BLAZARS AS ULTRA-HIGH-ENERGY COSMIC-RAY SOURCES: IMPLICATIONS FOR TeV GAMMA-RAY OBSERVATIONS

In the framework of one-zone leptonic synchrotron/SSC models, the double-humped SED of blazars and radio galaxies (their misaligned counterparts), plotted as log(νF_ν) versus log(ν), can be well described by a low-energy synchrotron component and a high-energy inverse Compton (IC) curve. Each component is characterized by the peak synchrotron flux νF^s_ν at peak synchrotron photon energy ε_s, and the peak IC flux νF^C_ν at peak IC energy ε_C, respectively. The spectrum of radio-loud AGNs is often highly variable, and rises and decays with variability time t_var. Here, we define the variability time as the shortest timescale in which the flux shows a significant factor-of-two change, and assume co-spatiality of the highly variable emissions in the different energy bands, as indicated by correlated variability (Ackermann et al. 2011).

Here we consider such a one-zone model. The emissions are produced by electrons in a spherical blob moving relativistically in a jetted geometry: the high-energy emission is produced via Compton scattering off the local synchrotron photons that are generated by the electrons in the blob. The synchrotron (IC) luminosities at peak energy are L^s(C)_γ ≈ 4πd_L²(ε_s(C)F^s(C)_ε) and the Compton dominance parameter is defined as A_C ≡ L^C_γ/L^s_γ ≈ (ε_CF^C_ε)/(ε_sF^s_ε). The electron distribution is typically assumed to consist of power-law segments. As long as both the νF^s_ν and νF^C_ν energy fluxes originate from electrons with the same mean comoving-frame energy γ'_bm_ec², the magnetic energy density can be rewritten in terms of A_C as B'²/8π ∼ L^s_γ/(4πR'²δ⁴cA_C). Then the Doppler factor is given in this relativistic spherical blob formalism by the expression (Ghisellini et al. 1996)

$$\begin{equation} \delta \sim \frac{{3^{1/2} \big(L_{\gamma }^{s} \big)}^{1/4} {\big(\varepsilon _C/m_e c^2 \big)}^{1/2}}{2^{3/4} c^{3/4} t_{\rm var}^{1/2} A_C^{1/4} B_{\rm Q}^{1/2} \big(\varepsilon _s/m_e c^2 \big)} \end{equation} \tag{ 1 }$$

and the comoving magnetic field is

$$\begin{eqnarray} B^\prime &\sim & (1+z) \frac{2^{11/4} c^{3/4} t_{\rm var}^{1/2} A_C^{1/4} B_{\rm Q}^{1/2} {\big(\varepsilon _s/m_ec^2 \big)}^3}{3^{3/2} {\big(L_{\gamma }^s \big)}^{1/4} {\big(\varepsilon _C/m_e c^2 \big)}^{3/2}} \nonumber \\ &\sim & (1+z) \frac{4 B_{\rm Q} \big(\varepsilon _s/m_e c^2 \big)}{3 \delta (\varepsilon _C/\varepsilon _s)} \end{eqnarray} \tag{ 2 }$$

provided the Compton scattering takes place in the Thomson regime, which applies when $\delta \gtrsim \delta _T = 2 \sqrt{3}\sqrt{\varepsilon _s \varepsilon _C/m_e^2 c^4} (1+z)$.⁷ Here the critical magnetic field is defined as B_Q ≡ m²_ec³/eℏ ≃ 4.4 × 10¹³ G (e.g., Brainerd & Petrosian 1987). The above equations are derived by using the common relations ε_s/m_ec² ≈ δ(B'/B_Q)γ'_b²/(1 + z) and ε_C/m_ec² ≈ (4/3)γ'_b²ε_s (i.e., the typical fluid-frame Lorentz factor of electrons radiating near the peak synchrotron and SSC frequencies is $\gamma _{b}^{\prime } \approx (\sqrt{3}/2) \sqrt{\varepsilon _C/\varepsilon _s}$ in the Thomson limit; e.g., Sikora et al. 2009).

For high-peaked BL Lac objects, the Compton scattering often occurs in the K-N regime, where more detailed modeling is required. Nevertheless, Equations (1) and (2) demonstrate that source parameters such as δ and B' can be determined from the double-humped SED.⁸

Table 1 gives measured and inferred properties for blazars and radio galaxies with good multiwavelength coverage that are well described by a synchrotron/SSC model, where derived values of magnetic fields and Doppler and Lorentz factors based on detailed synchrotron and SSC modeling are taken from the literature (rather than using Equations (1) and (2)). In Table 2, we also show parameters derived with Equations (1) and (2) (only when δ ≳ δ_T is satisfied). For blazars, we assume that Γ ≈ δ, whereas values of the angle of the jetted emission with respect to the observer inferred from observations are considered for radio galaxies. Also, R' ≈ ct'_var = cδt_var/(1 + z) is used.

