# Massive particle pair production and oscillation in Friedman Universe: dark
energy and matter interaction

She-Sheng Xue

###### Abstract

The classical Friedman equations of time-varying Hubble function $H$, dark-
energy and matter densities couple to quantised field equations for massive
modes $M\gg H$. Numerically solving these equations, we show the particle-
antiparticle pairs production and oscillation in microscopic time scale
${\mathcal{O}}(M^{-1})$. A massive pair plasma state is formed in macroscopic
time scale ${\mathcal{O}}(H^{-1})$. Its density and pressure introduce the
interaction of matter and dark energy densities in the Friedman equations.
Focusing on epochs after reheating, we show that the negative dark energy
tracks down the radiation energy in the radiation epoch. Such tracking
dynamics ends, and dark energy becomes positive in the matter epoch. The
matter converts to dark energy, and their present values are comparable,
explaining the cosmic coincidence. As a result, a class of effective
interacting dark energy models is advocated to confront cosmological
observations.

##  1 Introduction

The Universe’s evolution is gravitationally governed by matter and dark
energy. The latter can be represented by the cosmological $\Lambda$ 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 $\tilde{\Lambda}(t)$ 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. [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. 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 $\tilde{\Lambda}(t)$ and massive pairs $M\gg H$.

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 $M\gg H$.
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
${\mathcal{O}}(1/M)$ that is much smaller than classical Universe evolution
time scale ${\mathcal{O}}(1/H)$. To properly include the back-reaction of
particle production on Universe evolution, one should separate fast components
${\mathcal{O}}(1/M)$ from slow components ${\mathcal{O}}(1/H)$ 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 [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66].

In this article, we start with the Friedman equations for a flat Universe

$\displaystyle H^{2}=\frac{8\pi G}{3}\rho;\quad\dot{H}=-\frac{8\pi
G}{2}(\rho+p),$ (1.1)

where energy density $\rho\equiv\rho_{{}_{M}}+\rho_{{}_{\Lambda}}$ and
pressure $p\equiv p_{{}_{M}}+p_{{}_{\Lambda}}$. Equation of state
$p_{{}_{\Lambda}}=-\rho_{{}_{\Lambda}}$ is for the cosmological term (dark
energy), $p_{{}_{M}}=\omega_{{}_{M}}\rho_{{}_{M}}$ 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 $\rho_{{}_{\Lambda}}(t)\equiv\tilde{\Lambda}(t)/8\pi G$ [67], and it
reduces to the usual equation
$\dot{\rho}_{{}_{M}}+(1+\omega_{{}_{M}})H\rho_{{}_{M}}=0$ for time-constant
$\rho_{{}_{\Lambda}}$. We adopt the approach [42] to describe the
decomposition of slow and fast components: scale factor $a=a_{\rm slow}+a_{\rm
fast}$, Hubble function $H=H_{\rm slow}+H_{\rm fast}$, cosmological term and
matter density $\rho_{{}_{\Lambda,M}}=\rho^{\rm
slow}_{{}_{\Lambda,M}}+\rho^{\rm fast}_{{}_{\Lambda,M}}$ and pressure
$p_{{}_{\Lambda,M}}=p^{\rm slow}_{{}_{\Lambda,M}}+p^{\rm
fast}_{{}_{\Lambda,M}}$. 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 $\lambda$ of fast and slow components, the Friedman
equations (1.1) are decomposed into two sets. The slow components
${\mathcal{O}}(\lambda^{0})$ obey the same equations as usual Friedman
equations

$\displaystyle H_{\rm slow}^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3}(\rho_{{}_{M}}^{\rm slow}+\rho_{{}_{\Lambda}}^{\rm slow});$
$\displaystyle\quad\dot{H}_{\rm slow}$ $\displaystyle=$
$\displaystyle-\frac{8\pi G}{2}(\rho_{{}_{M}}^{\rm slow}+p_{{}_{M}}^{\rm
slow}),$ (1.2)

where $H_{\rm slow}=\dot{a}_{\rm slow}/a\approx\dot{a}_{\rm slow}/a_{\rm
slow}$, time derivatives $\dot{H}_{\rm slow}$ and $\dot{a}_{\rm slow}$ relate
to the macroscopic “slow” time variation scale ${\mathcal{O}}(1/H)$. The
faster components ${\mathcal{O}}(\lambda^{1})$ obey,

$\displaystyle H_{\rm fast}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{2\times 3H_{\rm slow}}(\rho_{{}_{M}}^{\rm fast}+\rho_{{}_{\Lambda}}^{\rm
fast});$ $\displaystyle\dot{H}_{\rm fast}$ $\displaystyle=$
$\displaystyle-\frac{8\pi G}{2}(\rho_{{}_{M}}^{\rm fast}+p_{{}_{M}}^{\rm
fast}),$ (1.3)

where $H_{\rm fast}=\dot{a}_{\rm fast}/a\approx\dot{a}_{\rm fast}/a_{\rm
slow}$, time derivatives $\dot{H}_{\rm fast}$ and $\dot{a}_{\rm fast}$ relate
to the microscopic “fast” time variation scale ${\mathcal{O}}(1/M)$, and slow
components are approximated as constants in “fast” time variation. For the
cosmological term, equation of state $p_{{}_{\Lambda}}=-\rho_{{}_{\Lambda}}$
becomes $p_{{}_{\Lambda}}^{\rm slow,fast}=-\rho_{{}_{\Lambda}}^{\rm
slow,fast}$ respectively at order ${\mathcal{O}}(\lambda^{0})$ and
${\mathcal{O}}(\lambda^{1})$. In due course we shall clarify the equation of
state $p_{{}_{M}}=\omega_{{}_{M}}\rho_{{}_{M}}$ for the matter term.

We adopt the approach [31] to describe the fast components of matter density
$\rho_{{}_{M}}^{\rm fast}$ and pressure $p_{{}_{M}}^{\rm fast}$, that are
attributed to the non-adiabatic production of particle and antiparticle pairs
in fast time variation $H_{\rm fast}=\dot{a}_{\rm fast}/a_{\rm slow}$. 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.

