#
The Hubble Tension, the M Crisis of Late Time H(z) Deformation Models and the Reconstruction of Quintessence Lagrangians^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- It provides a worse fit than $\mathrm{\Lambda}$CDM to low z geometric probes such as SnIa and BAO [6].
- As in the case of $\mathrm{\Lambda}$CDM, it favors a lower value of the SnIa absolute magnitude M than the local Cepheid calibrators.

## 2. Cosmological Data—Parameters

- h: The dimensionless Hubble parameter defined as: ${H}_{0}=100\phantom{\rule{0.166667em}{0ex}}\mathrm{h}\xb7{\mathrm{kms}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{Mpc}}^{-1}$.
- ${\mathrm{\Omega}}_{m,0}$: Present value of the matter density parameter.
- ${\mathrm{\Omega}}_{b}$: The baryon density parameter.
- $w\left(z\right)$: The equation-of-state parameter. It is also common to use two or more parameters $({w}_{0},{w}_{a},...)$ to define it. For example, in the CPL model [9,10]$w\left(z\right)={w}_{0}+{w}_{a}/(1+z)$. From the $w\left(z\right)$ parametrization, it is straightforward to obtain the dark energy density parameter ${\mathrm{\Omega}}_{DE}\left(z\right)$ (see Appendix A).
- M: In the present analysis, we also consider the SnIa absolute magnitude M. This parameter can be constrained using either a combination of cosmological data (SnIa, BAO and CMB) at $z>0.01$ or Cepheid calibrators at $z<0.01$. The root of the Hubble crisis lies in the mismatch of the values of M obtained by the above two distinct approaches, as discussed below.

#### 2.1. Supernovae as Distance Indicators

- The collapse of the core of a massive star. Such a star has a core mass higher than the Chandrasekhar limit, which is 1.4 solar masses ($1.4$${M}_{\odot}$).
- The abrupt re-ignition of nuclear fusion in a compact star (white dwarfs, neutron stars and black holes). In order to have re-ignition, additional energy is required to raise the temperature in the stars core. The star can obtain this energy either by a merger or by accretion.

#### Light Curve Features in Standard Gravity

- The light curve width depends on the optical opacity of the ejecta, the total ejected mass and the kinetic energy of the explosion [16], which is calculated as the difference between the energy produced by nuclear fusion and the gravitational binding energy of the white dwarf progenitor [15]. The latter contributes only weakly to the light curve width [16].

#### Light Curve Features in Modified Gravity

#### Observations

- They are the most common type of supernova in the Universe.
- They are extremely luminous, as at their peak luminosity they can reach an absolute magnitude of about: $M\approx -19$, which is about the absolute magnitude of a bright galaxy.
- They have a relatively small dispersion of peak absolute magnitude.
- Their explosion mechanism is fairly uniform and well understood and, according to known physics, has no cosmic evolution.
- There are a lot of local SNeIa that we can use to test their physics and calibrate the absolute magnitude for the distant ones.

#### 2.2. Baryonic Acoustic Oscillation Measurements (BAO)

- Recombination

- Photon Decoupling

- Beyond Photon Decoupling

- Structure Formation

- The epoch of recombination, which affects the drag epoch ${z}_{d}$.
- The expansion of the Universe, $H\left(z\right)$.
- The baryon-to-photon ratio, which affects ${c}_{s}$.

#### 2.2.1. BAO Measurements

#### 2.3. CMB Measurement

- Shift Parameter R

- The physical density parameters for matter ${\omega}_{m}$, baryons ${\omega}_{b}$, radiation ${\omega}_{r}$ and curvature ${\omega}_{k}$.
- The primordial fluctuation spectrum.
- The flat-universe comoving angular diameter distance to the recombination surface ${D}_{M}\left({z}_{\U0001f7c9}\right)$.

