Ultraviolet Freeze-In Mechanism

• Ultraviolet Freeze-In Mechanism is a non-thermal process producing dark matter through feeble interactions via high-dimensional operators.
• It operates at the highest temperatures in the early Universe where rare scattering and decay events accumulate the dark matter yield.
• The mechanism’s sensitivity to reheating conditions and cosmic evolution provides key insights into high-scale physics and dark sector dynamics.

The ultraviolet freeze-in mechanism is a non-thermal paradigm for producing the observed dark matter (DM) relic abundance. In this framework, DM particles (often called feebly interacting massive particles, or FIMPs) are generated via interactions between the visible sector and DM, but these interactions are so weak that DM never attains thermal equilibrium. In contrast to freeze-out—where the relic abundance is set by the thermal decoupling of particles initially in equilibrium—freeze-in predicts that the DM yield is incrementally accumulated over cosmic time, typically via out-of-equilibrium decay or scattering processes. Moreover, the mechanism can operate through renormalizable or non-renormalizable operators, with ultraviolet (UV) freeze-in specifically referring to the cases where non-renormalizable, higher-dimensional operators dominate production, rendering the DM yield highly sensitive to the highest temperatures in the early Universe.

1. Fundamental Principles of the Ultraviolet Freeze-In Mechanism

Ultraviolet freeze-in occurs when the visible and hidden sectors are connected only via non-renormalizable operators suppressed by a large mass scale $\Lambda$ (e.g., a GUT or Planck scale). The freeze-in process is characterized by the DM yield being dominated at the highest temperatures, typically near the maximum temperature $T_{\rm max}$ achieved during reheating. This contrasts with infrared (IR) freeze-in, where the production is IR-dominated (occurring primarily at temperatures near the mass thresholds of participating particles) and is largely independent of the UV details of reheating or inflation.

The generic production mechanism can be described as follows:

• The DM candidate—being weakly coupled—has a negligible initial abundance and is never in equilibrium with the plasma.
• Production occurs via rare processes, such as $2 \rightarrow 2$ or higher-point scatterings, governed by non-renormalizable operators:

$$\mathcal{L} \supset \frac{1}{\Lambda^n} \mathcal{O}_{n+4}$$

where the operator $\mathcal{O}_{n+4}$ has mass dimension $n+4$.

• The squared amplitude for the relevant process typically scales with temperature as $|{\cal M}|^2 \sim T^{2n}/\Lambda^{2n}$.
• The comoving yield scales as $Y \sim M_{\rm Pl} T_{\rm RH}^{2n-1}/\Lambda^{2n}$ (omitting order-one factors), indicating sensitivity of the final abundance to both $T_{\rm RH}$ and the operator dimension.

The sensitivity of UV freeze-in to the reheat (or maximum) temperature and the specifics of the high-scale operator structure are defining features (Elahi et al., 2014, Chen et al., 2017, Bernal et al., 2019).

2. UV Freeze-In through Non-Renormalizable Operators

In models where the connection between the visible and hidden sectors is mediated by non-renormalizable operators, the DM production rate is maximized at the highest available temperatures. An explicit example is a dimension-5 operator:

$$\mathcal{L} \supset \frac{1}{\Lambda} \phi \bar{\psi}_1 \psi_2 \varphi$$

where $\varphi$ is a DM scalar, $\psi_{1,2}$ are bath fermions, and $\phi$ is a bath scalar. The dominant scattering process $\phi \psi_1 \rightarrow \varphi \psi_2$ leads to a relic yield:

$$Y_{\rm UV} \sim \frac{M_{\rm Pl} T_{\rm RH}}{\Lambda^2}$$

More generally, for a dimension-$(n+4)$ operator, the yield scales as

$$Y \sim \frac{M_{\rm Pl} T_{\rm RH}^{2n-1}}{\Lambda^{2n}}$$

While the IR freeze-in scenario requires putting in ad hoc small couplings ($\lambda \ll 1$), UV freeze-in leverages the natural suppression associated with a high cutoff scale, leading to a "generic" realization of feeble interactions (Elahi et al., 2014).

