Black Hole Activity Imprints on the Internal Plateau and the Subsequent Sharp Decay

A stellar-mass black hole (BH) or a millisecond magnetar is believed to be born as the central engine of gamma-ray bursts (GRBs). The presence of plateaus in the X-ray extended emission or afterglow of GRBs is widely accepted as an indicator of a magnetar central engine, particularly those with a sharp decay (faster than t −3), the so-called internal plateau. However, an alternative model, by taking the evolution of the magnetic flux at the BH horizon into account, suggests that an internal plateau can also arise from a jet powered by the Blandford–Znajek (BZ) mechanism (hereafter, a BZ jet). In this study, we propose that a precessional BZ jet would manifest a quasiperiodic oscillation (QPO) signature on the internal plateau and the subsequent sharp decay. Such lightcurves cannot be readily explained by the activity of a short-lived, supermassive magnetar, thus favoring a Kerr BH as the central engine. The X-ray afterglow of GRB 050904, comprising nine flares, is characterized by a QPO-modulated plateau and sharp decay, which can be well reproduced by a precessional BZ jet model. Therefore, one potential clue for distinguishing between these two engines lies in whether the QPO signature is present throughout the entire plateau and the subsequent sharp decay, as the magnetar scenario suggests a collapse at the end of the plateau.


Introduction
Gamma-ray bursts (GRBs), known as the most energetic phenomena, release energy equivalent to the total energy of a Sun in a mere few seconds, leaving behind a mass of mist for scientists.There are two types of central engines suggested to be born during the burst of GRBs, one being a hyper-accreting stellar-mass black hole (BH; e.g., Popham et al. 1999) and the other being a rapidly rotating and highly magnetized neutron star (a so-called magnetar; e.g., Usov 1992).Unfortunately, the erratic prompt emission provides scant information for identifying the type of central engine.However, the discoveries of X-ray (Costa et al. 1997) andoptical (van Paradijs et al. 1997) afterglows offer a promising means to explore the properties of the central engine.The canonical X-ray afterglow lightcurve (Nousek et al. 2006;Zhang et al. 2006) suggests that both X-ray plateaus and X-ray flares are associated with a longlasting central engine.
A shallow decay (or plateau) followed by a normal decay is widely observed in GRB X-ray afterglows, and can be well interpreted by continuous energy injection (Dai & Lu 1998;Nousek et al. 2006;Zhang et al. 2006), geometric effect (e.g., offset observation of a structured jet; Peng et al. 2005), timedependent microphysics (Ioka et al. 2006;Wei & Fan 2007), or environment-influenced fireball radiation (Dereli-Bégué et al. 2022), etc.For a plateau followed by a rapid decay, with a decay index α ∼ −2.0, the most popular model is energy injection or the internal dissipation of a magnetar wind (Dai & Lu 1998;Zhang & Mészáros 2001).In some peculiar cases, the plateau is followed by a sharp decay α < −3.0 (e.g., Liang et al. 2007;Troja et al. 2007), dubbed an "internal" plateau (Lyons et al. 2010), where the break is associated with the disruption of the central engine, e.g., the collapse of a supermassive millisecond magnetar (Lyons et al. 2010;Rowlinson et al. 2010Rowlinson et al. , 2013)).Intriguingly, while the evolution of the magnetic flux at the BH horizon is taken into account, an internal plateau can also be derived from the Blandford-Znajek (BZ) mechanism (Blandford & Znajek 1977) powering the jet (the BZ jet; Kisaka & Ioka 2015;Tchekhovskoy & Giannios 2015), though this has yet to be extensively discussed.
It has been suggested that X-ray flares and prompt emission may share a common origin, with the former potentially being induced by the intermittent accretion of a long-lasting central engine (King et al. 2005;Perna et al. 2006).Zheng et al. (2021) established a connection between X-ray flares and the magnetar plateau, aiming to investigate the existence of a potential disk surrounding the central magnetar, where the successive flares upon the magnetar plateau were suggested to have periodicity.Quasiperiodic oscillation (QPO) behavior is a general phenomenon in astrophysics, observed in various systems, such as active galactic nuclei (AGN; Lu 1990), X-ray binary systems (Sarazin et al. 1980), and softer gamma-ray repeaters (Li et al. 2022), etc. Portegies Zwart et al. (1999) proposed that a jet precession induced by the Lense-Thirring effect (Lense & Thirring 1918) would leave a pattern on GRB prompt emission.Additionally, Lei et al. (2007) conducted a morphological study, while Li et al. (2023) initiated a study to explore the corresponding manifests in short GRBs.Recently, some potential QPO signals have been identified in both the prompt phase (Wang et al. 2022;Xiao et al. 2022;Zhang et al. 2022Zhang et al. , 2023) ) and the afterglow phase (Suvorov & Kokkotas 2020;Zou et al. 2021;Zou & Liang 2022).Therefore, a lightcurve exhibiting QPO characteristics can serve as a valuable tool for monitoring the activity of the central engine.
In this study, we propose that the precession of a BZ jet induced by the Lense-Thirring effect will imprint a QPO signature on the internal plateau and the subsequent sharp decay.Such lightcurves cannot be readily interpreted as the activities of a short-lived, supermassive magnetar, and instead suggest the presence of a Kerr BH as the central engine.The model successfully reproduces a distinctive X-ray lightcurve of GRB 050904, which consists of nine flares.This paper is organized as follows.In Section 2, we demonstrate that by taking into account the evolution of the magnetic field and the Lense-Thirring effect, the BZ jet can generate a lightcurve that is characterized by a QPO signature printed on the internal plateau and the subsequent sharp decay.The model is applied to the peculiar case of GRB 050904 and presented in Section 3. In Section 4, we test the periodicity of the observed lightcurve of GRB 050904.In Section 5, we give the conclusions and the discussions.The ΛCDM cosmology parameters in a flat Universe H 0 = 67.4km s −1 Mpc −1 and Ω M = 0.315 (Planck Collaboration et al. 2020) are adopted in this study.

