Massive particle pair production and oscillation in Friedman Universe: reheating energy and entropy, and cold dark matter

Suppose that the early Universe starts with a cosmological Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}-term (dark energy) originating from quantum spacetime at the Planck scale. Dark energy drives inflation and reheating by reducing its value for massive particle–antiparticle pairs production and oscillation, resulting in a holographic and massive pair plasma state. The back-and-forth reaction of dark energy and massive pairs slows inflation to its end and starts reheating by rapidly producing stable and unstable pairs. We introduce the Boltzmann-type rate equation describing the back-and-forth reaction. It forms a close set with Friedman equations and reheating equations for unstable pairs decay to relativistic particles. The numerical solutions show preheating, massive pairs dominated and genuine reheating episodes. We obtain the reheating temperature and entropy in terms of the tensor-to-scalar ratio 0<r<0.047\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< r < 0.047$$\end{document} consistently with observations. Stable massive pairs represent cold dark matter particles and weakly interact with dark energy. The resultant cold dark matter abundance Ωc∼10-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _c\sim 10^{-1}$$\end{document} is about a constant in time.


Introduction
In the standard model of modern cosmology (ΛCDM), the cosmological constant Λ, dark matter, inflation, reheating and coincidence problem have been long-standing basic issues for decades.The inflation [1][2][3][4][5][6][7] reheating [8][9][10][11][12][13][14][15][16][17] are fundamental processes.The latter transition the Universe from the cold and massive state left by inflation to the hot Big Bang, and then the standard cosmology follows.The evolution follows the Friedman equations of cosmological Λ, matter and radiation energy densities.The cosmological Λ and massive particle origin are still mysteries.One calls them "dark energy" and "cold dark matter".Moreover, their properties and interactions in the Universe's evolution are also in question.Why their present values are coincidentally in the same order of magnitude?
To get an insight into these issues, people have been intensively studying the gravitational particle production in Friedman Universe for decades [19][20][21][22][23][24][25][26][27][28][29][30][31][32].Based on the adiabatic and non-back-reaction approximation for a slowly time-varying Hubble function H, one adopted the semi-classical approaches to calculating the particle production rate.It is exponentially suppressed e −M/H for massive particles M H since the classical Universe evolution time scale O(1/H) is much larger than the quantum time scale O(1/M ) of particle production.However, the non-adiabatic back-reactions of massive particle productions on the Hubble function can be large.One has to take them into account.People have made many efforts [23,[33][34][35][36][37][38][39][40][41] to study non-adiabatic back-reaction and understand massive particle productions without exponential suppression.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.Here, the fast components represent fluctuating gravitational and particle fields at the time scale O(1/M ).The slow components represent slowly varying background fields and particle densities at the time scale O(1/H).
In Ref. [41], we assume the inflation epoch starts when the dark-energy density ρ Λ = Λ/(8πG) is the order of the Planck scale, dominates the Hubble function H 2 ≈ 8πGρ Λ /3 and the matter density is negligibly small.We study the Universe undergoes a Λ-driven inflation that slowly slows down to its end by producing heavy particles of mass M H. Inflation is a semi-classical dynamics of the time scale O(1/H), while massive particle production is a quantum-field dynamics of the time scale O(1/M ).We investigate how the massive particle production density ρ H M back reacts ρ Λ by separating the fast and slow components in the Friedman equation.Analysis shows H and ρ Λ slowly decrease, as ρ H M slowly increases in time, namely, ρ Λ slowly converts to ρ H M .It gives the quasi-de Sitter phase (slow-rolling dynamics) for inflation.The final results are consistent with observations.Here we turn to discuss the reheating epoch and possibly explain reheating energy and entropy, as well as cold dark matter, in comparison with current observations.We review the previous results of the fast and slow components' separation in Sec. 2, quantum massive pair production and oscillation in Sec. 3, and a massive pair plasma state in Sec. 4. We present in Secs. 5 and 6 the new studies of a complete set of differential equations and initial conditions for the reheating epoch.We numerically solve these equations and compare the results with observations in Secs.7 and 8.We present preliminary discussions on stable massive as cold dark matter candidate in Sec. 9. G = M −2 pl is the Newton constant, M pl is the Planck scale and reduced Planck scale m pl ≡ (8π) −1/2 M pl = 2.43 × 10 18 GeV.

Slow adiabatic and fast non-adiabatic components
We discuss such fast and slow separation in the ΛCDM scenario, where a time-varying cosmological Λ term in the Friedman equation represents such interacting dark energy.The Friedman equations for a flat Universe are [18] where energy density ρ ≡ ρ M + ρ R + ρ Λ and pressure p ≡ p M + p R + p Λ .The second Equation of (2.1) is the generalised conservation law (Bianchi identity) for including time-varying cosmological term ρ Λ (t) ≡ Λ/(8πG).It reduces to the usual Equation ρM + (1 + ω M )Hρ M + ρR + (1 + ω R )Hρ R = 0 for time-constant ρ Λ .The second Equation of (2.1) shows that Ḣ < 0 and H decreases in time, due to the matter's gravitational attractive nature.Separating fast components from slow ones [37], we describe the slow and fast components' decomposition: scale factor a = a slow +a fast , Hubble function H = H slow +H fast , cosmological Λ and matter densities ρ Λ,M,R = ρ slow Λ,M,R + ρ fast Λ,M,R and pressures p Λ,M,R = p slow Λ,M,R +p fast Λ,M,R .The fast components vary faster in time, but their amplitudes are much smaller than the slow ones.According to the order of small ratio λ of fast and slow components, the Friedman equations (2.1) decompose into two sets.The slow components O(λ 0 ) obey the same equations as usual Friedman equations ("macroscopic" O(H −1 slow ) 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 Equation of the state is p slow R,M = ω R,M ρ slow R,M for normal radiation and matter (including dark matter) components.They enter the usual dynamics of Universe evolution, i.e., inflation, reheating and standard cosmology.The faster components O(λ 1 ) obey "microscopic" O(M −1 ) equations 1 , where the fast components of matter density ρ fast M and pressure p fast M are due to the nonadiabatic production of massive particle and antiparticle pairs in fast time variation H fast = ȧfast /a slow and its time derivative Ḣfast .They relate to the microscopic "fast" time variation scale O(1/M ).Whereas all slow components approximate as constants "background" in "fast" time variation.The dark-energy equation of state p Λ = −ρ Λ splits into p slow 3 Quantum massive pair production and oscillation

