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DOE OSTI2026

Deep Spectroscopy with DESI for Photometric Redshift Training and Calibration

Deep spectroscopic samples can improve photometric redshift (photo-z) estimates and reduce uncertainties on redshift distributions. Such improvements can increase the cosmological constraining power of large imaging-based experiments such as the Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST) and mitigate what may be a limiting systematic effect. We present results from the “DESI-Deep pilot” program, which was designed to assess the capability of the Dark Energy Spectroscopic Instrument (DESI) on the 4m Mayall telescope to measure redshifts of galaxies as faint as expected lensing samples for early LSST data (m i ≤ 24.5). We find that DESI is remarkably efficient at this task, with redshift success rates comparable to the results of observations from 10 m class telescopes with only ∼2 × longer integration time (rather than ∼8 × longer as would be expected from aperture-area scaling), while simultaneously achieving ∼30 times larger multiplexing. We also find that the signal-to-noise ratio of the spectra scales as expected for background-limited observations even for the longest exposure times (∼7 hr) and faintest targets in the program. These results demonstrate that DESI could provide the definitive redshift sample for the early years of LSST with a modest investment of observing time. Based upon the results of this program, we provide updated predictions for the time required to collect benchmark samples for photo-z training and calibration using a variety of spectroscopic facilities. Finally, we describe a potential “DESI-Deep” survey designed to train and calibrate photo-z’s for imaging experiments, and provide forecasts of its impact on cosmological inference.

Dey, Biprateep [University of Toronto; University ↗
DOE OSTI2021

Using a Neural Network Classifier to Select Galaxies with the Most Accurate Photometric Redshifts

The Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) will produce several billion photometric redshifts (photo-z's), enabling cosmological analyses to select a subset of galaxies with the most accurate photo-z. For this work, we perform initial redshift fits on Subaru Strategic Program galaxies with deep grizy photometry using Trees for Photo-Z (TPZ) before applying a custom neural network classifier (NNC) tuned to select galaxies with (z phot – z spec )/(1 + z spec ) < 0.10. We consider four cases of training and test sets ranging from an idealized case to using data augmentation to increase the representation of dim galaxies in the training set. Selections made using the NNC yield significant further improvements in outlier fraction and photo-z scatter (σ z ) over those made with typical photo-z uncertainties. As an example, when selecting the best third of the galaxy sample, the NNC achieves a 35% improvement in outlier rate and a 23% improvement in σ z compared to using uncertainties from TPZ. For cosmology and galaxy evolution studies, this method can be tuned to retain a particular sample size or to achieve a desired photo-z accuracy; our results show that it is possible to retain more than a third of an LSST-like galaxy sample while reducing σ z by a factor of 2 compared to the full sample, with one-fifth as many photo-z outliers. For surveys like LSST that are not limited by shot noise, this method enables a larger number of tomographic redshift bins and hence a significant increase in the total signal to noise of galaxy angular power spectra.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Photometric Signature of Ultraharmonic Resonances in Barred Galaxies

Bars may induce morphological features, such as rings, through their resonances. Previous studies suggested that the presence of "dark gaps," or regions of a galaxy where the difference between the surface brightness along the bar major axis and that along the bar minor axis is maximal, can be attributed to the location of bar corotation. Here, using GALAKOS, a high-resolution N-body simulation of a barred galaxy, we test this photometric method's ability to identify the bar corotation resonance. Contrary to previous work, our results indicate that "dark gaps" are a clear sign of the location of the 4:1 ultraharmonic resonance instead of bar corotation. Measurements of the bar corotation can indirectly be inferred using kinematic information, e.g., by measuring the shape of the rotation curve. We demonstrate our concept on a sample of 578 face-on barred galaxies with both imaging and integral field observations and find that the sample likely consists primarily of fast bars.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Photometric Objects Around Cosmic Webs (PAC) Delineated in a Spectroscopic Survey. III. Accurate Measurement of Galaxy Stellar Mass Function with the Aid of Cosmological Redshift Surveys

