Gravitational waves represent tiny distortions in the fabric of spacetime, generated when some of the universe’s densest and most massive objects collide and coalesce. Traveling across billions of light-years, these waves arrive at Earth minutely altering space itself, detectable only by the most exquisitely sensitive instruments ever built. The LVK network, combining the US-based LIGO detectors, Italy’s Virgo, and Japan’s KAGRA observatories, listens intently to this faint cosmic choir. Their advanced interferometers split laser beams across multi-kilometer arms, measuring infinitesimal changes caused by passing gravitational waves to reveal the dynamic processes of celestial compact objects.

The newly published GWTC-4 catalog more than doubles the number of gravitational-wave candidates previously recorded, boasting 128 new detections from the latest observing run alone. These entries represent just a portion of approximately 300 mergers spotted during this period, but each candidate enriches our understanding of the range and nature of astrophysical objects in the cosmos. By analyzing these signals, scientists can reconstruct the masses, spins, and distances of the collisions, gaining unprecedented clarity on the evolutionary history of black holes and neutron stars.

At the heart of this progression is the remarkable improvement in the sensitivity and data processing capabilities of gravitational-wave detectors. Upgrades to LIGO’s interferometers now enable searches for signals from binary neutron stars as far as one billion light-years away, while heavier black holes can be observed at even greater distances. The delicate interplay of enhanced hardware and sophisticated computational algorithms is propelling gravitational-wave astronomy from its infancy into a mature, data-rich discipline providing rigorous tests of fundamental physics and astronomical phenomena.

These advancements have uncovered a remarkably diverse spectrum of merging compact objects. Not only do the new data reaffirm the predominance of binary black holes—pairs of black holes orbiting and eventually merging—but they also reveal extraordinary features: the heaviest black hole binaries ever detected, systems where component black holes spin at breathtaking fractions of the speed of light, and binaries with markedly unequal masses. Additionally, there are clear detections of collisions between black holes and neutron stars, expanding the catalog beyond “bread-and-butter” binary mergers and signaling a diversity of cosmic progenitors and evolutionary pathways.

Of particular note is the signal labeled GW231123_135430, originating from the merger of two remarkably massive black holes, each approximately 130 solar masses. This mass scale is significantly higher than that observed in most previous detections, challenging existing models of stellar collapse and black hole formation. The prevailing interpretation is that these black holes may themselves be products of previous mergers, a cosmic chain reaction amplifying their mass and spin properties. Such observations push the boundaries of theoretical astrophysics and hint at complex dynamical environments, such as dense star clusters, where repeated mergers may be commonplace.

On another front, the detection GW231028_153006 showcases unusually high spins in both black holes, with rotational velocities nearing 40% that of light speed. These rapid spins point toward formation scenarios involving previous binary mergers, where angular momentum is accrued through coalescence. Spin measurements are not mere curiosities—they provide crucial diagnostics for distinguishing between isolated stellar evolution and dynamic assembly in dense environments, offering clues about the population synthesis of compact objects.

The LIGO and Virgo detectors employ laser interferometry with unprecedented precision to discern gravitational-wave signals from a noisy background. Minute disturbances on the order of a thousandth the diameter of a proton are teased out, requiring exquisite calibration, data analysis, and cross-verification across multiple detectors. The stochastic nature of these signals means detection rates vary dramatically, with some days yielding multiple events and others none. This randomness reflects the turbulent and episodic nature of astrophysical compact object mergers across the universe.

Beyond identifying individual mergers, the expanding catalog permits population-level studies of black holes and neutron stars, improving statistical confidence in cosmological and astrophysical parameters. For instance, there is mounting evidence that black holes merging earlier in the universe’s history tend to possess higher spins than their more contemporary counterparts. This temporal evolution poses fascinating questions about the astrophysical conditions in the early universe, such as metallicity, stellar dynamics, and the role of environmental factors in black hole spin-up mechanisms.

Moreover, gravitational-wave detections form a novel means of probing the fundamental nature of gravity itself. The general theory of relativity posits gravity as a geometric property of spacetime, predicting specific waveforms for merging black holes’ gravitational-wave emissions. The LVK collaboration’s loudest signals, like GW230814_230901, have undergone rigorous scrutiny for deviations from theoretical predictions. Thus far, Einstein’s formulation has weathered these tests admirably, although improving sensitivity necessitates ever more refined modeling. Observational gravitational-wave astrophysics is thus becoming a critical arena where classical gravity is tested under its most extreme, nonlinear regimes.

