I think it's a good question. Basically it depends on a couple of variables.
Via the 4th Amendment to the US Constitution, a person has a right to be free of unreasonable searches and seizures. That's been interpreted to mean that if a person has "a reasonable expectation of privacy" in something, it's protected from a warrantless search. (A reasonable expectation of privacy would be a personal journal you keep in a safe in your house, whereas you have no reasonable expectation of privacy in something you broadcast on a billboard in Times Square.)
For the state to be able to intrude into those protected things (wire-tapping, searching your email, searching your house for personal letters or photographs, etc.) the state has to go to a judge and get a warrant.
The idea behind this is that the US Government is a system of checks and balances, and the executive branch (which includes the police) cannot write their own ticket insofar as to how far they can intrude into someone's life. Instead, they have to show the judicial branch of government (a judge) that they have "probable cause" to believe a crime has taken place or will be taking place. If the judge agrees that a reasonable person, on the basis of the facts presented, in their whole and proper context, would think it likely that a crime had occurred or would occur, the warrant will be issued.
The kicker is the "exclusionary rule" - basically it says if the prosecution violates the protections offered by the 4th, they can't use the evidence they've illegally seized against you OR any other evidence that flows from it (often referred to as "the fruit of the poisonous tree.")
There are a
ton of wrinkles in there though, at every single step. Including situations in which the 4th does not apply (usually called "an exception").
Wiki has a decent overview.
http://en.wikipedia.org/wiki/Fourth_Amendment_to_the_United_States_Constitution
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With AH, the question is basically, "
does he have a reasonable expectation of privacy in the jail calls?" If so, is there independent evidence (apart from the calls which were already heard) that would lead a prosecutor to believe the calls exist, and that they have some bearing on whether or not a crime was committed? If so, is the evidence of such weight that it amounts to probable cause upon which a warrant should be issued.
If AH does not have a reasonable expectation of privacy, there is no constitutional issue regarding using his recorded statements against him at trial. (Although there may be procedural objections.)
If AH does, and the prosecution does not go about obtaining the evidence "the right way," the risk is that the recorded statements and anything they lead to will not be able to be used against him at trial. For example, if AH implicates someone totally new - "Joe," and the investigation of "Joe" turns up the murder weapon and a videotape of the shooting, all that evidence (and any new charges against Joe) would be excluded - i.e., not useable. There are possible work-arounds to the exclusion of evidence, but it's better for the prosecution not to be fighting those battles.
So it sounds to me like the prosecution is trying to play it safe by getting a warrant, whether or not they can argue that AH didn't have a reasonable expectation of privacy, or whether or not the statements on the tapes fall into some kind of exception to the warrant requirement.