On high-energy particles in accretion disk coronae of supermassive black holes:
implications for MeV gamma rays and high-energy neutrinos from AGN cores

agn are powered by mass accretion onto \acpsmbh. They emit intense electromagnetic radiation in broad range of frequencies. Measurements of X-ray spectra of \acpagn allow us to study various aspect of \acpsmbh such as black hole spins (e.g., Reynolds, 2014), geometrical structures (e.g., Ramos Almeida & Ricci, 2017), and cosmological evolution (e.g., Ueda et al., 2014).

A key for understanding these phenomena is primary X-ray radiation of the accretion disk which arises from Comptonization of disk photons in moderately thick thermal plasma, namely coronae, above an accretion disk (see, e.g., Katz, 1976; Bisnovatyi-Kogan & Blinnikov, 1977; Pozdniakov et al., 1977; Galeev et al., 1979; Takahara, 1979; Sunyaev & Titarchuk, 1980). X-ray observations have indicated the coronal temperature of $\sim 10^{9}$ K and the Thomson scattering opacity of $\gtrsim 1$ (e.g. Zdziarski et al., 1994; Fabian et al., 2015). However, the nature of \acagn coronae is still veiled in mystery.

Very recently, Inoue & Doi (2018) has reported the detection of coronal radio synchrotron emission from two nearby Seyferts (e.g., Di Matteo et al., 1997; Inoue & Doi, 2014; Raginski & Laor, 2016) utilizing the \acalma. The inferred coronal magnetic field strength was $\sim 10$ G with a size of $40R_{s}$, where $R_{s}$ is the Schwartzschild radius, for both active \acpsmbh with a mass of $\sim 10^{8}M_{\odot}$. It is also found that coronae of Seyferts contain both thermal and non-thermal electrons. This implies that acceleration of high energy particles happens in AGN coronae.

High energy particles in the nuclei of Seyferts have been discussed for a long time¹¹1High energy particles in the coronae of X-ray binaries have been also discussed in literature (e.g., Bhattacharyya et al., 2003, 2006).. In the past, it was argued that primary X-ray emission comes from pair cascades induced by high energy particles accelerated in and/or around accretion flows (e.g., Zdziarski, 1986; Kazanas & Ellison, 1986; Ghisellini et al., 2004). In the pair cascade model, particles are accelerated by shock dissipation in accretion flows (e.g., Cowsik & Lee, 1982; Protheroe & Kazanas, 1983; Zdziarski, 1986; Kazanas & Ellison, 1986; Sikora et al., 1987; Begelman et al., 1990). However, the detection of the \acagn spectral cutoffs (e.g., Madejski et al., 1995; Zdziarski et al., 2000) and non-detection of Seyfert \acpagn in the gamma-ray band (e.g., Lin et al., 1993) ruled out the pair cascade scenario as a dominant source for the primary X-ray emission²²2TeV gamma rays are measured from the Galactic center (HESS Collaboration et al., 2016). This detection indicated possible particle acceleration in accretion flow, even though accretion rate in the Galactic center is several orders of magnitude lower than that in standard disks..

In this paper, we investigate the production mechanism of the observed high energy particles in \acagn coronae. As an example, we consider those high energy particles are supplied by \acdsa processes (e.g., Drury, 1983; Blandford & Eichler, 1987) in the coronae. Contrary to the previously discussed AGN accretion shock models, the required shock power is much lower in order to explain the observed non-thermal species and to be in concordance with the current picture of coronal X-ray emission. Moreover, previous studies of high energy particles in \acagn accretion disks have treated as free parameters corona size and magnetic field, which are important parameters for the understandings of particle acceleration. The \acalma observations allowed us to determine both of them (Inoue & Doi, 2018). Most critically, the observationally determined strength of the magnetic field appeared to be significantly smaller than the one previously considered in the literature. We take into account these newly determined coronal parameters.

