The following problem came up in one of my research works. Suppose that $C$ denotes the positive face of the $d$-dimensional unit sphere surface, i.e.
$$C := \{\mathbf{x} \in \mathbb{R}^d: x_1 >0,\ldots,x_d>0, \|\mathbf{x}\|_2 =1\}~.$$

I am given a fixed real number $\rho \in (0,1)$. Then, how many points can I have at maximum on $C$, such that all their pairwise inner products are bounded above by $\rho$? In other words, I am trying to fill up $C$ in a most crowded manner, such that the points are separated enough.

It seems to me that there should be some well known result in Mathematics about this, but I cannot figure it out. For my research purpose, I would be happy if the maximum number of such possible points is of the order $e^{\mathrm{poly}(d)}$, and if so, I would be grateful if someone can tell me the exact degree of this $\mathrm{poly}(d)$, assuming $\rho$ is fixed.