Einstein pioneered the era-defining concept of amalgamating the theory of gravity with the theory of geometry. Henceforth, all concepts of gravitation narrowed down to geometrical analysis. The success of the idea substantiated the foundation of researching the effects of gravity through geometry. The general theory ushered in the convention of considering spacetime as a metric structure [1], which is intrinsically related to differential geometry. As a concomitant outcome, the development of modified theories of gravity became a subject of the development of differential geometry [2]. Riemannian geometry [3] forms the fundamental structure of general relativity [4], [5], which has proven itself as the most successful theory of gravity, providing a pertinent explanation for the perihelion advance of Mercury, gravitational redshift [6], radar echo delay [7], [8], etc. Apart from that, general relativity successfully predicted [9] stellar phenomena like the orbital decay of the Hulse–Taylor binary pulsar due to gravitational wave damping, fully confirming the observationally weak-field validity of the theory [10].

Neutron Stars have become the cynosure among cosmic laboratories for evaluating gravitational interaction. Gravitational waves observed by the LIGO Scientific and Virgo Collaboration elucidated the shortcomings of general relativity in explaining the observations from massive neutron stars (). The gravitational waves from the events GW190814 and GW170817 strengthened the previous claims [11], [12], [13]. Gravitational-Wave Observatory (LIGO), gravitational waves were successfully detected [14], opening a window of opportunity to evaluate the success of the theory in the ultimate stage of black hole coalescence in the regime of strong gravitational fields. While general relativity (GR) proved to be a worthy contender at the solar system scale, at the grander scale of the universe, multiple drawbacks thwarted the path of establishing general relativity as the ultimate theory of gravitation. In particular, the GW 190814 Gravitational Wave signal directly indicated towards a Neutron Star- Black Hole merger, where the estimated results debated a 22.2– Black Hole merging with a 2.50–. Apart from that, general relativity fails to explain the observations in the case of neutron stars exceeding two solar masses; while the GW 170817 event on August 17, 2017, indicated an NS-NS merger where the estimated masses of the sources could go as high as . Further investigation into the GW 190814 signal exhibited no tidal deformations, apart from that, the absence of electromagnetic signals led to the identification of the secondary source as a possible neutron star with mass  [15]. The possible scenario was that the heavy neutron star was absorbed by a 23  BH, while the double merger scenario can be the result of a tight NS–NS scattering off a massive BH. Apart from LIGO VIRGO detections, recent observations have led to the discovery of heavy neutron stars like PSR J0952-0607 which has an estimated mass of . The field of heavy neutron stars are still evolving and the corresponding explanations are not pertinently explained by the general theory. Tangphati et al. [16] further evaluated the secondary component of the GW190814 event as a quark star (QS) obeying a colour-flavour locked (CFL) EOS within the framework of  gravity and by modifying the bag constant () the theory suggested the presence of Neutron Stars up to  with a radius of 12.43 km which is considered to be the heaviest neutron star to be ever accounted for, while the GW198014 signal supports this estimation. Hence, from the perspective of the evolution of fundamental physics, neutron stars become an intriguing system, taxing further research in the realm of modified gravity theories [17].

In addition to it, the results predicted by general relativity cannot properly explain the late-time acceleration of the universe [18], [19], [20], [21], and it also postulates a unique form of matter and energy that cannot be detected or seen [22]. The hypothetical dark matter and dark energy comprise 95% to 96% of the total mass-energy in the universe, while baryonic matter encompasses only 4% to 5% [23]. The observational results supported the expansion of the universe [24] proposed by the general theory, as multiple cosmological observations proved the precision of the predictions [25], [26], [27], while it failed to provide any feasible explanation of the aforementioned dark entities. In addition, the proposed singularities inside black holes might be resolved by extending the general theory into the quantum regime.

