⚛️ general relativity

Neutrino mass constraints in the Schwarzschild-de Sitter black-hole dark energy model with ACT DR6 and DESI DR2 data

Using recent CMB, DESI, and supernova data, this study finds that the Schwarzschild-de Sitter black-hole dark energy model prefers a positive neutrino mass due to parameter correlations, yet the standard $\Lambda$ CDM model remains strongly favored over this alternative framework.

Original authors: Sheng-Han Zhou, Tian-Nuo Li, Guo-Hong Du, Yi-Min Zhang, Zhao-Yu Li, Jing-Fei Zhang, Xin Zhang

Published 2026-07-07

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CC BY 4.0

Original authors: Sheng-Han Zhou, Tian-Nuo Li, Guo-Hong Du, Yi-Min Zhang, Zhao-Yu Li, Jing-Fei Zhang, Xin Zhang

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 had a very specific rulebook for how that balloon inflates, called the ΛCDM model. It's like a trusted recipe that has worked well for most ingredients, but recently, new measurements (from a massive telescope survey called DESI) have suggested the recipe might need a tweak. The data hints that the "dark energy" pushing the balloon apart isn't constant; it might be changing over time.

This paper asks a big question: If we change the recipe for dark energy, does it change what we think about the weight of neutrinos?

The Cast of Characters

Neutrinos: Think of these as the "ghost particles" of the universe. They are everywhere, zip through everything, and we know they have a tiny bit of mass, but we don't know exactly how heavy they are.
The Standard Recipe (ΛCDM): The current "gold standard" theory. In this version, when scientists look at the data, they usually find that neutrinos are so light they are basically at the minimum weight allowed by physics (or even zero).
The New Recipe (SdSDE): This is the "Schwarzschild–de Sitter Black-Hole Dark Energy" model. It's a fancy idea suggesting that the collective weight of all the black holes in the universe acts like a fluid that pushes the universe apart. It's a different way of explaining the "dark energy" force.
The Data: The scientists used a massive toolkit of cosmic measurements:
- CMB: The "baby picture" of the universe (Cosmic Microwave Background).
- DESI: A map of where galaxies are today.
- Supernovae: Exploding stars used as "standard candles" to measure distance.

The Experiment: Swapping the Recipe

The researchers ran a simulation. They took all the latest data and tried to fit it into two different boxes:

Box A: The Standard Recipe (ΛCDM) with flexible neutrino weights.
Box B: The New Black-Hole Recipe (SdSDE) with flexible neutrino weights.

The Surprising Result: The "Ghost" Gets Heavier

Here is the twist: The recipe you use changes the answer.

In the Standard Box (ΛCDM): The data says, "Neutrinos are very light. We can't prove they are heavy." The results push the neutrino mass down to the lowest possible limit.
In the Black-Hole Box (SdSDE): The data says, "Wait, if we use this new recipe, the neutrinos must have a positive, measurable weight."

When they used the Black-Hole recipe, the math strongly suggested that neutrinos have a mass of about 0.2 eV (a tiny number, but significant in physics). This is a "positive preference," meaning the data actively points to them having weight, rather than just saying "they could be anything."

Why Does This Happen? (The Analogy)

Imagine you are trying to balance a scale. On one side, you have the "expansion of the universe" (how fast the balloon is growing). On the other side, you have "gravity" (how much stuff is pulling it back).

The Problem: The new data (DESI) shows the universe is expanding in a way that is hard to explain with the old recipe.
The Compensation: In the Black-Hole recipe, the "dark energy" behaves differently. To make the scale balance with the new data, the computer simulation finds that it needs to add "weight" to the neutrinos.
The Trade-off: It's like a game of "Whac-A-Mole." If you change the shape of the hole (the dark energy model), the mole (the neutrino mass) pops up in a different spot. The paper suggests this isn't necessarily because neutrinos are heavy, but because the new recipe compensates for its own quirks by making the neutrinos heavier.

There is also a second character, $N_{eff}$ (the number of effective particle types). The Black-Hole recipe likes to lower this number. Because neutrino mass and this number are linked (like two ends of a seesaw), when one goes down, the other is pushed up, reinforcing the idea that neutrinos have mass.

The Verdict: A Better Fit?

You might think, "If the Black-Hole recipe makes the neutrino mass look more interesting, maybe it's the better theory!"

Not so fast.

The authors checked the "scorecard" (called $\chi^2$ ), which measures how well the theory matches the actual data.

The Result: The Standard Recipe (ΛCDM) actually fits the data better than the Black-Hole recipe.
The Problem: The Black-Hole recipe struggles to explain the distances to faraway galaxies and supernovae. It's too rigid; it can't bend enough to match the new observations.

The Bottom Line

This paper is a warning and a lesson in humility for cosmologists:

Context Matters: The answer to "How heavy are neutrinos?" depends entirely on which theory of dark energy you believe in.
The "Positive Mass" Might Be a Trick: The fact that the Black-Hole model suggests heavy neutrinos might just be the model trying to fix its own errors, not a discovery of new physics.
The Standard Model Still Wins (For Now): Even though the Black-Hole model gives a more exciting answer about neutrinos, it fails the overall test of fitting the universe's data better than the old standard model.

In short: Don't get too excited about heavy neutrinos just yet. It might just be the universe's way of balancing the books with a different accounting method. We need even better data to know for sure which recipe is the real one.