The catalog’s insights also enrich cosmology by offering independent measurements of the Hubble constant, the rate at which the universe expands today. By accurately gauging the luminosity distance to merging black hole binaries purely from gravitational-wave signals and combining this with redshift information—when available—astronomers derive key parameters governing cosmic expansion. Although still in early stages, results suggest a Hubble constant of about 76 kilometers per second per megaparsec, a figure that contributes to the ongoing debate between different cosmological probes and measurements.

In essence, each gravitational-wave detection recorded in the GWTC-4 catalog opens a new window into the universe’s most profound mysteries. As the sensitivity of detectors improves and the catalog grows, what started as a handful of detections a decade ago has blossomed into a flood of information challenging and refining our understanding of black holes, neutron stars, cosmology, and fundamental physics alike. Future observing runs promise yet greater discoveries, possibly revealing new classes of objects or unforeseen phenomena that could reshape astrophysics.

This scientific voyage is a testament to the convergence of innovation across experimental physics, computational science, and astrophysics. Enhanced data pipelines, machine learning algorithms, and global collaborations ensure that gravitational-wave astronomy remains at the forefront of discovery, providing a lens onto the violent, dynamic cosmos that no other method can offer. As the universe continues to churn out these cosmic ripples, humanity stands ready to listen, decode, and comprehend the grand celestial narrative encoded in waves traveling through spacetime itself.

Written by Jennifer Chu, MIT News

Subject of Research: Gravitational waves from compact object mergers, population properties of black holes and neutron stars, tests of general relativity, cosmological implications

Article Title: “GWTC-4.0: An Introduction to Version 4.0 of the Gravitational-Wave Transient Catalog”

Web References: http://dx.doi.org/10.3847/2041-8213/ae0c06

References: DOI 10.3847/2041-8213/ae0c06 (GWTC-4.0 Publication)

Image Credits: Ryan Nowicki / Bill Smith / Karan Jani

Keywords

Gravitational waves, Astrophysics, Black holes, Neutron stars, General relativity, Cosmic mergers, LIGO, Virgo, KAGRA, Hubble constant, Cosmology, Compact binaries

The elegance of teleparallel gravity lies in its radical departure from the conventional geometric interpretation of general relativity. Instead of focusing on the curvature of spacetime, teleparallel theories posit that gravity arises from the twisting and shearing of spacetime itself. This torsion, rather than curvature, dictates how objects move. Symmetric teleparallel gravity, a specific formulation within this broader class, introduces an additional layer of symmetry that simplifies the mathematical structure while retaining the ability to describe gravitational phenomena. The Minkowski background, representing a flat and empty universe devoid of matter and energy, serves as the ideal starting point for such investigations. It’s the theoretical equivalent of inspecting the pristine, undisturbed surface of a perfectly still lake before introducing any ripples; any deviations observed from this baseline are then directly attributable to the gravitational theory being tested, allowing for an unclouded examination of its fundamental properties and predictions.

Zhao’s investigation specifically targets “linear perturbations.” This is a crucial aspect of the research, akin to studying the minuscule vibrations of a bridge under a gentle breeze before considering a catastrophic earthquake. By examining how small, localized disturbances propagate and evolve within this theoretical framework, scientists can gain invaluable insights into the fundamental nature of gravity. These perturbations, when analyzed mathematically, can reveal the characteristic “modes” of the gravitational field, much like identifying the specific frequencies that a musical instrument can produce. The stability and behavior of these modes are critical indicators of a theory’s validity and its potential to describe the real universe. In essence, Zhao is performing a highly sophisticated diagnostic on the very language with which we describe the universe’s gravitational interactions.

The choice of the Minkowski background is not arbitrary; it represents the simplest possible spacetime. It is the bedrock upon which more complex gravitational structures are built. By understanding how a theory of gravity behaves in this pristine environment, one can then extrapolate its predictions to more intricate scenarios, such as those involving stars, galaxies, and the expansion of the universe itself. If a theory fails to accurately describe perturbations on a Minkowski background, it is highly unlikely to provide a correct description of gravity when matter and energy are present. Therefore, this foundational analysis is a critical gatekeeper for any proposed modification or alternative to Einstein’s venerable theory. Zhao’s meticulous work on this fundamental canvas provides a robust benchmark for evaluating the explanatory power of symmetric teleparallel gravity.

