Gravitino Thermal Abundance Calculator

Overview

This calculator estimates the thermal relic abundance of gravitinos produced in the hot early Universe after inflation. In supersymmetric theories, the gravitino is the spin-3/2 superpartner of the graviton, with interactions suppressed by the (reduced) Planck scale. Despite these weak couplings, the extremely high temperatures after reheating can generate an appreciable number density of gravitinos through scattering processes in the plasma.

The resulting gravitino abundance has important implications for cosmology and particle physics. If the gravitino is stable and sufficiently abundant, it can serve as a dark matter candidate. If it is unstable with a long lifetime, its decays can disrupt Big Bang nucleosynthesis (BBN) or leave imprints in the cosmic microwave background (CMB). This tool focuses on the simplest thermal production channel and provides an order-of-magnitude estimate of the present-day density parameter Ω_3/2 h² in terms of the gravitino mass and reheating temperature.

Key quantities and notation

The calculator works with the following parameters and cosmological quantity:

• Gravitino mass m_3/2 (in GeV): the physical mass of the gravitino. Depending on the supersymmetry-breaking scheme, this can range from the keV scale up to multi-TeV.
• Reheating temperature T_R (in GeV): the characteristic temperature of the Universe at the end of inflationary reheating, when the plasma first reaches approximate thermal equilibrium.
• Relic abundance Ω_3/2 h²: the present-day density of gravitinos divided by the critical density, multiplied by the squared reduced Hubble parameter h (where H₀ = 100 h km s⁻¹ Mpc⁻¹).

The observed dark matter abundance is approximately Ω_DM h² ≈ 0.12. Comparing the calculated Ω_3/2 h² to this value indicates whether thermally produced gravitinos could account for all, some, or too much of the dark matter in this simplified scenario.

Formulas used in the calculator

The thermally produced gravitino yield Y_3/2 is defined as the ratio of number density to entropy density, Y_3/2 = n_3/2 / s. For a minimal supersymmetric particle content with reheating temperatures well above the superpartner masses, a widely used approximation for the yield is

Y_3/2 ≈ 1.9 × 10⁻¹² (T_R / 10¹⁰ GeV).

This expression captures the leading, roughly linear dependence on T_R from gauge and gaugino scattering processes. The present-day contribution to the critical density can then be written as

Ω_3/2 h² ≈ 0.27 (m_3/2 / 100 GeV) (T_R / 10¹⁰ GeV).

The calculator evaluates this simple scaling formula using your inputs for m_3/2 and T_R.

MathML representation

The core relation implemented numerically is:

In this expression, masses and temperatures are in GeV. The numerical prefactor encompasses standard cosmological parameters (entropy density today, critical density, etc.) and details of the gauge interactions in a minimal supersymmetric model.

How to interpret the result

The output Ω_3/2 h² is a dimensionless number. To understand its meaning, compare it to the observed dark matter value Ω_DM h² ≈ 0.12:

• Ω_3/2 h² > 0.12 (overproduction): in this simple picture, thermally produced gravitinos would exceed the observed dark matter density. Such a parameter choice typically requires additional physics (entropy dilution, alternative production mechanisms, or modified cosmology) or is disfavored.
• Ω_3/2 h² ≈ 0.12 (dominant dark matter): gravitinos produced thermally could make up most of the dark matter. In a more detailed model, one would need to check consistency with structure formation, BBN, and CMB constraints.
• Ω_3/2 h² ≪ 0.12 (subdominant): gravitinos would contribute only a fraction of the dark matter, with the rest coming from other components (such as another weakly interacting massive particle or an axion-like field).

The scaling with m_3/2 and T_R is linear, so doubling the reheating temperature or doubling the gravitino mass doubles the predicted abundance. This makes it straightforward to see how sensitive a given scenario is to changes in these parameters.

Worked example

As an illustration, consider a gravitino with mass m_3/2 = 100 GeV and a reheating temperature T_R = 10¹⁰ GeV. Plugging these into the formula gives

Ω_3/2 h² ≈ 0.27 (100 GeV / 100 GeV)(10¹⁰ GeV / 10¹⁰ GeV) = 0.27.

This value is more than twice the observed dark matter density. In the simplest cosmological history with a stable gravitino, such a high reheating temperature would overproduce gravitino dark matter for this mass. A model builder might then:

• reduce the reheating temperature to lower the abundance,
• adjust the supersymmetric spectrum or couplings so that the effective yield is smaller, or
• introduce late-time entropy production (for example from decays of a heavy scalar) to dilute the gravitino density.

