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

Abstract

The primordial curvature power spectrum ${\mathcal{P}}_{\mathcal{R}}$ has been measured with high precision on large scales $10^{-4}\lesssim k\lesssim 3\mathrm{Mp}{\mathrm{c}}^{-1}$ based on observations of the cosmic microwave background, Lyman-$\alpha$ forest and large scale structure. On small scales $3\lesssim k\lesssim 10^{23}\mathrm{Mp}{\mathrm{c}}^{-1}$, constraints are primarily derived from studies on primordial black holes (PBHs). In particular, for very small scales $10^{17}\lesssim k\lesssim 10^{23}\mathrm{Mp}{\mathrm{c}}^{-1}$, 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 ($M_{\mathrm{PBH}}\lesssim 10^9\mathrm{g}$) 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 today. Based on recent theoretical advancements in black hole physics and existing constraints on the initial mass fraction of light PBHs with masses $10^3\lesssim M_{\mathrm{PBH}}\lesssim 2\times 10^9\mathrm{g}$, and especially the recent constraints on memory-burdened PBHs, we derive new and tighter upper limits on ${\mathcal{P}}_{\mathcal{R}}$ on small scales $3\times 10^{18}\lesssim k\lesssim 4.5\times 10^{21}\mathrm{Mp}{\mathrm{c}}^{-1}$, a regime that has been underexplored in previous literature. These constraints are derived under the specific assumption that the memory burden effect activates after the PBH loses half of its initial mass and subsequently halts further evaporation, and different assumptions on the memory burden parameters would lead to modified limits.

Comments: 7 pages, 2 figures. Added discussion and related formulas. Accepted for publication in EPJ Plus (European Physical Journal Plus)