Black hole growths in gamma-ray bursts driven by the Blandford-Znajek mechanism

The Blandford-Znajek (BZ) mechanism in stellar-mass black hole (BH) hyperaccretion systems is generally considered to power gamma-ray bursts (GRBs). Based on observational GRB data, we use the BZ mechanism driven by the BH hyperaccretion disc to investigate the evolution of the BH mass and spin after the jets break out from the progenitors. We find that the BH growths are almost independent of initial BH masses. Meanwhile, the BH growths will be more efficient with smaller initial spin parameters. We conclude that (i) the BZ mechanism is efficient for triggering BH growths for only 1 of 206 typical long-duration GRBs; (ii) the mean BH mass growths of ultra-long GRBs are marginal for all 7 samples collected; (iii) for the short-duration GRBs, the results that BHs show minimal growths is consistent with the mass supply limitation in the scenario of compact object mergers.

Moreover, the evolution of BH mass and spin may be violent in the hyperaccretion system (e.g., Liu et al. 2015b;Song et al. 2015;Qu & Liu 2022).If the initial BH mass is set as to approximately 3 M⊙, according to the neutrino annihilation process, the final BH mass is relatively large (> 5 M⊙) for LGRBs and small (3 − 5 M⊙) for SGRBs.The BZ mechanism is less efficient in triggering BH growths compared with the neutrino annihilation processes (e.g., Liu et al. 2015a;Qu & Liu 2022).However, in the scenarios of massive collapsars and compact object mergers, the neutrino annihilation process could only last for several or tens of seconds until the ignition accretion cannot be reached with the decreasing accretion rate.Then it might be quickly replaced by the BZ mechanism and jets could break out from the envelopes or ejecta, especially for collapsar model (e.g., Liu et al. 2018;Wei, Liu, & Xue 2021).Moreover, the typical luminosity of the neutrino annihilation process is lower than that of the BZ mechanism by about two orders of magnitude with the same high spin parameter and accretion rate (e.g., Liu et al. 2015a).If the disc outflows are considered, the above conclusion will be strengthened.Of course, one cannot rule out that the neutrino annihilation process will be dominated in the activities of GRB central engines (e.g., Liu, Gu, & Zhang 2017).Here we generally discussed the BH growths of GRBs powered by the BZ mechanism and set the initial BH properties at the moment that the jets break out form the envelopes of the massive collapsars or the ejecta of the compact object mergers.The growth caused by the BZ mechanism is small but not totally negligible, which is directly related to the observable GRBs as mentioned above.The growths of the BHs in the center of GRBs is an inevitable course in the evolution history of stellar-mass BHs, then effects the mass distribution of compact objects (e.g., Liu et al. 2021).
In this paper, combining with the GRB samples, we visit the BH hyperaccretion system with the BZ mechanism to test the suitability of the BZ mechanisam for GRBs and investigate the growths of newborn BHs in the center of GRBs.This paper is organised as follows.In Section 2, the model for the evolution of BH characteristic parameters is described.The results are shown in Section 3. Conclusions and discussion are presented in Section 4.

