The work, carried out by L. C. Garcia de Andrade of the Institute for Cosmology and Natural Philosophy, builds on the teleparallel equivalent of General Relativity, known as TEGR. Teleparallel gravity is mathematically equivalent to Einstein’s theory in its predictions, but it reformulates gravity in terms of torsion rather than curvature. Where general relativity describes gravity as the bending of spacetime, teleparallelism describes the same physics as a twisting of the internal frames that define direction and orientation at every point. Because the two pictures agree on observable outcomes, teleparallelism offers a complementary mathematical toolkit—one that makes certain questions about torsion, which are invisible in the standard formulation, suddenly tractable.

The heart of the new analysis lies in a class of topological defects called global textures. These are remnants of the early universe, formed when a quantum field that once filled all of space settled into its present state in a non-uniform way. Textures are thought to have seeded clumps of matter in galaxies and the voids between them, and they carry a characteristic energy scale set by the symmetry-breaking field. Crucially, the author treats textures not just as gravitational sources but as seeds of dynamical dark energy, drawing on earlier ideas that torsion could mimic the repulsive gravity associated with the universe’s accelerated expansion—a connection first explored by Gasperini in Einstein–Cartan cosmology and extended by Wanas to dark energy scenarios.

To extract a numerical constraint, the study matches an interior solution—a global texture obeying Einstein’s field equations—to an exterior flat spacetime, using the Israel–Lichnerowicz matching conditions generalized to Einstein–Cartan geometry. These conditions demand that the metric be continuous across the boundary and that the jump in its first derivatives be balanced by torsion. Working in null coordinates, the so-called light-cone approximation where the coordinate u equals t minus r, the author simplifies the matching equations dramatically. The texture metric used is the Notzold solution, characterized by a small parameter epsilon equal to eight pi G eta squared, where eta is the symmetry-breaking scale, bounded above by roughly ten to the eighteen GeV.

The payoff is striking. Applying the matching conditions to the angular part of the metric, and adopting the tachyonic approximation in which the radial coordinate exceeds the time coordinate, the torsion trace component T-squared-sub-1-2 comes out bounded below ten to the minus forty-four GeV. That is thirteen orders of magnitude tighter than the ten to the minus thirty-one GeV limit that Kostelecky and collaborators obtained in 2008 using dual maser experiments to search for torsion-induced Lorentz violation. The key to the estimate is identifying the texture correlation length with the inverse of the Hubble constant, roughly ten to the minus forty-two GeV in natural units, which converts the geometric matching condition into a cosmological bound. A second, independent component of the torsion trace, denoted K, is found to be approximately half of the first, confirming the internal consistency of the framework.

The analysis then turns to cosmic strings, the line-like cousins of textures. In teleparallel gravity, the connection splits into a Riemannian part and a contortion tensor, and with the teleparallel spin connection set to zero, the geodesic equation can be written entirely in terms of contortion. For a cosmic string metric with tension G mu, the contortion component K-sub-3-1-3 reduces to a simple expression involving the string tension and the radial coordinate. Plugging in the observational bound on string tension, eight G mu less than or equal to eight times ten to the minus seven, and taking a characteristic distance of one parsec, the author recovers a torsion trace bound of ten to the minus twenty-four per second—remarkably, the same order of magnitude that Nitsch derived in 1979 using metric-affine gravity and the physics of spinning particles on Earth.

Perhaps the most visually evocative result concerns what happens to particles near a torsion-carrying string. Solving the geodesic equations shows that a test particle moving in the plane orthogonal to the string follows a planar spiral, with angular velocity equal to one minus eight G mu divided by the radius. The author interprets this two-dimensional spiral as the projection of a three-dimensional Cartan spiral staircase, a structure previously explored by Alexander, Jenks, and Magueijo in the context of parity-violating cosmology. Because the radial acceleration is positive when the integration constant is positive, the teleparallel string is repulsive rather than attractive—reinforcing the picture of torsion-laden defects as candidates for the dark energy that drives cosmic acceleration. In this scenario, particles do not fall into the defect; they are flung outward while winding around it like threads on a screw.

