Primordial Black Hole Merger Rate in Ellipsoidal-Collapse Dark Matter Halo Models
aa r X i v : . [ a s t r o - ph . C O ] D ec Primordial Black Hole Merger Rate in Ellipsoidal Collapse Dark Matter Halos
Saeed Fakhry, ∗ Javad T. Firouzjaee,
2, 3, † and Mehrdad Farhoudi ‡ Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran Department of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran (Dated: December 6, 2020)We have studied the merger rate in the ellipsoidal collapse model of halo to explain the dark matterabundance by the primordial black holes (PBHs) estimated from the gravitational waves detectionsvia the LIGO detectors. We have indicated that the PBHs merger rate within each halo for theellipsoidal models is more significant than for the spherical models. We have specified that the PBHsmerger rate per unit time and per unit volume for the ellipsoidal collapse halo models is about oneorder of magnitude higher than the corresponding spherical models. Moreover, we have calculatedthe evolution of the PBHs total merger rate as a function of redshift. The results indicate that theevolution for the ellipsoidal halo models is more sensitive than spherical halo models, as expectedfrom the models. Finally, we have presented a constraint on the PBHs abundance within the contextof ellipsoidal and spherical models. By comparing the results with the LIGO sensitivity window, wehave shown that the merger rate in the ellipsoidal collapse halo models enters in the window for thePBHs fraction around 0 .
1, and hence, reinforces the multi-components scenario of dark matter.
PACS numbers: 97.60.Lf; 04.25.dg; 95.35.+d; 98.62.Gq.Keywords: Primordial Black Hole; Dark Matter; Merger Rate per Halo; Ellipsoidal Halo Collapse.
I. INTRODUCTION
The binary black holes detection with the LIGO [1–5]has opened a new epoch in probing the nature and behav-ior of compact objects in our cosmos. In the past years,the gravitational wave detectors have directly confirmedthe existence of black holes [2], and have provided pow-erful tests of general relativity [6]. These detectors arealso guided in the era of multi-messenger astronomy [7].While the gravitational wave observatories are continuedto probe the black holes population, another significantdiscovery arises as to whether mergers may provide directevidence for the existence of PBHs.It is known that the astrophysical objects are orig-inated from the early universe quantum fluctuationswhich became classical as were stretched to super-horizonscales in an exponentially expanding period. If thedensity perturbations of these fluctuations exceed somethreshold value, the PBHs might form. Since passingfrom the threshold value is the critical point of forma-tion, many numerical investigations have been done tostudy the threshold value for the density perturbations,see, e.g., Refs. [8–17]. There might be other formationchannels for PBHs, such as the gravitational collapse in adark sector [18], or the collapse of another compact objectdue to new physics [19], which can also result in observ-able black holes with the non-stellar beginning. Duringthe last two decades, many works have been done in thesubject of PBHs and related area, see, e.g., Ref. [20] and ∗ Electronic address: s˙[email protected] † Electronic address: fi[email protected] ‡ Electronic address: [email protected] references therein.Since the massive PBHs interact via gravitation, andsince a large set of black holes has fluid behavior on suf-ficiently large scales, the PBHs are a natural nominee fordark matter. Nowadays, though the possibility of PBHsexistence is yet neither proven nor refuted, the very ob-servational limits on its abundance represent themselvesa powerful and unique method of investigating the earlyuniverse at small scales, which cannot be tested by anyother method [21–23]. Most serious bounds on the PBHsabundance in the mass range around 10 − M ⊙ canbe obtained from the LIGO observations which are as-sumed in involving the merging of PBHs pairs. Shortlyafter the first observation of a binary black hole merger,in Refs. [24, 25], it has been stated that the merger rateby the LIGO discovery is potentially consistent with amass fraction of PBHs accounting for the total of darkmatter, assuming that the two black holes involved hada primordial origin and LIGO had detected dark matter.Since the PBHs merging happens in the halo and con-sists of a fraction of dark matter, the halo mass functioncan affect the PBHs merging rate. In addition, the con-centration parameter changes the relative velocity distri-bution of PBHs within each halo, which determines thePBHs merger rate R within each halo.On the other hand, there are different types of halocollapse models. The