How did Galadriel knew that Gandalf was back?

[Legolas]

There's not any authoritative statement required other than the fact that Galadriel, being of the Eldar, and Gandalf, an Ainu-incarnate can communicate through thought without any ring. Had they encountered each other (and they may have) in Valinor, they could've done the same thing.

They had the ability to do it without a Ring on, whether or not the Rings had anything to do with it, but the Rings didn't because never is there any mention of the Rings as a means of communication. Nowhere do they exhibit this power, nor is it hinted at. You're looking too far into the power and importance of the Rings.

An analogy for you...
Superman can fly. Everytime we see Superman flying, he has his cape on. Can he fly without his cape? Of course. Is it stated that he can fly without his cape, and that all beings of his kind can fly, caped or uncaped? Yes. Then why would someone wonder if the cape had anything to do with it? It's logically not related, and even if it gives normal humans the ability to fly, it's pointless to Superman because he can fly anyway.

If, as you say, they are a means to communicate thought, then Galadriel could easily have sensed Gandalf...especially if he had returned as Gandalf the White! He would be shining a little brighter, so to speak, in his thoughts.

This is part of the concept of osanwe. Elves and Ainur are capable of it without any aid from physical things. Osanwe can even be strengthened in three instances (in the event that it involves Incarnates - Ainur need no strengthening):

Strengthening can be by affinity, by urgency, or by authority.
Affinity may be due to kinship; for this may increase the likeness of hröa to hröa, and so of the concerns and modes of thought of the indwelling fëar, kinship is also normally accompanied by love and sympathy. Affinity may come simply from love and friendship, which is likeness or affinity of fëa to fëa.
Urgency is imparted by great need of the "sender" (as in joy, grief or fear); and if these things are in any degree shared by the "receiver" the thought is the clearer received. Authority may also lend force to the thought of one who has a duty towards another, or of any ruler who has a right to issue commands or to seek the truth for the good of others.

[ 11:05 PM November 25, 2003: Message edited by: Legolas ]