The man accused of the Pittsburgh synagogue shooting, Robert Bowers, legally purchased the guns he used to kill 11 people in what is believed to be the deadliest attack against the Jewish community in the United States, according to the federal authorities.

Officials have said Mr. Bowers used four guns — an AR-15 assault rifle and three Glock .357 handguns — in his shooting spree at the Tree of Life synagogue in Pittsburgh on Saturday morning.

An investigation has concluded that the guns were “acquired and possessed legally by Bowers,” the Philadelphia office of the Bureau of Alcohol, Tobacco, Firearms and Explosives said on Tuesday.

Mr. Bowers did not fall into any category barred from gun ownership under federal law, including felons, convicted domestic abusers, dishonorably discharged veterans, or people adjudicated to be mentally ill or subject to certain restraining orders.