Cosmology of Inelastic Self-Interacting Dark Matter: Linear Evolution and Observational Constraints

This paper investigates the linear cosmological evolution of inelastic self-interacting dark matter in a two-component sector, demonstrating how exothermic conversions generate pressure support that suppresses small-scale structure and produces observable cutoffs in the matter power spectrum constrained by Lyman-$\alpha$ forest data and high-redshift UV luminosity functions.

The Gist

The Big Picture: A Hidden, Complex Universe

Imagine the universe is a giant, dark ocean. For decades, scientists thought the "Dark Matter" floating in this ocean was just one type of invisible, silent rock that never bumped into anything else. It was boring, but it explained how galaxies held together.

However, this paper suggests the ocean is actually much more complex. Instead of just one type of rock, imagine the dark sector is a two-story building made of invisible matter:

1. The Heavy Floor (Heavy Dark Matter): The upper level, slightly heavier.
2. The Light Floor (Light Dark Matter): The lower level, slightly lighter.

These two floors are separated by a tiny gap (a "mass splitting"). The big discovery here is that particles can jump between these floors, but it's not a free-for-all. It's a specific kind of dance called "Inelastic Conversion."

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The Dance: Jumping Up and Down

Think of the heavy particles as people standing on a high diving board, and the light particles as people in the pool below.

• The Down Jump (Exothermic): When a heavy particle jumps down to the light floor, it doesn't just land; it releases a burst of energy (like a splash). This energy heats up the water (the light particles), making them move faster and push outward.
• The Up Jump (Endothermic): To jump back up, the light particles need a huge amount of energy to reach the diving board. If the water is too cold, they can't jump up at all.

The Key Insight: In the early universe, the "water" was hot, so particles jumped up and down freely. But as the universe expanded and cooled, the water got too cold for the light particles to jump back up. Suddenly, everyone started falling down to the light floor, releasing a massive amount of heat energy all at once.

The Consequence: The "Dark Acoustic Oscillation"

Here is where it gets interesting. When all those heavy particles suddenly jumped down and released their energy, they gave the light particles a massive kick.

Imagine a crowd of people in a stadium (the universe). If everyone suddenly gets a shove, they don't just move forward; they start bouncing back and forth, creating a wave.

• Standard Dark Matter: Just moves forward smoothly.
• This New Dark Matter: Because of the sudden energy injection, the light particles start bouncing (oscillating) like sound waves.

These are called Dark Acoustic Oscillations (DAOs). They are like sound waves traveling through the dark matter, but since there's no air, they are invisible to our ears. However, they leave a fingerprint on how galaxies form.

The Result: A "Cutoff" in the Galaxy Recipe

Because of this bouncing and the pressure from the heat, the dark matter resists clumping together on small scales.

• The Analogy: Imagine trying to build a sandcastle. If the sand is dry, you can make tiny, detailed towers. But if you pour water on the sand (the heat from the conversion), the sand becomes too loose to hold small shapes. You can still build a big castle, but the tiny turrets wash away.

In our universe, this means small galaxies and dwarf galaxies are suppressed or "washed out." The universe creates big galaxies fine, but it struggles to make the tiny ones. This paper calculates exactly how many tiny galaxies should be missing based on the rules of this "two-story" dark matter.

The Detective Work: Checking the Evidence

The authors used a super-computer (a "Boltzmann solver") to simulate this universe. They then compared their results to real-world data:

1. The Lyman-α Forest: This is like looking at the "fingerprint" of gas clouds between us and distant quasars. It tells us how much small-scale structure exists.
2. High-Redshift Galaxies: Looking at the very first, tiny galaxies formed after the Big Bang.

The Findings:
The paper found that this "two-story" dark matter model creates a very specific pattern. It doesn't just wipe out small galaxies; it creates a "sweet spot" where the effect is strongest.

• If the "jump" is too weak, nothing happens.
• If the "jump" is too strong, the heavy particles run out of energy too fast, and the effect stops.
• The Sweet Spot: There is a specific range of "jump strength" and "gap size" that perfectly matches the data we see today. This creates a closed "exclusion zone"—a specific set of rules that the dark matter must follow to be consistent with our observations.

Why Does This Matter?