- Photon Decoupling Epoch

## 3. Data Analysis

#### 3.1. Bayes Theorem

- $p\left(\theta \right|x)$: The posterior probability distribution $p\left(\theta \right|x)$ for the parameters $\theta $ and data x. It is the probability that the parameters will obtain certain values after completing the experiment and making some assumptions [36].
- $p\left(x\right|\theta )$: It is called likelihood, and we also refer to it as $\mathcal{L}(x;\theta )$
- $p\left(\theta \right)$: The prior probability distribution $p\left(\theta \right)$ for the parameters $\theta $. It expresses what we know about the parameters before performing the experiment, including the results of previous experiments or theory. For example, we know that the age of the Universe must be positive.In the absence of any previous information, it is common to adopt the principle of indifference and assume that all values of the parameters are equally likely and take $p\left(\theta \right)=constant$. As a bound, someone can either use some finite bounds or use infinite bounds and work with an unnormalized prior. This prior is called a flat prior.
- $p\left(x\right)$: The evidence.

#### 3.2. Maximum Likelihood

#### 3.2.1. Non-Independent Measurements

#### Covariance

#### Correlation

#### Covariance Matrix

#### ${\chi}^{2}$ Function

#### 3.2.2. Chi-by-Eye

#### 3.3. Likelihood Function

#### 3.3.1. Fisher Matrix

#### Hessian Matrix

#### Conditional Error

#### Fisher Matrix

#### 3.4. Marginalization

#### 3.5. Confidence Limits

#### 3.5.1. Constant ${\chi}^{2}$ Boundaries as Confidence Limits

#### 3.5.2. Errors

#### 3.6. ${\chi}^{2}$ for CMB Data

#### 3.7. ${\chi}^{2}$ for BAO Data

#### 3.7.1. 6dFGS and WiggleZ

#### 3.7.2. SDSS

#### 3.7.3. Ly-$\alpha $

#### 3.8. ${\chi}^{2}$ for SNIa Data

#### 3.9. Total ${\chi}^{2}$ Function

## 4. Dark Energy Models

#### 4.1. Spatially-Flat $\mathrm{\Lambda}$CDM Model

#### 4.1.1. Theoretical Challenges

#### Fine-Tuning Problem

#### Coincidence Problem

#### Anthropic Principle

#### 4.1.2. Observational Challenges

#### ${H}_{0}$ Tension

#### Growth Tension

#### 4.1.3. Fitting the $\mathrm{\Lambda}$CDM Parameters: Maximum Likelihood

#### BAO and CMB Data

#### SNIa Data

#### Combined Data

#### Results

#### 4.2. Spatially-Flat wCDM Model

#### Results

#### 4.3. Chevallier–Polarski–Linder (CPL) Parametrization

#### Results

#### 4.4. Adding the Local ${H}_{0}$ Determination

#### 4.4.1. Results

#### $\mathrm{\Lambda}$CDM

#### wCDM

#### CPL

## 5. Reconstruction of Dark Energy

#### 5.1. Scalar Field Dark Energy Models (Quintessence)

- Noisiness of measurements of the expansion.
- Translation from the measured quantity to ${\rho}_{DE}$ and w through one or two derivatives.
- Range of the scale factor or equivalently redshift coverage: $z=\frac{1}{a}-1$

#### 5.2. Reconstruction Equations

#### 5.3. Results

## 6. Discussion and Conclusions

- The best-fit value of ${H}_{0}$ in the context of all these models is not consistent with the local determination of ${H}_{0}$ shown in Equation (2).
- The best-fit value of the SnIa absolute magnitude M is not consistent with the value of M determined by the local Cepheid calibrators.
- The quality of fit of all these parameterizations becomes significantly worse when the local determination of the ${H}_{0}$ point is included.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FRW | Friedman–Robertson–Walker |

DE | dark energy |

SN | supernova |

SNe | supernovae |

SNIa | supernova of Type Ia |

SNeIa | supernovae of Type Ia |

BAO | baryonic acoustic oscillations |

CMB | cosmic microwave background |

CDM | cold dark matter |

CPL | Chevalier–Polarski–Linder |

PDL | phantom divide line |

## Appendix A. Notation—Cosmology Basics

#### Appendix A.1. FRW Metric

- The curvature parameter k can be −1, 0, 1.
- $a\left(t\right)$ is the cosmic scale factor.
- The cosmic time t is the proper time measured by a free-falling observer.
- The coordinates $r,\theta ,\varphi $ are comoving coordinates.

