FIMP Freeze-In Relic Density Calculator

Introduction

Freeze-in dark matter is a useful alternative to the more familiar freeze-out picture. Instead of starting in thermal equilibrium with the hot plasma of the early universe and then dropping out of equilibrium as the universe expands, a feebly interacting massive particle, or FIMP, is never abundant enough to thermalize in the first place. Its interactions are so weak that the dark matter population is built up slowly from rare decays or scatterings of ordinary bath particles. This calculator focuses on one of the cleanest cases: a heavier particle B in the thermal bath decays and occasionally produces a dark matter particle χ. Over time, those tiny contributions accumulate into a final relic abundance.

This page estimates two quantities that are central in freeze-in studies. The first is the asymptotic yield, written as Y_∞, which measures the final number density of dark matter relative to the entropy density. The second is the present-day relic density parameter Ω_χh², which is the quantity usually compared with the observed dark matter abundance of about 0.12. Because the freeze-in abundance is often proportional to the tiny coupling that controls the decay width, this kind of estimate is especially helpful when exploring models with extremely weak portals to the Standard Model.

While the freeze-out mechanism of weakly interacting massive particles has long dominated dark matter phenomenology, freeze-in has become increasingly important because it naturally describes hidden sectors that are almost invisible to conventional searches. 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 are slowly produced through the decays or scatterings of bath particles. The production rate is initially tiny 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.

How to Use

To use the calculator, enter the physical inputs in GeV where appropriate and then press the compute button. The form is intentionally compact, but each field has a specific meaning. The FIMP mass m_χ sets the mass of the dark matter particle whose relic density you want to estimate. The bath particle mass m_B is the mass of the heavier particle that is assumed to be in thermal equilibrium and to decay into the FIMP. The decay width Γ controls how often those decays happen; in freeze-in applications it is usually extremely small. The quantity g_B counts the internal degrees of freedom of the parent bath particle, and g_* represents the effective number of relativistic degrees of freedom in the plasma around the temperature scale T ≈ m_B.

After you click Compute Relic Density, the calculator returns the estimated yield Y_∞ and the corresponding Ω_χh². It also gives a simple interpretation: underabundant, overabundant, or roughly consistent with the observed dark matter density. That classification is only a quick guide. A result near 0.12 suggests that the chosen parameters could reproduce the measured abundance in this simplified decay-dominated freeze-in picture. A much smaller value means the model would need additional production channels or different parameters. A much larger value means the chosen setup would produce too much dark matter unless some later dilution or nonstandard cosmological effect reduces it.

In practical terms, the most sensitive input is often the decay width. Because the yield scales linearly with Γ in this approximation, changing the width by a factor of ten changes the abundance by the same factor. The FIMP mass also enters linearly in the final relic density conversion, so heavier dark matter gives a larger Ω_χh² for the same yield. By contrast, increasing the bath particle mass suppresses the yield through an m_B⁻² dependence. This means that a heavier parent particle generally produces less dark matter if all other inputs are held fixed.

Formula

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 the entropy density, is given by

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 expression assumes Maxwell–Boltzmann statistics and neglects inverse decays, which is an excellent approximation when the coupling is so feeble that produced FIMPs do not significantly 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. In the JavaScript implementation, these constants are combined into the numerical prefactor 0.173, which is simply a compact way of evaluating the same analytic approximation.

Once Y_∞ is known, the present-day density parameter 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 convenient relation converts the yield into a dimensionless density using the present-day entropy density and critical density. In plain language, the first formula tells you how many dark matter particles are produced relative to entropy, and the second tells you how much cosmic mass density that yield corresponds to today. The calculator implements these formulas entirely in client-side JavaScript, so the page works immediately in the browser without external computation.

