Constraining self-interacting ultrahigh-energy muon neutrinos by cosmic microwave background spectral distortion

This paper establishes stringent upper bounds on flavor-specific self-interactions of ultrahigh-energy muon neutrinos mediated by sub-GeV scalar bosons by analyzing how their energy injection into the early universe plasma induces $\mu$-type and $y$-type cosmic microwave background spectral distortions, demonstrating that future experiments like PIXIE could decisively probe neutrino physics beyond the Standard Model.

The Gist

The Big Picture: A Cosmic "Echo" in the Microwave Oven

Imagine the entire universe is filled with a faint, static hum of light left over from the Big Bang. Scientists call this the Cosmic Microwave Background (CMB). Think of it as the "afterglow" of the universe's birth, a perfect, smooth blanket of heat that has been cooling down for billions of years.

This paper asks a simple question: What if something bumped into that blanket and left a wrinkle?

The authors investigate a specific scenario where "ghostly" particles called neutrinos (specifically, ultra-high-energy muon neutrinos) might be colliding with each other in a way the Standard Model of physics doesn't fully explain. If these collisions happen, they would dump extra energy into the cosmic "soup," creating a tiny, detectable scar on that perfect blanket of light.

The Cast of Characters

1. The Ultra-High-Energy Neutrinos: Imagine these as cosmic bullets fired from a giant cannon. They are incredibly fast and energetic, possibly coming from the decay of super-heavy "dark matter" (the invisible stuff that holds galaxies together).
2. The Cosmic Neutrino Background (CνB): This is a sea of slow-moving, low-energy neutrinos that has been floating around since the universe was a baby. It's like a thick fog of invisible particles.
3. The Scalar Boson (The Messenger): This is a new, hypothetical particle that acts like a messenger. It allows the fast neutrinos to talk to the slow ones. The paper imagines this messenger is a "scalar" particle (a type of force carrier) with a specific mass.
4. The CMB (The Blanket): The background light we are trying to measure.

The Story: How the "Wrinkle" Happens

Here is the chain of events the paper describes, step-by-step:

1. The Collision
The fast, high-energy neutrinos (the bullets) fly through the universe and crash into the slow, background neutrinos (the fog). They don't just bounce off; they interact through our new "messenger" particle.

2. The Spark
When they collide, something interesting happens. Through a complex quantum process (involving a loop of particles like muons), this collision creates a burst of high-energy photons (light).

• Analogy: Imagine two cars crashing. Usually, they just crumple. But in this scenario, the crash is so energetic that it sparks a fire, shooting out bright flashes of light.

3. Heating the Soup
These new flashes of light get absorbed by the plasma (the hot, electric gas) of the early universe. This heats up the gas, injecting extra energy into the system.

4. The Scar on the Blanket
This extra energy messes up the perfect temperature of the CMB. Depending on when this happens in the history of the universe, it leaves a different kind of scar:

• The "µ" (Mu) Scar: If the crash happens when the universe is very young and hot (between 50,000 and 2 million years after the Big Bang), the energy gets trapped and creates a specific type of distortion called a µ-type.
• The "y" (Y) Scar: If the crash happens later, when the universe has cooled down a bit (less than 50,000 years after the Big Bang), the energy creates a y-type distortion.

The Detective Work: Measuring the Wrinkles

The authors used two "magnifying glasses" to look for these wrinkles:

1. COBE/FIRAS (The Old Camera): This is a past satellite mission that already took a picture of the CMB. It told us the blanket is very smooth, but it wasn't sensitive enough to see tiny wrinkles. It set a "safety limit" on how big a scar could be.
2. PIXIE (The Future Super-Camera): This is a proposed future mission. It is like upgrading from a standard camera to a high-definition microscope. It is designed to see wrinkles 1,000 times smaller than the old camera could.

What They Found (The Results)

The paper doesn't find the wrinkles yet (because we haven't built the super-camera), but it calculates how strong the neutrino interaction can be before it would create a scar big enough to be seen by these cameras.

