The density of ramified primes

Let F be a number field, O be a domain with fraction field K of characteristic zero and ρ : Gal(F/F) → GL_n (O) be a representation such that ρ ⊗ K is semisimple. If O admits a finite monomorphism from a power series ring with coefficients in a p-adic integer ring (resp. O is an affinoid algebra over a p-adic number field) and ρ is continuous with respect to the maximal ideal adic topology (resp. the Banach algebra topology), then we prove that the set of ramified primes of ρ is of density zero. If O is a complete local Noetherian ring over ℤ_p with finite residue field of characteristic p, ρ is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of ρ form a Zariski-dense subset of SpecO, then we show that the set of ramified primes of ρ is of density zero. These results are analogues, in the context of big Galois representations, of a result of Khare and Rajan, and are proved relying on their result.