The SED modeling is much improved thanks to the constantly increasing multiwavelength coverage, which also allows simultaneous multi-band observations. Nevertheless, there is still some degree of scatter among parameter sets obtained by different groups, partially related to unavoidable parameter degeneracy with respect to the observables. In some cases, there is even significant scatter between derived parameters for the same state of a specific source (Cen A) or between different states of the same source (Mrk 501). In particular, the whole SED of Cen A up to the TeV-band cannot be fitted with a unique parameter set. Abdo et al. (2010d) show different models and parameter values for the same data, which reflect the limitations of fitting single epoch, single component SEDs for derivations of source parameters. However, our conclusions are not affected by uncertainties of the poorly constrained parameters, as we discuss in more detail below.

The Hillas condition (Hillas 1984) limits the maximum accelerated energy of ions with charge Z to

$$\begin{equation} E_{A}^{\rm max} \approx Z e B^{\prime } \Gamma R^{\prime }, \end{equation} \tag{ 3 }$$

in order that the particle Larmor radius is smaller than the characteristic size scale R' ≲ ct'_var = cδt_var/(1 + z). The inequality is replaced by an equality in our estimates. Equation (3) can be rewritten using the Thomson-limit relations given above when δ ≈ Γ, in which case one gets

$$\begin{equation} E_{A}^{\rm max} \sim Z e \frac{4{\big(L_{\gamma }^{s} \big)}^{1/4} t_{\rm var}^{1/2} B_{\rm Q}^{1/2} \varepsilon _s}{3 c^{3/4} A_C^{1/4} \varepsilon _{C}^{1/2}}. \end{equation} \tag{ 4 }$$

Note that E^max_A is defined in the cosmic rest frame of UHECR production.

In Table 2, the maximum proton energy is estimated using Equation (3) and parameters given in Table 1, which in turn are based on the results of synchrotron/SSC model fits for these sources found in dedicated modeling papers (rather than using Equations (1) and (2)). Here, note that other losses and details of the acceleration process could limit the maximum particle energy further. For the cases considered in Table 1, it is barely possible to accelerate protons up to ∼10²⁰ eV, whereas Fe nuclei could easily each reach ≳ 10²⁰ eV provided that they can survive photodisintegration. Alternately, ≳ 10²⁰ eV proton acceleration could occur transiently during rare bursts or flares (Dermer et al. 2009; Murase & Takami 2009), though according to Table 2 it might still be difficult even for bright flares from Mrk 501 and PKS 2155−304. As noticed above, there is large scatter in parameters due to uncertainties. However, even with the allowed spread in the parameter values (see Table 2), this conclusion seems robust.

A similar conclusion is also reached when considering luminosity requirements for BL Lac objects and FR-I radio galaxies (Dermer & Razzaque 2010; Ghisellini et al. 2010). In Ghisellini et al. (2010), physical parameters were obtained via spectral modeling using their one-zone leptonic model of all blazars with known redshift detected by the Fermi satellite during its first three-month survey. The inferred magnetic luminosity of BL Lac objects is typically L_B ∼ 10⁴⁶δ²₁ erg s⁻¹ and almost all of them satisfy L_B ≲ 2 × 10⁴⁷δ²₁ erg s⁻¹, where δ₁ = δ/10. On the other hand, the required magnetic luminosity for UHECR acceleration to 10²⁰E^max_{A, 20} eV is L_B ≈ 2 × 10⁴⁷Γ²₁(E^max_{A, 20})²Z⁻² erg s⁻¹, where Γ₁ = Γ/10. Hence, it also suggests difficulties in acceleration of ≳ 10^20.5 eV protons in typical BL Lac objects, though the simple SSC model cannot be so simply applied to lower-peaked BL Lac objects where values of B' and δ are not well defined possibly due to external Compton scattering components.

If BL Lac objects and FR-I radio galaxies, as has often been considered (e.g., Tinyakov & Tkachev 2001; Berezinsky et al. 2002; Dermer et al. 2009), accelerate the UHECRs, then an ultra-high-energy (UHE) proton origin of the highest-energy cosmic rays is disfavored from spectral modeling if the standard synchrotron/SSC model is correct. Heavier nuclei can, however, be accelerated up to UHEs. The composition of UHECRs is an open question, with both proton and heavy-ion-dominated compositions having been claimed to be compatible with HiRes (Abbasi et al. 2010) and the Pierre Auger Observatory (PAO; Abraham et al. 2010) data, respectively. As seen here, the standard model for γ-ray emission from BL Lac objects and FR-I radio galaxies suggests a transition from proton to heavy-ion-dominated composition at ∼(10¹⁸–10¹⁹) eV.

We have assumed that γ-rays from TeV blazars and radio galaxies are from leptonic Compton scattering in order that synchrotron theory can be used to derive the various parameters. Thus the hadronic γ-ray flux must be considerably smaller for a consistent interpretation. Sufficiently high-energy protons and nuclei interact with synchrotron photons in the jet via the photomeson process, with photopion production efficiency f_pγ for cosmic-ray protons estimated to be (e.g., Murase & Beacom 2010a)

$$\begin{equation} f_{p \gamma } \simeq 2.3 \times {10}^{-4} \left(\frac{2.5}{1+\alpha } \right) L_{\gamma,46}^s t_{\rm var,4}^{-1} \delta _1^{-4} {\left(\frac{1\,\rm keV}{\varepsilon _s} \right)} {\left(\frac{E_p}{E_p^b} \right)}^{\alpha -1}, \end{equation} \tag{ 5 }$$

