Let $X_n$ be a chi-squared random variable with $n$ degrees of freedom.  What are the sharpest known lower bounds on the tails of its distribution?  Specifically, I am looking for the lower bounds in the form:

$$P(X_n-n\geq f_1(x)\sqrt{n}+g_1(x))\geq \exp(-x)$$
$$P(n-X_n\geq f_2(x)\sqrt{n}+g_2(x))\geq \exp(-x)$$

for some $f_1(x)>0$, $f_2(x)>0$, and $g_1(x)\geq0$, $g_2(x)\geq0$.

Basically, I am wondering if the lower-bound "equivalent" of the following upper bounds from [Massart and Laurent][1] (see Lemma 1 on page 1325) exist:

$$P(X_n-n\geq 2\sqrt{xn}+2x)\leq\exp(-x)$$
$$P(n-X_n\geq 2\sqrt{xn})\leq\exp(-x)$$

I tried Cramer-Chernoff Theorem (i.e. Theorem 1 in [here][2]), but the lower bound isn't sharp enough...


  [1]: https://projecteuclid.org/journals/annals-of-statistics/volume-28/issue-5/Adaptive-estimation-of-a-quadratic-functional-by-model-selection/10.1214/aos/1015957395.full
  [2]: https://web.archive.org/web/20160116161446/http://www.ifp.illinois.edu/~srikant/ECE567/Fall09/cramer-many-sources.pdf