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At least 451 records · Page 25

Statistics of density maxima and the large-scale matter distribution

High peaks in Gaussian noise display enhanced clustering. The enhancement takes two forms: on large scales one obtains a linear amplification of the correlation function which is independent of scale. On smaller scales, but larger than the mass scale of the peaks themselves, a nonlinear (exponential) enhancement of the number density of high peaks in overdense regions arises. The large-scale correlations of Abell's rich clusters can be understood as a manifestation of this phenomenon. If the formation of bright galaxies favors the high overdensity peaks then the number of galaxies (per unit mass) in clusters and groups may be considerably enhanced. Consequences of these ideas for the density parameter and the large-scale matter distribution are discussed.

Kaiser, N.↗

Application of Statistically Derived CPAS Parachute Parameters

The Capsule Parachute Assembly System (CPAS) Analysis Team is responsible for determining parachute inflation parameters and dispersions that are ultimately used in verifying system requirements. A model memo is internally released semi-annually documenting parachute inflation and other key parameters reconstructed from flight test data. Dispersion probability distributions published in previous versions of the model memo were uniform because insufficient data were available for determination of statistical based distributions. Uniform distributions do not accurately represent the expected distributions since extreme parameter values are just as likely to occur as the nominal value. CPAS has taken incremental steps to move away from uniform distributions. Model Memo version 9 (MMv9) made the first use of non-uniform dispersions, but only for the reefing cutter timing, for which a large number of sample was available. In order to maximize the utility of the available flight test data, clusters of parachutes were reconstructed individually starting with Model Memo version 10. This allowed for statistical assessment for steady-state drag area (CDS) and parachute inflation parameters such as the canopy fill distance (n), profile shape exponent (expopen), over-inflation factor (C(sub k)), and ramp-down time (t(sub k)) distributions. Built-in MATLAB distributions were applied to the histograms, and parameters such as scale (sigma) and location (mu) were output. Engineering judgment was used to determine the "best fit" distribution based on the test data. Results include normal, log normal, and uniform (where available data remains insufficient) fits of nominal and failure (loss of parachute and skipped stage) cases for all CPAS parachutes. This paper discusses the uniform methodology that was previously used, the process and result of the statistical assessment, how the dispersions were incorporated into Monte Carlo analyses, and the application of the distributions in trajectory benchmark testing assessments with parachute inflation parameters, drag area, and reefing cutter timing used by CPAS.

Romero, Leah M.↗

Primordial non-Gaussianity from the completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey II: measurements in Fourier space with optimal weights

ABSTRACT We present measurements of the local primordial non-Gaussianity parameter $f_{\mathrm{NL}}^\mathrm{local} $from the clustering of 343 708 quasars with redshifts 0.8 < z < 2.2 distributed over 4808 deg2 from the final data release (DR16) of the extended Baryon Acoustic Oscillation Spectroscopic Survey (eBOSS), the largest volume spectroscopic survey up to date. Our analysis is performed in Fourier space, using the power spectrum monopole at very large scales to constrain the scale-dependent halo bias. We demonstrate the robustness of our analysis pipeline with EZ-mock catalogues that simulate the eBOSS DR16 target selection. We carefully assess the impact of systematics on our measurement, exploiting a novel neural network scheme for cleaning the DR16 sample as well as test multiple contamination removal methods. We estimate the bias due to the overcorrection of the systematic removal to be sub-dominant compared to the statistical uncertainty. We find fNL = −12 ± 21 (68 per cent confidence) for the main clustering sample including quasars with redshifts between 0.8 and 2.2, after applying redshift weighting techniques, designed for non-Gaussianity measurement from large scales structure, to optimize our analysis, which improve our results by 37 per cent.

