# Holographic massive plasma state in Friedman Universe: cosmological fine-
tuning and coincidence problems

She-Sheng Xue

###### Abstract

Massive particle and antiparticle pair production and oscillation on the
horizon form a holographic and massive pair plasma state in the Friedman
Universe. Via this state, the Einstein cosmology $\Lambda$ term (dark energy)
interacts with matter and radiation in Universe evolution. It is determined by
a close set of ordinary differential equations and the initial conditions
given by observations. In inflation and reheating, dark energy density
decreases from the inflation scale, converting to matter and radiation energy
densities. In standard cosmology, matter and radiation energy densities
convert to dark energy density, reaching the present Universe. These results
can be the possible solutions for cosmological fine-tuning and coincidence
problems.

##  1 Introduction

Einstein’s equation describes the Universe’s evolution in terms of Newton’s
constant $G$, Ricci scalar $R$, cosmological constant $\Lambda$ and matter and
radiation energy densities. The $\Lambda$ was introduced to counterbalance
attractive gravity and achieve a static universe. It was abandoned after the
confirmation of Universe expansion. It has been revived since the discovery of
the Universe’s acceleration, which implies that the $\Lambda$ may have a
positive value. Owing to its mysterious origin and nature, the $\Lambda$
component is dubbed as “dark energy”. As a standard model of the Big Bang
cosmology, the simplest $\Lambda$CDM (Lambda cold dark matter) model provides
a reasonably good account of the observed properties of the cosmos and
accelerating Universe.

However, whatever the dark energy form and nature are, two cosmological
problems arise in $\Lambda$CDM. The cosmic fine-tuning problem [1]: how to
explain the present dark energy density $\sim 10^{-47}{\rm GeV}^{4}$ is about
$10^{-123}$ smaller than the energy density at the Planck scale $M_{\rm
pl}=G^{-1/2}\approx 1.2\times 10^{19}$GeV. It is a unique basic scale entering
Einstein’s equation. The cosmic coincidence problem [2, 3, 4]: how to explain
the coincidence of dark energy and matter densities at the present epoch after
a long cosmic evolution. Observe that throughout the cosmic evolution after
the Big Bang (reheating), dark energy density
$\rho_{{}_{\Lambda}}=\Lambda^{2}/(8\pi G)$ does not change. While radiation
and matter energy densities fall in powers of the scalar factor $a$ and change
many orders of magnitudes. In the recent epoch, their abundances
$\Omega_{\Lambda}\approx 0.7$ and $\Omega_{M}\approx 0.3$ not only coincide in
order of magnitude but also are just the correct values for forming structure,
galaxy and astrophysical objects. It is just the world where human being
creates and lives in. There are three possibilities. (i) We are subject to the
Anthropic Principle and happen to live with a peculiar $\Lambda$ value and in
a special epoch after the long history of Big Bang cosmology, (ii) Nature
fine-tunes in many orders of magnitude the $\Lambda$ value and the ratio of
dark energy and radiation densities at the beginning of Big Ban. (iii) Dark
energy is time-varying due to dynamically interacting with matter and
radiation.

The cosmic fine-tuning and coincidence problems have been studied in many
cosmological model extensions to $\Lambda$CDM, for example, quintessence [5,
6, 7, 8], interacting dark energy model [9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19], phenomenological model [20]. However, people have not yet found
consistent solutions to these two problems in the standard cosmology after Big
Ban. In addition, these beyond $\Lambda$CDM scenarios have not yet given an
overall consistent description of inflation, reheating, and standard cosmology
epochs. In the recently proposed $\tilde{\Lambda}$CDM model of time-varying
$\tilde{\Lambda}$ due to dark energy and matter interactions, we attempted to
consistently describe inflation epoch [21] and reheating epoch [22]. Following
these studies, in this article, we try to find and explain a self-consistent
dynamical solution for the cosmic fine-tuning and coincidence problems in the
standard cosmology epoch.

First, we briefly review the $\tilde{\Lambda}$CDM model and its applications
to the inflation and reheating epoch. Then, we present the Friedman equations
for the Hubble function $H$ and time-varying dark energy
$\rho_{{}_{\Lambda}}$, the cosmic rate equations for matter $\rho_{{}_{M}}$
and radiation $\rho_{{}_{R}}$ densities that interact with dark energy density
$\rho_{{}_{\Lambda}}$. These equations form a close set of ordinary
differential equations. The solution is unique, provided initial conditions
are given by observations. Numerically integrating these equations, we find
the $\tilde{\Lambda}$CDM dynamical solution: (i) in inflation and reheating,
dark energy converts to matter and radiation energies and vanishes at the end
of reheating; (ii) in standard cosmology, instead, matter and radiation
energies convert to dark energy. The conversion rate is proportional to $1/H$,
namely dark energy and matter interaction decreases as redshift $z$ increases.
These dynamical features yield a possible solution for the cosmic coincidence
problem in $\Lambda$CDM. The conversion rate $\propto 1/H$ is consistent with
the late-time interaction in dark matter converting dark energy, obtained by
data analysis in different $z$ value bins[23, 24].

We compare and contrast $\tilde{\Lambda}$CDM with $\Lambda$CDM and advocate an
approximate model for phenomenological studies and data analysis. Finally, we
discuss in the $\tilde{\Lambda}$CDM scenario the geometric and dynamic natures
of $\tilde{\Lambda}$ dark energy as a gravitational ground state,
asymptotically safe Einstein theory for the early and present Universe, and
$\tilde{\Lambda}$ dark energy solution for the cosmic fine-tuning problem.

##  2 Cosmological $\tilde{\Lambda}$CDM model

In such $\tilde{\Lambda}$CDM scenario, we have recently studied singularity-
free and large-scale anomaly issues, the spectral index and tensor-to-scalar
ratio relation in the inflation epoch [21], and calculated reheating energy
and entropy [22]. The results are consistent with observations. We briefly
recall three main features of the $\tilde{\Lambda}$CDM scenario.

###  2.1 Time-varying cosmological $\tilde{\Lambda}$ term

First, a time-varying cosmological $\tilde{\Lambda}$ term represents
interacting dark energy with matter and radiation. The Friedman equations for
a flat Universe of horizon $H$ are [25]

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3}(\rho_{{}_{M}}+\rho_{{}_{R}}+\rho_{{}_{\Lambda}}),$ (2.1)
$\displaystyle\dot{H}$ $\displaystyle=$ $\displaystyle-\frac{8\pi
G}{2}(\rho_{{}_{M}}+\rho_{{}_{R}}+\rho_{{}_{\Lambda}}+p_{{}_{M}}+p_{{}_{R}}+p_{{}_{\Lambda}}).$
(2.2)

Equations of States
$p_{{}_{M,R,\Lambda}}=\omega_{{}_{M,R,\Lambda}}\rho_{{}_{M,R,\Lambda}}$,
$\omega_{{}_{M}}\approx 0$ for massive particles and $\omega_{{}_{R}}\approx
1/3$ for massless radiation. The second equation of (2.2) is the generalised
conservation law (Bianchi identity) for including time-varying cosmological
term $\rho_{{}_{\Lambda}}(t)\equiv\tilde{\Lambda}/(8\pi G)$ and
$p_{{}_{\Lambda}}=-\rho_{{}_{\Lambda}}$. For constant $\Lambda$ and stable
particles, it reduces to the usual equations
$\dot{\rho}_{{}_{M}}+3(1+\omega_{{}_{M}})H\rho_{{}_{M}}=0$ and
$\dot{\rho}_{{}_{R}}+3(1+\omega_{{}_{R}})H\rho_{{}_{R}}=0$ respectively for
massive particle number and massless particle number (entropy) conservation.
The detailed discussions are in Secs. 7 and 9 of Ref. [25].

