Evolution of Cosmological Parameters and Fundamental Constants in a Flat
Quintessence Cosmology: A Dynamical Alternative to $\Lambda$CDM

Rodger I. Thompson

Department of Astronomy and Steward Observatory University of Arizona;
<EMAIL_ADDRESS>
##  1 Abstract

The primary purpose of this work is the provision of accurate, analytic,
evolutionary templates for cosmological parameters and fundamental constants
in a dynamical cosmology. A flat quintessence cosmology with a dark energy
potential that has the mathematical form of the Higgs potential is the
specific cosmology and potential addressed in this work. These templates,
based on the physics of the cosmology and potential are intended to replace
the parameterizations currently used to determine the likelihoods of dynamical
cosmologies. Acknowledging that, unlike $\Lambda$CDM, the evolutions are
dependent on both the specific cosmology and the dark energy potential the
templates are referred to as Specific Cosmology and Potential, SCP, templates.
The requirements set for the SCP templates are that they must be accurate,
analytic functions of an observable such as the scale factor or redshift. This
is achieved through the utilization of a modified beta function formalism that
is based on a physically motivated dark energy potential to calculate the beta
function. The methodology developed here is designed to be adaptable to other
cosmologies and dark energy potentials. The SCP templates are essential tools
in determining the relative likelihoods of a range of dynamical cosmologies
and potentials. An ultimate purpose is the determination whether dark energy
is dynamical or static in a quantitative manner. It is suggested that the SCP
templates calculated in this work can serve as fiducial dynamical templates in
the same manner as $\Lambda$CDM serves for static dark energy.

Keywords:

Cosmological Parameters ; Fundamental Constants; Quintessence

##  2 Introduction

This manuscript examines the evolutions in the late time, matter and dark
energy dominated, epoch between the scale factors of 0.1 and 1.0 for a flat
quintessence cosmology. This epoch is the primary focus of the upcoming Rubin
and Roman observatories observations. The study does not consider radiation
but only matter and dark energy and is therefore only relevant to late epochs
not under the influence of radiation. The farthest look back time considered
here is at a scale factor of 0.1 where radiation has no measurable effect.

Some preliminary aspects of areas covered in this publication are discussed in
(2022). This work, however, expands the study and is intended for both experts
in the field and those who wish an introduction to calculations of the
evolution of cosmological parameters and fundamental constants. The
methodology presented here is particular to the specific cosmology,
quintessence, and the evolutionary templates it calculates are for a specific
dark energy potential. The templates from the methodology are therefore
referred to as Specific Cosmology and Potential, SCP, templates.

The nature of dark energy has been declared one of the “grand challenges in
both physics and astronomy” by the Decadal Survey of Astronomy and
Astrophysics 2022 (National Academies of Sciences, Engineering, and Medicine,
2021). A major part of that challenge is the question of whether dark energy
is static or dynamic. An important aspect of the question is whether a
dynamical cosmology can fit the current and future observational data as well
or better than $\Lambda$CDM. This work explores a flat quintessence cosmology
with current parameter boundary conditions close but not equal to $\Lambda$CDM
to provide accurate predictions of parameter and fundamental constant
evolutions for comparison with data. The calculation of the evolutions is
described in detail, particularly the use of a modified beta function
formalism that produces accurate, analytic, functions of the parameters and
constants as a function of the observable scale factor. The dark energy
potential has a natural origin, having the same mathematical polynomial form
as the Higgs potential. It is, however, not the Higgs field and has none of
the rich physics of the Higgs. It is simply a rolling scalar field with
quintessence physics that is coupled to gravity. Since the potential has the
same mathematical form as the Higgs potential it is referred to as the Higgs
Inspired or HI potential.

The development of the SCP templates for a flat, minimally coupled,
Quintessence cosmology takes advantage of the property of minimally coupled
systems that the dark energy and matter density evolutions are independent of
each other. Each can be calculated separately, as is traditionally done for
Quintessence. (Scherrer and Sen, 2008; Copeland, 2006; Cicciarella and
Pieroni, 2017;  , 2018), and then combined when necessary for the calculation
of the cosmological parameters such as the Hubble parameter. As an example the
evolution of the matter density is simply $\frac{\rho_{m_{0}}}{a^{3}}$ where
$\rho_{m_{0}}$ is the current matter density and $a$ is the scale factor. The
evolution of the scalar and other dark energy functions are calculated in
sections 4 through 11 without reference to the matter density except in the
introduction of the Friedmann constraints. The matter density is incorporated
in section 12 where the Hubble parameter is calculated using the first
Friedmann constraint. An approximation is made in equation 16 where the
$\frac{\beta^{2}}{6}$ term is set to zero to achieve equation 17 which is only
a function of the dark energy potential and scalar. The approximation is valid
for all of the dark energy EoS values in this study but could lose accuracy
for high deviations of $w$ from minus one.

Beyond demonstrating the methodology for producing SCP templates for
cosmological parameter evolution this work also examines the role of
fundamental constants in setting constraints on both static and dynamical
cosmologies. Without invoking special symmetries it is difficult to prevent a
scalar field that couples with gravity from also coupling with other sectors
such as the weak, electromagnetic and strong forces (Carroll, 1998). The
values of the fundamental constants such as the fine structure constant
$\alpha$ and the proton to electron mass ratio $\mu$ are determined by the
Quantum Chromodynamic Scale $\Lambda_{QCD}$, the Higgs vacuum expectation
value $\nu$ and the Yukawa couplings $h$ (Coc, 2007). It is assumed here that
the HI scalar responsible for dark energy also interacts with these sectors.
The temporal evolution of the constants produced by the interactions is
examined in section 18 along with the connection to the dark energy Equation
of State, EoS, $w$.

The study utilizes natural units with $\hbar$, $c$ and $8\pi G$ set to one
where $G$ is the Newton gravitational constant. The mass units are reduced
Planck masses $m_{p}$. The constant $\kappa$ is the inverse reduced Planck
mass $1/m_{p}$. In the mass units of this study $\kappa=1$ but it is retained
in equations to display the proper mass units.

##  3 The Need for SCP Templates

The SCP templates generated in this study are candidates for a fiducial set of
templates to compare with the observations in the same way that the well known
static $\Lambda$CDM templates are currently used. Although accurate analytic
templates for the static $\Lambda$CDM cosmology exist similar templates for
dynamical cosmologies are exceedingly rare. Currently the primary tools for
analyzing dynamical cosmologies are parameterizations such as the Chevallier,
Polarski and Linder , CPL, (Chevallirer , 2001; Linder., 2003) linear
parameterization of the dark energy Equation of State, EoS,
$w(a)=w_{o}+(1-a)w_{a}$. Such parameterizations do not contain any of the
physics of the dynamical cosmology and its dark energy potential. Figure 1
shows the CPL fit to a quintessence cosmology $w$ with the dark energy
potential described in section 6.

Figure 1: The dashed line is the CPL linear fit to the $w(a)$ freezing
evolution for $w_{0}=-0.99$, solid line.

The linear CPL, dashed line, is a poor fit to the true evolution, solid line,
and is not an accurate measure of its likelihood. Beyond producing an
erroneous likelihood the CPL fit also produces erroneous conclusions. At scale
factors greater than 0.8 the CPL fit is in the phantom region, $w<-1$, even
though the true evolution has no phantom values. It is well known that it is
quite difficult for quintessence to cross the phantom boundary (Vikman, 2005).
Three recent analyses of observational data (Planck Collaboration, 2019; Di
Valentino, Melchiorri and Silk, 2020; Di Valentino, 2020) using a MCMC
analysis with a CPL dark energy template find phantom values of $w$ at low
redshift. The presence of phantom values of $w$ can be interpreted as strong
evidence against quintessence, however, figure 1 clearly shows that for a
quintessence cosmology, with no phantom values, a CPL fit erroneously produces
phantom values due to fitting a non-linear evolution with a linear
parameterization. The methodology described here produces templates based on
the action of the quintessence cosmology and a specific dark energy potential
for comparison to the observational data. SCP templates are essential to
properly compare accurate predictions to the observational data in
establishing the true likelihood of the cosmology and potential.

##  4 Quintessence

As one of the simplest dynamical cosmologies, flat quintessence provides a
straightforward example of a methodology for producing SCP templates.
Quintessence is a well studied (Copeland, 2006) and well known cosmology but
for easy reference some important aspects of its physics are given here.

The quintessence cosmology is defined by its action $S_{q}$.

$S_{q}=\int
d^{4}x\sqrt{-g}\left[m_{p}^{2}\frac{R}{2}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right],\hskip
14.22636ptS_{m}=\frac{\rho_{m_{0}}}{a^{3}}$ (1)

where $S_{q}$ is the dark energy action and $S_{m}$ is the matter (dust)
action. $R$ is the Ricci scalar, $g$ is the determinant of the metric
$g^{\mu\nu}$, $\phi$ is the scalar, and $V(\phi)$ is the dark energy
potential. The scalar $\phi$ is the true scalar with a value on the order of
$10^{-32}m_{p}$. A second scalar $\theta$ is introduced in section 6 for the
Ratra-Peebles form of the dark energy potential. This scalar is, therefore,
referred to as the Ratra-Peebles, or RP scalar and has a value on the order of
unity in units of $m_{p}$. In the matter action $\rho_{m_{0}}$ is the current
matter density and $a$ is the scale factor. The matter component of the total
action $S_{m}$ is separate from the quintessence dark energy component $S_{q}$
resulting in a total action of $S_{tot}=S_{q}+S_{m}$ as in equation (3.1) of
(Cicciarella and Pieroni, 2017) The matter component is introduced via the
first Friedmann constraint in section 12 in the calculation of the Hubble
parameter. In the following only dark energy is considered in the derivation
of the scalar since its evolution is not affected by matter. Matter is then
introduced in section 12 to derive the proper Hubble parameter that includes
both dark energy and matter.