Thermal coronal emission from Seyferts is known to explain the entire cosmic X-ray background radiation (e.g., Ueda et al., 2014). In contrast, the origin of the cosmic MeV background radiation from 0.1 MeV to several tens MeV is still unknown (see e.g., Inoue, 2014). Here, the non-thermal electrons in coronae seen by ALMA will invoke power-law MeV gamma-ray emission via Comptonization of disk photons. Such non-thermal emission is suggested as a possible explanation for the cosmic MeV gamma-ray background radiation (Inoue et al., 2008). However, non-thermal electron species in the previous work were included in an ad hoc way. In this work, we revisit the contribution of Seyferts to the MeV gamma-ray background radiation by considering the particle acceleration of non-thermal populations in coronae together with the latest X-ray luminosity function of Seyferts (Ueda et al., 2014).

High energy particles around accretion disks of \acpagn also generate intense neutrino emission through \acpp and \acpg interaction processes by interacting accreting gas and photon fields (e.g., Eichler, 1979; Begelman et al., 1990; Stecker et al., 1992; Alvarez-Muñiz & Mészáros, 2004). Although these originally predicted fluxes have been significantly constrained by high energy neutrino observations (The IceCube Collaboration, 2005), recent studies have revisited the estimated fluxes and found that \acagn core models are still viable (Stecker, 2005, 2013; Kalashev et al., 2015). However, normalization of neutrino fluxes from \acpagn and acceleration properties of high energy particles in those models are assumed to match with the observation. In this work, we also discuss the possible contribution from \acagn cores given our \acalma observations and investigate the required parameter spaces for the explanation of the IceCube diffuse neutrino fluxes.

We describe general particle acceleration processes in \acagn coronae in § 2. The broadband emission spectrum of the central region of \acpagn and physical properties of \acagn coronae are presented in § 3. Relevant timescales and steady-state particle spectra are discussed in § 4 and § 5, respectively. § 6 and § 7 present the results of the expected gamma-ray and neutrino fluxes from individual AGN cores and the cosmic gamma-ray and neutrino background fluxes from \acagn cores, respectively. Discussion including other possible particle acceleration mechanism is given in § 8, and conclusions are in § 9. Throughout this paper, we adopt the standard cosmological parameters of $(h,\Omega_{M},\Omega_{\Lambda})=(0.7,0.3,0.7)$.

As non-thermal coronal synchrotron emission is seen in nearby Seyferts (Inoue & Doi, 2018), particle acceleration should occur in AGN coronae, even though thermal populations are energetically dominant. Particle acceleration mechanism in the coronae is highly uncertain. Various acceleration mechanisms can take place in the coronae such as \acdsa mechanism (e.g., Drury, 1983; Blandford & Eichler, 1987), turbulent acceleration (e.g., Zhdankin et al., 2018), magnetosphere acceleration (e.g., Beskin et al., 1992; Levinson, 2000), and magnetic reconnection (e.g., Hoshino & Lyubarsky, 2012). In this work, for simplicity, we consider the \acdsa as the fiducial particle acceleration process. We discuss the other possible acceleration processes in § 8.3.

In order to investigate particle acceleration mechanism of the observed non-thermal electrons, we consider the interaction of locally injected relativistic particles with the matter, photons, and magnetic field in the infalling coronae. Although the location of shock sites is uncertain, for simplicity, we assume that shocks occur inside of the coronae. The shock accelerates a part of inflow plasma to high energies. As the energy loss timescale of high energy protons is in general longer than the free-fall timescale, a sufficiently high energy density of relativistic particles is maintained to provide pressure to support a standing shock around a \acsmbh (Protheroe & Kazanas, 1983).

Coronae are assumed to be spherical with a radius of $R_{c}\equiv r_{c}R_{s}$. $r_{c}$ is the dimensionless parameter of the corona size and $R_{s}=2GM_{\mbox{\tiny{BH}}}/c^{2}$, where $G$ is the gravitational constant, $M_{\mbox{\tiny{BH}}}$ is the mass of the central \acsmbh, $c$ is the speed of light. Coronae are also set to be in a steady state. We also do not consider positrons in coronae. Thus, the proton number density $n_{p}$ is equal to the electron density $n_{e}$ in this work, which gives the maximum number of protons in coronae. $n_{e}$ is defined through the Thomson scattering opacity in coronae, $\tau_{\mbox{\tiny{T}}}$ as

where $\sigma_{\mbox{\tiny{T}}}$ is the Thomson scattering cross section.

