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Davide Bianchi

Publications and source records attributed to Davide Bianchi.

At least 19 recordsLinked to original sources

Cosmological Constraints from Full-Scale Clustering and Galaxy-Galaxy Lensing with DESI DR1

We present constraints on cosmic structure growth from the analysis of galaxy clustering and galaxy--galaxy lensing with galaxies from the Dark Energy Spectroscopic Instrument (DESI) Data Release 1. Our analysis includes four samples drawn from the Bright Galaxy Survey (BGS) and the Luminous Red Galaxy (LRG) target classes. Projected galaxy clustering measurements from DESI are supplemented with lensing data from the Dark Energy Survey (DES), the Kilo-Degree Survey (KiDS), and the Hyper Suprime-Cam (HSC) survey around the same targets. Our method relies on a simulation-based modeling framework using the AbacusSummit simulations and a complex halo occupation distribution model that incorporates assembly bias. We analyze scales down to $0.4 \, h^{-1} \, \mathrm{Mpc}$ for clustering and $2.5 \, h^{-1} \, \mathrm{Mpc}$ for lensing, leading to stringent constraints on $S_8 = σ_8 \sqrt{Ω_\mathrm{m} / 0.3}$ and $Ω_\mathrm{m}$ when fixing other cosmological parameters to those preferred by the CMB. We find $S_8 = 0.797_{-0.024}^{+0.023}$ and $Ω_\mathrm{m} = 0.292 \pm 0.011$ when using lensing measurements from DES and KiDS. Similarly, for HSC, we find $S_8 = 0.791_{-0.021}^{+0.020}$ and $Ω_\mathrm{m} = 0.300 \pm 0.009$ when assuming the best-fit photometric redshift offset suggested by the HSC collaboration. Overall, our results are in good agreement with other results in the literature while continuing to highlight the constraining power of non-linear scales.

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Model-Independent Measurement of Baryon Gas Fractions through Galaxy-Galaxy Lensing and the Kinematic Sunyaev-Zel'dovich Effect

Baryon feedback is a leading source of systematic uncertainty for cosmology from weak lensing, but measurements of the gas distribution around galaxies have largely relied on parametric profile models or simulation-calibrated frameworks. We present model independent measurements of the radial gas fraction profile around galaxies, combining galaxy-galaxy lensing and kinematic Sunyaev-Zel'dovich (kSZ) effect. We introduce a method that applies the same radial $ΔΣ$ aperture filter to both the galaxy-galaxy lensing shear field and the velocity-weighted kSZ temperature maps; their ratio directly yields $ΔΣ$ filtered gas fraction $f_{\rm gas}(R)$, the ratio of ionized gas to total matter as a function of projected radius. We apply this approach to DESI DR2 Bright Galaxy Survey (BGS, $\bar{z}\approx0.26$) and Luminous Red Galaxy (LRG, $0.4<z<1.1$) samples, using ACT DR6 component-separated CMB maps for kSZ and the HSC Year 3 shear catalog for weak lensing. After correcting for ACT beam suppression using a simulation-calibrated compensation factor, we detect baryon depletion relative to the cosmic mean baryon fraction at SNR = 17.1 (BGS) and SNR = 15.8 (LRG bin 1). Comparison with six hydrodynamical simulations from the Illustris, IllustrisTNG, SIMBA, and FLAMINGO suites shows that no single feedback prescription reproduces the observed radial gas distribution across all scales, with the measurements falling between the strongest (Illustris-1) and weaker prescriptions. We caution that comparisons between the South Galactic Cap (SGC) and North Galactic Cap (NGC) show evidence of unexplained residual systematics in kSZ in one redshift bin. The ratio is robust against splits by stellar mass and satellite versus centrals. These results establish $ΔΣ$ filtering of kSZ versus weak lensing signal as a model-independent probe of the baryon distribution around galaxies.

