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Distributed Minimal Residual (DMR) method for acceleration of iterative algorithms

A new method for enhancing the convergence rate of iterative algorithms for the numerical integration of systems of partial differential equations was developed. It is termed the Distributed Minimal Residual (DMR) method and it is based on general Krylov subspace methods. The DMR method differs from the Krylov subspace methods by the fact that the iterative acceleration factors are different from equation to equation in the system. At the same time, the DMR method can be viewed as an incomplete Newton iteration method. The DMR method was applied to Euler equations of gas dynamics and incompressible Navier-Stokes equations. All numerical test cases were obtained using either explicit four stage Runge-Kutta or Euler implicit time integration. The formulation for the DMR method is general in nature and can be applied to explicit and implicit iterative algorithms for arbitrary systems of partial differential equations.

Lee, Seungsoo↗

Noncosmological signal contributions to the COBE DMR anisotropy maps

We examine the COBE Differential Microwave Radiometer (DMR) data for evidence of noncosmological source contributions. The DMR maps are cross-correlated with maps of rich clusters, extragalactic IRAS sources, HEAO 1 A-2 X-ray emission, and 5 GHz radio sources. We limit the rms contributions from these sources on a 7 deg angular scale to less than 10 micro-K (95 percent confidence level) in the DMR maps, although the LMC probably contributes about 50 micro-K to a limited region of the sky. Thus, our previous interpretation that the fluctuations in the COBE DMR data are most likely due to cosmic fluctuations at the surface of last scattering remains intact. The Comptonization parameter for hot electrons traced by rich clusters is limited to delta(y) less than 2 x 10 exp -6 (95 percent confidence level) averaged over the 7 deg DMR beam.

Bennett, C. L.↗

Correlated noise in the COBE DMR sky maps

The Cosmic Background Explorer Satellite Differential Radiometer (COBE DMR) sky maps contain low-level correlated noise. We obtain estimates of the amplitude and pattern of the correlated noise from three techniques: angular averages of the covariance matrix, Monte Carlo simulations of two-point correlation functions and direct analysis of the DMR maps. The results from the three methods are mutually consistent. The noise covariance matrix of a DMR sky maps is diagonal to an accuracy of better than 1%. For a given sky pixel, the dominant noise covariance occure with the ring of pixels at an angular separation of 60 deg due to the 60 deg separation of the DMR horns. The mean covariance at 60 deg is 0.45%((sup +0.18)(sub -0.14)) of the mean variance. Additionally, the variance in a given pixel is 0.7% greater than would be expected from a single beam experiment with the same noise properties. Autocorrelation functions suffer from a approximately 1.5 sigma positive bias at 60 deg while cross-correlations have no bias. Published COBE DMR results are not significantly affected by correlated noise.

Lineweaver, C. H.↗

Report on 3 and 4-point correlation statistics in the COBE DMR anisotrophy maps

As part of the work performed under NASA contract # NAS5-32648, we have computed the 3-point and 4-point correlation functions of the COBE-DNIR 2-year and 4-year anisotropy maps. The motivation for this study was to search for evidence of non-Gaussian statistical fluctuations in the temperature maps: skewness or asymmetry in the case of the 3-point function, kurtosis in the case of the 4-point function. Such behavior would have very significant implications for our understanding of the processes of galaxy formation, because our current models of galaxy formation predict that non-Gaussian features should not be present in the DMR maps. The results of our work showed that the 3-point correlation function is consistent with zero and that the 4-point function is not a very sensitive probe of non-Gaussian behavior in the COBE-DMR data. Our computation and analysis of 3-point correlations in the 2-year DMR maps was published in the Astrophysical Journal Letters, volume 446, page L67, 1995. Our computation and analysis of 3-point correlations in the 4-year DMR maps will be published, together with some additional tests, in the June 10, 1996 issue of the Astrophysical Journal Letters. Copies of both of these papers are attached as an appendix to this report.

Hinshaw, Gary↗

Preliminary DMR measurements of the CMB isotropy

The COBE Differential Microwave Radiometers (DMR) instrument has produced preliminary full-sky maps at frequencies 31.5, 53, and 90 GHz. The redundant channels and matched beams at three frequencies distinguish the DMR from previous large-scale surveys. Galactic emission is seen unambiguously at all three frequencies. The only large-scale anisotropy detected in the cosmic microwave background is the dipole anisotropy. There is no clear evidence for any other large-angular-scale feature in the maps. Without correcting for any systematic effects, we are able to place limits DeltaT/T sub 0 less than 3 x 10 exp -5 for the rms quadrupole amplitude, DeltaT/T sub 0 less than 4 x 10 exp -5 for monochromatic fluctuations, and DeltaT/T sub 0 less than 4 x 10 exp -5 for Gaussian fluctuations (all limits are 95 percent C.L. with TO = 2.735 K). The data limit DeltaT/T sub 0 less than 10 exp -4 for any feature larger than 7 deg. We briefly review the DMR and discuss some implications of these results in cosmology.