Emission from a Precessional BZ Jet
Currently, two models are involved in interpreting the observed internal plateau in the afterglow of GRBs.The first model involves an internal dissipation wind originating from a supermassive magnetar, in which the break of the plateau signifies an abrupt collapse of the magnetar.The second model pertains to an internal dissipation jet powered by the BZ mechanism (Kisaka & Ioka 2015;Tchekhovskoy & Giannios 2015), where the break of the plateau corresponds to the expansion of the magnetosphere.One distinguishing factor between these two models is that the latter has a continuously active central engine throughout the plateau and the subsequent sharp decay, and the QPO signature can help to record this continuous activity.

Jets Powered by BHs
The dimensionless spin parameter of a Kerr BH with mass M BH can be expressed as = a cJ M G BH BH 2 , where J BH is the angular momentum of the BH and c and G represent the speed of light and the gravitational constant, respectively.The magnetic flux passing through the event horizon of the BH is represented by , where r H is the radius of the outer horizon given by 2 and B H represents the magnetic field strength at the horizon.The power of the BZ jet is highly dependent on the magnetic flux, that is (e.g., Blandford & Znajek 1977;Tchekhovskoy et al. 2011) where k ≈ 0.05 and the angular frequency of the BH horizon Ω H = ac/2r H .For a ∼ 1, one has f = 1, r H ∼ GM BH /c 2 .The scenario with a ∼ 1 is utilized for all calculations conducted in this study.
According to the BH no-hair theorem, the magnetic field inherited from the progenitor is expected to naturally diffuse out of the BH, as the BH formed.The mass accretion rate of the magnetically arrested disk (MAD), however, is so high that the magnetic flux is squeezed at the BH horizon (Tchekhovskoy & Giannios 2015).The cross section of the magnetosphere, which the magnetic flux passes through, depends on the balance between the ram pressure of the accretion flow P f and the pressure exerted by the magnetic field P B .In the scenario of magnetic flux conservation, the magnetic pressure is expressed as where B H,0 is the characteristic magnetic field strength while the magnetic field is squeezed at the BH horizon, and one has ( ) The ram pressure of the accretion flow is denoted as  is the radial velocity and ò ∼ 10 −2 (e.g., Tchekhovskoy et al. 2011;Zamaninasab et al. 2014).The mass accretion rate  M increases with time in the early phase (t < t p ) and decreases with time in the later phase (t > t p ), where t p represents the characteristic transformation timescale of the mass accretion rate.For the later phase (Chevalier 1989), , the ram pressure drops below the magnetic pressure level.Consequently, the magnetosphere will move away from the BH horizon, leading to the diffusion of a portion of the magnetic flux beyond the horizon.The magnetic flux passing through the BH horizon decreases as the magnetosphere size (R m ) increases:


Therefore, the power of the BZ jet can be written as


The characteristic timescale for the diffusion of the magnetic field out of the horizon can be derived by balancing the ram pressure with the magnetic pressure:  Therefore, the time-evolving power acts as a plateau followed by a sharp decay: where the characteristic luminosity is expressed as 2 .A smooth broken-power-law function is employed to describe an observed internal plateau and the subsequent sharp decay: where α 2 represents the decay slope after the break time, with a typical value of 40/9, and the supplemented α 1 corresponds to the decay slope before the break time, which connects the evolution of the spin of the BH and the radius of the outer horizon.The radiative efficiency of the BZ jet is denoted by η BZ , with a typical value of approximately 0.1, as adopted from previous studies (Zhang et al. 2007;Kisaka & Ioka 2015).

Precession of BZ Jet
GRBs are believed to originate from a collimated jet.An ultrarelativistic BZ jet presents a promising possibility for yielding a beaming GRB, in the context of a BH central engine.
Portegies Zwart et al. (1999) have proposed that the intrinsic variability in certain GRB lightcurves may arise from the precession of a jet.
In the case of a Kerr BH, its strong angular momentum exerts a dragging effect on the neighbor space.This effect induces the inclined orbit plane of a particle to precess around the central BH, which is known as the Lense-Thirring effect (Lense & Thirring 1918).Moreover, the accretion disk surrounding the Kerr BH is also influenced by this effect.As discussed in Section 2.1, the MAD with a high mass accretion rate would squeeze the magnetic flux on the BH horizon.Consequently, the precessing MAD compels the direction of the BZ jet to deviate from the spin axis of the BH and to align with the axis of the MAD, as described by Portegies Zwart et al. (1999).Finally, the jet exhibits the same precession period as the MAD.The precession angular frequency of the inclined plane is expressed as (Wilkins 1972;Lei et al. 2013) where r is the radius of the precession orbit and the precession period is P = 2π/Ω.