The kinetic component of $S_{q}$ is

$X=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi=-\frac{\dot{\phi}^{2}}{2}.$
(2)

The common kinetic symbol $X$ for $-\frac{\dot{\phi}^{2}}{2}$ is used
throughout the manuscript. $X$ is a function of only time since the universe
is assumed to be spatially homogeneous.

The quintessence dark energy density and pressure are set by the action as

$\rho_{\phi}\equiv-X+V(\phi),\hskip 28.45274ptp_{\phi}\equiv-X-V(\phi)$ (3)

In natural units both the density and the pressure have units of $m^{4}_{p}$
and the time derivative of the scalar $\dot{\phi}$ has units of $m^{2}_{p}$.
The dark energy EoS $w(\phi)$ is

$w(\phi)=\frac{P_{\phi}}{\rho_{\phi}}=\frac{-X-V(\phi)}{-X+V(\phi)}.$ (4)

which is a pure number. Combining eqns. 3 gives

$P_{\phi}+\rho_{\phi}=-2X$ (5)

It follows that

$\frac{P_{\phi}+\rho_{\phi}}{\rho_{\phi}}=w+1=\frac{-2X}{\rho_{\phi}}$ (6)

giving $\dot{\phi}$ a relationship to the dark energy EoS and the dark energy
density.

$-2X=\rho_{\phi}(w+1).$ (7)

##  5 General Cosmological Constraints

Independent of the particular cosmology there are general constraints on the
evolution of the cosmological parameters. The first are the Friedmann
constraints.

###  5.1 The Friedmann constraints

The two Friedmann constraints play an important role in calculating the SCP
templates. The forms of the first and second Friedman constraints used here
are

$3\left(\frac{H(a)}{\kappa}\right)^{2}=\rho_{\phi}(a)+\rho_{m}(a)\hskip
14.22636pt,\hskip
14.22636pt3\left(\frac{\dot{H}(a)}{\kappa^{2}}+\left(\frac{H(a)}{\kappa}\right)^{2}\right)=-\frac{\rho(a)+3P(a)}{2}$
(8)

where $\rho_{\phi}(a)$ is the dark energy density, $\rho_{m}(a)$ is the matter
density, $\rho(a)$ is the sum of the matter and dark energy densities and
$P(a)$ is the dark energy pressure. In a universe with only dark energy the
Friedmann constraints are

$3\left(\frac{H_{\phi}(a)}{\kappa}\right)^{2}=3\left(\frac{\sqrt{\Omega_{\phi}}H(a)}{\kappa}\right)^{2}=\rho_{\phi}(a)\hskip
7.11317pt,\hskip
7.11317pt3\left(\frac{\dot{H}_{\phi}(a)}{\kappa^{2}}+\left(\frac{H_{\phi}(a)}{\kappa}\right)^{2}\right)=-\frac{\rho_{\phi}(a)+3P(a)}{2}$
(9)

where $H_{\phi}$ denotes the dark energy only Hubble parameter. The critical
density is $3H^{2}$ and the ratios of the dark energy density and matter
density to the critical density are $\Omega_{\phi}$ and $\Omega_{m}$ that add
to one in a flat universe. The dark energy only Hubble parameter is then
$\sqrt{\Omega_{\phi}}H$.

###  5.2 The Boundary Conditions

Since we are looking for solutions of differential equations the second set of
constraints is imposed by the boundary conditions, the current values of
certain cosmological parameters. Table 1 displays the cosmological boundary
conditions chosen for this study. The range of the scale factor is slightly
arbitrary but is set to include the range covered by the Rubin and Roman
observations. The range of $w_{0}$ is set close to minus one to be near but
not exactly minus one. The $H_{0}$ value is set to 73 consistent with the
current late time expectations (Riess et al., 2022). $\Omega_{m_{0}}$ and
$\Omega_{\phi_{0}}$ are the current concordance values. All of the boundary
conditions appear in the evolutionary functions of the SCP templates and are
therefore easily changed.

Table 1:  Boundary conditions and parameter values in this work. $H_{0}$ is the current value of the Hubble parameter in units of $\frac{km/sec}{Mpc}$. $\Omega_{m_{0}}$ and $\Omega_{\phi_{0}}$ are the current ratios of the matter and dark energy densities to the critical density. $w_{0}$ are the current values of the dark energy equation of state. $H_{0}$ | $\Omega_{m_{0}}$ | $\Omega_{\phi_{0}}$ |  | $w_{0}$ |  |  scale factor $a$
---|---|---|---|---|---|---
73 | 0.3 | 0.7 | -0.99 | -0.995 | -0.999 | 0.1 - 1.0

##  6 The Higgs Inspired Potential

The dark energy potential has the mathematical form of the Higgs potential
$V(\phi)\propto(\phi^{2}-\gamma^{2})^{2}$ a quartic polynomial with a constant
$\gamma$. It is chosen for two reasons. The first is that the mathematical
form is identical to the Higgs potential which gives rise to a scalar field
that is known to exist and is therefore physically motivated, hence the name
Higgs Inspired or HI potential. A second reason is that by varying the
constant term $\gamma$ it produces dark energy equations of state that are
freezing, thawing, and transitioning between freezing and thawing. This makes
it a good choice for a fiducial potential that covers a wide range of
evolutions.

The most convenient form for the potential is a modified Ratra Peebles format
(Ratra and Peebles, 1988; Peebles and Ratra, 1988) with a scalar field denoted
by $\theta$, the RP scalar introduced in section 4 that has units of mass in
$m_{p}$. The potential is then

$V(\kappa\theta)=M^{4}((\kappa\theta)^{2}-(\kappa\delta)^{2})^{2}=M^{4}((\kappa\theta)^{4}-2(\kappa\delta)^{2}(\kappa\delta)^{2}+(\kappa\delta)^{4})$
(10)

where the true scalar is $\phi=M\kappa\theta$. $M$ and $\delta$ are constants
with units of mass and both $(\kappa\theta)$ and $(\kappa\delta)$ are
dimensionless therefore all of the dimensionality is in the $M^{4}$ leading
term. Since both arguments are dimensionless there is no need for the $n$ in
the usual $M^{4-n}\phi^{n}$ Ratra Peebles format. The terms $\theta$ and
$\delta$ replace the scalar $\phi$ and $\gamma$ terms to differentiate them
from the true scalar $\phi$ and constant $\gamma$ which have values on the
order of $10^{-31}m_{p}$. The values of $\theta$ and $\delta$ are of the order
unity, The value of $\delta$ is chosen to be greater than the current scalar
$\theta_{0}$ to place the equilibrium point $\theta=\delta$ in the future.
This makes the constant $(\kappa\delta)^{4}$ the dominant term followed by the
two dynamical terms $-2(\kappa\theta)^{2}(\kappa\delta)^{2}$ and
$(\kappa\theta)^{4}$ in descending order.

##  7 The Quintessence Methodology

The methodology developed here is for the specific quintessence cosmology and
only applies to a cosmology whose action is given by equation 1. Among the
current plethora of dynamical cosmologies there are some with quite different
names that have the same action as quintessence where this methodology will
apply but not for cosmologies such as k-essence that has a different action.
The methodology is demonstrated with the modified Ratra Peebles HI potential
of equation 10.

###  7.1 The Modified Beta Function Formalism

The key to producing SCP templates that are accurate analytic functions of the
scale factor is the beta function formalism (Binetruy et al., 2015; Binetruy,
Mabillard $\&$ Pieroni, 2017; Cicciarella and Pieroni, 2017; Kohri and Matsui,
2017). The beta function is a differential equation relating the scalar
$\kappa\theta$ to the scale factor $a$ allowing the calculation of an analytic
form of $\kappa\theta(a)$. The analytic form for the scalar is achieved via
the approximation of a dominant potential component of the dark energy density
that allows the exclusion of the kinetic component to calculate the beta
function given in equation 17. This, in turn, enables the calculation of the
analytic function for the scalar. The cosmological parameters are analytic
functions of the scalar that are quite accurate but not exact. The
cosmological parameter templates do not contain any numerical calculations.

The primary beta function formalism papers relative to this work are (Binetruy
et al., 2015; Cicciarella and Pieroni, 2017). The work by (Binetruy et al.,
2015) considers a quintessence dark energy only universe while the work of
(Cicciarella and Pieroni, 2017) considers a quintessence universe with both
matter and dark energy which is the universe considered in this work. Both
(Binetruy et al., 2015) and (Cicciarella and Pieroni, 2017) consider the
general physics of the beta function formalism rather than the explicit
evolution of cosmological parameters. Their approach is therefore modified in
this work to provide analytic evolutionary templates for cosmological
parameters. These modifications are noted in the following discussion.

The generalized beta function (Binetruy, Mabillard $\&$ Pieroni, 2017) is
defined as

$\beta(\kappa\theta)\equiv\left(-\frac{\partial p}{\partial
X}\right)^{\frac{1}{2}}\frac{d\kappa\theta}{d\ln(a)}$ (11)

From equation. 3 for the quintessence dark energy pressure it is evident that
$(-\frac{\partial p}{\partial X})^{\frac{1}{2}}$ is one, giving a quintessence
beta function of

$\beta(\kappa\theta)\equiv\frac{d\kappa\theta}{d\ln(a)}=\frac{d\kappa\theta}{da}a.$
(12)

From equation 12 and the definition of the quintessence beta function and the
Hubble parameter

$\frac{d\kappa\theta}{da}=\frac{\beta}{a},\hskip
14.22636pt\kappa\dot{\theta}=\beta H_{\theta}.$ (13)

The dark energy only Hubble parameter $H_{\phi}$ is used in equation 13 to be
consistent with the dark energy only derivation of the scalar, however, when
the matter density is introduced in section 12 the Hubble parameter for both
dark energy and matter should be used since it sets the time evolution of the
scale factor $\frac{da}{dt}$. Equations 7 and 13 provide the useful relation

$\beta(\kappa\theta)=\sqrt{3\Omega_{\theta}(w+1)}$ (14)

Since the beta function formalism is developed for dark energy the first
Friedmann constraint in equation 9 applies and

$3H_{\theta}^{2}=\rho_{\theta}=-X+V(\theta)=\frac{(\beta
H_{\theta})^{2}}{2}+V(\theta)$ (15)

where now the subscript $\theta$ designates the dark energy density.

