GW190521 formation scenarios via relativistic accretion
Alejandro Cruz-Osorio, Fabio D. Lora-Clavijo, Carlos Herdeiro
PPrepared for submission to JCAP
GW190521 formation scenarios viarelativistic accretion
Alejandro Cruz-Osorio, a, Fabio D. Lora-Clavijo b, and CarlosHerdeiro c, a Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Straße 1, 60438 Frank-furt, Germany b Grupo de Investigación en Relatividad y Gravitación, Escuela de Física, Universidad Indus-trial de Santander A. A. 678, Bucaramanga 680002, Colombia c Departamento de Matemática da Universidade de Aveiro and CIDMA, Campus de Santiago,3810-183 Aveiro, PortugalE-mail: [email protected], [email protected], [email protected]
Abstract.
The recent gravitational wave transient GW190521 has been interpreted by theLIGO-Virgo collaboration (LVC) as sourced by a binary black hole (BH) merger. Accordingto the LVC parameter estimation, at least one of these progenitors falls into the so-calledpair-instability supernova mass gap. This raises the important question of how and whenthese progenitors formed . In this paper we analyse the scenario that the GW190521 originalprogenitors (OPs) formed at lower masses and grew to their estimated LVC parameters byrelativistic accretion. We consider the cosmic plasma, or the environment where the binarysystem is immersed, has density gradients as well as a dependence on the Mach number ofthe gas. Taking the LVC parameter estimation at z = 0 . as the endpoint of the accretionevolution, we estimate the initial masses of the progenitors at three different redshifts z =100 , , and , using the relativistic accretion formulas computed from general relativistichydrodynamics simulations. We found three distinct possible types of OPs: ( i ) 10 − M (cid:12) − M (cid:12) primordial BHs, for an early universe, z ∼ , origin; ( ii ) 3 M (cid:12) − M (cid:12) stellar mass BHs; ( iii ) 40 M (cid:12) − M (cid:12) BHs, which could originate from the collapse of high mass Pop III stars.The mass spread is due to varying the density gradient and the relativistic Mach number ofthe cosmic plasma; the variation of the masses due to the origin at different redshifts, on theother hand, is negligible, ∼ . We have also compared our results with previous studieswhere the Newtonian accretion model was used, finding relativistic corrections of ∼ forthe OPs masses. In particular, the relativistic model leads to smaller initial masses. Besidesthe initial masses, we have also computed the initial spins and bolometric bremsstrahlungluminosities for each of the GW190521 progenitors, according to our scenario. Corresponding author. a r X i v : . [ a s t r o - ph . H E ] J a n ontents The LIGO-Virgo scientific collaboration (LVC) has reported the detection of gravitationalwave transients since 2016 [1–8], opening up a new window towards highly relativistic systems,such as neutron star (NS) and black hole (BH) binaries, and providing new informationabout their populations in the Cosmos. The mass of the objects inferred from these eventsspans a considerable range; for the final object, for instance, the LVC reported . +0 . − . M (cid:12) ,for GW170817, interpreted as a NS-NS binary coalescence and +0 . − . M (cid:12) for GW150914,interpreted as a BH-BH binary merger. Yet, the masses of the progenitors in all eventsreported (until recently) correspond to the expected masses for NSs or stellar mass BHs(sBHs). Recently, the LVC reported the transient GW190521 [9], interpreted as the merger oftwo BHs with masses +21 − M (cid:12) and +17 − M (cid:12) , which implies at least one of these progenitorsis in the so called pair-instability supernova mass gap [10]. This led to a debate on how theGW190521 progenitor BHs formed, and even on the nature of the event, see e.g. [11, 12].A natural scenario is that the progenitors of GW190521 started off as lower mass BHs andgrew up to the LVC reported masses by accretion . A model to tackle gas dynamics around apoint mass was developed by Bondi, Hoyle and Lyttleton (BHL) [13–15]. This model assumesaccretion onto a central compact object moving in a homogeneously distributed gas and hasbeen widely used in different relativistic astrophysical scenarios, where the accreting objectcorresponds to a BH or a NS. Several numerical works have been dedicated to studying themorphological patterns in the vicinity of the BH as well as the mass accretion rates [16–20]. Additionally, many numerical simulations of the BHL mechanism have been developedin the context of relativistic astrophysics [21–24], which have been made evermore realistic(and complex) by including magnetic fields [25], radiative terms [26], density and velocitygradients [27, 28], small rigid bodies around the BH [29] and, more recently, by analyzingthe common envelope phase in the evolution of binary systems [30], where useful relativisticexpressions for the mass and angular momentum accretion rates, as well as luminosity, wereobtained in terms of the density gradient parameter and Mach number. These expressionscan be used to understand the behavior of the progenitors of the GW190521 event, at earlystages, and so establish constraints on the initial masses and luminosity of this binary systemin the relativistic regime.Other recent works have also attempted to understand the GW190521 progenitors atearly stages. For instance, Ref. [31] argues that GW190521 cannot have been sourced byprimordial BHs (PBHs) if PBHs do not accrete during their cosmological evolution. On theother hand, Ref. [32] proposes that the GW190521 binary system formation is possible dueto gas accretion onto BH remnants from Population III (Pop III) stars, born in high redshift– 1 –ini-halos. Moreover in Ref. [33] a gas accretion driven mechanism that can build up BHmasses rapidly in dense, gas-rich nuclear star clusters in presented. In Ref. [34], the authorsshow how the growth of sBH, embedded in a dense molecular cloud, can be associated withthe GW190521 event. It is worth mentioning that in all these works the gas around the BHis described by the Newtonian mass accretion formula. Another scenario is presented in Ref.