  In this paper, we prove a local limit theorem for the chi-square distribution
with $r > 0$ degrees of freedom and noncentrality parameter $\lambda \geq 0$.
We use it to develop refined normal approximations for the survival function.
Our maximal errors go down to an order of $r^{-2}$, which is significantly
smaller than the maximal error bounds of order $r^{-1/2}$ recently found by
Horgan & Murphy (2013) and Seri (2015). Our results allow us to drastically
reduce the number of observations required to obtain negligible errors in the
energy detection problem, from $250$, as recommended in the seminal work of
Urkowitz (1967), to only $8$ here with our new approximations. We also obtain
an upper bound on several probability metrics between the central and
noncentral chi-square distributions and the standard normal distribution, and
we obtain an approximation for the median that improves the lower bound
previously obtained by Robert (1990).
