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

To understand what is at stake, it helps to recall what torsion is and why anyone cares. In Einstein’s general relativity, spacetime geometry is described entirely by curvature, and the connection that tells parallel vectors how to move is symmetric. Einstein–Cartan theory relaxes that assumption, allowing spacetime to carry torsion, a twisting of the geometry sourced by the intrinsic spin of matter. Because ordinary matter carries very little net spin density, torsion is expected to be fantastically weak, and no experiment has yet measured its coupling to fermions directly. Any theoretical scheme that promises a concrete numerical bound therefore attracts attention, which is precisely why the original paper, authored by L. C. Garcia de Andrade and published in July 2026, seemed exciting.

The original argument ran roughly as follows. Consider a pair of domain walls, thin sheets of concentrated energy, in a spacetime whose geometry is described by a tetrad, the set of local orthonormal frames that generalizes the metric. In the teleparallel formulation, the spin connection is set to zero, so the torsion two-forms are simply the exterior derivatives of the tetrads. Garcia de Andrade computed these torsion components, derived equations for a wall-separation function h(t,z), solved them, and then studied Dirac fermions living on the walls. The effective interaction entering the Dirac equation combined the electromagnetic coupling eB with the torsion coupling g_T multiplied by an axial torsion field S. Setting that combination to zero in a thin-wall limit, the paper extracted g_T as a ratio of the magnetic field to the torsion field, arriving at a bound near ten to the minus four.

Castro’s comment dismantles this chain at its first link. Recomputing the torsion two-forms directly from the tetrad ansatz with vanishing spin connection, he obtains expressions that differ from the published ones by factors of two and by misplaced coefficients. For example, where the original paper reports a coefficient of minus four pi sigma times z h times the time derivative of h in one torsion component, the direct calculation gives minus eight pi sigma times the same combination. Since every downstream equation for the wall dynamics is built on these two-forms, the discrepancies are not cosmetic. They propagate into the differential equations governing h(t,z) and ultimately into the claimed wave equation and its effective propagation velocity, which had been interpreted as a signature of gravitational radiation from the walls.

A second, independent problem involves a subtle but crucial distinction between coordinate components and tetrad components. The original paper writes the torsion components in a form that requires reading the coefficients in the tetrad basis rather than directly in the coordinate basis of time and spatial differentials. When Castro performs the conversion carefully, the equation for the wall-separation function integrates to a logarithmic expression, h squared equals z squared plus an inverse constant times the logarithm of a linear function of z squared and t. This is not the polynomial solution printed in the original paper. Even taking the original’s own scalar equations at face value, Castro shows that their exact integration yields a logarithm, not the cubic polynomial reported, unless an unstated perturbative expansion is smuggled in. The small-h approximation invoked in the original work, he demonstrates, does not rescue the polynomial form either; it merely imposes extra consistency conditions on integration constants.

The consequences ripple outward. The wave equation for the wall separation, and with it the interpretation of an effective velocity and the possibility of gravitational waves emitted by the coalescing walls, does not follow from the corrected calculation. Castro is careful about the limits of his claim: he does not prove that no gravitational radiation or propagating torsional mode can arise in some corrected model. He shows only that the specific derivation in the original paper is invalid as written. He also notes that even within the original’s own reduced flatness constraint, the solution h squared equals two At plus B means the dramatic coalescence of the walls at time zero occurs only for a special choice of the integration constant B, not as a forced consequence of the equations.

The fermionic sector fares no better in Castro’s assessment. Solving the zero-mode Dirac equations with the effective combination A equal to eB plus g_T times S, he finds Gaussian solutions whose signs differ from those published. For positive A, one chiral component is Gaussian-damped and normalizable while the other grows without bound; for negative A, the roles reverse. This is the familiar chiral selection pattern of localized zero modes, and it matters enormously for the condensate. Castro further shows that the scalar Dirac bilinear used to construct the condensate was computed incorrectly: with the beta matrix as defined in the original paper, the bilinear mixes left- and right-handed components and, in any normalizable chiral sector, actually vanishes, even though the positive probability density remains localized. The original analysis, he argues, conflated a scalar condensate with a positive density, two mathematically distinct objects.

Then comes the dimensional problem, which may be the most damning point of all. The original bound rested on identifying the combination eB plus g_T S with a wall thickness or separation, a quantity with dimensions of length. But in natural units a magnetic field has dimensions of mass squared, while an axial torsion vector carries dimensions of mass, so the sum cannot equal a length without additional dimensional parameters. In Castro’s corrected reading, the combination A instead fixes a localization length, roughly one over the square root of the absolute value of A. A thin, tightly localized fermion mode corresponds to a large A, not to A equal to zero. Imposing the cancellation condition, as the original paper did, actually makes the localization length diverge, the opposite of a thin wall. Estimating g_T therefore requires specifying an independent physical scale, and the magnetic field and torsion field alone cannot determine the coupling.