Evolving Lightcurve Originates from a Precessing Jet
Since the jet is precessing, the angle between the observer's line of sight r obs and the direction of the jet r jet should evolve with time, and it can be written as (Lei et al. 2007;Liu et al. 2010) where θ jet and θ obs are the angles from the precession axis to the jet and to the observer, respectively.f jet = f jet,0 + 2πt/P is the phase of the jet, where f jet,0 corresponds to the initial phases, and the phase of the observer f obs is a constant.By setting a constant C = f jet,0 − f obs , a time-dependent ψ is written as Taking into account the Doppler effect, the observed flux of an off-axis observer is written as F(ψ) = D 3 F ψ=0 , and the Doppler factor , where Γ is the Lorentz factor.Finally, the emission observed from a precessing BZ jet can be summarized into The energetic GRB known as GRB 050904 was detected by the Swift Burst Alert Telescope at 01:51:44 on 2005 September 4 (UT; Cummings et al. 2005; hereafter T 0 ).The duration of its prompt emission was measured to be T 90 = 225 ± 10 s (Sakamoto et al. 2005).The Swift X-ray Telescope (XRT) instrument started to observe its X-ray emission since T 0 +161 s.The X-ray lightcurve, as observed, exhibits a total of nine flares from the end of the rapid decay phase (approximately 600 s) to 55,000 s, and the contour line of the lightcurve is characterized by a plateau followed by a sharp decay (with a decay index of α ∼ − 6; Evans et al. 2009).A speculative period for successive flares is approximately 5600 s.The confirmation of GRB 050904 as the first GRB with a spectroscopic redshift larger than 6 was provided by subsequent optical observations (z = 6.29;Kawai et al. 2006).The isotropic X-ray luminosity of GRB 050904 is derived with , where D L and f X (t) are the luminosity distance and the observed X-ray flux, respectively.The cosmological k-correction factor has k(z) = (1 + z) γ−2 (Bloom et al. 2001;Sasmaz Mus et al. 2019), where the average photon index of GRB 050904 has γ = 1.8. 4 The investigation of the jet opening angle for GRB050904 has been conducted extensively in previous studies (Tagliaferri et al. 2005;Frail et al. 2006;Wei et al. 2006;Gou et al. 2007), suggesting a potential range of values from 3°to 15°.In this study, we adopt a moderate value of θ j = 10°, which corresponds to a beaming factor q , assuming that the flares share the same jet opening angle as the prompt phase.Considering the small Lorentz factor required in the below analysis, the energy fraction into XRT windows η X = 0.1 (Xiao & Dai 2019) is adopted to calculate the afterglow luminosity L obs = L iso,X × f b /η X .The resulting lightcurve is presented in Figure 1.The observed flares are interpreted as a consequence of a precessing jet.A numerical fitting process based on radiation from a precessional BZ jet is carried out subsequently.Equation ( 14) demonstrates that the amplitude magnitude of the observed pulses is directly influenced by three parameters: the luminosity L 0 , Lorentz factor Γ, and angle ψ.While searching for the parameter distribution, we fix certain parameters based on observations to resolve the degeneracy of the above three parameters.The amplitude of the pulses spans a range of 2 orders of magnitude, as depicted in Figure 1, allowing us to build an equation like the one below: As the emission from a precessing jet can intermittently move in and out of the observer's line of sight, one expects that and y q q q = + > 2 j max jet obs . In this scenario, a pseudocolor plot (Figure 2) is constructed to visualize the range of θ jet and θ obs , where the color in the plot represents the corresponding values of Γ.The plot is configured with 10 < Γ < 200, but the majority of values fall within the range of 10-40.The given pseudocolor plot reveals a symmetrical distribution of θ jet and θ obs , with a viable range spanning from 0.02 rad to 0.18 rad.
Assuming θ jet < θ obs , we explore the parameter distribution by considering three groups of θ jet , θ obs , and Γ, as shown in Table 1.We employed the Markov Chain Monte Carlo method to explore the distribution of the parameters, and the results show good convergence with respect to the precession period.The median values of the parameters, along with their 1σ uncertainty, have been tabulated in Table 1.The results from the three different scenarios indicate that these three parameters have a negligible impact on the period of the precessing jet.corner plot in Figure 3 corresponds to the scenario where θ jet = 0.06 rad, θ obs = 0.11 rad, and Γ = 14.56.The resulting lightcurve is also shown in Figure 1.The results of the numerical fitting indicate that the period of the precessing jet and the characteristic timescale in the rest frame are P = 5620/(1 + z) = 771s and t b = 10 4.56 /(1 + z) = 4980s, respectively.Assuming a dimensionless spin parameter a ∼ 1 and a central BH mass of M BH = 3M e , the equivalent radius is estimated to be r = 1.13 × 10 8 cm.The derived characteristic luminosity suggests that the magnetic field at the BH horizon before it diffuses out is as high as B H,0 = 9.49 × 10 14 G.