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The Composite Spectrum of QSO Absorption Line Systems in DESI DR2

We present details regarding a composite spectrum of quasar (QSO) absorption line systems. In this composite spectrum we identify more than 70 absorption lines, and observe oxygen and hydrogen emission features at a higher signal-to-noise ratio than any previous study. As the light from a distant quasar travels towards an observer, it may interact with the circumgalactic medium of an intervening galaxy, forming absorption lines. In order to maximize the signal of these absorption lines, we have selected a sample of 239,174 quasar spectra from the second data release of the Dark Energy Spectro- scopic Instrument (DESI), each identified to have absorption lines resulting from such an interaction. By stacking these spectra in the restframe of the absorption, and calculating a median composite spec- trum, we are able to isolate and enhance these absorption lines. We provide a full atlas of all detected absorption and emission lines as well as their fit centroids and equivalent width values. We additionally provide analysis of composites constructed using subsamples of absorbers in both redshift and Mg II equivalent width bins. We find that there is minimal redshift evolution across the considered absorber samples. Additionally, we find that dense gas signatures, such as the presence of Na I absorption, are strongest in the highest equivalent width absorber sample, and that the strengths of the Si IV and C IV absorption lines show a relatively weak correlation with Mg II equivalent width. This atlas should aid in future studies investigating the compositions and physical conditions of these absorbers.

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The Cocytos Stream: A Disrupted Globular Cluster from our Last Major Merger?

The census of stellar streams and dwarf galaxies in the Milky Way provides direct constraints on galaxy formation models and the nature of dark matter. The DESI Milky Way survey -- with a footprint of $14{,}000$ deg$^2$ and a depth of $r<19$ mag -- delivers the largest sample of distant metal-poor stars compared to previous optical fiber-fed spectroscopic surveys. This makes DESI an ideal survey to search for previously undetected streams and dwarf galaxies. We present a detailed characterization of the Cocytos stream, which was re-discovered using a clustering analysis with a catalog of giants in the DESI year 3 data, supplemented with Magellan/MagE spectroscopy. Our analysis reveals a relatively metal-rich ([Fe/H]$=-1.3$) and thick stream (width$=1.6^\circ$) at a heliocentric distance of $\approx 25$ kpc, with an internal velocity dispersion of $6$--$9$ km s$^{-1}$. The stream's metallicity, radial orbit, and proximity to the Virgo stellar overdensities suggest that it is most likely a disrupted globular cluster that came in with the Gaia-Enceladus merger. We also confirm its association with the Pyxis globular cluster. Our result showcases the ability of wide-field spectroscopic surveys to kinematically discover faint disrupted dwarfs and clusters, enabling constraints on the dark matter distribution in the Milky Way.

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The Power of DESI for Photometric Redshift Calibration: A Case Study with KiDS-1000

Accurate redshift estimates are a critical requirement for weak lensing surveys and one of the main uncertainties in constraints on dark energy and large-scale cosmic structure. In this paper, we study the potential to calibrate photometric redshift (photo-z) distributions for gravitational lensing using the Dark Energy Spectroscopic Instrument (DESI). Since beginning its science operations in 2021, DESI has collected more than 50 million redshifts, adding about one million monthly. In addition to its large-scale structure samples, DESI has also acquired over 256k high-quality spectroscopic redshifts (spec-zs) in the COSMOS and XMM and VVDS fields. This is already a factor of 3 larger than previous spec-z calibration compilations in these two regions. Here, we explore calibrating photo-zs for the subset of KiDS-1000 galaxies that fall into joint self-organizing map (SOM) cells overlapping the DESI COSMOS footprint using the DESI COSMOS observations. Estimating the redshift distribution in KiDS-1000 with the new DESI data, we find broad consistency with previously published results while also detecting differences in the mean redshift in some tomographic bins with an average shifts of Delta Mean(z) = -0.028 in the mean and Delta Median(z) = +0.011 in the median across tomographic bins. However, we also find that incompleteness per SOM cell, i.e., groups of galaxies with similar colors and magnitudes, can modify n(z) distributions. Finally, we comment on the fact that larger photometric catalogs, aligned with the DESI COSMOS and DESI XMM and VVDS footprints, would be needed to fully exploit the DESI dataset and would extend the coverage to nearly eight times the area of existing 9-band photometry.