For scenarios in which the relevant fields carry vacuum expectation values (VEVs), the high-dimensional operator can project to lower-dimensional terms after symmetry breaking, thereby generating both IR and UV freeze-in contributions and introducing additional model dependence.

3. Dependence on Early-Universe Cosmology and Reheating

The total DM yield from UV freeze-in is highly sensitive to:

1. The maximum (not necessarily reheating) temperature, $T_{\rm max}$, achieved in the Universe (Bernal et al., 2019, Bernal et al., 2020).
2. The details of the cosmic equation of state and the duration and nature of reheating (parameterized by $\omega$ or equivalent exponents for the scale factor and temperature evolution).

This sensitivity arises because the DM production is dominated at the hottest stage. In standard radiation-dominated cosmology, one might simply equate $T_{\rm max}$ with the reheating temperature $T_{\rm RH}$. However, more general reheating scenarios (such as kination, early matter domination, or non-oscillatory inflation) can induce a substantial hierarchy $T_{\rm max} \gg T_{\rm RH}$, leading to an enhancement ("boost factor") in the DM relic density:

$$B \sim \left( \frac{T_{\rm max}}{T_{\rm RH}} \right)^\Delta$$

where the boost exponent $\Delta$ is determined by the operator dimension and the underlying cosmological history (Bernal et al., 2019, Bernal et al., 2020, Barman et al., 2022, Bernal et al., 8 Jan 2025). Moreover, a time-dependent inflaton decay width, as opposed to an instantaneous or constant width, modifies the temperature evolution and can either suppress or enhance $T_{\rm max}$ and consequently the DM yield (Barman et al., 2022, Bernal et al., 8 Jan 2025). The analytic structure of the yield and boost factor changes according to whether the operator is above or below a critical dimension determined by the expansion rate.

In scenarios with stiff fluids (kination, $w>1/3$), the DM yield can be especially boosted, with the enhancement factor scaling as $B \propto (T_{\rm max}/T_{\rm RH})^{n}$ for a cross section growing as $T^n$ (Bernal et al., 2020).

4. Model Realizations and Theoretical Embeddings

Ultraviolet freeze-in can be realized in a broad spectrum of theoretical settings:

• Axion/Peccei-Quinn portals: The connection between DM and the SM proceeds via higher-dimensional operators involving PQ-charged fields. After PQ-symmetry breaking, both UV and IR freeze-in components may be generated (Elahi et al., 2014).
• $Z'$ portals: Heavy $U(1)'$ gauge bosons are integrated out, yielding effective four-fermion operators with UV-dominated DM production (Elahi et al., 2014).
• Seesaw and leptogenesis scenarios: Operators involving right-handed neutrinos mediate UV freeze-in of DM and simultaneously enable baryogenesis via leptogenesis, especially when the inflaton decays exclusively into right-handed neutrinos (Barman et al., 2021).
• String moduli/modulinos: Weak-scale or lighter superpartners with high-scale suppressed couplings are well-motivated FIMP candidates (0911.1120).
• Dilaton portals in scale-invariant theories: Spontaneous breaking of conformal invariance at scale $f$ leads to a dilaton feebly connecting DM and the SM, with the yield controlled by the reheating history and $f$ (Ahmed et al., 2021).
• Gravitational portals: Gravitational freeze-in via Planck-suppressed operators (e.g., graviton or massive spin-2 exchange) produces DM relics with a strong $T^6$ or $T^8$ temperature dependence (Bernal et al., 2019, Bernal et al., 8 Jan 2025).

The dominant production channel at high temperature is generally via the operator with the lowest effective mass dimension, as the yield typically scales inversely with powers of $\Lambda$.