where E^b_p ≃ 1.6 × 10¹⁶ eV(ε_s/1 keV)⁻¹δ²₁(1 + z)⁻² is the typical energy of a proton that interacts with a photon with ε_s (where the proton energy is here defined in the observer frame). Also, α is the photon index at energies below or above ε_s. For α ≈ 1.5, which is typical of BL Lac objects at E_p > E^b_p, the photomeson production efficiency at ∼10¹⁹ eV becomes of the order of f_pγ ∼ 6 × 10⁻³, which suggests that the photomeson process is inefficient for this kind of blazar (though it could be more efficient for low-peaked BL Lac objects and FSRQs). The efficiency can also be higher if Γ is lower, provided that Γ is consistent with synchrotron/SSC model fits and minimum Lorentz factor estimates inferred from, e.g., γγ opacity arguments.

Roughly half of the pions produced by photomeson production are charged, and each neutrino carries ∼1/4 of the pion energy, so the total (isotropic-equivalent) neutrino luminosity at given E_ν ≈ 0.05E_p is estimated to be EL^ν_E ∼ (3/8)f_pγ(E_p)EL^CR_E ≃ 3.8 × 10⁴² erg s⁻¹f_{pγ, −2}(EL^CR_E/10⁴⁵ erg s⁻¹) for a source satisfying f_pγ ≲ 1 like BL Lac objects, where f_pγ = 10⁻²f_{pγ, −2} is used. Then, one finds that the neutrino flux from an individual source is typically too low to be detected with IceCube. One can also see that the cumulative background flux from high-peaked BL Lac objects is low. The UHECR energy input in the local universe is ∼5 × 10⁴³ erg Mpc⁻³ yr⁻¹ at 10¹⁹ eV (Murase & Takami 2009), so that assuming that such BL Lac objects and FR-I galaxies are the main UHECR sources, the expected cumulative muon neutrino background flux is estimated to be (Murase & Beacom 2010a)

$$\begin{eqnarray} E_{\nu }^2 \Phi _{\nu } & \sim & {10}^{-10}\,{\rm GeV}\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm sr}^{-1}\nonumber\\ && \times \left(\frac{f_{p \gamma } (20 E_\nu)}{{10}^{-2}} \right) E_{\nu,17.7}^{2-p} f_z, \end{eqnarray} \tag{ 6 }$$

where p is the cosmic-ray spectral index and f_z is a pre-factor coming from the redshift evolution of the sources. Low photomeson production efficiencies also follow if the UHECRs are heavy nuclei, whose losses are dominated by photodisintegration (see below). More luminous blazars, including low-peaked BL Lac objects, may, however, lead to higher photomeson production efficiencies, so that the cumulative neutrino background could be dominated by this class of AGNs (Mücke et al. 2003).

Our evaluation is based on the standard synchrotron/SSC model for BL Lac objects and FR-I radio galaxies. One could abandon the standard synchrotron/SSC model and consider a highly magnetized, ∼10–100 G jet model, which is needed in hadronic blazar models to accelerate protons to ≳ 10²⁰ eV (Aharonian 2000). Correspondingly, the minimum magnetic luminosity is estimated to be L_B ≈ 2 × 10⁴⁷ erg s⁻¹Γ²₁(E^max_{p, 20})², which is larger than the typical synchrotron luminosity of BL Lac objects, L^s_γ ∼ 10⁴⁶ erg s⁻¹. In the hadronic models, γ-ray emission is attributed to proton synchrotron radiation and/or proton-induced cascade emission, which leads to the requirement that the UHECR luminosity is L_UHECR ≳ L^C_γ = A_CL^s_γ. In the proton synchrotron blazar model (Aharonian 2000; Mücke & Protheroe 2001; Mücke et al. 2003), which is typically viable for high-peaked BL Lac objects, proton synchrotron radiation is emitted up to energies of ε^M_s ≈ 4δ₁/(1 + z) TeV (in the limit that the maximum energy is determined by the synchrotron cooling) in efficient Fermi acceleration scenarios. The photomeson production efficiency for protons is strongly dependent on δ, but the condition L_B ≫ L^s_γ suggests that the synchrotron energy loss is dominant at UHEs where the proton synchrotron radiation is typically prominent at ∼TeV energies (though the photohadronic cascade component may become relevant at lower energies). The strong magnetic field also suppresses electronic SSC emission because fewer electrons are needed to generate the same synchrotron flux.

