Density versions of the binary Goldbach problem

Let $\delta >1\mathrm{/}2$. 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 the sum of two primes in $A$. The constant $1\mathrm{/}2$ in the statement is best possible. Moreover we give an example to show that for any $\epsilon >0$ there exists a subset of the primes with relative density at least $1-\epsilon$ such that $A+A$ misses a positive proportion of even integers.