We present a novel method to accurately measure the galaxy stellar mass function (GSMF) based upon the Photometric objects Around Cosmic webs (PAC) method developed in our first paper (Paper I) of the series. The method allows us to measure the GSMF to a lower-mass end that is not accessible to the spectroscopic sample used in the PAC. Compared with Paper I, the current measurement of GSMF is direct and model independent. We measure the GSMFs in the redshift ranges of z$^{3}_{s}$ < 0.2, 0.2 < z s < 0.4, and 0.5 < z s < 0.7 down to stellar masses of M* = 10 8.2 , 10 10.6 , and 10 10.6 M ⊙ , using the data from the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Surveys and the spectroscopic samples of Slogan Digital Sky Survey (i.e., Main, LOWZ, and CMASS samples). Our results show that there is no evolution of GSMF from z s = 0.6 to z s = 0.1 for M* > 10 10.6 M ⊙ , and that there is a clear upturn at M* ≈ 10 9.5 M ⊙ toward smaller galaxies in the local GMSF at z s = 0.1. We provide an accurate double Schechter fit to the local GSMF for the entire range of M* and a table of our measurements at the three redshifts, which can be used to test theories of galaxy formation. Our method can achieve an accurate measurement of GSMF to the stellar mass limit where the spectroscopic sample is already highly incomplete (e.g., ~ 10 -3 ) for its target selection.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Hierarchical Inference of the Lensing Convergence from Photometric Catalogs with Bayesian Graph Neural Networks

Abstract We present a Bayesian graph neural network (BGNN) that can estimate the weak lensing convergence ( κ ) from photometric measurements of galaxies along a given line of sight (LOS). The method is of particular interest in strong gravitational time-delay cosmography (TDC), where characterizing the “external convergence” ( κ ext ) from the lens environment and LOS is necessary for precise Hubble constant ( H 0 ) inference. Starting from a large-scale simulation with a κ resolution of ∼1′, we introduce fluctuations on galaxy–galaxy lensing scales of ∼1″ and extract random sight lines to train our BGNN. We then evaluate the model on test sets with varying degrees of overlap with the training distribution. For each test set of 1000 sight lines, the BGNN infers the individual κ posteriors, which we combine in a hierarchical Bayesian model to yield constraints on the hyperparameters governing the population. For a test field well sampled by the training set, the BGNN recovers the population mean of κ precisely and without bias (within the 2 σ credible interval), resulting in a contribution to the H 0 error budget well under 1%. In the tails of the training set with sparse samples, the BGNN, which can ingest all available information about each sight line, extracts a stronger κ signal compared to a simplified version of the traditional method based on matching galaxy number counts, which is limited by sample variance. Our hierarchical inference pipeline using BGNNs promises to improve the κ ext characterization for precision TDC. The code is available as a public Python package, Node to Joy ⏬ .

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Bayesian Component Separation for DESI LAE Automated Spectroscopic Redshifts and Photometric Targeting

Lyα emitters (LAEs) are valuable high-redshift cosmological probes traditionally identified using specialized narrowband photometric surveys. In ground-based spectroscopy, it can be difficult to distinguish the sharp LAE peak from residual sky emission lines using automated methods, leading to misclassified redshifts. We present a Bayesian spectral component separation technique to automatically determine spectroscopic redshifts for LAEs while marginalizing over sky residuals. We use visually inspected spectra of LAEs obtained using the Dark Energy Spectroscopic Instrument (DESI) to create a data-driven prior and can determine redshift by jointly inferring sky residual, LAE, and residual components for each individual spectrum. We demonstrate this method on 881 spectroscopically observed z = 2–4 DESI LAE candidate spectra and determine their redshifts with >90% accuracy when validated against visually inspected redshifts. Using the Δχ$^{2}$ value from our pipeline as a proxy for detection confidence, we then explore potential survey design choices and implications for targeting LAEs with medium-band photometry. This method allows for scalability and accuracy in determining redshifts from DESI spectra, and the results provide recommendations for LAE targeting in anticipation of future high-redshift spectroscopic surveys.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2023

Photometry of Outer Solar System Objects from the Dark Energy Survey. I. Photometric Methods, Light-curve Distributions, and Trans-Neptunian Binaries

