In this Letter we present new results from extremely deep, high-resolution images of the eld arou... more In this Letter we present new results from extremely deep, high-resolution images of the eld around the double quasar QSO 0957+561. A possible gravitational arc system near the double quasar has recently been reported, which, if real, would set strong constraints on determinations of the Hubble constant from the time delay in the double quasar. We nd that both the morphology and the colors of the claimed arc systems suggest that they are chance alignments of three and two di erent objects, and not gravitationally lensed arcs. Hence, the constraints on H 0-determinations from the arcs are not valid. Also, a small group of galaxies at z = 0:5 near the line-of-sight which was required to have a very large mass in the physically interesting arc models, is most likely insigni cant. From our deep images we are able to use weak lensing of faint background galaxies in the eld to map the gravitational potential in the main cluster. This sets new constraints on determinations of H 0. We nd that the Hubble constant is constrained to be less than 70 kms 1 Mpc 1 , if the time delay between the two images of the QSO is equal to or larger than 1.1 years.
We report on the results from two independent but complementary statistical analyses of the WMAP ... more We report on the results from two independent but complementary statistical analyses of the 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 ◦), 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 fix our interest on the northern and southern hemispheres defined in ecliptic coordinates, and also find that the ratio of the ...
Following this astro-ph communication, we release a set of 10,000 WMAP ILC simulations, produced ... more Following this astro-ph communication, we release a set of 10,000 WMAP ILC simulations, produced as described by Eriksen et al. (2004). We strongly encourage that an analysis of these simulations accompanies any scientific analysis of the observed ILC map, in order to assess the stability of the results. For examples of such analyses, we refer the reader to our original paper. In particular, Figure 1c) and d) show the bias and uncertainty induced by the ILC method, and Figure 7 shows a sobering examples of a low-l analysis; the reconstructed quadrupole-octopole alignment is plotted against the true input alignment, and the correlation is underwhelming. It can hardly be emphasized too strongly -- the heavily processed full-sky WMAP maps are not reliable for quantitative analysis, neither at large nor small scales, and they should only, at best, be used as supportive evidence. We hope that these simulations will reduce some of the confusion regarding these maps that currently exists i...
We present a description of the pipeline used to calibrate the Planck Low Frequency Instrument (L... more We present a description of the pipeline used to calibrate the Planck Low Frequency Instrument (LFI) timelines into thermodynamic temperatures for the Planck 2015 data release, covering four years of uninterrupted operations. As in the 2013 data release, our calibrator is provided by the spinsynchronous modulation of the cosmic microwave background dipole, but we now use the orbital component, rather than adopting the Wilkinson Microwave Anisotropy Probe (WMAP) solar dipole. This allows our 2015 LFI analysis to provide an independent Solar dipole estimate, which is in excellent agreement with that of HFI and within 1σ (0.3% in amplitude) of the WMAP value. This 0.3% shift in the peak-to-peak dipole temperature from WMAP and a general overhaul of the iterative calibration code increases the overall level of the LFI maps by 0.45% (30 GHz), 0.64% (44 GHz), and 0.82% (70 GHz) in temperature with respect to the 2013 Planck data release, thus reducing the discrepancy with the power spectrum measured by WMAP. We estimate that the LFI calibration uncertainty is now at the level of 0.20% for the 70 GHz map, 0.26% for the 44 GHz map, and 0.35% for the 30 GHz map. We provide a detailed description of the impact of all the changes implemented in the calibration since the previous data release.
