  We consider the supercritical bond percolation on $\mathbb Z^d$ and study the
graph distance on the percolation graph called the chemical distance. It is
well-known that there exists a deterministic constant $\mu(x)$ such that the
chemical distance $\mathcal D(0,nx)$ between two connected points $0$ and $nx$
grows like $n\mu(x)$. Garet and Marchand (Ann. Prob., 2007) proved that the
probability of the upper tail large deviation event
$\left\{n\mu(x)(1+\varepsilon)<\mathcal D(0,nx)<\infty\right\} $ decays
exponentially with respect to $n$. In this paper, we prove the existence of the
rate function for upper tail large deviation when $d\ge 3$ and $\varepsilon>0$
is small enough. Moreover, we show that for any $\varepsilon>0$, the upper tail
large deviation event is created by space-time cut-points (points that all
paths from $0$ to $nx$ must cross after a given time) that force the geodesics
to consume more time by going in a non-optimal direction or by wiggling
considerably. This enables us to express the rate function in regards to
space-time cut-points.
