Massive particle pair production and oscillation in Friedman Universe: its effect on inflation

We study the classical Friedman equations for the time-varying cosmological term Λ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Lambda }$$\end{document} and Hubble function H, together with quantised field equations for the production of massive M≫H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\gg H$$\end{document} particles, namely, the Λ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Lambda }$$\end{document}CDM scenario of dark energy and matter interactions. Classical slow components O(H-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal O}(H^{-1})$$\end{document} are separated from quantum fast components O(M-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal O}(M^{-1})$$\end{document}. The former obeys the Friedman equations, and the latter obeys a set of nonlinear differential equations. Numerically solving equations for quantum fast components, we find the production and oscillation of massive particle-antiparticle pairs in microscopic time scale O(M-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal O}(M^{-1})$$\end{document}. Their density and pressure averages over microscopic time do not vanish. It implies the formation of a massive pair plasma state in macroscopic time scale O(H-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal O}(H^{-1})$$\end{document}, whose effective density and pressure contribute to the Friedman equations. Considering the inflation driven by the time-varying cosmological term and slowed down by the massive pair plasma state, we obtain the relation of spectral index and tensor-to-scalar ratio in agreement with recent observations. We discuss the singularity-free pre-inflation, the CMB large-scale anomaly, and dark-matter density perturbations imprinting on power spectra.


Introduction
The Universe's evolution is gravitationally governed by matter and dark energy.The latter can be represented by the cosmological Λ term in the Einstein equation.In addition to the mystery of its origin, people have not yet fully understood dark energy properties in Universe evolution.In particular, how dark energy and matter interact with each other in Universe evolution and why their present values are coincidentally at the same order of magnitude.Such interacting dark energy can be simply represented by a time-varying cosmological Λ(t) term in the Einstein equation or other modifications.Many theoretical ideas have been motivated for cosmology, and advocated to examine the H 0 tension recently observed Refs. .Here, we attempt to present a theoretical scenario to explain the dark energy and matter interaction by gravitational production and oscillation of particle-antiparticle pairs via quantum back and forth reaction processes between dark energy Λ(t) and massive pairs M H.The gravitational particle production in Friedman Universe expansion is an important theoretical issue [22][23][24][25] that has been intensively studied for decades [26][27][28][29][30]. Based on adiabaticity and non-back-reaction approximation for a slowly time-varying Hubble function H(t), one adopted the semi-classical WKB approaches to calculating the particle production rate, which is exponentially suppressed e −M/H for massive particles M H.However, the non-adiabatic back-reactions of particle creations on the Hubble function can be large and have to be taken into account.The non-adiabatic back-reactions of massive particle productions have a quantum time scale O(1/M ) that is much smaller than classical Universe evolution time scale O(1/H).To properly include the back-reaction of particle production on Universe evolution, one should separate fast components O(1/M ) from slow components O(1/H) in the Friedman equation.Many efforts [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] have been made to study non-adiabatic back-reaction and understand massive particle productions without exponential suppression.It is important for reheating, possibly accounting for massive dark matter and total entropy of the present Universe .
In this article, we start with the Friedman equations for a flat Universe where energy density ρ ≡ ρ M + ρ Λ and pressure p ≡ p M + p Λ .Equation of state p Λ = −ρ Λ is for the cosmological term (dark energy), p M = ω M ρ M for the matter terms representing relativistic (radiation) and/or non-relativistic components.The second equation is a gernalised conservation law for time-varying cosmological term ρ Λ (t) ≡ Λ(t)/8πG [67], and it reduces to the usual equation ρM + (1 + ω M )Hρ M = 0 for time-constant ρ Λ .We adopt the approach [42] to describe the decomposition of slow and fast components: scale factor a = a slow + a fast , Hubble function H = H slow + H fast , cosmological term and matter density ρ Λ,M = ρ slow Λ,M + ρ fast Λ,M and pressure p Λ,M = p slow Λ,M + p fast Λ,M .The fast components vary much faster in time, but their amplitudes are much smaller than the slow components.According to the order of small ratio λ of fast and slow components, the Friedman equations (1.1) are decomposed into two sets.The slow components O(λ 0 ) obey the same equations as usual Friedman equations where H slow = ȧslow /a ≈ ȧslow /a slow , time derivatives Ḣslow and ȧslow relate to the macroscopic "slow" time variation scale O(1/H).The faster components O(λ 1 ) obey, where H fast = ȧfast /a ≈ ȧfast /a slow , time derivatives Ḣfast and ȧfast relate to the microscopic "fast" time variation scale O(1/M ), and slow components are approximated as constants in "fast" time variation.For the cosmological term, equation of state Λ respectively at order O(λ 0 ) and O(λ 1 ).In due course we shall clarify the equation of state p M = ω M ρ M for the matter term.
We adopt the approach [31] to describe the fast components of matter density ρ fast M and pressure p fast M , that are attributed to the non-adiabatic production of particle and antiparticle pairs in fast time variation H fast = ȧfast /a slow .As a result, we find quantum pair production and oscillation and a macroscopic state of massive pair plasma.In radiation-and matter-dominated epochs after reheating, we study how it affects the Friedman equation (1.2) and introduces the interaction of dark-energy and matter densities.We show that the matter has converted to dark energy, and their present values are comparable, explaining the cosmic coincidence.

