i’ve got the same exact issue with my 2013 3.8 GT ‘69,000 miles’ it seems to be a metallic sound that could possibly be the timing chain or a related issue.. I’m not an expert on these cars but it’s definitely a metal rattle. looking for an answer or fix as well!
Rattle sound on cold start that lasts briefly until car warms up can be tbe timing chain or tensioners. 60K is about the recommended millage for most timing 'belts' to be changed, and so could be the tensioners. It's really a gamble. Some ppl claim they waited until the rattle lasted longer before they looked into it. Personally I would have it looked at. Change out the water pump as well since it's right there and mostly labor time.