Gravitinos from Heavy Scalar Decay

TL;DR

The study analyzes gravitino production from the decay of a heavy scalar X that can dominate the early Universe, showing the gravitino-producing coupling is controlled by the F-term vacuum expectation value of X and deriving its scaling in both spontaneous and explicit SUSY breaking. The partial width for X → gravitino pair is $\Gamma_{3/2} = \frac{d_{3/2}^2}{288 \pi} \frac{M_X^3}{M_P^2}$, with $d_{3/2}$ linked to the X F-term; the gravitino yield is $Y_{3/2} = Y_{3/2}^X + Y_{3/2}^{\rm TH}$, where $Y_{3/2}^X \simeq \frac{3}{2} B_{3/2} \frac{T_R}{M_X}$ and $Y_{3/2}^{\rm TH} \simeq 1.1 \times 10^{-12} (T_R/10^{10}\,\text{GeV})$. Cosmological constraints from BBN, LSP overproduction, and warm dark matter severely constrain or exclude many regions for unstable gravitinos, particularly in moduli-like X scenarios, while stable gravitinos can still accommodate gravitino DM or warm DM windows with tight bounds on $B_{3/2}$ and the reheating temperature $T_R$. The results have significant implications for inflation model building, suggesting that reducing $d_{3/2}$ or increasing $d_{\rm tot}$ can render inflaton decays cosmologically viable and may help address the cosmological moduli problem in certain SUSY-breaking frameworks.

Abstract

Cosmological issues of the gravitino production by the decay of a heavy scalar field $X$ are examined, assuming that the damped coherent oscillation of the scalar once dominates the energy of the universe. The coupling of the scalar field to a gravitino pair is estimated both in spontaneous and explicit supersymmetry breaking scenarios, with the result that it is proportional to the vacuum expectation value of the scalar field in general. Cosmological constraints depend on whether the gravitino is stable or not, and we study each case separately. For the unstable gravitino with $M_{3/2} \sim$ 100GeV--10TeV, we obtain not only the upper bound, but also the lower bound on the reheating temperature after the $X$ decay, in order to retain the success of the big-bang nucleosynthesis. It is also shown that it severely constrains the decay rate into the gravitino pair. For the stable gravitino, similar but less stringent bounds are obtained to escape the overclosure by the gravitinos produced at the $X$ decay. The requirement that the free-streaming effect of such gravitinos should not suppress the cosmic structures at small scales eliminates some regions in the parameter space, but still leaves a new window of the gravitino warm dark matter. Implications of these results to inflation models are discussed. In particular, it is shown that modular inflation will face serious cosmological difficulty when the gravitino is unstable, whereas it can escape the constraints for the stable gravitino. A similar argument offers a solution to the cosmological moduli problem, in which the moduli is relatively heavy while the gravitino is light.

Figures (8)

• Figure 1: Contour plot of $Y_{3/2}$ in the plane of $M_X$ and $d_{3/2}$ when $\Gamma_X$ is given by Eq. (\ref{['eq:GAMX_MODULI']}) with $d_{\rm tot}=1$. The solid lines correspond to $Y_{3/2} = 10^{-16}$, $10^{-15}$, $10^{-14}$ and $10^{-13}$ from left to right, respectively.
• Figure 2: Contour plot of $Y_{3/2}$ in the plane of $d_{3/2} M_X$ and $T_R$. The solid lines correspond to $Y_{3/2} = 10^{-16}$, $10^{-15}$, $10^{-14}$ and $10^{-13}$, from inside to outside, respectively. The dotted line is the lower bound on $T_R$ from $B_{3/2} \le 1$ for $d_{3/2} = 10^{-10}$.
• Figure 3: Present abundance of the wino LSP $\Omega_{\widetilde{W}}h^2$ when $\Gamma_X$ is given by Eq. (\ref{['eq:GAMX_MODULI']}) with $d_{\rm tot}=1$. The thick and thin solid lines are for $M_{\widetilde{W}} =100$ and 300 GeV, respectively. The horizontal dot-dashed line is the present dark matter density $\Omega_{\rm dm} h^2 \simeq 0.105$.
• Figure 4: Upper bounds on $M_X$ in terms of $M_{3/2}$, when $\Gamma_X$ is given by Eq. (\ref{['eq:GAMX_MODULI']}) with $d_{\rm tot}=1$. The dotted lines correspond to the upper bounds from $\Omega_{3/2} h^2 \le \Omega_{\rm dm} h^2 = 0.105$ for $B_{3/2} = 10^{-2}$, $10^{-4}$ and $10^{-6}$ from left to right, respectively. The solid line is the bound from $\Omega_{3/2} \le \Omega_{\rm dm}$ for $B_{3/2} = 3 \times 10^{-4}$, and the gravitino dark matter becomes viable above this line. The dot-dashed line is the upper bound on $M_X$ from $\Omega_{3/2}^X \le 0.12 \Omega_{\rm dm}$ avoiding the warm dark matter constraint in addition to $\Omega_{3/2} \le \Omega_{\rm dm}$, when $B_{3/2}=10^{-2}$.
• Figure 5: Upper bounds on $T_R$ in terms of $M_{3/2}$. The dotted lines correspond to the upper bounds from $\Omega_{3/2} h^2 \le \Omega_{\rm dm} h^2 = 0.105$ for $d_{3/2} B_{3/2} = 10^{-2}$, $10^{-4}$, $10^{-6}$ and $10^{-8}$ from left to right, respectively. The solid line is the bound from $\Omega_{3/2} \le \Omega_{\rm dm}$ for $d_{3/2} B_{3/2} = 3 \times 10^{-4}$, and the gravitino dark matter becomes viable above this line shown as the shaded region. The dot-dashed line is the upper bound on $T_R$ from $\Omega_{3/2}^X \le 0.12 \Omega_{\rm dm}$ avoiding the warm dark matter constraint in addition to $\Omega_{3/2} \le \Omega_{\rm dm}$, when $d_{3/2} B_{3/2}=10^{-2}$.