##  2 Quantum pair production and oscillation

A quantised massive scalar matter field inside the Hubble sphere volume $V\sim
H^{-3}_{\rm slow}$ of Friedman Universe reads

$\displaystyle\Phi({\bf x},t)$ $\displaystyle=$
$\displaystyle\sum_{n}A_{n}Y_{n}({\bf x})\psi_{n}(t),$ (2.1)

which exponentially vanishes outside the horizon $H^{-1}_{\rm slow}$, and
$\int_{V}Y_{n}({\bf x})Y^{\dagger}_{n^{\prime}}({\bf
x})h^{1/2}d^{3}x=\delta_{nn^{\prime}}$. The principal quantum number
“$n=0,1,2,\cdot\cdot\cdot$” stands for for quantum states of physical wave
vectors $k_{n}$, $n=0$ and $k_{0}=0$ for the ground state 111In Ref. [31], the
principal quantum number $n$ is the angular momentum number
“$\ell=0,1,2,\cdot\cdot\cdot$” and $Y_{n}({\bf x})=Y_{\ell,m}({\bf x})$ are
the four-dimensional spherical harmonics for the closed Robertson-Walker
metric. The ground state is $\ell=0$. . The $A_{n}$ and $A_{n}^{\dagger}$ are
time-independent annihilation and creation operators satisfying the
commutation relation $[A_{n}^{\dagger},A_{n}]=\delta_{n,n^{\prime}}$. The
time-separate equation for $\psi_{n}(t)$ is

$\displaystyle\partial_{t}^{2}\psi_{n}(t)+\omega_{n}(t)^{2}\psi_{n}(t)=0,\quad\omega_{n}(t)^{2}=k^{2}_{n}+M^{2},$
(2.2)

and Wronskian-type condition
$\psi_{n}(t)\partial_{t}\psi^{*}_{n}(t)-\psi^{*}_{n}(t)\partial_{t}\psi_{n}(t)=i$.
Expressing

$\displaystyle\psi_{n}(t)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{(2V\omega_{n})^{1/2}}\left(\alpha^{*}_{n}(t)e^{-i\int^{t}\omega_{n}dt}+\beta^{*}_{n}(t)e^{i\int^{t}\omega_{n}dt}\right)$
(2.3)

in terms of $\alpha_{n}(t)$ and $\beta_{n}(t)$, Equation (2.2) becomes

$\displaystyle\partial_{t}\alpha_{n}(t)$ $\displaystyle=$ $\displaystyle
C_{n}e^{-2i\int^{t}\omega_{n}dt}\beta_{n}(t);$
$\displaystyle\partial_{t}\beta_{n}(t)$ $\displaystyle=$ $\displaystyle
C_{n}e^{2i\int^{t}\omega_{n}dt}\alpha_{n}(t),$ (2.4)

and $|\alpha_{n}|^{2}-|\beta_{n}|^{2}=1$, where $C_{n}\equiv
3H\omega_{n}^{-2}[k_{n}^{2}/3+M^{2}/2]$. In an adiabatic process for slowly
time-varying $H=H_{\rm slow}$, the particle state $\alpha_{n}(0)=1$ and
$\beta_{n}(0)=0$ evolve to $|\alpha_{n}(t)|\gtrsim 1$ and
$|\beta_{n}(t)|\not=0$. Positive and negative frequency modes get mixed,
leading to particle productions of probability $|\beta_{n}(t)|^{2}\propto
e^{-M/H_{\rm slow}}$.

We will focus on studying particle production in non-adiabatic processes for
rapidly time-varying $H_{\rm fast}$, $\alpha_{n}$ and $\beta_{n}$ in the
ground state $n=0$ of the lowest lying massive mode $M\gg H$. First, we recall
that Parker and Fulling introduced the transformation [31],

$\displaystyle A_{0}=\gamma^{*}B+\delta B^{\dagger},\quad B=\delta
A^{\dagger}_{0}-\gamma A_{0},$ (2.5)

$[B,B^{\dagger}]=1$, and two mixing constants obeying
$|\gamma|^{2}-|\delta|^{2}=1$. For a given $A_{n}$ and its Fock space, the
state $|{\mathcal{N}}_{\rm pair}\rangle$ is defined by the conditions
$A_{n\not=0}|{\mathcal{N}}_{\rm pair}\rangle=0$ and

$\displaystyle B^{\dagger}B|{\mathcal{N}}_{\rm pair}\rangle={\mathcal{N}}_{\rm
pair}|{\mathcal{N}}_{\rm pair}\rangle.$ (2.6)

The $B^{\dagger}$ and $B$ are time-independent creation and annihilation
operators of the pair of mixed positive frequency $A_{0}$ particle and
negative frequency $A_{0}^{\dagger}$ antiparticle. The state
$|{\mathcal{N}}_{\rm pair}\rangle$ contains ${\mathcal{N}}_{\rm
pair}=1,2,3,\cdot\cdot\cdot$ pairs, and $|{\mathcal{N}}_{\rm pair}=0\rangle$
is the ground state of non-adiabatic interacting system of fast varying
$H_{\rm fast}$ and massive pair production and annihilation 222Discussions can
be applied for fermion fields. Analogously, we discussed the back and forth
processes of massive fermion and antifermion pairs production and annihilation
in spacetime ${\mathcal{S}}\Leftrightarrow\bar{F}+F$ in Refs. [45, 66]. It is
a coherent superposition of states of particle and anti-particle pairs. In
this coherent condensate state $|{\mathcal{N}}_{\rm pair}\rangle$ and
${\mathcal{N}}_{\rm pair}\gg 1$, neglecting higher mode $n\not=0$
contributions, they obtained the negative quantum pressure and positive
quantum density of coherent pair field, see Eqs. (59) and (60) of Ref. [31],

$\displaystyle p^{\rm fast}_{{}_{M}}$ $\displaystyle=$
$\displaystyle-\frac{M(2{\mathcal{N}}_{\rm pair}+1)}{2\pi^{2}V}\Big{\\{}{\rm
Re}[\gamma^{*}\delta(|\alpha|^{2}+|\beta|^{2})]$ (2.7) $\displaystyle+$
$\displaystyle(2|\delta|^{2}+1){\rm Re}(\alpha^{*}\beta e^{2iMt})\Big{\\}},$
$\displaystyle\rho^{\rm fast}_{{}_{M}}$ $\displaystyle=$
$\displaystyle\frac{M(2{\mathcal{N}}_{\rm pair}+1)}{\pi^{2}V}\Big{\\{}{\rm
Re}[\gamma\delta^{*}\alpha\beta)]$ (2.8) $\displaystyle+$
$\displaystyle(|\delta|^{2}+1/2)(|\beta|^{2}+1/2)\Big{\\}},$

where $\omega_{n=0}=M$, $\alpha_{n=0}=\alpha$ and $\beta_{n=0}=\beta$.
Equations (2.7) and density (2.8) were adopted for studying the avoidance of
cosmic singularity in curved Universe. Note that $p^{\rm fast}_{{}_{M}}$ (2.7)
and $\rho^{\rm fast}_{{}_{M}}$ (2.8) represent the quantum pressure and
density of massive coherent pair state (2.6) in short quantum time sales
${\mathcal{O}}(1/M)$. They do not follow an usual equation of state of
classical matter.

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 $M\gg H_{\rm slow}$ and large number
${\mathcal{N}}_{\rm pair}\gg 1$ pairs, and $M(2{\mathcal{N}}_{\rm pair}+1)$ in
(2.7) and (2.8) can be larger than the Planck mass $m_{\rm pl}$ so that higher
mode $(n\not=0)$ contributions can be neglected. Moreover, we adopt (2.7) and
(2.8) as the fast components $\rho^{\rm fast}_{{}_{M}}$ and $p^{\rm
fast}_{{}_{M}}$ in Eq. (1.3) to find their non-adiabatic back-reactions on
fast components $H_{\rm fast}$ and $\rho^{\rm fast}_{{}_{\Lambda}}$.

