Probing inflationary gravitational waves with cross-correlations: improved forecasting and validation with simulations

We present a follow-up study to the method recently proposed by Namikawa and Sherwin (2023) to probe gravitational waves using cross-correlations between two cosmic microwave background (CMB) $B$-modes and a large-scale structure tracer. We first improve on the previous forecast by including the impact of CMB component separation and find that, if the tensor-to-scalar ratio is $r=0$, we can achieve $\sigma_r\simeq3.6\times10^{-3}$ by combining upcoming experiments, i.e., LiteBIRD, CMB-S4 and the Advanced Simons Observatory. With a more futuristic experiment, we can achieve even tighter constraints on $r$ if improved delensing can be realized. Using a simulated analysis pipeline, we also explore possible biases from higher-order terms in the lensing potential, which were previously not examined in detail. We find that these bias terms are negligible compared to a detectable signal from inflationary gravitational waves. Our simulated results confirm that this new method is capable of obtaining powerful constraints on $r$. The method is immune to Gaussian Galactic foregrounds and has a different response to non-Gaussian Galactic foregrounds than the $B$-mode power spectrum, offering an independent cross-check of $r$ constraints from the standard power spectrum analysis.