Dark Matter Born from Gentle Interactions

While the freeze-out mechanism of weakly interacting massive particles (WIMPs) has long dominated dark matter phenomenology, recent years have witnessed growing interest in the complementary freeze-in paradigm. In this scenario, the dark sector is so feebly coupled to the Standard Model that it never attains thermal equilibrium with the primordial plasma. Instead, dark matter particles, often dubbed FIMPs for “feebly interacting massive particles,” are slowly produced through the decays or scatterings of bath particles. The production rate is initially negligible but accumulates over cosmic time, saturating once the temperature drops below the mass of the bath species. The resulting relic abundance is typically much smaller than that of a thermal WIMP for comparable masses, making freeze-in a compelling explanation for dark matter when experiments constrain interaction strengths to be minuscule.

The calculator above focuses on freeze-in via the decay of a heavy particle B into a lighter FIMP χ and other products. Assuming the decay width Γ is tiny compared to the Hubble rate at temperatures near m_B, the number of FIMPs produced per comoving volume can be approximated analytically. The final comoving abundance, or yield Y_∞ = n_χ/s (where s is entropy density), is given by

$Y_{\infty }\approx \frac{135\mathrm{g\_B\; \Gamma \; M\_\{Pl\}}}{8{\pi }^4{\mathrm{g\_*}}^{\frac{3}{2}}{\mathrm{m\_B}}^2}$

This expression assumes Maxwell–Boltzmann statistics and neglects inverse decays, an excellent approximation when the coupling is so feeble that produced FIMPs do not repopulate the bath. The key quantities are the bath particle’s internal degrees of freedom g_B, its mass m_B, the decay width Γ, the effective number of relativistic degrees of freedom g_* around T ≈ m_B, and the reduced Planck mass M_Pl ≈ 1.22×10¹⁹ GeV. Once Y_∞ is known, the present-day density parameter follows from

${\Omega }_{\chi }{h}^2\approx 2.742\times {10}^8⟨m_{\chi }/\text{GeV}⟩Y_{\infty }$

This convenient relation converts the yield into a dimensionless density using the present-day entropy density and critical density. The calculator implements these formulas entirely in client-side JavaScript, allowing rapid exploration of parameter space without external dependencies.

To get started, the user specifies the mass of the FIMP m_χ, the mass of the decaying bath particle m_B, its total decay width Γ, and the relevant degrees of freedom. The tool outputs both the yield and the relic density, along with a qualitative assessment of whether the predicted Ω_χh² matches the observed dark matter value (~0.12), falls short, or exceeds it. Because freeze-in abundances scale linearly with the tiny coupling controlling Γ, small adjustments to Γ can lead to large changes in the relic density. This sensitivity makes freeze-in models ideal targets for experiments seeking minuscule interaction strengths, such as long-lived particle searches at colliders or precision cosmological probes.

Unlike freeze-out, freeze-in typically occurs at temperatures near the mass of the parent particle. Consequently, the relic density is largely independent of the subsequent thermal history, provided entropy is conserved. The freeze-in mechanism can produce a wide spectrum of dark matter masses, from ultralight axions to superheavy particles, simply by tuning Γ. It also naturally yields a cold or warm dark matter velocity distribution depending on whether production is dominated by decays or scatterings at high temperature.

The table below demonstrates how varying parameters affects the predicted relic abundance:

mχ (GeV)	mB (GeV)	Γ (GeV)	Ωχh2	Outcome
1	100	1×10−20	0.12	Matches DM
0.1	150	1×10−24	0.004	Underabundant
10	1000	1×10−18	5.0	Overabundant

These examples highlight the linear dependence on Γ and m_χ. Lowering the coupling (hence Γ) or the FIMP mass reduces the relic density, while increasing them has the opposite effect. The steep m_B⁻² scaling underscores that heavier bath particles dilute the final abundance, as their number density is Boltzmann suppressed when decays become efficient.

Freeze-in theories open the door to rich phenomenology beyond the reach of traditional dark matter searches. Because the interactions are so weak, FIMP dark matter evades direct detection and most astrophysical constraints. Yet the same feebleness allows it to be produced at accelerators as long-lived states that decay outside detectors, or to influence cosmology through subtle alterations to Big Bang nucleosynthesis, structure formation, or the cosmic microwave background. Models inspired by supersymmetry, axion-like particles, sterile neutrinos, and hidden sectors often exhibit freeze-in behavior for certain parameter ranges.

The freeze-in mechanism also intersects with early-universe processes such as reheating and inflation. If the maximum temperature after inflation is lower than m_B, production can be significantly suppressed, altering the standard yield formula. Conversely, non-thermal production during reheating or from inflaton decays can enhance the abundance. Our calculator assumes a high reheating temperature and neglects these subtleties, but the long-form explanation here walks the user through such caveats, emphasizing when the simple expressions hold and when more elaborate Boltzmann solvers are required.

Understanding the freeze-in parameter space aids in planning experimental strategies. For instance, consider a portal coupling between the Standard Model Higgs boson and a scalar FIMP. If the portal coupling is around 10⁻¹², Higgs decays during the electroweak epoch could yield the observed relic density for m_χ around a few keV. Alternatively, a Z' gauge boson with mass of order TeV and width 10⁻¹⁸ GeV can populate a GeV-scale FIMP. The calculator enables quick estimates of such scenarios, helping researchers identify promising regions for detailed study.

In closing, the freeze-in paradigm illustrates how even the faintest whispers of interaction can sculpt the cosmic inventory. By providing a transparent numerical handle on the key quantities, this tool invites exploration of dark matter models that might otherwise be overlooked. Whether you are a student learning about early-universe thermodynamics, a phenomenologist scanning parameter spaces, or an experimentalist seeking novel signatures, the FIMP Freeze-In Relic Density Calculator offers a starting point for navigating the subtle frontier of feeble interactions.