Effective Potential for Quintessential Inflation Driven by Extrinsic Gravity

Abstract

We numerically study the evolution of the extrinsic energy density in the context of an inflationary regime at the background level in a five-dimensional model using a Bayesian analysis from a dynamic nested sampler (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

Cosmic inflation is one of the most fantastic mechanisms for understanding the evolution of the early universe, which underwent a rapid and exponential increase in the cosmic scale factor shortly after the Big Bang. It proposes a simple solution to the horizon problem by suggesting that far-reaching regions of the universe were in close proximity before inflation occurred. Then, it explains why the universe appears homogeneous and isotropic on large scales. Moreover, the so-called flatness problem that states, Why does the universe seem to have a nearly flat geometry? is solved in the context of inflation since it naturally leads to a flat universe [1]. It also provides an explanation of the formation of a large-scale structure in the universe, including galaxies and galaxy clusters, by quantum fluctuations in a peculiar scalar field dubbed as inflaton field during that cosmic inflationary period [2,3,4,5,6,7,8,9].

The nature of the inflaton field motivated the elaboration of a plethora of competing cosmological models [10,11,12,13,14,15,16,17,18,19] due to the fact that the form of the inflation field is not determined and the standard phenomenological model $\Lambda$ cold dark matter ($\Lambda$CDM) with the fluid parameter $w=-1$ rapidly decays at the end of the inflationary period [20]. Thus, the inflaton field cannot be directly regarded as a cosmological constant $\Lambda$ for the late acceleration of the universe. An immediate issue arises whether the inflaton field should be some form of a dark energy (DE) fluid: how does the equation of the state parameter w significantly deviate from $w=-1$ during the inflationary era? In this paper, we make the first steps to answer this question in the context of a modification of gravity by extra dimensions. Apart from traditional models, such as the Arkani-Hamed–Dvali–Dimopoulos (ADD) model [21], the Randall–Sundrum model [22,23], and/or the Dvali–Gabadadze–Porrati model (DPG) [24], we explore the dynamical embedding of space-times as the main mathematical structure [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. In Encyclopaedia Inflationaris [41], the reader can find various types and classifications of inflationary models, such as supergravity brane inflation, supersymmetry (SUSY), and brane inflation models.

The paper is organized into sections. The Section 2 and Section 3 summarize the essentials of previously published results on the construction of the four-dimensional embedded model and cosmological applications [26,27,28,31,39,40]. The Section 4 presents the fluid analogy necessary to make a comparison with phenomenological models. Moreover, we construct a model wherein the extrinsic curvature is thought as an inflaton field as shown in the Section 5. By means of numerical analysis generating synthetic random points using the 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

We define a model endowed with a gravitational action S in the presence of confined matter fields wherein a four-dimensional embedded space-time is embedded in five dimensions as

$$S=-\frac{1}{2{\kappa }_5^2}\int \sqrt{\left|\mathcal{G}\right|}\left({}^5\mathcal{R}+{\mathcal{L}}_m^{*}\right)d^{5}x,$$

(1)

where ${\kappa }_5^2$ is a fundamental energy scale on the embedded space, and the curly ${}^5\mathcal{R}$ denotes the five-dimensional Ricci scalar of the bulk given by the summation of the four-dimensional Ricci scalar R and the deformation scalar Q and has the simple form ${}^5\mathcal{R}=R+Q$. Moreover, ${\mathcal{L}}_m^{*}$ denotes the bulk source Lagrangian in five dimensions. Since gravity should only propagate in the bulk, ${\mathcal{L}}_m^{*}$ is localized in the four-dimensional embedded space and obeys the confinement condition

$$\begin{array}{c}\hfill {\kappa }_5^2{T}_{\mu \nu }^{*}=-8\pi G{T}_{\mu \nu },\\ \hfill {\kappa }_5^2{T}_{\mu a}^{*}=0,\\ \hfill {\kappa }_5^2{T}_{ab}^{*}=0,\end{array}$$