MODEL
The characteristic parameters of the BH will undergo drastic evolution if a rotating stellar BH surrounded by a hyperaccretion disc with a very high accretion rate is the central engine of GRBs (e.g., Liu et al. 2015b;Song et al. 2015;Qu & Liu 2022).Based on the conservation of the mass and angular momentum, considering that part of the BH rotational energy would be extracted by the Poynting jet for the BZ mechanism, the mass and angular momentum of a Kerr BH evolved with time read as (e.g., Lee, Wijers, & Brown 2000;Lee & Kim 2000) where MBH and JBH are the mass and angular momentum of the BH, Ṁ is the mass accretion rate, c is the speed of light, and ems and lms are the specific energy and mass growth/M ⊙ a *, 0 = 0.5 a *, 0 = 0.9 Figure 1.Influence of initial BH masses and spin parameters on BH mass growths in a typical GRB case with the luminosity L j = 10 49 erg s −1 and duration T 90 = 30 s.The circles and triangles represent the initial BH spin a * ,0 = 0.5 and 0.9, respectively.
angular momentum corresponding to the marginally stable orbit radius of the BH.They are defined as ems = is the dimensionless spin parameter of the BH and xms is the dimensionless marginally stable orbit radius of the BH, which can be written as xms ] and Z2 = 3a 2 * + Z 2 1 (e.g., Bardeen, Press, & Teukolsky 1972;Novikov 1998;Kato, Fukue, & Mineshige 2008).The last two terms on the right-hand side of Equations ( 1) and (2) are relevant to the BZ mechanism, where LBZ is the BZ jet power and ΩF is the magnetic field angular velocity at the marginally stable orbit radius.Here, we adopt the optimal mode ΩF = ΩH/2 (e.g., Lee, Wijers, & Brown 2000;Lee & Kim 2000), with ΩH ≡ a * c 3 /[2(1 + 1 − a 2 * )GMBH] being the angular velocity on the stretched horizon.
The mean luminosity of the GRB jet can be estimated as (e.g., Fan & Wei 2011;Liu et al. 2015b) where Eγ,iso is the isotropic radiated energy in the prompt emission phase, E k,iso is the isotropic kinetic energy powering long-lasting afterglow, z is the redshift, θj is the halfopening angle of the jet, and T90 can be approximately considered as the duration of the violent activity of the central engine for both LGRBs and SGRBs.Using these observational GRB data, the jet luminosity Lj can be obtained.
From the data of the prompt emission and afterglows, one can obtain the value of Eγ,iso and E k,iso , and the radiative efficiency in the fireball model is implied in the fitting process.Generally, E k,iso is larger than Eγ,iso in most of cases as shown in Tables 1, 2, and 3.Moreover, in fireball model (e.g., Rees & Mészáros 1992;Mészáros & Rees 1993), the energy is released by the central engine in a very short timescale.The collisions of the shells in the internal shock phase will trigger the prompt emission and the ex- ternal shocks sweeping the circumstellar media will power the long-lasting multiband afterglows.T90 is considered to reflect the activity timescale of the GRB central engines.
For a BH hyperaccretion system in the centre of GRBs with the BZ mechanism, the energy output is determined by the BZ jet power LBZ.It is related to the BH mass accre- tion rate and the spin parameter a * , which writes as (e.g., Liu et al. 2018;Du et al. 2021) where If we assume Lj = LBZ , the dimensionless mass accretion rate can be obtained as the function of Lj: (5) According to the above equations, the final BH characteristics can be described.With a given initial BH mass (e.g.MBH,0 = 3 M⊙) and spin (a * ,0 = 0.5 and 0.9) and the luminosity of a GRB jet Lj obtained by Equation (3), one can obtain the BZ jet power LBZ and the values of ṁ based on Equations ( 4) and ( 5), respectively.Incorporating ṁ and LBZ into the Equations ( 1) and ( 2), the mass and spin of the BH at the next time step can be solved.Step by step, the final BH mass M BH,f and spin a * ,f can be obtained when the time reaches T90.The fourth-order Runge-Kutta method, ode45 function of MATLAB, is used in our calculations.The main results are discussed below.

RESULTS
In this work, we adopt the data of 206 LGRBs, 7 ULGRBs, and 41 SGRBs to calculate the evolution of BH mass and spin in the BH hyperaccretion system with the BZ mechanism.The duration T90, redshifts z, half-opening angles θj, and isotropic radiated energy in the prompt emission phase Eγ,iso and isotropic kinetic energy E k,iso of the LGRBs, UL-GRBs and SGRBs are collected in Tables 1, 2, and 3, respectively.The GRBs associated with SNe are labelled by the superscript " * ".The GRB luminosity derived from Equation (3) is also presented.It should be noticed that the measurements of Eγ,iso and E k,iso are model dependent.Besides, the lower limits of θj are used in some bursts.
Before the discussion of the GRB cases collected, theoretical results of mass growths with different initial BH masses and spin parameters are illustrated in Figure 1 based on our model.In the figure, we adopt a typical GRB luminosity Lj = 10 49 erg s −1 and duration T90 = 30 s to calculate the BH mass growths with different initial BH masses, i.e., 3, 5, and 10 M⊙ and different initial spin parameters, i.e., a * ,0 = 0.5 and 0.9.One can find that the final BH mass is almost independent of the initial BH mass (≤ 10 M⊙) but depend on the initial spin and the BH growth will be more efficient with a smaller spin parameter.The conclusion can also be directly inferred from Equation ( 5).For a given luminosity, the initial BH mass has no effect on the accretion rate and then on the mass growth.However, the accretion rate is relevant to the BH spin, the larger initial spin parameter leads to a lower accretion rate, and thus a less final mass.As shown in Figure 1, one can see that the mass growth is about 6 × 10 −3 M⊙ even with a * ,0 = 0.5.There is almost no change for the evolution of BH mass with the BZ mechanism for the typical luminosity and duration, and thus one can expect that the BH growths will be obvious only for the case with high GRB luminosity and long accretion timesacle.
The BH mass before accretion might be around 3 M⊙ for the scenarios of the collapsars or the compact object mergers (e.g., Abbott et al. 2017Abbott et al. , 2021;;Belczynski et al. 2008;Liu et al. 2021;Wei, Liu, & Xue 2021;Qu & Liu 2022).In the collapsar scenario, if the NDAF phase lasts several to tens of seconds, the central BH should be fed to grow up, but the disc outflows will decrease the BH's appetite, and more important, the jets might be chocked in the envelope and no observable electromagnetic signals.For merger scenarios, the jet may also be chocked by the tidal and/or post-merger ejecta (e.g., Murguia-Berthier et al. 2021).Considering the dissipative energy by the chocked jets to build cocoon, the total GRB energy should be less than the real released energy and the BH mass growth will be more violent.Here we just investigate the BH evolution after the jets break out from the progenitors.Combining with the result that the initial BH mass has almost no effect on the mass growths as shown in Figure 1, we use MBH,0 = 3 M⊙ in the GRBs sample calculations.Besides, a * ,0 = 0.5 and 0.9 are adopted.

