Warm dark matter from freeze-in at stronger coupling

This paper investigates warm Higgs portal dark matter produced via freeze-in at stronger couplings within a low-temperature thermal bath, revealing that while significant couplings allow for collider detection via Higgs decay, Lyman-$\alpha$ constraints exclude masses below 50–100 keV and necessitate a non-thermal momentum distribution that defies standard parametrizations.

The Gist

The Big Picture: A "Warm" Mystery

Imagine the universe as a giant, expanding balloon. Inside this balloon, there is a hidden substance called Dark Matter (DM) that holds galaxies together, but we can't see it. For a long time, scientists thought this dark matter was made of heavy, slow-moving particles (like heavy boulders rolling slowly).

However, this new paper explores a different idea: Warm Dark Matter. Think of these particles not as heavy boulders, but as feather-light dust motes floating in a breeze. They are very light (thousands of times lighter than an electron) and move relatively fast.

The authors ask: How did these light particles get here, and could we catch them?

The Setup: A Universe That Never Got Hot

Usually, scientists imagine the early universe was a scorching hot furnace. In that furnace, particles would bounce around so much that they would "thermalize" (reach a perfect, balanced temperature).

This paper proposes a different scenario: The universe never got that hot.
Imagine the early universe was more like a lukewarm bath rather than a boiling pot. The temperature never rose above a certain point (specifically, below 100 million degrees, or 100 MeV).

Because the "bath" was never hot enough to boil the water, the light dark matter particles couldn't be created in huge numbers through normal collisions. Instead, they were created very slowly, like dripping water filling a bucket. This slow, steady accumulation is called the "Freeze-in" mechanism.

The Connection: The Higgs Portal

How do these invisible particles talk to the visible world? The paper uses a "Higgs Portal."

• The Higgs Field is like a thick, invisible soup that fills the universe.
• The Portal is a door connecting the visible world to the dark world.
• The authors suggest that the door is actually wide open (strong coupling). Usually, scientists think the door must be tiny (weak coupling) to explain why we haven't seen dark matter yet. But in this "lukewarm universe" scenario, the door can be wide open because the universe was too cold to push many particles through it anyway.

The Production Line: Pions and Muons

In this lukewarm universe, the main "machines" creating dark matter are pions and muons (types of subatomic particles).

• Imagine pions and muons as factory workers.
• They collide and, through the Higgs portal, occasionally spit out a pair of dark matter particles.
• Because the universe is cool, these workers are tired and slow. They don't produce dark matter often, but they do it steadily.

The Surprise: A Bumpy Distribution

Here is the most interesting part. When you create particles in a hot, boiling universe, their speeds are spread out evenly (like a smooth hill).

But in this "lukewarm" scenario, the distribution of speeds is weird and bumpy.

• The Analogy: Imagine a conveyor belt dropping boxes. In a normal factory, the boxes land in a neat pile. In this scenario, the conveyor belt is moving so fast that the boxes are thrown far apart, but the ones at the very front are missing.
• The Result: The dark matter particles have a very specific speed range. They are too fast to be "cold" (like boulders) but too slow to be "hot" (like light).
• The "Cut-off": Crucially, there are almost no very slow particles. The "slow lane" is empty. This is because the universe didn't have enough time to slow these particles down as it expanded.

Why This Matters: The Lyman-α Constraint

Scientists look at the "Lyman-alpha forest" (a pattern in the light from distant quasars) to see how dark matter clumps together.

• If dark matter is too "warm" (too fast), it smears out the structure of the universe, preventing small galaxies from forming.
• Because this paper's dark matter has a weird speed distribution with no slow particles, it is very "warm."
• The Verdict: The authors found that if the dark matter is too light (below 50 to 100 keV), it would have wiped out small galaxies. Therefore, the universe tells us the dark matter must be at least this heavy.

The Good News: We Can Detect It!

Usually, if dark matter interacts strongly with the Higgs, we would have seen it by now. But because the universe was so cold, the production was suppressed, so we missed it.

However, because the connection (coupling) is strong, there is a chance to see it today:

• The Invisible Decay: The Higgs boson (the particle associated with the Higgs field) might occasionally decay into these invisible dark matter particles.
• The Hunt: Experiments at the Large Hadron Collider (LHC) and future colliders (like the FCC) are looking for Higgs bosons that seem to disappear.
• The Prediction: This paper predicts that if we look closely enough, we might see the Higgs boson turning into dark matter about 3% to 0.3% of the time. This is right on the edge of what current and future machines can detect.