analytically simple model is thespherical collapse which has been found to over-predictthe abundance of small halos and under-predict for themassive ones. This issue is because the halo collapsesare generally triaxial rather than spherical. The Sheth-Tormen (S-T) model [26], uses the ellipsoidal collapsemodel and the obtained fitting functions provide a closermatch to the unconditional halo mass function in N-bodysimulations. Furthermore, the ellipsoidal collapse modelhas its mass-concentration which gives deep insight intothe formation and structure of halos [27].In this work, we propose to use the ellipsoidal collapsemodel to calculate the merger rate of the PBHs, whichare in the dark matter halo. In this respect, the outlineof the work is as follows. In Sec. II, we introduce the darkmatter halo model and its concentration. Furthermore,we discuss the halo mass function for the two sphericaland ellipsoidal collapse models. Then, in Sec. III, we cal-culate the PBHs merger rate in the ellipsoidal collapsemodel and compare it with the corresponding results ofthe spherical collapse model. Finally, we discuss the re-sults and summarize the findings in Sec. IV. II. MODELA. Halo Density Profile
Dark matter halos are considered as nonlinear cosmo-logical structures whose mass can be modeled by a radiusdependence quantity called density profile. Over recentyears, the analytical models and numerical simulationshave provided a clearer picture of the properties and be-havior of these structures. One of the important resultsof these studies is the extraction of various density pro-files, which include a lot of studies in this issue [28–34].Let us mention two of the most commonly used densityprofiles as follows. One of those is the Navarro-Frenk-White (NFW) profile, which is extracted from the N -body simulations and is well compatible with most ofthe rotation curve data [34]. The relation of this densityprofile is ρ ( r ) = ρ s r/r s (1 + r/r s ) . (1)The other one, which is derived from analytical models,is the Einasto profile that is also well consistent with theobservational data [29], and its density profile is ρ ( r ) = ρ s exp (cid:26) − α (cid:20)(cid:18) rr s (cid:19) α − (cid:21)(cid:27) . (2)In these relations, ρ s and r s are the scaled parametersthat vary from halo to halo, and α is the shape parameterfor the Einasto profile. It should be noted that for bothof the above forms, one has d ln ρ ( r ) d ln r = − r/r s = 1 , (3)i.e. the logarithmic slope of the density distribution is − C ≡ r vir r s , (4) where r vir is a viral radius considered as a radius withinwhich the average halo density reaches 200 to 500 timesthe critical density of the universe. Also, the N -bodysimulations show that the concentration parameter is adecreasing function of the halo mass and is a redshiftdependent function in the fixed mass [27, 35–37]. Wewill discuss about the mass distribution of dark matterhalos in the next subsection. B. Halo Mass Function
The existence of dark matter halos provides a con-venient and fundamental framework to study nonlineargravitational collapse in the universe. Hence, having aproper statistical view of the mass distribution of thesehalos can improve our understanding of the physics gov-erning those. With this argument, a function called thehalo mass function has been proposed, which describesthe mass distribution of these halos within a given vol-ume. In other words, the halo mass function describesthe mass of those structures whose overdensities exceedthe threshold overdensity, separate from the expansionof the universe, and collapse. Furthermore, in the stan-dard cosmology, one may define a linear quantity calleddensity contrast as δ ( x ) ≡ [ ρ ( x ) − ¯ ρ ] / ¯ ρ , where ρ ( x ) is thelocal density at any point x and ¯ ρ is the mean backgroundenergy density. As noted earlier, this quantity may growto the critical point while the universe expands, exceedslinear regimes, and enters into nonlinear regimes. Thissituation occurs when the overdensities separate from theexpansion of the universe, enter the turnaround phaseand collapse. That is, the structures are formed at thisstage. For an Einstein-de Sitter universe and a spheri-cally collapse halo model, the threshold overdensity hasbeen calculated to be δ sc = 1 . dndM = g ( σ ) ρ m M d ln( σ − ) dM , (5)where n ( M ) is the number density of halos with mass M , ρ m is the cosmological matter density, and g ( σ ) dependson the geometry of overdensities at the collapse time.The function σ ( M, z ) is the linear root mean square fluc-tuation of overdensities on mass M and redshift z , whichis defined as σ ( M, z ) ≡ π Z ∞ P ( k, z ) W ( k, M ) k dk. (6)In this relation, W ( k, M ) is the