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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Numerical modelling of impact seismic sources using the stress glut theory

SUMMARY Meteorite impacts have proved to be a significant source of seismic signal on the Moon, and have now been recorded on Mars by InSight seismometers. Understanding how impacts produce seismic signal is key to the interpretation of this unique data, and to improve their identification in continuous seismic records. Here, we use the seismic Representation Theorem, and particularly the stress glut theory, to model the seismic motion resulting from impact cratering. The source is described by equivalent forces, some resulting from the impactor momentum transfer, and others from the stress glut, which represents the mechanical effect of plasticity and non linear processes in the source region. We condense these equivalent forces into a point-source with a time-varying single force and nine-component moment tensor. This analytical representation bridges the gap between the complex dynamics of crater formation, and the linear point-source representation classically used in seismology. Using the multiphysics modelling software HOSS, we develop a method to compute the stress glut of an impact, and the associated point-source from hypervelocity impact simulations. For a vertical and an oblique impact at 1000 m s−1, we show that the moment tensor presents a significant deviatoric component. Hence, the source is not an ideal isotropic explosion contrary to previous assumptions, and draws closer to a double couple for the oblique impact. The contribution of the point force to the seismic signal appears negligible. We verify this model by comparing two signals: (1) HOSS is coupled to SPECFEM3D to propagate the near-source signal elastically to remote seismic stations; (2) the point-source model derived from the stress-glut theory is used to generate displacements at the same distance. The comparison shows that the point-source model is accurately simulating the low-frequency impact seismic waveform, and its seismic moment is in trend with Lunar and Martian impact data. High-frequencies discrepancies exist, which are partly related to finite-source effects, but might be further explained by the difference in mathematical framework between classical seismology and HOSS’ numerical modelling.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Analytic bounds on late-time axion-scalar cosmologies

The cosmological dynamics of multiple scalar/pseudoscalar fields are difficult to solve, especially when the field-space metric is curved. This presents a challenge in determining whether a given model can support cosmic acceleration, without solving for the on-shell solution. In this work, we present bounds on late-time FLRW-cosmologies in classes of theories that involve arbitrary numbers of scalar and pseudoscalar fields coupled both kinetically (leading to a curved field space metric) and through scalar potentials. Such bounds are proven analytically, independently of initial conditions, with no approximation in the field equations and without referring to explicit solutions. Besides their broad applications to cosmological model building, our bounds can be applied to studying asymptotic cosmologies of certain classes of string compactifications.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

It is not σ 8 : constraining the non-linear matter power spectrum with the Dark Energy Survey Year-5 supernova sample

The weak gravitational lensing magnification of Type Ia supernovae (SNe Ia) is sensitive to the matter power spectrum on scales $k\gt 1 h$ Mpc$^{-1}$, making it unwise to interpret SNe Ia lensing in terms of power on linear scales. We compute the probability density function of SNe Ia magnification as a function of standard cosmological parameters, plus an empirical parameter $A_{\rm mod}$ which describes the suppression or enhancement of matter power on non-linear scales compared to a cold dark matter only model. While baryons are expected to enhance power on the scales relevant to SN Ia lensing, other physics such as neutrino masses or non-standard dark matter may suppress power. Using the Dark Energy Survey Year-5 sample, we find $A_{\rm mod} = 0.77^{+0.69}_{-0.40}$ (68 per cent credible interval around the median). Although the median is consistent with unity there are hints of power suppression, with $A_{\rm mod} \lt 1.09$ at 68 per cent credibility.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1985

Statistical methods for astronomical data with upper limits. I - Univariate distributions

The statistical treatment of univariate censored data is discussed. A heuristic derivation of the Kaplan-Meier maximum-likelihood estimator from first principles is presented which results in an expression amenable to analytic error analysis. Methods for comparing two or more censored samples are given along with simple computational examples, stressing the fact that most astronomical problems involve upper limits while the standard mathematical methods require lower limits. The application of univariate survival analysis to six data sets in the recent astrophysical literature is described, and various aspects of the use of survival analysis in astronomy, such as the limitations of various two-sample tests and the role of parametric modelling, are discussed.