Quantum massive pair production
In this section, we briefly discuss the Parker and Fulling results [23] for the gravitational production of a large number of massive particles (M H slow ) via non-adiabatic processes.We will re-derive the results in the ΛCDM (2.1) and use them for the fast components of matter density ρ fast M and pressure p fast M in Eq. (2.4,2.5).In Ref. [23], authors discussed the results for boson fields.It is also valid for fermion fields.A quantised massive scalar matter field inside the Hubble sphere volume Here we consider a massive field M H slow and its modes well localise inside the horizon.The field exponentially vanishes outside the horizon H −1 slow , i.e., the particle horizon (a slow H slow ) −1 of comoving Hubble radius.The symbol "n" labels quantum states of physical wave vectors k n , n = 0 and k 0 = 0 for the ground state3 .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 (3.2) 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 evolves 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 study particle production in non-adiabatic processes of rapidly timevarying H fast , α n and β n .We focus only on the ground state n = 0 of the lowestlying massive mode M H. First, we recall that Parker and Fulling introduced transformation [23], [B, B † ] = 1, and two mixing constants obey |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 particles and negative frequency A † 0 antiparticle.The state |N pair contains N pair = 1, 2, 3, ••• pairs, and it is the ground state of non-adiabatic interacting system of fast varying H fast and massive pair production and annihilation.It is a coherent superposition of states of a large occupation number N pair of particle and anti-particle pairs.In Ref. [23], the authors compared it with the BCS condensate state in superconductivity theory and contrasted it with the normal single-particle state.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. [23], ) where ω n=0 = M , α n=0 = α and β n=0 = β.They satisfy the continuity equation of energy-momentum conservation.In addition to non-vanishing |β| 2 = 0, the large occupation number N pair 1 in the coherent state (3.6) is crucial for the significant gravitational production of massive pairs.It differs from adiabatic particle production in the vacuum state of zero particles.For a closed Universe case, they adopted the pressure (3.7) and density (3.8) for studying the avoidance of cosmic singularity at the beginning of the Universe.In their sequent article [43], the authors confirm Eqs.(3.7) and (3.8) by studying the regularisation of higher mode contributions to the energymomentum tensor of a massive quantized field of closed, flat, and hyperbolic spatial spaces.In the case of heavy particles produced near the Planck scale, the renormalization of high-energy contributions should not be the same as the case of produced light particles in low energies [39].To end this section, we emphasize two points.(i) The quantum pressure p fast M (3.7) oscillates, and its value can be positive or negative in oscillations of frequency 1/M , depending on modes' equation (3.4), superposition coefficients γ, δ (3.5) and mass M values.The negative value of microscopic time-averaged quantum pressure p fast M is crucial for forming the coherent condensate state |N pair (3.6) of a large occupation number N pair of massive particle and anti-particle pairs produced.(ii) Such coherent condensate state |N pair occurs only at the ground state k 0 = 0 and n = 0, i.e., the state of = 0 spherical S-wave.For high angular momentum states = 0, the timeaveraged pressure becomes non-negative classical values ∝ ( + 1) [23].Therefore, the coherent condensate state |N pair cannot form for high-energy states = 0.It gives us a lesson that the high-energy modes' renormalization or subtraction prescription for massive particles (M ∼ M pl H) production is not the same as light particles (M H M pl ) production.

Quantum massive pair oscillation
Following their approach for the ground state k n = 0, we arrive at the same quantum pressure (3.7) and density (3.8) in the ΛCDM.In our case, we consider the state (3.6) as a coherent condensate state of very massive M H slow and large number N pair 1 pairs.Therefore, M (2N pair + 1) in Eqs.(3.7,3.8)can be larger than the Planck mass, and higher mode (k n = 0, n = 0) contributions can be neglected.Their regularisation and corrections will be studied in future.In this article, we adopt p fast The real value γ * δ condition in Eqs.(3.9),3.10)leads to the time symmetry: a fast (t) = a fast (−t), α(t) = α * (−t) and β(t) = β * (−t) [23].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 non-linearly coupled Eqs.(3.4) and (3.9-3.11) with the initial condition (3.12).We report the results in Fig. 1 and details in Fig. 11 of Appendix.Similar to the previous results [41], we find that in the quantum period of microscopic time t, the negative quantum pressure P fast M < 0 and back-reaction effects lead to the quantum pair oscillation in a time characterised by the frequency ω ∼ M .The small a fast (t) varies around a slow at t slow ≡ 0. The massive pairs' density and pressure ( ) and massive particle pairs F F ( fast M , P fast M ), previously discussed [39].These results show the highly non-adiabatic and complex nature of massive pair-production processes and collective oscillations.Attributed to complex back-and-forth reactions at the scale O(1/M ), the quantum massive pair oscillation (3.13) cannot be described by oscillating scalar fields with polynomial potential.
As shown in Figs.> 0 is positive definite, leading to the decreasing h fast (t) (3.11).As a consequence, for positive time (t > 0) increasing (forward the future), the fast components h fast and fast Λ decrease, in order for pair production.Whereas for negative time (t < 0) increasing (backward the past), h fast and fast Λ increases, due to pair annihilation.In both situations, the ρ fast Λ is negative and ρ fast M is positive, as required by the energy conservation (3.11) in the massive pairs' production via fast oscillating h fast and ḣfast .Equations ρ Λ,M = ρ slow Λ,M + ρ fast Λ,M imply the dark energy ρ Λ decreases and matter ρ M increases, where the slow components ρ slow Λ,M are fixed values at t = t slow ≡ 0. In this sense, dark energy converts to massive pairs (matter) in a microscopic time scale, whereas the case for a macroscopic time scale will be studied in Sec. 5.It is our finding that the energy density and pressure (3.9,3.10) of the condensate state |N pair coherently interacts and exchanges energy with the fast components of spacetime variation (3.11).Such the back-and-forth reaction at the scale 1/M was not studied in Ref. [23] for the |N pair condensate state's energy density (3.8) and pressure (3.7).In our solutions, we find the quantum pressure P fast M and its time average are negative see Fig. 11 in Appendix.
It implies the possibility that the condensate state |N pair retains while it coherently interacts with the fast components of spacetime horizon variation (3.13).
This phenomenon is dynamically analogous to the plasma oscillation of electronpositron pair production in an external alternating electric field E [44].The pair production rate is not exponentially suppressed by e −πM 2 /E [45].The coherent plasma state of electron-positron pairs is analogous to the coherent pair state |N pair (3.6) and quantum pair oscillation, shown in Fig. 1.
Such massive and semi-classical state of large occupation number (3.6) and quantum pair oscillation (Fig. 1) well localises inside the horizon.They persist throughout the entire Universe's history, independent of slow components H slow and ρ slow Λ,M,R values in Friedman equations (2.2,2.3).However, the pair mass M (oscillating frequency ω ≈ M ) and pair number N pair depend on slow components' values in the Friedman equations (2.2,2.3).It is necessary and deserves to proceed with further studies.
4 Massive pair plasma state and holographic hypothesis H are very different.To deal with this difficulty, we adopt two approximate steps.First, due to nontrivial timeaveraged values of fast components over the microscopic time t 1/M , we assume the massive pair production and oscillation form a massive pair plasma state in the macroscopic time scale.We model such a semi-classical state as a perfect fluid by defining the effective density and pressure.Second, we discuss how it back-and-forth interacts and contributes to the slow components in the Friedman equations.
Figure 1 shows that massive pair quantum pressure P fast M (3.9) and density fast M (3.9) rapidly oscillate with the fast components h fast and fast Λ (3.11) in microscopic time.Their oscillating amplitudes are significantly large and not dampening in time.It, therefore, expects to form a massive pair plasma state in a macroscopic time and space.However, to study its effective impacts on the classical Friedman equations (2.3), we have to discuss two problems stemming from the scale difference M H slow .
(i) First, the different time scales.It is impossible to even numerically integrate slow and fast component coupled equations (2.3,2.5)due to their vastly different time scales.On this aspect, we consider the fast-component averages • • • over the microscopic time scale.time scale and differs from the oscillating one 1/M .Therefore, in principle, fastcomponent averages possibly affect the Friedman equation at the macroscopic time scale.In practice, the appropriate modelling of fast-component averages can avoid the difficulty of vastly different scale dynamics in calculations, and the scenario becomes tractable.However, we have to check its self-consistency with observation.
(ii) Second, the spatial distribution.We do not know the spatial distribution of the massive pair condensate state |N pair .Namely, we do not know the radial dependence of the quantum pressure and density (3.7,3.8) or (3.9,3.10),since the Ref. [23] authors obtained them by using the vacuum expectation value of field Φ(x, t) (3.1) energy-momentum tensor integrated over the entire space.There, they studied the cosmic singularity problem in the Universe beginning M H case, namely the massive mode wavelength M −1 is comparable with the horizon size H −1 .Here, we study the case M H ≈ H slow , namely the massive mode wavelength M −1 is much smaller than the horizon size H −1 slow .As a hypothesis, we speculate that the massive pair condensate state |N pair and the coherent oscillation (3.13) with H fast and Ḣfast spatially localize nearby the horizon following the holographic principle [46][47][48].The arguments are the following.(a) Such condensate state |N pair and oscillation (3.13) collectively couple with the fast components H fast and Ḣfast , which are quantum modes at short wavelengths (1/M ).These modes should associate with the horizon surface according to the holographic principle.(b) The massive pair condensate state |N pair is a ground state of a spherically symmetric S wave.(c) Such a very massive state of the radial size about M −1 H −1 slow is inside the horizon H −1 slow of the Friedman Universe, whose isotropic homogeneity extends up to the horizon.
Based on these hypotheses, we will introduce a holographic and massive pair plasma state that gives an effective description of the condensate state |N pair and coherent oscillation (3.13) at macroscopic space and time scales.

