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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Combination of cluster number counts and two-point correlations: validation on mock Dark Energy Survey

ABSTRACT We present a method of combining cluster abundances and large-scale two-point correlations, namely galaxy clustering, galaxy–cluster cross-correlations, cluster autocorrelations, and cluster lensing. This data vector yields comparable cosmological constraints to traditional analyses that rely on small-scale cluster lensing for mass calibration. We use cosmological survey simulations designed to resemble the Dark Energy Survey Year 1 (DES-Y1) data to validate the analytical covariance matrix and the parameter inferences. The posterior distribution from the analysis of simulations is statistically consistent with the absence of systematic biases detectable at the precision of the DES-Y1 experiment. We compare the χ2 values in simulations to their expectation and find no significant difference. The robustness of our results against a variety of systematic effects is verified using a simulated likelihood analysis of DES-Y1-like data vectors. This work presents the first-ever end-to-end validation of a cluster abundance cosmological analysis on galaxy catalogue level simulations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Improving Cosmological Constraints from Galaxy Cluster Number Counts with CMB-cluster-lensing Data: Results from the SPT-SZ Survey and Forecasts for the Future

Abstract We show the improvement to cosmological constraints from galaxy cluster surveys with the addition of cosmic microwave background (CMB)-cluster lensing data. We explore the cosmological implications of adding mass information from the 3.1 σ detection of gravitational lensing of the CMB by galaxy clusters to the Sunyaev–Zel’dovich (SZ) selected galaxy cluster sample from the 2500 deg 2 SPT-SZ survey and targeted optical and X-ray follow-up data. In the ΛCDM model, the combination of the cluster sample with the Planck power spectrum measurements prefers σ 8 Ω m / 0.3 0.5 = 0.831 ± 0.020 . Adding the cluster data reduces the uncertainty on this quantity by a factor of 1.4, which is unchanged whether the 3.1 σ CMB-cluster lensing measurement is included or not. We then forecast the impact of CMB-cluster lensing measurements with future cluster catalogs. Adding CMB-cluster lensing measurements to the SZ cluster catalog of the ongoing SPT-3G survey is expected to improve the expected constraint on the dark energy equation of state w by a factor of 1.3 to σ ( w ) = 0.19. We find the largest improvements from CMB-cluster lensing measurements to be for σ 8 , where adding CMB-cluster lensing data to the cluster number counts reduces the expected uncertainty on σ 8 by respective factors of 2.4 and 3.6 for SPT-3G and CMB-S4.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

The SRG/eROSITA All-Sky Survey: Constraints on the structure growth from cluster number counts

Recent advancements in methods used in wide-area surveys have demonstrated the reliability of the number density of galaxy clusters as a viable tool for precision cosmology. Beyond testing the current cosmological paradigm, cluster number counts can also be used to investigate the discrepancies currently affecting cosmological measurements. In particular, cosmological studies based on cosmic shear and other large-scale structure probes routinely find a value for the amplitude of the fluctuations in the universe S​ 8 = σ 8 (Ω m /0.3) 0.5 smaller than the one inferred from the primary cosmic microwave background. In this work, we investigate this tension by measuring structure evolution across cosmic time as probed by the number counts of massive halos with the first SRG/eROSITA All-Sky Survey cluster catalog in the western Galactic hemisphere, complemented with the overlapping Dark Energy Survey Year-3, Kilo-Degree Survey, and Hyper Suprime-Cam data for weak lensing mass calibration, by implementing two different parameterizations and a model-agnostic method. In the first model, we measured the cosmic linear growth index as γ = 1.19 ± 0.21, which is in tension with the standard value of γ = 0.55 but in good statistical agreement with other large-scale structure probes. The second model is a phenomenological scenario in which we rescale the linear matter power spectrum at low redshift to investigate a potential reduction of structure formation, and it provided similar results. Finally, in a third strategy, we considered a standard ΛCDM cosmology, but we separated the cluster catalog into five redshift bins, measuring the cosmological parameters in each and inferring the evolution of the structure formation, finding hints of a reduction. Interestingly, the S ​8 value inferred from the number counts of the cluster eRASS1 when we add a degree of freedom to the matter power spectrum recovers the value inferred by cosmic shear studies. The observed reduction in the growth rate or systematic uncertainties associated with various measurements may account for the discrepancy in the S 8 ​ values suggested between cosmic shear probes and eROSITA cluster number counts and Planck CMB measurements.

