Recent astrophysical observations, such as those from Type Ia supernovae (SNe-Ia) [1], [2], baryon acoustic oscillations (BAO) [3], [4], the Wilkinson microwave anisotropy probe (WMAP) [5], and cosmic microwave background (CMB) radiation [6], [7], have shown that the expansion of the universe is currently accelerating. These observations suggest that the universe is flat, homogeneous, isotropic, and can be described by a Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. The discovery of cosmic acceleration has spurred the scientific community to move beyond Einstein’s general relativity (GR) to explore other mechanisms that could explain this behavior. In the framework of GR, the cosmological constant  is introduced to account for the presence of dark energy (DE), a mysterious form of energy responsible for the accelerated expansion. However, this model faces several challenges, such as the fine-tuning problem, the coincidence problem, and the fact that DE is only observed at cosmological scales rather than at Planck scales [8], [9]. These issues have led to the development of alternative modified theories of gravity, which attempt to address these shortcomings and provide a more comprehensive understanding of cosmic phenomena.

Currently, two main approaches are widely used to explain the universe’s accelerated expansion. The first is the introduction of DE with negative pressure into Einstein’s field equations, which has led to the proposal of various models, such as phantom, quintessence, and k-essence [10], [11], [12]. The second approach modifies the gravitational theory itself, particularly at large scales, to account for the observed cosmic acceleration. Modified gravity theories have shown promising results in explaining both early-time inflationary behavior and late-time acceleration [13]. Among these modified theories,  gravity and  gravity have gained significant attention. In  gravity, the curvature scalar  in the Einstein–Hilbert action is replaced by an arbitrary function , and the theory is described using the Levi-Civita connection, which characterizes a spacetime with zero torsion and non-metricity but non-vanishing curvature [14], [15], [16]. On the other hand,  gravity generalizes  gravity by coupling the Ricci scalar  with the trace of the energy–momentum tensor  [17], [18], [19], [20]. Similarly,  gravity extends the theory by including the Gauss–Bonnet term  in the action [21], [22]. Recent developments in cosmology have highlighted limitations in describing gravitational interactions on cosmological scales with traditional Riemannian geometry, which performs well at smaller scales, such as within the solar system. This has led to the exploration of alternative geometric frameworks, including Weitzenböck spaces, introduced by Weitzenböck himself to extend the foundations of geometry for broader applications in gravitational theories [23]. A Weitzenböck manifold is characterized by a covariantly constant metric tensor, , a non-zero torsion tensor , and a vanishing Riemann curvature tensor . This combination yields a geometry in which the manifold reduces to Euclidean space in the absence of torsion, while torsion itself varies spatially across the manifold to enable alternative representations of gravitational effects. One unique feature of Weitzenböck geometry is its zero-curvature property, which supports the concept of distant parallelism, also known as teleparallelism. This idea attracted Einstein’s attention as a foundation for a unified theory of electromagnetism and gravity, leading to the development of teleparallel gravity, which describes gravity through torsion rather than curvature [24]. In teleparallel gravity, gravitational effects are captured using a tetrad field  instead of the traditional spacetime metric . The tetrads generate torsion, allowing gravity to be fully described in terms of torsion alone. This approach gave rise to the teleparallel equivalent of GR (TEGR) and eventually to  gravity, where torsion replaces curvature to yield a flat spacetime framework [25], [26], [27]. A significant advantage of  gravity lies in its second-order field equations, offering a simpler formulation compared to the fourth-order equations in  gravity. This theory has proven effective in explaining late-time cosmic acceleration without resorting to DE, making it a compelling alternative for addressing open questions in cosmology. The application of  gravity to cosmological and astrophysical phenomena has therefore gained considerable interest, as it provides a promising foundation for models of cosmic evolution and the dynamics of large-scale structures in the universe [28], [29], [30], [31], [32], [33], [34]. The  gravity model, introduced by Harko et al. [35], has garnered interest for its dual focus on the torsion scalar  and the energy–momentum trace . This theory builds upon teleparallel gravity, allowing for a richer description of gravitational dynamics by incorporating interactions between torsion and the energy–momentum contributions from matter