Influence of early dark matter halos on the primordial black holes merger rate

Primordial black hole (PBH) binaries forming in the early Universe may contribute to the merger events observed by the LIGO-Virgo-KAGRA collaborations. Moreover, the inferred merger rate constraints the fraction of PBH with masses $m \sim 10 \, M_{\odot}$ in the dark matter (DM) to $f_{PBH} \lesssim 10^{-3}$. This constraint assumes that after the formation of PBH binaries, they do not get destroyed or their parameters are not perturbed until the merger. However, PBHs themselves contribute to the formation of early DM structures in which the interactions between PBHs take place actively. This leads to the fact that the binaries can be perturbed in such a way that their lifetime becomes longer than the Hubble time $t_H$. In this work, we consider the effect of the initial spatial Poisson distribution of PBHs on the structure formation at the high redshifts $z \gtrsim 10$. Next, we explore the evolution of such halos due to the interaction of PBHs with each other and with DM particles. We show that the early halos evolve on timescales much shorter than the age of the Universe. Furthermore, for fractions of PBHs $f_{PBH}<1$, the internal dynamics of a halo is significantly accelerated due to the dynamical friction of PBHs against DM particles. As a result, a significant fraction of binaries will be perturbed in such structures, and the gravitational waves constraints on PBHs with masses $m \sim 10 \, M_{\odot}$ can be weakened to $f_{PBH} \sim 0.1$.

Despite the fact that PBHs can potentially be part ⋆ vdstasenko@mephi.ru† kmbelotskij@mephi.ru of gravitational wave events, the observed merger rate imposes the strongest constraints on the fraction of PBHs in DM fP BH 10 −3 at tens of solar masses (Sasaki et al. 2016;Ali-Haïmoud et al. 2017;Kavanagh et al. 2018;Pilipenko et al. 2022;Postnov & Mitichkin 2023;Jangra et al. 2023).This constraint suggests that in the early Universe, a pair of PBHs decouple from the Hubble flow and form a binary system (Nakamura et al. 1997;Ioka et al. 1998).Further, these binaries gradually undergo the stage of inspiral due to the emission of gravitational waves and eventually merge.However, from the moment of formation to the merger, the parameters of the binary system can be significantly perturbed so that the lifetime of the binary exceeds the Hubble time tH , as a result of which the merger rate can be reduced (Vaskonen & Veermäe 2020;Jedamzik 2020;De Luca et al. 2020).In addition, if PBHs are initially strongly clustered, the constraint on gravitational waves can be relaxed to fP BH ∼ 1 (Eroshenko & Stasenko 2023).
The discrete nature of PBHs leads to the early formation of structures due to their initial Poisson distribution in space (Meszaros 1975;Afshordi et al. 2003;Inman & Ali-Haïmoud 2019).The forming halos have a much larger mass than predicted in purely adiabatic inflationary fluctuations with nearly scale invariant power spectrum.The evolution of such DM halos containing some amount of PBHs is, in many re-spects, similar to that of globular star cluster; in particular, for very early halos, the phenomenon of core collapse takes place (see for review Spitzer (1987)), time of which is much less than the age of the Universe tH.If a PBH binary finds itself in a halo that experiences core collapse, then it is very likely that it will not contribute to the PBH merger rate (Vaskonen & Veermäe 2020).This is due to the fact that in the process of halo evolution, the probability of perturbing a pair of PBHs increases significantly.However, the analysis of Vaskonen & Veermäe (2020) is somewhat simplified, because it does not take into account the halo density profile and an influence of dark matter particles on the dynamics of PBHs in the halo for the case fP BH < 1. PBHs will experience dynamical friction against DM particles, which leads to a decrease in the core collapse time tcc.The main aim of this article is to find the fraction of binaries that will be in halos that experience core collapse in a time less than the age of the Universe.In this paper, to study the listed effects, the kinetic Fokker-Planck equation is solved.This equation describes the evolution of a self-gravitating system and is often used to study the dynamics of globular star clusters and galactic nuclei (Vasiliev 2017).
