Freeze-in and ultra-relativistic freeze-out during general reheating scenarios

This paper presents a general analytic framework that unifies freeze-in, ultra-relativistic freeze-out, and standard freeze-out mechanisms for dark matter production during non-instantaneous reheating, deriving critical temperature exponents and analytic relic yields that explain how varying reheating histories shift microscopic interactions between these distinct regimes.

The Gist

Imagine the early universe as a giant, bustling kitchen just after the "Big Bang" explosion. In this kitchen, there are two main chefs: the Standard Model (the known particles like electrons and quarks) and Dark Matter (the invisible stuff that holds galaxies together).

Usually, scientists assume that as soon as the kitchen cools down enough, the Dark Matter chef stops cooking and just sits there, keeping a steady amount of ingredients. This is the standard story.

But this paper asks: What if the kitchen didn't cool down smoothly? What if the main heat source (the "inflaton" field) was still dumping energy into the kitchen for a long time, changing how fast things cooled and how the ingredients mixed?

The author, Kuldeep Deka, creates a new "recipe book" to figure out how much Dark Matter we would end up with if the kitchen's cooling process was messy and slow.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Two "Knobs" of the Universe

The author describes this messy cooling period using two simple dials (parameters):

• Dial 1 (The Expansion Speed): How fast the universe stretches out.
• Dial 2 (The Cooling Speed): How fast the temperature drops as the universe stretches.

By turning these two dials, you can simulate different types of cosmic history. Some histories are like a slow, steady oven cooling down; others are like a fire that flares up and then dies out quickly.

2. The Three Ways Dark Matter "Leaves the Party"

The paper explains that Dark Matter can stop interacting with the rest of the universe in three different ways, depending on how hot the kitchen is and how fast it's cooling:

• Freeze-In (The Sneaky Guest): Dark Matter is so weakly connected to the other particles that it never really joins the party. It just slowly sneaks in a few particles at a time from the hot soup. The amount it gets depends on how hot the soup gets at its hottest point.
• Freeze-Out (The Regular Guest): Dark Matter joins the party, mixes well, and then leaves when the party gets too cold.
  • Ordinary Freeze-Out: It leaves when the party is already over and the room is cool.
  • Ultra-Relativistic Freeze-Out (UFO): This is the paper's special focus. The Dark Matter leaves the party while it's still super hot and energetic, but before the main "Radiation Dominated" era begins. It's like leaving a concert while the band is still playing the loudest song, but the crowd is already thinning out.

3. The "Critical Switches"

The author discovers that the outcome depends on two "critical switches" (mathematical numbers) that act like traffic lights:

• Switch 1: Decides if the Dark Matter leaves the party during the messy cooling phase or waits until the party is over.
• Switch 2: Decides which part of the cooling history matters most. Does the final amount depend on the very beginning (the hottest part), the very end (the coolest part), or the whole journey in between?

4. The "Entropy Dilution" Effect

This is a crucial metaphor. Imagine you bake a cake (Dark Matter) in a small pan (the early universe). Then, before you serve it, someone pours a giant bucket of water into the pan. The cake is still there, but it's now diluted in a huge amount of water.

In the universe, if the "messy cooling" phase produces a lot of extra heat (entropy), it dilutes the Dark Matter that was already made. The paper calculates exactly how much the "cake" gets diluted and how much new Dark Matter gets made after the dilution happens.

5. The Main Discovery: It's All About the History

The most important finding is that the same type of Dark Matter interaction can produce completely different amounts of Dark Matter depending on the universe's history.

• Scenario A (Standard Cooling): If the universe cools normally, a specific interaction might produce just enough Dark Matter to match what we see.
• Scenario B (Messy Cooling): If the universe had a "kination" phase (where energy moves differently) or a "quartic" phase, that exact same interaction might produce way too much or way too little Dark Matter.

