Here is the full list of all 25 Crystal Cosmos achievements.
Finish Level One
Finish Level Two
Finish Level Three
Finish Level Four
Finish Level Five
Finish Level Six
Finish Level Seven
Finish Level Eight
Get an S rank in Level One
Get an S rank in Level Two
Get an S rank in Level Three
Get an S rank in Level Four
Get an S rank in Level Five
Get an S rank in Level Six
Get an S rank in Level Seven
Get an S rank in Level Eight
Get Obsidious Ending
Get Alabastar Ending
Buy every upgrade
Get a 50 kill streak
Get a 100 kill streak
Get a 150 kill streak
Get a 200 kill streak
Finished Level Nine
Get an S rank in Level Nine