Let's say I have a function
\begin{align} r_t(\mu)=\sup\limits_{a<\sigma<b} \Bbb E_{N(\mu, \sigma^2)}\phi_t(x) -\inf\limits_{a<\sigma<b} \Bbb E_{N(\mu, \sigma^2)}\phi_t(x) \end{align}
where $0 \leq a < \sigma_0 < b \leq +\infty$
for two-sided symmetric t-Student test \begin{align} \phi_t(x)=\mathbb {1}\{|t|>t^{\{n-1\}}({\alpha})\}(x), \end{align}
with $\alpha$ and $t^{\{n-1\}}(\alpha)$ being a significance level and the critical value, respectively.
I would like to plot this function $r_t(\mu)$. Is there a way to do that? I know that
\begin{align} \Bbb E[ \mathbb {1}\{|t|>t^{\{n-1\}}({\alpha})\}(x)]=\Bbb P(|t|>t^{\{n-1\}}({\alpha})) =\Bbb P(t>t^{\{n-1\}}({\alpha}))+\Bbb P(t<-t^{\{n-1\}}({\alpha})) \overset{symmetry}{=} 2-2\cdot\Bbb P(t<t^{\{n-1\}}({\alpha})) \end{align}
and I know that $t$ follows t-Student distribution, but how can I proceed with supremum and infimum?