Primordial black holes save R 2 inflation

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.

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.

Recently, it has been shown that the primordial black holes (PBHs) produced by near critical collapse in the expanding universe have a scaling mass relation similar to that of black holes produced in asymptotically flat spacetime. Distinct from PBHs formed with mass about the horizon mass (Type I), the PBHs with the scaling relation (Type II) can be created with a range of masses at a given formation time. In general, only the case in which the PBH formation is concentrated at one epoch has been considered. However, it is expected that PBH formation is possible over a broad range of epochs if the density fluctuation has a rather large amplitude and smooth scale dependence. In this paper, we study the PBH formation for both types assuming the power spectrum of double inflationary models in which the small scale fluctuations could have large amplitudes independent of the CMBR anisotropy. The mass spectrum of Type II PBHs is newly constructed without limiting the PBH formation period. The double inflationary power spectrum is assumed to be of double simple power-law which are smoothly connected. Under the assumed power spectrum, the accumulation of small PBHs formed at later times is important and the mass range is significantly broadened for both Types. The PBH mass spectra are far smoother than the observed MACHO spectrum due to our assumption of a smooth spectrum. In order to fit the observation, a more spiky spectrum is required.

The formation of primordial black holes or other dark matter relics from amplified density fluctuations in the early Universe may also generate scalar-induced gravitational waves (GW), carrying vital information about the primordial power spectrum and the early expansion history of our Universe. We present a Bayesian approach aimed at reconstructing both the shape of the scalar power spectrum and the Universe’s equation of state from GW observations, using interpolating splines to flexibly capture features in the GW data. The optimal number of spline nodes is chosen via Bayesian evidence, aiming at balancing complexity of the model and the fidelity of the reconstruction. We test our method using both representative mock data and recent pulsar timing array measurements, demonstrating that it can accurately reconstruct the curvature power spectrum as well as the underlying equation of state, if different from radiation.

Observational constraints on the abundance of primordial black holes (PBHs) constrain the allowed amplitude of the primordial power spectrum on both the smallest and the largest ranges of scales, covering over 20 decades from 1−1020/ Mpc. Despite tight constraints on the allowed fraction of PBHs at their time of formation near horizon entry in the early Universe, the corresponding constraints on the primordial power spectrum are quite weak, typically \U0001d4abR≲ 10−2 assuming Gaussian perturbations. Motivated by recent claims that the evaporation of just one PBH would destabilise the Higgs vacuum and collapse the Universe, we calculate the constraints which follow from assuming there are zero PBHs within the observable Universe. Even if evaporating PBHs do not collapse the Universe, this scenario represents the ultimate limit of observational constraints. Constraints can be extended on to smaller scales right down to the horizon scale at the end of inflation, but where power spectrum constraints already exist they do not tighten significantly, even though the constraint on PBH abundance can decrease by up to 46 orders of magnitude. This shows that no future improvement in observational constraints can ever lead to a significant tightening in constraints on inflation (via the power spectrum amplitude). The power spectrum constraints are weak because an order unity perturbation is required in order to overcome pressure forces. We therefore consider an early matter dominated era, during which exponentially more PBHs form for the same initial conditions. We show this leads to far tighter constraints, which approach \U0001d4abR≲10−9, albeit over a smaller range of scales and are very sensitive to when the early matter dominated era ends. Finally, we show that an extended early matter era is incompatible with the argument that an evaporating PBH would destroy the Universe, unless the power spectrum amplitude decreases by up to ten orders of magnitude.

We propose a new, likelihood-free approach to inferring the primordial matter power spectrum and cosmological parameters from arbitrarily complex forward models of galaxy surveys where all relevant statistics can be determined from numerical simulations, i.e. black boxes. Our approach, which we call simulator expansion for likelihood-free inference (selfi), builds upon approximate Bayesian computation using a novel effective likelihood, and upon the linearization of black-box models around an expansion point. Consequently, we obtain simple ‘filter equations’ for an effective posterior of the primordial power spectrum, and a straightforward scheme for cosmological parameter inference. We demonstrate that the workload is computationally tractable, fixed a priori, and perfectly parallel. As a proof of concept, we apply our framework to a realistic synthetic galaxy survey, with a data model accounting for physical structure formation and incomplete and noisy galaxy observations. In doing so, we show that the use of non-linear numerical models allows the galaxy power spectrum to be safely fitted up to at least kmax = 0.5 h Mpc−1, outperforming state-of-the-art backward-modelling techniques by a factor of ∼5 in the number of modes used. The result is an unbiased inference of the primordial matter power spectrum across the entire range of scales considered, including a high-fidelity reconstruction of baryon acoustic oscillations. It translates into an unbiased and robust inference of cosmological parameters. Our results pave the path towards easy applications of likelihood-free simulation-based inference in cosmology. We have made our code pyselfi and our data products publicly available at http://pyselfi.florent-leclercq.eu.