Feigelson, E. D.↗
DOE OSTI2026

The Dark Energy Survey supernova program: a reanalysis of cosmology results and evidence for evolving dark energy with an updated Type Ia supernova calibration

We present improved cosmological constraints from a re-analysis of the Dark Energy Survey (DES) 5-year sample of Type Ia supernovae (DES-SN5YR). This re-analysis includes an improved photometric cross-calibration, recent white dwarf observations to cross-calibrate between DES and low-redshift surveys, retraining the salt3 light-curve model and fixing a numerical approximation in the host-galaxy colour law. Our fully recalibrated sample, which we call DES-Dovekie, comprises ~1600 likely Type Ia SNe from DES and ~200 low-redshift SNe from other surveys. With DES-Dovekie, we obtain Ω m = 0.330 ± 0.015 in flat Lambda-cold dark matter (⁠ΛCDM) which changes Ω m by –0.022 compared to DES-SN5YR. Combining DES-Dovekie with cosmic microwave background data from Planck, Atacama Cosmology Telescope, and South Pole Telescope and the DESI DR2 measurements in a flat CDM cosmology, we find ω 0 = –0.803 ± 0.054 and ω a = –0.72 ± 0.21⁠. Our results hold a significance of 3.2σ, reduced from 4.2σ for DES-SN5YR, to reject the null hypothesis that the data are compatible with the cosmological constant. This significance is equivalent to a Bayesian model preference odds of approximately 5:1 in favour of the flat ω 0 ω a CDM model. Using generally accepted thresholds for model preference, our updated data exhibits only a weak preference for evolving dark energy.

dark energy↗
DOE OSTI2025

The Dark Energy Survey Supernova Program: an updated measurement of the Hubble constant using the inverse distance ladder

We measure the current expansion rate of the Universe, Hubble’s constant $H_0$, by calibrating the absolute magnitudes of supernovae to distances measured by baryon acoustic oscillations (BAO). This ‘inverse distance ladder’ technique provides an alternative to calibrating supernovae using nearby absolute distance measurements, replacing the calibration with a high-redshift anchor. We use the recent release of 1829 supernovae from the Dark Energy Survey spanning $0.01\lt z\lt 1.13$ anchored to the recent baryon acoustic oscillation measurements from Dark Energy Spectroscopic Instrument (DESI) spanning $0.30 \lt z_{\mathrm{eff}}\lt 2.33$. To trace cosmology to $z=0$, we use the third-, fourth-, and fifth-order cosmographic models, which, by design, are agnostic about the energy content and expansion history of the universe. With the inclusion of the higher redshift DESI-BAO data, the third-order model is a poor fit to both data sets, with the fourth-order model being preferred by the Akaike Information Criterion. Using the fourth-order cosmographic model, we find $H_0=67.19^{+0.66}_{-0.64}\mathrm{~km} \mathrm{~s}^{-1} \mathrm{~Mpc}^{-1}$, in agreement with the value found by Planck without the need to assume Flat-$\Lambda$CDM. However, the best-fitting expansion history differs from that of Planck, providing continued motivation to investigate these tensions.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

