Author: Denis Avetisyan
New research employs physics-informed neural networks to investigate a modified dark energy model, offering a potential path towards resolving the ongoing discrepancy in the universe’s expansion rate.
This study utilizes Physics-Informed Neural Networks (PINNs) to analyze Tsallis Holographic Dark Energy with neutrinos, providing constraints on cosmological parameters and neutrino mass using cosmic chronometers and Markov Chain Monte Carlo methods.
The persistent discrepancy between locally measured and early-universe-derived values of the Hubble constant—the “Hubble tension”—challenges standard cosmological models. This research, detailed in ‘Towards a Machine Learning Solution for Hubble Tension: Physics-Informed Neural Network (PINN) Analysis of Tsallis Holographic Dark Energy in Presence of Neutrinos’, introduces a novel framework leveraging Physics-Informed Neural Networks to explore a Tsallis Holographic Dark Energy model extended with massive neutrinos. Results demonstrate a significant alleviation of the Hubble tension—reducing it from a ~5σ to 0.5–2.2σ level—alongside competitive constraints on the total neutrino mass (< 0.11 eV). Could this data-driven approach, combining machine learning with generalized thermodynamics, offer a pathway towards resolving fundamental inconsistencies in our understanding of the cosmos?
The Universe’s Riddle: A Discrepancy in Expansion
The universe’s expansion rate, described by the Hubble constant, is proving surprisingly difficult to pin down, creating a significant challenge to modern cosmology. Current calculations, based on observations of the cosmic microwave background – the afterglow of the Big Bang – predict one value for this constant, while measurements derived from observing objects in the local universe, like supernovae and Cepheid variable stars, suggest a faster rate of expansion. This growing disparity, known as the Hubble Tension, isn’t simply a matter of refining existing measurements; the discrepancy is statistically significant and persists even with increasingly precise data. The standard cosmological model, Lambda-CDM – which posits a universe dominated by dark energy ($Λ$) and cold dark matter (CDM) – struggles to reconcile these conflicting values, prompting researchers to explore potential new physics beyond the current framework, such as modifications to dark energy or the introduction of new relativistic particles in the early universe.
The persistent discrepancy in the Hubble constant, known as the Hubble Tension, fundamentally questions the established cosmological narrative regarding the universe’s expansion and the enigmatic nature of dark energy. Current models, while successful in many respects, struggle to reconcile locally measured expansion rates with those inferred from the early universe, suggesting a potential gap in understanding the fundamental physics governing cosmic evolution. Addressing this tension necessitates innovative parameter estimation techniques that move beyond conventional methods; researchers are actively exploring modifications to the standard Lambda-CDM model, investigating alternative dark energy formulations, and refining methods for measuring cosmic distances with greater precision. These efforts aren’t merely about refining a number; they represent a crucial endeavor to build a more complete and accurate picture of the universe’s past, present, and future, potentially revealing new physics beyond our current comprehension.
Efforts to reconcile differing measurements of the universe’s expansion rate, known as the Hubble Tension, are frequently hampered by limitations in conventional techniques. Current methods for determining the Hubble parameter – a value describing the rate at which the universe expands – often rely on complex statistical analyses and assumptions about the nature of dark energy. These analyses struggle to definitively isolate the source of the discrepancy, as many different dark energy models can produce similar predictions that align with existing data. Consequently, it remains difficult to ascertain whether the tension stems from systematic errors in local measurements, inaccuracies in early-universe predictions derived from the cosmic microwave background, or a fundamental misunderstanding of the underlying physics governing cosmic acceleration – necessitating the development of innovative approaches to both data analysis and theoretical modeling.
Mirroring Reality: Physics-Informed Neural Networks as a New Tool
Physics Informed Neural Networks (PINNs) represent a departure from traditional neural network applications by integrating governing physical equations into the network’s loss function. This is achieved by augmenting the standard mean squared error loss with terms representing the residuals of relevant partial differential equations, such as the Friedmann equations governing cosmological expansion. Consequently, the neural network is not solely trained on observational data; it is also constrained to produce solutions that satisfy established physical laws. This approach ensures that the network’s output adheres to fundamental cosmological principles, even in regions where observational data is sparse or unavailable, and can extrapolate beyond the training dataset while maintaining physical consistency. The network learns a solution $u(x)$ to the problem such that both the data and the physics are satisfied.
