Cherenkov plasmons emission by primordial neutrinos

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The Big Picture: Cooling Down a Cosmic Hot Spot

Imagine the early universe as a giant, boiling pot of soup. Inside this soup, there are tiny, ghostly particles called neutrinos. Usually, neutrinos are so shy and light that they just float around, barely interacting with anything.

However, this paper explores a fascinating "what if" scenario: What if these neutrinos clumped together to form a giant, dense ball (a cluster) held together by a mysterious, invisible glue (a hypothetical light scalar boson)?

When this ball forms, it gets squeezed tight. Just like when you pump up a bicycle tire and it gets hot, this neutrino ball gets incredibly hot. The problem? If it stays too hot, the particles will bounce around so violently that the ball will fly apart and disappear. To survive, the cluster needs a way to cool down quickly.

The author, Maxim Dvornikov, proposes a unique air-conditioning system for these cosmic clusters: Cherenkov Plasmon Emission.

The Analogy: The Sonic Boom of Light

To understand the cooling mechanism, let's break down the science into three parts:

1. The Ghost with a Temporary Charge

Neutrinos are electrically neutral (they have no charge), which usually means they can't create light or interact with electromagnetic fields. It's like a ghost walking through a room; it doesn't bump into the furniture.

The Twist: When a neutrino moves through a dense "soup" of other particles (the background plasma), it creates a tiny disturbance. Think of it like a ghost walking through a crowded dance floor; even though the ghost is invisible, the people on the dance floor react, creating a ripple. This ripple gives the neutrino a temporary, "induced" electric charge. Suddenly, the ghost can interact with the crowd.

2. The Speed Limit and the Sonic Boom

In a vacuum, nothing travels faster than light. But inside a dense medium (like water or this cosmic plasma), light slows down.

The Analogy: Imagine a jet plane flying faster than the speed of sound in the air. It creates a sonic boom (a shockwave) because it's outrunning the sound waves it creates.
The Physics: If our "ghost" neutrino moves through the plasma faster than the "light waves" (plasmons) can travel in that specific medium, it creates a Cherenkov shockwave. Instead of sound, it emits a burst of energy in the form of a "plasmon" (a collective vibration of the charged particles in the plasma).

3. The Cooling Effect

Every time the neutrino emits this shockwave, it loses a tiny bit of its own energy.

The Result: The neutrino slows down slightly, and the cluster as a whole loses heat. It's like a hot cup of coffee cooling down by releasing steam. If this happens fast enough, the cluster can shed its excess heat before the universe expands and cools everything else down, allowing the cluster to survive.

What Did the Author Actually Do?

The paper is a detailed mathematical proof that this cooling method works. Here is the step-by-step journey:

The Math of the Ghost: The author used complex quantum physics (Quantum Field Theory) to calculate exactly how much energy a neutrino loses when it creates these shockwaves. He had to account for the fact that the neutrinos have a temperature and a "chemical potential" (a measure of how crowded they are).
The Filter: He discovered that only one specific type of wave (called a longitudinal plasmon) contributes to this cooling. The other types of waves don't work in this scenario. It's like finding out that only a specific type of radio frequency can carry the signal.
The Test Drive: He applied his math to a specific model of a neutrino cluster formed in the early universe. He asked: "If this cluster forms at a certain temperature, will it cool down fast enough to survive?"
The Verdict:
- Yes, but with conditions. The cooling works efficiently if the cluster forms when the universe is still quite hot (around 220,000 degrees Kelvin).
- If the cluster forms later (when the universe is cooler), the cooling isn't fast enough, and the cluster might not survive.
- Interestingly, the "chemical potential" (how many neutrinos vs. anti-neutrinos are in the cluster) didn't change the result much. The cooling works regardless of that detail.

Why Does This Matter?

Dark Matter Mystery: We know the universe is full of "Dark Matter," but we don't know what it is. Neutrinos are a candidate, but they are usually too fast to clump together. This paper suggests that if they have a special interaction (the "glue" mentioned earlier), they could clump up and act as Dark Matter.
Survival of the Fittest: This study explains how these potential Dark Matter clusters could survive the violent, hot early universe without melting apart.
New Physics: It shows that even neutral particles like neutrinos can act like charged particles under the right conditions, creating light (or plasmons) in a way we hadn't fully calculated before.

The Bottom Line

The paper is essentially a recipe for a cosmic air conditioner. It proves that if neutrinos clump together in the early universe, they can use a "sonic boom" effect to radiate away their heat. This allows them to stay cool, stay together, and potentially become a major component of the invisible Dark Matter that holds our universe together.

1. Problem Statement

The paper addresses the stability and cooling mechanisms of neutrino clusters formed in the early universe.

Context: While standard massive neutrinos are too hot (relativistic) to be cold dark matter, they could form bound clusters (condensates) if an attractive interaction exists beyond the Standard Model. The proposed mediator is a hypothetical light scalar boson.
The Issue: When such a cluster forms, the neutrino gas is compressed, leading to a significant increase in temperature ( $T_{clust} > T_{background}$ ). Thermal motion threatens to destroy the cluster or prevent the formation of a superfluid state.
The Gap: Previous cooling mechanisms proposed for these clusters were found to be inefficient. While Cherenkov emission by neutrinos was previously studied, earlier works relied on ultrarelativistic plasma models or provided only estimatory calculations. There was a lack of rigorous treatment for nonrelativistic background plasmas, which is the relevant regime for neutrino decoupling ( $T \sim \text{MeV}$ ) and subsequent cluster evolution.