The final strand of the argument connects torsion to gravitational waves, bringing the analysis into contact with real detector data. Gravitational waves propagating through teleparallel spacetime induce torsion waves, and the relation between the wave amplitude and the torsion component takes a simple form: the strain is proportional to the torsion divided by the gravitational-wave frequency. For cosmological gravitational waves at frequencies around ten to the minus seven hertz, a torsion of ten to the minus twenty-four per second implies a strain of order ten to the minus seventeen, comfortably within the sensitivity band of modern interferometers. For black-hole merger signals at higher frequencies near ten to the minus five hertz, the predicted torsion-induced strain drops to about ten to the minus twenty-three, close to the ten to the minus twenty-one strains actually recorded by the LIGO–Virgo collaboration. Using Rumpf’s older terrestrial bound of ten to the minus fifteen per second instead would amplify the predicted strain to ten to the minus eight, illustrating how dramatically torsion can enhance gravitational-wave amplitudes and how future detectors might discriminate between torsion scenarios.

Beyond the numbers, the study sketches a broader vision in which topological defects serve as laboratories for physics that no particle accelerator can reach. The author notes that the torsion–Hubble connection hinted at in the texture bound may even bear on the Hubble tension, the persistent discrepancy between different measurements of the universe’s expansion rate, following recent proposals that torsion condensates could alleviate the conflict. Future work, the paper suggests, will extend the framework to Einstein–Cartan–Kalb–Ramond gravity, where a fully antisymmetric three-index field introduces axion-like torsion, and to hybrid defects such as domain walls threaded by cosmic strings, including strings threading teleparallel wormholes.

For now, the headline result stands: by treating cosmic textures as gravitational matching problems in null coordinates, teleparallel gravity delivers the most stringent constraint yet on a component of Cartan torsion, pushing it below ten to the minus forty-four GeV. Whether nature actually exploits torsion at any measurable level remains open, but the study demonstrates that the universe’s largest structures—defects stretched across cosmic horizons—can probe its subtlest geometric twists with a precision that rivals, and in this case surpasses, the finest instruments ever built. If torsion exists, the spiral staircases winding through spacetime just got a great deal harder to hide.

Subject of Research: Constraints on Cartan spacetime torsion from teleparallel topological defects and gravitational waves

Article Title: Torsion constraints driven by TEGR topological defects in null coordinates, GWs and matching conditions

Article References: de Andrade, L. C. G. (2026). Torsion constraints driven by TEGR topological defects in null coordinates, GWs and matching conditions. The European Physical Journal C, 86(9), Article 1038. https://doi.org/10.1140/epjc/s10052-026-15957-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-15957-w

Keywords: torsion, teleparallel gravity, TEGR, cosmic textures, cosmic strings, topological defects, dark energy, gravitational waves, LIGO, Einstein-Cartan theory, matching conditions, contortion

The problem with previous models lies in their simplifications. Population synthesis codes, which simulate millions of stellar binaries at once, typically treat stars as simplified one-dimensional objects and make sweeping assumptions about what happens when one star dumps its outer layers onto its companion. Whether the transfer of mass is stable or runaway, how much of the material is actually retained by the accreting star, and how the angular momentum of the orbit is redistributed inside the stellar interiors all remain deeply uncertain. These uncertainties have left the fate of the stable mass transfer channel, long considered a leading route to double black hole formation, frustratingly ambiguous. Earlier detailed calculations even suggested that the channel might produce far fewer merging systems than the observed gravitational-wave population requires.