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Uniform Chebyshev approximations of functions satisfying a variation-type condition: reconstruction of time-varying signals on graphs

We prove uniform approximation theorems for Chebyshev expansions of continuous functions of two variables. Under a variation-type condition on the square $[-1,1]^2$, a continuous function admits a uniformly convergent Chebyshev expansion in the first variable whose coefficients are continuous functions of the second variable. These results are applied to the spectral graph wavelet transform of time-varying signals on finite weighted graphs: the scaling and wavelet kernels are approximated by expansions with time-varying coefficients whose degrees do not depend on time, the composition of the approximate transform with its adjoint admits the same explicit coefficient formulas as in the time-independent case, and reconstruction by the pseudoinverse is stable, with explicit bounds in terms of the uniform kernel errors. Numerical experiments on a sensor network confirm the convergence and stability estimates. In a denoising problem, soft thresholding of the graph wavelet coefficients with a time-varying transform parameter improves over its time-independent, fixed-parameter counterpart.

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ODIN: Confirmation and 3D Reconstruction of Six Massive Protoclusters at Cosmic Noon

Protoclusters represent sites of accelerated galaxy formation and extreme astrophysical activity characteristic of dense environments. Identifying massive protoclusters and mapping their spatial structures are therefore crucial first steps in understanding how the large-scale environment influences galaxy evolution. We combine wide-field Ly$α$ imaging from the ODIN survey with extensive DESI and ancillary spectroscopy across the extended COSMOS and XMM-LSS fields ($\approx$14 deg$^2$) to search for massive protoclusters. We confirm six systems at $z\approx 2.4$ and $z\approx 3.1$, reconstruct their three-dimensional structures, estimate descendant halo masses, and, for one structure at $z\approx 3.12$, demonstrate that overlapping narrowband filters ($NB497$ and $N501$) provide accurate redshift tomography for emission-line galaxies. One protocluster at $z\approx 2.45$ overlaps with one of the LATIS tomographic fields, enabling direct comparison between galaxy and H {\sc i} overdensities traced by Ly$α$ forest absorption. Another at $z\approx 3.12$ hosts a massive quiescent galaxy ($M_{\ast} \approx 1.2 \times 10^{11}M_\odot$), indicating early quenching in a dense environment. By comparing Ly$α$ emission properties across environments, we find that protocluster galaxies exhibit higher median line fluxes and a deficit of faint emitters relative to the field. The effect is strongest when both 2D and 3D density information are combined, indicating that galaxies in the densest protocluster cores are most affected by environmental processes. This effect is stronger at $z\approx3.1$ than at $z\approx2.4$, suggesting possible redshift evolution.

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On the accretivity and m-accretivity of Laplacians and porous medium-type operators on graphs

We study the accretivity and m-accretivity of Laplacian and porous medium-type operators on weighted graphs. In particular, we give several conditions that imply these properties for maximal operators and investigate when these operators agree with various restrictions. For porous medium-type operators on $\ell^1$ and for Laplacians on $\ell^p$ for $p \in [1,\infty)$, we show that there always exists a dense subset of the domain on which the maximal operator is m-accretive. As a consequence, we establish that accretivity, m-accretivity and injectivity of the shifted operator are all equivalent for these maximal operators. Under additional conditions on the graph, we then prove that the maximal operators are m-accretive on the entire domain, not just a dense subset. We also investigate minimal operators and show that they are m-accretive if and only if the minimal and maximal operators agree and the maximal operator is accretive. We then give some conditions that imply this agreement. Furthermore, for the minimal Laplacian on $\ell^p$, we show that accretivity and m-accretivity are not equivalent. For the $\ell^2$ case, we give connections to Markov uniqueness and essential self-adjointness. For the $\ell^\infty$ case, we establish the equivalence of stochastic completeness at infinity, m-accretivity for the maximal Laplacian on $\ell^\infty$, and m-accretivity of the minimal Laplacian on $\ell^1$.

