Probing UHECR production in Centaurus A using secondary neutrinos and gamma-rays

In this paper, the production of neutrinos and photons by ultra high energy cosmic rays (UHECR) interacting with the extragalactic background radiation is studied. Centaurus A is assumed as the prime source of UHECR and the possibility to identify this source by detecting the secondary neutrinos and photons produced in the propagation of UHECR is investigated. Fifteen astrophysical models regarding three extragalactic magnetic fields (EGMF) and five composition abundances are simulated. The flux and arrival direction of neutrinos and photons are investigated. It is shown that the detection of a signal from Cen A with statistical significance is achievable by current observatories in a few years and by proposed experiments in the near future. The dependence of the results on the models is also presented.


I. INTRODUCTION
The unambiguous identification of an ultra high energy cosmic ray (UHECR) source would bring an enormous advance to astroparticle physics. Several unknowns related to the lack of knowledge of the source type, propagation effects and composition would be drastically reduced if one source is detected. The most famous candidate is Centaurus A (Cen A), the closest radio galaxy with a very rich morphology containing a radio jet extending over kiloparsecs and giant-lobes with kiloparsec size [1,2]. Several studies have shown that Cen A has the necessary conditions to accelerate particles at the highest energies [3-10].
Its relative size and proximity helps the detection and direct experimental evidences for acceleration of high energy particles have been published [11][12][13][14].
A multimessenger approach has been advocated as the only way to give definitive answers to UHECR production. Neutral particles (neutrons, neutrinos and photons) are considered to be the smoking gun of the UHECR source since they do not deviate on the way to Earth.
Intrinsic TeV gamma-rays have been detected from Cen A [14] however the connection to UHECR production depends on the unknown characteristics of the acceleration sites [15].
The uncertainties about the environment of production do not allow one to infer if high energetic photons and neutrinos are produced as a consequence of UHECR acceleration.
Models for the emission of primary neutrinos as a consequence of proton acceleration in Cen A have already been proposed to describe the core and jet emission [16,17] 1 . The original models were optimized in 2008 and 2009 using the first years of data measured by the Pierre Auger Observatory. A large uncertainty was introduced in the original proposals in the normalization of the neutrino flux which was taken to be proportional to the UHECR flux around a certain radius centered in Cen A. Figure 1 shows an update of these original models to the last data release of the Pierre Auger Observatory, see appendix A for details. The ignorance about the contribution of Cen A to the total flux of UHECR is shown as a shaded area that covers a range of contribution of Cen A to the total flux measured by the Pierre Auger Observatory between 1 and 100%. The 90% upper limits on the neutrino flux measured by the Pierre Auger Observatory and by the IceCube are also shown for comparison. The prediction of sensitivity of future experiments ARA [18], ARI-ANNA [19], POEMMA [20], IceCube Gen2 [21] and GRAND [22] are also shown. Note that the sensitivity of the future experiments are not corrected to the sensitivity in the direction of Cen A. They are shown here only for comparison of the overall flux. The primary flux of neutrinos for Cen A shown in this figure is of the same order of the flux predicted by cosmogenic neutrinos [23,24]. Future experiments will have sensitivity to detect intrinsic neutrinos from Cen A if: a) the models predicting the emission from the core are correct and b) Cen A contributes to more than roughly 50% of the total UHECR flux measured by the Pierre Auger Observatory.
Given that the intrinsic flux of photons and neutrinos at Cen A is or can be soon detected, the next sections delve into a possibly richer question concerning UHECR. The calculations presented here investigate for the first time if photons and neutrinos produced on the way from Cen A to Earth by UHECR could also be detected by current or future experiments.
An UHECR interacts with the photon field and produces, via photo-pion production and nuclear fragmentation, photons and neutrinos. The mechanisms of production are well known which leads to a very precise prediction within a certain astrophysical scenario. For this reason, the detection of secondary neutrinos and photons would correlate more clearly to UHECR rather than the intrinsic flux of photons and neutrinos.
In this paper, the arrival direction and flux of secondary photons and neutrinos are investigated for fifteen astrophysical scenarios. In the next section the astrophysical scenarios will be presented and discussed. In section III and IV the results concerning flux and arrival direction of neutrinos and photons are respectively presented. Section V presents the conclusions.