###  2.2 Massive pair production and oscillation

Second, we study vacuum pairs’ production [26] and oscillation [21] of the
large number (${\mathcal{N}}_{\rm pair}\gg 1$) and massive ($M\gg H$) pairs of
particles and anti-particles. They are attributed to the microscopic fast-
component $H_{\rm fast}$ in the Hubble function $H=H_{\rm fast}+H_{\rm slow}$.
The fast component $H_{\rm fast}$ oscillates coherently with the pairs’
oscillation, which relates to pairs’ production and annihilation from/into the
vacuum at the microscopic time scale $1/M$. It is consistent with recent
studies of vacuum fluctuation and “microcyclic universes” at small scales, see
local scale factor oscillation in Figure 1 of Refs. [27, 28].

The macroscopic slow component $H_{\rm slow}\gg H_{\rm fast}$, and $H_{\rm
slow}\approx H$ obeys the Friedman equation (2.2) at the “macroscopic” time
scale $1/H$. These microscopic and macroscopic processes couple with each
other. However, one cannot even numerically integrate their differential
equations due to the vast difference between the scale $1/M$ and $1/H$.
Therefore, at the macroscopic time scale $1/H$ given by the Friedman equation
(2.2), we have to model the “equilibrium” or “equipartition” state of the
microscopic fast component $H_{\rm fast}$ and pairs’ oscillation. It is the
method that we use to study the back-reactions of microscopic fast component
$H_{\rm fast}$ and pairs’ oscillation on macroscopic densities
$\rho_{{}_{\Lambda,M,R}}$ in the Friedman equation (2.2). Detailed discussions
are in Secs. 2-3 of Ref. [22].

###  2.3 Holographic and massive pair plasma state

Third, we assume the aforementioned “equilibrium” state is a holographic and
massive pair plasma state that contains a large number and massive pairs of
particles and antiparticles. We describe such a state as a perfect fluid state
of effective number $n^{H}_{{}_{M}}$ and energy $\rho^{H}_{{}_{M}}$ densities
of massive stable and unstable pairs,

$\displaystyle\rho^{H}_{{}_{M}}\equiv 2\chi m^{2}H^{2},\quad
n^{H}_{{}_{M}}\equiv\chi mH^{2}.$ (2.3)

The equation of state and pressure are
$p^{H}_{{}_{M}}=\omega^{H}_{{}_{M}}\rho^{H}_{{}_{M}}$. The lower limit
$\omega^{H}_{{}_{M}}\approx 0$ for $m\gg H$, and the upper limit
$\omega^{H}_{{}_{M}}\lesssim 1/3$ for $m\gtrsim H$. The
$m\propto{\mathcal{N}}_{\rm pair}M$ is the mass parameter of the massive pair
plasma state. The massive pair plasma state is a holographic layer near the
horizon. The layer width $\lambda_{m}=(\chi m)^{-1}\ll H^{-1}$ and the width
parameter $\chi\sim{\mathcal{O}}(10^{-3})$, $\chi\approx 1.85\times 10^{-3}$
is the reference value 111In Refs. [29, 30], we adopt the different
renormalization prescription at high energies $M\gg H$ from the usual
prescription (subtraction) at low energies $M\ll H$. We have consistently
obtained the mean density $n^{H}_{{}_{M}}\approx\chi mH^{2}$ (2.3) and
$\chi\approx 1.85\times 10^{-3}$ by studying massive fermion pair productions
in an exact De Sitter spacetime of constant $H$ and scaling factor
$a(t)=e^{iHt}$.. At a given $H$, the “macroscopic” condensation state (2.3)
represents all “microscopic” states of pair production and annihilation at the
time scale $1/M$.

We adopt Eq. (2.3) to study how the holographic and massive pair plasma state
interacts with the Friedman equation at the time scale $1/H$. Considering that
produced pairs’ mass $M$ and number ${\mathcal{N}}_{\rm pair}$ cannot be
exactly constant, we assume the parameter $m$ weakly depends on the horizon
$H$. The effective value $m_{\rm eff}$ for each evolution epoch is fixed by
observations. The detailed discussions are in Secs. 3-4 of Ref. [22].

###  2.4 Cosmic rate equations for matter and radiation densities

Fourth, we propose the cosmic rate equation to describe the interaction
between the massive pair plasma state (2.3) and usual matter and radiation
state $\rho_{{}_{M,R}}$. From the pair number density $n^{H}_{{}_{M}}$ (2.3)
and its time variation, we introduce average rate $\Gamma_{M}$ and time scale
$\tau_{{}_{M}}$

$\displaystyle\Gamma_{M}\approx\frac{\chi
m}{4\pi}\epsilon,\quad\tau_{{}_{M}}=\Gamma_{M}^{-1}.$ (2.4)

The rate $\Gamma_{M}$ is much smaller than the microscopic time scale $1/m$.
The $\tau_{{}_{M}}$ effectively describe the “relaxation” time scale of how
the massive pair plasma state varies in (response with) the macroscopic time,
as the Universe horizon $H(\rho_{{}_{M}},\rho_{{}_{R}},\rho_{{}_{\Lambda}})$
evolves. The Universe evolution $\epsilon$-rate is defined as,

$\displaystyle\epsilon$ $\displaystyle\equiv$
$\displaystyle-\frac{\dot{H}}{H^{2}}=\frac{3}{2}\frac{(1+\omega_{{}_{M}})\rho_{{}_{M}}+(1+\omega_{{}_{R}})\rho_{{}_{R}}+(1+\omega_{{}_{\Lambda}})\rho_{{}_{\Lambda}}}{\rho_{{}_{\Lambda}}+\rho_{{}_{M}}+\rho_{{}_{R}}}.$
(2.5)

The second equation comes from the Friedman equations (2.2). The asymptotic
values $\epsilon\approx 0$, $\epsilon\approx 2$, and $\epsilon\approx 3/2$
correspond to dark energy, radiation, and matter domination, respectively.

We note that the massive pair plasma state $\rho^{H}_{{}_{M}}$ (2.3) has a
different “relaxation” time scale $\tau_{{}_{M}}$ (2.4) from the one
$\tau_{{}_{H}}=H^{-1}$ of the usual matter/radiation state $\rho_{{}_{M,R}}$,
i.e., $\tau_{{}_{H}}>\tau_{{}_{M}}$. On the other hand, the massive pair
plasma state density $\rho^{H}_{{}_{M}}$ contributes to the usual
matter/radiation density $\rho_{{}_{M,R}}$. Moreover, the latter variation
affects the former. It implies the back-and-forth interaction between the
massive pair plasma state and the normal matter and radiation state during the
Universe’s evolution. Therefore, we propose the back-and-forth interaction
between the densities $\rho^{H}_{{}_{M}}$ and $\rho_{{}_{M,R}}$ follows the
cosmic rate equations of Boltzmann type,

$\displaystyle\dot{\rho}_{{}_{M}}+3(1+\omega_{{}_{M}})H\rho_{{}_{M}}$
$\displaystyle=$
$\displaystyle\Gamma_{M}(\rho_{{}_{M}}^{H}-\rho_{{}_{M}}-\rho_{{}_{R}})-\Gamma_{M}^{{}^{\rm
de}}\rho^{\rm de}_{{}_{M}},$ (2.6)
$\displaystyle\dot{\rho}_{{}_{R}}+3(1+\omega_{{}_{R}})H\rho_{{}_{R}}$
$\displaystyle=$
$\displaystyle\Gamma_{M}(\rho_{{}_{M}}^{H}-\rho_{{}_{M}}-\rho_{{}_{R}})+\Gamma_{M}^{{}^{\rm
de}}\rho^{\rm de}_{{}_{M}},$ (2.7)

where $\Gamma_{M}^{{}^{\rm de}}$ and $\tau_{{}_{R}}=(\Gamma_{M}^{{}^{\rm
de}})^{-1}$ are unstable massive pair decay rate and time. The term
$\Gamma^{{}^{\rm de}}_{M}\rho^{\rm de}_{{}_{M}}$ represents unstable massive
pairs decay to light particles, such as quarks and leptons, gauge bosons in SM
and other light sterile particles. The term
$3(1+\omega_{{}_{M,R}})H\rho_{{}_{M,R}}$ of the time scale
$[3(1+\omega_{{}_{M,R}})H]^{-1}$ represents the space-time expanding effect on
the density $\rho_{{}_{M,R}}$. While $\Gamma_{M}\rho_{{}_{M}}^{H}$ is the
source term and $\Gamma_{M}(\rho_{{}_{M}}+\rho_{{}_{R}})$ is the depletion
term. The detailed balance term
$\Gamma_{M}(\rho_{{}_{M}}^{H}-\rho_{{}_{M}}-\rho_{{}_{R}})$ indicates how
densities $\rho_{{}_{M}}^{H}$ and $\rho_{{}_{M,R}}$ of different time scales
couple together. The ratio $\Gamma_{M}/H>1$ indicates the coupled case, and
$\Gamma_{M}/H<1$ indicates the decoupled case. The detailed discussions are in
Secs. 4-5 of Ref. [22].

