk/kBTγ1+z+1,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≤Tγ, we find at T=Tγ the maximum value of ρF, that is, ρF~1.6×10-51 GeV4, which is of the same order of CMB energy. The state equation is wF~1/3. Condensates of heavier fermions give negligible contributions to the universe energy. Indeed, the number density of nonrelativistic particles (m≫Tγ) is exponentially suppressed by insufficient pair production due to low temperature. Only the condensates of fields with masses less than or equal to 10-4 eV, for example, neutrinos, may give relevant contributions.Taking into account such results, from (19) and (20), we compute the energy density and pressure for the thermal vacuum of the three neutrino fields at the cosmic neutrino background (CNB) temperature Tν=1.95K.1Forz=0 and neutrino masses mi~10-4 eV, the maximum value of the energy density turns out to be ρν~0.5×10-51 GeV4, with state equation wν~1/3. The thermal condensate of neutrinos with larger masses would give negligible contributions to ρν. Adopting as customary ρν≤ργ and taking the mass mν,1~10-4 eV (which leads to ρν≤ργ) by the lighter neutrino mass, one can derive mν,2 and mν,3 from the hierarchical neutrino model and Δm122=8×10-5 eV2 and Δm232=2.7×10-3 eV2. The result is mν,2=9×10-3 eV and mν,3=5.3×10-2 eV, and thus ∑mν=6×10-2 eV, as it should be in agreement with its estimated lower bound.We stress that in our discussion we consider the contribution of the condensate neutrino couples to the background energy. In this case, the condensate of neutrinos with larger masses gives negligible contributions to the energy density. Considering instead the particle contribution to the energy budget of the universe, a higher contribution should be obtained by nonrelativistic neutrinos (mν≫Tγ), with larger mass.


## 4. Fractal Structure of the Thermal States

Finally, we show that the thermal vacuum0θ has a fractal self-similar structure. Let us consider the time dependent case θ=θ(t). We will use the notation 0θt≡0t. The boson vacuum 0t provides the quantum representation of the system of couples of damped/amplified oscillators [23](21)mx¨+γx˙+κx=0,(22)my¨-γy˙+κy=0,(23)L=mx˙y˙+γ2xy˙-x˙y-κxy,where “dot” denotes time derivative, m, γ, and κ are positive real constants, and L is the Lagrangian from which (21) and (22) are derived.To see indeed how0t is obtained, one proceeds to the canonical quantization of the system described by (21)–(23) and assumes that the canonical commutation relations hold, x,px=iħ=y,py, x,y=0=px,py. The corresponding sets of annihilation and creation operators are (24)α≡12ħΩ1/2pxm-imΩx;α~≡12ħΩ1/2pym-imΩy,with [α,α†]=1=[α~,α~†],  [α,α~]=0=[α,α~†]. The canonical linear transformations a≡(1/2)(α+α~), b≡(1/2)(α-α~) are introduced. 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 ak and bk and their hermitian conjugates, so as to perform, as customary in QFT, the continuum momentum limit (or the infinite volume limit) by use of the relation ∑k→(V/(2π)3)∫d3k at the end of the computations. The Hamiltonian H of the system is found to be [23] H=H0+HI, with(25)H0=∑kħΩkak†ak-bk†bk,HI=i∑kħΓkak†bk†-akbk,where θk(t)=Γkt≡(γk/2m)t has been used for each k-mode. The group structure is the one of SU(1,1), [H0,HI]=0, and the Casimir operator C is given by C2=(1/4)(ak†ak-bk†bk)2. The initial condition of positiveness for the eigenvalues of H0 is thus protected against transitions to negative energy states. One then finds that the time evolution of the vacuum 0 for ak and bk is controlled by HI and given by 0θt=e-it(H/ħ)0=e-it(HI/ħ)0 which gives in fact (1).One also finds that0t 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 equation (3).Indeed, let us now introduce the functionalFa for the a-modes (26)Fa≡0tHa-1βSa0t,where Ha is the free Hamiltonian relative to the a-modes, Ha=∑kħΩkak†ak, and Sa is given by (27)Sa≡-∑kak†akln⁡sinh2⁡θ-akak†ln⁡cosh2⁡θ.Inspection of (26) and (27) suggests that Fa and Sa can be considered as free energy and the entropy, respectively. Minimization of the functional Fa, ∂Fa/∂θkt=0, ∀k [11, 12] (we consider ħ=c=1), then leads to (4) which is the Bose-Einstein distribution function for ak. The first principle of thermodynamics at constant temperature can be then expressed as (28)dFa=dEa-1βSa=0, where the change in time of the particle condensed in the vacuum turns out to heat dissipation dQ=(1/β)dS: (29)dEa=∑kħΩkN˙aktdt=1βdS=dQ, where N˙ak(t) denotes the time derivative of Nak(t).We now remark that the system of (21) and (22) possesses self-similarity properties. To see this, let us put (30)12z1t+z2∗-t=xt,12z1∗-t+z2t=ytwith z1(t)=r0e-iΩte-Γt and z2(t)=r0e+iΩte+Γt, Γ≡γ/2m and Ω2=(1/m)(κ-γ2/4m), κ>γ2/4m. Then, we see that (21) and (22) can be rewritten as [24](31)mz¨1+γz˙1+κz1=0,(32)mz¨2-γz˙2+κz2=0.Solutions of (31) and (32) are in fact z1(t)=r0e-iΩte-Γt and z2(t)=r0e+iΩte+Γt and they describe the parametric time evolution of clockwise and the anticlockwise logarithmic spirals, r=r0e-dα and r=r0edα, with α(t)=Γt/d and Ωt=Γt/d=α(t) [24].Thus, (21) and (22) (or equivalently (31) and (32)), 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 establishes the link between the SU(1,1) coherent states and fractal-like self-similarity [24]. The relation of the photon energy momentum tensor Tγμν with (21) and (22) can also be shown. For details, see [24]. Similar discussions can be done for the fermion vacuum.


## 5. Conclusions

We have studied the thermal vacuum structure at the temperature of the CMB. In the framework of TFD, 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.



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