In this section, we summarize the general observational properties of the central region of \acpagn related to high-energy particles in coronae.

Given the observed properties of AGN core regions, we can estimate the various timescales of high energy particles in the coronae. Figure 3 shows the cooling rates of electrons in the coronae for different energy-loss processes, together with the acceleration rate and the free-fall timescale following § 2 and parameters presented in § 3. We set $\eta_{g}=30$ in the figure, which reproduces the IceCube neutrino background fluxes as discussed later in § 7. Each panel corresponds to 2-10 keV X-ray luminosity of $10^{42}$, $10^{44}$, $10^{46}~{\rm erg\ s^{-1}}$.

Due to the intense broadband radiation field, the cooling is dominated by the Compton cooling. However, at higher energy regions, the main cooling channel is replaced by synchrotron cooling because of the \ackn effect. The more luminous AGNs tend to have more efficient \acic cooling effect, as the target photon density increases. When we assume $\eta_{g}=30$, electron acceleration up to $\gamma_{e}\sim 10^{5}$ ($\sim 50$ GeV) is feasible in \acagn coronae at various luminosities. Therefore, synchrotron radiation through coronal magnetic fields and gamma-ray emission by Comptonization of disk photons are naturally expected in AGN coronae.

alma spectra of two nearby Seyferts, whose X-ray luminosities are about $10^{44}\ {\rm erg\ s^{-1}},$ extends their radio synchrotron power-law spectra at least up to 230 GHz, which corresponds to $\gamma_{e}\sim 80$ given the magnetic field strength of $10$ G (Inoue & Doi, 2018)⁴⁴4This frequency limit is due to the instrumental coverage of the \acalma band-6 receiver. Therefore, the emission itself is likely to extend to higher frequencies, even though those emission signals would be buried in thermal dust emission.. As shown in the top right panel (the case of $\log L_{X}=44$) in Fig. 3, relativistic electrons with $\gamma_{e}\sim 80$ seen by \acalma can be easily accelerated in AGN coronae. Notably, such electrons can be accelerated even by a low efficiency acceleration process, e.g., with $\eta_{g}\sim 10^{6}$. For this energy, Compton cooling is the dominant energy loss process. As the cooling timescale for $\gamma_{e}\sim 80$ is about 100 s, flux variability in the radio synchrotron emission is expected, some Seyferts are already known to show a flux variation at least in day scales (Baldi et al., 2015). Further dense light curve observations may see shorter timescale variabilities.

Similar to Fig. 3 for electrons, Fig. 4 shows the timescales for high energy for various luminosities. As in Fig. 3, we set $\eta_{g}=30$. Since synchrotron and Compton cooling are not effective for protons in our case, we do not show these timescales in the figure.

It is evident that protons can be accelerated up to $\gamma_{p}\sim 10^{6}$ ($\sim 1$ PeV) in \acagn coronae for various luminosities. Maximum attainable energy is controlled by different processes for different luminosity AGNs due to SED and size dependence. For low-luminosity Seyferts ($L_{X}<10^{44}\ {\rm erg\ s^{-1}}$), acceleration is limited by the dynamical timescale rather than radiative cooling, while it becomes limited by the Bethe-Heitler cooling for higher luminosity objects. As the luminosity increases, \acpg and Bethe-Heitler cooling effects become more prominent. At higher luminosities, the Bethe-Heitler processes dominate the energy loss process of high energy particles. Therefore, in cases of high luminosity objects, resulting hadronic gamma-ray and neutrino spectra in the TeV band will show spectral suppression due to the Bethe-Heitler processes (see e.g., Murase, 2008, for the cases of gamma-ray burst).