###  2.5 Preliminary applications to inflation and reheating

The main aspects of the $\tilde{\Lambda}$CDM scenario are (a) the dark energy
and matter interacting Friedman Equations (2.1,2.2); (b) massive particle and
antiparticle pairs’ production and oscillation; (c) holographic massive pair
plasma state (2.3) and its variation rate (2.4); (d) cosmic rate equations
(2.6) and (2.7). They form a close set of first-order ordinary differential
equations for the densities $\rho_{{}_{M}},\rho_{{}_{R}},\rho_{{}_{\Lambda}}$
and Hubble function $H$. The solutions are completely determined, provided
initial or transition conditions are given.

In Ref. [21], we study the inflation when the dominant dark energy
$\rho_{{}_{\Lambda}}$ drives inflation and produces massive pairs’ plasma
$\rho^{H}_{{}_{M}}$, that slows down inflation
($\rho_{{}_{\Lambda}}\gg\rho^{H}_{{}_{M}}\approx\rho_{{}_{M}}$). Neglecting
Eqs. (2.6,2.7) for $\Gamma_{M}/H<1$, we use Eq. (2.2) and
$\rho_{{}_{M}}\approx\rho^{H}_{{}_{M}}$ to obtain an analytical solution
$H_{\rm end}=H_{*}\exp-(\epsilon^{*}\,N_{\rm end}).$ The $e$-folding numbers
$N_{\rm end}\approx(50~{}60)$ from the pivot inflation scale $H_{*}$ to the
inflation end $H_{\rm end}\approx\Gamma_{M}\approx(0.42,0.35)H_{*}$. We fix
the mass parameter $m_{*}$ by $\epsilon^{*}=\chi(m_{*}/m_{\rm
pl})^{2}=(1-n_{s})/2$ 222The reduced Planck mass $m_{\rm
pl}\equiv(8\pi)^{-1/2}M_{\rm pl}=2.43\times 10^{18}$GeV.. The obtained
relation of spectral index $n_{s}$ and tensor-to-scalar ratio $r$ agrees with
recent CMB observations. We discuss the singularity-free pre-inflation, the
CMB large-scale anomaly, and dark-matter density perturbations imprinting on
power spectra.

In Ref. [22], we study the reheating when the dark energy
$\rho_{{}_{\Lambda}}$ decreases, $\rho^{H}_{{}_{M}}$ and $\rho_{{}_{M}}$
increase. The competition between the Hubble function $H$, the rates
$\Gamma_{M}$ and $\Gamma_{M}^{{}^{\rm de}}$ play an important role in cosmic
rate equations (2.6) and (2.7). First, it appears the ${\mathcal{M}}$-episode
of massive pair domination when $\Gamma_{M}>H>\Gamma_{M}^{{}^{\rm de}}$. Then
it proceeds to the ${\mathcal{R}}$-episode of radiation domination when
$\Gamma_{M}^{{}^{\rm de}}>H>\Gamma_{M}$. The rate equation (2.7) becomes a
reheating equation for $\Gamma_{M}^{{}^{\rm de}}>\Gamma_{M}$. Unstable pairs
decay to light particles and the radiation energy $\rho_{{}_{R}}$ increases,
leading to reheating. We estimated the mass parameter $\hat{m}\gtrsim 20m_{\rm
pl}$ and obtained results agree with observations. Stable massive particles
remain as cold dark matter particles 333There, the strongly coupled case
$\Gamma_{M}/H\gg 1$ is assumed in the preliminary study of cold dark matter
abundance $\Omega_{M}$ evolution. We realise it should be the weakly coupled
case $\Gamma_{M}/H<1$ after studying the $\Omega_{M}$ evolution in this
article.. The detailed discussions are in Secs. 7.2-7.3 of Ref. [22].

##  3 $\tilde{\Lambda}$CDM solution to cosmological coincidence problem

At the reheating end, the radiation energy density $\rho_{{}_{R}}$ is
dominant, stable cold dark matter energy density
$\rho_{{}_{M}}\ll\rho_{{}_{R}}$ and the dark energy density vanishes
$\rho_{{}_{\Lambda}}\approx 0$, namely
$\rho_{{}_{R}}\gg\rho_{{}_{M}}\gg\rho_{{}_{\Lambda}}\approx 0$. These are the
initial conditions starting the standard cosmology. It then evolves to matter-
dominated, dark energy-dominated epochs. We study in this article the
$\tilde{\Lambda}$CDM solution after reheating, focusing on the problem of
cosmological coincidence between dark energy and matter.

###  3.1 Dark energy interaction with matter and radiation

To explicitly show dark energy and matter interaction, we recast the Friedman
equations (2.1,2.2), cosmic rate equations (2.6) and (2.7) as

$\displaystyle\dot{\rho}_{{}_{\Lambda}}+3(1+\omega_{{}_{\Lambda}})H\rho_{{}_{\Lambda}}$
$\displaystyle=$
$\displaystyle-2\Gamma_{M}\left(\rho^{H}_{{}_{M}}-\rho_{{}_{M}}-\rho_{{}_{R}}\right),$
(3.1) $\displaystyle\dot{\rho}_{{}_{M}}+3(1+\omega_{{}_{M}})H\rho_{{}_{M}}$
$\displaystyle=$
$\displaystyle+\Gamma_{M}\left(\rho^{H}_{{}_{M}}-\rho_{{}_{M}}-\rho_{{}_{R}}\right),$
(3.2) $\displaystyle\dot{\rho}_{{}_{R}}+3(1+\omega_{{}_{R}})H\rho_{{}_{R}}$
$\displaystyle=$
$\displaystyle+\Gamma_{M}\left(\rho^{H}_{{}_{M}}-\rho_{{}_{M}}-\rho_{{}_{R}}\right),$
(3.3)

where $\Gamma_{M}\left(\rho^{H}_{{}_{M}}-\rho_{{}_{M}}-\rho_{{}_{R}}\right)$
represents the interaction between dark energy and usual matter and radiation
via the massive pair plasma state $\rho^{H}_{{}_{M}}$. Equations (3.2) and
(3.3) are the cosmic rate equations (2.6) and (2.7). Here we neglect the decay
term $\pm\Gamma_{M}^{{}^{\rm de}}\rho^{\rm de}_{{}_{M}}$, assuming unstable
massive pairs $\rho^{\rm de}_{{}_{M}}$ has decay in reheating.

In inflation and reheating,
$\Gamma_{M}\left(\rho^{H}_{{}_{M}}-\rho_{{}_{M}}-\rho_{{}_{R}}\right)>0$ and
$\dot{\rho}_{{}_{\Lambda}}<0$, dark energy converts to matter and radiation
energies [21, 22]. After reheating, there are two cases:

  1. (a)

$\rho^{H}_{{}_{M}}<\rho_{{}_{M}}+\rho_{{}_{R}}$, matter and radiation converts
to dark energy $\dot{\rho}_{{}_{\Lambda}}>0$,

  2. (b)

$\rho^{H}_{{}_{M}}>\rho_{{}_{M}}+\rho_{{}_{R}}$, dark energy converts to
matter and radiation $\dot{\rho}_{{}_{\Lambda}}<0$.

These two cases are separated by
$\rho_{{}_{M}}+\rho_{{}_{R}}=\rho^{H}_{{}_{M}}$.