[35], where the GW190521 progenitors are proposed to be the result of the merger of thecentral BHs from two ultra–dwarf galaxies. Furthermore, the possibility that the GW190521binary system comes from the first generation of massive, metal-free, Pop III stars, is studiedin Ref. [36].Another interesting aspect of GW190521 is that the total mass of the GW190521 remnantis +28 − M (cid:12) , which can be considered as an intermediate massive BH (IMBH). IMBHs arean elusive class of BHs, albeit the logical intermediary between sBH and supermassive BHs(SMBH), with a mass range from M (cid:12) to M (cid:12) . If IMBHs are produced by accretiononto sBHs or mergers of massive stars or sBHs, then IMBHs should be abundant. On theother hand, if IMBHs are only produced from core-collapse of Pop III stars, they may havealready grown into SMBHs and be rare; in this scenario, IMBHs are the potential seeds ofSMBHs [37, 38]. IMBHs are excellent candidates to explain the observed Ultra LuminousX-ray (ULX) sources, even though many ULXs have been identified with low-mass X-raybinaries [39] and high-mass X-ray binaries [40, 41]. There seems to be, however, a goodcandidate for an IMBH in a ULX where using high-frequency QPOs (with 3:2 ratio), Pashamet al. [42] used inverse-scaling of sBH as well as a relativistic precession model to determinea mass of M BH (cid:39) M (cid:12) .IMBHs impact on several fields of astrophysics and are likely to grow as a focus ofresearch attention [37]. A standard scenario is that these BHs result from accretion ontoBH seeds or PBHs. The growth of PBHs could explain the masses of IMBH since these cangrow up to − M (cid:12) during the radiation dominated era [43]. Moreover, in the earlytime of BH formation, dark matter makes BHs grow and facilitates IMBHs formation [44].Ref. [45] studied the dynamical evolution of IMBHs in the nearby Universe, in the contextof popular hierarchical structure formation theories. This computational analysis estimatedthe number of BHs, their mass distribution and position within a galaxy. Additionally, Ref.[46] discussed the capability of a third-generation ground-based detector, such as the EinsteinTelescope (ET), to enhance the astrophysical knowledge through detections of gravitationalwaves emitted by IMBH binary systems.This article aims to explore the initial masses as well as the luminosity of the pro-genitors of the gravitational wave event GW190521, through a relativistic mass accretionformula, which includes the effects of the density gradients associated with the environmentwhere the massive binary system is immersed. Concretely, we have computed the masses,the bremsstrahlung bolometric luminosity, and the spin as a function of the redshift, for theindividual components of the GW190521 binary system estimated by the LVC collaboration.Our results exhibit, depending on the initial Mach number and density gradient parameter,a wide range of values for the initial masses, which consequently admit a variety of inter-pretations, from seed BHs formed from Pop III stars collapse, to PBHs. We also perform acomparison between the Newtonian and relativistic regimes, showing that relevant correctionsfor the initial masses are introduced by the relativistic treatment.This article is organized as follows: In section 2, we present the relativistic accretionmodel as well as the general relativistic equations used in our calculations. In section 3, wepresent the results. Finally, in section 4, we present our main conclusions and discussions.– 2 – General relativistic accretion model
BH mass and spin evolutions.
A Kerr BH with mass M and specific spin a may haveits mass and spin grow through the accretion of the surrounding plasma. As described in[47], each plasma mass element, dM , falling from the innermost stable circular orbit (ISCO)into the BH, induces a specific energy E = E ( a, M ) and a specific angular momentum L = L ( a, M ) , and produces a change in the total mass and angular momentum of the Kerr BHgiven by dM = E ( a, M ) dM and dJ = L ( a, M ) dM , where E ( a, M ) = (cid:114) − M r ISCO , (2.1) L ( a, M ) = 2 M √ (cid:32) (cid:114) r ISCO M − (cid:33) , (2.2)the ISCO radius as function of ( M, a ) of the Kerr BH being given by r ISCO = M (cid:104) Z − (cid:112) (3 − Z )(3 + Z + 2 Z ) (cid:105) , (2.3)and Z ≡ − a ) / (cid:2) (1 − a ) / + (1 + a ) / (cid:3) , Z ≡ (cid:112) a + Z . From the inducedquantities, the BH spin evolution is described by [48] ˙ a = (cid:18) L ( a, M ) M E ( a, M ) − a (cid:19) ˙ MM , (2.4)where ˙ M is the mass accretion rate and it is described below. Plasma rest mass accretion rates.