In this work, we generally consider the two-component dark matter, which consists of a PBH with mass m = 10 M⊙ (we assume a monochromatic mass spectrum) and some massive unknown particles that gravitationally interact with the PBHs.First, we show how PBHs influence the formation of early DM halos at high redshifts (z 10) due to Poisson noise.Then, the evolution of such halos is considered by solving the Fokker-Planck equation.We show that the timescale of the core collapse depends significantly on the PBH fraction in the DM composition fP BH .Next, we qualitatively consider the dynamics of PBH binaries that form in the early Universe and show that their parameters can be significantly perturbed during halo evolution.Eventually, we show how the modern PBH merger rate is changing and conclude that the constraints can be relaxed to fP BH ∼ 0.1.

EARLY DARK MATTER HALOS
The discrete nature of PBHs induces Poisson fluctuations with amplitude δ ∼ fP BH / √ N , where N is the average number of PBHs in the considered volume (Carr & Silk 2018).Note that these fluctuations are isocurvature perturbations and can have a much larger amplitude than the adiabatic inflationary fluctuations δ ad ∼ 10 −5 ÷ 10 −4 .As a result, on small scales, the matter power spectrum is modified as Here P ad = Ak n with n = 0.961 (Aghanim et al. 2020) and T ad are the matter power spectrum and the transfer function for adiabatic inflationary fluctuations (Mo & White 2002) and PP BH is the PBHs contribution (Mena et al. 2019;Hütsi et al. 2019): where nP BH is the comoving PBH number density and Tiso is the transfer function for isocurvature perturbations, which describes the linear growth of fluctuations (Peacock 1999;Gorbunov & Rubakov 2011) where g(0) ≈ 0.74 reflects the suppression of density fluctuation growth due to the Λ term (Mo & White 2002) and zeq ≈ 3402 is the redshift of matter-radiation equality.Thus, the power spectrum induced by the Poisson noise of PBHs has the form where ρcrit ≈ 126 M⊙ kpc −3 is the critical density.
The variance of fluctuations on the mass scale M = 4πR 3 ρM /3 as the function of the redshift z is given by standard expression where function D(z) = g(z)/[g(0)(1+z)] describes the growth of fluctuations and W (kR) is the window function in the Fourier space (Top-hat spherical filter) On the scales where the main contribution to the power spectrum is due to PBHs (particularly at high redshifts z), the variance of fluctuations will be Here, for simplicity, the contribution of baryons is neglected, i.e. it is assumed that ΩM = ΩDM and the value of the integral from the square window function Let us define the characteristic halo mass (such halos are also called 1σ halos) as σM (M * ) = δc, where δc = 1.69 is the critical fluctuation value for spherical collapse calculated in linear theory.
which approximates well at the redshifts z 10.It can be seen that the presence of PBHs leads to the production of more massive dark halos at high redshifts as compared with the case of purely adiabatic inflationary perturbations.
In the spherical top hat collapse model, after the formation of a structure with mass M , the average density of a halo will be ρH = ∆c ρM (z), where ∆c = 18π where Rvir is the virial radius.At high redshifts z 10 sufficiently dense halos are formed with a small velocity dispersion  5).Dashed lines are given by the approximate Eq. ( 9).
σ ∼ 1 km/s, which leads to active interaction of PBHs with each other.The dynamical evolution of such halos is similar to that of globular star clusters, in particular, the core collapse occurs for them (Lynden-Bell & Wood 1968;Cohn 1980) that is shrinking of the central region with increasing in density, it further increases the rate of PBH interactions.However, the structures are formed hierarchically from the bottom up, which can lead to the destruction of small halos in the process of their absorption by large structures.It is a difficult task to quantitatively take into account the dynamic internal evolution of halos and their interaction with each other during the structure formations.In this paper, we accept the common assumption that the dark halos are not destroyed with time and focus only on the internal dynamics of the halo and its influence on the PBH merger rate.