Quantum pair production and oscillation
A quantised massive scalar matter field inside the Hubble sphere volume V ∼ H −3 slow of Friedman Universe reads which exponentially vanishes outside the horizon H −1 slow , and The principal quantum number "n = 0, 1, 2, • • •" stands for for quantum states of physical wave vectors k n , n = 0 and k 0 = 0 for the ground state 1 .The A n and A † n are time-independent annihilation and creation operators satisfying the commutation relation and Wronskian-type condition in terms of α n (t) and β n (t), Equation (2.2) becomes and In an adiabatic process for slowly time-varying H = H slow , the particle state α n (0) = 1 and β n (0) = 0 evolve to |α n (t)| 1 and |β n (t)| = 0. Positive and negative frequency modes get mixed, leading to particle productions of probability |β n (t)|2 ∝ e −M/H slow .
We will focus on studying particle production in non-adiabatic processes for rapidly time-varying H fast , α n and β n in the ground state n = 0 of the lowest lying massive mode M H. First, we recall that Parker and Fulling introduced the transformation [31], [B, B † ] = 1, and two mixing constants obeying |γ| 2 − |δ| 2 = 1.For a given A n and its Fock space, the state |N pair is defined by the conditions A n =0 |N pair = 0 and The B † and B are time-independent creation and annihilation operators of the pair of mixed positive frequency A 0 particle and negative frequency A † 0 antiparticle.The state |N pair contains N pair = 1, 2, 3, • • • pairs, and |N pair = 0 is the ground state of non-adiabatic interacting system of fast varying H fast and massive pair production and annihilation2 .It is a coherent superposition of states of particle and anti-particle pairs.In this coherent condensate state |N pair and N pair 1, neglecting higher mode n = 0 contributions, they obtained the negative quantum pressure and positive quantum density of coherent pair field, see Eqs. ( 59) and (60) of Ref. [31], ) where Following their approach for the ground state n = 0, we arrive at the same quantum pressure (2.7) and density (2.8).We consider the state (2.6) as a coherent condensate state of very massive M H slow and large number N pair 1 pairs, and M (2N pair + 1) in (2.7) and (2.8) can be larger than the Planck mass m pl so that higher mode (n = 0) contributions can be neglected.Moreover, we adopt (2.7) and (2.8) as the fast components ρ fast Here, we study the epochs after reheating, when the Hubble scale and pair mass are very much smaller than the Planck mass, i.e., H slow < M m pl and N pair 1.Therefore, for a given H slow , we express in unit of the critical density ρ crit = 3m 2 pl H 2 slow the dimensionless quantum pressure (2.7) and density (2.8) as ) where M ≡ (2N pair + 1)(M/m pl ) and the reduced Planck mass m pl ≡ (8πG) −1/2 .The faster component equations (1.3) become, m pl and M m pl , a large amount of massive pairs N pair 1 is created for significant oscillating quantum pressure (2.9) and density (2.10).For details see Fig. 3