This paper is a biased review of the primordial black hole (PBH) formation and abundance estimation. We first review the three-zone model for PBH formation to help an intuitive understanding of the PBH formation process. Then, for more accurate analyses, we introduce necessary tools such as cosmological long-wavelength solutions, the definition of the mass and compaction function in a spherically symmetric spacetime and peak theory. Combining all these tools, we calculate the PBH mass spectrum for the case of the monochromatic curvature power spectrum as a demonstration.

Within canonical single field inflation models, we provide a method to reverse engineer and reconstruct the inflaton potential from a given power spectrum. This is not only a useful tool to find a potential from observational constraints, but also gives insight into how to generate a large amplitude spike in density perturbations, especially those that may lead to primordial black holes (PBHs). In accord with other works, we find that the usual slow-roll conditions need to be violated in order to generate a significant spike in the spectrum. We find that a way to achieve a very large amplitude spike in single field models is for the classical roll of the inflaton to over-shoot a local minimum during inflation. We provide an example of a quintic polynomial potential that implements this idea and leads to the observed spectral index, observed amplitude of fluctuations on large scales, significant PBH formation on small scales, and is compatible with other observational constraints. We quantify how much fine-tuning is required to achieve this in a family of random polynomial potentials, which may be useful to estimate the probability of PBH formation in the string landscape.

After the end of inflation, the inflaton field oscillates around a local minimum of its potential and decays into ordinary matter. These oscillations trigger a resonant instability for cosmological perturbations with wavelengths that exit the Hubble radius close to the end of inflation. In this paper, we study the formation of Primordial Black Holes (PBHs) at these enhanced scales. We find that the production mechanism can be so efficient that PBHs subsequently dominate the content of the universe and reheating proceeds from their evaporation. Observational constraints on the PBH abundance also restrict the duration of the resonant instability phase, leading to tight limits on the reheating temperature that we derive. We conclude that the production of PBHs during reheating is a generic and inevitable property of the simplest inflationary single-field models, and does not require any fine tuning of the inflationary potential.

In inflationary models with minimal amount of gravity waves, the primordial power spectrum of density fluctuations, $A_S^2(k)$, together with the basic cosmological parameters, completely specify the predictions for the cosmic microwave background (CMB) anisotropy and large scale structure. Here we show how we can strongly constrain both $A_S^2(k)$ and the cosmological parameters by combining the data from the Microwave Anisotropy Probe (MAP) and the galaxy redshift survey from the Sloan Digital Sky Survey (SDSS). We allow $A_S^2(k)$ to be a free function, and thus probe features in the primordial power spectrum on all scales. MAP and SDSS have scale-dependent measurement errors that decrease in opposite directions on astrophysically interesting scales; they complement each other and allow the measurement of the primordial power spectrum independent of inflationary models, giving us valuable information on physics in the early Universe, and providing clues to the correct inflationary model.

High-$z$ galaxy redshift surveys open up exciting possibilities for precision determinations of neutrino masses and inflationary models. The high-$z$ surveys are more useful for cosmology than low-$z$ ones owing to much weaker nonlinearities in matter clustering, redshift-space distortion, and galaxy bias, which allows us to use the galaxy power spectrum down to the smaller spatial scales that are inaccessible by low-$z$ surveys. We can then utilize the two-dimensional information of the linear power spectrum in angular and redshift space to measure the scale-dependent suppression of matter clustering due to neutrino free-streaming as well as the shape of the primordial power spectrum. To illustrate capabilities of high-$z$ surveys for constraining neutrino masses and the primordial power spectrum, we compare three future redshift surveys covering 300 square degrees at $0.5lzl2$, $2lzl4$, and $3.5lzl6.5$. We find that, combined with the cosmic microwave background data expected from the Planck satellite, these surveys allow precision determination of the total neutrino mass with the projected errors of $\ensuremath{\sigma}({m}_{\ensuremath{\nu},\mathrm{tot}})=0.059$, 0.043, and 0.025 eV, respectively, thus yielding a positive detection of the neutrino mass rather than an upper limit, as $\ensuremath{\sigma}({m}_{\ensuremath{\nu},\mathrm{tot}})$ is smaller than the lower limits to the neutrino masses implied from the neutrino oscillation experiments, by up to a factor of 4 for the highest redshift survey. The accuracies of constraining the tilt and running index of the primordial power spectrum, $\ensuremath{\sigma}({n}_{s})=(3.8,3.7,3.0)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and $\ensuremath{\sigma}({\ensuremath{\alpha}}_{s})=(5.9,5.7,2.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ at ${k}_{0}=0.05\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$, respectively, are smaller than the current uncertainties by more than an order of magnitude, which will allow us to discriminate between candidate inflationary models. In particular, the error on ${\ensuremath{\alpha}}_{s}$ from the future highest redshift survey is not very far away from the prediction of a class of simple inflationary models driven by a massive scalar field with self-coupling, ${\ensuremath{\alpha}}_{s}=\ensuremath{-}(0.8\char21{}1.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$.

