The last item in the article on Raikou as Legendary raid boss from GoHub is a table of capture rates for throw type and quality. It features a different chart for different badges (gold, silver, bronze, or none)
It doesn’t bother to mention which badge it’s referring to, though. Is it the Rocker badge (for catching electric-type Pokémon)? The gym badge for the gym where the raid occurred? The Battle Legend badge (for winning Legendary raids)?