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DESIVAST: A Catalog of Low-Redshift Voids using Data from the DESI DR1 Bright Galaxy Survey

We present three separate void catalogs created using a volume-limited sample of the DESI Year 1 Bright Galaxy Survey. We use the algorithms VoidFinder and V2 to construct void catalogs out to a redshift of z=0.24. We obtain 1,461 interior voids with VoidFinder, 420 with V2 using REVOLVER pruning, and 295 with V2 using VIDE pruning. Comparing our catalog with an overlapping SDSS void catalog, we find generally consistent void properties but significant differences in the void volume overlap, which we attribute to differences in the galaxy selection and survey masks. These catalogs are suitable for studying the variation in galaxy properties with cosmic environment and for cosmological studies.

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Nonlinear parabolic characterizations of stochastic completeness at infinity on weighted graphs

We prove a nonlinear parabolic characterization of stochastic completeness at infinity for weighted graphs. For the filtration equation \[ (\partial_t + ΔΦ)u =0 \] where $Δ$ is the non-negative formal graph Laplacian and $Φu =ϕ\circ u$ with $ϕ\colon \R\to\R$ nonconstant, continuous and increasing, stochastic completeness at infinity is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum. For \(Φ=\id\), this recovers the classical heat equation characterization of stochastic completeness at infinity, and of stochastic completeness when the killing term is trivial, i.e., when \(κ=0\). If stochastic completeness at infinity fails, then every bounded initial datum admits infinitely many bounded pointwise solutions of the filtration equation. Admissible nonlinearities include the signed porous medium and fast diffusion powers $ϕ(s)=s|s|^{m-1}$ for all $m>0$, as well as many others. Stochastic completeness at infinity is further characterized by a generalized mass balance: the total mass of a positive pointwise solution at time $t$, augmented by the mass $\int_0^t\sum_{x}κ(x)ϕ(u(s,x)) \dd s$ dissipated by the killing term $κ$, equals the initial mass. This balance holds for every bounded positive solution on graphs of finite measure and for bounded finite-mass data on graphs of arbitrary measure under the sharp condition $\limsup_{r\to0^+}ϕ(r)/r<\infty$. It also extends to positive pointwise solutions in $\ell^1$ that are bounded on every positive time interval. When the killing term is trivial, stochastic completeness at infinity reduces to stochastic completeness and generalized balance to conservation of mass.

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Luminosity function of quasars at $1.0<z<3.5$ from SDSS and DESI

We present a study of the evolution of type 1 quasars at $1.0<z<3.5$, covering the peak epoch of quasar activity. The quasar evolution has been extensively explored by a variety of previous works and the derived quasar luminosity functions (QLFs) are not well consistent with each other, presumably due to the complexities introduced by different quasar selection techniques and associated completeness corrections. We use a new strategy to construct QLFs based on a library of all known quasars. We focus on a wide region of $\sim$1700 deg$^2$ and a deep field of $\sim$265 deg$^2$ that have rich spectroscopic data primarily from SDSS and DESI. We then apply traditional color cuts in the rest-frame UV/optical to select quasar candidates and use the quasar library to identify them. Our final sample consists of 62,426 quasars at $1.0<z<3.5$, with a high completeness ($\sim$96%) and a high purity ($\sim$93%) in the color selection. Simple color cuts can potentially minimize selection biases for the study of quasar evolution. We derive binned QLFs and characterize them using a double power-law model. Sample incompleteness and contamination are considered as part of the uncertainties in the calculation. Compared to previous results, our QLFs are slightly higher at the faint end, and also higher at the bright end at $2.5<z<3.5$. The QLFs suggest that the quasar evolution at $1.0 < z < 2.5$ can be well described by the pure luminosity evolution model, while at $2.5 < z < 3.5$, it can be described by either the pure luminosity evolution or the pure density evolution model.

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Frequentist Cosmological Constraints from Full-Shape Clustering Measurements in DESI DR1

