Robust likelihoods for inflationary gravitational waves from maps of cosmic microwave background polarization

The $B$-mode polarization of the cosmic microwave background provides a unique window into tensor perturbations from inflationary gravitational waves. Survey effects complicate the estimation and description of the power spectrum on the largest angular scales. The pixel-space likelihood yields parameter distributions without the power spectrum as an intermediate step, but it does not have the large suite of tests available to power spectral methods. Searches for primordial $B$-modes must rigorously reject and rule out contamination. Many forms of contamination vary or are uncorrelated across epochs, frequencies, surveys, or other data treatment subsets. The cross power and the power spectrum of the difference of subset maps provide approaches to reject and isolate excess variance. We develop an analogous joint pixel-space likelihood. Contamination not modeled in the likelihood produces parameter-dependent bias and complicates the interpretation of the difference map. We describe a null test that consistently weights the difference map. Excess variance should either be explicitly modeled in the covariance or be removed through reprocessing the data.

Figure 1

Mask used in the simulations here ($N_{\text{side}}=8$, giving $f_{\text{sky}}\approx 0.5$), representative of the region accessible from the Atacama, as well as masking the galaxy through WMAP’s P06 map.

Figure 2

The Monte Carlo distribution of the maximum likelihood tensor-to-scalar ratio $r$ for several scenarios of detector noise. Solid curve: the distribution of $r$ when both the map and the covariance model have $10\mu \mathrm{K}\mathrm{arcmin}$ noise, and a true input $r=0.05$. Dotted curve: the biased distribution of $r$ when the map noise is $13\mu \mathrm{K}\mathrm{arcmin}$ while the likelihood’s covariance model assumes $10\mu \mathrm{K}\mathrm{arcmin}$. Dashed curve: the distribution of $r$ with a map noise level of $13\mu \mathrm{K}\mathrm{arcmin}$, and the likelihood models the detector noise amplitude jointly with the signal. Jointly fitting for the noise recovers the input $r$ value and represents the larger uncertainty in $r$ from $13\mu \mathrm{K}\mathrm{arcmin}$ noise vs the $10\mu \mathrm{K}\mathrm{arcmin}$ reference case.

Figure 3

The bias in ${\widehat{r}}_{\mathrm{ML}}$ from detector noise that is 30% higher than modeled in the likelihood, as a function of the true $B$-mode $r$ signal in the map. Light and dark gray regions are percentile regions equivalent to $2\sigma$ and $1\sigma$, respectively, about the median (solid line). The distribution is estimated as the Monte Carlo of the maximum likelihood value of $r$ over map realizations of both signal and detector noise. The dashed line shows bias in $r$ inferred from the quadratic estimate of $r$, as the first step of the Newton-Raphson approach to the maximum likelihood (Sec. 2a). The presence of true $B$-mode signal modulates the weighting of variance that contributes to the bias.

Figure 4

$S/(S+N)$ weight per eigenmode for three values of tensor amplitude: $r=0.01$ (solid curve), $r=0.1$ (dashed curve), and $r=1$ (dotted curve). The parameter dependence of the weight produces parameter dependence in the bias from contamination.

Figure 5

The likelihood of $r_{\text{bias}}$ and $r_{\text{true}}$ from the sum and difference of maps across a split in the data. Here $r_{\text{true}}=0.05$ is in common between the maps, and each map has an independent realization of contaminating $B$-modes at the level $r_{\text{bias}}=0.1$. This represents a scenario where there is time-varying contamination with structure indistinguishable from the $B$-modes. Solid lines: The regions of 68% and 95% probability enclosed ($1\sigma$, $2\sigma$ equivalent) in the likelihood of the sum map. It traces a degeneracy between interpretation of the data as all true $B$-mode signal or all bias. Dashed lines: The analogous regions for the likelihood of the difference map. This likelihood isolates residual variance at the level $r_{\text{bias}}=0.1$. The joint likelihood of sum and difference maps determines both the bias and the target $r$ amplitude with self-consistent errors. Both posterior distributions are from a single realization of data, so the maximum likelihood is not centered on the input parameters.

Figure 6

Same layout as Fig. 5, except that instead of adding variance in the form of $B$-modes (which are in the likelihood covariance model), this simulation has detector noise 30% higher than modeled and is not explained by any free term of the covariance. Mis-specification produces a parameter-dependent bias. Here we force the likelihood of the difference map to weight consistently with the covariance in the sum map likelihood. Without reweighting, the likelihood of the difference map gives $r_{\text{bias}}<0.004$ at 95% confidence.