  Let $F$ be a CM field and let $(\overline{r}_{\pi,\lambda})_{\lambda}$ be the
compatible system of residual $\mathcal{G}_n$-valued representations of
$\operatorname{Gal}_{F}$ attached to a RACSDC automorphic representation $\pi$
of $\operatorname{GL}_n(\mathbb{A})$, as studied by Clozel, Harris and Taylor
and others. Under mild assumptions, we prove that the fixed-determinant
universal deformation rings attached to $\overline{r}_{\pi,\lambda}$ are
unobstructed for all places $\lambda$ in a subset of Dirichlet density $1$,
continuing the investigations of Mazur, Weston and Gamzon. During the proof, we
develop a general framework for proving unobstructedness (which could be useful
for other applications in future) and an $R=T$-theorem, relating the universal
crystalline deformation ring of $\overline{r}_{\pi,\lambda}$ and a certain
unitary fixed-type Hecke algebra.