cosmological parameters↗
DOE OSTI2025

The SRG/eROSITA All-Sky Survey: Constraints on ultralight axion dark matter through galaxy cluster number counts

Ultralight axions are hypothetical scalar particles that influence the evolution of large-scale structures of the Universe. Depending on their mass, they can potentially be part of the dark matter component of the Universe as candidates commonly referred to as fuzzy dark matter. While strong constraints have been established for pure fuzzy dark matter models, the more general scenario where ultralight axions constitute only a fraction of the dark matter has been limited to only a few observational probes. In this work, we use the galaxy cluster number counts obtained from the first All-Sky Survey (eRASS1) of the SRG/eROSITA mission together with gravitational weak lensing data from the Dark Energy Survey, the Kilo-Degree Survey, and the Hyper Suprime-Cam to constrain the fraction of ultralight axions in the mass range 10 −32 eV to 10 −24 eV. We put upper bounds on the ultralight axion relic density Ω a in independent logarithmic axion mass bins by performing a full cosmological parameter inference. We find an exclusion region in the intermediate ultralight axion mass regime with the tightest bounds reported so far in the mass bins around m a = 10 −27 eV with Ω a < 0.0035 and m a = 10 −26 eV with Ω a < 0.0079; both are at a 95% confidence level. When combined with cosmic microwave background probes, these bounds are tightened to Ω a < 0.0030 in the m a = 10 −27 eV mass bin and Ω a < 0.0058 in the m a = 10 −26 eV mass bin, with both at a 95% confidence level. This is the first time that constraints on ultralight axions have been obtained using the growth of structure measured by galaxy cluster number counts. These results pave the way for large surveys, which can be utilized to obtain tight constraints on the mass and relic density of ultralight axions with better theoretical modeling of the abundance of halos.

cosmological parameters↗
DOE OSTI2020

Clusters of COVID ‐19 in long‐term care hospitals and facilities in Japan from 16 January to 9 May 2020

Aim To clarify the association of cluster number and size of coronavirus disease 2019 (COVID‐19) in long‐term care (LTC) hospitals/facilities, general medical/welfare facilities and non‐medical/welfare facilities with morbidity and mortality in 47 prefectures during 16 January to 9 May 2020 in Japan. Methods Information on COVID‐19 clusters ( n ≥2), and morbidity and mortality of COVID‐19 was collected. Results A total of 381 clusters with 3786 infected cases were collected, accounting for 23.9% of 15 852 cumulated cases on 9 May 2020. Although the cluster number (/10 7 subjects) in LTC hospitals/facilities was significantly smaller compared with those in the other two groups, the cluster size in LTC hospitals/facilities was significantly larger than that in non‐medical/welfare facilities. Cluster numbers in general medical/welfare facilities and in non‐medical/welfare facilities were significantly positively correlated with morbidity (/10 5 ), indicating relatively early identification of clusters in these facilities. Unlike in these facilities, cluster size in LTC hospitals/facilities was significantly positively correlated with morbidity, indicating that clusters in LTC hospitals/facilities were finally identified after already having grown to a large size in areas where infection was prevalent. Multivariate logistic regression analysis showed that both cluster number and cluster size only in LTC hospitals/facilities were independently associated with higher mortality (≥median 0.64/10 5 subjects) after adjustment. Conclusions Preventive efforts against COVID‐19 outbreaks even at the early phase of the epidemic are critically important in LTC hospitals/facilities, as both the larger number and size of clusters only in LTC hospitals/facilities were independently linked to higher mortality in prefectures in Japan. Geriatr Gerontol Int 2020; 20: 715–719 .