In (Binetruy et al., 2015; Cicciarella and Pieroni, 2017) the beta function is
defined as the negative of the logarithmic derivative of the dark energy
density. To achieve the desired analytic SCP templates equation 15 is
rearranged to a slightly modified density

$3H_{\theta}^{2}\left(1-\frac{\beta^{2}}{6}\right)=V(\theta)$ (16)

noting that $\frac{\beta^{2}}{6}\ll 1$ for all of the cases considered here .
Using the modified density the beta function is then the negative of the
logarithmic derivative of the analytic HI potential.

$\beta(\kappa\theta)=-\frac{\frac{\partial
V(\kappa\theta)}{\partial(\kappa\theta)}}{V(\kappa\theta)}=\frac{\partial(\kappa\theta)}{\partial\ln
a}.$ (17)

The leading constant, $M^{4}$, in the dark energy potential does not appear in
the logarithmic derivative defining the beta function leaving it as an
adjustable parameter to satisfy the Friedmann constraints. Note that the
approximation that $\frac{\beta^{2}}{6}\ll 1$ is equivalent to the statement
that the kinetic term $X=-\frac{(\beta H_{\theta})^{2}}{2}$ is small compared
to the dark energy potential. This is roughly equivalent to the slow roll
condition often used in evaluating dynamical cosmologies. In fact equation 17
is the negative of the first slow roll condition. The first slow roll
condition is often set to a small constant, eg. (Scherrer and Sen, 2008),
which is only valid for an exponential potential. Here the approximation of a
small value of $X$ is only used to calculate the analytic form of the scalar
and the non-constant time derivative of the scalar is used in all parameter
calculations. The approximation that $\beta H_{\theta}$ is set to zero in
determining the beta function also means that the Hubble parameter is not used
in the derivation of the scalar and that its value of either $H_{\theta}$ or
$H$ is not a factor.

Although the beta density $3H^{2}(1-\frac{\beta^{2}}{6})$ is slightly
different than the real density, application of the boundary conditions and
the Friedmann constraints produces evolutionary SCP templates of high accuracy
as illustrated in section 16.

####  7.1.1 The beta function for the HI potential

Using the HI potential in equation 10 the negative of the logarithmic
derivative is

$\beta(\kappa\theta)=\frac{-4\kappa\theta}{(\kappa\theta)^{2}-(\kappa\delta)^{2}}=\sqrt{3\Omega_{\theta}(w+1)}$
(18)

where the last term is from equation 14. Solving the equation formed by the
last two terms for the current time yields the current value $\theta_{0}$ for
the scalar which is an important boundary condition.

$\kappa\theta_{0}=-\frac{4-\sqrt{16+12\Omega_{\theta_{0}}(w_{0}+1)(\kappa\delta)^{2}}}{2\sqrt{3\Omega_{\theta_{0}}(w_{0}+1)}}$
(19)

Since the argument of the square root in the numerator is greater than 16
equation 19 is the positive solution of the quadratic equation.

###  7.2 The scalar as a function of the scale factor

An essential step in achieving SCP templates as analytic functions of the
scale factor $a$ is finding the scalar $\kappa\theta$ as a function of $a$.
From the definition of the beta function in equation 17 the differential
equation for the scalar as a function of the scale factor is

$\frac{\partial(\kappa\theta)}{\partial\ln
a}=\frac{-4\kappa\theta}{(\kappa\theta)^{2}-(\kappa\delta)^{2}}.$ (20)

Separating the scale factor and scalar terms gives

$4d(\ln(a))=-\frac{(\kappa\theta)^{2}-(\kappa\delta)^{2}}{\kappa\theta}d(\kappa\theta).$
(21)

An integral of both sides of equation 21 from 1 to $a$ for the left side and
from $\theta_{0}$ to $\theta$ on the right side gives

$8\ln(a)=2(\kappa\delta)^{2}\ln(\kappa\theta)-(\kappa\theta)^{2}-\left(2(\kappa\delta)^{2}\ln(\kappa\theta_{0})-(\kappa\theta_{0})^{2}\right).$
(22)

Equation 19 provides the value of $\kappa\theta_{0}$. The following
manipulations from ( , 2022) provide a solution for $\kappa\theta(a)$
involving the Lambert W function in terms of a constant
$c=2(\kappa\delta)^{2}\ln(\kappa\theta_{0})-(\kappa\theta_{0})^{2}$ and the
scale factor. Dividing both sides of equation 22 by $(\kappa\delta)^{2}$ gives

$\frac{8}{(\kappa\delta)^{2}}\ln(a)+\frac{c}{(\kappa\delta)^{2}}=2\ln(\kappa\theta)-\left(\frac{\theta}{\delta}\right)^{2}$
(23)

Taking the exponential of both sides of equation 23 and dividing again by
$(\kappa\delta)^{2}$ yields

$-\frac{a^{\frac{8}{(\kappa\delta)^{2}}}}{(\kappa\delta)^{2}}e^{\frac{c}{(\kappa\delta)^{2}}}=-\left(\frac{\theta}{\delta}\right)^{2}e^{-\left(\frac{\theta}{\delta}\right)^{2}}$
(24)

Equation 24 has the mathematical form of the Lambert W function that is the
solution to

$\chi=W(\chi)e^{W(\chi)}$ (25)

where

$\chi(a)=-\frac{a^{\frac{8}{(\kappa\delta)^{2}}}}{(\kappa\delta)^{2}}e^{\frac{c}{(\kappa\delta)^{2}}}\hskip
28.45274ptW(\chi)=-\left(\frac{\theta}{\delta}\right)^{2}.$ (26)

Equations 25 and 26 provide an analytic solution for $\kappa\theta)a)$

$\kappa\theta(a)=\kappa\delta\sqrt{-W\left(-\frac{a^{\frac{8}{(\kappa\delta)^{2}}}}{(\kappa\delta)^{2}}e^{\frac{c}{(\kappa\delta)^{2}}}\right)}$
(27)

that is the positive solution for the square root which is real since $W(x)$
is negative. The following variable changes produce a concise form for
$\kappa\theta(a)$.

$q=-\frac{e^{\frac{c}{(\kappa\delta)^{2}}}}{(\kappa\delta)^{2}}\hskip
4.26773pt,\hskip 4.26773ptp=\frac{8}{(\kappa\delta)^{2}}\hskip
4.26773pt,\hskip 4.26773pt\chi(a)=qa^{p}$ (28)

which yields

$\kappa\theta(a)=\kappa\delta\sqrt{-W(\chi(a))}$ (29)

Equation 29 provides the key to transforming evolutions that are a function of
the scalar into functions of the observable scale factor to produce the SCP
templates.

The term $(\kappa\theta)^{2}-(\kappa\delta)^{2}$ appears often in this
manuscript. In terms of the Lambert W function $W(\chi(a))$ it is

$(\kappa\theta)^{2}-(\kappa\delta)^{2}=-(\kappa\delta)^{2}(W(\chi(a))+1).$
(30)

The form of the HI potential is then

$V(a)=(M\kappa\delta)^{4}(W(\chi(a))+1)^{2}.$ (31)

The beta function also has a compact form in the W function format

$\beta(\kappa\theta)=\frac{-4\kappa\theta}{(\kappa\theta)^{2}-(\kappa\delta)^{2}}=\frac{4\sqrt{-W(\chi(a))}}{\kappa\delta(W(\chi(a))+1)}.$
(32)

###  7.3 Summary of the methodology

Although the methodology may appear to be complex the separate steps of the
quintessence beta function formalism are relatively simple. The quintessence
beta function is given in equation 12 which connects the scalar $\kappa\theta$
to the scale factor $a$. As stated in the text the beta function is defined as
the negative of the logarithmic derivative of the dark energy density but here
it is noted that the kinetic term in the density is small compared to the
potential term and a legitimate approximation is to ignore it and set the beta
function to the negative of the logarithmic derivative of the potential as
given in equation 17. This is the only place in the methodology that
$\frac{\beta^{2}}{6}$ is set to zero. The kinetic term $-X$, as shown in
equation 15, is used in all subsequent calculations of the templates. Equation
18 shows the beta function calculated from equation 17 and is shown in
differential form in equations 20 and 21. The solution for $\kappa\theta(a)$
is achieved through mathematical manipulation to the simple form in equation
29. This provides a solution for a cosmological parameter as a function of the
scale factor if the solution as a function of the scalar is known.

Since the matter (dust) action is separate from the dark energy action the
evolution dark energy scalar is calculated from the dark energy action as
described above. The matter is included in section 12 that derives the Hubble
parameter which is a function of the dark energy and matter. It is added to
the total density in equation 38 for the first Friedmann constraint and is
present in the Hubble parameter template in equation 40. This is the Hubble
parameter that is used in the calculation of the time derivative of the scalar
$\dot{\theta}=\beta H$. Adherence to the Friedmann constraints and the
inclusion of the matter density in the Hubble parameter that calculates
$\dot{\theta}$ produces accurate SCP templates that are functions of both the
matter and the dark energy densities and conforms to both Friedmann
constraints.

##  8 The Cosmology of $W(\chi)$

Before moving on to consider the evolution of $\kappa\theta(a)$, $\beta(a)$
and other parameters it is worthwhile to examine the cosmology embedded in the
evolution of $W(\chi)$. A thorough discussion of the mathematical properties
of the Lambert W function is in (Olver, F. W. F., Lozier, D. W., Boisvert,
R.F., and Clark, C.W, 2010). There it is shown that $W$ has negative values if
its argument is between $-\frac{1}{e}$ and zero which is true for all cases
considered here. This makes the argument of the square root in equation. 29
positive producing a real value of the scalar $\kappa\theta$. Figure 2 shows
the evolution of the principal branch of $W(\chi)$. The formal designation of
the principal branch is $W_{0}(\chi)$ but the subscript is dropped in the
following since only the principal branch is used in this work.

