Lens covariance effects on likelihood analyses of CMB power spectra

Non-Gaussian correlations induced in cosmic microwave background (CMB) power spectra by gravitational lensing must be included in likelihood analyses for future CMB experiments. We present a simple but accurate likelihood model which includes these correlations and use it for Markov Chain Monte Carlo parameter estimation from simulated lensed CMB maps in the context of $\mathrm{\Lambda }\mathrm{CDM}$ and extensions which include the sum of neutrino masses or the dark energy equation of state $w$. If lensing-induced covariance is not taken into account for a CMB-S4 type experiment, the errors for one combination of parameters in each case would be underestimated by more then a factor of two and lower limits on $w$ could be misestimated substantially. The frequency of falsely ruling out the true model or finding tension with other data sets would also substantially increase. Our analysis also enables a separation of lens and unlensed information from CMB power spectra, which provides for consistency tests of the model and, if combined with other such measurements, a nearly lens-sample-variance free test for systematics and new physics in the unlensed spectrum. This parameterization also leads to a simple effective likelihood that can be used to assist model building in case consistency tests of $\mathrm{\Lambda }\mathrm{CDM}$ fail.

Figure 1

Light gray circles show the correlated values of the binned power spectra $P_{1000,1300}^{BB}$ and $P_{1500,1800}^{BB}$ as determined from 2000 Lenspix simulations. Blue lines encompass regions of 68% and 95% confidence determined from these simulations. For comparison, dashed red lines show the same confidence intervals based on our theoretical model for covariance.

Figure 2

Distribution of ${\widehat{C}}_2^{BB},{\widehat{C}}_{30}^{BB}$ obtained from 2000 Lenspix simulations compared against a ${\chi }^2$ distribution with the Gaussian variance $({\sigma }_{\ell }^{BB}{)}^2$ (solid) and a normal distribution with the same variance centered on the expected mean of the data (dashed). The normal distribution becomes a good approximation for $\ell \gtrsim 30$.

Figure 3

Comparison of MCMC constraints on $\mathrm{\Lambda }\mathrm{CDM}$ parameters with analysis based on Gaussian (black curves) and non-Gaussian covariance (red shaded). Here and throughout contours enclose regions of 68% and 95% confidence intervals unless otherwise specified.

Figure 4

Histograms showing best fit values of $M$ determined from 2000 lensed CMB skies using an analytic approach (see text) for analysis based on non-Gaussian (left) and Gaussian covariance (right). The dashed red curves show the 1D posterior probability for $M$, as determined from a single CMB realization, shifted to zero for better comparison. Despite the Fisher forecast (solid blue curves) agreeing in both cases, the posterior probability does not reflect the much wider best-fit distribution in the Gaussian case.

Figure 5

Histogram of ${\chi }^2(\mathrm{\Delta }{\theta }_A)$ for the parameter deviations of the best fit from the true model (19), as determined from our simulations with Gaussian (blue) and non-Gaussian (red) covariance. For comparison, the solid line is proportional to probability density function of ${\chi }_6^2$. The long tail in the Gaussian case leads to anomalously frequent Type 1 errors where the true model is rejected at high confidence (see text).

Figure 6

Comparison of MCMC constraints on $\mathrm{\Lambda }\mathrm{CDM}+\sum m_{\nu }$ parameters with analysis based on Gaussian (black curves) and non-Gaussian covariance (red shaded).

Figure 7

Effect of the non-Gaussian covariance on constraints on a parameter combination $M^{\nu }$ (20) within $\mathrm{\Lambda }\mathrm{CDM}+\sum m_{\nu }$; the combination was chosen to maximize this effect. Solid line shows MCMC constraints with non-Gaussian covariance, dashed line with the Gaussian covariance.

Figure 8

Comparison of MCMC constraints on $\mathrm{\Lambda }\mathrm{CDM}+w$ parameters with analysis based on Gaussian (black curves) and non-Gaussian covariance (red shaded). The impact of non-Gaussian covariance is clearly apparent in constraints involving $w$.

Figure 9

Comparison of MCMC constraints on $w$ for our experimental setup; each panel represents a different $\mathrm{\Lambda }\mathrm{CDM}$ simulation of the CMB sky with that of Fig. 8 in the top left corner. The Gaussian analysis (dashed) increasingly deviates from the non-Gaussian analysis (solid) as the maximum likelihood value of $w$ decreases.