The Periodicity of Observed X-Ray Flares
A potential periodicity in the detected flares was noticed by Watson et al. (2006), but no significant QPO signals were identified in their analysis.In Section 3, the result of the model fitting indicates the existence of a period of about 5600 s in the observer frame.Therefore, it is valuable to investigate the possibility of identifying a periodic signal through a time-series analysis of the observed data.In this section, two methods are employed to test for periodicity.

Preprocessing of Observed Lightcurve
Section 2.3 indicates that the observed periodic component is amplified with a power factor resulting from the Doppler effect, i.e., ( ( ( ))) . The least-squares method, however, is based on the linear sine function Furthermore, the intrinsic luminosity follows a broken-power-law-described trend.Therefore, prior to testing for periodicity, the observed luminosity L EM is transformed into logarithmic form.The intrinsic mean luminosity has , where σ EM and σ T are the error of L EM and L T , respectively.Following Section 3, we adopt θ jet = 0.06 rad, θ obs = 0.11 rad, and Γ = 14.56, with the derived L T displayed in Figure 4(a).

Methods and Test Results
The least-squares method was proposed to extract periods from inhomogeneous astronomical lightcurves by Lomb (1976), and its reliability tests were contributed by Scargle (1982), so this method is now known as the Lomb-Scargle periodogram (LSP).Zechmeister & Kürster (2009) gave a generalized LSP (GLS) that takes errors into account and adds an offset constant to the sinusoidal function.The wavelet transform overcomes the limitations of the Fourier transform through mathematical advancements.Foster (1996) developed the weighted wavelet Z-transform (WWZ) to detect pseudoperiodic signals in irregular time series.Aydin (2017) developed a Python program for implementing the WWZ method.The false-alarm levels given by various methods are always based on statistics.However, the power-law component, red noise, always raises the low-frequency port to a lower false-alarm possibility.Vaughan (2005) introduced a method for estimating significance in the context of the

Conclusions and Discussions
Unlike a hairless BH, a magnetar possesses rich physical properties, such as rapid rotation, a high magnetic field, glitches, and a magnetosphere, making it more conducive to interpreting various astrophysical phenomena.AGN-powered jets, however, narrate a story of the interaction between a BH and its disk, facilitated by the presence of the magnetic field.The origin of plateaus in GRB lightcurves is still a subject of debate.A plateau, followed by either a shallow or steep decay, can be attributed to either internal or external origins.The internal plateau, however, is usually attributed to a short-lived, supermassive magnetar.A BH-based model, by taking the evolution of the magnetic field at the BH horizon into account, can also replicate a lightcurve featuring an internal plateau.In this study, we propose that a precessional BZ jet would imprint a QPO signature on the internal plateau and the subsequent sharp decay.Such lightcurves cannot be readily interpreted by the scenario of a short-lived magnetar, as it would have collapsed at the break time.This model successfully reproduces a peculiar X-ray afterglow of GRB 050904, which consists of nine flares, by assuming that the observed flares are intrinsic in nature.One suggests that GRB 050904 originates from a BH central engine, which powers a precessing jet with a period of 771 s.
Chromatic X-ray plateaus and optical afterglows have been extensively observed, such as GRB 070110 (Troja et al. 2007).In GRB 050904, the observed optical and near-infrared afterglow can be well explained by the standard afterglow model (e.g., Wei et al. 2006;Gou et al. 2007), while the X-ray flares are attributed to the internal dissipation of precessing jet.This fact implies a connection between X-ray flares and plateaus, in the afterglow of GRB 050904.The results of the periodicity test indicate that the confidence level for the periodicity of GRB 050904 might not exceed 99%, a limitation largely attributable to the observational qualities.GRB 050904 is not a singular example presenting a potential QPO in the extensive X-ray afterglows collected by Swift/XRT; for example, GRB 130925A is another source to exhibit possible QPO behavior (Hou et al. 2014).However, Earth's shielding limits the ability of the XRT to capture comprehensive lightcurves of these GRBs.The synergy between XRT and the forthcoming Einstein Probe (EP; Yuan et al. 2022) and Space-based Multiband Astronomical Variable Object Monitor (Wei et al. 2016) observations of the same GRB afterglow is anticipated to yield a holistic lightcurve of these GRB afterglows.Some researchers have suggested that the prompt emission possesses potential periodicity on timescales of seconds or less (Wang et al. 2022;Xiao et al. 2022;Zhang et al. 2022Zhang et al. , 2023)).Then whether these similar short-period activities exist in the X-ray plateau is very important for understanding the central engine of GRBs.However, the sensitivity of XRT is insufficient for adequately capturing photons to discern these short-period activities.NewAthena (see also Athena: Advanced Telescope for High-ENergy Astrophysics; Barcons et al. 2015) may have extraordinary potential for researching the periodicity in X-ray afterglows.The NewAthena Wide Field Imager detection capability surpasses the XRT by more than an order of magnitude, which will enable a detailed examination of the temporal evolution of X-ray afterglows.Furthermore, the EP Follow-up X-ray Telescope also exhibits enhanced performance in the lower energy band (0.5-2 keV) compared to XRT.
the initial time of accretion and  M p is the characteristic mass accretion rate at t p .As the mass accretion rate  M decreases and falls below a critical value of ~