The numbers illustrate how far the original estimate strays. Using the original paper’s own magnetic field of ten to the minus nine gauss and torsion scale of ten to the minus thirty gigaelectronvolts, Castro finds that the cancellation condition yields g_T of order minus six gigaelectronvolts, not ten to the minus four. With a localization length of one centimeter, the corrected formula gives a coupling of order hundreds of gigaelectronvolts; with one meter, the result stays near the cancellation value. The magnetic length associated with the field is about eight centimeters. No unique number emerges without an independently specified scale, which is exactly the kind of hidden assumption that can make a striking bound evaporate under scrutiny.

Castro closes with a measured summary of eight points and a constructive caveat: none of this rules out the possibility that a corrected model of magnetized, torsion-carrying double walls could support localized fermion modes or eventually yield genuine phenomenological constraints on axial torsion. But the specific conclusions and the numerical bound in the original paper, he concludes, require a full rederivation from the ground up. For a field where torsion remains one of gravity’s best-hidden secrets, the episode is a reminder that the path to measuring spacetime’s twist runs through careful bookkeeping as much as through bold ideas, and that in theoretical physics, the factors of two are never just details.

Subject of Research: Corrections to the derivation of axial torsion coupling bounds from fermion condensates in Einstein–Cartan double domain walls

Article Title: Comment on “Einstein–Cartan fermion condensates trapped in double walls induce axial torsion coupling bounds [Eur. Phys. J. C (2026) 86: 785]”

Article References: Castro, L. B. (2026). Comment on “Einstein–Cartan fermion condensates trapped in double walls induce axial torsion coupling bounds [Eur. Phys. J. C (2026) 86: 785]”. The European Physical Journal C, 86(10), Article 1148. https://doi.org/10.1140/epjc/s10052-026-16388-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16388-3

Keywords: Einstein–Cartan gravity, spacetime torsion, teleparallel gravity, fermion condensates, domain walls, Dirac zero modes, axial torsion coupling, gravitational waves, tetrad formalism, dimensional analysis, theoretical particle physics, general relativity

Torsion is not some exotic bolt-on to general relativity. It is the part of the affine connection that Einstein’s theory sets to zero by hand. In Einstein–Cartan gravity, torsion is sourced by the intrinsic, quantum-mechanical spin of matter, and its effects normally become competitive only at the most extreme fermion densities imaginable. The theory has a celebrated pedigree: in the 1970s, physicists showed that torsion could avert the initial singularity, replacing the Big Bang with a cosmic bounce. But a recent proposal by researchers Yun and Lee went much further, claiming that in Einstein–Cartan cosmology the scalar torsion mode could rescue a famously broken model of dark energy, produce late-time acceleration, and even allow the equation of state to cross the so-called phantom divide, with possible relevance to the DESI anomaly. The new paper shows, with brutal clarity, that the claim fails, and that it fails for reasons reaching well beyond this particular model.

The technical heart of the argument is a single scaling relation. If the matter sector of the universe is separately conserved, the homogeneous torsion scalar Phi must obey a dilution equation that forces it to fall as the inverse cube of the scale factor. When this mode enters the Friedmann constraint, it appears as a negative energy density scaling as the inverse sixth power of the scale factor, a stiff component with an effective equation of state equal to plus one. This combination is an old acquaintance in disguise: it is precisely how the spin–spin contact interaction of a Weyssenhoff spin fluid enters Einstein–Cartan cosmology, where it powers the nonsingular bounce at high density. The physics under scrutiny is therefore the classical Einstein–Cartan bounce, examined at the opposite end of cosmic history, and the question is how much room the data leave for it there.

The first casualty is the holographic dark energy component itself. Holographic dark energy ties the dark energy density to an infrared cutoff length scale, and the most natural choice, the Hubble radius, famously fails in ordinary general relativity: because the density is proportional to the square of the Hubble rate, it merely tracks the dominant component and can never drive acceleration. That failure is why the field migrated to the future event horizon, with all its causality troubles. Yun and Lee claimed torsion changes this verdict. The new analysis shows it does not. Because the holographic density is proportional to H squared, it drops out of the deceleration parameter identically: the deceleration parameter is the same function of matter and torsion with or without it. Whatever acceleration the model produces belongs to torsion alone, and the cancellation is structural, not an accident of the quadratic choice. Any cutoff built from the Hubble rate alone inherits its dynamics from the other components instead of supplying its own, torsion or no torsion.

So where does the acceleration actually come from? From the bounce, and only from the bounce. The constraint that the Hubble rate squared must remain nonnegative enforces a turning point at which the expansion rate vanishes, the classical Einstein–Cartan bounce. The entire accelerating regime is confined to a transient window around that bounce, spanning less than a factor of about 1.6 in scale factor. The universe exits the bounce super-accelerating, crosses the phantom divide partway through the window, stops accelerating at its edge, and then settles into behaving like cold matter with a rapidly fading memory of the bounce. The window through which torsion could accelerate the universe closes before any observer can look through it. Dragging that window to redshifts where acceleration is actually observed would force the Hubble rate to vanish in our recent past, and the measured expansion history says emphatically otherwise.