79 ASTRONOMY AND ASTROPHYSICS↗

Identifying geological structures through microseismic cluster and burst analyses complementing active seismic interpretation

At the Decatur carbon capture and storage site (IL, USA) CO 2 has been injected from 2011–2014 and from 2017 to present near the base of the Lower Mt. Simon Sandstone saline reservoir, resulting in microseismicity. Microseismicity is mainly located in the basement and distributed in distinct spatial clusters. The lack of significant impedance contrasts within the basement makes the interpretation of active-source seismic reflection data challenging, however, recent reprocessing allowed to resolve faults above and at the top of the basement. These faults generally do not coincide with the location of microseismic events and their continuation to the general depth of the seismic events cannot be assumed. This paper shows how the interpretation of the microseismicity can complement structural interpretations of active-source seismic reflection data. In particular, we analyze clusters and bursts (abrupt increases) of microseismicity, identify unresolved, smaller-scale weaknesses and extract statistical parameters. These parameters allow comparisons with the interpreted faults, and with fracture sets intercepted by boreholes. During injection at the Decatur site, the injection pressure was kept far below fracture pressure, nevertheless, seismic events were induced and spread far beyond the expected extent of the CO 2 plume. We argue that local stress transfers related to the CO 2 injection reactivated pre-existing fractures within the critically stressed basement. Finally, we conducted a slip tendency analysis for faults interpreted from active seismic, selected cluster, bursts and nodal planes from focal mechanisms to determine if the interpreted structures are optimally oriented with respect to the stress regime. Our results suggest that the orientation of fractures close to the injection well, generally shows slight deviations from the optimal orientation for slip. This might indicate either slight local deviations of the maximum horizontal stress azimuth from the average direction used in the analysis, or the lack of optimally oriented fractures at this location.

58 GEOSCIENCES↗

DESI peculiar velocity survey – Fundamental Plane

The Dark Energy Spectroscopic Instrument (DESI) peculiar velocity survey aims to measure the peculiar velocities of early- and late-type galaxies within the DESI footprint using both the Fundamental Plane and optical Tully–Fisher relations. Direct measurements of peculiar velocities can significantly improve constraints on the growth rate of structure, reducing uncertainty by a factor of approximately 2.5 at redshift 0.1 compared to the DESI Bright Galaxy Survey’s redshift space distortion measurements alone. We assess the quality of stellar velocity dispersion measurements from DESI spectroscopic data. These measurements, along with photometric data from the Legacy Survey, establish the Fundamental Plane relation and determine distances and peculiar velocities of early-type galaxies. During survey validation, we obtain spectra for 6698 unique early-type galaxies, up to a photometric redshift of 0.15. 64 per cent of observed galaxies (4267) have relative velocity dispersion errors below 10 per cent. This percentage increases to 75 per cent if we restrict our sample to galaxies with spectroscopic redshifts below 0.1. We use the measured central velocity dispersion, along with photometry from the DESI Legacy Imaging Surveys, to fit the Fundamental Plane parameters using a 3D Gaussian maximum likelihood algorithm that accounts for measurement uncertainties and selection cuts. In addition, we conduct zero-point calibration using the absolute distance measurements to the Coma cluster, leading to a value of the Hubble constant, H 0 = 76.05 ± 0.35 (statistical) ±0.49 (systematic Fundamental Plane) ±4.86 (statistical due to calibration) km s –1 Mpc –1 ⁠. This H 0 value is within 2σ of Planck cosmic microwave background results and within 1σ of other low-redshift distance indicator-based measurements.

cosmological parameters↗

New approaches to an old problem: addressing spatial gaps in the World Stress Map

Abstract A well-recognized characteristic of the World Stress Map (WSM) database is the continued presence of large spatial gaps in the distribution of the data records despite the more than 40-year development history of the database. The current release has more than 30 000 high-quality (A–C) data records (often referred to as ‘stress indicators’), but while some continental areas (such as Australia) have seen a significant increase in spatial converge with the latest release, other continental regions (Africa, central Asia, most of South America) remain markedly sparse. In this contribution we (1) review the current state of the spatial distribution of stress indicators in the continental regions (above sea-level); (2) quantify the clustering of the stress indicators in the latest WSM release using the Hopkins statistic as a way to explore the current spatial distribution of the indicators and assess future WSM releases; and (3) present three approaches (joint inversion, seismic anisotropy and InSAR) that provide a way to fill in the gaps (both in the S Hmax orientation and principal stress magnitudes) in regions that lack active seismicity and where borehole drilling is cost prohibitive. These three approaches have the potential to guide procedures for improving a priori estimates of the ambient stress field in the Earth's crust and reduce the uncertainty in predicting both the magnitude and orientation of the principal tectonic stresses.