1. It Solves a Mystery: It offers a new explanation for why we see fewer small galaxies than the standard model predicts (the "Small Scale Crisis").
2. It Hides from Detectors: Because the particles are so heavy and the gap is tricky, they don't bump into normal matter easily. This explains why we haven't found dark matter in underground labs yet.
3. It's a New Tool: It gives astronomers a new way to test dark matter. Instead of just looking for particles, we can look for the "sound waves" (oscillations) and the "missing sandcastles" (suppressed small galaxies) to prove this theory is real.

In a Nutshell

The universe's dark matter might not be a single, boring particle. It might be a complex system of two types of particles that jump between energy levels. When they jump down, they heat up the universe, creating sound waves that prevent tiny galaxies from forming. By measuring exactly which tiny galaxies are missing, we can figure out the rules of this invisible dance.

Technical Summary

1. Problem Statement

The standard $\Lambda$CDM model, assuming collisionless Cold Dark Matter (CDM), successfully explains large-scale cosmological observations but faces tensions on galactic and sub-galactic scales (e.g., the core-cusp problem, Too-Big-To-Fail problem). While Self-Interacting Dark Matter (SIDM) offers a solution by allowing heat transfer within halos, simple elastic SIDM models often conflict with direct detection limits and cluster merger observations.

Inelastic SIDM models propose a dark sector with two nearly degenerate states ($\chi_h$ and $\chi_l$) separated by a small mass splitting $\Delta m$. These models can naturally evade direct detection constraints (via endothermic up-scattering kinematic suppression) while maintaining large self-interaction cross-sections. However, the linear cosmological evolution of such a multi-component dark sector, specifically how inelastic conversion ($\chi_h\chi_h \leftrightarrow \chi_l\chi_l$) affects structure formation and the matter power spectrum, has not been systematically investigated. This paper aims to fill that gap by deriving the coupled background and perturbation equations and applying observational constraints.

2. Methodology

Theoretical Framework

• Model: A two-component dark sector with a light species ($\chi_l$, mass $m_l$) and a heavy species ($\chi_h$, mass $m_h$). The mass splitting is $\Delta m = m_h - m_l \ll m_l$.
• Interaction: The dominant process is inelastic conversion $\chi_h\chi_h \leftrightarrow \chi_l\chi_l$.
  • Exothermic ($\chi_h\chi_h \to \chi_l\chi_l$): Releases kinetic energy $\Delta m$ into the light component.
  • Endothermic ($\chi_l\chi_l \to \chi_h\chi_h$): Requires kinetic energy threshold, suppressed at low velocities.
• Cross-Section Parameterizations: The authors consider two forms for the velocity-dependent cross-section $\sigma(v_{rel})$:
  1. Power-Law: $\sigma \propto v_{rel}^{n-1}$ (covering $s$-wave, Sommerfeld enhancement, and resonant cases).
  2. Low-Velocity Saturation: A regularized form where $\sigma v_{rel}$ saturates to a constant at low velocities, mimicking unitarity limits or finite-range forces.

Mathematical Derivation

• Background Evolution: The authors derived coupled Boltzmann equations for the number densities ($n_l, n_h$) and temperatures ($T_l, T_h$).
  • The heavy fraction $r_h$ evolves based on the competition between conversion rates and Hubble expansion.
  • The light component temperature $T_l$ receives heating from exothermic conversion, deviating from standard adiabatic cooling.
• Linear Perturbations: Working in the Newtonian gauge, the authors derived linearized continuity and Euler equations for density ($\delta$) and velocity divergence ($\theta$) perturbations.
  • Key terms include conversion-induced friction (drag between species) and pressure support in the light component due to the heating-induced sound speed.
  • Second-order differential equations for density perturbations were derived to analyze physical regimes.
• Numerical Implementation: The equations were implemented into a modified version of the CLASS Boltzmann solver to compute linear power spectra.