The three Minkowski functionals and the recently defined length of the skeleton are estimated for... more The three Minkowski functionals and the recently defined length of the skeleton are estimated for the co-added first-year Wilkinson Microwave Anisotropy Probe (WMAP) data and compared with 5000 Monte Carlo simulations, based on Gaussian fluctuations with the a-priori best-fit running-index power spectrum and WMAP-like beam and noise properties. Several power spectrum-dependent quantities, such as the number of stationary points, the total length of the skeleton, and a spectral parameter, γ, are also estimated. While the area and length Minkowski functionals and the length of the skeleton show no evidence for departures from the Gaussian hypothesis, the northern hemisphere genus has a χ 2 that is large at the 95% level for all scales. For the particular smoothing scale of 3. • 40 FWHM it is larger than that found in 99.5% of the simulations. In addition, the WMAP genus for negative thresholds in the northern hemisphere has an amplitude that is larger than in the simulations with a significance of more than 3 σ. On the smallest angular scales considered, the number of extrema in the WMAP data is high at the 3 σ level. However, this can probably be attributed to the effect of point sources. Finally, the spectral parameter γ is high at the 99% level in the northern Galactic hemisphere, while perfectly acceptable in the southern hemisphere. The results provide strong evidence for the presence of both non-Gaussian behavior and an unexpected power asymmetry between the northern and southern hemispheres in the WMAP data. Subject headings: cosmic microwave background-cosmology: observations-methods: statistical
We report on the results from two independent but complementary statistical analyses of the WMAP ... more We report on the results from two independent but complementary statistical analyses of the 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 •) 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-DMR map, and the axis of maximum asymmetry is close to the one found in the WMAP data. Subject headings: cosmic microwave background-cosmology: observations-methods: statistical
We repeat and extend the analysis of Eriksen et al. and Hansen et al., testing the isotropy of th... more 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 10...
We study the Internal Linear Combination (ILC) method presented by the Wilkinson Microwave Anisot... more We study the Internal Linear Combination (ILC) method presented by the Wilkinson Microwave Anisotropy Probe (WMAP) science team, with the goal of determining whether it may be used for cosmological purposes, as a template-free alternative to existing foreground correction methods. We conclude that the method does have the potential to do just that, but great care must be taken both in implementation, and in a detailed understanding of limitations caused by residual foregrounds which can still affect cosmological results. As a first step we demonstrate how to compute the ILC weights both accurately and efficiently by means of Lagrange multipliers, and apply this method to the observed data to produce a new version of the ILC map. This map has 12% lower variance than the ILC map of the WMAP team, primarily due to less noise. Next we describe how to generate Monte Carlo simulations of the ILC map, and find that these agree well with the observed map on angular scales up to ℓ ≈ 200, using a conservative sky cut. Finally we make two comments to the ongoing debates concerning the large-scale properties of the WMAP data. First, we note that the Galactic southeastern quadrant is associated with notably different ILC weights than the other three quadrants, possibly indicating a foreground related anisotropy. Second, we study the properties of the quadrupole and octopole (amplitude, alignment and planarity), and reproduce the previously reported results that the quadrupole and octopole are strongly aligned and that the octopole is moderately planar. Even more interestingly, we find that the ℓ = 5 mode is spherically symmetric at about 3σ, and that the ℓ = 6 mode is planar at the 2σ level. However, we also assess the impact of residual foregrounds on these statistics, and find that the ILC map is not clean enough to allow for cosmological conclusions. Alternative methods must be developed to study these issues further. Subject headings: cosmic microwave background-cosmology: observations-methods: numerical
We study the Internal Linear Combination (ILC) method presented by the Wilkinson Microwave Anisot... more We study the Internal Linear Combination (ILC) method presented by the Wilkinson Microwave Anisotropy Probe (WMAP) science team, with the goal of determining whether it may be used for cosmological purposes, as a template-free alternative to existing foreground correction methods. We conclude that the method does have the potential to do just that, but great care must be taken both in implementation, and in a detailed understanding of limitations caused by residual foregrounds which can still affect cosmological results. As a first step we demonstrate how to compute the ILC weights both accurately and efficiently by means of Lagrange multipliers, and apply this method to the observed data to produce a new version of the ILC map. This map has 12% lower variance than the ILC map of the WMAP team, primarily due to less noise. Next we describe how to generate Monte Carlo simulations of the ILC map, and find that these agree well with the observed map on angular scales up to ℓ ≈ 200, using a conservative sky cut. Finally we make two comments to the ongoing debates concerning the large-scale properties of the WMAP data. First, we note that the Galactic southeastern quadrant is associated with notably different ILC weights than the other three quadrants, possibly indicating a foreground related anisotropy. Second, we study the properties of the quadrupole and octopole (amplitude, alignment and planarity), and reproduce the previously reported results that the quadrupole and octopole are strongly aligned and that the octopole is moderately planar. Even more interestingly, we find that the ℓ = 5 mode is spherically symmetric at about 3σ, and that the ℓ = 6 mode is planar at the 2σ level. However, we also assess the impact of residual foregrounds on these statistics, and find that the ILC map is not clean enough to allow for cosmological conclusions. Alternative methods must be developed to study these issues further. Subject headings: cosmic microwave background-cosmology: observations-methods: numerical
