HEALPix--the Hierarchical Equal Area isoLatitude Pixelization--is a versatile structure for the pixelization of data on the sphere. An associated library of computational algorithms and visualization ...software supports fast scientific applications executable directly on discretized spherical maps generated from very large volumes of astronomical data. Originally developed to address the data processing and analysis needs of the present generation of cosmic microwave background experiments (e.g., BOOMERANG, WMAP), HEALPix can be expanded to meet many of the profound challenges that will arise in confrontation with the observational output of future missions and experiments, including, e.g., Planck, Herschel, SAFIR, and the Beyond Einstein inflation probe. In this paper we consider the requirements and implementation constraints on a framework that simultaneously enables an efficient discretization with associated hierarchical indexation and fast analysis/synthesis of functions defined on the sphere. We demonstrate how these are explicitly satisfied by HEALPix.
We propose a method for CMB component separation based on standard Bayesian parameter estimation techniques. We assume a parametric spectral model for each signal component and fit the corresponding ...parameters pixel by pixel in a two-stage process. First we fit for the full parameter set (e.g., component amplitudes and spectral indices) in low-resolution and high signal-to-noise ratio maps using MCMC, obtaining both best-fit values for each parameter and the associated uncertainty. The goodness of fit is approximated by a j super(2) statistic. Then we fix all nonlinear parameters at their low-resolution best-fit values and solve analytically for high-resolution component amplitude maps. This likelihood approach has many advantages: the fitted model may be chosen freely, and the method is therefore completely general; all assumptions are transparent; no restrictions on spatial variations of foreground properties are imposed; the results may be monitored by goodness-of-fit tests; and, most importantly, we obtain reliable error estimates on all estimated quantities. We apply the method to simulated Planck satellite and 6 year WMAP data based on realistic models and show that separation at the microkelvin level is indeed possible in these cases. We also outline how the foreground uncertainties may be rigorously propagated through to the CMB power spectrum and cosmological parameters using a Gibbs sampling technique.
We report on the results from two independent but complementary statistical analyses of the Wilkinson Microwave Anisotropy Probe (WMAP) first- year data, based on the power spectrum and N-point ...correlation functions. We focus on large and intermediate scales (larger than about 3 degree ) and compare the observed data against Monte Carlo ensembles with WMAP-like properties. In both analyses, we measure the amplitudes of the large-scale fluctuations on opposing hemispheres and study the ratio of the two amplitudes. The power- spectrum analysis shows that this ratio for WMAP, as measured along the axis of maximum asymmetry, is high at the 95%-99% level (depending on the particular multipole range included). The axis of maximum asymmetry of the WMAP data is weakly dependent on the multipole range under consideration but tends to lie close to the ecliptic axis. In the N-point correlation-function analysis, we focus on the northern and southern hemispheres defined in ecliptic coordinates, and we find that the ratio of the large-scale fluctuation amplitudes is high at the 98%-99% level. Furthermore, the results are stable with respect to choice of Galactic cut and also with respect to frequency band. A similar asymmetry is found in the COBE Differential Microwave Radiometer (DMR) map, and the axis of maximum asymmetry is close to the one found in the WMAP data.
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.
ABSTRACT
We repeat and extend the analysis of Eriksen et al. and Hansen et al., testing the isotropy of the cosmic microwave background fluctuations. We find that the hemispherical power asymmetry ...previously reported for the largest scales ℓ = 2–40 extends to much smaller scales. In fact, for the full multipole range ℓ = 2–600, significantly more power is found in the hemisphere centered at (θ = 107° ± 10°, ϕ = 226° ± 10°) in galactic co-latitude and longitude than in the opposite hemisphere, consistent with the previously detected direction of asymmetry for ℓ = 2–40. We adopt a model selection test where the direction and amplitude of asymmetry, as well as the multipole range, are free parameters. A model with an asymmetric distribution of power for ℓ = 2–600 is found to be preferred over the isotropic model at the 0.4% significance level, taking into account the additional parameters required to describe it. A similar direction of asymmetry is found independently in all six subranges of 100 multipoles between ℓ = 2–600. None of our 9800 isotropic simulated maps show a similarly consistent direction of asymmetry over such a large multipole range. No known systematic effects or foregrounds are found to be able to explain the asymmetry.
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 describe and implement an exact, flexible, and computationally efficient algorithm for joint component separation and CMB power spectrum estimation, building on a Gibbs sampling framework. Two ...essential new features are (1) conditional sampling of foreground spectral parameters and (2) joint sampling of all amplitude-type degrees of freedom (e.g., CMB, foreground pixel amplitudes, and global template amplitudes) given spectral parameters. Given a parametric model of the foreground signals, we estimate efficiently and accurately the exact joint foreground-CMB posterior distribution and, therefore, all marginal distributions such as the CMB power spectrum or foreground spectral index posteriors. The main limitation of the current implementation is the requirement of identical beam responses at all frequencies, which restricts the analysis to the lowest resolution of a given experiment. We outline a future generalization to multiresolution observations. To verify the method, we analyze simple models and compare the results to analytical predictions. We then analyze a realistic simulation with properties similar to the 3 yr WMAP data, downgraded to a common resolution of 3 degree FWHM. The results from the actual 3 yr WMAP temperature analysis are presented in a companion Letter.
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 consider the issue of hemispherical power asymmetry in the third-year WMAP data, adopting a previously introduced modulation framework. Computing both frequentist probabilities and Bayesian ...evidences, we find that the model consisting of an isotropic CMB sky modulated by a dipole field gives a substantially better fit to the observations than the purely isotropic model, even when accounting for the larger prior volume. For the ILC map, the Bayesian log-evidence difference is similar to 1.8 in favor of the modulated model, and the raw improvement in maximum log likelihood is 6.1. The best-fit modulation dipole axis points toward (l, b) = (225 degree , -27 degree ), and the modulation amplitude is 0.114, in excellent agreement with the results from the first-year analyses. The frequentist probability of obtaining such a high modulation amplitude in an isotropic universe is similar to 1%. These results are not sensitive to data set or sky cut. Thus, the statistical evidence for a power asymmetry anomaly is both substantial and robust, although not decisive, for the currently available data. Increased sky coverage through better foreground handling and full-sky and high-sensitivity polarization maps may shed further light on this issue.
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