Reconstructing a non-linear interaction in the dark sector with cosmological observations

In this work we model two non-linear directly interacting scenarios in dark sector of the universe with the dimensionless parameter $\alpha$ and $\beta$, which dominate the energy exchange between dark energy and dark matter. The central goal of this investigation is to research the interacting model and discuss the cosmological implications based on the current observational datasets. The class of the interaction is generally characterized by a coupling function $Q\propto H(z)\rho_x$, $x$ denotes the energy density of dark matter or dark energy. The constrained results we obtained indicate that the direct interaction in cosmic dark sector is favored by various observational data and the key effects on CMB power spectrum and linear matter power spectrum appear compared to $\Lambda$CDM standard paradigm. Finally, we discuss in depth the effect of different neutrino mass hierarchy on matter power spectrum and the variation of the ratio of CMB temperature power spectrum $C_{\ell}^{TT}$ and matter power spectrum $P(k)$ when the $\Delta N_{eff}= N-3.046$ from 0.5 to 2, respectively.