If the standard synchrotron/SSC scenario holds for TeV blazars and their misaligned counterparts, then the protons can hardly reach 10²⁰ eV, as shown in Table 2 (see E^max_A obtained by detailed modeling in the literature). For BL Lac objects and FR-I galaxies to be the steady sources of UHECRs, therefore, UHECRs would primarily be heavier nuclei. In such a scenario, one has to examine whether ions can survive photodisintegration losses (cf. Murase et al. 2008a; Wang et al. 2008; Pe'er et al. 2009). The photodisintegration opacity is estimated similarly to the photomeson production efficiency. Approximating the photodisintegration cross section by the giant dipole resonance (GDR) cross section as $\sigma _{A \gamma } \sim \sigma _{\rm GDR} \delta (\varepsilon -\bar{\varepsilon }_{\rm GDR}) \Delta \bar{\varepsilon }_{\rm GDR}$, for a sufficiently soft photon spectrum (with α ≳ 1), we get (Murase & Beacom 2010a; see also Murase et al. 2008a for the non-GDR effect)

$$\begin{equation} \tau _{A \gamma } \approx \frac{t_{\rm var}^{\prime }}{t_{A \gamma }^{\prime }} \simeq \frac{2 \sigma _{\rm GDR}}{1+\alpha } \frac{\Delta \bar{\varepsilon }_{\rm GDR}}{\bar{\varepsilon }_{\rm GDR}} \frac{L_{\gamma }^s}{4 \pi \delta ^4 t_{\rm var} c^2 \varepsilon ^s} {\left(\frac{E_A}{E_A^b} \right)}^{\alpha -1}, \end{equation} \tag{ 7 }$$

where t'_Aγ is the photodisintegration interaction time, σ_GDR ≈ 1.45 × 10⁻²⁷A cm², $\bar{\varepsilon }_{\rm GDR} \approx 42. 65 A^{-0.21}$ MeV (for A > 4), $\Delta \bar{\varepsilon }_{\rm GDR} \sim 8$ MeV, and $E_A^b \approx 0.5 \delta ^2 (m_A c^2 \bar{\varepsilon }_{\rm GDR}/\varepsilon ^s) {(1+z)}^{-2}$ (in the observer frame). Then we numerically find

$$\begin{eqnarray} \tau _{A \gamma } (E_A) &\simeq & 0.16 \left(\frac{2.5}{1+\alpha } \right) L_{\gamma,46}^s t_{\rm var,4}^{-1} \delta _1^{-4}\nonumber\\ && \times {\left(\frac{\varepsilon _s}{1\,\rm keV} \right)}^{-1} {\left(\frac{E_A}{E_A^b} \right)}^{\alpha -1}, \end{eqnarray} \tag{ 8 }$$

where E^b_A ≃ 4.8 × 10¹⁶ eV(A/56)^0.79(ε_s/1 keV)⁻¹δ²₁(1 + z)⁻² is the energy of a nucleus that typically interacts with a photon with ε_s. Hence, heavy nuclei with E_A ∼ (Z/26)10^20.5 eV (given in the observer frame) undergo some photodisintegration reactions unless δ is high enough. The nucleus survival condition τ_Aγ(10²⁰ eV) ≲ 1 gives δ ≳ 17(Z/26)^0.1(A/56)^−0.079(L^s_{γ, 46})^1/5t_{var, 4}^−1/5(ε_s/1 keV)^−0.1(1 + z)^−1/5 (for α ∼ 1.5), but significant photodisintegration loss is easily avoided for reasonably large bulk outflow Doppler factors.

Recalling from Equation (5) that the photomeson production efficiency has the same dependence on δ, we can conclude that when heavy nuclei survive photodisintegration, the photomeson production efficiency is so low that the corresponding neutrino and γ-ray fluxes are not easily detected (Murase & Beacom 2010a, but see also Murase & Beacom 2010b).

SEDs of high-peaked BL Lac objects are generally well reproduced by the standard one-zone electronic synchrotron/SSC model. Among them, extreme TeV blazars have the hardest VHE γ-ray spectra at ∼1–10 TeV energies, as indicated by deabsorption of the measured γ-ray spectrum based on conventional EBL models (discussed below) and supported by non-detections of GeV γ-rays by Fermi. Also, in some cases (RGB J0152+017, 1ES 0229+200, and 1ES 0548−322), the optical/UV data show a rather steep spectrum which is thought to be the emission from the host galaxies (Tavecchio et al. 2011). Although the synchrotron component of extreme TeV blazars seems unremarkable at the optical/UV band, in these cases, comparison between optical/UV and X-ray data requires a strong roll-off of the nonthermal spectrum below the X-ray band, suggesting that F^s_ν∝ν^1/3 for 1ES 0229+200 (Tavecchio et al. 2011).

It is possible to explain such hard γ-ray spectra by the SSC model, but extreme parameters seem necessary compared to cases of typical, variable high-peaked BL Lac objects. It often suggests a very narrow-range energy distribution of electrons, and unusually large values of δ ∼ 10²–10³ may be necessary to avoid the K-N suppression. For 1ES 0229+200, PKS 0548−322, and 1ES 0347−121, extreme values of the electron minimum Lorentz factor of γ_{e, m} ∼ 10⁴–10⁵ are required from spectral modeling, where the hard SSC number spectrum, F_E∝E^−2/3 (in the Thomson regime) can be expected in the VHE range (Tavecchio et al. 2011).

There are several alternate blazar models that predict very hard ∼30–100 TeV γ-ray emission. In the hadronic model, the proton synchrotron process leads to multi-TeV emission if the outflow is ultrarelativistic, Γ ∼ 10²–10³. Then, further hardening may be caused by internal absorption due to some soft photon field outside the blob (Zacharopoulou et al. 2011). Another possibility to make hard TeV emission is electromagnetic radiation produced by nonthermal electrons in the vacuum gap of the black hole magnetosphere (e.g., Levinson 2000).

