Scalar–tensor theories have long been considered promising alternatives to the standard cosmological model because they introduce scalar degrees of freedom in addition to the metric tensor of general relativity. The Freund–Nambu formulation, introduced by Emil Freund and Yutaka Nambu in the late 1960s, augments the gravitational action with a real scalar field φ of mass μ that couples nonminimally to the Ricci scalar and, crucially, to matter itself. In the action written in geometrized units, the kinetic term of the scalar field is divided by the factor 1 + 2qφ, where the constant q governs the strength of the coupling between the scalar field and geometry.

Matter is coupled to the conformally rescaled metric (1 + 2qφ)g_αβ, so the scalar field does not merely ride along with gravity but actively reshapes how matter and spacetime talk to each other. Varying the total action with respect to the metric and the scalar field yields a set of coupled nonlinear differential equations: the Einstein tensor equals the sum of the scalar-field and matter energy-momentum tensors, while the scalar field obeys a wave equation with a source proportional to q times the total trace of the energy-momentum. To make analytic progress, the authors focused on the vacuum configuration, setting the scalar mass μ to zero and switching off the matter sector entirely.

This reduction isolates the purely geometric effects of a self-interacting massless scalar field on a curved background. Working with a static, spherically symmetric ansatz in which the product of the time and radial metric components equals minus one, the team reduced the field equations to a pair of simple differential equations for the metric functions. The radial function λ satisfies λ'' − 2 = 0, giving λ = r² + 2C₃r + C₄, while the second metric function follows from an integrable first-order relation.

After a coordinate transformation and careful identification of the integration constants with the mass M of the central object and a scalar parameter n, the final metric takes a strikingly compact form: ds² = −(1 − 2M/nr)ⁿ dt² + (1 − 2M/nr)⁻ⁿ dr² + r²(1 − 2M/nr)^(1−n)(dθ² + sin²θ dφ²). The associated scalar field profile is where the new physics lives. In Freund–Nambu theory the scalar field acquires a nonlinear self-interaction term proportional to q, so the solution reads φ = (√(1−n²)/2) ln(1 − 2M/nr) multiplied by a bracket containing an additional q-dependent logarithmic correction.

When q vanishes, the solution reduces exactly to the classic Janis–Newman–Winicour metric with its massless scalar field, confirming that the new spacetime is a natural parametrized extension of that important class. The metric itself retains the JNW form governed by n, while the new parameter q deforms only the scalar field's radial profile. This separation is elegant: the geometry is unchanged, but the scalar field that test particles can feel directly is modified, opening a clean observational window onto the coupling.

Naked singularities are the provocative backdrop for this entire program. In general relativity, a naked singularity is a region where curvature invariants such as the Kretschmann scalar diverge without being hidden behind an event horizon, meaning null and timelike geodesics originating arbitrarily close to the singularity can reach distant observers. Because the Freund–Nambu extension preserves the JNW metric while deforming the scalar profile, it offers a controlled laboratory for asking how scalar self-interaction changes strong-field phenomena without altering the underlying geometry.

The team then turned to test-particle dynamics, introducing a direct linear coupling g_s between a massive particle and the scalar field through an effective mass m* = m(1 + g_sφ). The authors are careful to note that this coupling is phenomenological: in minimally coupled models ordinary particles need not feel the scalar at all, while in scalar–tensor theories an effective coupling can arise through scalar-dependent masses or conformal-frame transformations. The parameters q and g_s therefore carry different physical meanings—q characterizes the background scalar configuration, while g_s characterizes the particle's scalar charge—and a specific relation between them would require a microscopic matter model beyond the scope of the analysis.

The innermost stable circular orbit, or ISCO, emerges as the key diagnostic. For g_s = 0 the team recovers the known analytic JNW result, r_ISCO = (3 + 1/n + √(5 − 1/n²))M. With nonzero coupling, the ISCO conditions become highly nontrivial because the nonlinear scalar contributions enter the effective potential multiplicatively, and numerical methods are required.

The results show a clear asymmetry with the sign of the coupling: for positive g_s the ISCO radius decreases, indicating that attractive scalar interactions enhance orbital stability at smaller radii, while for negative g_s the ISCO shifts outward, signaling a destabilizing effect. Radiative efficiency, defined as η = 1 − E_ISCO, follows suit. At n = 1 all curves converge to roughly 6 percent regardless of parameters, but for n ≠ 1 positive g_s enhances efficiency while negative g_s reduces it, and q mainly splits the curves apart.

In Kerr spacetime the ISCO radius depends on the black hole spin parameter a, shrinking from 6M for a Schwarzschild black hole toward M for a maximally rotating one. The authors demonstrate that certain combinations of q, n, and g_s can reproduce ISCO shifts typically attributed to spin, even when the intrinsic angular momentum is absent. Small values of the spacetime parameter n produce ISCO changes comparable to those of rapidly rotating Kerr black holes.

For certain parameter combinations, the effective ISCO radius corresponds to spin values comparable to those inferred for Sgr A* (a/M ≈ 0.65–0.9) and M87* (a/M ≈ 0.75–1). This means that interpreting X-ray observations in terms of spin alone could be misleading, and that multi-frequency quasi-periodic oscillation modeling is essential to break the degeneracy. That modeling is exactly what the team delivered.

By expanding the effective potential around circular orbits, they derived the radial and vertical epicyclic frequencies; by spherical symmetry the vertical frequency coincides with the orbital frequency. Within the epicyclic resonance model, the upper twin-peak QPO frequency is identified with the orbital frequency and the lower with the radial epicyclic frequency. The parameter n exerts the most pronounced influence, decreasing both frequencies across the radial range, while q modestly increases both and g_s raises the radial frequency while suppressing the orbital one.

The consistency of the fits with real data suggests that Freund–Nambu scalar–tensor gravity remains a viable and testable framework for strong-field physics beyond general relativity. The distinctive imprints of the parameters q, n, and g_s on orbital frequencies, ISCO locations, and accretion efficiency mean that current and future X-ray observations can discriminate between this class of theories and the Kerr paradigm.

Subject of Research: An exact spherically symmetric vacuum solution in Freund–Nambu scalar–tensor gravity and its astrophysical constraints from black hole X-ray observations

Article Title: Spherically symmetric vacuum solution in Freund–Nambu scalar–tensor gravity

Article References: Davlataliev, A., Turimov, B., Ahmedov, B., Vyblyi, Y., Yuan, C., & Zhou, C. (2026). Spherically symmetric vacuum solution in Freund–Nambu scalar–tensor gravity. The European Physical Journal C, 86(10), Article 1177. https://doi.org/10.1140/epjc/s10052-026-16324-5

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16324-5

Keywords: scalar-tensor gravity, Freund-Nambu theory, naked singularity, Janis-Newman-Winicour spacetime, black holes, ISCO, quasi-periodic oscillations, epicyclic resonance, Markov Chain Monte Carlo, microquasars, X-ray binaries, modified gravity

Published in The European Physical Journal C as a regular article in Theoretical Physics, the work examines the Bergmann-Wagoner-Nordtvedt class of scalar-tensor theories coupled to nonlinear electromagnetic fields through a Lagrangian of the form L(ψ,F), where ψ is a scalar field and F is the electromagnetic invariant. The framework permits arbitrary scalar self-interaction potentials and scalar fields that are canonical, phantom, or partly phantom in different regions of spacetime. By working in the Einstein conformal frame, the researchers derived a single Schrödinger-like equation for radial perturbation modes, together with a general effective potential intended to determine their behavior.

