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I would suggest going through the basics first. I too agree it could be something simple such as the exhaust rattling against something, a broken baffle is certainly a possibility. The exhaust is slung on rubber mounts, when the car is cold if you give it a shake you might be able to hear a broken baffle. When it's cold again with a thick glove you could also cover the exhaust (only for a moment) see if the sound stops. Anything inside the exhaust should in theory only rattle if there is airflow to move it.

I agree with Dan about the exhaust rattling against something as one possibility. It also sounds a bit like a loose baffle inside the silencer.

Your exhaust sounds like it's rattling on something. But there is a girdling noise too by the sounds of it. can you hear any noise from the air filter housing? Take the lid off and filter out. Start her up and see how she sounds there.