Fast Radio Bursts in the Disks of Active Galactic Nuclei

Fast radio bursts (FRBs) are luminous millisecond-duration radio pulses with extragalactic origin, which were discovered more than a decade ago. Despite the numerous samples, the physical origin of FRBs remains poorly understood. FRBs have been thought to originate from young magnetars or accreting compact objects (COs). Massive stars or COs are predicted to be embedded in the accretion disks of active galactic nuclei (AGNs). The dense disk absorbs FRBs severely, making them difficult to observe. However, progenitors ejecta or outflow feedback from the accreting COs interact with the disk material to form a cavity. The existence of the cavity can reduce the absorption by the dense disk materials, making FRBs escape. Here we investigate the production and propagation of FRBs in AGN disks and find that the AGN environments lead to the following unique observational properties, which can be verified in future observation. First, the dense material in the disk can cause large dispersion measure (DM) and rotation measure (RM). Second, the toroidal magnetic field in the AGN disk can cause Faraday conversion. Third, during the shock breakout, DM and RM show non-power-law evolution patterns over time. Fourth, for accreting-powered models, higher accretion rates lead to more bright bursts in AGN disks, accounting for up to 1% of total bright repeating FRBs.

In magnetar models, based on the difference in emission regions (Zhang 2020), it can be roughly divided into 'close-in' scenario ('pulsar-like' emission originates from the magnetosphere of magnetars, e.g., Yang & Zhang 2018;Kumar & Bošnjak 2020;Lu et al. 2020) and 'faraway' scenario (gamma-ray bursts like, 'GRB-like' emission originates from relativistic outflows, e.g., Lyubarsky 2014;Beloborodov 2017;Metzger et al. 2019).Alternatively, FRBs have been argued to arise possibly from collisions of pulsars with asteroids or asteroid belts (Geng & Huang 2015;Dai et al. 2016).Luo ★ E-mail:fayinwang@nju.edu.cnet al. (2020b) found that the polarization position angle (PA) of FRB 180301 swings across the pulse profiles, which is consistent with a magnetospheric origin.Recently, the swing of PAs has also been reported in simulations of relativistic magnetized ion-electron shocks (Iwamoto et al. 2024).For the FRB which has been active for ten years (Li et al. 2021a), the energy budget for the magnetar's magnetic energy becomes strained especially for the low-efficiency relativistic shock origin (Wu et al. 2020).One way to alleviate this problem is to seek other central engines.The close-in models are only applicable to magnetar engines, while the far-away models are applicable to a wider range of scenarios as long as energy can be injected into the surrounding medium from the central engine.Accreting-powered models from compact objects (COs), e.g., black holes (BHs) and neutron stars (NSs), have been proposed (Katz 2020;Deng et al. 2021;Li et al. 2021b;Sridhar et al. 2021;Sridhar & Metzger 2022).
COs are predicted to be embedded in the accretion disk of active galactic nuclei (AGN).Magnetars could form from the core-collapse (CC) of massive stars.Massive stars in AGN disks are more likely to evolve rapid rotation (Jermyn et al. 2021), and eventually more magnetars are produced via the dynamo mechanism (Raynaud et al. 2020).In some cases, a magnetar can also be formed in the following process: binary neutron star (BNS) mergers (Dai & Lu 1998;Rosswog et al. 2003;Dai et al. 2006;Price & Rosswog 2006;Giacomazzo & Perna 2013), binary white dwarf (BWD) mergers (King et al. 2001;Yoon et al. 2007;Schwab et al. 2016), accretion-induced collapse (AIC) of WDs (Nomoto & Kondo 1991;Tauris et al. 2013;Schwab et al. 2015) and neutron star-white dwarf (NSWD) mergers (Zhong & Dai 2020).The AGN disk provides a favorable environ-ment for these processes (e.g.McKernan et al. 2020;Perna et al. 2021;Zhu et al. 2021b;Luo et al. 2023).On the other hand, the accretion rate onto COs can be hyper-Eddington (e.g.Wang et al. 2021a;Pan & Yang 2021b;Chen et al. 2023), which would release enough inflow energy for single FRB or successive ones (Stone et al. 2017;Bartos et al. 2017).Meanwhile, the accreting COs can potentially launch relativistic jets (e.g.Tagawa et al. 2022Tagawa et al. , 2023;;Chen & Dai 2024), providing an optimistic channel to drive FRBs.Overall, FRBs are expected to be produced by both young magnetars and accreting COs in AGN disks.Alternatively, FRBs have been argued to arise possibly from collisions of pulsars with asteroids or asteroid belts.
In this paper, we investigate the generation and propagation of FRBs in AGN disks.Schematic diagrams of our model are shown in Figure 1, and related physical processes are shown in Figure 2.This paper is organized as follows.We assume that young magnetars or accreting COs in AGN disk can emit FRBs, and then progenitors' ejecta or outflows of the accretion disk interact with the disk material to form a cavity.The existence of the cavity can reduce the absorption of materials in the dense disk, making the FRB easier to observe.If the feedback of the ejecta or outflows is weak, the DM, absorption and RM from the disk are presented in Section 2. For FRBs from the magnetosphere of young magnetars, the AGN disk environments do not affect the radiation properties.The propagating effects from the cavity opened by progenitors' ejecta are presented in Section 3.For accreting-powered models, the burst luminosity depends on the accreting rate.The hyper-Eddington accreting of CO disks in AGN disks makes FRB brighter.In addition to burst luminosity, propagation effects from the cavity opened by outflows from CO disks are given in Section 4. The inflections on turbulent disk materials and magnetic field governed by a dynamo-like mechanism in AGN disk are discussed briefly in Section 5. Finally, conclusions are given in Section 6.In this work, we use the expression   = /10  in cgs units unless otherwise noted.

DM AND RM CONTRIBUTED BY AGN DISKS
In this work, we investigate the observational properties of FRBs in two different disk models, SG model (Sirko & Goodman 2003) and TQM model (Thompson et al. 2005).The disk structures of the SG/TQM model are presented in Appendix A. Laterally propagating signals are easily absorbed by dense materials.Therefore, we can only receive signals from the vertical direction.The vertical structure of AGN disks has a Gaussian density profile where  0 () is the mid-plane disk density and  is the scale height, which is given by the solution of the disk model (see Appendix A).
For the inner disk, the free-free absorption optical depth is extremely high at the midplane of the disk due to dense ionized gas.We can only detect FRBs from source sites at a few scale heights, e.g. for the source at ∼ 10 −3 pc, the disk becomes optically thin only for ℎ ≳ 3.9 (Perna et al. 2021).However, the temperature is lower for the outer disk.The temperature for an SG disk is ∼ 10 3 − 10 4 K when  > 10 4   , where   ≡ 2 / 2 is the gravitational radius of the SMBH.While for a TQM disk, it is just ∼ 10 2 − 10 3 K.If there is no extra ionization process (Ultraviolet/X-ray photons or shocks), gases are neutral at such temperatures.However, gases may also be ionized by radiation (Dyson & Williams 1980) or shocks associated with transients in AGN disk, such as supernova explosions (Grishin et al. 2021;Moranchel-Basurto et al. 2021;Li et al. 2023b), accretion-induced collapse of an NS (Perna et al. 2021) or a WD (Zhu et al. 2021b) and accretion outflow feedbacks of compact stars (Wang et al. 2021a;Chen et al. 2023), gamma-ray bursts/kilonovae (Zhu et al. 2021a;Ren et al. 2022;Yuan et al. 2022), and gravitationalwave bursts (Wang et al. 2021b).In this section, we assume that the radiation or shocks simply provide additional ionization and do not need to be associated with FRBs.In some models, FRBs are associated with young magnetars or accreting COs and we investigate the feedback (progenitors' ejecta or accreting outflows) in sections 3 and 4 in detail.
Considering the ionization process in detail is complicated, we simply discuss two extreme cases (Perna et al. 2021): the material is fully ionized and there is no extra ionization source in the vertical direction.FRBs are dispersed or absorbed by the disk plasma (see Section 2.1).The magnetic fields in the disk cause Faraday rotationconversion (see Section 2.2).