- The time is again the proper time measured by a free-falling observer, while $\chi $ is a new radial coordinate (in which $r,\theta ,\varphi $ is a comoving coordinate).

**Friedmann Equation:**

**Friedmann Acceleration Equation:**

- Constant Equation of State

**Figure A1.**Evolution of the energy densities in the Universe. Adapted from: [98].

- Dynamical Equation of State

#### Appendix A.1.1. Single Component Universe

#### Appendix A.1.2. Two-Component Universe

#### Appendix A.1.3. Multi-Component Universe

#### Appendix A.2. Redshift z

**Figure A2.**Diagram showing two null (photon) geodesics. In our case where the photons are emitted radially, we have ${\theta}_{E}={\theta}_{R}$ and ${\varphi}_{E}={\varphi}_{R}$. Adapted from [99].

#### Appendix A.3. Hubble’s Law

#### Appendix A.3.1. Edwin Hubble

#### Appendix A.3.2. Hubble Constant

#### Appendix A.3.3. Physical Density Parameters

#### Appendix A.3.4. Derivation

#### Appendix A.4. Distances in Cosmology

#### Appendix A.4.1. Theoretically Defined Distances

#### Metric Distance

**Figure A3.**Figure that helps with the definition of angular diameter distance. Adapted from: [98].

#### Comoving/Coordinate Distance

#### Appendix A.4.2. Observable Distances

#### Physical Distance

#### Angular Diameter Distance

**Figure A4.**Figure that helps with the definition of Luminosity distance. Adapted from: [98].

#### Luminosity Distance

- At the time ${t}_{0}$ that we observe the light from the object, the proper area of a sphere drawn around a supernova and passing through the Earth is $4\pi {d}_{m}^{2}$.
- The rate at which we detect photons from the object is reduced compared to the rate that they are emitted, by the redshift factor: $\frac{a\left({t}_{0}\right)}{a\left(t\right)}=1+z$.
- The energy of the photons is also being redshifted, so the energy that we observe them to have is reduced compared to the one they had when they were emitted by the same redshift factor: $1+z$.

#### Distance Modulus

## Appendix B. Theoretical Background

#### Appendix B.2. Dark Energy

**Figure A6.**The remnants of Supernova 1604 or Kepler’s Supernova. It was a Type Ia supernova that occurred in our galaxy in the constellation of Ophiuchus. It was observable by the naked eye and was named by Johannes Kepler who described it in: De Stella Nova. Adapted from: [102].

- A positive energy density $({\rho}_{X}>0)$, assuming that the universe is flat.
- A negative pressure $({p}_{X}<0)$, which can cancel out gravity and potentially lead to accelerating expansion.

- ${\mathrm{\Omega}}_{m}$ is the matter density parameter.
- ${\mathrm{\Omega}}_{X}\left(z\right)$ is the dark energy density parameter.

## Appendix C. Data Analysis

#### Appendix C.1. Useful Functions

#### Appendix C.1.1. Gamma Function

#### Incomplete Gamma Function

#### Error Function

#### Appendix C.1.2. Useful Distributions

#### Normal or Gaussian Distribution

#### Chi-Square or χ 2 Distribution

- Mean$\left\{{\chi}^{2}\left(\nu \right)\right\}=\nu $
- Var$\left\{{\chi}^{2}\left(\nu \right)\right\}=2\nu $

#### Appendix C.1.3. Derivation of Δχ 2 for Given Confidence Region in Parameter Space

**Figure A9.**Mathematica code for the derivation of Table 1, where n is the number of degrees of freedom and dchi is the value of $\mathrm{\Delta}{\chi}^{2}$.

## Appendix D. Proof of dg=gg ab dg ab

## Appendix E. Derivation of the Klein-Gordon Equation

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**Figure 1.**The SnIa absolute magnitudes obtained as ${M}_{i}={m}_{i}-\mu \left({z}_{i}\right)$ of the binned Pantheon sample [5] assuming Planck/$\mathrm{\Lambda}$CDM luminosity distance. The data are inconsistent with the ${M}^{\mathrm{R}20}$ of Equation (4), but they become consistent if there is a transition in the absolute magnitude with amplitude $\mathrm{\Delta}M\approx -0.2$. Note that in this model there is no transition in the Hubble function $H\left(z\right)$.