DYNAMICS OF PBH IN THE EARLY STRUCTURES
Two processes affect the dynamics of PBHs in a halo: the interaction of PBHs with each other and with DM particles.Both these processes can be characterized by diffusion coefficients using the Fokker-Planck equation (Binney & Tremaine 2008).They show the average rate of change in the body's velocity (PBH in our case) as a result of many weak gravitational encounters with other objects.The interaction between PBHs is characterized by the following coefficient where σ is the one-dimensional velocity dispersion, ρBH = ρHfP BH is the density of PBHs in the halo center and ln Λ ∼ 10 is the Coulomb logarithm.The characteristic timescale of PBH interactions is called the relaxation time tr, which is defined as tr = σ 2 / (∆v) 2 (see details in Binney & Tremaine (2008)) If the halo consisted only of PBHs, then its evolution is not different from the dynamics of the simplest globular star cluster and proceeds according to the scenario of a gravitational catastrophe (Lynden-Bell & Wood 1968): under the influence of pairwise interactions of PBHs, the central region of the halo (hereinafter core) is compressed to a sufficiently large density.The core collapse time is a certain number of relaxation times tcc = βtr, where the proportionality constant β is the number of those times which depends on the choice of the initial density profile (Quinlan 1996).Evolution after collapse is driven by three-body interactions with the formation of binary systems, which stop the collapse of the core and lead to the self-similar expansion of the cluster (Lee 1987;Takahashi 1996;Binney & Tremaine 2008).The merger rate of such binaries was estimated in Franciolini et al. (2022), but this channel is less efficient than mergers of binaries formed in the early Universe.
On the other hand, if a halo contains dark matter particles in addition to PBHs, then the interaction of PBHs with DM particles can be characterized by the dynamical friction coefficient (Merritt 2013) where ρDM = (1 − fP BH )ρH is the density of DM particles.Dynamical friction leads to the energy loss of PBHs, as a result of which they settle in the center of the halo.Similarly to the case considered above, the characteristic dynamical friction time can be defined as It is physically obvious that, for a small fraction of PBHs in the DM, their dynamics is determined only by the interaction with DM particles, and the scattering of PBHs on each other is negligible.In the intermediate case, the picture is as follows: in the early stages of evolution, dynamical friction is the dominant process, as a result of which PBHs will sink into the center of the halo.When a sufficiently high concentration of PBHs is reached, their further dynamics will already be determined by pair interactions with each other.From the condition that the diffusion coefficients are equal σ ∆v ∼ (∆v) 2 , one can find the density of PBHs, starting from which the dynamics will be determined only by the pairwise interaction of PBHs.It can be seen that this occurs at ρBH ∼ ρDM , which was obvious from physical considerations.PBH settling occurs on the scale of dynamical friction time Eq. ( 14).Further, as it was outlined, the evolution will occur due to interactions of PBHs with each other on the relaxation timescale Eq. ( 12), where ρBH ≈ ρDM .Therefore it is convenient to choose the following quantity as a characteristic time for the evolution of a halo consisting of both PBHs and DM.Here ρc is the central density of halo and σ is calculated using the Jeans formula (Binney & Tremaine 2008) where ρH = ρBH + ρDM is the DM halo density profile.
For further analysis, it is necessary to determine the halo core collapse time for different fractions of PBHs fP BH in the DM composition.For this, the Fokker-Planck kinetic equation is used, the procedure for numerical solution of which is described in Cohn (1980); Vasiliev (2017); Stasenko et In this work for definiteness, the Burkert profile is used as the initial density distribution (Burkert 1995) where r0 is the radius that determines the size of the halo core.We also assume that both PBHs and DM particles are equally distributed in space.That is, it is assumed that the fraction of PBHs matches fP BH .However, during the process of halo formation in the central region, the PBH number density can be higher, but this requires careful numerical simulation.In this case, the gravitational waves constraints will be relaxed slightly stronger.