Fourier spectrum of thetop-hat filter which depends on mass M and wavenumber k , and P ( k, z ) is the power spectrum of the fluctuations.There is a wide range of studies that have been con-ducted to extrapolate the halo mass function based onanalytical approaches and numerical simulations. Thepurpose of these studies is to provide the best fit forthe cosmic observations. The first model for the darkmatter halo mass function, assuming a homogeneousand isotropic collapse, was presented by Press andSchechter [40] as g ps ( σ ) = r π δ sc σ exp (cid:18) − δ σ (cid:19) , (7)which is called the Press-Schechter (P-S) mass function.This formalism is based on the assumption that every as-trophysical or cosmological object is formed via a gravi-tational collapse of overdensities. Moreover, although thefinal collapse is a nonlinear process, it is assumed that, inthe early universe, the density fluctuations had been verysmall and resulted in a linear approximation. As is clearfrom relation (7), at a fixed redshift, the mass functiondepends only on the mass of halos via σ ( M ), and it is ex-pected that no significant change can be observed. Thismass function has been proposed as the simplest modelfor the formation of dark matter halos, i.e. a sphericalcollapse model, and, in many cases, is consistent withthe observational data. Nevertheless, it quantitativelydeviates from the numerical results at some mass lim-its [39]. Therefore, some improvements have been madeto address this issue. One of the most successful im-provements was provided by Sheth and Tormen, which isbased on a more realistic model and fits simulation resultsbetter [26]. Their formalism was based on an ellipsoidalcollapse model with dynamical threshold density fluctu-ations, in contrast to an almost global threshold in theP-S model.As mentioned earlier, the threshold overdensity forspherical collapses, δ sc , has been introduced as a globalvalue. It means that, in about certain redshifts, all struc-tures with overdensities more than such a threshold cancollapse. Sheth and Tormen have proposed the idea thatdynamically considering the threshold overdensity for theellipsoidal collapses, δ ec , can provide a more realistic pic-ture of the halo mass function. With this assumptionand considering prolateness to be zero [26], they haveextracted this quantity as δ ec ( ν ) ≈ δ sc (1 + γ ν − β ) , (8)with γ = 0 . β = 0 .
615 and ν ≡ δ sc /σ ( M ). It isclear that this quantity not only implicitly depends onthe redshift, but also on the mass of the structure, and iscalled the moving barrier. With this assumption, one canfind the halo mass function for the ellipsoidal collapse,which is also called the S-T mass function, to be g st ( σ ) = a r bπ δ sc σ exp (cid:18) − aδ σ (cid:19) (cid:20) (cid:18) σ δ (cid:19) p (cid:21) , (9)with a = 0 . b = 0 .
707 and p = 0 .
3. This massfunction is expected to be more sensitive than the P-S mass function with redshift changes. Thus, we now have all the tools that one needs to study PBHs in darkmatter holes. In this regard, in the next section, we willtalk about the probability of encountering PBHs, theirbinary formation, and their merger rate within a certainvolume and time interval.
III. PRIMORDIAL BLACK HOLES MERGERRATEA. Merger Rate Within Each Halo
PBHs are a special type of black holes that are formedin the early universe due to the direct collapse of den-sity fluctuations or equivalently curvature perturbations.PBHs were not only able to form binaries when the uni-verse had been dominated by radiation but also couldencounter into each other in the late time universe dueto their random distribution.It is believed that the gravitational wave eventsrecorded by the LIGO detectors can be described by thePBHs scenario with 30 M ⊙ masses if these black holescould be considered as a component of dark matter. As aresult, dark matter halos are expected to contain a widemass spectrum of PBHs.The presence of PBHs with random distributions indark matter halos gives those a chance to form binariesvia the close encounter and emitting gravitational waves.In particular, the smallest dark matter halos, due to theirlower velocity distribution and higher density, are likelyto have the largest contribution to the formation of theblack hole binaries. That is why the probability of en-countering black holes in these halo mass ranges is moresignificant [24, 41].Let us suppose two PBHs with masses m i and m j andrelative velocity v rel = | v i − v j | in a dark matter haloform a gravitationally bound system. Physically at theclosest distance (i.e., at periastron), due to the maximumscattering amplitude, one can expect to have the mostgravitational radiation. The periastron can be estimatedto