Xu and his collaborators took a different approach. Rather than relying on approximate prescriptions, they constructed detailed binary evolution models using the MESA stellar evolution code, following two massive stars from the zero-age main sequence, the moment they begin burning hydrogen, all the way to the formation of their black holes. Crucially, their models include the internal differential rotation of the stars and the simultaneous transfer of both mass and angular momentum between the components. This level of physical fidelity allows the team to track the chemical structure of the initially less massive star in exquisite detail, first as it accretes material from its more massive partner and later as it, in turn, overflows its own Roche lobe and transfers mass back.

The chemical structure turns out to be the hidden protagonist of the story. When the initially more massive star, the primary, loses material to its companion, the accreting secondary is polluted with fresh hydrogen-rich gas that dilutes the chemically processed layers near its surface. By the time the secondary itself expands and begins its own episode of Roche-lobe overflow, the reverse mass transfer, its internal composition profile determines how the star responds. A star whose interior has been thoroughly mixed behaves very differently from one with a steep chemical gradient. In the models of Xu and colleagues, the accretion history imprinted on the secondary’s chemical structure strongly influences how much mass the star can stably donate, how tight the orbit becomes, and how much angular momentum the system retains.

To explore the consequences systematically, the team computed a grid of such models spanning a range of initial orbital separations and mass ratios, all with a fixed primary mass and at the metallicity of the Small Magellanic Cloud, a nearby dwarf galaxy whose metal-poor environment resembles the conditions in many star-forming regions across cosmic history. Metallicity matters because it controls the strength of stellar winds: at low metallicity, massive stars lose less mass, retain more of their heft, and are more likely to leave behind substantial black holes. The Small Magellanic Cloud thus serves as a natural laboratory for the kind of environments in which many of the observed gravitational-wave sources likely formed.

The outcome of the grid is striking. The models produce black hole binaries whose mass ratios and spin magnitudes fall squarely within the regime observed for gravitational-wave sources with primary black hole masses between roughly 10 and 25 solar masses. In particular, the stable mass transfer channel naturally generates moderately unequal-mass systems, binaries in which one black hole is noticeably heavier than the other, a configuration that has proven difficult to reproduce with some alternative formation scenarios. Because the accreting star is spun up by the material it receives, the resulting black holes can carry significant natal spins, and the predicted effective spin parameters of the merging binaries align well with the values inferred from the gravitational-wave waveforms in the fourth Gravitational-Wave Transient Catalog.

The physical chain of events deserves a closer look. In a close massive binary, the primary exhausts the hydrogen in its core and expands, spilling material through the inner Lagrangian point onto its companion. If the transfer is stable, meaning the donor shrinks in step with its Roche lobe rather than engulfing the companion in a common envelope, the system survives with its two stars intact but transformed. The primary, stripped of its envelope, evolves into a helium star and eventually collapses into a black hole. The secondary, now rejuvenated and rotating rapidly, continues its own evolution before undergoing a second episode of mass transfer, this time onto the newly formed black hole or, in some configurations, shedding mass from the system entirely. The final orbit must be tight enough that gravitational radiation, as described by the classic Peters formulas, drives the two black holes to coalescence within a Hubble time.

One of the most compelling aspects of the new work is that the ingredients for this channel are not exotic. As a large number of potential progenitor binaries are already known to exist among the observed population of massive binaries, the authors conclude that stable mass transfer might contribute considerably to the observed gravitational-wave source population, especially to mergers of moderately unequal-mass black hole binaries. Observational surveys of massive stars, including the Binarity at LOw Metallicity campaign, have revealed that a high fraction of massive stars live in close binaries and interact with their companions, providing an ample reservoir of systems that could follow exactly the evolutionary path traced by the new models.

The study also sharpens the contrast between competing formation scenarios. In the common envelope channel, a star engulfs its companion and the pair spirals together inside a shared gaseous shroud, a process so violent and poorly understood that its efficiency remains one of the great open problems in stellar astrophysics. In the chemically homogeneous evolution channel, very tight binaries rotate so fast that their interiors stay well mixed, allowing both stars to burn their fuel almost entirely before collapsing. The stable mass transfer channel now emerges as a third, quantitatively robust pillar, one that can be tested against the growing statistical power of the gravitational-wave catalogs. As the catalog expands with each observing run, the distributions of mass ratios and effective spins become increasingly precise fingerprints of the underlying astrophysics.