pop-cosmos : redshifts and physical properties of KiDS-1000 galaxies

ABSTRACT Principled Bayesian inference of galaxy properties has not previously been performed for wide-area weak-lensing surveys with millions of sources. We address this gap by applying the pop-cosmos generative model to perform spectral energy distribution (SED) fitting for 4 million KiDS (Kilo-Degree Survey)-1000 galaxies. Calibrated on deep COSMOS2020 photometric data, pop-cosmos specifies a physically motivated prior over the galaxy population up to $z \simeq 6$ in stellar population synthesis (SPS) parameter space. Using the Speculator SPS emulator with GPU (graphics processing unit)-accelerated Markov Chain Monte Carlo sampling, we perform full posterior inference at 8.2 GPU seconds per galaxy, obtaining joint constraints on galaxy redshifts and physical properties. We validate photometric redshifts against $\sim \!185\,\!000$ KiDS galaxies cross-matched to Dark Energy Spectroscopic Instrument Data Release 1 spectroscopic samples, achieving low bias ($2\times 10^{-3}$), scatter ($\sigma _{\mathrm{MAD}}=0.03$), and outlier fraction (3.2 per cent) for the Bright Galaxy Survey, with comparable performance (bias $3\times 10^{-2}$, $\sigma _{\mathrm{MAD}}=0.05$, 1.0 per cent outliers) for luminous red galaxies (LRGs). Within the LRG sample, we identify massive, dusty, star-forming contaminants at $z \simeq 0.4$ satisfying standard colour selections for quenched populations. We infer trends in stellar mass, star formation, metallicity, and dust across five tomographic redshift bins consistent with established scaling relations. Using specific star formation rate constraints, we identify $\sim$7 per cent of KiDS-1000 galaxies as quenched, versus 37 per cent implied by conservative colour cuts. This enables the construction of weak-lensing samples defined by physical properties while mitigating intrinsic alignment systematics and preserving statistical power. Our analysis validates pop-cosmos out of sample, establishing it as a scalable approach for galaxy evolution and cosmological analyses with photometric surveys.

Halder, Anik [Institute of Astronomy and Kavli Ins↗
DOE OSTI2025

Dark Energy Survey Year 6 results: cell-based coadds and METADETECTION weak lensing shape catalogue

We present the metadetection weak lensing galaxy shape catalogue from the 6-yr Dark Energy Survey (DES Y6) imaging data. This data set is the final release from DES, spanning 4422 deg 2 of the southern sky. We describe how the catalogue was constructed, including the two new major processing steps, cell-based image coaddition, and shear measurements with metadetection. The DES Y6 M etadetection weak lensing shape catalogue consists of 151 922 791 galaxies detected over riz bands, with an effective number density of n eff = 8.22 galaxies per arcmin 2 and shape noise of σ e = 0.29. We carry out a suite of validation tests on the catalogue, including testing for point spread function (PSF) leakage, testing for the impact of PSF modelling errors, and testing the correlation of the shear measurements with galaxy, PSF, and survey properties. In addition to demonstrating that our catalogue is robust for weak lensing science, we use the DES Y6 image simulation suite to estimate the overall multiplicative shear bias of our shear measurement pipeline. We find no detectable multiplicative bias at the roughly half-per cent level, with m = (3.4 ± 6.1) x 10 –3 , at 3σ uncertainty. This is the first time both cell-based coaddition and Metadetection algorithms are applied to observational data, paving the way to the Stage-IV weak lensing surveys.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2020

Diagnosing the sensitivity of grounding-line flux to changes in sub-ice-shelf melting

Abstract. Using a numerical ice flow model, we study changes in ice shelf buttressing and grounding-line flux due to localized ice thickness perturbations, a proxy for localized changes in sub-ice-shelf melting. From our experiments, applied to idealized (MISMIP+) and realistic (Larsen C) ice shelf domains, we identify a correlation between a locally derived buttressing number on the ice shelf, based on the first principal stress, and changes in the integrated grounding-line flux. The origin of this correlation, however, remains elusive from the perspective of a theoretical or physically based understanding. This and the fact that the correlation is generally much poorer when applied to realistic ice shelf domains motivate us to seek an alternative approach for predicting changes in grounding-line flux. We therefore propose an adjoint-based method for calculating the sensitivity of the integrated grounding-line flux to local changes in ice shelf geometry. We show that the adjoint-based sensitivity is identical to that deduced from pointwise, diagnostic model perturbation experiments. Based on its much wider applicability and the significant computational savings, we propose that the adjoint-based method is ideally suited for assessing grounding-line flux sensitivity to changes in sub-ice-shelf melting.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Evaluating cosmological biases using photometric redshifts for Type Ia Supernova cosmology with the Dark Energy Survey Supernova Program

