We propose a novel estimator of the polarization amplitude from a single measurement of its normally distributed (Q, U) Stokes components. Based on the properties of the Rice distribution and dubbed ...modified asymptotic, it meets several desirable criteria: (i) its values lie in the whole positive region; (ii) its distribution is continuous; (iii) it transforms smoothly with the signal-to-noise ratio (SNR) from a Rayleigh-like shape to a Gaussian one; (iv) it is unbiased and reaches its components' variance as soon as the SNR exceeds 2; and (v) it is analytic and can therefore be used on large data sets. We also revisit the construction of its associated confidence intervals and show how the Feldman-Cousins prescription efficiently solves the issue of classical intervals lying entirely in the unphysical negative domain. Such intervals can be used to identify statistically significant polarized regions and conversely build masks for polarization data. We then consider the case of a general Q, U covariance matrix and perform a generalization of the estimator that preserves its asymptotic properties. We show that its bias does not depend on the true polarization angle, and provide an analytic estimate of its variance. The estimator value, together with its variance, provide a powerful point estimate of the true polarization amplitude that follows an unbiased Gaussian distribution for an SNR as low as 2. These results can be applied to the much more general case of transforming any normally distributed random variable from Cartesian to polar coordinates.
We present constraints on the tensor-to-scalar ratio
r
using
Planck
data. We use the latest release of
Planck
maps, processed with the
NPIPE
code, which produces calibrated frequency maps in ...temperature and polarisation for all
Planck
channels from 30 GHz to 857 GHz using the same pipeline. We computed constraints on
r
using the
BB
angular power spectrum, and we also discuss constraints coming from the
TT
spectrum. Given
Planck
’s noise level, the
TT
spectrum gives constraints on
r
that are cosmic-variance limited (with
σ
r
= 0.093), but we show that the marginalised posterior peaks towards negative values of
r
at about the 1.2
σ
level. We derived
Planck
constraints using the
BB
power spectrum at both large angular scales (the ‘reionisation bump’) and intermediate angular scales (the ‘recombination bump’) from
ℓ
= 2 to 150 and find a stronger constraint than that from
TT
, with
σ
r
= 0.069. The
Planck
BB
spectrum shows no systematic bias and is compatible with zero, given both the statistical noise and the systematic uncertainties. The likelihood analysis using
B
modes yields the constraint
r
< 0.158 at 95% confidence using more than 50% of the sky. This upper limit tightens to
r
< 0.069 when
Planck
EE
,
BB
, and
EB
power spectra are combined consistently, and it tightens further to
r
< 0.056 when the
Planck
TT
power spectrum is included in the combination. Finally, combining
Planck
with BICEP2/Keck 2015 data yields an upper limit of
r
< 0.044.
We search for the signature of parity-violating physics in the cosmic microwave background, called cosmic birefringence, using the Planck data release 4. We initially find a birefringence angle of ...β=0.30°±0.11° (68% C.L.) for nearly full-sky data. The values of β decrease as we enlarge the Galactic mask, which can be interpreted as the effect of polarized foreground emission. Two independent ways to model this effect are used to mitigate the systematic impact on β for different sky fractions. We choose not to assign cosmological significance to the measured value of β until we improve our knowledge of the foreground polarization.
We present constraints on the tensor-to-scalar ratio r using a combination of BICEP/Keck 2018 (BK18) and Planck PR4 data allowing us to fit for r consistently with the six parameters of the ΛCDM ...model. We discuss the sensitivity of constraints on r to uncertainties in the ΛCDM parameters as defined by the Planck data. In particular, we are able to derive a constraint on the reionization optical depth τ and thus propagate its uncertainty into the posterior distribution for r. While Planck sensitivity to r is slightly lower than the current ground-based measurements, the combination of Planck with BK18 and baryon-acoustic-oscillation data yields results consistent with r=0 and tightens the constraint to r<0.032 at 95% confidence.
We present a cross-spectra-based approach for the analysis of cosmic microwave background data at large angular scales to constrain the reionization optical depth τ, the tensor to scalar ratio r and ...the amplitude of the primordial scalar perturbations A
s. With respect to the pixel-based approach developed so far, using cross-spectra has the unique advantage to eliminate spurious noise bias and to give a better handle over residual systematics, allowing to efficiently combine the cosmological information encoded in cross-frequency or cross-data set spectra. We present two solutions to deal with the non-Gaussianity of the
$\hat{C}_\ell$
estimator distributions at large angular scales: the first one relies on an analytical parametrization of the estimator distribution, while the second one is based on modification of the Hamimache and Lewis (HL) likelihood approximation at large angular scales. The modified HL method (oHL) is powerful and complete. It allows us to deal with multipole and mode correlations for a combined temperature and polarization analysis. We validate our likelihoods on numerous simulations that include the realistic noise levels of the Wilkinson Microwave Anisotropy Probe, Planck-Low Frequency Instrument and Planck-High Frequency Instrument experiments, demonstrating their validity over a broad range of cross-spectra configurations.
