  We present a comprehensive derivation of linear perturbation equations for
different matter species, including photons, baryons, cold dark matter, scalar
fields, massless and massive neutrinos, in the presence of a generic conformal
coupling. Starting from the Lagrangians, we show how the conformal
transformation affects the dynamics. In particular, we discuss how to
incorporate consistently the scalar coupling in the equations of the Boltzmann
hierarchy for massive neutrinos and the subsequent fluid approximations. We use
the recently proposed K-mouflage model as an example to demonstrate the
numerical implementation of our linear perturbation equations. K-mouflage is a
new mechanism to suppress the fifth force between matter particles induced by
the scalar coupling, but in the linear regime the fifth force is unsuppressed
and can change the clustering of different matter species in different ways. We
show how the CMB, lensing potential and matter power spectra are affected by
the fifth force, and find ranges of K-mouflage parameters whose effects could
be seen observationally. We also find that the scalar coupling can have the
nontrivial effect of shifting the amplitude of the power spectra of the lensing
potential and density fluctuations in opposite directions, although both probe
the overall clustering of matter. This paper can serve as a reference for those
who work on generic coupled scalar field cosmology, or those who are interested
in the cosmological behaviour of the K-mouflage model.
