Where did Irish "-acht" come from?

Question

Modern Irish has a suffix -acht (allomorphs -ocht, -eacht, -cht, probably others) that forms abstract nouns. For example, beo "alive" → beocht "life, vital spirit".

Since we also see Scottish Gaelic -achd and Manx Gaelic -aght, this suffix would seem to go back to Old Gaelic at least. But I haven't been able to find anything past that.

So, what is the full ancestry of -acht (at least as far as we can reconstruct)? Does it go all the way back to PIE, and if so, do we know what PIE form led to this? It doesn't look much like the Latin or Greek abstract-noun-forming suffixes, like -or or -ia.

Answer 1

A quick look at Stair na Gaeilge yields this (in Kim McCone’s chapter An tSean-Gaeilge agus a réamhstair — “Old Irish and its prehistory”)…

21.2 … It can be seen that use is made of the suffix *-(i)yā to make abstract nouns in IE itself (e.g., Gr. p^hil-ó-s ‘beloved’, p^hil-ía ’fondness’). The -e (MW -ed) that descended from it was a common way of forming feminine abstract nouns (13.5) from adjectives in Old Irish, e.g., dílis, díl^†s-e; maith ‘good’, maithe ‘goodness’; gor ‘fervent’ (MW gwar) < *g^war-os (< *g^wr̥-), goire ‘fervour’ (MW gwared) < *g^war-iyā (7.4). -tūt- was also available (17.1) and a couple of other suffixes were developed in (Insular?) Celtic itself to derive abstract nouns from other nouns, in particular *-axtā or *-yaxtā (OI -acht or -echt, MW -(i)aeth; [*-i-+] *-ak- + *-tā) and *-assu-s (OI -as, MW -as): e.g., OI marcach, MW marchawc ‘(horse-)rider’ and OI marcaigecht, MW marchogaeth ’horsemanship’, OI tigern(ae), MW teyrn ‘lord’ and OI tigern-as, MW teyrn-as ‘lordship’.

^{Kim McCone, Damian McManus, Cathal Ó hÁinle, Nicholas Williams, Liam Breatnach, eds. (1994). Stair na Gaeilge. II:21.2 “An tSean-Gaeilge agus a réamhstair: Gnéithe de dhíorthú na substainteach”, page 127.
My translation. OI = Old Irish, MW = Middle Welsh.}

A connection to aigid seems unlikely to me.

Note that beocht may be more complicated than it looks. The older form is béodacht and there was also a form using the *-assu-s suffix mentioned above — béodas (the modern beos probably only exists in the dictionary).