DM and the optical depth from the disk
If a FRB source is located at height ℎ, the DM from the disk for the ionized outer disk is where erfc is the complementary error function and ĥ = ℎ/.The mean molecular weight  = 0.62 is taken in this work.If the FRB source is located in the midplane of the AGN disk, DM can be estimated as The free-free absorption optical depth is () = ∫ LOS  ff ()d (Rybicki & Lightman 1986), where  ff () = 0.018 −3/2  2 i  e  i  −2 ḡff is the free-free absorption coefficient with ḡff ∼ 1 being the Gaunt factor.Here we assume  e ∼  i and  i ∼ 1.For the vertical direction, the free-free absorption optical depth is where  0 is the free-free absorption optical depth from the midplane of the AGN disk (5) DMs for different disk models are shown in Figure 3.The free-free absorption optically thick region for a signal with a frequency of 1 GHz is shown in gray.Due to absorption, FRBs can be observed only from source sited at a few scale heights.For the AGN disk of an SMBH with  = 10 8  ⊙ , the largest DM contributed by the disk is about 10 4 − 10 5 pc cm −2 .Although such value is larger than any DM of FRBs known to date, there are still possibilities that it can be detected by radio telescopes such as CHIME, which can detect DMs up to ∼13,000 pc cm −2 (CHIME/FRB Collaboration et al. 2018).For an SMBH with  = 4 × 10 6  ⊙ , the largest DM contributed by the disk is about 10 3 pc cm −2 .It is similar to the DM of FRB 20190520B (Niu et al. 2022).

Faraday rotation-conversion from the disk
The radiation transfer equation of the Stokes parameters is (Sazonov 1969;Melrose & McPhedran 1991) where   ,  = , , ,  are the emission coefficients,   ,  = , , ,  are the absorption coefficients and   ,  = , ,  are the Faraday rotation-conversion coefficients.If there are no extra emission and absorption processes during the propagation of FRBs, the total intensity  is conserved and can be assumed to be unity.Thus, the transfer equation can be simplified to The linear polarization degree is  = √︁  2 +  2 , the circle polarization degree is  and the total polarization degree is  =

√
2 +  2 .The Faraday rotation rate of the electromagnetic wave with the an-

Escape Absorbed
Strong Feedback

Escape Absorbed
Accreting COs

DM, Faraday rotationconversion
Long-term unchanged Time-dependent Figure 2. Related physical processes of FRBs in AGN disks.FRBs are thought to have originated from active magnetars or accreting compact COs.Progenitors' ejecta or accretion disk outflows interact with the disk material to form a cavity.If the feedback of the ejecta or outflows is weak, whether the FRB is absorbed or not depends only on the location of the source.In this case, the DM and RM from disk material are stable in the long term but fluctuate in the short term due to the turbulence.The toroidal magnetic field in the AGN disk can cause Faraday conversion.If the feedback is strong, as the cavity expands, at some point the optical depth drops below unity, allowing the FRB to be observed.The cavity expansion causes time-dependent DM and RM.For accreting-powered models, the burst luminosity depends on the accreting rate.The hyper-Eddington accreting of CO disks in AGN disks makes FRB brighter.gular frequency  is (Gruzinov & Levin 2019) and the Faraday conversion rate is where B , B , B are the three components of the magnetic field unit direction vector B.   = √︁ 4   2 / and   = / are plasma and Larmor frequencies, respectively.
Another description of the transfer Equation ( 7) is in the vector space: where  = (, , ) and  =   ,   ,   .The geometric interpretation of Equation ( 10) is the Faraday rotation-conversion (or generalized Faraday rotation) on the Poincaré sphere.The Faraday rotation or Faraday conversion angle is which are only relevant to the properties of the Faraday screen.The importance of FR and FC can be evaluated by the radio of Faraday conversion rate and Faraday rotation rate If   ∼   ∼   , we have   /  ≪ 1, which means that the Faraday rotation dominates.In most cases, the Faraday conversion is negligible unless the magnetic field has a significant vertical component (Melrose & Robinson 1994;Melrose 2010;Gruzinov & Levin 2019), e.g., the magnetic field reversal region in a binary system (Wang et al. 2022;Li et al. 2023a;Xia et al. 2023) or a quasi-toroidal magnetic field in a supernova remnant (SNR; Qu & Zhang 2023).
The magnetic field of the gas in the AGN disk can be estimated by the ratio of gas pressure to magnetic pressure For the very strong magnetization disk,  B ∼ 10, while for the very weak magnetization levels,  B ∼ 10 5 (Salvesen et al. 2016).The magnetic field of the AGN disk has both toroidal and poloidal components.If we set the direction of the toroidal field is the -axis and the direction perpendicular to the disk midplane (line of sight direction) is the -axis.In this work, we only consider Faraday rotation DMs form the AGN disk.The free-free absorption optically thick region for a signal with a frequency of 1 GHz is shown in gray.Due to absorption, FRBs can be observed only from source sited at a few scale heights.For the AGN disk of an SMBH with  = 10 8  ⊙ , the largest DM contributed by the disk is about 10 4 − 10 5 pc cm −2 .For an SMBH with  = 4 × 10 6  ⊙ , the largest DM contributed by the disk is about 10 3 pc cm −2 .It is almost the same as the DM of FRB 20190520B (Niu et al. 2022).
and Faraday conversion in the case where poloidal and toroidal fields dominate and we assume that  B does not change with height.
For the poloidal field, the parallel component is B , where  0 is the location of the source.If the outer disk is fully ionized, for an FRB emitted at a height ℎ from the midplane, its RM is where RM 0 is the RM from the midplane RMs from the weak magnetization AGN disk ( B ∼ 10 4 , Salvesen et al. 2016) are shown in Figure 4.The free-free absorption optically thick region for a signal with a frequency of 1 GHz is shown in gray.For the AGN disk of an SMBH with  = 10 8  ⊙ , the largest RM contributed by the disk is about 10 4 − 10 5 rad m −2 , which is in the same order of magnitude as the FRB with the extreme RM, e.g., FRB 20121102A (RM∼ 10 5 rad m −2 ; Michilli et al. 2018;Hilmarsson et al. 2021b) and FRB 20190520B (RM∼ 10 4 rad m −2 ; Anna-Thomas et al. 2023).For an SMBH with  = 4 × 10 6  ⊙ , the largest RM contributed by the disk is about 10 2 − 10 3 rad m −2 , which is consistent with the RM of most FRBs.
If there is no extra ionization source, whether the FRB is absorbed or not depends only on the optical depth of the disk.For the TQM disk, the disk becomes optical thin at the distance ∼ 10 3 − 10 4   , where the density is about 10 −10 g cm −3 and the temperature is about a few thousand Kelvin.At this temperature, the ionization degree of RMs contributed by the weak magnetization AGN disk ( B ∼ 10 4 , Salvesen et al. 2016).The free-free absorption optically thick region for a signal with a frequency of 1 GHz is shown in gray.For the AGN disk of an SMBH with  = 10 8  ⊙ , the largest RM contributed by the disk is about 10 4 − 10 5 rad m −2 , which is in the same order of magnitude as the FRB with the extreme RM, e.g., FRB 20121102A (RM∼ 10 5 rad m −2 ) and FRB 20190520B (RM∼ 10 4 rad m −2 ).For an SMBH with  = 4 × 10 6  ⊙ , the largest RM contributed by the disk is about 10 2 − 10 3 rad m −2 , which is consistent with the observed RMs of most FRB. the gas is extremely low.For example, for a gas at 3000 K, the ionization degree given by the Saha equation is  ∼ 10 −8 .Although the DM contribution from the disk is negligible (DM∼ 0.5 pc cm −2 ), FC may still occur for toroidal fields.By solving Equation (7), the polarization properties as a function of height are shown in the left panel of Figure 5.We assume that the intrinsic radiation of the source is 100% linearly polarized.When radiation passes through a disk with a toroidal magnetic field, the total polarization degree (, shown in the orange line) remains unchanged, and linear polarization (, shown in the blue line) and circular polarization (, shown in the green line) are converted into each other.In the 1-1.5 GHz band, the changes in linear and circular polarization are shown in the right panel of Figure 5, which is similar to the polarization properties of FRB 20201124A (Xu et al. 2022).
Magnetars could form from the CC of massive stars or compact binary mergers.Here we give the feedback of progenitors' ejecta in AGN disks.Considering that the shock propagation distance in the vertical direction may be much larger than the disk height, in addition to the disk material, the ejecta also interact with the surrounding The polarization properties vary with height (left panel) and different bands (1-1.5 GHz, right panel) when radiation passes through a disk with a toroidal magnetic field.The intrinsic radiation of the source is 100% linearly polarized.When radiation passes through a disk with a toroidal magnetic field, the total polarization degree (, shown in orange lines) remains unchanged, and linear polarization (, shown in blue lines) and circular polarization (, shown in green lines) are converted into each other.
Table 1.Disk Parameters of Different Models at  0 = 1 pc.material.Assuming the circum-disk material has a power-law density profile, the disk and circum-disk material can be modeled as (Zhou et al. 2023) where  0 () is the mid-plane disk density given in the solution of the disk model (see Appendix A), and  = 1.5 − 3 is the powerlaw index (Zhou et al. 2023).The critical height ℎ c represents the boundary between the disk and the circum-disk material.The critical density  c =  0 () exp −ℎ 2 c /2 2 is the boundary density at ℎ c .Observationally, the critical height is difficult to constrain.In this work, we assume that the transition between the disk and the circumdisk material occurs when (ℎ)/ 0 ∼ 10 −3 .

