We study the noise correlations of one-dimensional binary Bose mixtures, as a
probe of their quantum phases. In previous work, we found a rich structure of
many-body phases in such mixtures, such as paired and counterflow
superfluidity. Here we investigate the signature of these phases in the noise
correlations of the atomic cloud after time-of-flight expansion, using both
Luttinger liquid theory and the time-evolving block decimation (TEBD) method.
We find that paired and counterflow superfluidity exhibit distinctive features
in the noise spectra. We treat both extended and inhomogeneous systems, and our
numerical work shows that the essential physics of the extended systems is
present in the trapped-atom systems of current experimental interest. For
paired and counterflow superfluid phases, we suggest methods for extracting
Luttinger parameters from noise correlation spectroscopy.