One of the tantalizing possibilities that emerges from studying gravitational theories in this abstract setting is the potential to shed light on the mysterious phenomena that dominate our universe: dark matter and dark energy. While general relativity, in its standard form, requires the existence of these invisible components to explain galactic rotation curves and the accelerating expansion of the cosmos, these entities remain elusive and have resisted direct detection. Alternative theories of gravity, such as teleparallel gravity, offer the intriguing prospect of explaining these cosmic puzzles without invoking new, unseen substances. By modifying the way gravity itself interacts with spacetime, these theories might naturally account for the observed gravitational effects attributed to dark matter and dark energy, thus providing a more unified and parsimonious explanation for the universe’s grandest structures and its ongoing cosmic drama.

The mathematical rigor employed by Dr. Zhao is essential for this exploration. The equations governing gravitational perturbations can become incredibly complex, especially when dealing with modified gravity theories. Linearization, a technique that simplifies these equations by considering only small deviations from a background solution, allows for analytical or semi-analytical solutions that illuminate the fundamental properties of the theory. This process involves carefully expanding the gravitational field equations around the Minkowski background and then solving the resulting system of linearized equations. The solutions reveal the spectrum of possible gravitational waves and their characteristics, offering a precise framework for comparison with observational data, should such deviations be detectable in the future through sensitive gravitational wave observatories.

The implications of this research extend far beyond academic curiosity. If symmetric teleparallel gravity, or variations thereof, proves capable of explaining cosmological observations without recourse to dark matter or dark energy, it would represent a paradigm shift in our understanding of fundamental physics. It would necessitate a re-evaluation of our cosmological models and potentially open up new avenues for experimental and observational searches. The search for gravitational anomalies, even subtle ones predicted by modified theories, could guide future telescope designs and gravitational wave detector sensitivities. This work, therefore, acts as a theoretical compass, pointing physicists toward potentially fruitful areas of empirical investigation that could redefine our cosmic narrative.

The concept of symmetric teleparallel gravity offers a unique perspective on the gravitational interaction, proposing that it is not the curvature that governs motion, but rather the non-metricity of spacetime. Non-metricity essentially describes how the lengths of vectors change as they are parallel transported around a closed loop. In Einstein’s theory, spacetime is both curved and metric-compatible, meaning parallel transport preserves lengths. Teleparallel theories divorce these concepts, with gravity arising solely from torsion or, in the case of symmetric teleparallel gravity, a specific form of non-metricity that is constrained by symmetry conditions. This intricate interplay of geometric properties, explored through the lens of perturbations, is what Zhao meticulously dissects, seeking to understand its fundamental manifestations.

The rigorous mathematical framework of linear perturbations allows us to ask very specific questions about the nature of gravity. Are there new types of gravitational waves predicted by this theory that differ from those of general relativity? Do these perturbations exhibit any instabilities that would render the theory unphysical? Can these perturbations be excited by realistic astrophysical sources? By answering these questions, Zhao’s work provides a detailed spectral analysis of the gravitational field within this alternative framework. The study of these perturbations on the Minkowski background is akin to sending a ping through the theoretical structure of symmetric teleparallel gravity and listening for the echoes; these echoes reveal the inherent properties and limitations of the system.

The theoretical landscape of modified gravity theories is vast and often fraught with mathematical challenges. Many proposed alternatives to general relativity struggle to remain consistent with a wide range of observational data. However, teleparallel gravity, in its various forms, has shown promise in its ability to reproduce the successes of general relativity while offering potential explanations for cosmological conundrums. The focus on “symmetric” teleparallel gravity imbues the theory with specific properties that simplify its structure and make it amenable to detailed analysis, such as the perturbation study undertaken by Dr. Zhao. This particular formulation might strike a crucial balance between theoretical novelty and observational viability, making it a compelling subject for ongoing research and rigorous testing.

The act of perturbing a system, even a theoretical construct like spacetime, is a fundamental technique in physics. It allows us to understand the dynamics of that system in response to external influences or inherent instabilities. In the context of gravity, linear perturbations on a Minkowski background reveal the fundamental modes of the gravitational field. These modes are the basic building blocks of gravitational phenomena, from the propagation of gravitational waves to the formation of cosmic structures. By understanding how these modes behave within symmetric teleparallel gravity, scientists can ascertain whether this theory offers a compelling alternative to our current understanding of the universe’s gravitational behavior and its evolution.

