Editorial: Modified Theories of Gravity and Cosmological Applications—Topical Collection

General relativity (GR) has been remarkably successful in describing the gravitational interaction through the curvature of spacetime. It has provided an unparalleled theoretical framework that accurately explains a wide range of astrophysical and cosmological phenomena. The theory’s predictions, such as the perihelion precession of Mercury, the bending of light by massive bodies (gravitational lensing), and the redshift of light in a gravitational field (gravitational redshift), have been confirmed with high precision. Moreover, the detection of gravitational waves by LIGO and Virgo has provided direct observational evidence for the dynamic nature of spacetime predicted by GR, further solidifying its validity. In the cosmological context, GR underpins the standard $\Lambda$CDM model, successfully describing the large-scale structure of the universe, cosmic expansion, and the Cosmic Microwave Background (CMB) radiation. Despite these successes, GR is not without challenges [1,2]. It predicts singularities, such as those found in black holes and the Big Bang, where the theory itself breaks down and quantum effects are expected to become significant. Moreover, GR is non-renormalizable and hence it cannot be quantized according to the standard procedures. Additionally, the existence of dark matter and dark energy—introduced to reconcile observations with the theory—raises questions about whether GR remains valid on cosmic scales or requires modification. Finally, recently, there have been some tensions between theoretical predictions and observations, such as the $H_0$ and ${\sigma }_8$ ones, the possible effects of local anisotropy, etc. All these issues suggest that while GR is a powerful and elegant theory, it may not be the final description of gravitational phenomena, motivating ongoing research into alternative and extended theories of gravity.

Modified theories of gravity [3,4,5] extend the framework of general relativity through various methods, leading to different field equations and, consequently, distinct cosmological implications. Broadly, modified gravity theories can be constructed through several approaches. One of the most common methods is the extension of the Einstein–Hilbert action by introducing arbitrary functions of curvature invariants, leading to $f\left(R\right)$ gravity [6,7,8,9]. Another prominent approach is the formulation of gravity using torsion, such as in teleparallel gravity and its extensions like $f\left(T\right)$ gravity [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and $f(T,B)$ gravity [20,28], as well as non-metricity, such as in $f\left(Q\right)$ gravity [29,30,31,32,33,34,35] and in $f(Q,C)$ gravity [36]. An alternative class of modifications involves scalar–tensor theories, where additional scalar fields are coupled to gravity, such as in Brans–Dicke theory and Horndeski theories, offering a framework to study cosmological inflation and dark energy dynamics [37]. Another widely explored modification involves extra-dimensional theories, such as Kaluza–Klein and brane-world models, which attempt to explain gravity behavior at different scales by considering higher-dimensional spacetimes [38]. Furthermore, anisotropic cosmological Finsler and Finsler-like models have been developed when the underlying geometry of spacetime adopts a more generalized metric structure than the traditional Riemannian one [39,40,41,42,43,44,45,46,47,48,49,50]. In particular, Finsler-based cosmological scenarios may provide solutions to the anisotropy tension, since the internal geometric anisotropy changes the luminosity distance toward varying directions for a fixed redshift [51,52].

Modified gravity does not exclude standard general relativity but instead highlights its limitations. These modifications play an essential role and contribute significantly to modern cosmology, providing a foundation for understanding the physical phenomena of the Universe. Experiments involving gravitational wave detectors, such as LIGO, VIRGO, and KAGRA, amplify the scope of modified gravity investigations, extending models beyond general relativity. In the present Special Issue “Modified Theories of Gravity and Cosmological Applications”, the published articles contribute different approaches and consequences to the theoretical and observational treatment of fundamental open problems of cosmology, offering deeper understanding of gravitational phenomena beyond the bounds of GR. In these considerations, novel cosmological models play a significant role.

