Interpreting the Hubble tension with a cascade decaying dark matter sector

This paper proposes a cascade decaying dark matter model that incorporates both early- and late-time modifications to $\Lambda$CDM, finding that while it can yield a higher $H_0$ value, it ultimately fails to reduce the Hubble tension below the $3\sigma$ level, thereby revising previous claims in the literature.

The Gist

Imagine the universe is a giant, expanding balloon. For decades, scientists have been trying to measure exactly how fast this balloon is inflating. This speed is called the Hubble Constant ($H_0$).

Here's the problem: We have two very precise ways of measuring this speed, and they disagree.

1. The "Baby Photo" Method: We look at the Cosmic Microwave Background (CMB), which is like a baby photo of the universe taken 380,000 years after the Big Bang. Based on this photo and our standard rules of physics, the universe should be expanding at about 67 km/s per megaparsec.
2. The "Adult Photo" Method: We look at nearby stars and supernovae (the "adult" universe) to measure the speed directly. These measurements say the universe is expanding much faster, at about 73 km/s per megaparsec.

This disagreement is called the Hubble Tension. It's like if you measured your height as a child and as an adult, and the math said you should be 5 feet tall, but a ruler said you are 6 feet tall. Something is missing from our understanding of physics.

The Authors' Idea: A "Leaky" Dark Matter Cascade

The authors of this paper propose a new solution involving Dark Matter. In our standard model, Dark Matter is like a solid, unchanging brick that just sits there and helps hold galaxies together.

The authors suggest a different story: Dark Matter isn't a solid brick; it's more like a leaky, multi-stage water balloon.

They propose a "Cascade Decaying Dark Matter" (CDDM) model with two stages:

1. Stage 1 (The Early Leak): In the very early universe, a heavy, unstable particle (let's call it the "Parent") decays. It breaks apart and creates lighter, fast-moving particles (the "Children").

   • The Analogy: Imagine a heavy water balloon popping early in the race, splashing water everywhere. This splash adds extra energy to the early universe, making it expand a bit faster right from the start. This helps fix the "Baby Photo" measurement.

2. Stage 2 (The Late Leak): These lighter "Child" particles don't stay stable forever. Much later in the universe's life, they slowly decay into neutrinos (ghostly particles that barely interact with anything).

   • The Analogy: Imagine the water from the first splash slowly dripping out of a second container much later in the race. This changes how the universe expands in its "adult" years, helping to fix the "Adult Photo" measurement.

By combining these two effects—changing the early universe and the late universe—they hoped to bridge the gap between the two measurements and solve the tension.

What They Did: The Cosmic Math Check

The authors didn't just guess; they ran massive computer simulations (using a method called MCMC) to see if this "leaky balloon" idea actually fits the data. They fed their model into the latest data from:

• Planck: The satellite that took the "baby photo."
• DESI: A survey measuring the distances to galaxies.
• Pantheon & SH0ES: Data on supernovae and local expansion rates.

The Results: A Partial Fix, But Not a Miracle

Here is what they found, translated into plain English:

1. It helps, but not enough: Their model did push the calculated expansion rate higher, getting it closer to the "Adult Photo" measurement. They managed to get a value of about 69 km/s, which is better than the standard 67, but it's still not 73.
2. The "3-Sigma" Wall: In science, a "3-sigma" difference is a standard threshold for saying "this is likely a real problem, not just a fluke." The authors found that even with their fancy new model, the tension between the two measurements is still around 3.8 sigma.
   • The Takeaway: They cannot reduce the tension to the "safe" 3-sigma level or lower without breaking other rules of physics.
3. The Cost of "Tuning": They discovered that if they tweaked the settings (priors) of their model to force the number higher (closer to 73), the model started to fit the rest of the data much worse. It's like trying to force a square peg into a round hole; you might get the peg in, but you break the hole.
4. Checking the Other Rules: They also checked if their model broke other cosmic laws, like how elements formed in the first few minutes (Big Bang Nucleosynthesis) or how neutrinos behave. Fortunately, their "leaky balloon" model passes these tests. The numbers they need for the model are allowed by these other constraints.

The Bottom Line

The authors conclude that while their "Cascade Decaying Dark Matter" idea is a clever way to try to fix the Hubble Tension, it cannot solve the problem completely on its own.

They are essentially saying: "We tried a new, complex mechanism that changes both the beginning and the end of the universe's history. It helps a little, but it doesn't fix the disagreement enough to make it go away. If we try to force it to fix the whole problem, the model falls apart."

This revises earlier claims in the scientific community that suggested such models could easily solve the tension. The authors suggest that the Hubble Tension remains a stubborn mystery that likely requires a different kind of new physics, not just a tweak to how Dark Matter decays.