As a second example, fix T_R = 10⁹ GeV and m_3/2 = 10 GeV. The formula gives

Ω_3/2 h² ≈ 0.27 (10 / 100)(10⁹ / 10¹⁰) = 0.27 × 0.1 × 0.1 = 2.7 × 10⁻³.

Here the thermally produced gravitinos contribute only a few percent of the dark matter density, so they would be a subdominant component.

Typical parameter ranges

The tool accepts any positive values of m_3/2 and T_R (in GeV), but the underlying approximation is most relevant in the following broad ranges:

• Gravitino mass: from roughly keV to multi-TeV. Very light (≪ keV) or extremely heavy ( ≫ 100 TeV) gravitinos involve additional phenomenological considerations not captured by this simple scaling.
• Reheating temperature: roughly 10⁶–10¹² GeV, with the assumption that T_R lies comfortably above the superpartner masses so that production proceeds in a fully relativistic plasma.

Entering values far outside these ranges is allowed for exploration but should be interpreted cautiously, as the approximations may not hold.

Comparison of regimes

The table below summarizes how different combinations of m_3/2 and T_R qualitatively affect the thermal gravitino abundance in this simplified framework.

Assumptions and limitations

The simplicity of the implemented formula comes with a number of important assumptions and caveats. When using the output for phenomenological studies, keep in mind that the estimate is only intended as an order-of-magnitude guide.

• Minimal supersymmetric spectrum: the numerical coefficients assume a particle content close to the minimal supersymmetric standard model, with standard gauge couplings and typical gaugino masses. Non-minimal spectra can modify the production rate.
• High-temperature approximation: the formula assumes T_R is significantly larger than the masses of the supersymmetric particles participating in the scattering processes. If T_R is comparable to or below these masses, the production rate is suppressed and the linear scaling in T_R can break down.
• No late-time entropy production: it is assumed that there is no substantial injection of entropy after gravitino production (for example from decays of heavy moduli or other long-lived fields). Such entropy release would dilute the final gravitino abundance.
• Standard cosmological history: the estimate presumes a radiation-dominated Universe after reheating with the usual Friedmann–Robertson–Walker dynamics, and no exotic expansion phases (such as early matter domination or kination) that would alter the Boltzmann evolution.
• Stable gravitino when interpreting as dark matter: treating Ω_3/2 h² as a dark matter abundance assumes that the gravitino is effectively stable on cosmological timescales. If the gravitino decays, the relevant constraints come instead from its decay products and their impact on BBN and the CMB.
• Neglect of non-thermal production: additional gravitino production mechanisms (for example from inflaton or moduli decays) are ignored. In many models such non-thermal sources can dominate over the thermal contribution estimated here.
• Order-of-magnitude accuracy: uncertainties in the exact superpartner spectrum, gauge couplings at high scales, and higher-order corrections can change the predicted abundance by factors of a few. The calculator should not be used as a precision tool.

Cosmological implications

The thermal gravitino abundance connects inflationary physics to low-energy supersymmetry. A high reheating temperature is often favored for mechanisms such as thermal leptogenesis, but this simultaneously enhances gravitino production. This tension can be phrased as an upper bound on T_R once a specific gravitino mass and spectrum are chosen.

If the gravitino is the lightest supersymmetric particle (LSP) and stable, one aims for Ω_3/2 h² ≈ 0.12 while remaining compatible with structure formation and indirect constraints. If, instead, the gravitino is heavier and unstable, its decays into lighter superpartners and Standard Model particles can disrupt BBN or distort the CMB. In that case, cosmology typically imposes upper bounds on the initial abundance, and thus on T_R, which can range from about 10⁶ to 10⁹ GeV depending on the lifetime and decay channels.

Further reading

For more detailed and model-specific treatments of thermal gravitino production and cosmological constraints, see for example:

• Reviews on gravitino dark matter and cosmology that derive the thermal yield using full Boltzmann equations.
• Studies of reheating and leptogenesis that discuss the interplay between high reheating temperatures and gravitino overproduction.
• BBN and CMB analyses constraining late-decaying particles, including unstable gravitinos, via light-element abundances and spectral distortions.

The calculator is intended as a quick, transparent implementation of the standard approximate scaling used in these works, not as a substitute for a dedicated numerical analysis.