AHQ: lingual drift from a single language

First of all, I'm not a linguist, so I apologise in advance for any stupid assumptions I make.

starting from a PoD of middle antiquity or so, what would it take for a single language - spread out over large territories (like vulgar latin or arabic) but not unified under any single state - to fracture into visibly different languages? in addition, how much time must pass until these languages become completely non mutually-intelligible (like different branches of indo-european)? what part do political circumstances play here - would a language even fracture if the people speaking it remain unmolested?
 
It depends on how much integration and interaction there is across the whole region of the language. Limited interactions, especially before the advent of mass communications, print and then even more so radio and TV, allows for more rapid change. Easier and more frequent interaction slows change.
 
First of all, I'm not a linguist, so I apologise in advance for any stupid assumptions I make.

starting from a PoD of middle antiquity or so, what would it take for a single language - spread out over large territories (like vulgar latin or arabic) but not unified under any single state - to fracture into visibly different languages? in addition, how much time must pass until these languages become completely non mutually-intelligible (like different branches of indo-european)? what part do political circumstances play here - would a language even fracture if the people speaking it remain unmolested?

I'm very much an amateur linguist at best myself but in lieu of someone else more qualified responding I'll take a crack.

I would say that we actually have quite a few historical examples of this. By 842 old French had diverged sufficiently from Latin to have treaties written in it (oaths of Strasbourg), albeit it had been diverging since before the Roman empire fell and the spoken language had been different before the written form caught up. I would be inclined to take that to mean 200 years is a lower bound. Shorter changes and you're probably looking at language replacement as in England in the 600s and North Africa in the 700s. A counterpoint might be Africaans, which is at least half a seperate language from Dutch and only really started separating in 1700 or so in a much more connected world. Arabic by contrast is arguably still one mutually intelligible language, although there is the whole religious element plus the fact more recently Egyptian media has tended to be fairly dominant. Maltese might be an interesting one to look at - Arabic descendant, definitely not mutually intelligible.

I think what you need to ask is how much contact do the extremes have with each other and the cultural centre(s). Are there multiple cultural centres which are separated? How many other language groups are they in contact with? As traders if you're wanting to avoid conflicts? How much of a literary tradition is there (slowing down diveregence), or oral traditions (effect less clear)?

I would say that a people being completely unmolested politically by speakers of other languages seems unlikely if they aren't a more or less unified empire.

For numbers, this is my very unprofessional finger in the air judgement. A century or a bit more for meaningful divergence of dialects. Beyond that it all depends on mutual contact between speakers.
 
Arabic by contrast is arguably still one mutually intelligible language
In the spoken variety, it definitely is not.
Educated Moroccans and Iraqis may understand each other speaking Standard Arabic, which they would have learned at school and absorbed through the media, but in general they would not be able to do that in their respective everyday spoken languages, which are about as divergent as Romance languages if not more.
The religious and literary prestige of the Classical language and its Modern Standard derivation keeps the Arab World linguistically unified at the written and media level, but actual spoken language is different. This is somewhat changing as the Standard, with regional differences and somewhat of an Egyptian tinge, becomes more widespread thanks to modern media and increasingly widespread schooling, but still largely the case.
Educated people often mix varietes, and have been doing so for a long time.
Also, there's evidence pointing to spoken Arabic varieties diverging pretty quick, as as early as the tenth century Arab Grammarians were able to point out varieties in some areas. Scholars are not really in agreement on how to explain what happened, since we really lack vernacular written texts from early times outside some small exceptions, but there's increasing evidence pointing to "Arabic" having been a cluster of slightly different variaties as early as we can document it. Classical Arabic (and therefore Standard Arabic) would then be just the variety picked as the "official" one, in which the Qur'an was recited (it might be even more complicated than that, but nevermind) but it is possible that spoken Arabic in some places derives from variaties that have have diverged in pre-Islamic times.
Even the scant pre-Islamic evidence we have suggests huge dialectal difference within Arabic at the time.
Anyway, it is very likely that the rapid course of the Conquests led to a relatively quick linguistic replacement and change, as we find to be the case in Egypt for example, where Coptic probably turned into a minority language in about two centuries just to disappear sometime between the fourteenth and the seventeenth centuries (as the relatively low number of Coptic loanwords into spoken Egyptian Arabic would confirm; a longer period of widespread bilingualism would have led to a larger Coptic imprint). It seems that many people picked up Arabic, but at least at first, learned it imperfectly and this may have led to a partial "creolization" of the later spoken varieties. This model is supported by a prominent scholar in the field at least, but it is not universally accepted.
 
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