In [53], the authors present a review of energy–momentum squared gravity (EMSG). The inclusion of quadratic contributions from the energy–momentum components has intriguing cosmological implications, particularly during the Universe’s early epochs. These effects dominate under high-energy conditions, enabling EMSG to potentially address unresolved issues in general relativity (GR). The exploration of EMSG presents a novel framework for extending of understanding of gravitational phenomena beyond the limits of GR. This review offers to shed light on some of the fundamental questions in modern cosmology and gravitational theory.

In [54], the authors elucidate the back-reaction within macroscopic gravity in terms of the non-metricity of average geometry. With a 1 + 3 decomposition of the spacetime, they analyze how geometric flows are modified by deriving the Raychaudhuri and Sachs equations, which govern the expansion of the geodesic flows. Physically, these equations characterize the flow of the cosmic fluid and photon bundles, respectively. The work presents interesting results.

In [55], the authors propose a novel dark-energy equation-of-state parametrization, with a single parameter that quantifies the deviation from $\Lambda$CDM cosmology. Additionally, under this scenario, dark energy behaves like a cosmological constant at high redshifts, while the deviation becomes significant at low and recent redshifts, especially in the future. In light of this, the authors conduct a detailed analysis with observations, namely, with Hubble function (OHD), Pantheon, and baryon acoustic oscillations (BAO) data.

In [56], Palatini-like theories of gravity are studied in connection to models incorporating linear generalized uncertainty principles. In this consideration, the authors delve into the thermodynamics of systems comprising both Bose and Fermi gases. This analysis encompasses the equations of state for various systems and liquid helium. Additionally, the authors derive the energy and specific heat for photon and phonon gases.

In [57], the authors demonstrate reconstruction of the forms of $F\left(R\right)$ in the modified theory of gravity and the metric-compatible $F\left(T\right)$ together with the symmetric $F\left(Q\right)$ in alternative teleparallel theories of gravity, from different perspectives, primarily placing emphasis on a viable FLRW radiation era. Inflation has also been studied in this framework for a particular choice of scalar potential. The inflationary parameters are found to agree appreciably with the recently released observational data. This paper offers an interesting perspective of the corresponding research area.

In [58], the authors use the cosmography approach to study a particular self-tuning filter solution focused on a zero-curvature fixed point to study the H0 tension. In this approach, the equations restrict the universe’s evolution to certain scenarios, including radiation-like expansion, matter-like expansion, and late-time acceleration. Furthermore, they build the cosmographic series of the Fab-Four theory to obtain the kinematic parameters as the Hubble constant H0 and the deceleration parameter for all the scenarios mentioned. These results represent an interesting approach to the open problem of the current discrepancy of cosmological tensions.

In [59], the authors use generalized type attractor models of single-field inflation in order to accommodate the production of primordial black holes (PBHs) by adding a near-inflection point to the inflaton scalar potential at smaller scales, in good agreement with measurements of cosmic microwave background (CMB) radiation. Fine-tuning of the parameters in these models was needed not only to match the CMB measurements (inflation is robust in the E-models) but also to generate PBH production at lower scales during the USR phase, which would lead to a sizeable portion of PBHs in the current dark matter.

In [60], Newtonian fractional-dimension gravity (NFDG) is applied in an alternative gravitational model to some notable cases of galaxies with little or no dark matter. In the case of the ultra-diffuse galaxy AGC 114905, the authors show that NFDG methods can effectively reproduce the observed rotation curve using a variable fractional dimension D(R).

In [61], the authors investigate a general class of pseudo-Finsler spaces with $(\alpha ,\beta )$-metrics. They establish necessary and sufficient conditions such that these exhibit a Finsler spacetime structure. This means that the fundamental tensor has a Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future-pointing time-like vectors is ensured. The work offers an interesting consideration of applications of Finsler geometry to the gravitational theory.

In [62], Hořava–Lifshitz (HL) theory of gravity, in which equations governing cosmological evolution include a new term scaling similarly to the dark radiation term in the Friedmann equations, enabling a bounce of the universe instead of initial singularity. This review describes past works on the stability of such a bounce in different formulations of HL theory, an initial detailed balance scenario, and further projectable versions containing higher-than-quadratic terms to the original action.