58 GEOSCIENCES↗

Superclustering in the explosion scenario

A simple toy model is used to study the spatial distribution of rich clusters in a generic type of explosion scenario. The model, parameterized by the distribution of shell radii and the filling factor, places spherical shells at random and identifies each 'knot' as a cluster. The resulting cluster correlation function is close to a power law extending to the diameter of the largest spheres. Richer clusters form at the intersections of bigger shells and so have stronger correlations. Typical shell radii and filling factors are required to produce the observed number density of clusters. Models with a power-law radius distribution also reproduce the richness distribution of clusters in the Abell catalog. Supercluster multiplicity functions, void probabilities, number counts, topology statistics, and velocity correlations confirm the presence of strong superclustering and quantify the non-Gaussian nature of the model.

Weinberg, David H.↗

Collected Notes on the Workshop for Pattern Discovery in Large Databases

These collected notes are a record of material presented at the Workshop. The core data analysis is addressed that have traditionally required statistical or pattern recognition techniques. Some of the core tasks include classification, discrimination, clustering, supervised and unsupervised learning, discovery and diagnosis, i.e., general pattern discovery.

Buntine, Wray↗

Fire Induced Rock Spalls as Long-Term Traps for Ash

Severe fires accelerate rock weathering by spalling and exfoliation, creating abundant peels, flakes or spalls. In the following years, these spalls serve as physical traps which accommodate fine particles of dust, ash, organic matter, etc. We searched for traces of ash trapped under the spalls, after four major fires: 1989, 2005, 2010 and 2015 in Israel. Samples were collected beneath the spalls that formed on the rock outcrops, and in the immediate vicinity above and below them. Three laboratory analyses were performed: pH, EC and color. Five mineral/organic compounds were measured across the Mid-Infrared (MIR) spectral region at diffuse reflectance infrared Fourier transform mode: Hydroxylapatite (HAp), charcoal, organic carbon, montmorillonite and kaolinite. Several statistical analyses were performed: MANOVA, PCA and silhouette analysis on K-means clustering. The results show evidence of ash trapped under the spalls formed during the 2005 and 2010 fires, 6 to 11 years after the fires. Charcoal presence is evident, as well as increased amounts of HAp and organic carbon. In the exposed soil above or below the burned rock outcrop, these values are lower. Negligible amounts of ash were measured 27 years after the fire. In the 2015 burned outcrop, large amounts of charcoal were found above and below the outcrop, but not under the spalls. It seems that on the carbonate slopes of Israel and under Mediterranean climate, the time required for spalls to begin functioning as traps is longer than one rainy season, while ash traces are preserved in these traps for a period of two-three decades.

ash↗

The Experiment for Cryogenic Large-aperture Intensity Mapping (EXCLAIM)

Submillimeter and far-IR spectroscopy provides insight into galaxy evolution through measurement of atomic and molecular line emission. The EXperiment for Cryogenic Large-Aperture Intensity Mapping (EXCLAIM) is a cryogenic balloon-borne instrument designed to carry out intensity mapping to measure the cumulative redshifted line emission from carbon monoxide and singly-ionized carbon to probe star formation in windows from the present to z=3.5. During this time, the rate of star formation dropped dramatically, while dark matter continued to cluster. Intensity mapping permits a blind and complete survey of emitting gas through statistics of cumulative brightness fluctuations. EXCLAIM achieves high sensitivity using a cryogenic telescope coupled to six integrated spectrometers with spectral resolving power R=512 and employing microwave kinetic inductance detectors. Here we summarize the status of the mission.

Maryam Rahmani↗

The Experiment for Cryogenic Large-aperture Intensity Mapping (EXCLAIM)

Submillimeter and far-IR spectroscopy provides insight into galaxy evolution through atomic and molecular line emission. The EXperiment for Cryogenic Large-Aperture Intensity Mapping (EXCLAIM) is a cryogenic balloon-borne instrument designed to carry out an intensity mapping to measure the cumulative redshifted line emission from carbon monoxide and singly-ionized carbon to probe star formation in windows from the present to z=3.5. During this time, the rate of star formation dropped dramatically, while dark matter continued to cluster. Intensity mapping permits a blind and complete survey of emitting gas through statistics of cumulative brightness fluctuations. EXCLAIM achieves high sensitivity using a cryogenic telescope coupled to six integrated spectrometers with spectral resolving power R = 512 and employing kinetic inductance detectors. Here we summarize the status of the mission.