Iritani, Osamu↗
DOE OSTI2022

A Decentralized Approach for Modeling Organized Convection Based on Thermal Populations on Microgrids

Abstract In this study, a spectral model for convective transport is coupled to a thermal population model on a two‐dimensional horizontal “microgrid,” covering the typical gridbox size of general circulation models. The goal is to explore new ways of representing impacts of spatial organization in cumulus cloud fields. The thermals are considered the smallest building block of convection, with thermal life cycle and movement represented through binomial functions. Thermals interact through two simple rules, reflecting pulsating growth and environmental deformation. Long‐lived thermal clusters thus form on the microgrid, exhibiting scale growth and spacing that represent simple forms of spatial organization and memory. Size distributions of cluster number are diagnosed from the microgrid through an online clustering algorithm, and provided as input to a spectral multiplume eddy‐diffusivity mass flux scheme. This yields a decentralized transport system, in that the thermal clusters acting as independent but interacting nodes that carry information about spatial structure. The main objectives of this study are (a) to seek proof of concept of this approach, and (b) to gain insight into impacts of spatial organization on convective transport. Single‐column model experiments demonstrate satisfactory skill in reproducing two observed cases of continental shallow convection. Metrics expressing self‐organization and spatial organization match well with large‐eddy simulation results. We find that in this coupled system, spatial organization impacts convective transport primarily through the scale break in the size distribution of cluster number. The rooting of saturated plumes in the subcloud mixed layer plays a key role in this process.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Distribution Cutoff for Clusters near the Gel Point

The mechanical and dynamic properties of developing networks near the gel point are susceptible to the distribution of clusters coexisting with percolating networks. The distribution of cluster numbers follows a broad power law, wrapped by a cutoff function that rapidly decays at a characteristic size. The form of the cutoff function has been speculated based on known results from lattice percolation and, in certain cases, solved. We obtained this cutoff function from simulated dynamic clusters of polymeric precursor chains using a hybrid Monte Carlo algorithm. The results obtained from three different precursor chain lengths are consistent with each other and are consistent with the expectation from lattice percolation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Building an Efficient Cluster Cosmology Software Package for Modeling Cluster Counts and Lensing

We introduce a software suite developed for galaxy cluster cosmological analysis with the Dark Energy Survey Data. Cosmological analyses based on galaxy cluster number counts and weak-lensing measurements need efficient software infrastructure to explore an increasingly large parameter space, and account for various cosmological and astrophysical effects. Our software package is designed to model the cluster observables in a wide-field optical survey, including galaxy cluster counts, their averaged weak-lensing masses, or the cluster's averaged weak-lensing radial signals. To ensure maximum efficiency, this software package is developed in C++ in the CosmoSIS software framework, making use of the CUBA integration library. We also implement a testing and validation scheme to ensure the quality of the package. We demonstrate the effectiveness of this development by applying the software to the Dark Energy Survey Year 1 galaxy cluster cosmological data sets, and acquired cosmological constraints that are consistent with the fiducial Dark Energy Survey analysis.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2024

The SRG/eROSITA All-Sky Survey: Dark Energy Survey year 3 weak gravitational lensing by eRASS1 selected galaxy clusters