(2)

from the components of ${\mathcal{T}}_{AB}$ of the energy–momentum tensor of the bulk. The terms of $T_{\mu \nu }^{*}$, $T_{\mu a}^{*}$, and $T_{ab}^{*}$ denote the energy–momentum of the tangent (tensor), vector, and scalar components of ${\mathcal{T}}_{AB}$. Concerning notation, capital Latin indices are fixed to 5 to reinforce the notation of a five-dimensional space. Small-case Latin indices refer to the only one extra dimension fixed to 1. All Greek indices refer to the embedded space-time running from 1 to 4. Ordinary matter and gauge fields are represented by $T_{\mu \nu }$, and G is the gravitational Newtonian constant. From the set of equations in Equation (2), ${\mathcal{L}}_m^{*}$ reduces to the confined four-dimensional Lagrangian ${\mathcal{L}}_m$ related to $T_{\mu \nu }$ as

$$T_{\mu \nu }=-\frac{2}{\sqrt{\left|g\right|}}\frac{\delta \left(\sqrt{\left|g\right|}{\mathcal{L}}_m\right)}{\delta g^{\mu \nu }},$$

(3)

where $g_{\mu \nu }$ is the metric and $\left|g\right|$ is the absolute value of $g_{\mu \nu }$. Detailed demonstrations of the fundamental field equations, from a bulk of five or in arbitrary dimensions, can be found in Refs. [26,27,28,31,40]. In this work, in search of constructing a viable physical model, we consider the extrinsic curvature as a main character of cosmic dynamics. Mathematically speaking, besides the metric $g_{\mu \nu }$, such new curvature is a pivotal element in the embedding of geometries. As traditionally defined in [54], the nonperturbed extrinsic curvature $k_{\mu \nu }$ reads,

$$k_{\mu \nu }=-{\mathcal{X}}_{,\mu }^A{\eta }_{,\nu }^B{\mathcal{G}}_{AB}.$$

(4)

The definition in Equation (4) essentially means a measure of bending of the embedded geometry; i.e., it reflects how a normal unitary vector ${\eta }^A$ in the embedded space deviates from the tangent plane. Let the coordinate $\mathcal{X}$ define a regular map $\mathcal{X}:V_4\to V_5$. Such a local map states that a Riemannian manifold $V_4$ is isometrically embedded in a larger five-dimensional Riemannian bulk space $V_5$. The term ${\mathcal{G}}_{AB}$ is the bulk metric

$$\begin{array}{ccc}\hfill {\mathcal{G}}_{AB}& =& \left(\begin{array}{cc}{g}_{\mu \nu }& 0\\ 0& 1\end{array}\right).\hfill \end{array}$$

(5)

Another pivotal relation is Nash’s deformation formula given by

$$k_{\mu \nu }=-\frac{1}{2}\frac{\partial g_{\mu \nu }}{\partial y},$$

(6)

where y is an arbitrary spatial coordinate. Nash’s deformation formula is the pivot result of Nash’s embedding theorem of 1954 applied to non-Euclidean metrics [55]. In 1956, Nash generalized his former results to Riemannian metrics [56]. Decades later, Nash’s theorem was expanded to non-Riemannian metrics by Greene [57]. From a physical point of view, Equation (6) localizes gravity closer to the four-dimensional embedded space, imposing a strong bending (warping) geometrical constraint.

Nash’s results in Equation (6) is the capital element to obtain a solution to well-known Gauss and Codazzi equations given by

$$\begin{array}{c}{}^5{\mathcal{R}}_{ABCD}{\mathcal{Z}}^A{}_{,\alpha }{\mathcal{Z}}^B{}_{,\beta }{\mathcal{Z}}^C{}_{,\gamma }{\mathcal{Z}}^D{}_{,\delta }=R_{\alpha \beta \gamma \delta }+(k_{\alpha \gamma }{k}_{\beta \delta }-k_{\alpha \delta }{k}_{\beta \gamma }),\hfill \end{array}$$

(7)

$$\begin{array}{c}{}^5{\mathcal{R}}_{ABCD}{\mathcal{Z}}^A{}_{,\alpha }{\mathcal{Z}}^B{}_{,\beta }{\mathcal{Z}}^C{}_{,\gamma }{\eta }^D=k_{\alpha [\beta ;\gamma ]},\hfill & \end{array}$$

(8)

where ${}^5{\mathcal{R}}_{ABCD}$ denotes the five-dimensional Riemann tensor, and $R_{\alpha \beta \gamma \delta }$ is the four-dimensional Riemann tensor. The coordinate ${\mathcal{Z}}_{,\mu }^A$ is a perturbed coordinate in the sense that ${\mathcal{Z}}_{,\mu }^A={\mathcal{X}}_{,\mu }^A+\delta y{\eta }_{,\mu }^A$. The semicolon sign in Equation (8) represents the ordinary covariant derivative with respect to the metric $g_{\mu \nu }$ and $k_{\alpha [\beta ;\gamma ]}\equiv k_{\alpha \beta ;\gamma }-k_{\alpha \gamma ;\beta }$. The importance of Equations (7) and (8) is that they reflect the integrability conditions for the embedding explicitly relating the bulk and embedded space, and they are the starting point to obtain the induced field equations [26,27,28,31,40].

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

From integration of Equations (7) and (8), one obtains the induced four-dimensional nonperturbed field equations as

$$\begin{array}{ccc}\hfill G_{\mu \nu }-Q_{\mu \nu }& =& -8\pi G{T}_{\mu \nu },\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill k_{\mu [\nu ;\rho ]}& =& 0,\hfill \end{array}$$

(10)

where $T_{\mu \nu }$ is the energy–momentum tensor of the confined sources, G is the gravitational Newtonian constant, and $G_{\mu \nu }$ is the four-dimensional Einstein tensor. The deformation tensor $Q_{\mu \nu }$ is given by

$$Q_{\mu \nu }=k_{\mu }^{\rho }{k}_{\rho \nu }-k_{\mu \nu }h-\frac{1}{2}\left(K^2-h^2\right)g_{\mu \nu },$$

(11)

where the mean curvature is given by $h^2=h·h$ and $h=g^{\mu \nu }{k}_{\mu \nu }$. The quantity $K^2=k^{\mu \nu }{k}_{\mu \nu }$ is the Gaussian curvature. By direct derivation, Equation (11) is conserved in the sense that

$$Q_{\mu \nu ;\mu }=0.$$

(12)

Once the field equations of Equations (9) and (10) are set, we obtain the background cosmic evolution from the usual four-dimensional line element of the FLRW metric given by

$$d{s}^2=a^2\left(d{r}^2+r^{2}d{\theta }^2+r^2{sin}^2\theta d{\varphi }^2\right)-d{t}^2,$$

(13)

where $a\equiv a\left(t\right)$ is the scale factor and t is the physical time. As shown in previous works [27,31], the solutions of Equations (9) and (10) are summarized as

$$\begin{array}{cc}\hfill & k_{ij}=\frac{b}{a^2}{g}_{ij}\left(i,j=1,2,3\right),\hfill \end{array}$$

(14)

$$\begin{array}{c}{k}_{44}=\frac{-1}{\dot{a}}\frac{d}{dt}\frac{b}{a},\hfill \end{array}$$

(15)

$$\begin{array}{c}{k}_{44}=-\frac{b}{a^2}\left(\frac{B}{H}-1\right),\hfill \end{array}$$

(16)

$$\begin{array}{c}{K}^2=\frac{b^2}{a^4}\left(\frac{B^2}{H^2}-2\frac{B}{H}+4\right),h=\frac{b}{a^2}\left(\frac{B}{H}+2\right),\hfill \end{array}$$

(17)

$$\begin{array}{c}{Q}_{ij}=\frac{b^2}{a^4}\left(2\frac{B}{H}-1\right)g_{ij},Q_{44}=-\frac{3b^2}{a^4},\hfill \end{array}$$

(18)

$$\begin{array}{c}Q=-(K^2-h^2)=\frac{6b^2}{a^4}\frac{B}{H},\hfill \end{array}$$

(19)

where Q is the deformation scalar given by the contraction $g^{\mu \nu }{Q}_{\mu \nu }=Q$, and $H\equiv H\left(t\right)=\frac{\dot{a}}{a}$ is the Hubble parameter. Defined as an “extrinsic” copy of the Hubble parameter H, we have the cosmic bending parameter $B=B\left(t\right)\equiv \frac{\dot{b}}{b}$. The obtainment of these results is shown in detail in Refs. [27,31]. The warping or bending function $b\left(t\right)$ is given by

$$b\left(t\right)=\frac{b_0}{a_0^{{\beta }_0}}a{\left(t\right)}^{{\beta }_0},$$

(20)

where the parameter $b_0$ denotes the current value of $b\left(t\right)$ and $a_0$ is the current value of the scale factor.