TONS
The prediction of the flux and arrival direction of particles coming from Cen A depends significantly on the astrophysical scenarios taken into account. In this paper, fifteen astrophysical scenarios are considered representing three extragalactic magnetic field models and five composition abundances. For each scenario, the propagation of cosmic rays from Cen A to Earth is simulated and the numbers of secondary neutrinos and photons are calculated.
It is not the purpose of this study to build an astrophysical scenario compatible to current data. Instead, fifteen astrophysical published scenarios were chosen and the corresponding  The fractions of nuclei injected by the source in each composition scenario are presented in the table I, together with the mean chargeZ of the injection. These composition models were chosen to cover a wide range of possibilities, from very light to very heavy scenarios.

B. Galactic and Extragalactic Magnetic Field
The galactic magnetic field is known to have a large effect on the arrival direction of UHECR [29]. However, our galaxy is much smaller than the distance from Earth to Cen A, therefore, the large majority of secondary neutrinos and photons produced by UHECR on the way from Cen A are going to be produced outside the Milky Way. Because they are chargeless and are produced outside our Galaxy, the arrival direction and flux of secondary neutrinos and photons are independent of the galactic magnetic field.
The extragalactic magnetic field (EGMF) deviates the trajectory of charged cosmic rays.
The secondary neutrinos and photons generated in the interactions of the cosmic rays with the photon fields are produced in the direction of propagation of the charged cosmic ray therefore not all secondary neutrinos and photons point straight back to the source. On average, the direction of the secondary neutrinos and photons are deviated from the source position by an amount proportional to the path between the source and the production point of the secondary neutrino and photon (scattering should be taken into account.). This is the path traveled by the charged cosmic ray before producing the secondary neutrino or photon.
The total effect of EGMF in the trajectory of the cosmic rays depends on the magnetic field strenght and structure. Heavier nuclei can be largely deflected faking an almost isotropic sky even when a dominant nearby source is considered, as discusses in references [30][31][32][33][34].
The total effect in the secondary neutrino and photon point spread function is expected to be much smaller than the effect calculated for the charged particles. Nevertheless, the possibility to identify the source and the acceleration sites in the sources requires the treatment of this effect.
The EGMF models in reference [35] are used here. These modern computer simulations calculated the EGMF considering its evolution in an expanding universe imposing the current matter distribution as a constraint. Six models for the EGMF with different seed fields are able to describe the constrains. In this paper, three models were chosen to cover the widest range of field intensity within the six models. In reference [35] they were named Primordial (Prim), Primordial2R (Prim2R) and AstrophysicalR (AstroR). Prim and Prim2R models were constructed based on an initial magnetic field at redshift z = 60 uniform of strength 0.1 nG and a power-law distribution of spectral index -3, respectively. The AstroR was selected as one astrophysical origin EGMF exemplary, were the seed field is developed from cooling and AGN feedback. Figure 2 shows the field intensity (xy and yz-plane) in an 20 Mpc sphere around Earth.

