I am trying to understand the arguments given in [MAGOO](https://arxiv.org/abs/hep-th/9905111) in chapter 3.4.1(Hilbert Space of String Theory). The authors give descriptions of the Hilbert space of String Theory when we consider our theory in different energy regimes. There are 4 such regimes. Since each regime is enough material for one question I will spread this in 4 different questions. Let's start here with the first regime where we have

1: **Gas of Free Gravitons($E\ll m\_s$)**

In this low energy regime we can approxiamate the states of our Hilbert space by the Fock space of gravitons in $AdS\_5\times S^5$. Then the states are the stationary wave solutions as shown in section 2.2.2 which are the solutions of the scalar KG-equation in $AdS\_{p+2}$
$$
\phi=e^{i\omega\tau}G(\theta)Y\_l(\Omega\_p)
$$
with $G(\theta)=(\sin\theta)^l(\cos\theta)^{\lambda\_{\pm}} \;\_2F\_1(a,b,c;\sin\theta)$ and
$$
a=\frac{1}{2}(l+\lambda\_{\pm}-\omega R)\\
b=\frac{1}{2}(l+\lambda\_{\pm}+\omega R)\\
c=l+\frac{1}{2}(p+1)
$$

and
$$
\lambda\_\pm=\frac{1}{2}(p+1)\pm\frac{1}{2}\sqrt{(p+1)^2+4(mR)^2}
$$
Then they argue that stationary modes are quantised "in the unit set by R" so they view the supergravity particles as confined in a box of size R(the curvature radius). Why are they confined in this box of size R?

I understand that I'm probably missing something in the derivation of this, but the next point is very confusing to me.

They also state the entropy, as a function of density of states
$$
S(E)\sim(ER)^\frac{9}{10}
$$
where did this come from? How is this entropy calculated(estimated?)