A coherent over- or under-density contrast across a finite survey volume causes an upward- or downward fluctuation in the observed number of haloes. This fluctuation in halo number adds a significant ...co-variant scatter in the observed amplitudes of weak lensing power spectrum at non-linear, small scales – the so-called super-sample variance or the halo sample variance. In this paper, we show that by measuring both the number counts of clusters and the power spectrum in the same survey region, we can mitigate this loss of information and significantly enhance the scientific return from the upcoming surveys. First, using the halo model approach, we derive the cross-correlation between the halo number counts and the weak lensing power spectrum, taking into account the super-sample covariance effect, which well matches the distributions measured from 1000 realizations for a Λ-dominated cold dark matter model. Then we show that adding the observed number counts of massive haloes with M ≳ 1014 M⊙/h can significantly improve the information content of weak lensing power spectrum, almost recovering the Gaussian information up to l
max ≃ 1000, if the average mass profiles of the massive haloes are known, which can be estimated from stacked lensing. When combined with the halo number counts for M > 3 or 1 × 1014 M⊙ h
−1, the improvement is up to a factor of 1.4 or 2 at l
max ≃ 1000–2000, equivalent to a factor of 2 or 4 times larger survey volume, compared to the power spectrum measurement alone.
The Wilkinson Microwave Anisotropy Probe (WMAP) 5-year data provide stringent limits on deviations from the minimal, six-parameter Lambda cold dark matter model. We report these limits and use them ...to constrain the physics of cosmic inflation via Gaussianity, adiabaticity, the power spectrum of primordial fluctuations, gravitational waves, and spatial curvature. We also constrain models of dark energy via its equation of state, parity-violating interaction, and neutrino properties, such as mass and the number of species. We detect no convincing deviations from the minimal model. The six parameters and the corresponding 68% uncertainties, derived from the WMAP data combined with the distance measurements from the Type Ia supernovae (SN) and the Baryon Acoustic Oscillations (BAO) in the distribution of galaxies, are: Omega b h 2 = 0.02267+0.00058 -0.00059, Omega c h 2 = 0.1131 ± 0.0034, Omega Lambda = 0.726 ± 0.015, ns = 0.960 ± 0.013, tau = 0.084 ± 0.016, and at k = 0.002 Mpc-1. From these, we derive sigma 8 = 0.812 ± 0.026, H 0 = 70.5 ± 1.3 km s-1 Mpc-1, Omega b = 0.0456 ± 0.0015, Omega c = 0.228 ± 0.013, Omega m h 2 = 0.1358+0.0037 -0.0036, z reion = 10.9 ± 1.4, and t 0 = 13.72 ± 0.12 Gyr. With the WMAP data combined with BAO and SN, we find the limit on the tensor-to-scalar ratio of r < 0.22(95%CL), and that ns > 1 is disfavored even when gravitational waves are included, which constrains the models of inflation that can produce significant gravitational waves, such as chaotic or power-law inflation models, or a blue spectrum, such as hybrid inflation models. We obtain tight, simultaneous limits on the (constant) equation of state of dark energy and the spatial curvature of the universe: -0.14 < 1 + w < 0.12(95%CL) and -0.0179 < Omega k < 0.0081(95%CL). We provide a set of 'WMAP distance priors,' to test a variety of dark energy models with spatial curvature. We test a time-dependent w with a present value constrained as -0.33 < 1 + w 0 < 0.21 (95% CL). Temperature and dark matter fluctuations are found to obey the adiabatic relation to within 8.9% and 2.1% for the axion-type and curvaton-type dark matter, respectively. The power spectra of TB and EB correlations constrain a parity-violating interaction, which rotates the polarization angle and converts E to B. The polarization angle could not be rotated more than -59 < Delta alpha < 24 (95% CL) between the decoupling and the present epoch. We find the limit on the total mass of massive neutrinos of capital sigma m Delta < 0.67 eV(95%CL), which is free from the uncertainty in the normalization of the large-scale structure data. The number of relativistic degrees of freedom (dof), expressed in units of the effective number of neutrino species, is constrained as N eff = 4.4 ± 1.5 (68%), consistent with the standard value of 3.04. Finally, quantitative limits on physically-motivated primordial non-Gaussianity parameters are -9 < f local NL < 111 (95% CL) and -151 < f equil NL < 253 (95% CL) for the local and equilateral models, respectively.