be r p, max ≃ " π √ G / m i m j ( m i + m j ) / c v / , (10)where G is the gravitational constant and c is the velocityof light. Hence, the cross-section for such an event canbe found as [42, 43] ξ ( m i , m j , v rel ) ≃ πG ( m i + m j ) r p, max v . (11)Our focus is on the merger rate of the PBHsthat are compatible with the LIGO sensitivity, i.e. ∼ (30 M ⊙ − M ⊙ ) events in the galactic halos. Accord-ingly, we have normalized those masses to 30 M ⊙ withtheir relative velocities as the average velocities of darkmatter halos, i.e. 200 km / s.By inserting Eq. (10) into Eq. (11) and assuming m i = m j = M pbh and v rel = v pbh , one can reach an explicitform of the cross-section related to the normalized massand velocity of the PBHs as ξ ≃ π (cid:18) π (cid:19) / M G c / v / ! ≃ . × − (cid:18) M pbh M ⊙ (cid:19) (cid:18) v pbh / s (cid:19) − / in (pc) . (12)With these considerations, the PBHs merger rate pertime within each halo can be calculated via the formula[24] Φ = 2 π Z r vir r (cid:18) ρ halo ( r ) M pbh (cid:19) h ξv pbh i dr, (13)where ρ halo ( r ) is the halo density profile that can be con-sidered to be the NFW or the Einasto density profile,and h ξv pbh i represents an average over the PBHs relativevelocity distribution in the galactic halos.Moreover, the mass located within the virial radius ofthe halo, the virialized mass, can be found by M vir = Z r vir πr ρ ( r ) dr. (14)By inserting relation (1) into relation (14) and integrat-ing, one can find the virialized mass for the NFW densityprofile as M vir (NFW) = 4 πρ s r (cid:18) ln(1 + C ) − C C (cid:19) . (15)Similarly, by considering relation (2), the virialized massfor the Einasto density profile [44, 45] can be obtained as M vir (Ein) = 4 πρ s r l ( C, α ) . (16)In this relation, l ( C, α ) is a function of concentration andshape parameters and has the following form l ( C, α ) = exp(2 /α ) α (cid:16) α (cid:17) /α Γ( 3 α , α C α ) , where Γ( x, y ) = R y t x − e − t dt is the incomplete Gammafunction.To calculate the halo velocity dispersion, one can useits relation to the maximum velocity in a r max radius,which has been introduced in Ref. [35] as v disp = v max √ s GM ( r < r max ) r max . (17)In this work, we assume that the relative velocity dis-tributions of PBHs in a halo are random and follow the -16 -15 -14 -13 -12 -11 (NFW) M e r g e r R a t e P e r H a l o ( y r - ) M h (M sun /h)Okoli-Afshordi ConcentrationLudlow Concentration10 -16 -15 -14 -13 -12 -11 (Einasto) M e r g e r R a t e P e r H a l o ( y r - ) M h (M sun /h)Okoli-Afshordi ConcentrationLudlow Concentration FIG. 1: (color online) The PBHs merger rate in each haloconsidered with the NFW profile (top) and the Einasto pro-file (bottom). The solid (red) lines represent the merger ratefor the ellipsoidal collapse model with the O-A concentration-mass, and the dot-dashed (black) lines show the mergerrate for the spherical collapse model with the Ludlowconcentration-mass relation.
Maxwell-Boltzmann statistics. Hence, one can write thevelocity probability distribution function as P ( v pbh , v disp ) = A " exp − v v ! − exp − v v ! , (18)where A is determined by the normalization conditionand a cutoff is considered at the halo virial velocity.It is clear from Eqs. (13), (15) and (16) that, in or-der to calculate the merger rate in each halo, the mass-concentration relation, C ( M vir ), has to be determined.For this purpose, according to the initial conditions gov-erning the dark matter halos during the collapse, var-ious results can be found. In Ref. [24], the mergerrate has been performed using the two famous spheri-cal concentration-mass relations of Prada, et al. [35] andLudlow, et al. [37].In this research, we have employed the ellipsoidal col-lapse concentration-mass relation introduced in Ref. [27]that we refer to it as Okoli-Afshordi (O-A) concentration-mass relation. Furthermore, for the Einasto density pro-file, we have chosen the value of the shape parameterpresented in Ref. [46]. Also, we have set the mass ofPBHs to be 30 M ⊙ . In Fig. 1, we have indicated thePBHs merger rate per halo as a function of halo massby considering the O-A concentration-mass relation asan ellipsoidal model, and the Ludlow concentration-massrelation for a spherical model obtained in Ref. [24]. Theresults show that the PBHs merger rate grows with in-creasing halo mass for both models. A noteworthy pointis that in smaller mass halos for the case of the O-Amodel the merger rate is almost one order of magnitudelarger than in the Ludlow model.In the following, we propose to determine the effect ofthese changes on the total merger rate of PBHs in a givenvolume and time interval. B. Total Merger Rate
1. Present-Time Universe
