# Evolution of Cosmological Parameters and Fundamental Constants in a Flat Quintessence Cosmology: A Dynamical Alternative to ΛCDM

## Abstract

**:**

## 1. Introduction

## 2. The Need for SCP Templates

## 3. Quintessence

## 4. General Cosmological Constraints

#### 4.1. The Friedmann Constraints

#### 4.2. The Boundary Conditions

## 5. The Higgs Inspired Potential

## 6. The Quintessence Methodology

#### 6.1. The Modified Beta Function Formalism

#### The Beta Function for the HI Potential

#### 6.2. The Scalar as a Function of the Scale Factor

#### 6.3. Summary of the Methodology

## 7. The Cosmology of $\mathit{W}\left(\mathit{\chi}\right)$

#### 7.1. Past Evolution

#### 7.2. Future Evolution

## 8. The Evolution of the Scalar and the Beta Function

#### 8.1. The Evolution of $\kappa \theta \left(a\right)$

#### 8.2. The Evolution of $\beta \left(a\right)$

## 9. The Value of $\mathit{M}$ in the Dark Energy Potential

## 10. The Evolution of the HI Dark Energy Potential

## 11. The Hubble Parameter

#### 11.1. The Time Derivative of the Hubble Parameter

#### 11.2. The Evolution of the Hubble Parameter

#### 11.3. The Percentage Deviation from $\Lambda $CDM

## 12. The Scale Factor and Time Derivatives of the Scalar

## 13. The Dark Energy Density and Pressure

## 14. The Dark Energy Equation of State

## 15. The Accuracy of the Cosmology and Dark Energy Potential

#### 15.1. The Accuracy of the First Friedmann Constraint

#### 15.2. The Accuracy of the Second Friedmann Constraint

## 16. The HI Quintessence as a Fiducial Dynamical Cosmology

#### A $\Lambda $CDM-like Dynamical Cosmology

## 17. Temporal Evolution of Fundamental Constants

## 18. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Flat HI Quintessence Abridged Templates

**Units:**Natural units are utilized with ℏ, c, and $8\pi G$ set to 1. 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 $W\left(x\right)$ is used extensively in the templates. See [21] for a comprehensive description of the function.

**The Ratra–Peebles, RP scalar:**The RP scalar is used in all of the templates. Its functional form is

**The Higgs Inspired, HI, dark energy potential:**The dark energy potential is

**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.

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

The Ratra–Peebles scalar $\kappa \theta $ |

$\overline{)\kappa \theta \left(a\right)=\kappa \delta \sqrt{-W\left(\chi \right(a)}}$ |

$\overline{)\chi \left(a\right)=q{a}^{p}}$ |

$\overline{)c=2{\left(\kappa \delta \right)}^{2}ln\left(\kappa {\theta}_{0}\right)-{\left(\kappa {\theta}_{0}\right)}^{2}}$ |

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

$\overline{)p=\frac{8}{{\left(\kappa \delta \right)}^{2}}}$ |

$\overline{)\kappa {\theta}_{0}=-\frac{4-\sqrt{16+12{\mathsf{\Omega}}_{{\theta}_{0}}({w}_{0}+1){\left(\kappa \delta \right)}^{2}}}{2\sqrt{3{\mathsf{\Omega}}_{{\theta}_{0}}({w}_{0}+1)}}}$ |

$\overline{)\kappa \theta \left(1.0\right)=0.102202}$ |

The beta function $\beta $ |

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

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

$\overline{)\beta \left(1.0\right)=0.10247}$ |

The dark energy potential V |

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

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

$\overline{)M=\sqrt[4]{\frac{3{H}_{0}^{2}}{{({\left(\kappa {\theta}_{0}\right)}^{2}-{\left(\kappa \delta \right)}^{2})}^{2}}\left({\mathsf{\Omega}}_{{\theta}_{0}}-\frac{\beta {\left(1\right)}^{2}}{6}\right)}}$ |

$\overline{)V\left(1.0\right)=8.56409\times {10}^{-121}{m}_{p}^{4}}$ |

The Hubble parameter |

$\overline{)H\left(a\right)=\sqrt{\frac{{\left(M\delta \right)}^{4}{({\left(\kappa \theta \right)}^{2}-{\left(\kappa \delta \right)}^{2})}^{2}+\frac{{\rho}_{{m}_{0}}}{{a}^{3}}}{3\left(1-\frac{\beta {\left(a\right)}^{2}}{6}\right)}}}$ |

$\overline{)H\left(\chi \left(a\right)\right)=\sqrt{\frac{{\left(M\delta \right)}^{4}{(W\left(\chi \left(a\right)\right)+1)}^{2}+\frac{{\rho}_{{m}_{0}}}{{a}^{3}}}{3\left(1-\frac{{\left(\frac{4\sqrt{-W\left(\chi \right(a\left)\right)}}{\kappa \delta \left(W\right(\chi \left(a\right))+1)}\right)}^{2}}{6}\right)}}}$ |

$\overline{)H\left(1.0\right)=6.39403\times {10}^{-61}{m}_{p}}$ |

The derivative of the scalar with respect to the scale factor $\frac{d\theta}{da}$ |

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

$\overline{)\kappa \frac{d\theta \left(1.0\right)}{da}=0.10247}$ |

The derivative of the scalar with respect to time $\frac{d\theta}{dt}$ |

$\overline{)\frac{d\theta \left(a\right)}{dt}=\dot{\theta}\left(a\right)=\frac{d\theta \left(a\right)}{da}H\left(a\right)a}$ |