Following the disk model introduced in Ref. [48] weconsider that most of the gas is concentrated in a thin disc and that most of the accretionoccurs in the equatorial plane. Under this assumption, the accretion rates have been computedby performing systematic relativistic hydrodynamics simulations of BHL accretion, includingdensity and velocity gradients [27, 49]. The accretion model herein only considers a purebaryonic gas accretion where such a gas is described by an ideal gas equation of state p = ρ(cid:15) (Γ − , neglecting the radiation accretion. These simulations yield a semi-analyticalmodel for the evolution of the mass M , specific angular momentum (here denoted P φ ), dragforces F x , and bremsstrahlung luminosity L BR , as a function of the plasma initial density ρ ∞ ,velocity v ∞ , sound speed c s, ∞ , adiabatic index Γ , dimensionless density gradient parameter (cid:15) ρ , and BH mass M as follow log (cid:32) ˙ M ˙ M ref (cid:33) = µ + µ / (1 + µ (cid:15) ρ + µ (cid:15) ρ ) , ˙ M ref = 4 πλρ ∞ M / r / , (2.5) log (cid:32) ˙ P φ ˙ P ref (cid:33) = ℘ + ℘ (cid:15) ρ + ℘ (cid:15) ρ , ˙ P ref = ˙ M ref v ∞ (cid:112) − v ∞ /c , (2.6) F x = ˙ P ref (cid:0) ω + ω (cid:15) ρ + ω (cid:15) ρ (cid:1) , r acc := GMc s, ∞ + v ∞ , (2.7) log (cid:18) L BR L Edd (cid:19) = (cid:96) + (cid:96) (cid:15) ρ + (cid:96) (cid:15) ρ . (2.8) We are only considering redshifts up to z = 100 and the cosmological radiation era occurs for z (cid:38) . – 3 –ere, the normalization of accretion rates ˙M ref and ˙P ref are the relativistic spherical mass andangular momentum accretion rates, respectively [16]. In the above relations we introduced theaccretion radius r acc and the parameter λ := 0 .
71 (Γ = 4 /
3) = 0 .
25 (Γ = 5 / ; in our model weuse the adiabatic index Γ = 5 / . The luminosity is normalised by the Eddington luminosity L Edd = 1 . × (M / M (cid:12) ) erg / s [50]. The coefficients in the accretion formulas are: for themass accretion rate, µ i = (0 . , . , − . , . ; for the angular momentum accretion rate, ℘ i = (0 . , . , . ; for the drag forces in the x -direction, ω i = (2 . , . , − . ; andfor the bremsstrahlung luminosity, (cid:96) i = ( − . , . , . . The dimensionless densitygradient parameter range considered is (cid:54) (cid:15) ρ (cid:54) [27, 49]. Intergalactic density and sound speed evolution.
The evolution of the cosmic gasdensity and sound speed after the recombination ( z ∼ - assuming a pure hydrogen gas)can be defined as a function of the redshift z , in cgs unit, as follow ρ ( z ) = 200 m H (cid:18) z (cid:19) gcm , (2.9) c s ( z ) = 5 . (cid:114) z (cid:34) (cid:18) z dec z (cid:19) β (cid:35) − β ) kms , (2.10)where β = 1 . , z dec = 130 is the redshift when baryonic matter decouples from the radiationfluid and m H = 1 . × − g is the hydrogen mass. The sound speed is defined assumingthe plasma temperature decreases in an adiabatic way after z = 100 , when the electrontemperature decouples from the cosmic microwave background (CMB) [31, 51]. The densitydefined in (2.9) will be used in (2.5) and redefined as ρ ∞ ; furthermore, for the model where adensity gradient is also considered, such density will be representing the density at the positionof the BH, r , which corresponds to the center of the gradient direction (the density gradient isapplied in the perpendicular direction to the BH motion) ρ ( r ) = ρ ∞ exp[ − (cid:15) ρ ( r − r ) /r acc ] , seeRef. [49] for more details. Meanwhile, the sound speed in (2.10) is used to define the velocityof the plasma surrounding the binary BHs through the relativistic Mach number definition M := W v/W s c s , where W is the Lorentz factor and W s is the Lorentz factor of the soundspeed. The evolution of the plasma density, sound speed, and a few representative velocitiesas a function of the redshift z have been plotted in Figure 1, spanning a time interval fromthe early universe, z = 100 , to the LVC estimated redshift for GW190521, z = 0 . . Accretion into the individual BHs in the binary system.