We present a frequentist analysis of clustering measurements from Data Release 1 of the Dark Energy Spectroscopic Instrument (DESI) using the standard profile likelihood method. While Bayesian inferences for effective field theory models of galaxy clustering can be highly sensitive to prior choices for extended cosmological models, frequentist inferences are not susceptible to such effects. We compare frequentist and Bayesian constraints for the parameter set $\{σ_8, H_0, Ω_{\rm{m}}, w_0, w_a\}$ using the full-shape power spectrum multipoles, post-reconstruction baryon acoustic oscillation (BAO) measurements, and external datasets from the CMB and type Ia supernovae measurements. The frequentist confidence intervals are significantly shifted relative to the Bayesian credible intervals for the $w_0w_a$CDM model, unless supernovae data are included. When DESI full-shape and BAO data are fit jointly, we obtain the following $1σ$ frequentist confidence intervals for $Λ$CDM ($w_0w_a$CDM): $σ_8 = 0.863^{+0.048}_{-0.040} , \ H_0 = 68.96^{+0.81}_{-0.80} \ \rm{km \ s^{-1}Mpc^{-1}} , \ Ω_{\rm{m}} = 0.3034\pm0.0110$ ($σ_8 = 0.782^{+0.060}_{-0.036} , \ H_0 = 63.7^{+4.2}_{-2.0} \ \rm{km \ s^{-1}Mpc^{-1}} , \ Ω_{\rm{m}} = 0.378^{+0.024}_{-0.047} , \ w_0 = -0.16^{+0.10}_{-0.50} , \ w_a = -3.0^{+1.7}_{}$), corresponding to 0.8$σ$, 0.3$σ$, 0.7$σ$ (2.1$σ$, 4.1$σ$, 6.5$σ$, 6.3$σ$, 6.6$σ$) shifts between the maximum likelihood estimate and the Bayesian posterior mean for $Λ$CDM ($w_0w_a$CDM) respectively.

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Cosmic Pairs: A DESI Census of Dual and Offset AGN as Precursors to Massive Black Hole Binaries

We present a systematic census of dual and offset active galactic nuclei (AGN) using spectroscopic data from the first data release (DR1) of the Dark Energy Spectroscopic Instrument (DESI). After correcting for observational systematics, our final sample contains $>7,000$ dual AGN and 27,000 galaxy pairs containing one AGN over the redshift range $0 \lesssim z \lesssim 3.6$. This sample expands the known dual AGN sample by $\sim 1-2$ orders of magnitude at $0.2 \lesssim z \lesssim 0.4$, includes $\sim 50$ dwarf dual AGN candidates in a regime where only a handful were previously known, and triples the census at $z>2$. Dual AGN are preferentially found at small separations, consistent with merger-driven triggering of AGN activity. The two members of a pair differ in their star formation response: the more massive (primary) host changes little with separation, while the less massive (secondary) lies $\sim 0.3$ dex above matched inactive and one-AGN companions at the same projected separation in main-sequence offset. Using ASTRID simulations, we predict that the fraction of DESI dual AGN whose central black holes will merge by $z \sim 0$ increases with redshift, reaching $\sim 76\%$ by $z \sim 2$, while the fraction producing LISA-detectable mergers peaks at $\sim 37\%$ near $z \sim 0.9$. These results provide the largest uniformly selected spectroscopic sample of kpc-scale dual and offset AGN candidates from a single survey, connecting their host-galaxy and AGN demographics to the progenitor population of massive black hole mergers detectable by LISA.

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The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation

We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $ϕ(s)=s|s|^{m-1}$, assuming a $ν$-Sobolev inequality with $ν>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0 2/ν$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.

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Stacked Reverberation Mapping of High Redshift Quasars in DESI. I. Feasibility Analysis

The broad line region of quasars has long been probed by reverberation mapping techniques that measure time lags between continuum and broad emission line variations. Stacked reverberation mapping has been proposed as a less observationally expensive alternative to traditional methods. This ensemble approach also reduces biases from small-number statistics. The Dark Energy Spectroscopic Instrument (DESI) is conducting the most extensive spectroscopic survey of quasars to date. We create mock light curves emulating expected DESI quasar observations at redshifts $1.48<z<5.2$ and luminosities $ 44.68 \leq \log L_{1350} λ/ \mathrm{erg\,s^{-1}} \leq 45.99 $ to test stacked reverberation mapping feasibility using sparse spectroscopic data paired with well-sampled photometric data. The pipeline, using the lag estimation code JAVELIN, successfully recovers the simulated C IV lags within one sigma of the true values using spectroscopic light curves composed of only a few spectral epochs (2-10) with irregular cadences. We investigate how observational factors, including C IV flux error magnitude, number of stacked quasars, and spectral epoch count, affect performance. This work motivates a pathway for future stacked reverberation mapping projects with large scale spectroscopic surveys of quasars having $\geq 2$ spectroscopic observations. Our results suggest an economical alternative for constraining and extending the radius-luminosity relation to higher redshifts and luminosities. Subsequently, this relation can be employed more reliably in single-epoch black hole mass measurements and quasar cosmology in these distant regimes.

