Crossposted on MSE: https://math.stackexchange.com/questions/4261809/volume-of-a-geodesic-ball-in-sln-son

Question: What is the volume of a geodesic ball of radius $r$ in the symmetric space $SL(n) / SO(n)$?

Context: Let $M = SL(n) / SO(n)$. We can view $M$ as the Riemannian manifold of real $n\times n$ positive definite matrices with determinant 1 endowed with the so-called affine-invariant metric: $$g_P(U, V) = \mathrm{trace}(P^{-1} U P^{-1} V)$$ for $P \in M$ and $U, V \in \mathrm{Sym}(n)$ (symmetric $n \times n$ matrices).

Attempts/what I know:

(1) Theorem 2.4 of this paper (screenshot attached) provides a formula for the volume of a geodesic ball for any noncompact symmetric space. However, I am not very familiar with symmetric space theory, so I am unable to parse what this formula says for $SL(n) / SO(n)$. Example 3.4.6 of these lecture notes give a formula for the roots $\alpha_j$ used in Theorem 2.4, and it is well-known the rank is $q = n-1$. However, I do not understand how to determine the constant $c$. I have an idea what the domain $C_0$ might be (the intersection of a sphere and a Weyl chamber--expressions for the Weyl chambers can be found), but I am not confident if this is correct.

(2) When $n=2$, we know $SL(n) / SO(n)$ is isometric to the hyperbolic plane of curvature $-1/2$. However, I am looking for a formula for general $n$.

(3) It is well-known that $SL(n) / SO(n)$ contains a totally geodesic submanifold isometric to a hyperbolic space. This gives us a lower bound on the volume of a ball. However, this is not sufficient for my purposes.

(4) $SL(n) / SO(n)$ has constant Ricci curvature, so we can the Bishop-Gromov inequality to get an upper bound on the volume. However, this is not sufficient for my purposes.