Effective description of a massive pair plasma state
Based on these considerations, we assume a massive pair plasma state forms in a macroscopic time scale.We describe such macroscopic state as a perfect fluid state of effective number n H M and energy ρ 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 (3.5).The degeneracies g f d plays the same role of pair number N pair in Eqs.(3.7,3.8) or (3.9,3.10).The pair masses M f are smaller than the Planck mass M pl , but the mass parameter m can be larger than M pl for a large occupation number N pair 1 or degeneracy g f d 1.Note that the massive pair plasma state contains (i) unstable massive pairs that couple and decay to light particles; (ii) stable massive pairs with gravitational interaction only.Besides, one should differ the massive pair plasma state density ρ H M (4.1) from the normal matter or radiation (including dark matter) density ρ M,R ∝ (1/a) 3(1+ω M,R ) .The reason is that the massive pair plasma state (4.1) attributes to quantum pair production and oscillation, which couple to the oscillating spacetime fields S(h fast , ḣfast , fast Λ ) of Hubble function and dark energy.To some extent, we may consider the massive pair plasma state as an "equilibrium state" between quantum massive pairs and spacetime field oscillations.
Following the previous subsection discussions, we explain the reasons why the densities (4.1) are proportional to χmH 2 slow , rather than H 3 slow of 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 is the layer width λ m introduced as an effective parameter to describe the properties: (i) for m H slow the massive pair plasma state localizes as a spherical layer near to the horizon; (ii) the layer radial width λ m < H −1 slow depends on the massive pair plasma oscillation dynamics6 , rather than the H slow dynamics govern by the Friedman equations (2.3).The width parameter χ expresses the layer width Note that studying the prescription of high-energy modes' subtraction for m H, we approximately obtained the mean density n H M ≈ χmH 2 (4.1) and χ ≈ 1.85 × 10 −3 by studying massive fermion pair productions in a De Sitter spacetime of constant H and scaling factor a(t) = e iHt [39,40].We adopt this χ value for numerical calculations in the present article.
Since the 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 for the Friedman equations.However, the typical m and χm values should be different for Universe evolution epochs since the fast-component equations for massive pair productions and oscillations depend on the H slow value, see Sec. 3. We used the parameter m * for inflation, m for reheating and m M for the epochs after reheating.We will fix these parameter values by observations.
To end this section, we have to point out that (i) the pressure p H M and density ρ H M (4.1) are effective descriptions of the massive pair plasma state in macroscopic scales, that result from the coherence condensation state (3.6,3.7,3.8) and oscillating dynamics (Fig. 1) in microscopic scales; (ii) they contribute to the "slow" components ρ slow M and p slow M in the "macroscopic" O(H −1 slow ) Friedman equations (2.2,2.3).It means that in the Friedman equations (2.2,2.3), the matter density and pressure terms ρ slow M and p slow M contain (a) the normal matter state contributions and (b) the massive pair plasma state contributions.This will be clarified in the next Section.We shall study the massive pair plasma state effects on each epoch of the Universe's evolution.Here we investigate its impact on reheating.
After we adopt the effective description of massive pair plasma state (4.1), the quantum massive pair oscillation (Fig. 1 3), and their interacting equation (5.5).The final results depend only on the plasma state (4.1) with the mass m and width χ parameters.Henceforth we ignore the "fast" components, sub-script, and super-scripts "slow" will be dropped.

Back-and-forth process and cosmic rate equation
In Sec. 3, we show massive pairs' production and annihilation at time scale O(1/M ) via quantum pair oscillations, and dark energy effectively converts to massive pairs for the microscopic time t 1/M .These are the quantum back-and-forth process (3.13).By non-vanishing averages over microscopic time, these microscopic backreaction processes should impact the classical and slow components in Friedman's equations.Using massive pair plasma state (4.1), we will discuss how to effectively describe the back-and-forth process between massive pairs and spacetime fields (H, ρ Λ ) at a macroscopic time scale (1/H).