Furthermore, the study of linear perturbations can reveal whether a theory predicts phenomena that are observationally distinguishable from general relativity. For instance, modifications to gravity might lead to subtle differences in the strength or speed of gravitational waves, or alter the way light bends around massive objects. Identifying such unique signatures is the ultimate goal for experimentalists seeking to test these alternative theories. Zhao’s research lays the groundwork for such potential discoveries by providing a precise theoretical prediction of how gravity would behave under specific conditions within the symmetric teleparallel framework, offering a clear target for future observational campaigns.

The paper’s publication in the European Physical Journal C, a highly respected journal in the field of particle physics and cosmology, underscores the significance of this research. It signals that the work has undergone rigorous peer review and is considered a valuable contribution to the scientific literature. The accessibility of the DOI link further facilitates the rapid dissemination of these findings, allowing researchers worldwide to engage with the details of Zhao’s analysis and build upon its insights, fostering a global collaboration in the quest to unravel the universe’s deepest gravitational secrets and potentially revise our cosmic blueprint.

This research, by focusing on the fundamental behavior of gravity within a simplified yet critical theoretical context, provides a crucial stepping stone in the ongoing quest to understand the universe at its most fundamental level. The mathematical elegance and potential explanatory power of symmetric teleparallel gravity, as illuminated by Zhao’s meticulous analysis of linear perturbations, suggest that we may be on the cusp of a profound revision in our understanding of the force that shapes the cosmos. The potential to resolve long-standing mysteries like dark matter and dark energy through a modification of gravity itself, rather than the addition of unseen components, represents a deeply compelling prospect that will undoubtedly ignite further theoretical exploration and experimental pursuit.

Subject of Research: The fundamental properties and behavior of linear perturbations in symmetric teleparallel gravity on a Minkowski spacetime background. This involves exploring how minute disturbances propagate and evolve within this alternative framework for gravity, with the aim of understanding its implications for the structure and dynamics of spacetime.

Article Title: Linear perturbations of symmetric teleparallel gravity on Minkowski background

Article References: Zhao, D. Linear perturbations of symmetric teleparallel gravity on Minkowski background. Eur. Phys. J. C 85, 1396 (2025). https://doi.org/10.1140/epjc/s10052-025-15146-1

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-15146-1

Keywords: Symmetric teleparallel gravity, linear perturbations, Minkowski spacetime, modified gravity, spacetime geometry, gravitational theory, theoretical physics, cosmology

For over a century, Albert Einstein’s theory of General Relativity has stood as the bedrock of our understanding of gravity, describing it not as a force, but as the curvature of spacetime itself caused by mass and energy. This elegant framework has been a cornerstone of modern physics, accurately predicting phenomena from the bending of starlight around massive objects to the existence of black holes. However, as our observational capabilities have advanced, particularly with the recent groundbreaking detection of gravitational waves, physicists have begun to explore the frontiers and potential limitations of this venerable theory. A new study, published in the European Physical Journal C, ventures into this uncharted territory, proposing and investigating a fascinating modification to Einstein’s gravity that could reshape our comprehension of how cosmic ripples propagate across the universe. This research delves into the realm of “generalized hybrid metric-Palatini gravity,” a theoretical construct designed to reconcile some of the persistent enigmas encountered when attempting to unify gravity with other fundamental forces and to potentially explain the perplexing nature of dark energy and dark matter that dominate the cosmic landscape and influence the behavior of spacetime on the grandest scales, hinting at a universe far more complex than previously imagined.

The detection of gravitational waves – faint tremors in the fabric of spacetime predicted by Einstein and first directly observed by the LIGO and Virgo collaborations – has opened an entirely new window onto the cosmos. These waves, generated by cataclysmic events like the merger of black holes and neutron stars, carry pristine information about the most violent and energetic processes in the universe, unhindered by the electromagnetic interference that obscures light. While these detections have magnificently confirmed Einstein’s predictions, they also present an opportunity to scrutinize the theory under extreme conditions and to probe for subtle deviations that might hint at new physics. The Portuguese research team, led by Dr. Carlos Gomes and colleagues, has taken this opportunity to heart, developing a theoretical framework that extends General Relativity by incorporating additional gravitational degrees of freedom, thereby creating a more comprehensive model that could potentially address observations that currently fall outside the standard paradigm, and offering a fresh perspective on the dynamic evolution of the universe.

