Among single-field scalar-tensor theories, there is a special class called "cuscuton,"
which is represented as general relativity with a noncanonical scalar field. This theory
has a remarkable feature that scalar perturbations do not propagate on a cosmological
background. We specify a subclass of the (beyond) Horndeski class having this
property and thus extend the framework of cuscuton. We also discuss the stability of
cosmological perturbations in the extended cuscuton models.
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