Primordial Bounce-Inflation Scenario to Alleviate Cosmological Tensions and Lensing Anomaly

We put forward a primordial scenario to alleviate cosmological tensions, i.e. Hubble ($H_0$) tension and $S_8$ tension. Based on flat $\Lambda$CDM, the Bounce-Inflation (BI) scenario gives the results that $H_0 = 68.60^{+0.40}_{-0.45} \, \text{km}/\text{s}/\text{Mpc}$, $S_8 = 0.806 \pm 0.011$ by using \texttt{Planck 2018} data sets and $H_0 = 68.96 \pm 0.38 \, \text{km}/\text{s}/\text{Mpc}$, $S_8 = 0.797\pm 0.010$ by using \texttt{Planck 2018} + \texttt{SPT3G} data sets. These reduce the cosmological tensions slightly. We also take an extended $\Lambda$CDM model into account, $\Lambda$CDM (BI)+$A_L$, where $A_L$ is the gravitational lensing amplitude. The results are $H_0 = 69.38 \pm 0.49 \, \text{km}/\text{s}/\text{Mpc}$, $S_8 = 0.774 \pm 0.014$ fitted by \texttt{Planck 2018} data sets and $H_0 = 69.49 \pm 0.45 \, \text{km}/\text{s}/\text{Mpc}$, $S_8 = 0.771^{+0.013}_{-0.012}$ fitted by \texttt{Planck 2018} + \texttt{SPT3G} data sets, which reduce the Hubble tension to $\sim 3\sigma$ level and show no $S_8$ tension. The $A_L \approx 1.1$ is smaller than the result of the inflation scenario with a constraint of \texttt{Planck 2018} data sets. Besides, the spectral index of the bounce-inflation scenario $n_s$ is about $0.98$, with a trend to the Harrison-Zel'dovich spectrum.