The mass accretion ratefor the individual GW190521 progenitor BHs is measured using the global accretion rate ofthe total mass of the binary system, following the approximation from [52] ˙ M = ˙ M bin (cid:112) q ) , ˙ M = ˙ M bin √ q (cid:112) q ) , (2.11)where q = M /M (cid:54) is the mass ratio and ˙ M bin is the accretion rate of the binary system,neglecting the eccentricity contribution and assuming that the orbital period of the binaryis much smaller than the accretion time scale. For z (cid:54) the accretion time scale τ acc is λ comes from the mass accretion rate of relativistic spherical accretion (analytical solution). The adiabaticindex corresponds to an ionized hydrogen plasma with non-relativistic protons and non-relativistic electrons. – 4 – [ g / c m ] v , c s [ k m / s ] c s v ( = 1.1) v ( = 2) v ( = 5) v ( = 10) v ( = 15) Figure 1 . Rest mass density (left) and sound speed (right) profiles of the cosmic plasma as functionsof the redshift z , given by (2.9) and (2.10). The right panel also shows some representative cases forthe gas velocity evolution, assuming constant relativistic Mach numbers M = 1 . , , , , and .The x -range covers only the redshifts relevant for our analysis. smaller than the typical age of the universe, the accretion can therefore play an importantrole in the mass evolution of the BHs [51]; thus our calculation will starts at z = 100 .The procedure to compute the mass and spin evolution of the individual BHs in thebinary is as follows:• The binary mass accretion rate ˙ M bin is computed using the analytical formula given in(2.5) together with the density defined in (2.9), the sound speed presented in (2.10) andthe velocity from the relativistic Mach number.• The individual mass accretion rates, specific angular momentum, drag forces, and lu-minosity are estimated using the relations given in (2.11).• Spin rates for each BH are computed using (2.4).• The individual mass and spin evolution is computed at each time step as follows: M = M i, + ∆ t ˙ M , M = M i, + ∆ t ˙ M , (2.12) a = a i, + ∆ t ˙ a , a = a i, + ∆ t ˙ a , (2.13)where the index i indicates the values at the initial time, and ∆ t is the time step of theuniverse evolution written in terms of the redshift and the Hubble constant parameter H = 67 . − ) Mpc − .We investigate the binary BH evolution in the redshift range z ∈ [100 , . ; as alreadymentioned, z = 0 . corresponds to the GW190521 merger time estimated by the LVC, atwhich point the final masses and spins (following the LVC estimates [9]) are M = 85 +21 − M (cid:12) , M = 66 +17 − M (cid:12) , a = 0 . +0 . − . , a = 0 . +0 . − . . (2.14)Our resolution time step is defined by the step in the redshift, ∆ z = 10 − . This means thatthe mass and spin backward integration have been computed over ∼ time steps. Theparameters space analyzed in this paper consist in . × accretion scenarios varying therelativistic Mach number M ∈ [1 , and density gradient (cid:15) ρ ∈ [0 , in the plasma.– 5 – M , [ M ] m i,1 = 22.00Mm i,2 = 13.00Mm i,1 = 19.48Mm i,2 = 11.01M L u m i n o s i t y , [ e r g / s ] Sp i n a / a i +1a i,1 = 0.68951a i,2 = 0.72963 Figure 2 . Evolution of the BH mass (top left), bremsstrahlung bolometric luminosity (top right)and spin normalized by the initial spin (bottom) of the GW190521 progenitors, vs. the redshift, usingthe relativistic accretion model. Here, we assume that accretion process takes place between z = 50 and z = 10 . The blue and red lines shows the evolution that match the LVC estimated parametersat the end of the accretion ( M = 85 M (cid:12) , M = 66 M (cid:12) , a = 0 . +0 . − . and a = 0 . +0 . − . ). Thegray and magenta lines correspond to evolutions starting with the initial masses m i, = 22 M (cid:12) and m i, = 13 M (cid:12) that, according to the Newtonian accretion model [53], match the LVC estimatedparameters at the end of the accretion; such initial data clearly overshoots the GW190521 parameters,under a relativistic evolution. For this example we took the interstellar gas moving with twice thesound speed (Mach two) with respect to the BH frame. Comparison between the relativistic and Newtonian models.
In Figure 2 we showthe evolution of: the BH masses, in solar masses units (top left panel), bremsstrahlungbolometric luminosity (top right panel), and BH spins, normalized to the initial spins (bottompanel), of the individual BHs in the binary. To perform a comparison between the previousresults using the Newtonian model and our relativistic accretion model we proceded as follows.First, we have performed a forward integration (with our relativistic model) using the initialBH masses ( m i, = 22 M (cid:12) and m i, = 13 M (cid:12) ) previously obtained using the Newtonianmodel [53], that, according to the latter, lead to final masses at z = 10 coinciding with theLVC parameter estimation. For a fair comparison, we have only considered redshifts in therange z ∈ [50 , , gas Mach number M = 2 and no density gradients ( (cid:15) ρ = 0 ). The grayand magenta lines show such evolutions; the final masses are over 200% larger than thoseestimated for GW190521. Second, we have performed a backward integration using the final– 6 –VC parameters; the blue and red lines show such evolutions, from where we found the initialmasses m i, = 19 . M (cid:12) and m i, = 11 M (cid:12) ; these are ∼
13% smaller than the Newtonianestimations. Both models produce low luminosity emissions, less than erg/s . Small vs. large redshift accretion.
