*Summary:*

*According to physicists,**e-folding**is simply the amount of time it takes for space to expand by approximately 2.71828 times its original size. The number 2.71828 is eulers number.**In order to produce primordial black holes (PBH’s) from inflationary fluctuations, large deceleration of inflation is required.**Now, Inamoto and colleagues showed that a large enhancement of perturbations results when the inflation crosses a downward step in its potential in less than an e-fold.**In simple terms, during this step, inflation loses extra kinetic energy due to hubble friction and there are large enhancement of perturbations which produces primordial black holes and ultra-compact minihalos.*

Primordial black holes (PBHs) are one of the most intriguing topics in modern cosmology, owing to their potential to explain dark matter (DM) and the BHs detected by the LIGO-Virgo collaboration. Also, PBHs might be related to other observational results, such as the existence of supermassive black holes, the OGLE results, the recent NANOGrav results, and the anomalous excess of 511 keV photons. PBHs can be produced when very large density perturbations enter the horizon in the early universe. In particular, the PBH scenarios for DM or LIGO-Virgo events can be associated with the large power spectrum of primordial curvature perturbations, PR ∼ 10¯^{2}, on small scales.

Now, Inomata and colleagues, focused on single-field inflation models that can realize the large power spectrum on small scales for the PBH scenarios. Under the slow-roll approximation, the power spectrum is given by

where the subscript “∗” denotes evaluation at the horizon exit of the perturbation and

where N ≡ ∫ Hdt is the number of e-folds of inflationary expansion. From this relation, at first glance, the large power spectrum on small scales needed for the PBH scenarios seems to require a substantial decrease in ϵ, and hence the kinetic energy of the inflaton, from the horizon exit of CMB scales. This decrease is realized by a large negative value of η ≡ d ln ϵ/dN which violates the slow-roll assumption. This can be achieved with a very flat potential in a period of so called “ultra slow roll (USR)” when Hubble friction dominates over the potential slope. On the other hand since the slow-roll approximation must be violated, this invalidates the naive expectation of a decreased and leaves the possibility of alternative mechanisms.

“In our paper, we show that a decrease in the kinetic energy of the inflaton relative to that at CMB scales is not necessary for the large enhancement of perturbations required for the PBH scenarios. Equivalently, the inflation potential need not have a region that is flatter than it is at CMB scales. If the inflaton instead gains kinetic energy by rolling down a sufficiently sharp feature that it crosses in less than an e-fold, non-adiabatic particle production occurs.”— told Inomata, first author of the study.

They showed that a large enhancement of perturbations results when the inflaton crosses a downward step in its potential in less than an e-fold, which counter-intuitively allows a sizable amount of PBHs to form in a model wherein the inflaton always possesses a velocity higher than its value at the horizon exit of CMB scales. The enhancement can be interpreted as particle production due to the non-adiabatic transition whose curvature fluctuations are then adiabatically enhanced to large values as the inflaton loses the extra kinetic energy from the step due to Hubble friction.

Finally, they mentioned that, depending on the height and the location of the downward step, their enhancement mechanism can generate seeds not only for PBHs with a variety of masses, but also for ultra-compact minihalos. Additionally, the enhancement can be probed (constrained or discovered) by a range of complementary observables, such as the gravitational waves induced by the scalar perturbations, and CMB spectral distortions.

“Future observations of PBHs and these varied observable probes will enable us to probe this characteristic feature in the inflaton potential.”— told Inomata, first author of the study

* Featured image: The inflaton potential of Eq. (14) given in paper that realizes the large enhancement of perturbations, with the steplike transition at φ1 ≤ φ ≤ φ2 highlighted and an inset for the full range. The parameters are ns = 0.97, ϵ1 = 7.43 × 10¯^{10}, ϵ2 = 0.01, ϵ3 = 10¯^{9}, and ∆Nstep = 0.5. φend denotes the end of inflation (red vertical dotted line) and corresponds to 50 e-folds from φCMB.* ©

*Inomata et al.*

**Reference**: Keisuke Inomata, Evan McDonough, Wayne Hu, “Primordial Black Holes Arise When The Inflaton Falls”, Arxiv, pp. 1-6, 2021. https://arxiv.org/abs/2104.03972

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