Cosmic shear from scalar-induced gravitational waves

Weak gravitational lensing by foreground density perturbations generates a gradient mode in the shear of background images. In contrast, cosmological tensor perturbations induce a nonzero curl mode associated with image rotations. In this note, we study the lensing signatures of both primordial gravitational waves from inflation and second-order gravitational waves generated from the observed spectrum of primordial density fluctuations. We derive the curl mode for galaxy lensing surveys at redshifts of 1–3 and for lensing of the cosmic microwave background at a redshift of 1100. We find that the curl mode angular power spectrum associated with secondary tensor modes for galaxy lensing surveys dominates over the corresponding signal generated by primary gravitational waves from inflation. However, both tensor contributions to the shear curl mode spectrum are below the projected noise levels of upcoming galaxy and cosmic microwave background lensing surveys and therefore are unlikely to be detectable.

Figure 1

The power spectrum of primordial (dashed line) and secondary gravitational waves at $z=3$. For the secondary spectrum, we show results from two calculations in the literature: the solid line is from Baumann et al. [13] and the dotted-dashed line from Mollerach, Harari, and Matarrese [10]. The dotted part of the spectra corresponds to superhorizon scales at $z=3$.

Figure 2

The angular power spectra of the cosmic shear curl mode at $z=1$ (left panel) and at $z=3$ (right panel). The dotted line is the noise associated with a measurement of the shear curl mode power spectrum (see text for details). While the curl mode power spectrum is below the noise, the secondary gravitational waves produce a larger signal than the primordial tensor modes when $r<0.4$. For reference, the triple-dotted–dashed line on the left panel shows the angular power spectrum of secondary shear curl modes generated by the coupling of two lenses (lens-lens coupling) along the line of sight to background sources at $z=1$ [15].

Figure 3

The angular power spectrum of the curl mode at $z=1100$ for lensing of CMB anisotropies by foreground gravitational waves. The dotted line shows the expected noise from a cosmic-variance-limited reconstruction of the curl mode following Cooray, Kamionkowski, and Caldwell [14] using $E$- and $B$-mode CMB polarization maps. For CMB lensing, the primordial tensor modes dominate when $r\gtrsim {10}^{-6}$.