As an example, Fig.
(2) shows the evolution of PBHs distribution in the halo formed at z f = 20 (it corresponds t f ≈ 180 Myr).The PBH fraction was chosen to be fP BH = 0.1, and the parameter r0 was chosen so that Rvir/r0 = 5 and the halo mass M = 3•10 4 M⊙, which corresponds to a typical halo forming during this epoch, see Fig. (1).It can be seen that during the time tcc ∼ 1.2 Gyr the core of the halo shrinks to rc ∼ 1 pc while the density increases to ρ ∼ 100 M⊙ pc −3 .We stopped the calculations when the number of PBHs in the central region is N ≈ 20.Subsequent evolution within the framework of the Fokker-Planck equation would lead to further compression of the core to an infinite density, that is unphysical.In a realistic scenario, as noted above, the core collapse is terminated due to the formation of binaries, which act as a heat source and lead to the expansion of the cluster.However, post-collapse evolution may be more complex due to the dominant role of DM particles composing the dark halo.Nevertheless, it can be noted that the natural formation of PBH clusters is possible, namely, small dark structures with a significant concentration of PBHs in the central regions.fractions fP BH .The red dots in the graph are obtained as the result of the numerical solution of the Fokker-Planck equation and are well approximated by the following formula represented by a solid line on the graph tcc t ch = 15.9 1.3 e 2.1f P BH − 1 .
Due to computational difficulties, we considered only halos containing NP BH ≥ 30, so the Fokker-Planck equation was not solved in the region fP BH 10 −2 .In fact, for such fractions fP BH , the number of PBHs in dark halos will be less than 30 in this epoch, see Eq. ( 9).It was found that the core collapse time, expressed in characteristic times Eq. ( 15), is universal and does not depend on the profile parameter r0 and the moment of halo formation, which will be reflected in Fig. (4).However, it may depend on the specific choice of the initial density profile (Quinlan 1996).
The analysis of Vaskonen & Veermäe (2020) suggests that the core collapse time tcc does not depend on the PBH fraction fP BH and corresponds to a halo consisting only of PBHs, which in our case is tcc ≈ 150 t ch .However, as can be seen from Fig. (3) for the PBH fraction fP BH 0.1, such estimation is strongly overestimated.Halos dominated by dark matter evolve faster: as fP BH decreases, the core collapse time becomes constant tcc ≈ 5 t ch , which corresponds to the fact that the PBH dynamics in the halo is mainly determined by dynamical friction.
Let us estimate the characteristic time Eq. ( 15) for halos formed at different z.To do this, we set σ ∼ GM/Rvir and assume that the halo central density is α times bigger than the average matter density in the Universe ρc = αρM (z) where we set ln Λ = 10 and N -the number of PBHs in the halo.The choice of α ≈ 10 4 corresponds to Rvir/r0 = 5 in the density profile of Eq. ( 17).It can be seen that at the PBH fraction in DM fP BH ≈ 0.1, halos formed at z ≈ 10 experience core collapse within the time tcc tH.

PBH BINARIES
In the early Universe, two PBHs at some moment decouple from the Hubble expansion and form a bound binary system, which will subsequently experience the inspiral motion due to the emission of gravitational waves and eventually merge.
The lifetime of such a binary is given by Peters (1964) where m -mass of each PBH, a and j = √ 1 − e 2 are the semimajor axis and dimensionless angular momentum of the binary system and e is the eccentricity.The distribution of parameters a and e (which is easily converted to j) of binaries can be found in Sasaki et al. (2016); Nakamura et al. (1997); Eroshenko (2018).
The merger rate of the unperturbed PBH binaries is given by Vaskonen & Veermäe (2020); Raidal et al. ( 2019) where S is the suppression factor arising due to PBH interactions during the formation of a binary where σM ≈ 0.005 is the variance of matter fluctuations at the moment of PBH pair formation.