Figure 1: We show the quantum pair density and pressure oscillations in
microscopic time $t$ in unit of $M^{-1}$, using $H_{\rm slow}/M\approx
10^{-3}$, $M\simeq 10^{-10}m_{\rm pl}$, ${\mathcal{N}}_{\rm pair}\simeq
10^{25}$ and $\delta=1$. It is clear that for $H_{\rm slow}\ll m_{\rm pl}$ and
$M\ll m_{\rm pl}$, a large amount of massive pairs ${\mathcal{N}}_{\rm
pair}\gg 1$ is created for significant oscillating quantum pressure (2.9) and
density (2.10). For details see Fig. 3 in Supplemental Material.

Here, we study the epochs after reheating, when the Hubble scale and pair mass
are very much smaller than the Planck mass, i.e., $H_{\rm slow}<M\ll m_{\rm
pl}$ and ${\mathcal{N}}_{\rm pair}\gg 1$. Therefore, for a given $H_{\rm
slow}$, we express in unit of the critical density $\rho_{\rm crit}=3m_{\rm
pl}^{2}H^{2}_{\rm slow}$ the dimensionless quantum pressure (2.7) and density
(2.8) as

$\displaystyle{\mathcal{P}}^{\rm fast}_{{}_{M}}$ $\displaystyle=$
$\displaystyle-\frac{\bar{M}H_{\rm slow}}{6\pi^{2}m_{\rm pl}}\Big{\\{}{\rm
Re}[\gamma^{*}\delta(|\alpha|^{2}+|\beta|^{2})]+(2|\delta|^{2}+1){\rm
Re}(\alpha^{*}\beta e^{2iMt})\Big{\\}},$ (2.9) $\displaystyle{\varrho}^{\rm
fast}_{{}_{M}}$ $\displaystyle=$ $\displaystyle+\frac{\bar{M}H_{\rm
slow}}{3\pi^{2}m_{\rm pl}}\Big{\\{}{\rm
Re}[\gamma\delta^{*}\alpha\beta)]+(|\delta|^{2}+1/2)(|\beta|^{2}+1/2)\Big{\\}},$
(2.10)

where $\bar{M}\equiv(2{\mathcal{N}}_{\rm pair}+1)(M/m_{\rm pl})$ and the
reduced Planck mass $m_{\rm pl}\equiv(8\pi G)^{-1/2}$. The faster component
equations (1.3) become,

$\displaystyle h_{\rm fast}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\varrho_{{}_{M}}^{\rm
fast}+\varrho_{{}_{\Lambda}}^{\rm fast});$ $\displaystyle\dot{h}_{\rm fast}$
$\displaystyle=$ $\displaystyle-\frac{3}{2}(\varrho_{{}_{M}}^{\rm
fast}+{\mathcal{P}}_{{}_{M}}^{\rm fast}),$ (2.11)

where $h_{\rm fast}\equiv H_{\rm fast}/H_{\rm slow}$ and
$\varrho_{{}_{\Lambda}}^{\rm fast}\equiv\rho_{{}_{\Lambda}}^{\rm
fast}/\rho_{\rm crit}$.

Using negative ${\mathcal{P}}^{\rm fast}_{{}_{M}}$ (2.9) and positive definite
$\varrho^{\rm fast}_{{}_{M}}$ (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 $t\sim H^{-1}_{\rm fast}$, which is
around the macroscopic time $t_{\rm slow}\sim H^{-1}_{\rm slow}$, when the
slow components $a_{\rm slow}$, $H_{\rm slow}$, $\rho^{\rm
slow}_{{}_{M,\Lambda}}$ and $p^{\rm slow}_{{}_{M,\Lambda}}$ are valued,
following the Friedman equations (1.2). The integrals $\int^{t}\omega_{n}dt$
are over the microscopic time $t$ characterised by the Compton time scale
$1/M$. Its lower limit is $t=0$ by setting $t_{\rm slow}=0$ as a reference
time, when $a_{\rm fast}(0)=0$,

$\displaystyle H_{\rm fast}(0)=\dot{a}_{\rm fast}/a_{\rm
slow}=0;\quad\alpha(0)=1,\quad\beta(0)=0.$ (2.12)

The real value $\gamma^{*}\delta$ condition in Eqs. (2.9) and (2.10) leads to
the time symmetry: $a^{\rm fast}(t)=a^{\rm fast}(-t)$,
$\alpha(t)=\alpha^{*}(-t)$ and $\beta(t)=\beta^{*}(-t)$ [31]. When
$t\leftrightarrow-t$, positive and negative frequency modes interchange. Here
we use $a_{\rm slow}\not=0$, $H_{\rm slow}\not=0$ and co-moving radius
$(Ha)^{-1}\approx(H_{\rm slow}a_{\rm slow})^{-1}$ of Hubble volume $V\sim
H_{\rm slow}^{-3}$.

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 $C_{0}=(3/2)h_{\rm fast}(H_{\rm slow}/M)$ and verified condition
$|\alpha|^{2}-|\beta|^{2}=1$. In the quantum period of microscopic time $t$,
the negative quantum pressure ${\mathcal{P}}^{\rm fast}_{{}_{M}}<0$ and back-
reaction effects lead to the quantum pair oscillation characterised by the
frequency $\omega=M$ of massive quantised pair fields. The positive quantum
pair density $\varrho^{\rm fast}_{{}_{M}}>0$ indicates particle creations
without $e^{-M/H}$ suppression. It is consistent with increasing Bogoliubov
coefficient $|\beta(t)|^{2}$ that mixes positive and negative energy modes.
Observe that $\varrho^{\rm fast}_{{}_{M}}\gg|{\mathcal{P}}^{\rm
fast}_{{}_{M}}|$ and the sum $\varrho^{\rm fast}_{{}_{M}}+{\mathcal{P}}^{\rm
fast}_{{}_{M}}>0$ is positive definite, leading to the decreasing $h_{\rm
fast}(t)$ (1.3). As a consequence, for time $t>0$, the fast components $h_{\rm
fast}$ and $\varrho^{\rm fast}_{{}_{\Lambda}}$ decrease in time, in order for
pair production. Whereas for time $t<0$, $h_{\rm fast}$ and $\varrho^{\rm
fast}_{{}_{\Lambda}}$ increases, due to pair annihilation. The small $a_{\rm
fast}(t)$ varies around $a_{\rm slow}$ at $t_{\rm slow}\equiv 0$.

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 $e^{-\pi
M^{2}/E}$ [69]. The coherent plasma state of electron-positron pairs is
analogous to the coherent pair state $|{\mathcal{N}}_{\rm pair}\rangle$ (2.6).

