Observational constraints on the primordial curvature power spectrum

Abstract

CMB temperature fluctuation 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, [1-7, 11]. Luminous red galaxies and galaxy clusters probe the matter power spectrum on overlapping scales (0.02 Mpc−1 ≲ k ≲ 0.7 Mpc−1; [10, 12-20]), while the Lyman-alpha forest reaches slightly smaller scales (0.3 Mpc−1 ≲ k ≲ 3 Mpc−1; [22]). These observations indicate that the primordial power spectrum is nearly scale-invariant with an amplitude close to 2 × 10−9, [5, 23-28]. These observations strongly support Inflation and motivate us to obtain observations and constraints reaching to smaller scales on the primordial curvature power spectrum and by implication on Inflation. We are able to obtain limits to much higher values of k ≲ 105 Mpc−1 and with less sensitivity even higher k ≲ 1019− 1023 Mpc−1 using limits from CMB spectral distortions and other limits on ultracompact minihalo objects (UCMHs) and Primordial Black Holes (PBHs). PBHs are one of the known candidates for the Dark Matter (DM). Due to their very early formation, they could give us valuable information about the primordial curvature perturbations. These are complementary to other cosmological bounds on the amplitude of the primordial fluctuations. In this paper, we revisit and collect all the published constraints on both PBHs and UCMHs. We show that unless one uses the CMB spectral distortion, PBHs give us a very relaxed bounds on the primordial curvature perturbations. UCMHs, on the other hand, are very informative over a reasonable k range (3 ≲ k ≲ 106 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 DM that could be PBHs in reasonable models. 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 compare the UCMH bounds coming from those experiments which are sensitive to the nature of the DM, such as γ-rays, Neutrinos and Reionization, with those which are insensitive to the type of the DM, e.g. the pulsar-timing as well as CMB spectral distortion. We explicitly show that they lead to comparable results which are independent of the type of DM. These bounds however do depend on the required initial density perturbation, i.e. δmin. It could be either a constant or a scale-dependent function. As we will show, the constraints differ by three orders of magnitude depend on our choice of required initial perturbations.