I would like to have an estimate for the series $$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$ where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, $t$ represents the time and $t>0$, $m$ is a positive integer and $m>1$ (Obviously, $P(t)=1$ when $m=1$ since $P(t)$ is exactly the cdf of Poisson distribution with associated parameter $t$ in this case.).

I am interested in showing that the answer could look something like $$P(t) = O(t^{-\alpha m}),\alpha > 0.$$

It has come to my attention that in the paper “Rumors in a Network: Who's the Culprit?” D Shah and T Zaman proved that (Page 24-27) $$P(t) = O(\frac{1}{\sqrt{t}}),\text{if }m=2.$$

However, I find it difficult for me to extend their method to the cases when $m>2$.

I also tried to calculate $P(t)$ in Mathematica and it gave me the result like
$$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m = e^{-mt}\text{HypergeometricPFQ}[...,t^m]$$
It seems that the decreasing speed of $P(t)$ is much slower than that of $e^{-mt}$ as $t$ increases. But I could not find the closed-form bound when $m>2$ just like $O(1/\sqrt t)$ when $m=2$.

Does anyone know of a scale or a bound of $P(t)$ in the literature? Any comments and answers would be highly appreciated. Many thanks!