Smoot, G. F.↗

COBE DMR results and implications

This lecture presents early results obtained from the first six months of measurements of the Cosmic Microwave Background (CMB) by Differential Microwave Radiometers (DMR) aboard COBE and discusses significant cosmological implications. The DMR maps show the dipole anisotropy and some galactic emission but otherwise a spatially smooth early universe. The measurements are sufficiently precise that we must pay careful attention to potential systematic errors. Maps of galactic and local emission such as those produced by the FIRAS and DIRBE instruments will be needed to identify foregrounds from extragalactic emission and thus to interpret the results in terms of events in the early universe. The current DMR results are significant for Cosmology.

Smoot, George F.↗

Gaussian statistics of the cosmic microwave background: Correlation of temperature extrema in the COBE DMR two-year sky maps

We use the two-point correlation function of the extrema points (peaks and valleys) in the Cosmic Background Explorer (COBE) Differential Microwave Radiometers (DMR) 2 year sky maps as a test for non-Gaussian temperature distribution in the cosmic microwave background anisotropy. A maximum-likelihood analysis compares the DMR data to n = 1 toy models whose random-phase spherical harmonic components a(sub lm) are drawn from either Gaussian, chi-square, or log-normal parent populations. The likelihood of the 53 GHz (A+B)/2 data is greatest for the exact Gaussian model. There is less than 10% chance that the non-Gaussian models tested describe the DMR data, limited primarily by type II errors in the statistical inference. The extrema correlation function is a stronger test for this class of non-Gaussian models than topological statistics such as the genus.

Kogut, A.↗

Search for unresolved sources in the COBE-DMR two-year sky maps

We have searched the temperature maps from the Cosmic Background Explorer (COBE) Differential Microwave Radiometers (DMR) first two years of data for evidence of unresolved sources. The high-latitude sky (absolute value of b greater than 30 deg) contains no sources brighter than 192 microKelvin thermodynamic temperature (322 Jy at 53 GHz). The cumulative count of sources brighter than threshold T, N(greater than T), is consistent with a superposition of instrument noise plus scale-invariant spectrum of cosmic temperature fluctuations normalized to Q(sub rms-PS) = 17 microKelvin. We examine the temperature maps toward nearby clusters and find no evidence for any Sunyaev-Zel'dovich effect, delta y less than 7.3 x 10(exp -6) (95% CL) averaged over the DMR beam. We examine the temperature maps near the brightest expected radio sources and detect no evidence of significant emission. The lack of bright unresolved sources in the DMR maps, taken with anisotropy measurements on smaller angular scales, places a weak constraint on the integral number density of any unresolved Planck-spectrum sources brighter than flux density S, n(greater than S) less than 2 x 10(exp 4)/(S/1 Jy)(exp 2)/sr.

Kogut, A.↗

On determining the spectrum of primordial inhomogeneity from the COBE DMR sky maps: Method

The natural approach to a spectral analysis of data distributed on the sky employs spherical harmonic decomposition. A common problem encountered in practical astronomy is the lack of full sky coverage in the available data. For example, the removal of the Galactic plane data from the Cosmic Background Explorer (COBE) Differential Microwave Radiometer (DMR) sky maps compromises Fourier analysis of the cosmic microwave background (CMB) temperature distribution due to the loss of orthogonality of the spherical harmonics. An explicit method for constructing orthonormal functions on an incomplete (e.g., Galaxy-cut) sphere is presented. These functions should be used in the proper Fourier analysis of the COBE DMR sky maps to provide the correct input for the determination of the spectrum of primordial inhomogeneity. The results of such an analysis are presented in an accompanying Letter. A similar algebraic construction of appropriate functions can be devised for other astronomical applications.

Gorski, Krzysztof M.↗

On the rms anisotropy at 7 deg and 10 deg observed in the COBE-DMR two year sky maps

The frequency-independent rms temperature fluctuations determined from the Cosmic Background Explorer-Differential Microwave Radiometer (COBE-DMR) two-year sky maps are used to infer the parameter Q(sub rms-PS), which characterizes the normalization of power-law models of primordial cosmological temperature anisotropy, for a forced fit to a scale-invariant Harrison-Zel'dovich (n = 1) spectral model. Using a joint analysis of the 7 deg and 10 deg 'cross'-rms derived from both the 53 and 90 GHz sky maps, we find Q(sub rms-PS) = 17.0(sub -2.1 sup +2.5) micro Kelvin when the low quadrupole is included, and Q(sub rms-PS) = 19.4(sub -2.1 sup +2.3) micro Kelvin excluding the quadrupole. These results are consistent with the n = 1 fits from more sensitive methods. The effect of the low quadrupole derived from the COBE-DMR data on the inferred Q(sub rms-PS) normalization is investigated. A bias to lower Q(sub rms-PS) is found when the quadrupole is included. The higher normalization for a forced n = 1 fit is then favored by the cross-rms technique.