Stable massive pairs and cosmic rate equation
We discuss here how the massive pair plasma state (4.1) back-reacts and contributes to the slow components in Friedman equations.First, we introduce the mean pair production rate Γ M to describe the massive pair plasma state variation as the macroscopic time t varies.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 variation dt, It is in terms of the parameter χm (4.2) and Universe evolution -rate defined as, The second equation comes from the Friedman equations (2.2,2.3).The asymptotic values ≈ 0, ≈ 2 and ≈ 3/2 correspond to the dark-energy (inflation), radiation, and matter dominant epochs, respectively.The massive pair plasma state ρ H M (4.1) effectively represents an equilibrium state of quantum pair and spacetime field oscillations (3.13).It not only depends on the Hubble function H, but also contributes to the normal matter density ρ M .Back reactions must act on ρ H M , when H and ρ M vary in time following the Friedman equations.Moreover, the massive pair plasma state variation time scale τ M = Γ −1 M is smaller than the normal matter density 7 ρ M variation time scale τ H = 1/H.The difference τ H > τ M implies the back-and-forth interaction between the massive pair plasma state density and the normal matter density during the Universe's evolution.The process is induced by quantum pair oscillation coherently with fast oscillating components of the Hubble function and dark energy.
To model such dynamics (5.3), we recall the rate equation for the back-and-forth process e + e − ⇔ γγ [49][50][51][52]: where n e + e − (t) is the electron and positron pair density governed by the macroscopic time scale H −1 evolution.While n e + e − eq is the density of electrons and positrons in equilibrium with two photons n γγ eq in microscopic time scale ( σv n e + e − ) −1 , namely n e + e − eq ≈ n γγ eq .The RHS represents the averaged interacting rate dN/dt ≈ σv n e + e − for microscopic detail balance between n e + e − (t) and n e + e − eq .They are coupled for n e + e − eq ≈ n e + e − and decoupled for n e + e − eq n e + e − .We make the following analogies: n e + e − ↔ ρ M , n e + e − eq ↔ ρ H M and photons n γγ eq correspond to fast oscillating components of the Hubble function and dark energy.This analogy motivates us to propose an effective cosmic rate equation where g Y is the Yukawa coupling between the massive pairs ( F F ) and relativistic particles.It is important to note that the decay rate Γ de M (5.6) depends not only on the Yukawa coupling g Y but also on the phase space of final states.While for stable massive pairs, the decay rate Γ de M is zero.The combination of cosmic rate equation (5.5) and Friedman equations (2.3) , representing the interaction and exchange between dark energy and normal matter via the massive pair plasma state ρ H M .To discuss this in some more detail, we ignore the decay term Γ de M ρ M .It is negligible for unstable pairs, provided Γ M Γ de M .We point out four particular cases: (i) Recall the results [41]   is the case for epochs after reheating.Dark energy converts to matter and reduces to its minimal value in reheating, and matter and radiation become dominant over (much larger than) dark energy.However, dark energy weakly couples to matter and radiation, i.e., δQ ≈ 0 and ρΛ ≈ 0. Its variation is much more slowly than matter/radiation decrease.Then it dominates over matter and radiation today.We present the preliminary discussions in Ref. [53].
The inclusion of decay terms Γ de M ρ M and transitions from one case to another are complex and need numerical studies.

Unstable massive pair decay and reheating equation
Coming from massive unstable pairs' decay, the radiation energy density ρ R of relativistic particles ¯ (5.6) obeys the energy conservation law, see for example Ref. [49], (5.10) As a result, we have a close set of four ordinary differential equations to uniquely determine the time evolution of the Hubble rate H, dark-energy density ρ Λ , massive particles' energy density ρ M and relativistic particles' energy density ρ R .They are generalised Friedman equations (2.2,2.3) for H and ρ Λ , the cosmic rate equation (5.5) for ρ M , and the reheating equation (5.10) for ρ R .In addition, there are four algebraic relations: the massive pair plasma density ρ H M (4.1), the pair-production rate Γ M (5.1), the Universe evolution -rate (5.2) and the pair-decay rate Γ de M (5.6).We will numerically solve these equations, provided initial conditions are known.

Initial conditions and basic equations for reheating
The inflation epoch (H > Γ M ) ends, and the reheating epoch (H < Γ M ) starts.The transitioning process must be very complex due to the back reactions of microscopic and macroscopic processes.We assume the transition to be instantaneous at the inflation end a end and H end when H Γ M .For the inflation epoch from a * to a end , see Figure 2, we obtain [41] where the inflation scale H * and a * correspond to the pivot scale k * = 0. 05 (Mpc) −1 crossed the horizon (k * = H * a * ) for CMB observations [55].The observed spectral index n s and scalar amplitude A s determine H * = 3.15 × 10 −5 (r/0.1) 1/2 m pl and the * = (1 − n s )/2 ≈ 0.0175 of the -rate (5.2) in inflation.The e-folding number N end and the tensor-to-scalar ratio r are related by H end Γ M r 7.97 × 10 4 χ(1 − n s ) 3 e (1−ns)N end , ( and χ > 0 implies r > 0. The observational constraint on the tensor-to-scalar ratio and spectra index (r, n s ), see Fig. 2 of Ref. [41], gives χ O(10 −3 ).The small H variation implies at the inflation end and We approximately adopt the value for which (Γ M /H) end ≈ 1.These are the reheating epoch initial conditions.Using the characteristic scale H end and density ρ end c , we normalize h ≡ H/H end , Here we introduce the mass parameter m or χ( m/m pl ) 2 as a typical scale parameter for reheating and will fix its value by observations.Thus, we recast the Friedman equations (2.2) and (2.3), the cosmic rate equation (5.5) and reheating equation (5.10) as, ) ) The ratios are and the -rate (5.2) becomes Instead of the cosmic time t, here we adopt the cosmic e-folding variable x = ln(a/a end ) and d(• • •)/dx = d(• • •)/(Hdt) for the sake of simplicity and significance in physics.In the next sections, we will numerically integrate these basic equations (6.6-6.11) for the reheating epoch by using the inflation end (6.4) as the initial condition.
We have to emphasize that the differential equations (6.6-6.9)represent a macroscopic back-and-forth reaction system characterized by the scales τ H , τ M and τ R .It describes the processes: inflation, reheating and standard cosmology.It differs from the differential equations (2.4-2.5) and those in Sec. 3 for a microscopic back-and-forth reaction system of fast-oscillating components characterized by the scale O(1/M ).The fast components' contributions are effectively represented by the massive pair plasma state ρ H M (4.1) that enters the cosmic rate equation (6.8).

Different episodes in reheating epoch
In the reheating epoch, generally speaking, the horizon h and the dark energy Ω Λ decreases, as the matter content Ω M or Ω R increases, meanwhile the ratio Γ M /H (6.10) and the -rate (6.11) increase.To gain insight into the physics first, we use the -rate values (6.11) to characterize the different episodes in the reheating epoch.In each episode, the rate slowly varies in time, we approximately have the time scale of the spacetime expansion H −1 ≈ t.In the transition from one episode to another, the -rate significantly changes its value.Using the characteristic values 1, ≈ 3/2, ≈ 2, we identify the following three different episodes P-episode, M-episode and R-episode in the reheating epoch.

Preheating P-episode: dark energy ρ Λ converting into matter ρ M
The short preheating P-episode is the transition from the inflation end to the reheating start.In this episode, the pair production rate Γ M (5.1) is larger than the Hubble rate H, that is still much larger than the pair decay rate Γ de M (5.6), The radiation energy density is completely negligible ρ R ≈ 0. We neglect massive pairs decay to light particles Γ de M ≈ 0 (5.6).The reheating equation (6.9) is then not relevant, and the basic equations (6.6), (6.7) and (6.8) reduce to where the ratio Γ M /H > 1 (6.10) increases as the -rate (6.11) In the P-episode, these equations uniquely determine the evolution of the Hubble rate H, pairs' energy densities ρ M and dark-energy density ρ Λ .2).We plot these solutions with the initial condition (6.4) and parameter ( m/m pl ) = 27.7.

