“Probable cause,” as Bricker said, is the standard. This is something of a term of art, though. There are a number of elements that might go to demonstrate “probable cause” to believe that person X committed crime Y. Among them:
A sworn information or complaint – Person A is willing to go on record, under oath and subject to penalties for perjury or reasonable facsimile, to state that X did Y.
Evidence clearly linking X to the circumstances under which Y was committed, both in physical/temporal location (opportunity) and motive.
Circumstantial evidence strongly suggesting the sort of situation in #3. E.g., if crime Y was committed under a certain modus operandi, and someone convicted of crimes under that M.O. is in the same town and cannot account for his/her whereabouts at the time of the crime, and has recently changed his/her behavior pattern in ways suggestive of having benefitted from crime Y, this may constitute probable cause.
People with more background in criminal law can probably define this far better than I. My point is that there has to be a nexus linking X to Y, but not necessarily rock-solid proof; “beyond a reasonable doubt” is a question for a trial, not for issuance of a warrant.
Just to note in passing that (a) police may arrest without a warrant only under specific circumstances, (b) a judge, not the police, issues the warrant (and requires that “probable cause” nexus), but © the police, for obvious reasons, are loath to seek a warrant before having established probable cause to the standard they know the judge will accept.
Further, the judge is entitled to treat any of the above skeptically if circumstances call for it. A person filing a complaint charging that G.W.B. has sent the FBI to attempt to capture him so that they can transport him to Area 51 so the aliens can install tracking devices in his body, is probably not going to get a warrant requiring apprehension of G.W.B. for this evil plot!