Banday, A. J.↗

COBE DMR-normalized open inflation cold dark matter cosmogony

A cut-sky orthogonal mode analysis of the 2 year COBE DMR 53 and 90 GHz sky maps (in Galactic coordinates) is used to determine the normalization of an open inflation model based on the cold dark matter (CDM) scenario. The normalized model is compared to measures of large-scale structure in the universe. Although the DMR data alone does not provide sufficient discriminative power to prefer a particular value of the mass density parameter, the open model appears to be reasonably consistent with observations when Omega(sub 0) is approximately 0.3-0.4 and merits further study.

Gorski, Krzysztof M.↗

Report on 3- and 4-Point Correlation Statistics in COBE DMR Anisotropy Maps

As part of the work performed under this contract, we have computed the 3- and 4-point correlation functions of the COBE-DMR 2-year and 4-year anisotropy maps. The results of our work showed that the 3-point correlation function is consistent with zero and that the 4-point function is not a very sensitive probe of non-Gaussian behavior in the COBE-DMR data.

Hinshaw, Gary↗

Design and Results of Differential Microwave Radiometers (DMR) on COBE

The Differential Microwave Radiometer (DMR) experiment on the Cosmic Background Explorer is in the final year of a scheduled four years of operation to measure large- and intermediate-scale anisotropies in the Cosmic Microwave Background (CMB). The DMR instrument comprises two independent radiometers at each of three frequencies, 31.5 53, and 90 GHz, where the frequencies were chosen to best separate the CMB from the foreground emissions from galactic dust and electrons...

Kogut, Alan J.↗

COBE Differential Microwave Radiometer (DMR) data processing techniques

The purpose of the Differential Microwave Radiometer (DMR) experiment on the Cosmic Background Explorer (COBE) satellite is to make whole-sky maps, at frequencies of 31.5, 53, and 90 GHz, of any departures of the Cosmic Microwave Background (CMB) from its mean value of 2.735 K. An elaborate software system is necessary to calibrate and invert the differential measurements, so as to make sky maps free from large scale systematic errors to levels less than a millionth of the CMB.

Jackson, P. D.↗

On determining the spectrum of primordial inhomogeneity from the COBE DMR sky maps: Results of two-year data analysis

A new technique of Fourier analysis on a cut sky has been applied to the two-year Cosmic Background Explorer (COBE) Differential Microwave Radiometer (DMR) 53 and 90 GHz sky maps. The Bayesian power spectrum estimation results are consistent with the Harrison-Zel'dovich n = 1 model. The maximum likelihood estimates of the usual parameters defining the power spectrum of primordial perturbations are n = 1.22 (1.02) and Q(sub rms-PS) = 17 (20) microK including (excluding) the quadrupole. A spectral-index-independent normalization is naturally expressed for the two-year maps in terms of the multipole amplitude a(sub 9) = 8.2 (8.3) microK (to approximately 12 sigma significance). The marginal likelihood function on n obtained by intergration with respect to a(sub 9) renders n = 1.17 +/- 0.31 (0.96 +/- 0.36).

Gorski, K. M.↗

Limits on three-point correlations in the COBE DMR first-year anisotropy maps

We compute the three-point temperature correlation function of the Cosmic Background Explorer (COBE) Differential Microwave Radiometer (DMR) first-year sky maps to search for non-Gaussian temperature fluctuations. The level of fluctuations seen in the computed correlation function are too large to be attributable solely to instrument noise. However the fluctuations are consistent with the level expected to result from a superposition of istrument noise and sky signal arising from a Gaussian power-law model of initial fluctuations, with a quadrupole normalized amplitude of 17 micro K and a power-law spectral index n = 1. We place limits on the amplitude of intrinsic three-point correlations with a variety of predicted functional forms.

Hinshaw, G.↗

Three-Point Correlations in the COBE DMR 2 Year Anisotropy Maps

We compute the three-point temperature correlation function of the COBE Differential Microwave Radiometer (DMR) 2 year sky maps to search for evidence of non-Gaussian temperature fluctuations. We detect three-point correlations in our sky with a substantially higher signal-to-noise ratio than from the first-year data. However, the magnitude of the signal is consistent with the level of cosmic variance expected from Gaussian fluctuations, even when the low-order multipole moments, up to l = 9, are filtered from the data. These results do not strongly constrain most existing models of structure formation, but the absence of intrinsic three-point correlations on large angular scales is an important consistency test for such models.

Hinshaw, G.↗