When attempting to delegate SharePoint 2013 Search Administration to a small team/group, we were able to delegate everything except for the "Result Sources" and "Query Rules" pages/links, as seen in yellow highlight below.
Clicking either page resulted in the "Sorry, this site hasn't been shared with you" (Access Denied/ Access Required) message despite having done "all the right things."
If we added the delegated admin group to the farm administrators group, it all works of course. But we we're trying to taking it that far.
This was resolved in the October 2013 Cumulative Update for SP2013 (and higher). Link: http://technet.microsoft.com/en-us/sharepoint/jj891062.aspx