• The "Speed Limit" for Neutrinos: The authors calculated the maximum strength (coupling) that the interaction between these neutrinos can have. If the interaction is too strong, the "crash" would create a scar so big that the COBE camera would have already seen it. Since we didn't see it, the interaction must be weaker than a certain limit.
• The "Sweet Spot" (The Kink): They found a weird behavior. If the "messenger" particle (the scalar boson) is very light, the limit is strict. But as the messenger gets heavier, the limit changes. There is a specific point (a "kink") where the messenger's mass matches the energy of the collision. At this exact moment, the interaction is strongest, and the limit on how strong the force can be changes dramatically.
• The Muon Connection: The paper focuses specifically on muon neutrinos. Why? Because there are other mysteries in physics involving muons (like the "muon g-2" anomaly) that suggest muons might be interacting with new physics. This study checks if the same new physics could explain both the muon mystery and the neutrino behavior.

The Bottom Line

The paper concludes that CMB spectral distortion is a powerful new way to test neutrino physics.

• If we build the PIXIE telescope, we will be able to rule out (or find) much weaker interactions than we can today.
• The authors provide a "map" showing exactly how strong the neutrino self-interaction can be for different masses of the messenger particle.
• Essentially, they are saying: "If neutrinos are talking to each other this strongly, we would have seen a scar on the cosmic background light. Since we haven't (yet), they must be talking a bit more quietly than that."

This work doesn't claim to cure diseases or build new technology; it is purely about understanding the fundamental rules of the universe by looking for tiny ripples in the oldest light we can see.

Technical Summary

Technical Summary: Constraining Self-Interacting Ultrahigh-Energy Muon Neutrinos by Cosmic Microwave Background Spectral Distortion

Problem Statement
The existence of ultrahigh-energy (UHE) neutrinos, detected by observatories such as IceCube and KM3NeT, suggests potential physics beyond the Standard Model (BSM), particularly regarding neutrino self-interactions (νSI). While strong νSI mediated by light bosons has been proposed to explain anomalies like the Hubble tension, the muon $g-2$ anomaly, and spectral features in IceCube data, these interactions must be constrained to avoid conflict with cosmological observations. Specifically, if UHE neutrinos (potentially originating from the decay of superheavy dark matter) undergo radiative scattering with the cosmic neutrino background (CνB), they inject energy into the primordial plasma. This energy injection can distort the Cosmic Microwave Background (CMB) spectrum, providing a sensitive probe to constrain the coupling strength and mass of the mediating scalar boson.

Methodology
The authors investigate a theoretical framework where a scalar boson ($\phi$) mediates flavor-specific self-interactions between muon neutrinos ($\nu_\mu$) and the CνB. The study focuses on the following physical processes:

1. Source Mechanism: UHE neutrinos are assumed to originate from the decay of superheavy dark matter ($m_\chi \geq \text{PeV}$), with a fraction $f_\chi$ of dark matter converting into UHE neutrinos.
2. Interaction Model: The interaction is described by a Lagrangian involving Yukawa-type couplings ($g_{\nu\mu}$ for neutrinos and $g_\mu$ for muons) to a scalar mediator. The scattering process $\nu_\mu + \nu_{\text{C}\nu\text{B}} \to \nu_\mu + \nu_{\text{C}\nu\text{B}} + \gamma$ occurs via a one-loop diagram involving a muon, leading to radiative scattering and photon production.
3. Energy Injection: The cross-section for this radiative scattering is calculated, accounting for the center-of-mass energy ($s \approx 2E_\nu E_{\nu\mu}$) and the mediator mass ($m_\phi$). The resulting energy injection rate into the plasma is derived as a function of redshift ($z$).
4. CMB Distortion Calculation: The energy injection is modeled to determine its impact on CMB spectral distortions. The authors distinguish between two regimes based on the redshift of energy injection:
   • $\mu$-type distortion: Occurs for $5 \times 10^4 \lesssim z \lesssim 2 \times 10^6$, where double Compton scattering is inefficient, leading to a Bose-Einstein distribution with a non-zero chemical potential.
   • $y$-type distortion: Occurs for $z \lesssim 5 \times 10^4$, where only elastic Compton scattering is efficient, leading to a Compton $y$-parameter distortion.
5. Constraints: Theoretical predictions for $\mu$ and $y$ distortions are compared against observational limits from the COBE/FIRAS experiment and projected sensitivities of the future Primordial Inflation Explorer (PIXIE).

