# A Loop Quantum-Corrected Family of Chiral Cosmology Models

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## Abstract

**:**

## 1. Introduction

## 2. Effective LQC Dynamics

- $\mathrm{Range}(\rho )=(0,\phantom{\rule{4pt}{0ex}}{\rho}_{\mathrm{c}}]$;
- $\mathrm{Range}(\beta )=\left(\right)open="("\; close=")">0,\phantom{\rule{4pt}{0ex}}\frac{\pi}{\lambda}$;
- The volume, V, reaches a minimum at $\beta =\frac{\pi}{2\lambda}$;
- $\beta $ is monotonic (in particular, decreasing).

## 3. Effective Dynamics with a Simple $\mathit{k}$-Essence Field

#### 3.1. A Further Geometrical Interpretation of the (k-Essence) Chiral Cosmology Scenario

#### 3.2. Classical Solutions in Connection Variables

#### 3.2.1. $f(\varphi )=w{\varphi}^{m}$ (Sáez–Ballester)

#### 3.2.2. $f(\varphi )={e}^{m\varphi}$

#### 3.3. Holonomized Hamiltonian

#### 3.3.1. $f(\varphi )=w{\varphi}^{m}$ (Sáez–Ballester)

#### 3.3.2. $f(\varphi )={e}^{m\varphi}$

## 4. Chiral Fields in Effective LQC

#### 4.1. Standard (Semi-)Classical and Quantum Treatments

#### 4.2. Holonomized Chiral Cosmology

- The energy density function,$$\begin{array}{}& \begin{array}{cc}\hfill {\rho}_{\mathrm{q}}& =\frac{1}{2}\left(\right)open="("\; close=")">{\dot{\varphi}}^{2}-{\dot{\psi}}^{2}+2{m}_{12}\dot{\varphi}\dot{\psi}+U(\varphi ,\psi )\hfill \end{array}\hfill \end{array}\mathrm{(108)}\hfill & \begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\frac{3{\beta}^{2}}{8\pi G{\gamma}^{2}},\hfill \end{array}\hfill $$
- $V\to 0$ is not forbidden; in fact, it is observed that at $V=0$, ${\rho}^{\mathrm{q}}\to \infty $ (this is, of course, a cosmological singularity);
- $\beta =0$ corresponds to either a maximum of V or a minimum of V, according to whether the full chiral kinetic term is positive or negative at $\beta =0$, respectively;
- If the potential function, $U(\varphi ,\psi )$, is negative at $\beta =0$, then the volume function reaches a maximum at $\beta =0$.

- The energy density function,$$\begin{array}{}& \begin{array}{cc}\hfill {\rho}_{\mathrm{q}}^{\mathrm{eff}}& =\frac{1}{2}\left(\right)open="("\; close=")">{\dot{\varphi}}^{2}-{\dot{\psi}}^{2}+2{m}_{12}\dot{\varphi}\dot{\psi}+U(\varphi ,\psi )\hfill \end{array}\hfill \end{array}\mathrm{(121)}\hfill & \begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\frac{3{sin}^{2}\left(\right)open="("\; close=")">\lambda \beta}{}8\pi G{\gamma}^{2}{\lambda}^{2},\hfill \end{array}\hfill $$
- The function $\beta $ is a monotonic function of time provided that$${p}_{\varphi}^{2}-{p}_{\psi}^{2}+2{m}_{12}{p}_{\varphi}{p}_{\psi}\ge 0\phantom{\rule{1.em}{0ex}}\mathrm{or}\phantom{\rule{1.em}{0ex}}{p}_{\varphi}^{2}-{p}_{\psi}^{2}+2{m}_{12}{p}_{\varphi}{p}_{\psi}\le 0$$
- In the case at hand, a simple sufficient condition for the first relation in (122) to be satisfied is ${V}_{1},{V}_{2}<0$;
- Provided (122) is fulfilled during evolution, the volume function reaches a minimum at $\beta =\frac{\pi}{2\lambda}$—which corresponds to ${\rho}_{c}$—and it is attained only once, given that the image of $\beta $ is in the interval $(0,\phantom{\rule{0.166667em}{0ex}}\pi /\lambda )$.