Figure 10

Effect of the non-Gaussian covariance on constraints on a parameter combination $M^w$ (21) within $\mathrm{\Lambda }\mathrm{CDM}+w$; the combination was chosen to maximize this effect locally around the fiducial model. Solid line shows MCMC constraints with non-Gaussian covariance, dashed line with the Gaussian covariance.

Figure 11

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

Figure 12

Joint posterior distribution for PCs ${\mathrm{\Theta }}^{(1–5)}$ (black) as a product of the individual posteriors from 50 independent all-sky simulations. The joint posterior is unbiased to a small fraction of the width of the distribution of a single simulation (blue) and its Fisher prediction (red dashed).

Figure 13

Correlation matrix for ${\mathrm{\Theta }}^{(i)}$ averaged over 50 MCMC analyses. Black squares represent ones on the diagonal. The tilded parameters affect only the unlensed CMB, as explained in the text.

Figure 14

MCMC constraints on the lens and unlensed parameters ${\mathrm{\Theta }}^{(i)},{\tilde{\theta }}_A$ in a typical simulation with a Gaussian (black curves) and non-Gaussian covariance (red shaded) analysis.

Figure 15

Lines show how ${\mathrm{\Theta }}^{(1)}$ and ${\mathrm{\Theta }}^{(2)}$ change when we increase $M$ (black dashed), $M^{\nu }$ (green) or $M^w$ (black solid); length of the lines is arbitrary. For comparison, in the background we show typical constraints on these two parameters in an MCMC analysis based on non-Gaussian (red) and Gaussian (blue) covariance.

Figure 16

Consistency mode ${\mathrm{\Psi }}^{(1)}$ as determined from posterior mean values in 50 simulated lensed power spectra through MCMC analysis against values determined from known realizations of the lensing potential (see text for details). The dashed line represents points where the two determinations are equal.

Figure 17

Same as Fig. 16 but for the consistency mode ${\mathrm{\Psi }}^{(2)}$.

Figure 18

Joint posterior of the consistency parameters ${\mathrm{\Psi }}^{(1)},{\mathrm{\Psi }}^{(2)}$ (black) as the product of 50 individual posterior distributions from independent all-sky simulations. Compared against the width of a single posterior (blue) there is no indication of bias with respect to the fiducial value ${\mathrm{\Psi }}^{(i)}=0$ at a fraction of the standard deviation.

Figure 19

Comparison of constraints on $\mathrm{\Lambda }\mathrm{CDM}+w$ parameters from the standard analysis (red shaded) with results of an approximate analysis based on the effective likelihood of $\{{\mathrm{\Theta }}^{(i)},{\tilde{\theta }}_A\}$ instead of the raw CMB data (black).

Figure 20

Bias $\xi$ caused by the interpolation part of the lensing algorithm and corresponding runtime for various values of the precision parameter interp_factor. Comparison of the Lenspix interpolation routine (gray squares, from right interp_factor values 2, 2.5 and 3) and our modifications (black dots, values 2, 2.5, 3, 3.5, 4, 5 and 6).

Figure 21

Lensed power spectra bias $b_{\ell }^{XY}$ for several values of $\ell$, averaged over 2000 lensed CMB simulations calculated with the precision settings used in this work (black). In red the same quantities determined from 400 lensed CMB simulations calculated with original Lenspix interpolation algorithm with $\mathtt{\text{interp}}\_\mathtt{\text{factor}}=2$. Error bars represent errors on the mean estimated from the simulated values.

Figure 22

Lensed $C_{\ell }^{BB}$ calculated with lensing potential increased by $\mathrm{\Delta }{C}_{\ell }^{\varphi \varphi }$ which corresponds to parameter shifts listed in Tab. 2 and its representation in terms of the first $N\in 0\dots 4$ PCs (top: absolute; bottom percent error between the two).

Figure 23

Dependence of ${\chi }_{\mathrm{PC},N}^2$ (c2), measure of error caused by approximating the lensing potential using first $N$ principal component, on $N$. In this case $\mathrm{\Delta }{C}_{\ell }^{\varphi \varphi }$ corresponds to parameter shifts listed in Table 2.