We report the methods of and initial scientific inferences from the extraction of precision photometric information for the >800 trans-Neptunian objects (TNOs) discovered in the images of the Dark Energy Survey (DES). Scene-modeling photometry is used to obtain shot-noise-limited flux measures for each exposure of each TNO, with background sources subtracted. Comparison of double-source fits to the pixel data with single-source fits are used to identify and characterize two binary TNO systems. A Markov Chain Monte Carlo method samples the joint likelihood of the intrinsic colors of each source as well as the amplitude of its flux variation, given the time series of multiband flux measurements and their uncertainties. A catalog of these colors and light-curve amplitudes A is included with this publication. We show how to assign a likelihood to the distribution q(A) of light-curve amplitudes in any subpopulation. Using this method, we find decisive evidence (i.e., evidence ratio <0.01) that cold classical (CC) TNOs with absolute magnitude 6 < H$_{r}$ < 8.2 are more variable than the hot classical (HC) population of the same H$_{r}$, reinforcing theories that the former form in situ and the latter arise from a different physical population. Resonant and scattering TNOs in this H$_{r}$ range have variability consistent with either the HCs or CCs. DES TNOs with H$_{r}$ < 6 are seen to be decisively less variable than higher-H$_{r}$ members of any dynamical group, as expected. More surprising is that detached TNOs are decisively less variable than scattering TNOs, which requires them to have distinct source regions or some subsequent differential processing.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2023

Binning is Sinning: Redemption for Hubble Diagram Using Photometrically Classified Type Ia Supernovae

Bayesian Estimation Applied to Multiple Species (BEAMS) is implemented in the BEAMS with Bias Corrections (BBC) framework to produce a redshift-binned Hubble diagram (HD) for Type Ia supernovae (SNe Ia). BBC corrects for selection effects and non–SN Ia contamination, and systematic uncertainties are described by a covariance matrix with dimension matching the number of BBC redshift bins. For spectroscopically confirmed SN Ia samples, a recent "Binning is Sinning" article showed that an unbinned HD and covariance matrix reduces the systematic uncertainty by a factor of ~1.5 compared to the binned approach. Here we extend their analysis to obtain an unbinned HD for a photometrically identified sample processed with BBC. To test this new method, we simulate and analyze 50 samples corresponding to the Dark Energy Survey (DES) with a low-redshift anchor; the simulation includes SNe Ia, and contaminants from core-collapse SNe and peculiar SNe Ia. The analysis includes systematic uncertainties for calibration and measures the dark energy equation of state parameter (w). Compared to a redshift-binned HD, the unbinned HD with nearly 2000 events results in a smaller systematic uncertainty, in qualitative agreement with BHS21, and averaging results among the 50 samples we find no evidence for a w-bias. To reduce computation time for fitting an unbinned HD with large samples, we propose an HD-rebinning method that defines the HD in bins of redshift, color, and stretch; the rebinned HD results in similar uncertainty as the unbinned case, and shows no evidence for a w-bias.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Improving Photometric Redshift Estimates with Training Sample Augmentation

Abstract Large imaging surveys will rely on photometric redshifts (photo- z 's), which are typically estimated through machine-learning methods. Currently planned spectroscopic surveys will not be deep enough to produce a representative training sample for Legacy Survey of Space and Time (LSST), so we seek methods to improve the photo- z estimates that arise from nonrepresentative training samples. Spectroscopic training samples for photo- z 's are biased toward redder, brighter galaxies, which also tend to be at lower redshift than the typical galaxy observed by LSST, leading to poor photo- z estimates with outlier fractions nearly 4 times larger than for a representative training sample. In this Letter, we apply the concept of training sample augmentation, where we augment simulated nonrepresentative training samples with simulated galaxies possessing otherwise unrepresented features. When we select simulated galaxies with ( g - z ) color, i -band magnitude, and redshift outside the range of the original training sample, we are able to reduce the outlier fraction of the photo- z estimates for simulated LSST data by nearly 50% and the normalized median absolute deviation (NMAD) by 56%. When compared to a fully representative training sample, augmentation can recover nearly 70% of the degradation in the outlier fraction and 80% of the degradation in NMAD. Training sample augmentation is a simple and effective way to improve training samples for photo- z 's without requiring additional spectroscopic samples.

