The vigor of Roman Empire began when what happened? The prince- ( is the reason when roman empire started to hire Goths as mercenaries)

I think you mean the vigor of the Roman Empire
began to decline when what happened. Although the sheer
size of the Empire caused problems, the one event which marked its eventual decline was
the decision to allow the Goths to cross over the Danube into the Empire in 376--one
hundred years before the Empire finally came to an end. At that point, the Goths had
been pushed against the Danube, the Eastern border of the Empire by the Huns. They
requested and received permission to enter the Empire, but were treated by the Romans
with great hostility and disdain. The Romans made no provision for their care or upkeep
which led to a rebellion by the Goths. During the resulting Battle of Adrianople, the
Goths defeated the Roman army, and the Eastern Emperor Valens was killed when the Goths
set fire to a farm house in which he had been carried after being wounded. Incidentally,
there is no evidence that they knew he was there or targeted him. At any rate, the loss
of that Battle made the Goths a permanent fixture in the Empire. They did fight as
mercenaries from time to time as did other Germanic Tribes, because (1) the Empire was
too large to defend and (2) competing factions within the Empire often fought with each
other and employed mercenaries to do so. So, if one had to pick a date and event, I
would say it was the Gothic crossing of 376 A.D. An excellent resource is Christopher
Kelly, The End of Empire: Attila the Hun and the Fall of
Rome.