**Figure 4.**Type Ia light curves for different values of Chandrasekhar mass showing that the peak luminosity decreases as the Chandrasekhar mass increases and thus the Gravitational constant decreases. Adapted from: [15].

**Figure 5.**(

**a**) Light curves of some local Type Ia supernovae as measured. (

**b**) The light curves of the same supernovae corrected in terms of their width. Adapted from: [14].

**Figure 6.**A cartoon produced by the BOSS project showing the spheres of baryons around the initial dark matter clumps. Adapted from: [23].

**Figure 7.**Perturbation evolution: (

**a**) Early Universe: initial perturbations. (

**b**) Early Universe: neutrinos spread out, acoustic waves form. (

**c**) Early Universe: acoustic waves propagate outwards, dark matter perturbation grows. (

**d**) Recombination: photons do not scatter efficiently, sound speed drops. (

**e**) Photon decoupling: photons spread out, sound wave stalls. (

**f**) Before structure formation: dark matter and baryons attract each other and they mix up, with dark matter dominating due to much higher mass. Adapted from: [25], while the original animation can be found here: [26].

**Figure 10.**The CMB power spectrum versus the multipole moment l and the angular size $\theta $. The curve shows the theoretical prediction of the power spectrum, while the red points represent the Planck data as of March 2013. Adapted from: [34].

**Figure 11.**Probability distribution for the case M = 2 (two parameters ${a}_{0}$ and ${a}_{1}$). Furthermore, there are three different confidence regions all at the same confidence level. The first region is defined by the vertical lines and represents a $68\%$ confidence interval for the variable ${a}_{0}$ without regard to the value of ${a}_{1}$, while the second one is defined by the horizontal lines and represents a $68\%$ confidence interval for the variable ${a}_{1}$ without regard to the value of ${a}_{0}$. The third one is the ellipse, which shows a $68\%$ confidence interval for ${a}_{0}$ and ${a}_{1}$, jointly. Adapted from: [41].

**Figure 12.**Confidence regions derived using a constant $\mathrm{\Delta}{\chi}^{2}$ region as a boundary. Here, the shape of the confidence region is chosen to be an ellipse. It is important to note that the intervals AA’, BB’, CC’ are the ones that contain the percentage of the normally distributed data that correspond to the respective $\mathrm{\Delta}{\chi}^{2}$. Adapted from: [41].

**Figure 13.**Standard errors for the parameters a and b found as the projections of the ellipse, which is the confidence region boundary with $\mathrm{\Delta}{\chi}^{2}=1$, on the a and b axes. Adapted from: [41].

**Figure 14.**Plot of the inferred SNIa absolute magnitudes ${M}_{B,i}={m}_{B,i}-\mu \left({z}_{i}\right)$ of the binned Pantheon data [5] under the assumption of a Planck/$\mathrm{\Lambda}CDM$ luminosity distance. Furthermore, there is a straight line at the value: ${M}_{B}^{p18}=-19.4$, which is the value of the SNIa absolute magnitude calibrated via BAO and CMB measurements found in: [20].

**Figure 15.**The $1\sigma -3\sigma $ confidence contours in the parametric space $(h-{\mathrm{\Omega}}_{0m})$ for the BAO data and the BAO + CMB data combination. Notice the dramatic improvement of the constraints when the CMB data points are included.

**Figure 16.**The $1\sigma -3\sigma $ confidence contours in the parametric space $(\mathcal{M}-{\mathrm{\Omega}}_{0m})$.The red contour corresponds to the full Pantheon dataset while the blue contour corresponds to the Pantheon+CMB+BAO data combination. As expected, when the CMB+BAO data are included the constraints on ${\mathrm{\Omega}}_{0m}$ improve dramatically.

