CMB lens sample covariance and consistency relations

Gravitational lensing information from the two and higher point statistics of the cosmic microwave background (CMB) temperature and polarization fields are intrinsically correlated because they are lensed by the same realization of structure between last scattering and observation. Using an analytic model for lens sample covariance, we show that there is one mode, separately measurable in the lensed CMB power spectra and lensing reconstruction, that carries most of this correlation. Once these measurements become lens sample variance dominated, this mode should provide a useful consistency check between the observables that is largely free of sampling and cosmological parameter errors. Violations of consistency could indicate systematic errors in the data and lens reconstruction or new physics at last scattering, any of which could bias cosmological inferences and delensing for gravitational waves. A second mode provides a weaker consistency check for a spatially flat universe. Our analysis isolates the additional information supplied by lensing in a model-independent manner but is also useful for understanding and forecasting CMB cosmological parameter errors in the extended $\mathrm{\Lambda }$ cold dark matter parameter space of dark energy, curvature, and massive neutrinos. We introduce and test a simple but accurate forecasting technique for this purpose that neither double counts lensing information nor neglects lensing in the observables.

Figure 1

Comparison of the lensing potential power spectra $C_{\ell }^{\varphi \varphi }$ (solid) with the reconstruction noise forecast in this work (dashed; see the text for details). The forecast is lens sample variance limited for $\ell \lesssim 10^3$.

Figure 2

Correlation matrix $R_{{\ell }_{XY},{\ell }_{\varphi \varphi }}^{XY,\varphi \varphi }$ (8) between the $C_{\ell }^{XY}$ CMB power spectra and the power spectra of the reconstructed lensing potential $C_{\ell }^{\varphi \varphi }$. Barely visible features for ${\ell }_{XY}={\ell }_{\varphi \varphi }\lesssim 50$ in the first three panels represent contributions from the Gaussian terms due to nonzero $C_{\ell }^{T\varphi },C_{\ell }^{E\varphi }$.

Figure 3

Forecasts for two parameter extensions to $\mathrm{\Lambda }\mathrm{CDM}$: $w\text{−}\sum m_{\nu }$ (left) and ${\mathrm{\Omega }}_K\text{−}\sum m_{\nu }$ (right). Black curves show $\mathrm{\Delta }{\chi }^2=1$ constraints considering the full covariance (solid) and with covariances ${\mathrm{Cov}}^{XY,\varphi \varphi }$ neglected (dashed); $\mathrm{\Lambda }\mathrm{CDM}$ parameters are marginalized over. The blue curves show the same constraints with $\mathrm{\Lambda }\mathrm{CDM}$ parameters fixed to their fiducial values.

Figure 4

Accuracy of the independent lensing information model of (17) (red dashed) compared with the full Fisher forecast for the cases from Fig. 3 (black solid).

Figure 5

KL components of the lensing potential most affected by the covariances ${\mathrm{Cov}}^{XY,\varphi \varphi }$ of CMB fields with the reconstructed lensing potential. By neglecting these covariances, constraints on the corresponding amplitude ${\mathrm{\Psi }}^{(k)}$ would be overly optimistic due to double counting of lensing information.

Figure 6

Five principal components $K_{\ell }^{(i)}$ of the lensing potential best measured by the lensed power spectra.

Figure 7

Impact of eliminating the lens information associated with the KL consistency mode ${\mathrm{\Psi }}^{(1)}$ (dashed line). Solid lines represent the full Fisher forecast from Fig. 3. The consistency mode carries a substantial amount of the total information, especially in cases where low $\ell$ lens information breaks parameter degeneracies.

Figure 8

Derivatives of $\ln C_{\ell }^{\varphi \varphi }$ with respect to cosmological parameters $w,\sum m_{\nu },{\mathrm{\Omega }}_K,{\mathrm{\Omega }}_c{h}^2,\ln A_s$ normalized at $\ell =1000$ to highlight degeneracies. These derivatives are taken at fixed acoustic scale $\theta$.

Figure 9

New SLA forecasting approximation (solid blue) compared with the full forecast (black solid) and the frequently used approach of using the unlensed spectra and $\varphi$ lensing power spectrum (green dashed). The unlensed approach is simple but errs in assuming the unlensed spectra are directly observable. The SLA approach employs the lensed CMB spectra but omits both their lensing information and covariances, making it both accurate and simple.