NoiseCovarMatrices

Introduction

The Planck low resolution frequency maps are provided with a pixel-pixel noise covariance matrix. The matrix is required for maximum likelihood analysis of CMB anisotropies in the maps but also for low resolution component separation.

The first step in calculating the noise matrices is estimating the detector noise. We do this for each detector at roughly one day intervals. The process is

1. Time-ordered information (TOI) is cleaned of signal by interpolating the full frequency, full mission map to the sample positions on the sky.
2. Estimate the auto covariance function of the TOI without samples that either have their quality flag raised or that fall within the galactic or point source mask
3. Fourier transform the auto covariance function into a power spectral density (PSD)
4. Reduce realization noise in the PSD by fitting and analytical model

The noise PSDs feed into two noise covariance estimation codes. The LFI noise covariance is estimated using a module integrated in the Madam mapmaking code. It directly estimates the posterior pixel-pixel covariance in the presence of correlated noise residuals after destriping. The HFI noise covariance approach is different because of the longer baseline (offset) used in HFI destriping. We break the HFI residual noise covariance into offset-offset (between rings) and short time scale (within ring) components. The short time scale component is readily available as the noise PSD while the offset-offset covariance requires calculating over all of the offset-offset crossing points and inverting and binning the resulting offset covariance matrix.

Noise Covariance Methods

The methods of evaluating the pixel-pixel noise covariance matrix, [math]N[/math] depend on the mapmaking method and the features of the noise PSD. LFI uses short baselines and a noise prior to reduce correlated noise up to roughly 1Hz. HFI maps are destriped with long baselines, leaving much of the correlated noise above 1mHz unchanged.

LFI

The LFI noise covariance follows from the destriping principle, where the correlated noise is approximated by correlated baseline offsets:

[math] d = Pm + Fa + n, [/math]

where the data vector, [math]d[/math], is a linear combination of the sky signal, [math]Pm[/math], baseline offsets, [math]Fa[/math], and white noise, [math]n[/math]. Matrices [math]P[/math] and [math]F[/math] scan the sky map, [math]m[/math] and the baseline offsets, [math]a[/math], into time domain.

The residual pixel-pixel noise covariance under the generalized destriping principle (MADAM) is then

[math] N^{-1}_\text{LFI} = P^T \mathcal N_w^{-1} P - P^T \mathcal N_w^{-1} F ( \mathcal N_a^{-1} + F^T\mathcal N_w^{-1}F ) F^T N_w^{-1} P^T.[/math]

The prior offset-offset covariance, [math]\mathcal N_a[/math], is evaluated from the measured noise PSD and is also used as a prior when destriping the data. The diagonal white noise covariance matrix, [math]\mathcal N_w,[/math] reflects as accurately as possible the white noise sample variance, not including the correlated noise.

HFI

The short time scale noise correlations are unaffected by the long baseline destriping used in HFI mapmaking. This means that the resulting pixel-pixel covariance can be evaluated simply by binning their time domain covariance, [math]\mathcal N_\text{in-ring}[/math], into pixel domain:

[math] N_\text{in-ring} = B \mathcal N_\text{in-ring} B^T [/math],

where [math]B[/math] is the map binning operator, typically constructed from the pointing matrix, [math]P,[/math] and the detector noise weights contained in the diagonal white noise covariance matrix, [math]\mathcal N_w[/math]:

[math] B = (P^T \mathcal N_w^{-1} P)^{-1} P^T \mathcal N_w^{-1}[/math].

In this HFI formalism, the long time scales are treated differently. We evaluate the posterior (destriped) offset-offset covariance that includes destriping errors from sky template uncertainties:

[math]\mathcal N_a^\text{post} = ( F^T \mathcal N_w^{-1} Z F)^{-1},[/math]

where [math]Z[/math] is a projection matrix that removes sky-synchronous degrees of freedom from the TOI:

[math] Z = I - P(P^T \mathcal N_w^{-1} P)^{-1} P^T \mathcal N_w.[/math]

The offset-offset, or low frequency, part of the HFI pixel-pixel covariance is then binned from the posterior offset-offset covariance by expanding the matrix into time domain and binning it using the same binning operator, [math]B,[/math] as in regular mapmaking and the in-ring noise covariance:

[math] N_a = B F \mathcal N_a^\text{post} F^T B^T.[/math]

The white noise weights used in the construction of the offset-offset covariance are somewhat arbitrary due to the shape of the HFI noise PSD. Therefore we take [math]N_a[/math] only to be proportional to the actual low frequency covariance and use the FFP8 simulations to measure a scaling coefficient between [math]N_a[/math] and the true low frequency covariance. The fit is done by least squares fitting a full covariance matrix of the form

[math]N_\text{HFI} = N_\text{in-ring} + \alpha N_a[/math]

to a sample covariance matrix built from the noise simulations.

Smoothing

Direct downgrading of the maps is problematic due to aliasing of the small small scale signal. Band-limiting the content of the low resolution maps is desirable also for their use in likelihood evaluation. We apply a smoothing cosine filter to the maps and the corresponding noise matrices to band-limit the signal to angular scales that are supported by the low resolution:

[math]W_\ell = \begin{cases} 1,\quad\ell\leq\ell_1\\ {1 \over 2}[1+\cos((\ell-\ell_1)\pi/(\ell_2-\ell_1))],\quad\ell_1\lt \ell\leq\ell_2\\ 0,\quad\ell\gt \ell_2 \end{cases}[/math]