Unitarity Bound on Dark Matter in Low-temperature Reheating Scenarios

This paper derives model-independent unitarity bounds on thermal dark matter mass in low-temperature reheating scenarios, demonstrating that while kination-like evolution tightens the limit to a few TeV, an early matter-dominated era allows dark matter to reach masses up to $\sim 10^{10}$ GeV due to significant entropy dilution.

The Gist

Here is an explanation of the paper "Unitarity Bound on Dark Matter in Low-temperature Reheating Scenarios," translated into everyday language with some creative analogies.

The Big Picture: The "Speed Limit" of the Universe

Imagine the universe is a giant, bustling highway. On this highway, there are invisible cars called Dark Matter. We know they exist because they have gravity (they hold galaxies together), but we can't see them, and we don't know how heavy they are.

For decades, physicists have had a "speed limit" sign for these dark matter cars. This sign is based on a fundamental rule of quantum mechanics called Unitarity.

Think of Unitarity like a law of conservation for probability. It basically says: "You cannot have a collision so violent that the odds of it happening exceed 100%." If a particle is too heavy, the math says it would have to collide with other particles so frequently to maintain its population that it would break the laws of physics.

The Old Rule: In the standard story of the universe, this speed limit meant Dark Matter couldn't be heavier than about 130 TeV (a trillion electron-volts). If it were heavier, it would have to be "colliding" too hard, which is impossible.

The Twist: The Universe's "Reheating" Party

The authors of this paper ask a simple question: What if the standard story of the universe isn't the whole story?

Usually, we think the universe cooled down smoothly after the Big Bang, like a hot cup of coffee sitting on a table. But what if, instead, the universe went through a weird phase where it got heated up again, or expanded in a weird way, before it cooled down for good?

The paper looks at two specific "weird phases" (scenarios) where the universe was reheated by a decaying particle, but the temperature wasn't very high (Low-temperature Reheating).

Scenario 1: The "Kination" Rush (The Fast Lane)

Imagine the universe is a race car. In the standard story, the car coasts at a steady speed. In this Kination scenario, the car slams the accelerator and zooms forward incredibly fast.

• What happens: Because the universe expands so fast, the Dark Matter particles get separated from each other too quickly. They stop interacting (they "freeze out") much earlier than usual.
• The Consequence: To get the right amount of Dark Matter left over today, these particles would have needed to be super efficient at colliding with each other before they got separated.
• The Result: The "speed limit" gets stricter. Because they had to be so efficient, they can't be very heavy.
  • New Limit: Instead of 130 TeV, the limit drops to just a few TeV. It's like saying, "If you have to run a marathon in 10 minutes, you can't be a heavy, slow person; you have to be light and fast."

Scenario 2: The "Early Matter" Dilution (The Watering Down)

Now, imagine a different scenario. The universe is like a giant soup. In the standard story, the soup is just hot water. In this Early Matter Domination scenario, someone dumps a massive amount of extra ingredients (a heavy particle) into the soup, which then decays into a huge amount of new broth.

• What happens: The Dark Matter particles freeze out (stop interacting) early, just like in the first scenario. But then, boom, a massive amount of new energy (entropy) is injected into the universe.
• The Consequence: This new energy acts like pouring a gallon of water into a cup of coffee. It dilutes everything. The Dark Matter particles that were already frozen out get spread out so thin that their density drops drastically.
• The Result: To fix this, the Dark Matter particles must have been massive to begin with. They needed to be so heavy that even after being diluted by the "gallon of water," there was still enough of them left to make up the Dark Matter we see today.
  • New Limit: The speed limit is lifted! Dark Matter could be incredibly heavy, up to 10,000,000,000 GeV (10 billion GeV). It's like saying, "If you are going to get diluted, you better start out as a giant."

The Two Types of "Freeze-Out"

The paper also looks at how these particles stop interacting. They compare two methods:

1. Visible Freeze-out (The Standard WIMP): Dark Matter particles crash into normal matter (like protons) and bounce off. This is the classic "Weakly Interacting Massive Particle" idea.
2. Dark Freeze-out (The Cannibal): Dark Matter particles only crash into each other. They eat their own kind (3 particles turn into 2). This is common in theories where Dark Matter has its own secret society.

The authors found that the "Fast Lane" (Kination) makes the mass limit stricter for both types, while the "Dilution" (Early Matter) makes the mass limit much more relaxed, especially for the "Cannibal" type.