LGRB case
The distributions of final BH masses M BH,f and spins a * ,f for LGRB cases are presented in Figures 2(a) and 3(a), respectively.The dark and light colors denote the initial BH spins a * ,0 = 0.5 and 0.9.As mentioned above, the mass growth under the BZ mechanism is inefficient.Thus, high GRB luminosity and long accretion timescale are required in this case.Moreover, a smaller initial BH spin parameter is favored.
Figure 2(a) shows the final BH masses distribution after the accretion phase for 206 LGRB events.If the initial BH mass MBH,0 is 3 M⊙, the mean final BH masses are about 3.203 M⊙ for a * ,0 = 0.5 and 3.059 M⊙ for a * ,0 = 0.9.One can find that a smaller a * ,0 leads to a higher M BH,f and that most of the final BH masses are below 5 M⊙.With relatively high jet luminosity and long accretion time, one bursts, namely, GRB 050401, has a final BH mass larger than 5 M⊙, which are 7.992 M⊙ for a * ,0 = 0.5 and 7.172 M⊙ for a * ,0 = 0.9, respectively, for which the BZ mechanism is a plausible prescription.
In Figure 3(a), we show the distribution of the final BH spins for the LGRB events.The mean final BH spin parameters are about 0.60 for a * ,0 = 0.5 and 0.91 for a * ,0 = 0.9.The upper limit of the BH spin is set as 0.998 (e.g., Kato, Fukue, & Mineshige 2008).The bursts with larger final BH masses have higher final BH spin parameters, and several bursts including the GRB 050401 mentioned above, can reach up to the spin limitation.
In our calculation GRB 050401 is a special case with high luminosity, which is mainly caused by the relatively high kinetic energy.Actually, some other bursts have the similar or even higher kinetic energy but with lower luminosity because of other parameters, such as the opening angle and T90.Besides, it should be mentioned that the fitting of the kinetic energy is model-dependent (e.g., Zhang et al. 2007b).We noticed that Kamble et al. (2009) used a doublejet model to fit the X-ray afterglow of GRB 050401, the kinetic energies of the narrow and wide jets are ∼ 5.46 × 10 53 erg and > 5.07 × 10 51 erg, respectively, which are much less than the value adopted here.
It is widely accepted that LGRBs signal the collapse of massive stars, which usually end their lives companied with SNe.The conclusion is confirmed by the observation evidence, such as the appearance of SN-like bumps in the optical afterglow light curves of several bursts (e.g., Hjorth et al. 2003;Zhang et al. 2009).There is competition on the lumi-nosity of LGRBs and those of corresponding SN bumps (e.g., Song & Liu 2019;Liu et al. 2021).Thus, the typical luminosity of LGRBs associated with SNe is lower than that of LGRBs without observable SNe, which will further decrease the values of the BH growths.We indeed find that the BH growths are less obviously for LGRBs associated with SNe in most cases.Besides, it should be noticed that the SN explosions may exist in the cases without SNe observed in result of that they are too weak (even failed) or too distant to be observed.

ULGRB case
ULGRBs are characterised by gamma-ray emission lasting longer than ∼ 1000 s.Although their duration are different as typical LGRBs, the evidence for a new GRB classification is still unclear (e.g., Zhang et al. 2014;Perna, Lazzati, & Cantiello 2018).Even so, given by the long accretion time, we expect correspondingly large BH growths and we thus separate them in Table 2.
The distribution of the final BH masses after the accretion phase for ULGRB events is shown in Figure 2(b).The figure shows that the mass growths are marginal.None of the bursts in the sample has a final BH mass larger than 5 M⊙ even with a * ,0 = 0.5 for the initial BH mass MBH,0 = 3M⊙.In this case, the mean final BH masses are about 3.193 M⊙ and 3.039 M⊙ for a * ,0 = 0.5 and 0.9, respectively.
Figure 3(b) presents the distribution of the final BH spins for ULGRB events.The evolution of spin parameters is also inefficient.The mean final BH spin parameters are about 0.61 for a * ,0 = 0.5 and 0.92 for a * ,0 = 0.9.
One can find that the BH growths are inefficient for ULGRBs.This is because the long accretion time cannot offset the effect caused by the low burst luminosity.