The cavity evolution
After SN explosions or the merger of two COs, forward and reverse shock are generated during the interaction between the ejecta and the circumstellar medium (CSM).The DM and RM from the ejecta and the shocked CSM have been well studied before (Yang & Zhang 2017;Piro & Gaensler 2018;Zhao et al. 2021;Zhao & Wang 2021).
When the swept mass is much smaller than the ejected mass, the shock evolution is in the free expansion (FE) phase with the velocity The shock velocity (top panel) and radius (bottom panel) evolution for  0 = 1 pc,  0 = 10 51 erg,  ej = 10 ⊙ and  = 2.The disk parameters are taken from Model c in Table 1.Timescales of FE duration, breakout and leaving the disk boundary are shown in gray vertical solid, dashed, and dashdotted lines, respectively.The height expansion (blue lines) goes through the Free expansion phase (phase I,  ≤  FE ), the ST phase (decelerated by disk medium, phase II,  FE <  ≤  bre ), the Sakurai accelerating phase (phase III,  bre <  ≤  c ) and the ST phase (decelerated by circum-disk medium, phase IV,  c <  ≤  cav ) in sequence.The width expansion (red lines) goes through the Free expansion phase (phase I,  ≤  FE ), the ST phase (phase II,  FE <  ≤  bre ) and the SP phase (phase V,  >  bre ) in sequence.
sh =  0 ≡ √︃  0 / ej , where  0 and  ej are the explosion energy and the ejecta mass, respectively.When the swept mass is comparable to the ejected mass, the shock evolution enters the Sedov-Taylor (ST) phase  sh ∝  −3/5 (Taylor 1946;Sedov 1959).The duration of the FE phase is In previous studies, the contribution to DM and RM from both shocked ejecta and shocked ISM has been considered because the age of the source is unknown (Piro & Gaensler 2018;Zhao et al. 2021;Zhao & Wang 2021).However, in the AGN disk, since the FE phase (ejecta-dominated phase) lasts only a few years, we only consider the evolution of the forward shock.
The conditions for shock breakout can be roughly estimated when the vertical propagation distance is comparable to the height of the disk  H ≃  (Moranchel-Basurto et al. 2021;Chen et al. 2023).After the breakout, the shock propagates in a region where the density drops sharply, and the shock is accelerated (Sakurai accelerating phase  sh ∝  −  , see Sakurai 1960).All the above processes can be described by the following equation (Matzner & McKee 1999) where  = 0.19 (Matzner & McKee 1999), is the swept mass in the vertical direction.The FE phase, the ST phase and the Sakurai accelerating phase work when  sw ≪  ej ,  sw ≫  ej and (ℎ) ≫ (0), respectively.The breakout timescale is When the shock reaches a critical height ℎ c , the acceleration stops and is decelerated by the circum-disk material.The shock evolution re-enters the ST phase for  >  c , where  c and  c is the shock velocity and time at ℎ = ℎ c .
Although the size of the radial propagation of the shock is much smaller than that of the vertical propagation, the radial propagation determines how long the cavity exists.When  ≤  bre , the shock also experiences the FE and ST phase in the radial direction The radial distance  is usually much smaller than the location of the FRB source  0 , which makes radial density changes across the disk negligible.Therefore, the swept mass in the radial direction is  sw () = 4/3 3  d ( 0 ).Also, since the density does not change much, the Sakurai accelerating phase is not considered in the radial direction.When the shock breaks out, the cavity depressurizes and the SNR becomes the shape of a ring-like shell.Then, the shock evolution enters a momentum-conserving snowplow (SP) phase When the shock decelerates to the speed of the local sound, the shock evolution ends.The radial width of the cavity in the AGN disk is The cavity formation timescale can be calculated from Equation ( 22) In summary, shock velocity is given by and The shock radius can be obtained by solving Equations ( 25) and ( 26).The radial and vertical shock evolution for the SNe explosion model with  0 = 1 pc,  0 = 10 51 erg,  ej = 10 ⊙ and  = 2 is shown in Figure 6.The disk parameters are taken from Model c in Table 1.Timescales of FE duration, breakout and leaving the disk boundary are shown in gray vertical solid, dashed, and dash-dotted lines, respectively.The height expansion (blue lines) goes through the Free expansion phase (phase I,  ≤  FE ), the ST phase (decelerated by disk medium, phase II,  FE <  ≤  bre ), the Sakurai accelerating phase (phase III,  bre <  ≤  c ) and the ST phase (decelerated by circum-disk medium, phase IV,  c <  ≤  cav ) in sequence.The width expansion (red lines) goes through the Free expansion phase (phase I,  ≤  FE ), the ST phase (phase II,  FE <  ≤  bre ) and the SP phase (phase V,  >  bre ) in sequence.
Finally, the cavity is refilled by the AGN disk material with the speed of the local sound, and the refill timescale is estimated by

DM and RM Variations
The long-term monitoring of DM and RM of repeating FRBs can reveal environments of the magnetar, so it is necessary to show the time-dependent DM and RM.The contributions from the disk are given before and are unimportant after the shock breakouts.Thus, we consider the contributions of DM mainly from two regions: the unshocked cavity and the shocked shell.In the shocked region, some shock energy converts into magnetic field energy.Therefore, RM is only contributed by the shocked shell.Before the shock breakouts, the difference in expansion between the radial and vertical directions can be ignored, so the cavity can be regarded as spherical (see Figure 6).However, after the shock breakouts, because the density of gas in the AGN disk decreases rapidly in the vertical direction, the vertical propagation of the shock becomes easier.At this point, the shape of the cavity deviates from a spherical shape.For simplicity, we assume that the cavity is always a cylinder during expansion and the cavity volume is  cav =  H  2 W , where  H and  W are the solutions on Equations ( 25) and ( 26).The time-dependent DM from the cavity is where  is the thickness of the shock and  is the ionization fraction.The shock thickness 1 −  ∼ 0.9 is taken, which is consistent with results given by self-similar solutions of the SNR (Chevalier 1982).From Equation (28), we find that DM cav only depends on the radial expansion.For the FE phase, the cavity size is much smaller than the disk height ( W ≈  H ≪ ).At this time, the disk density can be regarded as a constant (see Equation ( 1)).Thus, DM from the cavity evolves as DM cav ∝  −2 W ∝  −2 .The approximate scaling laws for the remaining two phases in Equation ( 28) can be obtained in the same way.
The time-dependent free-free absorption optical depth from the cavity is   3.2.Solid and dashed lines represent the case of SG and TQM disk models, respectively. ff,sh = 1 are shown in gray dashed horizontal lines.For SG-NSWD and SG-BNS models with  = 10 8  ⊙ , the optical depth is still greater than unity when the cavity stops expanding.In other cases, the cavity becomes transparent sometime after the magnetar is born.For binary mergers or AIC progenitors, this time is hundreds to thousands of years for the SG disk model, while it only takes a few decades for the TQM disk model.For massive star progenitors, the time become transport is about a thousand years for the SG disk model, while it is shortened to a few hundred years for the TQM disk model.
For the unshocked ejecta, a low ionization fraction  = 0.1 and temperature  ej = 10 4 K is taken (Zhao et al. 2021).Except for the Sakurai accelerating phase ( bre <  ≤  c ), where  H depends on the numerical solution of Equation ( 25), the approximate scaling laws for the remaining phases is given in Equation ( 29).
For the shock region, the temperature can be estimated from where  B is the Boltzmann constant and the shock velocity  sh is given in Equation ( 25).The high temperature makes gas fully ionized.The density of the shocked matter is  sh = 4 cd for strong shock waves.The time-dependent DM from the shocked shell is The time-dependent free-free absorption optical depth from the shocked shell is The magnetic field in the shocked region is where  B is the magnetic energy density fraction and  th = 9 cd  2 sh /8.The RM from the shocked shell is

