In the main series of Pokémon games, since Generation V, there has been a mechanic related to capturing Pokémon called a “critical capture.” When you get a critical capture, the thrown ball will make a distinct whistling sound, fall to the ground, then shake only once to determine whether the Pokémon is captured or escapes. This greatly increases one’s chances of successfully capturing a Pokémon during an encounter.
As mentioned in my answer to this question, the critical capture mechanic has been implemented in Pokémon Go as of version 0.57.2 for Android and 1.27.2 for iOS.
What is the likelihood of getting a critical capture during a Pokémon encounter? Is it affected by anything such as badges earned or species caught (similar to the main series)?
This is entirely random. There is no way to trigger a Critical Catch, and the chance isn’t based on anything to do with your progress in the game.
Source 1 (Emphasis mine):
Critical catches happen completely randomly, and it’s triggered by the game, so if you don’t see the effects described in other posts, it’s not critical.
Source 2 (emphasis mine):
Critical Catch is a mechanic in Pokemon Go. It is a random occurrence that guarantees that the Pokemon is caught
As to what exactly this chance is, that information has not been released, but as far as I can tell from all my experience in-game, and online research, this is entirely random.
The only other piece of information I have found about this is the fact that the “critical capture” is entirely client side, and doesn’t actually mean anything, other than a “faster animation”
System efficiency requires that as much work as possible is delegated to the client, and that there is minimal data transfer. Therefore it’s likely that the server only decides and reports whether the catch was successful, leaving the client to decide how to animate that result.