#### 4.2.1. The First Case

#### 4.2.2. The Second Case

## 5. Summary and Discussion

- In preparation for the main parts of the manuscript, in Section 3, a holonomization of a FLRW background with a particularly simple k-essence field was performed, and some exact solutions were found (which were shown to reduce to the standard classical ones in the limit of a vanishing area gap).
- In Section 4.2, we constructed a family of holonomy-corrected chiral quintom cosmology models. We established that the key single big bounce in the standard LQC paradigm is achieved provided the full chiral kinetic energy does note change sign during its evolution (which is ensured, in particular, by taking parameters ${V}_{1},{V}_{2}<0$). We also analyzed the corresponding equation-of-state parameter.
- The main objective of Section 4.2.1 and Section 4.2.2 was to exemplify the general remarks given for the holonomy-corrected Hamiltonian system in Section 4.2 via particular numerical solutions.

#### A Larger Family of Loop Quantum-Corrected Chiral Models

- A negative definite K together with an identically zero U is not consistent with (131), and such a situation is therefore forbidden.
- If U is identically zero, $K\ge 0$ must be satisfied during evolution, and hence the single big bounce takes place.
- The more restrictive situation in which K is positive and definite is, of course, a particular case of the result established above.
- If $U\le 0$ is satisfied during evolution, then the big bounce occurs. This includes the more restrictive situation in which U is negative and definite. Indeed, if $U\le 0$, then from (131), it follows that $K\ge 0$ during evolution.
- The more restrictive situation in which K is negative and definite (in which case it is necessary that $U>0$ during evolution) is, of course, a particular case of the result established above.

- Standard models with holonomy corrections:
- −
- A standard quintom scenario in effective LQC is obtained by considering two scalar fields, $\phi $ and $\psi $, with ${m}_{ab}=\mathrm{diag}(1,-1)$ (see, e.g., [53]);
- −
- Inflationary scenarios within effective LQC are obtained by considering one scalar field, $\phi $, with ${m}_{11}=1$ and suitable forms for $U(\phi )$ (see, e.g., [54]);
- −
- The original effective scheme of LQC is achieved by considering one free scalar field, $\phi $, with ${m}_{11}=1$.

- Standard models without holonomy corrections: The replacement $sin(\lambda \beta )\to \lambda \beta $ is to be performed in the Hamiltonian (130), and $\beta $ is no longer restricted to take values only in the range $(0,\phantom{\rule{0.166667em}{0ex}}\pi /\lambda )$.
- −
- The standard quintom scenario is obtained by considering only two scalar fields, $\phi $ and $\psi $, and taking $\left[{m}_{ab}\right]=\mathrm{diag}(1,-1)$ with suitable potentials, $U(\phi )$ and $V(\psi )$. Relevant potentials are reported in [1].
- −
- The standard quintessence scenario is obtained by considering only one scalar field, $\phi $, and taking ${m}_{11}=1$ with a suitable potential, $U(\phi )$. Relevant potentials are reported in [1,55]. The standard ΛCDM model can, of course, be regarded as a limiting case of the quintessence scenario.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | In cosmology, one is usually restricted to homogenous (metric and scalar) fields. |

2 | The spatial integration featured in the action, when carried over the whole spacelike slice, diverges in the flat case. Due to homogeneity, we can restrict the action to a compact region $\mathcal{V}$ of the spacelike slice. |