C. Energy spectrum at the source
First-order Fermi acceleration is the paradigm for particle acceleration [36,37]. Diffusive shock acceleration has been proposed as the main acceleration mechanism in AGN jets [10].
Shock waves in the backflow of the jets have been measured in Cen A [38,39] in which particle acceleration is expected to occur. The energy spectrum can be written as a power law combined with a charge dependent exponential cutoff [40] that accounts for the limitation in the maximal energy that a source can accelerate particles where L CR is the cosmic-ray luminosity, γ is the spectral index, Z is the charge of the particle and R cut is the rigidity cutoff such as that E cut = ZR cut . Continuous emission by the source was considered.
The cosmic-ray luminosity and rigidity cutoff of Cen A was estimated based on the approach developed in reference [26]. The fundamental hypothesis, in agreement with Fermi mechanism, is that L cr must be a fraction of the total kinetic energy in the jet, Q jet , which was shown to be proportional to the radio luminosity [41] L ν at frequency ν = 1.1 GHz the values of A and β L can be obtained indirectly from data or from theoretical models [42].
Reference [26] uses values obtained for FRII radio galaxies. However, Cen A is classified as a FRI and the values of β L can be different for FRI and FRII [42]. Reference [43] used the cavity method, in which the kinetic power of the jet is determined by the radio cavity volume, to calculate A = 8.6 × 10 43 and β L = 0.75 for FRI-type radio galaxies.
The measured radio luminosity at 1.  45]. In this model, g cr contains the relation between the energy stored in hadrons and in the magnetic field. If a source satisfies the minimum energy condition, reference [46] proposed g cr = 4/7. In this paper, instead of fixing the value of g cr , it will be fitted to the energy spectrum measured by the Pierre Auger Observatory as explained below.
The rigidity cutoff is related to the escape time from the source. Reference [26] relates g ac depends on plasma physics details in the acceleration site and, is given by (assuming shock acceleration) g ac = 8β 2 sh f 2 β j , with β sh the typical shock velocity responsible for the particle acceleration (in speed of light units), β j the jet velocity, and f provide specific plasma properties, with 1 ≤ f ≤ 8 for shocks with typical geometries and turbulent magnetic fields [25]. If Cen A has a jet with velocity v j ∼ 0.5c [47] and a typical velocity of the shock waves v sh ∼ 0.2c [10]: 0.1 ≤ g ac ≤ 0.8.
For each particle type Z, the energy spectrum of Cen A used here is determined as a function of three parameters: γ, g cr and g ac . For each of the extragalactic magnetic fields models and composition abundances described in the previous sections, γ, g cr and g ac are going to be fitted to the energy spectrum of cosmic rays measured by the Pierre Auger Observatory.

