Cross-Correlating Probes of Primordial Gravitational Waves

TL;DR

The paper investigates how primordial gravitational waves imprint B-modes in both the CMB polarization and the curl mode of cosmic shear, and it evaluates the utility of cross-correlating these probes. It derives semi-analytic expressions for the GW-induced B-mode auto-spectra in polarization and in lensing, and introduces a cross-correlation framework with a correlation coefficient $\alpha_l$ that depends on redshift and angular scale. A Fisher-matrix analysis shows that lensing can modestly improve GW amplitude constraints only if the cross-correlation is small; more importantly, cross-correlation can strengthen the case for detection by providing an independent, systematics-resilient check. The work suggests that all-sky lensing surveys could be valuable for confirming a CMB B-mode detection, even if they rarely sharpen amplitude bounds themselves.

Abstract

One of the most promising ways of detecting primordial gravitational waves generated during inflation is to observe B-modes of polarization, generated by Thomson scattering after reionization, in the cosmic microwave background (CMB). Large scale foregrounds though are expected to be a major systematic issue, so -- in the event of a tentative detection -- an independent confirmation of large scale gravitational waves would be most welcome. Previous authors have suggested searching for the analogous mode of cosmic shear in weak lensing surveys but have shown that the signal to noise of this mode is marginal at best. This argument is reconsidered here, accounting for the cross-correlations of the polarization and lensing B-modes. A lensing survey can potentially strengthen the argument for a detection of primordial gravitational waves, although it is unlikely to help constrain the amplitude of the signal.

Figures (8)

• Figure 1: The double integral which determines the B-mode signal due to reionization. The outer integral (Eq. (\ref{['eq:qpmu']})) is along the line of sight $D\hat{n}$ out to the time/distance of reionization, $D_{\rm reion}$. The quadrupole in the integrand is determined by an integral along $\Delta D'\hat{n}'$ (Eq. (\ref{['eq:quad']})) out to the surface of last scattering (a distance $\Delta D$ away) to capture the contribution of the gravitational waves.
• Figure 2: The $m=2$ moment of the B-mode transfer function in polarization for $l=2-6$ as a function of wavenumber $k$. Here reionization is assumed to occur instantaneously at $z=10$.
• Figure 3: The $m=2$ moment of the B-mode lensing transfer function for $l=2-6$. Sources are all assumed to be at $z=1$.
• Figure 4: Cross-correlation window function for lowest moments when lensing galaxies are all at $z=1$ and reionization occurs instantaneously at $z_{\rm reion}=10$.
• Figure 5: Cross-correlation coefficient of CMB and weak lensing B-modes for the lowest two moments as a function of the redshift of the galaxies in the lensing survey.