The $\chi^2_{n-1}$ null distribution for the sample-variance test statistic requires normality

Suppose $\{X_1,X_2,\dots,X_n\}$ is a random sample from the distribution of X with mean $\mu$ and variance $\sigma^2$. The test statistic to test $H_0:\sigma^2=\sigma_0^2$ against $H_1:\sigma^2>\sigma_0^2$ is given by $T_n=\sum(X_i-\bar X)^2/\sigma_0^2$. Then the null distribution of $T_n$ is: