Bouncing Scenario in the $f(T)$ Modified Gravity Model with Dynamical System Analysis

The $f(T)$ modified gravity model is a theoretical framework that extends the standard theory of gravity by modifying the gravitational action based on the torsion scalar $T$ instead of the Ricci scalar $R$ used in General Relativity [1, 2, 3, 4, 5, 6, 7, 8]. This approach is part of a broader category of modified gravity theories that aim to address various cosmological and astrophysical phenomena, including the accelerated expansion of the universe and the nature of dark energy [6, 7, 8]. In the $f(T)$ framework, the gravitational dynamics are described using the torsion scalar, which arises in the context of teleparallel gravity. This scalar is related to the torsion of spacetime, providing an alternative geometric description of gravity [1]. The model is expressed as $f(T)$, where $f$ is a function that can take various forms depending on the physical scenario being modeled [2, 3, 4]. This flexibility allows for a wide range of behaviors and predictions. The $f(T)$ models have been explored for their implications in cosmology, particularly in explaining the late-time acceleration of the universe [7, 8]. They can potentially provide a unified description of cosmic evolution without invoking dark energy. The $f(T)$ model has been applied to various astrophysical scenarios, including the study of compact objects like gravastars and wormholes, as well as in the context of inflationary cosmology [9]. The $f(T)$ modified gravity model introduces several key features that distinguish it from traditional theories of gravity, particularly General Relativity. The field equations derived from the $f(T)$ action are second-order differential equations, which helps avoid issues related to ghost instabilities that can arise in higher-order theories [9]. The $f(T)$ model provides a framework to understand dark energy phenomena by modifying the gravitational interaction at large scales, which can lead to effects similar to those attributed to dark energy in standard cosmological models [3, 4]. Ongoing research aims to compare the predictions of $f(T)$ models with observational data from cosmic microwave background radiation, galaxy distributions, and other cosmological measurements to assess their viability [7, 8]. In summary, the $f(T)$ modified gravity model represents a significant area of research in theoretical physics, offering new insights into the nature of gravity and its role in the universe’s evolution [10].
The bouncing scenario in cosmology presents an alternative to the traditional Big Bang model, suggesting that the universe undergoes a ”bounce” rather than originating from a singularity [11, 12]. This concept has gained traction as researchers explore the implications of a universe that can contract and then expand, potentially resolving several fundamental cosmological issues. The bouncing model posits that the universe can transition from a contracting phase to an expanding phase without encountering a singularity. This non-singular bounce avoids the infinite densities and temperatures associated with the Big Bang, providing a smoother transition between phases [12]. Some bouncing models propose a cyclic universe, where the universe undergoes repeated cycles of contraction and expansion. Each bounce represents a new cycle, allowing for a universe that has no definitive beginning or end [13]. This idea contrasts with the traditional view of a singular beginning at the Big Bang. The bouncing scenario addresses several longstanding issues in cosmology, such as the horizon problem, flatness problem, and inhomogeneity problem. By providing a mechanism for a non-singular transition, it can potentially explain the uniformity observed in the cosmic microwave background radiation [14]. Theoretical frameworks that incorporate quantum gravity suggest that quantum effects could play a significant role during the bounce. These effects might prevent the universe from collapsing into a singularity, allowing for a bounce instead. Various mathematical models have been developed to describe bouncing cosmologies. These models often utilize tools from numerical relativity to simulate the dynamics of the universe during the bounce, helping to predict observable consequences and test the viability of the scenario [15, 16]. Some bouncing models can be made compatible with current observational data, such as the cosmic microwave background and large-scale structure. For instance, after the bounce, the universe can enter a slow-roll inflationary phase, which aligns with observations of cosmic expansion [17]. While promising, bouncing cosmologies face challenges in terms of theoretical development and empirical validation. Critics argue that these models need to be as robust and well-developed as inflationary models to gain wider acceptance in the cosmological community.
On the other hand, Dynamical system analysis is a powerful mathematical tool used in cosmology to study the behavior of various cosmological models [18]. This approach allows researchers to understand the qualitative dynamics of the universe without necessarily solving the complex differential equations that govern these models. In dynamic system analysis, cosmological models are represented in a phase space, where each point corresponds to a state of the system. The evolution of the universe can be visualized as trajectories in this space, providing insights into the stability and behavior of different cosmological scenarios [18]. A central concept in dynamic systems is the identification of fixed points, which represent equilibrium states of the cosmological model. By analyzing the stability of these fixed points, researchers can determine whether small perturbations will lead to a return to equilibrium or diverge away, indicating the model’s behavior under various conditions [19]. The dynamical systems approach enables the study of the qualitative behavior of cosmological models, such as the existence of attractors, repellers, and limit cycles [19]. This qualitative analysis is crucial for understanding the long-term evolution of the universe, including scenarios like inflation, dark energy domination, and bouncing cosmologies. Dynamic system analysis has been applied to a wide range of cosmological models, including those involving scalar fields, perfect fluids, and modified gravity theories. For instance, it has been used to explore the dynamics of models with exponential potentials and to investigate the implications of dark energy on cosmic expansion [20, 21, 22]. Researchers employ both numerical simulations and analytical techniques to study the dynamics of cosmological models. This dual approach allows for a comprehensive understanding of the models, including their global behavior and local stability properties. The analysis can reveal how different models respond to initial conditions and how they align with observational data. Dynamic system analysis is particularly useful in studying bouncing and cyclic cosmological models. By examining the phase space of these models, researchers can identify conditions under which a non-singular bounce occurs and explore the implications for the universe’s evolution.
However, according to the above discussions, we organize the structures of the paper as follows: in Section 2, a review of the original $f(T)$ model present. In Section 3, we discuss dynamic analysis of gravity model, and stability analysis of model, in Section 4, we check the bouncing scenario in the model. Finally, in Section 5, the conclusion and summary offer.

In this model, we consider the action as a function of the space-time torsion, which is minimally coupled with the scalar field, with kinetic and potential terms as [7, 23]

$$S=\int d^{4}x(f(T)-\partial_{\mu}\varphi\partial^{\mu}\varphi-V(\varphi)),$$

(1)

where, $T$ is torsion scalar, and $\varphi$ is scalar field. For the scalar field, pressure and density are defined as follows

$$\rho_{\varphi}=\frac{1}{2}\dot{\varphi}^{2}+V(\varphi),$$

(2)

$$P_{\varphi}=\frac{1}{2}\dot{\varphi}^{2}-V(\varphi).$$

(3)

We assume that the space-time torsion is in the form of a perfect fluid. Therefore, we have the following relations for the torsion density and pressure:

$$\rho_{T}=\,\alpha_{1}H^{2},$$

(4)

$$P_{T}=\alpha_{2}H^{2},$$

(5)

where $\alpha_{1},\alpha_{2}$ are arbitrary constants. With this assumption, we write the Friedman equations for action (1) as follows

$$3H^{2}=\rho_{\varphi}+\rho_{T},$$

(6)

$$2\dot{H}=-(\rho_{\varphi}+P_{\varphi})-(\rho_{T}+P_{T}),$$

(7)

Also, the evolution equation of the scalar field is as follows

$$\ddot{\varphi}+3H\dot{\varphi}+V^{\prime}(\varphi)=0$$

(8)

In this relation, the prime represents the derivative of $V$ with respect to the scalar field.

By analyzing the dynamics of cosmological models, researchers can identify stable states and trajectories that the universe might follow. For example, they can determine whether the universe is likely to continue expanding, enter a phase of contraction, or undergo a bounce. This understanding helps in predicting the general fate of the universe over long timescales. Also, dynamic system analysis allows for the identification of fixed points in the phase space, which correspond to equilibrium states of the universe. By studying the stability of these points, cosmologists can infer whether the universe will settle into a particular state or if it will evolve towards a different scenario, such as dark energy domination or a bouncing model. While dynamic system analysis can outline possible future scenarios based on current models, it does not provide precise predictions of specific events (like supernovae or galaxy collisions). Instead, it helps in understanding the conditions under which certain events might occur, such as the onset of inflation or the transition to a dark energy-dominated phase. Therefore, by applying dynamic system analysis to various cosmological models, researchers can compare their predictions and assess which models are more consistent with observational data. This comparative analysis can guide future research and observational efforts to test the viability of different cosmological theories. On the other hand, while dynamic system analysis can indicate trends and potential outcomes, it cannot predict specific cosmic events with high precision. The universe is influenced by numerous factors, including quantum fluctuations and chaotic dynamics, which can complicate predictions. Therefore, while it can provide a framework for understanding possible futures, it does not yield exact forecasts of future cosmic phenomena.
However, because the cosmic field equations are non-linear and complex, we can analyze the qualitative behavior of cosmic models, including the bouncing scenario, by using dynamic systems analysis. Therefore, to consider the cosmic equations by analyzing dynamic systems, we first determine the variables of the dynamic system as follows [23]

$$x=\frac{\varphi}{H},$$

(9)

$$y=\frac{\sqrt{V}}{H},$$

(10)

$$z=\frac{1}{\varphi}.$$

(11)