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

$\overline{)\frac{d\theta \left(1.0\right)}{dt}=6.55193\times {10}^{-62}{m}_{p}^{2}}$ |

The kinetic term X = $-\frac{{\dot{\theta}}^{2}}{2}$ |

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

$\overline{)X\left(\chi \left(a\right)\right)={\left(\kappa \delta \frac{p\sqrt{-W\left(\chi \right(a\left)\right)}}{2(1+W(\chi \left(a\right)\left)\right)}\right)}^{2}\left(\frac{{\left(M\delta \right)}^{4}{(W\left(\chi \left(a\right)\right)+1)}^{2}+\frac{{\rho}_{{m}_{0}}}{{a}^{3}}}{3\left(1-\frac{{\left(\frac{4\sqrt{-W\left(\chi \right(a\left)\right)}}{\kappa \delta \left(W\right(\chi \left(a\right))+1)}\right)}^{2}}{6}\right)}\right)}$ |

The dark energy density |

$\overline{){\rho}_{\theta}\left(a\right)=\frac{{\dot{\theta}}^{2}\left(a\right)}{2}+{\left(M\delta \right)}^{4}{({\left(\kappa \theta \left(a\right)\right)}^{2}-{\left(\kappa \delta \right)}^{2})}^{2}}$ |

$\overline{){\rho}_{\theta}\left(\chi \left(a\right)\right)=-X\left(\chi \left(a\right)\right)+{\left(M\delta \right)}^{4}{(W(\chi \left(a\right)+1)}^{2}}$ |

$\overline{){\rho}_{\theta}\left(1\right)=8.58555\times {10}^{-121}{m}_{p}^{4}}$ |

The matter density |

$\overline{){\rho}_{m}\left(a\right)=\frac{{\rho}_{{m}_{0}}}{{a}^{3}}}$ |

${\rho}_{m}$ is not a function of $W\left(\chi \right(a\left)\right)$ |

$\overline{){\rho}_{m}\left(1\right)={\rho}_{{m}_{0}}=3.67852\times {10}^{-121}{m}_{p}^{4}}$ |

The dark energy pressure |

$\overline{){p}_{\theta}\left(a\right)=\frac{{\dot{\theta}}^{2}\left(a\right)}{2}-{\left(M\delta \right)}^{4}{({\left(\kappa \theta \left(a\right)\right)}^{2}-{\left(\kappa \delta \right)}^{2})}^{2}}$ |

$\overline{){p}_{\theta}\left(\chi \left(a\right)\right)=-X\left(\chi \left(a\right)\right)-{\left(M\delta \right)}^{4}{(W(\chi \left(a\right)+1)}^{2}}$ |

$\overline{){p}_{\theta}\left(1\right)=-8.54262\times {10}^{-121}{m}_{p}^{4}}$ |

The dark energy equation of state w |

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

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

$\overline{)w\left(1\right)=-0.995}$ |

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

**Figure 2.**The figure shows 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 $\chi $ 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 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.

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

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

**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.

**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 ${w}_{0}$.

**Figure 9.**The (

**left**) panel shows the evolution of ${M}^{2}{\kappa}^{2}\dot{\theta}$ for all cases and the (

**right**) panel shows the evolution at scale factors between 0.4 and 1.0 in more detail.

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

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

**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.

**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 $.

**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.

**Table 1.**Boundary conditions and parameter values in this work. ${H}_{0}$ is the current value of the Hubble parameter in units of $\frac{km/sec}{Mpc}$. ${\mathsf{\Omega}}_{{m}_{0}}$ and ${\mathsf{\Omega}}_{{\varphi}_{0}}$ 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.

${\mathit{H}}_{0}$ | ${\mathbf{\Omega}}_{{\mathit{m}}_{0}}$ | ${\mathbf{\Omega}}_{{\mathit{\varphi}}_{0}}$ | ${\mathit{w}}_{0}$ | Scale Factor a | ||
---|---|---|---|---|---|---|

73 | 0.3 | 0.7 | −0.99 | −0.995 | −0.999 | 0.1–1.0 |

**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 7.2, are in the last column.

The Values of $\mathit{\chi}$ | |||||
---|---|---|---|---|---|

Scale Factor $\mathit{a}$ | |||||

$\mathbf{\delta}$ | ${\mathit{w}}_{\mathbf{0}}$ | 0.1 | 0.5 | 1.0 | ${\mathit{a}}_{\mathit{eq}}$ |

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 |

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**MDPI and ACS Style**

Thompson, R.I.
Evolution of Cosmological Parameters and Fundamental Constants in a Flat Quintessence Cosmology: A Dynamical Alternative to ΛCDM. *Universe* **2023**, *9*, 172.
https://doi.org/10.3390/universe9040172

**AMA Style**

Thompson RI.
Evolution of Cosmological Parameters and Fundamental Constants in a Flat Quintessence Cosmology: A Dynamical Alternative to ΛCDM. *Universe*. 2023; 9(4):172.
https://doi.org/10.3390/universe9040172

**Chicago/Turabian Style**

Thompson, Rodger I.
2023. "Evolution of Cosmological Parameters and Fundamental Constants in a Flat Quintessence Cosmology: A Dynamical Alternative to ΛCDM" *Universe* 9, no. 4: 172.
https://doi.org/10.3390/universe9040172