Worked Example

Suppose you choose a FIMP mass of 1 GeV, a bath particle mass of 100 GeV, a decay width of 1×10⁻²⁰ GeV, one internal degree of freedom for the parent particle, and g_* = 100. These are the default values in the form because they provide a simple benchmark. When you compute the result, the yield comes out to a very small number, as expected for freeze-in, but after multiplying by the dark matter mass and the cosmological conversion factor, the final Ω_χh² lands near the observed dark matter abundance. That makes this benchmark a useful reference point for understanding how the parameters interact.

Now imagine changing only one input at a time. If you lower Γ by four orders of magnitude, the yield and relic density also drop by four orders of magnitude, and the model becomes underabundant. If instead you keep Γ fixed but raise m_χ from 1 GeV to 10 GeV, the relic density increases by a factor of ten because each produced particle contributes more mass today. If you increase m_B while holding the other inputs fixed, the abundance falls because the parent particle is heavier and the analytic yield scales as 1/m_B². These trends are exactly what the calculator is designed to make visible.

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

These examples highlight the linear dependence on Γ and m_χ. Lowering the coupling, and therefore Γ, or lowering 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. For a student or researcher scanning a model, this kind of quick benchmark is often enough to identify whether a parameter region is promising before moving on to a full Boltzmann-equation treatment.

Interpretation and Physical Context

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 later thermal history, provided entropy is conserved after production ends. The mechanism can produce a wide spectrum of dark matter masses, from very light candidates to much heavier ones, simply by tuning the tiny interaction strength encoded here through the decay width. It also naturally leads to different momentum distributions depending on whether production is dominated by decays or by high-temperature scatterings, although this calculator is specifically built for the decay-dominated case.

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 often evades direct detection and many astrophysical constraints. Yet the same feebleness can make the parent or mediator particles long-lived, which is why collider searches for displaced vertices or detector-stable particles are often discussed in the same breath as freeze-in models. Hidden sectors, sterile neutrinos, scalar portals, supersymmetric states, and axion-like constructions can all exhibit freeze-in behavior in suitable regions of parameter space.

Understanding the freeze-in parameter space also helps with model building. For instance, a portal coupling between the Standard Model Higgs boson and a scalar FIMP can generate the observed relic density if the coupling is tiny enough that the dark matter never thermalizes. Similarly, a heavy Z′ boson or another exotic mediator can populate a dark sector through rare decays. This calculator does not replace a full phenomenological study, but it does provide a transparent first estimate that is often valuable when checking whether a proposed scenario is even in the right ballpark.

Limitations and Assumptions

This calculator is intentionally simple, so it is important to understand what is being assumed. First, it treats freeze-in as being dominated by decays of a single bath particle species. If scatterings, multiple parent particles, resonant effects, or threshold corrections are important, the true abundance can differ substantially. Second, it assumes a standard radiation-dominated cosmological history with sufficiently high reheating temperature, so that the bath particle was thermally populated and the usual analytic approximation applies. If reheating never reached temperatures near m_B, the abundance may be strongly suppressed.

Third, the calculation assumes that entropy is conserved after production and that there is no late dilution from decays of heavy fields or other exotic cosmological events. Fourth, it uses a fixed value of g_* rather than tracking its temperature dependence in detail. That is often adequate for order-of-magnitude work, but precision studies may need a more careful treatment. Fifth, the result is only meaningful in the true freeze-in regime, where the coupling is weak enough that the dark matter population never approaches equilibrium. If Γ is too large, the assumptions behind the formula break down and a freeze-out or mixed regime analysis becomes necessary.

There is also a practical modeling limitation: the calculator reports whether the result is underabundant, overabundant, or near the observed dark matter density, but that label should not be interpreted as a full viability test. A model that matches Ω_χh² can still be excluded by structure formation, Big Bang nucleosynthesis, collider bounds, or stability requirements. Conversely, an underabundant result may still be interesting if the FIMP is only one component of the dark matter or if additional production channels exist. In other words, this tool is best used as a fast analytic estimator and teaching aid, not as a substitute for a complete cosmological and particle-physics analysis.