The steady state particle distributions $n=dN/d\gamma$ can be derived from the solution of the transport equation (Ginzburg & Syrovatskii, 1964)

where $\dot{\gamma}_{\rm cool}$ is the total cooling rate, $Q(\gamma)$ is the injection function, which describes phenomenologically some acceleration process, e.g., \acdsa. The injection function for non-thermal protons and electrons is set as $Q(\gamma)=Q_{0}\gamma^{-p_{\rm inj}}\exp(-\gamma/\gamma_{\rm max})$. Here, $\gamma_{\rm max}$ is the maximum Lorentz factor determined by balancing the acceleration and cooling time scales (Figures. 3 and 4). The corresponding solution is

where

By solving Equation. 26, we obtain a steady-state spectrum of the non-thermal particles.

Fig. 5 shows the steady-state non-thermal electron spectrum obtained for the injection spectral index of $p_{\rm inj}=2.0$ together with the observationally determined electron spectral distribution for IC 4329A (Inoue & Doi, 2018). \acalma observed non-thermal synchrotron radiation between 90.5 GHz and 231 GHz which corresponds to the electron Lorentz factors between 50 and 80, respectively. The corresponding region is shown as the shaded region in the Fig. 5.

For the calculation of the steady-state spectrum, we set $M_{\mbox{\tiny{BH}}}=10^{8}M_{\odot}$, $r_{c}=40$, $B=10$ G, $kT_{e}=100$ keV, $\tau_{\mbox{\tiny{T}}}=1.1$, and $\eta_{g}=30$. The synthetic electron distribution obtained for $p_{\rm inj}=2.0$ nicely reproduces the observationally determined electron spectrum in the energy range constrained by the observations. This injection index is naturally expected in a simple \acdsa scenario for a strong shock.

The resulting particle spectrum at $\gamma_{e}>10^{4}$ becomes softer than observationally determined index at $50\lesssim\gamma_{e}\lesssim 80$. This is because of the influence of the cutoff imposed by the particle cooling. Therefore, if we consider the high energy synchrotron or \acic spectral shapes, the cooling effects should be taken into account accurately. Even though the electron spectrum extends down to lower energies, it is hard to see the corresponding synchrotron emission due to synchrotron self-absorption effect (Inoue & Doi, 2014).

The calculated electron spectrum is renormalized to agree with the observationally determined spectrum, which is achieved if the non-thermal electrons contains $f_{\rm nth}=0.03$ of the energy in thermal leptons. We note that, in order to define the energy content in the non-thermal particles, we formally integrate above $\gamma_{e}=1$ in this study. We keep this fraction for non-thermal electron energy fixed in calculations below for all Seyferts.

The energy fraction of non-thermal electrons was fixed to $\xi_{\rm nth}=0.04$ in Inoue & Doi (2018). $\xi_{\rm nth}$ is defined beyond the break electron Lorentz factor, while $f_{\rm nth}$ is above $\gamma_{e}=1$. That amount of non-thermal electrons overproduces the MeV background flux given the measured electron spectral index (see §. 7). To be consistent with the observed cosmic MeV gamma-ray background flux, we set $\xi_{\rm nth}=0.015$ in this work, which corresponds to $f_{\rm nth}=0.03$. The obtained best fit parameters with this fraction for the radio spectrum of IC 4329A is $p=2.9\pm 0.9$, $B=11.4\pm 5.6$ G, and $r_{c}=42.7\pm 7.8$, which are very similar to those obtained for the case of $\xi_{\rm nth}=0.04$. We adopt these parameters for the observationally determined electron distribution in the Fig. 5. Fitting results for the other parameters were also the same as those with $\xi_{\rm nth}=0.04$.