##  3 Massive pair plasma state

As shown in Fig. 1, massive pair quantum pressure ${\mathcal{P}}^{\rm
fast}_{{}_{M}}$ (2.9) and density $\varrho^{\rm fast}_{{}_{M}}$ (2.9) can be
significantly large and rapidly oscillate with the fast components $h_{\rm
fast}$ and $\varrho^{\rm fast}_{{}_{\Lambda}}$ (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 $M\gg H_{\rm slow}$. 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 $\langle\cdot\cdot\cdot\rangle$ over
the microscopic period in time. Figure 1 shows $\langle\varrho^{\rm
fast}_{{}_{M}}+{\mathcal{P}}^{\rm fast}_{{}_{M}}\rangle$ and other averages of
fast oscillating components do not vanish. Second the spatial dependence of
pair quantum pressure ${\mathcal{P}}_{\rm fast}$ (2.7) and density
$\varrho_{\rm fast}$ (2.7) are unknown, since they are obtained by using the
vacuum expectation value of field $\Phi({\bf x},t)$ energy-momentum tensor
over entire space. For the case $M\gg H_{\rm slow}$, the Compton length
$M^{-1}$ of ground state $n=0$ is much smaller than the Hubble horizon
$H^{-1}_{\rm slow}$. 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
$\rho^{H}_{{}_{M}}$ densities as,

$\displaystyle\rho^{H}_{{}_{M}}\equiv 2\chi m^{2}H^{2}_{\rm slow},\quad
n^{H}_{{}_{M}}\equiv\chi mH^{2}_{\rm slow};\quad
m^{2}\equiv\sum_{f}g_{d}^{f}M^{2}_{f},$ (3.1)

and pressure $p^{H}_{{}_{M}}=\omega^{H}_{{}_{M}}\rho^{H}_{{}_{M}}$. The
$\omega^{H}_{{}_{M}}\approx 0$ for $m\gg H_{\rm slow}$ and its upper limit is
$1/3$. The introduced mass parameter $m$ represents possible particle masses
$M_{f}$, degeneracies $g_{d}^{f}$ and the mixing coefficient $\delta$ (2.5).
The degeneracies $g_{d}^{f}$ plays the same role of pair number
${\mathcal{N}}_{\rm pair}$ in Eq. (2.8), namely
$\sum_{f}g_{d}^{f}\approx(2{\mathcal{N}}_{\rm pair}+1)$. We explain the
reasons why the densities (3.1) are proportional to $\chi mH^{2}_{\rm slow}$,
rather than $H^{3}_{\rm slow}$ from the entire Hubble volume $V$. The “surface
area” factor $H^{2}_{\rm slow}$ is attributed to the spherical symmetry of
Hubble volume. The “radial size” factor $\chi m$ comes from the layer width
$\lambda_{m}$ introduced as an effective parameter to describe the properties
(i) for $m\gg H_{\rm slow}$ the massive pair plasma is localised as a
spherical layer and (ii) its radial width $\lambda_{m}<H^{-1}_{\rm slow}$
depends on the massive pair plasma oscillation dynamics 333It may also include
self-gravitating dynamics, due to pair plasma are very massive., rather than
the $H_{\rm slow}$ dynamics govern by the Friedman equations (1.2). The width
parameter $\chi$ expresses the layer width $\lambda_{m}=(\chi m)^{-1}\gg 1/m$
in terms of the effective Compton length $1/m$,

$\displaystyle\lambda_{m}=(\chi m)^{-1}<H^{-1}_{\rm slow},\quad
1\gg\chi>(H_{\rm slow}/m).$ (3.2)

Because parameters $m$ and $\chi m$ represent time-averaged values over fast
time oscillations of massive pair plasma state, we consider $m$ and $\chi m$
as approximate constants in slowly varying macroscopic time. However, the $m$
and $\chi m$ values, namely the $M_{f}$ and $g_{d}^{f}$ 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 $H_{\rm slow}$ value, see Sec. 2.
We will duly come back to this point how characteristic value $\chi m$ relates
to the Hubble function $H_{\rm slow}$ in a given evolution epoch.

We have to point out that (i) the pressure $p^{H}_{{}_{M}}$ and density
$\rho^{H}_{{}_{M}}$ (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 $p_{{}_{M}}=\omega_{{}_{M}}\rho_{{}_{M}}$ and (ii) the
massive pair plasma state of pressure and density
$p^{H}_{{}_{M}}=\omega^{H}_{{}_{M}}\rho^{H}_{{}_{M}}$. 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.

##  4 Cosmic rate equation

Up to macroscopic time $H^{-1}$, we estimate the total number of particles
produced inside the Hubble sphere $N\approx n^{H}_{{}_{M}}H^{-3}/2$ and mean
pair production rate w.r.t. macroscopic time

$\displaystyle\Gamma_{M}\approx\frac{dN}{2\pi dt}\approx\frac{\chi
m}{4\pi}\epsilon,\quad\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{3}{2}\frac{(1+\omega_{{}_{M}})\rho_{{}_{M}}}{\rho_{{}_{\Lambda}}+\rho_{{}_{M}}}.$
(4.1)

It is not a theoretical derivation, but modelling parameterized by $\chi m$
and Universe evolution rate $\epsilon$. The asymptotic values $\epsilon\approx
2$ and $\epsilon\approx 3/2$ are respectively for radiation and matter epoch.
Here we neglect the back-reactions of slow time-varying components $H$,
$\rho_{{}_{\Lambda,M}}$ and $p_{{}_{\Lambda,M}}$ on fast components $H_{\rm
fast}$, $\rho^{\rm fast}_{{}_{M}}$ and $p^{\rm fast}_{{}_{M}}$.

We turn to study how the massive pair plasma density interacts with the matter
density $\rho_{{}_{M}}$ that governs the Universe evolution,

$\displaystyle\rho^{H}_{{}_{M}}\Leftrightarrow\rho_{{}_{M}}.$ (4.2)

Recall the rate equation for the back and forth process
$e^{+}e^{-}\Leftrightarrow\gamma\gamma$ [70, 71, 72, 73]:

$\displaystyle\frac{dn_{e^{+}e^{-}}(t)}{dt}+3Hn_{e^{+}e^{-}}(t)=\langle\sigma
v\rangle\Big{(}n^{2}_{e^{+}e^{-}}\big{|}_{\rm eq}-n^{2}_{e^{+}e^{-}}\Big{)},$
(4.3)

where $n_{e^{+}e^{-}}(t)$ is the density governed by macroscopic evolution and
$n_{e^{+}e^{-}}\big{|}_{\rm eq}$ is the density in an equilibrium with
photons. The RHS represents the averaged interacting rate
$dN/dt\approx\langle\sigma v\rangle n_{e^{+}e^{-}}$ for microscopic detail
balance between $n_{e^{+}e^{-}}$ and $n_{e^{+}e^{-}}\big{|}_{\rm eq}$. They
are coupled for $n_{e^{+}e^{-}}\big{|}_{\rm eq}\approx n_{e^{+}e^{-}}$ and
decoupled $n_{e^{+}e^{-}}\big{|}_{\rm eq}\ll n_{e^{+}e^{-}}$. This motivates
us to propose an effective cosmic rate equation,

