Primordial black hole dark matter from single field inflation (1709.05565v2)

Abstract: We propose a model of inflation capable of generating a population of light black holes (about $10^{-16}$ - $10^{-14}$ solar masses) that might account for a significant fraction of the dark matter in the Universe. The effective potential of the model features an approximate inflection point arising from two-loop order logarithmic corrections in well-motivated and perturbative particle physics examples. This feature decelerates the inflaton before the end of inflation, enhancing the primordial spectrum of scalar fluctuations and triggering efficient black hole production with a peaked mass distribution. At larger field values, inflation occurs thanks to a generic small coupling between the inflaton and the curvature of spacetime. We compute accurately the peak mass and abundance of the primordial black holes using the Press-Schechter and Mukhanov-Sasaki formalisms, showing that the slow-roll approximation fails to reproduce the correct results by orders of magnitude. We study as well a qualitatively similar implementation of the idea, where the approximate inflection point is due to competing terms in a generic polynomial potential. In both models, requiring a significant part of the dark matter abundance to be in the form of black holes implies a small blue scalar tilt with a sizable negative running and a tensor spectrum that may be detected by the next-generation probes of the cosmic microwave background. We also comment on previous works on the topic.

• The paper presents a single-field inflation model that creates an inflection point to amplify primordial fluctuations for PBH production.
• The paper employs detailed numerical analysis beyond slow-roll approximations to compute PBH mass spectra in the 10⁻¹⁶ to 10⁻¹³ solar mass range.
• The paper discusses how non-minimal coupling and precise parameter tuning are vital to reconciling the model with CMB observations and potential gravitational signals.

Primordial Black Hole Dark Matter from Single Field Inflation

The exploration of primordial black holes (PBHs) as potential constituents of dark matter (DM) has pervasive implications in cosmology and particle physics. The paper under discussion proposes a theoretical framework wherein inflation driven by a single scalar field can produce a population of primordial black holes. These black holes may constitute a significant fraction of dark matter. The model articulates a complex interconnection between early universe dynamics, quantum field theory effects, and astrophysical observations, setting the stage for a phenomenologically rich arena.

Theoretical Framework

The authors develop a model within the spectrum of scalar field inflationary theories, specifically focusing on the potential:

$$V(\phi) = \frac{\lambda_0}{4!}\left(1-2(1+b_1)\log\frac{\phi^2}{\phi_0^2}+2(1+b_2)\left(\log\frac{\phi^2}{\phi_0^2}\right)^2\right)\phi^4$$

This potential incorporates two-loop order logarithmic corrections, thereby introducing an approximate inflection point—a pivotal feature decelerating the inflaton at certain epochs, significantly influencing PBH production by amplifying the primordial scalar power spectrum.

The coupling between the inflaton field and curvature, described by a non-minimal interaction, is crucial. It modifies the effective potential, leading to a plateau at large field values, necessary for a successful inflationary period that complies with Cosmic Microwave Background (CMB) observations. This coupling is parametrized in the equations by a field-dependent term:

$\Omega^2 = 1 + \frac{\xi \phi^2}{M_P^2}$

Primordial Black Hole Formation

The formation of PBHs is tied to the inflationary mechanism modulating initial density fluctuations. The authors employ the Press-Schechter formalism and Mukhanov-Sasaki equation to precisely compute the PBHs' mass spectrum and abundance. They critically evaluate the limits of the slow-roll approximation, which is shown to fail near the inflection point, necessitating exact numerical solutions to capture the dynamics accurately.

The numerical analysis reveals that the potential accommodates significant spikes in the primordial power spectrum at scales corresponding to PBH masses in the range of $10^{-16} - 10^{-13}$ solar masses. This mass range is particularly interesting due to its capacity to evade stringent astrophysical constraints while potentially explaining a sizable fraction of DM.

Implications and Challenges

The proposed mechanism for PBH formation aligns with the observational constraints on CMB anisotropies, scalar spectral index, and tensor-to-scalar ratio. However, achieving the desired PBH abundance requires precise and sensitive tuning of the model parameters. The inflaton's trajectory around the plateau undergoes substantial slowdowns, enhancing the scalar perturbations to the necessary amplitudes for PBH production.

The research underscores multiple unresolved aspects and challenges, emphasizing the sensitivity of PBH abundance to the collapse threshold and primordial spectrum peak. This sensitivity amplifies the need for precise theoretical and observational frameworks to constrain these parameters more rigorously.

Future Prospects

The presented model sets a viable groundwork for further investigation into the dual nature of inflation and PBH formation. Future research can potentially incorporate non-Gaussian perturbations and explore broader parameter spaces using stochastic inflationary dynamics. The paper also brushes on the emerging prospect of gravitational wave signals sourced by these PBHs, presenting another exciting developmental trajectory.

In summary, the paper enriches our understanding of PBHs as dark matter candidates within a single scalar field inflation paradigm. It adds theoretical depth to the understanding of early universe phenomena intersecting with observational cosmology's continuing quest to unravel the elusive nature of DM.