How Primordial Black Holes Forms From Inflation With Solo or Multiple Bumps? (Cosmology / Quantum Physics)

Ruifeng Zheng and colleagues investigated the formation of primordial black holes (PBHs) from inflation model with bumpy potential, which has multiple bumps. They found that, the potential can give rise to power spectrum with single or multiple peaks in small scales, which can in turn predict the production of primordial black holes. Their study recently appeared in Arxiv.

There are several inflation models which discuss the possibility of the production of primordial black holes in inflation process of the early universe. But, the production of PBHs requires the violation of slow-roll condition. For this reason, slow-roll violating models become an interesting alternative, including ultra-slow-roll (USR) inflation, inflation with inflection points or bumps and others.

Previous studies have already discussed the case of potential with one bump. Now, Ruifeng and colleagues discussed the extended case of multiple bumps, which can generate PBHs at different mass ranges.

Fig 1: The evolution of φ with the parameter N, which enters the USR-like stage three times near N = 20, N = 35 and N = 50. © Zheng et al.

Specifically, they considered the power-law potential as the basic potential, and add one or several bumps of Gaussian type which makes the inflaton roll from the slow-roll stage to USR-like stage.

“These multiple bumps are quite different from the solo-bumpy ones, we have to take care of not only the shape of each bump like height, width, etc., but also the relative distance of the bumps. This is important because, when the inflaton field passes through one bump, it will lose kinetic energy, and if the bumps are far from each other, it may not have enough energy to pass through the next ones. For this reason, we set the bumps close to each other.”

With the potential, they constructed the power spectrum with single or multiple peaks in small scales, while keeping the large scale power spectrum consistent with CMB data. Later, they numerically calculated the abundances of PBHs (fraction to dark matter) at the mass range given by the solo-bumpy potential, as well as three mass ranges given by the multi-bumpy potential. Finally, they found that, PBHs can be formed at different mass ranges, including asteroid mass range (10¯16 − 10¯14M), planet mass range (10¯6 − 10¯3M) and solar mass range (around 1M), some of which can reach significant abundance.

FIG. 2: They plot fP BH for potential (21) given in paper with p = 2 using different threshold densities δc, where the yellow line corresponds to δc = 0.41, the purple line corresponds to δc = 0.46, and the blue line corresponds to δc = 0.486. Their results are consistent with the constraints from current observations. © Zheng et al.

They also found that, the larger threshold energy density (δc) is, the smaller the abundance will be, and this is easy understanding: the larger the threshold energy is, the more difficult it is to form black holes. For the small value of threshold energy density, the abundance of PBHs can reach around 10% of dark matter. The mass range of the PBHs formed is around 10¯15 M, namely the asteroid mass.

Moreover, they also considered the possibility of formation of primordial black holes (PBHs) in the early universe, through ellipsoidal collapse instead of spherical collapse. The difference between these two collapse models is that the threshold density for forming PBH is different. Because compared with the spherical collapse, the PBHs formed by the ellipsoidal collapse will increase the ellipticity of the formed PBHs, which will lead to the correction of the threshold density. Thus, abudance of ellipsoidal PBHs is lower than that of spherical PBHs, due to difference in their threshold densities.

FIG. 3: The figures above show the constraints on primordial black holes acting as dark matter, in which the colored region is excluded by various observations. The blue line correspond to fPBH, and the red line correspond to fe-PBH. The plot is for potential and δc = 0.465. From left to right, the masses of PBHs are 3.6975 × 10¯27 M, 5.8601 × 10¯16 M and 3.6975 × 10¯3 M respectively. Constraints are obtained from the publicly available Python code PBHbounds. © Zheng et al.

Finally, it has been suggested that, considering the age of the universe, PBHs with initial mass less than 1015g (∼ 10¯18 M) has been completely evaporated today. But, the PBHs of mass 3.6975 × 10¯27 (as shown in figure 3 above) may actually be vanishing, and cannot explain the dark matter today. But, although they can’t explain today’s dark matter they may still have a significant impact on the early universe, such as the process of Big Bang Nucleosynthesis, reheating, baryogenesis and so on.

Ruifeng and colleagues suggested that, we can be able to detect the traces left by such PBHs with future observation techniques, to find more evidence of their existence. Meanwhile, for other mass ranges, the PBHs are hardly evaporated till now and thus can act as dark matter.

“We will explore further details on the influences of the PBHs in our model in the future work.”

— concluded authors of the study

Reference: Ruifeng Zheng, Jiaming Shi, Taotao Qiu, “On Primordial Black Holes generated from inflation with solo/multi-bumpy potential”, Arxiv, pp. 1-14, 2021.

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