High efficiency of dark energy converting into matter
Using the values H end (6.1) and Ω end M (6.4) at the inflation end as the initial conditions for the P-episode, we numerically integrate Eqs.(6.6), (6.7) and (6.8), by selecting values of the mass parameter m/m pl .In Figs. 3 and 4, the numerical solutions are plotted in terms of the e-folding variable x = ln(a/a end ).These solutions show an important result that the dark-energy density ρ Λ is significantly converted to the matter-energy density ρ M , as the pair-production rate Γ M increases and becomes much larger than the Hubble rate H.In more detail, we list that in the P-episode the physical quantities vary in time as follows, (i) the Hubble rate h decreases rapidly in a few e-folding number, as ρ M becomes dominate over ρ Λ , see Fig. 3 (a); (ii) the ρ M increases at the expense of the ρ Λ , eventually ρ M exceeds and dominates ρ Λ , see Fig. (iii) the ratio Γ M /H (7.1) increases and becomes much larger than unity (Γ M /H 1), see Fig. 3 (c); (iv) the H varying rate (7.5) increases from 1 to ∼ O(1), indicating the transition from the inflation end to the preheating P-episode, and it then approaches to an asymptotic value, see Fig. 3 (d).
In Figure 4 (left), we plot the energy densities Ω Λ and Ω M as functions of the horizon h 2 , corresponding to Figures (a) and (b) in Fig. 3.It shows two branches of asymptotic solutions Ω Λ (orange) respectively, Ω M +Ω Λ = h 2 and the turning point is about h 2 ≈ 0.98, at which Ω M exceeds Ω Λ and the rapid ρ Λ ⇒ ρ M converting process takes place.The characteristic behaviour Ω Λ ∝ h 2 (7.6) is the same as that in the pre-inflation and inflation epochs.Correspondingly, two branches of asymptotic solutions for the matter Ω M (blue) are and where Ω max M ≈ 0.85 when h 2 ≈ 0.95.The coefficients α 1,2 Λ and α 1,2 M in Eqs.(7.6) and (7.7) can be numerically obtained.As h 2 → 0 and (a/a end ) increases, 4 (right) shows that in the preheating P-episode, the dark energy density ρ Λ converts to the matter-energy ρ M .The ratio ρ M /ρ Λ rapidly increases in a few e-folding numbers from ρ M /ρ Λ 1 at the inflation end to a value ρ M /ρ Λ O(1).Then ρ M becomes dominant.The Hubble rate H rapidly decreases and becomes much smaller than the pair-production rate Γ M .We define the P-episode end at 1.11a end by ρ Λ ≈ ρ M from Figs. 3 and 4 (right).It shows that the preheating P-episode is a very brief transition episode.
We ought to discuss the condition for high efficiency of dark energy converting into massive pairs' energy in this preheating P-episode.Figure 3 (b) shows ρ Λ decreases and ρ M increases rapidly, the efficiency of dark energy converting matter-energy is large.The reasons are the following.(i) The ratio Γ M /H 1 increases rapidly as H decreases rapidly, see Fig. 3 (c) and (a), respectively.(ii) The rate (5.2) rapidly increases to the order of unity, see Fig. 3 (d).(iii) The mass parameter m (5.1) is large, see Fig. 4 (right), which implies heavy mass M and large number N pair of pairs produced in the massive pair plasma state (4.1).If the mass parameter m is small, the dark energy and matter conversion efficiency is small, see the ratio Ω M /Ω Λ in Fig. 4 (right).A priori, we do not have theoretical arguments about how much m value is.Instead, we select m values a posteriori in comparison with observations, and the Universe does not stay cold state of ρ Λ > ρ M .The necessary condition is the existence of a threshold mthresh for which ρ M > ρ Λ of efficient conversion in reheating.The reason is that the number N pair (or effective degeneracy g d ) of massive pairs produced in the reheating epoch has to be large enough so that Γ M H and the conversion from ρ Λ to ρ M is efficient.It corresponds to the physical situation that the most radiation and matter of the Universe is generated in the reheating epoch.
(b) For small mass parameters m/m pl < 20, the massive pairs' energy density ρ M never exceeds the dark-energy density ρ Λ , namely ρ M /ρ Λ < 1.The conversion from ρ Λ to ρ M is inefficient.This case corresponds to the unrealistic situation that the Universe inflation would never have completely ended, i.e., the cosmological term ρ Λ always dominates H 2 .
Observe that the mass parameter m of the reheating epoch is larger than the mass parameter m * of the inflation epoch.From the viewpoint of pair production, the pair mass scale in reheating should be smaller than that in inflation since the reheating horizon H is smaller than the inflation one.Therefore, it implies that the effective numbers N pair of massive pairs produced in reheating, Γ M /H > 1 is much larger than that of massive pairs produced in inflation Γ M /H < 1.These massive pairs contain both stable and unstable pairs.Equation (7.5) shows that the asymptotic value of the Horizon variation -rate (5.2) relates to the ratio ρ M /ρ Λ asymptotic value, see Figs. 3 (d) and 4 (right).For large mass parameter m/m pl 27.7, the ratio ρ M /ρ Λ 19 , the -rate (7.5) approaches to the asymptotic value ≈ M = 3/2.It shows the episode of massive pairs domination: M-episode.

Minimal comoving radius (Ha) −1 location
Before discussing the M-episode, we would like to mention the turning point at which the Universe acceleration vanishes ä = 0, which is obtained from the 1−1 component of the Einstein equation At this turning point, the Universe stops acceleration ä > 0 and starts deceleration ä < 0. The turning point occurs at ρ Λ = ρ M /2 for ω M ≈ 0 and ρ R ≈ 0. It tells us the balance point of the competition between ρ Λ and ρ M in the P-episode.
On the other hand, the minimal value of the comoving radius (aH) −1 locates at From Friedman equations (2.1), we obtain coinciding with the turning point (7.9).Namely at the minimal comoving radius (aH) −1 , the Universe stops acceleration ä > 0 and begins deceleration ä < 0, starting the reheating epoch and standard cosmology.This is indeed the case for large mass parameter ( m/m pl ) > 20 and the ratio ρ M /ρ Λ becomes larger than 2. The numerical results (Fig. 3) show that this turning/minimal point min = 1 locates at x min ≈ 1.7 × 10 −2 and a min ≈ a end × exp (1.7 × 10 −2 ) = 1.02 a end .While the turning/minimal point min = 1 is never reached, for the cases of the small mass parameter ( m/m pl ) < 20 and the ratio ρ M /ρ Λ is always smaller than 2, see Fig. 4 (right).The reason is that dark energy converting to matter is inefficient, the massive pairs energy is not large enough to balance the dark energy and slow down the Universe's acceleration.The Universe keeps acceleration ä > 0 and does not run into the reheating epoch.Therefore, the mass parameter range below the threshold mthresh (7.8) ( m/m pl ) < 20 should be excluded.We plot these solutions with the initial condition (6.4) and parameter ( m/m pl ) = 27.7.