###  3.2 Cosmic rate equations for cosmic abundance

We define the cosmic abundance

$\displaystyle\Omega_{{}_{\Lambda,M,R}}\equiv\frac{\rho_{{}_{\Lambda,M,R}}}{\rho_{\rm
tot}},\quad\rho_{\rm tot}\equiv\frac{8\pi G}{3}H^{2},$ (3.4)

and $\Omega_{{}_{\Lambda}}+\Omega_{{}_{M}}+\Omega_{{}_{R}}=1$ (2.1). The
“time” variable $x$ relates to the scale factor $a=a(t)$,

$\displaystyle
x=\ln(a/a_{0})+\ln(a_{0}/a_{{}_{R}})=-\ln(1+z)+\ln(a_{0}/a_{{}_{R}}).$ (3.5)

The derivative $dx=Hdt$ and $dx=-dz/(1+z)$, where $z$ is the redshift.
Equation (2.5) becomes

$\displaystyle\epsilon=(1+z)\frac{dH}{Hdz}=\frac{3}{2}~{}\left[(1+\omega_{{}_{M}})\Omega_{{}_{M}}+(1+\omega_{{}_{R}})\Omega_{{}_{R}}+(1+\omega_{{}_{\Lambda}})\Omega_{{}_{\Lambda}}\right].$
(3.6)

Whereas, Equations (3.1-3.3) becomes

$\displaystyle-(1+z)\frac{d\Omega_{{}_{\Lambda}}}{dz}+3(1+\omega_{{}_{\Lambda}})\Omega_{{}_{\Lambda}}$
$\displaystyle=$
$\displaystyle-2\frac{\Gamma_{M}}{H}\left(\Omega^{H}_{{}_{M}}-\Omega_{{}_{M}}-\Omega_{{}_{R}}\right),$
(3.7)
$\displaystyle-(1+z)\frac{d\Omega_{{}_{M}}}{dz}+3(1+\omega_{{}_{M}})\Omega_{{}_{M}}$
$\displaystyle=$
$\displaystyle+\frac{\Gamma_{M}}{H}\left(\Omega^{H}_{{}_{M}}-\Omega_{{}_{M}}-\Omega_{{}_{R}}\right),$
(3.8)
$\displaystyle-(1+z)\frac{d\Omega_{{}_{R}}}{dz}+3(1+\omega_{{}_{R}})\Omega_{{}_{R}}$
$\displaystyle=$
$\displaystyle+\frac{\Gamma_{M}}{H}\left(\Omega^{H}_{{}_{M}}-\Omega_{{}_{M}}-\Omega_{{}_{R}}\right),$
(3.9)

where $\Omega^{H}_{{}_{M}}=(2/3)\chi(\bar{m}/M_{\rm pl})^{2}$ and the
$\bar{m}$ is the mass parameter. The dark energy and matter interacting rate
$\Gamma_{M}/H$ is characterized by the ratio

$\displaystyle\frac{\Gamma_{M}}{H}=\frac{\chi\epsilon}{(4\pi)}\frac{(\bar{m}/H_{0})}{(H/H_{0})},$
(3.10)

and $H_{0}$ is the Hubble constant at the present time $a_{0}=0$ and $z=0$. We
define the dark energy and matter exchanging amount $\delta Q$

$\displaystyle\delta
Q\equiv\frac{\Gamma_{M}}{H}\left(\Omega^{H}_{{}_{M}}-\Omega_{{}_{M}}-\Omega_{{}_{R}}\right).$
(3.11)

Both rate $\Gamma_{M}/H$ and amount $\delta Q$ are functions of redshift $z$.

These dynamical equations (3.1-3.3) are reminiscent of general modelling
interacting dark energy and matter based on total mass-energy conservation,
$\delta Q_{\rm int}\propto H\rho$, where $\rho$ relates to energy density, see
review [12, 14]. In comparison, we find the crucial differences between the
$\tilde{\Lambda}$CDM and interacting dark energy model are: (i) the
interacting term $\delta Q$ (3.11) changes its sign depending on the dark
energy converting to matter or inverse process, while the dark energy
interacting model $\delta Q_{\rm int}\propto H\rho$ does not change sign in
evolution; (ii) the interacting rate (3.10) is proportional to $1/H$, while
$\delta Q_{\rm int}\propto H$ in dark energy interacting models. The
difference (ii) shows that the dark energy and matter interacting rate is
small in the early time of large redshift and becomes large in small redshift
for the later time. As will be shown, it is an important feature for solving
the cosmological coincidence problem.

To solve these dynamical equations (3.6-3.9), we adopt the initial conditions
given by observations today ($a_{0}=1$)

$\displaystyle\Omega^{0}_{{}_{\Lambda}}\approx
0.7,\quad\Omega^{0}_{{}_{M}}\approx 0.3,\quad\Omega^{0}_{{}_{R}}\approx
3\times 10^{-5},$ (3.12)

and $H_{0}$. Therefore the first-order ordinary differential equations
(3.6-3.9) form a close set, uniquely determine the solutions
$\Omega_{{}_{\Lambda}}$, $\Omega_{{}_{M}}$, $\Omega_{{}_{R}}$ and $H$ as
functions of the redshift $z$ from the today $z=0$ to the past $z_{{}_{R}}$
and to the future $z\rightarrow-1$.

We adopt (3.12) as initial conditions, because
$\Omega_{{}_{R,M,\Lambda}}(z_{{}_{R}})$ values are unknown. The redshift
$z_{{}_{R}}$ at the reheating is given by

$\displaystyle(1+z_{{}_{R}})=a_{0}/a_{{}_{R}}\approx(g_{*}/2)^{1/3}(T_{\rm
RH}/T_{\rm CMB}),$ (3.13)

depending on the degeneracy $g^{*}$ of relativistic particles, the reheating
temperature $T_{\rm RH}$ and CMB temperature $T_{\rm CMB}$ [31, 22]. The
$z_{{}_{R}}\gg 1$ for $T_{\rm RH}\gg T_{\rm CMB}$. The Hubble scale variation
is huge after reheating, $H_{\rm RH}\sim T_{\rm RH}\gg H_{0}$. The Universe
evolves from radiation-, matter- and dark-energy-dominated epochs. The mass
parameter can vary in time. Therefore, we treat the mass parameter
$\bar{m}>H_{0}$ as an effective value for qualitatively studying
$\Omega_{{}_{\Lambda}}$, $\Omega_{{}_{M}}$, $\Omega_{{}_{R}}$ and $H$
evolution.

Figure 1: Numerically solving Eqs. (3.6-3.9) and (3.12), $\bar{m}/H_{0}=10$,
we present the $\tilde{\Lambda}$QCD solutions as functions of redshift $z$,
from today to the future (the left column) and to the past (the right column).
See text for detailed discussions.

###  3.3 Solution to cosmological coincidence problem

We present the numerical solutions in Figure 1, the left column for the future
$0>z>-1$ and the right column for the past $z_{{}_{R}}>z>0$. This solution is
unique and independent of where the initial condition $z=0$ or $z=z_{{}_{R}}$
is implemented. The discussions on the solution are in order:

  1. (i)

Figures 1 (a) and (b) show $\Omega_{{}_{R,M,\Lambda}}$ evolution in time (or
inverse time): (a) from the today $z=0$ (3.12) to the future ($z=-1$) when
$\Omega_{R}\rightarrow 0,\Omega_{M}\rightarrow 0$ and
$\Omega_{\Lambda}\rightarrow 1$; (b) from $z\approx z_{{}_{R}}$ radiation
domination $\Omega_{R}\approx 1,\Omega_{M}\ll 1$ and $\Omega_{\Lambda}\approx
0$ to the today $z=0$ (3.12). From $z=z_{{}_{R}}$ to $z=0$, the matter
increases, radiation decreases and $\Omega_{R}=\Omega_{M}$ occurs around
$z\sim(10^{3}-10^{4})$ for the effective mass parameter
$\bar{m}/H_{0}\sim(5-10)$. The equality $\Omega_{\Lambda}=\Omega_{M}$ occurs
around $z\approx 0.2\sim 0.4$, which is not sensitive to the parameter
$\bar{m}/H_{0}$ value. Figures 1 (c) and (d) show the Universe evolution
$\epsilon$ rate (2.5) varies from $\epsilon\approx 2$ (radiation) to
$\epsilon\approx 3/2$ (matter), $\epsilon\approx 0.45$ (today), and then to
$\epsilon\approx 0$ (dark energy) domination. The quantitative results mildly
depend on the parameter value $\bar{m}/H_{0}$. All these behaviours are
qualitatively following the $\Lambda$CDM model.

