If I am not mistaken, the probability distitribution pdf and the cummulative probability distribution cdf of $S^2$ should be given by
pdf = σ^2/n PDF[ChiSquareDistribution[n - 1]][t/n σ^2];
cdf = CDF[ChiSquareDistribution[n - 1]][t/n σ^2];
Some wuick consistency tests for the scalings:
Integrate[pdf, {t, 0, ∞}, Assumptions -> {σ > 0, n >= 2}] == 1
Limit[cdf, t -> ∞, Assumptions -> {σ > 0, n >= 2}] == 1
D[cdf, t] == pdf // Simplify
True
True
True