Noise immediately after engine start

[franic]

I have an automatic Yaris sedan 2007 with 64500 miles. Every morning when I start the engine I can hear a chirp noise immediately after the engine start (just for a second). I’ve already replaced the water pump but the problem persists. I made a test removing the serpentine belt and start the engine without that belt and the noise continue. Browsing on the internet I found a video with the same exact problem http://www.youtube.com/watch?v=9a9wYUKvako . The problem is only when the engine is off for more than 8 or 10 hrs. When I start the engine less than 2 or 3 hrs after the last on period I did not hear the sound. Did anyone has experience that? Help is appreciate