  2. (ii)

Figures 1 (e) and (f) show the solution of how the Hubble function in the unit
of $H_{0}$ evolves from $z=z_{{}_{R}}$ to $z=0$ then to $z=-1$. The
approximate constancy $H\approx H_{0}$ since $z\approx 0.1$ shows the Universe
acceleration and dark energy $\Omega_{\Lambda}$ domination over matter
$\Omega_{M}$ and radiation $\Omega_{R}$. Towards the future $z<0$,
$H^{2}\approx(8\pi/3)G\rho_{{}_{\Lambda}}$ slowly varies, asymptotically
approaching to constant $\Omega_{\Lambda}\lesssim 1$. The evolution is
analogous to inflation.

  3. (iii)

Figures 1 (g) and (h) show the dark energy and matter interacting rate
$\Gamma_{M}/H$ is small and exchanging amount $\delta Q$ (3.11) is negative.
Therefore, the matter and radiation energy slowly convert to dark energy from
the reheating end $z=z_{{}_{R}}$ to today $z=0$, then to the future $0>z>-1$.
The dark energy $\Omega_{{}_{\Lambda}}(z)$ increases from
$\Omega_{{}_{\Lambda}}(z_{{}_{R}})\approx 0$ to
$\Omega_{{}_{\Lambda}}(0)\approx 0.7$ and then to
$\Omega_{{}_{\Lambda}}(-1)\approx 1$. For values $\bar{m}\sim(5\sim 10)H_{0}$
and $H_{0}\ll M_{\rm pl}$, we can neglect the
$\Omega^{H}_{{}_{M}}=(2/3)\chi(\bar{m}/M_{\rm pl})^{2}\ll 1$, and the
$\chi\bar{m}/H_{0}$ is a unique parameter for differential equations
(3.7-3.9).

The novelty is the $\tilde{\Lambda}$CDM results of Fig. 1 show a natural
solution to the cosmic coincidence problem. It requires an incredible fine-
tuning on the initial value $\Omega_{{}_{\Lambda}}(z_{{}_{R}})\approx 0$ many
orders of magnitude, so as to achieve
$\Omega_{{}_{\Lambda}}(0)\sim\Omega_{{}_{M}}(0)\sim{\mathcal{O}}(1)$ for an
extreme long period from the reheating era $z_{{}_{R}}\gg 1$ to the present
time $z=0$. We explain below how the solution works.

The dark energy density vanishes ($\rho_{{}_{\Lambda}}\approx 0$) at the
reheating end $z_{{}_{R}}$. However, for a long period ($5\lesssim z\lesssim
z_{{}_{R}}$), it slowly increases ($\rho_{{}_{\Lambda}}\gtrsim 0$) and closely
follows up with radiation $\rho_{{}_{R}}$ and matter energy $\rho_{{}_{M}}$
densities’ evolution, see Figures 1 (b) and 2 (c) for large $z>5$. The reason
is that the dark-energy, matter, and radiation interacting rate
$\Gamma_{M}/H\ll 1$ and exchanging amount $\delta Q\ll 1$ are very small, but
not zero, see Figure 1 (h) and (j). It is crucial for the dark energy density
$\rho_{{}_{\Lambda}}$ following up the energy densities $\rho_{{}_{R}}$ and
$\rho_{{}_{M}}$ since they had been varying many orders of magnitudes in the
$5\lesssim z\lesssim z_{{}_{R}}$ period 444It is a very long period, and that
is why the $\Lambda$CDM has a fine-tuning problem.. To indicate the
$\rho_{{}_{\Lambda}}$ increase following $\rho_{{}_{\Lambda,M}}$, we here
adopt the word “following up”, which has a similar sense as the word “track
down” used in Ref. [2].

When the redshift $z\lesssim 5$, the interacting rate $\Gamma_{M}/H$ and
exchanging amount $\delta Q<0$ increase significantly because the Hubble
function $H$ becomes smaller and smaller, see Figure 1 (h) and (j). The dark
energy significantly increases after $z\approx 5$ when the matter $\Omega_{M}$
domination begins 555This is consistent with the discussions that dark energy
evolution follows first radiation then matter in different ways [29, 30]., see
Fig. 1 (b). These features are consistent with the late-time interaction in
the dark sector observed by data analysis [23]. As a result, in a short period
from $z\approx 5$ to $z\approx 0$, dark energy $\Omega_{{}_{\Lambda}}$
increases from $\Omega_{{}_{\Lambda}}\ll 1$ to the order of unit
${\mathcal{O}}(1)$. The $\Omega_{{}_{\Lambda}}$ and $\Omega_{{}_{M}}$ coincide
$\Omega_{{}_{\Lambda}}\approx\Omega_{{}_{M}}$ at $z\approx 0.5$, and are in
the same order of magnitude up to $z\approx 0$. In this short period
$0.5\lesssim z\lesssim 5$, the energy densities $\rho_{{}_{M}}$ and
$\rho_{{}_{R}}$, and Hubble function $H$ vary only a few orders of magnitudes,
see Fig. 1 (b) and (h), in contrast with their variations in many orders of
magnitudes since $z=z_{{}_{R}}$. In other words, the recent $\rho_{{}_{M}}$
and $\rho_{{}_{\Lambda}}$ evolution are insensible to their initial values at
$z_{{}_{R}}$. Nature does not need to fine-tune the initial ratios of dark
energy, matter and radiation densities at the Big Bang beginning (the
reheating end).

The dark energy, matter, and radiation interacting rate $\Gamma_{M}/H$ (3.10)
and exchanging amount $\delta Q<0$ (3.11) are small at high redshift $z$, and
large at low redshift $z$. Due to such redshift $z$ dependence of the
interaction, $\tilde{\Lambda}$CDM gives a dynamical solution to the cosmic
coincidence problem of $\Lambda$CDM in the following scenes. Without any fine-
tuning, the dynamical solution uniquely determines the evolution from
$\Omega_{{}_{M}}(0)\sim\Omega_{{}_{\Lambda}}(0)\gg\Omega_{{}_{R}}(z_{{}_{R}})$
to
$\Omega_{{}_{R}}(z_{{}_{R}})\gg\Omega_{{}_{M}}(z_{{}_{R}})\gg\Omega_{{}_{\Lambda}}(z_{{}_{R}})\approx
0$, and vice versa. Namely, if we would know the initial conditions
$\Omega_{{}_{R,M,\Lambda}}(z_{{}_{R}})$ at $z\approx z_{{}_{R}}$ (3.13), we
would have obtained the same dynamical solutions (Fig. 1) and the present
values (3.12) by without fine-tuning. It is the main result presented in this
article.

We have made numerical verification that the $\tilde{\Lambda}$CDM solution
from the maximal $\Omega_{{}_{M}}$ ($z\approx 5$) to
$\Omega_{{}_{M}}\approx\Omega_{{}_{\Lambda}}$ ($z\approx 0.5$) does not
sensitively depend on the effective value of mass parameter $\bar{m}$. It
implies that the $\tilde{\Lambda}$CDM solution should similarly function if we
adopt the mass parameter $\bar{m}$ weakly depending on $H$.

