We construct an analytically solvable simplified model that captures the essential features for primordial black hole (PBH) production in most models of single-field inflation. The construction makes use of the Wands duality between the constant-roll (or slow-roll) and the preceding ultra-slow-roll phases and can be realized by a simple inflaton potential of two joined parabolas. Within this framework, it is possible to formulate explicit inflationary scenarios consistent with the CMB observations and copious production of PBHs of arbitrary mass. We quantify the variability of the shape of the peak in the curvature power spectrum in different inflationary scenarios and discuss its implications for probing PBHs with scalar-induced gravitational wave backgrounds. We find that the COBE/Firas μ-distortion constraints exclude the production of PBHs heavier than 104 M ⊙ in single-field inflation.

/pdf/anatomy-of-single-field-inflationary-models-for-primordial-3c88b1xw.pdf

Anatomy of single-field inflationary models for primordial black holes

Inflationary models predicting a scale-dependent large amplification of the density perturbations have recently attracted a lot of attention because the amplified perturbations can seed a sizable amount of primordial black holes (PBHs) and stochastic background of gravitational waves (GWs). While the power spectra in these models are computed based on the linear equation of motion, it is not obvious whether loop corrections are negligible when such a large amplification occurs during inflation. In this paper, as a first step to discuss the loop corrections in such models, we use the in-in formalism and calculate the one-loop scalar power spectrum numerically and analytically in an illustrative model where the density perturbations are resonantly amplified due to oscillatory features in the inflaton potential. Our calculation is technically new in that the amplified perturbations are numerically taken into account in the in-in formalism for the first time. In arriving at our analytical estimates, we highlight the role that the Wronskian condition of perturbations, automatically satisfied in our model, plays in obtaining the correct estimates. In addition, the analytical estimates show that the contribution originating from the quantum nature of the perturbations in the loop can be dominant. We also discuss the necessary conditions for subdominant loop corrections in this model. We find that, for the typical parameter space leading to the 𝒪(107) amplification of the power spectrum required for a sufficient PBH production, the one-loop power spectrum dominates over the tree-level one, indicating the breakdown of the perturbation theory.

/pdf/questions-on-calculation-of-primordial-power-spectrum-with-1jmg6kyy.pdf

Questions on calculation of primordial power spectrum with large spikes: the resonance model case

We investigate the two-stage inflation regime in the theory of hybrid cosmological α-attractors. The spectrum of inflationary perturbations is compatible with the latest Planck/BICEP/Keck Array results, thanks to the attractor properties of the model. However, at smaller scales, it may have a very high peak of controllable width and position, leading to a copious production of primordial black holes (PBH) and generation of a stochastic background of gravitational waves (SGWB).

/pdf/hybrid-a-attractors-primordial-black-holes-and-gravitational-kguxrw8c.pdf

Hybrid α-attractors, primordial black holes and gravitational wave backgrounds

Recent population studies have searched for a subpopulation of primordial black holes (PBHs) in the gravitational-wave (GW) events so far detected by LIGO/Virgo/KAGRA (LVK), in most cases adopting a phenomenological PBH mass distribution. When deriving such population from first principles in the standard scenario, however, the equation of state of the Universe at the time of PBH formation may strongly affect the PBH abundance and mass distribution, which ultimately depend on the power spectrum of cosmological perturbations. Here we improve on previous population studies on several aspects: (i) we adopt state-of-the-art PBH formation models describing the collapse of cosmological perturbations across the QCD epoch; (ii) we perform the first Bayesian multi-population inference on GW data including PBHs and directly using power spectrum parameters instead of phenomenological distributions; (iii) we critically confront the PBH scenario with LVK phenomenological models describing the GWTC-3 catalog both in the neutron-star and in the BH mass ranges, also considering PBHs as subpopulation of the total events. Our results confirm that LVK observations prevent the majority of the dark matter to be in the form of stellar mass PBHs. We find that the best fit PBH model can comprise a small fraction of the total events, in particular it can naturally explain events in the mass gaps. If the lower mass-gap event GW190814 is interpreted as a PBH binary, we predict that LVK should detect up to a few subsolar mergers and one to $\approx 30$ lower mass gap events during the upcoming O4 and O5 runs. Finally, mapping back the best-fit power spectrum into an ultra slow-roll inflationary scenario, we show that the latter predicts detectable PBH mergers in the LVK band, a stochastic GW background detectable by current and future instruments, and may include the entirety of dark matter in asteroid-mass PBHs. 

/pdf/from-inflation-to-black-hole-mergers-and-back-again-2ci96haf.pdf

From inflation to black hole mergers and back again: Gravitational-wave data-driven constraints on inflationary scenarios with a first-principle model of primordial black holes across the QCD epoch

We construct $\alpha$-attractor versions of hybrid inflation models. In these models, the potential of the inflaton field $\varphi$ is uplifted by the potential of the second field $\chi$. This uplifting ends due to a tachyonic instability with respect to the field $\chi$, which appears when $\varphi$ becomes smaller than some critical value $\varphi_{c}$. In the large $N$ limit, these models have the standard universal $\alpha$-attractor predictions. In particular, $n_{s }= 1- {2 \over N}$ for the exponential attractors. However, in some special cases the large $N$ limit is reached only beyond the horizon, for $N \gtrsim 60$. This may change predictions for the cosmological observations. For any fixed $N$, in the limit of large uplift $V_{\rm up}$, or in the limit of large $\varphi_{c}$, we find another attractor prediction, $ n_s = 1$. By changing the parameters $V_{\rm up}$ and $\varphi_{c}$ one can continuously interpolate between the two attractor predictions $n_{s }= 1- {2 \over N}$ and $n_{s} = 1$. This provides significant flexibility, which can be very welcome in view of the rapidly growing amount and precision of the cosmological data. Our main result is not specific to the hybrid inflation models. Rather, it is generic to any inflationary models where the inflaton potential, for some reasons, is uplifted, and inflation ends prematurely. 

/pdf/hybrid-cosmological-attractors-1449txgw.pdf

Hybrid cosmological attractors

Abstract Cosmological α -attractors stand out as particularly compelling models to describe inflation in the very early universe, naturally meeting tight observational bounds from cosmic microwave background (CMB) experiments. We investigate α -attractor potentials in the presence of an inflection point, leading to enhanced curvature perturbations on small scales. We study both single- and multi-field models, driven by scalar fields living on a hyperbolic field space. In the single-field case, ultra-slow-roll dynamics at the inflection point is responsible for the growth of the power spectrum, while in the multi-field set-up we study the effect of geometrical destabilisation and non-geodesic motion in field space. The two mechanisms can in principle be distinguished through the spectral shape of the resulting scalar power spectrum on small scales. These enhanced scalar perturbations can lead to primordial black hole (PBH) production and second-order gravitational wave (GW) generation. Due to the existence of universal predictions in α -attractors, consistency with current CMB constraints on the large-scale spectral tilt implies that PBHs can only be produced with masses smaller than 10 8 g and are accompanied by ultra-high frequency GWs, with a peak expected to be at frequencies of order 10 kHz or above. 

https://iopscience.iop.org/article/10.1088/1475-7516/2022/06/007/pdf

Davide Amendola

Papers

Revisiting small-scale fluctuations in α-attractor models of inflation