Figure 2: The figure show the valid region with negative $\chi$ and invalid
regions with positive $\chi$ of the W function for this work The thin vertical
line at $\chi=0$, $a=0$, is the big bang. The X on the track is the present
day location and the O at the end of the track is when the acceleration goes
to zero. A more detailed description of the figure is given in the text.

Figure 2 shows the evolution of $W(\chi$) as a solid line for negative $\chi$
and a dashed line for positive $\chi$. The negative portion of the Lambert W
function terminates at $\chi=-\frac{1}{e}$ while the positive portion
continues indefinitely. Only the negative portion has real solutions for the
scalar. Equation 26 shows that $\chi$ is negative for all positive values of
the scale factor $a$. Evolution in figure 2 proceeds from right to left as the
top arrow indicates. The variable $\chi$ is zero when $a=0$ therefore the big
bang is at $\chi=0$ shown by the vertical thin line. The dashed vertical line
just to the left of the big bang shows the maximum excursion of the greatest
evolution case, $\delta=3$ and $w_{0}=-0.99$. All of the cases considered here
have $-W(\chi)$ values much less than one which means that $\theta\ll\delta$.

Figure 3 shows the detail of the evolution region of figure 2. The black solid
and dashed lines are the same as in figure 2 but only for the evolution region
between the thick dashed and thin solid vertical lines. The first case,
$\delta=3$ $w_{0}=-0.99$ in red, with the highest $\delta$ and largest
deviation of $w_{0}$ from minus one has the most extensive evolution with the
right end of the evolution, $a=0.1$, significantly after the big bang. Note
that the evolutions which actually overlap the black lines have been shifted
downward for visibility. The third case with the lowest value of $\delta$ and
the least deviation of $w_{0}$ from minus one has the least evolution with its
$a=0.1$ start $\chi$ at $-1.312\times 10^{-12}$ and its maximum $\chi$ of
$1.3119\times 10^{-4}$ barely visible on the diagram. The middle excursion
with $\delta$ = 2 and $w_{0}$ = -0.995 starts with $\chi$ = $2.604\times
10^{-5}$ and a current $\chi$ of $2.6404\times 10^{-3}$. The similar numbers
are due to the end scale factor being ten times the beginning scale factor.
Recall that $\chi$ is not time so that a small value of $\chi$ does not mean
that the scale factor of 0.1 is very near the big bang. As hinted at in figure
2 the evolution region’s small extent make the evolutionary tracks in $\chi$
to appear linear. Figure 3 is an indication, verified later on, that the value
of $\delta$ has a strong influence on the SCP templates. The upper left of the
figure shows the color code of the $\delta$ values and the line styles of the
$w_{0}$ values maintained through out the manuscript.

Figure 3: The figure shows the region between the big bang and the furthest
evolution of any of the cases in this study. This figure initiates a code
continued throughout the paper. The $\delta$ = 1, 2 and three cases are
displayed in red, green and blue. The $w_{0}$ = -0.99, -0.995 and-0.999 cases
are displayed with solid, dashed and long dashed line styles as shown in the
upper left of the figure. The figure is an expanded view of the region between
the thin black vertical line and the thick dashed line in Figure 2. In this
figure the big bang is the black vertical line marked with an asterisk near
the bottom. The right and left ends of the evolutions are the start point at
$a=0.1$ and the end point at $a=1.0$ respectively. Further discussion of the
figure is in the text.

The black sloped solid and dashed line is the same as in Figure 2 for the
small expanded region. The red solid and green dashed line are the evolutions
of the $\delta=3$, $w_{0}=-0.99$ and the $\delta=2$ $w_{0}=0.995$ cases. The
long dashed blue evolution for the $\delta=1$ $w_{0}=-0.999$ is so short that
it appears only as a small dot near $\chi=0$ below the $\delta=3$ and 2
evolutions. The evolutions are offset downward from the black track for
visibility.

To aid comprehension of Figure 3 table 2 gives the value of $\chi$ at scale
factors of 01., 0.5 and 1.0 for all values of $\delta$ and $w_{0}$. All values
of $\chi$ are negative The magnitude of $\chi$ increases with increasing
values of $\delta$ and higher deviations of $w_{0}$ from minus one. The time
evolution of the tracks in Figure 3 is from right to left, the same as in
Figure 2. The time extents for all tracks are the same, $a=0.1$ to 1.0, but
the evolution of $\chi$ has a large variation.

Table 2:  The values of $\chi$ for all values of $\delta$ and $w_{0}$ for scale factors of 0.1, 0.5 and 1.0. The barely visible Figure 3 blue $\delta=1.$ and $w_{0}=-0.999$ $\chi$ values are given by the last row of the $\delta=1$ $\chi$ values in the table. The equilibrium values of the scale factor $a_{eq}$, discussed in section 8.2, are in the last column. |  |  The Values of $\chi$ |
---|---|---|---
|  |  scale factor $a$ |
$\delta$ |  $w_{0}$ . | 0.1 | 0.5 | 1.0 | $a_{eq}$
1. | -0.99 | $-1.307\times 10^{-11}$ | $-5.107\times 10^{-6}$ | $-0.00131$ | 2.024
1. | -0.995 | $-6.550\times 10^{-12}$ | $-2.558\times 10^{-6}$ | $-0.000655$ | 2.206
1. | -0.999 | $-1.312\times 10^{-12}$ | $-5.125\times 10^{-7}$ | $-0.000131$ | 2.698
2. | -0.99 | $-0.0000517$ | $-0.00129$ | $-0.00517$ | 8.437
2. | -0.995 | $-0.0000260$ | $-0.000651$ | $-0.00260$ | 11.885
2. | -0.999 | $-5.243\times 10^{-6}$ | $-0.000131$ | $-0.000524$ | 26.492
3. | -0.99 | $-0.00147$ | $-0.00616$ | $-0.0114$ | 49.776
3. | -0.995 | $-0.000750$ | $-0.00313$ | $-0.00580$ | 106.479
3. | -0.999 | $-0.000152$ | $-0.000636$ | $-0.00118$ | 640.859

###  8.1 Past Evolution

The main content of this manuscript is the past evolution from the present to
a past scale factor of 0.1 which is a redshift of nine. This encompasses a
large fraction of the history of the universe in the matter and dark energy
dominant epochs. An important question is how far back can the SCP templates
be utilized. A hard limit is the onset of the radiation dominated epoch since
the radiation density is not included in the present work. A reasonable limit
is when the radiation density is $1\%$ of the matter density. The present
matter density for $H_{0}=73$ and $\Omega{m_{0}}$ of 0.3 is $3.68\times
10^{-121}$ $m_{p}^{4}$ and a present radiation density of $6.17\times
10^{-125}$ $m_{p}^{4}$. The radiation density is $1\%$ of the matter density
at a scale factor of 0.0168 or a redshift of 58.5. This is strictly a physics
limit on the validity of the templates. Figures 2 and 3 plus table 2 indicate
that the template for the scalar $\theta$ is valid back to this limit but the
templates have not been tested for mathematical stability at scale factors
smaller than 0.1. Inclusion of the radiation density is beyond the scope of
this work but it can probably be included in the same manner as the matter
density.

###  8.2 Future Evolution

A perhaps even more intriguing question is how far in the future can the
templates be extended. The solution of $\theta$ is analytic at scale factors
greater than one which is all of the region to the left of the dashed vertical
line in Figure 2. There is a limit however to the principal branch of the
Lambert W function at $\chi=-\frac{1}{e}$. Figure 2 marks this location with a
O at $W(\chi)=-1$. Equation 29 shows that at $W(\chi)=-1$ $\theta=\delta$
which is an equilibrium point where the dark energy potential is zero. The
scale factor, $a_{eq}$, where this occurs is given by

$a_{eq}=(\frac{-1}{qe})^{\frac{1}{p}}.$ (33)

where $p$ and $q$ are the same as in equation 28. The values of $a_{eq}$ are
listed in the last column of Table 2. It is beyond the scope of this
manuscript to determine whether this is a stable equilibrium point. If it is a
stable equilibrium, with $\dot{\theta}$ also zero, then it would be the end of
dark energy acceleration. The universe would return to a matter dominated
evolution. with $3H^{2}=\frac{\rho_{m_{0}}}{a^{3}}$ making a graceful exit
from acceleration. The speed of expansion would be

$\dot{a}=aH=\sqrt{\frac{\rho_{m_{0}}}{a}}.$ (34)

The universe would then evolve in a classical manner, slowing down to zero
expansion at infinity. Given the past history of the universe it is reasonable
that the lowering level of total density might reveal a new source of
accelerated expansion whose density is below that of the current density.

##  9 The Evolution of the Scalar and the Beta Function

The scalar $\kappa\theta(a)$ and the beta function $\beta(a)$ influence the
evolution of all of the cosmological parameters in this study. The sections
below document their evolution.

###  9.1 The Evolution of $\kappa\theta(a)$

Figure 4 shows the evolution of the scalar $\kappa\theta$ over the scale
factors considered in this work. The colors and line styles are consistent
with the previous figure.

Figure 4: The figure plots the evolution of the scalar for all of the $\delta$
and $w_{0}$ values in this study.

The $w_{0}$ values for the $\delta=3$ case are labeled on the plot. The order
and line styles are the same for the other two $\delta$ values. The evolution
is relatively small consistent with a slow roll. As expected the scalar values
are monotonically increasing. The most striking feature is that the second
derivative of the evolution changes from positive for $\delta=1$ to almost
zero for $\delta=2$ to negative for $\delta=3$. The three, seemingly
arbitrary, delta values were chosen to illustrate this transition. The
transition has a large effect on some parameters, such as the dark energy EoS,
$w$, but relatively little effect on the Hubble parameter as is shown later in
Figure 7.