## References

- Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D
**2006**, 15, 1753–1936. [Google Scholar] [CrossRef] - Cai, Y.F.; Saridakis, E.N.; Setare, M.R.; Xia, J.Q. Quintom Cosmology: Theoretical implications and observations. Phys. Rep.
**2010**, 493, 1–60. [Google Scholar] [CrossRef] - Vikman, A. Can dark energy evolve to the phantom? Phys. Rev. D
**2005**, 71, 023515. [Google Scholar] [CrossRef] - Chervon, S.V. On the chiral model of cosmological inflation. Russ. Phys. J.
**1995**, 38, 539–543. [Google Scholar] [CrossRef] - Chervon, S.V. Chiral Cosmological Models: Dark Sector Fields Description. Quantum Matter
**2013**, 2, 71–82. [Google Scholar] [CrossRef] - Socorro, J.; Pérez-Payán, S.; Hernández-Jiménez, R.; Espinoza-García, A.; Díaz-Barrón, L.R. Quintom Fields from Chiral K-essence Cosmology. Universe
**2022**, 8, 548. [Google Scholar] [CrossRef] - Armendariz-Picon, C.; Damour, T.; Mukhanov, V.F. k—Inflation. Phys. Lett. B
**1999**, 458, 209–218. [Google Scholar] [CrossRef] - Armendariz-Picon, C.; Mukhanov, V.; Steinbardt, P.J. Dynamical solution to the problem of a small cosmological constant and late-time cosmic acceleration. Phys. Rev. Lett.
**2000**, 85, 4438. [Google Scholar] [CrossRef] - Armendariz-Picon, C.; Mukhanov, V.; Steinbardt, P.J. Essentials of k-essence. Phys. Rev. D
**2001**, 63, 103510. [Google Scholar] [CrossRef] - De Felice, A.; Tsujikawa, S. f(R) theories. Living Rev. Relativ.
**2010**, 13, 3. [Google Scholar] [CrossRef] - Kaiser, D.I. Conformal transformations with multiple scalar fields. Phys. Rev. D
**2010**, 81, 084044. [Google Scholar] [CrossRef] - Birrell, N.D.; Davies, P.C.W. Quantum Fields in Curved Space; Cambridge University Press: New York, NY, USA, 1982. [Google Scholar]
- Paliathanasis, A.; Leon, G.; Pan, S. Exact Solutions in Chiral Cosmology. Gen. Relativ. Gravit.
**2019**, 51, 106. [Google Scholar] [CrossRef] - Socorro, J.; Pérez-Payán, S.; Hernández-Jiménez, R.; Espinoza-García, A.; Díaz-Barrón, L.R. Classical and quantum exact solutions for a FRW in chiral like cosmology. Class. Quant. Grav.
**2021**, 38, 135027. [Google Scholar] [CrossRef] - Paliathanasis, A. Bianchi I Spacetimes in Chiral–Quintom Theory. Universe
**2022**, 8, 503. [Google Scholar] [CrossRef] - Socorro, J.; Pérez-Payán, S.; Hernández-Jiménez, R.; Espinoza-García, A.; Díaz-Barrón, L.R. Quintom fields from chiral anisotropic cosmology. Gen. Relativ. Gravit.
**2023**, 55, 75. [Google Scholar] [CrossRef] - Ashtekar, A.; Lewandowski, J. Background independent quantum gravity: A status report. Class. Quant. Grav.
**2004**, 21, R53–R152. [Google Scholar] [CrossRef] - Thiemann, T. Introduction to Modern Canonical Quantum General Relativity; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Arnowitt, R.; Deser, S.; Misner, C.W. Republication of: The dynamics of general relativity. Gen. Relativ. Gravit.
**2008**, 40, 1997–2027. [Google Scholar] [CrossRef] - Rovelli, C. Quantum Gravity; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Ashtekar, A.; Singh, P. Loop quantum cosmology: A status report. Class. Quant. Grav.
**2011**, 28, 213001. [Google Scholar] [CrossRef] - Bojowald, M. Loop quantum cosmology. Living Rev. Relativ.
**2005**, 8, 11. [Google Scholar] [CrossRef] - Bojowald, M. Absence of singularity in loop quantum cosmology. Phys. Rev. Lett.
**2001**, 86, 5227–5230. [Google Scholar] [CrossRef] - Diener, P.; Gupt, B.; Megevand, M.; Singh, P. Numerical evolution of squeezed and non-Gaussian states in loop quantum cosmology. Class. Quant. Grav.
**2014**, 31, 165006. [Google Scholar] [CrossRef] - Ashtekar, A.; Bojowald, M.; Lewandowski, J. Mathematical structure of loop quantum cosmology. Adv. Theor. Math. Phys.
**2003**, 7, 233–268. [Google Scholar] [CrossRef] - Bojowald, M. Canonical Gravity and Applications; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Ashtekar, A.; Corichi, A.; Singh, P. Robustness of key features of loop quantum cosmology. Phys. Rev. D
**2008**, 77, 024046. [Google Scholar] [CrossRef] - Sotiriou, T.P. Covariant effective action for loop quantum cosmology from order reduction. Phys. Rev. D
**2009**, 79, 044035. [Google Scholar] [CrossRef] - Ribeiro, A.R.; Vernieri, D.; Lobo, F.S.N. Effective f(R) Actions for Modified Loop Quantum Cosmologies via Order Reduction. Universe
**2023**, 9, 181. [Google Scholar] [CrossRef] - Socorro, J.; Pimentel, L.O.; Espinoza-García, A. Classical Bianchi type I cosmology in K-essence theory. Adv. High Energy Phys.
**2014**, 2014, 805164. [Google Scholar] [CrossRef] - Corben, H.C.; Stehle, P. Classical Mechanics; Dover Publications: Mineola, NY, USA, 1994. [Google Scholar]
- Goldstein, H.; Poole, C.; Safko, J. Classical Mechanics; Addison-Wesley: Reading, MA, USA, 2002. [Google Scholar]
