The BICEP2 collaboration reports a detection of primordial cosmic microwave background (CMB) B-mode with a tensor-scalar ratio $r=0.20^{+0.07}_{-0.05}$ (68% C.L.). However, this result is in tension with the recent Planck limit, $r<0.11$ (95% C.L.), on constraining inflation models. In this Letter we consider an inflationary cosmology with a preceding nonsingular bounce which gives rise to observable signatures on primordial perturbations. One interesting phenomenon is that both the primordial scalar and tensor modes can have a step feature on their power spectra, which nicely cancels the tensor excess power on the CMB temperature power spectrum. By performing a global analysis, we obtain the 68% C.L. constraints on the parameters of the model from the Planck+WP and BICEP2 data together: the jump scale $log_{10}(k_{rm b}/{rm Mpc}^{-1})=-2.4pm0.2$ and the spectrum amplitude ratio of bounce-to-inflation $r_Bequiv P_{rm m} / A_{rm s} = 0.71pm0.09$. Our result reveals that the bounce inflation scenario can simultaneously explain the Planck and BICEP2 observations better than the standard $Lambda$CDM model, and can be verified by the future CMB polarization measurements.
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