⚛️ general relativity

Late-Time Alleviation of the Hubble Tension in CPL Cosmology with Massive Neutrinos via Bayesian Physics-Informed Neural Networks

This study employs Bayesian Physics-Informed Neural Networks to demonstrate that combining a time-evolving dark energy equation of state (CPL) with massive neutrinos effectively alleviates the Hubble tension, reconciling early- and late-Universe measurements of $H_0$ to within a $\sim1$ - $2\sigma$ range.

Original authors: Muhammad Yarahmadi

Published 2026-07-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Muhammad Yarahmadi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, expanding balloon. For decades, scientists have been trying to figure out exactly how fast this balloon is inflating right now. This speed is called the Hubble constant. The problem is, we have two very different ways of measuring it, and they don't agree. One way looks at the "baby pictures" of the universe (the Cosmic Microwave Background) and says the balloon is inflating at a certain slow pace. The other way looks at "adult" stars and supernovae nearby and says, "Nope, it's inflating much faster!" This disagreement is known as the "Hubble Tension," and it's like two experts measuring the same room with different tape measures and getting different lengths. If the tape measures are both right, it means our understanding of the rules of the universe (the physics) might be missing a piece of the puzzle.

To solve this, scientists look for hidden ingredients that might be changing how the universe expands. Two big suspects are "Dark Energy" (a mysterious force pushing the universe apart) and "Neutrinos" (tiny, ghost-like particles that have a tiny bit of mass). Usually, we assume Dark Energy is a constant, unchanging force, but maybe it's actually a shapeshifter that changes its strength over time. And maybe those ghostly neutrinos are heavier than we thought, acting like a subtle brake on the universe's expansion. The question is: if we let these ingredients wiggle and change, can we make the two different tape measures agree?

In this paper, a researcher named Muhammad Yarahmadi tackles this cosmic disagreement using a very modern tool: a "smart" computer program called a Bayesian Physics-Informed Neural Network (PINN). Think of this program not just as a calculator, but as a student who is forced to learn the laws of physics (like the rules of how the universe expands) while simultaneously studying the actual data from telescopes. Instead of just guessing numbers, the program is "informed" by the universe's rulebook, ensuring that every answer it gives makes physical sense.

The study tests three different versions of the universe's rulebook. The first is the standard model, where Dark Energy is a boring, unchanging constant. The second allows Dark Energy to be a "chameleon," changing its behavior over time (using a model called CPL). The third adds a twist: it lets the ghostly neutrinos have a specific, free-floating mass. The researcher feeds this smart program data from three main sources: the "baby pictures" of the universe (Planck 2018), the "adult" supernovae (Pantheon+), and the "echoes" of sound waves from the early universe (DESI DR2 BAO measurements).

Here is what the smart program found. When they stuck to the boring, unchanging Dark Energy model, the disagreement between the early and late universe measurements remained stubborn. When they allowed Dark Energy to change over time (the CPL model), the results shifted. The program suggested that Dark Energy might be slightly different today than it was in the past, which pushed the calculated expansion rate higher. This helped the "late universe" measurements agree better with the nearby star measurements, reducing the tension in some specific cases, but it didn't fully resolve the issue on its own and often made the disagreement with the early universe data worse.

The real magic happened when they added the massive neutrinos to the mix. By allowing the neutrinos to have a mass (specifically, a sum of masses less than about 0.16 to 0.28 eV depending on the data combination), the model stabilized. The smart program found a "Goldilocks" zone where the expansion rate settled into a range of roughly 69.7 to 71.6 km s⁻¹ Mpc⁻¹. In this specific scenario (CPL with massive neutrinos), the tension with the local measurements dropped significantly, in several cases falling below the 1σ level, while the tension with the early universe data settled into a more manageable 1–2σ range. This suggests that a universe with a slightly changing Dark Energy and some heavy neutrinos is a better fit than the standard, boring one.

The paper also did a crucial check to make sure their "smart student" wasn't just making things up. They compared the PINN's answers against a traditional, very slow, and very careful method called MCMC (Markov Chain Monte Carlo). The results matched almost perfectly, with the PINN finding the same answers but doing it thousands of times faster. This proves that the new method is a reliable, speedy way to solve these complex cosmic puzzles.

Ultimately, the paper suggests that the Hubble Tension isn't just a measurement error; it might be a clue that our universe is more dynamic than we thought. It hints that Dark Energy might be evolving and that neutrinos might be heavier than the "massless" ghost we imagined. While this doesn't completely solve the mystery (a small disagreement remains), it points the way toward a more complex and interesting universe, and it shows that using AI to learn physics directly from the data is a powerful new way to explore the cosmos.

Technical Summary: Late-Time Alleviation of the Hubble Tension via Bayesian PINNs

Problem Statement
The paper addresses the persistent "Hubble tension," a statistically significant discrepancy between the Hubble constant ( $H_0$ ) inferred from early-Universe probes (primarily Planck 2018 CMB data under the $\Lambda$ CDM model, yielding $H_0 \approx 67\text{--}68$ km s $^{-1}$ Mpc $^{-1}$ ) and direct late-time measurements (e.g., SH0ES/R22 using the cosmic distance ladder, yielding $H_0 \approx 73\text{--}74$ km s $^{-1}$ Mpc $^{-1}$ ). While the standard $\Lambda$ CDM model fits a wide range of observations, its assumption of a strictly constant dark energy equation of state ( $w=-1$ ) is theoretically ad hoc. The authors investigate whether extending the model to include dynamical dark energy (via the Chevallier–Polarski–Linder or CPL parametrization) and massive neutrinos ( $\Sigma m_\nu$ ) can alleviate this tension. Furthermore, the paper seeks to demonstrate the efficacy of Bayesian Physics-Informed Neural Networks (PINNs) as a computationally efficient alternative to traditional Markov Chain Monte Carlo (MCMC) methods for such complex, high-dimensional cosmological inference.

