Density versions of the binary Goldbach problem

Let δ > 1/2. We prove that if A is a subset of the primes such that the relative density of A in every reduced residue class is at least δ, then almost all even integers can be written as the sum of two primes in A. The constant $1/2$ in the statement is best possible. Moreover we give an example to show that for any ε > 0 there exists a subset of the primes with relative density at least 1 - ε such that $A+A$ misses a positive proportion of even integers.