We do not discuss the dark energy density perturbations
($\delta\rho_{{}_{\Lambda}},\delta p_{{}_{\Lambda}}$) caused by its time-
varying interaction with matter and radiation. However, we speculate that dark
energy undergoes transitions and becomes dominant from $z\approx 5$ to
$z\approx 0.1$ should impact matter density perturbation, leading to the
effect on the formation of large-scale structures and clusters. In addition,
it should induce the peculiar fluctuations of the gravitational field,
possibly imprinting on observations, for instance, the integrated Sachs-Wolfe
effect or galaxy positions. The reason is that the dark energy $\Lambda$
results on the gravitation field are rather different from the gravitational
potential of matter.

To end this section, we mention that in the remote future ($z\rightarrow-1$)
of dominant dark energy, radiation and matter $\Omega_{R}+\Omega_{M}$
continually decrease until the exchanging amount $\delta Q$ (3.11) changes
sign from negative $\delta Q<0$ to negative $\delta Q>0$. The dark energy
density decreases, converting to matter and radiation energy densities. It
shows the possibility that the Universe ends the current acceleration and
starts recycling again. The topic is not the scope of this article. We do not
present figures and discussions for this situation in the remote future
$z\rightarrow-1$.

Figure 2: We present the $\tilde{\Lambda}$CDM solution (Fig. 1) in comparison
and contrast with the canonical $\Lambda$CDM model (4.1), 4.2) and (3.12). See
text for detailed discussions.

##  4 Approximated $\tilde{\Lambda}$CDM solution for data analysis

###  4.1 Comparison and contrast with $\Lambda$CDM model

In this section, we present the $\tilde{\Lambda}$CDM solution in comparison
and contrast with the $\Lambda$CDM results. In the $\Lambda$CDM model, we
define the cosmic abundance of radiation, matter, and dark energy

$\displaystyle\Omega_{{}_{R}}=\Omega^{0}_{{}_{R}}\frac{(1+z)^{4}}{E(z)^{2}},\quad\Omega_{M}=\Omega^{0}_{{}_{M}}\frac{(1+z)^{3}}{E(z)^{2}},\quad\Omega_{{}_{\Lambda}}=\Omega^{0}_{{}_{\Lambda}}\frac{(1+z)^{0}}{E(z)^{2}},$
(4.1)

and the dimensionless Hubble function $E(z)\equiv(H/H_{0})$

$\displaystyle
E(z)^{2}=\Omega^{0}_{{}_{R}}(1+z)^{4}+\Omega^{0}_{{}_{M}}(1+z)^{3}+\Omega^{0}_{{}_{\Lambda}}(1+z)^{0},$
(4.2)

where
$E(0)^{2}=\Omega^{0}_{{}_{R}}+\Omega^{0}_{{}_{M}}+\Omega^{0}_{{}_{\Lambda}}=1$.
The evolution $\epsilon$-rate is given by Eq. (3.6). The values
$\Omega^{0}_{{}_{R}},\Omega^{0}_{{}_{M}},\Omega^{0}_{{}_{\Lambda}}$ and
$H_{0}$ at $z=0$ are the same as the initial conditions (3.12). Here we use
the same notations for quantities of the $\tilde{\Lambda}$CMD and $\Lambda$CMD
models. The former is the dark energy and matter interacting solutions to Eqs.
(3.6-3.9). The latter is (4.1) and (4.2) for the constant dark energy density.
We have implemented only one observed data point (3.12) for both models.

In Figure 2, we compare $\tilde{\Lambda}$CMD solutions (Fig. 1) with the
$\Lambda$CMD (4.1-4.2) results. The discussions are in order.

  1. (i)

Figures 2 (a) and (b) show $\Omega_{{}_{R,M,\Lambda}}$ and expansion rate
$\epsilon$ (3.6) evolution for $\Lambda$CDM and $\tilde{\Lambda}$CDM. Overall
they are consistent and agree with each other, particularly for $z<10$. Two
$\Omega_{{}_{\Lambda}}$ curves overlap in (a) and the point
$\Omega_{{}_{\Lambda}}=\Omega_{{}_{M}}$ is about the same. The main
differences are $\Omega_{{}_{R,M}}$ and $\epsilon$-rate are in the range
$z\sim 10^{2}-10^{5}$, and the crossing point
$\Omega_{{}_{R}}=\Omega_{{}_{M}}$. These differences mildly depend on the
$\tilde{\Lambda}$CDM parameter value
$\chi\bar{m}/H_{0}\sim{\mathcal{O}}(10^{-2})$.

  2. (ii)

Figures 2 (c) and (d) shows $\Omega_{{}_{\Lambda}}$ and $H/H_{0}$ evolution
for $\tilde{\Lambda}$CDM and $\Lambda$CDM. The crucial difference in
$\Omega_{{}_{\Lambda}}$ appears for $z>10$. Due to the nature of constant dark
energy density, the $\Lambda$CDM $\Omega_{{}_{\Lambda}}\propto 1/H^{2}$ for
$z\rightarrow z_{{}_{R}}\gg 1$. It leads the fine-tuning problem for achieving
$\Omega^{0}_{{}_{\Lambda}}\propto 1/H^{2}_{0}\sim{\mathcal{O}}(1)$ from
$\Omega_{{}_{\Lambda}}\propto 1/H^{2}_{\rm RH}\ll{\mathcal{O}}(1)$. Whereas,
it is not the case for the $\tilde{\Lambda}$CDM model, as discussed in the
previous section. The Hubble function $H/H_{0}$ discrepancy between
$\tilde{\Lambda}$CDM and $\Lambda$CDM is significant at high red-shift
$(z>1000)$. These comparisons and contrasts imply that the
$\tilde{\Lambda}$CDM could relieve the $H_{0}$ and $S_{8}$ tensions between
the values measured today and calculated in the $\Lambda$CDM model based on
measurements at high red-shifts $z$.

These discussions show that (i) apart from solving the cosmic coincidence
problem, the $\tilde{\Lambda}$CDM’s quantities slightly deviate from the
$\Lambda$CDM counterparts for $z<10^{3}$; (ii) the $\tilde{\Lambda}$CDM
represent a one-parameter ($\chi\bar{m}$) extension to the $\Lambda$CDM model.
These are based on numerical solutions to non-linearly coupled differential
equations (3.6-3.9). Therefore, it is not convenient in practice for
quantitatively comparing the $\tilde{\Lambda}$CDM with observation data.

###  4.2 Approximated $\tilde{\Lambda}$CDM solution for phenomenological
studies

The dynamic system formed by four differential equations (3.6-3.9) should have
a fixed point at low redshifts ($z\ll 1$), where the $\Lambda$CDM realizes.
From high redshifts ($z\gg 1$), the $\tilde{\Lambda}$CDM quantities approach
this fixed point in scaling laws, namely, scaling factors $(1+z)^{\delta}$
corrected $\Lambda{\rm CDM}$ counterparts. The scaling indexes $|\delta|\ll
1$, because the $\tilde{\Lambda}$CDM approaches $\Lambda$CDM for low
redshifts, as shown in Fig. 2. In Ref. [25], we expect these dynamics and
approximately derive analytical solutions (4.6,4.7) in the spirit of
asymptotic safety of gravitational theories [32].

