FIMP Freeze-In Relic Density Calculator

Introduction to FIMP freeze-in relic density

FIMP freeze-in relic density is one of the cleanest alternatives to the better-known freeze-out picture of dark matter. In a freeze-out model, the dark matter species starts in thermal equilibrium with the hot plasma of the early universe and later falls out of equilibrium as the expansion rate wins over the interaction rate. In a freeze-in model, the opposite idea is used: the interaction is so feeble that the dark matter never thermalizes at all. A feebly interacting massive particle, or FIMP, is instead populated slowly and sparsely by rare processes in the thermal bath. This calculator focuses on a standard benchmark channel in which a heavier bath particle B decays and occasionally creates the dark matter particle χ. Those tiny injections add up over cosmic time and leave a fixed final abundance once the universe cools below the bath-particle mass scale.

This FIMP freeze-in relic density page estimates the two quantities that usually matter most in a first-pass model scan. The first is the asymptotic yield Y_∞, which is the final comoving abundance of dark matter expressed as number density divided by entropy density. The second is the present-day density parameter Ω_χh², which is the number normally compared with the observed dark matter abundance near 0.12. Because the freeze-in abundance often scales directly with the tiny coupling responsible for production, even a compact analytic calculator like this can be surprisingly useful when you are trying to understand whether a hidden-sector model is plausibly in the right range.

Freeze-in dark matter has become increasingly important because it naturally describes sectors that are nearly invisible to conventional searches. When the dark sector talks to the Standard Model through an extremely weak portal, direct detection can be ineffective, collider signals can become long-lived or displaced, and the thermal history can look very different from the classic weakly interacting massive particle story. That makes the freeze-in mechanism both theoretically appealing and experimentally subtle. A good calculator should therefore do more than print a number. It should help you see which inputs matter most, which assumptions are being made, and how far the result can be trusted before a full Boltzmann-equation treatment is required.

How to use the FIMP freeze-in relic density calculator

This FIMP freeze-in relic density calculator is designed for a quick browser-based estimate, so the form stays short while each input still has a precise physical meaning. Enter the FIMP mass m_χ in GeV, the bath-particle mass m_B in GeV, the decay width Γ in GeV, the bath-particle internal degrees of freedom g_B, and the effective relativistic degrees of freedom g_* near the production era. Then press Compute Relic Density. Scientific notation such as 1e-20 is accepted, which is especially convenient because realistic freeze-in widths are often tiny.

Each of those inputs maps directly onto the analytic approximation used below. The FIMP mass m_χ determines how much mass density each produced dark matter particle contributes today. The bath-particle mass m_B sets the parent scale and, in this decay-dominated approximation, suppresses the yield through an inverse-square dependence. The decay width Γ controls how often B decays into channels containing χ, so it is commonly the most sensitive model parameter. The factor g_B counts internal states of the parent particle, while g_* summarizes how many relativistic species share the energy density of the universe around the temperature where production is most effective.

After you compute, the result panel reports Y_∞ and Ω_χh² and also labels the point as underabundant, overabundant, or roughly compatible with the observed dark matter density. That classification is only a fast interpretation aid, not a full viability verdict. A value near 0.12 suggests that the chosen parameters reproduce the observed abundance in this simplified single-channel decay picture. A much smaller value means this channel by itself does not make enough dark matter. A much larger value means the same setup would overproduce dark matter unless some other effect, such as dilution or a modified cosmological history, reduces the final abundance.

There are also useful scaling lessons built into the calculator. If you increase Γ by a factor of ten and leave the other inputs alone, the freeze-in yield and relic density also rise by a factor of ten in this approximation. If you keep the yield fixed but increase m_χ, then Ω_χh² increases proportionally because each particle is heavier. By contrast, making the bath particle heavier tends to suppress production because the formula carries an m_B⁻² dependence. These are exactly the trends researchers look for when they scan a model’s parameter space before doing a more exact numerical study.

The freeze-in decay formula for Y∞ and Ωχh²

This freeze-in decay formula section describes the approximation implemented by the calculator. The setup assumes that a bath particle B, which is itself in thermal equilibrium, occasionally decays into a final state containing the FIMP χ. In the regime where the coupling is tiny and inverse decays remain negligible, the final yield can be estimated analytically. The asymptotic yield Y_∞ = n_χ/s is

Formula: Y_∞ ≈ (135 g_B Γ M P_l) / (8 π^4 g_*^3/2 m_B^2)

$Y_{\infty }\approx \frac{135g_B\Gamma M{P}_l}{8{\pi }^4{g_{*}}^{\frac{3}{2}}{m_B}^2}$

This FIMP freeze-in expression assumes Maxwell–Boltzmann statistics and the usual radiation-dominated cosmology. The important ingredients are the parent’s internal degrees of freedom g_B, the bath-particle mass m_B, the decay width Γ, the plasma degrees of freedom g_*, and the Planck-scale normalization through M_Pl ≈ 1.22×10¹⁹ GeV. In the JavaScript calculation, these factors are collected into the numerical prefactor 0.173 so the browser can evaluate the analytic estimate directly and instantly.

