Weapon Restrictions
Here are the rules in Idaho. I bet Utah is similar. You could hunt with a .357 - .38 special. rifle.
Rifle and Shotgun
In any hunt, including any-weapon seasons, it is unlawful to
pursue or kill big game animals:
• By any means other than approved firearms,
muzzleloaders and archery methods.
• With any electronic device attached to, or incorporated
on, the firearm or scope; except scopes containing
battery powered or tritium lighted reticles are allowed.
• With any firearm that, in combination with a scope, sling
and/or any attachments, weighs more than 16 pounds.
• With any fully automatic firearm.
• With any shotgun using shot smaller than #00 buck.
• With any rimfire rifle, rimfire handgun, or muzzleloading
handgun, except for mountain lion.