4. The Fluid Analogy

We obtain the hydrodynamical equations for a perfect fluid in comoving coordinates as the same way as in the usual general relativity (GR) framework. Thus, the stress energy tensor $T_{\mu \nu }$ has the standard form

$$T_{\mu \nu }=\left(\rho +p\right)u_{\mu }{u}_{\nu }+p{g}_{\mu \nu };u_{\mu }={\delta }_{\mu }^4,$$

(21)

and the conservation equation

$$\rho +3H\left(\rho +p\right)=0,$$

(22)

from $T_{\mu \nu ;\mu }=0$. Hence, one obtains the resulting Friedmann equation as

$$H^2=\frac{8}{3}\pi G\rho +\frac{b^2}{a^4},$$

(23)

where $\rho \equiv {\rho }_m\left(t\right)$ is the nonperturbed matter density. Using Equations (20) and (23), we have

$$H^2=\frac{8}{3}\pi G{\rho }_{m\left(0\right)}{a}^{-3}+\frac{b_0^2}{a_0^{2{\beta }_0}}{a}^{2{\beta }_0-4},$$

(24)

where ${\rho }_{m\left(0\right)}$ is the current value of matter energy density. With respect to the cosmological parameters ${\Omega }_{\mathbf{i}}=\frac{8\pi G}{3H_0^2}{\rho }_{i\left(0\right)}$, $\mathbf{i}$ identifies the density species, and considering only matter and extrinsic species, we write the Hubble evolution $H\left(z\right)$ simply as

$$H\left(z\right)=H_0\sqrt{{\Omega }_m\left(z\right)+{\Omega }_{ext}\left(z\right)},$$

(25)

where $H\left(z\right)$ is the Hubble parameter in terms of the redshift $z=\frac{1-a}{a}$. The matter density parameter is denoted by ${\Omega }_m\left(z\right)={\Omega }_{m\left(0\right)}{(1+z)}^3$. The term ${\Omega }_{ext}\left(z\right)={\Omega }_{ext\left(0\right)}{(1+z)}^{4-2{\beta }_0}$ stands for the density parameter associated with the extrinsic curvature. The cosmological parameters ${\Omega }_{m\left(0\right)}$ and ${\Omega }_{ext\left(0\right)}=b_0^2/H_0^2{a}_0^{2{\beta }_0}$ denote the present value of the matter and extrinsic contributions, respectively. The current warp (bending) of the universe denoted by $b_0$ has the same dimension as $H_0$ that is the current value of a Hubble constant in units of km·s${}^{-1}$ Mpc${}^{-1}$. If ${\Omega }_{ext}\left(z\right)$ vanishes with $b_0\to 0$, one obtains GR limit as a background with the recovery of Einstein equations. For a flat universe, we have ${\Omega }_{ext\left(0\right)}=1-{\Omega }_{m\left(0\right)}$.

To obtain an effective cosmological model in order to make tests with real data, an effective “extrinsic fluid parameter” $w_{ext}$ must be set. We define a “fluid analogy” by means of an effective equation of state (EoS) as

$$w_{ext}=-1+\frac{1}{3}\left(4-2{\beta }_0\right),$$

(26)

in which the equality $w_{ext}=w$ happens if ${\beta }_0=\frac{1}{2}\left(1-3w\right)$, where the dimensionless fluid parameter w defines an EoS related to pressure p and energy density $\rho$ as $w=\frac{p}{\rho }$. Thus, one defines the dimensionless Hubble parameter $E\left(z\right)\equiv \frac{H}{H_0}$ that reads

$$E^2\left(z\right)={\Omega }_{m\left(0\right)}{(1+z)}^3+{\Omega }_{ext\left(0\right)}{(1+z)}^{3(1+w)},$$

(27)

which resembles an XCDM fluid [58].

5. Extrinsic Curvature as an Effective Inflaton Field

Besides Equations (26) and (27), we can explore the effective fluid approach rewriting Equation (9) in the form

$$R_{\mu \nu }-\frac{1}{2}R{\overline{g}}_{\mu \nu }=-8\pi G{\overline{T}}_{total},$$

(28)

where ${\overline{T}}_{\mu \nu }^{total}={\overline{T}}_{\mu \nu }+\frac{1}{8\pi G}{\overline{T}}_{\mu \nu }^{ext}$. The quantity ${\overline{T}}_{\mu \nu }^{ext}\doteq Q_{\mu \nu }$ denotes the extrinsic contribution. Mimicking Equation (21), we write ${\overline{T}}_{\mu \nu }^{ext}$ as

$${\overline{T}}_{\mu \nu }^{ext}=({\overline{p}}_{ext}+{\overline{\rho }}_{ext})u_{\mu }{u}_{\nu }+{\overline{p}}_{ext}{\overline{g}}_{\mu \nu },u_{\mu }={\delta }_{\mu }^4,$$