Here, the total shock power $P_{\mbox{\tiny{sh}}}$ can be estimated as

For objects with $L_{\rm X}=10^{44}~{\rm erg\ s^{-1}}$, $f_{\rm nth}=0.03$ corresponds to $\sim 5$% of the shock power is injected into acceleration of electrons. This high value implies that if \acdsa is responsible for particle acceleration in \acagn coronae then processes regulating injection of electrons into \acdsa are very efficient. For example in the case of \acdsa in supernovae remnants non-thermal electrons obtain only $\sim 1$% of energy transferred to non-thermal protons (Ackermann et al., 2013). Detailed consideration of the reasons of this unusually high efficiency of electron acceleration is beyond the scope of this paper, however we note that a significant presence of positrons may affect the ratio (see, e.g., Park et al., 2015). Given these uncertainties, for protons we set that the same energy injection rate is achieved as for electrons. This power appears to be sufficient to explain the observed IceCube neutrino fluxes.

For the other object, NGC 985, the observed electron spectral index is $2.11\pm 0.28$ (Inoue & Doi, 2018), which is hard considering the radiative cooling effect. Cascade components would have such a hard spectrum below the threshold energy (see, e.g., Aharonian & Plyasheshnikov, 2003). In addition, due to the quality of data at low frequencies, we could not precisely determine the other components such as free-free emission and synchrotron emission from star formation activity, and synchrotron emission from the jet. Those uncertainties may resulted in a less reliable measurement of the corona emission spectrum slope. Further observations are required to determine the radio spectral properties in NGC 985 precisely.

Accelerated electrons and protons in AGN coronae generate gamma-ray and neutrino emission through \acic scattering, $pp$ interaction, and $p\gamma$ interaction. Adopting a steady-state particle spectrum, we calculate the resulting gamma-ray and neutrino spectra from AGN coronae. We follow Blumenthal & Gould (1970) for the gamma-ray emission due to the \acic scattering by non-thermal electrons. We calculate the gamma-ray and neutrino emission induced by hadronic interactions following Kelner et al. (2006) for \acpp interactions and Kelner & Aharonian (2008) for \acpg interactions. For simplicity, we do not take into account \acic scattered emission by secondary electrons and positrons. For the thermal Comptonization spectra, we adopt the AGN SED shown in Fig. 1 which takes into account reflection components but does not account for attenuation by torus. The torus attenuation is mainly relevant for $\lesssim 30$ keV, which is below the range of our interest.

Figure 6 shows the resulting gamma-ray and neutrino spectra for two cases. The neutrino flux is shown in the form of per flavour. The left panel of the figure shows the case with a 2-10 keV luminosity of $10^{43}~{\rm erg\ s^{-1}}$ at a distance of 14 Mpc, while the right panel shows the case with a luminosity of $10^{44}~{\rm erg\ s^{-1}}$ at a distance of 69 Mpc. The former and the latter roughly corresponds to NGC 4151 and IC 4329A, respectively. NGC 4151 is the brightest Seyfert in the X-ray sky (Oh et al., 2018). For the comparison, the overall fluxes of both panels are renormalized to match with the Swift/BAT flux of NGC 4151 and IC 4329A, respectively, at 14-195 keV (Oh et al., 2018). We note that we do not calculate the detailed X-ray spectra of each objects, which is beyond the scope of this paper.

We set the injection spectral index of $p_{\rm inj}=2.0$ and the gyrofactor of $\eta_{g}=30$ for both electrons and protons (See § 5). We also set the same injection power into protons and electrons as described in §. 5. The target photon density for \acic scatterings and \acpg interactions is defined as $U_{\rm ph}(\epsilon)$ (See § 3.1). Since we assume a uniform spherical source, gamma-ray photons are attenuated by internal photon field by a factor of $3u_{\rm int}(\tau_{\rm int})/\tau_{\rm int}$, where $u_{\rm int}(\tau)=1/2+\exp(-\tau)/\tau-[1-\exp(-\tau)]/\tau^{2}$ (See Sec. 7.8 in Dermer & Menon, 2009), where $\tau_{\rm int}$ is the internal gamma-ray optical depth (See §. 3.3). Gamma rays are also attenuated by the \acebl during the propagation in the intergalactic space. We adopt Inoue et al. (2013a) for the \acebl attenuation.