fields. The  model has been applied in various cosmological and astrophysical contexts, where its implications for cosmic expansion, structure formation, and compact objects have been actively explored. The cosmological relevance of  gravity has been examined extensively. Junior et al. [36] analyzed the thermodynamics, stability, and reconstruction of the CDM model within this framework, showing that it aligns with observational data while perhaps providing information about how classical and quantum gravity behaves. Furthermore, Harko et al. [37] extended  gravity by introducing a non-minimal coupling between torsion and matter, enhancing the model’s ability to capture the intricate interactions in cosmological dynamics. Similarly, Momeni and Myrzakulov [38] explored the cosmological reconstruction of , shedding light on its potential to describe diverse evolutionary scenarios in the universe. Studies of structure formation in  gravity have further deepened our understanding of cosmic evolution. Farrugia and Said [39] analyzed the growth factor of cosmic structures in this framework, while Pace and Said [40] employed a perturbative approach to investigate the behavior of compact objects like neutron stars. These studies collectively emphasize the potential of  gravity to provide a unified framework for explaining cosmic expansion, structure formation, and compact object dynamics, marking it as a promising candidate for addressing key questions in modern cosmology.

The inflationary model suggests that the universe experienced a rapid expansion in its earliest moments, smoothing out initial irregularities and setting the stage for its observed large-scale structure. However, this model does not fully account for the origin of the universe, as it assumes a pre-existing singularity prior to inflation. To address this limitation, the matter bounce scenario has emerged as a compelling alternative, proposing that the universe initially underwent a contraction phase before experiencing a bounce, which then led to the current expansion phase [41], [42], [43]. This approach allows for a Universe that avoids a singularity, replacing it with a bounce that generates causal fluctuations, potentially seeding structure formation. The matter bounce scenario introduces an initial matter-dominated contraction phase, which eventually reverses into expansion through a non-singular bounce, challenging the need for a singular beginning. This non-singular cosmological model, however, often requires a violation of the null energy condition (NEC), which has been demonstrated in certain modified gravity theories, such as generalized Galileon models [44]. The concept of a big bounce replacing the Big Bang singularity has drawn substantial interest in the field of modified gravity [45], [46], [47], [48], [49], [50], inspiring new ways to model the early universe and the transition from contraction to expansion. Numerous studies have explored the implications of bouncing cosmologies within various modified gravity frameworks, including  gravity. For instance, Cai et al. [51] investigated a matter bounce cosmology within the framework of  gravity, while Rodrigues and Junior [52] examined black-bounce solutions within  gravity, analyzing how these configurations can address singularities in black hole models. Further work by Amorós et al. [53] explored matter bounce scenarios in loop quantum cosmology derived from  gravity, and Skugoreva and Toporensky [54] studied bouncing solutions using the power-law  model. Other studies have looked into bouncing models addressing potential future singularities, collectively underscoring the versatility of modified gravity in addressing early-universe phenomena and advancing our understanding of the Universe’s origin and evolution. Despite the successes of inflationary models, they are not without challenges, namely, the trans-Planckian problem [55], fine-tuning of initial conditions [9], and difficulty in embedding in quantum gravity frameworks. Bouncing cosmologies provide an alternative paradigm where the universe undergoes a contraction phase followed by a regular bounce into expansion, avoiding the initial singularity. In this context,  gravity offers a compelling framework: it allows for NEC violation through matter-geometry coupling without introducing ghost instabilities typical in higher-order curvature theories. Furthermore, since torsion-based theories are dynamically second-order, they are more tractable for bounce analysis compared to  [17] or loop quantum cosmology (LQC) scenarios [56]. In this work, we explore the possibility of a non-singular bouncing universe within the  gravity framework. The structure of this paper is as follows: In Section 2, we introduce the theoretical framework of  gravity and the basics of the FLRW model. Section 3 discusses the parameterization of bounce cosmology within the context of  gravity. In Section 4, we analyze the energy conditions, examining their implications in gravitational theory. Finally, Section 5 provides concluding remarks.