Try a new fanbelt and cleaning the pulley wheel on the alternator and bottom pully on engine that drives it.
It is worse when it is cold because it is putting back current used to start the car so under more load, if not that it may well just be noisy bearings but these are more of a chiiinging type noise, like a musical chain is the best way to describe it.
You could even try temporarily disconnecting the alternator when you start it first thing to see if it stops the noise then you know for sure.
tabetha