$\displaystyle\dot{\rho}_{{}_{M}}+3(1+\omega_{{}_{M}})H\rho_{{}_{M}}$
$\displaystyle=$ $\displaystyle\Gamma_{M}(\rho_{{}_{M}}^{H}-\rho_{{}_{M}})$
(4.4)

for the the back and forth $\rho_{{}_{M}}$ and $\rho^{H}_{{}_{M}}$ interaction
(4.2) in the Universe evolution. It actually represents a general conservation
law of all matter including massive pair plasma density $\rho_{{}_{M}}^{H}$
(3.1) with the production rate (4.1). The term
$3(1+\omega_{{}_{M}})H\rho_{{}_{M}}$ of the time scale $(3H)^{-1}$ represents
the space-time expanding effect on the density $\rho_{{}_{M}}$. While
$\Gamma_{M}\rho_{{}_{M}}^{H}$ is the source term and $\Gamma_{M}\rho_{{}_{M}}$
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
$\Gamma_{M}/H>1$ indicates the coupled case, and $\Gamma_{M}/H<1$ indicate the
decoupled case.

We see how the massive pair plasma density (3.1) and cosmic rate equation
(4.4) affect on the Friedman equations (1.2). The cosmic rate equation (4.4)
combines with Eqs. (1.2), yielding

$\displaystyle\dot{\rho}_{{}_{\Lambda}}$ $\displaystyle=$
$\displaystyle-\Gamma_{M}\left(\rho^{H}_{{}_{M}}-\rho_{{}_{M}}\right).$ (4.5)

Equations (4.4) and (4.5) is reminiscent of a generally modeling interacting
dark energy and matter $\delta Q=\Gamma_{M}(\rho_{{}_{M}}^{H}-\rho_{{}_{M}})$,
based on the total mass-energy conservation, see review [74, 75] and [76, 77,
78]. It shows that the cosmological constant (dark energy)
$\rho_{{}_{\Lambda}}$ and matter energy $\rho_{{}_{M}}$ interact via the
massive pair plasma $\rho^{H}_{{}_{M}}$ produced by massive particle
production and oscillation in the Friedman space. Two cases: (i) dark energy
converts to matter energy when $\rho_{{}_{M}}^{H}>\rho_{{}_{M}}$ and (ii)
matter energy converts to dark energy when $\rho_{{}_{M}}^{H}<\rho_{{}_{M}}$.

Equations (1.2) and (4.4) are a set of first-order ordinary differential
equations, numerical solutions for $\rho_{{}_{M}}$ and $\rho_{{}_{\Lambda}}$
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.

##  5 Radiation and matter dominate epochs

Suppose that all radiation $\rho_{{}_{R}}$ and matter $\rho_{{}_{M}}$
densities were created in the reheating epoch, and they were much larger than
the dark energy density $\rho_{{}_{\Lambda}}$ 444The reheating epoch is
discussed in a separated article Ref. [66]. The standard cosmology started
with the radiation dominated epoch and proceeded to the matter-dominated
epoch. We adopt an analytical way to reveal approximate
$\rho_{{}_{\Lambda}}-\rho_{{}_{R}}$ and $\rho_{{}_{\Lambda}}-\rho_{{}_{M}}$
relations.

In radiation dominate epoch, we replace
$\rho_{{}_{M}}\rightarrow\rho_{{}_{R}}$ and
$\omega_{{}_{M}}\rightarrow\omega_{{}_{R}}\approx 1/3$ in Eqs. (4.4) and
(4.5). Neglecting dark-energy and non-relativistic matter densities
$H^{2}\approx\rho_{{}_{R}}/(3m^{2}_{\rm pl})$, and
$\rho_{{}_{M}}^{H}\approx(2\chi/3)\bar{m}_{{}_{R}}^{2}\rho_{{}_{R}}$, we
recast Eqs. (4.4) and (4.5) as

$\displaystyle\frac{d\rho_{{}_{R}}}{dx}+4\rho_{{}_{R}}$ $\displaystyle=$
$\displaystyle+\langle\Gamma_{M}/H\rangle_{{}_{R}}[\chi\bar{m}_{{}_{R}}^{2}-1]\rho_{{}_{R}}$
(5.1) $\displaystyle\frac{d\rho_{{}_{\Lambda}}}{dx}$ $\displaystyle=$
$\displaystyle-\langle\Gamma_{M}/H\rangle_{{}_{R}}[\chi\bar{m}_{{}_{R}}^{2}-1]\rho_{{}_{R}}$
(5.2)

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
$\chi m\propto H$ (3.2) vary as well, which we cannot go to details. Thus we
introduce the average mass parameter $m_{{}_{R}}=\langle m\rangle_{{}_{R}}$
and average rate $\langle\Gamma_{M}/H\rangle_{{}_{R}}$ over the entire
radiation epoch, assuming they vary much slowly than $\rho_{{}_{R}}$ and
$\rho_{{}_{\Lambda}}$. The dimensionless average mass parameter
$\bar{m}_{{}_{R}}\equiv(2/3)m_{{}_{R}}/m_{\rm pl}$ and
$\chi\bar{m}_{{}_{R}}^{2}<1$.

The asymptotic solutions are

$\displaystyle\rho_{{}_{R}}$ $\displaystyle=$ $\displaystyle\rho^{\rm
RH}_{{}_{R}}\left(\frac{a_{{}_{R}}}{a}\right)^{4-\gamma_{{}_{R}}},\quad\gamma_{{}_{R}}\equiv\langle\Gamma_{M}/H\rangle_{{}_{R}}(\chi\bar{m}^{2}_{{}_{R}}-1)$
(5.3) $\displaystyle\rho_{{}_{\Lambda}}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{{}_{R}}}{4-\gamma_{{}_{R}}}\rho_{{}_{R}}+\tilde{\mathcal{C}}_{{}_{\Lambda}},\quad\rho_{{}_{\Lambda}}=\frac{\gamma_{{}_{R}}}{4-\gamma_{{}_{R}}}\rho_{{}_{R}}.$
(5.4)

The dark-energy and matter coupling parameter $\gamma_{{}_{R}}<0$
($|\gamma_{{}_{R}}|<1$) represents the $\rho_{{}_{\Lambda}}-\rho_{{}_{R}}$
interaction and $\rho_{{}_{R}}$ conversion to $\rho_{{}_{\Lambda}}$. The
initial values $\rho^{\rm RH}_{{}_{R}}$ and $\tilde{C}_{{}_{\Lambda}}$ are
given at the reheating end $a=a_{{}_{R}}$. In this article, to study dark
energy and radiation interaction, we select the initial condition
$\tilde{C}_{{}_{\Lambda}}=0$, consistently with $\rho^{\rm
RH}_{{}_{\Lambda}}\propto\rho^{\rm RH}_{{}_{R}}$ and $\rho^{\rm
RH}_{{}_{\Lambda}}\ll\rho^{\rm RH}_{{}_{R}}$ at reheating end. The reasons are
that the dark energy $\rho_{{}_{\Lambda}}$ converts to massive pair plasma
energy $\rho^{H}_{{}_{M}}$ (3.1), and massive pairs decay to relativistic
particles, producing radiation energy $\rho_{{}_{R}}$ [66]. With such an
initial condition $\tilde{C}_{{}_{\Lambda}}=0$, the dark-energy in radiation
epoch is negative $\rho_{{}_{\Lambda}}<0$ because of $\gamma_{{}_{R}}<0$. The
detailed discussions about negative dark energy can be found in Refs. [79, 80,
81, 82, 83, 84, 85]. It requires more studies of the transition from reheating
to radiation epochs to determine $\tilde{C}_{{}_{\Lambda}}$.

