  We provide a sublinear bound on the cop throttling number of a connected
graph. Related to the graph searching game Cops and Robbers, the cop throttling
number, written $\mathrm{th}_c(G)$, is given by
$\mathrm{th}_c(G)=\min_k\{k+\mathrm{capt}_k(G)\}$, in which
$\mathrm{capt}_k(G)$ is the $k$-capture time, or the length of a game of Cops
and Robbers with $k$ cops on the graph $G$, assuming both players play
optimally.
  No general sublinear bound was known on the cop throttling number of a
connected graph. Towards a question asked by Breen et al., we prove that
$\mathrm{th}_c(G)\leq \frac{(2+o(1))n\sqrt{W(\log(n))}}{\sqrt{\log(n)}},$ where
$W=W(x)$ is the Lambert W function.