In [63], the authors focus on the theoretical research and application of strong lenses and weak lenses. The authors begin by introducing the basic principles of geometric gravitational lensing, the representation of deflection angles, and the curvature relationship in different geometric spaces. In addition, they propose interesting views regarding the detection of dark matter and research on dark energies such as PLANCK and BIGBOSS.

In [64], the author reviews the construction of the deformed quaplectic group that is given by the semidirect product of $U(1,3)$ with the deformed (non-commutative) Weyl–Heisenberg group corresponding to non-commutative fiber coordinates and momenta. This construction leads to more general algebras being given by a two-parameter family of deformations of the quaplectic algebra. This analysis allows the author to find a direct link between non-commutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions.

In [65], the authors explore a bouncing cosmology with modified generalized Chaplygin Gas in a bulk viscosity framework. In addition, the authors study the equation of state parameter under various cases and review the stability of the models through the sign of the squared speed of sound. Additionally, the authors propose a bouncing scale factor of a special form. The bouncing point appeared at $t=0$. This model also realizes big bounce in the framework of Einstein gravity as well as $f\left(T\right)$-modified gravity, which violates the null energy condition.

In [66], the author provides a pedagogical introduction to the degenerate Hamiltonian systems, showing both very simple mechanical examples and general arguments about how such systems work. For the familiar field theory models, he explains why the gauge freedom there “hits twice” in the sense of producing twice as many first-class constraints as gauge symmetries, and why primary, and only primary, constraints should be put into the total Hamiltonian. The discussion of the author on the role of constraints and degrees of freedom in the Hamiltonian Formalism offers useful information.

In [67], the authors consider a Friedmann–Lemâitre–Robertson–Walker cosmological model dominated by bulk viscous cosmic fluid in $f\left(Q\right)$ gravity, where Q represents the non-metricity scalar. They find an interesting result: that the $f\left(Q\right)$ cosmological model efficiently describes the observational data. In addition, the authors present the evolution of their deceleration parameter with redshift, and show that it properly predicts a transition from decelerated to accelerated phases of the universe’s expansion.

In [68], a simple cooling model of white dwarf stars is re-analyzed in Palatini $f\left(R\right)$ gravity. Modified gravity affects the white dwarf structures and, consequently, their ages. The author finds that the resulting super-Chandrasekhar white dwarfs need more time to cool down than sub-Chandrasekhar ones. In addition, he discusses the basic formalism of Palatini $f\left(R\right)$ gravity and briefly reviews how it modifies the hydrostatic balance equations which he employs to derive the modified temperature equation in this framework.

In [69], the authors study a perturbation theory for embedding gravity equations in a background for which corrections to the embedding function are linear with respect to corrections to the flat metric. It is shown that this function can be chosen in such a way that the gravitational potential is in a good agreement with the observed distribution of dark matter in a galactic halo. In this approach, the problem of constructing a perturbation theory for Regge–Teitelboim equation to provide background information is considered, and an additional energy–momentum tensor is proposed. It is induced by this embedding, with additional fictitious embedding matter simulated with dark matter. This is a novel and interesting approach.

In [70], the authors explore wormhole geometry in spiral galaxies under third-order Lovelock gravity. Using the cubic spline interpolation technique, they find the rotational velocity of test particles in the halo region of our spiral galaxy from observed values of radial distances and rotational velocities. Taking this value of the rotational velocity, they are able to show that it is possible to present a mathematical model regarding the viable existence of wormholes in the galactic halo region of the Milky Way under Lovelock gravity. An important result that they obtain from the present investigation is that the galactic wormhole in the halo region can exist with normal matter as well as exotic matter.

