Density versions of the binary Goldbach problem

Let $\delta \gt 1/2$. 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 $\delta $, then almost all even integers can be written as a 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 $\varepsilon \gt 0$ there exists a subset of the primes with relative density at least $1 - \varepsilon $ such that $A+A$ misses a positive proportion of even integers.