###  9.2 The Evolution of $\beta(a)$

The beta function evolution is shown in figure 5

Figure 5: The figure shows the evolution of the beta function for all cases in
this study.

Although the general nature of the evolution of the beta function is different
from the scalar it shows the same change in the second derivative of the
evolution, positive for $\delta=1$, almost zero for $\delta=2$ and negative
for $\delta=3$. The absolute value of $\beta$ is small and decreases as
$w_{0}$ approaches minus one. The current value of beta, $\beta_{0}$ is
identical for a given value of $w_{0}$ due to equation 14 which sets
$\beta_{0}$ at $\sqrt{3\Omega_{\theta_{0}}(w_{0}+1)}$ where the subscript 0
indicates the current values. The beta function appears in many cosmological
parameters due to equation 13 that links $\kappa\dot{\theta}$ and the Hubble
parameter.

##  10 The Value of $M$ in the Dark Energy Potential

At this point the value of $M$ in equation 10 has not been calculated. The
Friedmann constraints and equation. 3 provide the means of calculating $M$.
The dark energy density is

$\rho_{\theta}=\frac{(\kappa\dot{\theta})^{2}}{2}+M^{4}((\kappa\theta)^{2}-(\kappa\delta)^{2})^{2}=3\Omega_{\theta}H^{2}.$
(35)

Using $\kappa\dot{\theta}=\beta H$

$3\Omega_{\theta}H^{2}=\frac{(\beta
H)^{2}}{2}+M^{4}((\kappa\theta)^{2}-(\kappa\delta)^{2})^{2}.$ (36)

Since $M$ is a constant it can be set using the current boundary conditions
which insures adherence to the first Friedmann constraint at a scale factor of
one. This eliminates any constant offsets due to the approximation in equation
17 further improving the accuracy of the templates.

Rearranging equation 36 and using the current values of the parameters gives

$3H_{0}^{2}(\Omega_{\theta_{0}}-\frac{\beta_{0}^{2}}{6})=M^{4}((\kappa\theta_{0})^{2}-(\kappa\delta)^{2})^{2},\hskip
14.22636ptM=\sqrt[4]{{\frac{3H_{0}^{2}}{((\kappa\theta_{0})^{2}-(\kappa\delta)^{2})^{2}}\left(\Omega_{\theta_{0}}-\frac{\beta_{0}^{2}}{6}\right)}}.$
(37)

##  11 The Evolution of the HI Dark Energy Potential

The evolution of the dark energy potential can now be calculated. Figure 6
shows the evolution of the HI dark energy potential for all of the cases.

Figure 6: The figure shows the evolution of the HI dark energy potential for
all cases in this study.

Figure 6 shows that there is only a small evolution of the potential between
scale factors of 0.1 and 1.0, again consistent with a slow roll. The maximum
evolution is $1.7\%$ for the $\delta=3$ $w_{0}=-0.99$ case. The $\delta=1$
evolutions are essentially constant, mimicking $\Lambda$CDM, until a scale
factor of $\approx 0.7$ and then decrease slightly to the $a=1.0$ value for
$w_{0}=-0.99$ and -0.995. The $\delta=2$ cases deviate from constant evolution
earlier than the $\delta=1$ cases, The $\delta=3$ cases have almost linear
evolution and have the highest values, particularly at small scale factors.
All of the $w_{0}=-0.999$ cases have a very flat evolution. For a given value
of $w_{0}$ the current value of the potential is the same for all $\delta$
values. From eqns. 10 and 37 the potential at a scale factor of one is
$3H_{0}(\Omega_{\theta_{0}}-\frac{\beta_{0}^{2}}{6})$. Equation 14 shows that
the value of the beta function at a scale factor of one is
$\sqrt{3\Omega_{\theta_{0}}(w_{0}+1)}$ making $V_{0}$ the same for a given
$w_{0}$.

##  12 The Hubble Parameter

The calculation of the Hubble parameter for the real universe requires the
inclusion of both dark energy and matter. In (Cicciarella and Pieroni, 2017)
matter is introduced via a differential equation involving the Hubble
parameter and the beta function. Here the Friedmann constraints are the
primary tools for deriving the Hubble parameter in a universe with both matter
and dark energy. The first Friedmann constraint gives

$3H^{2}(a)=\rho_{\theta}+\rho_{m}=\frac{(\kappa\dot{\theta})^{2}}{2}+M^{4}((\kappa\theta)^{2}-(\kappa\delta)^{2})^{2}+\frac{\rho_{m_{0}}}{a^{3}}.$
(38)

Here $\rho_{m_{0}}$ is the current matter density and
$\frac{\rho_{m_{0}}}{a^{3}}$ is the mass density as a function of the scale
factor. Using equation 13 $\beta H$ is substituted for $\kappa\dot{\theta}$ in
equation 38 to obtain

$3H^{2}(a)(1-\frac{\beta(a)^{2}}{6})=M^{4}((\kappa\theta)^{2}-(\kappa\delta)^{2})^{2}+\frac{\rho_{m_{0}}}{a^{3}}.$
(39)

The Hubble Parameter is therefore

$H(a)=\sqrt{\frac{M^{4}((\kappa\theta)^{2}-(\kappa\delta)^{2})^{2}+\frac{\rho_{m_{0}}}{a^{3}}}{3\left(1-\frac{\beta(a)^{2}}{6}\right)}}$
(40)

###  12.1 The Time Derivative of the Hubble Parameter

The second Friedmann constraint in eqns. 9 provides the method for calculating
$\dot{H}$.

$\dot{H}=-\left(\frac{\rho_{\theta}(a)+\rho_{m}(a)+3P(a)}{6}+H^{2}\right)=-\frac{1}{2}\left((M\kappa)^{4}\dot{\theta}^{2}+\frac{\rho_{m_{0}}}{a^{3}}\right)$
(41)

where $\dot{\phi}=M^{2}\kappa^{2}\dot{\theta}$ in units of the reduced Planck
mass.

###  12.2 The Evolution of the Hubble Parameter

Figure 7 shows the evolution of the Hubble parameter for all values of
$\delta$ and $w_{0}$ plus $\Lambda$CDM. At the resolution of the figure all of
the tracks overlap each other to the thickness of the line.

Figure 7: The figure shows the evolution of the Hubble parameter for all cases
in this study and $\Lambda$CDM. The evolution of the time derivative of the
scale factor is also shown in the dashed line to indicate the onset of the
acceleration of the expansion of the universe. The scale of its evolution has
been magnified by 10 to make it visible in the plot.

The dashed line on figure 7 shows the time derivative of the scale factor
$\dot{a}$ to show the transition to acceleration of the expansion of the
universe. It occurs at a scale factor of $\approx 0.6$ consistent with current
observations eg. (Dahiya , 2022). The $\dot{a}$ track has been multiplied by
ten to remove its overlap with the H parameter track. The following section
shows the percentage deviation of the HI Hubble parameters from $\Lambda$CDM
for all of the cases

###  12.3 The Percentage Deviation from $\Lambda$CDM

The fractional deviation of the HI Hubble parameters from $\Lambda$CDM is
given by

$dev=\frac{H_{HI}-H_{\Lambda CDM}}{H_{\Lambda CDM}}$ (42)

Figure 8 shows the percentage deviation of the HI H parameters from the
$\Lambda$CDM H parameter.

Figure 8: The percentage deviation from $\Lambda$CDM for the HI Hubble
parameter. The negative numbers at the peaks and valleys in each panel are the
values of $w_{0}$.

The figure readily shows that the percentage deviation of the HI Hubble
parameter from $\Lambda$CDM is exceedingly small. The highest deviation is
$0.17\%$ for the $\delta=3$, $w_{0}=-0.99$ and the smallest deviation is
$0.005\%$ for the $\delta=1$, $w_{0}=-0.999$ case. All of the $\delta=1$ cases
have a negative deviation indicating that the HI Hubble parameter is slightly
less than $\Lambda$CDM while the other $\delta$ values have positive
deviations with the HI Hubble parameter slightly higher than $\Lambda$CDM. The
maximum deviations occur at scale factors between 0.6 for $\delta=3$ and 0.8
for $\delta=1$ where dark energy begins to dominate. The overall shape of the
deviations are reasonable. The deviation is zero at $a=1$ since it set by the
$H_{0}$ boundary condition. After the peak the deviation drops again as the
density becomes matter dominated. Currently the deviations of the
$w_{0}=-0.999$ cases are impossible to detect observationally and the highest
deviation is below the detection limit of the proposed near future facilities.
Further discussion of the implications of the HI quintessence cosmology
appears in section 17.

##  13 The Scale Factor and Time Derivatives of the Scalar

The scale factor and time derivatives of the scalar are not observables but
are essential for the calculation of the SCP templates. The starting point is
the derivative of the Lambert W function in equation 43

$\frac{dW(x)}{dx}=\frac{W(x)}{x(1+W(x))}.$ (43)

From this base the derivative of the scalar $(\kappa\theta)$ with respect to
the scale factor $a$ is

$\frac{d\kappa\theta}{da}=\frac{d(\kappa\delta\sqrt{-W(qa^{p})})}{da}=\kappa\delta\frac{p\sqrt{-W(qa^{p})}}{2a(1+W(qa^{p}))}.$
(44)

The derivative of the scalar with respect to time is then

$\frac{d\kappa\theta}{dt}=\frac{d\kappa\theta}{da}\frac{da}{dt}=\frac{d\kappa\theta}{da}Ha=\kappa\delta\frac{p\sqrt{-W(qa^{p})}}{2(1+W(qa^{p}))}H.$
(45)

Equation 45 gives the same answer as equation 13.

Figure 9 shows the evolution of $\dot{\phi}=M^{2}\kappa^{2}\dot{\theta}$ in
units of $m_{p}^{2}$ for all of the cases along with a more detailed plot of
the region between a scale factor of 0.4 and 1.0.

