We make a perturbative analysis of the number of degrees of freedom in a large class of metric theories respecting spatial symmetries, of which the Lagrangian includes kinetic terms of both the spatial metric and the lapse function. We show that, as long as the kinetic terms are degenerate, the theory propagates a single scalar mode at the linear order in perturbations around a Friedmann-Robertson-Walker background. Nevertheless, an unwanted mode will reappear pathologically, either at nonlinear orders around the Friedmann-Robertson-Walker background, or at linear order around an inhomogeneous background. In both cases, it turns out that a consistency condition has to be imposed in order to remove the unwanted mode. This perturbative approach provides an alternative and also complementary point of view of the conditions derived in a Hamiltonian analysis. We also discuss the relation under field redefinitions, between theories with and without the time derivative of the lapse function. *

We investigate the inhomogeneous inflation, in which the space exponentially expands with inhomogeneities, and its cosmological perturbations. The inhomogeneous inflation is realized by introducing scalar fields with spacelike gradients that break the spatial symmetry. We find that the space can expand uniformly in different direction with the same rate. By using the perturbative method, we calculate the corrections to the power spectra of gravitational waves and curvature perturbation up to the linear order in the background inhomogeneities. Since the background is inhomogeneous, perturbations modes with different wave numbers get correlated. We show that generally the power spectra of perturbations depend on the ratio and the angle of wave numbers of the two correlated modes. In particular, the two circular polarization modes of the gravitational waves gain different powers when the background inhomogeneity is of vector or tensor type.