Physics Informed Neural Networks (PINNs) utilize data from Cosmic Chronometers and the Cosmic Microwave Background (CMB) to refine estimations of the Hubble parameter, $H_0$. Cosmic Chronometers, which include observations of passively evolving galaxies, provide direct measurements of redshift and luminosity distance, establishing a baseline for cosmological distance calculations. The CMB, representing the earliest observable light in the universe, offers a snapshot of the universe at a high redshift. By training neural networks on these datasets – and crucially, incorporating the physics governing their relationships – PINNs can effectively interpolate between data points and extrapolate beyond observed redshifts. This process yields a more accurate reconstruction of $H_0$ compared to methods relying solely on statistical analysis, as the network is constrained by established cosmological principles embedded within its architecture.
Traditional statistical methods for estimating cosmological parameters often rely heavily on data fitting without explicitly enforcing adherence to established physical laws. Physics Informed Neural Networks (PINNs) address this limitation by integrating governing equations – such as those derived from general relativity – directly into the network’s loss function. This constraint ensures that the predicted cosmological parameters yield solutions consistent with known physics, reducing the likelihood of statistically plausible but physically unrealistic outcomes. Consequently, PINNs provide a more robust estimation process, particularly in scenarios with limited or noisy observational data, and offer improved consistency between estimated parameters and the underlying cosmological model. The incorporation of physical constraints minimizes the risk of overfitting and enhances the generalizability of the estimated parameters across different datasets and cosmological scenarios.
Dissecting the Darkness: PINNs and the Nature of Dark Energy
Physics-Informed Neural Networks (PINNs) were employed to discriminate between several theoretical models of dark energy. The tested models included the Cosmological Constant, characterized by a constant energy density; Quintessence, a dynamic scalar field; Holographic Dark Energy, based on the holographic principle and event horizon; and the more intricate Tsallis Holographic Dark Energy, a modification incorporating Tsallis entropy. The differentiation process involved training PINNs on observational data – specifically, measurements of cosmic expansion – and assessing their ability to accurately reconstruct the Equation of State parameter, $w$, for each model. Successful discrimination indicates the potential for using PINNs to distinguish between competing dark energy theories based on observational evidence.
To ensure the reliability of parameter estimation, Physics-Informed Neural Networks (PINNs) were integrated with Markov Chain Monte Carlo (MCMC) methods. The PINN served as a surrogate model to efficiently evaluate the likelihood function, which is computationally expensive to calculate directly from the underlying cosmological equations. MCMC sampling was then employed to explore the parameter space, using the PINN to rapidly assess the goodness-of-fit for each sampled parameter set. This combination allowed for robust validation of inferred parameters – such as the Equation of State parameter $w$ – and facilitated the quantification of uncertainties through the generation of posterior distributions. The MCMC chains were assessed for convergence using standard diagnostics, ensuring that the inferred parameter distributions accurately represent the underlying probability distribution.
Analysis utilizing Physics-Informed Neural Networks (PINNs) successfully constrained the Equation of State parameter, $w$, for dark energy, yielding a value consistent with current cosmological observations. This parameter directly influences the rate of the universe’s expansion and provides differentiation between various dark energy models. Through this constraint, the research team determined a Hubble constant, $H_0$, of 71.89 ± 2.00 km/s/Mpc, a value that aligns with, though slightly differs from, some existing measurements and contributes to ongoing efforts to refine the cosmological parameter landscape.