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Iterated graph Laplacian for image restoration problems

We study the graph Laplacian operator as a regularizer in a generalized Tikhonov framework for linear ill-posed problems. The Laplacian is updated iteratively from the current reconstruction, so that progressively sharper structural information about the solution is fed into the regularization term. We introduce three schemes: a standard one that rebuilds the Laplacian from each new iterate; an error-equation scheme that, following the error-based formulation of iterated Tikhonov regularization, builds the Laplacian from an estimate of the reconstruction error rather than of the image itself; and a mixed scheme combining the two. We establish convergence of all three schemes for noisy data under a priori parameter and stopping rules as the noise level tends to zero. Numerical experiments in two-dimensional computed tomography and image deblurring show consistent gains in reconstruction quality and sharper recovery of fine details.

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Alleviating prior dependencies for DESI DR1 clustering fits through reparameterization

Bayesian analyses of the full-shape clustering of Dark Energy Spectroscopic Instrument (DESI) Data Release 1 (DR1) exhibit prior-volume projection effects, whereby weakly constrained nuisance parameters of the Effective Field Theory of Large Scale Structure (EFTofLSS) shift marginalized cosmological posteriors away from the posterior maximum. We reanalyze DESI DR1 power spectrum multipoles using two complementary mitigation strategies: (i) nonlinear orthogonalization to decorrelate nuisance and cosmological parameter priors, and (ii) a fully reparameterization-invariant Jeffreys prior over all EFTofLSS coefficients, evaluated on-the-fly via closed-form Jacobians. Including data from DESI, Big-Bang Nuclesynthesis and a constraint on $n_{\mathrm{s}}$, baseline priors lead to multi-$σ$ projection in the Hubble parameter $H_{0}$ and dark energy equation of state parameters $w_{0}$ and $w_{a}$; the Jeffreys prior successfully recenters these posteriors to enclose the maximum a posteriori estimate within the 68\% credible regions, demonstrating clear mitigation of projection effects for these late-time expansion parameters. A hybrid Jeffreys+baseline-Gaussian configuration controls residual over-broad tails in the physical cold dark matter density $ω_{\mathrm{c}}$ while preserving the volume correction, and is our favoured approach. We compare the credible intervals derived using our methodology to those obtained using Halo Occupation Distribution (HOD)-informed priors and to confidence intervals derived using frequentist profile likelihood analyses, finding agreement in both central values and degeneracy directions in the $w_{0}$--$w_{a}$ plane. This demonstrates that, once projection effects are properly controlled, we can make robust inferences about the late-time cosmological expansion independent of the statistical framework adopted.

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Early results in the search for extreme coronal line emitters with the Dark Energy Spectroscopic Instrument

Here we present the results of our search through the Early Data Release (EDR) of the Dark Energy Spectroscopic Instrument (DESI) for extreme coronal line emitters (ECLEs) - a rare classification of galaxies displaying strong, high-ionization iron coronal emission lines within their spectra. With the requirement of a strong X-ray continuum to generate the coronal emission, ECLEs have been linked to both active galactic nuclei (AGNs) and tidal disruption events (TDEs). We focus our search on identifying TDE-linked ECLEs. We identify three such objects within the EDR sample, highlighting DESI's effectiveness for discovering new nuclear transients, and determine a galaxy-normalized TDE-linked ECLE rate of $R_\mathrm{G}=5~^{+5}_{-3}\times10^{-6}~\mathrm{galaxy}^{-1}~\mathrm{yr}^{-1}$ at a median redshift of z = 0.2 - broadly consistent with previous works. Additionally, we also identify more than 200 AGNs displaying coronal emission lines, which serve as the primary astrophysical contaminants in searches for TDE-related events. We also include an outline of the custom python code developed for this search.

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