Gauge Invariant Second Order Gravitational Waves

Abstract

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.

Major revisions. We further analyze the relation between the gauge invariant synchronous gravitational waves and the gravitational waves fixed in the Newtonian gauge. In this way, we could explain why one obtained correctly the energy density spectrum from the Newtonian gauge in previous studies (our method is available to other gauge fixings). Additional citations are added. Typos are corrected

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