I think Libby's secret has something to do with Oxford. She supposedly works there as a researcher now, right? I have a feeling she's no longer there. When Libby and Denise were talking on the bench outside, Denise asked her about Oxford. Libby looked a little anxious and not especially convincing when she said she liked it there. Then Denise went onto say how proud she was to have such an intelligent daughter.
This made me think that perhaps Libby was falsifying research findings, or maybe she was dismissed due to plagarism or something. I'm not entirely sure what that would mean for Denise, unless her alcoholism is going to be revisited. Libby is her pride and joy, so finding this out could lead to Denise turning to booze for comfort again.