  Usually, we assume that there is no inhomogeneity isotropic in terms of our
location in the universe. This assumption has not been observationally
confirmed yet in sufficient accuracy and we need to consider a method to
restrict isotropic inhomogeneities more strongly. If there are isotropic
inhomogeneities in the universe, the gravitational wave background is induced
by nonlinear interactions between isotropic inhomogeneities and anisotropic
inhomogeneous density perturbations. In this paper, we calculate the power
spectrum and the relative energy density of this gravitational wave background
to discuss observability. We show that, at the decoupling time, the relative
energy density of this gravitational wave background is comparable to that of
the primordial inflationary gravitational wave background whose
tensor-to-scalar ratio is 0.001, if 6%-level isotropic deviations from
homogeneity exist. Therefore, there are possibilities of observing this
gravitational wave background and restricting isotropic inhomogeneities by
using the future cosmic microwave background observation project.
