# Effective Potential for Quintessential Inflation Driven by Extrinsic Gravity

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## Abstract

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`DYNESTY`) code. By means of the Nash–Greene embedding theorem, we show that the corresponding model provides an effective potential driven by the influence of extrinsic geometry. We obtain a quintessential inflation that defines a model with a potential $V\left(\varphi \right)={e}^{-{\alpha}_{1}\varphi}(1-{\alpha}_{2}{\varphi}^{2})$, where ${\alpha}_{1}$ and ${\alpha}_{2}$ are dimensionless parameters. Using some known phenomenological parameterizations, such as Chevallier–Polarski–Linder (CPL) and Barboza–Alcaniz (BA) parameterizations, we show that the model reflects a slow-varying inflation preferring a thawing behavior, suggesting an optimistic scenario for further research on the unification of inflation with late cosmic acceleration.

## 1. Introduction

`DYNESTY`[42,43] Python code, we constrain the parameters of our toy model. To analyze how the related equation of state (EoS) evolves, we use Chevallier–Polarski–Linder (CPL) [44,45] and Barboza–Alcaniz (BA) [46] parameterizations to distinguish the present model as a thawing [47,48,49] or freezing [50,51,52,53] pattern. Thawing models are conceived to be those that the fluid parameter as a function of the redshift $w\left(z\right)$ in EoS is moving away from $-1$, which means that DE density decreases over time, gradually allowing the universe to accelerate more rapidly. On the other hand, in freezing models, DE density remains approximately constant as the universe expands, with $w\left(z\right)$ approaching the value $-1$. Finally, the conclusion and prospects are presented in the Section 6.

## 2. Essentials on Embeddings

## 3. Friedmann–Lemaître–Robertson–Walker (FLRW) Embedded Cosmology

## 4. The Fluid Analogy

## 5. Extrinsic Curvature as an Effective Inflaton Field

#### 5.1. The Extrinsic Inflaton Field

#### 5.2. Numerical Parameter Estimation

`DYNESTY`[42,43] Python code for estimating Bayesian posteriors and evidences, we adopt priors for the “true model” set as $\{({\alpha}_{1\left(true\right)},{\alpha}_{2\left(true\right)},{f}_{\left(true\right)}=(4,2,1.7)\}$, where the quantity f is defined as a mechanism to control the variance in the resulting Gaussian distributions. To estimate the values of the parameters ${\alpha}_{1}$ and ${\alpha}_{2}$ on how they can approximate to the “true” values, we define a prior transform ${u}_{\theta}=({u}_{af1},{u}_{af2},{u}_{f})$ as

`DYNESTY`dynamically allocates points in the runs. Each point also carries information of their covariances. It allows for creating a variable number of ${K}_{i}$ live points at each iteration i to realize a change in a prior volume ${X}_{i}$. From

`Numpy`5000 random seeds and the

`celerite`[69] library for a scalable Gaussian process, we obtained a total of 144,659 interactions in the final of the runs. We have that the prior volume and evidence are controlled when the variation $\Delta ln{X}_{i}\approx {K}_{i}$, as shown in Figure 2.

#### 5.3. Dynamical System and Comparison with the Potentials

## 6. Remarks

`DYNESTY`code. Establishing a quintessential potential from a reference model of Equation (44) used in Ref. [68], we have obtained the related autonomous system of equations for forecasting models compatible with the universe evolution. Moreover, we have shown a comparison with CPL and BA parameterizations of our model that prefers a thawing cosmic scenario.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Comparison between the potentials from Equation (44) (blue line) and Equation (43) (black line) that indicates a close overall pattern of such potentials. The blue line was produced by adopted parameters $\{({\beta}_{1},{\beta}_{2})=(5,1.2)\}$, as shown in Ref. [68]. The black line was produced by the adopted priors that define our “true” model with the values $\{({\alpha}_{1},{\alpha}_{2})=(4,2)\}$.

**Figure 2.**In the left panels, we have the control of the prior volume $ln{X}_{i}$ of random points that shrinks exponentially over time to determine the evidence. The right panels show the posteriors (Gaussian distribution of the parameters). In both sets of panels, the red lines denote the priors (“true” values) adopted in the code.

**Figure 3.**Contour plot for the marginalized posteriors for $({\alpha}_{1},{\alpha}_{2},lnf)$ parameters at $10\%$, $40\%$, $65\%$, and $85\%$ confidence levels. Vertical dashed lines mark the $2\sigma $ region, while horizontal lines indicate the mean values of the marginalized parameters.

**Figure 4.**Comparison between the potentials from Equation (44) (black lines) and Equation (43) (red lines) that shows the evolution of the related autonomous systems. The outer continuous blue line and the dashed blue line semicircles represent the x-axis with $x\sim \sqrt{{\Omega}_{ext}}\sim 0.82$ and ${\Omega}_{ext}={\Omega}_{\varphi}=0.68$, respectively.

**Figure 5.**Behavior of numerical curves (dotted line) produced from the related autonomous system of the model preferring a thawing pattern of CPL (blue line) and BA (red line) parameterizations.

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**MDPI and ACS Style**

Capistrano, A.J.S.; Cabral, L.A.
Effective Potential for Quintessential Inflation Driven by Extrinsic Gravity. *Universe* **2023**, *9*, 497.
https://doi.org/10.3390/universe9120497

**AMA Style**

Capistrano AJS, Cabral LA.
Effective Potential for Quintessential Inflation Driven by Extrinsic Gravity. *Universe*. 2023; 9(12):497.
https://doi.org/10.3390/universe9120497

**Chicago/Turabian Style**

Capistrano, Abraão J. S., and Luís Antonio Cabral.
2023. "Effective Potential for Quintessential Inflation Driven by Extrinsic Gravity" *Universe* 9, no. 12: 497.
https://doi.org/10.3390/universe9120497