Context. Number counts of galaxy clusters across redshift are a powerful cosmological probe if a precise and accurate reconstruction of the underlying mass distribution is performed – a challenge called mass calibration. With the advent of wide and deep photometric surveys, weak gravitational lensing (WL) by clusters has become the method of choice for this measurement. Aims. We measured and validated the WL signature in the shape of galaxies observed in the first three years of the Dark Energy Survey (DES Y3) caused by galaxy clusters and groups selected in the first all-sky survey performed by SRG (Spectrum Roentgen Gamma)/eROSITA (eRASS1). These data were then used to determine the scaling between the X-ray photon count rate of the clusters and their halo mass and redshift. Methods. We empirically determined the degree of cluster member contamination in our background source sample. The individual cluster shear profiles were then analyzed with a Bayesian population model that self-consistently accounts for the lens sample selection and contamination and includes marginalization over a host of instrumental and astrophysical systematics. To quantify the accuracy of the mass extraction of that model, we performed mass measurements on mock cluster catalogs with realistic synthetic shear profiles. This allowed us to establish that hydrodynamical modeling uncertainties at low lens redshifts (z < 0.6) are the dominant systematic limitation. At high lens redshift, the uncertainties of the sources’ photometric redshift calibration dominate. Results. With regard to the X-ray count rate to halo mass relation, we determined its amplitude, its mass trend, the redshift evolution of the mass trend, the deviation from self-similar redshift evolution, and the intrinsic scatter around this relation. Conclusions. The mass calibration analysis performed here sets the stage for a joint analysis with the number counts of eRASS1 clusters to constrain a host of cosmological parameters. We demonstrate that WL mass calibration of galaxy clusters can be performed successfully with source galaxies whose calibration was performed primarily for cosmic shear experiments, opening the way for the cluster cosmological exploitation of future optical and NIR surveys like Euclid and LSST.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Velocity dispersion and dynamical mass for 270 galaxy clusters in the Planck PSZ1 catalogue

Here, we present the velocity dispersion and dynamical mass estimates for 270 galaxy clusters included in the first Planck Sunyaev-Zeldovich (SZ) source catalogue, the PSZ1. Part of the results presented here were achieved during a two-year observational program, the ITP, developed at the Roque de los Muchachos Observatory (La Palma, Spain). In the ITP we carried out a systematic optical follow-up campaign of all the 212 unidentified PSZ1 sources in the northern sky that have a declination above –15° and are without known counterparts at the time of the publication of the catalogue. We present for the first time the velocity dispersion and dynamical mass of 58 of these ITP PSZ1 clusters, plus 35 newly discovered clusters that are not associated with the PSZ1 catalogue. Using Sloan Digital Sky Survey archival data, we extend this sample, including 212 already confirmed PSZ1 clusters in the northern sky. Using a subset of 207 of these galaxy clusters, we constrained the M SZ –M dyn scaling relation, finding a mass bias of (1 – B) = 0.83 ± 0.07(stat) ± 0.02(sys). We show that this value is consistent with other results in the literature that were obtained with different methods (X-ray, dynamical masses, or weak-lensing mass proxies). This result cannot dissolve the tension between primordial cosmic microwave background anisotropies and cluster number counts in the Ω M –σ 8 plane.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Extending empirical constraints on the SZ–mass scaling relation to higher redshifts via HST weak lensing measurements of nine clusters from the SPT-SZ survey at z ≳ 1

We present a Hubble Space Telescope (HST) weak gravitational lensing study of nine distant and massive galaxy clusters with redshifts 1.0 ≲ z ≲ 1.7 (z median = 1.4) and Sunyaev Zel’dovich (SZ) detection significance ξ > 6.0 from the South Pole Telescope Sunyaev Zel’dovich (SPT-SZ) survey. We measured weak lensing galaxy shapes in HST/ACS F606W and F814W images and used additional observations from HST/WFC3 in F110W and VLT/FORS2 in U HIGH to preferentially select background galaxies at z ≳ 1.8, achieving a high purity. We combined recent redshift estimates from the CANDELS/3D-HST and HUDF fields to infer an improved estimate of the source redshift distribution. We measured weak lensing masses by fitting the tangential reduced shear profiles with spherical Navarro-Frenk-White (NFW) models. We obtained the largest lensing mass in our sample for the cluster SPT-CL J2040—4451, thereby confirming earlier results that suggest a high lensing mass of this cluster compared to X-ray and SZ mass measurements. Combining our weak lensing mass constraints with results obtained by previous studies for lower redshift clusters, we extended the calibration of the scaling relation between the unbiased SZ detection significance ζ and the cluster mass for the SPT-SZ survey out to higher redshifts. We found that the mass scale inferred from our highest redshift bin (1.2 < z < 1.7) is consistent with an extrapolation of constraints derived from lower redshifts, albeit with large statistical uncertainties. Thus, our results show a similar tendency as found in previous studies, where the cluster mass scale derived from the weak lensing data is lower than the mass scale expected in a PlanckνΛCDM (i.e. νΛ cold dark matter) cosmology given the SPT-SZ cluster number counts.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