Multiple attempts have been made to modify the theory of gravitation; many classical approaches were introduced to modify aspects of the theory, while arriving at a universally successful explanation remained a distant dream. One of the many attempts involved modifying the standard Hilbert action used in general relativity to account for the anomalies found in the observational data by considering an arbitrary function of the Ricci Scalar () in the gravitational action [28], [29] to make it more general than the one proposed by Hilbert. The action  which provided a window for finding a geometric solution to the dark matter problem [30], [31], [32], [33]. Further approaches developed through cultivating the concept proposed two geometries that focused on relating matter with geometry by proposing a non-minimal coupling between the two entities. The first branch is called  gravity [34] which is based on the action . The second branch is the famous  gravity [35], bearing an action represented by , where  refers to the trace of the Energy-Momentum tensor. One of the theoretical approaches for constructing a modified gravity theory focused on establishing a modified gravitational Lagrangian by combining the metric and the Palatini formalism [36], [37]. This approach is known as hybrid metric-Palatini gravity [38], [39], [40], [41], [42], [43]. In a recent paper [44], it has been explored how exotic matter influences gravitational collapse and stellar structure using Palatini  gravity. They have developed a theoretical relation between gravitational waves and dark matter which can potentially help in solving the mysteries of this hidden matter in the universe.

 gravity is an intriguing theory with various implications that have been extensively discussed by many research groups. An interesting feature of this theory is its ability to explain many quantum gravity phenomena. It is suggested that a non-perturbative approach for the quantization of the gravitational metric could result in a new version of the  gravity solely due to the quantum fluctuations of the metric tensor. The action is represented as , where  is a constant. This idea unambiguously proposes a connection between the theoretical description of the quantum field in the curved spacetime background, which represents particle creation in the gravitational field, and the effective classical description within the  gravity system [45], [46], [47], [48], [49], [50], [51], [52], [53], [54].

Weyl developed a new school of thinking, paving the path for a more general geometry [55] than the Riemannian framework, which can interplay between mathematics and physics. Weyl’s study focused on creating a unified theory encompassing gravitation and electromagnetism. The metric-compatible Levi-Civita connection played a pivotal role in Riemannian geometry as it was the tool that provided the ability to compare lengths. Weyl’s revolutionary idea was the removal of the metric field from the geometry. He introduced a series of conformally equivalent metrics that could be connected through a parallel transport that did not require any information about the length of the vector. In addition to this, the length connection was introduced, which in practical application is associated with the electromagnetic field. The length connection can fix or gauge the conformal factor, but it does not seek information on the vector length or direction of the vectors participating in the parallel transport. This resulted in the non-zero divergence of the metric tensor in Weyl’s theory, which required a new geometric quantity for its explanation, coining the term “Non-metricity”.

Further generalization of the theory was proposed by Dirac [56], who aimed to create a generalized Weyl’s theory that included both the physically undetectable metric , which varies in length during transformation, and the measurable one , which remains invariant. While  is altered during the transformation,  remains a conformally invariant atomic metric. Though Einstein’s criticism [57] thwarted the progress of research on this topic, its mathematical elegance remains something to be praised. Cartan [58] played an important role in the creation of a new class of generalized geometrical theory [59], [60], [61] as an extension to Einstein’s work, which is famously known as the Einstein–Cartan theory [62]. This theory involves the torsion field, an element introduced by Cartan that finds its source in spin density, but it can be easily amalgamated into Weyl’s concept, forming the Weyl-Cartan geometry [63], [64], [65], [66], which is a widely explored domain. Weyl’s theory finds its natural progression in the torsion field and hence the inclusion is a natural extension of the system and exists in the Weyl–Dirac geometry [67], generating an action integral for the general relativistic massive electrodynamics. Multiple researchers have explored the physics applications as well as the geometries [68].