The paper draws "maps" (contour plots) showing that if you change the cooling history, the "safe zone" for Dark Matter moves around. An interaction that looks perfect in one universe might fail in another.

6. Why This Matters

The paper doesn't propose a new particle or a new way to detect Dark Matter in a lab. Instead, it provides a universal translator.

It tells physicists: "If you find a Dark Matter particle with specific properties, you can't just assume the universe cooled normally. You have to check if the universe went through a 'messy cooling' phase, because that changes the math entirely."

In summary:
The paper builds a general framework to calculate how much Dark Matter exists based on how the early universe cooled down. It shows that the "recipe" for the universe's ingredients is sensitive to the cooking method. If the universe cooled differently than we thought, our current theories about Dark Matter might need to be rewritten, even if the Dark Matter particles themselves haven't changed.

Technical Summary

Technical Summary: Freeze-in and Ultra-Relativistic Freeze-out During General Reheating Scenarios

Problem Statement
The thermal history of the Universe prior to Big Bang Nucleosynthesis (BBN) is weakly constrained. Standard relic density calculations typically assume an instantaneous transition to radiation domination (RD) following inflation, where the Hubble rate and plasma temperature follow standard RD scalings. However, if the energy stored in the inflaton (or another dominant component) is transferred to the Standard Model (SM) bath over an extended period, the expansion rate and temperature evolution can deviate significantly from RD behavior. This "low-temperature reheating" (LTR) era alters the production mechanisms of dark matter (DM). Specifically, it modifies the viable parameter space for non-relativistic thermal relics, affects freeze-in (FI) scenarios where production rates grow with temperature, and introduces the possibility of ultra-relativistic freeze-out (UFO) occurring during reheating rather than after. Existing literature has treated these regimes (matter-like reheating, UV freeze-in, UFO) separately, lacking a unified analytic framework to describe how a single microscopic interaction transitions between pure freeze-in, UFO during reheating, and ordinary freeze-out as the pre-BBN expansion history varies.

Methodology
The paper develops a general analytic framework to unify DM production mechanisms during non-instantaneous reheating. The approach relies on the following parametrizations:

1. Background Evolution: The reheating background is described by two effective parameters: the equation-of-state parameter $\omega$ of the dominant post-inflationary component and the cooling index $\alpha$, defined by the temperature scaling $T \propto a^{-\alpha}$. The Hubble rate during LTR is derived as $H(T) \propto T^{\gamma_2}$, where $\gamma_2 = 3(1+\omega)/2\alpha$.
2. Dark Matter Interactions: DM ($\chi$) is produced via relativistic $2 \leftrightarrow 2$ processes from the SM bath. The thermally averaged cross-section is parametrized as $\langle \sigma v \rangle \propto T^n / \Lambda^{n+2}$, where $\Lambda$ is an effective interaction scale and $n$ characterizes the temperature dependence (typically $n \geq 0$ for higher-dimensional contact interactions).
3. Boltzmann Analysis: The authors derive the Boltzmann equation in a comoving formulation, tracking the number density $N = n_\chi a^3$. They distinguish between thermalization (where the interaction rate $\Gamma_{eq} > H$) and decoupling.
4. Critical Indices: The dynamics are organized by two critical temperature exponents derived from the background and interaction parameters:
   • $n_c = \gamma_2 - 3$: Determines whether a thermalized relativistic species can decouple (freeze-out) during LTR or must wait until RD begins.
   • $n^* = \gamma_2 - 6 + 3/\alpha$: Determines the dominance of the freeze-in production integral (Infrared (IR) dominated, logarithmic, or Ultraviolet (UV) dominated).
5. Validation: Analytic results are cross-checked against numerical solutions of the full Boltzmann equation using the massive Maxwell-Boltzmann equilibrium density to capture the transition from ultra-relativistic to non-relativistic decoupling.