By using these variables, the field equations (6,7,8) are transformed into the equations of the independent system in the following form

$$\frac{dx}{dN}=x(-3+\frac{\alpha_{1}+\alpha_{2}}{2})+\frac{x^{3}}{2}-\frac{V^{\prime}}{V}y^{2},$$

(12)

$$\frac{dy}{dN}=\frac{y}{2}(\frac{V^{\prime}}{V}+\alpha_{1}+\alpha_{2}+x^{2}),$$

(13)

$$\frac{dz}{dN}=z^{2}x,$$

(14)

with $N=\ln a$. If we define the exponential potential function as

$$V(\varphi)=e^{\mu\varphi},$$

(15)

where $\mu$ is an arbitrary parameter. The above relations are rewritten in the following form

$$\frac{dx}{dN}=x(-3+\frac{\alpha_{1}+\alpha_{2}}{2})+\frac{x^{3}}{2}-\mu y^{2},$$

(16)

$$\frac{dy}{dN}=\frac{y}{2}(\mu+\alpha_{1}+\alpha_{2}+x^{2}),$$

(17)

$$\frac{dz}{dN}=z^{2}x.$$

(18)

It is also possible to obtain the equation of state in terms of deceleration parameter

$$w=\frac{P}{\rho}=\frac{-\rho-2\dot{H}}{\rho}=$$

$$-1-2(\frac{\dot{H}}{3H^{2}})=-1-\frac{2}{3}(-1-q)=\frac{-1}{3}-\frac{2}{3}q.$$

(19)

According to the equilibrium points in the phase space and the paths that enter or leave these points, we can draw the display of changes in the two-dimensional phase space (Figure 1). Considering the selected initial conditions and its closeness to the fixed points and the analysis of the paths in the vicinity of the fixed points of the phase space, we examine the bouncing scenario.

Bouncing according to the Hubble parameter occurs when $H=0$ with $\dot{H}>0$ or $\dot{H}<0$. According to pressure and density equations, if $\rho+3P<0$, Hubble parameter will be positive, and if $\rho+3P>0$, Hubble parameter will be negative [23]. Also, according to phase space variables, and the previous conditions for pressure and density, if $x^{2}<y^{2}$ Hubble parameter will be increasing. On the other hand, if the relation $x^{2}>y^{2}$ is established, Hubble parameter will be decreasing. If the selected initial conditions are close to the fixed point $P1$, considering that this point is stable, the paths in the phase space enter to this point and the system reaches a stable equilibrium.
If we choose the initial conditions in the vicinity of fixed points $P2,P3,P4,P5$, because the relation $x^{2}>y^{2}$ is established in these areas, therefore Hubble parameter are decreasing and the scale factor are negative. Starting from the vicinity of these points, if Hubble parameter is positive ($H>0$) (expanding universe) then the paths reach to infinity in the compressed phase space [23](In the Minkowski limit $H\rightarrow 0$). These paths at infinity leave the saddle points and go to the point stable equilibrium ends at the origin. Due to the negative values of the Hubble parameter, $H$ reaches negative values and the universe enters the contraction phase. Therefore, by establishing the necessary conditions, the bouncing will happen. Upon the onset of the universe, the quantum effects lead to a contraction phase and bouncing, resulting in the creation of fermions. This fermion creation induces a space-time torsion, aligning with our gravitational model $f(T)$ as described in the reference [26].

This study explores how modifications to gravitational action can lead to unique cosmological behaviors, particularly focusing on stability and the evolution of the universe’s scale factor. We delve into the main equations of the $f(T)$ model, analyze its dynamics, and investigate the intriguing bouncing scenario. In this regard, the $f(T)$ modified gravity model extends traditional gravitational theories by incorporating a function of the torsion scalar $T$, allowing for a different treatment of gravity, particularly in cosmological contexts. This model provides a framework to explore various cosmological phenomena, including the accelerated expansion of the universe and the nature of dark energy. Also, after applying dynamical system analysis, the analysis reveals the bouncing scenario in this model. In summary, this study concludes that the $f(T)$ modified gravity model, through its unique approach to gravitational dynamics, presents significant insights into the nature of cosmic evolution. The bouncing scenario, in particular, highlights the potential for a cyclic universe that avoids singularities, offering a compelling alternative to traditional cosmological models. This work underscores the importance of dynamical system analysis in exploring the stability and future trajectories of the universe, paving the way for deeper insights into the fundamental nature of gravity and cosmic evolution.

Data availability No new data were generated or analysed in support of this research.

Declarations Conflict of interest The authors declare that they have no conflict of interest.