  (abridged) The Galactic Center is one of the most promising targets for
indirect detection of dark matter with gamma rays. We investigate the
sensitivity of the upcoming Cherenkov Telescope Array (CTA) to dark matter
annihilation and decay in the Galactic Center. As the inner density profile of
the Milky Way's dark matter halo is uncertain, we study the impact of the slope
of the Galactic density profile, inwards of the Sun, on the prospects for
detecting a dark matter signal with CTA. We find that the sensitivity achieved
by CTA to annihilation signals is strongly dependent on the inner profile
slope, whereas the dependence is more mild in the case of dark matter decay.
Surprisingly, we find that the optimal choice of signal and background regions
is virtually independent of the assumed density profile. For the fiducial case
of a Navarro-Frenk-White profile, we find that CTA will be able to probe
annihilation cross sections well below the canonical thermal relic value for
dark matter masses from a few tens of GeV up to $\sim 5$ TeV for annihilation
to $\tau^{+}\tau^{-}$, and will achieve only a slightly weaker sensitivity for
annihilation to $b\bar{b}$ or $\mu^{+}\mu^{-}$. CTA will improve significantly
on current sensitivity to annihilation signals for dark matter masses above
$\sim 100$ GeV, covering parameter space that is complementary to that probed
by searches with the Fermi Large Area Telescope. The interpretation of apparent
excesses in the measured cosmic-ray electron and positron spectra as signals of
dark matter decay will also be testable with CTA observations of the Galactic
Center. We demonstrate that both for annihilation and for decay, including
spectral information for hard channels (such as $\mu^{+}\mu^{-}$ and
$\tau^{+}\tau^{-}$) leads to enhanced sensitivity for dark matter masses above
$m_{\rm DM}\sim 200$ GeV.
