We revisit the problem of exact CMB likelihood and power spectrum estimation with the goal of minimizing computational cost through linear compression. This idea was originally proposed for CMB purposes by Tegmark et al. (1997), and here we develop i
t into a fully working computational framework for large-scale polarization analysis, adopting WMAP as a worked example. We compare five different linear bases (pixel space, harmonic space, noise covariance eigenvectors, signal-to-noise covariance eigenvectors and signal-plus-noise covariance eigenvectors) in terms of compression efficiency, and find that the computationally most efficient basis is the signal-to-noise eigenvector basis, which is closely related to the Karhunen-Loeve and Principal Component transforms, in agreement with previous suggestions. For this basis, the information in 6836 unmasked WMAP sky map pixels can be compressed into a smaller set of 3102 modes, with a maximum error increase of any single multipole of 3.8% at $ellle32$, and a maximum shift in the mean values of a joint distribution of an amplitude--tilt model of 0.006$sigma$. This compression reduces the computational cost of a single likelihood evaluation by a factor of 5, from 38 to 7.5 CPU seconds, and it also results in a more robust likelihood by implicitly regularizing nearly degenerate modes. Finally, we use the same compression framework to formulate a numerically stable and computationally efficient variation of the Quadratic Maximum Likelihood implementation that requires less than 3 GB of memory and 2 CPU minutes per iteration for $ell le 32$, rendering low-$ell$ QML CMB power spectrum analysis fully tractable on a standard laptop.
In a recent preprint (CCC-predicted low-variance circles in the CMB sky and LCDM), Gurzadyan and Penrose (2011) claim for the second time to find evidence for pre-Big Bang activity in the form of concentric circles of low variance in the WMAP data. T
he same claim was made in November 2010, but quickly shown to be false by three independent groups. The culprit was simply that Gurzadyan and Penroses simulations were based on an inappropriate power spectrum. In the most recent paper, they now claim that the significance is indeed low if the simulations are based on the realization-specific WMAP spectrum (ie., the one directly measured from the sky maps and affected by cosmic variance), but not if the simulations are based on a theoretical LCDM spectrum. In this respect, we note that the three independent reanalyses all based their simulations on the LCDM spectrum, not the observed WMAP spectrum, and this alone should suffice to show that the updated claims are also incorrect. In fact, it is evident from the plots shown in their new paper that the spectrum is still incorrect, although in a different way than in their first paper. Thus, Gurzadyan and Penroses new claims are just as wrong as those made in the first paper, and for the same reason: The simulations are not based on an appropriate power spectrum. Still, while this story is of little physical interest, it may have some important implications in terms of scienctific sociology: Looking back at the background papers leading up to the present series by Gurzadyan and Penrose, in particular one introducing the Kolmogorov statistic, we believe one can find evidence that a community based and open access referee process may be more efficient at rejecting incorrect results and claims than a traditional journal based approach.
We develop a new Bayesian method for estimating white noise levels in CMB sky maps, and apply this algorithm to the 5-year WMAP data. We assume that the amplitude of the noise RMS is scaled by a constant value, alpha, relative to a pre-specified nois
e level. We then derive the corresponding conditional density, P(alpha | s, Cl, d), which is subsequently integrated into a general CMB Gibbs sampler. We first verify our code by analyzing simulated data sets, and then apply the framework to the WMAP data. For the foreground-reduced 5-year WMAP sky maps and the nominal noise levels initially provided in the 5-year data release, we find that the posterior means typically range between alpha=1.005 +- 0.001 and alpha=1.010 +- 0.001 depending on differencing assembly, indicating that the noise level of these maps are biased low by 0.5-1.0%. The same problem is not observed for the uncorrected WMAP sky maps. After the preprint version of this letter appeared on astro-ph., the WMAP team has corrected the values presented on their web page, noting that the initially provided values were in fact estimates from the 3-year data release, not from the 5-year estimates. However, internally in their 5-year analysis the correct noise values were used, and no cosmological results are therefore compromised by this error. Thus, our method has already been demonstrated in practice to be both useful and accurate.
