Gravitino Thermal Abundance Calculator

Thermal gravitino abundance formula

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 commonly used approximation for the yield is

Y_3/2 ≈ 1.9 × 10⁻¹² (T_R / 10¹⁰ GeV).

This calculator uses that nearly linear dependence on T_R, which is the main reason reheating temperature has such a strong effect on the final abundance. 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 scaling relation directly from your inputs for m_3/2 and T_R.

MathML form of the gravitino abundance scaling

The core relation implemented numerically is:

In this expression, masses and temperatures are in GeV. The numerical prefactor collects the standard cosmological factors needed to convert a thermal yield into today’s density parameter, together with the usual gauge-interaction assumptions of the minimal setup.

How to interpret the gravitino abundance result

For the gravitino thermal abundance calculator, Ω_3/2 h² is easiest to read by comparing it with the observed dark matter density Ω_DM h² ≈ 0.12:

• Ω_3/2 h² > 0.12 (overproduction): the thermal estimate exceeds the dark matter budget in this simplified scenario. A viable model would then need some combination of a lower reheating temperature, entropy dilution, a different production channel, or a nonstandard cosmological history.
• Ω_3/2 h² ≈ 0.12 (dominant dark matter): thermally produced gravitinos could account for most of the dark matter, provided the rest of the model remains consistent with BBN, CMB, and structure-formation limits.
• Ω_3/2 h² ≪ 0.12 (subdominant): the thermal gravitino component would be only a slice of the dark matter abundance, so some other particle or field would need to make up the remainder.

Because the formula is linear in both m_3/2 and T_R, doubling either input doubles Ω_3/2 h². That makes it easy to see whether the reheating temperature or the gravitino mass is driving the final result in a given scenario.

Worked gravitino abundance example

As a concrete gravitino abundance check, 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, that reheating temperature would overproduce gravitino dark matter for this mass. A model builder might then:

• reduce the reheating temperature to lower the thermal gravitino yield,
• adjust the supersymmetric spectrum or couplings so that the effective production rate is smaller, or
• introduce late-time entropy production, such as decay of a heavy scalar, to dilute the gravitino density.

As a second gravitino abundance 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.

Gravitino abundance across common parameter regimes

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

Assumptions behind the gravitino estimate

The simplicity of the implemented gravitino formula comes with several important assumptions. When you use the output for phenomenology, treat it as an order-of-magnitude guide rather than a precision prediction.

• Minimal supersymmetric spectrum: the coefficients assume a particle content close to the minimal supersymmetric standard model, with standard gauge couplings and typical gaugino masses. Non-minimal spectra can shift the thermal production rate.
• High-temperature approximation: the formula assumes T_R is well above the masses of the supersymmetric particles involved in the scattering processes. If T_R is comparable to or below those masses, thermal production is suppressed and the linear scaling can fail.
• No late-time entropy production: the estimate assumes no substantial entropy injection after gravitino production, such as decay of heavy moduli or other long-lived fields. Such a process would dilute the final gravitino abundance.
• Standard cosmological history: the calculation presumes a radiation-dominated Universe after reheating with ordinary Friedmann–Robertson–Walker expansion. Exotic eras such as early matter domination or kination would change the Boltzmann evolution.
• Stable gravitino when interpreting as dark matter: reading Ω_3/2 h² as a dark matter abundance assumes the gravitino is effectively stable on cosmological timescales. If it decays, the relevant observables are instead its decay products and their impact on BBN and the CMB.
• Neglect of non-thermal production: inflaton decays, moduli decays, or other non-thermal sources are ignored here. In many models those sources can dominate over the thermal contribution estimated by this calculator.
• Order-of-magnitude accuracy: uncertainties in the high-scale spectrum, gauge couplings, and higher-order corrections can shift the abundance by factors of a few. Use the result as a quick check, not as a precision constraint.

Gravitino inputs and abundance notation

The calculator uses the following gravitino parameters and cosmological quantity:

• Gravitino mass m_3/2 (in GeV): the physical mass of the gravitino. Depending on the supersymmetry-breaking scheme, it can range from the keV scale to multi-TeV values.
• 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 shows whether thermally produced gravitinos could account for all, some, or too much of the dark matter in this simplified setup.

Typical parameter ranges for this calculator

The tool accepts any positive values of m_3/2 and T_R (in GeV), but the 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 phenomenology that this simple scaling does not capture.
• Reheating temperature: roughly 10⁶–10¹² GeV, with the assumption that T_R lies comfortably above the superpartner masses so production proceeds in a hot plasma.

Entering values far outside these ranges is allowed for exploration, but the approximations should be treated cautiously because the simple reheating-era formula may no longer hold.

Cosmological implications of the thermal gravitino abundance

The thermal gravitino abundance links inflationary reheating to low-energy supersymmetry. A high reheating temperature is often attractive for mechanisms such as thermal leptogenesis, but the same high temperature also boosts gravitino production. In practice, that tension is often expressed as an upper bound on T_R once a specific gravitino mass and superpartner spectrum are chosen.

If the gravitino is the lightest supersymmetric particle (LSP) and stable, one aims for Ω_3/2 h² ≈ 0.12 while staying compatible with structure formation and indirect constraints. If 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 usually places upper bounds on the initial thermal abundance, and therefore on T_R, which can range from about 10⁶ to 10⁹ GeV depending on the lifetime and decay channels.

Further reading on gravitino thermal production

For more detailed and model-specific treatments of gravitino thermal 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, through light-element abundances and spectral distortions.

This calculator is meant as a quick, transparent implementation of the standard approximate scaling used in those studies, not as a replacement for a dedicated numerical analysis.