As shown in the previous section, early DM halos evolve on timescales smaller than the age of the Universe.Therefore, it can be expected that the parameters of the binaries will be perturbed as a result of scattering with other PBHs.Moreover, these PBH binaries formed in the early Universe are highly eccentric j ≪ 1.The scattering of such binaries with a single PBH will lead to a decrease in eccentricity (Jedamzik 2020), which ultimately leads to a significant increase of the lifetime.Taking this into account, equation of the merger rate (21) will be modified where S f 1 is the suppression factor that shows the fraction of binaries remaining unperturbed and is calculated in the next of a binary system with other PBHs can be divided into the following: encounters with large impact parameters, which are of a tidal nature; and close hard scatterings.
Let us estimate the first.The change in energy in a binary system as a result of such tidal interaction will be (Binney & Tremaine 2008) where b is the impact parameter and v is the relative velocity (further, for estimates, it is assumed that v = σ).The rate of energy change is obtained by the standard procedure of multiplying Eq. ( 24) by a factor nBH σ 2πb db and integration over the impact parameter from bmin = a to ∞ The lower limit of integration was chosen as a based on the consideration that for b < a, it is assumed that the scattering is already strong.It can be shown that an order of magnitude estimate of the change of j is (Ali-Haïmoud et al. 2017) then, the rate of change in j will be Let us estimate the change time of the dimensionless angular momentum by a value of the order of j itself (then the lifetime of binary will increase by the factor ∼ 2 7 , see Eq. ( 20)) as At the beginning of the halo evolution, this time is very small (due to the smallness of ρBH), but then the PBH density begins to dominate and the quantity 1/ √ GρBH becomes about the dynamical time in the center of the halo t dyn ∼ rc/σ, which for ρBH = 100 M⊙ pc −3 (see Fig. ( 2)) is t dyn ∼ 1 Myr.Thus, on the time scales of halo evolution, the characteristic time of the change of j can be estimated as that is several dynamical times, since j ∼ 10 −2 and a ∼ 100 a.u.(Ali-Haïmoud et al. 2017) and rc ∼ 1 pc.Thus, the efficiency of tidal perturbations of binaries regarding changing j becomes important on the timescale of halo evolution.
Another process that may be responsible for the perturbation of the binary parameters is strong scattering with a single PBH.We will assume that the pericenter (the distance of closest approach) for such scattering is rp ∼ a.Then, the cross section of such a process will be Since the binding energy of the binaries is greater than the characteristic kinetic energy of the PBHs in the halo, the gravitational focusing approximation is valid (i.e., the second term dominates in Eq. ( 30)), then the rate of close scattering can be estimated as So, it can be seen that strong scatterings are less efficient than long-range tidal interactions, but, nevertheless, they can also make some contribution to the pertu rbation of binary parameters.Thus, on the timescales of halo core collapse tcc, the parameters of the binaries can be significantly perturbed, leading to an increase in the lifetime of the PBH pairs and a reduction in the merger rate.
An important remark should be made.In addition to the merger of binaries formed in the early Universe, direct mergers of PBHs in the dark halo are also possible (Bird et al. 2016;Clesse & García-Bellido 2017).Since halo evolution leads to an increase in the PBHs number density in the central region, direct mergers of PBHs can make a significant contribution to gravitational wave events.However, accurate accounting for these mergers in the modern era requires knowledge of the abundance of such clusters.As an example, it can be seen from Fig.
(2) that the maximum PBH number density in the halo is reached by redshift z ∼ 4.However, in the further process of structures formation, such clusters can be destroyed, which will reduce the rate of direct PBH mergers.We leave this analysis for the next work.