Massive pairs domination: M-episode
After the P-episode transition, it is the M-episode of massive pair domination characterised by The radiation energy density ρ R is negligible in the basic equations (6.6-6.11).The H variation -rate M ≈ 3/2 in Fig. 3 (d) for ρ M /ρ Λ 1 in Fig. 4 (right).In this episode, the Hubble rate H and scale factor a(t) vary as h 2 ≈ Ω M and the pair energy density Ω M ∝ (a/a end ) −2 M .Moreover, the back-and-forth processes (5.3) are important, as described by the cosmic rate equation (7.4) with the detailed balance term D M , and we define its characteristic time scale Note that τ D differs from τ M = Γ −1 M (5.1).The microscopic time scale τ M is much smaller than the macroscopic expansion time scale τ H = H −1 , τ M τ H . Therefore, the back-and-forth (5.3) can build an energy equipartition ρ M ≈ ρ H M , and the detailed balance term (7.15) vanish in its time-averaged over the macroscopic time τ H τ M .The cosmic rate equation becomes approximately whose solution is ρ M ∝ a −3 .It is consistent with the matter-dominated solution to Eq. (2.3), yielding H 2 ∼ ρ M ∝ a −3 .It is also self-consistent with the pair plasma density (4.1) ρ H M = 2χ m2 H 2 ∝ a −3 .In order to verify these discussions and ρ M ≈ ρ H M , we check the solution (7.17) or (7.18) analytically and numerically.The ρ H M = 2χ m2 H 2 averaged over the time τ H consistently obeys the same equation (7.18) where ≈ M = 3/2.Numerical results quantitatively show the same conclusion ρ M ≈ ρ H M , see Fig. 5 (a), and the detailed balance term (7.15) vanishes, see Fig. 5 (b).Thus we conclude that in the M-episode, due to Γ M H and τ M τ H , the massive pairs plasma state ρ H M tightly couples with the mass density ρ M in the H evolution.

Relativistic particles domination: R-episode of genuine reheating
At the end of the M-episode, the massive pairs' decay term Γ de M ρ M in equations (6.8,6.9)starts to dominate, when the time t τ R .The τ R (5.6) is the characteristic time scale of massive pairs decay to relativistic particles.It represents the reheating period of producing tremendous amounts of entropy.The reheating epoch starts its genuine reheating episode, i.e., R-episode.

Massive and unstable pairs decay to relativistic particles
To study the R-episode, we numerically solve the closed set of the basic equations (6.6-6.9) with the radiation energy density Ω R and the decay term, and we define the characteristic time scale of the decay term R M , which is the same as τ R (5.6).The initial condition of the radiation energy density is Ω end R = 0 at the inflation end a end , in addition to the initial conditions (6.1) and (6.4).We report the numerical results in Figures 6.It shows that in the H 2 (6.6), the radiation energy density Ω R increases and becomes dominant, compared with Ω M and Ω Λ .
We explain this phenomenon by comparing the decay term R M (7.20) with the detailed balance term D M (7.15) in the cosmic rate equation (6.8).Two different dynamics D M and R M compete with each other in the process.The ρ R is negligible when D M > R M , while the ρ R is dominant when R M > D M , and the transition from one to another occurs approximately at R M D M , where ρ R h 2 , as shown in Figs.7  (a) and (b).More precisely, it is the comparison between the characteristic time scale τ D (7.16) of the back-and-forth process (5.3) and the characteristic time scale τ R (7.20) of the pair decay process (5.6).When τ D < τ R , the faster back-and-forth process (5.3) dominates, whereas τ R < τ D , the faster decay process (7.21) dominates.
In Figs. 7 (c) and (d), two-time scales τ D and τ R are plotted as dimensionless quantities τ D /τ H and τ R /τ H to show two episodes: the stiffness system of step size being effective zero.However, the analytical solution to these basic equations is studied in the next section.

Energy densities of massive pairs and relativistic particles
The R-episode is characterised by and Γ de M /H > 1, as shown in Figs. 6.As a result, Equations (6.6) and (6.7) or Eq.(6.11) give Following the line presented in Ref. [49], we discuss how the massive pairs transfer their mass energy to relativistic particles, and calculate the radiation energy density ρ R , entropy S and temperature T of relativistic particles.
Since unstable massive pairs predominately decay to relativistic particles, the detailed balance term D M (7.15) is negligible, and the cosmic rate equation (6.8) reduces to, ) The reheating equation (6.9) becomes In theory, it requires the the time integration from the initial time t i (a i ) τ R when ρ R (a i ) = 0 to the final time t f τ R to obtain the radiation energy density ρ R , Through their gauge and other induced interactions, these relativistic particles ¯ including sterile particles and other particles beyond the SM interact with each other.They are quickly thermalised at a very high temperature T , due to their high number and energy densities.The local thermalisation time scale is much shorter than the expansion time scale τ H = H −1 , and thus the local thermal equilibrium is built.

Reheating temperature and entropy
We follow the approach [49] to calculate the reheating temperature and entropy.The second law of thermodynamics applied to a comoving volume element yields where dQ is the pair mass energy and dS is the entropy of relativistic particles produced from massive pairs decay.Therefore, in a comoving volume, the entropy and energy densities of relativistic particles at the thermal state of temperature T are given by, (7.28) The appropriately time-averaged degeneracy g * over the decay period τ R counts for the total number of effectively massless degrees of freedom, those species share a common temperature T .Using the entropy (7.28), one writes Eq. (7.27) as, Integrating this equation over the decay period τ R from the initial scale factor a i to the reheating scaling factor a R > a i leads to an approximate solution Here one adopts Eq. (7.24) and S i (a i ) ≈ 0, namely, the initial massive pairs' entropy is approximately zero.In principle, it requires integrating from the initial time t i (a i ) τ R to the final time t f (a R ) τ R , when the entropy significantly increases.In practice, a i a R and t f τ R are approximately adopted in Eq. (7.30), since the all-important entropy S R mainly produces in the reheating period τ R (a R ).
At the scale factor a R , the reheating scale H RH can be obtained by τ R from the Friedmann equation (6.6) or the reheating temperature T RH ≡ T (t = τ R ) from the thermalization (7.28) [49]: It leads to the reheating temperature and the all-important entropy per comoving volume, Y m (5.6).Our numerical calculations show the consistency of the approximation a i a R used in Eq. (7.30) and the agreement with the analytical solutions (7.31) and (7.33).From Figs. 6 and 7, we find that the reheating predominately takes place around a R , at which τ R ∼ τ D , and the ratios τ R /H RH ≈ τ D /H RH ∼ O(1).Moreover, from Fig. 6 (a) we obtain the reheating scale H RH ≈ 3.16 × 10 −4 H end ≈ 6.1 × 10 9 GeV, a R ≈ 20.1a end (7.35) for the case g 2 Y = 10 −9 and m = 27.7mpl .We also obtain H RH ∼ 10 −1 H end = 1.9 × 10 12 GeV (the plot is not present), and a R ≈ 1.8 a end for the case g 2 Y = 10 −6 .
To estimate the the scale factor change a R /a end in the reheating epoch, we approximately use the conservation law (7.18)for the massive pair domination

