Ultra-Relativistic Freeze-Out (UFO)

Updated 8 December 2025

Ultra-Relativistic Freeze-Out (UFO) is a dynamical decoupling mechanism defined by matching the local expansion rate to microscopic scattering or annihilation rates.
It governs the transition from interacting fluid dynamics to free-streaming particles in both heavy-ion collisions and cosmological dark matter environments.
UFO captures spatial-temporal fluctuations in freeze-out conditions, offering a more physically motivated alternative to fixed-temperature prescriptions.

Ultra-Relativistic Freeze-Out (UFO) is the designation for a dynamical decoupling mechanism applicable to both heavy-ion collisions and cosmological dark matter, where freeze-out occurs in an environment characterized by ultra-relativistic expansion or particle kinematics. Unlike phenomenological prescriptions based on constant temperature or energy density, the UFO scenario bases the freeze-out locus on the competition between macroscopic expansion and microscopic scattering rates, or, in the context of dark matter, on the interplay between annihilation rates and Hubble expansion during radiation or reheating domination. This kinetic or chemical decoupling criterion captures crucial dynamical gradients and generically leads to a freeze-out surface closely tied to physical rates rather than ad hoc threshold values.

1. Dynamical Freeze-Out Criterion and Physical Foundations

The central tenet of the ultra-relativistic freeze-out prescription is that particles decouple when the local expansion rate of the medium equals the relevant microscopic scattering or annihilation rate. Quantitatively, this condition is often expressed as a Knudsen-number criterion,

$K(x) = \frac{\theta(x)}{\Gamma(x)} \simeq 1,$

where $\theta(x) = \partial_\mu u^\mu(x)$ is the local hydrodynamic expansion scalar and $\Gamma(x)$ the species-dependent microscopic collision or scattering rate. In the context of hadronic fireballs (QGP to HRG transition), $\Gamma(x)$ is typically the pion or nucleon scattering rate, with explicit dependence on local temperature, chemical potential, and hadro-resonance content (Holopainen et al., 2012, 0710.4476, Holopainen et al., 2013, Ahmad et al., 2016).

The analogous criterion in cosmological or new-physics settings (e.g., WIMP, FIMP, UFO dark matter production) is

$\Gamma(T_\mathrm{fo}) = \langle \sigma v \rangle n_\mathrm{eq}(T_\mathrm{fo}) = H(T_\mathrm{fo}),$

where $H(T)$ is the Hubble rate, and the structure of $\langle \sigma v \rangle$ (cross section scaling with $T$ ) and the background domination (radiation, reheating) set the dynamical scaling (Henrich et al., 7 May 2025, Henrich et al., 3 Nov 2025, Henrich et al., 3 Dec 2025, Lebedev et al., 2023).

This kinetic criterion generalizes to chemical freeze-out (when inelastic rates drop below expansion) as well as to full kinetic decoupling (when all scattering becomes subdominant).

2. Implementation in Heavy-Ion Collision Hydrodynamics and Transport

In numerical hydrodynamics, the UFO criterion is operationalized as follows:

The local expansion rate $\theta(x)$ is computed from the gradient of the fluid four-velocity $u^\mu(x)$ ; in $(2+1)$ D Milne (boost-invariant) coordinates,

$\theta = \partial_\tau u^\tau + \partial_x u^x + \partial_y u^y + \frac{u^\tau}{\tau}$

(Holopainen et al., 2012, Ahmad et al., 2016).

The microscopic scattering rate

$\Gamma = \frac{1}{n_\pi} \sum_i \int \frac{d^3p_\pi}{(2\pi)^3} \frac{d^3p_i}{(2\pi)^3} f_\pi f_i v_{\rm rel} \sigma_{\pi i}(s)$

is evaluated using thermal phase-space densities $f_i$ , relative velocities $v_{\rm rel}$ , and resonance-dominated cross sections (e.g., from UrQMD tables or analytic parameterizations) (0710.4476).

The freeze-out hypersurface $\Sigma$ is constructed numerically as the set of cells satisfying $K(x)=K_\mathrm{fo}$ , often with $K_\mathrm{fo}$ varied in $[0.5, 2]$ for systematics.

Event-by-event simulations reveal that fluctuations in initial energy deposition (from Glauber or Monte-Carlo initializations) lead to a freeze-out surface with enhanced spatial structure, but integrated observables remain robust (Holopainen et al., 2012, Holopainen et al., 2013).