The view of scaling-law $(1+z)^{\delta}$ corrections is further supported by
the smallness parameter $\chi\bar{m}/H_{0}$ in the dark energy and matter
interacting rate (3.10). For low redshifts, we approximately decouple Eqs.
(3.1-3.3) into

$\displaystyle\dot{\rho}_{{}_{\Lambda}}+0H\rho_{{}_{\Lambda}}$
$\displaystyle\approx$
$\displaystyle+\delta_{{}_{\Lambda}}H\rho_{{}_{\Lambda}},$ (4.3)
$\displaystyle\dot{\rho}_{{}_{M}}+3H\rho_{{}_{M}}$ $\displaystyle\approx$
$\displaystyle-\delta^{G}_{{}_{R}}H\rho_{{}_{M}},$ (4.4)
$\displaystyle\dot{\rho}_{{}_{R}}+4H\rho_{{}_{R}}$ $\displaystyle\approx$
$\displaystyle-\delta^{G}_{{}_{M}}H\rho_{{}_{R}}.$ (4.5)

Three new dimensionless parameters $\delta^{G}_{{}_{R}}$,
$\delta^{G}_{{}_{M}}$ and $\delta_{{}_{\Lambda}}$ are proportional to the
primal parameter $\chi\bar{m}/H_{0}$. They are much smaller than the unity.
Equations (4.3-4.5) yield the effectively corrected densities

$\displaystyle\rho_{{}_{R}}\approx\rho^{0}_{{}_{R}}(1+z)^{4-\delta^{{}^{R}}_{G}},\quad\rho_{{}_{M}}\approx\rho^{0}_{{}_{M}}(1+z)^{3-\delta^{{}^{M}}_{G}},\quad\rho_{{}_{\Lambda}}\approx\rho^{0}_{{}_{\Lambda}}(1+z)^{\delta_{{}_{\Lambda}}},$
(4.6)

and the Hubble function

$\displaystyle
E^{2}(z)=\Omega^{0}_{{}_{R}}(1+z)^{4-\delta^{{}^{R}}_{G}}+\Omega^{0}_{{}_{M}}(1+z)^{3-\delta_{G}^{{}^{M}}}+\Omega^{0}_{{}_{\Lambda}}(1+z)^{\delta_{{}_{\Lambda}}}.$
(4.7)

In the view of Eqs. (4.6), we consider the equations of states effectively
modify: $\omega^{\rm eff}_{{}_{R}}\approx 1/3(1-\delta^{{}^{R}}_{G})$,
$\omega^{\rm eff}_{{}_{M}}\approx-(1/3)\delta^{{}^{M}}_{G}$ and $\omega^{\rm
eff}_{{}_{\Lambda}}\approx-1+(1/3)\delta_{{}_{\Lambda}}$, see also Ref. [33].
Equations (4.6,4.7) are $\tilde{\Lambda}$CDM approximate solutions, giving
scaling-law $(1+z)^{\delta}$ corrections to $\Lambda$CDM results. The fourth
independent equation (2.5) gives the constraint of parameters
$\delta^{G}_{{}_{R}}$, $\delta^{G}_{{}_{M}}$ and $\delta_{{}_{\Lambda}}$,

$\displaystyle\delta_{\Lambda}$ $\displaystyle\approx$
$\displaystyle(\Omega^{0}_{M}\delta^{{}^{M}}_{G}+\Omega^{0}_{R}\delta^{{}^{R}}_{G})/\Omega^{0}_{{}_{\Lambda}},\
$ (4.8)

and two parameters are independent. They depend actually on one primal
parameter $\chi m$ (2.3). The approximate solutions (4.6,4.7,4.8) facilitate
data analysis for comparing $\tilde{\Lambda}$CDM with observational data.

References [34, 35] presents detailed numerical studies and data analysis
based on the approximated $\tilde{\Lambda}$CDM solutions (4.6,4.7) and
numerous data sets of observations. It shows that both $\Lambda$CDM $H_{0}$
and $S_{8}$ tensions reduces to $2\sigma$ level with constraint parameters
$\delta^{{}^{R}}_{G}\approx-1.5\times 10^{-2}$,
$\delta^{{}^{M}}_{G}\approx-5.0\times 10^{-4}$ and
$\delta_{{}_{\Lambda}}\approx-2.0\times 10^{-4}$. The negative parameter
values support the scenario of energy conversion from radiation and matter to
dark energy, as discussed in the previous section. The negative
$\delta_{{}_{\Lambda}}\lesssim 0$ implies that due to interactions, dark
energy slightly behaves as if it was a phantom energy $\omega^{\rm
eff}_{{}_{\Lambda}}\approx-1+(1/3)\delta_{{}_{\Lambda}}\lesssim-1$. It differs
from the situation in inflation and reheating when dark energy converts to
matter and radiation energies [22] see Fig. 3, dark energy behaves as if it
was a quintessence energy $\omega^{\rm eff}_{{}_{\Lambda}}>-1$.

##  5 Discussions on Einstein cosmological $\Lambda$ term

We end this article with some discussions and speculations on the
gravitational (geometric) and dynamical natures of the cosmological
$\tilde{\Lambda}$ term and dark-energy density
$\rho_{{}_{\Lambda}}=\tilde{\Lambda}/(8\pi G)$ of the Einstein theory. We
discuss the dynamical solution to the cosmic fine-tuning problem.

###  5.1 Geometric nature of $\tilde{\Lambda}$ dark energy as gravitational
ground state

The $\tilde{\Lambda}$ term possibly represents [36, 37, 38, 39, 40, 25] the
non-trivial ground state (Wheeler spacetime foam [41, 42]) of the spacetime.
The perturbative quantum gravitational field fluctuates upon such a ground
state, and the classical gravitational field varies in such a ground state.
They are effectively described by the gravitation coupling $G$, the
$\tilde{\Lambda}$ and the Ricci scalar $R$ terms in Einstein’s theory. Such a
ground state is probably a coherent state of the long-ranged holonomy field
(see Eq. (133) of Ref. [38]). It is a condensate state due to strongly violent
quantum gravity at the Planck scale. The spacetime foam structure of such a
ground state is most intriguing. It could be an interacting gas of
gravitational instantons (wormholes), whose effective equation of state
behaves as $p_{{}_{\Lambda}}=-\rho_{{}_{\Lambda}}$, see Sec. X of Ref. [43].
We are proceeding with further studies on these aspects.

The $\tilde{\Lambda}$ ($\xi\sim 1/\tilde{\Lambda}^{1/2}$) is the
characteristic scale (correlation length) of such non-trivial geometric ground
state [40, 25]. It represents the intrinsic scale for effective gravitational
field theories realized in the scaling domains of fixed points of effective
gravitational coupling $g\sim GM^{2}_{\rm pl}$ to matter and radiation. The
$\tilde{\Lambda}$ and $g$ varying from one fixed point to another render its
dynamic nature. It is nontrivial to demonstrate these dynamical features.
However, as analogies, we mention fundamental field theories of interactions
(i) the electroweak scale $v\sim 10^{2}$GeV for electroweak field theory
realized in the scaling domain of infrared (IR) fixed point; (ii) the scale
$\Lambda_{\rm QCD}\sim 10^{2}$MeV for perturbative QCD field theory realized
in the scaling domain of ultraviolet (UV) fixed point; (iii) the low-energy
hadron scale for non-perturbative QCD field theory realized in the scaling
domain of IR fixed point.

###  5.2 Asymptotically safe Einstein theory for early and present Universe

On the one hand, in early Universe of $\tilde{\Lambda}$ dark energy dominated
inflation, $H^{2}\sim(8\pi G/3)\rho_{{}_{\Lambda}}$ and
$\rho_{{}_{\Lambda}}=\tilde{\Lambda}/(8\pi G)$ asymptotically give $\xi\sim
1/\tilde{\Lambda}^{1/2}\sim H^{-1}$. Namely, the correlation length $\xi$ is
the size of the horizon. The $\tilde{\Lambda}^{1/2}$ slowly varies from the
inflation scale $H^{*}$ to the scale $H_{\rm end}\approx(0.42,0.35)H_{*}$ at
inflation end $a_{\rm end}$. The $H_{*}\sim 10^{-6}M_{\rm pl}$ is obtained
from the CMB data, see Eqs. (6.5) and (6.10) of Ref. [21]. The inflation scale
$H_{*}$ is much smaller than the Planck scale. How quantum gravitation field
theory with the intrinsic scale $\tilde{\Lambda}^{1/2}\sim M_{\rm pl}$ runs to
the effective Einstein theory at the scale $\tilde{\Lambda}^{1/2}\sim H_{*}\ll
M_{\rm pl}$. How does the quantum gravity ground state evolve to the
$\tilde{\Lambda}$ ground state of effective Einstein theory? One could study
it in the context of the asymptotically safe and effective theories of
gravitation [32] and the scaling domain of a UV unstable fixed point [25].