It incorporates the scalar field’s kinetic function, the electromagnetic Lagrangian and its derivatives, background metric functions, electric and magnetic charges, and the spatial variation of the scalar field. After separating a perturbation into a time-dependent factor and a radial mode, the master equation takes the form Yzz + [Ω² − Veff(z)]Y = 0, where z is a tortoise-like radial coordinate and Ω is the mode frequency. A nontrivial solution with Im Ω less than or equal to zero that satisfies physically reasonable boundary conditions, including the absence of ingoing waves at the ends of the coordinate range, indicates instability.

If no such solution exists, the system is stable in the linear approximation. Stability therefore has to be assessed as a boundary-value problem with conditions appropriate to the background space-time. A striking result emerges when the nonlinear electromagnetic Lagrangian has the correct Maxwell limit at weak fields.

In the zero-charge limit, every trace of the nonlinear theory disappears from the effective potential, so the perturbation dynamics coincide with those of the corresponding vacuum scalar-tensor solution. This conclusion also holds for the exceptional Brans-Dicke wormhole studied by the authors: it remains stable under the radial perturbations considered when it carries nonzero electric charge, magnetic charge, or both, but is unstable in the uncharged, symmetric case. The finding means that nonlinear electrodynamics, under the assumptions examined, does not remove the instability of that neutral configuration.

The same result applies to the family of scalar-tensor electrovacuum configurations in which the electromagnetic invariant vanishes everywhere and the electric and magnetic charge magnitudes are equal. Under a Maxwell-like weak-field condition, L(F) and its first derivative at F = 0 reduce to the corresponding Maxwell values. The electromagnetic stress-energy tensor and the effective potential governing monopole perturbations are then identical to those of Maxwell-coupled scalar-tensor gravity.

Consequently, established stability results for these Maxwell configurations extend directly to their nonlinear-electrodynamics counterparts without modification. This equivalence is an analytical result under the stated conditions rather than a numerical approximation. The authors also considered cases in which the nonlinear electromagnetic field is not trivial.

Their worked example lies within general relativity rather than a nonminimally coupled scalar-tensor theory: a massless scalar field and truncated Born-Infeld electrodynamics are minimally coupled to gravity. The Born-Infeld Lagrangian includes a parameter b that controls the departure from Maxwell behavior, while the scalar field’s radial flux supplies an integration constant. The researchers numerically solved the coupled background equations and then evaluated the radial effective potential.

The study was conducted in units with the Schwarzschild mass fixed at m = 1, and the numerical integration began at a large radius using asymptotic expansions whose neglected terms were estimated to be no greater than 10^−12. For these example configurations, the nonlinear theory changes the central matter distribution without erasing the characteristic instability inherited from the corresponding scalar-Maxwell system. With only electric charge or only magnetic charge, the electromagnetic stress-energy diverges near the center more slowly than in Maxwell electrodynamics, and the background develops a scalar-type naked singularity.

When both charge types are present but unequal, the balance between scalar and electromagnetic contributions produces regions with either scalar-type or Reissner-Nordström-type singularities. The division between the two depends on the Born-Infeld parameter and the total charge, while the results remain symmetric under exchanging electric and magnetic charge. The equal-charge case has F = 0 and coincides with the previously described Penney solution.

Numerically, the effective potential approaches zero at large radius but develops an infinitely deep well close to the center. For scalar-type singularities it approaches −1/(4z²), while for Reissner-Nordström-type singularities it approaches −2/(9z²). These are the same leading forms found for the scalar-Maxwell configurations.

Changing the Born-Infeld parameter altered the potential only slightly: at one representative radius, the reported value shifted from −2.049 in the Maxwell limit to −2.072 at b = 10, a difference of about one percent. The authors therefore interpret the Born-Infeld result as a quantitative refinement rather than a qualitative change in the radial stability properties. The implication is that radial perturbations in the studied nonlinear-electrodynamics configurations can exhibit the same catastrophic linear instability as their Maxwell counterparts, even when the initial perturbations are small.

Such growth makes a linear calculation alone inadequate for describing the subsequent evolution and requires a nonperturbative treatment. These results do not amount to a universal verdict on nonlinear-electrodynamics black holes or other compact objects: the analysis is restricted to static spherical backgrounds and radial perturbations, and its Born-Infeld example is a singular general-relativistic system rather than a broad survey of scalar-tensor black holes. The authors identify quasinormal modes, shadows and other observational properties as subjects for subsequent work.

Subject of Research: Linear radial stability of scalar-tensor gravity coupled to nonlinear electrodynamics

Article Title: Nonlinear electrodynamics and stability of spherically symmetric space-times in scalar-tensor gravity

Article References: Bronnikov, K. A., Ibadov, R., Shaymanova, F. Y., & Shukurullokhon, N. M. (2026). Nonlinear electrodynamics and stability of spherically symmetric space-times in scalar-tensor gravity. The European Physical Journal C, 86(10), Article 1190. https://doi.org/10.1140/epjc/s10052-026-16469-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16469-3

Keywords: scalar-tensor gravity, nonlinear electrodynamics, radial perturbations, linear stability, effective potential, Born-Infeld electrodynamics, Bergmann-Wagoner-Nordtvedt theory, Brans-Dicke gravity, spherical symmetry, gravitational instability, Nonlinear, electrodynamics

Brans-Dicke theory, first proposed in the 1960s, is one of the oldest and most celebrated extensions of Einstein’s general relativity. In it, the strength of gravity itself is not fixed but governed by a scalar field that varies across space and time. The original theory contains a single adjustable parameter, the Brans-Dicke parameter, which controls how strongly this scalar field interacts with the curvature of spacetime. Solar system experiments, notably radio tracking of the Cassini spacecraft, constrain this parameter to be enormous, roughly greater than ten thousand, which pushes ordinary Brans-Dicke theory very close to general relativity and limits its usefulness as a cosmological alternative. The new work revives the framework by generalising it in two directions at once: the scalar field is non-minimally coupled to gravity through a function of the field, and its kinetic term is non-canonical, meaning the field’s motion contributes to the cosmic energy budget in a field-dependent way rather than in the standard quadratic form.