Technical Summary: Neutrino Mass Constraints in the Schwarzschild-de Sitter Black-Hole Dark Energy Model

Problem Statement
Recent observations from the Dark Energy Spectroscopic Instrument (DESI) DR2, when combined with Cosmic Microwave Background (CMB) data, have revealed a tension within the standard $\Lambda$ CDM model. Specifically, DESI data combined with CMB yields a 95% confidence upper limit on the sum of neutrino masses ( $\sum m_\nu$ ) of $< 0.064$ eV, which approaches the lower bound ( $\gtrsim 0.059$ eV) required by neutrino oscillation experiments for the normal hierarchy. Furthermore, DESI data shows a preference for dynamical dark energy (DE) over a cosmological constant. This paper investigates whether a specific physical model of dark energy—the Schwarzschild–de Sitter black-hole dark energy (SdSDE) model—alters the cosmological inference of neutrino parameters ( $\sum m_\nu$ and $N_{\text{eff}}$ ) compared to the standard $\Lambda$ CDM framework. The study addresses the model dependence of neutrino mass constraints and whether the SdSDE scenario can resolve or shift the current tensions.

Methodology
The authors employ a Bayesian inference framework using Markov Chain Monte Carlo (MCMC) sampling via the Cobaya code and the Boltzmann solver CAMB. The analysis utilizes the following datasets:

CMB: Planck PR4 (temperature, polarization, and lensing) and Atacama Cosmology Telescope (ACT) DR6 data.
BAO: DESI DR2 measurements using various tracers (bright galaxies, LRGs, ELGs, quasars, Lyman- $\alpha$ forest).
Supernovae: DES-Dovekie (a recalibrated sample of 1820 Type Ia SNe) and PantheonPlus (1550 spectroscopically confirmed SNe).

The theoretical framework is the SdSDE model proposed by Hayashi [228]. In this model, the effective dark energy density $\rho_{\text{DE}}(z)$ is driven by the redshift evolution of the cosmic black-hole mass density, parameterized by a fixed polynomial fit $F(z)$ . This results in a specific, non-dynamical equation of state $w_{\text{DE}}(z)$ that is distinct from the phenomenological $w_0w_a$ CDM parameterization. The model is implemented with a smooth cutoff at $z \approx 10$ to ensure it approaches $\Lambda$ CDM at high redshifts.

The study compares four model configurations:

Baseline SdSDE.
SdSDE + free $\sum m_\nu$ .
SdSDE + free $N_{\text{eff}}$ .
SdSDE + free $\sum m_\nu$ and $N_{\text{eff}}$ .

These are contrasted against corresponding extensions of the $\Lambda$ CDM model. The analysis uses flat priors for all cosmological parameters, including $\sum m_\nu \in [0, 5]$ eV and $N_{\text{eff}} \in [0, 7]$ .

Key Contributions and Results
The primary finding is that the SdSDE framework induces a robust preference for a positive neutrino mass, a result that contrasts with the upper limits typically found in $\Lambda$ CDM analyses with the same datasets.

Neutrino Mass Preference: When analyzing CMB + DESI + DES-Dovekie data, the SdSDE model with a free $\sum m_\nu$ yields $\sum m_\nu = 0.207^{+0.047}_{-0.052}$ eV, representing a $\sim 4.0\sigma$ preference for a non-zero mass. When $N_{\text{eff}}$ is also allowed to vary, the constraint becomes $\sum m_\nu = 0.162^{+0.055}_{-0.056}$ eV (a $\sim 2.9\sigma$ preference).
Parameter Compensation: The preference for a positive mass is linked to a systematic shift in $N_{\text{eff}}$ . The SdSDE model prefers $N_{\text{eff}}$ values below the standard model prediction ( $N_{\text{eff}} \approx 2.72$ vs. $3.044$). Since $\sum m_\nu$ and $N_{\text{eff}}$ are positively correlated, a lower $N_{\text{eff}}$ typically weakens the mass preference; however, the SdSDE background evolution drives the mass parameter upward to compensate.
Impact on $H_0$ and $S_8$ : The SdSDE model naturally favors a higher $H_0$ ( $\sim 71.85$ km s $^{-1}$ Mpc $^{-1}$ in the baseline) and a lower $S_8$ when massive neutrinos are included. The inclusion of a positive $\sum m_\nu$ suppresses structure growth, lowering $S_8$ and partially alleviating the $S_8$ tension, while the model's background evolution helps maintain a higher $H_0$ , reducing the tension with SH0ES measurements to $\sim 1.5\sigma$ in the CMB+DESI case.
Goodness-of-Fit: Despite the interesting parameter shifts, the SdSDE model does not provide a better global fit than the extended $\Lambda$ CDM model. The $\chi^2$ comparison shows that SdSDE is strongly disfavored relative to $\Lambda$ CDM extensions ( $\Delta \chi^2 \approx 18.65$ for CMB+DESI, increasing to $\sim 66$ with supernova data). The degradation in fit is driven primarily by BAO and Supernova data, suggesting the fixed SdSDE equation of state cannot mimic the phantom-like or quintom-like evolution ( $w(z)$ crossing $-1$) preferred by current late-time distance measurements.

Significance and Claims
The paper concludes that the positive neutrino mass preference observed in the SdSDE model is likely a result of parameter compensation rather than an indication of improved physical reality. The authors argue that the "preference" arises because the rigid SdSDE equation of state forces the cosmological parameters to shift (specifically lowering $N_{\text{eff}}$ and raising $\sum m_\nu$ ) to accommodate the data, rather than the model providing a superior description of the universe.

The authors modestly state that while the SdSDE scenario offers a concrete physical template for dark energy linked to black-hole populations, its current form is insufficient to resolve the tensions in a way that improves the overall fit. They suggest that future high-precision data is required to test whether this mass preference is a genuine physical signal or an artifact of the specific model assumptions. The work highlights the critical model dependence of neutrino mass constraints, reinforcing the need to test such inferences across diverse dark energy scenarios.