The Takeaway

This paper is a reminder that context matters.

• If the universe expanded normally, Dark Matter has a strict weight limit (it can't be too heavy).
• If the universe had a "Fast Lane" phase, Dark Matter must be even lighter.
• If the universe had a "Dilution" phase, Dark Matter can be as heavy as we can imagine (up to 10 billion GeV).

In simple terms: The rules of the game change depending on the history of the playground. If the playground is chaotic and expanding fast, the players have to be light. If the playground gets flooded with new players, the original players can be giants.

This doesn't prove Dark Matter is heavy or light, but it tells us: If you find Dark Matter that is 100 TeV heavy, you know the universe didn't go through a "Fast Lane" phase. If you find it's 10 billion GeV heavy, you know the universe definitely had a "Dilution" phase. It helps us narrow down the history of our universe by weighing the invisible stuff.

Technical Summary

Here is a detailed technical summary of the paper "Unitarity Bound on Dark Matter in Low-temperature Reheating Scenarios" by Bernal, Konar, and Show.

1. Problem Statement

The paper addresses the theoretical upper limit on the mass of thermal Dark Matter (DM) particles. In the standard cosmological scenario (radiation-dominated universe), the unitarity of the S-matrix imposes a strict upper bound on the DM mass. For standard Weakly Interacting Massive Particles (WIMPs) undergoing $2 \to 2$annihilation, this limit is approximately **130 TeV**. For Strongly Interacting Massive Particles (SIMPs) undergoing$3 \to 2$ processes, the limit is around 1 GeV.

However, these bounds rely on the assumption that the early universe followed the standard radiation-dominated expansion history between the end of inflation and Big Bang Nucleosynthesis (BBN). The authors investigate how these bounds change in non-standard cosmological scenarios characterized by low-temperature reheating. Specifically, they examine two distinct phases where the energy density evolution deviates from the standard radiation scaling ($\rho \propto a^{-4}$):

1. Kination-like scenarios: A component dominates the energy density with a redshift behavior faster than radiation ($\rho \propto a^{-(4+n)}$ with $n>0$).
2. Early Matter Domination (EMD): A component scales like non-relativistic matter ($\rho \propto a^{-3}$) and decays into Standard Model (SM) particles, leading to entropy injection.

The core question is: How do these modified expansion histories and entropy production mechanisms alter the required thermally-averaged cross-sections to match the observed relic density, and consequently, how do they shift the unitarity mass bounds?

2. Methodology

The authors employ a rigorous theoretical framework combining Quantum Field Theory (S-matrix unitarity) with Cosmological Boltzmann equations.

• S-Matrix Unitarity Derivation:

  • They derive the maximum inelastic cross-section for general number-changing processes ($r \to 2$, where $r \ge 2$) using the optical theorem and partial-wave unitarity.
  • They establish the maximum thermally-averaged cross-section $\langle \sigma_{r \to 2} v^{r-1} \rangle_{\text{max}}$ for both $2 \to 2$(WIMP) and$3 \to 2$ (SIMP) scenarios.
  • The derivation accounts for identical vs. non-identical initial states and includes the exponential phase-space suppression factor $e^{-(r-2)x}$ for $r > 2$.

• Cosmological Evolution Models:

  • Kination-like: The universe is dominated by a fluid $\phi$ with $\rho_\phi \propto a^{-(4+n)}$. The Hubble rate scales as $H \propto T^{2+n/2}$ during this era. The SM entropy is conserved.
  • Early Matter Domination: The universe is dominated by a massive particle $\phi$ ($\rho_\phi \propto a^{-3}$) which decays into SM particles. This leads to a non-adiabatic epoch where SM entropy is not conserved, and the temperature scales differently ($T \propto a^{-3/8}$ during decay).

• Freeze-out Dynamics:

  • They solve the Boltzmann equations for the DM number density (or yield $Y$) in these modified backgrounds.
  • They distinguish between Visible Freeze-out (DM annihilates to SM particles) and Dark Freeze-out (DM annihilates within the dark sector, e.g., $3 \to 2$).
  • They calculate the freeze-out temperature ($x_{fo} = m/T_{fo}$) and the required cross-section to satisfy the observed relic density ($\Omega_{DM} h^2 \approx 0.12$) for various reheating temperatures ($T_{rh}$).