3. Particle Decoupling, Observables, and Equivalence to Isothermal Freeze-Out

Once $\Sigma$ is obtained, final-state particle spectra are computed via the Cooper–Frye integral,

$E \frac{dN}{d^3p} = \int_\Sigma f(x,p) p^\mu d \Sigma_\mu.$

Under the UFO prescription, $f(x,p)$ is typically the equilibrium distribution appropriate to the local temperature and chemical potential, possibly extended with viscous $\delta f$ corrections (rarely in ideal hydro studies) (Holopainen et al., 2012).

Empirical comparisons between UFO-based and constant- $T$ freeze-out reveal the following (0710.4476, Holopainen et al., 2012, Ahmad et al., 2016):

Integrated single-particle spectra and flow coefficients (e.g., $v_2(p_T)$ , $v_3(p_T)$ ) are virtually indistinguishable when $\langle T \rangle$ is matched, with typical differences $\lesssim 2-3\%$ in $0 < p_T < 2$ GeV.
The geometric shape of the freeze-out surface is markedly different: UFO surfaces decouple the dilute (high-flow) "corona" at higher $T$ and the fireball core at lower $T$ , whereas isothermal surfaces freeze out all zones at the same temperature.
For event-by-event initial states, UFO surfaces exhibit strong local fluctuations (horns/fins), but these average out in final observables.

These results underscore the limited phenomenological difference between dynamical and isothermal approaches for bulk observables, though the UFO prescription is physically motivated and provides improved treatment of coupling between hydrodynamics and hadron cascades (Holopainen et al., 2012, Holopainen et al., 2013, Ahmad et al., 2016). For observables sensitive to emission duration or spatial characteristics (e.g., HBT radii, event-by-event $v_n$ distributions, small-system flows), distinctions may be more pronounced.

4. Chemical and Kinetic Freeze-Out: Criteria and Systematics

Chemical decoupling (fixing relative hadron yields) and kinetic decoupling (fixing momentum spectra) are generally non-simultaneous. In the UFO framework, the chemical freeze-out surface is defined analogously by the local breakdown of inelastic scattering: $\Gamma_\mathrm{inel}(x) \lesssim \theta(x),$ with mapping to experimental data via comparison to observed particle ratios (Reichert et al., 2020, Uddin et al., 2014, Assis et al., 2012). Grand-canonical statistical hadron gas models, both ideal and with van der Waals corrections, confirm that the extracted freeze-out parameters cluster along a universal line in $(T, \mu_B)$ space, corresponding closely with the QCD crossover from lattice calculations (Din et al., 15 Aug 2024, Uddin et al., 2014).

Key systematics include:

The energy per particle, $\langle E \rangle / \langle N \rangle \approx 1$ GeV, is a robust criterion across a wide range of collision energies.
Entropy- and density-based criteria (e.g., $s/T^3 \simeq 7$ , $n_B+n_{\bar B} \simeq 0.12\,\mathrm{fm}^{-3}$ ) are less universal and tend to fail at low collision energies (Reichert et al., 2020).

Freeze-out hierarchies among hadron species (earlier decoupling of multi-strange hyperons, later for pions) naturally emerge in UFO-based descriptions incorporating species-dependent cross sections (Uddin et al., 2014).

5. Ultra-Relativistic Freeze-Out in Cosmology and Dark Matter

UFO is also the correct formalism for chemical decoupling of cosmological relics whenever the decoupling temperature satisfies $T_\mathrm{fo} \gg m$ , as seen for SM neutrinos or for DM candidates that freeze out while relativistic or during reheating (Henrich et al., 3 Nov 2025, Henrich et al., 7 May 2025, Henrich et al., 3 Dec 2025, Lebedev et al., 2023). The central freeze-out equation,

$\langle \sigma v \rangle n_\mathrm{eq}(T_\mathrm{fo}) = H(T_\mathrm{fo}),$

has solutions in the relativistic regime when the temperature-scaling of the interaction, $n$ in $\langle \sigma v \rangle \propto T^n / \Lambda^{n+2}$ , is sufficiently steep: specifically, $d \ln \Gamma / d \ln T > d \ln H/ d \ln T$ ; this is realized for interactions mediated by heavy species with $n \ge 2$ . The UFO regime is thus generic for heavy ( $\sim$ TeV-PeV) mediator models (" $Z'$ portals"), which dominate the available DM parameter space for $M_{Z'} \gtrsim$ TeV and $m_\chi \lesssim M_{Z'}$ (Henrich et al., 3 Dec 2025, Henrich et al., 3 Nov 2025).