The core of the new research lies in the concept of “generalized hybrid metric-Palatini gravity.” Historically, Einstein’s theory relates spacetime curvature directly to the distribution of matter and energy. However, alternative theories have explored variations by introducing additional fields or modifying the fundamental equations. The Palatini formulation, for instance, treats the gravitational connection and the metric as independent variables, leading to different equations of motion compared to the standard metric formulation. The “hybrid” aspect suggests a combination of these approaches, while “generalized” implies that this combination is not a simple addition but a more intricate interplay designed to capture a wider range of gravitational phenomena. This sophisticated theoretical edifice aims to achieve a more robust description of gravity, particularly in regimes where it might deviate from Einstein’s predictions, such as at very high energies or during the universe’s earliest moments, and offers a path to potentially resolving some of the outstanding cosmological puzzles.

One of the most significant motivations for exploring such modified gravity theories stems from the persistent mysteries of dark energy and dark matter. These enigmatic components are inferred from their gravitational effects on visible matter and the expansion of the universe, yet their fundamental nature remains elusive. Standard General Relativity, as it stands, requires the existence of these invisible entities to explain observed cosmic phenomena, such as the accelerated expansion of the universe attributed to dark energy. However, generalized hybrid metric-Palatini gravity offers an alternative. Instead of invoking entirely new substances, this theoretical framework suggests that the observed cosmological effects might be a consequence of gravity itself behaving differently under certain conditions, effectively mimicking the presence of dark energy or dark matter through modifications to the gravitational interaction. This, in turn, could provide a more parsimonious explanation for the universe’s accelerating expansion and the formation of large-scale structures without the need for exotic, unseen matter.

The new study particularly focuses on how gravitational waves propagate within this generalized hybrid metric-Palatini gravity framework. In standard General Relativity, gravitational waves travel at the speed of light. However, modifications to the gravitational action can introduce new polarization modes and alter the propagation speed of these waves. The research team meticulously derived the equations of motion for gravitational waves within their proposed theory. They found that the presence of the additional terms and fields inherent in the generalized hybrid metric-Palatini formulation can lead to deviations in the expected behavior of gravitational waves, potentially impacting their speed and their polarization properties. This is a crucial aspect, as future observations of gravitational waves from distant sources could, in principle, detect such deviations and provide direct evidence for the validity of these modified gravity theories, acting as a powerful diagnostic tool for probing the fundamental nature of gravity.

The implications of these potential deviations in gravitational wave propagation are profound. If gravitational waves were found to travel at a speed different from the speed of light, it would be a definitive smoking gun for physics beyond Einstein’s General Relativity. Furthermore, the existence of additional polarization modes beyond the two predicted by General Relativity (plus and cross polarizations) would also signal a departure from the standard model of gravity. Such discoveries would necessitate a revision of our cosmological models and could offer vital clues about the underlying structure of spacetime and the fundamental forces that govern it. The research meticulously explores these possibilities, presenting the mathematical machinery for calculating these effects and setting the stage for future observational tests that could confirm or refute their theoretical predictions, pushing the boundaries of our cosmic understanding.

The study delves into the specifics of how different terms within the generalized hybrid metric-Palatini action influence the gravitational wave solutions. They explore scenarios where the interaction coupling constants, which dictate the strength of these additional gravitational effects, are varied. By analyzing the equations, they can determine the conditions under which these modifications become significant and observable. This detailed theoretical exploration is essential, as it provides concrete predictions that astronomers and experimental physicists can aim to verify. The precision of current and future gravitational wave detectors, such as LIGO, Virgo, KAGRA, and the upcoming LISA mission, offers a realistic prospect of probing these subtle effects, transforming theoretical speculation into observable cosmology and potentially revolutionizing our understanding of the fundamental forces shaping the universe.

This research represents a significant step in the ongoing quest to develop a more complete and accurate description of gravity that aligns with all available observational data, from the microscopic realm of particle physics to the macroscopic expanse of the cosmos. General Relativity, while incredibly successful, faces theoretical challenges, particularly in its inability to incorporate quantum mechanics or fully explain phenomena like dark energy. Modified gravity theories, like the one proposed here, offer potential avenues to bridge these gaps. By exploring how gravitational waves behave in these alternative frameworks, scientists are not just testing Einstein’s legacy but actively building the next chapter of gravitational physics, creating a more comprehensive picture of the universe’s intricate workings and dynamic evolution, and opening up new avenues for scientific inquiry.

