Vanishing lines in chromatic homotopy theory

We show that at the prime 2, for any height $h$ and any finite subgroup $G \subset \mathbb{G}_h$ of the Morava stabilizer group, the $RO(G)$-graded homotopy fixed point spectral sequence for the Lubin--Tate spectrum $E_h$ has a strong horizontal vanishing line of filtration $N_{h, G}$, a specific number depending on $h$ and $G$. It is a consequence of the nilpotence theorem that such homotopy fixed point spectral sequences all admit strong horizontal vanishing lines at some finite filtration. Here, we establish specific bounds for them. Our bounds are sharp for all the known computations of $E_h^{hG}$. Our approach involves investigating the effect of the Hill--Hopkins--Ravenel norm functor on the slice differentials. As a result, we also show that the $RO(G)$-graded slice spectral sequence for $(N_{C_2}^{G}\bar{v}_h)^{-1}BP^{(\!(G)\!)}$ shares the same horizontal vanishing line at filtration $N_{h, G}$. As an application, we utilize this vanishing line to establish a bound on the orientation order $\Theta(h, G)$, the smallest number such that the $\Theta(h, G)$-fold direct sum of any real vector bundle is $E_h^{hG}$-orientable.