The Weight of Evidence: Refining Our Cosmological Narrative
Evaluating the effectiveness of each proposed dark energy model required a rigorous statistical approach, and therefore, metrics quantifying both goodness-of-fit and model complexity were essential. Reduced Chi-Square assessed how well a model’s predictions aligned with observed data, while the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) penalized models for unnecessary complexity, preventing overfitting. Lower values for these criteria indicate a preferable model; a strong fit and parsimony. These metrics allowed for a comparative analysis, enabling researchers to discern whether the increased complexity of a particular dark energy model – such as incorporating neutrino masses – was statistically justified by a significantly improved fit to cosmological observations. The combined use of $χ^2_{reduced}$, AIC, and BIC provided a robust framework for model selection, ultimately strengthening the evidence supporting the most viable explanations for the accelerating expansion of the universe.
Integrating elementary particles, specifically neutrinos, into the Tsallis Holographic Dark Energy model demonstrates a marked improvement in its capacity to align with current observational data. This enhancement stems from the model’s ability to account for the contribution of neutrino mass to the overall energy density of the universe. Analyses reveal a total sum of neutrino masses – represented as $Σmν$ – less than 0.12 electron volts, a value consistent with both cosmological constraints and particle physics expectations. This incorporation not only refines the model’s predictive power but also offers a potential pathway towards resolving discrepancies between theoretical predictions and observed cosmic expansion rates, suggesting neutrinos play a crucial, though subtle, role in the dynamics of dark energy.
Statistical analyses reveal compelling support for Tsallis Holographic Dark Energy as a potential resolution to the persistent Hubble Tension in cosmology. The model demonstrates a superior fit to observational data, as quantified by the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). Specifically, Tsallis Holographic Dark Energy achieves an AIC of 3624.274 and a BIC of 3655.217, representing a noticeable improvement over the standard $\Lambda$CDM model’s values of 3654.447 and 3673.013, respectively. These lower values suggest a more parsimonious and statistically favorable explanation for the universe’s expansion rate, hinting that Tsallis Holographic Dark Energy could offer a more accurate representation of dark energy’s influence than current prevailing models.
The pursuit of cosmological parameters, as demonstrated by this research into Tsallis Holographic Dark Energy and neutrino mass, often feels akin to charting the unchartable. Any model constructed to resolve the Hubble tension—a discrepancy stubbornly resisting explanation—is inherently provisional. As Stephen Hawking once observed, “The best equations are those that are simple and beautiful.” This study’s application of Physics-Informed Neural Networks, while complex in execution, strives for that same elegance—a parsimonious explanation for observed phenomena. Yet, like any attempt to model the universe, it remains susceptible to the limitations of its underlying assumptions, a reminder that even the most sophisticated theories may ultimately vanish beyond the event horizon of observational reality.
What Lies Beyond the Horizon?
The application of Physics-Informed Neural Networks to Tsallis Holographic Dark Energy, as demonstrated, presents a superficially elegant avenue for addressing the Hubble tension. However, the very success of such modeling should prompt circumspection. The network, trained on cosmic chronometers and other observational data, merely maps a solution space; it does not, in itself, explain. The fundamental assumptions inherent in the Tsallis entropy formulation, and the choice of holographic boundary, remain largely unconstrained by the process. The model’s capacity to yield acceptable values for cosmological parameters should not be mistaken for deeper understanding.
Future iterations must move beyond parameter estimation. Investigating the stability of the Tsallis parameter under varying observational constraints is critical. Furthermore, the incorporation of independent datasets – particularly those probing the early universe – could reveal internal inconsistencies currently masked by the flexibility of the network. A rigorous assessment of the model’s predictive power, beyond the training dataset, is paramount. The accretion disk of observational data exhibits anisotropic emission; spectral line variations demand thorough consideration of relativistic Lorentz effects and strong spacetime curvature.
Ultimately, the true test lies not in refining the model, but in confronting its limitations. The universe, after all, is under no obligation to conform to any particular mathematical framework. The pursuit of a ‘solution’ to the Hubble tension may prove to be a transient illusion, a local minimum in a vast, unexplored landscape of possibilities. The darkness at the event horizon is not merely a lack of information; it is a reminder of the limits of knowledge.
Original article: https://arxiv.org/pdf/2511.09706.pdf
Contact the author: https://www.linkedin.com/in/avetisyan/
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2025-11-16 01:59