Böttcher et al. (2008) suggested that hard VHE emission originates from CMB photons Compton-upscattered by relativistic electrons that are accelerated in the extended jet. In this model, if the electron spectrum is hard, p ∼ 1.5, the resulting number spectrum of the IC emission has F_E∝E^{−(1 + p)/2} ∼ E^−5/4, which is compatible with the observed VHE γ-ray spectrum. The same process could be important for recollimation shocks (e.g., Bromberg & Levinson 2009; Nalewajko & Sikora 2009), or acceleration at knots and hot spots, noting that variations on much longer timescales can be expected in these models.

When VHE γ-rays are emitted from a source, they induce an electromagnetic cascade in intergalactic space. This cascade unavoidably accompanies spectral production of extreme TeV blazars as long as the IGMF in voids is weak enough. To demonstrate this, we show in Figures 1 and 2 the VHE γ-ray-induced cascade emission for sources at various redshifts. In Figure 1, primary source photons with F_E∝E^−β with β = 2/3 and E^max = 100 TeV (in the source rest frame) are assumed. One sees that the observed cutoff due to the EBL becomes lower for more distant sources since the γγ pair-creation opacity increases. In Figure 2, a different photon index (β = 5/4) and/or a different maximum energy (E^max = 10^1.5 TeV) are assumed for comparison, which causes slight differences in spectra. As indicated by Equation (9), the intergalactic cascade emission induced by primary γ-rays will be slowly variable or almost steady.

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Another way to have γ-ray-induced emission involves UHE γ-rays produced in blazar jets or radio galaxies. Such a case is shown in Figure 3 assuming much higher injected photon energies than before. Here we assume an injection spectrum with −2 number index centered at 10 EeV spanning one decade. The photomeson production by UHE protons, which can be expected in hadronic models, leads to UHE photons with energy E_γ ≈ 0.1E_p ≃ 10¹⁹ eVE_{p, 20}. In the synchrotron source in which UHE protons are accelerated, one may expect that the synchrotron self-absorption cutoff curtails the number of low-energy photons impeding UHE photon escape from the emission region (Murase 2009). A caveat of this model in our case is that generation of UHE γ-rays in the source requires acceleration of UHE protons and moderately efficient photomeson production. As noted before, the photomeson production in the source may not be too efficient in high-peaked BL Lac objects, which implies that the required UHECR luminosity has to be very large. As can be seen, there is some notable differences at low redshifts z ≲ 0.1 due to the longer effective energy-loss length of UHE γ-rays, but the received spectra are not strongly sensitive to the energy at which the photons are injected for higher redshift sources.

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Note that the intergalactic cascade scenario makes a non-variable or slowly variable component, even when the γ-ray emission made in the jet contributes to a separate highly variable component. Although there is no strong evidence of time variability for several extreme TeV blazars, future sensitive observations by CTA (Actis et al. 2011), HAWC (Sandoval et al. 2009), LHAASO (Cao 2010), or SCORE (Hampf et al. 2011) will be crucial for identifying a slowly variable γ-ray emission component.

In the previous subsection, we considered cascade emission induced by primary γ-rays. VHE γ-rays at ≲ 100 TeV, or UHE γ-rays with energies ≳ 3 EeV, where the opacity of the background radiation is not so large, can leave structured regions of the universe, whereas cosmic rays should feel structured EGMFs in clusters and filaments. The deflection of cosmic rays by the structured region with size l ∼ Mpc, magnetic field of B_EG ∼ 10 nG, and coherence length of λ_coh ∼ 0.1 Mpc (which may be typical of filaments; Ryu et al. 2008) is estimated to be (Takami & Murase 2011)

$$\begin{equation} \theta _{\rm CR} \approx \frac{\sqrt{2 \lambda _{\rm coh} l}}{3 r_L} \simeq 8^{\circ } Z E_{A,19}^{-1} B_{\rm EG,-8} {\left(\frac{\lambda _{\rm coh}}{0.1\,\rm Mpc} \right)}^{1/2} {\left(\frac{l}{\rm Mpc} \right)}^{1/2}. \end{equation} \tag{ 10 }$$

Therefore, the deflection by the structured EGMFs is not negligible for cosmic rays with energies ≲ 10¹⁹ eV, since the deflection angle is larger than the typical jet opening angle of θ_j ∼ 0.1 ∼ 6°. The corresponding time spread due to a structured EGMF around the source (that is comparable to the time delay) is expected to be

$$\begin{eqnarray} \frac{\Delta t_{\rm CR}}{1+z} &\approx & \frac{1}{4} \theta _{\rm CR}^2 \frac{l}{c} \simeq 2 \times {10}^{4}\,{\rm yr}\; Z^2 E_{A,19}^{-2} B_{\rm EG,-8}^2\nonumber \\ && \times {\left(\frac{\lambda _{\rm coh}}{0.1\,\rm Mpc} \right)} {\left(\frac{l}{\rm Mpc} \right)}^{2}, \end{eqnarray} \tag{ 11 }$$

which is unavoidable as long as cosmic rays pass through the structured region around the source, and it implies that the resulting cascade emission is essentially regarded as steady emission. Note that the total time spread ΔT_CR could generally be longer than Δt_CR due to additional time spread by intervening structured EGMFs and the void IGMF.