We present the two-and three-point real space correlation functions of the five-year WMAP sky map... more We present the two-and three-point real space correlation functions of the five-year WMAP sky maps and compare the observed functions to simulated ΛCDM concordance model ensembles. In agreement with previously published results, we find that the temperature correlation functions are consistent with expectations. However, the pure polarization correlation functions are acceptable only for the 33GHz band map; the 41, 61, and 94 GHz band correlation functions all exhibit significant large-scale excess structures. Further, these excess structures very closely match the correlation functions of the two (synchrotron and dust) foreground templates used to correct the WMAP data for galactic contamination, with a cross-correlation statistically significant at the 2σ − 3σ confidence level. The correlation is slightly stronger with respect to the thermal dust template than with the synchrotron template. Subject headings: cosmic background radiation-cosmology: observations-methods: numerical
We compute the two-, three-and four-point correlation functions from the Wilkinson Microwave Anis... more We compute the two-, three-and four-point correlation functions from the Wilkinson Microwave Anisotropy Probe (WMAP) first-year data, and compare these to a Monte Carlo ensemble of 5000 realizations, based on the best-fit WMAP running-index spectrum of Gaussian fluctuations. The analysis is carried out in three steps, covering small (< 72 ′), intermediate (< 5 •) and large scales (up to 180 •). On the largest scales our results are consistent with the previously reported hemisphere power asymmetries: the northern ecliptic hemisphere is practically devoid of large scale fluctuations, while the southern hemisphere show relatively strong fluctuations. We also detect excess correlations in W-band difference maps as compared to the detailed noise simulations produced by the WMAP team, possibly indicative of unknown systematics. While unlikely to affect any temperature based results, this effect could potentially be important for the upcoming polarization data. On intermediate angular scales we find hints of a similar anisotropic distribution of power as seen on the very largest scales, but not to the same extent. In general the model is accepted on these scales. Finally, the same is also true on the smallest scales probed in this paper. Subject headings: cosmic microwave background-cosmology: observations-methods: statistical
We apply the recently defined multipole vector framework to the frequency-specific first-year WMA... more We apply the recently defined multipole vector framework to the frequency-specific first-year WMAP sky maps, estimating the low-ℓ multipole coefficients from the high-latitude sky by means of a power equalization filter. While most previous analyses of this type have considered only heavily processed (and foreground-contaminated) full-sky maps, the present approach allows for greater control of residual foregrounds, and therefore potentially also for cosmologically important conclusions. The low-ℓ spherical harmonics coefficients and corresponding multipole vectors are tabulated for easy reference. Using this formalism, we reassess a set of earlier claims of both cosmological and non-cosmological low-ℓ correlations based on multipole vectors. First, we show that the apparent ℓ = 3 and 8 correlation claimed by Copi et al. (2004) is present only in the heavily processed map produced by Tegmark et al. (2003), and must therefore be considered an artifact of that map. Second, the well-known quadrupoleoctopole correlation is confirmed at the 99% significance level, and shown to be robust with respect to frequency and sky cut. Previous claims are thus supported by our analysis. Finally, the low-ℓ alignment with respect to the ecliptic claimed by Schwarz et al. (2004) is nominally confirmed in this analysis, but also shown to be very dependent on severe a-posteriori choices. Indeed, we show that given the peculiar quadrupole-octopole arrangement, finding such a strong alignment with the ecliptic is not unusual. Subject headings: cosmic microwave background-cosmology: observations-methods: numerical
We propose a method for CMB component separation based on standard Bayesian parameter estimation ... more 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 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. Subject headingg s: cosmic microwave background-cosmology: observations-methods: numerical
We develop, implement and test a set of algorithms for estimating N-point correlation functions f... more We develop, implement and test a set of algorithms for estimating N-point correlation functions from pixelized sky maps. These algorithms are slow, in the sense that they do not break the O(N N pix) barrier, and yet, they are fast enough for efficient analysis of data sets up to several hundred thousand pixels. The typical application of these methods is Monte Carlo analysis using several thousand realizations, and therefore we organize our programs so that the initialization cost is paid only once. The effective cost is then reduced to a few additions per pixel multiplet (pair, triplet etc.). Further, the algorithms waste no CPU time on computing undesired geometric configurations, and, finally, the computations are naturally divided into independent parts, allowing for trivial (i.e., optimal) parallelization.