The influence of the progenitors
The DM,  ff and RM evolution of the SNe model are shown in Figure 7.The parameters are the same as Figure 6. ).Although the material in the AGN disk is very dense, as the cavity expands, it can become transparent around a thousand years after the birth of the magnetar.
Although the merger dynamics are complicated, we can think of mergers as an explosion like a supernova in the long term, ejecting a certain amount of energy and material into the surroundings (Zhao et al. 2021).Due to the lack of observations, the mass and energy of the ejecta after compact binary mergers are taken from the results of the numerical simulation (Dessart et al. 2007;Bauswein et al. 2013;Radice et al. 2018;Zenati et al. 2019; see Table 3.2).
The DM, free-free absorption optical depth and RM evolution of different magnetar formation channels are shown in Figure 8.The parameters and the evolution timescale of each model are listed in Table 3.2.Solid and dashed lines represent the case of SG and TQM disk models, respectively. ff,sh = 1 are shown in gray dashed horizontal lines.For SG-NSWD and SG-BNS models with  = 10 8  ⊙ , the optical depth is still greater than unity when the cavity stops expanding.In other cases, the cavity becomes transparent sometime after the magnetar is born.For binary mergers or AIC progenitors, this time is hundreds to thousands of years for the SG disk model, while it only takes a few decades for the TQM disk model.For massive star progenitors, the time become transport is about a thousand years for the SG disk model, while it is shortened to a few hundred years for the TQM disk model.

ACCRETING COMPACT OBJECTS IN AGN DISKS
The close-in models are only applicable to magnetar engines, while the far-away models apply to a wider range of scenarios as long as energy can be injected into the surrounding medium from the central engine.Motivated by the periodic repeating FRBs (Chime/Frb Collaboration et al. 2020;Rajwade et al. 2020;Cruces et al. 2021), accreting-powered models from COs in ULX-like binaries have been proposed (Sridhar et al. 2021).In this model, FRBs are generated via synchrotron maser emission from the short-lived relativistic outflows (or "flares") decelerated by the pre-existing (or "quiescent") jet.For the BH engine, if the spin axis is misaligned with the angular momentum axis of the accretion disk, Lens-Thirring (LT) precession makes the FRB periodic.FRB from the precession jet encounters the disk wind which contributes variable DMs and RMs.The synchrotron radio emission from the ULX hypernebulae has been proposed to explain the persistent radio source (PRS) of FRBs (Sridhar & Met-zger 2022), e.g., FRB 20121102A (Chatterjee et al. 2017) and FRB 20190520B (Niu et al. 2022).
To explain some most luminous FRBs, the COs should be undergoing hyper-Eddington mass transfer from a main-sequence star companion.We would like to point out that mass inflow rates of COs in the AGN disks are also possible to be extremely hyper-Eddington (Chen et al. 2023).In this section, we investigate the accretingpowered models of FRBs in the AGN disks.In this work, we focus on the influence of the AGN disk environment, such as the burst properties and variable DMs and RMs from the disk wind (ignore the precession).Other properties can be found in Sridhar et al. (2021); Sridhar & Metzger (2022).

Burst properties
In this section, we briefly outline the intrinsic properties of FRBs from accreting COs (taking a BH as an example) in AGN disks.The extremely high accreting rate of COs in the AGN disk environment (see Section 4.2 in detail) has a great impact on the burst luminosity and peak frequency.When a BH is rotating, the spin energy can be extracted by forming ultra-relativistic jets (i.e., Blandford-Znajek (BZ) mechanism Blandford & Znajek 1977).The BZ luminosity is (Tchekhovskoy et al. 2011) where  BZ is the jet efficiency,  = /  Edd is the the dimensionless accretion rate,  Edd =  Edd / 2 is the Eddington limit accretion rate and  Edd = 4   /  = 1.26 × 10 39 erg s −1 /10 ⊙ is the Eddington limit accretion luminosity.The maximum jet efficiency is  max ∼ 1.4 for an extreme rotating BH with the dimensionless spin parameter  = 0.99 (Tchekhovskoy et al. 2011).For an accreting NS, the jet power can also be extracted from the rotational energy via a BZ-like mechanism (Parfrey et al. 2016).Taking  ∼  max ≃ 1, the maximum isotropic-equivalent FRB luminosity can be estimated by where  CO is the CO mass,   is the radio emission efficiency (for synchrotron maser scenarios,   ∼ 10 −3 , e.g., Sridhar et al. 2021) and   is the beaming fraction. acc is the efficient accretion timescale of COs (see Section 4.3).As mentioned before, FRBs are powered by the sudden accretion flare with the luminosity  f =   •  2 .The flare ejecta propagates into the cavity of the quiescent jet with the luminosity and bulk Lorentz factor being  q =  q  •  2 and Γ q , respectively.For FRBs to escape, the quiescent jet should have a large bulk Lorentz factor Γ q ≳ 100 and a low jet efficiency  q ≪ 1 (Sridhar et al. 2021).During the interaction between the flare ejecta and the gas in the quiescent jet, a forward shock generates with a radius (Sari & Piran 1995) where  f is the duration of burst.The blast Lorentz factor Γ sh is given by pressure balance (Beloborodov 2017) Electrons at the FRB emission radius  FRB in the shocked gas gyrate with the Larmor radius  L = Γ q  e  2 / q , where  q ≃  q  q / 2 FRB 1/2 is the lab-frame measured magnetic field in high magnetization ( q ≳ 1) upstream.The peak frequency from synchrotron maser emission is (Gallant et al. 1992;Plotnikov & Sironi 2019) FRBs have been detected from 110 MHz (Pleunis et al. 2021) to 8 GHz (Gajjar et al. 2018), which is about the same as the estimated frequency for Γ q ∼ 100 and  q ∼ 0.01.When  <  acc , the CO accretion rate in AGN disks is extreme hyper-Eddington (  ∼ 10 5 − 10 10 , see Chen et al. 2023), which is enough to drive the most luminous FRBs till now.If the disk wind cavity can become optical thin before the efficient accretion stops (see Section 4.3), high-frequency bright bursts are generated.However, when  >  acc , the accretion is weak in the low-density cavity (  ∼ 10, Chen et al. 2023).At this time, we can only receive low-frequency faint bursts.The intrinsic burst properties for different accretion CO models are shown in Table 3, and the disk parameters are taken from the TQM disk model in Table 1.