Key Contributions

• Radiative Scattering Mechanism: The paper explicitly calculates the energy injection rate resulting from the radiative scattering of UHE muon neutrinos off the CνB mediated by a scalar boson, a process that generates high-energy photons capable of distorting the CMB.
• Flavor-Specific Constraints: Unlike previous studies focusing on flavor-universal interactions, this work derives stringent upper bounds specifically for muon-flavored neutrino self-interactions, motivated by anomalies in muon-related observables.
• Mediator Mass Dependence: The analysis characterizes the behavior of the coupling bounds as a function of the mediator mass ($m_\phi$), identifying a resonance region where the center-of-mass energy matches the mediator mass ($\sqrt{s} \approx m_\phi$).
• Comparison with Existing Bounds: The derived constraints are systematically compared with limits from Big Bang Nucleosynthesis (BBN), CMB anisotropies, rare kaon decays ($K \to \mu\nu\phi$), and solutions to the Hubble tension and muon $g-2$ anomalies.

Results

• Coupling Bounds: Using PIXIE projected sensitivities ($\mu \sim 5 \times 10^{-8}$, $y \sim 10^{-8}$), the authors derive upper bounds on the self-interaction coupling $g_{\nu\mu}$.
  • For $E_\nu = 1 \text{ PeV}$ and $f_\chi = 1$, the bounds are $g_{\nu\mu} \lesssim 1.0 \times 10^{-3}$ ($\mu$-type) and $g_{\nu\mu} \lesssim 3.8 \times 10^{-4}$ ($y$-type).
  • These represent an improvement of approximately one order of magnitude over current COBE/FIRAS constraints.
  • In scenarios where the coupling to muons is fixed to explain the $g-2$ anomaly ($g_\mu = 5 \times 10^{-4}$), the PIXIE bounds on $g_{\nu\mu}$ improve by two orders of magnitude compared to FIRAS limits.
• Mass Dependence: The bounds on $g_{\nu\mu}$ remain constant for mediator masses $m_\phi \ll \sqrt{s}$. As $m_\phi$ approaches $\sqrt{s}$, a "kink" appears in the constraints due to the resonant enhancement of the cross-section. For $m_\phi \gg \sqrt{s}$, the upper bound on the coupling relaxes and becomes proportional to $m_\phi$.
• Energy Scaling: Increasing the UHE neutrino energy (e.g., to 100 PeV) shifts the resonance kink to higher mediator masses and generally relaxes the coupling bounds.
• Dark Matter Fraction: Reducing the fraction of dark matter decaying into UHE neutrinos ($f_\chi$) from 1 to 0.1 relaxes the bounds by a factor of approximately 1.8.

Significance and Claims
The paper claims that CMB spectral distortions serve as a decisive and complementary probe for exploring neutrino physics beyond the Standard Model. The authors assert that future missions like PIXIE will provide valuable insights, potentially improving sensitivity by three orders of magnitude over current limits. This enhanced sensitivity could decisively test scenarios involving flavor-specific neutrino self-interactions mediated by sub-GeV scalar bosons, particularly those motivated by the muon $g-2$ anomaly and the Hubble tension. The study concludes that these cosmological constraints are robust and can effectively rule out or constrain specific regions of the parameter space for self-interacting neutrinos that are not excluded by laboratory searches or other cosmological observations.