5. Quantitative Features and Boltzmann Formalism

The DM evolution is governed by the Boltzmann equation:

$\frac{dn}{dt} + 3 H n = \gamma(T)$

where $\gamma(T)$ encodes the production rate and scales with temperature as $\gamma(T) \propto T^k/\Lambda^{k-4}$ for an operator yielding a reaction rate with $k$ powers of $T$ (Bernal et al., 8 Jan 2025). The yield $Y \equiv n/s$ (with $s$ the entropy density) is often computed by changing variables to the scale factor or the temperature and integrating from the maximal temperature down to the point when production becomes negligible.

For scattering-dominated UV freeze-in:

$$Y \sim \frac{M_{\rm Pl} T_{\rm RH}^{k-5}}{\Lambda^{k-4}}$$

Production is IR-dominated if $k$ is below a cosmology-dependent critical value $k_c$, marginal ($\sim \log(T_{\rm max}/T_{\rm RH})$) if $k = k_c$, and UV-dominated ($\propto (T_{\rm max}/T_{\rm RH})^{k-k_c}$) for $k>k_c$ (Bernal et al., 8 Jan 2025).

For DM production via inflaton decays, the structure is analogous, although the dependence on the equation-of-state parameter $\omega$ and $T_{\rm RH}$ can differ.

When DM possesses strong self-interactions, post-freeze-in phase thermalization and number-changing reactions can further boost the relic density, as the initially injected energy is redistributed over more quanta, enhancing the yield by a calculable boost factor $B$ depending on the dark sector properties (Bernal, 2020).

6. Phenomenological Implications and Experimental Considerations

UV freeze-in models have distinctive experimental signatures and cosmological implications:

• Collider signals: Since UV freeze-in does not require ad hoc small couplings to the visible sector, but rather relies on high-scale suppression, the associated new heavy states (gauged $Z'$, scalars, etc.) may evade collider limits if their mass scale is sufficiently high, but some scenarios (e.g., those with long-lived LOSPs) can yield observable displaced vertices at the LHC (0911.1120).
• Direct/Indirect detection: DM produced via UV freeze-in generally has no detectable direct or indirect signals via Standard Model interactions, as their couplings are typically far below experimental reach. However, in specific cases (e.g., strong coupling with Boltzmann suppression), parameter regions may be tested in future experiments (Cosme et al., 7 Feb 2024).
• Cosmological constraints: Nonstandard cosmological eras during reheating (e.g., kination, early matter domination) can modify or relax bounds on the DM mass and couplings, alter the allowed reheating temperature, and potentially allow UV freeze-in scenarios to explain baryogenesis (UV freeze-in baryogenesis) (Goudelis et al., 2022, Dalianis et al., 2023). The sensitivity to inflationary e-folds links UV freeze-in scenarios to CMB observables (Dalianis et al., 2023).
• Gravitational wave background: High-frequency gravitational wave spectra from graviton bremsstrahlung during UV freeze-in may offer a future indirect window into early universe DM production, though signals are currently beyond projected sensitivities (Wang et al., 14 Aug 2025).

7. Consequences for Model Building and Directions for Future Research

UV freeze-in enlarges the dark matter model-building landscape, enabling the construction of models that are (a) robust against UV uncertainties for renormalizable operators, or (b) directly sensitive to early-universe cosmology and heavy-physics scales for non-renormalizable cases. This mechanism accommodates FIMPs as DM candidates spanning a mass range from keV to superheavy scales, with signals often intertwined with baryogenesis, hidden symmetry breaking, and new gauge sectors (0911.1120, Barman et al., 2021, Ahmed et al., 2021, Goudelis et al., 2022).

The interplay of freeze-in with nonstandard reheating, UV-complete portals, and strong dark sector interactions continues to motivate cross-disciplinary studies, especially in the context of cosmological probes, indirect detection, and collider searches. Since the UV freeze-in yield is inextricably linked to high-scale dynamics—including the reheat process, cosmic equation of state, and operator structure—progress in these areas offers a route toward indirectly constraining physics at otherwise inaccessible energy scales. Future work will further clarify the relationship between UV freeze-in, baryogenesis, inflationary observables, and the microphysics of dark sectors, as well as extend the minimal Hilbert space for viable DM theories.