In order to model the structured EGMFs, we have assumed a simplified two-zone model with structured EGMF and IGMF in voids (see Takami & Murase 2011 for details). We model a cluster of galaxies by a sphere with the radius of 3 Mpc, and B_EG(r) = B₀(1 + r/r_c)^−0.7, with B₀ = 1 μG and r_c = 378 kpc. The magnetic field direction is assumed to be turbulent with the Kolmogorov spectrum and the maximum length of λ_max = 100 kpc. In addition to the EBL, the infrared background in the cluster is considered as the superposition of the SEDs of 100 giant elliptical galaxies calculated by GRASIL (Silva et al. 1998), using a fitting formula for the gas distribution (Rordorf et al. 2004). Filaments are modeled by a cylinder with a radius of 2 Mpc (Ryu et al. 2008) and a height of 25 Mpc. The magnetic field is assumed to be turbulent, which is described by the Kolmogorov spectrum with B_EG = 10 nG and λ_max = 100 kpc, although these values are very uncertain. Some numerical simulations imply a large-scale coherent component of the magnetic field in filaments (e.g., Brüggen et al. 2005), which may deflect cosmic-ray trajectories even more effectively. UHECRs are injected from the center of the filament toward a direction perpendicular to the cylindrical axis in order to examine a relatively conservative case. Throughout this work, the IGMF in voids is assumed to be weak enough to be less important for cosmic-ray deflections.

In Figure 4, we show our numerical results for the case E^max_p = 10¹⁹ eV expected in the standard synchrotron/SSC model of typical, variable BL Lac objects and FR-I galaxies. One sees that the structured EGMFs play an important role by suppressing the resulting γ-ray flux by more than one order of magnitude compared to the case without them. In Figure 5, we show the case of E^max_p = 10²⁰ eV, which can be achieved in the hadronic model. While the Bethe–Heitler pair-creation process provides the dominant electromagnetic component in Figure 4, contribution of photomeson production is more important in Figure 5. In the filament case, the deflection angle of UHECRs around 10²⁰ eV is still less than the jet opening angle, so that the γ-ray flux is diluted by only a small factor. On the other hand, in the cluster case, because UHECRs cannot be beamed, the γ-ray flux becomes almost isotropic and the corresponding flux is reduced according to the jet beaming factor (1 − cos θ_j) ≃ 1/200 for θ_j = 0.1. The effects of the structured EGMFs are illustrated in Figure 6, where the relative contributions are calculated from two-dimensional Gaussian fits. Note that if we express the isotropic-equivalent cosmic-ray luminosity where cosmic rays leave the structured region as EL^CR_E, then the relative contributions are (1 − cos θ_j)(EL^CR_E)/(EL^{CR, j}_E). In the filament case, isotropization becomes significant at ∼10¹⁹ eV rather than at ∼10²¹ eV for the cluster case.

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In Figure 7, we show resulting γ-ray spectra for various redshifts. Owing to the Bethe–Heitler process with energy-loss length ∼Gpc, UHE protons continue to supply electron–positron pairs for a longer distance than the photomeson energy-loss length of ∼100 Mpc. As a result, the dependence of the proton-induced γ-ray fluxes on distance is much gentler than γ-ray-induced fluxes. Indeed, one sees that the relative importance of the proton-induced γ-ray flux to the γ-ray-induced flux increases with distance (compare Figure 7 with Figures 1 and 3). Importantly for distant sources, the proton-induced cascade spectrum is much harder than the γ-ray-induced spectrum, especially above TeV energies. Future VHE observations by CTA and HAWC are important to identify the origin of UHECRs through detection of high-energy γ-rays, as we demonstrate for 1ES 0229+200 in the next subsection.

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In this work, we are interested in cases where IC cascade emission in voids is important in the VHE range, since it can explain hard VHE spectra of extreme TeV blazars as suggested by Essey et al. (2010). When pairs are mainly supplied via the Bethe–Heitler process, the timescale of secondary photons produced by a proton beam roughly becomes

$$\begin{equation} {\Delta t}_{\rm IGV} \simeq 14\,{\rm yr}\; E_{\gamma,11}^{-2} B_{\rm IGV,-17}^2 {(\lambda _{\rm BH}/\rm Gpc)} {(1+z)}^{-1}, \end{equation} \tag{ 12 }$$

which is more relevant than ΔT_CR when the void IGMF is so strong that ΔT_CR < Δt_IGV is satisfied. Here, λ_BH is the Bethe–Heitler energy-loss length. One should also keep in mind that the proton-induced GeV–TeV synchrotron emission from the structured region itself, where the EGMFs are stronger, should also be expected (see Gabici & Aharonian 2005; Kotera et al. 2009, 2011 and references therein). For a weak IGMF that is of interest in this work, its relative importance is somewhat smaller when the volume filling fraction of the magnetized region is taken into account.