In general relativity, the metric  plays a pivotal role in explaining gravitational theories by utilizing a set of tetrad vectors. An attempt to replace metricity must nullify the requirement of the metric tensor in explanations, and this is when the torsion field can be used to its fullest potential for explaining gravitational effects. The introduction of the Weitzenbock manifold [69] provided a solution to the problem posed. The Weitzenbock space possesses the characteristics: The metric tensor reduces to null divergence , while the torsion tensor becomes non-zero , and the curvature tensor reduces to zero, . If the Weitzenbock manifold becomes torsion-free, i.e., , then it reduces to a Euclidean manifold. An important property of this manifold is its null divergence of the metric tensor, which is attributed to distant parallelism—often referred to as absolute parallelism or teleparallelism. Utilizing the properties of the Weitzenbock manifold, Einstein introduced a teleparallel theory of gravitation and electromagnetism [70]. Furtherance of this research led to TEGR (Teleparallel Equivalent of General Relativity) [71], [72], [73], which is popularly known as the  gravity theory. The primary feature of this theory is a flat spacetime where torsion is utilized to represent curvature; hence, it does not require the fourth-order metric structure like  gravity. This theory is extensively used to explain the late expansion of the universe without requiring dark energy [74]. The Weitzenbock manifold was further connected to the Weyl-Cartan spacetime, which advanced teleparallel gravity theories. The newly formed WCW (Weitzenbock–Cartan–Weyl) theory deals with 4D curved spacetime and provides a pertinent geometrical description of dark energy, incorporating the late expansion of the universe as an intrinsic property [75], [76], [77], [78], [79].

Extensive research suggests that the geometric description of gravitation can be approached in two distinct ways: the curvature representation, which involves the vanishing of torsion and non-metricity, and the teleparallel description of the manifold, which corresponds to the nullification of curvature and non-metricity. Hence, a generalized theory can be introduced based on the metric’s non-metricity () nature, which represents the length variation during parallel transport as a purely geometric effect. This is known as symmetric teleparallel gravity [80]. This theory benefits from covariantizing the usual coordinate calculations in general relativity, and further development led to the celebrated gravity, also known as non-metric gravity. This theory has been extensively researched, with a detailed exploration available in [81].

Symmetric teleparallel gravity (STG) provides a unique framework for formulating a theory of gravitation that relies neither on torsion nor on curvature but instead originates from non-metricity. In this framework, the manifestation of gravity arises directly from the non-metricity scalar , which is based on the concept that vectors remain parallel over long distances on a manifold [82]. The quantity  represents the magnitude of the non-metricity present in a metric tensor solution and corresponds to the Ricci scalar in general relativity (GR) [83], [84], [85]. By defining the affine connection in general frames where curvature vanishes and the connection is torsionless gravitational phenomena are encoded in terms of contributions from non-metricity [86]. While the idea of torsionless symmetric indices can lead to a new form of gravitation, in STG, the metricity condition of general relativity is neglected, giving rise to the teleparallel equivalent of general relativity (TEGR) scalar [87]. A key advantage of teleparallel gravity and STG is their ability to differentiate gravitational phenomena from inertial phenomena-something the general theory of relativity cannot achieve [88].

Apart from its applications in cosmology,  gravity has versatile applications in astrophysical objects. One of the most promising arenas for testing this theory is compact stellar bodies [89], [90], [91], [92], [93], which serves as natural laboratories for examining high-density nuclear matter. In this paper, we focus on neutron stars (NS), which provide a wealth of observational data in various forms, such as gravitational wave (GW) events from the collaboration between the Laser Interferometer Gravitational-Wave Observatory (LIGO) and the Virgo Gravitational-Wave Interferometer (Virgo), as well as massive pulsar data from the Neutron star Interior Composition Explorer (NICER). Hence, neutron stars play a pivotal role in evaluating modified gravity theories [94], [95], [96], [97], [98]. By analysing the pressure, density, and redshift, we can derive the equation of state (EoS) of compact matter. Additionally, using mass–radius relations, tidal deformations, and rotational dynamics, we can evaluate the effect of non-metricity coupling on the structure of neutron stars.