- Dirac, P.A.M. Lectures on Quantum Mechanics; Dover Publications: Mineola, NY, USA, 2001. [Google Scholar]
- Wipf, A. Hamilton’s Formalism for Systems with Constraints. In Canonical Gravity: From Classical to Quantum; Ehlers, J., Friedright, H., Eds.; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar]
- Christodoulakis, T.; Dimakis, N.; Terzis, P.A. Lie point and variational symmetries in minisuperspace Einstein gravity. J. Phys. A Math. Theor.
**2014**, 47, 095202. [Google Scholar] [CrossRef] - Saez, D.; Ballester, V.J. A simple coupling with cosmological implications. Phys. Lett. A
**1986**, 113, 467–470. [Google Scholar] [CrossRef] - Shi, J.; Wu, J. Dynamics of k-essence in loop quantum cosmology. Chin. Phys. C
**2021**, 45, 045104. [Google Scholar] [CrossRef] - Abraham, R.; Marsden, J.E. Foundations of Mechanics; American Mathematical Society: Providence, RI, USA, 2008. [Google Scholar]
- De Witt, B.S. Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev.
**1967**, 160, 1113. [Google Scholar] [CrossRef] - Ryan, M.P., Jr.; Turbiner, A.V. The conformally invariant Laplace–Beltrami operator and factor ordering. Phys. Lett. A
**2004**, 333, 30–34. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Course of Theoreitcal Physics. In Quantum Mechanics (Non-Relativistic Theory); Elsevier: Amsterdam, The Netherlands, 1977; Volume 3. [Google Scholar]
- Kaminski, W.; Kolanowski, M.; Lewandowski, J. Dressed metric predictions revisited. Class. Quant. Grav.
**2020**, 37, 095001. [Google Scholar] [CrossRef] - Kim, S.A.; Liddle, A.R. Nflation: Multi-field inflationary dynamics and perturbations. Phys. Rev. D
**2006**, 74, 023513. [Google Scholar] [CrossRef] - Lazkoz, R.; Leon, G.; Quiros, I. Quintom cosmologies with arbitrary potentials. Phys. Lett. B
**2007**, 649, 103–110. [Google Scholar] [CrossRef] - Leon, G.; Paliathanasis, A.; Morales-Martínez, J.L. The past and future dynamics of quintom dark energy models. Eur. Phys. J. C
**2018**, 78, 753. [Google Scholar] [CrossRef] - Dimakis, N.; Paliathanasis, A. Crossing the phantom divide line as an effect of quantum transitions. Class. Quant. Grav.
**2021**, 38, 075016. [Google Scholar] [CrossRef] - Elizalde, E.; Nojiri, S.; Odintsov, S.D.; Saez-Gomez, D.; Faraoni, V. Reconstructing the universe history, from inflation to acceleration, with phantom and canonical scalar fields. Phys. Rev. D
**2008**, 77, 106005. [Google Scholar] [CrossRef] - Christodoulidis, P.; Roest, D.; Sfakianakis, E.I. Scaling attractors in multi-field inflation. J. Cosmol. Astropart. Phys.
**2019**, 12, 59. [Google Scholar] [CrossRef] - Beesham, A.; Chervon, S.V.; Maharaj, S.D.; Kubasov, A.S. An Emergent Universe with Dark Sector Fields in a Chiral Cosmological Model. Quantum Matter
**2013**, 2, 388–395. [Google Scholar] [CrossRef] - Chervon, S.V.; Abbyazov, R.R.; Kryukov, S.V. Dynamics of Chiral Cosmological Fields in the Phantom-Canonical Model. Russ. Phys. J.
**2015**, 58, 597–605. [Google Scholar] [CrossRef] - Socorro, J.; Núñez, O.E. Scalar potentials with Multi-scalar fields from quantum cosmology and supersymmetric quantum mechanics. Eur. Phys. J. Plus
**2017**, 132, 168. [Google Scholar] [CrossRef] - Socorro, J.; Núñez, O.E.; Hernández-Jiménez, R. Classical and Quantum Exact Solutions for a FRW Multiscalar Field Cosmology with an Exponential Potential Driven Inflation. Adv. Math. Phys.
**2018**, 2018, 3468381. [Google Scholar] [CrossRef] - Wei, H.; Zhang, S.N. Dynamics of quintom and hessence energies in loop quantum cosmology. Phys. Rev. D
**2007**, 76, 063005. [Google Scholar] [CrossRef] - Barboza, L.N.; Graef, L.L.; Ramos, R.O. Warm bounce in loop quantum cosmology and the prediction for the duration of inflation. Phys. Rev. D
**2020**, 102, 103521. [Google Scholar] [CrossRef] - Tsujikawa, S. Quintessence: A review. Class. Quant. Grav.
**2013**, 30, 214003. [Google Scholar] [CrossRef] - Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. Oscillatory approach to a singular point in relativistic cosmology. Adv. Phys.
**1970**, 19, 525. [Google Scholar] [CrossRef] - Chiou, D.W. Loop quantum cosmology in Bianchi Type I Models: Analytical Investigation. Phys. Rev. D
**2007**, 75, 024029. [Google Scholar] [CrossRef] - Chiou, D.W. Effective dynamics, big bounces and scaling symmetry in Bianchi I loop quantum cosmology. Phys. Rev. D
**2007**, 76, 124037. [Google Scholar] [CrossRef] - Ashtekar, A.; Wilson-Ewing, E. Loop quantum cosmology of Bianchi type I models. Phys. Rev. D
**2009**, 79, 083535. [Google Scholar] [CrossRef] - Wilson-Ewing, E. Loop quantum cosmology of Bianchi type IX models. Phys. Rev. D
**2010**, 82, 043508. [Google Scholar] [CrossRef]

