My Elise was making the same noise until just a few minutes ago. I just fixed it by tightening up two of my rear diffuser bolts. My diffuser bolts looked and felt tight, but they were not. The diffuser drops down as the bolts loosen and there is Locktite on the threads, so the bolts may not be as tight as they seem. Push up on the diffuser about 6 inches or so past the exhaust pipe (towards the front of the car) and look at the bolts to the right and left. The bolts have large washers, so they are easy to recognize. There should not be any play between the diffuser and the washers. If the bolts are loose, the diffuser panel will rattle against the washers when the engine vibrates during a cold start. Tighten the bolts and apply Locktite as needed.