Up to here, the merger rate has been considered withineach dark matter halo. However, as the gravitationalwave detectors statistically receive the cumulative events,it is necessary to calculate the total merger event rateper unit volume and per unit time. For this purpose,convolving the merger rate per halo, Φ( M h ), with thehalo mass function, dn/dM h , leads to the total mergerevent rate per unit volume and per unit time as R = Z dndM h Φ( M h ) dM h , (19)where M h is the halo mass, which can be estimated as thevirialized mass, M vir . As is clear from Eqs. (5), (7) and(9), the exponential decay of the mass function meansthat the upper limit of the integral does not affect thefinal result. Instead, the lower limit plays a crucial role.It should be noted that the merger time of PBHs is afunction of the velocity of halos. On the other hand, dueto the instantaneous merger time compared to the age ofthe universe for binary formations, one can neglect thethree-body collisions because these binary systems mergeat a time longer than the Hubble time. That is, why thereshould not be a significant effect on the expected mergerrate in the present-time universe.It is known that the smaller-mass halos are more con-centrated, for these type of halos have already becomevirialized and naturally evaporate faster than the larger-mass halos. For instance, the evaporation time for a halowith a mass of 400 M ⊙ has been estimated to occur atabout 3 Gyr [24]. Naturally, halos with lower mass evap-orate at a lower time scales at which the compensatoryfactors of evaporation become very slow due to the pre-dominance of dark energy effects. Accordingly, we con-sider the smallest halos to be 400 M ⊙ . -8 -7 -6 -5 -4 -3 -2 -1 (NFW) M e r g e r R a t e ( G p c - y r - ) M vir (M sun /h)Ellipsoidal Halo ModelSpherical Halo Model -8 -7 -6 -5 -4 -3 -2 -1 (Ein a(cid:0)(cid:1)(cid:2)(cid:3)M(cid:4)(cid:5)(cid:6) e r R(cid:7)(cid:8)(cid:9) ( G p c - y r - ) M vir (M sun /h)Ellipsoidal Halo ModelSpherical Halo Model FIG. 2: (color online) The PBHs merger event rate per unitvolume and per unit time for the spherical and ellipsoidalcollapse models with the NFW profile (top) and the Einastoprofile (bottom). The solid (red) lines represent the ellip-soidal halo model with the S-T mass function and the O-Aconcentration-mass relation, and the dot-dashed (black) linesshow the spherical halo model with the P-S mass function andthe Ludlow concentration-mass relation.
To quantify the total merger rate introduced inEq. (19), two crucial quantities, namely the halo massfunction and the concentration-mass relation, must bespecified in proportion to the dark matter halo formationconditions. The idea is to look at the PBHs merger ratefor the ellipsoidal collapse halo models. For this purpose,we use the S-T mass function and the O-A concentration-mass relation which have been introduced for the ellip-soidal collapse halo models.Fig. 2 shows the merger rate of the PBHs for the ellip-soidal halo models per unit time and per unit volume, andcompares it with the results of the spherical model, whichhas been evaluated in Ref. [24], while taking into accountthe NFW density profile (top) and the Einasto densityprofile (bottom). In the ellipsoidal model, the S-T massfunction and the ellipsoidal O-A concentration-mass rela-tion have been considered, while in the spherical model,
TABLE I: General information on the total merger rate for the ellipsoidal and spherical models in terms of the two densityprofiles, the NFW and the Einasto, at the present-time universe.Halo Density Profile Halo Mass Function C ( M ) Lower Limit Halo Mass Total Merger Rate( M ⊙ ) (Gpc − yr − )NFW P-S Ludlow 400 1 . . . . the P-S mass function and the Ludlow concentration-mass relation are used. As expected, the total mergerrate of the PBHs for ellipsoidal models, like the sphericalmodels, increases with decreasing halo mass due to thesignificance of merger events in the smallest halos. Forthe halo masses larger than M h > − M ⊙ , themerger rate is approximately the same in both models.However, for masses smaller than M h < − M ⊙ with the ellipsoidal model, it is prominently increased byabout one order of magnitude compared with the spher-ical model. The total merger rate has been obtained byintegrating over the surface below the curves, and the re-sults have been presented in Table. I for the present-timeuniverse.
2. Redshift Evolution of PBHs Merger Rate
Given that the PBHs formed in the early universe andhad the chance to form binaries during the age of theuniverse, make an attractive option to study the mergerrate in small non-zero redshifts. On the other side, thesensitivity of the LIGO detectors can observe the binariesup to z ∼ . − .