**Figure 1.**Behavior of the scalar field (Equation (55)) for different values of m. For the left top panel, $m=1$; for the right top panel, $m=0$; and for the bottom panel, $m=-1$.

**Figure 2.**Behavior of the energy density, ${\rho}_{k}$, and the volume function, V, (Equation (55)). We can immediately spot the cosmological singularity.

**Figure 3.**Behavior of the scalar field given by Equation (56).

**Figure 4.**Plot of the scalar field specified by Equation (58) for a given set of values.

**Figure 6.**Energy density for the scalar field given by Equation (72). Also, the volume function is depicted. Notice the bouncing scenario.

**Figure 7.**Overall behavior of the energy density for the scalar field given by Equation (72) (arbitrary units).

**Figure 8.**Energy density for the scalar field given by Equation (73). We have choosen B and C to be imaginary.

**Figure 9.**Generic behavior of the energy density for the scalar field given by Equation (75) (arbitrary units).

**Figure 10.**Numerical solution of the Hamilton Equations (101)–(106): (

**a**) $\beta (t)$, (

**b**) $V(t)$, (

**c**) $\varphi (t)$, (

**d**) ${p}_{\varphi}(t)$, (

**e**) $\psi (t)$, and (

**f**) ${p}_{\psi}(t)$. We use arbitrary units, namely $\lambda =1$, ${V}_{1}=0.1$, ${V}_{2}=1\times {10}^{-5}$, ${\lambda}_{1}=100$, and ${\lambda}_{2}=50$, and the initial condition $\beta (0)=V(0)=\pi /2\lambda $, $\varphi (0)={p}_{\varphi}(0)=\psi (0)={p}_{\psi}(0)=0.1$.

**Figure 11.**Numerical solution for the energy density, ${\rho}_{\mathrm{q}}(t)$ (Equation (108)). We use arbitrary units, namely $\lambda =1$, ${V}_{1}=0.1$, ${V}_{2}=1\times {10}^{-5}$, ${\lambda}_{1}=100$, and ${\lambda}_{2}=50$, and the initial condition $\beta (0)=V(0)=\pi /2\lambda $, $\varphi (0)={p}_{\varphi}(0)=\psi (0)={p}_{\psi}(0)=0.1$.

**Figure 12.**Numerical solution for the equation-of-state variable, ${\omega}_{\mathrm{q}}(t)$, (Equation (112)). We use arbitrary units, namely $\lambda =1$, ${V}_{1}=0.1$, ${V}_{2}=1\times {10}^{-5}$, ${\lambda}_{1}=100$, and ${\lambda}_{2}=50$, and the initial condition $\beta (0)=V(0)=\pi /2\lambda $, $\varphi (0)={p}_{\varphi}(0)=\psi (0)={p}_{\psi}(0)=0.1$.

**Figure 13.**Numerical solution for the e-folds, $N(t)=lnV{(t)}^{1/3}$. We use arbitrary units, namely $\lambda =1$, ${V}_{1}=0.1$, ${V}_{2}=1\times {10}^{-5}$, ${\lambda}_{1}=100$, and ${\lambda}_{2}=50$, and the initial condition $\beta (0)=V(0)=\pi /2\lambda $, $\varphi (0)={p}_{\varphi}(0)=\psi (0)={p}_{\psi}(0)=0.1$.

