Exploring non-equilibrium effects in sequential freeze-in

This paper investigates non-equilibrium effects in a two-scalar sequential freeze-in dark matter model, demonstrating that full phase-space treatments can yield relic abundance predictions deviating by up to an order of magnitude from traditional number-density approaches, thereby underscoring the necessity of advanced numerical tools for accurate multi-component dark sector modeling.

The Gist

The Big Picture: The Mystery of the "Ghost" Universe

Imagine the universe is a giant, bustling city. We know about the "citizens" (normal matter like stars and planets), but we also know there is a massive, invisible population living in the shadows called Dark Matter. We know they exist because of their gravity, but we have no idea what they look like or how they were born.

For decades, scientists have had a favorite story about how Dark Matter was made. They call it "Freeze-in."

The Old Story (The "Thermal Bath" Assumption):
Imagine the early universe as a giant, boiling hot bath of energy (the "thermal plasma"). The scientists assumed that Dark Matter particles were like little swimmers who jumped into this hot bath. They quickly got used to the water temperature, swam around, and then, as the universe cooled down, they "froze" in place, becoming the Dark Matter we see today.

The old math assumed these swimmers were always perfectly happy with the water temperature. If the water was hot, they were hot; if it cooled, they cooled instantly. This made the math easy, but the authors of this paper asked: "What if the swimmers are actually too shy to get comfortable with the water? What if they stay cold while the water is hot, or hot while the water is cold?"

The New Story: The "Secret Society" (Sequential Freeze-in)

The authors propose a more complex scenario involving a two-step process (Sequential Freeze-in).

1. The Messenger: First, the hot bath creates a "Messenger" particle (let's call him Phi).
2. The Dark Matter: Then, Phi meets another particle and they combine to create the actual Dark Matter particle (let's call her S).

The problem is that Phi is a bit of a loner. He doesn't hang out with the hot bath enough to get warm. He stays "out of equilibrium." Because he is cold (or has a weird energy distribution), he doesn't pass the right amount of energy to create Dark Matter.

The Analogy:
Think of the hot bath as a giant factory churning out raw materials (energy).

• The Old View: The factory sends materials to a warehouse (Dark Matter) at a steady, predictable rate.
• The New View: The factory sends materials to a middleman (Phi), who then passes them to the warehouse. But the middleman is distracted. He only passes the materials when he has a specific, high-energy burst of motivation. If the factory is just "warm," the middleman is too lazy to work. He only works when the factory is really hot, or when he gets a specific type of energy packet.

The "Traffic Jam" of Energy

The paper uses a concept called Phase-Space (which sounds scary, but is simple). Imagine a highway.

• The Old Math (nBE): Assumes all cars (particles) are driving at the exact same speed limit (temperature). It's a smooth, orderly traffic flow.
• The Real Math (fBE): The authors realized that in this "Secret Society" scenario, the cars aren't driving at the same speed. Some are speeding (high energy), and most are crawling (low energy).

Because the Dark Matter creation process is like a bouncer at a club, it only lets in the "speeding cars" (high-energy particles).

• If you assume everyone is driving at the average speed (the old math), you think the bouncer lets in a lot of people.
• But in reality, because the cars are actually moving slowly (non-equilibrium), the bouncer lets in very few people.

The Result: The old math predicted there would be 10 times more Dark Matter than there actually is in some scenarios. The new math shows that because the particles are "out of sync" with the universe's temperature, the production of Dark Matter is much less efficient than we thought.

Why Should You Care? (The Detective Work)

The authors didn't just do math; they looked for clues to find these particles.

1. The Long-Lived Messenger: The "Messenger" particle (Phi) is unstable. It eventually decays into normal particles (like light or electrons). Because it's shy, it lives a long time before dying.

   • Analogy: Imagine a spy who stays in the city for a long time before slipping away.
   • Detection: New experiments like FASER or MATHUSLA (huge detectors built to catch long-lived spies) might catch this messenger.