We have demonstrated the likely importance of the structured EGMFs for proton-induced intergalactic cascade emission. They are also important for UHE nuclei. Since nuclei with energy ZE_p have the same deflection angle as protons with energy E_p, our results indicate that Fe nuclei should be significantly isotropized for all observed UHECR energies. For UHE nuclei, the photodisintegration energy-loss length is ∼100 Mpc, for which the energy fraction carried by γ-rays and neutrinos is small as long as E^max_A is not too high. On the other hand, UHE nuclei supply high-energy pairs via the Bethe–Heitler process, whose effective cross section is κ_{BH, A}σ_{BH, A} ∼ κ_{BH, p}σ_{BH, p}(Z²/A), which induces cascades in the same manner as UHE protons. Therefore, the intergalactic cascade signal, which is generated outside the source, is also important for sources of primary UHE nuclei.⁹

In a wide range of EBL models, deabsorption of measured TeV blazar spectra leads to hard excesses at >TeV energies in, e.g., 1ES 1101−232, 1ES 0229+200, and 1ES 0347−121 (see, e.g., Figure 8 in Finke et al. 2010). These unusual TeV spectral emission components are conventionally explained by (either leptonic or hadronic) emissions at the source, but they could also be explained by intergalactic cascade emissions. Non-simultaneous TeV excesses are also seen above the extrapolation of the GeV flux in NGC 1275 (Abdo et al. 2009b) and the core of Cen A (Abdo et al. 2010d), but because of their proximity, these excesses are unlikely to be UHECR-induced emissions made in intergalactic space.

Figure 8 demonstrates that 1ES 0229+200 can be fit by both the γ-ray-induced cascade and proton-induced cascade emissions. Because of the uncertainty in EBL models, it is not easy to distinguish between the two possibilities at ∼0.1–1 TeV energies. At higher energies, however, our calculations show that UHECR-induced cascade emission becomes harder than γ-ray-induced cascade emission resulting from attenuation of hard γ-ray source photons for a given EBL model. More importantly, the emission spectrum measured as a result of the injection of VHE/UHE photons at the source is strongly suppressed above ∼10 TeV for a wide range of EBL models, whereas a cosmic-ray-induced cascade displays a significantly harder spectrum above this energy, and detection of >25 TeV γ-rays from 1ES 0229+200 is only compatible if the γ-rays are hadronic in origin. This is because UHE protons (and UHE nuclei) can inject high-energy pairs over the Bethe–Heitler energy-loss length (λ_BH ∼ (A/Z²) Gpc at E_A ∼ A10¹⁹ eV) that is typically longer than the effective loss length of VHE/UHE photons. For steady, non-variable γ-ray sources, this intergalactic cascade signal induced by UHECRs provides a crucial probe of UHECR sources. Its identification would demonstrate that a distant blazar is an UHECR source through electromagnetic channels, which provides another important clue besides γ-ray variability. Identifying this feature by future Cerenkov detectors such as CTA or HAWC is possible, and the differential sensitivity goal of CTA is shown (Actis et al. 2011). Note that this is a differential sensitivity curve with the requirement of 5σ significance for 50 hr observations per bin, with 4 bins per decade. This is a much more stringent requirement than detection of a source with 5σ based on integrated flux, which can be divided into three data points with ≈3σ significance each. Given the differential CTA sensitivity for a 50 hr observation, the spectral hardening associated with hadronic cascade development can be clearly detected.

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It is theoretically expected that cosmic-ray-induced and γ-ray-induced cascade emissions are more easily discriminated in higher redshift sources. For the γ-ray-induced cascade, there should be a cutoff because of γγ pair creation by the EBL, while spectra of the cosmic-ray-induced cascade emission are hardened by the continuous injection through the Bethe–Heitler process. Hence, deep observations at ≳TeV energies by CTA or HAWC for moderately high-redshift blazars will also be important to resolve this question, along with detailed theoretical calculations for individual TeV blazars.

Now that IceCube has been completed, it has started to give important insights into the origin of UHECRs by itself and with GeV and VHE γ-ray observations. But detection of neutrino signals produced outside the source seems difficult for high-peaked BL Lac objects and FR-I galaxies because the point source flux sensitivity at >10 PeV is of the order of ∼10⁻¹¹ TeV cm⁻² s⁻¹ (Spiering 2011; Abbasi et al. 2011), which is typically larger than the expected neutrino fluxes, as shown in Figure 9. On the other hand, the cumulative (diffuse and stacked) background neutrino flux may be detectable especially for E^max_p ≳ 10²⁰ eV (cf. Anchordoqui et al. 2007; Takami et al. 2009 and references therein), which is possible in hadronic models with large magnetic fields in jets. For E^max_p = 10¹⁹ eV, however, protons mostly interact with the EBL, and the expected flux is lower than the proton case even if nuclei can be accelerated up to E_A = ZE_p (e.g., Anchordoqui et al. 2007). If BL Lac objects and FR-I galaxies are the main sources of UHECRs made mainly of ions with E_A ≲ Z10¹⁹ eV, the cumulative neutrino background would be difficult for IceCube to detect.