As a result, the asymptotic solution (5.4) shows that $\rho_{{}_{\Lambda}}$
linearly tracks down (follows) $\rho_{{}_{R}}$. Here we adopt the terminology
“track down” used in the discussions of Ref. [86]. Such
$\rho_{{}_{\Lambda}}-\rho_{{}_{R}}$ tracking dynamics continues in the entire
radiation epoch. We will show that the tracking dynamics ends and
$\rho_{{}_{\Lambda}}$ 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
$a_{\rm tr}$ to characterize the transition time scale, and discuss two
extremal cases:

  1. (i)

$a_{\rm tr}\sim a_{\rm eq}$ transition occurs at the radiation-matter
equilibrium moment;

  2. (ii)

$a_{\rm tr}>a_{\rm eq}$ 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
$\rho_{{}_{M}}\rightarrow\rho_{{}_{M}}$ and
$\omega_{{}_{M}}\rightarrow\omega_{{}_{M}}\approx 0$ in Eqs. (4.4) and (4.5).
Analogously to the approach in radiation epoch, neglecting dark-energy and
radiation-energy density, $H^{2}\approx\rho_{{}_{M}}/(3m^{2}_{\rm pl})$, and
$\rho_{{}_{M}}^{H}\approx\chi\bar{m}_{{}_{M}}^{2}\rho_{{}_{M}}$, we recast
Eqs. (4.4) and (4.5) as

$\displaystyle\frac{d\rho_{{}_{M}}}{dx}+3\rho_{{}_{M}}$ $\displaystyle=$
$\displaystyle+\langle\Gamma_{M}/H\rangle_{{}_{M}}(\chi\bar{m}_{{}_{M}}^{2}-1)\rho_{{}_{M}},$
(5.5) $\displaystyle\frac{d\rho_{{}_{\Lambda}}}{dx}$ $\displaystyle=$
$\displaystyle-\langle\Gamma_{M}/H\rangle_{{}_{M}}(\chi\bar{m}_{{}_{M}}^{2}-1)\rho_{{}_{M}},$
(5.6)

where $\bar{m}_{{}_{M}}\equiv(2/3)m_{{}_{M}}/m_{\rm pl}$ and
$\chi\bar{m}_{{}_{M}}^{2}<1$. Here we introduce the average mass parameter
$m_{{}_{M}}$ and rate $\langle\Gamma_{M}/H\rangle_{{}_{M}}$ over the matter
epoch, assuming they vary much slowly than $\rho_{{}_{M}}$. The asymptotic
solutions are

$\displaystyle\rho_{{}_{M}}$ $\displaystyle=$ $\displaystyle\rho^{\rm
eq}_{{}_{M}}\left(\frac{a_{\rm
eq}}{a}\right)^{3-\gamma_{{}_{M}}},\quad\gamma_{{}_{M}}\equiv\langle\Gamma_{M}/H\rangle_{{}_{M}}(\chi\bar{m}^{2}_{M}-1),$
(5.7) $\displaystyle\rho_{{}_{\Lambda}}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{{}_{M}}}{3-\gamma_{{}_{M}}}\rho_{{}_{M}}+\tilde{\mathcal{C}}^{\rm
eq}_{{}_{\Lambda}},\quad\rho_{{}_{\Lambda}}\rightarrow\rho^{0}_{{}_{\Lambda}}\approx\tilde{\mathcal{C}}^{\rm
eq}_{{}_{\Lambda}}.$ (5.8)

The coupling parameter $\gamma_{{}_{M}}<0$ ($|\gamma_{{}_{M}}|<1$) represents
the $\rho_{{}_{\Lambda}}-\rho_{{}_{M}}$ interaction and $\rho_{{}_{M}}$
conversion into $\rho_{{}_{\Lambda}}$. Here we adopt the case (i) $a_{\rm
tr}\sim a_{\rm eq}$ for discussions. Namely, the
$\rho_{{}_{\Lambda}}-\rho_{{}_{R}}$ tracking continues until the Universe
reaches the radiation-matter equilibrium $\rho^{\rm eq}_{{}_{M}}=\rho^{\rm
eq}_{{}_{R}}$ at $(a_{\rm eq}/a_{{}_{R}})=(T_{\rm RH}/T_{\rm eq})\sim
10^{15}{\rm GeV}/10\,{\rm eV}\sim 10^{23}$, where $T_{\rm RH}(T_{\rm eq})$ is
the reheating (equilibrium) temperature. The initial value $\rho^{\rm
eq}_{{}_{M}}$ is given at the radiation-matter equilibrium $\rho^{\rm
eq}_{{}_{M}}=\rho^{\rm eq}_{{}_{R}}$ at $a=a_{\rm eq}$, where the solutions
(5.4) and (5.8) should match, yielding

$\displaystyle\tilde{\mathcal{C}}^{\rm eq}_{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{3\gamma_{{}_{R}}-4\gamma_{{}_{M}}}{(4-\gamma_{{}_{R}})(3-\gamma_{{}_{M}})}\rho^{\rm
eq}_{{}_{M}},\quad\rho^{\rm
eq}_{{}_{M}}\approx\rho^{0}_{{}_{M}}\left(\frac{a_{0}}{a_{\rm
eq}}\right)^{3},$ (5.9)

where $\rho^{0}_{{}_{M}}$ and $\rho^{0}_{{}_{\Lambda}}$ are the values at the
present time $a_{0}=(1+z)\sim 10^{4}a_{\rm eq}$. The solution (5.8) shows that
the term $(\gamma_{{}_{M}}/3)\rho_{{}_{M}}$ decreases as $a^{-3}$,
$\rho_{{}_{\Lambda}}$ fails to track down $\rho_{{}_{M}}$, and becomes
positive value approaching to the constant
$\rho^{0}_{{}_{\Lambda}}\approx\tilde{\mathcal{C}}^{\rm eq}_{\Lambda}$. These
results (5.7,5.8,5.9) depend on the transition period from radiation to matter
epoch. As for the case (ii) that $\rho_{{}_{\Lambda}}-\rho_{{}_{R}}$ tracking
dynamics ends and $\rho_{{}_{\Lambda}}$ becomes positive value at sometime
$a_{\rm tr}$ around/after the last scattering surface. Discussions and results
are similar with substitutions: $a_{\rm eq}\rightarrow a_{\rm tr}$, $\rho^{\rm
eq}_{{}_{M}}\rightarrow\rho^{\rm tr}_{{}_{M}}$ and $\tilde{\mathcal{C}}^{\rm
eq}_{\Lambda}\rightarrow\tilde{\mathcal{C}}^{\rm tr}_{\Lambda}$ in Eqs.
(5.7-5.9).

