If we shift the set $\mathbb P=\{p_1,...,p_n,...\}$ of all prime numbers by some natural number $2a$ to obtain a set $\mathbb P+2a=\{p_1+2a,...,p_n+2a,...\}$ then I expect that $\mathbb P +2a$ contains an infinite number of prime numbers and that it contains an infinite number of composite numbers.

I would like to know are there any conjectures about density of primes in the set $\mathbb P+2a$, that is, what is known about the limit $$\lim_{n \to + \infty} \dfrac{nop\{p_1+2a,...,p_n+2a\}}{n}$$, where $nop$ stands for the "number of primes", that is, if $S$ is any set then $nop(S)$ gives as a number of primes in the set $S$.

We can denote $\lim_{n \to + \infty} \dfrac{nop\{p_1+2a,...,p_n+2a\}}{n}=f(a)$, and, I would also like to know what is known about $f$, for example, is it reasonable to expext that $f$ is a constant function?

More particularly, is there any evidence that we could have $f(a)=0$ for every $a \in \mathbb N$, that is, that shifting of the set of primes by some even number $2a$ gives us a set where "almost all" numbers are composite numbers?

More generally, I am also interested in every conjecture about this topic that you know of.