If a person is charged with a capital crime and police recover evidence in that person's trash, can it be used as evidence in their trial even without a warrant?
Yes. Evidence that is found in a person's trash is admissible even if the police do not have a warrant. That is true whether the case is a capital one or not.
The reason for this rests with the idea of the expectation of privacy. Basically, if you have good reason to think that something of yours will be kept private, the police need a warrant to search it. But if you have no reason to think it will be kept private, they do not need a warrant. The Court has ruled that there is no good reason to expect that your trash is private. This is stuff you have put out at the curb where anyone can come and look at it. Dogs can knock over your trash and spill it. The garbage collectors can see it when they dump it and so on.
For these reasons, you have no presumption that your trash will remain private and the police can search it even without a warrant.