If a handful of anti ship cruise missiles did end up hitting it, that could disable the facilities needed for planes to take off and land and force the ship to get back for repairs. You don't need to sink it. And if you manage to hit propulsion system with a torpedo, it will have to be towed.
Well, multiple AEGIS ship carries hundreds of interceptors (with different ranges and altitudes) and each can simultaneously guide dozens of them. That of course, if you can get close enough to launch some missiles to begin with. However, Carrier's air wing supported by AEWCS and growler will prevent your aircrafts and ships from getting into AShM release points. (So you have to fight your way through carrier's air wing.)
Also, SSK cannot keep up with nuclear powered carrier's speed. Even with a SSK that is equipped with latest lithium ion batteries, if you try to cruise at 20 knots continuously, you wont last more than 36 hours. (instead of three weeks)
To effectively threaten a carrier, you need at least one (preferably two) ultra modern SSN (like Astute, Virginia, Seawolf, or Suffren) which can cruise at 20 knots without the limitation of running out charge.
That is actually the most realistic way for a smaller navy to partially deter/deny US naval supremacy. By the very nature of subsurface warfare, a quite and modern SSN gives you an asymmetric advantage that you cannot obtain otherwise.
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