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Cosmological constraints on extra radiation and early dark energy with lepton asymmetry and varying quark masses

This paper investigates cosmological constraints on extra radiation and early dark energy by incorporating primordial lepton asymmetry and varying quark masses into a combined analysis of CMB, BAO, supernovae, and primordial abundance data, revealing that lepton asymmetry can reconcile helium measurements with higher Hubble constants in extra-radiation models while varying quark masses are required to relax deuterium-based restrictions on early dark energy.

Original authors: Yo Toda

Published 2026-10-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yo Toda

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 able to measure how fast this expansion is happening. However, a stubborn disagreement has emerged between two ways of measuring this speed. One method looks at the oldest light in the cosmos, the cosmic microwave background, which is a faint glow left over from the moment the universe was born. The other method looks at nearby stars and galaxies to measure the expansion directly. When scientists use the standard model of cosmology to interpret the ancient light, they get a slower speed than what is measured locally. This mismatch, known as the Hubble tension, suggests that our current understanding of the universe's history might be missing a piece of the puzzle. To solve this, researchers have proposed that something extra might have existed in the very early universe, perhaps a hidden form of radiation or a burst of energy that sped up the expansion just before the first atoms formed.

A new study by Yo Toda at Kochi University of Technology investigates whether these proposed additions can actually fix the speed discrepancy without breaking the rules of how the first elements of the universe were created. The early universe was a hot, dense soup where protons and neutrons fused to form the first nuclei of hydrogen, helium, and deuterium in a process called big bang nucleosynthesis. This process is incredibly sensitive to the conditions of the time. If the universe expanded too quickly, or if there were too many types of particles, the amount of helium and deuterium produced would change. Recent, highly precise measurements of these ancient elements now agree perfectly with the standard model, making it very difficult for new theories to sneak in without being noticed. Toda's work asks a specific question: can we introduce extra radiation or early energy, and perhaps tweak other hidden variables like the balance of particles or the mass of tiny building blocks, to raise the expansion speed while still matching the observed amounts of these ancient elements?

The researcher combined the latest data from the Planck satellite, which mapped the cosmic microwave background, with new measurements of galaxy distributions from the DESI survey and observations of distant supernovae. Crucially, the study incorporated the most precise recent measurements of primordial helium and deuterium. The team tested several scenarios. In one, they allowed for extra radiation and a specific imbalance between matter and antimatter in the form of neutrinos, known as lepton asymmetry. In another, they looked at a form of early dark energy, a temporary energy field that briefly accelerated the universe's growth. They also tested whether the mass of the lightest quarks, the particles that make up protons and neutrons, might have been slightly different during the formation of the first elements.

The results revealed a clear divide in how these different ideas hold up. When the team looked at the extra radiation model, they found that it could indeed help resolve the tension, but only if they allowed for that specific particle imbalance. Without this imbalance, the extra radiation would have produced too much helium, contradicting the new precise measurements. However, when they included the imbalance, the extra radiation could be higher, and the inferred speed of the universe's expansion rose to 69.24 kilometers per second per megaparsec. This value is much closer to the local measurements than the standard model's prediction, reducing the statistical disagreement between the two methods from nearly six standard deviations down to just over three. This suggests that a universe with a bit more radiation and a slight tilt in the balance of neutrinos is a viable path forward, provided the helium measurements remain as they are.

The story was different for the early dark energy model. This theory proposes a temporary energy field that speeds up expansion, but it tends to favor a higher density of ordinary matter in the early universe. A higher matter density, in turn, leads to less deuterium being produced. The new, precise measurements of deuterium are very strict about this amount. The study found that adding lepton asymmetry did not help the early dark energy model; the imbalance affected helium but did not fix the deuterium problem. However, a different tweak worked. When the researchers allowed the mass of the light quarks to vary slightly during the formation of the first elements, the model could accommodate a larger fraction of early dark energy. A small increase in quark mass boosts the production of deuterium, compensating for the reduction caused by the higher matter density. This adjustment allowed the early dark energy model to reach a slightly higher expansion speed, though it still did not resolve the tension as effectively as the extra radiation model with lepton asymmetry.