Figure 3 shows evolutions for three different scenariosfor the origin of the GW190521 progenitors, all obtained by backwards integration fromthe LVC parameters at redshift z = 0 . , and assuming that the gas is moving supersonicvelocity, M = 2 . The first scenario traces the progenitors back to the early universe, at z = 100 , obtaining initial masses m i, = 2 . M (cid:12) and m i, = 0 . M (cid:12) (orange and greensolid lines); the second scenario traces the progenitors only back to z = 50 , still obtainingsimilar initial masses (to the first scenario), m i, = 2 . M (cid:12) and m i, = 0 . M (cid:12) (blue andred dashed lines); the third scenario only evolves back to z = 20 , but still obtains similarmasses, m i, = 2 . M (cid:12) and m i, = 0 . M (cid:12) (purple and gray dotted lines). Thus, the BHs M , [ M ] m i,1 = 2.319Mm i,1 = 2.329Mm i,1 = 2.366M m i,2 = 0.182Mm i,2 = 0.185Mm i,2 = 0.195M z = 100z = 50z = 20 L u m i n o s i t y , [ e r g / s ] Sp i n a / a f a i,1 = 0.68938a i,1 = 0.68938a i,1 = 0.68938 a i,2 = 0.72955a i,2 = 0.72955a i,2 = 0.72955 Figure 3 . Evolution of the BH mass (top left), bremsstrahlung bolometric luminosity (top right) andspin normalized by the initial spin (bottom) of the GW190521 progenitors, vs. the redshift, using therelativistic accretion model. Here the evolutions assume the final data given by the LVC at z = 0 . for GW190521. The backwards evolution goes back to z = 20 (purple and gray dotted lines), z = 50 (green and orange dashed lines) or z = 100 (blue and solid lines). The results shows that the BHsgain less than of their masses from redshifts − . gain most of their masses in the final stage of the evolution, for redshifts z < , where anexponential increment of their masses occurs. Mass (top left), luminosity (top right) and spin(bottom) show the same behavior; even more dramatic, the spins are almost constant up to z ∼ for all three scenarios. The putative different redshift origin of the progenitors leadsto only to differences of ∼
2% in the initial masses and no differences in the initial spins, for– 7 –ormation at high ( z = 100 ), moderate ( z = 50 ) or low ( z = 20) redshifts. This suggests threedifferent possible origins for the GW190521 progenitors: as primordial BHs at high redshift,from the collapse of dense clumps at moderate redshift, or as core-collapse of Pop III stars atlow redshift. (m i, 1 ) [M ] . . . . . (m i, 2 ) [M ] - . - . . . . . i, 1 0.72925 0.72950 0.72975 0.73000 i, 2 Figure 4 . GW190521 OPs masses (top) and spins (bottom) at z = 100 in logarithmic scale asfunction of relativistic Mach number M and density gradient in the interstellar medium (cid:15) ρ . All pointsare obtained integrating back from the LVC estimated final masses and spins at redshift z = 0 . [9]( M = 85M (cid:12) , M = 66M (cid:12) , a = 0 . and a = 0 . ). The plots shows . × solutions varyingthe interstellar gas velocity and density gradients. Initial BHs mass and spin dependence on the gas velocity and density gradients.
The previous analysis established the insensitivity of the OPs parameters to the cosmological– 8 –poch of formation, as long as such formation occurs for z (cid:38) . Let us now consider thedependence on the details of the accretion. In Figure 4 we show initial masses in solarmasses m i , (top left) and m i , (top right) in logarithmic scale integrating back from the LVCestimated parameters at z = 0 . back to redshift z = 100 . The bottom panels exhibit thecorresponding spins a i , and a i , . The plots were obtained from . × accretion models,varying the relativistic Mach number of the gas, considering supersonic relative motion withMach numbers in the range of M ∈ [1 , , and density gradient in the plasma (cid:15) ρ ∈ [0 , .The backward integration of the relativistic accretion equations reveals that some modelsare self-constrained. Observe the white regions in the right bottom region of the plots; thisexcluded region is delimited by small mass BHs, from a few to − solar masses. Such BHsare only allowed if formed in the early universe as PBHs. On the other hand, for low-densitygradients ρ < . and high Mach number M > we have a family of high mass BHs; in suchmodels the progenitors keep an almost constant mass and spin from their formation up tothe merger at z = 0 . , accreting only a very small fraction of their mass. The largest initialmasses found are m i, = 51 . M (cid:12) and m i, = 37 . M (cid:12) , around of the final masses.In Table 1 we display nine representative models, varying the density gradient andMach number for each formation scenario. Our relativistic accretion model also allows sBHsas the OPs, for some range of density gradients and Mach numbers. Such binary systemscan originate from the collapse of dense clumps and core-collapse of hyper-massive stars,gaining their masses through accretion and reaching the observed masses. Our analysis showsonly small variations of BH spins; we found that the maximum changes are around . regardless