*Summary:*

*Fei and Rui proposed to introduce non-minimal gravitational coupling of AD field to Ricci scalar and discuss its impacts on the Q-balls formation and the associated gravitational wave production.**They obtained new form of scalar potential for AD field in the einstein frame.**Through simulations, they showed that the AD condensate can fragment efficiently into Q-balls when the fluctuations produced by linear parametric resonance become significant and the dynamics become highly nonlinear.**They also found that unless the effects of non-minimal gravitational couplings are taken into account, the Q-balls can not form with positive K parameter.**Moreover, they discussed the associated gravitational wave (GW) productions as well as their dependences on the ξ parameter and showed that the peak frequency fP is not sensitive to the choices of ξ while the peak value of Ω (sys. gw, stat. 0)h² depends on the choices of ξ. Larger ξ will lead to larger peak value of Ω (sys. gw, stat. 0)h². The peak frequency for GW power spectrum lies around a few KHz with their choice of m3/2 = 100TeV. Therefore, the stochastic GW backgrounds can not be observed by the current and upcoming interferometer based GW experiments.*

Observations of the gravitational waves (GWs) from black hole mergers by the Advanced LIGO/VIGO detectors open a new era in astrophysics and cosmology. Many experiments plan to further explore GWs, including both the transient GW signals and stochastic GW backgrounds, in a broader range of frequencies and with more accuracy in the coming decades. The stochastic GW backgrounds could reveal certain interesting properties of the very early universe, including the information of the baryogenesis stage, because the relevant dynamics can be potential sources of the stochastic GW backgrounds.

The origin of the baryon asymmetry, which make up 5 percent of the total energy density of the universe, is still an unsolved puzzle today. Sakharov suggested that current baryon density might be understandable in terms of microphysical laws instead of some sort of initial condition. His famous three conditions include the baryon number violation, C and CP violation, the departure from thermal equilibrium. The coherent production of baryons or leptons by the Affleck-Dine (AD) fields can satisfy Sakharov’s conditions and act as a viable baryogenesis mechanism other than the electroweak baryogenesis, the leptogenesis etc. According to the AD mechanism, the AD field starts oscillating around its origin and gives rise to rotational motion when the Hubble parameter becomes as small as the AD scalar mass after inflation, making the baryon number of the universe. The instability of AD field oscillations under small perturbations, which are inevitably introduced by quantum fluctuations of the field, will drive the condensate to fragment into non-topological solitons called Q-balls. The existence and stability of such nontopological solitons are guaranteed by the conserved charge related to a global symmetry. Many numerical studies simulate the formation of Q-balls.

The formation of AD condensate is fairly generic, relying only on the assumptions of inflation and flat directions in supersymmetry (SUSY). SUSY is widely regarded as one of the most promising candidates for new physics beyond the standard model. SUSY gauge theories often possess a remarkable vacuum degeneracy at the classical level, which can be parameterized by some gauge invariant combinations of squarks and sleptons. The flat directions can be lifted by SUSY breaking terms and higher order terms in the superpotential. Such flat directions can act as the AD fields for baryogenesis.

It was shown that significant GWs can be emitted during the Q-balls formation associated with AD mechanism of baryogenesis because the formation of Q-balls is inhomogeneous and not spherical. The formation of Q-balls depend on the effective potential of the AD field, which had been discussed for different SUSY breaking types. On the other hand, it is possible for the AD field to have a direct coupling to the Ricci scalar R, which can change the shape of the effective potential and amend the GW signals. So, it is interesting to discuss the impact of such non-minimal couplings of AD fields to gravity in the AD mechanism and know the new predictions of this scenario and thats what Fei and Rui have discussed in their recent paper.

They proposed to introduce non minimal gravitational coupling of AD field to Ricci scalar and discuss its impacts on the Q-balls formation and the associated gravitational wave productions.

“The AD field can develop a large Vaccum expectation value (VEV) during inflation, and it starts to oscillate after inflation when the cosmic expansion rate becomes comparable to its mass. Soon after the onset of oscillations, the AD field experiences spatial instabilities and deforms into clumpy Q-ball. From the equations of motion for the homogeneous modes and the fluctuations, one can check if the fluctuations can grow exponentially so as to go nonlinear and eventually form Q-balls, given the explicit form of the AD scalar potential.”— wrote authors of the study

The non-linear property of the Q-ball formation and the subsequent evolution necessitates a numerical simulation. So, they used the public code HLATTICE to simulate numerically the fragmentation of AD fields and the formation of Q-balls with the potential of the form given below.

They also solved, the evolution of the equation of motion for the homogeneous mode and the small perturbations δR and δΘ numerically, on a three dimensional cubic N3 lattice with N = 128. In the HLATTICE package, the equations of motion are re-cast in a different form in order to make use of more accurate, stable symplectic integrators.

The initial small fluctuations of the AD condensate come dominantly from the primordial quantum fluctuations, which exited the horizon during inflation and re-entered the horizon afterwards. From the inflationary cosmology, it can be expected to be |δφ/φ| ∼ 10¯^{5}. The final amplitude of the gravity waves is proved to be independent of the size of the initial perturbation in the flat direction.

In addition, they discussed the Q-balls formation process and the emitted GWs with non-vanishing ξ, especially the most interesting K ≥ 0, ξ > 0 case. Their numerical results for fiducial points K = 0, ξ = 10 and K = 0.01, ξ = 10 are shown in Fig.1. In the simulation of this fiducial points, the gravitino mass m3/2 in the scalar potential is chosen to be 100 TeV and the initial value of the homogeneous mode Φ0 is chosen to be 10^{17} GeV, which may correspond to the flat direction lifted by n = 9. **They found that the AD condensate can fragment efficiently into Q-balls when the fluctuations produced by linear parametric resonance become significant and the dynamics become highly nonlinear.**

They showed the process of the Q-balls formation in the panels of Fig.1. From the panels, we can see the stages for AD fragmentation, the emerging of Q-balls and the further evolution of Q-balls, respectively. The maximum values of the energy density within the formed Q-balls are not large for positive K. **It can be understood that, unless the effects of non-minimal gravitational couplings are taken into account, the Q-balls can not form with positive K. **As such effects are not large, the maximum values of the energy density in the Q-balls should not be large either.

“We find that, with non-minimal gravitational coupling to AD field, Q-balls can successfully form even with the choice of non-negative K parameter for ξ > 0”— wrote authors of the study

They also showed that the fragmentation process is not isotropic and non-spherical motions of the condensate can generate a quadrupole moment, which will emit GWs during Q-balls formation. The corresponding GW productions associated with the fragmentation of AD field in the case K > 0 and K = 0 can reveal the information of evolution. The evolution of GWs as a function of the evolution time m3/2 t are shown in the upper and lower right panels of Fig.1, respectively.

Moreover, they have also given the associated GW productions as well as their dependences on the ξ parameter. They showed the form of the final-stage massive Q-balls and the frequencies of the GW power spectrum in the case K = −0.1 and ξ = 3 in Fig.2. The GW power spectrum for K = −0.1 and various value of ξ are also shown in Fig. 2. We can see that the peak frequency fP is not sensitive to the choices of ξ while the peak value of Ω (sys. gw, stat. 0)h² depends on the choices of ξ. Larger ξ will lead to larger peak value of Ω (sys. gw, stat. 0)h². The peak frequency for GW power spectrum lies around a few KHz with their choice of m3/2 = 100TeV. Therefore, the stochastic GW backgrounds can not be observed by the current and upcoming interferometer based GW experiments. It was shown in Shuang-Yong Zhou that the peak position of the GW power spectrum depends on the value of m3/2 . Larger values of m3/2 lead to higher GW frequencies. Choosing a lower value of m3/2 can shift the spectrum to lower frequency. However, they found that such a small-shifted GW power spectrum can still not be detected by the upcoming interferometer based GW experiments. As the gravitino mass is given by m3/2 = F/√ 3MP , low SUSY breaking scale F may cause low GW frequencies. If such stochastic GW signal are detected, it may give interesting information on SUSY breaking scale.

**Reference***: Fei Wang, Rui Wang, “Q-balls Formation and the Production of Gravitational Waves With Non-minimal Gravitational Coupling”, ArXiv, pp. 1-11, 2021. **https://arxiv.org/abs/2104.04682*

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