Figure 9: The left panel shows the evolution of $M^{2}\kappa^{2}\dot{\theta}$
for all cases and the right panel shows in more detail the evolution at scale
factors between 0.4 and 1.0.

The units of $\dot{\theta}$ are $m_{p}^{2}$ since $\theta$ has the units of
mass and time has units of inverse mass. In figure 9 $\kappa^{2}\dot{\theta}$
is multiplied by $M^{2}$ to show the value of the time derivative of the true
scalar $\dot{\phi}$. In the left panel the full evolution of $\dot{\theta}$ is
shown for the scale factors between 0.1 and 1.0 Unlike the scalar the
magnitude of $\dot{\theta}$ is decreasing for $\delta=3$ but increasing for
$\delta=1$ with corresponding differences in the second derivative. The right
hand panel shows the evolution between scale factors of 0.4 and 1.0 in more
detail. Close inspection of the $\delta=2$ and $w_{0}=-0.99$ track show that
it was initially decreasing but is currently increasing. This non-monotonic
evolution is also present in the dark energy EoS described later.

##  14 The Dark Energy Density and Pressure

Several cosmological parameters depend on the evolution of the dark energy
density and pressure. Equation 3 gives the functions for them in terms of the
kinetic term $X$ and the dark energy potential. Figure 10 shows the evolution
of the dark energy density for all of the cases.

Figure 10: The evolution of the dark energy density for all of the cases in
this study.

As usual the $\delta=1$ evolutions have a different character from the other
two. The density evolution for $w_{0}=-0.999$ is essentially flat and the
highest evolution case, $w_{0}=-0.99$, only changes by $0.3\%$. For all values
of $w_{0}$ the $\delta=1$ density has a slight rise near a scale factor of
one. For scale factors less than 0.6 the densities are essentially constant
acting like a cosmological constant in the matter dominated epoch. For
$\delta=2$ and 3 the density is monotonically decreasing with increasing scale
factor. The second derivative of the decrease of density for the $\delta=2$
case changes from negative to positive with increasing scale factor similar to
the scalar. The decrease in density for the $\delta=3$ case is larger than for
the other two cases but is still only on the order of $20\%$ for the maximum
case. Unlike the $\delta=2$ case the second derivative of the evolution is
negative at all scale factors.

Figure 11 shows the evolution of the dark energy pressure.

Figure 11: The evolution of the dark energy pressure for all of the cases in
this study.

The dark energy pressures have their characteristic negative values and,
although more than the density, the absolute evolution is relatively small.
The $\delta=1$ and 3 evolutions are monotonically positive and negative
respectively but the $\delta=2$ pressure evolutions have stronger transitions
from negative to positive than the density. As in the dark energy density the
$w_{0}=-0.999$ evolution is quite flat as would be expected for a $w_{0}$
value so close to minus one.

##  15 The Dark Energy Equation of State

By definition the dark energy EoS is the ratio of the dark energy pressure to
the dark energy density. Figure 12 shows the evolution of
$w=\frac{p_{\theta}}{\rho_{\theta}}$.

Figure 12: The evolution of the dark energy equation of state for all of the
cases in this study.

The $\delta=1$ $w$ evolution is the classic thawing evolution where $w$ is
initially near minus one and then thaws to the less negative values of
$w_{0}$. The $\delta=3$ is the classic freezing case where $w$ starts at
values less negative than minus one and then freezes toward minus one. The
$\delta=2$ evolution, however, is non-monotonic, starting as a freezing
solution, and then transitioning to a thawing evolution. These evolutions
mirror the evolution of the dark energy pressure in figure 11 since the
magnitude of the pressure evolution is greater than the evolution of the
density. The author does not know of any similar case in the literature but
suggests that it may be called the freeze and thaw evolution. Figure 12
demonstrates the motivation for the simple choices of one, two and three for
the $\delta$ values. The $\delta=3$ cases have late time evolutions very
similar to $\Lambda$CDM but significant and observable deviations at early
times. The $\delta=1$ evolutions of $w$ are indistinguishable from
$\Lambda$CDM at early times and only slightly deviant from $\Lambda$CDM at
late times due to the purposely chosen $w_{0}$ values very near minus one. The
$w_{0}=-0.999$ evolution of $w$ would not be distinguishable from $\Lambda$CDM
with current analysis techniques. These aspects are discussed more thoroughly
in section 17 that considers the HI quintessence as a candidate for a fiducial
dynamical cosmology in the same way that $\Lambda$CDM is a fiducial static
cosmology.

##  16 The Accuracy of the Cosmology and Dark Energy Potential

At this point the SCP evolutionary templates of all of the cosmological
parameters considered in this work are calculated. It is appropriate then to
consider the accuracy of the cosmology and HI potential as a whole. The metric
for the accuracy utilized here is the accuracy of the two Friedmann
constraints which contain the Hubble parameter and its time derivative, the
dark energy density and pressure plus the matter density and HI potential.
Other parameters such as the dark energy equation of state are functions of
the parameters in the Friedmann constraints. The first and second Friedmann
constraints are given in equation 8. The two constraints are considered
separately below.

###  16.1 The Accuracy of the First Friedmann Constraint

The left and right sides of the first Friedmann constraint should be equal
therefore the accuracy, $fracerr$, is determined by

$fracerr=\frac{3H^{2}-(\rho_{\theta}+\rho_{m})}{3H^{2}}.$ (46)

The results for all of the cases are similar in the overall magnitude but of
course dissimilar in detail. The results for $w_{0}=-0.995$ and the three
values of $\delta$ are shown in figure 13.

Figure 13: The fractional error for the first Friedmann constraint with
$w=-0.995$ and $\delta$ = 1, 2, and 3 .

It is obvious that the first Friedmann constraint is satisfied to better than
one part in $10^{16}$. This is on the order of the digital accuracy of the
Mathematica code used in the calculation.

###  16.2 The Accuracy of the Second Friedmann Constraint

The second Friedmann constraint explicitly covers more parameters including
the time derivative of the Hubble parameter. The fractional error for the
second Friedmann constraint is given by

$fracerr=\frac{3(\dot{H}+H^{2})+\frac{(\rho_{\theta}+\rho_{m}+3P)}{2}}{3(\dot{H}+H^{2})}$
(47)

Unlike the first Friedmann constraint where all of the terms are positive the
second Friedmann constraint has a mixture of positive and negative terms. Both
the left and right hand term in the numerator of the constraint transition
between positive and negative values. The left hand term is the denominator in
equation 47 which means that it goes through zero making the fractional error
infinite. The transition occurs at a scale factor of approximately 0.6. Since
the calculations in Mathematica are digital true zero rarely occurs but the
fractional error does spike at the transition point. Figure 14 shows the
fractional error for the same cases considered in the first Friedmann
constraint.

Figure 14: The fractional error for the second Friedmann constraint with
$w=-0.995$ and $\delta$ = 1, 2, and 3 . The spikes at $a\approx 0.6$ is due to
the denominator passing through zero.

The regions away from the spike have similar fractional errors as for the
first Friedmann constraint but build slightly before the spike. Even including
the spike the second Friedmann constraint is satisfied to a high accuracy
indicating that the SCP evolutionary templates also have a high degree of
accuracy, exceeding the accuracy of the observations by a high degree.

##  17 The HI Quintessence as a Fiducial Dynamical Cosmology

In most likelihood examinations of cosmological data $\Lambda$CDM is
considered the fiducial static cosmology. A fiducial dynamical cosmology,
however, has not been identified. This may be due in part to the multitude of
dynamical cosmologies and the number of possible dark energy potentials. This
leads to the use of parameterizations and their incumbent pitfalls as
discussed in section 3. HI quintessence may be a good candidate for a fiducial
dynamical cosmology for comparison with $\Lambda$CDM. This confronts the
question of whether dark energy is static or dynamic with the canonical
scientific method of comparing physics based predictions to the data to
measure the likelihood of the predictions.

There are compelling reasons for picking HI quintessence as one of perhaps
several fiducial dynamical cosmologies. A particularly compelling reason is
that the HI potential has a natural physical basis since its mathematical form
is the same as the only confirmed isotropic and homogenous field, the Higgs
field. It should be emphasized again here that the HI scalar is not the Higgs
field. It is just a quintessence scalar field with the mathematical form of
the Higgs potential. Another compelling reason is that, unlike monomial
potentials, the HI potential covers a wide range of possible evolutions by
simply varying the value of $\delta$ in the potential. Section 15 showed that
both freezing and thawing evolutions of the dark energy EoS, $w$, are easily
obtained as well as evolutions that transition between freezing and thawing.
The SCP templates for all of these evolutions are physics based and test real
predictions for discriminating between static and dynamic dark energy plus
determining the nature of a dynamical dark energy.

An additional reason for utilizing HI quintessence as a fiducial cosmology is
that it comes arbitrarily close to $\Lambda$CDM by varying the constant in the
HI potential and adjusting the boundary conditions such as $w_{0}$ without
invoking a cosmological constant. The best example of a $\Lambda$CDM type of
evolution in this work is the $\delta=1$ and $w_{0}=-0.999$ case examined more
closely in the next section.

###  17.1 A $\Lambda$CDM like dynamical cosmology

Due to the many successes of the $\Lambda$CDM cosmology in matching the
observational data the dark energy EoS $w_{0}$ values were purposely set close
to but not equal to minus one. The $\delta=1$ and $w_{0}=-0.999$ case is the
closest one to $\Lambda$CDM. All of the $\delta=1$ cases are thawing which
means that the maximum value of $w$ is $w_{0}$ and the early time values of
$w$ are very close to minus one. It is this dynamical case of all studied in
this work that has the best chance of having a likelihood close to that of
$\Lambda$CDM. In earlier plots that show evolutions for all cases the
evolution of this case is often hard to discern since it is much smaller than
the maximum evolution case in the plots. To better illustrate the $\delta=1$,
$w_{0}=-0.999$ evolutions figure 15 plots the fractional deviations from
$\Lambda$CDM of this case only for the Hubble parameter, the dark energy
density and the dark energy EoS.

