Constraints on the primordial curvature power spectrum at small scales between $3\times {10}^{18}$ and $4.5\times {10}^{21}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$

The primordial curvature power spectrum ${\mathcal{P}}_{\mathcal{R}}$ has been measured with high precision on large scales ${10}^{-4}\lesssim k\lesssim 3\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$ based on observations of the cosmic microwave background, Lyman-$\alpha$ forest and large scale structure. On small scales $3\lesssim k\lesssim {10}^{23}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$, constraints are primarily derived from studies on primordial black holes (PBHs). In particular, for very small scales ${10}^{17}\lesssim k\lesssim {10}^{23}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$, current limits come exclusively from investigations of the lightest supersymmetric particles produced by PBH radiation and the stable Planck-mass relics after their evaporation. Recent findings also indicate that the evaporation of light PBHs ($M_{\mathrm{P}\mathrm{B}\mathrm{H}}\lesssim {10}^9\mathrm{g}$) can modify the expansion rate of the Universe and the baryon-to-photon ratio, thereby affecting the primordial abundance of light nuclei. Moreover, it has been proposed that the ``memory burden’’ effect can slow down the mass loss rate of black holes, allowing light PBHs to survive until the present day. Based on recent theoretical advancements in black hole physics and existing constraints on the initial mass fraction of light PBHs with masses ${10}^3\lesssim M_{\mathrm{P}\mathrm{B}\mathrm{H}}\lesssim 2\times {10}^9\mathrm{g}$, we derive new constraints on ${\mathcal{P}}_{\mathcal{R}}$ on small scales $3\times {10}^{18}\lesssim k\lesssim 4.5\times {10}^{21}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$, a regime that has been underexplored in previous literature.