Observations to fix reheating temperature and entropy
We use the method proposed by Ref. [54] to fix the reheating temperature by the CMB observations.The cosmological evolution of the physical wavelength λ(a) and wavenumber k(a) is where the present time a 0 = 1, the comoving wavenumber k(a 0 ) and wavelength λ 0 = 1/k 0 are constants in the evolution, see Figure 2. The total increase of the scale factor from the horizon crossing a * (6.1) to a 0 is given by At the CMB pivot scale λ(a * ) = H −1 * = k −1 * , the scalar spectrum gives On the other hand, as illustrated in Fig. 2 and Ref. [54], ∆ tot = ∆ 3 ∆ 2 ∆ 1 ∆ 0 , ∆ 1 = (a rec /a R ) = (g * /2) 1/3 (T RH /T rec ) and ∆ 0 = (a 0 /a rec ) = 1 + z rec are given in terms of the temperature T rec = T CMB (1 + z rec ) and redshift z rec at the recombination, (Color Online).Fixed the observed spectral index n s = 0.965, in terms of the tensor-to-scalar ratio r, we plot the inflation end e-folding number N end (6.2) and the reheating temperature T RH (8.6).These plots refer to their lower limits due to the nature of inequality (6.2).The real values of N end and T RH should be slightly above the curves for a given r value.
Equations (8.3) and (8.5) are independent of the effective reheating degeneracy g * and yield the reheating temperature in terms of the CMB observations T CMB = 2.725 K = 2.348 × 10 −4 eV and k * = 0.05Mpc −1 ( Mpc −1 = 6.39 × 10 −30 eV), as well as N end (6.2) and H end (6.1),whose values depend on the CMB measurements A s , n s and r, see Sec. 6.
8.1 Reheating temperature and entropy vs tensor-to-scalar ratio r < 0.048 Given the observed the scalar amplitude A s = 2.1×10 −9 and spectral index n s = 0.965, the inflation ending e-folding number N end (6.1) and the reheating temperature T RH (8.6) are plotted in Fig. 8 as functions of the tensor-to-scalar ratio r without any free parameter.Figure 8 shows that their values are N end ≈ (50, 60) and T RH /M pl ≈ (5.5 × 10 −13 , 1.1 × 10 0 ) in the range r ≈ (0.037, 0.052).After obtaining the reheating temperature at the end a R of reheating, we calculate the entropy S patch produced within the physical patch of the volume H −3 RH , which evolves from the initial patch of the volume H −3 * at the start a * = 1 of the inflation.The patch grows by a scale factor of Eqs.(6.1) and (7.36),In Fig. 10, we numerically plot the constraints (8.10) and (8.11) as a function of the tensor-to-scalar ratio r.Recall that in Sec.7.1.2we point out the theoretical threshold mthresh : m > 20m pl = 4M pl (7.8) for ρ M ρ Λ .We conclude that the nontrivial threshold mthresh demands the tensor-to-scalar ratio r > 0, see also Eq. (6.2).Applying the obtained theoretical threshold mthresh = 4M pl to numerical results Fig. 10, we find that the tensor-to-scalar ratio r 0.044.However, it depends on the parameter χ ≈ 1.85 × 10 −3 adopted and other approximations used for numerical calculations.To obtain the lower limit r min = 0, it requires more elaborated calculations, for example, separating in Eq. (4.1) unstable modes' contribution ρ H M | unstable from stable modes' contribution ρ H M | stable to more accurately determine the mass parameter threshold mthresh .Nevertheless, the obtained r-range (0 < r < 0.047) is relevant to the measurements by the next generation CMB observations, such as CMB-S4 which measures r 10 −3 [60].
We report the numerical results in the r range [0.042, 0.048].The effective pair mass parameter is in 10 −1 m/M pl 10 4 and the effective Yukawa coupling is in 10 −14 (g 2 Y /g 1/2 * ) 10 −9 .If g * 10 2 for the standard model of particle physics, including sterile neutrinos, the Yukawa coupling is in the range of 10 −13 g 2

Stable massive pairs and cold dark matter abundance
To further clarify dark energy and matter interaction, we study stable massive pairs after reheating.These pairs neither decay into light particles nor contribute to reheating.Thus, they remain and can be candidates for massive cold dark matter (CDM) particles.Here, we qualitatively discuss their evolution after reheating.

Summary
We make a summary to close this lengthy article.In the ρ Λ -dominated inflation H > Γ M , where the massive pair plasma density ρ H M is small and normal matter density ρ M is negligible.The reheating epoch starts Γ M > H, ρ H M and ρ M become large, and their back reaction and decay to relativistic particles are important.Therefore, the cosmic rate equation (5.5) governing the processes ρ H M ⇔ ρ M (5.3) and unstable massive pairs' decay F F → ¯ are relevant.It is an additional dynamical equation to two Friedman equations (2.2,2.3) and the reheating equation (5.10) from energy conservation.
Using the massive pair plasma density ρ H M (6.5), production rate Γ M and decay rate Γ de M (6.10), we study the reheating epoch by a close system of four dynamical (ordinary differential equations) equations (6.6-6.9) for the horizon H and three densities ρ Λ,M,R .The initial conditions are given by the inflation end.Numerically solving this system, we find three characteristic episodes: (i) the P-episode of the transition from the inflation end to the reheating start, when the pair-production rate is much larger than the Hubble rate (Γ M H), the dark energy density ρ Λ quickly decreases and converts to the matter-energy density ρ M .As a consequence ρ Λ ρ M ; (ii) the M-episode of massive pairs domination, where the back-and-forth interaction of the cosmic rate equation (5.5) plays an essential role, and dark energy density slowly varies; (iii) the R-episode of the genuine reheating ρ R ρ M , when unstable massive pairs predominately decay to relativistic particles that quickly thermalised.
We emphasise the pair mass threshold m > mthresh (7.8) that at the pre-reheating start ρ Λ ρ M ρ R , the rapid converting process ρ Λ ⇒ ρ M ⇒ ρ R leads to ρ Λ ρ M ρ R .The most relevant mass-energy and entropy of Universe are produced by the end of reheating.Such dynamic processes should lead to the emission of primordial gravitational waves [61,62].
The initial conditions for the reheating epoch are the Hubble scale H end and energy densities ρ end Λ,M at the end of inflation after e-folding number N end .They are determined by the CMB measurements of scalar amplitude A s and spectral index n s at pivot scale k * = 0.05(Mpc) −1 and the Hubble scale H * [41].We obtain the reheating Hubble scale H RH , temperature T RH and entropy S patch at the genuine reheating episode.They are functions of the tensor-to-scalar ratio r, and their numerical values are in accordance with the CMB observations.Moreover, from purely theoretical viewpoints, we preliminarily limit the r values in the range 0 < r < 0.047.Among massive pairs gravitationally produced, (i) unstable pairs decay to relativistic particles, accounting for reheating; (ii) stable pairs couple only to gravity and are candidates for cold dark matter.The resultant cold dark matter abundance Ω c ∼ 10 −1 is about a constant in time.They should play the role in the formation of primordial black holes.

