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Register a new userReconstruction of the primordial power spectrum of curvature perturbations using multiple data sets
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Detailed knowledge of the primordial power spectrum of curvature perturbations is essential both in order to elucidate the physical mechanism (`inflation) which generated it, and for estimating the cosmological parameters from observations of the cosmic microwave background and large-scale structure. Hence it ought to be extracted from such data in a model-independent manner, however this is difficult because relevant cosmological observables are given by a convolution of the primordial perturbations with some smoothing kernel which depends on both the assumed world model and the matter content of the universe. Moreover the deconvolution problem is ill-conditioned so a regularisation scheme must be employed to control error propagation. We demonstrate that `Tikhonov regularisation can robustly reconstruct the primordial spectrum from multiple cosmological data sets, a significant advantage being that both its uncertainty and resolution are then quantified. Using Monte Carlo simulations we investigate several regularisation parameter selection methods and find that generalised cross-validation and Mallows $C_p$ method give optimal results. We apply our inversion procedure to data from the Wilkinson Microwave Anisotropy Probe, other ground-based small angular scale CMB experiments, and the Sloan Digital Sky Survey. The reconstructed spectrum (assuming the standard $Lambda$CDM cosmology) is emph{not} scale-free but has an infrared cutoff at $k lesssim 5 times 10^{-4}; mathrm{Mpc}^{-1}$ (due to the anomalously low CMB quadrupole) and several features with $sim 2 sigma$ significance at $k/mathrm{Mpc}^{-1} sim$ 0.0013--0.0025, 0.0362--0.0402 and 0.051--0.056, reflecting the `WMAP glitches. To test whether these are indeed real will require more accurate data, such as from the Planck satellite and new ground-based experiments.
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The properties of primordial curvature perturbations on small scales are still unknown while those on large scales have been well probed by the observations of the cosmic microwave background anisotropies and the large scale structure. In this paper, we propose the reconstruction method of primordial curvature perturbations on small scales through the merger rate of binary primordial black holes, which could form from large primordial curvature perturbation on small scales.
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.
In the model where Primordial Black Holes (PBHs) form from large primordial curvature (C) perturbations, i.e., CPBHs, constraints on PBH abundance provide in principle constraints on the primordial curvature power spectrum. This connection however depends necessarily on the details of PBH formation mechanism. In this paper we provide, for the first time, constraints on the primordial curvature power spectrum from the latest limits on PBH abundance, taking into account all the steps from gravitational collapse in real space to PBH formation. In particular, we use results from numerical relativity simulations and peak theory to study the conditions for PBH formation for a range of perturbation shapes, including non-linearities, perturbation profile and a careful treatment of smoothing and filtering scales. We then obtain updated PBH formation conditions and translate that into primordial spectrum constraints for a wide range of shapes and abundances. These updated constraints cover a range of scales not probed by other cosmological observables. Our results show that the correct and accurate modelling of non-linearities, filtering and typical perturbation profile, is crucial for deriving meaningful cosmological implications.
In the context of transient constant-roll inflation near a local maximum, we derive the non-perturbative field redefinition that relates a Gaussian random field with the true non-Gaussian curvature perturbation. Our analysis shows the emergence of a new critical amplitude $zeta_*$, corresponding to perturbations that prevent the inflaton from overshooting the local maximum, thus becoming trapped in the false minimum of the potential. For potentials with a mild curvature at the local maximum (and thus small non-Gaussianity), we recover the known perturbative field redefinition. We apply these results to the formation of primordial black holes, and discuss the cases for which $zeta_*$ is smaller or of the same order than the critical value for collapse of spherically symmetric overdensities. In the latter case, we present a simple potential for which the power spectrum needs an amplitude 10 times smaller that in the Gaussian case for producing a sizeable amount of primordial black holes.
The primordial power spectrum describes the initial perturbations in the Universe which eventually grew into the large-scale structure we observe today, and thereby provides an indirect probe of inflation or other structure-formation mechanisms. Here, we introduce a new method to estimate this spectrum from the empirical power spectrum of cosmic microwave background (CMB) maps. A sparsity-based linear inversion method, coined textbf{PRISM}, is presented. This technique leverages a sparsity prior on features in the primordial power spectrum in a wavelet basis to regularise the inverse problem. This non-parametric approach does not assume a strong prior on the shape of the primordial power spectrum, yet is able to correctly reconstruct its global shape as well as localised features. These advantages make this method robust for detecting deviations from the currently favoured scale-invariant spectrum. We investigate the strength of this method on a set of WMAP 9-year simulated data for three types of primordial power spectra: a nearly scale-invariant spectrum, a spectrum with a small running of the spectral index, and a spectrum with a localised feature. This technique proves to easily detect deviations from a pure scale-invariant power spectrum and is suitable for distinguishing between simple models of the inflation. We process the WMAP 9-year data and find no significant departure from a nearly scale-invariant power spectrum with the spectral index $n_s = 0.972$. A high resolution primordial power spectrum can be reconstructed with this technique, where any strong local deviations or small global deviations from a pure scale-invariant spectrum can easily be detected.
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