Summary

1. Scenario: The early universe was cooler than we thought.
2. Mechanism: Dark matter was created slowly ("freeze-in") by pions and muons, not by a hot explosion.
3. Result: The dark matter is "warm" and has a strange speed distribution with no slow particles.
4. Constraint: It must be heavier than 50–100 keV, or it would have destroyed the structure of the universe.
5. Discovery: Because the connection to the Higgs is strong, we might detect it by watching the Higgs boson disappear at particle colliders.

Technical Summary

1. Problem Statement

The paper addresses the Dark Matter (DM) puzzle, specifically focusing on Warm Dark Matter (WDM) within the Higgs portal framework. Traditional Weakly Interacting Massive Particle (WIMP) scenarios are under pressure from null results in direct detection experiments. Consequently, alternative mechanisms like freeze-in (where DM is never in thermal equilibrium) are being explored.

However, standard freeze-in models often suffer from two issues:

1. Gravitational Overproduction: In high-temperature scenarios, gravitational particle production during inflation can lead to DM over-abundance, requiring extremely small couplings ($\lambda \ll 10^{-10}$) to avoid this, making detection impossible.
2. Thermalization: Standard freeze-in usually assumes high reheating temperatures ($T_R$), leading to thermal-like momentum distributions.

The authors investigate a specific regime: Freeze-in at stronger coupling combined with a low reheating temperature ($T_R \lesssim 100$ MeV). In this scenario, the Standard Model (SM) thermal bath temperature is always lower than the mass of the SM particles responsible for DM production (pions and muons). This suppresses the production rate via Boltzmann factors, allowing for significantly larger DM-Higgs couplings ($\lambda_{hs} \sim 10^{-4} - 10^{-2}$) without overproducing DM, while keeping the DM "warm" (mass $\sim$ 100 keV).

2. Methodology

Theoretical Framework

• Model: A real scalar singlet $S$ coupled to the SM Higgs doublet $H$ via the interaction $\mathcal{L} \supset \frac{\lambda_{hs}}{2} H^\dagger H S^2$. The DM mass $m_s$ is assumed to be very light ($< 1$ MeV).
• Cosmology: The authors consider post-inflationary scenarios where the SM sector is produced via the decay of a spectator field (or inflaton) into intermediate states, leading to a low maximum temperature ($T_{max} \approx T_R$). They analyze two distinct temperature evolution profiles:
  1. Temperature Peak at $T_R$: $T$ rises to $T_R$ and then decreases (standard reheating).
  2. Flat Temperature Profile: $T$ remains constant at $T_R$ for a period before the final reheating epoch (e.g., via a decay chain $\phi \to \nu_R \nu_R \to \text{SM}$).
• Production Mechanism: DM is produced via the annihilation of SM particles mediated by the Higgs boson ($SM + SM \to S + S$). At $T < 100$ MeV, the dominant channels are pion ($\pi$) and muon ($\mu$) annihilation. Electron annihilation dominates only at very low $T$ ($\sim 10$ MeV).

Analytical and Numerical Techniques

• Kinematics: The authors perform a detailed kinematic analysis of the production process. They demonstrate that while the center-of-mass energy is near $2m_\pi$, the resulting DM particles can have a wide range of momenta. Crucially, producing low-momentum DM requires highly boosted initial states, which are exponentially suppressed by the Boltzmann factor.
• Boltzmann Equation: They solve the Boltzmann equation for the DM momentum distribution function $f(p, t)$.
  $$\frac{\partial f}{\partial t} - H|p|\frac{\partial f}{\partial |p|} = \frac{1}{p^0} C(p, t)$$
  The collision integral $C(p, t)$ is computed using Maxwell-Boltzmann statistics for the initial SM states.
• Distribution Analysis: They derive asymptotic limits for the comoving momentum distribution $f(q)$ (where $q = p/T$).
• Constraints:
  • Lyman-$\alpha$ Forest: They use the CLASS code to compute the matter power spectrum and apply the "area criterion" to constrain the DM mass based on structure formation.
  • Collider Searches: They calculate the invisible Higgs decay width ($h \to SS$) and compare it with current LHC limits and future prospects (HL-LHC, FCC).

3. Key Contributions

A. Non-Thermal Momentum Distribution

The most significant theoretical contribution is the characterization of the DM momentum distribution in low-$T_R$ freeze-in scenarios.