• Constraint Analysis:

  • By equating the required cross-section (to fit observations) with the maximum unitarity-allowed cross-section, they derive the new upper bounds on the DM mass ($m_{DM}$) as a function of the reheating temperature ($T_{rh}$) and the expansion parameter ($n$).

3. Key Contributions

1. Generalization of Unitarity Bounds: The paper extends the classic Griest-Kamionkowski unitarity bound to general $r \to 2$ processes and, crucially, to non-standard cosmological backgrounds.
2. Dual Impact of Low-Temperature Reheating: The authors demonstrate that low-temperature reheating has opposite effects on the unitarity bound depending on the specific cosmological scenario:
   • Kination-like: Tightens the bound (lowers the maximum allowed mass).
   • Early Matter Domination: Relaxes the bound (allows for significantly heavier DM).
3. Analytical Solutions: They provide analytical expressions for the freeze-out temperature and required cross-sections in these modified cosmologies, offering a tool for future model building.

4. Results

A. Kination-like Scenarios

• Mechanism: The universe expands faster than in the standard radiation era ($H$ is larger). This causes DM to freeze out earlier (at higher temperatures).
• Consequence: To achieve the observed relic density with an earlier freeze-out, a larger annihilation cross-section is required.
• Unitarity Bound: Since the required cross-section increases, it hits the unitarity limit at a lower mass.
  • Result: The upper bound on DM mass becomes stricter.
  • Quantitative: For $T_{rh} \approx T_{BBN} \approx 4$ MeV, the standard 130 TeV bound for WIMPs ($2 \to 2$) drops to **$\sim 3$TeV** (for$n=2$) or **$\sim 1.2$TeV** (for$n=4$). For SIMPs ($3 \to 2$), the bound drops from 1 GeV to **$\sim 0.4$ GeV**.
  • Implication: Heavy WIMPs are more strongly excluded in these scenarios.

B. Early Matter Domination (EMD) Scenarios

• Mechanism: The universe expands rapidly, but the decay of the matter-dominated component injects a massive amount of entropy into the SM bath after DM freeze-out.
• Consequence: This entropy injection dilutes the DM relic abundance. To compensate for this dilution and still match the observed $\Omega_{DM}$, the DM must overproduce initially. This requires a smaller annihilation cross-section (allowing DM to survive longer before decoupling).
• Unitarity Bound: Since the required cross-section is smaller, the unitarity limit is reached at much higher masses.
  • Result: The upper bound on DM mass is relaxed significantly.
  • Quantitative: For $T_{rh} \sim 10^6$ GeV, WIMP masses up to $\sim 10^{10}$ GeV become viable. For SIMPs, masses up to $\sim 10^8$ GeV are allowed.
  • Implication: Super-heavy WIMPs (Super-WIMPs) are no longer ruled out by unitarity in these scenarios.

C. Visual Summary of Constraints

• Figure 2 (Kination): Shows the excluded region (gray) in the $[m, T_{rh}]$ plane. The bound tightens as $T_{rh}$ decreases and $n$ increases.
• Figure 4 (EMD): Shows the allowed region expanding to higher masses as $T_{rh}$ increases. The "No freeze-out" region (where chemical equilibrium cannot be reached) is also mapped.

5. Significance

• Model Independence: The results are largely model-independent regarding the specific particle physics of the DM, relying only on the S-matrix unitarity and the cosmological expansion history.
• Re-evaluating Exclusion Limits: The paper warns that the standard "130 TeV" exclusion limit for WIMPs is not absolute; it is contingent on the standard cosmological history. If the early universe underwent an early matter-dominated phase, much heavier DM candidates remain theoretically viable.
• Guidance for Experiments:
  • In Kination scenarios, direct detection experiments should focus on lighter WIMPs, as heavy ones are theoretically excluded.
  • In EMD scenarios, the search space for heavy DM (up to $10^{10}$ GeV) is reopened, suggesting that indirect detection or gravitational wave signatures from such heavy relics might be worth investigating.
• Cosmological Probes: The tight correlation between the DM mass bound and the reheating temperature ($T_{rh}$) suggests that future constraints on DM mass could provide indirect bounds on the reheating temperature of the early universe, and vice versa.

In conclusion, the paper establishes that the "Unitarity Bound" is not a fixed number but a dynamic constraint heavily dependent on the thermal history of the early universe. Low-temperature reheating can either severely restrict or dramatically expand the viable parameter space for thermal Dark Matter.