Relic abundances in the UFO regime are given by analytic yield expressions of the form

$Y_\infty = \frac{135\,\zeta(3)}{8\pi^4} \frac{g_\chi}{g_{*s}},$

with the present-day density

$\Omega_\chi h^2 = 2.75 \times 10^8 \frac{m_\chi}{\mathrm{GeV}} Y_\infty,$

subject to corrections from entropy dilution after freeze-out and duration of reheating.

A critical observation is that, in scenarios with freeze-out during reheating, post-decoupling production (an out-of-equilibrium "freeze-in" tail) and continued redshifting ensure DM is sufficiently cold for structure formation, even for very light species ( $m_\chi \sim \mathcal{O}(\mathrm{keV})$ ) (Henrich et al., 7 May 2025, Henrich et al., 3 Nov 2025). The UFO mechanism bridges the WIMP and FIMP regimes, with parameter space spanning $m_\chi$ from sub-eV to PeV, $T_\mathrm{RH}$ from MeV to $10^{15}$ GeV, and BSM scales from TeV to $10^{14}$ GeV.

6. Comparison to Isothermal and Energy Density Freeze-Out Prescriptions

Traditional hydrodynamics and hybrid models frequently employ constant temperature or energy-density freeze-out surfaces. UFO-based analyses reveal that such prescriptions approximate the kinetic decoupling surface only if the microscopic rate's $T$ dependence is steep (e.g., $\Gamma(T) \propto T^3$ ), which is generally not the case in more realistic hadronic environments (Ahmad et al., 2016, Holopainen et al., 2012). Local velocity gradients and highly non-uniform expansion lead to decoupling at a range of temperatures across the system—a fact that can only be captured via dynamical, rate-based freeze-out. Microscopic transport calculations (e.g., UrQMD coarse-graining) confirm that the true freeze-out surface is not isothermal but is determined locally by the expansion/scattering competition (Inghirami et al., 2021, Reichert et al., 2020).

A representative comparison:

Freeze-Out Criterion	Definition	Phenomenological Consequence
Constant-T ( $T_\mathrm{fo}$ )	$T(x) = \mathrm{const}$	Simple, matches integrated spectra; spatially uniform surface
Constant- $\epsilon$	$\epsilon(x) = \mathrm{const}$	Analogous, but with energy density
UFO (Knudsen/Rates)	$\theta(x)/\Gamma(x) = K_\mathrm{fo}$	Locally adaptive, captures spatial/temporal heterogeneity

The equivalence of integrated observables between UFO and isothermal freeze-out arises once quantitative matching of $\langle T \rangle$ is performed, but differential and space-time sensitive probes can discern the underlying dynamical structure (Ahmad et al., 2016, Holopainen et al., 2012).

7. Outlook, Extensions, and Limitations

UFO provides a physically transparent, local criterion for freeze-out, eliminating the need for arbitrary global parameters and naturally accommodating event-by-event and centrality-dependent effects. It supports systematic analyses of sequential freeze-out, non-equilibrium partitioning (e.g., species-dependent relaxation rates), and mapping to transport afterburners.

Known limitations include:

Precise knowledge of all relevant cross sections, proper treatment of chemical nonequilibrium, and hadron-resonance feed-down, especially below $T_\mathrm{chem}$ .
Viscous (non-ideal) corrections to both expansion and scattering rates, which remain an active area of model development (Holopainen et al., 2012, Ahmad et al., 2016, Holopainen et al., 2013).
Microphysical timescales and finite-width effects in rapid expansion environments.

Continued refinement will require systematic inclusion of non-equilibrium effects, improved data-to-model interfacing, and application to small-system and low-energy collision programs.

References:

(Holopainen et al., 2012, 0710.4476, Ahmad et al., 2016, Holopainen et al., 2013, Reichert et al., 2020, Uddin et al., 2014, Assis et al., 2012, Din et al., 15 Aug 2024, Becattini et al., 2012, Uddin et al., 2011, Xu et al., 2017, Inghirami et al., 2021, Henrich et al., 7 May 2025, Henrich et al., 3 Nov 2025, Henrich et al., 3 Dec 2025, Lebedev et al., 2023).