The work, carried out by Yerlan Myrzakulov of L. N. Gumilyov Eurasian National University in Astana, Saddam Hussain of Zhejiang University of Technology in Hangzhou, and Mohd Shahalam of Integral University in Lucknow, focuses on a class of models known as quintessence. In quintessence scenarios, dark energy is not the unchanging cosmological constant of the standard Lambda-CDM model but a dynamical scalar field that rolls down a potential energy landscape over cosmic time. The specific potential studied here is a double exponential, first proposed in the context of a recently introduced scenario called Quintessence-Driven Slow-Contraction Cold Dark Matter, or Q-SC-CDM. In that scenario, dark energy gradually loses strength, cosmic acceleration eventually ceases, and the universe drifts into a period of slow contraction that terminates in a Big Crunch.

The authors’ key innovation is to add a non-minimal coupling between the scalar field and gravity. In standard quintessence, the field interacts with gravity only through its energy and pressure. In the non-minimally coupled version, the field multiplies the Ricci scalar, the mathematical object that encodes spacetime curvature, through a coupling function of the form F(phi) = 1 minus kappa squared xi phi squared. The dimensionless parameter xi controls the strength of this coupling. When xi equals zero, ordinary minimal coupling is recovered; the value xi equal to one sixth corresponds to the theoretically motivated conformal coupling; and large values of xi can radically reshape cosmic evolution. This kind of coupling has a distinguished pedigree, tracing back to Brans-Dicke theory in 1961, and it has recently gained attention because a strongly coupled Higgs field can drive cosmic inflation.

Technically, the coupling changes everything. The Friedmann equation governing the expansion rate acquires extra terms proportional to xi, including contributions proportional to the Hubble parameter times the field and its derivative, and the field’s equation of motion picks up a curvature-driven term xi R phi. The researchers solved these coupled equations numerically on a spatially flat Friedmann-Lemaitre-Robertson-Walker background, starting from a matter-dominated early universe in which the field is frozen by Hubble damping. Their numerical experiments, sweeping the coupling across the range from 0.01 to 1 and varying the potential parameters alpha and beta, revealed a striking bifurcation. For couplings below roughly 0.2, and for sufficiently large potential parameters, the universe passes through decelerated expansion, then acceleration, and finally a slow-contraction phase in which the Hubble parameter falls to zero and the scale factor collapses. For couplings above roughly 0.2, the contraction phase vanishes entirely: the universe accelerates today and settles into de Sitter expansion, an exponentially expanding state with a constant Hubble rate, in the far future.

But a model that avoids the Big Crunch is worthless if it cannot reproduce the universe we actually observe. The crucial test is whether the model produces a sufficiently long matter-dominated epoch at high redshift, the era in which galaxies, clusters, and the cosmic microwave background patterns formed. The authors found that many parameter combinations producing a future slow-contraction phase fail this test badly: the matter era is cut short, the field’s fractional energy density turns negative or exceeds unity at high redshift, or the dynamics diverge. Such solutions, however dramatic their future, are physically non-viable. This observation alone begins to undermine the Q-SC-CDM picture, since the very trajectories that lead to a Big Crunch cannot sustain a consistent past.

To make the case quantitative, the team confronted three versions of the model with a formidable battery of data. The datasets included 32 cosmic chronometer measurements of the Hubble parameter spanning redshifts from 0.07 to nearly 2; the Pantheon+ compilation of 1701 Type Ia supernovae; the Dark Energy Survey five-year sample of 1829 supernovae; the Union 3 compilation of 2087 supernovae; and seven baryon acoustic oscillation data points from the second data release of the Dark Energy Spectroscopic Instrument, DESI, with the sound horizon at the drag epoch treated as a free parameter. Using Markov chain Monte Carlo sampling with the dynesty nested sampler, they computed posterior distributions and Bayesian evidence for each model relative to Lambda-CDM, interpreting evidence differences on the revised Jeffreys scale.

The results were unambiguous. Model I, with the coupling fixed at xi equal to 0.3, yields a Hubble constant near 70 kilometers per second per megaparsec and a matter density around 0.295, but its Bayesian evidence falls far below that of Lambda-CDM across every dataset combination. Model II, with xi equal to 0.5, fares even worse: it prefers a matter density of only about 0.255, produces a sound horizon near 145 megaparsecs, and shows the largest deviations in the Om(z) diagnostic, a low-redshift test that flags dynamical dark energy by measuring departures from constancy in a normalized combination of Hubble measurements. Both fixed-coupling models predict Hubble constants above 70 kilometers per second per megaparsec, which might superficially look like a route to easing the Hubble tension, but the authors caution that the severe evidence penalty means this should not be read as a genuine resolution of that conflict.