We revisit a recently introduced power spectrum estimation technique based on Gibbs sampling, wit... more We revisit a recently introduced power spectrum estimation technique based on Gibbs sampling, with the goal of applying it to the high-resolution WMAP data. In order to facilitate this analysis, a number of sophistications have to be introduced, each of which is discussed in detail. We have implemented two independent versions of the algorithm to cross-check the computer codes, and to verify that a particular solution to any given problem does not affect the scientific results. We then apply these programs to simulated data with known properties at intermediate (N side = 128) and high (N side = 512) resolutions, to study effects such as incomplete sky coverage and white vs. correlated noise. From these simulations we also establish the Markov chain correlation length as a function of signal-to-noise ratio, and give a few comments on the properties of the correlation matrices involved. Parallelization issues are also discussed, with emphasis on real-world limitations imposed by current super-computer facilities. The scientific results from the analysis of the first-year WMAP data are presented in a companion letter. Subject headings: cosmic microwave background-cosmology: observations-methods: numerical
We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission t... more We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission temperature anisotropy measurements, combined with the WMAP large-angle polarization, constrain the scalar spectral index to be n s = 0.9603 ± 0.0073, ruling out exact scale invariance at over 5σ. Planck establishes an upper bound on the tensor-to-scalar ratio of r < 0.11 (95% CL). The Planck data thus shrink the space of allowed standard inflationary models, preferring potentials with V < 0. Exponential potential models, the simplest hybrid inflationary models, and monomial potential models of degree n ≥ 2 do not provide a good fit to the data. Planck does not find statistically significant running of the scalar spectral index, obtaining dn s /dln k = −0.0134 ± 0.0090. We verify these conclusions through a numerical analysis, which makes no slowroll approximation, and carry out a Bayesian parameter estimation and model-selection analysis for a number of inflationary models including monomial, natural, and hilltop potentials. For each model, we present the Planck constraints on the parameters of the potential and explore several possibilities for the post-inflationary entropy generation epoch, thus obtaining nontrivial data-driven constraints. We also present a direct reconstruction of the observable range of the inflaton potential. Unless a quartic term is allowed in the potential, we find results consistent with second-order slow-roll predictions. We also investigate whether the primordial power spectrum contains any features. We find that models with a parameterized oscillatory feature improve the fit by ∆χ 2 eff ≈ 10; however, Bayesian evidence does not prefer these models. We constrain several single-field inflation models with generalized Lagrangians by combining power spectrum data with Planck bounds on f NL. Planck constrains with unprecedented accuracy the amplitude and possible correlation (with the adiabatic mode) of non-decaying isocurvature fluctuations. The fractional primordial contributions of cold dark matter (CDM) isocurvature modes of the types expected in the curvaton and axion scenarios have upper bounds of 0.25% and 3.9% (95% CL), respectively. In models with arbitrarily correlated CDM or neutrino isocurvature modes, an anticorrelated isocurvature component can improve the χ 2 eff by approximately 4 as a result of slightly lowering the theoretical prediction for the < ∼ 40 multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.
