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Ultrahigh energy protons and nuclei from extragalactic cosmic ray sourcesinitiate intergalactic electromagnetic cascades, resulting in observable fluxesof $\gamma$-rays in the GeV-TeV energy domain. The total spectrum of suchcascade $\gamma$-rays of hadronic nature is significantly harder than the oneusually expected from blazars. The spectra of some sources known as "extremeTeV blazars" could be well-described by this "intergalactic hadronic cascademodel" (IHCM). We calculate the shape of the observable point-like spectrum, aswell as the observable angular distibution of $\gamma$-rays, for the first timetaking into account the effect of primary proton deflection in filaments andgalaxy clusters of the extragalactic magnetic field assuming the model of Dolaget al. (2005). We present estimates of the width of the observable $\gamma$-rayangular distribution derived from simple geometrical considerations. We alsoemploy a hybrid code to compute the observable spectral and angulardistributions of $\gamma$-rays. The observable point-like spectrum at multi-TeVenergies is much softer than the one averaged over all values of the observableangle. The presence of a high-energy cutoff in the observable spectra ofextreme TeV blazars in the framework of the IHCM could significantly facilitatefuture searches of new physics processes that enhance the apparent $\gamma$-raytransparency of the Universe (for instance, $\gamma \rightarrow ALP$oscillations). The width of the observable angular distribution is greater thanor comparable to the extent of the point spread function of next-generation$\gamma$-ray telescopes.

monthly notices of the royal astronomical society

Observable spectral and angular distributions of $\gamma$-rays from  extragalactic ultrahigh energy cosmic ray accelerators: the case of extreme  TeV blazars