Technically, the authors work with an action in which the Ricci scalar is multiplied by a function f of the scalar field, taken to be one plus a coupling constant xi times the field, while the scalar sector takes the form of a kinetic term divided by that same function, minus a potential. This particular structure is inspired by Noether symmetry arguments, a mathematical technique for selecting physically consistent Lagrangians. The construction is deliberately broad: setting the coupling xi to zero and choosing a constant potential recovers standard LambdaCDM, while other choices reproduce quintessence and the original Brans-Dicke theory. The non-canonical kinetic term, in which the usual kinetic energy is scaled by the inverse of the gravitational coupling function, is what distinguishes this model from most scalar-tensor studies and, as the authors show, noticeably changes the resulting cosmic dynamics even when the dynamical variables are defined in ways similar to those used in simpler models.

To analyse the cosmology, the team assumes a flat, homogeneous and isotropic universe filled with pressureless matter and radiation, then rewrites the modified Friedmann equations and the scalar field equation as an autonomous system of first-order differential equations. This dynamical systems approach, a staple of modern theoretical cosmology, replaces the difficult task of solving the equations outright with the more tractable problem of finding equilibrium states, called critical points, and determining whether they are stable. The dimensionless variables capture the kinetic and potential energy of the scalar field relative to the total energy density, along with rescaled matter and radiation densities. Because of the non-minimal coupling, these rescaled densities are not the physical ones, but they relate to them through the coupling function, a subtlety the authors handle carefully throughout.

A key constraint shapes the whole analysis. For the phase space of solutions to be bounded, and therefore mathematically well-behaved, the coupling parameter omega must be positive; if it were negative, the allowed region of the phase space would become a hyperbola stretching to infinity, and the phantom-like field it describes would require more elaborate compactification techniques. Positivity of omega also guarantees the absence of ghosts and Laplacian instabilities, the pathological modes that render many modified gravity theories unphysical. With these restrictions in place, the authors investigate three widely used potentials in turn: a constant potential, a power-law potential and an exponential potential, visualising the flow of cosmic histories with bounded phase portraits.

For the constant potential, the analysis uncovers a sequence of critical points that mirrors the broad architecture of LambdaCDM. A radiation-dominated saddle point gives way to a mixed matter-and-scalar-field epoch, and the universe finally settles into a scalar-field-dominated attractor that can drive accelerated expansion, provided omega exceeds half of xi squared, a condition easily satisfied when deviations from general relativity are small. Intriguingly, the matter-dominated point in this model always carries some scalar field contribution, a feature absent from simpler quintessence models, though tuning the couplings can suppress it. Numerical evolutions show that choosing omega at least an order of magnitude larger than xi shifts matter-radiation equality to a redshift of order one thousand, matching observations, whereas smaller values of omega push equality to redshifts around one hundred, which is ruled out by data.

The power-law potential, written in a form that keeps the key dynamical variable constant, produces richer behaviour with two additional saddle points at which the scalar field mixes with radiation or matter. These mixed epochs can even exhibit transient acceleration for a narrow range of negative exponents, though the authors caution that this does not guarantee a consistent later cosmology. For the illustrative inverse-square case, viability requires the matter-dominated point to be a saddle and the late-time attractor to exist, which constrains the ratio of omega to xi squared to lie either below one half or above six and a half. Notably, the team finds no tracking behaviour of the kind familiar from minimally coupled quintessence, which they attribute to the non-minimal coupling, since tracking solutions were never designed for such theories. They also verify that their critical points reduce exactly to those of standard Brans-Dicke cosmology under an appropriate limit, a reassuring consistency check.

The exponential potential brings the most surprising result. Because of the non-minimal coupling, the slope variable of the potential becomes dynamical rather than fixed, and this generates an entirely new critical point: a fully scalar-field-dominated, dark and accelerating attractor that cannot exist in standard quintessence with an exponential potential. This point is stable for all relevant values of the coupling and serves as a genuine late-time dark energy attractor. However, the model also exhibits an early epoch of kination, in which the kinetic energy of the scalar field dominates, which does not appear in the standard cosmic history and must be suppressed through careful choice of initial conditions. Fitting the physical density parameters to a LambdaCDM-like evolution suggests that couplings of order one require an essentially flat potential, while smaller couplings allow a gentle exponential slope of order one tenth.

Across all three potentials, a consistent picture emerges: viable cosmologies exist, but they demand that deviations from general relativity be modest, with the coupling xi comfortably below one, and that the parameters satisfy specific stability and acceleration conditions summarised by the authors in a compact table. The dynamical behaviour differs noticeably from other scalar-tensor models despite superficially similar variables, underlining how the combination of non-minimal coupling and non-canonical kinetics reshapes the cosmic phase portrait. None of the models provides an early inflationary starting point from the dynamics alone, meaning these scenarios are best understood as descriptions of the late universe, though the exponential case offers a saddle point that could act as an inflationary transition for suitable parameters.

The study is a background-level analysis, and the authors are explicit about what comes next. Cosmological perturbations in these models must be evolved and confronted with observations of structure growth and the cosmic microwave background before any claim can be made that they ease the Hubble tension or the clustering tension. The team also flags the Palatini formalism, an alternative variational approach in which Brans-Dicke parameters can take much lower values, as a promising direction. Still, the message of the work is striking: a sixty-year-old alternative to Einstein’s gravity, generalised with a non-canonical twist, can reproduce the entire background history of our universe, from radiation domination through matter domination to today’s accelerated expansion, and may yet hold clues to the anomalies that currently trouble precision cosmology.

Subject of Research: Cosmological dynamics of a non-canonical generalised Brans-Dicke scalar-tensor theory as an alternative to LambdaCDM dark energy

Article Title: Cosmological dynamics of a non-canonical generalised Brans-Dicke theory

Article References: Debono, M., Said, J. L., & Farrugia, G. (2026). Cosmological dynamics of a non-canonical generalised Brans-Dicke theory. General Relativity and Gravitation, 58(10), Article 116. https://doi.org/10.1007/s10714-026-03616-0

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03616-0

Keywords: Brans-Dicke theory, dark energy, cosmology, dynamical systems, scalar-tensor gravity, Hubble tension, LambdaCDM, modified gravity, quintessence, critical points, non-canonical scalar field, General Relativity and Gravitation

The phenomenon at stake is called spontaneous scalarization, a concept first discovered in the 1990s for compact stars by Thibault Damour and Gilles Esposito-Farèse. In scalar-tensor gravity, matter couples to a scalar field through a function that vanishes in weak fields, so the Solar System looks exactly like general relativity. Near dense objects, however, the coupling can switch on, and the scalar field grows abruptly from an apparently stable, scalar-free configuration. The onset of this growth is a linear instability: the scalar perturbation obeys an equation with a tachyonic effective mass, meaning small disturbances grow exponentially in time. Scalarization has since been studied extensively for neutron stars and for black holes surrounded by smooth distributions of matter such as dark-matter halos, where the trace of the matter stress tensor gives the scalar a position-dependent effective mass.

What makes the new work striking is its choice of minimal laboratory. Instead of a smooth halo with an arbitrary density profile, Kaminaga modeled the surrounding matter as two infinitely thin, concentric spherical shells at radii beyond the event horizon. In the thin-layer limit, each shell contributes a delta-function interaction to the radial equation governing the scalar perturbation, with a strength fixed not by hand but by an invariant quantity: the trace of the shell’s surface stress tensor, a combination of its surface energy density and tangential pressure. This choice matters. A trace-free shell, where the surface density equals twice the tangential pressure, does not couple to the scalar field at all, so the instability is controlled by the physics of the matter, not by an adjustable potential parameter.

