I found the following bound for $X \sim \mathcal{N}(\mu, \sigma^2)$:
$$\mathbb{P}(X-\mu\geq t) \leq \exp\left(-\frac{t^2}{2\sigma^2}\right).$$ $$\mathbb{P}(|X-\mu|\geq t) \leq 2 \exp\left(-\frac{t^2}{2\sigma^2}\right).$$
Do you know if this bound has a name? I use it in my thesis and I need name or some kind of source. I didn't found any unfortunately, so maybe you can help me. (I do not need a proof.)