**Figure 14.**Numerical solution of the Hamilton Equations (114)–(119): (

**a**) $\beta (t)$, (

**b**) $v(t)$, (

**c**) $\varphi (t)$, (

**d**) ${p}_{\varphi}(t)$, (

**e**) $\psi (t)$, and (

**f**) ${p}_{\psi}(t)$. The bounce is clearly noticeable in the behavior of the volume function. It was set to take place at ${t}_{c}=0$. We use arbitrary units, namely $\lambda =1$, ${V}_{1}=0.1$, ${V}_{2}=1\times {10}^{-5}$, ${\lambda}_{1}=100$, and ${\lambda}_{2}=50$, and the initial condition $\beta (0)=V(0)=\pi /2\lambda $, $\varphi (0)={p}_{\varphi}(0)=\psi (0)={p}_{\psi}(0)=0.1$.

**Figure 15.**Numerical solution for (

**a**) energy density, ${\rho}_{\mathrm{q}}^{\mathrm{eff}}(t)$ (Equation (121)), and (

**b**) the equation-of-state parameter, ${\omega}_{\mathrm{q}}^{\mathrm{eff}}$ (Equation (126)). We use arbitrary units, namely $\lambda =1$, ${V}_{1}=0.1$, ${V}_{2}=1\times {10}^{-5}$, ${\lambda}_{1}=100$, and ${\lambda}_{2}=50$, and the initial condition $\beta (0)=V(0)=\pi /2\lambda $, $\varphi (0)={p}_{\varphi}(0)=\psi (0)={p}_{\psi}(0)=0.1$. Observe that the energy density function is bounded, with its maximum value being achieved precisely at the bounce.

**Figure 16.**Numerical solution of the Hamilton Equations (114)–(119): (

**a**) $\beta (t)$, (

**b**) $v(t)$, (

**c**) $\varphi (t)$, (

**d**) ${p}_{\varphi}(t)$, (

**e**) $\psi (t)$, and (

**f**) ${p}_{\psi}(t)$. The bounce is clearly noticeable in the behavior of the volume function. It was set to take place at ${t}_{c}=0$. We use arbitrary units, namely $\lambda =1$, ${V}_{1}=-0.1$, ${V}_{2}=-1\times {10}^{-5}$, ${\lambda}_{1}=100$, and ${\lambda}_{2}=50$, and the initial condition $\beta (0)=V(0)=\pi /2\lambda $, $\varphi (0)={p}_{\varphi}(0)=\psi (0)={p}_{\psi}(0)=0.1$.

**Figure 17.**Numerical solution for (

**a**) energy density, ${\rho}_{\mathrm{q}}^{\mathrm{eff}}(t)$ (Equation (121)), and (

**b**) the equation-of-state parameter, ${\omega}_{\mathrm{q}}^{\mathrm{eff}}$ (Equation (126)). We use arbitrary units, namely $\lambda =1$, ${V}_{1}=-0.1$, ${V}_{2}=-1\times {10}^{-5}$, ${\lambda}_{1}=100$, and ${\lambda}_{2}=50$, and the initial condition $\beta (0)=V(0)=\pi /2\lambda $, $\varphi (0)={p}_{\varphi}(0)=\psi (0)={p}_{\psi}(0)=0.1$. Observe that the energy density function is bounded, with its maximum value being achieved precisely at the bounce.

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**MDPI and ACS Style**

Díaz-Barrón, L.R.; Espinoza-García, A.; Pérez-Payán, S.A.; Socorro, J.
A Loop Quantum-Corrected Family of Chiral Cosmology Models. *Universe* **2024**, *10*, 88.
https://doi.org/10.3390/universe10020088

**AMA Style**

Díaz-Barrón LR, Espinoza-García A, Pérez-Payán SA, Socorro J.
A Loop Quantum-Corrected Family of Chiral Cosmology Models. *Universe*. 2024; 10(2):88.
https://doi.org/10.3390/universe10020088

**Chicago/Turabian Style**

Díaz-Barrón, Luis Rey, Abraham Espinoza-García, Sinuhé Alejandro Pérez-Payán, and J. Socorro.
2024. "A Loop Quantum-Corrected Family of Chiral Cosmology Models" *Universe* 10, no. 2: 88.
https://doi.org/10.3390/universe10020088