Technical Summary

Technical Summary: Interpreting the Hubble Tension with a Cascade Decaying Dark Matter Sector

Problem Statement
The paper addresses the "Hubble tension," a $\sim 5.8\sigma$ discrepancy between the early-universe measurement of the Hubble constant ($H_0 = 67.36 \pm 0.54$ km s$^{-1}$ Mpc$^{-1}$) from Planck 2018 CMB data and the late-universe direct measurement ($H_0 = 73.01 \pm 0.92$ km s$^{-1}$ Mpc$^{-1}$) from the SH0ES collaboration. Within the standard $\Lambda$CDM framework, this tension is widely regarded as an indication of new physics. Previous attempts to resolve this have focused on either early-time modifications (e.g., Early Dark Energy) or late-time modifications. However, the authors note that single-effect modifications generally fail to reduce the tension to the acceptable $3\sigma$ level, with Early Dark Energy (EDE) being a notable exception. The paper investigates whether a unified model incorporating both early- and late-time effects can achieve a higher $H_0$ value while remaining consistent with other cosmological constraints.

Methodology
The authors propose and analyze a Cascade Decaying Dark Matter (CDDM) model. In this framework, the standard Cold Dark Matter (CDM) is replaced by a sector involving two particle species, $\chi_M$ and $\chi_m$:

1. Early-Time Production: A heavy parent particle $\chi_M$ (mass $M$) decays into a lighter particle $\chi_m$ (mass $m$) and a Standard Model (SM) final state $X$ during the early universe ($\tau_M \le 10^4$ s). The resulting $\chi_m$ particles are initially relativistic, contributing to the effective neutrino number ($N_{\text{eff}}$), and later become non-relativistic.
2. Late-Time Decay: The $\chi_m$ particles, acting as dark matter, subsequently decay into SM neutrinos ($\chi_m \to \nu + \bar{\nu}$) at late times ($\tau_m \ge 100$ Gyr).

The model is parametrized by three independent parameters: the mass ratio $M/m$, the lifetime of the parent particle $\tau_M$, and the decay rate of the daughter particle $\Gamma_m$ (related to $\tau_m$). The authors implement the background and linear perturbation equations for this sector into the Boltzmann solver CLASS and perform a Markov Chain Monte Carlo (MCMC) analysis using Cobaya.

The analysis fits the model to the latest datasets:

• CMB: Planck 2018 low-$\ell$ temperature/polarization, high-$\ell$ TT/TE/EE spectra, and lensing potential.
• BAO: DESI DR2 data (isotropic and anisotropic distance constraints).
• Supernovae: Pantheon+ compilation.
• Local Prior: SH0ES $H_0$ measurement ($73.04 \pm 1.04$ km s$^{-1}$ Mpc$^{-1}$).

The study systematically varies parameter priors and includes/excludes the local $H_0$ prior to assess the robustness of the results. Finally, the favored parameter regions are tested against complementary constraints: Big Bang Nucleosynthesis (BBN), neutrino flux limits from various telescopes, and structure formation (free-streaming length).

Key Results

1. Hubble Constant Values:

   • When fitting Planck 2018 + DESI BAO + Pantheon + SH0ES with specific priors (Priors III), the model yields $H_0 = 69.05^{+0.31}_{-0.27}$ km s$^{-1}$ Mpc$^{-1}$ (68% CL).
   • Without the local SH0ES prior, the value drops to $H_0 = 68.76 \pm 0.35$ km s$^{-1}$ Mpc$^{-1}$.
   • These values reduce the tension to approximately $3.8\sigma$.

2. Impact of Priors and $\Delta\chi^2$:

   • The study reveals a strong dependence on parameter priors. While adjusting priors can yield slightly higher $H_0$ values, this comes at the cost of a significantly increased $\Delta\chi^2$ (worsening the statistical fit).
   • For the best-fitting scenario (Priors III), $\Delta\chi^2 = +16.0$ relative to $\Lambda$CDM, indicating a statistical preference for the standard model over the CDDM model given the datasets used.
   • The authors observe a trend where reducing the tension further (below $3\sigma$) requires priors that result in a prohibitively large $\Delta\chi^2$.

3. Complementary Constraints:

   • BBN: The parameter regions favored by cosmological data (specifically $\tau_M$ and $M/m$) are consistent with BBN limits on light element abundances (D/H and $^7$Li/H).
   • Neutrino Flux: The late-time decay into neutrinos implies a mass range for $\chi_m$ of $m \sim 1-10$ MeV, which remains compatible with constraints from Borexino, KamLAND, and Super-Kamiokande.
   • Structure Formation: The free-streaming length ($\lambda_{fs}$) induced by the early decay kick velocity is calculated to be $\sim 10^{-2}$ Mpc, well within the observational limit of $\lambda_{fs} < 0.1$ Mpc.

Significance and Claims
The paper claims to revise earlier literature results that suggested the Hubble tension could be reduced below the $3\sigma$ level within dark matter modification scenarios. The authors argue that previous studies may have suffered from inaccurate or incomplete MCMC analyses.

The primary conclusion is that while the CDDM model successfully incorporates both early-time (relativistic) and late-time (decay) effects to raise $H_0$, it cannot reduce the Hubble tension below the $\sim 3\sigma$ level without incurring a significant statistical penalty (large $\Delta\chi^2$). The authors assert that the tension remains a significant challenge, as the model does not offer a "free" solution that satisfies all datasets and statistical criteria simultaneously. The work serves as a rigorous test of cascade decaying dark matter, demonstrating its compatibility with current limits while highlighting the difficulty of resolving the tension solely through this mechanism.