A machine learning study on spinodal clumping in heavy ion collisions

Possible observables of baryon number clustering due to the instabilities occurring at a first order QCD phase transition are discussed. The dynamical formation of baryon clusters at a QCD phase transition can be described by numerical fluid dynamics, augmented with a gradient term and an equation of state with a mechanically unstable region. It is shown that the dynamical description of this phase transition, in nuclear collisions, will lead to the formation of dense baryon clusters at the phase boundary. State-of-the-art machine learning methods find that the coordinate space clumping leaves characteristic imprints on the spatial net density distribution in almost every event. On the other hand the momentum distributions do not show any clear event-by-event features. Lastly, it is shown that the 'third order' cumulant, the skewness, shows a peak at the beam energy where the system, created in the heavy ion collision, reaches the deconfinement phase transition.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Affinity propagation clustering of full-field, high-spatial-dimensional measurements for robust output-only modal identification: A proof-of-concept study

Determination of the model order is a challenging problem in system identification, especially in output-only or operational modal identification where some modes are weakly excited. While existing methods such as the stabilization diagram method (spectral information) are effective, they do not scale to high-dimensional data, which is usually needed for high-fidelity characterization of structural dynamics and has been made available in the emerging full-field measurement techniques using optical methods such as photogrammetry and laser vibrometers. In this proof-of-concept study we present a new non-parametric, data-driven approach for robust output-only identification of high-spatial-dimensional modal parameters of basic structures by efficiently processing and interactively exploiting the full-field measurement (i.e., very dense spatial measurement points). Specifically, we first over-estimate the system model once, producing a pool of candidate modes associated with their modal frequencies and full-field, high-spatial-dimensional mode shapes. This is accomplished by a data-driven method termed affinity propagation clustering (APC), where the active clusters, which are the active modes in our formulations, emerge from the "message-passing" procedure and does not require a pre-determination of the cluster number (mode or model order). Next, rather than using the spectral information to distinguish the physical and spurious modes in the stabilization diagram method, we exploit and visualize the spatial, full-field mode shape associated with each candidate mode to do so. We conduct extensive experiments on basic structural models with comparisons to a few existing methods. The results indicate that the new method is computationally efficient for identifying high-spatial-dimensional modal parameters, and robust to identify weak modes by exploiting the full-field measurement. We also discuss its applicability and limitations for structures with complex geometry (shapes).

42 ENGINEERING↗
DOE OSTI2023

Application of unsupervised deep learning to image segmentation and in-situ contact angle measurements in a CO 2 -water-rock system