Key Contributions and Results
The paper provides a unified classification of DM production regimes and derives compact analytic expressions for the relic yield ($Y_0$) and the required interaction scale ($\Lambda_{relic}$) to match the observed abundance ($\Omega_{DM}h^2 \approx 0.12$).

• Regime Classification: The interaction power $n$ relative to $n_c$ and $n^*$ defines four distinct regimes (Table 1):
  • Regime I ($n \leq n_c$): No LTR freeze-out; decoupling occurs in RD.
  • Regime II ($n_c < n < n^*$): LTR UFO is possible; post-freeze-out production is IR dominated (controlled by $T_{RH}$).
  • Regime III ($n = n^*$): LTR UFO possible; production is logarithmic.
  • Regime IV ($n > n^*$): LTR UFO possible; production is UV dominated (controlled by $T_{max}$ or $T_{FO}$).
• Analytic Yields: The authors derive explicit formulas for the relic yield in these regimes. A key finding is that the final abundance in the LTR UFO branch consists of two parts: an entropy-diluted contribution inherited from the freeze-out temperature ($T_{FO}$) and a post-freeze-out production term.
  • For IR-dominated cases ($n < n^*$), the yield is controlled by the reheating temperature $T_{RH}$.
  • For UV-dominated cases ($n > n^*$), the yield is controlled by the maximum temperature $T_I$ (for pure FI) or the freeze-out temperature $T_{FO}$ (for UFO).
• Scaling Laws: The paper elucidates how the relic density contours ($\Lambda$ vs. $m_\chi$ or $T_{RH}$) change slope depending on the regime.
  • In matter-like reheating ($\omega=0, \alpha=3/8$), the $n=4$ branch is IR-dominated, causing deep UFO and pure FI to connect smoothly with the same scaling.
  • In UV-dominated reheating (e.g., quartic potentials or kination), the UFO and FI branches are controlled by different scales ($T_{FO}$ vs. $T_I$), leading to visibly different contour slopes.
  • In kination ($\omega=1, \alpha=1$), the entropy dilution factor vanishes ($d=0$), causing the UFO branch to resemble an undiluted relativistic thermal relic rather than generating a new mass-scaling.
• Threshold Effects: The analysis accounts for the transition where the dark matter mass $m_\chi$ exceeds the relevant production temperature ($T_{RH}$ or $T_{FO}$). This "threshold-controlled" production modifies the scaling of IR-dominated contours when $m_\chi \gtrsim T_{RH}$, introducing a dependence on $m_\chi$ that differs from the relativistic limit.

Significance and Claims
The paper claims to provide the first unified analytic classification that maps a single microscopic interaction across the full spectrum of production mechanisms (freeze-in, ultra-relativistic freeze-out during reheating, post-reheating freeze-out, and non-relativistic freeze-out) as a function of the reheating history.

The significance lies in demonstrating that the relic density target associated with a given interaction scale is cosmology-dependent. Specifically:

1. Reheating History Sensitivity: The same microscopic interaction (fixed $n, \Lambda$) can yield vastly different relic abundances depending on the reheating parameters $(\omega, \alpha)$.
2. Contour Interpretation: The shape of relic-density contours in parameter planes (e.g., $m_\chi$ vs. $\Lambda$) is not universal but is dictated by whether the production is IR or UV dominated and whether the reheating epoch involves entropy production ($d > 0$) or not ($d=0$).
3. Model Embedding: The derived phase diagrams and scaling laws provide the necessary "cosmology-side map" for embedding UFO production into explicit particle physics models. By combining these results with UV completions (which specify mediator masses and couplings), future studies can simultaneously test dark matter interactions and the pre-BBN expansion history.

The authors remain modest regarding model-dependent constraints (such as direct detection or collider limits), noting that their EFT-based framework isolates the early-Universe production mechanism, while specific UV completions would determine the applicability of other experimental bounds. The results are validated against numerical Boltzmann solutions, confirming the accuracy of the analytic scalings in the relevant regimes.