Gauge-Invariant Tensor Perturbations Induced from Baryon-CDM Relative Velocity and the B-mode Polarization of the CMB

At second-order, scalar perturbations can source traceless and transverse perturbations to the metric, called induced gravitational waves (IGW). The apparent gauge-dependence of the IGW obscures the interpretation of the stochastic gravitational-wave signal. To elucidate the gauge dependence, we study the IGW from manifestly gauge-invariant scalar perturbations, namely, the relative velocity between the baryon and cold dark matter. From this relative velocity perturbation, we compute the dimensionless gravitational wave power spectrum and the corresponding expected angular power spectrum of the B-mode polarization of the cosmic microwave background. Although the effect turns out to be unobservably small, the calculation demonstrates both the importance of using observable quantities to remove the gauge ambiguity and the observable consequences of tensor perturbations which are not propagating gravitational waves.