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 system at the center of the analysis is the Euler-Heisenberg anti-de Sitter (AdS) black hole, a solution of general relativity coupled to a nonlinear version of electromagnetism. Nonlinear electrodynamics arises naturally when quantum effects modify Maxwell’s classical theory, and the Euler-Heisenberg Lagrangian is the canonical example, originally derived to describe how vacuum fluctuations soften extremely strong electric fields. In the black hole spacetime, the strength of these nonlinear corrections is controlled by a single parameter, usually denoted a, which appears in the metric function as a term proportional to a times the fourth power of the charge divided by the sixth power of the radius. That seemingly modest correction turns out to reshape the entire thermodynamic landscape, altering the equation of state and generating qualitatively different phase behavior depending on the parameter’s value.

The framework in which this behavior is interpreted is known as extended phase space thermodynamics, or black hole chemistry. In this picture, the cosmological constant of the spacetime is no longer treated as a fixed background quantity but as a genuine thermodynamic variable identified with pressure. Once pressure enters the first law, charged AdS black holes begin to mimic van der Waals fluids, exhibiting first-order phase transitions between small and large black hole phases that terminate at critical points, exactly as liquid and gas phases terminate at the critical point of a real fluid. For the Euler-Heisenberg black hole, the parameter a modulates this analogy: for negative values the system shows the standard van der Waals-like transition, for intermediate values up to a threshold set by the charge squared it develops reentrant phase transitions, and for sufficiently large values the transition disappears altogether, leaving a single thermodynamic phase.

The new study focuses on a regime where the mathematics becomes especially intricate. Solving the standard criticality conditions, that the first and second derivatives of the Hawking temperature with respect to the horizon radius vanish simultaneously, yields not one but two distinct sets of critical parameters. At fixed pressure, the first critical point sits at a horizon radius of about 0.59 with a critical temperature near 0.079, while the second lies at a radius of roughly 4.46 with a temperature near 0.024. Below these critical values, the thermodynamic behavior splits into four separate black hole branches, which the authors label the Smallest, Small, Intermediate, and Large black hole phases. The temperature curve possesses three extrema that carve the parameter space into these four sectors, and the specific heat diverges at three locations, marking the boundaries between branches where stability flips sign. Branches with positive specific heat are locally stable; those with negative specific heat are unstable and cannot persist against small fluctuations.

Even more striking is what happens when the pressure is tuned. As pressure increases, the two critical points migrate toward one another and eventually coalesce at a unique set of parameters, a horizon radius of about 0.296, a charge of roughly 0.148, and a pressure near 0.171. At this degenerate critical point, the first three derivatives of the Hawking temperature vanish simultaneously, producing a higher-order stationary inflection point in the temperature profile. The four-phase structure collapses into just two branches, one stable and one unstable, and the separate free-energy curves that distinguished the multiple phases merge into a simplified structure. The degenerate point therefore acts as the merger of two independent criticalities, a higher-order critical phenomenon that has no analogue in the ordinary van der Waals system.

To probe what happens above and beyond these critical points, the authors turned to the Lee-Yang theory of phase transitions, formulated in 1952 by Tsung-Dao Lee and Chen-Ning Yang. The central insight of that theory is that phase transitions are encoded in the zeros of a system’s partition function, the mathematical object that sums over all possible configurations. For finite systems these zeros live in the complex plane, but as the system grows toward the thermodynamic limit they can pinch the real axis, and where they do, the Gibbs free energy becomes non-analytic: a phase transition. In black hole thermodynamics, the partition function is related to the Euclidean gravitational action, and the Gibbs free energy equals minus the temperature times the logarithm of the partition function. Consequently, the Lee-Yang zeros of the black hole system can be located by finding where the specific heat diverges, since the specific heat is proportional to the second derivative of the free energy.

Crucially, the complex horizon radius in this construction is not a physical claim that horizons occupy imaginary positions in spacetime. It is an analytic continuation, a mathematical device for tracking where the thermodynamic functions develop singularities. In the critical region, the relevant singularities sit on the real axis, corresponding to spinodal boundaries where the specific heat blows up. Once the system enters the supercritical regime, the singularities lift off the real axis and travel as complex-conjugate pairs. The authors found that in the two-critical-point regime, the singularities organize into two separate families, each anchored at one of the critical points, and that as the degenerate point is approached, these families coalesce into a single configuration whose branches intersect the real axis at exactly the degenerate critical radius.

Projecting these complex singularity trajectories onto the real temperature-pressure plane produces the Widom line, a crossover boundary that divides the supercritical region into sectors with liquid-like and gas-like character in ordinary fluids, or small-black-hole-like and large-black-hole-like character in gravitational ones. The surprise lies in what the projections mean for the Euler-Heisenberg system. At the degenerate critical point, no conventional coexistence curve exists, because there are not two stable phases with equal free energy to coexist. Yet a well-defined Widom line still emerges from the complex singularity structure, and it does real thermodynamic work: the specific heat changes sign across it, so the projected line acts as an effective stability boundary separating locally stable from locally unstable sectors of the supercritical parameter space. In the two-critical-point regime, the situation becomes richer still. One critical point carries the conventional story, with a coexistence curve terminating on it and a Widom line continuing that curve into the supercritical region. The second critical point, by contrast, has no coexistence line at all, but its complex singularities still generate a second, independent Widom line. Two qualitatively different crossover structures thus coexist within the same black hole system, demonstrating that supercritical crossovers need not share a single geometric origin.

