How much is the temperature at a place like inside the ear related to core temperature for example?
With a short search I found that ear thermometers aren't accurate, and the shown temperature might be +-1 degree different from the true temperature. You will likely need more accuracy there, so an ear thermometer isn't ideal.
If not what other physiological changes can be externally detected to know if someone's asleep?
The next thing that comes to my mind is movement. Movements are reduced during sleep (but not absent).
You could put one or more movement detectors. Good places are the limbs, and (if possible) the eyelid.
The thing is, if the person stands still without sleeping, then the results may be faulty. In addition, some people move a lot during sleep, but I think most of the times you will be able to distinguish the sleeping from the awake phase. Especially if you use an eyelid sensor.
Another thing is heart rate. During sleep heart rate falls. The decrease is usually around 10-20 (for example from 70 while resting, to 55 while sleeping) but it might me more or less.
Again there are exceptions, for example if the person drinks coffee before he sleeps, the measurement might be faulty.
Well, another thing you could do is make all the measurements at once, to decrease the chance of error
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If it's possible, it would be very helpful if, before the tests, you measured the person's temperature-movement-heart rate while knowing his state. Then, at future measurements, use the previous values to determine if the person is sleeping or awake.