Acknowledgment
The author thanks the EPJC Editor Alexei Starobinsky and anonymous referees for their reviews and reports that give chances to improve the manuscript.
12 Appendix: Quantum pair oscillation details In microscopic time, we plot the Bogoliubov coefficient |β| 2 , the quantum pair density

M ( 3 . 7 )
and ρ fast M and (3.8) as the fast components in Eqs.(2.4,2.5) to find their non-adiabatic backreactions on fast components H fast and ρ fast Λ .We will study the reheating epoch when the Hubble scale and pair mass are very much smaller than the Planck scale, i.e., H slow < M m pl and N pair 1.Therefore, in the unit of the mass M and the critical density ρ crit = 3m

4. 1
Reasons for a macroscopic descriptionWe see the non-adiabatic and back-reacting phenomena of massive pairs' production and oscillation at a time scale O(1/M ).How the fast oscillating components P fast M 10) in Eqs.(2.4,2.5)couple to the slow components in Friedman equations (2.2,2.3).It is a difficult task to simultaneously analyze O(1/M ) and O(1/H) backreaction dynamics even numerically since two scales M

Figure 1
which does not oscillate alternatively between negative and positive values.Its time average fast M + P fast M does not vanish.Other fast-component averages do not vanish as well.Fast-component averages have another time scale τ M (5.1) in response to the slow horizon variations H slow in macroscopic time.It is a kind of "relaxation" ) of the Boltzmann type for the the back-and-forth ρ M and ρ H M interaction (5.3) in the Universe's evolution.It represents a general conservation law of dark energy and matter, including massive pair plasma state ρ H M (4.1) with the production rate (5.1).The term 3(1 + ω M )Hρ M of the time scale [3(1 + ω M )H] −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 detailed balance term Γ M (ρ H M − ρ M ) indicates how two densities ρ H M and ρ M of different time scales couple together.The ratio Γ M /H > 1 indicates the coupled case, and Γ M /H < 1 indicates the decoupled case.The last term Γ de M ρ M represents unstable massive pairs' decay to relativistic particle pairs ¯ , and the decay rate and time are given by

)Figure 2 .
Figure2.We make this figure by modifying Fig.1in Ref.[54].Schematic evolution of the Hubble radius H −1 and the physical length scale λ(a), where physically interested scale λ 0 = λ(a 0 ) at the present time a 0 = 1 crossed the Hubble horizon H * at the early time a * , fixed by the CMB pivot scale λ 0 = λ * = k −1 * .The pre-inflation a > a * , the inflation a * < a < a end , the reheating a end < a < a R , and the recombination at a rec .

Figure 3 .
Figure 3. (Color Online).In a few e-folding number x = ln(a/a end ), (a) the Hubble rate drops rapidly; (b) the massive pair energy density exceeds the dark-energy density; (c) the ratio Γ M /H > 1 increases rapidly; (d) the -rate of H variation increases in the transition from 1 (inflation epoch) to the asymptotic value ∼ O(1) (M-episode, see Sec. 7.2).We plot these solutions with the initial condition (6.4) and parameter ( m/m pl ) = 27.7.

Figure 5 .
Figure 5. (Color Online).In a few e-folding number x = ln(a/a end ), (a) the energy density ρ H M (6.5) (orange) and the solution ρ M (blue) to the cosmic rate equation (7.4), showing ρ M approaches ρ H M ; (b) the detailed balance term D M (7.15) vanishes.We plot these solutions with the initial condition (6.4) and parameter ( m/m pl ) = 27.7.

Y 10 − 8 .
We check back the parameters used in Figs.6 and 7are ( m/m pl ) = 27.7 and g 2 Y = 10 −9 .It indicates that the ΛCDM scenario is self-consistent and self-contained.
fast Λ and pressure P fast Λ , as well as the fast components of Hubble function h fast , and cosmological term fast Λ .Recall P fast Λ ≈ − fast Λ .
The natures of the massive coherent pair state |N pair (3.6) of the pressure (3.7) and density (3.8) are rather generic for non-adiabatic production of massive particles in curved spacetime.The coherent state |N pair (3.6) and (3.7,3.8)should be valid also for M H, provided the pair occupation number N pair 1.Note that p fast and density of massive coherent pair state (3.6) in short quantum time sales O(1/M ).They do not follow the usual equation of the state of classical matter.
M (3.7) and ρ fast M (3.8) represent the quantum pressure [41]how the quantum pair density and pressure oscillations in microscopic time t in the unit of M −1 , by using H slow /M ≈ 10 −3 , M 10 −5 m pl , N pair 10 12 and δ = 1, with C 0 = (3/2)h fast (H slow /M ) and verified condition |α| 2 − |β| 2 = 1.It shows that a large number of massive pairs creates significantly oscillating quantum pressure P fast fast and ḣfast , see Fig.11in Appendix.Note that the pair number N pair , mass scales M and H slow values differ from those used for inflation see Fig.1in Ref.[41].
Λ, h ) of fast components ( fast M , P fast M , h fast , ḣfast , fast details at the scale O(1/M ) average out and become irrelevant for the macroscopic scale O(τ H ) and O(τ M ) processes: inflation, reheating and standard cosmology.The relevant quantities and equations are massive pair plasma state p H Λ for the inflation epoch, when the radio Γ M /H ∝ * production costs dark energy, but it adds into matter-energy ρM + 3(1 + ω M )Hρ M ≈ Γ M ρ H M 0. The exchange rate δQ ≈ Γ M ρ H M 0 ispositive and small.It yields ρΛ 0 and Ḣ 0, i.e., slow-rolling dynamics for inflation.Dark energy converts slowly to matter till inflation ends when Γ M /H ≈ 1. (ii) In a short pre-reheating episode, when Γ M /H 1 and ρ Λ > ρ H M > ρ M , the exchange rate δQ 1 is large.Dark energy rapidly converts to matter till ρ H M ≈ ρ M > ρ Λ .The conversion is very efficient.Details will be in Sec.7.1.(iii) The coupled case is Γ M /H > 1 and ρ H M ≈ ρ M , when ρ H M tightly couples with ρ M in the Universe evolution of the Hubble time scale τ H . Dark energy and matter exchange rate δQ = Γ M ρ H M − ρ M ≈ 0 is very small.Dark energy is almost constant in time ρΛ ≈ 0. It is the case for the matter-dominated episode in reheating, see Sec. 7.2, and for stable massive pairs (cold dark matter) evolution, see Sec. M m/2 the Hubble rate is in the same order as the pair decay rate; (ii) H 2 RH ≈ ρ R /3m 2 pl the radiation energy is predominate.These results depend on the effective degeneracy g * (7.28) and the decay rate τ −1 de M /2 = g 2 Y R = g 2 4 [59]he ΛCDM scenario, there are two parameters to describe the properties of the reheating epoch: (i) the effective mass parameter m/M pl physically represents pairs' masses and numbers; (ii) the effective Yukawa coupling (g 2 RH (7.33) and the ratio T RH / m ≈ 3.3×10 −7 , obtained by the baryon number-to-entropy ratio n B /S R = 0.864 +0.016 −0.015 × 10 −10 [58], theoretical relation n B /S R ≈ CP (T RH / m)[49]and CP ≈ 2.6 × 10 −4 , see Eq. (5.3) of Ref.[59].