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I'd first try removing the auxiliary drive belt and turning every pulley and ancillary ( power steering pump , alternator etc) by hand. All should turn easily (with the exception of the AC pump) with no grittyness or wobbly pulleys- If you do find a sticky pump / alternator then it should be replaced or repaired
If the engine is stone cold you can briefly run it with no belts to see if the noise changes or has gone away, this will help show if an ancillary is the issue
Examine carefully the condition of the bottom / crank pulley as these are composite construction with a rubber element and its possible this rubber is breaking up and allowing the two metal portions to move.
Another possibility is the variable valve timing system as this does rely on oil pressure to correctly synchronise your vvti pulley on the end of the camshaft. If the valve has become blocked up then its possible the pulley is rattling about and producing the noise.
If all of this is OK then its possible the issue is with the dual mass flywheel on the other end of the engine, these are constructed in the same way I mentioned for the front pulley but are more prone to problems due to greater heat and larger forces.
If your gearstick is rattling this is another pointer to this being the problem.
if you faulty VVT then the car will feel very 'flat' in performance until over 4000rpm , so if it feels lacking in performance then this is another reason for checking out the VVT system