On approximating the distribution of the distance between two random points on an $n$-sphere

Let $tauin(0,2)$ a given threshold value. What is the probability that the Euclidean distance $D$ between two points $mathbf{x}$ and $mathbf{y}$ selected uniformly at random on the $n$-sphere $mathcal{S}_n={mathbf{v}inmathbb{R}^n:|mathbf{v}|_2=1}$, is greater than or equal to $tau$?

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It is well known that the probability density function of the Euclidean distance $d$ between two points on the $n$-sphere
$mathcal{S}_n={mathbf{v}inmathbb{R}^n:|mathbf{v}|_2=1}$ is

$$f(d)=frac{Gamma(n/2) d^{n-2}}{sqrt{pi},Gamma((n-1)/2)}left(1-left(frac{d}{2}right)^2right)^{frac{n-3}{2}}~.$$

It is also well known that, as $n$ increases, $f(d)$ approaches the normal distribution $mathcal{N}left(sqrt{2}, frac{1}{2n}right)$.

However, even with this knowledge, it is not clear to me how to obtain an analytical expression of $Pr(dgetau)$ for a finite number of dimensions $ngg 1$, because it seems hard to evaluate the corresponding integral. Hence, I want to find an approximation of $Pr(dgetau)$. More precisely I want to obtain a meaningful upper bound of $Pr(dgetau)$.