What did the Prophets hope to accomplish by activating the Halo array?
This question will be spoiling key plot details from the prequel novel Halo: Contact Harvest, as well as some story elements from the Halo video game trilogy.
First, a little background. Contact Harvest takes place a few decades before the events of the main Halo trilogy. At this point in the timeline, humanity has colonized many planets, but it is in the midst of the civil war, where a faction known as "Insurrectionists" want independence from the UNSC. At the same time, the UNSC has noticed numerous human spacecraft mysteriously disappearing near the outer farming colony Harvest. The UNSC initially believes this activity to be the responsibility of the Insurrectionists, which results in the UNSC deploying Marines to Harvest in order to train a militia for the purpose of the colony. Unbeknownst to the UNSC, at first, is that the aforementioned freighters have actually been intercepted by Covenant vessels (commanded and operated by <PERSON>, a.k.a. Jackals). The situation quickly escalates to eventually begin humanity's desperate war against the Covenant. [SPOILERS begin here]
! The Covenant have been strongly interested in the technology of the Forerunners, an ancient race of aliens that ruled the galaxy millennia before. The Covenant are devoted followers of a religion that reveres the Forerunners, believing that they ascended into godhood through the construction, and subsequent activation, of several long-lost Halo rings located throughout the galaxy; and believe that they, too, can follow in Forerunner's footsteps by activating the Halo Array. The Covnenant refer to this as "The Great Journey."
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! The aforementioned Kig-Yar scavengers responsible for the disappearing human vessels were conducting searches for Forerunner relics, utilizing a Covenant device known as a "Luminary," which they reverse-engineered from Forerunner technology for this exact purpose. 46
The Luminary would automatically mark any humans that it detected with glyphs, and the Covenant mistakenly thought that the glyphs indicated "Reclamations" (in other words, relics). The Luminary notes the presence of thousands on Harvest, hence their interest in the colony.
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! Three Covenant leaders (each a member of the San 'Shyuum species, a.k.a. Prophets) - the Minister of Fortitude, the Vice-Minister of Tranquility and the Philologist - are finally informed by the long-dormant Oracle/Monitor of the Forerunner dreadnought Mendicant Bias that they have actually been misinterpreting the Luminaries, and other Forerunner technology, for a long time, stating that they have actually discovered "Reclaimers" (humans) rather than "Reclamations."
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! From what the Oracle tells them, the three San'Shyuum conclude that the humans are the living descendants of the Forerunners, which strongly contradicts virtually everything that their religion says about the Forerunners' fate. The Covenant had long believed that the reason they never discovered the remains of any of the Forerunners was that they had already ascended into godhood thanks to the Great Journey. Fearing that this information would be severely damaging to the very pious Covenant society, the three San'Shyuum quickly proceed with a conspiracy to take over the throne and to initiate a military campaign to exterminate humanity. Thus beginning the Human-Covenant War.
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! Part of the <PERSON>' plan in the war involved deceiving the rest of the Covenant about the nature of humanity and the humans' relation to the Forerunners. They convinced the Covenant that the humans were a blasphemous species that had to be destroyed, and that the reason that no Forerunner relics were found wherever humans were found (as the Luminaries would always mark the humans with the glyphs that they mistranslated a "Reclamations") was because the humans were purposefully destroying the relics to prevent the Covenant from obtaining them. !
! From this, we can establish that A) The Prophets are well aware that their religious beliefs are actually not true, and B) They are undoubtedly aware that activating the Halo Array will not send them on the Great Journey as they previously thought it would. Despite this, they still try to activate Halo during the events of both Halo 2 and Halo 3. I'm skeptical that they actually knew what activating the rings would accomplish, since it would have resulted in the deaths of all sentient life in the galaxy, and their attempt at activating them at the end of Halo 2 was clearly not a half-hearted attempt. So, had the attempt succeeded, it would have not only killed all of humanity, but also all of the Covenant, including the <PERSON> and their fleet (as they had not yet made it safely to the Ark).
So, through this grand plan of theirs, what did the Prophets believe they were going achieve by activating the Halo Array? 45