Non-minimal Coupling Inflation and Dark Matter under the $\mathbb{Z}_{3}$ Symmetry

We study the cosmological inflation and dark matter (DM) in a unified way within a $Z_3$ complex scalar model. The real and imaginary parts of the complex scalar act as the inflaton and DM respectively. The slow-rolling inflation with non-minimal coupling in both the metric and Palatini formalisms can be realized. We examine the whole parameters space by fully considering the theoretical and experimental constraints. We find that in the low-energy scale, the DM relic density and the DM-nucleon direct scattering experiments favor the mixing angle $|\theta| \lesssim 0.25$, the DM mass $m_\chi \gtrsim 80\rm{GeV}$, and the mass of Higgs-like scalar $m_{h_2} \gtrsim 300\rm{GeV}$. In the high-energy scale, after further considering the cosmological constraints of the scalar spectral index and the tensor-to-scalar ratio for the two forms of inflation, the scalar spectral indices are both $\sim 0.965$, the non-minimum coupling coefficients are $\sim 10^4$ and $\sim 10^9$, and the tensor-to-scalar ratios are $\sim 10^{-3}$ and $\lesssim 10^{-11}$ respectively, which suggests that the inflation under the two formalisms can be distinguished by measuring the tensor-to-scalar ratio with higher precision.