A well-tested and validated Gibbs sampling code, that performs component separation and CMB power spectrum estimation, was applied to the {it WMAP} 5-yr data. Using a simple model consisting of CMB, noise, monopoles and dipoles, a ``per pixel low-fre
quency power-law (fitting for both amplitude and spectral index), and a thermal dust template with fixed spectral index, we found that the low-$ell$ ($ell < 50$) CMB power spectrum is in good agreement with the published {it WMAP}5 results. Residual monopoles and dipoles were found to be small ($lesssim 3 mu$K) or negligible in the 5-yr data. We comprehensively tested the assumptions that were made about the foregrounds (e.g. dust spectral index, power-law spectral index prior, templates), and found that the CMB power spectrum was insensitive to these choices. We confirm the asymmetry of power between the north and south ecliptic hemispheres, which appears to be robust against foreground modeling. The map of low frequency spectral indices indicates a steeper spectrum on average ($beta=-2.97pm0.21$) relative to those found at low ($sim$GHz) frequencies.
(Abridged)Motivated by the recent results of Hansen et al. (2008) concerning a noticeable hemispherical power asymmetry in the WMAP data on small angular scales, we revisit the dipole modulated signal model introduced by Gordon et al. (2005). This mo
del assumes that the true CMB signal consists of a Gaussian isotropic random field modulated by a dipole, and is characterized by an overall modulation amplitude, A, and a preferred direction, p. Previous analyses of this model has been restricted to very low resolution due to computational cost. In this paper, we double the angular resolution, and compute the full corresponding posterior distribution for the 5-year WMAP data. The results from our analysis are the following: The best-fit modulation amplitude for l <= 64 and the ILC data with the WMAP KQ85 sky cut is A=0.072 +/- 0.022, non-zero at 3.3sigma, and the preferred direction points toward Galactic coordinates (l,b) = (224 degree, -22 degree) +/- 24 degree. The corresponding results for l <~ 40 from earlier analyses was A = 0.11 +/- 0.04 and (l,b) = (225 degree,-27 degree). The statistical significance of a non-zero amplitude thus increases from 2.8sigma to 3.3sigma when increasing l_max from 40 to 64, and all results are consistent to within 1sigma. Similarly, the Bayesian log-evidence difference with respect to the isotropic model increases from Delta ln E = 1.8 to Delta ln E = 2.6, ranking as strong evidence on the Jeffreys scale. The raw best-fit log-likelihood difference increases from Delta ln L = 6.1 to Delta ln L = 7.3. Similar, and often slightly stronger, results are found for other data combinations. Thus, we find that the evidence for a dipole power distribution in the WMAP data increases with l in the 5-year WMAP data set, in agreement with the reports of Hansen et al. (2008).
We introduce a new CMB temperature likelihood approximation called the Gaussianized Blackwell-Rao (GBR) estimator. This estimator is derived by transforming the observed marginal power spectrum distributions obtained by the CMB Gibbs sampler into sta
ndard univariate Gaussians, and then approximate their joint transformed distribution by a multivariate Gaussian. The method is exact for full-sky coverage and uniform noise, and an excellent approximation for sky cuts and scanning patterns relevant for modern satellite experiments such as WMAP and Planck. A single evaluation of this estimator between l=2 and 200 takes ~0.2 CPU milliseconds, while for comparison, a single pixel space likelihood evaluation between l=2 and 30 for a map with ~2500 pixels requires ~20 seconds. We apply this tool to the 5-year WMAP temperature data, and re-estimate the angular temperature power spectrum, $C_{ell}$, and likelihood, L(C_l), for l<=200, and derive new cosmological parameters for the standard six-parameter LambdaCDM model. Our spectrum is in excellent agreement with the official WMAP spectrum, but we find slight differences in the derived cosmological parameters. Most importantly, the spectral index of scalar perturbations is n_s=0.973 +/- 0.014, 1.9 sigma away from unity and 0.6 sigma higher than the official WMAP result, n_s = 0.965 +/- 0.014. This suggests that an exact likelihood treatment is required to higher ls than previously believed, reinforcing and extending our conclusions from the 3-year WMAP analysis. In that case, we found that the sub-optimal likelihood approximation adopted between l=12 and 30 by the WMAP team biased n_s low by 0.4 sigma, while here we find that the same approximation between l=30 and 200 introduces a bias of 0.6 sigma in n_s.
