nt.number theory - Primes p for which p - 1 has a large prime factor

What are the best known density results and conjectures for primes p where p - 1 has a large prime factor q, where by "large" I mean something greater than $sqrt{p}$.

The most extreme case is that of a safe prime (Wikipedia entry), which is a prime p such that $(p - 1)/2$ is also a prime (the smaller prime is called a Sophie Germain prime). I believe it is conjectured (and not yet proved) that infinitely many safe primes exist, and that the density is roughly $c/log^2 n$ for some constant $c$ (as it should be from a probabilistic model).

For the more general setting, where we are interested in the density of primes p for which p - 1 has a large prime factor, the only general approach I am aware of is the prime number theorem for arithmetic progressions, and some of its strengthenings such as the Bombieri-Vinogradov theorem (conditional to the GRH), the (still open) Elliott-Halberstam conjecture, Chowla's conjecture on the first Dirichlet prime, and some partial results related to this conjecture. All of these deal with the existence of primes $p equiv a pmod q$ for arbitrary q and arbitrary a that is coprime to q.

My question: can we expect qualitatively better results for the situation where q is prime and $a = 1$? Also, I am not interested in specifying q beforehand, so the existence of a p such that there exists any large prime q dividing $p - 1$ would be great. References to existing conjectures, conditional results, and unconditional results would be greatly appreciated.