(29)

where $u_{\mu }$ is the comoving four velocities. It is worth noting that the deformation tensor $Q_{\mu \nu }$ is independently conserved as shown in Equation (12), so is ${\overline{T}}_{\mu \nu }^{ext}$. We stress that the dynamical embedding naturally warrants the influence of the bulk over the embedded space, such as quantities $Q_{\mu \nu }$ with a mixture of intrinsic and extrinsic terms, as shown in Equation (11). As a consequence, ${\overline{T}}^{ext}$ is conserved unless an exotic matter-energy source is added. This result differs from a DE fluid in standard rigid brane-world models [22,23,24], where ${\overline{T}}^{ext}$ is not conserved due to the necessity of the bulk–brane energy exchange.

From the conservation of Equation (29), we obtain

$$\frac{d{\overline{\rho }}_{ext}}{dt}+3H\left({\overline{\rho }}_{ext}+{\overline{p}}_{ext}\right)=0.$$

(30)

In a perfect fluid analogy with Equation (22), ${\overline{\rho }}_{ext}$ denotes the nonperturbed “extrinsic” energy density given by

$${\overline{\rho }}_{ext}\left(a\right)={\rho }_{ext\left(0\right)}{a}^{-3(1+w)},$$

(31)

where the current “extrinsic” energy density is ${\rho }_{ext\left(0\right)}=\frac{3b_0^2}{8\pi G{a}_0^{1-3w}}$. Therefore, the “extrinsic” pressure ${\overline{p}}_{ext}$ is straightforwardly calculated by Equation (30) to obtain

$${\overline{p}}_{ext}\left(a\right)=w{\rho }_{ext\left(0\right)}{a}^{-3(1+w)}.$$

(32)

When $b_0\to 0$, the extrinsic curvature ceases, and ${\overline{T}}_{\mu \nu }^{ext}$, ${\overline{\rho }}_{ext}\left(a\right)$, and ${\overline{p}}_{ext}\left(a\right)$ vanish, and the GR limit is reached. Alternatively, one writes the related Friedman equation in the form

$$H^2=\frac{8}{3}\pi G\left({\overline{\rho }}_m+{\overline{\rho }}_{ext}\right).$$

(33)

5.1. The Extrinsic Inflaton Field

Since ${\overline{\rho }}_{ext}$ plays a role in driving the late accelerated expansion as explored in [27,31], the same energy density ${\overline{\rho }}_{ext}$ should have the dynamics of a scalar field potential energy density $V\left(\varphi \right)$ during early inflationary periods generated by the “extrinsic scalar” field $\varphi$. Hence, the related Lagrangian ${\mathcal{L}}_{\varphi }$ is defined as

$${\mathcal{L}}_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2-V\left(\varphi \right).$$

(34)

The form written in Equation (34) shows that the field $\varphi$ is spatially homogeneous; then it only depends on the cosmic time t. In addition, the related energy momentum of such a field is written as

$$T_{\mu \nu }^{\varphi }={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +g_{\mu \nu }\left(\frac{1}{2}{\dot{\varphi }}^2-V\left(\varphi \right)\right).$$

(35)

From the conservation of Equation (35), i.e., $T_{\mu \nu ;\mu }^{\varphi }=0$, one obtains the correspondence relations

$$\begin{array}{ccc}& & {\overline{\rho }}_{ext}={\overline{\rho }}_{\varphi }=\frac{{\dot{\varphi }}^2}{2}+V\left(\varphi \right),\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& & {\overline{p}}_{ext}={\overline{p}}_{\varphi }=\frac{{\dot{\varphi }}^2}{2}-V\left(\varphi \right),\hfill \end{array}$$

(37)

where the dot symbol denotes an ordinary derivative with respect to the cosmic time t. The forms of Equations (36) and (37) couple with the inflation dynamics given by $V\left(\varphi \right)$ to the background evolution. Then, the quantities ${\overline{\rho }}_{ext}$ and ${\overline{p}}_{ext}$ can be written as a function of the extrinsic scalar field $\varphi$.