Measurements of the cosmic microwave background (CMB) spectral distortions (SDs) will open a new window on the very early universe, providing new information complementary to that gathered from CMB temperature and polarization anisotropies. In this paper, we study their synergy as a function of the characteristics of the considered experiments. In particular, we examine a wide range of sensitivities for possible SD measurements, spanning from FIRAS up to noise levels 1000 times better than PIXIE, and study their constraining power when combined with current or future CMB anisotropy experiments such as Planck or LiteBIRD plus CMB-S4. We consider a number of different cosmological models such as the ΛCDM, as well as its extensions with the running of the scalar spectral index, the decay or the annihilation of dark matter (DM) particles. While upcoming CMB anisotropy experiments will be able to decrease the uncertainties on inflationary parameters such as As and ns by about a factor 2 in the ΛCDM case, we find that an SD experiment 100 times more sensitive than PIXIE (comparable to the proposed Super-PIXIE satellite) could potentially further contribute to constrain these parameters. This is even more significant in the case of the running of the scalar spectral index. Furthermore, as expected, constraints on DM particles decaying at redshifts probed by SDs will improve by orders of magnitude even with an experiment 10 times worse than PIXIE as compared to CMB anisotropies or Big Bang Nucleosynthesis bounds. On the contrary, DM annihilation constraints will not significantly improve over CMB anisotropy measurements. Finally, we forecast the constraints obtainable with sensitivities achievable either from the ground or from a balloon.

It has now been widely accepted that many inflation models can account for the formation of primordial black holes (PBHs). In particular, it has been fully realized that the PBH formation depends crucially on the tails of the probability distribution function of the curvature perturbation. I will review some models of inflation that give rise to the PBH formation, and their observational implications, in particular, for the blossoming field of gravitational-wave cosmology.

The observables of the perturbed universe, cosmic microwave background (CMB) anisotropy and large structures depend on a set of cosmological parameters, as well as the assumed nature of primordial perturbations. In particular, the shape of the primordial power spectrum (PPS) is, at best, a well-motivated assumption. It is known that the assumed functional form of the PPS in cosmological parameter estimation can affect the best-fit-parameters and their relative confidence limits. In this paper, we demonstrate that a specific assumed form actually drives the best-fit parameters into distinct basins of likelihood in the space of cosmological parameters where the likelihood resists improvement via modifications to the PPS. The regions where considerably better likelihoods are obtained allowing free-form PPS lie outside these basins. In the absence of a preferred model of inflation, this raises a concern that current cosmological parameter estimates are strongly prejudiced by the assumed form of PPS. Our results strongly motivate approaches toward simultaneous estimation of the cosmological parameters and the shape of the primordial spectrum from upcoming cosmological data. It is equally important for theorists to keep an open mind towards early universe scenarios that produce features in the PPS.

Although the standard $\mathrm{\ensuremath{\Lambda}}+\mathrm{cold}$ dark matter ($\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$) model is well tested on large scales, the primordial power spectrum may deviate from the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ spectrum on small scales due to specific dark matter properties or alternative inflationary models. These deviations affect the formation of dark matter structure, which subsequently leads to different observable properties of galaxies. In this work, we study the impact of a blue and red tilted power spectrum on the central density of dwarf galaxies. To do this, we model densities of dwarf galaxies using a combination of high-resolution numerical simulations and galaxy formation model. The model galaxies in $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ are consistent with observations of 41 faint dwarf satellite galaxies of the Milky Way. The deviations from the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ power spectrum are constrained by the central matter densities of dwarf galaxies, which set stringent constraints on the possible small-scale tilt of the primordial power spectrum, improving on the current limits. Moreover, similar analysis can be applied to test any feature in the power spectrum at small scales between $k\ensuremath{\sim}10--100\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$.