If $A\subseteq\mathbb{N}$ is recursively enumerable, then there is a $\Delta^0_0$ set $B\subseteq\mathbb{N}^2$ such that $A=\{x|\exists y\;(x,y)\in B\}$. $\Delta^0_0$ consists of exactly the sets in the linear time hierarchy. Are there weaker complexity classes like $L$, $NL$, or some finite level of the linear time hierarchy, that have the same relation to recursively enumerable sets? What about complexity classes that are independent of $LTH$, like (probably) $P$? Of course, I know we couldn't have a proof that the answers are no for these examples, since we don't even know that $L\subsetneq LTH$ or $LTH\nsubseteq P$, but conceivably some widely-believed conjecture could be shown to imply a negative answer.

In general, is there a known weakest natural complexity class with that property? (I don't actually know exactly what I mean by "natural" complexity class, but if arbitrary classes of sets of tuples of integers are considered as complexity classes, then it seems like the answer would inevitably be no for boring reasons.)