SUPPRESSION FACTOR AND THE PBH MERGER RATE
To estimate the suppression factor S f in Eq. ( 23), we use the formalism of Vaskonen & Veermäe (2020), the idea of which is to calculate the fraction of PBH binaries that are in halos experiencing core collapse.As shown in the previous section, it is assumed that the lifetime of binaries becomes longer than the Hubble tH and, therefore, they will not contribute to the merger rate.If at some redshift z f a halo is formed containing Nc PBHs and experiencing core collapse to the redshift zc, then a halo containing a smaller number of PBHs N < Nc (but forming at the same redshift z f ) will collapse to redshifts z > zc.The fraction of PBH binaries that will not be perturbed as a result of the evolution of halos formed at z f is estimated as Here pN is the distribution function of the number of PBHs N in the halo which from the Press-Schechter mass function and it is valid at high redshifts (Hütsi et al. 2019) and N * (z) = M * (z)/m is the characteristic number of PBHs in the halo.
The physical meaning of the terms in Eq. ( 32) is as follows: the second term is the probability that the binary is in a halo containing N PBHs, the third term is the probability that the binary is in substructures of the halo with the number of PBHs N ′ > Nc (but inside this large halo there will be substructures with N < Nc).As noted earlier, binaries which are in halos collapsing to the redshift z will not contribute to the merger rate.Therefore, the fraction of these binaries should be subtracted, which is reflected in Eq. ( 32).The further idea is to find the halo formation redshift z f at which the suppression factor will be minimal to the redshift z.This is done as follows: some redshift is taken at which halos are formed.Then, the critical value of PBHs Nc in the halo, which will experience the core collapse to the redshift z, is found.Next, we calculate the suppression factor using Eq. ( 32), then we shift along axis z f in the direction of decreasing S f until its minimum value is reached.To implement this algorithm, it is necessary to calculate the halo core collapse time.This can be done either directly by solving the Fokker-Planck equation or using Eq. ( 18).Since we are interested in the modern merger rate, it is further assumed that z = 0.  18) and by calculating directly the core collapse time using the Fokker-Planck equation.
numerical solution of the Fokker-Planck equation, while the solid curves were obtained using the expression for the core collapse time in Eq. ( 18).As expected, with the increase of the halo density (a decrease of the parameter r0), the suppression factor S f decreases, which is due to the fact that the evolution of the halo proceeds more rapidly, see Eq. ( 19).It should be repeated that we do not take into account the influence of hierarchical structure formation, which can lead to tidal disruption of the halo.However, in such process, the central part of the halo will probably continue to evolve, and, in addition, acceleration of the core collapse is possible (Nishikawa et al. 2020;Quinlan 1996).
It is also important to note that the suppression factor S f does not depend on the PBH mass m.This is due to the fact that the characteristic time t ch depends only on the number of PBHs, see Eq. ( 19).On the other hand, the characteristic number of PBHs in the halo N * also does not depend on the mass of PBH, as can be seen from Eq. ( 9), and also from the fact that Poisson fluctuations are determined only by the number of PBHs δ ∼ 1/ √ N .Fig. ( 5) shows the PBH merger rate with and without the suppression factor (given by Eq. ( 21)), where the parameter r0 was chosen as Rvir/r0 = 5 for density profile in Eq. ( 17).The shaded area corresponds to the measurements of the LIGO-Virgo-KAGRA collaborations.It can be seen that the fraction of PBHs in DM allowed from the point of view of observations of the BH merger rate is relaxed weakened to fP BH ∼ 0.1, which corresponds to the constraints on PBHs at masses ∼ 10 M⊙ in the modern Universe on dwarf galaxies (Brandt 2016;Koushiappas & Loeb 2017) and lensing (Alcock et al. 2000;Oguri et al. 2018).