Since that very memorable day at the Beijing 2008 Olympics, a big question on every sports commentators mind has been What would the 100 meter dash world record have been, had Usain Bolt not celebrated at the end of his race? Glen Mills, Bolts coach
suggested at a recent press conference that the time could have been 9.52 seconds or better. We revisit this question by measuring Bolts position as a function of time using footage of the run, and then extrapolate into the last two seconds based on two different assumptions. First, we conservatively assume that Bolt could have maintained Richard Thompsons, the runner-up, acceleration during the end of the race. Second, based on the race development prior to the celebration, we assume that he could also have kept an acceleration of 0.5 m/s^2 higher than Thompson. In these two cases, we find that the new world record would have been 9.61 +/- 0.04 and 9.55 +/- 0.04 seconds, respectively, where the uncertainties denote 95% statistical errors.
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 multi-resolution observations. To verify the method, we analyse 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.
Precision measurement of the scalar perturbation spectral index, n_s, from the Wilkinson Microwave Anisotropy Probe temperature angular power spectrum requires the subtraction of unresolved point source power. Here we reconsider this issue. First, we
note a peculiarity in the WMAP temperature likelihoods response to the source correction: Cosmological parameters do not respond to increased source errors. An alternative and more direct method for treating this error term acts more sensibly, and also shifts n_s by ~0.3 sigma closer to unity. Second, we re-examine the source fit used to correct the power spectrum. This fit depends strongly on the galactic cut and the weighting of the map, indicating that either the source population or masking procedure is not isotropic. Jackknife tests appear inconsistent, causing us to assign large uncertainties to account for possible systematics. Third, we note that the WMAP teams spectrum was computed with two different weighting schemes: uniform weights transition to inverse noise variance weights at l = 500. The fit depends on such weighting schemes, so different corrections apply to each multipole range. For the Kp2 mask used in cosmological analysis, we prefer source corrections A = 0.012 +/- 0.005 muK^2 for uniform weighting and A = 0.015 +/- 0.005 muK^2 for N_obs weighting. Correcting WMAPs spectrum correspondingly, we compute cosmological parameters with our alternative likelihood, finding n_s = 0.970 +/- 0.017 and sigma_8 = 0.778 +/- 0.045 . This n_s is only 1.8 sigma from unity, compared to the ~2.6 sigma WMAP 3-year result. Finally, an anomalous feature in the source spectrum at l<200 remains, most strongly associated with W-band.
Using a Gibbs sampling algorithm for joint CMB estimation and component separation, we compute the large-scale CMB and foreground posteriors of the 3-yr WMAP temperature data. Our parametric data model includes the cosmological CMB signal and instrum
ental noise, a single power law foreground component with free amplitude and spectral index for each pixel, a thermal dust template with a single free overall amplitude, and free monopoles and dipoles at each frequency. This simple model yields a surprisingly good fit to the data over the full frequency range from 23 to 94 GHz. We obtain a new estimate of the CMB sky signal and power spectrum, and a new foreground model, including a measurement of the effective spectral index over the high-latitude sky. A particularly significant result is the detection of a common spurious offset in all frequency bands of ~ -13muK, as well as a dipole in the V-band data. Correcting for these is essential when determining the effective spectral index of the foregrounds. We find that our new foreground model is in good agreement with template-based model presented by the WMAP team, but not with their MEM reconstruction. We believe the latter may be at least partially compromised by the residual offsets and dipoles in the data. Fortunately, the CMB power spectrum is not significantly affected by these issues, as our new spectrum is in excellent agreement with that published by the WMAP team. The corresponding cosmological parameters are also virtually unchanged.
