Production of Alternate Realizations of DESI Fiber Assignment for Unbiased Clustering Measurement in Data and Simulations (2024)

1 INTRODUCTION

Large scale spectroscopic sky surveys have been a primary way of studying the universe over the past twenty five years [1, 2, 3]. They obtain redshifts of many millions of galaxies in order to measure their position in three dimensional space and extract information regarding the clustering of galaxies and the dark matter fields which they trace throughout cosmic time. These measurements not only provide insight into the composition of the universe and the physics which drive the evolution of its expansion, but also the formation of the galaxies themselves.

The Dark Energy Spectroscopic Instrument (DESI) survey, like the extended Baryon Oscillation Spectroscopic Survey (eBOSS) and the Sloan Digital Sky Survey (SDSS) before it, utilizes a multi object spectrograph (MOS) which enables multiplexing of spectroscopic observations. However, DESI is able to multiplex at a scale $\sim$5x larger than that of eBOSS, capturing $\gtrsim$4000 spectra of astronomical objects per pointing. This increase in efficiency is largely due to the fact that SDSS and eBOSS hand-placed the fibers which fed their spectrographs into plates, while the fibers which feed the DESI spectrographs are controlled robotically. These positioners place all 5000 fibers on targets within the focal plane [4]. DESI also utilizes a dynamic exposure time calculator [5, 6] which takes information on observing conditions including seeing, transparency, and sky brightness to estimate the true signal-to-noise accumulated during an exposure. The capability to optimize exposure times and to construct new "plates" as needed, combined with a fivefold increase in available fibers and a $\sim 2.5x$ increase in the mirror area, allowed DESI to obtain $\sim$10x as many extragalactic redshifts in its first year as eBOSS did in its entire 5 year survey.

While robotically controlled fibers have provided many gains to the DESI survey, they have not eliminated the problem of fiber collisions. A fiber collision occurs when two targets are close enough on the sky that two fibers and their plugs (in SDSS and eBOSS) or their positioners (in DESI) would have collided were a survey to attempt to simultaneously observe both [7]. DESI also has a limitation similar to that of fiber collisions where only one fiber may be placed within a single fiber patrol radius, with a small amount of overlap towards the edges of the patrol radii of neighboring fibers. This enforces uniform areal observation density within a single field of view which can only begin to reflect the non-uniform density of targets after multiple passes. Since they have similar effects on the ability to observe close pairs, we will refer to both effects collectively as ’fiber collisions.‘

Additionally, robotic fiber positioners can suffer from time-variable issues which were not present in the hand-plugged fiber systems used by SDSS, such as losses to positioner power and software-based failures which interrupt the positioning process. Fiber-collision issues combine with more traditional problems such as sky subtraction failures, atmospheric variability, and redshift determination errors, to create incompleteness effects which vary spatially and temporally across the DESI survey.

Fiber collisions affect the smallest scale clustering since they are limited to angular scales on the sky of approximately $<1.48$arcminutes which projects to a distance of 0.99 Mpc $h^{-1}$ at an intermediate redshift of z = 1.0. Real time hardware events like power failures to the devices which control groups of fiber positioners (CAN busses) can also cause missing observations at larger sets of separations. These smaller scales are critical to analyses of the galaxy-halo connection. These analyses, whether they use Halo Occupation Distribution (HOD) methods [8, 9, 10, 11, 12, 13] or SubHalo Abundance Matching (SHAM) methods [14, 15, 16], rely on clustering measurements at small scales to determine the placement and quantity of satellite galaxies.

Prior to the usage of pairwise inverse probability weighting, (e)BOSS and SDSS used have used several different methods to correct for incompleteness. The BOSS DR9 BAO analysis [17] used the method developed by Guo, Zehavi, and Zheng in 2012 [18] to correct for fiber collisions. These weights are determined by dividing the galaxy sample into those which are not subject to any fiber collision effects and those which are subject to those effects. The relative fraction of galaxies within the collided sample which are observed is calculated. Then the pair counts which include galaxies from the collided sample are upweighted based on that fraction. This method does not account for the fact that unobserved pairs are not statistically equivalent to those which are observed. The Reid et al 2014 [19] analysis of the growth of structure using small scale clustering utilizes a combination of angular upweighting on small scales, and nearest neighbor upweighting, which accounts for unobserved targets by adding one to a "close pair" weight for an observed target for each unobserved target which is closer to it than any other observed member of its target class, at large scales, and a truncation of the measured multipoles at small separation perpendicular to the line of sight. Other standard SDSS and eBOSS analyses [20, 21, 22, 23, 24] have used nearest neighbor upweighting. This method assumes that all fiber collisions occur between galaxies within clusters or are otherwise at similar redshifts.

To address this issue of measuring small scale clustering, Bianchi and Percival 2017 (hereafter BP17; [25]) developed a new method of correction for incompleteness by upweighting each pair of galaxies rather than individual galaxies or the total number of paircounts. They devise a method of Pairwise Inverse Probability (PIP) weights which maintains consistent numbers of both weighted pair counts and weighted individual galaxy counts on all scales as long as there is some area of the survey footprint which is observed multiple times.

Bianchi and Percival note that, despite the nominal possibility of computing such pairwise weights analytically, for modern surveys like DESI which will observe $\mathcal{O}(10^{7})$ galaxies, such calculations will be practically impossible. They do note that these weights can be effectively estimated by randomly repeating the galaxy selection process a sufficient number of times and estimating the probability of each pair based on the frequency at which each pair is selected. The remainder of this paper will show a new method which allows us to estimate those probabilities, show its effect on the clustering measurements, and validate the weights using mocks.

Mohammad et al 2018 [26] showed the first application of the BP17 PIP weighting to data using the VIPERS survey. Additionally, they compute angular upweighting as described in Percival and Bianchi 2017 (hereafter PB17; [27]). This application to a slit-based survey is not strictly the same as the DESI application, but showed the validity of the BP17 PIP + PB17 angular upweighting method when applied to mocks and data. Mohammad et al 2020 [28] showed the first application of the BP17 PIP weighting + PB17 angular upweighting method to MOS data using the eBOSS DR16 catalogs. They used a version of the eBOSS tiling software which was modified to enable easier parallelization of fiber assignment across 1860 realizations of the eBOSS survey in which they varied only the initial random number generation used by the software. They store the results of the realizations as bitweights as recommended in BP17 and then use them to compute pairwise inverse probability weights. The authors then use EZ Mocks to validate that their application of PIP weighting successfully recovers one-halo clustering down to a scale of 0.1 Mpc $h^{-1}$ .