The most interesting case is Model III, in which the coupling is allowed to float between 0.01 and 0.7. Here the data consistently pull the coupling down to a small value, xi of roughly 0.12, close to the conformal value of one sixth. The best-fit evolution closely mimics Lambda-CDM: the matter density stays near unity at high redshift, the effective equation of state remains at zero through the matter era and saturates at minus one in the future, and the potential-related combination y squared minus A squared stays positive throughout, meaning the effective potential never turns negative. Because a negative potential is precisely what triggers recollapse in these models, the observationally favored parameter space simply never enters the Big Crunch regime. Even for this modest coupling, well below the threshold of 0.2 that mechanically eliminates contraction, the data steer the model into the corner of parameter space where the potential parameters are small enough that slow contraction never occurs.

An independent phase-space analysis confirms the picture. Rewriting the field equations as a first-order autonomous system, the authors identified five stationary points and classified them by the eigenvalues of the linearized flow. Points A1, A2, and A4 are saddle points corresponding to transient matter-dominated or non-accelerating phases; A3 is a non-hyperbolic point with de Sitter characteristics; and A5 is a one-parameter family of equilibrium points that acts as a stable attractor. For the observationally preferred parameters, with alpha near ten to the minus four, beta near 0.26, and xi near 0.12, the three transverse eigenvalues at A5 are negative, and phase portraits show trajectories from across the phase space converging onto the equilibrium curve. At that attractor, the field freezes, the Hubble derivative vanishes, the dark energy density parameter approaches one, and the effective equation of state approaches minus one, exactly the de Sitter endpoint implied by the observational fits.

The study leaves the standard cosmological model standing, at least at the level of background evolution: every variant considered was strongly disfavored relative to Lambda-CDM, with evidence differences exceeding five on the logarithmic scale. Yet the work delivers a substantive negative result with real physical content. The coupling strengths that would spare the universe a Big Crunch through brute force are ruled out by data, while the couplings the data prefer steer the universe away from collapse more subtly, by selecting potential shapes that never go negative. The framework also remains phantom-free, with a positive kinetic term that avoids ghost instabilities even as DESI measurements hint at evolving dark energy in other models. The authors note that the real test lies ahead: extending the analysis to cosmological perturbations, confronting the model with full cosmic microwave background data and large-scale structure surveys, and checking its impact on the matter clustering amplitude sigma eight. For now, the verdict is reassuring. The same observations that map the expansion history of the cosmos appear to be quietly vetoing the universe’s most dramatic possible ending, pointing instead toward a future of gentle, eternal expansion.

Subject of Research: Nonminimally coupled quintessence dark energy models and observational constraints on the future fate of the universe

Article Title: Nonminimally coupled quintessence with double exponential potential: observational evidence against big crunch singularity

Article References: Myrzakulov, Y., Hussain, S., & Shahalam, M. (2026). Nonminimally coupled quintessence with double exponential potential: observational evidence against big crunch singularity. The European Physical Journal C, 86(9), Article 1055. https://doi.org/10.1140/epjc/s10052-026-16306-7

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16306-7

Keywords: dark energy, quintessence, nonminimal coupling, Big Crunch singularity, de Sitter expansion, cosmology, scalar field, Bayesian evidence, DESI, supernovae, Lambda-CDM, phase-space analysis

Unimodular gravity has long intrigued theorists precisely because of how it handles the cosmological constant problem, arguably the most notorious discrepancy between theory and observation in physics. Quantum field theory predicts a vacuum energy vastly larger than what astronomers measure, yet in unimodular gravity the situation changes fundamentally. By restricting the symmetry of Einstein’s theory so that transformations must preserve the four-dimensional volume element, the theory yields only the trace-free part of Einstein’s field equations. The cosmological constant then ceases to be a parameter fixed by the microscopic physics and instead appears as a constant of integration, much like the energy of a pendulum, whose value is determined by initial conditions rather than by quantum loops.

The price of this elegant restructuring is a modified conservation law. In standard general relativity, the energy-momentum tensor of matter is covariantly conserved, which forces the cosmological constant to be truly constant. In unimodular gravity, that constraint is relaxed, and the divergence of the energy-momentum tensor is balanced by the gradient of the Lagrange multiplier that enforces the volume restriction. Solving the resulting equation reveals that the cosmological term splits into two pieces: a fixed integration constant and an arbitrary function of time, which the authors call the energy diffusion function. This function quantifies exactly how much local energy conservation fails, and it acts as a continuous source or sink of energy for the cosmic fluids filling the universe.

Physically motivated guesses for this diffusion function have been proposed before. One striking example draws on the continuous spontaneous localization model of quantum collapse, in which energy is genuinely created during wavefunction collapse, suggesting a diffusion term proportional to the energy density of the fluid itself. Earlier work showed that when only dark matter diffuses, the resulting cosmology closely reproduces the standard Lambda-CDM model, and related diffusion scenarios have even been explored as a way to ease the Hubble tension, the persistent disagreement between different measurements of the universe’s expansion rate. The trouble, as the Chilean team emphasizes, is that no widely accepted diffusion function follows from an established physical process, so most proposals rest on phenomenological convenience or mathematical simplicity rather than principle.