**Figure 17.**Confidence contours for the wCDM model: (

**a**) The $1\sigma -3\sigma $ confidence contours in the parametric space $(\mathcal{M}-{\mathrm{\Omega}}_{0m})$. (

**b**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $(h-{\mathrm{\Omega}}_{0m})$ and $(h-\mathcal{M})$. (

**c**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $({w}_{0}-{\mathrm{\Omega}}_{0m})$, $({w}_{0}-\mathcal{M})$ and $({w}_{0}-h)$.

**Figure 18.**Confidence contours for the CPL model: (

**a**) The $1\sigma -3\sigma $ confidence contour in the parametric space $(\mathcal{M}-{\mathrm{\Omega}}_{0m})$. (

**b**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $(h-{\mathrm{\Omega}}_{0m})$ and $(h-\mathcal{M})$. (

**c**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $({w}_{0}-{\mathrm{\Omega}}_{0m})$, $({w}_{0}-\mathcal{M})$ and $({w}_{0}-h)$.

**Figure 19.**The $1\sigma -3\sigma $ confidence contours in the parametric spaces: $({w}_{a}-{\mathrm{\Omega}}_{0m})$, $({w}_{a}-M)$, $({w}_{a}-h)$ and $({w}_{a}-{w}_{0})$ for the CPL model.

**Figure 20.**Confidence contours for the $\mathrm{\Lambda}$CDM model with the addition of the Riess point: (

**a**) The $1\sigma -3\sigma $ confidence contour in the parametric space $(\mathcal{M}-{\mathrm{\Omega}}_{0m})$ for SNIa data, SNIa+BAO+CMBand SNIa+BAO+CMB+Riess data. (

**b**) The $1\sigma -3\sigma $ confidence contour in the parametric space $(\mathcal{M}-{\mathrm{\Omega}}_{0m})$ for the SNIa+BAO+CMB and SNIa+BAO+CMB+Riess data.

**Figure 21.**Confidence contours for the wCDM model with the addition of the Riess point: (

**a**) The $1\sigma -3\sigma $ confidence contour in the parametric space $(\mathcal{M}-{\mathrm{\Omega}}_{0m})$. (

**b**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $(h-{\mathrm{\Omega}}_{0m})$ and $(h-\mathcal{M})$. (

**c**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $({w}_{0}-{\mathrm{\Omega}}_{0m})$, $({w}_{0}-\mathcal{M})$ and $({w}_{0}-h)$.

**Figure 22.**Confidence contours for the CPL model with the addition of the Riess point: (

**a**) The $1\sigma -3\sigma $ confidence contour in the parametric space $(\mathcal{M}-{\mathrm{\Omega}}_{0m})$. (

**b**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $(h-{\mathrm{\Omega}}_{0m})$ and $(h-\mathcal{M})$. (

**c**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $({w}_{0}-{\mathrm{\Omega}}_{0m})$, $({w}_{0}-\mathcal{M})$ and $({w}_{0}-h)$. (

**d**) The $1\sigma -3\sigma $ confidence contours in the parametric spaces $({w}_{a}-{\mathrm{\Omega}}_{0m})$, $({w}_{a}-\mathcal{M})$, $({w}_{a}-h)$ and $({w}_{a}-{w}_{0})$.

**Figure 23.**Plots for the field reconstruction: (

**a**) Plot of the equation of state w in terms of the redshift z for the three models considered. (

**b**) Plot of the energy density ${\rho}_{DE}$ in terms of the redshift z for the three models considered. (

**c**) Plot of the fraction $\frac{H\left(z\right)}{1+z}$ in terms of the redshift z for the three models considered.

**Figure 24.**Plots for the field reconstruction: (

**a**) Plot of the potential $V\left(z\right)$ in terms of the redshift z for the three models considered. (

**b**) Plot of the kinetic term ${\varphi}^{2}/2$ in terms of the redshift z for the three models considered. (

**c**) Plot of the field $\varphi $(z) in terms of the redshift z or the three models considered. (

**d**) Plot of the potential of the $w$CDM model in terms of the field $\varphi $. (

**e**) Plot of the potential CPL model in terms of the field $\varphi $.

**Table 1.**Table containing $\mathrm{\Delta}{\chi}^{2}$ values as a function of confidence level p and the number of parameters M. The process used to derive these values can be seen in Appendix C.1.3.