of the density gradient, Mach number or epoch of formation. The bolometricluminosity generated by the accretion process is of the order of erg / s in all cases, atthe merger. This low value is consistent with the LVC conclusion that no electromagneticemission was detected. Table 1 . Representative values for the mass and spin of the GW190521 OPs at z = 100 , , and ,by assuming that the final quantities match the LVC values at z = 0 . . As illustrations, we showthe models for density gradients of the interstellar medium (cid:15) ρ = 0 , . and . , and three relativisticMach numbers M = 2 , , and (the minimum allowed for (cid:15) ρ = 0 . is slightly higher). (cid:15) ρ M m i , l [M (cid:12) ] m i , [M (cid:12) ] a i , l a i , z = 100 z = 50 z = 20 z = 100 z = 50 z = 20 z = 100 z = 50 z = 20 z = 100 z = 50 z = 20 Figure 5 shows the evolution diagrams for the BH spins and luminosity for density gra-dients (cid:15) ρ = 0 (top row), 0.2 (middle row) and 0.4 (bottom row). The BH spins and accretionluminosity evolution shows a tiny increment for low Mach numbers, M < , occurring only(and exponentially) in the last stage of the evolution. For high Mach numbers M > , weobserve a smooth growth, when increasing the density gradients the accretion at the begin-ning of the evolution is small for all cases (see the second and third rows) and only in the laststage we observe an exponential growth ( . < z < ).– 9 – . . . . Spin1 . . . Spin2 z . . . . . . . . . . . . . . log (L br ) [erg/s] log (L br ) [erg/s] Figure 5 . Evolution of the spin and luminosity for the two component of the binary BH system.Each panel shows the evolution of the physical quantities in the redshift range z ∈ [100 , . varyingthe relativistic Mach number. The rows corresponds to constant density gradient in the medium: (cid:15) ρ = 0 (top panel), (cid:15) ρ = 0 . (middle panel) and (cid:15) ρ = 0 . (bottom panel). Motivated by the recently reported GW190521 gravitational wave transient, and the factthat the estimated masses of the two BH progenitors for this event fall into the so-calledpair instability supernova gap, in this paper we have studied the role of accretion, in arelativistic treatment, in the evolution of these progenitors. Thus, we have explored thepossible initial masses for the progenitors, as well as the luminosity of the IMBH detectedfrom the gravitational wave event GW190521, by using a general relativistic mass accretionformula. This formula allows to include the effects of the density gradients associated withthe environment where the massive binary system is immersed.Concretely, we have computed the masses, the bremsstrahlung bolometric luminosity,and the spin as a function of the redshift of the epoch of formation of the progenitor BHs.The results allow a wide range of values for the initial masses, depending on the initial Machnumber and density gradient parameter, which can thus originated from different astrophys-ical scenarios at different redshifts. We have carried out a comparison between the initialmasses of the GW190521 binary system, predicted using the Newtonian and relativistic ex-pressions of the mass accretion rates. For this comparison, we assume that the BHs formedat z = 50 and merged at z = 10 with final masses M = 85M (cid:12) and M = 66M (cid:12) . In theNewtonian case, the initial masses found for the individual progenitors were m i , = 22M (cid:12) and m i , = 13M (cid:12) , while using the relativistic formula the initial masses were m i , = 19 . (cid:12) – 10 –nd m i , = 11M (cid:12) , which are smaller than in the Newtonian case. Moreover, if we usethe relativistic accretion formulas to evolve the initial masses previously estimated with theNewtonian treatment, m i , = 22M (cid:12) and m i , = 13M (cid:12) [31], the final masses of the systemare larger than the LVC estimated ones. These calculations mean that important cor-rections are introduced by considering the relativistic formula [49], as expected, since theseare strong gravity systems. It is worth mentioning that in order to perform this comparisonwe did not include density gradients ( (cid:15) ρ = 0 ), since Newtonian mass accretion rate does notinclude this kind of information about the environment wherein the massive binary system isimmersed.The main features of our results have been quantified by analyzing the initial masses,the bremsstrahlung bolometric luminosity and the spins of the GW190521 OPs, as functionsof the relativistic Mach number and of the density gradients of the environment were thesystem is immersed. Here we assume that the observed masses M = 85M (cid:12) and M = 66M (cid:12) were measured at z = 0 . consistently with the LVC parameter estimation for GW190521.We have found from relativistic accretion calculations that the BHs detected from GW190521could originate at considerably different redshifts, say, z = 20 , , and , with only smallvariations in their mass, of ∼ . The analysis also shows that the individual spin