Intergalactic magnetic field lower limits up to the redshift $z\approx3$

Large-scale intergalactic magnetic fields may contain a mixture of galactic and cosmogenic contributions, that can be probed via observations of $\gamma$-ray "echo" - a delayed emission from electromagnetic cascades initiated by the highest energy photons from the sources at cosmological distances. While these fields contributions may be disentangled based on the difference in their redshift evolution, thus far indications of non-negligible magnetic field have been found only at low redshifts. This work aims at extending the intergalactic magnetic field constraints to redshifts $z\gtrsim 1$ using 17-year long all-sky observations of high-redshift active galactic nuclei with Fermi/LAT $\gamma$-ray telescope. Combing the Fermi/LAT measurements in the 0.1 GeV - 1 TeV energy range with the Monte Carlo simulations of the $\gamma$-ray "echo", it is shown that the zero field strength hypothesis at $z = [0.5; 3]$ is disfavoured at the $\approx 8.6\sigma$ significance level, yielding the lower limit of $B \gtrsim 1\times10^{-18}$ cG for the magnetic field correlation length above 1 Mpc. It is further shown that the same data put a lower limit on the volume-filling fraction of this field of $f\gtrsim 90\%$ in the redshift range considered. It is also demonstrated that the derived limits are not substantially affected by either source flux variability or the assumed $\gamma$-ray emission attenuation model. Implications of these limits for intergalactic magnetic field origin are discussed.