The new study attacks this problem from the opposite direction. Rather than postulating a diffusion law and studying its consequences, the authors recast the cosmological equations as an autonomous dynamical system and ask what diffusion functions the phase-space structure itself demands. The key move is to introduce a dimensionless variable called the diffusion slope, defined as minus the logarithmic derivative of the diffusion function with respect to the e-fold number, a natural clock for cosmic expansion. For the system to close, this slope must be invertible along the trajectories — a condition satisfied whenever it evolves monotonically. Remarkably, when the slope is held constant, the formalism automatically spits out the power-law diffusion functions that had previously been introduced by hand, suggesting that the dynamical systems perspective can rediscover and organize known models rather than merely accommodate them.

With the autonomous system in hand, the team mapped out the fixed points of cosmic evolution and their stability. The familiar matter-dominated era appears as a saddle point, not an attractor, meaning the universe inevitably passes through it rather than lingering. The de Sitter solution, where expansion accelerates at a constant rate under the influence of the integration constant, emerges as a genuine late-time attractor for positive diffusion slopes. More intriguingly, the analysis uncovered a novel matter-diffusion scaling solution in which the diffusion term tracks the dark matter density, maintaining a constant fractional share of the cosmic energy budget. Within a specific range of parameters, this scaling regime itself drives accelerated expansion — entirely without a cosmological constant — although it represents a transient stage rather than a final destiny.

Perhaps the most provocative result concerns the purely diffusion-dominated configuration. When the diffusion function becomes constant, it behaves exactly like a cosmological constant, producing exponential expansion with an effective equation-of-state parameter of minus one. In that regime, the diffusion sector alone can power late-time acceleration, with no fundamental vacuum energy required at all. The stability analysis revealed subtlety here: linear theory alone cannot settle the fate of this point because one eigenvalue vanishes, so the researchers supplemented their analytic work with numerical integration of the phase-space flow, showing that trajectories are attracted along one direction but repelled along another. The diffusion-dominated solution is therefore a saddle, a waystation the cosmos may visit but not a permanent home — the true endpoint remains the de Sitter state governed by the integration constant.

Beyond the asymptotic regimes, the authors built a full reconstruction machinery analogous to potential reconstruction in scalar-field cosmology. By specifying a curvature function that controls how the diffusion slope bends in logarithmic space, one can integrate the slope evolution and then reconstruct the diffusion function itself through a simple exponential integral. The constant-curvature family already displays rich behavior: curvature equal to one recovers power-law diffusion, curvature greater than one drives the diffusion term smoothly to zero as the universe expands, and curvature below one produces a finite-time singularity in the slope, signaling a breakdown of the description. More elaborate curvature functions that cross unity allow the slope to station at multiple values, enabling trajectories that interpolate between a rapidly decaying diffusion contribution in the early universe and an asymptotically constant component at late times — precisely the kind of transition a viable dark energy candidate might need.

The framework also clarifies what thermodynamics demands. Using the Gibbs relation, the team showed that the second law of thermodynamics requires the diffusion function to be non-increasing as the universe expands, guaranteeing positive entropy production. Notably, this condition constrains only the trend, not the sign, of the diffusion term, which can push the effective cosmological constant either up or down. The reconstruction formalism respects this automatically: because the diffusion function is built from a strictly positive exponential factor, it cannot change sign dynamically, keeping every reconstructed model consistent with the thermodynamic constraint from the outset.

The implications reach well beyond formal elegance. Because the diffusion sector exchanges energy with dark matter, it should alter the growth of cosmic structure, leaving fingerprints in the matter power spectrum and in the growth rate of galaxies that could distinguish diffusion cosmologies from Lambda-CDM. The authors identify this perturbative analysis, together with a direct confrontation of the reconstructed models with observational data, as the natural next step of their program. If the coming generation of surveys continues to hint that dark energy is not perfectly constant — as recent measurements have suggested — then a framework that generates and classifies diffusion models from first principles, rather than by trial and error, may prove exactly what cosmologists need to make sense of a universe that refuses to sit still.

Subject of Research: Cosmological diffusion models and energy non-conservation in unimodular gravity, analyzed through dynamical systems reconstruction

Article Title: Unimodular gravity with arbitrary diffusion function: a dynamical system reconstruction approach

Article References: Unimodular gravity with arbitrary diffusion function: a dynamical system reconstruction approach. (n.d.). https://doi.org/10.1140/epjc/s10052-026-16363-y

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16363-y

Keywords: unimodular gravity, cosmological constant problem, dark energy, energy diffusion function, dynamical systems, phase-space fixed points, de Sitter expansion, scaling solutions, entropy production, Lambda-CDM, Hubble tension, theoretical cosmology