The authors also verified the framework’s consistency in limiting cases. When the system possesses a single ordinary critical point, the Lee-Yang singularities intersect the real axis exactly once and reproduce the textbook picture of one coexistence curve and one Widom line, here separating intermediate and large black hole regimes, with the coexistence curve bending in accordance with the Clapeyron relation as entropy and thermodynamic volume vary along it. And when the Euler-Heisenberg parameter is large enough that no criticality exists, the complex-conjugate singularity trajectories remain forever detached from the real axis, the Gibbs free energy stays analytic, and no phase transition signature appears. The approach of Lee-Yang zeros toward the real axis is thus a faithful diagnostic: when they reach it, criticality follows; when they do not, the thermodynamic landscape remains smooth.

The broader significance of the work is twofold. First, it establishes the complex phase diagram as a practical tool for mapping black hole thermodynamics beyond the critical point, a regime that has received far less attention than the subcritical territory where most known transitions live. Second, and perhaps more conceptually provocative, it shows that Widom lines can exist and carry physical meaning even in the complete absence of a first-order coexistence curve, suggesting that the complex singularity structure of the free energy encodes crossover phenomena that conventional real-valued analysis misses entirely. As theoretical physicists continue to catalog the phase structures of increasingly exotic black hole solutions, from Gauss-Bonnet to Born-Infeld to noncommutative geometries, the message of this study is that the richest information may reside not on the real thermodynamic plane but in the complex manifold just beyond it, where the shadows of vanished phase transitions still leave visible traces.

Subject of Research: Supercritical thermodynamics and Lee-Yang phase transition theory of Euler-Heisenberg AdS black holes

Article Title: Complex phase structure and Widom line for Euler–Heisenberg black holes

Article References: Awal, M. B., & Phukon, P. (2026). Complex phase structure and Widom line for Euler–Heisenberg black holes. The European Physical Journal C, 86(10), Article 1146. https://doi.org/10.1140/epjc/s10052-026-16405-5

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16405-5

Keywords: black hole thermodynamics, Euler-Heisenberg black holes, AdS black holes, Lee-Yang zeros, Widom line, phase transitions, critical points, supercritical regime, nonlinear electrodynamics, extended phase space, Gibbs free energy, spinodal lines

The AdS/CFT correspondence, proposed by Juan Maldacena in the late 1990s, asserts that a gravitational theory in a volume of Anti-de Sitter spacetime is exactly equivalent, or dual, to a conformal field theory living on the boundary of that space. It is a kind of holographic dictionary: every black hole in the bulk corresponds to a hot quantum fluid on the boundary, and thermodynamic quantities such as mass, charge, temperature and entropy map onto energy, electric potential and the central charge of the field theory. The most famous example is the Hawking-Page transition, in which a black hole becomes thermodynamically favored over empty AdS space, interpreted holographically as a confinement-deconfinement transition in the boundary gauge theory. The new work, by Abhishek Baruah, Amijit Bhattacharjee and Prabwal Phukon, takes a deliberately restricted but sharp question: when a black hole is built from nonlinear electromagnetism and its entropy is deformed by generalized statistics, does the pattern of critical points on the gravity side still match the pattern on the field theory side?

The team examined three distinct classes of charged AdS black holes arising from nonlinear electrodynamics, a family of theories that modify Maxwell’s equations in strong electromagnetic fields. The first is the ModMax black hole, based on a theory remarkable for preserving both conformal invariance and electromagnetic duality while introducing a single dimensionless deformation parameter gamma. The second is a generic nonlinear electrodynamics model whose Lagrangian is suppressed by hyperbolic functions of the field invariant, producing magnetically charged black holes with a finite nonlinear field energy. The third, and richest, is the Euler-Heisenberg black hole, rooted in the genuine quantum correction to electrodynamics first calculated by Heisenberg and Euler in 1936, which captures vacuum polarization effects that become important near the horizon. In each case the nonlinear corrections alter the metric function of the black hole, and therefore reshape the temperature-entropy relation and the response functions that govern stability.

On top of these electromagnetic deformations, the authors layered a second axis of modification: the choice of entropy functional. Alongside the standard Bekenstein-Hawking area law, they employed two generalized entropies drawn from non-extensive statistical mechanics. The Renyi entropy, a one-parameter generalization of the von Neumann entropy familiar from quantum information theory, arises naturally in replica methods and probes the full entanglement spectrum. The Kaniadakis entropy, by contrast, emerges from a kappa-deformed relativistic statistical mechanics built on generalized logarithms and exponentials, providing a consistent non-Gibbsian extension of equilibrium thermodynamics. Both reduce to the Bekenstein-Hawking entropy in appropriate limits, but their deformation parameters can substantially alter the phase structure, shifting or even multiplying the critical points at which a black hole changes stability.

