Since the min_score is currently just an absolute value. What would be more useful would be a relative min score. When sorting, it would really nice to limit the results to those that score above a minimum threshold, say 30% for example.
You can do it today with the Scroll API . Execute your query of choice, note the number of matching documents in the first response and then keep scrolling until you've seen n% of the docs that you're interested in 
I doubt this will ever be added as a "regular search" feature for performance reasons. E.g. asking for docs above 30% score == return the top 70% of the results. Unfortunately, if you have 1 billion documents, that means we return 700 million documents in a single request