What turns this qualitative observation into hard numbers is a requirement almost embarrassing in its modesty: the universe must contain its own past. Because the torsion term grows as the sixth power of one plus redshift going backward in time, faster than matter, radiation, or anything else, demanding that the expansion rate remain real up to some redshift the universe demonstrably reached yields a ladder of nested upper bounds on the torsion density parameter. A fit to the official DESI DR2 baryon acoustic oscillation likelihood gives a bound of 8.7 times ten to the minus four at 95 percent confidence. The mere existence of the spectroscopically confirmed galaxy JADES-GS-z14-0 at redshift 14.32 tightens this to 8.4 times ten to the minus five. The cosmic microwave background pushes it to 3.1 times ten to the minus ten. And Big Bang nucleosynthesis crushes it to 5 times ten to the minus twenty-four. Each rung rests on strictly weaker assumptions than the one before, and the last one settles the question beyond any reasonable doubt.

The DESI fit itself is a model of care. The authors used the released data vector and covariance exactly as distributed with the collaboration likelihood, thirteen entries spanning the brightest galaxy sample and six anisotropic tracers, and validated their pipeline by reproducing the published DESI Lambda-CDM fit to full precision before opening the torsion parameter. The result: the data express no preference whatsoever for torsion, with a negligible improvement in chi-squared. Even saturating the weakest bound would place the torsion-induced turning point at a redshift of roughly six, leaving no universe for the quasars and galaxies we observe beyond that epoch. The baryon acoustic data run out of constraining power exactly where the existence bounds take over, a striking convergence of independent lines of evidence.

The imprint on today’s dark energy equation of state makes the irrelevance concrete. The torsion contribution shifts the effective equation of state by an amount bounded by twice the torsion density parameter divided by the dark energy density. Even the weakest rung of the ladder caps this shift at a few parts in a thousand, on the phantom side of minus one, where DESI does not want it. The DESI preference for evolving dark energy asks for deviations of order a quarter on the quintessence side. That is a gap of two orders of magnitude at best, and twenty-two at worst, with the wrong sign throughout. In the Granda–Oliveros cutoff, an alternative infrared choice that includes the Hubble derivative, the verdict inverts in a darkly comic way: torsion does not prevent the big-rip singularity, it deepens it, adding to the super-acceleration instead of moderating it as it becomes ever more irrelevant on approach to the rip.

Every no-go theorem is a list of assumptions read backwards, and this one has a short list. The load-bearing assumption is adiabaticity: the requirement that matter be separately conserved, which forces the inverse-cube scaling. The authors show in an appendix that this scaling is nothing deeper than dilution, the kinematics of a comoving spin fluid whose particle number falls as the volume grows. Escaping the no-go therefore costs real physics. One must let matter and torsion exchange energy, turning the conservation identity into the definition of an interacting dark sector, and demand a spin density that refuses to dilute, through progressive spin alignment, condensation, or some other mechanism operating at late times. In Einstein–Cartan gravity torsion is algebraic and cannot outlive its source, so the mechanism must live in the matter sector explicitly, making a fermionic dark matter component with evolving spin alignment the natural candidate. Either sign of the spin contact term lands on the same verdict: no late-time acceleration from the adiabatic mode, with the sign merely deciding whether the early universe gets a bounce in compensation.

The authors are careful to delimit their target. Torsion cosmology at large survives this paper. Dynamical torsion in Poincaré gauge theory, where propagating modes can oscillate and drive late-time acceleration, remains untouched, as do interacting matter–torsion scenarios and the steady-state vectorial mode, which has already been confronted with DESI, supernova and microwave background data with results consistent with zero at sub-percent precision. But the message for the holographic dark energy program is stark. The Hubble cutoff’s Einstein–Cartan incarnation inherits, intact, the defect identified two decades ago: proportionality to the Hubble rate squared makes it a spectator. Torsion adds a bounce, and the bounce brings a brief phantom display, but every part of it stays locked behind a wall at high redshift. The window through which torsion could have accelerated the universe sits, and always sat, on the far side of that wall. If torsion has something to say about dark energy, it will have to say it where adiabaticity breaks, and that is precisely where the authors say their work is heading next.

Subject of Research: A no-go result for torsion-based holographic dark energy in Einstein–Cartan cosmology

Article Title: No late-time role for adiabatic torsion: a no-go result for Hubble-cutoff holographic dark energy in Einstein–Cartan cosmology

Article References: Izaurieta, F., Lepe, S., & Quinzacara, C. (2026). No late-time role for adiabatic torsion: a no-go result for Hubble-cutoff holographic dark energy in Einstein–Cartan cosmology. The European Physical Journal C, 86(9), Article 1088. https://doi.org/10.1140/epjc/s10052-026-16337-0

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16337-0

Keywords: dark energy, torsion, Einstein–Cartan gravity, holographic dark energy, DESI, cosmology, no-go theorem, phantom divide, Big Bang nucleosynthesis, cosmic microwave background, baryon acoustic oscillations, Hubble radius