In terms of fluid analogy given by Equation (26) and the direct sum of Equations (36) and (37), and using Equations (31) and (32), one obtains

$$\dot{\varphi }\left(a\right)=\sqrt{{\rho }_{ext\left(0\right)}(1+w)}{a}^{-\frac{3}{2}(1+w)}.$$

(38)

Using the relation $\dot{\varphi }=aH\frac{\varphi }{da}$ and the inflation condition $H^2\left(a\right)\sim \frac{8\pi G}{3}{\rho }_{ext}\left(a\right)$, we integrate Equation (38) to obtain the potential $\varphi \left(a\right)$ as

$$\varphi \left(a\right)=\sqrt{|\frac{3(1+w)}{8\pi }|}{M}_{pl}lna,$$

(39)

where we denote the Planck mass as $M_{pl}=\frac{1}{\sqrt{G}}$. We consider the initial conditions ${\varphi }_{ini}=0$ and $a_{ini}=1$. Putting the solution of Equation (39) in Equation (36), one obtains a potential $V\left(\varphi \right)$ in the form

$$V\left(\varphi \right)=V_0{e}^{-{\alpha }_0\varphi /M_{pl}}.$$

(40)

For the sake of notation, we denote the quantity ${\alpha }_0$ in Equation (40) as ${\alpha }_0=\sqrt{\left|24\pi \right(1+w\left)\right|}$. The models of such exponential form produce a power law inflation (PLI), first introduced in Refs. [59,60,61]. For the time being, this type of potential has strong constraints imposed by Planck data [62] and is ruled out at more than 2-$\sigma$ confidence regions. Moreover, PLI inflation requires a graceful exit in the slow-roll approximation. In such a model, the first slow-roll Hubble parameter is constant that leads to an eternal inflation [41]. Such potential was also studied in the context of M-theory [63] and RSII scenarios [64]. Due to the strong constraints on PLI, based on Equation (40), our numerical study will take an effective quintessential dimensionless potential $V{\left(\varphi \right)}_{eff}$ to generalize Equation (40), such as

$$V{\left(\varphi \right)}_{eff}=V_{PLI}\left(\varphi \right)+V_{LMI}\left(\varphi \right),$$

(41)

where $V_{PLI}\left(\varphi \right)$ denotes our PLI potential given by Equation (40). The quantity $V_{LMI}\left(\varphi \right)$ denotes a phenomenological potential that defines the so-called logamediate inflation (LMI) [65,66] in the form

$$V_{LMI}\left(\varphi \right)=V_0{\gamma }_3{\left(\frac{\varphi }{M_{pl}}\right)}^{{\gamma }_2}{e}^{-{\gamma }_1{\left(\frac{\varphi }{M_{pl}}\right)}^{{\gamma }_4}},$$

(42)

in which $({\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4)$ are free dimensionless parameters. For our purposes, we need to impose on Equation (40) $V_0={\tilde{V}}_0=1$, ${\alpha }_0={\alpha }_1$, and $\varphi =\tilde{\varphi }{M}_{pl}$. A similar adaptation is applied to Equation (42) with $V_0={\tilde{V}}_0=1$, ${\gamma }_1={\alpha }_1$, ${\gamma }_2=2$, ${\gamma }_3=-\left|{\alpha }_2\right|$$({\alpha }_2>0)$, ${\gamma }_4=1$, and $\varphi =\tilde{\varphi }{M}_{pl}$. Hence, the resulting potential $V{\left(\varphi \right)}_{eff}$ is given by

$$V{\left(\varphi \right)}_{eff}={\tilde{V}}_0{e}^{-{\alpha }_1\tilde{\varphi }}(1-{\alpha }_2{\tilde{\varphi }}^2).$$

(43)

We point out that $({\alpha }_1,{\alpha }_2)$ are dimensionless parameters. Assuming that the field should roll in a direction wherein $\frac{\partial V{\left(\varphi \right)}_{eff}}{\partial \varphi }=0$, starting at $\varphi ={\varphi }_{min}=1$, one obtains the condition ${\alpha }_2=\left|\frac{{\alpha }_1}{2-{\alpha }_1}\right|>0$. For our purposes, the positivity of ${\alpha }_2$ warrants a logistic equation set in the domain of real numbers. Therefore, the model turns again as being a one-parameter model from an extrinsic origin. This guarantees a reheating phase without relying on any approximation in a future work. It is worth noting that when an extrinsic curvature vanishes ${\alpha }_1={\alpha }_0=0$, then ${\alpha }_2=0$, and inflation ceases.