2). The integrals
t ω n dt are over the microscopic time t characterised by the Compton time scale 1/M .Its lower limit is t = 0 by setting t slow = 0 as a reference time, when a fast (0) = 0, (2.12) The real value γ * δ condition in Eqs.(2.9) and (2.10) leads to the time symmetry: [31].When t ↔ −t, positive and negative frequency modes interchange.Here we use a slow = 0, H slow = 0 and co-moving radius (Ha) −1 ≈ (H slow a slow ) −1 of Hubble volume V ∼ H −3 slow .In microscopic time t of unit M −1 , we numerically solve coupled Eqs.(2.4) and (2.9-2.11) with the initial condition (2.12).Figure 1 shows results for C 0 = (3/2)h fast (H slow /M ) and verified condition |α| 2 − |β| 2 = 1.In the quantum period of microscopic time t, the negative quantum pressure P fast > 0 is positive definite, leading to the decreasing h fast (t) (1.3).As a consequence, for time t > 0, the fast components h fast and fast Λ decrease in time, in order for pair production.Whereas for time t < 0, h fast and fast Λ increases, due to pair annihilation.The small a fast (t) varies around a slow at t slow ≡ 0.
The quantum pair oscillation phenomenon is dynamically analogous to the plasma oscillation of electron-positron pair production in an external electric filed E [68] and pair production rate is not exponentially suppressed by e −πM 2 /E [69].The coherent plasma state of electron-positron pairs is analogous to the coherent pair state |N pair (2.6).

Massive pair plasma state
As shown in Fig. 1, massive pair quantum pressure P fast M (2.9) and density fast M (2.9) can be significantly large and rapidly oscillate with the fast components h fast and fast Λ (2.11) in microscopic time and space.Their oscillating amplitudes are not dampen in time, and it is therefore expected to form a massive pair plasma state in a long macroscopic time.However, to study their effective impacts on the classical Friedman equations (1.2) evolving in macroscopic time and space, we have to discuss two problems coming from scale difference M H slow .First, it is impossible to even numerically integrate slow and fast component coupled equations (1.2,1.3)due to their vastly different time scales.On this aspect, we consider their non-vanishing averages • • • over the microscopic period in time.Figure 1 shows fast and other averages of fast oscillating components do not vanish.Second the spatial dependence of pair quantum pressure P fast (2.7) and density fast (2.7) are unknown, since they are obtained by using the vacuum expectation value of field Φ(x, t) energy-momentum tensor over entire space.For the case M H slow , the Compton length M −1 of ground state n = 0 is much smaller than the Hubble horizon H −1 slow .Therefore, the massive coherent pair state (2.6-2.8) and quantum plasma oscillation of Fig. 1 well localise inside the Hubble sphere.We speculate that their location should be nearby the horizon because of isotropic homogeneity extending up to the horizon.
Based on these considerations and non-vanishing averages of fast oscillating components (Fig. 1) over macroscopic time, we assume the formation of massive pair plasma state in macroscopic time scale.We describe such macroscopic state as a perfect fluid state of effective number n H M and energy ρ H M densities as, and pressure p The ω H M ≈ 0 for m H slow and its upper limit is 1/3.The introduced mass parameter m represents possible particle masses M f , degeneracies g f d and the mixing coefficient δ (2.5).The degeneracies g f d plays the same role of pair number N pair in Eq. (2.8), namely f g f d ≈ (2N pair +1).We explain the reasons why the densities (3.1) are proportional to χmH 2 slow , rather than H 3 slow from the entire Hubble volume V .The "surface area" factor H 2 slow is attributed to the spherical symmetry of Hubble volume.The "radial size" factor χm comes from the layer width λ m introduced as an effective parameter to describe the properties (i) for m H slow the massive pair plasma is localised as a spherical layer and (ii) its radial width λ m < H −1 slow depends on the massive pair plasma oscillation dynamics 3 , rather than the H slow dynamics govern by the Friedman equations (1.2).The width parameter χ expresses the layer width λ m = (χm) −1 1/m in terms of the effective Compton length 1/m, Because parameters m and χm represent time-averaged values over fast time oscillations of massive pair plasma state, we consider m and χm as approximate constants in slowly varying macroscopic time.However, the m and χm values, namely the M f and g f d values (3.1) cannot be unique in entire Universe evolution, and should depend on Universe evolution epochs.One of the reasons is the fast-component equations for massive pair productions and oscillations depend on the H slow value, see Sec. 2. We will duly come back to this point how characteristic value χm relates to the Hubble function H slow in a given evolution epoch.
We have to point out that (i) the pressure p H M and density ρ H M (3.1) are effective descriptions of the massive pair plasma state in macroscopic scales, that may result from the coherence condensation state (2.6,2.7,2.8) and oscillating dynamics (Fig. 1) in microscopic scales; (ii) they play the role of "slow" components contributing to the Friedman equations (1.1) or (1.2).It means that in the Friedman equations, there are two sets of the matter: (i) the normal matter state of pressure and density p M = ω M ρ M and (ii) the massive pair plasma state of pressure and density p H M = ω H M ρ H M .These two sets interact with each other, shown below.We shall study the massive pair plasma state effects on each epoch of Universe evolution.Here we start to study its effects on the epoch after reheating.Henceforth sub-and super-scripts "slow" are dropped.