We present cosmological parameter constraints based on the final nine-year Wilkinson Microwave Anisotropy Probe (WMAP) data, in conjunction with a number of additional cosmological data sets. The ...WMAP data alone, and in combination, continue to be remarkably well fit by a six-parameter ?CDM model. We compare recent Planck measurements of the Sunyaev-Zel'dovich effect with our seven-year measurements, and show their mutual agreement. Our analysis of the polarization pattern around temperature extrema is updated. This confirms a fundamental prediction of the standard cosmological model and provides a striking illustration of acoustic oscillations and adiabatic initial conditions in the early universe.
This paper focuses on cosmological constraints derived from analysis of WMAP data alone. A simple Lambda CDM cosmological model fits the five-year WMAP temperature and polarization data. The basic ...parameters of the model are consistent with the three-year data and now better constrained: Omega b h 2 = 0.02273 ± 0.00062, Omega c h 2 = 0.1099 ± 0.0062, Omega Lambda = 0.742 ± 0.030, ns = 0.963+0.014 -0.015, tau = 0.087 ± 0.017, and sigma 8 = 0.796 ± 0.036, with h = 0.719+0.026 -0.027. With five years of polarization data, we have measured the optical depth to reionization, tau >0, at 5 sigma significance. The redshift of an instantaneous reionization is constrained to be z reion = 11.0 ± 1.4 with 68% confidence. The 2 sigma lower limit is z reion > 8.2, and the 3 sigma limit is z reion > 6.7. This excludes a sudden reionization of the universe at z = 6 at more than 3.5 sigma significance, suggesting that reionization was an extended process. Using two methods for polarized foreground cleaning we get consistent estimates for the optical depth, indicating an error due to the foreground treatment of tau ~ 0.01. This cosmological model also fits small-scale cosmic microwave background (CMB) data, and a range of astronomical data measuring the expansion rate and clustering of matter in the universe. We find evidence for the first time in the CMB power spectrum for a nonzero cosmic neutrino background, or a background of relativistic species, with the standard three light neutrino species preferred over the best-fit Lambda CDM model with N eff = 0 at >99.5% confidence, and N eff > 2.3(95%confidence limit (CL)) when varied. The five-year WMAP data improve the upper limit on the tensor-to-scalar ratio, r < 0.43(95%CL), for power-law models, and halve the limit on r for models with a running index, r < 0.58(95%CL). With longer integration we find no evidence for a running spectral index, with dns /dln k = -0.037 ± 0.028, and find improved limits on isocurvature fluctuations. The current WMAP-only limit on the sum of the neutrino masses is capital sigma m Delta < 1.3 eV(95%CL), which is robust, to within 10%, to a varying tensor amplitude, running spectral index, or dark energy equation of state.
We derive a fast way for measuring primordial non-Gaussianity in a nearly full-sky map of the cosmic microwave background. We find a cubic combination of sky maps combining bispectrum configurations ...to capture a quadratic term in primordial fluctuations. Our method takes only N super(3/2) operations rather than N super(5/2), as taken by the bispectrum analysis (1000 times faster for l = 512), retaining the same sensitivity. A key component is a map of the underlying primordial fluctuations, which can be more sensitive to the primordial non-Gaussianity than a temperature map. We also derive a fast and accurate statistic for measuring non-Gaussian signals from foreground point sources. The statistic is 10 super(6) times faster than the full bispectrum analysis and can be used to estimate contamination from the sources. Our algorithm has been successfully applied to the Wiikinson Microwave Anisotropy Probe sky maps by Komatsu and coworkers.
