Gauge Invariant Second Order Gravitational Waves
  We investigate the gauge invariance of the second order gravitational waves
induced by the first order scalar perturbations by following the Lie derivative
method. It is shown explicitly that the second order gravitational waves are
gauge invariant in the synchronous frame. In the gauge invariant framework, we
derive the equation of motion of the second order gravitational waves and show
that the second order gravitational waves are sourced from the first order
scalar perturbations described well in the gauge invariant Newtonian frame.
Since the observables of gravitational waves are measured in the synchronous
frame, we define the energy density spectrum of the second order gravitational
waves in terms of the gauge invariant synchronous variables. This way
guarantees no fictitious tensor perturbations. It is shown that the gauge
invariant energy density spectrum of the second order gravitational waves
coincides with the one in the Newtonian gauge.