Cosmic rate equation
Up to macroscopic time H −1 , we estimate the total number of particles produced inside the Hubble sphere N ≈ n H M H −3 /2 and mean pair production rate w.r.t.macroscopic time where n e + e − (t) is the density governed by macroscopic evolution and n e + e − eq is the density in an equilibrium with photons.The RHS represents the averaged interacting rate dN/dt ≈ σv n e + e − for microscopic detail balance between n e + e − and n e + e − eq .They are coupled for n e + e − eq ≈ n e + e − and decoupled n e + e − eq n e + e − .This motivates us to propose an effective cosmic rate equation, Equations (4.4) and (4.5) is reminiscent of a generally modeling interacting dark energy and matter δQ = Γ M (ρ H M − ρ M ), based on the total mass-energy conservation, see review [74,75] and [76][77][78].It shows that the cosmological constant (dark energy) ρ Λ and matter energy ρ M interact via the massive pair plasma ρ H M produced by massive particle production and oscillation in the Friedman space.Two cases: (i) dark energy converts to matter energy when ρ H M > ρ M and (ii) matter energy converts to dark energy when ρ H M < ρ M .Equations (1.2) and (4.4) are a set of first-order ordinary differential equations, numerical solutions for ρ M and ρ Λ can be studied, provided that initial or transition conditions from one epoch to another are known.In this article, we approximately find asymptotic solutions of specific epochs to gain a qualitative insight into how dark energy and matter interact in Universe evolution.

