How to grab the attention of a new area's big names to read my work?
It's generally acceptable to e-mail senior scientists a copy of your paper. Say who you are and who your advisor is, describe your paper very briefly, and tell them that if they have any comments you'd be grateful to hear them.

Most will ignore you, don't take it personally -- it's not something that will "count against you", unless you do it too often, or to people who's research has nothing to do with your paper. Usually this just means they're too busy or have nothing in particular to say. But you might get a couple of interesting and/or supportive replies. I did, when I was a grad student.