The central result is a deceptively simple algebraic condition. In the probe limit, where the shells are gravitationally light and the background geometry is exactly Schwarzschild, the onset of instability is determined by the equation 1 minus a1, minus a2, plus one minus chi, times a1 a2, equal to zero. Here a1 and a2 are the attractive strengths of the two shells, each normalized by its own single-shell threshold on the same background, and chi is a purely geometric overlap factor, the ratio of two logarithms of the Schwarzschild metric function at the shell radii. The mathematics behind this formula borrows from the theory of Schrödinger operators with delta interactions supported on spheres, converting the instability question into a two-by-two boundary determinant problem. The determinant’s off-diagonal term represents static scalar exchange between the separated shells, and it is precisely this term that permits collective destabilization.

The consequence is dramatic. There exists an open region of parameter space in which both shells individually lie below their scalarization thresholds, with normalized strengths less than one, yet the pair together produces a growing s-wave mode. For equal normalized strengths, the critical attraction is one divided by one plus the square root of chi, a number that always falls between one half and one. When the two shells are nearly coincident, each needs only about half of its isolated critical strength, exactly as if the two layers had merged into a single double-strength shell. When they are far apart, the overlap factor chi shrinks toward zero and each shell must approach its own separate threshold. For a representative configuration with shells at four and eight Schwarzschild radii, chi equals about 0.415, and the collective threshold drops to roughly 0.61 of the individual critical strength. Notably, an inner shell pushed toward the horizon becomes weakly coupled to an outer layer in these normalized units, a suppression that is a distinctly black-hole effect arising from the horizon boundary condition and the logarithmic stretching of the tortoise coordinate.

The analysis goes further than a threshold. Kaminaga derived the critical scalar cloud at onset in closed form: a static, node-free profile that is regular at the future horizon, changes slope as it crosses each shell, and decays like one over r at infinity. This cloud is a zero-energy resonance rather than a square-integrable bound state, sitting at the edge of the continuous spectrum. When the critical curve is crossed, the resonance descends into a genuine growing mode. A numerical check confirmed the analytic prediction: solving the full frequency-domain determinant with high-precision integration showed no growing mode below threshold, exactly one above it, and a growth rate that vanishes continuously as the threshold is approached from the unstable side. The study also proves that the s-wave sector is always the first to destabilize, and that each angular momentum channel contains at most two radial growth rates, consequences of a comparison principle between angular sectors and a finite-rank structure of the shell interactions.

Crucially, the result survives beyond the probe approximation. Kaminaga constructed an exact scalar-free background consisting of three Schwarzschild regions of different masses, joined across the two shells by the Israel junction conditions that general relativity imposes on surface layers. On this spacetime, the same finite-rank determinant governs the onset, with the shell strengths computed from the exact surface traces and a static scalar resistance function determined by the lapse and curvature factors. A comparison for concrete shell radii showed that the exact critical coupling differs from the probe value by a correction linear in the total shell mass ratio, confirming that the collective effect is not an artifact of neglecting the geometry’s response to the matter. The formal framework is a careful double scaling: the shells are light enough to leave the metric essentially Schwarzschild, while their coupling to the scalar field, multiplied by the scalar-matter coupling constant, remains finite.

The exact Israel calculation also delivered a surprise with geometric flavor. The surface trace of a static shell changes sign as the shell moves radially, and the transition condition, that the geometric mean of the metric factors on the two sides equals one third, reduces in the light-shell limit to the Schwarzschild photon sphere at radius three times the black-hole mass. A shell sitting just inside the photon sphere, between two and a half and three Schwarzschild radii, can satisfy the surface dominant energy condition while carrying a positive trace, which flips the sign of the scalar coupling needed for attraction. This connects the abstract stability analysis to well-known features of black-hole optics, hinting that the geography of strong gravity itself shapes when matter can seed scalar hair.

The authors are careful about physical scope, and so should readers be. The analysis establishes the linear onset only; the nonlinear scalarized branch beyond threshold, the mechanical equation of state of the shells, and their radial stability remain open problems. The model is spherical and static, so accretion disks, rotation, and the superradiant channels of Kerr black holes lie outside it. Moreover, the large coupling constants that light pressureless shells would require to reach the threshold, of order several hundred in the worked examples, are far above the values permitted by binary-pulsar constraints in the standard Damour-Esposito-Farèse model, although the positive-coupling regime associated with pressurized shells near the photon sphere has a different phenomenological status that has not yet been assessed. What the paper delivers instead is a clean, exactly solvable template for a genuinely collective effect: an analytic demonstration that the whole can be unstable when the parts are not, governed by a single geometric overlap number. As physicists increasingly treat black holes as environmental objects, embedded in halos, disks, and stratified clouds, such solvable benchmarks will be essential for deciding when the environment, and its internal cooperation, writes new hair on the horizon.

Subject of Research: Collective matter-induced scalarization instability of a black hole surrounded by two thin shells in scalar-tensor gravity

Article Title: Collective onset of matter-induced scalarization around a black hole with two thin shells

Article References: Kaminaga, M. (2026). Collective onset of matter-induced scalarization around a black hole with two thin shells. General Relativity and Gravitation, 58(10), Article 115. https://doi.org/10.1007/s10714-026-03620-4

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03620-4

Keywords: black holes, scalarization, scalar-tensor gravity, thin shells, Israel junction conditions, tachyonic instability, Schwarzschild spacetime, linear stability, Birman-Schwinger principle, general relativity, no-hair theorem, compact objects

The stakes come from a subtle but consequential feature of the DESI data. When the latest baryon-acoustic-oscillation measurements are combined with cosmic microwave background constraints and Type Ia supernova distances, the data can be summarized by a phenomenological form in which the dark energy equation of state, written as w(a) = w0 + wa(1 − a), prefers w0 slightly greater than −1 and wa negative. That combination implies that the effective equation of state crosses the so-called phantom divide at w = −1 at some point in the recent expansion history. For theorists, this crossing is notoriously difficult. A canonical scalar field with a standard kinetic term can never push w below −1, while a field with a wrong-sign kinetic term can do so but at the cost of ghost instabilities, quantum states of negative norm that signal a breakdown of the theory. Stable phantom-crossing models therefore demand extra structure: additional degrees of freedom, effective-field-theory operators, modified gravity, or scalar-tensor dynamics.