Rock surface wettability is a critical property that regulates multiphase flows in porous media, which can be quantified using the surface contact angle (CA). X-ray micro-computed tomography (μCT) provides an effective approach to in-situ measurements of surface CAs. However, the CA measurement accuracy depends significantly on the quality of CT image segmentation, which is the clustering of CT pixels into separate phases. Inspired by this, we developed a deep learning (DL)-based CA measurement workflow. Motivated by the recent tremendous progress in unsupervised learning techniques and aiming to avoid expensive manual data annotations, an unsupervised DL pipeline for CT image segmentation was proposed and implemented, which includes unsupervised model training and post-processing. The unsupervised model training was driven by a novel loss function constrained with feature similarity and spatial continuity and implemented by iterative forward and backward paths; the former clustered the pixel-wise feature vectors extracted by convolution neural networks, whereas the latter updated the parameters using gradient descent. An over-segmentation strategy was adopted for model training. The post-processing steps based on agglomerative hierarchical clustering (AHC) were implemented to further merge the over-segmented model output to the desired cluster number, which is intended to improve the efficiency of image segmentation. The developed unsupervised DL pipeline was compared with other commonly-used image segmentation methods using pixel-wise and physics-based evaluation metrics on a synthetic raw-image dataset, which had a known ground truth. The unsupervised DL pipeline showed the best performance. Next, the segmented images were input to an automatic CA measurement tool, and the results were validated by comparisons with manual measurements. The CA values from the manual and automatic measurements showed similar distributions and statistical properties. The automatic measurement demonstrated a wider spectrum because of the much larger number of measurement data points. The primary novelty of the unsupervised DL pipeline developed in this study lies in the novel loss function and the over-segmentation strategy associated with AHC post-processing. Finally, the workflow has been proven an efficient tool for pore-scale wettability characterization, which has a wide range of applications in fundamental studies of multiphase flows in natural porous media, which have critical implications to geological carbon sequestration, hydrocarbon energy recovery, and contaminant transport in groundwater.

42 ENGINEERING↗
DOE OSTI2022

Extracting dark-matter velocities from halo masses: A reconstruction conjecture

Increasing attention has recently focused on nontraditional dark-matter production mechanisms which result in primordial dark-matter velocity distributions with highly nonthermal shapes. Here, in this paper, we undertake an assessment of how the detailed shape of a general dark-matter velocity distribution impacts structure formation in the nonlinear regime. In particular, we investigate the impact on the halo-mass and subhalo-mass functions, as well as on astrophysical observables such as satellite and cluster-number counts. We find that many of the standard expectations no longer hold in situations in which this velocity distribution takes a highly nontrivial, even multimodal shape. For example, we find that the nominal free-streaming scale alone becomes insufficient to characterize the effect of free-streaming on structure formation. In addition, we propose a simple one-line conjecture which can be used to “reconstruct” the primordial dark-matter velocity distribution directly from the shape of the halo-mass function. Although our conjecture is completely heuristic, we show that it successfully reproduces the salient features of the underlying dark-matter velocity distribution even for nontrivial distributions which are highly nonthermal and/or multimodal, such as might occur for nonminimal dark sectors. Moreover, since our approach relies only on the halo-mass function, our conjecture provides a method of probing dark-matter properties even for scenarios in which the dark and visible sectors interact only gravitationally.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

An Integrated View of Hydraulic Induced Fracture Geometry in Hydraulic Fracture Test Site 2