grow onlyaround from the initial time at z = 100 up to the merger at z = 0 . , where the finalspins were chosen to match the LVC estimates, a = 0 . and a = 0 . , respectively [9].These conclusions hold regardless of the density gradients and the relativistic Mach numberof the cosmic plasma, see Table 1 and Figure 5. Thus, these results indicate that each BHsourcing GW190521 was formed with almost the same spin as that detected at the merger.This indication is also consistent with what is expected from a relativistic accretion modelfor low-density plasma, where the accreted angular momentum is proportional to the plasmadensity and mass accretion rate (see Section 2), and thus the growth of the spin due to theinduced angular momentum onto the BH is also small [48].A similar trend occurs for the luminosity, which showed an increment from z = 100 upto z = 0 . , of around three orders of magnitudes (see left panels of Figure 5). Still, themaximum bolometric luminosity generated by the relativistic accretion process is of the orderof erg/s, which correspond to low energetic emission. This result is consistent with theLVC observation of GW190521 [9], where no electromagnetic emission was detected.We also found self-constrained models, corresponding to the white regions in Figure 4and 5, where the backwards integration gives forbidden solutions, i.e., negative masses. Suchconstrained region is correlated with the density gradient: increasing the density gradient theconstrained regions also increases, allowing only the models where the Mach numbers are high.This region is bounded by OPs with small masses, of m i , ∼ − M (cid:12) and m i , ∼ − M (cid:12) ,at z = 100 , , and . Thus, PBHs that grow through accretion could be the origin of theGW190521 progenitors. The growth of PBHs corresponds to a possible scenario to explainthe masses of IMBH since these can grow up to − M (cid:12) during the radiation dominatedera [43]. On the other hand, for low values of the density gradient parameter, (cid:15) ρ < . andhigh values of the Mach number, M > , we found a family of massive initial BH seeds, withmasses of m i , = 51 . (cid:12) and m i , = 37 . (cid:12) . In this case, the accretion rates are smalldue to the high Mach numbers. The early formation of BHs, with these initial masses, canbe associated with the collapse of Pop III stars that occurs at redshift < z < [54, 55].Several solutions found correspond to the sBHs seeds, M ∼ − (cid:12) , which could form atearly stages by direct collapse, and grow in mass due to accretion until reaching the finalvalues at the merger compatible with the LVC parameter estimates.– 11 – cknowledgments ACO gratefully acknowledges support from the COST Action CA16214 “PHAROS", theLOEWE-Program in HIC for FAIR, and the EU Horizon 2020 Research ERC Synergy Grant“Black-HoleCam: Imaging the Event Horizon of Black Holes" (grant No. 610058). F.D.L-Cwas supported in part by VIE-UIS, under Grant No. 2493 and by COLCIENCIAS, Colom-bia, under Grant No. 8863. This work is also supported by the Center for Research andDevelopment in Mathematics and Applications (CIDMA) through the Portuguese Founda-tion for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), refer-ences UIDB/04106/2020 and UIDP/04106/2020. We acknowledge support from the projectsPTDC/FIS-OUT/28407/2017, CERN/FIS-PAR/0027/2019 and PTDC/FIS-AST/3041/2020.This work has further been supported by the European Union’s Horizon 2020 research andinnovation (RISE) programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. Theauthors would like to acknowledge networking support by the COST Action CA16104.
References [1] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams,T. Adams, P. Addesso, R. X. Adhikari, et al., Phys. Rev. Lett. , 061102 (2016), .[2] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams,T. Adams, P. Addesso, R. X. Adhikari, et al., Phys. Rev. Lett. , 241103 (2016), .[3] The LIGO Scientific Collaboration and the Virgo Collaboration, Phys. Rev. Lett. , 221101(2016), .[4] The LIGO Scientific Collaboration, the Virgo Collaboration, B. P. Abbott, R. Abbott, T. D.Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, et al. (LIGO ScientificCollaboration and Virgo Collaboration), Astrophys. J. Lett. , L35 (2017), URL http://stacks.iop.org/2041-8205/851/i=2/a=L35 .[5] The LIGO Scientific Collaboration, the Virgo Collaboration, B. P. Abbott, R. Abbott, T. D.Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, et al. (LIGO ScientificCollaboration and Virgo Collaboration), Phys. Rev. Lett. , 141101 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.119.141101 .[6] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams,P. Addesso, R. X. Adhikari, V. B. Adya, et al. (LIGO Scientific Collaboration and VirgoCollaboration), Phys. Rev. Lett. , 161101 (2017), .[7] B. P. Abbott, R. Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, C. Adams, R. X.Adhikari, V. B. Adya, C. Affeldt, et al., Astrophys. J. Lett. , L3 (2020), .[8] The LIGO Scientific Collaboration, the Virgo Collaboration, R. Abbott, T. D. Abbott,S. Abraham, F. Acernese, K. Ackley, C. Adams, R. X. Adhikari, V. B. Adya, et al., arXive-prints arXiv:2006.12611 (2020), .[9] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. , 101102 (2020), .[10] R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. Lett. , L13 (2020), .[11] J. Calderón Bustillo, N. Sanchis-Gual, A. Torres-Forné, J. A. Font, A. Vajpeyi, R. Smith,C. Herdeiro, E. Radu, and S. H. Leong (2020), .[12] J. Sakstein, D. Croon, S. D. McDermott, M. C. Straight, and E. J. Baxter (2020), . – 12 –