To end this section, we mention $\rho_{{}_{\Lambda}}$-dominate epoch in
future, when $H^{2}\approx\rho_{{}_{\Lambda}}/(3m^{2}_{\rm pl})$, and
$\rho_{{}_{M}}^{H}\approx\chi\bar{m}_{{}_{\Lambda}}^{2}\rho_{{}_{\Lambda}}$,
we recast Eqs. (4.4) and (4.5) as

$\displaystyle\frac{d\rho_{{}_{M}}}{dx}+3\rho_{{}_{M}}$ $\displaystyle=$
$\displaystyle+\gamma_{{}_{\Lambda}}(\chi\bar{m}_{{}_{\Lambda}}^{2}\rho_{{}_{\Lambda}}-\rho_{{}_{M}}),$
(5.10) $\displaystyle\frac{d\rho_{{}_{\Lambda}}}{dx}$ $\displaystyle=$
$\displaystyle-\gamma_{{}_{\Lambda}}(\chi\bar{m}_{{}_{\Lambda}}^{2}\rho_{{}_{\Lambda}}-\rho_{{}_{M}}),$
(5.11)

where positive
$\gamma_{{}_{\Lambda}}\equiv\langle\Gamma_{M}/H\rangle|_{\Lambda}$,
$\bar{m}_{{}_{\Lambda}}\equiv(2/3)m_{{}_{\Lambda}}/m_{\rm pl}$ and
$\chi\bar{m}_{{}_{\Lambda}}^{2}<1$. Here we introduce the average mass
parameter and rate for this epoch. For
$\chi\bar{m}_{{}_{\Lambda}}^{2}\rho_{{}_{\Lambda}}>\rho_{{}_{M}}$, namely
$\rho^{H}_{{}_{M}}>\rho_{{}_{M}}$ in the cosmic rate equation (4.4) or (4.5),
asymptotic solutions are slowly time varying

$\displaystyle\rho_{{}_{\Lambda}}\approx\rho^{0}_{{}_{\Lambda}}\left(\frac{a_{0}}{a}\right)^{\chi\bar{m}_{{}_{\Lambda}}^{2}\gamma_{{}_{\Lambda}}};\quad\rho_{{}_{M}}\approx\chi\bar{m}_{{}_{\Lambda}}^{2}\gamma_{{}_{\Lambda}}\rho_{{}_{\Lambda}}.$
(5.12)

It shows that dark energy decreases in time and converts to matter, and the
latter tracks down the former.

Figure 2: The ratio $\rho_{{}_{\Lambda}}/\rho_{{}_{M}}$ (6.1) is plotted as a
function of $\ln(a/a_{\rm eq})$, where the scaling factor $a$ runs from the
reheating end $a_{{}_{R}}$, through transition period $a_{\rm tr}$ to the
present time $a_{0}$, $a_{{}_{R}}<a_{\rm tr}<a_{0}$. It shows that (a) the
tracking-down behavior: the ratio $\rho_{{}_{\Lambda}}/\rho_{{}_{R}}$ is a
small negative constant $\gamma_{{}_{R}}/4$ for $\ln(a/a_{\rm tr})<0$; (b)
tracking-down failure occurs and dark-energy density $\rho_{{}_{\Lambda}}$
becomes positive around $\ln(a/a_{\rm tr})=0$; (c) $\rho_{{}_{M}}\sim(a/a_{\rm
eq})^{-3}$ (5.7) and $\rho_{{}_{\Lambda}}\approx\rho^{0}_{{}_{\Lambda}}$ (5.8)
constant, the ratio $\rho_{{}_{\Lambda}}/\rho_{{}_{M}}$ increases to
${\mathcal{O}}(1)$. The upper panel:
$|\gamma_{{}_{M}}|\approx|\gamma_{{}_{R}}|\sim 10^{-11}$ for the case (i)
$a_{\rm tr}\sim a_{\rm eq}\sim 10^{4}a_{0}$ and the present time
$\ln(a_{0}/a_{\rm eq})\approx 9.2$. The lower panel:
$|\gamma_{{}_{M}}|\approx|\gamma_{{}_{R}}|\sim 10^{-3}$ for the case (ii)
$a_{\rm tr}\sim 30a_{0}$ and the present time $\ln(a_{0}/a_{\rm tr})\approx
3.4$.

##  6 Cosmic coincidence of present dark and matter energies

To discuss the cosmic coincidence, we use the ratio
$\rho_{{}_{\Lambda}}/\rho_{{}_{M}}$ which is independent of the characteristic
scales in different epochs. We separately discuss two extremal cases: (i)
$a_{\rm tr}\sim a_{\rm eq}\sim 10^{4}a_{0}$ or (ii) $a_{\rm tr}\sim
10^{2}a_{0}$, when the $\rho_{{}_{\Lambda}}-\rho_{{}_{M}}$ tracking ends and
$\rho_{{}_{\Lambda}}$ becomes positive. In radiation epoch, solution (5.4)
shows the ratio $\rho_{{}_{\Lambda}}/\rho_{{}_{R}}\approx\gamma_{{}_{R}}/4$
keeps constant, as $\rho_{{}_{\Lambda}}$ tracks down $\rho_{{}_{R}}$ from the
reheating end $a_{{}_{R}}$ to (i) the radiation-matter equilibrium $a_{\rm
eq}\sim 10^{23}a_{{}_{R}}$ or (ii) sometime after the last scattering surface
$a_{\rm tr}\sim 10^{25}a_{{}_{R}}$. This tracking dynamics avoids the fine
tuning cosmic $\rho_{{}_{\Lambda}}$ and $\rho_{{}_{R}}$ coincidence of the
order of (i) $(a_{\rm eq}/a_{{}_{R}})^{4}\sim 10^{92}$ or (ii) $(a_{\rm
tr}/a_{{}_{R}})^{4}\sim 10^{100}$. Whereas, from the transition time (i)
$a_{\rm tr}\sim a_{\rm eq}=(1+z_{\rm eq})^{-1}a_{0}\sim 10^{-4}a_{0}$ or (ii)
$a_{\rm tr}\sim(1+z_{\rm tr})^{-1}a_{0}\sim 10^{-2}a_{0}$ to the present time
$a_{0}$, solutions (5.7) and (5.8) give