For the comparison, we also show the expected sensitivity curve of planned MeV missions: COSI-X (300 days)⁵⁵5COSI collaboration website (The Compton Spectrometer and Imager http://cosi.ssl.berkeley.edu/, e-ASTROGM (3 yrs, De Angelis et al., 2017)⁶⁶6e-ASTROGAM collaboration website (enhanced ASTROGAM http://eastrogam.iaps.inaf.it/, GRAMS (35 days, Aramaki et al., 2019), and GRAMS (3 yrs, Aramaki et al., 2019). 10-yr sensitivity of Fermi/LAT⁷⁷7Fermi/LAT collaboration website (The Large Area Telescope http://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm is also shown. We also plot the sensitivity of neutrino detectors: IceCube⁸⁸8IceCube collaboration website (https://icecube.wisc.edu/ and IceCube-Gen2 (van Santen & IceCube-Gen2 Collaboration, 2017). For the left panel, we assume the declination $\delta$ of $30^{\circ}$, while $-30^{\circ}$ for the right panel.

Since the spectral index of electrons is $\sim 3$ after radiative cooling, the resulting non-thermal gamma-ray spectrum is flat in $\nu F_{\nu}$ in the MeV band which appears after the thermal cutoff. Given the cooling limited maximum energy $\gamma_{e}\sim 10^{5}$, the intrinsic \acic spectrum can extend up to $\sim 100$ GeV. However, due to the strong internal gamma-ray attenuation effect, the spectra will have a cutoff around 100 MeV in both cases. In the sub-MeV band, the spectrs show super-thermal tails due to the combination of thermal and non-thermal components and a spectral hardening at $\sim 1$ MeV. These superthermal and flat spectral tails should be tested by future MeV gamma-ray missions. Ballon flights with such as GRAMS (Aramaki et al., 2019) and SMILE (Takada et al., 2011; Komura et al., 2017)⁹⁹9SMILE collaboration website (The Sub-MeV gamma-ray Imaging Loaded-on-balloon Experiment http://www-cr.scphys.kyoto-u.ac.jp/research/MeV-gamma/wiki/wiki.cgi?page=Top_en may be able to catch this superthermal tail. And, satellite-class MeV missions such as e-ASTROGAM (De Angelis et al., 2017), AMEGO¹⁰¹⁰10AMEGO collaboration website (The All-sky Medium Energy Gamma-ray Observatory https://asd.gsfc.nasa.gov/amego/, and GRAMS (Aramaki et al., 2019) will be able to see also the non-thermal power-law tail. For the case of NGC 4151, Fermi/LAT may be able to see the signature with its 10 yrs survey. However, the expected flux is almost at the sensitivity limit. Thus, it may need further exposures for Fermi/LAT to see the coronal emission.

The \acpp and \acpg production efficiency is given by the ratio between the dynamical timescale (Eq. 2) and the interaction timescales (Eqs. 2.2 and 10). The \acpp production efficiency is analytically given as

Gamma rays and neutrinos induced by hadronic interactions carry $1/3$ and $1/6$ of those interacted hadron powers. Therefore, hadronic gamma-ray and neutrino luminosity is expected to be $\sim 5$% and $\sim 3$% of the intrinsic proton luminosity, respectively. Since we assume the same energy injection to electrons and protons and the coronal Thomson scattering optical depth is 1.1, before the attenuation, we have hadronic gamma-ray and neutrino fluxes are $\sim 5$% and $\sim 3$% of the \acic gamma-ray fluxes.

The \acpp and \acpg induced gamma rays are also mostly attenuated by the internal photon fields. Thus, we do not expect any $\gtrsim$GeV gamma-ray emission from Seyferts. Moreover, the intrinsic gamma-ray energy fluxes due to hadronic interactions is about a factor of 10 less than that by primary electrons because of radiative efficiency differences between protons and electrons. This implies that gamma rays produced by secondary pairs should not significantly alter the resulting spectra. Therefore, we can safely ignore the cascade contribution.

