Constraints on the small scale curvature perturbation using Planck-2015 data

The particles emitted from PBHs through the Hawking radiation have interactions with the particles present in the Universe. Due to the interactions, the evolution of the intergalactic medium (IGM) is changed and the changes have imprints on the anisotropies of the cosmic microwave background (CMB). In this paper, we focus on the PBHs with the lifetime in the range of $10^{13}s \lesssim \tau_{\rm PBH} \lesssim 10^{17}s$, corresponding to the mass range of $2.8\times 10^{13}\mathrm{g} \lesssim M_{\mathrm{PBH}} \lesssim 2.5\times 10^{14}\mathrm{g}$. We update the constraints on the initial mass fraction of PBHs using the Plank-2015 data. We find that the optimistic upper limits are $4\times 10^{-29}\lesssim \beta(M_{\mathrm{PBH}}) \lesssim 5\times 10^{-28}$, depending on the mass of PBH. The formation of PBHs is related to the primordial curvature perturbations. Therefore, using the constraints on the initial mass fraction of PBHs, we get the upper limits on the power spectrum of primordial curvature perturbation. For the investigated mass range of PBHs, corresponding to the range of scales $8.9 \times 10^{15}\ \mathrm{Mpc^{-1}}\lesssim k \lesssim 2.8 \times 10^{16}\ \mathrm{Mpc^{-1}}$, we find that the upper limits change slightly with a value of $\mathcal{P}_\mathcal{R}(k) \sim 0.0045$, and the limits are slightly stronger compared with the previous results.