Radiation and matter dominate epochs
Suppose that all radiation ρ R and matter ρ M densities were created in the reheating epoch, and they were much larger than the dark energy density ρ Λ 4 .The standard cosmology started with the radiation dominated epoch and proceeded to the matterdominated epoch.We adopt an analytical way to reveal approximate ρ Λ − ρ R and ρ Λ − ρ M relations.
As a result, the asymptotic solution (5.4) shows that ρ Λ linearly tracks down (follows) ρ R .Here we adopt the terminology "track down" used in the discussions of Ref. [86].Such ρ Λ − ρ R tracking dynamics continues in the entire radiation epoch.We will show that the tracking dynamics ends and ρ Λ becomes positive during a continuous transition period from radiation epoch to matter epoch.We use an analytical approach to asymptotic solutions in different epochs.Therefore we cannot precisely determine the transition period.Therefore, we introduce the scale factor a tr to characterize the transition time scale, and discuss two extremal cases: (i) a tr ∼ a eq transition occurs at the radiation-matter equilibrium moment; (ii) a tr > a eq transition occurs at sometime around/after the last scattering surface.
More details of the transition behave and period need numerical studies of massive pair plasma (3.1,4.1),Friedman equation (1.2) and cosmic rate equation (4.4).
In the matter dominate epoch, we identify ρ M → ρ M and ω M → ω M ≈ 0 in Eqs.(4.4) and (4.5).Analogously to the approach in radiation epoch, neglecting darkenergy and radiation-energy density, H 2 ≈ ρ M /(3m 2 pl ), and ρ H M ≈ χ m2 M ρ M , we recast Eqs.(4.4) and (4.5) as ) where mM ≡ (2/3)m M /m pl and χ m2 M < 1.Here we introduce the average mass parameter m M and rate Γ M /H M over the matter epoch, assuming they vary much slowly than ρ M .The asymptotic solutions are The coupling parameter γ M < 0 (|γ M | < 1) represents the ρ Λ − ρ M interaction and ρ M conversion into ρ Λ .Here we adopt the case (i) a tr ∼ a eq for discussions.Namely, the ρ Λ − ρ R tracking continues until the Universe reaches the radiation-matter equilibrium ρ eq M = ρ eq R at (a eq /a R ) = (T RH /T eq ) ∼ 10 15 GeV/10 eV ∼ 10 23 , where T RH (T eq ) is the reheating (equilibrium) temperature.The initial value ρ eq M is given at the radiationmatter equilibrium ρ eq M = ρ eq R at a = a eq , where the solutions (5.4) and (5.8) should match, yielding where ρ 0 M and ρ 0 Λ are the values at the present time a 0 = (1 + z) ∼ 10 4 a eq .The solution (5.8) shows that the term (γ M /3)ρ M decreases as a −3 , ρ Λ fails to track down ρ M , and becomes positive value approaching to the constant ρ 0 Λ ≈ Ceq Λ .These results (5.7,5.8,5.9)depend on the transition period from radiation to matter epoch.As for the case (ii) that ρ Λ − ρ R tracking dynamics ends and ρ Λ becomes positive value at sometime a tr around/after the last scattering surface.Discussions and results are similar with substitutions: a eq → a tr , ρ eq M → ρ tr M and Ceq Λ → Ctr Λ in Eqs.(5.7-5.9).To end this section, we mention ρ Λ -dominate epoch in future, when H 2 ≈ ρ Λ /(3m ) in the cosmic rate equation (4.4) or (4.5), asymptotic solutions are slowly time varying (5.12) It shows that dark energy decreases in time and converts to matter, and the latter tracks down the former.

Cosmic coincidence of present dark and matter energies
To discuss the cosmic coincidence, we use the ratio ρ Λ /ρ M which is independent of the characteristic scales in different epochs.We separately discuss two extremal cases: (i) a tr ∼ a eq ∼ 10 4 a 0 or (ii) a tr ∼ 10 2 a 0 , when the ρ Λ − ρ M tracking ends and ρ Λ becomes positive.In radiation epoch, solution (5.4) shows the ratio ρ Λ /ρ R ≈ γ R /4 keeps constant, as ρ Λ tracks down ρ R from the reheating end a R to (i) the radiationmatter equilibrium a eq ∼ 10 23 a R or (ii) sometime after the last scattering surface a tr ∼ 10 25 a R .This tracking dynamics avoids the fine tuning cosmic ρ Λ and ρ R coincidence of the order of (i) (a eq /a R ) 4 ∼ 10 92 or (ii) (a tr /a R ) 4 ∼ 10 100 .Whereas, from the transition time (i) a tr ∼ a eq = (1 + z eq ) −1 a 0 ∼ 10 −4 a 0 or (ii) a tr ∼ (1 + z tr ) −1 a 0 ∼ 10 −2 a 0 to the present time a 0 , solutions (5.7) and (5.8) give This ratio consistently approaches the constant −γ R /4 when scale factor a traces back to the reheating end a R .Using (i) γ M ≈ γ R ∼ 10 −11 for the case a tr ∼ a eq ∼ 10 4 a 0 ; (ii) γ M ≈ γ R ∼ 10 −3 for the case a tr ∼ 10 2 a 0 , we plot in Fig. 2 the ratio ρ Λ /ρ R,M varying from −γ R /4 to O(1) as a function of the scale factor ln(a/a tr ) from the reheating to present time.It shows that the cosmic coincidence of the present ρ 0 Λ and ρ 0 M values appear naturally without any extremely fine-tuning their values at the transition time.Namely, in Eq. ??6.1) the ratio ρ Λ /ρ M ∼ (a/a tr ) 3 variation is about (i) O(10 −12 ) or (ii) O(10 −6 ), see the right column of Fig. 2. The reason is that the matter-dominated epoch of (i) z eq ∼ 10 4 or (ii) z tr ∼ 10 2 is much shorter than the radiation dominated epoch of (i) (a eq /a R ) ∼ 10 23 or (ii) (a tr /a R ) ∼ 10 25 , when the ρ Λ tracks down ρ R and the ratio ρ Λ /ρ R is a constant, see the left column of Fig. 2. Otherwise, to reach present ρ Λ and ρ M observational values of the same order of magnitude, we would have the cosmic coincidence problem of incredibly fine-tuning their reheating values ρ RH