In this paper we present a new method of calculating PIP weights to correct the DESI SV3 and Y1 clustering measurements for incompleteness. This method utilizing Alternate MTLs (AMTLs) is a nearly-maximally realistic monte carlo estimation of the likelihood that any target or collection of targets would be observed over an infinite repetition of the DESI experiment. This method incorporates more realism than prior treatements in eBOSS by utilizing real observation results, hardware states, and sequential changes in priority as overlapping tiles are observed. This method will enable unbiased measurement of clustering of galaxies down to scales $\leq 0.1$ Mpc $h^{-1}$ .

In section 2 we describe the DESI survey strategy and how this affects the observation of different classes of targets, in Section 3 we briefly discuss the mock galaxy catalogs used to validate this method, in Section 4 we describe both the regular and alternate MTLs and the procedure for how we create and propagate the alternate MTLs, in Section 5 we describe the DESI clustering code, Pycorr and how it implements these weights, in Section 6 we show results from the data and mocks to validate the method and show the differences with other methods of incompleteness determination, and finally in Section 7 we discuss the results and their applicability to other analyses within DESI.

2 Survey

The Dark Energy Spectroscopic Instrument (DESI: [29, 30, 31]) is the Stage IV successor to large scale sky surveys such as the Sloan Digital Sky Survey (SDSS: [1]) and the (extended) Baryon Oscillation Spectroscopic Survey ((e)BOSS: [3]). The DESI experiment is currently in its third year of a projected five year program and has already recorded redshifts for over 30 million galaxies out to redshifts over $z\sim 3.5$. The primary goal of this survey is to further constrain cosmological parameters such as the dark energy equation of state parameter ($w$) and the Hubble Constant ($H_{0}$). The survey plans to do this by performing the most precise measurement of the Baryon Acoustic Oscillation (BAO) feature and of Redshift Space Distortions (RSD) in its clustering measurements. However, the science goals of DESI are not limited to dark energy. Already-published analyses from DESI include explorations of the Galaxy Halo Connection, measurements of the local stellar population in the Milky Way, studies of Quasistellar Object (QSO) astrophysics, and many tests of the survey and pipeline performances. This paper is part of a set of analyses surrounding the first Data Release of DESI (DR1, [32]) which includes two point clustering measurements and validation [33], measurements of the BAO feature in Galaxies, QSOs, and the Lyman-$\alpha$ forest [34, 35], full shape measurements of Galaxy and QSO clustering [36], as well as cosmological constraints measured from the above features including tests of primordial non-gaussianity [37, 38, 39].

DESI selects the galaxies and stars that it observes from a parent population of Milky Way targets (MWS; [40]), Bright, low redshift, Galaxies (BGS; [41]), Luminous Red Galaxies (LRGs; [42]), Emission Line Galaxies (ELGs; [43]), and Quasistellar Objects (Quasars/QSOs; [44]). Targets are obtained via prior imaging of the DESI footprint[45]. These targets are divided into two concurrent programs with BGS and MWS targets being observed during bright times or in other sub-optimal conditions while LRG, ELG, and QSO targets are observed only in sufficiently good dark time [5].

During the One-Percent Survey (also referred to as SV3), these targets were observed within 20 rosettes [46]. SV3 was designed to optimize the observation of complete samples of tracers over an area approximately equal to one percent of the final survey area. This survey also provided a final test for the survey operations procedures and software prior to the main survey. These rosettes were selected to be evenly distributed between the northern and southern galactic caps as well as to overlap with various fields of interest from CFHTLS, COSMOS, DES, GAMA, Euclid, ELAIS, GOODS, HSC, KiDS, VVDS, XBootes, and XDEEP as well as overlapping with the Coma cluster and the ecliptic pole. For each rosette, 12/10 (dark/bright) tiles were designed with tile centers spaced evenly around the rosette center at a distance of $0.12^{\circ}$ as shown in Figure1. Observation of these rosettes followed a sequence where, for each rosette, one tile was designed and fiberassigned in the afternoon before observing or earlier, observed at night, and then the Merged Target Ledgers (MTLs¹¹1MTLs will be further explained in §4.3 and more details can be found in §6 of [5].) containing the targets which were observed in that tile²²2MTLs are divided into HEALPixels (https://healpix.jpl.nasa.gov/) with nside = 32 (pixel size $\sim 1.8^{\circ}$) for ease of reading and storage were updated. Then the next overlapping tile on the same rosette was designed, incorporating the information from the previous night’s observations, and observed in the same manner until all 12/10 tiles were observed. A flowchart of this process is shown in Figure2.

This strategy enabled high completeness of 96.2% for MWS targets, 98.9% for BGS targets, 98.6% for LRG targets, 95.2% for ELG targets, and 99.4% for QSO targets within a range of rosette radii of 0.2 and 1.45 degrees[46]. Despite the nearly 100% completeness in that range, there is still an inner and outer region in which we can study the incompleteness measurements and weights from the AMTL method. Additionally, some regions within the central complete area are still subject to fiber collision effects due to high target number density and nightly focal plane positioning issues.

The observed SV3 spectra were processed by the DESI spectroscopic pipeline [47] in a hom*ogeneous processing run denoted as ‘Fuji’, with the resulting redshift and large-scale structure (LSS) catalogs used in this paper released in the early data release (EDR [48]). The results of the AMTL pipeline we describe in this paper were included in the EDR LSS catalogs.

The main survey maintained the strategy of not observing overlapping areas of sky before updating the MTLs, but is no longer confined to rosettes. The tiles for the main survey were chosen based on the Hardin, Sloane, and Smith 2001 icosahedral tiling³³3http://neilsloane.com/icosahedral.codes/[49] and are often fiber assigned "on-the-fly" as survey observations are occurring. This allows for greater flexibility in where one can point the telescope during a night of observing due to changing conditions including wind and clouds.The main survey is ongoing, but the DESI data release 1 (DR1 [32]) will include data from just over the first year of its operations, May 14, 2021 through June 14, 2022. Analyses of this dataset are ongoing and the resulting LSS catalogs have completeness 62% in BGS, 69% in LRG, 35% in ELG, and 88% in QSOs [33]. While we do not focus this analysis on the DR1 data, we do discuss some of the Y1 pipeline outputs and validation that the pipeline is working as expected. The relative incompleteness of the DR1 sample makes the AMTL pipeline a vital piece in the analysis, especially for the creation of realistic mocks [50, 51] and analyses of small-scale clustering.