2. Methodology

The author employs Finite Temperature Quantum Field Theory (FTQFT) to derive the energy emission rate of neutrinos via Cherenkov plasmon emission.

Physical Process: The study focuses on the process $\nu \to \nu + \gamma^*$ , where a neutrino emits a longitudinal plasmon (a collective excitation of the background plasma). Although neutrinos are electrically neutral, they acquire an induced electric charge in a medium due to loop effects involving charged leptons.
Theoretical Framework:
- Matrix Element Calculation: The scattering amplitude is calculated using the Feynman diagram for neutrino-plasmon interaction. The calculation involves the generalized polarization tensor ( $\Pi_{\mu\nu}$ ) of the background plasma.
- Imaginary Time Formalism: The polarization tensor is computed using imaginary time perturbation theory, explicitly accounting for both the temperature ( $T$ ) and chemical potential ( $\mu$ ) of the background plasma.
- Plasmon Properties: The paper derives plasmon form factors ( $\Pi_L, \Pi_T$ ) and dispersion relations for a nonrelativistic background plasma (charged leptons are nonrelativistic).
- Emissivity Derivation: The energy emission rate ( $\dot{E}$ ) is obtained by averaging the squared matrix element over the Fermi-Dirac distribution functions of the incoming and outgoing neutrinos.
Cluster Modeling: The results are applied to a specific neutrino cluster model (based on numerical simulations from previous work) characterized by:
- Neutrino mass $m_\nu = 0.1$ eV.
- Present-day radius $R_{now} \approx 5 m_s^{-1}$ (where $m_s$ is the scalar boson mass).
- Adiabatic evolution assumptions to relate formation temperature to present-day parameters.

3. Key Contributions

Rigorous Nonrelativistic Treatment: Unlike previous studies that assumed ultrarelativistic plasmas, this work derives the emissivity specifically for a nonrelativistic background plasma ( $T \ll m_{lepton}$ ). This is crucial for accurately modeling the epoch of neutrino decoupling and cluster formation.
Inclusion of Chemical Potential: The derivation explicitly includes the neutrino chemical potential ( $\mu_\nu$ ) in the distribution functions and the polarization tensor, addressing a limitation of the Hard Thermal Loop (HTL) approximation which often neglects chemical potential contributions.
Longitudinal vs. Transverse Modes: The analysis proves that for nonrelativistic plasmas, only longitudinal plasmons contribute to Cherenkov emission. The condition $k > \omega$ (required for Cherenkov radiation) is satisfied for longitudinal modes but never for transverse modes in this regime.
Renormalization and Damping: The paper provides detailed calculations for the renormalization of the electric charge in the medium and estimates the plasmon damping (Landau damping) to ensure the emitted plasmons escape the cluster rather than being reabsorbed.

4. Key Results

Emissivity Formula: A general expression for the neutrino gas emissivity ( $\dot{E}$ ) is derived (Eq. 2.8 and 3.4), dependent on the cluster temperature $T_{clust}$ , chemical potential $\mu_\nu$ , and plasma frequency $\omega_p$ .
Cooling Efficiency:
- The cooling time $t_{cool}$ is compared to the Hubble time ( $H^{-1}$ ). The cooling is effective if the parameter $\Xi = N_\nu T_{clust} H / \dot{E} < 1$ .
- Temperature Threshold: For a cluster with specific parameters (radius $\sim 5 m_s^{-1}$ , scalar mass $m_s = 10^{-4}$ eV), the Cherenkov cooling mechanism is efficient if the cluster forms at temperatures $T \gtrsim 220$ keV.
- Chemical Potential Independence: The study finds that the neutrino chemical potential (asymmetry) has a negligible effect on the cooling rate; the curves for $\xi = \pm 3.9 \times 10^{-3}$ and $\xi = 0$ overlap almost perfectly.
Propagation Length: The plasmon propagation length $L$ is calculated to be much larger than the cluster radius ( $L \gg R$ ) for the relevant temperature range. This validates the assumption that plasmons carry energy away from the entire cluster volume rather than being reabsorbed locally.
Flavor Dependence: The mechanism is efficient for electron neutrinos ( $\nu_e$ ) due to their larger vector coupling constant ( $c_V$ ). It is inefficient for muon and tau neutrinos ( $\nu_\mu, \nu_\tau$ ) because their coupling constants are significantly smaller.

5. Significance

Viability of Neutrino Dark Matter: The paper demonstrates that Cherenkov plasmon emission is a viable and efficient cooling mechanism for neutrino clusters formed in the early universe. This supports the hypothesis that neutrinos could constitute a component of dark matter if they form condensates via scalar interactions.
Phenomenological Constraints: By establishing the temperature range ( $T \gtrsim 220$ keV) where cooling is effective, the paper provides constraints on the formation epoch of such clusters. This range is consistent with the neutrino decoupling temperature ( $\sim 2-3$ MeV), suggesting such clusters could survive thermal destruction.
Theoretical Utility: The appendices provide a comprehensive set of technical tools (polarization tensors, form factors, dispersion relations, and sum rules) for finite-temperature field theory in nonrelativistic media, which can be applied to other problems in astrophysics and cosmology involving neutrino-plasma interactions.

In conclusion, Dvornikov provides a robust theoretical foundation showing that primordial neutrino clusters can cool down rapidly enough to survive the expansion of the universe, provided they form in a specific temperature window and interact with a light scalar boson.