Discussions
Massive pair productions and oscillations on the cosmic horizon lead to a massive pair plasma (3.1,4.1).It back reacts on Friedman equation (1.2) with matter ρ M and dark energy ρ Λ , via cosmic rate equation (4.4).As a consequence, matter and dark energy interact with each other in Universe evolution.The induced dark-energy and matter ρ Λ − ρ M interacting strength Γ M /H depends on evolution epochs.We study asymptotic solutions for radiation and matter epochs, starting from the reheating end.Because of different epoch transitions, ρ Λ − ρ M tracking dynamics proceeds in the radiation epoch and ends in matter one.As a result, a slowly varying dark-energy density is of the same order of matter-energy density today.We can avoid the extremal fine-tuning problem of cosmic coincidence.
Due to the lack of enough knowledge, we have not been able to determine the details of epoch transitions.However, asymptotic solutions (5.3), (5.7) and (5.12) show modified scaling laws in contrast with the counterparts of ΛCDM.Therefore we consider the following phenomenological model of dark energy and matter interaction.The Hubble function E(z) 2 = H 2 /H 2 0 can be parametrized Here energy densities ρ R,M,Λ are in units of the critical density ρ 0 c = 3m 2 pl H 2 0 today, and Ω R,M,Λ are the present values and Ω R + Ω M + Ω Λ = 1.Inserting E(z) (7.1) into the dark-energy and matter interacting equation (1.2), the dark energy term can be obtained as, 2) Equations (7.1) and (7.2) give a class of effective interacting dark energy models with two parameters δ M G and δ R G .These modified scaling laws (7.1) were also proposed from the view point that time-varying cosmological term Λ(t) and gravitational coupling G(t) obey scaling laws approaching to their present values (G, Λ), where Ricci scalar term R and cosmological term Λ of classical Einstein gravity are realized [67] in the spirit of Weinberg asymptotic safety [87] for the quantum field theory of gravity.Based on observational data, the model is examined and parameters are constrained in Refs.[88] and [89], showing it greatly relieves the H 0 tensions of the standard cosmology model ΛCDM.
We end this article with some remarks.In radiation dominate epoch, negative dark-energy density ρ Λ ≈ γ R m 2 pl H 2 (5.4) follows the "area law" ∝ H 2 .In matter dominate epoch, it changes sign at ρ Λ = 0 in Eq. (5.8), and approaches a positive constant ρ 0 Λ ≈ Ctr Λ (5.9).The dark energy undergoes these transitions and becomes dominant, converting to matter, and matter density ρ M , in turn, tracks down dark energy density (5.12).We speculate that such ρ Λ -transitions should induce the peculiar fluctuations of the gravitational field that possibly imprint on the CMB and matter spectrum, analogously to the integrated Sachs-Wolfe effect.

Supplemental Material: quantum pair oscillation details
In microscopic time, we plot the Bogoliubov coefficient |β| 2 , the quantum pair density ρ fast

M
and p fast M in Eq. (1.3) to find their non-adiabatic back-reactions on fast components H fast and ρ fast Λ .