Maryam Rahmani↗

The Experiment for Cryogenic Large-aperture Intensity Mapping (EXCLAIM)

Submillimeter and far-IR spectroscopy provides insight into galaxy evolution through measurement of atomic and molecular line emission. The EXperiment for Cryogenic Large-Aperture Intensity Mapping (EXCLAIM) is a cryogenic balloon-borne instrument designed to carry out intensity mapping to measure the cumulative redshifted line emission from carbon monoxide and singly-ionized carbon to probe star formation in windows from the present to z=3.5. During this time, the rate of star formation dropped dramatically, while dark matter continued to cluster. Intensity mapping permits a blind and complete survey of emitting gas through statistics of cumulative brightness fluctuations. EXCLAIM achieves high sensitivity using a cryogenic telescope coupled to six integrated spectrometers with spectral resolving power R=512 and employing microwave kinetic inductance detectors. Here we summarize the status of the mission.

Maryam Rahmani↗

Hydrogen Leak Modeling for Development of Smart Distributed Monitoring Under Unintended Releases

Hydrogen is a versatile and clean energy carrier that can be produced from various renewable sources such as wind, solar, and hydropower. Hydrogen has the potential to play a crucial role in decarbonizing industrial processes that are currently reliant on fossil fuels and provide long-duration and/or seasonal energy storage to enable electricity decarbonization. Hydrogen can also be used as a fuel for fuel cell vehicles, providing a zero-emission alternative to traditional internal combustion engines. DOE launched the Hydrogen Energy Earthshot (Hydrogen Shot) in June 2021 to reduce the cost of clean hydrogen by 80% to $1 per 1 kilogram in 1 decade ("1 1 1"). While promising, Hydrogen is highly-flammable, and in the presence of oxygen, it can form explosive mixtures. . Therefore, understanding leak scenarios is essential to evaluate and mitigate the safety risks associated with potential hydrogen leaks. An increased understanding of leak behavior, and having tools to model leaks, can help assess how hydrogen would disperse in different environments, influencing emergency response plans and safety measures, and identify potential issues with materials and design systems that can withstand the challenges posed by hydrogen. Recently, researchers have attempted to study hydrogen leaks for development of risk management strategies. However, the focus has been on closed or semi-closed spaces like storage rooms, vehicles, garages, and fueling stations - all promising locations for future hydrogen infrastructure. In this presentation, the modeling environment extends the span of research further by modeling hydrogen leak in an outdoor, open space. We will present the key challenges with modeling hydrogen leaks in an uncontrollable environment, how they were handled, and how modeling results informed sensor selection and placement. A Hydrogen research facility at the National Renewable Energy Laboratory (NREL) was used as a case study to model hydrogen leaks. In the future, Hydrogen wide area detection methodologies will be developed and tested at this site to monitor for unintended and operational hydrogen releases. The data generated from modeling will be used to develop a predictive model to detect hydrogen leak location based on concentration measured by sensors in this open space. Furthermore, the facility was also chosen because controlled hydrogen releases can be performed. A computational fluid dynamics (CFD) based modeling approach was taken to model hydrogen leak. The full-scale hydrogen facility was modeled with a large ambient domain. The electrolyzer at the facility can produce a controlled release rate of 27 kg-H2/hr. Site-specific atmospheric and weather condition data such as wind direction, wind speed at various altitudes, and temperature were used as inputs to the model. To capture the variability of weather conditions, a subset of the weather conditions experienced during daytime hours without precipitation over the course of three months was generated; using established data clustering techniques, a total of 100 condition sets were chosen. The results show statistical distributions and ranges of hydrogen concentrations at locations throughout the domain. These distributions are compared to experimental data from a constant mass flow, controlled hydrogen release at the facility. The stochastic wind conditions of the release make direct validation difficult, therefore, statistical comparison approaches were used. Wind conditions are found to significantly impact the release behavior, including direction and concentration. Sensor selection and placement is proposed for the facility and is now based on release behavior predicted for the facility given its weather patterns; this is much more informed than without the modeling results. The methodology and analysis procedure can be translated to other facilities using modified geometries and site-specific weather conditions. Hydrogen holds great promise as a renewable energy fuel, but ensuring safety in its production, storage, and use is paramount. Studying potential leak scenarios in an open space will help develop sensors to detect hydrogen on a large spectrum of concentration and eventually build a smart distributed monitoring system.

