⚛️ general relativity

DESI DR2 Constraints on Coupled Hu--Sawicki $f(R)$ Gravity with Interacting Neutrinos: Implications for the Hubble and $S_8$ Tensions

This study demonstrates that a coupled Hu--Sawicki $f(R)$ gravity model with interacting neutrinos, constrained by DESI DR2 and other cosmological data, significantly alleviates both the Hubble and $S_8$ tensions while providing a statistically superior fit to the standard $\Lambda$ CDM model.

Original authors: Muhammad Yarahmadi

Published 2026-09-15

📖 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

The universe is expanding, and for decades, astronomers have been trying to measure exactly how fast. This rate, known as the Hubble constant, is a fundamental number that tells us the age and history of everything around us. However, a stubborn problem has emerged: when scientists look back at the very beginning of the universe using the afterglow of the Big Bang, they calculate one speed. When they look at the nearby universe today, using exploding stars and other cosmic markers, they measure a different, faster speed. This disagreement, known as the Hubble tension, suggests that our standard model of cosmology might be missing a piece of the puzzle. Complicating matters further is a second discrepancy regarding how clumpy the universe is; the matter in space seems to be grouping together less than our theories predict. These two conflicts have left physicists searching for new physics that could explain why the early and late universe seem to be telling different stories.

A researcher has proposed a new way to bridge this gap by rethinking the force of gravity itself and how it interacts with invisible particles called neutrinos. In their study, they explored a specific version of modified gravity, a theory that suggests the rules governing gravity change slightly over vast cosmic distances, rather than staying exactly the same as they do in our solar system. They combined this idea with the behavior of neutrinos, which are tiny, ghostly particles that zip through the universe almost at the speed of light. The researcher hypothesized that these two elements might be linked, allowing energy to flow between the fabric of space and the neutrino particles. By testing this idea against a massive collection of modern astronomical data, they found that this interaction could nudge the universe's expansion rate and the clumping of matter in just the right direction to ease the tension between early and late measurements.

To test their idea, the researcher built a complex computer model that simulates the history of the universe from the Big Bang to the present day. They used a framework based on the Hu–Sawicki model, a popular and mathematically sound version of modified gravity that behaves like standard gravity in dense areas like our solar system but changes its behavior in the vast emptiness between galaxies. They then added a new ingredient: a mechanism that allows the curvature of space to exchange energy with the neutrino fluid. This interaction was not just a guess; it was a specific mathematical adjustment designed to see if it could alter the universe's expansion history and the way matter clumps together. The researcher then fed this model into a powerful analysis system, feeding it data from the cosmic microwave background, the distribution of galaxies, the light from distant supernovae, and the ticking clocks of aging stars.

The results of this analysis were promising. When the researcher ran their simulations with the interacting model, the universe's expansion rate shifted upward, moving closer to the faster speed measured by local astronomers. At the same time, the amount of clumping in the universe decreased slightly, bringing the theoretical prediction closer to what telescopes actually see in the nearby universe. Specifically, the model suggested a Hubble constant of 70.21 kilometers per second per megaparsec, a value that sits comfortably between the conflicting measurements. Simultaneously, the clustering parameter, which measures how tightly matter is grouped, settled at 0.796, a figure that aligns much better with recent observations of galaxy surveys than the standard model does. This dual improvement suggests that the interaction between modified gravity and neutrinos could be the missing link that resolves both tensions at once.

The study also looked at the total mass of the neutrinos, a quantity that has been difficult to pin down. The new model placed an upper limit on this mass, suggesting it is less than 0.122 electronvolts. This constraint is consistent with what other experiments have found and does not break any known laws of physics. Furthermore, the researcher compared their new model against the standard model of cosmology, which assumes gravity works exactly as Einstein described and neutrinos do not interact with the expansion of space. Using statistical tools to weigh the fit of the data against the complexity of the theory, the interacting model came out as the better description of the universe. It provided a significantly better match to the observations, reducing the statistical error by a large margin compared to the standard model.

While the findings are compelling, the researcher is careful to note that this is not a final solution. The model suggests a path forward, showing that a universe where gravity and neutrinos talk to each other can resolve the current conflicts without contradicting existing data. The interaction parameter they found is positive, indicating that energy is indeed flowing in the way they proposed, but the exact nature of this flow remains a topic for future investigation. The study does not prove that modified gravity is definitely real, but it demonstrates that such a scenario is viable and worth pursuing. As new data from upcoming telescopes and surveys arrive, scientists will be able to test these predictions with even greater precision, potentially confirming whether the universe is indeed governed by these subtle, interconnected forces.

Technical Summary: DESI DR2 Constraints on Coupled Hu–Sawicki f(R) Gravity with Interacting Neutrinos

Problem Statement
The paper addresses two persistent discrepancies in modern precision cosmology: the Hubble tension ( $H_0$ ), representing the conflict between early-Universe (CMB) and late-Universe (local distance ladder) measurements of the expansion rate, and the $S_8$ tension, which involves a systematic difference in the amplitude of matter clustering inferred from CMB data versus weak-lensing and galaxy surveys. While the standard $\Lambda$ CDM model successfully describes a broad range of observables, it fails to reconcile these tensions without invoking unexplained systematics. The author investigates whether a specific extension of General Relativity (GR)—Hu–Sawicki $f(R)$ gravity—coupled with an effective interaction between the curvature scalar and the neutrino sector, can simultaneously alleviate both tensions while remaining consistent with current observational data.