In the case of PBH clustering stronger than the Poisson noise predicted in works Rubin et al. (2001); Khlopov et al. (2005) (see also the review of Belotsky et al. (2019)), the constraints are likely to be weakened more.Since in this case, the PBH structures will be formed in a younger Universe.These clusters will experience core collapse on smaller timescales (Stasenko et al. 2022), as a result of which most of the binaries will be perturbed compared to pure Poisson clustering.However, in this case, PBH binaries will already be formed in these clusters through dynamical channels: due to the emission of gravitational waves during close approaches (Bird et al. 2016;Clesse & García-Bellido 2017;Stasenko & Kirillov 2021;García-Bellido et al. 2022) and as a result of three-body interactions (Franciolini et al. 2022).The merger rate of these binaries may dominate over early ones, which requires a separate analysis.

CONCLUSION
PBHs with the masses ∼ 10 M⊙ are the subject of active discussion regarding strongest constraints on their contribution to the composition of DM (fP BH 10 −3 ) due to the observation of gravitational wave signals.This constraint assumes that binaries forming in the early Universe are not perturbed and/or destroyed with time.However, the Poisson initial space distribution of PBHs leads to the early formation of a dark halos.In such halos, the probability of perturbing and/or destroying a pair of PBHs is significant.In this work, the dynamics of PBHs in early dark matter structures and its influence on the merger rate were considered.
To study the dynamics of PBHs in early dark halos, the Fokker-Planck kinetic equation was solved numerically.We have shown that the halo core collapse time essentially depends on the fraction of PBHs in the DM composition fP BH .For fP BH < 1, the halo evolves much faster than in the case when the entire DM consists of PBHs, which is due to dynamical friction against DM particles.Halo evolution leads to an increase in the density of PBHs in the central region of the halo, which leads to the perturbation of PBH binaries, as a result of which their lifetime can increase significantly.
The result was obtained under the assumption that early halos are not destroyed during the structure formation and their internal dynamics leads to core collapse due to the in-teraction of PBHs both with each other and with dark matter particles.By calculating the core collapse time, we estimated the suppression factor for the modern merger rate.Namely, it is assumed that if a finds itself in a halo that experiences core collapse to the redshift z = 0, then it will no longer contribute to the merger rate.This is due to the fact that in the processes of interactions of a binary with other PBHs, its parameters will be significantly perturbed, leading to a binary lifetime will exceed the age of the Universe.
Ultimately, we showed that the constraints on the fraction of PBHs in DM can be relaxed to fP BH 0.1, which is compatible with the constraints obtained in the modern Universe from dwarf galaxies and lensing.Moreover, in the case of initial PBH clustering, the constraints will probably be weakened more strongly due to the fact that an even larger fraction of binaries will be perturbed.
Fig. (1) shows the redshift dependencies of the characteristic halo mass calculated for different PBH contributions to DM fP BH , where the dotted lines are obtained with the formula

Figure 1 .
Figure1.The characteristic halo mass as a function of the redshift for different fractions of PBHs.The solid lines correspond to the solution of the equation σ M (M, z) = δc, where σ M is defined by Eq. (5).Dashed lines are given by the approximate Eq. (9).

Figure 2 .
Figure 2. PBH density profile evolution (solid lines) in the halo with mass M = 3 • 10 4 M ⊙ formed at z f = 20 and the PBH fraction f P BH = 0.1.The dashed line is the density profile of DM particles.

Figure 3 .
Figure3.Core collapse time of the characteristic halo mass formed at z f = 10.The red dots are given by the direct numerical solution of the Fokker-Planck equation.The solid curve is given by Eq. (18).

Figure 4 .
Figure4.The suppression factor at z = 0 calculated for different values R vir /r 0 : curves from bottom to top correspond to a decrease of the r 0 .The solid lines and dots are obtained, respectively, by estimating the lifetime from Eq. (18) and by calculating directly the core collapse time using the Fokker-Planck equation.

Figure 5 .
Figure5.The PBH merger rate with the suppression factor (solid line) and without it (dashed line).The shaded area is the PBH merger rate inferred by the LIGO-Virgo-KAGRA collaboration R = 17.9 ÷ 44 Gpc −3 yr −1(Abbott et al. 2023)