Cosmological analyses with Type Ia Supernovae (SNe Ia) have traditionally been reliant on spectroscopy for both classifying the type of supernova and obtaining reliable redshifts to measure the distance–redshift relation. While obtaining a host-galaxy spectroscopic redshift for most SNe is feasible for small-area transient surveys, it will be too resource intensive for upcoming large-area surveys such as the Vera Rubin Observatory Legacy Survey of Space and Time, which will observe on the order of millions of SNe. Here, we use data from the Dark Energy Survey (DES) to address this problem with photometric redshifts (photo-z) inferred directly from the SN light curve in combination with Gaussian and full p(z) priors from host-galaxy photo-z estimates. Using the DES 5-yr photometrically classified SN sample, we consider several photo-z algorithms as host-galaxy photo-z priors, including the Self-Organizing Map redshifts (SOMPZ), Bayesian Photometric Redshifts (BPZ), and Directional-Neighbourhood Fitting (DNF) redshift estimates employed in the DES 3 × 2 point analyses. With detailed catalogue-level simulations of the DES 5-yr sample, we find that the simulated w can be recovered within ±0.02 when using SN+SOMPZ or DNF prior photo-z, smaller than the average statistical uncertainty for these samples of 0.03. With data, we obtain biases in w consistent with simulations within ~1σ for three of the five photo-z variants. We further evaluate how photo-z systematics interplay with photometric classification and find classification introduces a subdominant systematic component. This work lays the foundation for next-generation fully photometric SNe Ia cosmological analyses.

(cosmology:) dark energy↗
DOE OSTI2026

Discovering Strong Gravitational Lenses in the Dark Energy Survey with Interactive Machine Learning and Crowd-sourced Inspection with Space Warps

We conduct a search for strong gravitational lenses in the Dark Energy Survey (DES) Year 6 imaging data. We implement a pre-trained Vision Transformer (ViT) for our machine learning (ML) architecture and adopt interactive machine learning to construct a training sample with multiple classes to address common types of false positives. Our ML model reduces ∼236 million DES cutout images to 22,564 targets of interest, including ∼85% of previously reported galaxy–galaxy lens candidates discovered in DES. These targets were visually inspected by citizen scientists, who ruled out ∼90% as false positives. Of the remaining 2618 candidates, 149 were expert-classified as “definite” lenses and 516 as “probable” lenses, for a total of 665 systems, with 147 of these candidates being newly identified. Additionally, we trained a second ViT to find double-source plane lens systems, finding at least one double-source system. Our main ViT excels at identifying galaxy–galaxy lenses, consistently assigning high scores to candidates with high expert assessments. The top 800 ViT-scored images include ∼100 of our “definite” lens candidates. This selection is an order of magnitude higher in purity than previous convolutional neural-network-based lens searches and demonstrates the feasibility of applying our methodology for discovering large samples of lenses in future surveys.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Comparing the DES-SN5YR and Pantheon+ SN cosmology analyses: investigation based on ‘evolving dark energy or supernovae systematics’?

Recent cosmological analyses measuring distances of type Ia supernovae (SNe Ia) and baryon acoustic oscillations (BAO) have all given similar hints at time-evolving dark energy. To examine whether underestimated SN Ia systematics might be driving these results, Efstathiou (2025) compared overlapping SN events between Pantheon+ and DES-SN5YR (20 per cent SNe are in common), and reported evidence for an $\sim$0.04 mag offset between the low- and high-redshift distance measurements of this subsample of events. If this offset is arbitrarily subtracted from the entire DES-SN5YR sample, the preference for evolving dark energy is reduced. In this paper, we show that this offset is mostly due to different corrections for Malmquist bias between the two samples; therefore, an object-to-object comparison can be misleading. Malmquist bias corrections differ between the two analyses for several reasons. First, DES-SN5YR used an improved model of SN Ia luminosity scatter compared to Pantheon+ but the associated scatter-model uncertainties are included in the error budget. Secondly, improvements in host mass estimates in DES-SN5YR also affected SN standardized magnitudes and their bias corrections. Thirdly, and most importantly, the selection functions of the two compilations are significantly different, hence the inferred Malmquist bias corrections. Even if the original scatter model and host properties from Pantheon+ are used instead, the evidence for evolving dark energy from CMB, DESI BAO Year 1 and DES-SN5YR is only reduced from 3.9$\sigma$ to 3.3$\sigma$, consistent with the error budget. Finally, in this investigation, we identify an underestimated systematic uncertainty related to host galaxy property uncertainties, which could increase the final DES-SN5YR error budget by 3 per cent. In conclusion, we confirm the validity of the published DES-SN5YR results.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows

Herein we demonstrate that the reformulation of renormalization group (RG) flow equations as nonlinear heat equations has severe implications on the understanding of RG flows in general. We demonstrate by explicitly constructing an entropy function for a zero-dimensional Z 2 -symmetric model that the dissipative character of generic nonlinear diffusion equations is also hard-coded in the functional RG equation. This renders RG flows manifestly irreversible, revealing the semigroup property of RG transformations on the level of the flow equation itself. Additionally, we argue that the dissipative character of RG flows, its irreversibility and the entropy production during the RG flow may be linked to the existence of a so-called C– / A-function. In total, this introduces an asymmetry in the so-called RG time—in complete analogy to the thermodynamic arrow of time—and allows for an interpretation of infrared actions as equilibrium solutions of dissipative RG flows equations. The impossibility of resolving microphysics from macrophysics is evident in this framework. Furthermore, we directly link the irreversibility and the entropy production in RG flows to an explicit numerical entropy production, which is manifest in diffusive and non-linear partial differential equations (PDEs) and a standard mathematical tool for the analysis of PDEs. Using exactly solvable zero-dimensional Z 2 -symmetric models, we explicitly compute the (numerical) entropy production related to the total variation nonincreasing property of the PDE during RG flows toward the infrared limit. Finally, we discuss generalizations of our findings and relations to the C– / A-theorem as well as how our work may help to construct truncations of RG flow equations in the future, including numerically stable schemes for solving the corresponding PDEs.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

How to Obtain the Redshift Distribution from Probabilistic Redshift Estimates

Abstract A reliable estimate of the redshift distribution n ( z ) is crucial for using weak gravitational lensing and large-scale structures of galaxy catalogs to study cosmology. Spectroscopic redshifts for the dim and numerous galaxies of next-generation weak-lensing surveys are expected to be unavailable, making photometric redshift (photo- z ) probability density functions (PDFs) the next best alternative for comprehensively encapsulating the nontrivial systematics affecting photo- z point estimation. The established stacked estimator of n ( z ) avoids reducing photo- z PDFs to point estimates but yields a systematically biased estimate of n ( z ) that worsens with a decreasing signal-to-noise ratio, the very regime where photo- z PDFs are most necessary. We introduce Cosmological Hierarchical Inference with Probabilistic Photometric Redshifts ( CHIPPR ), a statistically rigorous probabilistic graphical model of redshift-dependent photometry that correctly propagates the redshift uncertainty information beyond the best-fit estimator of n ( z ) produced by traditional procedures and is provably the only self-consistent way to recover n ( z ) from photo- z PDFs. We present the chippr prototype code, noting that the mathematically justifiable approach incurs computational cost. The CHIPPR approach is applicable to any one-point statistic of any random variable, provided the prior probability density used to produce the posteriors is explicitly known; if the prior is implicit, as may be the case for popular photo- z techniques, then the resulting posterior PDFs cannot be used for scientific inference. We therefore recommend that the photo- z community focus on developing methodologies that enable the recovery of photo- z likelihoods with support over all redshifts, either directly or via a known prior probability density.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Boundary correlators and the Schwarzian mode

The effective low temperature dynamics of near-extremal black holes is governed by the quantum fluctuations of the Schwarzian mode of JT gravity. Utilizing as a proxy a planar charged black hole in asymptotically Anti-de-Sitter spacetime, we investigate the effects of these fluctuations on a probe scalar field. The corresponding holographic real-time boundary correlators are computed following a holographic renormalization procedure, using the dubbed gravitational Schwinger-Keldysh geometry (grSK) and known exact results of boundary correlators from the near-horizon region. This analysis gives rise to a retarded Green’s function that decays as a power law for late Lorentzian times. Its analytic structure indicates the presence of a branch cut in the complex frequency domain at finite temperature. These features are a non-perturbative hallmark that prevails as long as the planar transverse space is kept compact.