The study and interpretation of the differences and similarities between radio-loud and radio-qui... more The study and interpretation of the differences and similarities between radio-loud and radio-quiet quasars (RLQs and RQQs, hereafter) has kept astronomers busy for a long time. The spectral energy distributions of RLQs and RQQs
In this Letter we present new results from extremely deep, high-resolution images of the eld arou... more In this Letter we present new results from extremely deep, high-resolution images of the eld around the double quasar QSO 0957+561. A possible gravitational arc system near the double quasar has recently been reported, which, if real, would set strong constraints on determinations of the Hubble constant from the time delay in the double quasar. We nd that both the morphology and the colors of the claimed arc systems suggest that they are chance alignments of three and two di erent objects, and not gravitationally lensed arcs. Hence, the constraints on H 0-determinations from the arcs are not valid. Also, a small group of galaxies at z = 0:5 near the line-of-sight which was required to have a very large mass in the physically interesting arc models, is most likely insigni cant. From our deep images we are able to use weak lensing of faint background galaxies in the eld to map the gravitational potential in the main cluster. This sets new constraints on determinations of H 0. We nd that the Hubble constant is constrained to be less than 70 kms 1 Mpc 1 , if the time delay between the two images of the QSO is equal to or larger than 1.1 years.
We report on the results from two independent but complementary statistical analyses of the WMAP ... more We report on the results from two independent but complementary statistical analyses of the 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 ◦), 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 fix our interest on the northern and southern hemispheres defined in ecliptic coordinates, and also find that the ratio of the ...
Following this astro-ph communication, we release a set of 10,000 WMAP ILC simulations, produced ... more Following this astro-ph communication, we release a set of 10,000 WMAP ILC simulations, produced as described by Eriksen et al. (2004). We strongly encourage that an analysis of these simulations accompanies any scientific analysis of the observed ILC map, in order to assess the stability of the results. For examples of such analyses, we refer the reader to our original paper. In particular, Figure 1c) and d) show the bias and uncertainty induced by the ILC method, and Figure 7 shows a sobering examples of a low-l analysis; the reconstructed quadrupole-octopole alignment is plotted against the true input alignment, and the correlation is underwhelming. It can hardly be emphasized too strongly -- the heavily processed full-sky WMAP maps are not reliable for quantitative analysis, neither at large nor small scales, and they should only, at best, be used as supportive evidence. We hope that these simulations will reduce some of the confusion regarding these maps that currently exists i...
We present a description of the pipeline used to calibrate the Planck Low Frequency Instrument (L... more We present a description of the pipeline used to calibrate the Planck Low Frequency Instrument (LFI) timelines into thermodynamic temperatures for the Planck 2015 data release, covering four years of uninterrupted operations. As in the 2013 data release, our calibrator is provided by the spinsynchronous modulation of the cosmic microwave background dipole, but we now use the orbital component, rather than adopting the Wilkinson Microwave Anisotropy Probe (WMAP) solar dipole. This allows our 2015 LFI analysis to provide an independent Solar dipole estimate, which is in excellent agreement with that of HFI and within 1σ (0.3% in amplitude) of the WMAP value. This 0.3% shift in the peak-to-peak dipole temperature from WMAP and a general overhaul of the iterative calibration code increases the overall level of the LFI maps by 0.45% (30 GHz), 0.64% (44 GHz), and 0.82% (70 GHz) in temperature with respect to the 2013 Planck data release, thus reducing the discrepancy with the power spectrum measured by WMAP. We estimate that the LFI calibration uncertainty is now at the level of 0.20% for the 70 GHz map, 0.26% for the 44 GHz map, and 0.35% for the 30 GHz map. We provide a detailed description of the impact of all the changes implemented in the calibration since the previous data release.