13] F. Hoyle and R. A. Lyttleton, in
Proceedings of the Cambridge Philosophical Society (1939),vol. 35, p. 405.[14] H. Bondi and F. Hoyle, Mon. Not. R. Astron. Soc. , 273 (1944).[15] H. Bondi, Mon. Not. R. Astron. Soc. , 195 (1952).[16] L. I. Petrich, S. L. Shapiro, R. F. Stark, and S. A. Teukolsky, Astrophys. J. , 313 (1989).[17] J. A. Font and J. M. Ibáñez, Astrophys. J. , 297 (1998).[18] J. A. Font and J. M. Ibáñez, Mon. Not. R. Astron. Soc. , 835 (1998).[19] J. A. Font, J. M. Ibáñez, and P. Papadopoulos, Astrophys. J. , L67 (1998).[20] J. A. Font, J. M. Ibáñez, and P. Papadopoulos, Mon. Not. R. Astron. Soc. , 920 (1999), arXiv:astro-ph/9810344 .[21] O. Dönmez, O. Zanotti, and L. Rezzolla, Mon. Not. R. Astron. Soc. , 1659 (2011), .[22] A. Cruz-Osorio, F. D. Lora-Clavijo, and F. S. Guzmán, Mon. Not. R. Astron. Soc. , 732(2012), .[23] F. D. Lora-Clavijo and F. S. Guzmán, Mon. Not. R. Astron. Soc. , 3144 (2013), .[24] A. Cruz-Osorio, F. D. Lora-Clavijo, and F. S. Guzmán, in
American Institute of PhysicsConference Series , edited by L. A. Urenña-López, R. Becerril-Bárcenas, and R. Linares-Romero(2013), vol. 1548 of
American Institute of Physics Conference Series , pp. 323–327, ,URL https://ui.adsabs.harvard.edu/abs/2013AIPC.1548..323C .[25] A. J. Penner, Mon. Not. R. Astron. Soc. p. 490 (2011), .[26] O. Zanotti, C. Roedig, L. Rezzolla, and L. Del Zanna, Mon. Not. R. Astron. Soc. , 2899(2011), .[27] F. D. Lora-Clavijo, A. Cruz-Osorio, and E. Moreno Méndez, Astrophys. J., Supp. , 30(2015), .[28] A. Cruz-Osorio and F. D. Lora-Clavijo, Mon. Not. R. Astron. Soc. , 3193 (2016), .[29] A. Cruz-Osorio, F. J. Sanchez-Salcedo, and F. D. Lora-Clavijo, Mon. Not. Roy. Astron. Soc. , 3127 (2017), , URL https://doi.org/10.1093/mnras/stx1815 .[30] A. Cruz-Osorio, S. Gimeno-Soler, and J. A. Font, Mon. Not. R. Astron. Soc. , 5730 (2020), .[31] V. De Luca, V. Desjacques, G. Franciolini, P. Pani, and A. Riotto, arXiv e-printsarXiv:2009.01728 (2020), .[32] M. Safarzadeh and Z. Haiman,
Formation of gw190521 via gas accretion onto population iiistellar black hole remnants born in high-redshift minihalos (2020), .[33] P. Natarajan,
A new channel to form imbhs throughout cosmic time (2020), .[34] J. R. Rice and B. Zhang,
Growth of stellar mass black holes in dense molecular clouds andgw190521 (2020), .[35] A. Palmese and C. J. Conselice,
Gw190521 from the merger of ultra-dwarf galaxies (2020), .[36] B. Liu and V. Bromm,
The population iii origin of gw190521 (2020), .[37] H. Vandeven, Journal of Scientific Computing , 159 (1991), ISSN 0885-7474, URL .[38] J. E. Greene, J. Strader, and L. C. Ho, arXiv e-prints arXiv:1911.09678 (2019), . – 13 –
39] M. Bachetti, F. A. Harrison, D. J. Walton, B. W. Grefenstette, D. Chakrabarty, F. Fürst,D. Barret, A. Beloborodov, S. E. Boggs, F. E. Christensen, et al., Nature , 202 (2014), .[40] C. Motch, M. W. Pakull, R. Soria, F. Grisé, and G. Pietrzyński, Nature , 198 (2014), .[41] J.-F. Liu, J. N. Bregman, Y. Bai, S. Justham, and P. Crowther, Nature , 500 (2013), .[42] D. R. Pasham, T. E. Strohmayer, and R. F. Mushotzky, Nature , 74 (2014), .[43] F. D. Lora-Clavijo, F. S. Guzmán, and A. Cruz-Osorio, Journal of Cosmology andAstroparticle Physics , 015 (2013), .[44] F. D. Lora-Clavijo, M. Gracia-Linares, and F. S. Guzmán, Mon. Not. R. Astron. Soc. ,2242 (2014), , URL https://ui.adsabs.harvard.edu/abs/2014MNRAS.443.2242L .[45] M. Volonteri and R. Perna, Monthly Notices of the Royal Astronomical Society , 913(2005).[46] J. R. Gair, I. Mandel, M. C. Miller, and M. Volonteri, General Relativity and Gravitation ,485 (2011).[47] J. M. Bardeen, Nature , 64 (1970).[48] K. S. Thorne, Astrophys. J. , 507 (1974).[49] A. Cruz-Osorio and L. Rezzolla, Astrophys. J. , 147 (2020), .[50] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics (Oxford University Press, Oxford, UK,2013), ISBN 9780198528906.[51] M. Ricotti, J. P. Ostriker, and K. J. Mack, Astrophys. J. , 829 (2008), .[52] V. De Luca, G. Franciolini, P. Pani, and A. Riotto, Journal of Cosmology and AstroparticlePhysics , 044 (2020), .[53] V. De Luca, G. Franciolini, P. Pani, and A. Riotto, Journal of Cosmology and AstroparticlePhysics , 052 (2020), .[54] M. Coleman Miller and E. J. Colbert, International Journal of Modern Physics D , 1 (2004).[55] J. E. Greene, J. Strader, and L. C. Ho, Annual Review of Astronomy and Astrophysics , 257(2020)., 257(2020).