What specific limitation does science impose on the types of phenomena that can be studied?

There are no limitations imposed on any phenomena. Science means knowledge. A scientists job is to pursue knowledge regardless of where it takes him. They are required to be honest and non biased in their research and their statements must be verifiable. It simply cannot be because 'they said so'. There must also be acceptance of not knowing the answers as well along with the possibility that they may never find the answers. This does not give them the right to develop their own answers.

There are an infinite number of questions in the Universe. But given an infinite timeline, answers will be found for all of them.