Figure 1 .
Figure 1.A demonstration of a QPO lightcurve.The observed lightcurve is shown in the green scatter, while the lightcurve predicted by the precessional BH jet is represented by the red dashed line.Figure 2. The pseudocolor plot illustrates values within the exclusive range of 10 < Γ < 200.Intriguingly, in the majority of scenarios, the values of Γ fall between 10 and 40, implying that the model favors a lower Γ.Additionally, the plot reveals a symmetrical distribution of θ jet and θ obs , with the viable range spanning from 0.02 rad to 0.18 rad.

Figure 2 .
Figure 1.A demonstration of a QPO lightcurve.The observed lightcurve is shown in the green scatter, while the lightcurve predicted by the precessional BH jet is represented by the red dashed line.Figure 2. The pseudocolor plot illustrates values within the exclusive range of 10 < Γ < 200.Intriguingly, in the majority of scenarios, the values of Γ fall between 10 and 40, implying that the model favors a lower Γ.Additionally, the plot reveals a symmetrical distribution of θ jet and θ obs , with the viable range spanning from 0.02 rad to 0.18 rad.

Figure 3 .
Figure 3.The contour plot for the posterior distributions of the parameters corresponds to the scenario where θ jet = 0.06 rad, θ obs = 0.11 rad, and Γ = 14.56.
background noise following a chi-square distribution.Furthermore, the background of the periodogram always exhibits two smooth broken-power-law behaviors, which complicate the estimation of the background noise.By fitting a first-order autoregressive procedure,Schulz & Mudelsee (2002) developed a Fortran program (REDFIT). 5Panels (b) and (c) in Figure 4 represent the REDFIT-based periodogram and the WWZ-based wavelet transform, respectively.The REDFIT program offers confidence estimates for the corresponding LSP.The confidence based on the chi-square distribution of the spectrum is more stringent, with the signal at P ∼ 5600 s having a confidence level of Pr > 95%, while the Monte Carlo ensemble provides a more flexible limit of Pr > 99%.The wavelet transform reveals a robust and periodic signal at a frequency of approximately f ∼ 1.8 × 10 −4 Hz (corresponding to a period of P ∼ 5600 s) .

Figure 4 .
Figure 4. (a) The derived lightcurve of L T .(b) The periodogram generated by REDFIT.The significance levels of 95% and 99% are shown by the dashed lines.(c) The periodogram generated by WWZ.A clear periodic signal is located at f = 1.8 × 10 −4 Hz.

Table 1
Fitting Results with the Precessing Jet Model Based on Figure2, we select three groups of θ jet and θ obs to test their impact on other parameters, and the Γ is determined by Equation (15).