2. The Ghostly Signal: When two Dark Matter particles (S) meet, they might turn back into Messengers, which then turn into light. This could create a faint glow in the center of our galaxy.

   • Analogy: Two ghosts bumping into each other and flashing a light.
   • Detection: Telescopes like CTA (Cherenkov Telescope Array) are looking for this specific flash.

The Takeaway

This paper is a warning to scientists: "Don't assume everything is in equilibrium."

If you assume the Dark Matter particles are just "happy swimmers" in the hot bath, you might get the wrong answer about how much Dark Matter exists, and you might look for it in the wrong places.

By using a more sophisticated "phase-space" approach (tracking the exact speed and energy of every particle, not just the average), the authors found that the amount of Dark Matter can be drastically different—sometimes by a factor of 10!

In short: The universe is more chaotic and less orderly than we thought. The "swimmers" are actually cold, distracted, and moving at weird speeds, and we need to update our maps to find them.

Technical Summary

1. Problem Statement

The paper addresses a critical gap in the calculation of Dark Matter (DM) relic abundance in freeze-in scenarios, specifically within multi-component dark sectors.

• The Assumption: Standard calculations often assume that even in freeze-in (where DM never reaches chemical equilibrium with the Standard Model plasma), the DM momentum distribution $f(p)$ retains a thermal shape (Maxwell-Boltzmann) characterized by the temperature of the SM bath ($T_{SM}$).
• The Issue: In sequential freeze-in scenarios, where DM ($S$) is produced via an intermediate mediator ($\phi$) rather than directly from the SM, the dynamics of the mediator dictate the production of DM. If the mediator $\phi$ does not maintain kinetic equilibrium with the SM bath, its momentum distribution becomes distorted (non-thermal).
• The Consequence: These distortions, particularly in the high-momentum tail of the distribution, can significantly alter the rate of conversion processes ($\phi\phi \to SS$). Consequently, standard "number density" Boltzmann equation (nBE) approaches may yield inaccurate predictions for the final relic abundance, potentially deviating by orders of magnitude from the true value.

2. Methodology

The authors perform a comprehensive study using a minimal two-scalar model coupled to the SM via a Higgs portal.

• The Model:
  • Two real scalars: A stable DM candidate $S$ and an unstable mediator $\phi$.
  • $\phi$ mixes with the Higgs boson ($h$) with a small mixing angle $\theta$, allowing it to decay into SM particles.
  • Interactions include Higgs decays ($h \to \phi\phi, SS$) and conversions ($\phi\phi \leftrightarrow SS$).
• Phenomenological Scan:
  • A six-dimensional parameter space scan ($m_\phi, m_S, \lambda_{h\phi}, \lambda_{hS}, \lambda_{S\phi}, \sin\theta$) was performed to identify points satisfying the observed relic density ($\Omega h^2 \approx 0.12$).
  • Constraints from Indirect Detection (CMB, Fermi-LAT, CTA) and Long-Lived Particle searches (CHARM, LHCb, FASER, MATHUSLA) were applied.
• Computational Framework (Three Levels of Approximation):
  To quantify non-equilibrium effects, the authors compare three distinct treatments of the Boltzmann equation:
  1. nBE (Number Density Boltzmann Equation): The standard approach. Assumes all particles share the SM temperature ($T_{SM}$) and have equilibrium distributions. Tracks only yields ($Y$).
  2. cBE (Coupled Boltzmann Equations): Tracks both yields ($Y$) and effective temperatures ($T_i$) for each species. Assumes distributions remain thermal but at a distinct temperature ($T_i \neq T_{SM}$).
  3. fBE (Full Boltzmann Equation): The most rigorous approach. Solves integro-differential equations for the full phase-space distribution functions $f(p, t)$ without assuming a thermal shape. This captures genuine non-thermal distortions (e.g., high-momentum tails).
  • Implementation: The fBE and cBE calculations were performed using a modified version of the DRAKE code.