When combining cosmological and oscillations results to constrain the neutrino sector, the question of the propagation of systematic uncertainties is often raised. We address this issue in the ...context of the derivation of an upper bound on the sum of the neutrino masses (Σmν) with recent cosmological data. This work is performed within the ΛCDM model extended to Σmν, for which we advocate the use of three mass-degenerate neutrinos. We focus on the study of systematic uncertainties linked to the foregrounds modelling in cosmological microwave background (CMB) data analysis, and on the impact of the present knowledge of the reionisation optical depth. This is done through the use of different likelihoods built from Planck data. Limits on Σmν are derived with various combinations of data, including the latest baryon acoustic oscillations (BAO) and Type Ia supernovae (SNIa) results. We also discuss the impact of the preference for current CMB data for amplitudes of the gravitational lensing distortions higher than expected within the ΛCDM model, and add the Planck CMB lensing. We then derive a robust upper limit: Σmν< 0.17 eV at 95% CL, including 0.01eV of foreground systematics. We also discuss the neutrino mass repartition and show that today’s data do not allow one to disentangle normal from inverted hierarchy. The impact on the other cosmological parameters is also reported, for different assumptions on the neutrino mass repartition, and different high and low multipole CMB likelihoods.
We present cosmological parameter constraints using maps from the last
Planck
data release (PR4). In particular, we detail an upgraded version of the cosmic microwave background likelihood,
HiLLiPoP
..., that is based on angular power spectra and relies on a physical modeling of the foreground residuals in the spectral domain. This new version of the likelihood retains a larger sky fraction (up to 75%) and uses an extended multipole range. Using this likelihood, along with low-
ℓ
measurements from
LoLLiPoP
, we derived constraints on ΛCDM parameters that are in good agreement with previous
Planck
2018 results, but with smaller uncertainties by 10% to 20%. We demonstrate that the foregrounds can be accurately described in the spectral domain, with a negligible impact on ΛCDM parameters. We also derived constraints on single-parameter extensions to ΛCDM, including
A
L
, Ω
K
,
N
eff
, and ∑
m
ν
. Noteworthy results from this updated analysis include a lensing amplitude value of
A
L
= 1.039 ± 0.052, which is more closely aligned with theoretical expectations within the ΛCDM framework. Additionally, our curvature measurement, Ω
K
= −0.012 ± 0.010, is now fully consistent with a flat universe and our measurement of
S
8
is closer to the measurements derived from large-scale structure surveys (at the 1.5
σ
level). We also added constraints from PR4 lensing, making this combination the most tightly constrained data set currently available from
Planck
. Additionally, we explored the addition of baryon acoustic oscillation data, which tightens the limits on some particular extensions to the standard cosmology.
The quadratic maximum likelihood estimator can be used to reconstruct the cosmic microwave background (CMB) power spectra with minimal error bars. Still, it requires an accurate estimate of the data ...set’s noise covariance matrix in order to be corrected for spurious bias. We describe an extension of this method to cross-correlation, thus removing noise bias and mitigating the impact of systematic effects, providing that they are uncorrelated. This estimator is tested on two simulation surveys at large and intermediate angular scales, respectively corresponding to satellite and ground-based CMB experiments. The analysis focuses on polarization maps, over a wide range of noise levels from 0.1 to 50 μK arcmin. We show how this estimator minimizes the increase of variance due to polarization leakage between E and B modes. We compare this method with the pure pseudospectrum formalism, which is computationally faster but less optimal, especially on large angular scales.
nsions in cosmological parameters measurement motivate a revisit of the effects of instrumental systematics. In this article, we focus on the Pearson's correlation coefficient of the cosmic microwave ...background temperature and polarization E modes RℓTE, which has the property of not being biased by multiplicative instrumental systematics. We build a RℓTE-based likelihood for the Planck data and present the first constraints on Λ CDM (Lambda cold dark matter) parameters from the correlation coefficient. Our results are compatible with parameters derived from a power-spectra-based likelihood. In particular, the value of the Hubble parameter H0 characterizing the expansion of the Universe today, 67.5 ± 1.3 km / s / Mpc , is consistent with the ones inferred from standard cosmic microwave background analysis. We also discuss the consistency of the Planck correlation coefficient with the one computed from the most recent ACTPol power spectra.
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