Moskowitz, Irene (ORCID:0000000222068589)↗
DOE OSTI2025

Identification and Photometric Classification of Extragalactic Transients in the Vera C. Rubin Observatory’s Data Preview 1

The Vera C. Rubin Observatory will soon survey the southern sky, delivering a depth and sky coverage that is unprecedented in time-domain astronomy. As part of commissioning, Data Preview 1 (DP1) has been released. It comprises a Legacy Survey of Space and Time (LSST) Commissioning Camera observing campaign between 2024 November and December with multiband imaging of seven fields, covering roughly 0.4 deg 2 each, providing a first glimpse into the data products that will become available once the LSST begins. In this work, we search three fields for extragalactic transients. We identify eight new likely supernovae (SNe), and three known ones from a sample of 369,644 difference image analysis objects. Photometric classification using Superphot+ assigns subclasses with >95% confidence to only one SN Ia and one SN II in this sample. Our findings are in agreement with SN detection rate predictions of 15 ± 4 SNe from simulations using simsurvey. The SN detection rate in the data is possibly affected by the lack of suitable templates. Nevertheless, this work demonstrates the quality of the data products delivered in DP1 and indicates that the Rubin Observatory’s LSST is well placed to fulfill its discovery potential in time-domain astronomy.

Freeburn, James [University of North Carolina, Cha↗
DOE OSTI2022

Photometric Properties of Jupiter Trojans Detected by the Dark Energy Survey

The Jupiter Trojans are a large group of asteroids that are coorbiting with Jupiter near its L4 and L5 Lagrange points. The study of Jupiter Trojans is crucial for testing different models of planet formation that are directly related to our understanding of solar system evolution. In this work, we select known Jupiter Trojans listed by the Minor Planet Center from the full six years data set (Y6) of the Dark Energy Survey (DES) to analyze their photometric properties. The DES data allow us to study Jupiter Trojans with a fainter magnitude limit than previous studies in a homogeneous survey with griz band measurements. We extract a final catalog of 573 unique Jupiter Trojans. Our sample include 547 asteroids belonging to L5. This is one of the largest analyzed samples for this group. By comparing with the data reported by other surveys we found that the color distribution of L5 Trojans is similar to that of L4 Trojans. We find that L5 Trojans’ g − i and g − r colors become less red with fainter absolute magnitudes, a trend also seen in L4 Trojans. Both the L4 and L5 clouds consistently show such a color–size correlation over an absolute magnitude range 11 < H < 18. We also use DES colors to perform taxonomic classifications. C- and P-type asteroids outnumber D-type asteroids in the L5 Trojans DES sample, which have diameters in the 5–20 km range. This is consistent with the color–size correlation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Selection of high-redshift Lyman-Break Galaxies from broadband and wide photometric surveys

In this paper, we investigate the possibility of selecting high-redshift Lyman-Break Galaxies (LBG) using current and future broadband wide photometric surveys, such as the Ultraviolet Near Infrared Optical Northern Survey (UNIONS) or the Vera C. Rubin Legacy Survey of Space and Time (LSST), using a Random Forest algorithm. This work is conducted in the context of future large-scale structure spectroscopic surveys like DESI-II, the next phase of the Dark Energy Spectroscopic Instrument (DESI), which will start around 2029.We use deep imaging data from the Hyper Suprime Camera (HSC) and the Canada-France-Hawaii Telescope Large Area U-band Deep Survey (CLAUDS) on the COSMOS and XMM-LSS fields. To predict the selection performance of LBGs with image quality similar to UNIONS, we degrade the u,g,r,i and z bands to UNIONS depth.The Random Forest algorithm is trained with the u,g,r,i and z bands to classify LBGs in the 2.5 < z < 3.5 range.We find that fixing a target density budget of 1,100 deg$^{-2}$, the Random Forest approach gives a density of z > 2 targets of 873 deg$^{-2}$, and a density of 493 deg$^{-2}$ of confirmed LBGs after spectroscopic confirmation with DESI. This UNIONS-like selection was tested in a dedicated spectroscopic observation campaign of 1,000 targets with DESI on the COSMOS field, providing a safe spectroscopic sample with a mean redshift of 3. This sample is used to derive forecasts for DESI-II, assuming a sky coverage of 5,000 deg$^{2}$. We predict uncertainties on Alcock-Paczynski parameters α$_{⊥}$ and α$_{∥}$ to be 0.7% and 1% for 2.6 < z < 3.2, resulting in a potential 2% measurement of the dark energy fraction at high redshift. Additionally, we estimate the uncertainty in local non-Gaussianity and predict σ$_{fNL}$ ≈ 7, which would be comparable to the current best precision achieved by Planck. The latter forecast suggests that achieving the precision required to place stringent constraints on inflationary models (σ$_{fNL}$ ≈ 1) using spectroscopic galaxy surveys necessitates the development of a next-generation (Stage V) spectroscopic survey.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