D. Energy spectrum at Earth
The energy spectrum and composition leaving the source is modified by the interactions with the photon and magnetic fields. These interactions, producing secondary neutrinos and photons, were simulated with the CRPropa 3 framework [48]. The relevant energy loss mechanisms and interactions were considered in the propagation of the cosmic rays: adiabatic losses, photodisintegration, photopion production, pair production, and nuclei decay. Only adiabatic losses was considered in the propagation of the secondary neutrinos.
Secondary photon were allowed to generate electromagnetic cascades, taking in account inverse Compton scattering, pair production, double and triple pair production. The background photon fields considered were the cosmic microwave background, the extragalactic background light [49] and radio [50]. The photon fields have an important effect on the propagation of high energetic cosmic and gamma rays even for close sources. For instance, the pair-production mean free path of a EeV gamma ray is of the order of Mpc, therefore important for propagation from Cen A. However, the differences between different models of extragalactic background light is important mainly to energies below the interest of this paper [51] and for distances larger than the source considered here [52]. The interaction of UHECR with the gas surrounding Cen A can be safely neglect given the baryon content estimated in the filaments of massive galaxy cluster to have density ≈ 10 −5 cm −3 [53], the mean free path for proton-proton interactions in this medium is of the order of 1 Gpc.
Cosmic rays were injected with energies between 10 18 and 10 21 eV and propagated down to 10 17 eV. The calculations include 10 9 events of 1 H, 4 He, 14 N , 28 Si and 56 F e nuclei for each magnetic field configuration. The particles were injected with a spectral index −1 to guarantee equal statistical fluctuations at all energies. The energy spectrum shape and composition fractions were introduced as weights according to the procedure explained in reference [26]. The observational radius was considered to be r obs = 100 kpc. An observational radius of 10 kpc was also simulated and the conclusion presented here remains the same.
The simulated energy spectrum arriving on Earth has three parameters γ, g cr and g ac which were fitted to the energy spectrum measured by the Pierre Auger Observatory [54].
The parameters γ, g cr and g ac were allowed to vary between [2.00, 2.30], [0.10, 1.00] and [0.01, 1.00] in steps of 0.02, respectively. These parameter limits brackets the possible values within the classical model used here, which guarantees the coherent use of the theory.
However, it is important to notice that some sophisticated models includes, for instance, harder energy spectrum [55]. In this fit, 100% of the flux measured by the Pierre Auger Observatory was considered to be generated by Cen A. This is probability a simplification because other sources beyond Cen A are expected to generate a non negligible flux of cosmic rays arriving at Earth. However, this simplification is acceptable because the fit is used only to fine tune the shape of the energy spectrum predicated by acceleration models. By bracketing the limits of the parameters with the theoretical prediction, no inconsistency is propagated to the further analysis. The contribution of Cen A to the total flux of cosmic rays will be allowed to vary in the next section to remove this simplification when needed.
Equation 1 was fitted to the energy spectrum data using the standard χ 2 method including uncertainties for energies above 10 18.7 eV. Table II and figure 3 shows the results of the fits.
It is beyond the scope of this paper to evaluate how well each astrophysical model describes the data measured by the Pierre Auger Observatory. For that purpose, the fit would have to include other measured quantities (i.e. Xmax distribution) and more free parameters in the models. The fit implemented in this study has a limited focus on improving the calculation of g cr and g ac normalizing the simulations to the flux measured by the Pierre Auger Observatory. The values of these parameters have been estimated based on theoretical assumptions as shown above. In the fit presented here, the theoretical estimations are used as limits and first guesses to search the best values of g cr and g ac that describes the energy spectrum. Note that some models produces a flux higher than the one measured by the Pierre Auger Observatory, which in principal could be used to invalidate the model, however small changes to the models could ameliorate the agreement with the data. Composition γ g ac g cr χ 2 /dof γ g ac g cr χ 2 /dof γ g ac g cr χ 2 /dof  GRAND [56], POEMMA [20] and IceCube 2Gen [21] are also shown. The effect of the injection composition is clearly seen. Heavier compositions (C4 and C5) produce less neutrinos at the highest energies because nuclear fragmentation is more likely than photo-pion production. The flux of secondary neutrinos from Cen A is much smaller than the sensitivity of the future experiments for all scenarios tested here. The possibility of detection of an anisotropic signal of secondary neutrinos from Cen A was calculated using the Li-Ma significance [57,58]. The background given by cosmogenic neutrino flux, null hypothesis in the statistical test, was taken from references [23] and [24]. The same calculations regarding flux and arrival direction were done for the secondary photons produced by UHECR on the way from Cen A to Earth. Figure 8 shows the flux of secondary photons arriving at Earth in comparison to the sensitivity of current and future gamma-ray observatories. The SWGO [59] experiment will have the highest probability to detect a signal because its maximum sensitivity is at the energy range (from 0.5 to 1×10 14 eV) in which the highest flux of photons arrives on Earth. Figure 9 shows the calculated integral photon flux of secondary photons in comparison to current measurements [60][61][62] and future predictions [22]. If compositions similar to C1, C2 and C3 models are emitted by Cen A and the magnetic field is not so extreme as Prim model, the Pierre Auger Observatory will measure a photon signal from Cen A by 2025 and the GRAND experiment will confirm this measurement after three years of operation.  [4] C. Dermer, S. Razzaque, J. Finke, A. Atoyan, Ultra-high-energy cosmic rays from black hole jets of radio galaxies, New Journal of Physics 11 (6) (2009) 065016.  [63] and Ξ 2008 = 9000 km 2 sr yr [64]. ξ is a factor to convert from the point source coverage to full sky coverage. ξ = 0.11 for the window considered in reference [17] and ξ = 0.20 for the window considered in reference [16]. AstroR, Prim2R and Prim. Each curve corresponds to a composition abundance: C1-C5.

Prim2R
Prim AstroR  A to the total UHECR flux measured by the Pierre Auger Observatory for energies above 60 EeV. The significance was calculated supposing 80 000 m 2 detection area observatory taking data during five years. Full (Dashed-dotted) lines represents the calculations when the 90% lower (upper) limit flux of cosmogenic background photons estimated in reference [23] was considered as the null hypothesis. The three EGMF models (AstroR,