The methodology employed by Gomes and his colleagues involves advanced theoretical calculations within the framework of differential geometry and field theory. They start with the generalized action for hybrid metric-Palatini gravity, which includes terms that modify the standard Einstein-Hilbert action. From this action, they derive the field equations and then specifically analyze the linearized perturbation equations that describe gravitational waves. This perturbation analysis allows them to extract information about the dispersion relations and polarization properties of these waves. The mathematical rigor ensures that the predictions made by the theory are derived from sound physical principles, providing a robust foundation upon which future observational tests can be built and offering a clear path for scientific verification.

The potential to unify gravity with quantum mechanics is another driving force behind the exploration of modified gravity theories. While General Relativity describes gravity on large scales, quantum mechanics governs the universe at subatomic levels. A major unresolved problem in physics is the lack of a consistent theory of quantum gravity. Some extensions to General Relativity might offer a glimpse into how gravity behaves at the quantum level, and observing deviations in gravitational wave propagation could provide experimental hints towards such a unified theory, shedding light on the very nature of reality from the smallest to the largest scales, and connecting two seemingly disparate domains of physics.

The “generalized hybrid metric-Palatini gravity” theory, as explored in this paper, is not merely an abstract mathematical exercise; it is a tangible proposal with potential observable consequences that can be tested against the universe’s own phenomena. The precise measurements of gravitational waves are rapidly advancing, and future observatories are being designed with enhanced sensitivity and broader frequency coverage. This technological progress means that the subtle signatures predicted by modified gravity theories may soon be within our reach. The research team’s work, therefore, serves as a vital theoretical guide, pointing experimentalists toward specific observable features that could confirm or necessitate a revision of our fundamental understanding of gravity and the cosmos.

The paper’s contribution lies in providing a consistent theoretical framework to explore these possibilities. It systematically lays out the mathematical structure of generalized hybrid metric-Palatini gravity and derives the specific predictions for gravitational wave propagation. This detailed analysis makes the theory accessible to further investigation by the broader physics community and provides a concrete foundation for designing future experiments and interpreting their results, fostering a collaborative environment where theoretical insights can directly inform observational endeavors, accelerating the pace of discovery in fundamental physics.

In essence, this research is about pushing the boundaries of our knowledge. It acknowledges the immense success of Einstein’s General Relativity but also recognizes the unanswered questions and the ongoing evolution of our understanding. By proposing and investigating a modified theory of gravity, the scientists are not discarding Einstein’s legacy but building upon it, seeking a more complete picture of the universe. The propagation of gravitational waves in these new theoretical landscapes offers what could be the ultimate testbed for discerning the true nature of gravity, potentially leading to a paradigm shift in our understanding of the cosmos and its most fundamental constituents.

The study serves as a powerful testament to the dynamic nature of scientific inquiry. It highlights how dedicated theoretical work, coupled with advancements in observational technology, can lead to profound insights into the nature of reality. The exploration of generalized hybrid metric-Palatini gravity and its impact on gravitational waves is a prime example of this synergistic process, promising to unveil deeper secrets of the universe and potentially redefine our place within it, pushing the frontiers of human knowledge ever outwards.

Subject of Research: The propagation characteristics of gravitational waves within a modified theory of gravity known as generalized hybrid metric-Palatini gravity. This research explores how deviations from standard Einsteinian gravity might affect the speed, polarization, and other properties of these cosmic ripples.

Article Title: Gravitational wave propagation in generalized hybrid metric-Palatini gravity.

Article References:
Gomes, C., Rosa, J.L. & Pinto, M.A.S. Gravitational wave propagation in generalized hybrid metric-Palatini gravity.
Eur. Phys. J. C 85, 1359 (2025). https://doi.org/10.1140/epjc/s10052-025-15085-x

DOI: https://doi.org/10.1140/epjc/s10052-025-15085-x

Keywords**: Modified gravity, General Relativity, Gravitational waves, Palatini gravity, Hybrid gravity, Spacetime curvature, Cosmology, Dark energy, Dark matter, Astrophysical phenomena, Theoretical physics, Observational cosmology.