The three Minkowski functionals and the recently defined length of the skeleton are estimated for... more The three Minkowski functionals and the recently defined length of the skeleton are estimated for the co-added first-year Wilkinson Microwave Anisotropy Probe (WMAP) data and compared with 5000 Monte Carlo simulations, based on Gaussian fluctuations with the a-priori best-fit running-index power spectrum and WMAP-like beam and noise properties. Several power spectrum-dependent quantities, such as the number of stationary points, the total length of the skeleton, and a spectral parameter, γ, are also estimated. While the area and length Minkowski functionals and the length of the skeleton show no evidence for departures from the Gaussian hypothesis, the northern hemisphere genus has a χ 2 that is large at the 95% level for all scales. For the particular smoothing scale of 3. • 40 FWHM it is larger than that found in 99.5% of the simulations. In addition, the WMAP genus for negative thresholds in the northern hemisphere has an amplitude that is larger than in the simulations with a significance of more than 3 σ. On the smallest angular scales considered, the number of extrema in the WMAP data is high at the 3 σ level. However, this can probably be attributed to the effect of point sources. Finally, the spectral parameter γ is high at the 99% level in the northern Galactic hemisphere, while perfectly acceptable in the southern hemisphere. The results provide strong evidence for the presence of both non-Gaussian behavior and an unexpected power asymmetry between the northern and southern hemispheres in the WMAP data. Subject headings: cosmic microwave background-cosmology: observations-methods: statistical
We report on the results from two independent but complementary statistical analyses of the WMAP ... more We report on the results from two independent but complementary statistical analyses of the 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 •) 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-DMR map, and the axis of maximum asymmetry is close to the one found in the WMAP data. Subject headings: cosmic microwave background-cosmology: observations-methods: statistical
We repeat and extend the analysis of Eriksen et al. and Hansen et al., testing the isotropy of th... more 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 10...
We study the Internal Linear Combination (ILC) method presented by the Wilkinson Microwave Anisot... more We study the Internal Linear Combination (ILC) method presented by the Wilkinson Microwave Anisotropy Probe (WMAP) science team, with the goal of determining whether it may be used for cosmological purposes, as a template-free alternative to existing foreground correction methods. We conclude that the method does have the potential to do just that, but great care must be taken both in implementation, and in a detailed understanding of limitations caused by residual foregrounds which can still affect cosmological results. As a first step we demonstrate how to compute the ILC weights both accurately and efficiently by means of Lagrange multipliers, and apply this method to the observed data to produce a new version of the ILC map. This map has 12% lower variance than the ILC map of the WMAP team, primarily due to less noise. Next we describe how to generate Monte Carlo simulations of the ILC map, and find that these agree well with the observed map on angular scales up to ℓ ≈ 200, using a conservative sky cut. Finally we make two comments to the ongoing debates concerning the large-scale properties of the WMAP data. First, we note that the Galactic southeastern quadrant is associated with notably different ILC weights than the other three quadrants, possibly indicating a foreground related anisotropy. Second, we study the properties of the quadrupole and octopole (amplitude, alignment and planarity), and reproduce the previously reported results that the quadrupole and octopole are strongly aligned and that the octopole is moderately planar. Even more interestingly, we find that the ℓ = 5 mode is spherically symmetric at about 3σ, and that the ℓ = 6 mode is planar at the 2σ level. However, we also assess the impact of residual foregrounds on these statistics, and find that the ILC map is not clean enough to allow for cosmological conclusions. Alternative methods must be developed to study these issues further. Subject headings: cosmic microwave background-cosmology: observations-methods: numerical
We study the Internal Linear Combination (ILC) method presented by the Wilkinson Microwave Anisot... more We study the Internal Linear Combination (ILC) method presented by the Wilkinson Microwave Anisotropy Probe (WMAP) science team, with the goal of determining whether it may be used for cosmological purposes, as a template-free alternative to existing foreground correction methods. We conclude that the method does have the potential to do just that, but great care must be taken both in implementation, and in a detailed understanding of limitations caused by residual foregrounds which can still affect cosmological results. As a first step we demonstrate how to compute the ILC weights both accurately and efficiently by means of Lagrange multipliers, and apply this method to the observed data to produce a new version of the ILC map. This map has 12% lower variance than the ILC map of the WMAP team, primarily due to less noise. Next we describe how to generate Monte Carlo simulations of the ILC map, and find that these agree well with the observed map on angular scales up to ℓ ≈ 200, using a conservative sky cut. Finally we make two comments to the ongoing debates concerning the large-scale properties of the WMAP data. First, we note that the Galactic southeastern quadrant is associated with notably different ILC weights than the other three quadrants, possibly indicating a foreground related anisotropy. Second, we study the properties of the quadrupole and octopole (amplitude, alignment and planarity), and reproduce the previously reported results that the quadrupole and octopole are strongly aligned and that the octopole is moderately planar. Even more interestingly, we find that the ℓ = 5 mode is spherically symmetric at about 3σ, and that the ℓ = 6 mode is planar at the 2σ level. However, we also assess the impact of residual foregrounds on these statistics, and find that the ILC map is not clean enough to allow for cosmological conclusions. Alternative methods must be developed to study these issues further. Subject headings: cosmic microwave background-cosmology: observations-methods: numerical
