  The energy density spectrum is an observable of gravitational waves.
Divergence has appeared in the energy density spectra of the scalar induced
gravitational waves for different gauge fixings. To resolve the discrepancy, we
investigate the gauge invariance of the scalar induced gravitational waves. It
is shown that the gauge invariant induced gravitational waves can be obtained
by subtracting the fictitious tensor perturbations via introducing the counter
term composed of the first order scalar perturbations. The kernel function
uniquely determines the energy density spectrum of the scalar induced
gravitational waves. We explicitly calculate the gauge invariant kernel
functions in the Newtonian gauge and the uniform density gauge, respectively.
The discrepancy between the energy density spectra upon the Newtonian gauge and
the uniform density gauge is shown to be eliminated in the gauge invariant
framework. In fact, the gauge invariant approach is also available to other
kinds of gauge fixings.