Until relatively recently, most of our understanding about the geometry of hydraulically induced fractures in unconventional reservoirs has been inferred by patterns observed via microseismic monitoring. Important information, such as fracture-azimuth, fracture-length, height-growth and other geometric data has been determined from microseismic “event clouds”, which in turn has been used to interpret a stimulated reservoir volume (SRV). It is important to note that relatively limited microseismic results have been acquired in conjunction with other diagnostics and validated in a consistent manner. The Hydraulic Fracture Test Site 2 (HFTS2) in the Permian Delaware Basin provides a unique opportunity to compare the frac geometry interpretations derived from multiple frac diagnostic tools and at multiple scales. The integrated view that emerges from all the frac diagnostic tools is that in HFTS2 the geometries of hydraulically induced fractures are not random and with a highly complex branching architecture. In fact, the fracs are mostly vertical, parallel plana domains, and both vertically and horizontally asymmetrical. The hydraulically induced fractures have a very consistent azimuth with a strike that matches SHmax. At the distance corresponding to the spacing between nearby wells in HFTS2, ~ 660 ft, the pattern of strain interceptions observed from Low Frequency Distributed Acoustic Sensing (LF-DAS) shows far-field dimensions that are consistent with the stage length dimensions at the stimulated well. In general, larger stages with higher number of clusters create larger number of interceptions and wider stimulated intervals than that of shorter stages with the same cluster to cluster spacing and fewer clusters. The hydraulically induced fractures also show a clear tendency to preferentially grow upwards as shown by the microseismic data, by the relative lower number of LF-DAS interceptions in nearby deeper wells and finally by the observed LF-DAS strain patterns in the vertical observation well. In the near field, the geometries of the fracture domains are also consistent. As shown by the distributed strain measurements obtained during production via the monitoring of Rayleigh Frequency Shift (RFS), each cluster has a separate, non-overlapping, frac-zone-domain that are generally centered around the locations of each perforation cluster. Overall, the hydraulically induced fractures in HFTS2 are interpreted as occurring in swarms associated with individual clusters, each with its own unique geometry but with similar predictable geometric and propagation tendencies. The comprehensive frac diagnostic program in HFTS2, the well-pad layout that permits the investigation of the Downloaded from http://onepetro.org/URTECONF/proceedings-pdf/21URTC/1-21URTC/D011S018R003/2477511/urtec-2021-5396-ms.pdf/1 by Carol Worster on 28 February 2022 URTeC 5396 2 impact that nearby producing wells has on frac geometry, together with an innovative Design of Experiment (DofE) that includes sequential-fracing, consistent cluster-to-cluster distances and few single-perforation cluster stages, has allowed us to gain a unique perspective about the geometry of the induced hydraulic fractures on this pad. Finally, the HFTS2 dataset has also provided us with a better understanding of the applicability and limitations of the different diagnostic tools.

58 GEOSCIENCES↗
DOE OSTI2023

The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

The hypersimplex Δ <#comment/> k + 1 , n \Delta _{k+1,n} is the image of the positive Grassmannian G r k + 1 , n ≥ <#comment/> 0 Gr^{\geq 0}_{k+1,n} under the moment map. It is a polytope of dimension n − <#comment/> 1 n-1 in R n \mathbb {R}^n . Meanwhile, the amplituhedron A n , k , 2 ( Z ) \mathcal {A}_{n,k,2}(Z) is the projection of the positive Grassmannian G r k , n ≥ <#comment/> 0 Gr^{\geq 0}_{k,n} into the Grassmannian G r k , k + 2 Gr_{k,k+2} under a map Z ~ <#comment/> \tilde {Z} induced by a positive matrix Z ∈ <#comment/> M a t n , k + 2 > 0 Z\in Mat_{n,k+2}^{>0} . Introduced in the context of scattering amplitudes , it is not a polytope, and has full dimension 2 k 2k inside G r k , k + 2 Gr_{k,k+2} . Nevertheless, there seem to be remarkable connections between these two objects via T-duality , as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes —images of positroid cells of G r k + 1 , n ≥ <#comment/> 0 Gr^{\geq 0}_{k+1,n} under the moment map—translate into sign conditions characterizing the T-dual Grasstopes —images of positroid cells of G r k , n ≥ <#comment/> 0 Gr^{\geq 0}_{k,n} under Z ~ <#comment/> \tilde {Z} . Moreover, we subdivide the amplituhedron into chambers , just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]: a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron A n , k , 2 ( Z ) \mathcal {A}_{n,k,2}(Z) for all Z Z . Moreover, we prove Arkani-Hamed–Thomas–Trnka’s conjectural sign-flip characterization of A n , k , 2 \mathcal {A}_{n,k,2} , and Łukowski–Parisi–Spradlin–Volovich’s conjectures on m = 2 m=2 cluster adjacency and on positroid tiles for A n , k , 2 \mathcal {A}_{n,k,2} (images of 2 k 2k -dimensional positroid cells which map injectively into A n , k , 2 \mathcal {A}_{n,k,2} ). Finally, we introduce new cluster structures in the amplituhedron.

Parisi, Matteo↗