# Primordial black holes in linear and non-linear regimes

## Abstract

We revisit the formation of primordial black holes (PBHs) in the radiation-dominated era for both linear and non-linear regimes, elaborating on the concept of an apparent horizon. Contrary to the expectation from vacuum models, we argue that in a cosmological setting a density fluctuation with a high density does not always collapse to a black hole. To this end, we first elaborate on the perturbation theory for spherically symmetric space times in the linear regime. Thereby, we introduce two gauges. This allows to introduce a well defined gauge-invariant quantity for the expansion of null geodesics. Using this quantity, we argue that PBHs do not form in the linear regime irrespective of the density of the background. Finally, we consider the formation of PBHs in non-linear regimes, adopting the spherical collapse picture. In this picture, over-densities are modeled by closed FRW models in the radiation-dominated era. The difference of our approach is that we start by finding an exact solution for a closed radiation-dominated universe. This yields exact results for turn-around time and radius. It is important that we take the initial conditions from the linear perturbation theory. Additionally, instead of using uniform Hubble gauge condition, both density and velocity perturbationsmore »

- Authors:

- Department of Physics, Sharif University of Technology, Tehran (Iran, Islamic Republic of)
- School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran (Iran, Islamic Republic of)

- Publication Date:

- OSTI Identifier:
- 22676147

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2017; Journal Issue: 06; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; BLACK HOLES; DENSITY; DISTURBANCES; EXACT SOLUTIONS; EXPANSION; FLUCTUATIONS; GAUGE INVARIANCE; NONLINEAR PROBLEMS; PERTURBATION THEORY; SOUND WAVES; SPACE-TIME; SYMMETRY; UNIVERSE; VELOCITY

### Citation Formats

```
Allahyari, Alireza, Abolhasani, Ali Akbar, and Firouzjaee, Javad T., E-mail: allahyari@physics.sharif.edu, E-mail: j.taghizadeh.f@ipm.ir.
```*Primordial black holes in linear and non-linear regimes*. United States: N. p., 2017.
Web. doi:10.1088/1475-7516/2017/06/041.

```
Allahyari, Alireza, Abolhasani, Ali Akbar, & Firouzjaee, Javad T., E-mail: allahyari@physics.sharif.edu, E-mail: j.taghizadeh.f@ipm.ir.
```*Primordial black holes in linear and non-linear regimes*. United States. doi:10.1088/1475-7516/2017/06/041.

```
Allahyari, Alireza, Abolhasani, Ali Akbar, and Firouzjaee, Javad T., E-mail: allahyari@physics.sharif.edu, E-mail: j.taghizadeh.f@ipm.ir. Thu .
"Primordial black holes in linear and non-linear regimes". United States.
doi:10.1088/1475-7516/2017/06/041.
```

```
@article{osti_22676147,
```

title = {Primordial black holes in linear and non-linear regimes},

author = {Allahyari, Alireza and Abolhasani, Ali Akbar and Firouzjaee, Javad T., E-mail: allahyari@physics.sharif.edu, E-mail: j.taghizadeh.f@ipm.ir},

abstractNote = {We revisit the formation of primordial black holes (PBHs) in the radiation-dominated era for both linear and non-linear regimes, elaborating on the concept of an apparent horizon. Contrary to the expectation from vacuum models, we argue that in a cosmological setting a density fluctuation with a high density does not always collapse to a black hole. To this end, we first elaborate on the perturbation theory for spherically symmetric space times in the linear regime. Thereby, we introduce two gauges. This allows to introduce a well defined gauge-invariant quantity for the expansion of null geodesics. Using this quantity, we argue that PBHs do not form in the linear regime irrespective of the density of the background. Finally, we consider the formation of PBHs in non-linear regimes, adopting the spherical collapse picture. In this picture, over-densities are modeled by closed FRW models in the radiation-dominated era. The difference of our approach is that we start by finding an exact solution for a closed radiation-dominated universe. This yields exact results for turn-around time and radius. It is important that we take the initial conditions from the linear perturbation theory. Additionally, instead of using uniform Hubble gauge condition, both density and velocity perturbations are admitted in this approach. Thereby, the matching condition will impose an important constraint on the initial velocity perturbations δ {sup h} {sub 0} = −δ{sub 0}/2. This can be extended to higher orders. Using this constraint, we find that the apparent horizon of a PBH forms when δ > 3 at turn-around time. The corrections also appear from the third order. Moreover, a PBH forms when its apparent horizon is outside the sound horizon at the re-entry time. Applying this condition, we infer that the threshold value of the density perturbations at horizon re-entry should be larger than δ {sub th} > 0.7.},

doi = {10.1088/1475-7516/2017/06/041},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 06,

volume = 2017,

place = {United States},

year = {Thu Jun 01 00:00:00 EDT 2017},

month = {Thu Jun 01 00:00:00 EDT 2017}

}