(Quasi-) de Sitter solutions across dimensions and the TCC bound

In this work, we investigate the existence of string theory solutions with a $d$-dimensional (quasi-) de Sitter spacetime, for $3\le d\le 10$. Considering classical compactifications, we derive no-go theorems valid for general $d$. We use them to exclude (quasi-) de Sitter solutions for $d\ge 7$. In addition, such solutions are found unlikely to exist in $d=6,5$. For each no-go theorem, we further compute the $d$-dependent parameter $c$ of the swampland de Sitter conjecture, $M_p\frac{\mathrm{\mid }\mathrm{\nabla }V\mathrm{\mid }}{V}\ge c$. Remarkably, the TCC bound $c\ge \frac{2}{\sqrt{(d-1)(d-2)}}$ is then perfectly satisfied for $d\ge 4$, with several saturation cases. However, we observe a violation of this bound in $d=3$. We finally comment on related proposals in the literature, on the swampland distance conjecture and its decay rate, and on the so-called accelerated expansion bound.