If it isn’t instantaneous, it’s so close I can’t tell the difference.
As for time between triggers, there doesn’t seem to be a minimum (either in battery or mains powered setups). The configuration choices are simply for sensitivity, and how long after the last motion detection to wait before clearing the status.
To be honest, the battery powered Fibaro motion sensor has similar configuration options, and is also near instantaneous on the default settings. I’ve got one of each, and the only meaningful difference is that the Fibaro doesn’t support humidity measurements. At their defaults, both are slightly more sensitive than my house alarm’s sensors.
I’ve made Arduino based multisensor with PIR, light, temp and humidity. PIR turnaround time is about 1 sec to see the status in Web UI. Not military class, but more than good for me. BTW PIR itself is always on and wakes Arduino with interrupt. Arduino talks to Hass via nRF24 radio.
Like @Tinkerer I own both, the Fibaro FGMS and the Aeon Labs Multisensor.
They trigger instantly and their battery consumption is minimal with halfway conservative settings.
I have had the Fibaro since end of October 2016 and the Aeotec since mid March 2017.
Both are still at 100% battery power.
The Aeotec sensor has humidity and can be powered via USB, so you might prefer that one.
But I don’t think it’s a necessity to have AC power.
The batteries are relatively expensive, though and the Aeotec sensor ships without batteries while the Fibaro includes some.