3. Key Contributions

1. Derivation of Coupled Equations: The paper provides the first complete set of background and linear perturbation equations for a two-component inelastic SIDM sector, explicitly accounting for the thermodynamic feedback of exothermic conversion on the light component's temperature and sound speed.
2. Identification of Physical Regimes: The authors classify the linear evolution into six distinct regimes:
   • Superhorizon & Early Subhorizon: Standard CDM-like growth.
   • Conversion-Driven Transition: Onset of deviation due to heating/depletion.
   • Acoustic Oscillation: Pressure support in the light component causes $\delta_l$ to oscillate.
   • Acoustic Damping: Friction from conversion suppresses oscillation amplitudes.
   • Late-Time Gravitational Growth: Recovery of standard growth once conversion ceases.
3. Non-Monotonic Constraints: The study reveals that observational constraints on these models are non-monotonic. There is an "allowed" window of cross-section normalizations; too weak a cross-section yields no effect, while too strong a cross-section depletes the heavy component too rapidly, cutting off the heating source before it can significantly suppress structure.

4. Key Results

Linear Structure Formation

• Dark Acoustic Oscillations (DAOs): The pressure support generated by exothermic heating in the light component leads to acoustic oscillations in the matter power spectrum ($P(k)$).
• Small-Scale Cutoff: The combined effect of acoustic oscillations and damping creates a cutoff in the power spectrum at scales $k \gtrsim 1 \, h \, \text{Mpc}^{-1}$.
• Spectral Shape: Unlike Warm Dark Matter (WDM), which produces a smooth cutoff, inelastic SIDM produces a cutoff with distinct DAO features (wiggles) and a shallower suppression tail at high $k$. This is because suppression is driven by conversion-induced heating rather than collisionless free-streaming.
• Component Behavior:
  • The light component ($\chi_l$) exhibits strong suppression and oscillations.
  • The heavy component ($\chi_h$) is suppressed due to friction and the weakening of the gravitational potential caused by the light component's suppression, despite having negligible pressure.

Observational Constraints

Using recast constraints from Lyman-$\alpha$ forest data and high-redshift UV luminosity functions, the authors mapped the allowed parameter space:

• Exclusion Regions: The constraints form closed, non-monotonic contours in the $(\sigma, \Delta m/m_l)$ and $(\sigma, m_l)$ planes.
  • Power-Law ($n=-2$): Excluded $\sigma_{PL} \sim 10^{-48} - 10^{-40} \, \text{cm}^2$.
  • Power-Law ($n=-1$): Excluded $\sigma_{PL} \sim 10^{-37} - 10^{-35} \, \text{cm}^2$.
  • Low-Velocity Saturation: Excluded $\sigma_{sat} \sim 10^{-30} - 10^{-22} \, \text{cm}^2$.
• Mass Splitting Limits: For optimistic parameters ($m_l \approx 100$ MeV), Lyman-$\alpha$ data constrains the mass splitting to $\Delta m/m_l \lesssim \mathcal{O}(10^{-5})$ for $n=-2$ and $\mathcal{O}(10^{-3})$ for $n=-1$.
• Mechanism of Exclusion:
  • Weak Conversion: Insufficient heating to suppress small-scale power.
  • Strong Conversion: Rapid depletion of the heavy species ($r_h \to 0$) stops the energy injection, preventing significant suppression.
  • Large Mass Splitting: Boltzmann suppression of the initial heavy fraction ($r_h(z_{init}) \approx 0$) prevents the process from starting.

5. Significance

• Probing Secluded Sectors: The results demonstrate that the internal thermodynamics of a secluded, multi-component dark sector leave distinct, observable imprints on structure formation, independent of Dark Matter-Standard Model couplings.
• Model Discrimination: The presence of Dark Acoustic Oscillations and the specific shape of the suppression tail (shallower than WDM) provide a unique signature to distinguish inelastic SIDM from other small-scale structure solutions like WDM or standard SIDM.
• Complementarity: These cosmological constraints are complementary to direct detection and collider searches, probing regions of parameter space (specifically small mass splittings and specific velocity dependencies) that are difficult to access otherwise.
• Future Directions: The paper sets the stage for future N-body simulations to capture non-linear halo formation and suggests that next-generation probes (21 cm cosmology, CMB spectral distortions) could further refine these constraints.

In summary, this work establishes that inelastic self-interactions in a two-component dark sector can significantly alter the linear matter power spectrum through a mechanism of conversion-induced heating and acoustic oscillations, leading to testable predictions that are currently constrained by Lyman-$\alpha$ forest data.