The miniJPAS survey quasar selection – II. Machine learning classification with photometric measurements and uncertainties

Astrophysical surveys rely heavily on the classification of sources as stars, galaxies, or quasars from multiband photometry. Surveys in narrow-band filters allow for greater discriminatory power, but the variety of different types and redshifts of the objects present a challenge to standard template-based methods. In this work, which is part of a larger effort that aims at building a catalogue of quasars from the miniJPAS survey, we present a machine learning-based method that employs convolutional neural networks (CNNs) to classify point-like sources including the information in the measurement errors. We validate our methods using data from the miniJPAS survey, a proof-of-concept project of the Javalambre Physics of the Accelerating Universe Astrophysical Survey (J-PAS) collaboration covering ∼1 deg 2 of the northern sky using the 56 narrow-band filters of the J-PAS survey. Due to the scarcity of real data, we trained our algorithms using mocks that were purpose-built to reproduce the distributions of different types of objects that we expect to find in the miniJPAS survey, as well as the properties of the real observations in terms of signal and noise. We compare the performance of the CNNs with other well-established machine learning classification methods based on decision trees, finding that the CNNs improve the classification when the measurement errors are provided as inputs. The predicted distribution of objects in miniJPAS is consistent with the putative luminosity functions of stars, quasars, and unresolved galaxies. Our results are a proof of concept for the idea that the J-PAS survey will be able to detect unprecedented numbers of quasars with high confidence.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Photometric sensing of heavy metal ions using a naphthoquinodimethyl- bis -thioamide dye: Selectivity & photophysics of the metal organic complexes

As Mother Nature is experiencing catastrophic environmental pollutions from both natural sources and anthropogenic activities, many scientists have been working around the clock to develop environmentally benign and cost-effective, yet sensitive, detection techniques for pollutants, especially heavy metal species (e.g. lead, cadmium, mercury, among others) that are deleterious to human health. Herein, we report a novel sulfur-containing small organic dye/sensor naphthoquinodimethyl-bis-thioamide (QDM), which was found to be particularly selective toward mercury ion (Hg 2+ ). Using a combination of UV-vis absorption, photoluminescence, and time-resolved pump-probe techniques, we established that QDM and Hg 2+ can form stable complex(es) due to the strong affinity of sulfur toward mercury. In this investigation, while a higher ratio of QDM:Hg 2+ was necessary to fully quench the fluorescence emission of QDM, only 1 equiv of the Hg 2+ ion was necessary to observe the sensing effect on the excited state photo-behavior(s) of QDM. Furthermore, the present results highlight a synergy between molecular sensors' selectivity/sensitivity and sensor-analytes dynamics.

