Generally speaking it didn't need one. A nearly-identical question was asked some time ago, and while it deals with the late Republic the answers there (and the further links given) are generally pertinent to the Principate, in which not much changed besides the existence of dedicated instruments of suppression of popular revolt. Roman law was pretty self-help, with accusations being brought by private individuals into the courts. Organizations like the vigiles might arrest people, though it was not their primary function, for being caught red-handed in a crime, but the identification, apprehension, and accusation of criminals was left mostly the private individual or his patrons. With no modern science of criminal forensics an investigative police force would have been absurd, and the Principate had methods of maintaining public order at least as effective as the police--the emperors maintained an entire private army within the city to quell popular unrest.