A simple cosmological model with only six parameters (matter density, Omega sub(m)h super(2), baryon density, Omega sub(b)h super(2), Hubble constant, H sub(0), amplitude of fluctuations, sigma ...sub(8), optical depth, tau , and a slope for the scalar perturbation spectrum, n sub(s)) fits not only the 3 year WMAP temperature and polarization data, but also small-scale CMB data, light element abundances, large-scale structure observations, and the supernova luminosity/distance relationship. Using WMAP data, only, the best-fit values for cosmological parameters for the power-law flat Lambda cold dark matter ( Lambda CDM) model are ( Omega sub(m)h super(2), Omega sub(b)h super(2),h,n sub(s), tau , sigma sub(8)) = (0.1277 super(+0.0080)-0.0079,0.02229 plus or minus 0.00073,0.732 super(+0.031)-0.032,0.958 plus or minus 0.016,0.089 plus or minus 0.030,0.761 super(+0.049)-0.048). The 3 year data dramatically shrink the allowed volume in mis six-dimensional parameter space. Assuming that the primordial fluctuations are adiabatic with a power-law spectrum, the WMAP data alone require dark matter and favor a spectral index that is significantly less than the Harrison-Zel'dovich-Peebles scale-invariant spectrum (n sub(s) = 1, r = 0). Adding additional data sets improves the constraints on these components and the spectral slope. For power-law models, WMAP data alone puts an improved upper limit on the tensor-to-scalar ratio, r sub(0.002) < 0.65 (95% CL) and the combination of WMAP and the lensing-normalized SDSS galaxy survey implies r sub(0.002) < 0.30 (95% CL). Models that suppress large-scale power through a running spectral index or a large-scale cutoff in the power spectrum are a better fit to the WMAP and small-scale CMB data than the power-law Lambda CDM model; however, the improvement in the fit to the WMAP data is only Delta chi super(2) = 3 for 1 extra degree of freedom. Models with a running-spectral index are consistent with a higher amplitude of gravity waves. In a flat universe, the combination of WMAP and the Supernova Legacy Survey (SNLS) data yields a significant constraint on the equation of state of the dark energy, w = -0.967 super(+0.073)-0.072. If we assume w = -1, then the deviations from the critical density, Omega sub(K) are small: the combination of WMAP and the SNLS data implies Omega sub(k) = -0.011 plus or minus 0.012. The combination of WMAP 3 year data plus the HST Key Project constraint on H sub(0) implies Omega sub(k) = -0.014 plus or minus 0.017 and Omega sub( Lambda ) = 0.716 plus or minus 0.055. Even if we do not include the prior that the universe is flat, by combining WMAP, large-scale structure, and supernova data, we can still put a strong constraint on the dark energy equation of state, w = -1.08 plus or minus 0.12. For a flat universe, the combination of WMAP and other astronomical data yield a constraint on the sum of the neutrino masses, capital sigma m sub( upsilon ) < 0.66 eV (95%CL). Consistent with the predictions of simple inflationary theories, we detect no significant deviations from Gaussianity in the CMB maps using Minkowski functionals, the bispectrum, trispectrum, and a new statistic designed to detect large-scale anisotropies in the fluctuations.
We present new full-sky temperature and polarization maps in five frequency bands from 23 to 94 GHz, based on data from the first five years of the Wilkinson Microwave Anisotropy Probe (WMAP) sky ...survey. The new maps are consistent with previous maps and are more sensitive. The five-year maps incorporate several improvements in data processing made possible by the additional years of data and by a more complete analysis of the instrument calibration and in-flight beam response. We present several new tests for systematic errors in the polarization data and conclude that W-band polarization data is not yet suitable for cosmological studies, but we suggest directions for further study. We do find that Ka-band data is suitable for use; in conjunction with the additional years of data, the addition of Ka band to the previously used Q- and V-band channels significantly reduces the uncertainty in the optical depth parameter, tau . Further scientific results from the five-year data analysis are presented in six companion papers and are summarized in Section 7 of this paper. With the five-year WMAP data, we detect no convincing deviations from the minimal six-parameter Lambda CDM model: a flat universe dominated by a cosmological constant, with adiabatic and nearly scale-invariant Gaussian fluctuations. Using WMAP data combined with measurements of Type Ia supernovae and Baryon Acoustic Oscillations in the galaxy distribution, we find (68% CL uncertainties): Omega b h 2 = 0.02267+0.00058 -0.00059, Omega c h 2 = 0.1131 ± 0.0034, Omega Lambda = 0.726 ± 0.015, ns = 0.960 ± 0.013, tau = 0.084 ± 0.016, and at k = 0.002 Mpc-1. From these we derive sigma 8 = 0.812 ± 0.026, H 0 = 70.5 ± 1.3 km s-1 Mpc-1, Omega b = 0.0456 ± 0.0015, Omega c = 0.228 ± 0.013, Omega m h 2 = 0.1358+0.0037 -0.0036, z reion = 10.9 ± 1.4, and t 0 = 13.72 ± 0.12 Gyr. The new limit on the tensor-to-scalar ratio is r < 0.22(95%CL), while the evidence for a running spectral index is insignificant, dns /dln k = -0.028 ± 0.020 (68% CL). We obtain tight, simultaneous limits on the (constant) dark energy equation of state and the spatial curvature of the universe: -0.14 < 1 + w < 0.12(95%CL) and -0.0179 < Omega k < 0.0081(95%CL). The number of relativistic degrees of freedom, expressed in units of the effective number of neutrino species, is found to be N eff = 4.4 ± 1.5 (68% CL), consistent with the standard value of 3.04. Models with N eff = 0 are disfavored at >99.5% confidence. Finally, new limits on physically motivated primordial non-Gaussianity parameters are -9 < f local NL < 111 (95% CL) and -151 < f equil NL < 253 (95% CL) for the local and equilateral models, respectively.