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

Measuring photometric redshifts for high-redshift radio source surveys

With the advent of deep, all-sky radio surveys, the need for ancillary data to make the most of the new, high-quality radio data from surveys like the Evolutionary Map of the Universe (EMU), GaLactic and Extragalactic All-sky Murchison Widefield Array survey eXtended, Very Large Array Sky Survey, and LOFAR Two-metre Sky Survey is growing rapidly. Radio surveys produce significant numbers of Active Galactic Nuclei (AGNs) and have a significantly higher average redshift when compared with optical and infrared all-sky surveys. Thus, traditional methods of estimating redshift are challenged, with spectroscopic surveys not reaching the redshift depth of radio surveys, and AGNs making it difficult for template fitting methods to accurately model the source. Machine Learning (ML) methods have been used, but efforts have typically been directed towards optically selected samples, or samples at significantly lower redshift than expected from upcoming radio surveys. This work compiles and homogenises a radio-selected dataset from both the northern hemisphere (making use of Sloan Digital Sky Survey optical photometry) and southern hemisphere (making use of Dark Energy Survey optical photometry). We then test commonly used ML algorithms such as k-Nearest Neighbours (kNN), Random Forest, ANNz, and GPz on this monolithic radio-selected sample. We show that kNN has the lowest percentage of catastrophic outliers, providing the best match for the majority of science cases in the EMU survey. We note that the wider redshift range of the combined dataset used allows for estimation of sources up to z = 3 before random scatter begins to dominate. When binning the data into redshift bins and treating the problem as a classification problem, we are able to correctly identify ≈ 76% of the highest redshift sources—sources at redshift z > 2.51 —as being in either the highest bin (z > 2.51) or second highest (z = 2.25).

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

photoD with Rubin ’s Data Preview 1: First stellar photometric distances and faint blue star deficits

Aims. We investigate the utility of Rubin’s Data Preview 1 (DP1) for estimating stellar number density profiles across the Milky Way halo. Methods. We used stellar broad-band near-UV to near-IR ugrizy photometry released in Rubin’s DP1 to estimate distance and metallicity for blue main sequence stars brighter than r = 24 in three ~1.1 sq. deg. fields at southern Galactic latitudes. Results. Compared to TRILEGAL simulations of the Galaxy’s stellar content, we found a likely deficit of blue main sequence turn-off stars with 22 < r < 24. We interpreted this discrepancy as a signature of a steeper halo number density profile at galactocentric distances 10–50 kpc than the canonical ~1/r 3 profile assumed in TRILEGAL simulations. Conclusions. This interpretation is consistent with earlier suggestions based on observations of more luminous, but much less numerous, evolved stellar populations, along with a few pencil beam surveys of blue main sequence stars in the northern sky. These results bode well for the future Galactic halo exploration with Rubin’s Legacy Survey of Space and Time (LSST).

Galaxy: fundamental parameters↗
DOE OSTI2020

From the inner to outer Milky Way: a photometric sample of 2.6 million red clump stars

ABSTRACT Large pristine samples of red clump stars are highly sought after given that they are standard candles and give precise distances even at large distances. However, it is difficult to cleanly select red clumps stars because they can have the same Teff and log g as red giant branch stars. Recently, it was shown that the asteroseismic parameters, $\rm {\Delta }$P and $\rm {\Delta \nu }$, which are used to accurately select red clump stars, can be derived from spectra using the change in the surface carbon to nitrogen ratio ([C/N]) caused by mixing during the red giant branch. This change in [C/N] can also impact the spectral energy distribution. In this study, we predict the $\rm {\Delta }$P, $\rm {\Delta \nu }$, Teff, and log g using 2MASS, AllWISE, Gaia, and Pan-STARRS data in order to select a clean sample of red clump stars. We achieve a contamination rate of ∼20 per cent, equivalent to what is achieved when selecting from Teff and log g derived from low-resolution spectra. Finally, we present two red clump samples. One sample has a contamination rate of ∼20 per cent and ∼405 000 red clump stars. The other has a contamination of ∼33 per cent and ∼2.6 million red clump stars that includes ∼75 000 stars at distances >10 kpc. For |b| > 30 deg, we find ∼15 000 stars with contamination rate of ∼9 per cent. The scientific potential of this catalogue for studying the structure and formation history of the Galaxy is vast, given that it includes millions of precise distances to stars in the inner bulge and distant halo where astrometric distances are imprecise.

Lucey, Madeline↗