Once the yield is known, the present-day relic density follows from

Formula: Ω_χ h^2 ≈ 2.742 × 10^8 m_χ / GeV Y_∞

${\Omega }_{\chi }{h}^2\approx 2.742\times {10}^8\frac{m_{\chi }}{\text{GeV}}{Y}_{\infty }$

This second relation converts a microscopic abundance into a cosmological density using the present-day entropy density and critical density. In plain language, the first formula tells you how many dark matter particles were produced relative to entropy, while the second tells you how much of the universe’s energy density those particles represent today. Together they explain the basic shape of freeze-in parameter scans: Γ and m_χ push the answer upward, while larger m_B and larger g_* pull it downward.

One subtle but important point is that the formula is intended for the true freeze-in regime. If Γ becomes too large, the FIMP population can approach equilibrium and the simple analytic result stops being reliable. That does not mean the calculation is useless; it means the interpretation changes. In practice, this quick estimate is most valuable as a first diagnostic: it tells you whether the required coupling is tiny, whether your chosen mass scales can plausibly hit Ω_χh² ≈ 0.12, and whether a more complete treatment is worth pursuing.

Worked example: a 1 GeV FIMP from 100 GeV bath-particle decays

This worked example uses parameters close to the defaults so the scaling is easy to follow. Suppose the FIMP mass is 1 GeV, the bath-particle mass is 100 GeV, the decay width is 1×10⁻²⁰ GeV, the parent has one internal degree of freedom, and the effective relativistic degrees of freedom are g_* = 100. When you compute the result, the yield is tiny, which is exactly what freeze-in predicts. Yet even a tiny yield can translate into a meaningful modern relic density once the particle mass and cosmological conversion factor are included.

Now vary one parameter at a time and the physics becomes intuitive very quickly. Lower Γ by four orders of magnitude and the yield drops by the same factor, so the model becomes strongly underabundant. Increase m_χ by a factor of ten at fixed yield and Ω_χh² rises by ten because each produced particle is heavier. Increase m_B while holding everything else fixed and the abundance falls because the analytic yield scales like 1/m_B². That inverse-square dependence is one of the fastest ways to suppress production in this decay-driven picture.

The table below offers a few benchmark points for the same freeze-in calculator. It is not meant as a substitute for a full scan, but it shows how quickly the output responds to the key parameters and why the decay width is often the first lever model builders reach for.

Those benchmarks illustrate the same lesson from another angle. Lower masses and tiny widths can leave you far below the cosmological target, while a heavier FIMP or a larger width can overshoot quickly. Because many freeze-in models live in parameter regions that are hard to probe experimentally, analytic examples like this remain useful for teaching, model triage, and back-of-the-envelope checks before one commits to a full numerical integration of coupled Boltzmann equations.

Interpretation and physical context for freeze-in dark matter

This freeze-in dark matter interpretation matters because a numerical match to Ω_χh² ≈ 0.12 is only one part of the story. In a standard decay-dominated freeze-in scenario, production is usually most efficient around temperatures of order m_B, and after that era the abundance effectively freezes into its final value as long as entropy is conserved. That makes the relic density comparatively insensitive to much of the later thermal history, which is one reason the mechanism is so attractive in hidden-sector model building.

The same feeble interactions that make FIMPs hard to detect also create distinctive phenomenology elsewhere. Parent particles or mediators can become long-lived at colliders, portals can be too weak for classic direct-detection signatures, and the dark sector can remain thermally decoupled from the visible sector throughout cosmic history. Sterile-neutrino models, Higgs-portal scalars, dark photons, supersymmetric states, and other exotic constructions can all realize some version of this behavior in appropriate corners of parameter space.

There is also an important conceptual difference between using this calculator as a teaching tool and using it as a model-building filter. As a teaching tool, it shows the hierarchy of sensitivities very clearly and makes the relationship between microscopic and cosmological quantities tangible. As a model-building filter, it tells you whether a proposal is plausibly in the right ballpark before you start worrying about details like momentum distributions, structure-formation limits, or multiple competing production channels. In other words, it is best thought of as a sharp first estimate rather than a final cosmological verdict.

Limitations and assumptions of this freeze-in estimate

This freeze-in estimate is intentionally simple, so it rests on assumptions that should be kept in view. First, it treats the dark matter abundance as being dominated by decays of a single thermal bath particle species. If scatterings, several parent states, threshold effects, or resonant enhancements matter, then the true relic density can shift significantly. Second, it assumes a standard radiation-dominated cosmological history with a reheating temperature high enough that the bath particle was thermally populated in the first place. If reheating never reached temperatures near m_B, the abundance could be much smaller than this formula predicts.

This freeze-in calculator also assumes entropy conservation after production ends and ignores late dilution from decays of heavy moduli, inflaton remnants, or other exotic sectors. It uses a fixed g_* input rather than a temperature-dependent table, which is often acceptable for order-of-magnitude reasoning but not for precision work. It further assumes the true freeze-in regime, meaning the coupling is weak enough that χ never approaches equilibrium. If that assumption fails, one must transition to a fuller Boltzmann treatment and possibly even a mixed freeze-in or freeze-out analysis.

Finally, a label such as underabundant, overabundant, or matching observed dark matter should not be confused with a complete model viability statement. A point that lands near 0.12 can still be ruled out by collider limits, stability issues, structure-formation constraints, or Big Bang nucleosynthesis. An underabundant point can still be interesting if the FIMP is only one component of the dark matter or if additional channels contribute later. The safest way to use this page is exactly what analytic calculators do best: as a transparent, rapid, and physically interpretable estimate that helps you decide what to investigate next.