3 Mocks

In order to validate the AMTL method, we use a set of 25 mock realizations processed in a similar manner to the data, reproducing the SV3 processing. They are built from the dedicated AbacusSummit⁴⁴4https://abacussummit.readthedocs.io/en/latest/ N-body simulations [52, 53] and include different ensembles for LRG, ELG and QSO. These are then combined into a single file that mimics the structure of the observation data and that are suitable to be processed through fiber assignment and LSS pipelines.

Mocks are made from Abacus realizations with fiducial cosmology (Planck 2015 $\Lambda$CDM cosmology [54]). For each tracer type (ELG, LRG and QSO) we build cubic mocks of 2 Gpc $h^{-1}$ size each, at different primary redshifts following a HOD modelling [55, 56] tuned to the clustering amplitude of SV3 data. In SV3, we chose the following Abacus snapshots: LRGs at z=0.8, ELGs at z=1.1 and QSOs at z=1.4.

Once the cubic box mocks are made, we transform them to CutSky mocks, translating from Cartesian coordinates to sky coordinates and including the effect of redshift-space distortions, resulting in galaxy catalogs already in redshift space. These catalogs are resampled to match the same n(z) distribution and footprint as SV3 data and prepared such that we add the columns needed to go through fiber assignment. In addition, we also make random catalogs following the same footprint and redshift distribution as the mock catalogs.

These mock catalogs are then processed through the AMTL pipeline (describe in §4) and LSS pipeline (described in §4 in [48]), creating clustering catalogs analogous to those of the data. Not all the observational effects are modelled, and therefore, we adapt the AMTL and LSS pipelines to accommodate these differences.

The only significant difference in the AMTL pipeline is that we do not pull any information from the redshift-fitting process [47] and the effect of redshift failures is ignored through the entire AMTL loop (fixing ZWARN = 0 in all steps). In summary, we assume that all targets assigned have a good redshift. In future releases we expect to model this effect more robustly, although we expect negligible effect on the clustering signal, since redshift failures will depend on the individual physical properties of the targets and is assumed to be independent of the underlying clustering.

Finally, the output products of the mock AMTL method are run through the LSS pipeline to create clustering ready catalogs with the same completeness properties as the data. Together with the mock data, we process the associated randoms. Here we have modified the pipeline to ignore any effect coming from low data quality (assuming a 100% success rate) and ignoring any angular mask applied to the data. These implied subtle changes in the pipeline at the full catalog stage (see §4.2.1 in [48]).

4 Alternate Merged Target Ledgers

5 Clustering Measurements

where $XY(s_{\perp},s_{||})$ correspond to the weighted number of pairs of objects $X$ and $Y$ in a $s_{\perp},s_{||}$ bin divided by the total weighted number of pairs. Specifically, $DD(s_{\perp},s_{||})$ is the weighted number of data (galaxy and quasar) pairs and $RR(s_{\perp},s_{||})$ is the weighted number of pairs of objects in the random catalog which samples the selection function. For the autocorrelation (single-tracer) measurements performed in our analysis, by symmetry $DR(s_{\perp},s_{||})=RD(s_{\perp},s_{||})$. For standard estimates pair weights are the product of the total individual weights of the two objects in the pair $w_{1}$ and $w_{2}$, themselves obtained as the product of systematic correction weights and FKP weights. We use 48 logarithmically spaced $s_{\perp}$ bins between 0.01 and 100 $\mathrm{Mpc}/h$ and 40 linearly spaced $s_{||}$-bins between 0 and $40\;\mathrm{Mpc}/h$.

To optimize computing time we use the technique of Keihänen et al 2019[62], which consists in summing $DR$, $RD$ and $RR$ pair counts obtained using several random catalogs of a size approximately matching that of the data catalog. In practice, we use the maximum number available for the SV3 LSS catalogs, which is 18 and corresponds to 28, 67, 220, and 54$\times$ the data number for ELGs, LRGs, QSOs, and BGS_BRIGHT respectively.

the projected correlation function $w_{p}$ integrates the full two point correlation function $\xi$ over a range in the line-of-sight distance spanning 40 Mpc $h^{-1}$ . This distance is chosen to sufficiently remove all rsd effects while not introducing significant noise.

Correlation function measurements are performed with pycorr⁹⁹9https://github.com/cosmodesi/pycorr which wraps a version of the Corrfunc package[63, 64] modified to also run on GPU and support alternative lines-of-sight (first-point, end-point) and weight definitions (PIP and angular upweights) and the $\theta$-cut, along with jackknife error estimation.

5.1 Weights

The set of weights applied in the DESI SV3 TPCFs include the "default“ (IIP) weights, FKP weights, angular upweights, and bitwise (PIP) weights. The PIP and IIP weights are never applied simultaneously since they are derived from the same set of AMTLs and correct for the same incompleteness.

The ”default“ IIP weights are calculated from the bitweights output from the AMTL pipeline with the ’efficient estimator‘ of Bianchi and Verde 2020 [65].

$$w_{i,\rm IIP}=\frac{N_{\rm real}+1}{({\rm popcnt}\>w^{(b)}_{i})+1}$$

(5.3)

FKP weights were derived in Feldman, Kaiser, and Peaco*ck 1994[66] as a variance-minimizing weight for a given scale of a power spectrum. These FKP weights are given by

$$w_{FKP}(z)=\frac{1}{1+n(z)P_{0}}$$

(5.4)

where $n(z)$ is the number density of galaxies at redshift $z$, and $P_{0}$ is the clustering power at the scale in the power spectrum at which we desire to minimize the variance. In DESI SV3, we chose the following $P_{0}$values for each tracer:

LRG: 10000ELG: 4000QSO: 6000BGS: 7000

Angular upweighting, as first outlined in PB17 [27] and applied in Mohammad et al 2018 [26] and Mohammad et al 2020[28], is derived from the angular clustering of the parent catalog of all potential targets of a tracer type and the angular clustering of the targets which have been observed. These pairwise weights are generated separately for the Data-Data pairs and the Data-Random pairs with the bitweights factoring into both sets of counts as shown below.

$$w^{\rm DD}_{\rm ANG}=\frac{DD^{\rm par}(\theta)}{DD^{\rm fib}_{\rm PIP}(\theta)}$$

(5.5)

$$w^{\rm DR}_{\rm ANG}=\frac{DR^{\rm par}(\theta)}{DR^{\rm fib}_{\rm IIP}(\theta)}$$

(5.6)

In equations 5.5 and 5.6, "par“ and "fib“ refer to the parent sample and fiber-assigned samples respectively. The fiber-assigned Data-Data pairs are weighted by the inverse of the probability that the pair would be observed (PIP). Similarly, the fiber-assigned Data-Random pairs are weighted by the inverse of the probability that the data galaxy would be observed (IIP).

Finally, the bitwise PIP weights are calculated using the efficient estimator from Bianchi and Verde 2020 [65]:

$$w_{ij}=\frac{N_{\rm real}+1}{{\rm popcnt}\>{w^{(b)}_{i}{\rm and}\>w^{(b)}_{j}}$$

(5.7)

where $w_{ij}$ is the weighting applied to the pair of galaxies $i$ and $j$, $N_{\rm real}$ is the number of AMTL realizations, and popcnt $w^{(b)}_{i}andw^{(b)}_{j}$ is the count of the bitwise "and“ operation between the bitweights of galaxies i and j. In the case of PIP weighting, the count of DD pairs is also normalized as in Equation 22 of Bianchi and Verde 2020 [65].

These weights are provided to pycorr by multiplicatively combining all non-bitwise weights and then adding them to a (list of) bitweight(s).

Figure5 shows the projected clustering results using the above methods for all tracer types which have a ’clustering‘ catalog produced from DESI SV3 Data.

Figure6 shows the clustering amplitude of four tracers (BGS_ANY, LRG, ELG, and QSO) as a function of their separation in bins of distance parallel and perpendicular to the line of sight ( $s_{||}$ and $s_{\perp}$ respectively).

6 Results

In this section we display results from the first two runs of the AMTL pipeline, one run covering SV3, and the other run covering Y1. Statistics which we will show include a comparison of the number of targets in each class observed in the data and averaged over the AMTL runs, the estimation of completion from the AMTL pipeline, the difference in the completeness weights between the AMTL pipeline and methods which bootstrap completeness estimation from only the data, and finally the effects of weighting using PIP weights from this pipeline and other weighting methods on clustering statistics.

6.3 SV3 2PCF results

Figures 19, 20, and 21 show the mean projected TPCF over 25 SV3 Abacus mocks with errorbar coming from the standard deviation over the 25 mocks in four cases. The first case, which we treat as truth is the projected TPCF parent catalog of all possible mock targets and is shown in black. The remaining cases are different weights on what was chosen to be the "observed“ set of targets. The "default“ weight (orange) includes the IIP completeness weight, a combination of the default weights and the angular upweighting of PB17 (green; [27] and §5.1 for more details), and the combination of the angular weights with the bitwise PIP weights discussed here (blue).

The inset on each plot shows the fractional difference between each weighting scheme with the "truth“ with the blue (PIP weighted) line having error bars equal to the ratio of the "truth“ errorbar and its value in each bin.

These plots show that the bitweights produced by the AMTL method are able to recover the parent clustering well within $1-\sigma$ for all tracers down to 0.05 Mpc $h^{-1}$ . Furthermore, the PIP weighting improves the recovery of the parent catalog clustering in all but a few separation bins.

The case of ELGs is most informative, since ELGs are the least complete target class. Clustering measured using only the default weighting is off by 20% uniformly down to a scale of nearly 1Mpc $h^{-1}$ . Additionally, the addition of angular upweights causes a spurious shift which pushes the 20% discrepancy down to a scale over 10 Mpc $h^{-1}$ . The PIP weights when compbined with the above schemes shows agreement within a few percent between scales of 0.05 Mpc $h^{-1}$ and 25 Mpc $h^{-1}$ with agreement within $1-\sigma$ at all scales shown.

7 Discussion

Comparing the statistics of the data AMTL with the data shows a small, but persistent $\sim 1\%$ excess in the number of observed targets. The mocks also show a similar excess. This is likely due to the AMTL pipeline not fully accurately sampling observation results from the data for the alternate targets. As mentioned in section §4.4, we take observation results in the AMTL from the exact same fiber in the observations without regard to the target type which is assigned in the alternate realizations. In a future release of the AMTL pipeline we will have a fiber twin method implemented which will select observation results from a nearby fiber with nearly identical observational properties including target type.While the number of total observations shows some disagreements, the shape of the PROB_OBS distribution is recovered very well. For Rosette 16, the data PROB_OBS distributions in Figure7 and mock PROB_OBS distributions in Figure13 show nearly identical structure with the mocks showing much less fluctuation due to higher statistics and the data showing a small decrease in PROB_OBS towards the lower right of the rosette not present in the mocks due to a high target number density in that region. Rosette 1 also shows good agreement between Figures9 and 15 with the significantly higher PROB_OBS in the top left of the data rosette as compared to the mock coming from the very low data target number density.

The difference between PROB_OBS and the more generic FRACZ_TILELOCID in figures11, 12, 17, and 18 show similar patterns. There is very little difference between the two completeness metrics in the centers of both rosettes since completeness is at or near 100% at all locations. Towards the edges, we see a significant amount of fluctuation between the PROB_OBS and FRACZ_TILELOCID suggesting that completeness is somewhat different when measured through more approximate methods than an AMTL method.

Figures19, 20, and 21 show the validation of the AMTL method by using the bitweights obtained by using the method on mocks to recover the clustering of the parent mocks. Since the true redshift of all targets in the mock are known, this enables us to validate against a known "truth“ projected two point correlation function.