3. Constraint on PBHs Fraction
As the last part, let us concentrate on the expectedPBHs fraction, f pbh , extracted from the ellipsoidal col-lapse halo model. The problem of PBHs abundance hasbeen an important issue since the beginning of the emer- (cid:18)(cid:19)(cid:20) (cid:21)(cid:22)(cid:23) (cid:24)(cid:25)(cid:26) (cid:27)(cid:28)(cid:29) (cid:30)(cid:31) (Spherical Model) T!" e r )*+, ( G p c - y r - ) z Einasto
P-.f/45N67 89:;<=>
70 0 ?@A BCD EFG HIJ KLO (Ellipsoidal Model)
QSUVWXYZ[ e r \]^_ ( G p c - y r - ) ‘ Einasto bcdeghijkl mnopqrs
FIG. 3: (color online) The PBHs total merger event rate forthe spherical (top) and the ellipsoidal (bottom) collapse mod-els as a function of redshift. The solid (blue) lines indicatethe calculations considered the Einasto density profile, andthe dashed (red) lines are for the NFW density profile. gence of the PBHs scenario. Moreover, one of the mostimportant constraints imposed on PBHs is their abun-dance in the late-time universe. The fraction of PBHsdetermines their contribution from dark matter. Manystudies have been performed in this area, and today it isbelieved that this fraction is lower than one [23, 25, 51–56]. It means that the dark matter consists of severalcomponents, one of which is the PBHs. For this purpose,one of the best references to investigate suitable mod- -3 -2 -1 -2 -1 tuvwx M e r g e r R a t e ( G p c - y r - ) y{|} Ellipsoidal Model (Einasto)Ellipsoidal Model ~(cid:127)(cid:128)(cid:129)(cid:130)
Spherical Model (Einasto)
FIG. 4: (color online) The PBHs total merger event rate forthe ellipsoidal and the spherical models with respect to thePBHs fraction, f pbh . The solid (red) line indicates the to-tal merger rate for the ellipsoidal model with the S-T massfunction and the Einasto density profile, while the dot-dashed(blue) line shows the same model for the NFW density profile.The dotted (black) line represents the spherical model withthe P-S mass function and the Einasto density profile. Theshaded (orange) band is the estimated merger rate from theLIGO detectors, i.e. (0 . −
12) Gpc − yr − . els of dark matter halos is to compare the merger rateobtained from those models with the merger rate rangedetermined by the LIGO detectors.In Fig. 4, we have depicted the total merger rate asa function of f pbh for the ellipsoidal halo model whileconsidering the two density profiles NFW and Einasto,and have compared the results with the spherical modelused the Einasto density profile [49]. In this figure.the shaded band is the estimated merger event rate(0 . −
12) Gpc − yr − by the LIGO detectors [57]. Theresults indicate that the merger rate of the PBHs for theellipsoidal model is within the wider range of the LIGOsensitivity band than for the spherical model. In otherwords, considering the ellipsoidal model for the collapsedhalos, the abundance of the PBHs is significantly de-creased, which is in agreement with the results obtainedin Ref. [58]. This result means that the ellipsoidal halomodel reinforces the multi-components paradigm of darkmatter. IV. CONCLUSIONS
In this work, we have investigated the PBHs mergerrate by focusing on the ellipsoidal collapse dark matterhalo model. Specifically, to perform this task, we haveconsidered two crucial components that have been calcu-lated for the case of ellipsoidal dark matter halos, namelythe S-T mass function, and the ellipsoidal concentration-mass relation obtained in Ref. [27]. The main idea behind the extraction of these two important components in theellipsoidal collapse halos has been to propose a dynam-ical threshold overdensity, δ ec ( ν ), instead of a constantthreshold one, δ sc = 1 . . −
12) Gpc − yr − . This evaluation is important be-cause it can estimate the contribution of PBHs in darkmatter. We have shown that the total merger rate ofPBHs in the ellipsoidal halo models enters the LIGOsensitivity window for values of about 0.1 of the PBHsfraction. Therefore, the PBHs total merger rate in theellipsoidal model predicts a smaller number for the lowerlimit of the abundance constraint of the PBHs comparedwith the total merger rate of the spherical halo models.This result reinforces the scenario that dark matter iscomposed of several components, one of which can bethe PBHs. Acknowledgments
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