  Among single-field scalar-tensor theories, there is a special class called
"cuscuton," which is represented as some limiting case of k-essence in general
relativity. This theory has a remarkable feature that the number of propagating
degrees of freedom is only two in the unitary gauge in contrast to ordinary
scalar-tensor theories with three degrees of freedom. We specify a general
class of theories with the same property as the cuscuton in the context of the
beyond Horndeski theory, which we dub as the extended cuscuton. We also study
cosmological perturbations in the presence of matter in these extended cuscuton
theories.