We present the two-and three-point real space correlation functions of the five-year WMAP sky map... more We present the two-and three-point real space correlation functions of the five-year WMAP sky maps and compare the observed functions to simulated ΛCDM concordance model ensembles. In agreement with previously published results, we find that the temperature correlation functions are consistent with expectations. However, the pure polarization correlation functions are acceptable only for the 33GHz band map; the 41, 61, and 94 GHz band correlation functions all exhibit significant large-scale excess structures. Further, these excess structures very closely match the correlation functions of the two (synchrotron and dust) foreground templates used to correct the WMAP data for galactic contamination, with a cross-correlation statistically significant at the 2σ − 3σ confidence level. The correlation is slightly stronger with respect to the thermal dust template than with the synchrotron template. Subject headings: cosmic background radiation-cosmology: observations-methods: numerical
We compute the two-, three-and four-point correlation functions from the Wilkinson Microwave Anis... more We compute the two-, three-and four-point correlation functions from the Wilkinson Microwave Anisotropy Probe (WMAP) first-year data, and compare these to a Monte Carlo ensemble of 5000 realizations, based on the best-fit WMAP running-index spectrum of Gaussian fluctuations. The analysis is carried out in three steps, covering small (< 72 ′), intermediate (< 5 •) and large scales (up to 180 •). On the largest scales our results are consistent with the previously reported hemisphere power asymmetries: the northern ecliptic hemisphere is practically devoid of large scale fluctuations, while the southern hemisphere show relatively strong fluctuations. We also detect excess correlations in W-band difference maps as compared to the detailed noise simulations produced by the WMAP team, possibly indicative of unknown systematics. While unlikely to affect any temperature based results, this effect could potentially be important for the upcoming polarization data. On intermediate angular scales we find hints of a similar anisotropic distribution of power as seen on the very largest scales, but not to the same extent. In general the model is accepted on these scales. Finally, the same is also true on the smallest scales probed in this paper. Subject headings: cosmic microwave background-cosmology: observations-methods: statistical
We apply the recently defined multipole vector framework to the frequency-specific first-year WMA... more We apply the recently defined multipole vector framework to the frequency-specific first-year WMAP sky maps, estimating the low-ℓ multipole coefficients from the high-latitude sky by means of a power equalization filter. While most previous analyses of this type have considered only heavily processed (and foreground-contaminated) full-sky maps, the present approach allows for greater control of residual foregrounds, and therefore potentially also for cosmologically important conclusions. The low-ℓ spherical harmonics coefficients and corresponding multipole vectors are tabulated for easy reference. Using this formalism, we reassess a set of earlier claims of both cosmological and non-cosmological low-ℓ correlations based on multipole vectors. First, we show that the apparent ℓ = 3 and 8 correlation claimed by Copi et al. (2004) is present only in the heavily processed map produced by Tegmark et al. (2003), and must therefore be considered an artifact of that map. Second, the well-known quadrupoleoctopole correlation is confirmed at the 99% significance level, and shown to be robust with respect to frequency and sky cut. Previous claims are thus supported by our analysis. Finally, the low-ℓ alignment with respect to the ecliptic claimed by Schwarz et al. (2004) is nominally confirmed in this analysis, but also shown to be very dependent on severe a-posteriori choices. Indeed, we show that given the peculiar quadrupole-octopole arrangement, finding such a strong alignment with the ecliptic is not unusual. Subject headings: cosmic microwave background-cosmology: observations-methods: numerical
We propose a method for CMB component separation based on standard Bayesian parameter estimation ... more 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 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. Subject headingg s: cosmic microwave background-cosmology: observations-methods: numerical
We develop, implement and test a set of algorithms for estimating N-point correlation functions f... more We develop, implement and test a set of algorithms for estimating N-point correlation functions from pixelized sky maps. These algorithms are slow, in the sense that they do not break the O(N N pix) barrier, and yet, they are fast enough for efficient analysis of data sets up to several hundred thousand pixels. The typical application of these methods is Monte Carlo analysis using several thousand realizations, and therefore we organize our programs so that the initialization cost is paid only once. The effective cost is then reduced to a few additions per pixel multiplet (pair, triplet etc.). Further, the algorithms waste no CPU time on computing undesired geometric configurations, and, finally, the computations are naturally divided into independent parts, allowing for trivial (i.e., optimal) parallelization.
