Twin prime gaps

Question

I realize that because it is only conjectured that there are infinitely many twin primes, we can't say, "There will always be a twin prime between _____ and _____." Like we can for primes. Still, do we even have any conjectures regarding twin primes and their distance apart?

There seems to always be a twin prime between the square of one prime and the square of the next. Is a tighter bound conjectured to exist?

Answer 1

See the strong twin primes conjecture for the conjectured asymptotics of the number of twin primes $\le x$. If this is the case, then for any $\epsilon > 0$ there should be twin primes between $x$ and $x^{1+\epsilon}$ for large enough $x$.