Gravitino Freeze-In Dark Matter with an Additional Scalar Field

Technical Summary: Gravitino Freeze-In Dark Matter with an Additional Scalar Field

Problem Statement
The gravitino is a leading candidate for freeze-in dark matter (DM), where its relic abundance is generated through rare scatterings and decays in the thermal bath rather than thermal freeze-out. In standard cosmology, the gravitino abundance ($\Omega_{3/2}h^2$) depends critically on the reheating temperature ($T_{reh}$) and supersymmetry-breaking parameters, specifically the universal gaugino mass ($M_{1/2}$) and the gravitino mass ($m_{3/2}$).

A significant phenomenological tension arises in this framework:

1. Collider Constraints: Future collider searches are expected to push the lower bounds on $M_{1/2}$ higher (e.g., from $\sim 1$ TeV to $\sim 2$ TeV or more).
2. Cosmological Requirements: Successful thermal leptogenesis, a mechanism to explain the baryon asymmetry of the universe, typically requires $T_{reh} \gtrsim 2 \times 10^9$ GeV.
3. The Conflict: In the standard radiation-dominated scenario, the maximum reheating temperature consistent with the observed DM abundance ($T_{reh}^{peak}$) decreases as $M_{1/2}$ increases. Consequently, higher collider bounds on $M_{1/2}$ force $T_{reh}^{peak}$ down, potentially below the threshold required for thermal leptogenesis. Furthermore, any additional source of gravitino production would further lower this limit.

Methodology
To address this tension, the authors investigate a non-standard cosmological scenario where the thermal bath is supplemented by an additional scalar field, $\phi$. The study adopts a model-independent approach, treating $\phi$ phenomenologically without specifying its microscopic origin (e.g., moduli, saxions, or hidden-sector scalars).

The methodology involves:

• Cosmological Framework: The universe is modeled with three components: radiation, gravitinos, and the scalar $\phi$. The evolution is governed by coupled Boltzmann equations for the energy densities of these components.
• Phenomenological Parameters: The dynamics of $\phi$ are parametrized by:
  • Initial energy density ratio at reheating: $r_\phi \equiv \rho_\phi(T_{reh})/\rho_R(T_{reh})$.
  • Equation-of-state parameter: $w_\phi$ (ranging from matter-like $w_\phi=0$ to kination-like $w_\phi=1$).
  • Decay width: $\Gamma_\phi$ (determining the lifetime and the onset of entropy injection).
• Numerical Analysis: The authors solve the Boltzmann equations numerically from $T_{reh}$ down to the present epoch. They calculate the modified gravitino yield ($Y_{3/2}^\phi$) and relic density ($\Omega_{3/2}^\phi h^2$) for various benchmark values of $M_{1/2}$ ($1, 2, 5, 10$ TeV).
• Dilution Factor: A key metric is the dilution factor $\Delta_\phi = \Omega_{3/2}/\Omega_{3/2}^\phi$, comparing the standard scenario to the modified one. $\Delta_\phi > 1$ indicates suppression (dilution), while $\Delta_\phi < 1$ indicates enhancement.

Key Contributions and Results

1. Equation of State Dependence:

   • Matter-like ($w_\phi < 1/3$): If the scalar behaves like matter (or near-matter, e.g., $w_\phi = 0.1$), its energy density redshifts slower than radiation. It can temporarily dominate the expansion history before decaying. The subsequent decay injects significant entropy into the thermal bath, diluting the previously generated gravitino abundance.
   • Kination-like ($w_\phi > 1/3$): If the scalar behaves like kination ($w_\phi = 1$) or radiation ($w_\phi = 1/3$), it redshifts faster than or similarly to radiation. In these regimes, the modification of the Hubble expansion rate during gravitino production leads to an enhancement of the gravitino abundance rather than dilution.

2. Impact on Reheating Temperature ($T_{reh}^{peak}$):

   • Dilution Regime: For matter-like scenarios with small decay widths ($\Gamma_\phi \lesssim 10^{-16}$ GeV) and sufficient initial abundance ($r_\phi$), the dilution factor can reach $\Delta_\phi \sim 10^7 - 10^8$. This allows the reheating temperature to be significantly higher while maintaining the correct DM abundance. For $M_{1/2} = 1$ TeV, $T_{reh}^{peak}$ can increase by nearly two orders of magnitude (e.g., from $\sim 10^9$ GeV to $\sim 10^{11}$ GeV or higher).
   • Enhancement Regime: For kination-like scenarios ($w_\phi = 1$), the gravitino abundance is enhanced, forcing $T_{reh}^{peak}$ to be lower than in the standard scenario to satisfy the DM constraint.

3. Parameter Space Constraints:

   • The authors impose a conservative upper bound $T_{reh}^{peak} \lesssim 10^{16}$ GeV, as higher values approach the energy scale of inflation and challenge the validity of the effective cosmological description.
   • In the exact matter-like case ($w_\phi = 0$), the dilution is so efficient that almost the entire explored parameter space yields $T_{reh}^{peak} > 10^{16}$ GeV, rendering it phenomenologically excluded in this specific analysis.
   • For near-matter cases ($w_\phi = 0.1$), viable regions exist where $T_{reh}^{peak}$ is elevated but remains below $10^{16}$ GeV, provided $\Gamma_\phi$ is not too small or $r_\phi$ is not too large.

Significance and Claims
The paper claims that the presence of an additional scalar component during the freeze-in era can substantially alter the cosmological predictions for gravitino dark matter. Specifically:

• Alleviating Tension: By inducing entropy dilution, the non-standard scenario allows for significantly larger reheating temperatures consistent with the observed DM abundance. This offers a potential resolution to the tension between stringent future collider bounds on gaugino masses and the high reheating temperatures required for thermal leptogenesis.
• Robustness: The effect is driven by general thermodynamic and expansion history arguments (redshift behavior and entropy injection) rather than specific model details, making the conclusions applicable to a broad class of supergravity and string-inspired frameworks.
• Dual Nature: The study highlights that the impact of extra scalars is not universally dilutive; depending on the equation of state, they can either relax or tighten the constraints on the reheating temperature.

The authors conclude that while specific microscopic realizations require further study, the general mechanism of entropy dilution by a long-lived scalar provides a viable pathway to accommodate high-scale reheating in gravitino DM scenarios.