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

Ultrahigh energy protons and nuclei from extragalactic cosmic ray sources initiate intergalactic electromagnetic cascades, resulting in observable fluxes of $gamma$-rays in the GeV-TeV energy domain. The total spectrum of such cascade $gamma$-rays of hadronic nature is significantly harder than the one usually expected from blazars. The spectra of some sources known as extreme TeV blazars could be well-described by this intergalactic hadronic cascade model (IHCM). We calculate the shape of the observable point-like spectrum, as well as the observable angular distibution of $gamma$-rays, for the first time taking into account the effect of primary proton deflection in filaments and galaxy clusters of the extragalactic magnetic field assuming the model of Dolag et al. (2005). We present estimates of the width of the observable $gamma$-ray angular distribution derived from simple geometrical considerations. We also employ a hybrid code to compute the observable spectral and angular distributions of $gamma$-rays. The observable point-like spectrum at multi-TeV energies is much softer than the one averaged over all values of the observable angle. The presence of a high-energy cutoff in the observable spectra of extreme TeV blazars in the framework of the IHCM could significantly facilitate future searches of new physics processes that enhance the apparent $gamma$-ray transparency of the Universe (for instance, $gamma rightarrow ALP$ oscillations). The width of the observable angular distribution is greater than or comparable to the extent of the point spread function of next-generation $gamma$-ray telescopes.