Li and Liu’s proposal builds on the running-curvaton framework, in which a light scalar field called the curvaton contributes to the primordial curvature perturbation that seeded all structure in the universe. In the original, minimally coupled version of the model, the same field was expected to survive to the present day and act as dark energy, but its late-time behavior resembled thawing quintessence, a form of dynamical dark energy that cannot naturally reproduce the DESI-motivated phantom-crossing region. The authors’ key move is to add a Jordan-frame non-minimal coupling, a term of the form ξχ²R that ties the curvaton field directly to the Ricci scalar, a measure of spacetime curvature. This coupling is not exotic by the standards of modern field theory; scalar fields coupled to curvature have been studied for decades in inflation and scalar-tensor gravity, and such theories belong to the well-studied Horndeski class. The novelty lies in embedding the coupling within the running-curvaton scenario in a way that preserves the model’s early-universe successes.

That preservation is the first technical hurdle the paper clears. During slow-roll inflation, the Ricci scalar is approximately 12H², where H is the Hubble expansion rate, so the non-minimal coupling shifts the effective mass of the curvaton fluctuation from its original value to m_eff² = 6H²(g0 + 2ξ), where g0 parametrizes the inflaton-curvaton interaction. Crucially, the observable spectral tilt of the curvaton perturbations depends only on the combination g0 + 2ξ. The authors show that this shift can be absorbed, at leading order, by retuning the mass parameter: one simply defines an observed coupling g0_obs = g0 + 2ξ and adjusts g0 accordingly. The predicted spectral index n_χ − 1 ≈ 4g0_obs then matches the minimally coupled model exactly. Similarly, the local non-Gaussianity parameter f_NL, which measures departures from Gaussian statistics in the primordial perturbations, is controlled mainly by the curvaton’s energy fraction at decay, with f_NL ≈ 5/(4r_dec) in the sudden-decay approximation. Because spacetime is nearly flat during radiation domination, R ≈ 0, the curvature-induced mass correction is suppressed precisely when it matters most for non-Gaussianity, and the Planck satellite’s observational bounds remain satisfied. Tensor perturbations fare equally well: since the curvaton is a subdominant spectator during inflation, the correction to the primordial gravitational-wave amplitude is of order ξχ²/M_P², negligible for sub-Planckian field values.

The second hurdle is the late-time dynamics, and this is where the non-minimal coupling earns its keep. After the inflaton decays, the surviving part of the curvaton potential, a phenomenological form V(χ) = V1(1 − V0 e^(−λχ/M_P)), drives the field’s dark-energy-like evolution. When the modified Friedmann equations are recast in the standard Einstein form, the effective dark-energy pressure acquires geometric terms involving the coupling, such as −ξχ²(2Ḣ + 3H²) and −2ξ(χχ̈ + χ̇²). The sum of the effective dark-energy density and pressure, which determines whether w is above or below −1, becomes ρ+p = (1 − 2ξ)χ̇² + 2ξ(Hχχ̇ − χχ̈ − χ²Ḣ). The second group of terms can become sufficiently negative during late-time evolution to push the effective equation of state across the phantom divide, all without introducing a phantom kinetic term. The theory retains a canonical kinetic structure, and because the Horndeski function G4 depends on the field but not on its kinetic variable, the speed of gravitational waves remains exactly luminal, c_T² = 1, consistent with the landmark constraint from the neutron-star merger GW170817.

To test whether these theoretical trajectories are actually compatible with observations, the authors performed a Markov-chain Monte Carlo likelihood analysis using numerical solutions of the model’s homogeneous background equations. The observational input combined three pillars: a reduced CMB prior on the acoustic angular scale and the physical baryon and total matter densities, the official DESI DR2 baryon-acoustic-oscillation data vector with its full 13-by-13 covariance matrix, and the Pantheon+ supernova compilation with 1580 objects, deliberately excluding the SH0ES Cepheid calibration to avoid entangling the analysis with the Hubble-tension debate. The sampled parameters were the dimensionless Hubble parameter h, the baryon and cold dark matter densities, and four microscopic quantities of the curvaton sector: the coupling ξ, the potential slope λ, the potential normalization V0, and the initial field value χ_i. The late-time normalization V1 was fixed for each sample by a shooting condition requiring the normalized expansion rate to equal unity today.

The results are encouraging but carefully circumscribed. The combined data yield H0 = 67.88 km/s/Mpc with an uncertainty of roughly half a percent, a matter density Ωm = 0.3072, and dark energy parameters w0 = −0.922 and wa = −0.205, placing the posterior squarely in the DESI-motivated phantom-crossing region. The best-fit point improves the chi-square relative to a flat ΛCDM reference fit by Δχ² = −6.56, but because the curvaton model carries four additional microscopic parameters, the Akaike information criterion actually favors ΛCDM by ΔAIC = +1.44. The authors are explicit about this: the model is compatible with the distance data and can modestly improve the best fit, but the analysis does not constitute a decisive statistical preference over the standard cosmological constant. The posterior bands show the median equation of state starting above −1 today and evolving toward a mildly phantom regime at intermediate redshift, while the expansion-rate history H(z)/H0 remains tightly constrained, demonstrating that the phantom-crossing mechanism does not spoil the distance information.

Equally instructive is what the data cannot yet pin down. The derived background quantities, including H0, Ωm, the sound horizon r_d, w0 and wa, are compressed sharply by the likelihood, but the microscopic parameters ξ, λ, V0 and χ_i remain broad, skewed and long-tailed in the marginalized posterior. A quantitative compression diagnostic makes the point vivid: the 68 percent posterior width of ξ spans about 61 percent of its allowed prior range, whereas the corresponding ratios for h and the cold dark matter density are only about 2 to 3 percent. A supplementary run widening the prior on ξ from 5 to 8 broadens the high-ξ tail further without the chain running away, confirming that the current bound is prior-conditional. This degeneracy is not a failure of the mechanism but a genuine limitation of background-level data, which constrain the expansion history far more tightly than the underlying microphysics that produces it.

The authors are equally candid about the analysis’s boundaries. The CMB information was compressed to three early-background numbers, so the calculation does not include the non-minimal coupling’s effects on the full temperature and polarization spectra, CMB lensing, the late integrated Sachs-Wolfe effect, or scalar perturbation evolution. A complete test will require implementing the model in a Boltzmann code, the standard machinery used to compute CMB anisotropies from first principles. Local gravity constraints also demand a separate post-Newtonian or environmental analysis: in high-density regions the scalar mass receives a contribution proportional to the local matter density, suggesting possible Yukawa suppression of fifth-force effects analogous to chameleon or symmetron screening, but no definitive Solar-System viability claim is made. Within those limits, the study delivers a clean proof of concept: the same field that may have shaped the seeds of galaxies in the infant universe could, through its intimate coupling to spacetime curvature, be steering the accelerated expansion we observe today, and it can do so while crossing the phantom divide that has long frustrated simpler models of dark energy.

