# Observational Constraints on the Primordial Curvature Power Spectrum

## TL;DR

This paper reviews observational constraints on the primordial curvature power spectrum across a wide range of scales, emphasizing the roles of UCMHs and PBHs in probing inflationary models and dark matter composition.

## Contribution

It consolidates existing bounds on primordial fluctuations from various astrophysical observations, highlighting the significance of UCMHs in constraining small-scale power spectrum and PBHs as dark matter candidates.

## Key findings

- UCMHs provide strong bounds on the small-scale primordial spectrum.
- PBHs offer relaxed constraints unless spectral distortions are considered.
- UCMH bounds are consistent across different dark matter models.

## Abstract

CMB observations provide a precise measurement of the primordial power spectrum on large scales, corresponding to wavenumbers $10^{-3}$ Mpc$^{-1}$ < k < 0.1 Mpc$^{-1}$, [1-8]. Luminous red galaxies and galaxy clusters probe the matter power spectrum on overlapping scales (0.02 Mpc$^{-1}$ < k < 0.7 Mpc$^{-1}$ [9-18]), while the Lyman-alpha forest reaches slightly smaller scales (0.3 Mpc$^{-1} < k < 3$ Mpc$^{-1}$; [19]). These observations indicate that the primordial power spectrum is nearly scale-invariant with amplitude close to $2 \times 10^{-9}$, [5, 20-25]. They also strongly support Inflation and motivate us to obtain constraints reaching to smaller scales on the primordial curvature power spectrum and by implication on Inflation. One could obtain limits to much higher values of $k < 10^5$ Mpc$^{-1}$ and with less sensitivity even higher to $k < 10^{19}- 10^{23}$ Mpc$^{-1}$ using limits from CMB spectral distortions(SD)and on ultracompact minihalo objects(UCMHs)and Primordial Black Holes(PBHs). In this paper, we revisit and collect all the known constraints on both PBHs and UCMHs. We show that unless one uses SD, PBHs give us very relaxed bounds on the primordial curvature perturbations. UCMHs are very informative over a reasonable $k$ range($3 < k < 10^6$ Mpc$^{-1}$)and lead to significant upper-bounds on the curvature spectrum. We review the conditions under which the tighter constraints on the UCMHs could imply extremely strong bounds on the fraction of Dark Matter that could be PBHs. Failure to satisfy these conditions would lead to over production of the UCMHs, which is inconsistent with the observations. Therefore, we can almost rule out PBH within their overlap scales with the UCMHs. We consider the UCMH bounds from experiments such as $\gamma$-rays, Neutrinos, Reionization, pulsar-timing and SD. We show that they lead to comparable results independent of the form of DM.