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The transverse-traceless gauge and the gauge problem of second order gravitational waves

The gauge problem arises in the second order gravitational waves due to the mode mixing. Here, we introduce the transverse-traceless (TT) gauge to cosmological backgrounds, and find that if we choose the TT gauge at first order, the second order tensor mode would be gauge invariant. Analogous to the Ricci flat spacetime, the vacuum condition is the key to guarantee the existence of the TT gauge on cosmological backgrounds. When we have the vacuum condition, the Poisson gauge, the uniform curvature gauge, the synchronous gauge and the total matter gauge are all equivalent to the TT gauge. Once the vacuum condition is approximately satisfied, the Poisson gauge would reduce to the TT gauge at the same order of approximation. With the sub-horizon limit, the vacuum condition could be obtained approximately, and the Poisson gauge, the uniform curvature gauge and the synchronous gauge are all approximated TT gauge. Our findings explain several existing results in the literature and indicate that the proposed TT gauge is useful to discuss higher order gravitational waves.