How Production Of Primordial Black Holes Take Place In NonMinimal Derivative Coupling Inflation? (Cosmology / Quantum)

Heydari and Karami investigated the generation of Primordial Black Holes (PBHs) with the aid of gravitationally increased friction mechanism originated from the NonMinimal field Derivative Coupling (NMDC) to gravity framework, with the quartic potential. By assigning a coupling parameter as a two-parted function of inflation field and fine-tuning of 4 parameter cases of the model they showed that they could acquire an epoch of ultra slow-roll inflation on scales smaller than CMB scale making the inflaton slow down, sufficient to generate PBHs. Their study recently appeared in Arxiv.

You may have heard the concept of primordial black holes (PBHs) generation from the over-dense regions of the early universe. For the generation of primordial black holes (PBHs) during Radiation Dominated (RD) era, production of a large enough amplitude of primordial curvature perturbations (R) during inflationary epoch is necessary. Overdense regions can be formed when superhorizon scales associated with the large amplitude of R become subhorizon during RD era, and gravitationally collapse of these overdensities generate PBHs.

Meaning, PBHs generation requires an enhancement in the power spectrum of R to order 10¯2 at scales smaller than CMB scales. Several techniques have been employed for multiplying the amplitude of the power spectrum of R at small scale by seven orders of magnitude in comparison with CMB scales. One of the proper ways to achieve a rise in the scalar power spectrum is a brief period of Ultra Slow-Roll (USR) inflation due to declining the speed of inflaton field via gravitationally enhanced friction. The framework of NonMinimal Derivative Coupling to gravity (NMDC) beside the fine-tuning of the parameters of the model can give rise to increase friction gravitationally.

The nonminimal derivative coupling model is a subclass of a generic scalar-tensor theory with second-order equations of motion namely Horndeski theory, which prevents the model from negative energy and pertinent instabilities. A characteristic of the nonminimal field derivative coupling to gravity is that the gravitationally increased friction mechanism can be applied for generic steep potentials such as quartic potential.

Now, Heydari and Karami investigated the generation of Primordial Black Holes (PBHs) with the aid of gravitationally increased friction mechanism originated from the NonMinimal field Derivative Coupling (NMDC) to gravity framework, with the quartic potential.

© Heydari and Karami

By assigning the coupling parameter as a two-parted function of inflaton field, and fine-tuning of the four parameter cases (A, B, C, and D) of the model (see Table I), they showed that, we could acquire an epoch of ultra slow-roll inflation on scales smaller than CMB scale making the inflaton slow down due to high friction. This enables them to achieve enough enhancement in the amplitude of curvature perturbations power spectra to generate PBHs with masses of order 10 M for Case A (stellar mass), 10¯6 M for Case B (earth mass), 10¯13 M for Case C, and 10¯15 M for Case D (asteroid mass).

Their results indicated that PBHs of case A is suitable to describe GWs and LIGO events, Case B can be useful to expound microlensing events in OGLE data, and PBHs of cases C and D can be interesting candidates for composing around 98.32% and 99.11% of dark matter (DM) content of the universe (see Table II and Fig. 1)

FIG. 1. The PBHs abundance (fPBH) in terms of PBHs mass (M) for Case A (purple line), Case B (green line), Case C (red line), and Case D (blue line). The red spots indicate the upper limit on the PBH abundance owing to the upper limit on the LIGO event merger rate. The brown shadowy zone signifies the permitted zone of PBH abundance from the ultrashort-timescale microlensing events in the OGLE data. The other shadowy areas demonstrate the current observational restrictions on the fractional abundance of PBHs comprising extragalactic gamma rays from PBH evaporation (EGγ), galactic center 511 keV γ-ray line (INTEGRAL), white dwarf explosion (WD), microlensing events with Subaru HSC (Subaru HSC), with the Kepler satellite (Kepler), with EROS/MACHO (EROS/MACHO), and accretion constraints from CMB © Heydari and Karami

Additionally, they inquired generation of the induced GWs subsequent to PBHs formation for all cases of their model. Their calculation of current density parameter spectra (ΩGW0) indicated that, all cases have climaxes at contrasting frequencies with nearly identical heights of order 10¯8. The climaxes of ΩGW0 for cases A and B have placed at frequencies 10¯10Hz and 10¯7 Hz, respectively, and both cases can be traced via the SKA detector. Moreover, the spectra of ΩGW0 for Cases A and B have climaxes localized at mHz and cHz bands which are tracked down by LISA, TaiJi, and TianQin (see Fig. 2). Hence, validity of their model can be assessed in view of GWs via the extricated data of these detectors.

Fig 2. The present induced GWs energy density parameter (ΩGW0) with respect to frequency. The solid purple, green, red, and blue lines associate with Cases A, B, C, and D of Table I, respectively. The power-law form of ΩGW0 is illustrated by black dashed line for Case D. © Heydari and Karami

Finally, they demonstrated that in the vicinity of climaxes, the spectra of density parameter behave as a power-law function with respect to frequency (ΩGW0 (f) ∼ (f/fc)^n). Also, in the infrared regime f<<fc, the power index satifies the relation n = 3 – 2/ ln(fc/f).


Reference: Soma Heydari, Kayoomars Karami, “Primordial black holes in nonminimal derivative coupling inflation driven by quartic potential”, Arxiv, pp. 1-28, 2021.
arXiv:2107.10550


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