The Aryan Invasion Theory (AIT) says that a set of invasions of people(s) from Central Asia around 3500-3200 years ago, first introduced the Indo-European (IE) language family to India. Since the archaeological record does not show any discontinuity (e.g., as is evident in parts of Europe where an invasion did take place), sometimes AIT is moderated to AMT (Aryan Migration Theory), and instead of outright conquest and extermination, elite dominance is the mechanism by which the large native population utterly forgot its original language and place-names.
Please note, 4600-3900 years ago is the "mature period" of the Saraswati-Sindhu civilization; and per AIT/AMT this civilization is not of the IE family; and so the postulated period of invasion/migration cannot be pushed back very much in time. That is, if for example, an IE incursion occurred 4200 years ago, it would no longer be AIT/AMT. The linguistic basis of the AIT/AMT theory also suffers if the incursion is pushed back in time. For example, the Indo-Aryan words found in the written records of the Mitanni (a people of Northwestern Syria, eastern Anatolia) date to 1400 BC, and were, per AIT, part of the expansion of Indo-Aryan speakers into India. (BTW, if you examine it closely, "Indo-Aryan" is simply a fancy and obscuring word for archaic Sanskrit.)
The counter-narrative to AIT/AMT is OIT (Out-of-India Theory). OIT, I think covers a wide gamut of possibilities, but the basis of it is:
1. The Saraswati river that was an abundant river in the time of the Rg Veda, as mentioned there, is the same Saraswati river whose dried up course is described in the Mahabharata; the Mahabharata river corresponds to the dried river-bed along whose banks the vast majority of Saraswati-Sindhu civilization sites have been found - very much more than along the Indus. Hence the "Saraswati-Sindhu" rather than "Indus" civilization.
2. If the above is valid, then IE speakers were in India while the Saraswati still flowed, and that throws AIT/AMT out-of-whack. And then the initial seeds of the IE language might have been spread by migrations out of India much before 4000 years ago. Nothing precludes round-tripping, of course.
Anyway, there's also an increasing amount of genetic evidence and more to come (Indian populations have been vastly undersampled, most of the studies having people from Pakistan or Gujarat as representative of the whole of India). And according to Prem Kumar, "There is Genetic Evidence against the Aryan Invasion Myth".
The Indian population is a mixture of two ancestral populations, termed ANI and ASI. Except for some isolated tribes in the Andaman islands, everyone else in India is a mixture of these two. A diagram from one of the papers is shown below. This shows a family tree, but not the time depth. Other papers claim to establish that the Europe-ANI split is at least 12,500 years old. The ASI-ANI split is theorized to have arisen 60-55,000 years ago. The great volcanic eruption of Toba in Indonesia, some 75,000 years ago, has left deep deposits of ash in parts of India; it is theorized to have greatly depopulated India. Incursions of new peoples, some 55000 years ago led to ASI. With that background, you can read Prem Kumar's article.
Please note, 4600-3900 years ago is the "mature period" of the Saraswati-Sindhu civilization; and per AIT/AMT this civilization is not of the IE family; and so the postulated period of invasion/migration cannot be pushed back very much in time. That is, if for example, an IE incursion occurred 4200 years ago, it would no longer be AIT/AMT. The linguistic basis of the AIT/AMT theory also suffers if the incursion is pushed back in time. For example, the Indo-Aryan words found in the written records of the Mitanni (a people of Northwestern Syria, eastern Anatolia) date to 1400 BC, and were, per AIT, part of the expansion of Indo-Aryan speakers into India. (BTW, if you examine it closely, "Indo-Aryan" is simply a fancy and obscuring word for archaic Sanskrit.)
The counter-narrative to AIT/AMT is OIT (Out-of-India Theory). OIT, I think covers a wide gamut of possibilities, but the basis of it is:
1. The Saraswati river that was an abundant river in the time of the Rg Veda, as mentioned there, is the same Saraswati river whose dried up course is described in the Mahabharata; the Mahabharata river corresponds to the dried river-bed along whose banks the vast majority of Saraswati-Sindhu civilization sites have been found - very much more than along the Indus. Hence the "Saraswati-Sindhu" rather than "Indus" civilization.
2. If the above is valid, then IE speakers were in India while the Saraswati still flowed, and that throws AIT/AMT out-of-whack. And then the initial seeds of the IE language might have been spread by migrations out of India much before 4000 years ago. Nothing precludes round-tripping, of course.
Anyway, there's also an increasing amount of genetic evidence and more to come (Indian populations have been vastly undersampled, most of the studies having people from Pakistan or Gujarat as representative of the whole of India). And according to Prem Kumar, "There is Genetic Evidence against the Aryan Invasion Myth".
The Indian population is a mixture of two ancestral populations, termed ANI and ASI. Except for some isolated tribes in the Andaman islands, everyone else in India is a mixture of these two. A diagram from one of the papers is shown below. This shows a family tree, but not the time depth. Other papers claim to establish that the Europe-ANI split is at least 12,500 years old. The ASI-ANI split is theorized to have arisen 60-55,000 years ago. The great volcanic eruption of Toba in Indonesia, some 75,000 years ago, has left deep deposits of ash in parts of India; it is theorized to have greatly depopulated India. Incursions of new peoples, some 55000 years ago led to ASI. With that background, you can read Prem Kumar's article.