Condition to form a primordial black hole that I don't understand

[icalle3]

Homework Statement

Primordial black hole formation

Relevant Equations

no equations

Hi guys, I'm new here.
I am doing my final degree project and it's hard for me to understand what this paragraph means in one of the papers that I'm reading, it's about primordial black hole formation.
[Talking about a spherically density perturbation]

The rarefaction wave starts at the surface χ = χa of the overdense region at η = 0 and propagates inwardly to the center. The compression wave also propagates from the center to the surface outwardly, if there is any inhomogeneity within the overdensity. Since the region is initially expanding and the pressure gradient force generally pushes the fluid outwardly, if the sound wave crosses over the overdense region before the maximum expansion, the dynamics of the overdense region may be strongly affected due to the pressure gradient force so that it may not reach the maximum expansion but continue expanding. We can at least expect that the pressure gradient force significantly delays the collapse in this case. This expectation motivates us to adopt the criterion that if and only if the sound wave crosses from the center to the surface outwardly or from the surface to the center inwardly before the maximum expansion, the pressure gradient force prevents the overdense region from becoming a black hole.
What I don't understand is the criterion that they are adopting. Can someone explain me this? Thank you!

 

[mark726]

Hi there, welcome to the forum! I am a scientist and I would be happy to help explain the paragraph you are having trouble with.

So, this paragraph is discussing the formation of primordial black holes, which are thought to have formed in the early universe. The author is specifically talking about a spherically dense perturbation, which is essentially a localized area of high density compared to the surrounding region.

The first part of the paragraph is describing the behavior of the fluid within this overdense region. The rarefaction wave, which is a decrease in density, starts at the surface of the overdense region and moves inward towards the center. At the same time, a compression wave, which is an increase in density, moves outward from the center towards the surface. This is due to the initial expansion of the region and the pressure gradient force, which is a force that pushes the fluid outward.

Now, the key point here is that if the sound wave (a type of compression wave) crosses over the overdense region before the maximum expansion of the region, the dynamics of the region may be affected by the pressure gradient force and it may not reach the maximum expansion. In other words, the pressure gradient force can delay the collapse of the overdense region.

Based on this understanding, the author is proposing a criterion for determining if an overdense region will become a black hole. If the sound wave crosses from the center to the surface or vice versa before the maximum expansion, then the pressure gradient force is strong enough to prevent the region from collapsing into a black hole.

I hope this helps clarify the paragraph for you. Let me know if you have any other questions!