Figure 15: The evolution of the $\delta=1$, $w_{0}=-0.999$. Hubble parameter,
dark energy density and dark energy EoS. Note that unlike figure 8 the
fractional deviation of the HI Hubble parameter from $\Lambda$CDM is shown
rather than the percentage.

The left panel shows the fractional, not percentage as in figure 8, deviation
of the HI Hubble parameter from $\Lambda$CDM. The maximal fractional deviation
is only -0.00005 which is below any current or expected near term detectable
limit. The center panel shows the ratio of the dark energy density to the
cosmological constant, black line at 1.0, with a maximum deviation of 0.00025
at small scale factors. The boundary conditions set the deviation at $a=1$ to
zero. The first panel of figure 10 indicates that the deviation smaller scale
factors than 0.6 is constant at the $a=0.1$ value. It is unlikely that the
small deviation produces any detectable effects. The right hand panel shows
the dark energy EoS $w$ which has a maximum deviation from minus one of 0.001.
This also is below current or expected near term detection limits. It is clear
that any constraint on the deviation of $w$ can be met by moving $w_{0}$
closer to minus one, which also lowers the deviations of the other two
parameters. This indicates that it is very difficult to falsify a dynamical
cosmology or to confirm $\Lambda$CDM. On the other hand a confirmed deviation
from the $\Lambda$CDM predictions can falsify it but not necessarily confirm a
dynamical cosmology. It would, however, produce a higher likelihood for a
dynamical cosmology than for $\Lambda$CDM.

##  18 Temporal Evolution of Fundamental Constants

Constraints on the temporal and spatial variance of fundamental constants are
excellent, but seldom used, discriminators between static and dynamic dark
energy. They are also sensitive tests of the validity of the standard model of
physics. Fundamental constants are dimensionless numbers whose values
determine the laws of physics. Primary examples are the fine structure
constant $\alpha$ and the proton to electron mass ratio $\mu$ that are
cosmological observables. Both are measured by spectroscopic observation of
atomic and molecular transitions respectively. As discussed briefly in the
introduction the same scalar that produces the late time inflation by
interacting with gravity most likely interacts with other sectors producing
changes in the values of the fundamental constants. Since the same scalar is
determining the value of the dark energy EoS and the values of the constants
there is a relationship that makes the fundamental constants $w$ meters in the
universe. The summation of the interactions of the scalar with the Quantum
Chromodynamic Scale $\Lambda_{QCD}$, the Higgs vacuum expectation value $\nu$
and the Yukawa couplings $h$ produce a net coupling constant $\zeta_{x}$ where
$x$ can be either $\mu$ or $\alpha$. In the absence of any knowledge of the
coupling it is assumed to be linear as in equation 48 which can be thought of
as the first term of a Taylor series of the real coupling.

$\frac{\Delta x}{x}=\zeta_{x}(\kappa\theta-\kappa\theta_{0})$ (48)

Current limits on the temporal variation of the constants are
$\Delta\alpha/\alpha=-(1.3\pm 1.3_{stat}\pm 0.4_{sys})\times 10^{-6}$,
$1\sigma$ (Murphy, 2021) at $z=1.15$ and $\Delta\mu/\mu\leq\pm 1.1\times
10^{-7}$ $2\sigma$ (Bagdonaite, 2013; Kanekar, 2015) at $z=0.89$. The
redshifts for both of these measurements are look back times greater than half
the age of the universe. The $\alpha$ constraints are from optical
spectroscopy of multiple atomic fine structure lines and the $\mu$ constraints
are from radio observations of methanol absorption lines in cold molecular
clouds along the line of sight to a quasar. In the following the tighter
constraint on the temporal variation of $\mu$ is used as an example.

Figure 16 shows the evolution of $\frac{\Delta\mu}{\mu}$ for $\delta=1$ and 3
for all three $w_{0}$ values and $\zeta_{\mu}=10^{-6}$.

Figure 16: The evolution of $\frac{\Delta\mu}{\mu}$ for $\delta=1$ and 3 for
all three values of $w_{0}$. The error bar at a scale factor of 0.5303 is the
$2\sigma$ constraint on the temporal variation of $\mu$.

The $\delta=1$ cases represent the least evolution and the $\delta=3$ cases
represent the most evolution with the $\delta=2$ lying between the two. All of
the cases satisfy the constraint, mainly because of the small deviations of
$w_{0}$ from minus one. Only a small tightening of the constraint would start
to eliminate some of the $\delta=3$ cases. The proposed fiducial case of
$\delta=1$ and $w_{0}=-0.999$ is well within the observational constraint.
Additionally restrictive observational constraints can always be
cosmologically accommodated by making $w_{0}$ closer to minus one or by
lowering the value of the particle physics parameter $\zeta_{\mu}$.

The last sentence in the above paragraph indicates that a constraint on the
temporal variance of either $\mu$ or $\alpha$ is a constraint on a cosmology-
particle physics plane defined by $w$ and $\zeta_{\mu}$. An earlier analysis
(Thompson, 2013) determined the relationship between $\zeta_{\mu}$ and $w$ as

$\zeta_{\mu}=\frac{\pm\frac{\Delta\mu}{\mu}}{\kappa\theta(a_{ob},w_{0},\Omega_{\theta_{0}})-\kappa\theta_{o}(1,w_{0},\Omega_{\theta_{0}})}$
(49)

where $a_{ob}$ is the scale factor of the observation. Equation 19 for
$\theta_{0}$ is the source of the $\Omega_{\theta_{0}}$ term in equation 49.
Equation 49 defines regions in the $\zeta_{\mu}$-$(w_{0}+1)$ plane that
satisfy the observational constraint and those that don’t. Figure 17 shows the
allowed and forbidden area for the $\frac{\Delta\mu}{\mu}$ constraint.

Figure 17: The forbidden and allowed regions in the $w_{0}$ \- $\zeta_{\mu}$
plane determined by the constraints on the temporal deviation of the proton to
electron mass ratio $\mu$. Areas inside the two boundary lines for a case are
allowed while areas outside the two boundary lines are forbidden.

The positive and negative tracks for each $\delta$ case are due to the
positive and negative values for $\zeta_{\mu}$ in equation 49. The $\delta=3$
more restricted allowed area is due to the greater evolution of
$\frac{\Delta\mu}{\mu}$. as shown in figure 16, that requires a smaller
$\zeta_{\mu}$ to meet the constraint than the $\delta=1$ case. The
$\Lambda$CDM point in the figure is the 0,0, origin where $\zeta_{\mu}=0$ and
$(w_{0}+1)=0$. A confirmed observation of $(w_{0}+1)\neq 0$ with no detected
variance of $\mu$ would place a hard limit on the particle physics parameter
$\zeta_{\mu}$ but also would require new physics to account for a deviation of
$w$ from minus one.

##  19 Conclusions

This study addresses the question of whether dark energy is static or dynamic
by first pointing out that the use of parameterizations to represent dynamical
cosmologies results in not only erroneous likelihoods but also in erroneous
conclusions about the validity of dynamical cosmologies such as quintessence.
The study then presents a methodology for creating accurate analytic templates
of the evolution of cosmological parameters and fundamental constants for the
flat quintessence dynamical cosmology. The methodology utilizes a modified
beta function formalism to determine the evolution of the quintessence scalar
as a function of the observable scale factor. Solutions for the evolution of
parameters and constants that were previously only functions of the
unobservable scalar are then translated to templates that are functions of the
scale factor for direct comparison with the current and expected cosmological
observations. Recognizing that dynamical cosmologies can have a multitude of
dark energy potentials the study introduced the concept of Specific Cosmology
and Potential, SCP, templates to replace the parameterizations with SCP
evolutionary templates based on the physics of the cosmology and dark energy
potential.For this reason the study concentrated on the methodology to produce
SCP templates that can embrace a broad range of analytic physics based
potentials..

To demonstrate the formalism the study then calculated SCP templates of
several observable and some necessary but not observable parameters such as
the time derivative of the scalar that appear in the functions of many
observable parameters. An important aspect of the study is the example quartic
polynomial dark energy HI potential. The modified beta function formalism
applied to flat quintessence with the HI potential resulted in a scalar that
is a simple function of the Lambert W function. This step provided the means
to produce accurate analytic SCP templates as a function of the scale factor.

Given the many observational successes of the $\Lambda$CDM static cosmology
the study chose boundary conditions close to $\Lambda$CDM. In particular
$w_{0}$ values close to, but not equal to, minus one were adopted. This choice
produced a simplification of the beta function formalism where the beta
function for quintessence is the negative of the logarithmic derivative of a
slightly modified dark energy density. The beta function is then accurately
approximated by the logarithmic derivative of the potential. Care should be
taken in using the formalism for $w$ values significantly different than minus
one. Equation 7 shows that the kinetic term $X$ is directly proportional to
$(w+1)$. If $w+1)$ becomes too large the approximation can break down and
other means must be employed. The SCP templates are calculated by imposing the
Friedmann constraints on the parameters. Since the studied epoch included only
the matter dominated and dark energy dominated epochs, radiation is not
included in the calculations. This precludes utilization of the templates for
scale factors smaller than 0.016, such as the CMB dominated epoch, that have
significant radiation densities.

The polynomial HI potential provided a significantly larger range of
evolutions than the often use monomial potentials. In particular small changes
of the constant term $\delta$ in the potential produced dark energy EoS
evolution that were both freezing and thawing plus evolutions that
transitioned from freezing to thawing. Giving the naturalness of the HI
potential and the large range of evolutions the study suggests that the HI SCP
templates become a fiducial dynamical cosmology in the same way as
$\Lambda$CDM is for static cosmologies. Several of the cases studied are
indistinguishable from $\Lambda$CDM with the accuracy of the present and near
future observations even though their dark energy density arises from a
dynamical scalar field rather than a cosmological constant. Given this and the
relative rigidity of the predicted evolutions it appears that $\Lambda$CDM is
easy to falsify but hard to confirm and that flat HI quintessence is hard to
falsify but easy to confirm if new observations confirm predictions such as a
dynamical dark energy EoS.