What makes the result resonate beyond the specialist community is the elegance of the mechanism. Two stars exchanging material back and forth over millions of years, with the chemistry of their interiors quietly dictating the outcome, ultimately produce the most violent events since the Big Bang, events that humanity now detects as fleeting chirps in laser interferometers. The work of Xu and colleagues demonstrates that resolving the mystery of merging black holes requires not grander telescopes but deeper fidelity in the physics of stars themselves. By following the rotation, the angular momentum and the chemical stratification of massive binaries from birth to collapse, the study transforms a channel once considered marginal into a credible, perhaps dominant, factory of the gravitational-wave sources that now illuminate the dark side of the Universe.

Subject of Research: Formation of merging binary black holes through stable mass transfer in massive binary star evolution

Article Title: The formation of merging black holes via stable mass transfer in massive binary stars

Article References: Xu, X.-T., Langer, N., Klencki, J., Wang, C., & Li, X.-D. (2026). The formation of merging black holes via stable mass transfer in massive binary stars. Nature Astronomy. https://doi.org/10.1038/s41550-026-03000-7

Image Credits: AI Generated

DOI: 10.1038/s41550-026-03000-7

Keywords: binary black holes, gravitational waves, stable mass transfer, massive binary stars, stellar evolution, black hole spins, MESA, Small Magellanic Cloud, Roche-lobe overflow, common envelope, LIGO, Nature Astronomy

The work is rooted in the effective-one-body (EOB) framework, a celebrated theoretical tool originally developed by Thibault Damour and Alessandra Buonanno. The central idea is elegant: instead of solving the fiendishly complicated two-body problem of general relativity directly, one maps it onto the motion of a single effective particle traveling through a deformed version of spacetime. If the mapping is done correctly, the geodesic motion of that fictitious particle reproduces the full conservative dynamics of the real binary. The catch is that the deformation, encoded in an effective spacetime metric, is not unique. Theorists must decide how quantities in the real two-body system correspond to quantities in the effective one, and different choices of this dictionary lead to different effective metrics that nonetheless describe the same physics.

Historically, the EOB framework was built on the post-Newtonian expansion, which works well when the orbiting bodies move slowly and gravity is weak, as in the early inspiral of a binary. But the final moments before merger are neither slow nor weak-field, and that is where the post-Minkowskian (PM) expansion enters. Rather than expanding in velocity, the PM expansion treats gravity’s coupling constant G as the small parameter while keeping velocities fully relativistic. Recent breakthroughs in scattering-amplitude calculations, notably by Zvi Bern and collaborators in 2019, have supplied the explicit conservative Hamiltonian for a nonspinning two-body system at third post-Minkowskian order, opening the door to pushing EOB models into the fast-motion regime.

Most previous PM-EOB constructions relied on scattering problems: two bodies fly past each other and deflect, and theorists match the scattering angle computed in the real system to the one computed in the effective spacetime. That approach has produced impressive results, but it requires an additional conceptual step, the Kalin-Porto map, to translate scattering information into predictions for bound orbits. The new study takes a more direct route. Because the most common gravitational-wave sources detected so far are bound, inspiraling binaries rather than unbound scattering events, the authors work exclusively with quantities intrinsic to closed orbits: the radial action variable and the precession angle of an elliptical trajectory.

The radial action is an adiabatic invariant, a quantity that stays constant as the orbit slowly evolves, computed by integrating the square root of an effective radial potential between the periastron and apastron turning points. Using the 3PM Hamiltonian from scattering-amplitude research, the team evaluated this integral for the real two-body system and, in parallel, for a test particle moving in a deformed Schwarzschild-like metric. A subtle technical point emerged: the direct third-order contribution to the radial action vanishes, but the full 3PM information is not lost. Because the radial action is a nonlinear functional of the potential, cross-terms appear at fourth order in G divided by the cube of angular momentum, and these terms carry exactly the missing information. Matching the real and effective radial actions at this level yields a set of correspondence relations without ever introducing genuine fourth-post-Minkowskian dynamics.