The study also examined the total mass of neutrinos, the ghostly particles that zip through the universe. By allowing the lepton asymmetry to vary, the researchers found that the upper limit on the combined mass of these neutrinos could be relaxed slightly, moving from a very tight constraint to a slightly more flexible one. This means that if the universe does contain extra radiation, it might be able to support a bit more neutrino mass than previously thought without contradicting the data.

Ultimately, the paper concludes that the path to solving the Hubble tension depends heavily on which specific physical changes are introduced. The extra radiation model, when paired with a specific particle imbalance, emerges as the most promising candidate for raising the expansion speed while staying consistent with the precise chemical fingerprints of the early universe. The early dark energy model can also be improved, but only if the fundamental masses of the particles themselves were slightly different in the past. These findings do not declare the mystery solved, but they provide a clearer map of where the solution might lie. They show that the universe's history is likely more complex than the standard model suggests, and that the key to unlocking the speed of expansion may be hidden in the subtle chemistry of the first few minutes after the Big Bang.

Technical Summary: Cosmological Constraints on Extra Radiation and Early Dark Energy with Lepton Asymmetry and Varying Quark Masses

Problem Statement
The standard $\Lambda$ CDM cosmological model faces the "Hubble tension," a significant discrepancy between the Hubble constant ( $H_0$ ) inferred from Cosmic Microwave Background (CMB) observations and that measured via the local distance ladder. Proposed solutions often involve modifying the early Universe to increase the expansion rate prior to recombination, thereby reducing the sound horizon and allowing for a higher inferred $H_0$ . Two primary classes of such extensions are extra relativistic energy density (parameterized by the effective number of neutrino species, $N_{\text{eff}}$ ) and axion-like Early Dark Energy (EDE). However, these modifications must remain consistent with Big Bang Nucleosynthesis (BBN) predictions for primordial light-element abundances, specifically helium ( $Y_P$ ) and deuterium (D/H). Recent high-precision measurements, particularly the LBT helium measurement ( $Y_P = 0.2458 \pm 0.0013$ ) and the 2026 Particle Data Group (PDG) deuterium compilation ( $10^5(\text{D/H})_p = 2.508 \pm 0.029$ ), impose stringent constraints on these models. This paper investigates whether primordial lepton asymmetry ( $\xi$ ) and variations in light-quark masses ( $\delta_q$ ) can relax these BBN constraints, thereby allowing these extensions to alleviate the Hubble tension while remaining consistent with updated observational data.

Methodology
The author performs Markov Chain Monte Carlo (MCMC) analyses using the Cobaya framework with theoretical CMB spectra computed via CAMB. The analysis combines:

CMB Data: Planck 2018 low-multipole temperature and E-mode polarization, high-multipole NPIPE CamSpec TTTEEE likelihood, and Planck PR4 lensing.
Large Scale Structure: DESI DR2 Baryon Acoustic Oscillation (BAO) measurements.
Supernovae: Pantheon+ light-curve sample.
Primordial Abundances: Independent Gaussian likelihoods for helium ( $Y_P = 0.2458 \pm 0.0013$ ) and deuterium ( $10^5(\text{D/H}) = 2.508 \pm 0.029$ ).

The study evaluates ten distinct cosmological models extending the flat $\Lambda$ CDM baseline:

Extra Radiation Models: Variations of $N_{\text{eff}}$ combined with lepton asymmetry ( $\xi$ ) and/or summed neutrino mass ( $\sum m_\nu$ ). The lepton asymmetry modifies the neutron-to-proton ratio via charged-current weak reactions, affecting $Y_P$ and D/H.
Early Dark Energy Models: Axion-like EDE characterized by a potential $V(\phi) = m^2 f^2 [1 - \cos(\phi/f)]^n$ (with $n=3$ ). The EDE fraction $f_{\text{EDE}}$ is constrained by its indirect effect on the baryon density ( $\Omega_b h^2$ ) favored by cosmological data, which in turn alters predicted D/H.
Quark Mass Variation: A phenomenological parameter $\delta_q$ representing the variation of the average light-quark mass during BBN. This modifies nuclear binding energies and scattering lengths, altering primordial abundances.

The BBN predictions are calculated using PRIMAT interpolation tables, with corrections applied for $\xi$ and $\delta_q$ based on established fitting relations and nuclear response coefficients. The analysis fixes the summed neutrino mass at $0.06$ eV in most models, except where explicitly sampled.