Subject of Research: A non-minimally coupled running curvaton model linking primordial perturbations to phantom-crossing dynamical dark energy constrained by DESI, CMB and supernova data

Article Title: Non-minimally coupled running curvaton for DESI-motivated dynamical dark energy

Article References: Li, B., & Liu, L.-H. (2026). Non-minimally coupled running curvaton for DESI-motivated dynamical dark energy. The European Physical Journal C, 86(10), Article 1156. https://doi.org/10.1140/epjc/s10052-026-16396-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16396-3

Keywords: dark energy, curvaton, non-minimal coupling, DESI, phantom divide, cosmology, scalar-tensor gravity, baryon acoustic oscillations, Pantheon+, cosmic microwave background, Horndeski theory, MCMC

The problem the authors set out to address is the so-called Hubble tension. When the Planck satellite measures the cosmic microwave background, the afterglow of the Big Bang, and interprets the data through the standard Lambda-CDM model, it infers a present-day expansion rate of about 67.4 kilometers per second per megaparsec. When astronomers in the SH0ES collaboration measure the same quantity directly using Cepheid variable stars and Type Ia supernovae in the local universe, they obtain roughly 73.0 in the same units. The two numbers disagree by far more than their combined uncertainties, and after a decade of increasingly precise measurements, the discrepancy can no longer be dismissed as a statistical fluke. Something, somewhere, is missing from our picture of the cosmos.

The new framework belongs to a class of theories known as scalar-vector-tensor gravity. In addition to the familiar spin-two graviton of general relativity, the gravitational sector contains a scalar degree of freedom, denoted by a dimensionless field Xi, and a massive vector field. The crucial ingredient is a conformal coupling: ordinary matter does not respond to the bare Einstein-frame metric but to a rescaled version of it, multiplied by the factor one plus Xi. Because atoms, rulers and clocks all live in this rescaled Jordan frame, the expansion rate that matter observers actually measure differs from the Einstein-frame rate. In the small-field regime relevant to the model, the observed Hubble rate receives an additive correction proportional to the scalar field’s velocity, meaning a slowly rolling scalar can systematically boost the expansion rate inferred at low redshift without touching the early universe.

Here the authors make a subtle but decisive observation that elevates the paper above many previous attempts. The Hubble constant itself is not a conformal invariant: it depends on which metric you quote it in. However, the acoustic angular scale measured by the cosmic microwave background, the ratio of the sound horizon at recombination to the comoving distance to the last-scattering surface, is exactly invariant under conformal transformations, because the conformal factor cancels between time and distance. This means the Hubble tension is a genuine physical discrepancy between invariants, not an artifact of bookkeeping, and it imposes a strict integral condition that any viable modification of the expansion history must satisfy. The team proves that a scalar field whose velocity keeps a single sign throughout cosmic history necessarily violates this condition, shifting the acoustic angle by about 2.5 percent, roughly eighty times the observational uncertainty.

The escape route is a two-epoch scalar evolution in which the field’s velocity changes sign. In the late universe, a positive scalar velocity enhances the observed Hubble rate toward the locally measured value of 73. Before recombination, an earlier component with the opposite sign drives the scalar field to a small positive value that freezes in, effectively rescaling the gravitational constant by about 2.4 percent during the acoustic era. This shrinks the sound horizon by just the right amount to compensate for the late-time enhancement, allowing the model to match the observed acoustic scale exactly while still delivering a present-day Hubble rate of 73 kilometers per second per megaparsec. Crucially, the microphysics of recombination is untouched: because particle masses are constant in the Jordan frame, hydrogen recombines at exactly the same temperature as in the standard model, and only the expansion rate at that epoch is altered.

The framework also enforces a constraint that simpler treatments would miss. Because the conformal coupling exchanges energy between matter and the scalar sector, the scalar is not separately conserved; its continuity equation carries a source term proportional to the baryon density. Where the early scalar velocity is negative, this term drains the scalar’s energy density as one looks back in time. Requiring the scalar energy density to remain positive at all epochs bounds the transition redshift of the early component to values below roughly 13, carving a well-defined window in the model’s parameter space. Even within that window, the authors are candid about costs: the total matter density emerges as an output rather than an input and lands slightly above the Planck-preferred value, and residual deviations of a few percent remain in baryon acoustic oscillation observables at intermediate redshifts, larger than current DESI uncertainties.

Perhaps the most striking feature is that the same scalar field doubles as a dynamical dark-energy sector. Its self-interaction potential, reconstructed self-consistently from the expansion history rather than assumed in advance, reproduces the late-time accelerated expansion, with a characteristic energy scale of about one millielectronvolt to the fourth power. The model predicts a present-day dark-energy equation-of-state parameter that deviates from minus one at the level of roughly one percent, with the deviation tied directly to the same quantity that controls the Hubble enhancement. This is a falsifiable correlation: future measurements of the time evolution of dark energy, of the kind now being pursued by the Dark Energy Spectroscopic Instrument, whose latest results already mildly favor evolving dark energy, will directly test whether this link holds.

The vector sector completes the unified dark sector with two distinct contributions. The temporal component of the massive vector field is not a propagating particle at all but an auxiliary variable fixed algebraically by the conserved baryon number current. Its energy is an apparent, matter-like contribution to the background expansion, a manifestation of interaction energy rather than a true fluid, locked to the baryons and unable to cluster independently. The propagating spatial modes, by contrast, can form a coherent condensate that, when the vector mass greatly exceeds the Hubble rate, oscillates rapidly and time-averages into a collisionless, pressureless component scaling exactly like cold dark matter. Unlike the baryon-locked interaction energy, this condensate clusters on its own and can supply the non-oscillating gravitational wells that the observed acoustic peak structure demands, its abundance set by primordial initial conditions.

One formidable obstacle remains: local gravity tests. An unscreened scalar with the order-unity coupling adopted in the numerical analysis would renormalize Newton’s constant by fifty percent in the Solar System, grossly violating the Cassini constraint on light deflection. The authors show that a chameleon-type screening mechanism, in which the scalar acquires a large effective mass in dense environments and becomes short-ranged, is essential rather than optional, and that for the chosen normalization it requires the scalar potential to contain a runaway piece. Alternatively, the coupling can be weakened by pushing the mass scale two hundred times beyond the Planck mass, a rescaling the background cosmology accommodates without changing its predictions. Either way, the survival of the model hinges on screening physics that must be verified in realistic astrophysical settings.

The authors are appropriately measured about what they have achieved. Matching the acoustic scale exactly is by construction, not prediction; what is nontrivial is that a consistent solution exists at a fixed high Hubble constant with only a modest pre-recombination gravitational shift. A full statistical confrontation with cosmic microwave background priors, baryon acoustic oscillations, supernovae and growth-of-structure data, together with a complete perturbation analysis in a Boltzmann code, is the necessary next step. But if the framework survives that scrutiny, cosmology may have gained something remarkable: a single gravitational extension in which dark energy, dark matter and the Hubble tension are not three separate puzzles, but three faces of one underlying field.

Subject of Research: A scalar-vector-tensor gravity model that unifies the dark sector and alleviates the Hubble tension through conformal scalar coupling and a vector condensate acting as cold dark matter.

