  We consider a nonminimally coupled scalar field as a potential cold dark
matter candidate. These models are natural extensions of the ultralight axion
models which are based on minimally coupled scalar fields. Such ultralight
scalar fields are motivated by string theory and, in particular, have been
studied in the context of the axiverse scenario. For a nonminimally coupled
field, the scalar-field energy density behaves as radiation at early times,
which yields a bound on the coupling constant, $\xi \lesssim 10$, from the
primordial nucleosynthesis theory. The first-order perturbations of the
nonminimally coupled field with adiabatic initial conditions cause the
gravitational potential to decay on large scales. A comparison of the
cosmological data with the theoretical matter power spectrum yields the
following constraint on the coupling constant: $\xi \lesssim 0.01$. We also
consider isocurvature modes in our analysis. We argue that a mix of adiabatic
and isocurvature initial conditions for a nonminimally coupled scalar field
might allow one to obtain the usual adiabatic CDM power spectrum.