In [71], the authors present an analysis of a chiral cosmological scenario from the perspective of K-essence formalism. In this setup, several scalar fields interact within the kinetic and potential sectors. In particular, in this framework, the authors consider a flat Friedmann–Robertson–Lamaître–Walker universe coupled minimally to two quintom fields: one quintessence and one phantom. In addition, they present an interesting explanation of the “big-bang” singularity by means of a “big-bounce”. Moreover, for the quantum counterpart, the Wheeler–DeWitt equation is analytically solved for various cases.

In [72], the authors argue that the Barrow entropy should run based on black hole thermodynamics (i.e., it should be energy-scale dependent), which is reasonable given that quantum gravitational corrections are expected to be important only in the high-energy regime. When applied to the Friedmann equations, the authors suggest that such a running Barrow entropy index could give rise to a dynamical effective dark energy, which is asymptotically positive and vanishing, but negative at the big bang. Such a sign switching dark energy could help to alleviate the Hubble tension. Other cosmological implications are also discussed.

In [73], two different Hamiltonian formulations of the metric gravity are discussed in this work and applied to describe a free gravitational field in d-dimensional Riemann spacetime. The theory of canonical transformations, which relates equivalent Hamiltonian formulations of metric gravity, is investigated in detail and the theory of integral invariants and its applications to the Hamiltonian metric gravity are also discussed. In this approach, Dirac’s modifications of the classical Hamilton method contain several crucial advantages, which indicate the obvious superiority of this method in order to develop various non-contradictory Hamiltonian theories of many physical fields.

In [74], the authors review the history of, as well as recent progress in, the composite view of the gravitational constant. This a useful approach which could be applied in future research aimed at understanding and discussing Newton’s gravitational constant. This work has provided an overview of much of the work that has been conducted in relation to the composite view of the gravitational constant. The implications of this work may open doors of insight into relationships between macroscopic gravity phenomena and the Planck scale.

In [75], the authors present a brief description of non-compactified higher-dimensional theories from the perspective of general relativity. More concretely, the space–time–matter theory, or induced matter theory, and the reduction procedure used to construct the modified Brans–Dicke theory and the modified Sáez–Ballester theory are briefly explained. Finally, the authors apply the latter to the Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological models in arbitrary dimensions and analyze the corresponding solutions.

The remarkable success of general relativity in describing gravitational interactions has been firmly established through numerous experimental and observational confirmations. However, the challenges it faces at both theoretical and observational levels have spurred the exploration of modified gravity theories as viable extensions. These alternative theories, such as $f\left(R\right)$, $f\left(T\right)$, and $f\left(Q\right)$ gravity, scalar–tensor theories, extra-dimensional models, Finsler-based constructions, etc., offer new insights into possible cosmological problems, including the nature of dark matter and dark energy, the resolution of cosmic tensions like $H_0$ and ${\sigma }_8$, possible anisotropies in the CMB spectrum, and the quest for a consistent quantum theory of gravity.

The contributions in this Special Issue focus on the latest advancements in the field of modified gravity and cosmological applications. The diverse range of approaches presented highlights the growing interdisciplinary efforts to accumulate theoretical developments with observational data. The articles present novel approaches to some fundamental questions, using investigations which extend upon $\Lambda$CDM cosmology, proposing new relations between macroscopic gravity phenomena and the Planck scale, anisotropic extensions based on the Finsler geometry for studying of the gravitational field, and higher-dimensional theories from the perspective of general relativity to construct the modified Brans–Dicke theory. Palatini $f\left(R\right)$, Hořava–Lifshitz, and Lovelock gravity techniques are also applied to the cosmological framework. Furthermore, Barrow entropy is also examined in relation to the Friedmann equations. The investigation of Nature as a whole, and not only its specific parts, seeks to reveal the truth—namely the physical laws that govern objective reality—and this is a hard and arduous procedure. It is not a solitary or instant pursuit, but demands collective effort from the scientific community in many generations and different research directions. Nature is out there, it is knowable, and the physical laws, first principles and the highest causes, are waiting to to be discovered and explored further. We believe that the works published in this Issue will contribute proportionally to deepening our understanding of the fundamental nature of spacetime and the universe.