Article Title: Unified dark sector and Hubble-tension alleviation in scalar–vector–tensor gravity

Article References: Jusufi, K., Khodahami, A. A., Sheykhi, A., Said, J. L., & Saridakis, E. N. (2026). Unified dark sector and Hubble-tension alleviation in scalar–vector–tensor gravity. The European Physical Journal C, 86(10), Article 1149. https://doi.org/10.1140/epjc/s10052-026-16406-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16406-4

Keywords: Hubble tension, dark energy, dark matter, scalar-tensor gravity, vector field, cosmology, conformal coupling, cosmic microwave background, Lambda-CDM, modified gravity, recombination, DESI

The research, carried out by B. Yildirim and A. A. Coley of the Department of Mathematics and Statistics at Dalhousie University in Halifax, Canada, approaches the problem through the mathematics of scalar-tensor gravity. This class of theories extends Einstein’s general relativity by adding a scalar field that dynamically couples to the curvature of spacetime. Scalar-tensor gravity has long attracted attention in cosmology because it provides a natural mathematical setting in which nonsingular bouncing universes can be realized. In such models, the scalar field’s evolution can drive a contraction to a halt and trigger a rebound, replacing the dreaded big bang singularity with a smooth, finite transition. But while bouncing cosmologies have been studied extensively in homogeneous settings, the inclusion of localized objects such as black holes makes the field equations dramatically harder to solve.

To make progress, the authors adopted a perturbative strategy, treating the black hole as a small deviation from an otherwise perfectly uniform cosmos. At leading order in their perturbative scheme, controlled by a small parameter epsilon, the solution is a spatially flat Friedmann-Lemaître-Robertson-Walker, or FLRW, spacetime undergoing a bounce, sourced by a perfect fluid of radiation. This background captures the essence of a bouncing cosmology in the simplest possible form: the universe contracts, reaches a minimum size at a moment identified with a conformal time coordinate eta equal to zero, and then re-expands. This leading-order solution respects what the authors call the parabolic structure of the bounce, a smoothness condition on how the scale factor behaves at the turning point.

At the next order in the expansion, the team embedded a central inhomogeneity into the bouncing background using a generalized McVittie geometry. The original McVittie solution, constructed in 1933, is a celebrated exact solution of general relativity that describes a mass concentrated at the center of an expanding universe, providing a mathematically tractable bridge between black hole physics and cosmology. By generalizing this construction to the scalar-tensor setting and treating it perturbatively, Yildirim and Coley encoded the gravitational imprint of a localized compact object within the contracting and rebounding cosmos. The perturbations appear as first-order corrections to the metric and to the scalar field, and the coupled field equations were solved as a series expansion carried up to fourth order in the parameter eta near the bounce.

A central technical challenge arose from the nature of the matter content near the inhomogeneity. In the vicinity of a concentrated mass, the stress-energy generically becomes anisotropic, meaning that pressure differs along the radial direction compared with the tangential directions. The authors therefore first allowed an anisotropic fluid with separate radial and tangential pressures, whose diagonal components suffice to solve the diagonal components of the field equations. They then imposed the physically motivated condition that the stress-energy reduce to a perfect fluid, one with a single isotropic pressure, far from the center. The resulting perfect fluid solution contains three arbitrary functions, which are constrained by demanding that the spacetime smoothly asymptote to the homogeneous FLRW background as the radial coordinate tends to infinity, ensuring that the black hole’s influence fades with distance as it must.

With suitable initial conditions chosen to preserve the parabolic structure of the bounce, a remarkable simplification emerged. The solution’s integration constants consolidate into a single quantity, denoted d0, which the authors identify as the true perturbative parameter of the problem. When d0 is set to zero, every perturbation vanishes and the spacetime reverts exactly to the homogeneous bouncing FLRW universe. When d0 is small but nonzero, a localized inhomogeneity, and with it a small evolving horizon, appears in the geometry. This clean parametrization means the entire structure of the black hole embedding is controlled by one number, allowing the authors to track precisely how the compact object’s gravitational field responds to the cosmic contraction and rebound.

The key result of the analysis concerns the fate of that horizon. Yildirim and Coley find a small evolving horizon whose radius scales linearly with the perturbative parameter, roughly as d0, and which they interpret as the horizon of the central inhomogeneity. Crucially, this horizon persists through the bounce at eta equals zero, supporting the interpretation that a black hole present before the cosmological transition survives it and continues to exist in the expanding universe on the other side. Intriguingly, the evolution is not symmetric about the bounce: the horizon’s behavior on the contracting side differs from its behavior on the expanding side, suggesting that the cosmic reversal leaves a subtle imprint on the object even as it survives. In a cosmological context, such findings resonate with long-standing speculations about black holes from a previous cosmic epoch, sometimes discussed in connection with ideas about the origin of supermassive black holes and the possible relics of a pre-bounce universe.

The authors are careful about what their construction does and does not establish. The small horizon they track is a future outer trapping horizon, a locally defined surface characterized by the convergence properties of outgoing and ingoing light rays, and it behaves as such in a two-sided neighborhood of the bounce. Because the entire construction is local and perturbative, it does not demonstrate the existence of a global event horizon, the teleological boundary beyond which nothing can escape to infinity. The distinction matters in a dynamical spacetime, where global horizons are notoriously difficult to define and can depend on the entire future evolution of the cosmos. Still, the persistence of a local trapping horizon through a nonsingular bounce is a nontrivial and suggestive result, indicating that the mechanisms of black hole formation and survival may be more robust under extreme cosmological conditions than simpler arguments had implied.

The work also fits into a broader effort to understand how inhomogeneities behave in bouncing scenarios. Previous studies have explored whether structure formation is possible through a bounce, whether primordial black holes can survive ekpyrotic contractions, and how numerical simulations of nonsingular bouncing spacetimes handle black holes and their horizons. By providing an analytic, perturbative solution in scalar-tensor gravity, the Dalhousie team adds a complementary tool to this mostly numerical and heuristic literature. The mathematical framework, combining generalized McVittie geometries with a controlled expansion around a radiation-dominated bounce, offers a concrete laboratory in which questions about horizons, matter anisotropies, and scalar-field dynamics can be addressed with precision. While the analysis is idealized, assuming spherical symmetry, a radiation fluid, and a small perturbation strength, it demonstrates that black hole persistence through a bounce is a mathematically consistent possibility in a well-motivated class of gravitational theories, bringing the speculative picture of black holes bridging cosmic epochs one step closer to rigorous footing.

Subject of Research: Perturbative scalar-tensor cosmology modeling black hole survival through a nonsingular bouncing universe

Article Title: Black hole persistence in scalar-tensor theories

Article References: Yildirim, B., & Coley, A. A. (2026). Black hole persistence in scalar-tensor theories. General Relativity and Gravitation, 58(9), Article 111. https://doi.org/10.1007/s10714-026-03610-6

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03610-6

Keywords: black holes, scalar-tensor gravity, bouncing cosmology, general relativity, McVittie metric, trapping horizon, FLRW spacetime, cosmic bounce, perturbation theory, theoretical cosmology, nonsingular cosmology, gravitational physics

The idea that (G) might vary is not new. It has appeared in theories inspired by Dirac’s large-number hypothesis, scalar–tensor gravity, Brans–Dicke theory, and several approaches to modified gravity. In ordinary general relativity, (G) sets the strength of the coupling between matter and spacetime curvature. If it changes, the gravitational response of the universe changes as well, potentially affecting the expansion rate, the growth of cosmic structure, stellar evolution, planetary motion, and the behavior of compact objects. The cosmological term (\Lambda), meanwhile, is usually interpreted as a constant vacuum-energy density that drives the late-time acceleration of cosmic expansion. Allowing both quantities to evolve introduces a new degree of freedom into the background dynamics and may provide a way to describe observations that do not fit perfectly within a rigid constant-(G), constant-(\Lambda) framework.