Figure 1 .
Figure1.We show the quantum pair density and pressure oscillations in microscopic time t in unit of M −1 , using H slow /M ≈ 10 −3 , M 10 −10 m pl , N pair 10 25 and δ = 1.It is clear that for H slow m pl and M m pl , a large amount of massive pairs N pair 1 is created for significant oscillating quantum pressure (2.9) and density (2.10).For details see Fig.3in Supplemental Material.

M < 0
and back-reaction effects lead to the quantum pair oscillation characterised by the frequency ω = M of massive quantised pair fields.The positive quantum pair density fast M > 0 indicates particle creations without e −M/H suppression.It is consistent with increasing Bogoliubov coefficient |β(t)| 2 that mixes positive and negative energy modes.Observe that fast M |P fast M | and the sum fast M + P fastM

2 )
by using the new variable x = ln a and dx = Hdt.Because the radiation epoch is very long and the H varies a lot, the mass m (3.1) and width parameter χm ∝ H (3.2) vary as well, which we cannot go to details.Thus we introduce the average mass parameter m R = m R and average rate Γ M /H R over the entire radiation epoch, assuming they vary much slowly than ρ R and ρ Λ .The dimensionless average mass parameter mR ≡ (2/3)m R /m pl and χ m2 R < 1.The asymptotic solutions are

. 4 )
The dark-energy and matter coupling parameterγ R < 0 (|γ R | < 1) represents the ρ Λ − ρ R interaction and ρ R conversion to ρ Λ .The initial values ρ RHR and CΛ are given at the reheating end a = a R .In this article, to study dark energy and radiation interaction, we select the initial condition CΛ = 0, consistently with ρ RH Λ ∝ ρ RH R and ρ RH Λ ρ RH R at reheating end.The reasons are that the dark energy ρ Λ converts to massive pair plasma energy ρ H M (3.1)

Λ
and pressure p fast Λ , as well as the fast components of Hubble function H fast , and cosmological term ρ fast Λ .

Figure 3 .
Figure 3. Corresponding to Fig. 1, the details of quantum pair oscillation are shown in microscopic time t in unit of M −1 .The oscillatory |β(t)| 2 , h fast and fast Λ structures are too small to see.
in Supplemental Material.whereh fast ≡ H fast /H slow and fast Λ ≡ ρ fast Λ /ρ crit .Using negative P fast M (2.9) and positive definite fast M (2.10), we search for a solution of fast component equation (2.11) and quantum fluctuating mode equations (2.4) in the period [−t, t] of the microscopic time t ∼ H −1 fast , which is around the macroscopic time t slow ∼ H −1 slow , when the slow components a slow , H slow , ρ slow M,Λ and p slow M,Λ are valued, following the Friedman equations (1.
We turn to study how the massive pair plasma density interacts with the matter density ρ M that governs the Universe evolution, .1) It is not a theoretical derivation, but modelling parameterized by χm and Universe evolution rate .The asymptotic values ≈ 2 and ≈ 3/2 are respectively for radiation and matter epoch.Here we neglect the back-reactions of slow time-varying components H, ρ Λ,M and p Λ,M on fast components H fast , ρ fast M and p fast M .+ e − ⇔ γγ [70-73]: dn e + e − (t) dt + 3Hn e + e − (t) = σv n 2 e + e − eq − n 2 e + e − , .4) for the the back and forth ρ M and ρ H M interaction (4.2) in the Universe evolution.It actually represents a general conservation law of all matter including massive pair plasma density ρ H M (3.1) with the production rate (4.1).The term 3(1 + ω M )Hρ M of the time scale (3H) −1 represents the space-time expanding effect on the density ρ M .While Γ M ρ H M is the source term and Γ M ρ M is the depletion term.The time-varying horizon H and massive pair plasma state are coupled via the back and forth processes (4.2).The ratio Γ M /H > 1 indicates the coupled case, and Γ M /H < 1 indicate the decoupled case.