Key Contributions and Results

Extra Radiation and Lepton Asymmetry ( $N_{\text{eff}} + \xi$ ):
When $N_{\text{eff}}$ is varied alone, BBN constraints (specifically the LBT helium measurement) force $N_{\text{eff}}$ back toward the standard value ( $3.058 \pm 0.075$ ), suppressing the inferred $H_0$ to $68.10 \pm 0.51$ km s $^{-1}$ Mpc $^{-1}$ . However, allowing a positive lepton asymmetry ( $\xi$ ) compensates for the helium increase caused by extra radiation. The joint analysis yields:
- $N_{\text{eff}} = 3.27 \pm 0.16$
- $\xi = 0.017 \pm 0.011$
- $H_0 = 69.24 \pm 0.92$ km s $^{-1}$ Mpc $^{-1}$
  This combination reduces the Gaussian tension with the SH0ES measurement ( $73.17 \pm 0.86$ km s $^{-1}$ Mpc $^{-1}$ ) from $5.74\sigma$ (in $\Lambda$ CDM) to $3.12\sigma$ . Crucially, this result is achieved without requiring a helium abundance lower than the standard BBN prediction.
  Furthermore, allowing lepton asymmetry relaxes the 95% upper limit on the summed neutrino mass from $0.0640$ eV to $0.0777$ eV in the extra-radiation model.
Early Dark Energy (EDE):
In the standard EDE model, BBN constraints (driven by the deuterium abundance) tighten the upper limit on the EDE fraction ( $f_{\text{EDE}}$ ) from $0.0171$ (without BBN) to $0.0120$, reducing $H_0$ to $68.50^{+0.25}_{-0.70}$ km s $^{-1}$ Mpc $^{-1}$ .
- Lepton Asymmetry in EDE: Adding $\xi$ to the EDE model does not significantly relax these constraints ( $f_{\text{EDE}} < 0.0113$ , $H_0 = 68.45^{+0.24}_{-0.64}$ ). The helium measurement restricts $\xi$ to values that do not sufficiently compensate for the deuterium reduction caused by the higher baryon density associated with EDE.
- Quark Mass Variation in EDE: Introducing a positive variation in light-quark mass ( $\delta_q$ $δ_{q}$ ) effectively compensates for the deuterium reduction. The model yields:
  - $\delta_q = 0.0072 \pm 0.0044$ (a $\sim 0.72\%$ increase in quark mass during BBN)
  - $f_{\text{EDE}} < 0.0187$
  - $H_0 = 68.85^{+0.33}_{-0.96}$ km s $^{-1}$ Mpc $^{-1}$
    This result brings the tension down to $4.02\sigma$ , significantly improving upon the standard EDE model.
Quark Mass Variation in Extra Radiation:
In contrast to the EDE case, allowing $\delta_q$ in the extra-radiation model ( $N_{\text{eff}} + \delta_q$ ) leaves the constraints on $N_{\text{eff}}$ and $H_0$ nearly unchanged compared to the $N_{\text{eff}}$ -only model.

Significance and Claims
The paper claims that the effectiveness of lepton asymmetry and quark-mass variation in alleviating the Hubble tension depends critically on the specific BBN mechanism constraining the model:

Lepton Asymmetry is effective for extra radiation because it directly compensates the helium production induced by a higher expansion rate, allowing for larger $N_{\text{eff}}$ and $H_0$ consistent with the precise LBT helium measurement.
Quark-Mass Variation is effective for EDE because it compensates the deuterium reduction caused by the higher baryon density favored by EDE models.
The study demonstrates that the extra-radiation model with lepton asymmetry provides the most significant reduction in Hubble tension ( $3.12\sigma$ ) among the tested scenarios, without requiring deviations from standard BBN helium predictions.
The results regarding quark-mass variation are noted as conditional on the adopted nuclear response coefficients; the author emphasizes that the modest displacement of $\delta_q$ from zero suggests that small modifications to BBN nuclear physics could be beneficial for EDE-based solutions to the Hubble tension.

The author concludes that precision measurements of primordial abundances, combined with improved nuclear response calculations, are essential for assessing these extensions to the standard cosmological model.

Technical Summary: Cosmological Constraints on Extra Radiation and Early Dark Energy with Lepton Asymmetry and Varying Quark Masses

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