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 $\mathrm{\Lambda }$CDM, the Bounce-Inflation (BI) scenario gives the results that $H_0=68.60_{-0.45}^{+0.40}\text{km}\mathrm{/}\text{s}\mathrm{/}\text{Mpc}$, $S_8=0.806\pm 0.011$ by using $\mathtt{\text{Planck}}\mathtt{\text{2018}}$ data sets and $H_0=68.96\pm 0.38\text{km}\mathrm{/}\text{s}\mathrm{/}\text{Mpc}$, $S_8=0.797\pm 0.010$ by using $\mathtt{\text{Planck}}\mathtt{\text{2018}}$ + $\mathtt{\text{SPT3G}}$ data sets. These reduce the cosmological tensions slightly. We also take an extended $\mathrm{\Lambda }$CDM model into account, $\mathrm{\Lambda }$CDM (BI)+$A_L$, where $A_L$ is the gravitational lensing amplitude. The results are $H_0=69.38\pm 0.49\text{km}\mathrm{/}\text{s}\mathrm{/}\text{Mpc}$, $S_8=0.774\pm 0.014$ fitted by $\mathtt{\text{Planck}}\mathtt{\text{2018}}$ data sets and $H_0=69.49\pm 0.45\text{km}\mathrm{/}\text{s}\mathrm{/}\text{Mpc}$, $S_8=0.771_{-0.012}^{+0.013}$ fitted by $\mathtt{\text{Planck}}\mathtt{\text{2018}}$ + $\mathtt{\text{SPT3G}}$ 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 $\mathtt{\text{Planck}}\mathtt{\text{2018}}$ 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.