• Deviation from $\alpha\beta\gamma$ Parametrization: Standard freeze-in distributions are often approximated by $f(q) \propto q^\alpha e^{-\beta q^\gamma}$. The authors show this fails for low-$T_R$ scenarios.
• IR Suppression: Due to the low bath temperature, the production of low-momentum DM is kinematically suppressed.
  • In the Peak Temperature model, low momenta are exponentially cut off: $f(q) \propto \sqrt{q} \exp(-m_\pi^2 / q T_R^2)$ for small $q$.
  • In the Flat Temperature model, low momenta scale as $f(q) \propto q^2$.
• Result: The distribution is "shock-wave-like" with a sharp cutoff at low momenta and a peak around $q \sim m_\pi/T_R$. This makes the DM effectively "warmer" (higher average velocity) than a thermal distribution with the same mass would suggest.

B. Viability of Stronger Coupling

The paper demonstrates that a low reheating temperature ($T_R \sim 50-80$ MeV) allows for a much larger Higgs-DM coupling ($\lambda_{hs}$) compared to high-$T_R$ scenarios.

• The Boltzmann suppression of the production rate ($e^{-m_{SM}/T}$) compensates for the large coupling, preventing DM over-abundance.
• This opens a viable parameter space where $\lambda_{hs} \sim 10^{-2}$, which is orders of magnitude larger than typical freeze-in couplings.

C. Lyman-$\alpha$ Bounds on WDM Mass

The authors derive stringent constraints on the DM mass using the Lyman-$\alpha$ forest.

• Because the momentum distribution is non-thermal and peaked at higher momenta (relative to the temperature), the free-streaming length is larger.
• Result: They exclude DM masses below 50–100 keV (depending on the specific temperature profile and $T_R$). This is a stronger bound than in standard high-$T_R$ freeze-in models.

4. Results

• Allowed Parameter Space:
  • DM Mass ($m_s$): $50 \text{ keV} \lesssim m_s \lesssim 1 \text{ MeV}$.
  • Coupling ($\lambda_{hs}$): $10^{-4} \lesssim \lambda_{hs} \lesssim 10^{-2}$.
  • Reheating Temperature ($T_R$): $4 \text{ MeV} \lesssim T_R \lesssim 80 \text{ MeV}$ (limited by Big Bang Nucleosynthesis and the QCD phase transition).
• Collider Signatures:
  • The large coupling implies a significant branching ratio for invisible Higgs decay ($h \to SS$).
  • Current LHC: Excludes $\lambda_{hs} \gtrsim 10^{-2}$ (Branching Ratio $\text{BR}_{inv} \gtrsim 10\%$).
  • Future Colliders: The HL-LHC (target $\text{BR}_{inv} \sim 3\%$) and FCC (target $\text{BR}_{inv} \sim 0.3\%$) can probe the entire viable parameter space for this model.
• Temperature Profile Impact:
  • Models with a flat temperature profile before reheating allow for slightly lower couplings and slightly relax the Lyman-$\alpha$ mass bound (down to $\sim 50$ keV) compared to the peak temperature model, but the qualitative features (non-thermal distribution, strong bounds) remain.

5. Significance

1. Testability: This work revitalizes the Higgs portal WDM scenario. By linking the low reheating temperature to stronger couplings, it transforms a "dark" sector model (undetectable) into one that is highly testable at current and future colliders via invisible Higgs decays.
2. Cosmological Insight: It highlights that the thermal history of the early universe (specifically the reheating temperature and temperature evolution profile) drastically alters the momentum distribution of freeze-in DM. This necessitates moving beyond standard $\alpha\beta\gamma$ parametrizations when interpreting structure formation data.
3. Resolution of Tension: It provides a consistent framework where WDM can explain small-scale structure issues (via free-streaming) while satisfying Lyman-$\alpha$ constraints, provided the mass is in the specific 50–100 keV window and the coupling is strong enough to be detected.
4. Methodological Rigor: The paper provides a rigorous derivation of the collision integrals and momentum distributions for low-temperature freeze-in, offering a template for analyzing similar non-standard cosmological scenarios.

In conclusion, the paper establishes that Warm Higgs Portal Dark Matter produced via freeze-in at low temperatures is a robust, testable hypothesis. It predicts a specific mass range ($>50$ keV) and a coupling strength that makes invisible Higgs decay the primary discovery channel, bridging the gap between early universe cosmology and high-energy collider physics.