Similar to
LGRBs, the evolution of BH characteristic parameters is investigated for SGRBs.For the merger scenario, the ejecta hardly stops the SGRB jets, but the limited accretion matter can only support the central engine activity for no more than seconds.Thus, one can expect that the growths of BHs in the centre of SGRBs are undoubtedly very limited.
The distributions of the final BH masses and spin parameters of 41 SGRB events are presented in Figures 2(c) and 3(c), respectively.We find that if the initial BH mass MBH,0 = 3M⊙, the mean final BH masses are about 3.0173 M⊙ for a * ,0 = 0.5 and 3.0026 M⊙ for a * ,0 = 0.9.Besides, the mean final BH spin parameters are about 0.512 for a * ,0 = 0.5 and 0.901 for a * ,0 = 0.9.It shows that the effect on the evolution of BH characteristic parameters is not significant and the final BH masses for all of the bursts are between 3-5 M⊙.
As the results, the BH growths in SGRBs is consistent with the mass supply limitation in the scenario of compact object mergers.

CONCLUSIONS AND DISCUSSION
Based on the observational data, we test the capacity of the BZ mechanism to power GRBs and present the growths of BHs in the center of GRBs.The BZ mechanism is capable of powering all types of GRBs with reasonable parameters of BH hyperaccretion systems and the main conclusions are summarised as follows: (i) By assuming GRBs powered by the BZ jets, for LGRBs with typical durations, the mean BH mass growth is about 0.203 M⊙ for a * ,0 = 0.5.Only GRB 050401 has a final BH mass larger than 5 M⊙ for the initial BH mass ∼ 3 M⊙.At the same time, the BH spin parameter for GRB 050401 almost reaches up to 0.998.
It should be noticed that the jet half-opening angle θj is usually estimated by the jet break time in X-ray afterglows and it is hard to constrain since the absence of jet break observations.The lower limits of θj are widely used (e.g., Liu et al. 2015b;Yi et al. 2017), and therefore one can see from Equation (3) that the BH growths are lower limits in this work.
The assumption of a Keplerian disc is adopted in this work.Actually, Gammie, Shapiro, & McKinney (2004) investigated the BH spin evolution for the thick torus accretion, and they found that the spin-down effects will be resulted in case of large initial spins for some physical processes.Besides, Janiuk, Moderski, & Proga (2008) studied the evolution of the BH spin in LGRB cases and found that the spin-down of the BH is possible due to the low specific angular momentum of the accretion materials.
The Swift satellite observations have revealed the diverse morphology of light curves in X-ray afterglows (e.g., Gehrels et al. 2004;Burrows et al. 2005).Among them, Xray flares are important signatures, which may be caused by the restart of central engine (e.g., Burrows et al. 2005;Liu, Gu, & Zhang 2017).The BZ mechanism is a promising candidate to explain the GRBs accompanied by longlasting X-ray flares (e.g., Luo et al. 2013).If the effect of flares is considered, the evolution of BH will be more violent.In addition, the long-lasting plateau phases are observed in some GRB afterglows, and energy injection is one of the most widely accepted explanations (e.g., Fan & Xu 2006;Zhang et al. 2006).In this case, the external shock will be refreshed by the huge energy contribution from the central engine.As well as X-ray flares, if the plateau phase induced by the energy injection is originated from the long-lasting BH hyperaccretion (e.g., Huang & Liu 2021), the BH will undergo more drastic evolution to promote its growths.
Moreover, in collapsar and merger scenarios, the outflows from the BH hyperaccretion disc are definitely strong (e.g., Liu et al. 2018) and will inject matter and energy into the shock to power luminous SNe and kilonovae (e.g., Surman, McLaughlin, & Hix 2006;Song, Liu, & Li 2018;Song & Liu 2019;Metzger 2019;Qi et al. 2022).Once the disc outflows are strong enough, the accretion rate in the inner region of the disc is too low to ignite NDAFs, which  should weaken the domination of the neutrino annihilation process to power GRBs.

Figure 2 .
Figure 2. Distributions of the final BH masses M BH,f .The initial BH mass M BH,0 is 3 M ⊙ .The dark and light colors denote a * ,0 = 0.5 and 0.9, respectively.

Figure 3 .
Figure 3. Distributions of the final BH spin parameters a * ,f .The dark and light colors denote a * ,0 = 0.5 and 0.9, respectively.

Table 3 .
SGRB Data GRB 200826A is a short LGRB, which is associated with SNe.