Methodology
The study employs a Bayesian Physics-Informed Neural Network (BPINN) framework to perform joint parameter inference.

Architecture: The Hubble expansion history $H(z)$ is represented by a fully connected feed-forward neural network with Softplus activation functions. To enable Bayesian inference and uncertainty quantification, dropout layers are retained during training to approximate Bayesian marginalization.
Physics Constraints: The Friedmann equation for a spatially flat universe is embedded directly into the loss function as a soft physical constraint. This ensures the reconstructed expansion history remains consistent with the background cosmology defined by the CPL parametrization and massive neutrinos.
Parameter Space: The model simultaneously infers cosmological parameters $\Theta = \{H_0, \Omega_m, w_0, w_a, \Sigma m_\nu\}$ alongside the neural network weights. Priors are set to be physically motivated but broad (e.g., $H_0 \in [60, 80]$ , $w_0 \in [-2.5, -0.2]$ ).
Data: The analysis combines late-time observational probes:
- DESI DR2 BAO: 13 data points providing geometric constraints on transverse and radial expansion rates.
- Pantheon+: 1701 Type Ia supernovae for distance modulus constraints.
- Cosmic Chronometers (CC): 32 direct $H(z)$ measurements.
- Planck 2018: Compressed CMB distance priors (shift parameter $R$ , acoustic scale $l_A$ , and $\omega_b$ ) are used to anchor early-Universe geometry without requiring full Boltzmann solvers for the late-time analysis.
Training: The total loss function minimizes the negative log-posterior, comprising the data likelihood ( $\chi^2_{data}$ ), the physics-informed term ( $L_{phys}$ ), and prior terms ( $L_{prior}$ ).

Key Contributions

Methodological Validation: The paper validates the BPINN framework against standard MCMC analyses (performed using Cobaya) under identical data configurations. It demonstrates that BPINNs reproduce MCMC posteriors within $1\text{--}2\sigma$ while reducing inference times by orders of magnitude.
Model Comparison: A systematic comparison is conducted across three models:
- $\omega$ CDM (constant $w$ ).
- CPL (dynamical $w(z) = w_0 + w_a \frac{z}{1+z}$ ).
- CPL + $\Sigma m_\nu$ (dynamical $w$ plus free summed neutrino mass).
Quantification of Tension: The study rigorously quantifies the Hubble tension ( $T_{Planck}$ and $T_{SH0ES}$ ) for various dataset combinations, specifically leveraging the new DESI DR2 BAO data.

Results

$\omega$ CDM Baseline: In the standard model, dataset combinations involving BAO systematically favor lower $H_0$ values, reducing tension with Planck but increasing tension with SH0ES. The full combination yields $H_0 \approx 69.03 \pm 1.35$ km s $^{-1}$ Mpc $^{-1}$ , leaving residual tensions of $\sim 1.2\sigma$ (Planck) and $\sim 2.1\sigma$ (SH0ES).
CPL Dynamics: Allowing for dynamical dark energy shifts the posterior of $H_0$ toward higher values ( $H_0 \gtrsim 70$ ). The data favor a present-day equation of state $w_0 \gtrsim -1$ (quintessence-like) with negative $w_a$ . This significantly reduces the tension with SH0ES (in some cases to $<1\sigma$ ) but correspondingly increases the tension with Planck.
CPL + $\Sigma m_\nu$ : The inclusion of massive neutrinos stabilizes the inferred $H_0$ range ( $69.7\text{--}71.6$ km s $^{-1}$ Mpc $^{-1}$ ). This scenario yields a more balanced reconciliation, with tensions typically confined to the $1\text{--}2\sigma$ range for both Planck and SH0ES. Neutrino mass bounds are consistent with current limits ( $\Sigma m_\nu \lesssim 0.16\text{--}0.28$ eV).
Model Selection: Using $\chi^2$ , AIC, and BIC statistics, the extended models (CPL and CPL+ $\Sigma m_\nu$ ) are statistically preferred over $\omega$ CDM even after penalizing for additional free parameters. The improvement in fit is confirmed not to be an artifact of overfitting.
BPINN vs. MCMC: The BPINN results for $H_0$ , $w_0$ , $w_a$ , and $\Sigma m_\nu$ agree with MCMC results within $1\text{--}2\sigma$ . The BPINN approach offers comparable accuracy with significantly reduced computational cost, particularly for the posterior exploration phase.

Significance and Claims
The paper claims that the interplay between dynamical dark energy and massive neutrinos offers a viable phenomenological extension to $\Lambda$ CDM that can substantially alleviate the Hubble tension without introducing pathological instabilities. Specifically, the CPL+ $\Sigma m_\nu$ model provides a "balanced reconciliation" where the tension with local measurements is reduced to below $2\sigma$ while maintaining consistency with CMB constraints.

Methodologically, the work establishes Bayesian PINNs as an accurate, efficient, and versatile tool for precision cosmology. By embedding physical laws directly into the learning process, the framework handles parameter degeneracies (common in dynamical dark energy and neutrino mass analyses) effectively. The authors conclude that while late-time modifications alone may not fully resolve the Hubble tension (as a residual $\sim 2\sigma$ discrepancy with Planck often persists), the combination of evolving dark energy and neutrino mass represents a statistically favored direction for future research. The study emphasizes that the BPINN framework is a robust alternative to MCMC, capable of delivering high-precision constraints with orders-of-magnitude faster inference times.

Technical Summary: Late-Time Alleviation of the Hubble Tension via Bayesian PINNs

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