Gauge Dependence of Gravitational Waves Generated from Scalar Perturbations

A tensor-type cosmological perturbation, defined as a transverse and traceless spatial fluctuation, is often interpreted as gravitational waves. While decoupled from the scalar-type perturbations in linear order, the tensor perturbations can be sourced from the scalar-type in nonlinear order. The tensor perturbations generated by the quadratic combination of a linear scalar-type cosmological perturbation are widely studied in the literature, but all previous studies are based on a zero-shear gauge without proper justification. Here, we show that, being second order in perturbation, such an induced tensor perturbation is generically gauge dependent. In particular, the gravitational wave power spectrum depends on the hypersurface (temporal gauge) condition taken for the linear scalar perturbation. We further show that, during the matter-dominated era, the induced tensor modes dominate over the linearly evolved primordial gravitational wave amplitude for K ≳ 10^-2(h/Mpc)even for the gauge that gives the lowest induced tensor modes with the optimistic choice of primordial gravitational waves (r = 0.1). The induced tensor modes, therefore, must be modeled correctly specific to the observational strategy for the measurement of primordial gravitational waves from large-scale structure via, for example, the parity-odd mode of weak gravitational lensing, or clustering fossils.