The validation results show that the bitwise weights calculated from the AMTL method are a strict improvement over the default completeness weights at small scales. For all three dark time tracers, the (blue) PIP+ANG upweighted mock projected two point correlation functions are an improvement of the default completeness weights at nearly all small ( $\leq 1$Mpc $h^{-1}$ ) scales. Furthermore, down to scales of 0.02, 0.01, and 0.1 Mpc $h^{-1}$ (for LRGs, ELGs, and QSOs respectively), the PIP+ANG upweighted clustering from the mocks is consistent with the parent mock clustering within the scatter among the 25 mocks. These results, especially the ELG clustering, show that the PIP weighting and angular upweighting are both necessary to correct the clustering measurements for incompleteness. For a sample like SV3 which is highly complete, IIP weights function similarly to nearest neighbor weights, indicating that those weights would be insufficient for incompleteness correction at small scales.

The larger scale clustering ($s\geq 1$ Mpc $h^{-1}$ and especially $s\geq 10$Mpc $h^{-1}$ ), while statistically consistent with the parent clustering within the mock variance, is not as well recovered. However, there are several reasons why this should be the case. First, the SV3 rosette design strongly limits the number of pairs which we can observe at separations nearing 100 Mpc $h^{-1}$ . The size of an individual rosette at a redshift z=1.0 is only $\sim 70$Mpc $h^{-1}$ . While DESI did place some rosettes adjacent to each other enabling the observation of some pairs at those separations, the vast majority of rosettes were isolated.

In conclusion, we have demonstrated the generation of 128 realizations of DESI SV3 and Y1 of the main survey using a nearly-maximally-realistic method utilizing alternative merged target ledgers. We have validated the AMTL method on 256 realizations of DESI SV3-like mocks showing the recovery of clustering down to scales of 0.01 Mpc $h^{-1}$ for the least complete tracer, Emission Line Galaxies.

Acknowledgements

JL and RK acknowledge support of the U.S. Department of Energy (DOE) in funding grant DE-SC0010129 for the work in this paper. Computational resources for JL were provided by SMU’s Center for Research Computing. This work made use of Astropy:¹¹¹¹11http://www.astropy.org a community-developed core Python package and an ecosystem of tools and resources for astronomy [68, 69, 70]ADM was supported by the U.S.Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0019022.

This material is based upon work supported by the U.S. Department of Energy (DOE), Office of Science, Office of High-Energy Physics, under Contract No. DE–AC02–05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Humanities, Science and Technology of Mexico (CONAHCYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: https://www.desi.lbl.gov/collaborating-institutions. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U. S. National Science Foundation, the U. S. Department of Energy, or any of the listed funding agencies.

The authors are honored to be permitted to conduct scientific research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation.

8 Data Availability

The SV3 data used in this analysis is currently availabile to the public at https://data.desi.lbl.gov/doc/releases/edr/ .The Y1 data used in this analysis will be made public along the Data Release 1 (details in https://data.desi.lbl.gov/doc/releases/)