Accretion and outflow of COs
We adopt descriptions of the accretion and outflow of COs in AGN disks from Chen et al. 2023.Here, we list the relative equations briefly.The mass inflow rate of the CO accretion disk at the outer boundary ( obd ) can be described based on the Bondi-Holye-Lyttleton (BHL) accretion rate (see Edgar 2004 for a review) where  CO is the gas density near the CO, and  rel is the relative velocity between the CO and the gas in the AGN disk.However, in the AGN disk, taking into account the influence of SMBH gravity and the finite height of the AGN disk, the modified gas inflow rate is (Kocsis et al. 2011) where  Hill = ( CO /3) 1/3  CO is the Hill radius with  CO being the radial location of the CO and  BHL =   CO /  2 rel +  2 s is BHL radius.
The outer boundary radius of the circum-CO disk can be approximated to the circularization radius ( obd ∼  cir ).Under the assumption of the angular momentum conservation, the circularization radius of infalling gas is where  rel = min { BHL ,  Hill } is the radius of CO gravity sphere.Due to the differential rotation of the AGN disk, the captured gas exhibits varying velocity relative to the CO, which can be estimated as where  K is the Keplerian velocity of the AGN disk.In the above Equation, we use the following approximation:  CO ≫  rel , which is obviously true for the CO location we are interested in.Though simplified, the resulting  cir approximately matches the values obtained from numerical simulations (e.g.Tanigawa et al. 2012;Li et al. 2022).
Besides capturing the nearby gas, an embedded CO can also exert a gravitational torque to repel the ambient gas, resulting in a reduction of gas density in the AGN disk annulus at  CO (e.g. Ward 1997).Consequently, the gas density around the CO is set as where the reduced density and half-width of the reduced region are (Kanagawa et al. 2015(Kanagawa et al. , 2016;;Tanigawa & Tanaka 2016) and The outer region of the circum-CO accretion disk becomes selfgravity unstable if the inflow mass rate is very high, leading to the reduction in the rate because of gas gravitational fragmentation (Pan & Yang 2021b;Tagawa et al. 2022).The Toomre parameter of the circum-CO disk is  CO = Ω K   /Σ ∼ 2 CO ℎ 3  3 K /  inflow , where  CO , ℎ =  CCOD / and  K = √︁   CO / is the viscosity parameter, the disk height ratio and the Keplerian velocity of the circum-CO disk, respectively.The modified mass inflow rate is The values of the inflow mass rate  inflow and the outer boundary radius of the circum-CO disk  obd depend on Equations ( 40)-( 47).It isn't easy to express with a simple analytical formula.The circum-CO accretion disk's self-gravity unstable conditions for different SMBH mass, CO locations and specific accreting models are studied in Chen et al. (2023).In this work, the value of  CO of different accreting models is shown in Table 3.For all the models we chose, the accretion of circum-CO disk is stable.
The initial mass inflow rates of COs in the AGN disk are given by Equation ( 47), which should be hyper-Eddington (Chen et al. 2023).Photons are trapped for the extremely hyper-Eddington inflow and the trapping radius is  tr = 3 ℎ  (Kocsis et al. 2011).The mass inflow rate is reduced inside the trapping radius due to the efficient outflow driven by the disk radiation pressure.In summary, the radiusdependent mass inflow rate is (Blandford & Begelman 1999) where  is the power-law index.The numerical simulations of Yang et al. (2014) show that  ∼ 0.4 − 1, but the smaller value is also possible (Kitaki et al. 2021).
The outflow of the circum-CO disk driven by radiation pressure can take away a fraction of the viscous heating.The luminosity of disk outflow is given by (Chen et al. 2023) where  w is the fraction of the heat taken away and   ≡ 2  CO / 2 is the gravitational radius of the CO.In this work, we take a constant  w = 0.5 (Chen et al. 2023). out is the mass inflow rate at the outer boundary of the circum-CO disks  out = min{ obd ,  tr }.The inner boundary of the circum-CO disk is The inner boundary for BHs is ∼ 10  .But for NSs, the strong magnetic field and the hard surface should also be considered (Takahashi et al. 2018).The circum-NS disk is truncated due to the magnetic stress of the NS magnetosphere at a radius where the typical value of  is 0.5−1 (Ghosh & Lamb 1979;Chashkina et al. 2019).When the accretion flow hits the hard surface of the NS, additional energy with luminosity  acc ≃   in ( in )  NS / NS is released, so the total energy injected is  out +  acc (Chashkina et al. 2019).The velocity of the disk wind is where the total luminosity the disk wind takes away is In our calculations, the following typical values are used:  NS = 1.4  ⊙ ,  BH = 10  ⊙ ,  NS = 10 km and  = 0.5.

The cavity evolution
The disk wind interacts with the disk material and forms a shocked shell, which is analogous to the evolution of the stellar-wind-driven interstellar bubbles (Castor et al. 1975;Weaver et al. 1977).At first, the wind expands freely until the mass released out  out =  w  in the wind is comparable to the swept mass in the disk  sw = 4/3( w ) 3  d (in this phase, the cavity is almost spherical).Thus, the duration of the FE phase is For  ≫  FE , the shell evolution also goes through the adiabatic expansion phase and the radiative cooling phase in sequence.In both phases, the shock wave radius evolves as follows (Weaver et al. 1977): where  ≈ 0.88 for the adiabatic expansion phase, but in the radiative cooling phase, swept gas collapses into a thin shell and makes  ≈ 0.76 (Weaver et al. 1977).In this work, we ignore the FE and radiative cooling phases because they have less impact on the shock evolution (see Equations ( 54) and ( 55)).The shock velocity in the adiabatic expansion phase is (Weaver et al. 1977) The adiabatic expansion phase lasts until efficient accretion of CO stops.The accretion timescale can be approximated by the viscous timescale of the accretion disk (Chen et al. 2023) Similar to Section 3, the propagation of shock in the AGN disk also needs to consider the vertical and radial directions respectively.The breakout timescale is After the shock breaks out, the shock is accelerated in the vertical direction (Sakurai 1960).When the shock enters the circum-disk material, the shock evolution re-enters the adiabatic expansion phase.When the efficient accretion of CO stops, the shock transitions to the ST phase.
In the radial direction, the shock evolution enters a momentumconserving snowplow phase (see Equation 22) after the shock breaks out.The radial width of the cavity can be obtained when the shock velocity equates to the local sound speed Inspired by Equation ( 18), we describe the evolution of the cavity as in the vertical direction, and  60) and ( 61) for BH1.The model parameters are listed in the BH1 model in Table 3.The shock velocity and radius evolution are shown in the top and bottom panels, respectively.Timescales of breakout, leaving the disk boundary and effective accretion are shown in gray vertical dashed, dash-dotted and solid lines, respectively.The height expansion (blue lines) goes through the adiabatic expansion phase in the disk (phase I,  ≤  bre ), the Sakurai accelerating phase (phase II,  bre <  ≤  c ), the adiabatic expansion phase in the circum-disk medium (phase III,  c <  ≤  acc ) and the ST phase (phase IV,  acc <  ≤  cav ) in sequence.The width expansion (red lines) goes through the adiabatic expansion phase in the disk (phase I,  ≤  bre ) and sp phase (phase V,  >  bre ) in sequence.
in the radial direction.Equation ( 60) means that the shock leaves the disk boundary before the efficient accretion stops ( c ≤  acc ).But if  c >  acc , the diabatic expansion phase (the second line in Equation ( 60)) will be skipped.Solutions of Equations ( 60) and ( 61) for BH1 are shown in Figure 9.The model parameters are listed in the BH1 model in Table 3.The shock velocity and radius evolution are shown in the top and bottom panels, respectively.Timescales of breakout, leaving the disk boundary and effective accretion are shown in gray vertical dashed, dash-dotted and solid lines, respectively.The height expansion (blue lines) goes through the adiabatic expansion phase in the disk (phase I,  ≤  bre ), the Sakurai accelerating phase (phase II,  bre <  ≤  c ), the adiabatic expansion phase in the circum-disk medium (phase III,  c <  ≤  acc ) and the ST phase (phase IV,  acc <  ≤  cav ) in sequence.The width expansion (red lines) goes through the adiabatic expansion phase in the disk (phase I,  ≤  bre ) and sp phase (phase V,  >  bre ) in sequence.
When expansion ceases, the AGN disk material will refill the cavity.The refilled timescale is In the case of progenitor ejecta, the cavity can only be opened once.66) and ( 67) for the NS2 model.The model parameters are listed in the NS2 model in Table 3.When  bre >  acc , the shock evolution is in analogy with that of the SN explosion (see Figure 6).The shock velocity and radius evolution are shown in the top and bottom panels, respectively.Timescales of breakout and leaving the disk boundary are shown in gray vertical dashed and dash-dotted lines, respectively.The height expansion (blue lines) goes through the ST phase in the disk (phase I,  ≤  bre ), the Sakurai accelerating phase (phase II,  bre <  ≤  c ) and the ST phase in the circum-disk medium (phase III,  c <  ≤  cav ) in sequence.The width expansion (red lines) goes through the ST phase in the disk (phase I,  ≤  bre ) and sp phase (phase IV,  >  bre ) in sequence.
But for the case of accretion outflow, the accretion rate can be restored to hyper-Eddington after the cavity is refilled.Then, the strong disk outflow forms the cavity again and the evolution process is circular.
In Equation ( 61), we assume that the accretion timescale is much longer than the breakout timescale ( acc ≫  bre ).If  bre >  acc , we can treat the short-duration accretion as an injection with the energy  w ∼  w  acc .The radius of the shock shell expands as (Ostriker & McKee 1988) and the shock velocity is In this case, the breakout timescale can be recalculated by Equation ( 63) The shock evolution equation for  bre >  acc is in the vertical direction, and in the radial direction.Solutions of Equations ( 66) and ( 67) for the NS2 model are shown in Figure 10.The model parameters are listed in the NS2 model in Table 3.When  bre >  acc , the shock evolution is in analogy with that of the SN explosion (see Figure 6).The shock velocity and radius evolution are shown in the top and bottom panels, respectively.Timescales of breakout and leaving the disk boundary are shown in gray vertical dashed and dash-dotted lines, respectively.The height expansion (blue lines) goes through the ST phase in the disk (phase I,  ≤  bre ), the Sakurai accelerating phase (phase II,  bre <  ≤  c ) and the ST phase in the circum-disk medium (phase III,  c <  ≤  cav ) in sequence.The width expansion (red lines) goes through the ST phase in the disk (phase I,  ≤  bre ) and sp phase (phase IV,  >  bre ) in sequence.
For the SG model, weaker outflows and a shorter cavity formation timescale are expected because the sound speed is higher.For a accreting BH in the SG disk with  = 4 × 10 6  ⊙ and  0 = 1 pc (model a in Table 1), the wind luminosity is  w = 10 40 erg s −1 and the efficient accretion timescale is  acc ≈ 0.15 yr.From Equation (65), the breakout timescale is  bre ≈ 3600 yr.However, before the breakout happens, the shock is decelerated to the local speed of sound.Let  s =  sh in Equation ( 64), the cavity formation timescale is  cav ≈ 2630 yr.This means that for the SG model, the shock is less likely to break out.In the following section, we choose the TGM model to discuss the evolution of the cavity.The evolution timescales of cavities in the AGN disk for different accretion CO models are listed in Table 3.The disk parameters are taken from the TQM disk model in Table 1.

