Isotropy constraints on powerful sources of ultrahigh-energy cosmic rays at $10^{19}$ eV

Anisotropy in the arrival direction distribution of ultrahigh-energy cosmic rays (UHECRs) produced by powerful sources is numerically evaluated. We show that, taking account of the Galactic magnetic field, nondetection of significant anisotropy at $\approx 10^{19}$ eV at present and in future experiments imposes general upper limits on UHECR proton luminosity of steady sources as a function of source redshifts. The upper limits constrain the existence of typical steady sources in the local universe and limit the local density of $10^{19}$ eV UHECR sources to be $\gtrsim 10^{-3}$ Mpc$^{-3},$ assuming average intergalactic magnetic fields less than $10^{-9}$ G. This isotropy, which is stronger than measured at the highest energies, may indicate the transient generation of UHECRs. Our anisotropy calculations are applied for extreme high-frequency-peaked BL Lac objects 1ES 0229+200, 1ES 1101-232, and 1ES 0347-121, to test the UHECR-induced cascade model, in which beamed UHECR protons generate TeV radiation in transit from sources. While the magnetic-field structure surrounding the sources affects the required absolute cosmic-ray luminosity of the blazars, the magnetic-field structure surrounding the Milky Way directly affects the observed anisotropy. If both of the magnetic fields are weak enough, significant UHECR anisotropy from these blazars should be detectable by the Pierre Auger Observatory unless the maximum energy of UHECR protons is well below $10^{19}$ eV. Furthermore, if these are the sources of UHECRs above $10^{19}$ eV, a local magnetic structure surrounding the Milky Way is needed to explain the observed isotropy at $\sim 10^{19}$ eV, which may be incompatible with large magnetic structures around all galaxies for the UHECR-induced cascade model to work with reasonable jet powers.