On the contrary to gamma rays, neutrinos induced by hadronic interactions can escape from the system without any attenuation. Since we adopt the same $p_{\rm inj}=2$ for protons as for electrons, we expect a flat $\nu F_{\nu}$ spectrum for neutrinos, to which \acpp makes dominant contribution. At higher energies, especially in the case of IC 4329A, \acpp and \acpg spectra are suppressed due to the Bethe-Heitler cooling process. The exact position of the cutoff energy depends on the assumed $\eta_{g}$. Here, as described later, we set $\eta_{g}=30$ in order to be consistent with the IceCube background flux measurements. This gyrofactor results in a neutrino spectral cutoff around 100 TeV. Although it is difficult to see neutrino signals from individual Seyferts with the current generation of IceCube, it would be possible to see bright Seyferts in the northern hemisphere in the era of IceCube-Gen2 (see also Murase & Waxman, 2016, for more general arguments). Therefore, even though Seyferts are faint in the GeV gamma-ray band, future MeV gamma-ray and TeV neutrino observations can test our scenario.

In this section, we calculate the cosmic gamma-ray and neutrino background spectra from AGN coronae. For the cosmological evolution of \acpagn, we follow Ueda et al. (2014) in which the evolutionary functions are defined at 2–10 keV intrinsic X-ray luminosity. We briefly review their formalism here.

Based on the luminosity-dependent density evolution model, the \acagn X-ray luminosity function at a given luminosity $L_{X}$ and a given redshift $z$ is defined as

where ${d\Phi_{\rm X}(L_{\rm X},0)}/{d{\rm log}L_{\rm X}}$ is the luminosity function in the local universe defined as

where $A$ is the normalization and $L_{*}$ is the break luminosity. $e(z,L_{\rm X})$ is the evolution factor represented as

Here the luminosity dependence for the $p1$ parameter is considered as

where we set ${\rm log}L_{\rm p}=44$. Both cutoff redshifts are given by power law functions of $L_{\rm X}$ as

and

The parameters are summarized in Table. 4 in Ueda et al. (2014). There is also a substantial fraction of Compton-thick AGNs in the universe (e.g., Ueda et al., 2003; Ricci et al., 2015). In order to take into account this population, we multiply the normalization factor by a factor of 1.5 (see Ueda et al., 2014, for details).

The cosmic gamma-ray background fluxes are calculated as

where $E^{\prime}=(1+z)E$ and $L_{\gamma}(E,L_{\rm X})$ is the gamma-ray luminosity at energy $E$ for a given X-ray luminosity of $L_{\rm X}$. The redshift and luminosity ranges are selected to be the same as in Ueda et al. (2014). $\tau_{\rm int}$ and $\tau{}_{\mbox{\tiny{EBL}}}$ is the gamma-ray optical depth due to the internal photon field and the \acebl. We do not consider the cascade gamma-ray photons (e.g., Inoue & Ioka, 2012) because the gamma-ray energy fluxes due to hadronic interactions is already subdominant compairing to that by primary electrons.

The neutrino background fluxes can be also calculated in the same manner ignoring the gamma-ray attenuation terms and replacing $L_{\gamma}(E,L_{\rm X})$ with $L_{\nu}(E,L_{\rm X})$. $L_{\nu}(E,L_{\rm X})$ is the neutrino intensity at an energy of $E$ for a given X-ray luminosity of $L_{\rm X}$.

Figure 7 shows the cosmic X-ray/gamma-ray and neutrino background spectra from \acagn coronae assuming the case of $p_{\rm inj}=2.0$ and $\eta_{g}=30$. We also plot the observed background spectrum data by HEAO-1 A2 (Gruber et al., 1999), INTEGRAL (Churazov et al., 2007), HEAO-1 A4 (Kinzer et al., 1997), Swift-BAT (Ajello et al., 2008), SMM (Watanabe et al., 1997), Nagoya–Ballon (Fukada et al., 1975), COMPTEL (Weidenspointner et al., 2000), Fermi-LAT (Ackermann et al., 2015), and IceCube (Aartsen et al., 2015).

