New Window into Stochastic Gravitational Wave Background

A stochastic gravitational wave background (SGWB) would gravitationally lens the cosmic microwave background (CMB) photons. We correct the results provided in existing literature for modifications to the CMB polarization power spectra due to lensing by gravitational waves. Weak lensing by gravitational waves distorts all four CMB power spectra; however, its effect is most striking in the mixing of power between the $E$ mode and $B$ mode of CMB polarization. This suggests the possibility of using measurements of the CMB angular power spectra to constrain the energy density (${\Omega }_{\mathrm{GW}}$) of the SGWB. Using current data sets (QUAD, WMAP, and ACT), we find that the most stringent constraints on the present ${\Omega }_{\mathrm{GW}}$ come from measurements of the angular power spectra of CMB temperature anisotropies. In the near future, more stringent bounds on ${\Omega }_{\mathrm{GW}}$ can be expected with improved upper limits on the $B$ modes of CMB polarization. Any detection of $B$ modes of CMB polarization above the expected signal from large scale structure lensing could be a signal for a SGWB.

Figure 1

The figure depicts the constraints on the spectral energy density of GW ${\Omega }_{\mathrm{GW}}$ defined in Eq. (7), provided by probes of the SGWB at different frequency bands. The red band pertains to constraints from CMB angular power spectra measurements up to $l_{\max }\simeq 3300$ for GWs sourced at redshift $z=1$ (see Fig. 2 for details.). The sensitivity of this new probe will extend to GWs with higher frequencies either for probing GWs sourced at lower redshifts or by using the higher resolution measurements of the CMB angular power spectra.

Figure 2

The constraints on ${\Omega }_{\mathrm{GW}}$ obtained from the current measurements of CMB intensity and polarization spectra for GWs sourced at redshifts $z_s=1100$, 10, and 1 (upper, middle, and bottom panel, respectively). The constraints from the same multipole range ($l_{\max }\le 3300$) constrain the energy density in relatively low frequency (small $k$) for GWs which are sourced further away (larger redshift). The projected, ultimate constraints on ${\Omega }_{\mathrm{GW}}$ are evaluated by demanding that the measurements of the lensed ${\tilde{C}}_l^{BB}$ spectra be consistent with signal predicted by the $\Lambda \mathrm{CDM}$ model.

Figure 3

These are the lensed $C_l^{BB}$ constructed by calculating the curl deflection power spectra arising from power in different bins of the tensor power spectra. In this figure, we use the constraints derived from the $\mathrm{WMAP}+\mathrm{ACT}$ likelihood analysis ($\Delta {\chi }_{\mathrm{eff}}^2=4$) for GWs sourced at $z_s=1100$ (see Fig. 2).