On the other hand, in the recent Universe of $\tilde{\Lambda}$ dark energy
dominated acceleration, $H^{2}_{0}\sim(8\pi G/3)\rho^{0}_{{}_{\Lambda}}$
asymptotically gives the scale $\xi\sim 1/\tilde{\Lambda}^{1/2}\sim
H^{-1}_{0}$ and density $\rho^{0}_{{}_{\Lambda}}\approx H_{0}^{2}/(8\pi G)$
[44]. Based on the same spirit of asymptotic safety of effective gravitational
theories [32], we study its realization in the scaling domain of a UV stable
fixed point, where is the effective Einstein theory of relevant operators
$R/G$ and $\tilde{\Lambda}/G$, and gravitational coupling $G$ and cosmological
$\tilde{\Lambda}$ approach their values today [40, 25]. However, due to the
dark energy, radiation and matter interactions, as well as pair production of
massive particles and antiparticles on the horizon, it is nontrivial to find
the scaling laws for operators $R/G$ and $\tilde{\Lambda}/G$ by using the
asymptotic safety principle. The questions are how the $\tilde{\Lambda}$ dark
energy varies from the inflation scale $H_{*}$ to the recent Hubble scale
$H_{0}\ll H_{*}$. How the dark energy density changes from
$\rho^{*}_{{}_{\Lambda}}\approx H_{*}^{2}/(8\pi G)$ to
$\rho^{0}_{{}_{\Lambda}}\ll\rho^{*}_{{}_{\Lambda}}$ in many orders of
magnitudes. We use the $\tilde{\Lambda}$CDM solutions in inflation, reheating
and standard cosmology to explain the possible solution to such cosmic fine-
tuning problem.

Figure 3: We reproduce the $\tilde{\Lambda}$CDM results of the Hubble function
$h^{2}=H^{2}/H^{2}_{\rm end}$ (left), $\Omega_{M}$, $\Omega_{R}$ and
$\Omega_{\Lambda}$ (right) in reheating, see Figure 6 (a) and (b) in Ref.
[22]. They are in the unit of $\rho^{\rm end}_{c}=3m^{2}_{\rm pl}H^{2}_{\rm
end}$ and the Hubble scale $H_{\rm end}\approx(0.42,0.35)H_{*}$ at inflation
end $a_{\rm end}$, see Eq. (6.10) of Ref. [21]. The parameters $\hat{m}/m_{\rm
pl}\approx 27.7$ and $\chi\approx 1.85\times 10^{-3}$. It shows that in a few
numbers $x=\ln(a/a_{\rm end})$, dark energy $\Omega_{\Lambda}$ decreases from
one to zero because of rapidly converting to matter and radiation. Matter
$\Omega_{M}$ increases and dominates over $\Omega_{\Lambda}$. Then
$\Omega_{M}$ decreases because of decaying to radiation $\Omega_{R}$. As a
result, $\Omega_{R}$ increases and becomes dominant. At the reheating end
$(a_{{}_{R}}/a_{0})=(1+z_{{}_{R}})^{-1}$, the radiation abundance is about
one, and the dark energy and matter abundances are about zero.

###  5.3 Dynamical nature of $\tilde{\Lambda}$ dark energy solving fine-tuning
problem

After the inflation end, the Universe undergoes reheating. Based on dynamical
equations (3.6-3.9), we show [22] that due to strong coupling
($\Gamma_{M}/H\gg 1$) between $\tilde{\Lambda}$ dark energy and matter energy
densities, dark energy rapidly converts into massive matter, and the latter
decays to radiation energy. As a result, dark energy density decreases from
$\rho^{\rm end}_{{}_{\Lambda}}\approx 3m^{2}_{\rm pl}H^{2}_{\rm end}$ to
$\rho^{R}_{{}_{\Lambda}}\approx 0$, where $\rho^{R}_{{}_{\Lambda,M,R}}$ stand
for the dark energy, matter and radiation densities at the reheating end
$a_{{}_{R}}/a_{0}=(1+z_{{}_{R}})^{-1}$. We illustrate in Fig. 3 the dynamical
reheating process from the inflation end $\rho^{\rm
end}_{{}_{\Lambda}}\gg\rho^{\rm end}_{{}_{M}}\gg\rho^{\rm end}_{{}_{R}}\approx
0$ to the reheating end
$\rho^{R}_{{}_{R}}\gg\rho^{R}_{{}_{M}}\gg\rho^{R}_{{}_{\Lambda}}\approx 0$.
The radiation energy density $\rho^{R}_{{}_{R}}$ becomes dominant, initiating
the standard cosmology. There is no fine-tuning in this process.

Then how the standard cosmology dynamically evolves to the coincidence
$\rho^{0}_{{}_{\Lambda}}\sim\rho^{0}_{{}_{M}}\gg\rho^{0}_{{}_{R}}\approx 0$ in
the recent epoch. It is the issue addressed in this article. The initial
values of the scale factor $a_{{}_{R}}$, Hubble constant $H_{\rm RH}$ and
energy densities $\rho^{R}_{{}_{R,M,\Lambda}}$ cannot be completely
determined. Therefore we cannot uniquely solve ordinary differential equations
(3.6-3.9) from the reheating end $z_{{}_{R}}$ to the present epoch $z=0$.
However, we use the present values (3.12) to uniquely solve ordinary
differential equations (3.6-3.9) from today $z=0$ back to the reheating end
$z_{{}_{R}}\gg 0$. As shown in Fig. 1 (b) and Fig. 2 (c), the dynamical
solutions asymptotically approach the same initial conditions
$\rho^{R}_{{}_{R}}\gg\rho^{R}_{{}_{M}}\gg\rho^{R}_{{}_{\Lambda}}\approx 0$ for
$z\rightarrow z_{{}_{R}}\gg 1$ without any fine-tuning.

Such qualitative matching implies a consistent dynamical solution for the
cosmic fine-tuning problem in the following way. Converting to matter and
radiation $\delta Q\gg 1$ (3.11), the dark energy density decreases from the
inflation scale $\rho^{*}_{{}_{\Lambda}}\approx 3m^{2}_{\rm pl}H^{2}_{*}$ to
inflation end $\rho^{\rm end}_{{}_{\Lambda}}\approx 3m^{2}_{\rm pl}H^{2}_{\rm
end}$, then to reheating end $\rho^{R}_{{}_{\Lambda}}\approx 0$. Since the
standard cosmology starts, converted from matter and radiation $\delta
Q\lesssim 0$ (3.11), the dark energy density increases from the reheating end
value $\rho_{{}_{\Lambda}}^{R}\approx 0$ to the present value
$\rho^{0}_{{}_{\Lambda}}\approx H_{0}^{2}/(8\pi G)$ [44]. Such a dynamical
evolution is free from fine-tuning. It can be the solution to the cosmic
coincidence problem. The basic reasons are that in evolution dark energy and
matter conversion $\delta Q$ (3.11) changes sign and is proportional to the
interacting rate $\Gamma_{M}/H\propto\chi m\epsilon/H$ and $m>H$.

Nonetheless, we have not yet found the complete and quantitative solution to
the cosmic fine-tuning problem since we separately adopt the effective values
of mass parameter $m_{*}/H_{*}$ for inflation, $\hat{m}/H_{\rm end}$ for
reheating and $\bar{m}/H_{0}$ for standard cosmology. The mass parameter $m$
is proportional to the mass $M$ and number ${\mathcal{N}}_{\rm pair}$ of
massive particle and antiparticle pairs in the holographic massive plasma
state. Therefore, its value should depend on the horizon $H$, i.e., $m=m(H)$.
Its time-varying should be slower than the $H$ so that the dark energy and
matter interacting rate $\Gamma_{M}/H\propto m(H)/H\propto m_{\rm eff}/H$
decreases (increase) as $H$ increases (decreases). We have not been able to
determine $m(H)$ in theory. Instead, we fix its effective values $m_{\rm eff}$
by observations.

We are studying the aforementioned natures of the Einstein cosmological
$\Lambda$ term. It is also worthwhile to investigate the $\tilde{\Lambda}$CDM
scenario in the quintessence framework with an effective potential $V(\phi)$.
On the other hand to understand these fundamental issues in cosmology, further
observations of the cosmos are necessary [45, 46].

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