References

• York etal. [2000]DonaldG. York, J.Adelman, Jr. Anderson, JohnE., and SDSSCollaboration.The Sloan Digital Sky Survey: Technical Summary.AJ, 120(3):1579–1587, September 2000.doi: 10.1086/301513.
• Dawson etal. [2013]KyleS. Dawson, DavidJ. Schlegel, ChristopherP. Ahn, etal.The Baryon Oscillation Spectroscopic Survey of SDSS-III.AJ, 145(1):10, January 2013.doi: 10.1088/0004-6256/145/1/10.
• Dawson etal. [2016]KyleS. Dawson, Jean-Paul Kneib, WillJ. Percival, etal.The SDSS-IV Extended Baryon Oscillation Spectroscopic Survey:Overview and Early Data.AJ, 151(2):44, February 2016.doi: 10.3847/0004-6256/151/2/44.
• DESI Collaboration etal. [2022]DESI Collaboration etal.Overview of the Instrumentation for the Dark Energy SpectroscopicInstrument.AJ, 164(5):207, November 2022.doi: 10.3847/1538-3881/ac882b.
• Schlafly etal. [2023]EdwardF. Schlafly, David Kirkby, DavidJ. Schlegel, etal.Survey Operations for the Dark Energy Spectroscopic Instrument.AJ, 166(6):259, December 2023.doi: 10.3847/1538-3881/ad0832.
• Kirkby et al. [2024]Kirkby et al., in preparation, 2024.
• Blanton etal. [2003]MichaelR. Blanton, Huan Lin, RobertH. Lupton, F.Miller Maley, NealYoung, Idit Zehavi, and Jon Loveday.An Efficient Targeting Strategy for Multiobject SpectrographSurveys: the Sloan Digital Sky Survey “Tiling” Algorithm.AJ, 125(4):2276–2286, April 2003.doi: 10.1086/344761.
• Gao etal. [2023]Hongyu Gao, Y.P. Jing, Shanquan Gui, etal.The DESI One-Percent Survey: Constructing Galaxy-Halo Connectionsfor ELGs and LRGs Using Auto and Cross Correlations.ApJ, 954(2):207, September 2023.doi: 10.3847/1538-4357/ace90a.
• Pearl etal. [2023]AlanN. Pearl, AndrewR. Zentner, JeffreyA. Newman, etal.The DESI One-Percent Survey: Evidence for Assembly Bias fromLow-Redshift Counts-in-Cylinders Measurements.arXiv e-prints, art. arXiv:2309.08675, September 2023.doi: 10.48550/arXiv.2309.08675.
• Rocher etal. [2023]Antoine Rocher, Vanina Ruhlmann-Kleider, Etienne Burtin, etal.The DESI One-Percent survey: exploring the Halo OccupationDistribution of Emission Line Galaxies with ABACUSSUMMIT simulations.J. Cosmology Astropart. Phys, 2023(10):016, October 2023.doi: 10.1088/1475-7516/2023/10/016.
• Smith etal. [2023]A.Smith, C.Grove, S.Cole, etal.Generating mock galaxy catalogues for flux-limited samples like theDESI Bright Galaxy Survey.arXiv e-prints, art. arXiv:2312.08792, December 2023.doi: 10.48550/arXiv.2312.08792.
• Yuan etal. [2023a]Sihan Yuan, Hanyu Zhang, AshleyJ. Ross, etal.The DESI One-Percent Survey: Exploring the Halo OccupationDistribution of Luminous Red Galaxies and Quasi-Stellar Objects withAbacusSummit.arXiv e-prints, art. arXiv:2306.06314, June2023a.doi: 10.48550/arXiv.2306.06314.
• Yuan etal. [2023b]Sihan Yuan, RisaH. Wechsler, Yunchong Wang, etal.Unraveling emission line galaxy conformity atz~1 with DESI early data.arXiv e-prints, art. arXiv:2310.09329, October2023b.doi: 10.48550/arXiv.2310.09329.
• Yu etal. [2024]Jiaxi Yu, Cheng Zhao, Violeta Gonzalez-Perez, etal.The DESI One-Percent Survey: exploring a generalized SHAM formultiple tracers with the UNIT simulation.MNRAS, 527(3):6950–6969, January 2024.doi: 10.1093/mnras/stad3559.
• Gao etal. [2024]Hongyu Gao, Y.P. Jing, Kun Xu, etal.The DESI One-Percent Survey: A Concise Model for the GalacticConformity of Emission-line Galaxies.ApJ, 961(1):74, January 2024.doi: 10.3847/1538-4357/ad09d6.
• Prada etal. [2023]F.Prada, J.Ereza, A.Smith, etal.The DESI One-Percent Survey: Modelling the clustering and halooccupation of all four DESI tracers with Uchuu.arXiv e-prints, art. arXiv:2306.06315, June 2023.doi: 10.48550/arXiv.2306.06315.
• Anderson etal. [2012]Lauren Anderson, Eric Aubourg, Stephen Bailey, etal.The clustering of galaxies in the SDSS-III Baryon OscillationSpectroscopic Survey: baryon acoustic oscillations in the Data Release 9spectroscopic galaxy sample.MNRAS, 427(4):3435–3467, December 2012.doi: 10.1111/j.1365-2966.2012.22066.x.
• Guo etal. [2012]Hong Guo, Idit Zehavi, and Zheng Zheng.A New Method to Correct for Fiber Collisions in Galaxy Two-pointStatistics.ApJ, 756(2):127, September 2012.doi: 10.1088/0004-637X/756/2/127.
• Reid etal. [2014]BethA. Reid, Hee-Jong Seo, Alexie Leauthaud, JeremyL. Tinker, andMartin White.A 2.5 per cent measurement of the growth rate from small-scaleredshift space clustering of SDSS-III CMASS galaxies.MNRAS, 444(1):476–502, October 2014.doi: 10.1093/mnras/stu1391.
• Anderson etal. [2014]Lauren Anderson, Éric Aubourg, Stephen Bailey, etal.The clustering of galaxies in the SDSS-III Baryon OscillationSpectroscopic Survey: baryon acoustic oscillations in the Data Releases 10and 11 Galaxy samples.MNRAS, 441(1):24–62, June 2014.doi: 10.1093/mnras/stu523.
• Reid etal. [2016]Beth Reid, Shirley Ho, Nikhil Padmanabhan, etal.SDSS-III Baryon Oscillation Spectroscopic Survey Data Release 12:galaxy target selection and large-scale structure catalogues.MNRAS, 455(2):1553–1573, January 2016.doi: 10.1093/mnras/stv2382.
• Ross etal. [2020]AshleyJ. Ross, Julian Bautista, Rita Tojeiro, etal.The Completed SDSS-IV extended Baryon Oscillation SpectroscopicSurvey: Large-scale structure catalogues for cosmological analysis.MNRAS, 498(2):2354–2371, October 2020.doi: 10.1093/mnras/staa2416.
• Lyke etal. [2020]BradW. Lyke, AlexandraN. Higley, J.N. McLane, etal.The Sloan Digital Sky Survey Quasar Catalog: Sixteenth DataRelease.ApJS, 250(1):8, September 2020.doi: 10.3847/1538-4365/aba623.
• Raichoor etal. [2021]Anand Raichoor, Arnaud de Mattia, AshleyJ. Ross, etal.The completed SDSS-IV extended Baryon Oscillation SpectroscopicSurvey: large-scale structure catalogues and measurement of the isotropic BAObetween redshift 0.6 and 1.1 for the Emission Line Galaxy Sample.MNRAS, 500(3):3254–3274, January 2021.doi: 10.1093/mnras/staa3336.
• Bianchi and Percival [2017]Davide Bianchi and WillJ. Percival.Unbiased clustering estimation in the presence of missingobservations.MNRAS, 472(1):1106–1118, November 2017.doi: 10.1093/mnras/stx2053.
• Mohammad etal. [2018]F.G. Mohammad, D.Bianchi, W.J. Percival, etal.The VIMOS Public Extragalactic Redshift Survey (VIPERS). Unbiasedclustering estimate with VIPERS slit assignment.A&A, 619:A17, November 2018.doi: 10.1051/0004-6361/201833853.
• Percival and Bianchi [2017]WillJ. Percival and Davide Bianchi.Using angular pair upweighting to improve 3D clusteringmeasurements.MNRAS, 472(1):L40–L44, November 2017.doi: 10.1093/mnrasl/slx135.
• Mohammad etal. [2020]FaizanG. Mohammad, WillJ. Percival, Hee-Jong Seo, etal.The completed SDSS-IV extended baryon oscillation spectroscopicsurvey: pairwise-inverse probability and angular correction for fibrecollisions in clustering measurements.MNRAS, 498(1):128–143, October 2020.doi: 10.1093/mnras/staa2344.
• Levi etal. [2013]Michael Levi, Chris Bebek, Timothy Beers, Robert Blum, Robert Cahn,Daniel Eisenstein, Brenna Flaugher, Klaus Honscheid, Richard Kron,Ofer Lahav, Patrick McDonald, Natalie Roe, David Schlegel, andrepresenting the DESI collaboration.The DESI Experiment, a whitepaper for Snowmass 2013.arXiv e-prints, art. arXiv:1308.0847, August 2013.
• DESI Collaborationetal. [2016a]DESI Collaboration etal.The DESI Experiment Part I: Science,Targeting, and Survey Design.arXiv e-prints, art. arXiv:1611.00036, October2016a.
• DESI Collaborationetal. [2016b]DESI Collaboration etal.The DESI Experiment Part II: Instrument Design.arXiv e-prints, art. arXiv:1611.00037, October2016b.
• DESI Collaboration [2025]DESI Collaboration.DESI 2024 I: Data Release 1 of the Dark Energy SpectroscopicInstrument.in preparation, 2025.
• DESI Collaboration [2024a]DESI Collaboration.DESI 2024 II: Sample definitions, characteristics and two-pointclustering statistics.in preparation, 2024a.
• DESI Collaboration etal. [2024a]DESI Collaboration etal.DESI 2024 III: Baryon Acoustic Oscillations from Galaxies andQuasars.arXiv e-prints, page arXiv:2404.03000, 2024a.doi: 10.48550/arXiv.2404.03000.