DM and RM variations
In this section, we investigate the DM and RM from the disk wind in the cavity and the shock shell.The mass released from the disk wind is Before the shock breakouts, the difference in expansion between the radial and vertical directions can be ignored, so the cavity can be regarded as spherical.However, after this, because the density of gas in the AGN disk decreases rapidly in the vertical direction, the vertical propagation of the shock becomes easier.At this point, the shape of the cavity deviates from a spherical shape.For simplicity, we assume that the cavity is always a cylinder during expansion.When  bre <  acc , the DM from disk wind in the cavity can be estimated by The shock thickness 1 −  = 0.86 is taken from Weaver et al. 1977.
The free-free absorption optical depth from disk wind is For the shocked region, gas is fully ionized because of the high temperature (see Equation ( 30)).The DM from the shock shell is The free-free absorption optical depth from shock shell is In this work, we assume that only the shocked region is magnetized, and the magnetic field in the shocked shell is given in Equation ( 33).The RM from the shocked shell is For the case  bre >  acc , the DM from the disk wind in the cavity is and the free-free absorption optical depth from disk wind is The peak frequency of FRBs are calculated for Γ q ∼ 100 and  q ∼ 10 −3 (Sridhar et al. 2021).
t\U  60) and ( 61).Timescales of the breakout, leaving the disk boundary and effective accretion are shown in gray vertical dashed, dash-dotted and solid lines, respectively.For BH1, the cavity and the shocked shell are opaque for FRBs with  = 1 GHz for a few decades (see the black dashed line for  ff = 1).After the shock breakouts, the DM from the shocked shell and the free-free absorption become neglected compared to the disk wind.Bottom panels: The time-dependent results are based on numerical solutions of Equations ( 66) and ( 67).Timescales of the breakout and leaving the disk boundary are shown in gray vertical dashed and dash-dotted lines, respectively.For NS2, the cavity and the shocked shell are opaque for FRBs with  = 1 GHz for a few decades (see the black dashed line for  ff = 1).
The results when  acc <  ≤  bre do not satisfy the approximation expression given by Equations ( 76) -(78).For NS2, the cavity radius is close to  when  ∼  acc , then it is no longer possible to assume that the density is constant as in the previous discussion (see Equation ( 1)).Thus, we only show the approximate expression when  >  c for the shocked shell.
The same as the case  bre <  acc , we can only give the time evolution scaling law for the phases I and III, which is given in Equations ( 74) and ( 75).
The DM, free-free absorption optical depth and RM from the shocked shell evolve as and The DM, free-free absorption optical depth and RM evolution of accretion COs model for BH1 (top panel, based on numerical solutions of Equations ( 60) and ( 61)) and NS2 (bottom panel, based on numerical solutions of Equations ( 66) and ( 67)) are shown in Figure 11.The model parameters are listed in Tabel 3. The contributions from the unshocked cavity and the shocked shell are shown in blue and red lines, respectively.Same as the young magnetar models, DM  ff and RM show non-power-law evolution patterns over time in the Sakurai accelerating phase, which is different from other environments, e.g., SNR (Yang & Zhang 2017;Piro & Gaensler 2018), compact binary merger remnant (Zhao et al. 2021), PWN/MWN (Margalit & Metzger 2018;Yang & Dai 2019;Zhao & Wang 2021).For BH1, timescales of the breakout, leaving the disk boundary and effective accretion are shown in gray vertical dashed, dash-dotted and solid lines, respectively.The cavity and the shocked shell are opaque for FRBs with  = 1 GHz for a few decades (see the black dashed line for  ff = 1).After the shock breakouts, the DM from the shocked shell and the free-free absorption become neglected compared to the disk wind.For NS2, the time-dependent results are based on numerical solutions of Equations ( 66) and (67).The cavity and the shocked shell are opaque for FRBs with  = 1 GHz for a few decades (see the black dashed line for  ff = 1).The results when  acc <  ≤  bre do not satisfy the approximation expression given by Equations ( 76) -(78).For NS2, the cavity radius is close to  when  ∼  acc , then it is no longer possible to assume that the density is constant as in the previous discussion (see Equation (1)).Thus, we only show the approximate expression when  >  c for the shocked shell.
The total DM,  ff and RM (including the contributions of the cavity and the shocked shell) from different accretion CO models are shown in Figure 12.The model parameters are listed in Tabel 3.For the model BH1, BH2, BH3, BH4 and NS1, the effective accretion timescale is much longer than the shock breakout timescale, whose cavity evolution is governed by Equations ( 60) and (61).However, for NS2, the duration of effective accretion is very short compared to the shock breakout timescale, whose cavity evolution is governed by Equations ( 66) and (67).The condition under which FRBs with  = 1 GHz can be observed ( ff = 1) is represented by the gray horizontal dashed line.For  = 4 × 10 6  ⊙ (shown in blue lines), the cavity formation timescale is about tens of thousands of years.For the model of accreting BHs (BH1 and BH2), the effective accretion lasts about a few thousand years.When  c <  ≤  acc , the DM sh decrease as ∝  2(1−) 5− but DM w increase as ∝  1/3 .Thus, the total DM (DM = DM sh +DM w ∼ 10 2 −10 3 pc cm −2 ) shows a relatively stable evolutionary trend at this phase.When  >  acc , the DM is dominated by the disk wind in the cavity and decreases as ∝  −2/3 .For NS1, the effective accretion only lasts about a few hundred years.Owing to less disk outflow materials accumulating in the cavity, the DM and  ff mainly come from the shocked shell.After the shock breakouts, the shocked shell becomes transparent.The DM (DM ≃ DM sh ) evolves as ∝  for  acc <  ≤  cav .For  = 10 8  ⊙ ( shown in red lines), the cavity formation timescale is about several thousands of years.For the model of accreting BHs (BH3 and BH4), the DM,  ff and RM profiles are similar to the case of  = 4 × 10 6  ⊙ , but the value is greater.For NS2, the DM and  ff are dominated by the shocked shell (see Figure 11).For the accreting CO models, the RM is large and decreases with time, which is similar to the RM of FRB 20121102A (Michilli et al. 2018;Hilmarsson et al. 2021b).

DISCUSSION
In addition to the DM, absorption, Faraday rotation-conversion and burst luminosity discussed above, other possible observation properties should also be studied and tested in the future in the frame of the AGN disk.

The distribution of COs in AGN disk
Both magnetars and accreting COs are expected to be embedded in AGN disks.The distribution of FRB sources in AGN disk depends on the formations and migration of COs.COs can be formed via the core collapse of massive stars or AIC/compact binary merger.The outer disk is self-gravity unstable and is ongoing effective star formation.Thus, the SNe formation channel mainly occurs in the outer disk.The AIC of NSs preferentially occurs in the outer disk, while the binary NS merger preferentially occurs at  ≲ 10 −3 pc from the SMBH (Perna et al. 2021).The migration of COs in the disk is subject to large uncertainty (e.g.Bellovary et al. 2016;Pan & Yang 2021a;Grishin et al. 2023).Migration traps caused by gas torque are thought to occur at tens to hundreds of times the gravitational radius (Bellovary et al. 2016).However, migration traps can also exist at larger distances (∼ 10 3−5  g ) if thermal torques are taken into consideration (Grishin et al. 2023).
A detailed study of CO distribution is beyond the scope of this article.Although the radial distribution of COs is uncertain, FRBs can be generated across the entire disk.If the FRB sources are in the inner disk, we can only detect signal sites at a few scale heights because of the absorption of the dense gas.However, most COs locate close to the midplane of the disk (Perna et al. 2021).The feedback (progenitors' ejecta or accreting outflows) of magnetars or accreting COs should be considered for emission from the midplane (see Sections 3 and 4).The local sound speed for the inner disk is too high for the shock to break out.However, the feedback cavity in the outer disk can reduce the absorption by the dense disk materials, making FRBs detectable.

Event rate
The event rate in AGN disks can be estimated as (Tagawa et al. 2020)  60) and ( 61).However, for NS2, the duration of effective accretion is very short compared to the shock breakout timescale, whose cavity evolution is governed by Equations ( 66) and (67).The condition under which FRBs with  = 1 GHz can be observed ( ff = 1) is represented by the gray horizontal dashed line.For  = 4 × 10 6  ⊙ (shown in blue lines), the cavity formation timescale is about tens of thousands of years.
For the model of accreting BHs (BH1 and BH2), the effective accretion lasts about a few thousand years.When  c <  ≤  acc , the DM sh decrease as ∝  2(1−) 5− but DM w increase as ∝  1/3 .Thus, the total DM (DM = DM sh + DM w ∼ 10 2 − 10 3 pc cm −2 ) shows a relatively stable evolutionary trend at this phase.When  >  acc , the DM is dominated by the disk wind in the cavity and decreases as ∝  −2/3 .For NS1, the effective accretion only lasts about a few hundred years.
Owing to less disk outflow materials accumulating in the cavity, the DM and  ff mainly come from the shocked shell.After the shock breakouts, the shocked shell becomes transparent.The DM (DM ≃ DM sh ) evolves as ∝ 5− for  acc <  ≤  cav .For  = 10 8  ⊙ (shown in red lines), the cavity formation timescale is about several thousands of years.For the model of accreting BHs (BH3 and BH4), the DM,  ff and RM profiles are similar to the case of  = 4 × 10 6  ⊙ , but the value is greater.For NS2, the DM and  ff are dominated by the shocked shell (see Figure 11).For the accreting CO models, the RM is large and decreases with time, which is similar to the RM of FRB 20121102A (Michilli et al. 2018;Hilmarsson et al. 2021b).
The AGN disk is also found to be inhomogeneous.Sonic-scale magneto-rotational and gravitational instabilities would commonly occur in AGN disks (e.g.Balbus & Hawley 1998;Gammie 2001;Goodman 2003;Chen & Lin 2023), both can excite inhomogeneous turbulence with locally chaotic eddies, leading to stochastic variations on DMs and RMs of FRBs.

RM reversals
Recently, RM reversals have been reported for some FRBs, such as FRB 20201124A (Xu et al. 2022), FRB 20190520B (Anna-Thomas et al. 2023) and FRB 20180301A (Kumar et al. 2023).Possible explanations of magnetic field reversals are that the FRB source is in a massive binary system (Wang et al. 2022;Zhao et al. 2023) or a magnetized turbulent environment (Anna-Thomas et al. 2023).
The large-scale toroidal magnetic fields of a magnetized accretion disk reverse with the magnetorotational instability (MRI) dynamo cycles (Salvesen et al. 2016).For the FRB sources in the AGN disks, the magnetic field reversal can be explained naturally.

CONCLUSIONS
In this work, we investigate the observational properties of FRBs to occur in the disks of AGNs.Two mainstream types of radiation mechanism models are considered, such as close-in models from the magnetosphere of magnetars and far-away models from relativistic outflows from accreting COs.Progenitors' ejecta or accretion disks' outflows interact with the disk material to form a cavity.The cavity makes the FRB easier to escape.The propagation of FRBs in the disk or cavity causes dispersion, free-free absorption, Faraday rotation and Faraday conversion.Our conclusions are summarized as follows.
• If the feedback of the ejecta or outflows is weak and there is no extra ionization source, whether the FRB is absorbed or not depends only on the optical depth of the disk.For an SG disk, the disk becomes optically thin at  > 10 4 − 10 5  g .For a TQM disk, the disk becomes optical thin at the distance ∼ 10 3 − 10 4   .
• For the inner disk or the outer disk ionized by the radiation or shocks, FRBs can be observed only from sources located at a few scale heights.The largest DM and RM contributed by the disk is about 10 3 − 10 5 pc cm −2 and 10 4 − 10 5 rad m −2 , respectively.
• If the magnetic field in the AGN disks is toroidal field dominated, FC occurs.For the intrinsic 100% linearly polarized radiation, FC converts linear polarization into circular polarization when radiation passes through the disk.
• For close-in models of young magnetar born in SN explosions or the merger of two compact stars, the AGN disk environments do not affect the radiation properties.For accreting-powered models, the burst luminosity depends on the accretion rate.The event rate of magnetar-origin FRBs in AGN disks is ∼ 5 − 30 Gpc −3 yr −1 .For accreting CO models, the FRB rate in AGN disks is ∼ 3.5 − 35 Gpc −3 yr −1 .The hyper-Eddington accretion of COs in AGN disks makes FRB brighter, accounting for about 0.1% − 1% of total bright repeating FRBs.
• Shock is generated during the interaction between the ejecta or outflows and the AGN disk materials.The shock is quenched by dense disk materials in the radial direction, but can break out in the vertical direction.The DM and RM during shock breakout show a non-power law evolution pattern over time, which is completely different from other environments (such as supernova remnants).
• The gas in AGN disk is inhomogeneous and turbulent, resulting in stochastic DM and RM variations.The multipath propagation in turbulent magnetized plasma also causes frequency-dependent depolarization (Bochenek et al. 2020;Yang et al. 2022;Lu et al. 2023).
• The large-scale toroidal magnetic fields of a magnetized accretion disk reverse with the magnetorotational instability (MRI) dynamo cycles (Salvesen et al. 2016).For the FRB sources in the AGN disks, the magnetic field reversal can be explained naturally.

