If one did not know the name of “Peter Brown” (above, photo), then it is likely that one would have a radically limited understanding of the religious landscape of Late Antiquity. Should one do any real digging, or suffer any serious historical training in this period, it would be impossible to totally avoid him, much less to fail to take notice of his work.
His work is in academia. Brown currently enjoys the title of Professor Emeritus at Princeton University, having previously been a lecturer at All Souls College in Oxford and having enjoyed a position at the University of California, Berkeley. He lectures widely, and just finished a major book. Together with Robert A. Markus, he has largely pioneered the study of the period known as “Late Antiquity” — some would say he inaugurated it (wresting it away from the narrative of decline still presided over by the ghost of Edward Gibbon). (One of Markus’ books, The End of Ancient Christianity, was dedicated to Brown.) Brown is the recipient of numerous awards (some of which pay quite substantially), and a member of many societies. He can read in more languages than most professional translators can (26, if Wikipedia is correct, and his books seem to bear this out).
As for these books of his, which bear out his familiarity with a high number of languages: they are many. His books include staple introductions to Late Antiquity, and standard surveys of Church history from A.D. 200-1000, primary syllabus texts for topics such as marriage and sexual renunciation and the cult of the saints (as well as the way that wealth might be marshaled to aid the departed), the highly-acclaimed biography of Augustine of Hippo that seems to reign as king, and his recent, magnificent book on the economic dimensions of cultural transformation –specifically religious ideas surrounding wealth and poverty– in the Roman West between A.D. 200-550. He has returned several times to at least three of his major earlier works to update them, decades later, in the light of more recent scholarship.
I noted that he lectures widely. In a paper he delivered in 2013 at Saint Vladimir’s Orthodox Seminary titled “Constantine, Eusebius, and the Future of Christianity” (audio here), Brown addressed the question:
what did Constantine himself and the Christians of his age think that the future of Christianity would be and should be? What were for them the horizons of the possible? And so, what would they settle for as the measure of success? 
In examining the matter, Brown looks at the writings of Eusebius of Caesarea, bishop there “from 313 to 339/340.” As “Eusebius put himself out to be eminently representative”, and as “he represented at least one major constellation of opinion among the bishops of the Greek East” , it is worth looking at his writings to take the pulse of Christians during the age of Constantine. In short, the answer to the above question is that
The exaltation of Christianity –not the end of paganism, not the Christianization of the entire Roman Empire– was what deeply concerned both [Eusebius and Constantine]. It was this which they hoped would last. 
For a meatier answer than this, we should return to the opening question, above, concerning Constantine and his cohort’s understanding of the future of Christianity. Amusingly, the “measure of success” in the opening question, and the “horizons of” possibility were very much “like the map which can be found on the website of Starbucks.” 
On this map, huge stretches of our planet (not least, the whole of China) are marked green. For, in these regions, Starbucks has arrived. What the map does not assert […] was that every inhabitant of China was going or soon would go to a Starbucks. The map conveys […] the world-wide extension […] of the miraculous possibility of a good cup of coffee. 
That is, what was important is that anyone anywhere have the possibility of becoming Christian — not Christian majoritarianism.
What [Eusebius] wanted was the “exaltation” of the church. His mental furniture did not include the idea of anything as grand as the eventual creation of a majoritarian Christianity through the Christianization of the Roman empire. […] Victory was enough. 
What victory was this? The victory of freedom against the tyranny of the stars, against the empire of the gods. In another piece on mirroring, I wrote about the sway of the stars, how pagans and Christians alike thought that astral overseers had apportioned the rites and customs of the various nations, to whom these overseers were also apportioned. The freedom to shake these rites and customs off, and follow another way — the Church’s presence afforded that, and her children understood her presence in any place to show that the power of the stars had been broken (Christians thought that the stars were fallen powers, given the presence of things like human sacrifice). This sign was itself liberating for the world, so it was thought by these early fourth-century Christians: it showed that, “[c]reated by free will, the customs of societies could be undone by free will.”  “It did not matter how many”  were liberated as signs of freedom in any place, only that some were, “in any corner of the world”. Indeed, it “was to conquer these powers that Christ had descended to earth” , either replacing fallen angels with good ones, or else ruling directly. “For Eusebius, the empire of the Devil was over. The gods had begun to die. […] The invisible empire of the gods had been swept away”. 
Eusebius conveys the sense that the present progress of Christianity is no more than the aftershock of a massive earthquake that had changed the substratum of the invisible world many centuries before. The progress of the Church on earth was the realization of a far greater triumph already won in heaven. 
The spread of the Church is thus the “visibilization”  on earth of this heavenly victory. “[I]n the church, the invisible persistently and triumphantly attained visibility.”  (Though Brown does not say it, it does seem that in this model, Church history, and not any other history, is sacred history — despite the fact that Church and Empire, both having their origins in God, “come from the same pool” .) Building bigger and better basilicas is desirable, for, as Constantine writes, in them “the divine power has been made clear to all”.  Persecution wreaks “de-visibilization” , but as “paganism […] was an empty shell”  after the victory of Christ, it posed no substantial threat. The eradication of paganism was not necessary, as it “was enough to “take out” the shrines of a few gods […] to show the bankruptcy of them all.”  “It was enough to humble the gods by repeating on earth Christ’s previous victory over the invisible empire of the demons.”  This is not a story about “what we would call historical causality — a chronological succession of events linked to each other by cause and effect” ; it is, rather, one about “the charged pairing of invisible and visible, and the direct flow of the one into the other.” 
And so, “to a surprisingly large extent” this was “the same view of the future as that which Constantine made his own in the years after 312” . Christians had a sense of taboo surrounding pagan sacrifices, which they were to avoid, and “it was enough for [Eusebius] and for his fellow-Christians to avoid it as much as possible. […] Eusebius expected Christians to have a healthy sense of pollution when faced with the relics of the demonic Ancien Régime. 
But the corollary of this sharp sense of pollution was that pollution was not everywhere. One did not need to get hay fever allergies every time that one read a pagan poet, attended an imperial audience, viewed a classical statue, or even when one sat with one’s fellows at the games. 
This creates a very specific ritual buffer between the two communities, pagan and Christian. Constantine “was a good Christian in the blunt manner that Eusebius and others had promoted. He avoided the “pollution” of strictly pagan rituals.” 
Protected by a strict, old-fashioned code of avoidance, he passed through murderous wars, visited ancient cities, crowded with temples, and presided over high ceremonies of state, surrounded all the time by non-Christians, but sheathed, as if by a second skin, by the knowledge that, as a Christian, he had remained untouched by “alien”, unholy things. 
This model is worth recounting here, because, during the course of the fourth century, Eusebius and Constantine’s model “very rapidly became out of date.”  The model that appears later in the fourth century –during the age of the Emperor Theodosius, in the writings of figures such as Ambrose of Milan, Prudentius and Augustine of Hippo– envisions a Christian majority, and then soon after even appeals to a Christian majority. The conceived relationship of Church to Empire –already connected in the above quote of Eusebius, if not centuries earlier in Paul of Tarsus– changes, as the Christianized Empire becomes explicitly part of sacred history.
If Robert Markus is to be believed, it was this idea of the Church and of Christian history (and sacred Empire) that Augustine rejected in his mature thought, the rejection of which saw the birth of Augustine’s invention of the category of the secular.