$\mathit{\sigma}$ | Probability | M | |||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | ||

1 | $68.27\%$ | 1.00 | 2.30 | 3.53 | 4.72 |

2 | $95.45\%$ | 4.00 | 6.18 | 8.02 | 9.72 |

3 | $99.73\%$ | 9.00 | 11.8 | 14.2 | 16.3 |

**Table 2.**Table containing the best-fit parameters for the $\mathrm{\Lambda}CDM$ model for different data combinations.

$\mathbf{\Lambda}\mathit{CDM}$ | |||
---|---|---|---|

BAO + CMB | SNIa | Combined | |

${\mathrm{\Omega}}_{0m}$ | $0.3178\pm 0.0059$ | $0.299\pm 0.022$ | $0.3169\pm 0.0057$ |

$\mathcal{M}$ | − | $23.809\pm 0.011$ | $23.817\pm 0.049$ |

h | $0.6718\pm 0.0039$ | − | $0.6724\pm 0.0038$ |

${\chi}^{2}$ | 6.3927 | 1025.63 | 1032.7 |

**Table 3.**Table containing the best-fit values for the $\mathrm{\Lambda}CDM$, $wCDM$ and $CPL$ models using both the CMB and BAO measurements and data from Type Ia supernovae.

$\mathbf{\Lambda}\mathit{CDM}$ | wCDM | CPL | |
---|---|---|---|

Combined | |||

${\mathrm{\Omega}}_{0m}$ | $0.3169\pm 0.0057$ | $0.315\pm 0.008$ | $0.315\pm 0.013$ |

${w}_{0}$ | $-1$ | $-1.01\pm 0.03$ | $-1.07\pm 0.15$ |

${w}_{a}$ | − | − | $0.24\pm 0.47$ |

$\mathcal{M}$ | $23.812\pm 0.006$ | − | − |

M | − | $-19.42\pm 0.02$ | $-19.43\pm 0.02$ |

h | $0.6724\pm 0.0038$ | $0.675\pm 0.008$ | $0.674\pm 0.011$ |

${\chi}^{2}$ | 1032.7 | 1032.6 | 1031.97 |

**Table 4.**Table containing the best-fit values for the $\mathrm{\Lambda}CDM$, $wCDM$ and $CPL$ models using both the CMB and BAO measurements, data from Type Ia supernovae and the Riess point.

$\mathbf{\Lambda}\mathit{CDM}$ | wCDM | CPL | |
---|---|---|---|

${\mathrm{\Omega}}_{0m}$ | $0.3120\pm 0.0055$ | $0.299\pm 0.007$ | $0.298\pm 0.013$ |

${w}_{0}$ | $-1$ | $-1.07\pm 0.03$ | $-1.12\pm 0.16$ |

${w}_{a}$ | − | − | $0.19\pm 0.56$ |

$\mathcal{M}$ | $23.814\pm 0.049$ | − | − |

M | − | $-19.39\pm 0.02$ | $-19.39\pm 0.02$ |

h | $0.6757\pm 0.0037$ | $0.693\pm 0.008$ | $0.693\pm 0.011$ |

${\chi}^{2}$ | 1054.48 | 1047.86 | 1047.59 |

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**MDPI and ACS Style**

Theodoropoulos, A.; Perivolaropoulos, L.
The Hubble Tension, the *M* Crisis of Late Time *H*(*z*) Deformation Models and the Reconstruction of Quintessence Lagrangians. *Universe* **2021**, *7*, 300.
https://doi.org/10.3390/universe7080300

**AMA Style**

Theodoropoulos A, Perivolaropoulos L.
The Hubble Tension, the *M* Crisis of Late Time *H*(*z*) Deformation Models and the Reconstruction of Quintessence Lagrangians. *Universe*. 2021; 7(8):300.
https://doi.org/10.3390/universe7080300

**Chicago/Turabian Style**

Theodoropoulos, Anastasios, and Leandros Perivolaropoulos.
2021. "The Hubble Tension, the *M* Crisis of Late Time *H*(*z*) Deformation Models and the Reconstruction of Quintessence Lagrangians" *Universe* 7, no. 8: 300.
https://doi.org/10.3390/universe7080300