  We show that there is no $2^{o(k^2)} n^{O(1)}$ time algorithm for Independent
Set on $n$-vertex graphs with rank-width $k$, unless the Exponential Time
Hypothesis (ETH) fails. Our lower bound matches the $2^{O(k^2)} n^{O(1)}$ time
algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010]
and it answers the open question of Bergougnoux and Kant\'{e} [SIAM J. Discret.
Math., 2021]. We also show that the known $2^{O(k^2)} n^{O(1)}$ time algorithms
for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set
parameterized by rank-width $k$ are optimal assuming ETH. Our results are the
first tight ETH lower bounds parameterized by rank-width that do not follow
directly from lower bounds for $n$-vertex graphs.