To study the evolution of the effective fluid parameter w and the characteristics of the potential $V{\left(\varphi \right)}_{eff}$, we adopt a dynamical system approach as in the works of Clemson and Liddle [67] and Pantazis, Nesseris, and Perivolaropoulos [68]. In the latter, they use a quintessence potential of the form

$$V\left(\varphi \right)=e^{-{\beta }_1\varphi }(1+{\beta }_2\varphi ),$$

(44)

where ${\beta }_1$ and ${\beta }_2$ are free dimensionless parameters with ${\beta }_1>{\beta }_2$. For comparison purposes, such potential will be also taken into account as a reference in contrast with Equation (43).

5.2. Numerical Parameter Estimation

In order to obtain the parameter estimation of our toy model by the potential in Equation (43), we define a “true” model that produces a close curve pattern with the adopted priors to the reference model in Equation (44).

As shown in Figure 1, we have a comparison between the potentials from Equation (44) (blue line) and our model defined by Equation (43) (black line). They present a similar pattern with the same peak height, and the latter model presents a narrower basis of its curve (black line). Both cases positively respond to a remote future passing through an accelerated phase.

In order to use the 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

$$\begin{array}{ccc}\hfill {\alpha }_1& =& 5u_{af1}+0.5;\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill {\alpha }_2& =& 2.5u_{af2}+0.5;\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill lnf& =& 2.8u_f-1.\hfill \end{array}$$

(47)

We start generating n = 30000 synthetic random points and 8000 live points for better efficiency. On the other hand, under the circumstances we have defined, during the runs, we noted that generating larger random points does not change the results. To estimate both evidence and posteriors, 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.

In Figure 3, we show the posteriors in the contour plots with the values of the parameters $({\alpha }_1,{\alpha }_2,f)$. In the lower panels, we have confidence regions for marginalized posteriors. In the upper panels, the estimated values of the parameters and their Gaussian distributions are shown. In both panels, the vertical lines indicate the values of the “true” model, defined by the priors, and horizontal lines indicate the estimated values for the model parameters.

Using the logarithmic Jeffrey’s scale [70] to input a qualitative classification of the evidence, our results indicate barely worth-mentioning evidence against the model as compared with the “true” model with roughly $X\sim {10}^{-7}$ for the Bayes factor that is well accommodated in the range $0\le X<1.1$. As traditionally known, Jeffrey’s scale classifies it as weak evidence against the model with the interval $1.1\le X<3$. The range $3\le X<5$ indicates strong evidence disfavoring the competing model. Finally, very strong evidence against the competing model is achieved at $X\ge 5$.

5.3. Dynamical System and Comparison with the Potentials

Following Refs. [67,68], one constructs a dynamical system with the coordinates

$$\begin{array}{ccc}& & x=\frac{\dot{\varphi }}{\sqrt{6}H},\hfill \end{array}$$

(48)

$$\begin{array}{ccc}& & y=\sqrt{\frac{V}{3H^2}},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & \lambda =\frac{-V^{\prime }}{V},\hfill \end{array}$$

(50)

where the prime sign ${(}^{\prime })$ indicates an ordinary derivative with respect to the scalar field $\varphi$. The variables $(x,y,\lambda )$ can be written in the form of an autonomous system as

$$\begin{array}{c}\frac{dx}{dN}=-3x+\sqrt{\frac{3}{2}}\lambda y^2+\frac{3}{2}x(1+x^2-y^2),\hfill \end{array}$$

(51)

$$\begin{array}{c}\frac{dy}{dN}=-\sqrt{\frac{3}{2}}\lambda xy+\frac{3}{2}y(1+x^2-y^2),\hfill \end{array}$$

(52)

$$\begin{array}{ccc}& & \frac{d\lambda }{dN}=-\sqrt{6}{\lambda }^2(\Gamma -1)x.\hfill \end{array}$$

(53)

The quantity N is the number of e-foldings of inflation that is commonly defined as the logarithm of the scale factor $a\left(t\right)$ such as $N=N_0+lna$. The term $N_0$ denotes the number of the e-folds set by the current time. The quantity $\Gamma$ is defined in terms of the related potential as

$$\Gamma =\frac{V{V}^{″}}{V^{\prime 2}},$$

(54)

in which the double prime symbol ${(}^{″})$ denotes an ordinary second derivative with respect to the scalar field $\varphi$. From Equation (36), and using Equations (48) and (49), one obtains

$${\Omega }_{ext}(x,y)=x^2+y^2,$$

(55)

for a locally universe flat space curvature. Defining a corresponding effective equation of the state ${\gamma }_{ext}$, one writes

$${\gamma }_{ext}=1+w=\frac{2x^2}{x^2+y^2}.$$

(56)