3. Key Contributions

• Quantification of Non-Thermal Effects: The paper demonstrates that the assumption of thermal distributions in sequential freeze-in is often invalid. The deviation between the full phase-space treatment (fBE) and the standard number-density approach (nBE) can reach an order of magnitude in the predicted relic abundance.
• Mechanism Identification: The authors identify that the discrepancy arises primarily because:
  1. The mediator $\phi$ often has a temperature $T_\phi < T_{SM}$ (captured by cBE).
  2. The distribution of $\phi$ develops a non-thermal high-momentum tail (captured only by fBE). Since the conversion $\phi\phi \to SS$ is often kinematically suppressed (requiring high energy), this tail is the dominant driver of DM production. Standard nBE overestimates the population in this tail by assuming a thermal equilibrium at $T_{SM}$.
• Benchmark Analysis: Six benchmark points (BM1–BM6) were selected to illustrate different regimes:
  • Direct Freeze-in: nBE, cBE, and fBE agree (BM1).
  • Sequential Freeze-in: Significant deviations observed; fBE yields lower abundance than nBE due to the depletion of the high-momentum tail (BM2, BM3).
  • Dark Freeze-out: Strong self-interactions/conversions lead to internal thermalization; cBE becomes accurate, but nBE still fails (BM5, BM6).
• Experimental Relevance: The model predicts observable signatures in forward physics experiments (due to long-lived $\phi$) and indirect detection (via $SS \to \phi\phi \to 4\gamma$), highlighting the complementarity of these search strategies.

4. Results

• Abundance Deviations:
  • In sequential freeze-in regimes (e.g., BM2, BM3), the nBE approach overestimates the DM abundance compared to fBE.
  • For BM2, the nBE prediction is roughly 3 times higher than the fBE result.
  • The deviation is driven by the fact that nBE assumes a thermal bath at $T_{SM}$, which artificially replenishes the high-momentum tail of $\phi$ needed for the kinematically suppressed conversion $\phi\phi \to SS$. The fBE correctly shows this tail is depleted.
• Temperature Evolution:
  • The mediator $\phi$ often cools down relative to the SM bath ($T_\phi < T_{SM}$) after the Electroweak Phase Transition (EWPT) due to Higgs decays producing particles with specific kinematics.
  • In regimes with strong conversion rates (BM5, BM6), the dark sector particles thermalize with each other ($T_\phi \approx T_S$) but remain decoupled from the SM bath.
• Parameter Dependence:
  • The magnitude of the non-thermal effect depends heavily on the conversion coupling $\lambda_{S\phi}$ and the mass hierarchy $m_S/m_\phi$.
  • Larger mass ratios ($m_S \gg m_\phi$) increase kinematic suppression, making the process more sensitive to the high-momentum tail and thus increasing the discrepancy between nBE and fBE.

5. Significance

• Theoretical Impact: This work challenges the standard practice of using thermal distribution functions for freeze-in calculations in multi-component models. It establishes that phase-space level computation is essential for accurate predictions in sequential freeze-in scenarios.
• Phenomenological Impact:
  • Relic Density: Incorrect assumptions can lead to wrong conclusions about which parameter space is viable for a given DM model.
  • Indirect Detection: Since the relic abundance determines the expected signal flux, errors in abundance calculation directly impact the interpretation of indirect detection limits (e.g., from CTA or CMB).
  • Experimental Strategy: The paper provides a concrete framework for interpreting signals in forward physics experiments (like FASER, MATHUSLA) where long-lived mediators are produced.
• Tool Development: The study motivates and lays the groundwork for the inclusion of two-component freeze-in dynamics in public numerical tools (specifically the upcoming release of DRAKE), enabling the broader community to perform accurate non-equilibrium calculations.

In summary, the paper proves that for sequential freeze-in, the "thermal approximation" is insufficient. Accurate modeling requires tracking the full momentum distribution to correctly predict the relic abundance and associated experimental signatures.