To check this result independently, the authors turned to the precession angle, the amount by which a bound orbit fails to close after each revolution, the same relativistic effect famously first observed in the orbit of Mercury. For the real system, the calculation reduces to an integral that evaluates to a complete elliptic integral of the first kind, which can be expanded analytically in the weak-binding limit. For the effective system, the team worked in isotropic coordinates, where the radial momentum squared takes a particularly clean form terminating at the inverse-cube term, dramatically simplifying the analysis. Remarkably, matching the precession angles produced exactly the same correspondence relations as the radial action method, a consistency check that strongly supports the validity of the construction.

With the correspondence established, the remaining freedom in the effective metric had to be fixed through a choice of parametrization. The authors adopted an isotropic gauge with a Schwarzschild-like parametrization, setting the first-order coefficient of the spatial part to unity and higher-order coefficients to zero, an alternative to the Finsler-type corrections used in Damour’s earlier scattering-based approach. The result is an explicit effective metric whose second- and third-order coefficients, a2 and a3, are expressed entirely in terms of the physical parameters of the real binary, including its total relativistic energy. Intriguingly, the metric depends on the conserved energy of the orbit, meaning each bound orbit is described by its own member of a family of effective geometries rather than by a single universal spacetime. The authors emphasize that this energy dependence is physically sensible: different initial configurations emit different gravitational waves, and since the energy is conserved, the corresponding effective metric remains fixed along any given conservative orbit.

The team validated the construction by comparing the binding energy of quasi-circular orbits predicted by their new 3PM Hamiltonian against existing PM-EOB models, using both a recent fourth-post-Minkowskian Hamiltonian and numerical relativity simulations as benchmarks. The new formulation tracked the high-precision baselines more closely, and it did so without introducing Finsler-type terms or post-Newtonian input, which the authors highlight as a key advantage. The agreement with existing results at second post-Minkowskian order further confirms internal consistency, while the differences between approximants grow as the binary moves from inspiral toward merger, precisely the regime where improved modeling matters most for gravitational-wave astronomy.

The implications extend beyond formal elegance. EOB models calibrated to numerical relativity are the workhorses of LIGO and Virgo data analysis, and every improvement in the underlying analytical description translates into more accurate templates, better parameter estimation, and sharper tests of general relativity in the strong-field regime. By grounding the PM-EOB program directly in bound-state observables, the new work removes a conceptual detour and provides a practical tool for modeling the gravitational waves emitted by compact binaries. The authors outline clear next steps: extending the energy map and effective metric to higher post-Minkowskian orders and generalizing the framework to binaries with spin, the configuration relevant for most of the black hole mergers actually observed. As detectors grow more sensitive and begin resolving fainter, more distant mergers, the demand for theoretical templates of ever-greater fidelity will only intensify, and constructions like this one form the mathematical backbone of that effort.

Subject of Research: Effective-one-body modeling of bound two-body dynamics in general relativity at third post-Minkowskian order

Article Title: Effective metric for bound state in an effective-one-body theory based on the third-post-Minkowskian approximation

Article References: Zou, H., Long, S., He, X., & Cao, Z. (2026). Effective metric for bound state in an effective-one-body theory based on the third-post-Minkowskian approximation. The European Physical Journal C, 86(9), Article 1128. https://doi.org/10.1140/epjc/s10052-026-16354-z

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16354-z

Keywords: gravitational waves, effective-one-body, post-Minkowskian, two-body problem, black hole binaries, general relativity, precession angle, radial action, scattering amplitudes, numerical relativity, LIGO, theoretical physics