  We propose a method to study and constrain modified gravity theories for dark
matter using CMB temperature anisotropies and polarization. We assume that the
theories considered here have already passed the matter power-spectrum test of
large-scale structure. With this requirement met, we show that a modified
gravity theory can be specified by parametrizing the time evolution of its
dark-matter density contrast, which is completely controlled by the dark matter
stress history. We calculate how the stress history with a given
parametrization affects the CMB observables, and a qualitative discussion of
the physical effects involved is supplemented with numerical examples. It is
found that, in general, alternative gravity theories can be efficiently
constrained by the CMB temperature and polarization spectra. There exist,
however, special cases where modified gravity cannot be distinguished from the
CDM model even by using both CMB and matter power spectrum observations, nor
can they be efficiently restricted by other observables in perturbed
cosmologies. Our results show how the stress properties of dark matter, which
determine the evolutions of both density perturbations and the gravitational
potential, can be effectively investigated using just the general conservation
equations and without assuming any specific theoretical gravitational theory
within a wide class.