By direct calculations of Equations (48)–(50), with the initial value of ${\lambda }_{ini}={\alpha }_1$, we obtain the logistic equation

$$\begin{array}{ccc}& & \frac{d\lambda }{dN}=2\sqrt{6}{(\lambda -(\sqrt{{\alpha }_2}+{\alpha }_1))}^{2}x.\hfill \end{array}$$

(57)

In Figure 4, we show the resulting curves from the models of Equation (44) (black lines) and our model defined by Equation (43) (red lines), showing the evolution for the dynamical variable ${\lambda }_i$ running from 0.1 to 1, denoted by the outer and inner curves, respectively. The curves were obtained, setting the values of ${\Omega }_{ext}={\Omega }_{\varphi }=0.68$ and the thawing initial conditions to both models as $N_0=-15,x_i={10}^{-5},y_i={10}^{-3}$. As a result, the evolution of the curves of our model begins earlier than the reference model in Equation (44) shown by the red lines. In both cases, the semicircles reach the x-axis at ∼0.82. The curves close to this value are compatible with the models of late cosmic acceleration of the universe. The origin represents an initial point in an early matter domination universe.

Henceforth, we use the estimated values of the parameters of out model to evaluate how a resulting EoS of an effective $w\left(z\right)$ evolves. For this matter, we adapt the publicly code provided in [68]. The set of equations of Equations (48)–(50) has the initial conditions $N_0=-15,x_i={10}^{-5};y_i={10}^{-3},\lambda \left[N_0\right]={\alpha }_1$ for thawing models. For freezing models, one defines $N_0=Log\left[{10}^{-6}\right],x_i=-0.5;y_i=0.1,\lambda \left[N_0\right]=0.1$. We use CPL parameterization as a function of the redshift written as

$$w\left(z\right)=w_0+w_a\frac{z}{1+z},$$

(58)

where $(w_0,w_a)$ are parameters used to evaluate how $w\left(z\right)$ tends to depart away or converge to $\Lambda$CDM values. In conjunction with CPL, we use BA parameterization, which is given by

$$w\left(z\right)=w_0+w_a\frac{z(1+z)}{1+z^2}.$$

(59)

We adopt the maximum value for the redshift as $z_{max}=2$ and the maximum number of iterations i as $i_{max}=50$.

In Figure 5, we have comparisons between CPL and BA parameterizations of the effective model in Equation (43) (with the values of Figure 5). Thus, we have a test adopting generic thawing and freezing models by means of Equations (48)–(50), and we obtain similar results as in Ref. [68]. The model prefers a thawing behavior from CPL (blue line) and BA (red line) parameterizations that are closer to numerical points (dotted line). When plotting the freezing pattern for the model, we obtained curves out of range with a large difference from the numerical curve, and they were omitted.

6. Remarks

We investigate the extrinsic curvature as an inflaton scalar field. The starting point lies in the correlation of the model parameter that is constrained for both accelerated regimes. For instance, the dark energy accelerated regime relies on $w<-1/3$; then we expect $w>-1/3$ for inflationary regimes in accordance with Figure 5. We also present a fluid approach in order to provide a better understanding of the related dynamics with a relation between the model parameter ${\beta }_0$ and the fluid parameter w, as shown in Equation (26). Our intent is to construct the basis of a geometrical model that fully relies on gravitational interactions. In this paper, we focus our attention on the inflationary period obtaining a logarithm inflaton field that generates an exponential potential in Equation (40). This form of potential creates a PLI mechanism that has strong constraints imposed by Planck data [62]. Then, we have proposed an effective form for a new potential defined in Equation (43) to generalize Equation (40). We have numerically restrained our model parameters by means of a Bayesian analysis obtained from random points using the 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.

In this paper, we have considered all the cosmic components to be immutable. The numerical analysis gave us an optimistic scenario for further studies on the unification of inflation with late cosmic acceleration. The stability of the present model in this numerical evaluation allows us to use more realistic contexts to produce red spectra with a scalar spectral tilt and a small tensor-to-scalar ratio. As a subject of a future work, we will consider the full perturbation equations of the model in a scenario with a matter-sourced anisotropic stress and a series of latest astronomical data from cosmic microwave background radiation (CMB) experiments, such as Planck collaboration.