Mandal and colleagues focus on an isotropic and homogeneous universe, described by the Friedmann–Lemaître–Robertson–Walker metric. This approximation treats the cosmos on very large scales as spatially uniform and directionally equivalent, making it possible to connect theoretical equations with measurements of the Hubble expansion. The authors adopt a power-law parametrization for the gravitational constant, allowing (G) to change in a controlled mathematical form rather than introducing an entirely arbitrary function. They then solve the corresponding cosmological equations to derive the model’s expansion rate. Because a time-dependent gravitational coupling modifies the Friedmann equations and their consistency conditions, the model must be examined as a coupled system in which the evolution of (G), (\Lambda), matter, and the scale factor are not independent ingredients.

The observational engine of the study is DESI’s second data release, which provides baryon acoustic oscillation measurements across a broad range of cosmic epochs. Baryon acoustic oscillations are relics of sound waves that travelled through the hot plasma of the early universe before atoms formed. Their characteristic scale became imprinted in the distribution of galaxies, quasars, and intergalactic matter. Today, that scale functions as a cosmic ruler. By measuring its apparent size across different redshifts, astronomers can infer combinations of the expansion history, including the Hubble parameter (H(z)) and distance–redshift relations. DESI’s measurements are especially valuable because they probe the period when dark energy began to dominate the universe, precisely the era in which departures from a constant cosmological term could become visible.

To strengthen the analysis, the researchers combine DESI BAO observations with two independent classes of data. Cosmic Chronometers estimate the expansion rate directly through the differential ageing of passively evolving galaxies. Rather than relying primarily on a distance ladder, this method uses galaxies as cosmic clocks: the change in their ages over redshift provides an estimate of (dz/dt), which can be converted into (H(z)). The Pantheon+SH0ES supernova dataset contributes luminosity-distance information from Type Ia supernovae, stellar explosions whose calibrated brightness allows them to be used as standardisable candles. Together, the three datasets test the model through complementary observables—cosmic distances, direct expansion rates, and supernova brightness—reducing the risk that a preferred result is driven by a single observational technique.

The team estimates the parameters of the varying-(G), varying-(\Lambda) framework using Markov chain Monte Carlo analysis. MCMC methods explore the multidimensional parameter space by generating chains of possible cosmological models, assigning greater statistical weight to those that provide better agreement with the data. The resulting likelihood analysis allows the authors to constrain the parameters controlling the gravitational variation and the cosmic expansion while accounting for observational uncertainties and parameter correlations. The study also compares parameter estimates obtained from different dataset combinations, an important test of robustness. If independent observations point toward compatible regions of parameter space, confidence in the model’s viability increases; if they disagree, the apparent preference for evolving physics may instead reflect statistical fluctuations or hidden systematic effects.

Beyond fitting the expansion data, the researchers reconstruct the evolution of several quantities that describe the universe’s physical behavior. The Hubble parameter (H(z)) tracks the expansion rate at different times, while the deceleration parameter (q(z)) indicates whether expansion is slowing or accelerating. A transition from positive (q) in the matter-dominated past to negative (q) in the recent universe signals the onset of accelerated expansion. The authors also examine an effective equation-of-state parameter, (w_{\rm eff}), which summarizes the pressure-to-density behavior of the total cosmic contents. In standard terminology, values near (-1) resemble a cosmological constant, while departures from that value can indicate evolving dark energy or an effective modification of gravity. In a model where (G) and (\Lambda) vary, these reconstructed parameters provide an intuitive view of how the new physics reshapes cosmic history.

A central motivation for the work is the continuing debate over whether dark energy is truly constant. Several recent analyses of large-scale structure and supernova data have reported hints that the properties of dark energy may evolve, although the statistical significance and interpretation remain under discussion. A varying cosmological term could mimic some of the observational signatures associated with dynamical dark energy, while a changing (G) could alter the inferred expansion history without behaving like a conventional dark-energy fluid. This distinction matters because cosmological parameters are not measured in isolation. Inferences about the amount and nature of dark energy depend on assumptions about gravity, the standard ruler, the calibration of supernovae, and the conservation of matter and energy. By varying (G) and (\Lambda) together, the study explores a broader theoretical landscape in which apparent tensions may arise from interactions between gravitational physics and cosmic acceleration.

The authors also test the late-time behavior of the model by examining the present-day rate of change of the gravitational constant, expressed as (\dot G/G), and by applying information criteria to compare the model with (\Lambda)CDM. The quantity (\dot G/G) is especially important because local experiments place stringent limits on any present variation in (G), using lunar laser ranging, planetary dynamics, pulsar timing, and other precision measurements. A cosmological model can fit large-scale observations yet remain physically problematic if it predicts a modern rate of change that violates these local bounds. Information criteria provide a complementary perspective: they penalize models for adding parameters, asking whether improved agreement with observations is substantial enough to justify the extra complexity. The study therefore treats a successful cosmology not simply as one capable of matching data, but as one that remains compatible with local gravitational tests and earns its additional freedom statistically.

The broader significance of the research lies in its attempt to connect two major questions in contemporary cosmology: why the universe is accelerating and whether the laws governing gravity are immutable. The analysis does not overturn general relativity or establish that (G) and (\Lambda) definitely vary. Instead, it supplies observational constraints on a specific phenomenological framework and evaluates how its predicted expansion history compares with the established cosmological model. As DESI continues mapping the universe and future supernova, galaxy, and gravitational-wave surveys expand the available data, models with evolving constants will face increasingly precise tests. If their parameters converge toward zero variation, the result will reinforce (\Lambda)CDM and place tighter limits on alternatives. If consistent evidence for evolution emerges across independent datasets, cosmology could be forced to reconsider whether the constants written into its foundational equations are truly constant—or only appear so across the limited cosmic history we have measured.

Subject of Research: Varying gravitational constant and cosmological term in observational cosmology

Article Title: Observational constraints on varying-((G,\Lambda)) cosmology with DESI DR2

Article References: Mandal, S., Singh, A. & Chaubey, R. Astrophysics and Space Science 371, 44 (2026). https://doi.org/10.1007/s10509-026-04576-2

Image Credits: AI Generated

DOI: 10.1007/s10509-026-04576-2

Keywords: Varying dark energy, gravitational constant, cosmological constant, DESI DR2, Cosmic Chronometers, Pantheon+SH0ES, cosmology, cosmic acceleration