Odd-parity bipolar spherical harmonics

Bipolar spherical harmonics (BiPoSHs) provide a general formalism for quantifying departures in the cosmic microwave background (CMB) from statistical isotropy and from Gaussianity. However, prior work has focused only on BiPoSHs with even parity. Here we show that there is another set of BiPoSHs with odd parity, and we explore their cosmological applications. We describe systematic artifacts in a CMB map that could be sought by measurement of these odd-parity BiPoSH modes. These BiPoSH modes may also be produced cosmologically through lensing by gravitational waves (GWs), among other sources. We derive expressions for the BiPoSH modes induced by the weak lensing of both scalar and tensor perturbations. We then investigate the possibility of detecting parity-breaking physics, such as chiral GWs, by cross-correlating opposite-parity BiPoSH modes with multipole moments of the CMB polarization. We find that the expected signal-to-noise of such a detection is modest.

Figure 1

Here we plot the autocorrelation power spectrum $C_L^{\varphi \varphi }$ of the gradient-type $\varphi$ modes of cosmic shear. In green squares we show the autocorrelation of the $\varphi$ modes from lensing by scalar perturbations, and in blue circles that of the $\varphi$ modes induced by tensor perturbations. We use the WMAP-7 cosmological parameters, and assume the maximum allowable tensor-to-scalar ratio $r=0.24$ from the WMAP-7 data combined with BAO and the $H_0$ measurement [42], to calculate the tensor contribution. The error with which these power spectra could be measured using the parameters of the Planck satellite is shown as red $+$s.

Figure 2

Here we plot the autocorrelation power spectrum $C_L^{\Omega \Omega }$ of the curl-type $\Omega$ modes of the weak lensing of the CMB temperature field. These modes can only be induced by tensor perturbations. We show the signal in blue circles and the error with which they could be measured using the parameters of the Planck satellite as red $+$s.

Figure 3

Here we plot the cross-correlation $C_L^{\varphi B}$ between the gradient $\varphi$ modes of the weak lensing of cosmic shear with the curl-type $B$ modes of the CMB-polarization in blue circles, and the noise on this measurement due to cosmic variance and Planck satellite instrumental noise in red $+$s. Since these quantities are of opposite parity, in the absence of parity-breaking physics we expect this cross-correlation to vanish. However, if we assume, for example, that the entire allowable GW background is right-circularly-polarized, such a cross-correlation could occur. The cross-correlation is linearly proportional to the chirality parameter $A$, defined such that $A=1$ denotes a completely right-circularly-polarized GW background, $A=-1$ denotes completely left-circularly-polarized, and $A=0$ denotes an unpolarized background. Here we assume the maximum allowable tensor-to-scalar ratio $r=0.24$, the limit from WMAP-7 data combined with BAO and the $H_0$ measurement [42]. Cusps in the absolute value of the correlation function correspond to sign changes of the correlation function.

Figure 4

Here we plot the cross-correlation $C_L^{\Omega E}$ between the curl-type $\Omega$ modes of cosmic shear with the gradient-type $E$-modes of the CMB-polarization in blue circles, and the noise on this measurement due to cosmic variance and Planck-satellite instrumental noise in red $+$s. As with the $\varphi -B$ correlation, we assume a completely right-circularly-polarized GW background, with the maximum currently permitted tensor-to-scalar ratio.