2D Gravity↗
DOE OSTI2022

Analytical Modeling of Exoplanet Transit Spectroscopy with Dimensional Analysis and Symbolic Regression

Abstract The physical characteristics and atmospheric chemical composition of newly discovered exoplanets are often inferred from their transit spectra, which are obtained from complex numerical models of radiative transfer. Alternatively, simple analytical expressions provide insightful physical intuition into the relevant atmospheric processes. The deep-learning revolution has opened the door for deriving such analytical results directly with a computer algorithm fitting to the data. As a proof of concept, we successfully demonstrate the use of symbolic regression on synthetic data for the transit radii of generic hot-Jupiter exoplanets to derive a corresponding analytical formula. As a preprocessing step, we use dimensional analysis to identify the relevant dimensionless combinations of variables and reduce the number of independent inputs, which improves the performance of the symbolic regression. The dimensional analysis also allowed us to mathematically derive and properly parameterize the most general family of degeneracies among the input atmospheric parameters that affect the characterization of an exoplanet atmosphere through transit spectroscopy.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Phase-curve Pollution of Exoplanet Transit Depths

The next generation of space telescopes will enable transformative science to understand the nature and origin of exoplanets. In particular, transit spectroscopy will reveal the chemical composition of the exoplanet atmospheres with unprecedented detail thanks to precise measurements of the visible-to-infrared transit depths down to 10 parts per million. Such a level of instrumental precision raises the challenge to obtain even more precise astrophysical models so as not to significantly influence the interpretation of the observed data. We must therefore critically revisit some of the commonly accepted assumptions that were adequate for analyzing past and current observations. A common approximation in the analysis of exoplanetary primary transits is that the planet does not contribute to the recorded flux, so-called dark planet hypothesis. In this paper, we investigate the impact of the dark planet hypothesis on the parameters obtained from the analysis of transits with particular attention to the transit depth. We develop mathematical formulae and release new software to estimate the magnitude of the potential bias. These tools will be useful in the preparation of observing proposals, as well as within the scientific consortia of the James Webb Space Telescope (JWST) and the Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL) missions. We probe the accuracy of the mathematical formulae through the analysis of synthetic observations with the JWST Mid-InfraRed Instrument. We find that self-blending from nightside emission attenuates the transit depth by >3σ for some of the known exoplanet systems, in agreement with previous work. An additional unreported effect caused by the nightside rotating into view can also impart a significant effect, but in the opposite direction (increasing the transit depth); this effect can largely be removed with conventional detrending practices, at the expense of a slight increase in noise, and mixing astrophysical variations and instrumental drifts.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Studying the effects of overlapping objects in dark energy

Observing the clustering of galaxies allows us to calculate cosmological parameters necessary for understanding dark energy. However, as the density of observed objects increases, the probability of these objects blending likewise increases, causing multiple galaxies to be observed as one. This affects the inferred values of parameters such as the galaxy bias (b) and the matter energy density (ΩM). To see whether the bias from incorrectly inferring the galaxy count is significant, we compare the correlation functions in simulated data for “true” and “observed” data sets with one-to-one and multiple-to-one correspondences, respectively. For each data set, we create two correlation functions: one “measured” function directly relying on the galaxies’ positions using the TreeCorr python library, and one “model” derived mathematically from the galaxies’ power spectrum using the Cosmological Core Library (CCL). By minimizing the residual between these two functions, we compute the ideal values for b and ΩM across the various possible redshifts that position the galaxies in three dimensional space. This minimization is done with an Markov chain Monte Carlo (MCMC) estimate that finds one value of ΩM and ten values for b corresponding to the ten redshift bins ranging from z = 0.2 to z = 1.2. We find that neither b nor ΩM is particularly affected by inclusion of blended galaxies. Though there is room for improvement, the data suggests that the fluctuations we found are a result of noise or limitations on the modeling rather than blending explicitly.

79 ASTRONOMY AND ASTROPHYSICS↗