$\displaystyle\frac{\rho_{{}_{\Lambda}}}{\rho_{{}_{M}}}\approx\frac{\gamma_{{}_{M}}}{3-\gamma_{{}_{M}}}+\frac{(3\gamma_{{}_{R}}-4\gamma_{{}_{M}})}{(4-\gamma_{{}_{R}})(3-\gamma_{{}_{M}})}\,\left(\frac{a}{a_{\rm
tr}}\right)^{3-\gamma_{{}_{M}}}.$ (6.1)

This ratio consistently approaches the constant $-\gamma_{{}_{R}}/4$ when
scale factor $a$ traces back to the reheating end $a_{{}_{R}}$. Using (i)
$\gamma_{{}_{M}}\approx\gamma_{{}_{R}}\sim 10^{-11}$ for the case $a_{\rm
tr}\sim a_{\rm eq}\sim 10^{4}a_{0}$; (ii)
$\gamma_{{}_{M}}\approx\gamma_{{}_{R}}\sim 10^{-3}$ for the case $a_{\rm
tr}\sim 10^{2}a_{0}$, we plot in Fig. 2 the ratio
$\rho_{{}_{\Lambda}}/\rho_{{}_{R,M}}$ varying from $-\gamma_{{}_{R}}/4$ to
${\mathcal{O}}(1)$ as a function of the scale factor $\ln(a/a_{\rm tr})$ from
the reheating to present time. It shows that the cosmic coincidence of the
present $\rho^{0}_{{}_{\Lambda}}$ and $\rho^{0}_{{}_{M}}$ values appear
naturally without any extremely fine-tuning their values at the transition
time. Namely, in Eq. LABEL:(6.1) the ratio
$\rho_{{}_{\Lambda}}/\rho_{{}_{M}}\sim(a/a_{\rm tr})^{3}$ variation is about
(i) ${\mathcal{O}}(10^{-12})$ or (ii) ${\mathcal{O}}(10^{-6})$, see the right
column of Fig. 2. The reason is that the matter-dominated epoch of (i) $z_{\rm
eq}\sim 10^{4}$ or (ii) $z_{\rm tr}\sim 10^{2}$ is much shorter than the
radiation dominated epoch of (i) $(a_{\rm eq}/a_{{}_{R}})\sim 10^{23}$ or (ii)
$(a_{\rm tr}/a_{{}_{R}})\sim 10^{25}$, when the $\rho_{{}_{\Lambda}}$ tracks
down $\rho_{{}_{R}}$ and the ratio $\rho_{{}_{\Lambda}}/\rho_{{}_{R}}$ is a
constant, see the left column of Fig. 2. Otherwise, to reach present
$\rho_{{}_{\Lambda}}$ and $\rho_{{}_{M}}$ observational values of the same
order of magnitude, we would have the cosmic coincidence problem of incredibly
fine-tuning their reheating values $\rho^{\rm RH}_{{}_{\Lambda}}$ and
$\rho^{\rm RH}_{{}_{M}}$ at order of (i)
$(10^{-23})^{4}\times(10^{-4})^{3}\sim 10^{-104}$ or (ii)
$(10^{-25})^{4}\times(10^{-2})^{3}\sim 10^{-106}$.

##  7 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 $\rho_{{}_{M}}$ and dark energy $\rho_{{}_{\Lambda}}$, 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
$\rho_{{}_{\Lambda}}-\rho_{{}_{M}}$ interacting strength
$\langle\Gamma_{M}/H\rangle$ depends on evolution epochs. We study asymptotic
solutions for radiation and matter epochs, starting from the reheating end.
Because of different epoch transitions, $\rho_{{}_{\Lambda}}-\rho_{{}_{M}}$
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
$\Lambda$CDM. 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

$\displaystyle
E(z)^{2}=\Omega_{{}_{R}}(1+z)^{4-\delta^{G}_{{}_{R}}}+\Omega_{{}_{M}}(1+z)^{3-\delta^{G}_{{}_{M}}}+\Omega_{{}_{\Lambda}}(1+z)^{\delta_{{}_{\Lambda}}}.$
(7.1)

Here energy densities $\rho_{{}_{R,M,\Lambda}}$ are in units of the critical
density $\rho^{0}_{c}=3m_{\rm pl}^{2}H_{0}^{2}$ today, and
$\Omega_{R,M,\Lambda}$ are the present values and
$\Omega_{{}_{R}}+\Omega_{{}_{M}}+\Omega_{{}_{\Lambda}}=1$. Inserting $E(z)$
(7.1) into the dark-energy and matter interacting equation (1.2), the dark
energy term can be obtained as,

$\displaystyle\Omega_{{}_{\Lambda}}(1+z)^{\delta_{\Lambda}}=\Omega_{{}_{\Lambda}}+\delta^{M}_{{}_{G}}\frac{\Omega_{M}}{3}\left[(1+z)^{3-\delta^{M}_{{}_{G}}}-1\right]+\delta^{R}_{{}_{G}}\frac{\Omega_{R}}{4}\left[(1+z)^{4-\delta^{R}_{{}_{G}}}-1\right].$
(7.2)

Equations (7.1) and (7.2) give a class of effective interacting dark energy
models with two parameters $\delta^{M}_{\rm G}$ and $\delta^{R}_{\rm G}$.
These modified scaling laws (7.1) were also proposed from the view point that
time-varying cosmological term $\tilde{\Lambda}(t)$ and gravitational coupling
$\tilde{G}(t)$ obey scaling laws approaching to their present values
($G,\Lambda$), where Ricci scalar term $R$ and cosmological term $\Lambda$ 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 $\Lambda$CDM.

We end this article with some remarks. In radiation dominate epoch, negative
dark-energy density $\rho_{{}_{\Lambda}}\approx\gamma_{{}_{R}}m^{2}_{\rm
pl}H^{2}$ (5.4) follows the “area law” $\propto H^{2}$. In matter dominate
epoch, it changes sign at $\rho_{{}_{\Lambda}}=0$ in Eq. (5.8), and approaches
a positive constant $\rho^{0}_{\Lambda}\approx\tilde{\mathcal{C}}^{\rm
tr}_{\Lambda}$ (5.9). The dark energy undergoes these transitions and becomes
dominant, converting to matter, and matter density $\rho_{{}_{M}}$, in turn,
tracks down dark energy density (5.12). We speculate that such
$\rho_{{}_{\Lambda}}$-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.

##  8 Supplemental Material: quantum pair oscillation details

In microscopic time, we plot the Bogoliubov coefficient $|\beta|^{2}$, the
quantum pair density $\rho^{\rm fast}_{{}_{\Lambda}}$ and pressure $p^{\rm
fast}_{{}_{\Lambda}}$, as well as the fast components of Hubble function
$H_{\rm fast}$, and cosmological term $\rho^{\rm fast}_{{}_{\Lambda}}$.

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
$|\beta(t)|^{2}$, $h_{\rm fast}$ and $\varrho^{\rm fast}_{{}_{\Lambda}}$
structures are too small to see.

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