Methodology
The study employs a comprehensive theoretical and statistical framework:

Theoretical Formulation: The author starts with the metric formulation of $f(R)$ gravity, specifically the Hu–Sawicki model, which asymptotically recovers GR in high-curvature regimes (satisfying local constraints via the Chameleon mechanism) but introduces deviations at low curvatures to drive late-time acceleration. They extend this by introducing a phenomenological interaction between the scalar degree of freedom ( $f_R$ ) and massive neutrinos. This interaction modifies the neutrino continuity equation, allowing for energy exchange between the curvature and neutrino sectors.
Numerical Implementation: The modified background and perturbation equations are implemented within the hi_class Boltzmann solver (an extension of CLASS for scalar-tensor theories) and interfaced with MontePython for Bayesian inference. The implementation rigorously includes the interaction source terms in both the background evolution and the linear perturbation hierarchy (monopole and dipole moments) for neutrinos.
Data Analysis: A Bayesian Markov Chain Monte Carlo (MCMC) analysis is performed using a combination of datasets:
- Planck 2018 CMB temperature, polarization, and lensing data.
- DESI DR2 Baryon Acoustic Oscillation (BAO) measurements.
- Pantheon+ Type Ia supernovae.
- Cosmic Chronometers (CC) for direct $H(z)$ measurements.
  The analysis compares three models: standard flat $\Lambda$ CDM, non-interacting Hu–Sawicki $f(R)$ , and the interacting Hu–Sawicki $f(R)$ scenario.

Key Contributions

Novel Coupling Mechanism: The paper derives and implements a covariant interaction between the $f(R)$ scalaron and neutrinos, treating the interaction rate $\Gamma$ as a free parameter constrained by data rather than fixed by a specific microscopic Lagrangian.
Integrated Constraints: It provides the first joint constraints on this specific coupled model using the latest DESI DR2 BAO data combined with Planck, supernovae, and cosmic chronometers.
Dual Tension Mitigation: The study demonstrates that the coupled model offers a unique mechanism to address both the $H_0$ and $S_8$ tensions simultaneously, a feat often difficult to achieve with single-modification models (which typically resolve one tension at the expense of the other).

Results
The analysis yields the following key findings:

Hubble Constant ( $H_0$ ): The interacting Hu–Sawicki model shifts the inferred Hubble constant to higher values compared to $\Lambda$ CDM. For the full dataset combination, the model yields $H_0 = 70.21 \pm 1.60$ km s $^{-1}$ Mpc $^{-1}$ , compared to $69.19 \pm 1.42$ km s $^{-1}$ Mpc $^{-1}$ for $\Lambda$ CDM. This reduces the tension with the local SH0ES (R22) measurement from $2.3\sigma$ to $1.4\sigma$ .
Clustering Amplitude ( $S_8$ ): The model simultaneously lowers the clustering amplitude to $S_8 = 0.796 \pm 0.009$ , reducing the tension with DES-Y6 weak-lensing data to $0.9\sigma$ (below the $1\sigma$ threshold) and KiDS to $1.2\sigma$ .
Model Comparison: The interacting model provides the best statistical fit among the scenarios tested. It achieves a minimum chi-squared improvement of $\Delta\chi^2 \simeq -45$ relative to $\Lambda$ CDM. According to the Akaike Information Criterion (AIC), the interacting model is strongly preferred; while the Bayesian Information Criterion (BIC) penalizes the additional complexity, the interacting model still retains the lowest BIC value, indicating the likelihood gain outweighs the complexity penalty.
Neutrino Mass: The model remains consistent with current neutrino mass bounds, yielding an upper limit of $\sum m_\nu < 0.122$ eV for the full dataset, which is compatible with both cosmological limits and laboratory constraints (e.g., KATRIN).
Parameter Constraints: The interaction parameter is constrained to a positive range of $\Gamma \approx 0.063 - 0.069$ , suggesting an effective energy exchange within the dark sector. The Hu–Sawicki parameters ( $C_1, C_2, f_0$ ) remain close to the GR limit, indicating that current data only allows for moderate deviations from standard gravity.

Significance and Claims
The author claims that the interacting Hu–Sawicki $f(R)$ scenario offers a viable route to moderately easing both the $H_0$ and $S_8$ tensions simultaneously without violating current observational constraints. The paper emphasizes that while the results do not constitute definitive evidence for modified gravity or dark-sector interactions, they demonstrate that such extensions are statistically competitive with $\Lambda$ CDM and capable of reconciling early- and late-Universe discrepancies. The work suggests that the interplay between modified gravity and massive neutrinos provides a physical mechanism to adjust the sound horizon and structure growth, thereby offering a coherent explanation for the observed cosmological tensions. The author concludes that these models warrant further investigation with higher-precision data from future surveys like Euclid and Rubin LSST.

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