Figure 1 .
Figure 1.Schematic diagram of the generation and propagation of FRBs in AGN disk.(a)If the feedback of the source is weak, FRBs only can be detected from source sites at a few scale heights due to the absorption from the dense disk materials.If the feedback of the source is strong, a shock is formed during the interaction between the ejecta/outflows and the disk materials.The shock punches a cavity in the AGN disk.When  ≤  bre , the shock shell is almost spherical.(b) After the shock breaks out, the vertical propagation of the shock becomes easier because the density of gas in the AGN disk decreases rapidly.At this point, the shape of the cavity becomes ringlike.(c) When the shock decelerates to the speed of the local sound, the shock evolution ends.Finally, the cavity is refilled by the AGN disk material.In the case of progenitor ejecta, the cavity can only be opened once.But for the case of accretion outflows, the accretion rate can be restored to hyper-Eddington after the cavity is refilled.Then, the strong disk outflow forms the cavity again and the evolution process is periodic.(d) The 'close-in' models.FRB originates from the magnetosphere of magnetars.The AGN disk environment only affects the propagation effect but not the radiation properties for magnetospheric origins.(e) The 'far-away' models.FRB originates from relativistic outflows of accreting COs.In addition to the propagation effect, high accretion rates in the AGN disk lead to more bright bursts for accreting-powered origins.

Figure 8 .
Figure8.The DM, free-free absorption optical depth and RM evolution for different magnetar formation channels.The parameters and the evolution timescale of each model are listed in Table3.2.Solid and dashed lines represent the case of SG and TQM disk models, respectively. ff,sh = 1 are shown in gray dashed horizontal lines.For SG-NSWD and SG-BNS models with  = 10 8  ⊙ , the optical depth is still greater than unity when the cavity stops expanding.In other cases, the cavity becomes transparent sometime after the magnetar is born.For binary mergers or AIC progenitors, this time is hundreds to thousands of years for the SG disk model, while it only takes a few decades for the TQM disk model.For massive star progenitors, the time become transport is about a thousand years for the SG disk model, while it is shortened to a few hundred years for the TQM disk model.

WFigure 9 .
Figure 9. Solutions of Equations (60) and (61) for BH1.The model parameters are listed in the BH1 model in Table3.The shock velocity and radius evolution are shown in the top and bottom panels, respectively.Timescales of breakout, leaving the disk boundary and effective accretion are shown in gray vertical dashed, dash-dotted and solid lines, respectively.The height expansion (blue lines) goes through the adiabatic expansion phase in the disk (phase I,  ≤  bre ), the Sakurai accelerating phase (phase II,  bre <  ≤  c ), the adiabatic expansion phase in the circum-disk medium (phase III,  c <  ≤  acc ) and the ST phase (phase IV,  acc <  ≤  cav ) in sequence.The width expansion (red lines) goes through the adiabatic expansion phase in the disk (phase I,  ≤  bre ) and sp phase (phase V,  >  bre ) in sequence.

Figure 10 .
Figure 10.Solutions of Equations (66) and (67) for the NS2 model.The model parameters are listed in the NS2 model in Table3.When  bre >  acc , the shock evolution is in analogy with that of the SN explosion (see Figure6).The shock velocity and radius evolution are shown in the top and bottom panels, respectively.Timescales of breakout and leaving the disk boundary are shown in gray vertical dashed and dash-dotted lines, respectively.The height expansion (blue lines) goes through the ST phase in the disk (phase I,  ≤  bre ), the Sakurai accelerating phase (phase II,  bre <  ≤  c ) and the ST phase in the circum-disk medium (phase III,  c <  ≤  cav ) in sequence.The width expansion (red lines) goes through the ST phase in the disk (phase I,  ≤  bre ) and sp phase (phase IV,  >  bre ) in sequence.

Figure 11 .
Figure 11.The DM, free-free absorption optical depth and RM evolution of accretion COs model for BH1 (top panel) and NS2 (bottom panel).The model parameters are listed in Tabel 3. The contributions from the unshocked cavity and the shocked shell are shown in blue and red lines, respectively.Same as the young magnetar models, DM  ff and RM show non-power-law evolution patterns over time in the Sakurai accelerating phase, which is different from other environments, e.g., SNR (Yang & Zhang 2017; Piro & Gaensler 2018), compact binary merger remnant (Zhao et al. 2021), PWN/MWN (Margalit & Metzger 2018; Yang & Dai 2019; Zhao & Wang 2021).Top panels: The time-dependent results are based on numerical solutions of Equations (60) and (61).Timescales of the breakout, leaving the disk boundary and effective accretion are shown in gray vertical dashed, dash-dotted and solid lines, respectively.For BH1, the cavity and the shocked shell are opaque for FRBs with  = 1 GHz for a few decades (see the black dashed line for  ff = 1).After the shock breakouts, the DM from the shocked shell and the free-free absorption become neglected compared to the disk wind.Bottom panels: The time-dependent results are based on numerical solutions of Equations (66) and (67).Timescales of the breakout and leaving the disk boundary are shown in gray vertical dashed and dash-dotted lines, respectively.For NS2, the cavity and the shocked shell are opaque for FRBs with  = 1 GHz for a few decades (see the black dashed line for  ff = 1).The results when  acc <  ≤  bre do not satisfy the approximation expression given by Equations (76) -(78).For NS2, the cavity radius is close to  when  ∼  acc , then it is no longer possible to assume that the density is constant as in the previous discussion (seeEquation (1)).Thus, we only show the approximate expression when  >  c for the shocked shell.

Table 2 .
(Margalit & Metzger 2018;Yang & Dai 2019;odel.The parameters are the same as Figure6.Blue and red lines represent the contribution from the cavity and shock shell, respectively.Timescales of FE duration, breakout and leaving the disk boundary are shown in gray vertical solid, dashed, and dash-dotted lines, respectively.Except for the Sakurai accelerating phase ( bre <  ≤  c ), the approximate scaling laws for the remaining phases are given.In the Sakurai accelerating phase, DM  ff and RM show non-power law evolution patterns over time, which is different from other environments, e.g., SNR(Yang & Zhang 2017;Piro & Gaensler 2018), compact binary merger remnant(Zhao et al. 2021), PWN/MWN(Margalit & Metzger 2018;Yang & Dai 2019; Zhao & Wang  2021).Although the material in the AGN disk is very dense, as the cavity expands, it can become transparent around a thousand years after the birth of the magnetar.Evolution timescale of cavities in the AGN disk for different progenitor models Radice et al. (2018)3)t al. (2013);Radice et al. (2018) 10 6 M (Margalit & Metzger 2018;Yang & Dai 2019;tion from the cavity and shock shell, respectively.Timescales of FE duration, breakout and leaving the disk boundary are shown in gray vertical solid, dashed, and dashdotted lines, respectively.Except for the Sakurai accelerating phase ( bre <  ≤  c ), the approximate scaling laws for the remaining phases are given.In the Sakurai accelerating phase, DM  ff and RM show non-power-law evolution patterns over time, which is different from other environments, e.g., SNR(Yang & Zhang 2017;Piro & Gaensler 2018), compact binary merger remnant(Zhao et al. 2021), PWN/MWN(Margalit & Metzger 2018;Yang & Dai 2019; Zhao &  Wang 2021

Table 3 .
The intrinsic burst properties and evolution timescales of cavities in the AGN disk for different accretion CO models The total DM,  ff and RM (including the contributions of the cavity and the shocked shell) from different accretion CO models.The model parameters are listed in Tabel 3.For the model BH1, BH2, BH3, BH4 and NS1, the effective accretion timescale is much longer than the shock breakout timescale, whose cavity evolution is governed by Equations ( (Tagawa et al. 2020;Perna et al. 2021) in AGN disks,  AGN is the average lifetime of AGN disks and  FRB is the fraction of CO that can produce FRB.Using the fitting results of AGN number density  AGN / fromBartos et al. (2017), the event rate in AGN disk is given by(Tagawa et al. 2020;Perna et al. 2021)t\U