Thermal Production of Gravitinos in the Early Universe
In supersymmetric theories, the gravitino is the spin-3/2 superpartner of the graviton, arising from the gauging of supersymmetry. Although its couplings are suppressed by the Planck scale, the vast densities and temperatures of the early universe can thermally produce an appreciable number of gravitinos. The abundance of these particles carries profound implications for cosmology and particle physics. If gravitinos are stable, they may constitute dark matter; if unstable, their late decays can disrupt nucleosynthesis or distort the cosmic microwave background. This calculator offers an order-of-magnitude estimate of the relic density of thermally produced gravitinos based on the reheating temperature after inflation and the gravitino mass.
The thermal production rate of gravitinos is governed by scattering processes involving gauge and gaugino fields in the hot plasma. Detailed calculations involve integrating Boltzmann equations with thermally averaged cross sections, yielding a yield that scales roughly linearly with the reheating temperature. A commonly used approximation, valid for temperatures well above the superpartner masses, is . This formula captures the leading dependence on the reheating temperature under the assumption of minimal supersymmetric spectra.
The present-day contribution to the critical density can then be estimated via , where m3/2 is in GeV and TR in GeV. This expression assumes that gravitinos are stable and that entropy production after reheating is negligible. Comparing the computed value to the observed dark matter density provides a quick check of whether a given parameter set yields overclosure or an acceptable dark matter candidate.
Gravitino cosmology is rich with scenarios. If the gravitino is not the lightest supersymmetric particle (LSP) and decays into other superpartners, its lifetime can be long due to the weak coupling. Decays occurring during or after Big Bang nucleosynthesis can alter the abundances of light elements or inject high-energy photons that distort the microwave background. These considerations place upper bounds on TR for unstable gravitinos, often around 106–109 GeV depending on the mass and decay channels. Conversely, if the gravitino is the LSP, it may serve as dark matter provided its relic density matches observations and it does not conflict with structure formation constraints.
The calculator’s result includes a qualitative classification: if exceeds 0.12, the parameter set overproduces gravitinos relative to the observed dark matter density. Values below this threshold are labeled subdominant, indicating that gravitinos would constitute only a fraction of dark matter. Such classifications are simplistic but provide guidance when scanning parameter space.
To offer context, the following table lists sample abundances for selected masses and reheating temperatures:
| m3/2 (GeV) | TR (GeV) | Ωh² |
|---|---|---|
| 1 | 10⁶ | 2.7×10⁻⁵ |
| 100 | 10⁹ | 0.27 |
| 1000 | 10¹⁰ | 2.7 |
The wide range illustrates how sensitively the relic density depends on both mass and reheating temperature. Adjusting TR by an order of magnitude scales the abundance by the same factor, underscoring why inflationary reheating is tightly constrained in supersymmetric cosmologies.
While the calculator focuses on thermal production, gravitinos can also arise from nonthermal processes such as the decay of heavier particles (e.g., inflatons, moduli, or scalar condensates). Nonthermal production can dominate in scenarios with low reheating temperatures or when certain couplings enhance the branching ratio into gravitinos. Accounting for these channels typically requires model-specific analysis beyond the scope of this simple estimate.
The theoretical foundations of gravitino production involve the interplay between supersymmetry breaking and supergravity. In the early universe, the goldstino component of the gravitino is effectively massless and participates in scatterings. As the universe cools and supersymmetry breaks, the gravitino acquires a mass and its interactions become suppressed. The decoupling temperature is usually so high that the yield remains frozen thereafter. The abundance therefore encodes information about both the high-energy reheating process and the supersymmetry-breaking scale.
From an observational standpoint, detecting gravitino dark matter directly is extraordinarily challenging due to its feeble interactions. Indirect searches might look for decay products if the gravitino is metastable on cosmological timescales. Alternatively, collider experiments like the LHC can search for signatures of gravitino production in decays of heavier superpartners, such as missing energy from a neutralino NLSP decay. However, connecting such observations to the cosmological abundance requires careful consideration of the production mechanisms encapsulated in this calculator.
The early literature on gravitino cosmology dates to the late 1970s and 1980s, when researchers realized that even extremely weakly interacting particles could have significant cosmological consequences. Subsequent refinements incorporated thermal field theory, precise cross sections, and the effects of phase transitions like the QCD crossover. Modern analyses also include lattice-determined degrees of freedom for the plasma and potential contributions from axions or other exotic sectors. Despite these complexities, the simple proportionality between yield and reheating temperature remains a useful rule of thumb, enabling quick estimates that guide more detailed studies.
In summary, this calculator translates the core physics into an accessible tool: input the gravitino mass and reheating temperature, and obtain an estimate of the relic density. Whether one is exploring models of supersymmetric dark matter, setting constraints on inflationary reheating, or studying the cosmic consequences of Planck-suppressed interactions, having a fast way to evaluate the gravitino abundance is invaluable. The lengthy explanation provided here aims to demystify the underlying assumptions and highlight the broader implications of gravitino production in the early universe.