I suggest "Islamic spread".
In fact, with some little adjustment, your map fits like Abbasid Caliphate + Ottoman Empire + some other historic islamic states (like Sokoto Caliphate in Nigeria, Kingdom of Baguirmi in Chad, Songhai Empire in Mali).
Maybe "the ummah" is the better name but, as you told, it isn't a commonly known word. "Islamic spread" is vague as the previous but is easy to understand and to find.
* Here you can see a map of Abbasid Caliphate (http://en.wikipedia.org/wiki/File:Abbasid_Caliphate_most_extant.png
* This is Ottoman Empire (http://en.wikipedia.org/wiki/File:OttomanEmpireIn1683.png