A measure $\mu$ on $S^{n-1}$ is called isotropic if $$\int_{S^{n-1}} \langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$ for all $\theta\in S^{n-1}$.
I want to know if there are any known lower bounds on the $\mu$-area measure of spherical caps $$C_{\theta,t}:= \{u \in S^{n-1}: \langle\theta,u\rangle \geq t\},$$ for $t\in \mathbb{R}$ and $\theta\in S^{n-1}$, when $\mu$ is isotropic. The particular cases I am interested in are when $t=0$ or $t=\epsilon$ for some small $\epsilon>0$. For example, from this paper https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.891.1575&rep=rep1&type=pdf, one can infer that for any $\theta$, $$\mu(C_{\theta,0})\geq \mu(S^{n-1})/n.$$
But I cannot seem to find anything about spherical caps $\mu(C_{\theta,\epsilon})$. I wonder if one can use a result on the concentration of measures for this case, and show something like $\mu(C_{\theta,\epsilon})\geq \mu(S^{n-1})e^{-n \epsilon^2/2}/n$?