CFD↗

Using Machine Learning to Infer Material Properties of Debris Fragments from X-ray Images in the DebriSat Project

The DebriSat project is a collaboration effort with the NASA Orbital Debris Program Office, the U.S. Space Force Space Systems Command Center, The Aerospace Corporation, and the University of Florida. To date, over 200,000 fragments from this ground-based, hypervelocity impact experiment have been collected, and processing is underway to determine their physical characteristics, such as material, shape, color, characteristic length, and average cross-sectional area. The x-ray process is primarily used to identify the location of the fragments and estimated size for extraction, so that these physical characteristics can be assessed. This paper proposes a machine learning-based approach to characterize materials from x-ray images of debris fragments embedded in soft-catch foam used in the DebriSat project. The novel methodology discussed in this paper will highlight the use of x-ray imagery data to characterize these fragments without extraction or a human-in-the-loop. Both supervised and unsupervised machine learning techniques are utilized with this approach to infer the physical parameters of the fragments embedded in the soft-catch foam panels used in the impact experiment based on x-ray images of the foam panels. Additionally, 3D reconstructions of the extracted fragments are created with images taken from two different angles using the structure from motion (SfM) method. The characteristic lengths and shape from the 3D reconstruction, alongside the physical characteristics of the debris, are used in the inference of the material type. To develop and test the approach, a dataset of x-ray images of debris fragments of varying sizes and materials is collected. Supervised learning methods such as convolutional neural networks (CNNs), support vector machines (SVM), decision trees, and random forest classifiers are used due to the high-dimensional feature spaces of the debris and nonlinear decision boundaries for material categorization. Given the limited pre-labeled data of embedded debris materials smaller than 10 mm, unsupervised machine learning techniques such as clustering algorithms and autoencoders are used, in addition to supervised learning methods. The clustering algorithms group similar fragments together based on their physical properties, and autoencoders reduce the dimensionality of the x ray images and extract relevant features. The performance of the proposed approach's is analyzed using a range of statistical methods, including confusion matrices, receiver operating characteristic curves, and precision-recall curves. The results are compared with those obtained using a baseline approach that relies on manual identification and classification of debris fragments. To evaluate the effectiveness of different machine learning methods, statistical tests such as t-tests, ANOVA, and cross-validation are performed, comparing the performance of CNNs, SVMs, clustering algorithms, and autoencoders. Additional analysis needs to be conducted to identify any sources of bias or variability that may affect the results, such as variations in imaging conditions or fragmentation patterns. Other topics explored are limitations, refinements, and the potential use of semi-supervised learning techniques, such as self-training to label unlabeled datasets and co-training using x-ray images taken from two different angles as two different models.

Saik Anam Siam↗

Using Machine Learning to Infer Material Properties of Debris Fragments from X-ray Images in the DebriSat Project

The DebriSat project is a collaboration effort with the NASA Orbital Debris Program Office, the U.S. Space Force Space Systems Command Center, The Aerospace Corporation, and the University of Florida. To date, over 200,000 fragments from this ground-based, hypervelocity impact experiment have been collected, and processing is underway to determine their physical characteristics, such as material, shape, color, characteristic length, and average cross-sectional area. The x-ray process is primarily used to identify the location of the fragments and estimated size for extraction, so that these physical characteristics can be assessed. This paper proposes a machine learning-based approach to characterize materials from x-ray images of debris fragments embedded in soft-catch foam used in the DebriSat project. The novel methodology discussed in this paper will highlight the use of x-ray imagery data to characterize these fragments without extraction or a human-in-the-loop. Both supervised and unsupervised machine learning techniques are utilized with this approach to infer the physical parameters of the fragments embedded in the soft-catch foam panels used in the impact experiment based on x-ray images of the foam panels. Additionally, 3D reconstructions of the extracted fragments are created with images taken from two different angles using the structure from motion (SfM) method. The characteristic lengths and shape from the 3D reconstruction, alongside the physical characteristics of the debris, are used in the inference of the material type. To develop and test the approach, a dataset of x-ray images of debris fragments of varying sizes and materials is collected. Supervised learning methods such as convolutional neural networks (CNNs), support vector machines (SVM), decision trees, and random forest classifiers are used due to the high-dimensional feature spaces of the debris and nonlinear decision boundaries for material categorization. Given the limited pre-labeled data of embedded debris materials smaller than 10 mm, unsupervised machine learning techniques such as clustering algorithms and autoencoders are used, in addition to supervised learning methods. The clustering algorithms group similar fragments together based on their physical properties, and autoencoders reduce the dimensionality of the x ray images and extract relevant features. The performance of the proposed approach's is analyzed using a range of statistical methods, including confusion matrices, receiver operating characteristic curves, and precision-recall curves. The results are compared with those obtained using a baseline approach that relies on manual identification and classification of debris fragments. To evaluate the effectiveness of different machine learning methods, statistical tests such as t-tests, ANOVA, and cross-validation are performed, comparing the performance of CNNs, SVMs, clustering algorithms, and autoencoders. Additional analysis needs to be conducted to identify any sources of bias or variability that may affect the results, such as variations in imaging conditions or fragmentation patterns. Other topics explored are limitations, refinements, and the potential use of semi-supervised learning techniques, such as self-training to label unlabeled datasets and co-training using x-ray images taken from two different angles as two different models.