We revisit a recently introduced power spectrum estimation technique based on Gibbs sampling, wit... more We revisit a recently introduced power spectrum estimation technique based on Gibbs sampling, with the goal of applying it to the high-resolution WMAP data. In order to facilitate this analysis, a number of sophistications have to be introduced, each of which is discussed in detail. We have implemented two independent versions of the algorithm to cross-check the computer codes, and to verify that a particular solution to any given problem does not affect the scientific results. We then apply these programs to simulated data with known properties at intermediate (N side = 128) and high (N side = 512) resolutions, to study effects such as incomplete sky coverage and white vs. correlated noise. From these simulations we also establish the Markov chain correlation length as a function of signal-to-noise ratio, and give a few comments on the properties of the correlation matrices involved. Parallelization issues are also discussed, with emphasis on real-world limitations imposed by current super-computer facilities. The scientific results from the analysis of the first-year WMAP data are presented in a companion letter. Subject headings: cosmic microwave background-cosmology: observations-methods: numerical
We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission t... more We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission temperature anisotropy measurements, combined with the WMAP large-angle polarization, constrain the scalar spectral index to be n s = 0.9603 ± 0.0073, ruling out exact scale invariance at over 5σ. Planck establishes an upper bound on the tensor-to-scalar ratio of r < 0.11 (95% CL). The Planck data thus shrink the space of allowed standard inflationary models, preferring potentials with V < 0. Exponential potential models, the simplest hybrid inflationary models, and monomial potential models of degree n ≥ 2 do not provide a good fit to the data. Planck does not find statistically significant running of the scalar spectral index, obtaining dn s /dln k = −0.0134 ± 0.0090. We verify these conclusions through a numerical analysis, which makes no slowroll approximation, and carry out a Bayesian parameter estimation and model-selection analysis for a number of inflationary models including monomial, natural, and hilltop potentials. For each model, we present the Planck constraints on the parameters of the potential and explore several possibilities for the post-inflationary entropy generation epoch, thus obtaining nontrivial data-driven constraints. We also present a direct reconstruction of the observable range of the inflaton potential. Unless a quartic term is allowed in the potential, we find results consistent with second-order slow-roll predictions. We also investigate whether the primordial power spectrum contains any features. We find that models with a parameterized oscillatory feature improve the fit by ∆χ 2 eff ≈ 10; however, Bayesian evidence does not prefer these models. We constrain several single-field inflation models with generalized Lagrangians by combining power spectrum data with Planck bounds on f NL. Planck constrains with unprecedented accuracy the amplitude and possible correlation (with the adiabatic mode) of non-decaying isocurvature fluctuations. The fractional primordial contributions of cold dark matter (CDM) isocurvature modes of the types expected in the curvaton and axion scenarios have upper bounds of 0.25% and 3.9% (95% CL), respectively. In models with arbitrarily correlated CDM or neutrino isocurvature modes, an anticorrelated isocurvature component can improve the χ 2 eff by approximately 4 as a result of slightly lowering the theoretical prediction for the < ∼ 40 multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.
The study and interpretation of the differences and similarities between radio-loud and radio-qui... more The study and interpretation of the differences and similarities between radio-loud and radio-quiet quasars (RLQs and RQQs, hereafter) has kept astronomers busy for a long time. The spectral energy distributions of RLQs and RQQs
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Papers by Per Lilje