The study concluded with an examination of the role of fundamental constants
in the discrimination between static and dynamical cosmologies. The scalar in
a dynamical dark energy that interacts with gravity will most likely interact
with other sectors which produces temporal variations in the fundamental
constants. To date no confirmed variations of either $\alpha$ or $\mu$ have
been found at the one part in $10^{7}$ level. All of the cases in this study
predict variations that are less than the current limits. Future observations
may, however, lower the limit that would make it difficult to meet the
constraints or find a variation that is consistent with the dynamical
predictions.

##  20 Appendix 1, Flat HI quintessence abridged templates

This is an abridged set of evolutionary templates for flat HI quintessence.
The unabridged template set contains significantly more information including
code for implementing the templates. The templates developed in the main text
are gathered here to provide a convenient listing for community use. The
appendix repeats information provided in the text to gather most of the
relevant material in a single location.

Units: Natural units are utilized with $\hbar$, $c$, and $8\pi G$ set to one.
The units of mass are the reduced Planck mass $m_{p}$.

General constants: The constant $\kappa=\frac{1}{m_{p}}$. In the mass units
utilized here $\kappa=1$ but it is retained to provide the proper mass units
for the templates.

Primary variable: The primary variable is the scale factor $a$. All templates
are functions of the observable scale factor.

Special functions: The Lambert W function $W(x)$ is used extensively in the
templates. See (Olver, F. W. F., Lozier, D. W., Boisvert, R.F., and Clark,
C.W, 2010) for a comprehensive description of the function.

The Ratra-Peebles, RP scalar: The RP scalar is used in all of the templates.
Its functional from is
$\kappa\theta(a)=\kappa\delta\sqrt{-W(qa^{p})}=\kappa\delta\sqrt{-W(\chi(a))}$
in terms of the Lambert W function with $q$ and $p$ as constants given below.

The Higgs Inspired, HI, dark energy potential: The dark energy potential is
$V(\kappa\theta)=M^{4}((\kappa\theta)^{2}-(\kappa\delta)^{2})^{2}=M^{4}((\kappa\theta)^{4}-2(\kappa\delta)^{2}(\kappa\delta)^{2}+(\kappa\delta)^{4})$
where $M$ is a constant with units of mass in $m_{p}$. The constant $\delta$
also has units of $m_{p}$. Both $V(a)$ and $\kappa\theta(a)$ will be repeated
below along with the definitions of the constants $q$, $p$ and $M$.

Assigned constants: The HI potential constant $\delta$ is assigned the
constants 1.0, 2.0, and 3.0 in this work

Changeable cosmological constants: These constants are assigned values in this
work and appear in the templates, thus they can be assigned different values
according the the desired boundary conditions for the cosmological parameters.
The boundary conditions are set at the current epoch hence the subscript 0 on
their designations.

$H_{0}$ the Hubble parameter

$\Omega_{\theta_{0}}=\frac{\rho_{\theta_{0}}}{3H_{0}^{2}}$

$\Omega_{m_{0}}=\frac{\rho_{m_{0}}}{3H_{0}^{2}}$

$w_{0}$ The dark energy equation of state

Cosmological parameter templates: The cosmological parameter template formats
include, where possible, the parameter first in terms of the RP scalar
$(\kappa\theta)$, second the parameter in terms of the Lambert W function,
third its magnitude at a scale factor of one for $H_{0}=73$
$\frac{kmsec^{-1}}{Mpc}$, $w_{0}=-0.995$ and $\kappa\delta=2$ and fourth any
associated constants.  The Ratra Peebles scalar $\kappa\theta$

$\kappa\theta(a)=\kappa\delta\sqrt{-W(\chi(a)}$

$\chi(a)=qa^{p}$

$c=2(\kappa\delta)^{2}\ln(\kappa\theta_{0})-(\kappa\theta_{0})^{2}$

$q=-\frac{e^{\frac{c}{(\kappa\delta)^{2}}}}{(\kappa\delta)^{2}}$

$p=\frac{8}{(\kappa\delta)^{2}}$

$\kappa\theta_{0}=-\frac{4-\sqrt{16+12\Omega_{\theta_{0}}(w_{0}+1)(\kappa\delta)^{2}}}{2\sqrt{3\Omega_{\theta_{0}}(w_{0}+1)}}$

$\kappa\theta(1.0)=0.102202$ The beta function $\beta$

$\beta(a)=-\frac{4\kappa\theta(a)}{(\kappa\theta(a))^{2}-(\kappa\delta)^{2}}$

$\beta(\chi(a))=\frac{4\sqrt{-W(\chi(a))}}{\kappa\delta(W(\chi(a))+1)}$

$\beta(1.0)=0.10247$

The dark energy potential $V$

$V(a)=(M\delta)^{4}((\kappa\theta(a))^{2}-(\kappa\delta)^{2})^{2}$

$V(\chi(a))=(M\delta)^{4}(W(\chi(a)+1)^{2}$

$M=\sqrt[4]{\frac{3H_{0}^{2}}{((\kappa\theta_{0})^{2}-(\kappa\delta)^{2})^{2}}\left(\Omega_{\theta_{0}}-\frac{\beta(1)^{2}}{6}\right)}$

$V(1.0)=8.56409\times 10^{-121}$ $m_{p}^{4}$
The Hubble parameter

$H(a)=\sqrt{\frac{(M\delta)^{4}((\kappa\theta)^{2}-(\kappa\delta)^{2})^{2}+\frac{\rho_{m_{0}}}{a^{3}}}{3\left(1-\frac{\beta(a)^{2}}{6}\right)}}$

$H(\chi(a))=\sqrt{\frac{(M\delta)^{4}(W(\chi(a))+1)^{2}+\frac{\rho_{m_{0}}}{a^{3}}}{3\left(1-\frac{\left(\frac{4\sqrt{-W(\chi(a))}}{\kappa\delta(W(\chi(a))+1)}\right)^{2}}{6}\right)}}$

$H(1.0)=6.39403\times 10^{-61}$ $m_{p}$
The derivative of the scalar with respect to the scale factor
$\frac{d\theta}{da}$

$\frac{d\theta(a)}{da}=\kappa\delta\frac{p\sqrt{-W(\chi(a))}}{2a(1+W(\chi(a)))}$

$\kappa\frac{d\theta(1.0)}{da}=0.10247$
The derivative of the scalar with respect to time $\frac{d\theta}{dt}$

$\frac{d\theta(a)}{dt}=\dot{\theta}(a)=\frac{d\theta(a)}{da}H(a)a$

$\frac{d\theta(\chi(a))}{dt}=\kappa\delta\frac{p\sqrt{-W(\chi(a))}}{2(1+W(\chi(a)))}\sqrt{\frac{(M\delta)^{4}(W(\chi(a))+1)^{2}+\frac{\rho_{m_{0}}}{a^{3}}}{3\left(1-\frac{\left(\frac{4\sqrt{-W(\chi(a))}}{\kappa\delta(W(\chi(a))+1)}\right)^{2}}{6}\right)}}$

$\frac{d\theta(1.0)}{dt}=6.55193\times 10^{-62}$ $m_{p}^{2}$
The kinetic term X = $-\frac{\dot{\theta}^{2}}{2}$

$X(a)=-\frac{\dot{\theta}(a)^{2}}{2}=-\frac{1}{2}(\frac{d\theta(a)}{da})^{2}H(a)^{2}a^{2}$

$X(\chi(a))=\left(\kappa\delta\frac{p\sqrt{-W(\chi(a))}}{2(1+W(\chi(a)))}\right)^{2}\left(\frac{(M\delta)^{4}(W(\chi(a))+1)^{2}+\frac{\rho_{m_{0}}}{a^{3}}}{3\left(1-\frac{\left(\frac{4\sqrt{-W(\chi(a))}}{\kappa\delta(W(\chi(a))+1)}\right)^{2}}{6}\right)}\right)$
The dark energy density

$\rho_{\theta}(a)=\frac{\dot{\theta}^{2}(a)}{2}+(M\delta)^{4}((\kappa\theta(a))^{2}-(\kappa\delta)^{2})^{2}$

$\rho_{\theta}(\chi(a))=-X(\chi(a))+(M\delta)^{4}(W(\chi(a)+1)^{2}$

$\rho_{\theta}(1)=8.58555\times 10^{-121}$ $m_{p}^{4}$
The matter density

$\rho_{m}(a)=\frac{\rho_{m_{0}}}{a^{3}}$

$\rho_{m}$ is not a function of $W(\chi(a))$

$\rho_{m}(1)=\rho_{m_{0}}=3.67852\times 10^{-121}$ $m_{p}^{4}$
The dark energy pressure

$p_{\theta}(a)=\frac{\dot{\theta}^{2}(a)}{2}-(M\delta)^{4}((\kappa\theta(a))^{2}-(\kappa\delta)^{2})^{2}$

$p_{\theta}(\chi(a))=-X(\chi(a))-(M\delta)^{4}(W(\chi(a)+1)^{2}$

$p_{\theta}(1)=-8.54262\times 10^{-121}$ $m_{p}^{4}$
The dark energy equation of state $w$

$w(a)=\frac{\frac{\dot{\theta}^{2}(a)}{2}-(M\delta)^{4}((\kappa\theta(a))^{2}-(\kappa\delta)^{2})^{2}}{\frac{\dot{\theta}^{2}(a)}{2}+(M\delta)^{4}((\kappa\theta(a))^{2}-(\kappa\delta)^{2})^{2}}$

$w(\chi(a))=\frac{X(\chi(a))-(M\delta)^{4}(W(\chi(a)+1)^{2}}{X(\chi(a))+(M\delta)^{4}(W(\chi(a)+1)^{2}}$

$w(1)=-0.995$

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