• DESI Collaboration etal. [2024b]DESI Collaboration etal.DESI 2024 IV: Baryon Acoustic Oscillations from the Lyman AlphaForest.arXiv e-prints, page arXiv:2404.03001, 2024b.doi: 10.48550/arXiv.2404.03001.
• DESI Collaboration [2024b]DESI Collaboration.DESI 2024 V: Analysis of the full shape of two-point clusteringstatistics from galaxies and quasars.in preparation, 2024b.
• DESI Collaboration etal. [2024c]DESI Collaboration etal.DESI 2024 VI: Cosmological Constraints from the Measurements ofBaryon Acoustic Oscillations.arXiv e-prints, page arXiv:2404.03002, 2024c.doi: 10.48550/arXiv.2404.03002.
• DESI Collaboration [2024c]DESI Collaboration.DESI 2024 VII: Cosmological constraints from full-shape analyses ofthe two-point clustering statistics measurements.in preparation, 2024c.
• DESI Collaboration [2024d]DESI Collaboration.DESI 2024 VIII: Constraints on Primordial Non-Gaussianities.in preparation, 2024d.
• Allende Prieto etal. [2020]Carlos Allende Prieto, AndrewP. Cooper, Arjun Dey, etal.Preliminary Target Selection for the DESI Milky Way Survey (MWS).Research Notes of the American Astronomical Society,4(10):188, October 2020.doi: 10.3847/2515-5172/abc1dc.
• Ruiz-Macias etal. [2020]Omar Ruiz-Macias, Pauline Zarrouk, Shaun Cole, etal.Preliminary Target Selection for the DESI Bright Galaxy Survey(BGS).Research Notes of the American Astronomical Society,4(10):187, October 2020.doi: 10.3847/2515-5172/abc25a.
• Zhou etal. [2020]Rongpu Zhou, JeffreyA. Newman, KyleS. Dawson, etal.Preliminary Target Selection for the DESI Luminous Red Galaxy (LRG)Sample.Research Notes of the American Astronomical Society,4(10):181, October 2020.doi: 10.3847/2515-5172/abc0f4.
• Raichoor etal. [2020]Anand Raichoor, DanielJ. Eisenstein, Tanveer Karim, etal.Preliminary Target Selection for the DESI Emission Line Galaxy (ELG)Sample.Research Notes of the American Astronomical Society,4(10):180, October 2020.doi: 10.3847/2515-5172/abc078.
• Yèche etal. [2020]Christophe Yèche, Nathalie Palanque-Delabrouille, Charles-AntoineClaveau, etal.Preliminary Target Selection for the DESI Quasar (QSO) Sample.Research Notes of the American Astronomical Society,4(10):179, October 2020.doi: 10.3847/2515-5172/abc01a.
• Dey etal. [2019]Arjun Dey, DavidJ. Schlegel, Dustin Lang, etal.Overview of the DESI Legacy Imaging Surveys.AJ, 157(5):168, May 2019.doi: 10.3847/1538-3881/ab089d.
• DESI Collaboration etal. [2023a]DESI Collaboration etal.Validation of the Scientific Program for the Dark EnergySpectroscopic Instrument.arXiv e-prints, art. arXiv:2306.06307, June2023a.doi: 10.48550/arXiv.2306.06307.
• Guy etal. [2023]J.Guy, S.Bailey, A.Kremin, etal.The Spectroscopic Data Processing Pipeline for the Dark EnergySpectroscopic Instrument.AJ, 165(4):144, April 2023.doi: 10.3847/1538-3881/acb212.
• DESI Collaboration etal. [2023b]DESI Collaboration etal.The Early Data Release of the Dark Energy Spectroscopic Instrument.arXiv e-prints, art. arXiv:2306.06308, June2023b.doi: 10.48550/arXiv.2306.06308.
• Hardin, R. H., Sloane, N. J. A., and Smith, W. D. 2001, Tables ofSpherical Codes with Icosahedral Symmetry [Florham Park: AT&T ShannonLab.]Hardin, R. H., Sloane, N. J. A., and Smith, W. D. 2001, Tables of SphericalCodes with Icosahedral Symmetry (Florham Park: AT&T Shannon Lab.).Accessed 02-12-2024.
• C.Zhao et al. [2024]C.Zhao et al.Mock catalogues with survey realism for the DESI DR1.in preparation, 2024.
• D.Bianchi et al. [2024]D.Bianchi et al.Characterization of DESI fiber assignment incompleteness effect on2-point clustering and mitigation methods for 2024 analysis.in preparation, 2024.
• Maksimova etal. [2021]NinaA Maksimova, LehmanH Garrison, DanielJ Eisenstein, etal.AbacusSummit: a massive set of high-accuracy, high-resolution N-bodysimulations.Monthly Notices of the Royal Astronomical Society,508(3):4017–4037, 09 2021.ISSN 0035-8711.doi: 10.1093/mnras/stab2484.URL https://doi.org/10.1093/mnras/stab2484.
• Garrison etal. [2021]LehmanH Garrison, DanielJ Eisenstein, Douglas Ferrer, etal.The abacus cosmological N-body code.Monthly Notices of the Royal Astronomical Society,508(1):575–596, 09 2021.ISSN 0035-8711.doi: 10.1093/mnras/stab2482.URL https://doi.org/10.1093/mnras/stab2482.
• Planck Collaboration etal. [2016]Planck Collaboration etal.Planck 2015 results - xiii. cosmological parameters.A&A, 594:A13, 2016.doi: 10.1051/0004-6361/201525830.URL https://doi.org/10.1051/0004-6361/201525830.
• Rocher etal. [2023]Antoine Rocher, Vanina Ruhlmann-Kleider, Etienne Burtin, etal.The desi one-percent survey: exploring the halo occupationdistribution of emission line galaxies with abacussummit simulations.Journal of Cosmology and Astroparticle Physics, 2023(10):016, oct 2023.doi: 10.1088/1475-7516/2023/10/016.URL https://dx.doi.org/10.1088/1475-7516/2023/10/016.
• Yuan etal. [2023c]Sihan Yuan, Hanyu Zhang, AshleyJ. Ross, etal.The DESI One-Percent Survey: Exploring the Halo OccupationDistribution of Luminous Red Galaxies and Quasi-Stellar Objects withAbacusSummit.arXiv e-prints, art. arXiv:2306.06314, June2023c.doi: 10.48550/arXiv.2306.06314.
• Bianchi etal. [2018]Davide Bianchi, Angela Burden, WillJ. Percival, etal.Unbiased clustering estimates with the DESI fibre assignment.MNRAS, 481(2):2338–2348, December 2018.doi: 10.1093/mnras/sty2377.
• Smith etal. [2019]Alex Smith, Jian-hua He, Shaun Cole, etal.Correcting for fibre assignment incompleteness in the DESI BrightGalaxy Survey.MNRAS, 484(1):1285–1300, March 2019.doi: 10.1093/mnras/stz059.
• Myers etal. [2023]AdamD. Myers, John Moustakas, Stephen Bailey, etal.The Target-selection Pipeline for the Dark Energy SpectroscopicInstrument.AJ, 165(2):50, February 2023.doi: 10.3847/1538-3881/aca5f9.
• Raichoor et al. [2024]Raichoor et al.2024.in preparation.
• Landy and Szalay [1993]StephenD. Landy and AlexanderS. Szalay.Bias and Variance of Angular Correlation Functions.ApJ, 412:64, July 1993.doi: 10.1086/172900.
• Keihänen etal. [2019]E.Keihänen, H.Kurki-Suonio, V.Lindholm, etal.Estimating the galaxy two-point correlation function using a splitrandom catalog.A&A, 631:A73, November 2019.doi: 10.1051/0004-6361/201935828.
• Sinha and Garrison [2020]Manodeep Sinha and LehmanH. Garrison.CORRFUNC - a suite of blazing fast correlation functions on theCPU.MNRAS, 491(2):3022–3041, Jan 2020.doi: 10.1093/mnras/stz3157.
• Sinha and Garrison [2019]Manodeep Sinha and Lehman Garrison.Corrfunc: Blazing fast correlation functions with avx512f simdintrinsics.In Amit Majumdar and Ritu Arora, editors, Software Challengesto Exascale Computing, pages 3–20, Singapore, 2019. Springer Singapore.ISBN 978-981-13-7729-7.URL https://doi.org/10.1007/978-981-13-7729-7_1.
• Bianchi and Verde [2020]Davide Bianchi and Licia Verde.Confronting missing observations with probability weights: Fourierspace and generalized formalism.MNRAS, 495(1):1511–1529, June 2020.doi: 10.1093/mnras/staa1267.
• Feldman etal. [1994]HumeA. Feldman, Nick Kaiser, and JohnA. Peaco*ck.Power-Spectrum Analysis of Three-dimensional Redshift Surveys.ApJ, 426:23, May 1994.doi: 10.1086/174036.
• A.Ross et al. [2024]A.Ross et al.Construction of Large-scale Structure Catalogs for the Dark EnergySpectroscopic Instrument.in preparation, 2024.
• Astropy Collaboration etal. [2013]Astropy Collaboration etal.Astropy: A community Python package for astronomy.A&A, 558:A33, October 2013.doi: 10.1051/0004-6361/201322068.
• Astropy Collaboration etal. [2018]Astropy Collaboration etal.The Astropy Project: Building an Open-science Project and Status ofthe v2.0 Core Package.AJ, 156(3):123, September 2018.doi: 10.3847/1538-3881/aabc4f.
• Astropy Collaboration etal. [2022]Astropy Collaboration etal.The Astropy Project: Sustaining and Growing a Community-orientedOpen-source Project and the Latest Major Release (v5.0) of the Core Package.ApJ, 935(2):167, August 2022.doi: 10.3847/1538-4357/ac7c74.