Figure 8 shows the cosmic MeV gamma-ray background spectrum only from Figure 7. By setting $f_{\rm nth}=0.03$, the gamma-ray fluxes from \acpagn coronae due to \acic scattering by thermal and non-thermal electrons can nicely explain the observed cosmic MeV gamma-ray background radiation in an extension from the cosmic X-ray background radiation, which is known to be explained by Seyferts (Ueda et al., 2014). Since the spectral index of non-thermal electrons in the coronae is $\sim 3$, the resulting MeV gamma-ray background spectrum becomes flat in $E^{2}dN/dE$ (See Fig. 8). Here, the cosmic X-ray background spectrum by Seyferts has a spectral cutoff above $\sim 300$ keV because of temperature of thermal electrons $\sim 100$ keV (Ueda et al., 2014). By summing up these two thermal and non-thermal components, superthermal tail appears in the sub-MeV band as observed by Fukada et al. (e.g., 1975); Kinzer et al. (e.g., 1997); Watanabe et al. (e.g., 1997). Since the dominant \acic contributors switches from thermal electrons to non-thermal electrons at around $1$ MeV, the MeV background spectrum may have spectral hardening feature at $\sim 1$ MeV. In the figure, we set $\eta_{g}=30$. The result does not significantly change as far as $\eta_{g}<1000$. If $\eta_{g}>1000$, we may require lower $f_{\rm nth}$.

Due to the internal gamma-ray attenuation effect, these non-thermal gamma rays can not contribute to the emission above GeV. Because of the same reason, most of hadronic gamma-ray photons are attenuated by internal photon fields, resulting in generation of multiple secondary particles. Since calculation of those populations are beyond the scope of this paper, we ignore those populations in our estimate. Moreover, as we describe above, the intrinsic hadronic fluxes are already an order of magnitude below the leptonic fluxes. Thus, pairs induced by hadronic cascades will not significantly change our results.

Here, \acic emission due to non-thermal electrons also contribute in the X-ray band. Their contribution is about $\sim 5$% at 30 keV of the observed cosmic X-ray background flux, which may reduce the required number of the Compton-thick population of AGNs.

The model curve at $\sim 10$ keV slightly overproduces the measured background spectrum. This is because we do not take into account X-ray attenuation by torus. However, the treatment of those soft X-ray photons does not affect our results at all.

For neutrinos, the combination of \acpp and \acpg interactions can nicely reproduce the IceCube fluxes below 100–300 TeV by assuming $\eta_{g}=30$ and about 5% of the shock power into proton acceleration, same as electrons. \acpp interactions dominate the flux at $\lesssim 10$ TeV, while \acpg interactions prevail above this energy. Because of the target photon field SED, \acpg is subdominant in the GeV-TeV band. If we inject more powers into protons, it inevitably overproduces the IceCube background fluxes. As $\gtrsim$ GeV gamma rays are internally attenuated, AGN coronae emission will not be seen in GeV gamma-rays, even though they can make the IceCube neutrino fluxes. Such hidden cosmic-ray accelerators are suggested as a possible origin of the IceCube neutrinos (see Murase et al., 2016, for a general argument).

Figure 9 shows the cosmic neutrino background spectra from \acagn cores with various gyro factors ranging from 1 (Bohm limit) to $10^{3}$. It is clear that if $\eta_{g}\ll 30$, the resulting neutrino fluxes overproduce the measured fluxes. On the contrary, if $\eta_{g}\gg 30$, AGN coronae can not significantly contribute to the observed neutrino background fluxes. Thus, in order to explain the IceCube neutrino background fluxes by AGN cores, $\eta_{g}\sim 30$ is required. However, we note that these estimates are based on the assumed energy injection fraction to protons. Recent particle-in-cell simulations of proton-electron plasma considering radiatively inefficient accretion flows (RIAFs) showed that protons will carry have several factors more energies than electrons (Zhdankin et al., 2018). If this is the case, larger $\eta_{g}$ is favored.