Saik Anam Siam↗

Topological Interpretability for Deep Learning

With the growing adoption of AI-based systems across everyday life, the need to understand their decision-making mechanisms is correspondingly increasing. The level at which we can trust the statistical inferences made from AI-based decision systems is an increasing concern, especially in high-risk systems such as criminal justice or medical diagnosis, where incorrect inferences may have tragic consequences. Despite their successes in providing solutions to problems involving real-world data, deep learning (DL) models cannot quantify the certainty of their predictions. These models are frequently quite confident, even when their solutions are incorrect. This work presents a method to infer prominent features in two DL classification models trained on clinical and non-clinical text by employing techniques from topological and geometric data analysis. We create a graph of a model's feature space and cluster the inputs into the graph's vertices by the similarity of features and prediction statistics. We then extract subgraphs demonstrating high-predictive accuracy for a given label. These subgraphs contain a wealth of information about features that the DL model has recognized as relevant to its decisions. We infer these features for a given label using a distance metric between probability measures, and demonstrate the stability of our method compared to the LIME and SHAP interpretability methods. This work establishes that we may gain insights into the decision mechanism of a DL model. This method allows us to ascertain if the model is making its decisions based on information germane to the problem or identifies extraneous patterns within the data.

Spannaus, Adam↗

Global radar units on Venus derived from statistical analysis of Pioneer Venus Orbiter radar data

The classification of surface radar units on Venus using an unsupervised cluster analysis of Pioneer Venus radar reflectivity and root-mean-square (rms)-slope data is described. The advantages of the unsupervised analysis are discussed. F tests are utilized to evaluate the numerical significance of the clusters. The derived rms-slope data and reflectivity for 15 radar units are presented. The relations between radar data bases and elevation are studied. The lowlands, rolling plains, highlands, and mountainous surface of Venus are examined. The geology of Venus landing sites and radar properties, and the surface radar reflectivity images and earth-based images are compared. The spatial relations between classification units are calculated. It is concluded that the unsupervised analysis data correlate well with Head et al. (1985b) data and produce more detailed classification images.

Davis, P. A.↗

Measuring the Redshift Dependence of The Cosmic Microwave Background Monopole Temperature With Planck Data

We study the capability of Planck data to constrain deviations of the cosmic microwave background (CMB) blackbody temperature from adiabatic evolution using the thermal Sunyaev-Zeldovich anisotropy induced by clusters of galaxies. We consider two types of data sets depending on how the cosmological signal is removed: using a CMB template or using the 217 GHz map. We apply two different statistical estimators, based on the ratio of temperature anisotropies at two different frequencies and on a fit to the spectral variation of the cluster signal with frequency. The ratio method is biased if CMB residuals with amplitude approximately 1 microK or larger are present in the data, while residuals are not so critical for the fit method. To test for systematics, we construct a template from clusters drawn from a hydro-simulation included in the pre-launch Planck Sky Model. We demonstrate that, using a proprietary catalog of X-ray-selected clusters with measured redshifts, electron densities, and X-ray temperatures, we can constrain deviations of adiabatic evolution, measured by the parameter α in the redshift scaling T (z) = T0(1 + z)(sup 1−alpha), with an accuracy of sigma(sub alpha) = 0.011 in the most optimal case and with sigma alpha = 0.018 for a less optimal case. These results represent a factor of 2-3 improvement over similar measurements carried out using quasar spectral lines and a factor 6-20 with respect to earlier results using smaller cluster samples.

cosmic background radiation↗