Appendix A Author Affiliations

¹Department of Physics, Southern Methodist University, 3215 Daniel Avenue, Dallas, TX 75275, USA

²Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206, La Laguna, Tenerife, Spain

³Instituto de Astrofísica de Canarias, C/ Vía Láctea, s/n, E-38205 La Laguna, Tenerife, Spain

⁴Department of Physics & Astronomy, University of Wyoming, 1000 E. University, Dept.3905, Laramie, WY 82071, USA

⁵Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA

⁶Department of Astronomy, The Ohio State University, 4055 McPherson Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA

⁷The Ohio State University, Columbus, 43210 OH, USA

⁸Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy

⁹University of Michigan, Ann Arbor, MI 48109, USA

¹⁰IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

¹¹Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada

¹²Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada

¹³Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada

¹⁴Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA

¹⁵Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA

¹⁶Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK

¹⁷Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK

¹⁸Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México

¹⁹Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

²⁰Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Menlo Park, CA 94305, USA

²¹SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA

²²Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio Ip, CP 111711, Bogotá, Colombia

²³Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio H, CP 111711 Bogotá, Colombia

²⁴Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

²⁵Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK

²⁶Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain

²⁷Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA

²⁸Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA

²⁹School of Mathematics and Physics, University of Queensland, 4072, Australia

³⁰NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA

³¹Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), FR-75005 Paris, France

³²Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

³³Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain

³⁴Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain

³⁵Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA

³⁶Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, U.K

³⁷National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Rd., Chaoyang District, Beijing, 100012, P.R. China

³⁸Departamento de Física, Universidad de Guanajuato - DCI, C.P. 37150, Leon, Guanajuato, México

³⁹Instituto Avanzado de Cosmología A.C., San Marcos 11 - Atenas 202. Magdalena Contreras, 10720. Ciudad de México, México

⁴⁰Korea Astronomy and Space Science Institute, 776, Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea

⁴¹Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA

⁴²University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, USA

⁴³Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain

⁴⁴Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA

⁴⁵Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea

⁴⁶CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain

⁴⁷Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

⁴⁸Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA