Dark energy and matter interacting scenario can relieve $H_0$ and $S_8$ tensions

In this work, we consider a new cosmological model (named $\tilde\Lambda$CDM) in which the vacuum energy interacts with matter and radiation, and test this model using the current cosmological observations. We find that this model can significantly relieve the $H_0$ tension, and at the same time it can also slightly reduce the $S_8$ tension, which cannot be easily observed in other cosmological models. Using the CMB+BAO+SN (CBS) data to constrain the model, we obtain the results of $H_0=70.6^{+1.4}_{-1.7}~\rm{km~s^{-1} Mpc^{-1}}$ and $S_8=0.820\pm 0.011$, and thus the $H_0$ and $S_8$ tensions are relieved to $1.28\sigma$ and $2.67\sigma$, respectively. However, in this case the $\tilde\Lambda$CDM model is not favored by the data, compared with $\Lambda$CDM. We find that when the $H_0$ and $S_8$ data are added into the data combination, the situation is significantly improved. In the CBS+$H_0$ case, we obtain the result of $H_0=72.2\pm 1.2$ ${\rm km~s^{-1}~Mpc^{-1}}$, which relieves the $H_0$ tension to $0.53\sigma$, and in this case the model is favored over $\Lambda$CDM. In the CBS+$H_0$+$S_8$ case, we get a synthetically best situation, $H_0=71.9\pm 1.1$ ${\rm km~s^{-1}~Mpc^{-1}}$ and $S_8=0.8071\pm 0.0099$, in which the $H_0$ and $S_8$ tensions are relived to $0.75\sigma$ and $2.09\sigma$, respectively. In this case, the model is most favored by the data. Therefore, such a cosmological model can greatly relieve the $H_0$ tension, and at the same time it can also effectively alleviate the $S_8$ tension.Comment: 7 pages, 4 figure