The analytical machinery that ties these threads together is geometrothermodynamics, a framework developed by Hernando Quevedo that applies differential geometry to the space of thermodynamic equilibrium states. Earlier metrics proposed by Weinhold and Ruppeiner suffered from a crucial flaw: they were not invariant under Legendre transformations, so different choices of thermodynamic potential yielded different curvatures. The geometrothermodynamic construction repairs this by building a Legendre-invariant metric on an extended phase space whose contact structure encodes the first law of thermodynamics. The payoff is a scalar curvature, R_GTD, with a clean physical interpretation: flat geometry corresponds to a non-interacting system such as an ideal gas, while curvature singularities mark phase transitions. For black holes, the promise is that the curvature diverges exactly where the heat capacity diverges and where the temperature-entropy curve reaches an extremum, providing a geometric, potential-independent diagnostic of criticality.

The results are strikingly consistent. For the ModMax AdS black hole with Bekenstein-Hawking entropy, the temperature profile shows two extrema at entropy values of 16.454 and 88.265, the specific heat diverges at precisely those points, and the GTD curvature develops singularities at exactly the same locations. Under Renyi entropy the two critical points persist at shifted positions, while Kaniadakis entropy introduces a third critical point that is absent in the other two descriptions. The same triple-diagnostic pattern, temperature extrema coinciding with heat capacity divergences and curvature singularities, repeats for the generic NED black hole and, most dramatically, for the Euler-Heisenberg black hole, which exhibits three critical points under Bekenstein-Hawking entropy, three under Renyi, and four under Kaniadakis statistics. The Euler-Heisenberg case is the richest of the three models, a fact the authors attribute to the higher-order QED vacuum polarization corrections that reorganize the competition between gravitational attraction and electromagnetic repulsion near the horizon.

The crucial test comes when the bulk black holes are mapped to their holographic duals. Using the restricted phase space thermodynamics framework, in which the cosmological constant is held fixed and the thermodynamic variables are reconstructed in terms of boundary quantities such as the central charge C and the CFT volume, the authors computed the energy, temperature and heat capacity of the dual field theories. The mapping is implemented through a conformal scaling factor relating bulk mass, temperature and charge to their boundary counterparts, with the central charge tied to the AdS radius and Newton’s constant. Remarkably, the number, ordering and structure of the critical points are preserved across the holographic dictionary. The ModMax black hole with two critical points maps to a CFT with two critical points; the Euler-Heisenberg system with three maps to a boundary theory with three; and the Kaniadakis-deformed descriptions carry their extra critical point from bulk to boundary in every case.

The authors are careful about what this does and does not prove. The numerical entropy values of the critical points do not coincide between bulk and boundary, and they are not expected to, since the generalized entropies are nonlinear functions of the underlying variables. What is preserved is the qualitative critical organization: the same number of thermodynamic branches, the same ordering of instabilities, and the same one-to-one correspondence between curvature singularities and heat capacity divergences. The extra Kaniadakis critical point, they argue, is statistical rather than gravitational in origin, arising because the kappa-deformed entropy functional modifies the derivatives that enter the temperature and heat capacity, rather than from any new bulk field or conserved charge. The analysis is therefore a nontrivial consistency check of bulk-boundary thermodynamics under simultaneous deformations of the gauge sector and the entropy prescription, not a full dynamical proof of AdS/CFT.

The study leaves open a tantalizing next step: extending the comparison from the locations of critical points to the near-critical scaling behavior itself, including critical exponents and universality classes. That undertaking is considerably harder within restricted phase space thermodynamics, which lacks the Van der Waals-style equation of state that makes exponent extraction straightforward in the extended phase space framework, and it would need to be performed consistently on both sides of the duality. For now, the message of this work is quietly profound. Even when the electromagnetic self-interaction is deformed by quantum vacuum polarization and the entropy law is rewritten by non-extensive statistics, the holographic mirror does not crack. The phase structure of a black hole, encoded in the curvature of an abstract thermodynamic manifold, is faithfully reflected in the thermodynamics of a quantum field theory it never touches, a consistency that continues to justify the strange and beautiful conviction that spacetime itself may be a hologram.

Subject of Research: Thermodynamic geometry and holographic consistency checks of nonlinear electrodynamics AdS black holes with generalized entropies

Article Title: Testing the AdS/CFT correspondence through thermodynamic geometry of nonlinear electrodynamics AdS black holes with generalized entropies

Article References: Baruah, A., Bhattacharjee, A., & Phukon, P. (2026). Testing the AdS/CFT correspondence through thermodynamic geometry of nonlinear electrodynamics AdS black holes with generalized entropies. The European Physical Journal C, 86(10), Article 1144. https://doi.org/10.1140/epjc/s10052-026-16330-7

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16330-7

Keywords: AdS/CFT correspondence, black hole thermodynamics, geometrothermodynamics, nonlinear electrodynamics, ModMax, Euler-Heisenberg, Renyi entropy, Kaniadakis entropy, phase transitions, holography, conformal field theory, critical points