Im a gothic guy, that loves this kind of stuff, great work me likes, might try to give this challenege a try.

More AA! use the 2 medium tones, and make it more progressive (longer on the near horizontal/vertical parts). Also the inside of the bright ring.

Edit: edit  X)

Smoothed the circles (a bit jaggy and not symmetric), and refined AA (not perfect yet).
Another option would be to dither, so that the colors (pretend to) blend into each other and into the dark bg, and make the rainbow look less solid.

Nice colors btw.

 Edited with aa Cure.Thanks for the tip.

{BR Amei legend!

some aa around the lightest ring of the rainbow would smooth out those jaggies