The Merger Rate of Black Holes in a Primordial Black Hole Cluster

Round 1

Reviewer 1 Report

- The authors consider PBH clusters formed in the early universe (at redshift z>=10^4).
I would like to understand which formation mechanism they rely on, as in the standard scenario,
it was shown with a (cosmological) N-body simulation, at those redshift PBHs are mainly isolated,
see arXiv:1907.08129.

- In Eq. 9, they consider a cluster density profile which scales like r^{-1} for r<r^{-6} for r>>r_0. I would like to ask the authors to justify this choice in the text, as it
appears to be in sharp contrast with the results of numerical simulations (arXiv:1907.08129 and 
their Ref. 13).

- The authors consider a PBH mass function described by a power law in the range (0.01 M_sun , 10M_sun), with also the additional presence of an intermediate mass PBH in the center of the cluster. 
I would like the authors to justify this choice, as the presence of two features for the PBH mass
distribution seems rather difficult to obtain in realistic models of PBH formation.

- I would like the authors to clarify the meaning of the index \alpha in Eq. 5. As they discuss in
the text leading to Eq. 5, the cross section has a scaling with respect to the velocity which goes like v^\alpha. 
I think however, the index of such a scaling is uniquely defined by Eq.~(1). 
I would also suggest to change the following notation for the spectral index of the mass function
in Eq.(10), as it generates confusion for the reader.

- While the authors consider the direct capture mechanism generating BH binaries, it was shown 
that other mechanisms (like three body interactions) may lea to larger merger rates in globular clusters, see for example 2007.02968 and 2012.03585. I would suggest the authors to discuss this scenario as well.

Author Response

Please see the attachment. 

Author Response File: Author Response.pdf

Reviewer 2 Report

Dear Editors,

The authors investigate the merger rate of primordial black holes (PBHs) in the clusters. Although formation of such clusters and the presence of the massive black hole at the center of the cluster are not the universal phenomenon in the PBH scenario, clarifying the merger rate quantitatively in such cases will be useful to constrain some models of the early universe. I do not find technical inconsistency and the results presented look sound. Thus, I recommend the manuscript for publication in the journal.

There is one thing I strongly recommend the authors to rectify. Between eq.(4) and eq.(5), the parameter “alpha” is introduced to represent the dependence of the cross section on velocity. The symbol “alpha” is also used in eq.(10) for the completely different quantity. Different symbols should be used to avoid unnecessary confusion.

Sincerely yours,

Author Response

We are grateful to the referee for the review report. In the revised manuscript, we have taken into account the mentioned recommendation.

Reviewer 3 Report

The authors study the merger rate of primordial black holes in primordial black hole clusters using the Fokker-Planck approach. While common in the study of globular clusters, the latter approach has not been extensively used to study the evolution of PBH structures and its impact on the PBH binary merger rate. This is of the main novelties of the study. The following points should be addressed before the manuscript may be accepted for publication.

1. The article needs moderate language editing. For example, the authors use “merge rate”, “merging rate”, and “mergers rate”, while the commonly used term in the literature is “merger rate”.

2. The introduction is missing a discussion about the origins of PBH clusters. It seems that the authors consider initial clustering instead of the later clustering of PBHs that takes place during structure formation. It should be clearly stated which PBH scenarios are relevant for their study.

3. The PBH merger rate is expected to be dominated by binaries formed in the early universe from compact PBH pairs. This paper considers only binary formation via radiative capture, which is usually subdominant but may be enhanced by (initial) clustering. The authors should compare their results to the common PBH binary formation channels studied in the literature. In particular, they claim that their results can restrict the PBH abundance using LIGO/Virgo data, but there are no computations to back up this claim. Typically, mergers of PBH binaries formed in clusters tend to be too infrequent to constrain the PBH abundance.

4. The cross-section in Eq.~1 describes PBH capture through radiation of gravitational waves, that is, the formation of PBH binaries (not their mergers). Thus, the subsequent treatment assumes that these binaries will merge instantly after formation. This is justified if the dynamical timescales are much larger than the typical merger times. It is known that the presence of binaries in the core of the cluster can significantly alter its dynamics, e.g., stop core-collapse. Can the contribution of PBH binaries, i.e. binary-PBH collisions, be neglected in such clusters? Is the timescale of binary-PBH collisions much larger than the typical merger time?

5. Finally, the presence of primordial binaries can affect the evolution of the cluster. It is known that a relatively large fraction of PBH can form binaries during the first stages of structure formation. How do the results depend on the initial fraction of binaries? Can the initial/primordial binaries be neglected?

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have positively addressed my queries, and improved their manuscript.

I only point out that in Eqs. 5 and 8 it appears to be an undefined $\alpha$, that may have remained from the previous version. I suggest the authors to double check and provide a clarification for that expression.

Author Response

We are grateful to the referee for the review report. In the revised manuscript, we made the correction. 

Reviewer 3 Report

The changes to the manuscript are satisfactory with a few exceptions:

1. The manuscript still needs some language editing, e.g. it still contains sentences like "the question of the black holes merge was not studied".

3 and 5. In the early universe, most binaries form from nearby PBH pairs (see eg 1709.06576 and your refs [10] and [11]) and this mechanism should remain intact if PBHs appear initially in clusters (it is discussed e.g. in 1707.01480 and also in your ref [11]). Many of these binaries collide with other PBH in the cluster, but they can survive (as they are hard) and contribute to the present merger rate (see e.g. 1908.09752). Please include a discussion on the potential effects of such binaries on cluster evolution and on the merger rate. I agree that providing a detailed estimate is complicated and beyond the scope of this study.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 3

Reviewer 3 Report

The authors have addressed my questions and clarified their underlying assumptions and potential caveats. Although some text editing is still required, the manuscript is now suitable for publication.

P.S. There is a missing reference below eq. 6.