Universal behavior of the BCS energy gap

We consider the BCS energy gap $\mathrm{\Xi }(T)$ (essentially given by $\mathrm{\Xi }(T)\approx \mathrm{\Delta }(T,\sqrt{\mu })$, the BCS order parameter) at all temperatures $0\le T\le T_c$ up to the critical one, $T_c$, and show that, in the limit of weak coupling, the ratio $\mathrm{\Xi }(T)\mathrm{/}{T}_c$ is given by a universal function of the relative temperature $T\mathrm{/}{T}_c$. On the one hand, this recovers a recent result by Langmann and Triola (Phys. Rev. B 108.10 (2023)) on three-dimensional $s$-wave superconductors for temperatures bounded uniformly away from $T_c$. On the other hand, our result lifts these restrictions, as we consider arbitrary spatial dimensions $d\in \{1,2,3\}$, discuss superconductors with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of temperatures close to $T_c$.