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Next, let us discuss the UHECR luminosity required to explain such extreme TeV blazars in the intergalactic UHECR-induced cascade scenario. In Figure 8, with E^max_p = 10¹⁹ eV and p = 2, the inferred isotropic-equivalent UHECR luminosities (at the source) are L_UHECR ≃ 10⁴⁶ erg s⁻¹ (for the filament) and L_UHECR ≃ 5 × 10⁴⁶ erg s⁻¹ (for the cluster), respectively, while L_γ ≃ 10⁴⁵ erg s⁻¹ when primary γ-rays are injected.

If no structured EGMFs are there, the required isotropic-equivalent UHECR luminosity (at the source) is LUHECR ∼ 1045–1046 erg s−1 (i.e., the corresponding absolute, beaming-corrected cosmic-ray luminosity LUHECR, j ∼ 1043 erg s−1), which is consistent with Essey et al. (2011) and Razzaque et al. (2012), and the calculation for 1ES 0229+200 shown here. It is also consistent with the proton power needed in hadronic models for typical, variable BL Lac objects. Where structured EGMFs play a role, the required UHECR luminosity becomes much larger. In fact, we obtained LUHECR ∼ 1046–1047 erg s−1 (for filaments) and LUHECR ∼ 1047–1048 erg s−1 (for clusters), depending on the EBL model and spectral indices, when we assume Emaxp = 1019 eV. Such UHECR luminosities, obtained with the structured EGMFs, seem rather extreme, since the total cosmic-ray luminosity including low-energy cosmic rays is at least ∼20 times larger for p ≳ 2 and Eminp = 10 GeV. In the cluster EGMF case, this means an absolute luminosity of LCR, j ≡ ∫EmaxpEpmindE LCR, jE ≳ 1046–1047 erg s−1, which is comparable to the Eddington luminosity of a ∼108–109 M☉ black hole,

$$\begin{equation} L_{\rm Edd,abs} = \frac{4 \pi G M_{\rm BH} m_p c}{\sigma _T} \simeq 1.3 \times {10}^{46}\,{\rm erg}\,{\rm s}^{-1} \left(\frac{M_{\rm BH}}{{10}^{8}\, M_{\odot }} \right), \end{equation} \tag{ 13 }$$

where M_BH is the black hole mass. Therefore, the intergalactic UHECR-induced cascade interpretation becomes problematic if runaway UHECRs are significantly isotropized and/or the spectral index of cosmic rays is steep enough. Such isotropization may be realized by some plasma instability, or by the structured EGMFs and/or magnetic fields in radio bubbles or lobes accompanied by radio-loud AGNs (see below).

Note that our conclusion from Table 2 does not hold in the intergalactic hadronic cascade interpretation of extreme blazars since the SSC model is here abandoned. But one may adopt E^max_p ∼ 10¹⁹ eV, motivated by results of the SSC modeling for typical, variable blazars (see Table 2). On the other hand, higher E^max_p is also possible and a proton spectrum with higher E^max_p is favored in view of smaller deflections and relaxed luminosity requirement to fit TeV data (see Figure 6). However, similar to the proton synchrotron blazar model for variable BL Lac objects, the proton synchrotron component is expected as well as the intergalactic hadronic cascade component.

It is useful to compare those luminosities with the required UHECR energy budget indicated from UHECR observations. From recent PAO observations, the local UHECR energy budget above 10^18.5 eV is a few × 10⁴⁴ erg Mpc⁻³ yr⁻¹. For the local blazar density, n_s ∼ 10^−6.5 Mpc⁻³ (Padovani & Urry 1990), the inferred isotropic-equivalent UHECR luminosity is L_UHECR ∼ 10^43.5 erg s⁻¹ (regarding blazars as radio-loud AGNs pointing toward us). This is much smaller than the cosmic-ray luminosity required for explaining extreme TeV blazars, and implies that those distant radio-loud AGNs with hard VHE spectra should be rarer and more powerful in cosmic rays than nearby AGNs responsible for the observed UHECRs. For 1ES 0229+200, the single-source flux is ∼10% of the observed UHECR flux so that the anisotropy can be used as a useful probe.

We demonstrated the importance of structured EGMFs that help isotropize the trajectories of UHECRs, though the EGMF strengths are still uncertain. In addition, there are other causes that can diminish the beaming of UHECRs and resulting cascade fluxes. One arises from plasma instabilities induced by cosmic rays (K. Murase et al. 2011, in preparation). Second, radio lobes of powerful radio-loud AGNs, like in the case of Cen A with B ∼ 1 μG, would also isotropize UHECRs (Dermer et al. 2009), as might radio bubbles from the jets of typical FR-I radio galaxies and aligned counterparts. These magnetic fields seem relevant in order that cosmic rays from relativistic jets of radio-loud AGNs to contribute to the observed flux of UHECRs. Indeed, for nearby radio-loud AGNs, the UHECRs must be significantly isotropized, since there is no blazar (i.e., aligned radio-loud AGNs) within ∼100 Mpc¹⁰ and no evidence of cross-correlation with nearby blazars such as Mrk 501 and Mrk 421 (Dermer et al. 2009). The isotropic-equivalent UHECR luminosity (at the source) L_UHECR ≳ 10⁴⁵ erg s⁻¹ at ∼100 Mpc will lead to overproduction of the observed UHECR spectral flux.