The only time they need a warrant is for a search, and the reasons needed can be as little as your refused consent, or they have reason to believe you will refuse. They do not need to ask you. That is black-letter law.
That may not be entirely true. The judge will still have to decide on what those reasons are so its not a slam dunk on that warrant. There has to be more than just refusal....imagine how much power that is and how much it can be abused otherwise.
Would they really go out of their way and spend this much time just purely on a neigbour who happens to be walking by who may have seen something that appear to be a gun? They may send someone to check it out but not immediately banging in with a warrant.