The Casimir effect for rectangular boxes has been studied for several decades. But there are still some points unclear. Recently, there are new developments related to this topic, including the demonstration of the equivalence of the regularization m
ethods and the clarification of the ambiguity in the regularization of the temperature-dependent free energy. Also, the interesting quantum spring was raised stemming from the topological Casimir effect of the helix boundary conditions. We review these developments together with the general derivation of the Casimir energy of the $p$-dimensional cavity in ($D+1$)-dimensional spacetime, paying special attention to the sign of the Casimir force in a cavity with unequal edges. In addition, we also review the Casimir piston, which is a configuration related to rectangular cavity.
We compute the string production rate at the end of inflation, using the string spectrum obtained in lss in a near-de Sitter space. Our result shows that highly excited strings are hardly produced, thus the simple slow-roll inflation alone does not offer a cosmic string production mechanism.