The combination of seven-year data from WMAP and improved astrophysical data rigorously tests the standard cosmological model and places new constraints on its basic parameters and extensions. By ...combining the WMAP data with the latest distance measurements from the baryon acoustic oscillations (BAO) in the distribution of galaxies and the Hubble constant (H 0) measurement, we determine the parameters of the simplest six-parameter Delta *LCDM model. The power-law index of the primordial power spectrum is ns = 0.968 ? 0.012 (68% CL) for this data combination, a measurement that excludes the Harrison-Zel'dovich-Peebles spectrum by 99.5% CL. The other parameters, including those beyond the minimal set, are also consistent with, and improved from, the five-year results. We find no convincing deviations from the minimal model. The seven-year temperature power spectrum gives a better determination of the third acoustic peak, which results in a better determination of the redshift of the matter-radiation equality epoch. Notable examples of improved parameters are the total mass of neutrinos, Delta *Sm Delta *n < 0.58 eV(95%CL), and the effective number of neutrino species, N eff = 4.34+0.86 --0.88 (68% CL), which benefit from better determinations of the third peak and H 0. The limit on a constant dark energy equation of state parameter from WMAP+BAO+H 0, without high-redshift Type Ia supernovae, is w = --1.10 ? 0.14 (68% CL). We detect the effect of primordial helium on the temperature power spectrum and provide a new test of big bang nucleosynthesis by measuring Yp = 0.326 ? 0.075 (68% CL). We detect, and show on the map for the first time, the tangential and radial polarization patterns around hot and cold spots of temperature fluctuations, an important test of physical processes at z = 1090 and the dominance of adiabatic scalar fluctuations. The seven-year polarization data have significantly improved: we now detect the temperature-E-mode polarization cross power spectrum at 21 Delta *s, compared with 13 Delta *s from the five-year data. With the seven-year temperature-B-mode cross power spectrum, the limit on a rotation of the polarization plane due to potential parity-violating effects has improved by 38% to (68% CL). We report significant detections of the Sunyaev-Zel'dovich (SZ) effect at the locations of known clusters of galaxies. The measured SZ signal agrees well with the expected signal from the X-ray data on a cluster-by-cluster basis. However, it is a factor of 0.5-0.7 times the predictions from 'universal profile' of Arnaud et al., analytical models, and hydrodynamical simulations. We find, for the first time in the SZ effect, a significant difference between the cooling-flow and non-cooling-flow clusters (or relaxed and non-relaxed clusters), which can explain some of the discrepancy. This lower amplitude is consistent with the lower-than-theoretically expected SZ power spectrum recently measured by the South Pole Telescope Collaboration.
We show how the correlation between the thermal Sunyaev Zel'dovich effect (tSZ) from galaxy clusters and dust emission from cosmic infrared background (CIB) sources can be calculated in a halo model ...framework. Using recent tSZ and CIB models, we find that the size of the tSZ×CIB cross-correlation is approximately 20 per cent at 150 GHz. The contribution to the total angular power spectrum is around -2 μK2 at ℓ = 3000; however, this value is uncertain by a factor of 2-3, primarily because of CIB source modelling uncertainties. We expect the large uncertainty in this component to degrade upper limits on the kinematic Sunyaev Zel'dovich effect (kSZ), due to similarity in the frequency dependence of tSZ×CIB and kSZ across the frequency range probed by current cosmic microwave background missions. We also find that the degree of tSZ×CIB correlation is higher for mm×sub-mm spectra than mm×mm, because more of the sub-mm CIB originates at lower redshifts (z ≲ 2), where most tSZ clusters are found.
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