Luminosity and Stellar Mass Functions of Faint Photometric Satellites around Spectroscopic Central Galaxies from DESI Year-1 Bright Galaxy Survey Wenting Wang 1,2aa, Xiaohu Yang 1,3aa, Yipeng Jing 1,3aa, Ashley J. Ross 4 , Malgorzata Siudek 5,6aa, John Moustakas 7aa, Samuel G. Moore 8 , Shaun Cole 8aa, Carlos Frenk8aa, Jiaxi Yu9, Sergey E. Koposov 10,11aa, Jiaxin Han 1aa, Zhenlin Tan 1 , Kun Xu 12aa, Yizhou Gu 3 , Yirong Wang 1aa, Oleg Y. Gnedin 13aa, Jessica Nicole Aguilar 14aa, Steven Ahlen 15aa, Davide Bianchi 16,17aa, David Brooks 18aa, Todd Claybaugh 14 , Axel de la Macorra 19,20,21,22 , Arjun Dey 23aa, Peter Doel18aa, Jaime E. Forero-Romero 24,25 , Enrique Gaztañaga 26,27 , Satya Gontcho A Gontcho 14 , Gaston Gutierrez 28 , Klaus Honscheid 4,29,30aa, Mustapha Ishak 31aa, Theodore Kisner 14 , Martin Landriau 14aa, Laurent Le Guillou 32aa, Marc Manera 33,34 , Aaron Meisner 23 , Ramon Miquel 33,35aa, Seshadri Nadathur27aa, Claire Poppett 14,36,37aa, Francisco Prada 38,39,40 , Ignasi Pérez-Ràfols 41aa, Graziano Rossi 42 , Eusebio Sanchez 43aa, David Schlegel 14aa, Hee-Jong Seo 44aa, Joseph Harry Silber 14aa, David Sprayberry 23aa, Gregory Tarlé 45aa, Benjamin Alan Weaver 23,46 , and Hu Zou 47aa 1 Department of Astronomy, School of Physics and Astronomy, and Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China; wenting.wang@sjtu.edu.cn, xyang@sjtu.edu.cn, ypjing@sjtu.edu.cn 2 State Key Laboratory of Dark Matter Physics, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China 3 Tsung-Dao Lee Institute, and Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China 4 Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA 5 Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain 6 Instituto AstroFsica de Canarias, Av. Via Lactea s/n, 38205 La Laguna, Spain 7 Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12110, USA 8 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK 9 Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 10 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK 11 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 12 Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 13 Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA 14 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA 15 Physics Department, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA 16 Dipartimento di Fisica “Aldo Pontremoli,” Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy 17 INAF—Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy 18 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK 19 Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, Av. Politécnico 2508, Col. Sn. Pedro Zacatenco, Del. Gustavo A. Madero, 07010 CDMX, México 20 Departamento de Física, Instituto Nacional de Investigaciones Nucleares, Carreterra México-Toluca S/N, La Marquesa, Ocoyoacac, Edo. de México C. P. 52750, México 21 Departamento de Física, DCI-Campus León, Universidad de Guanajuato, Loma del Bosque 103, León, Guanajuato C. P. 37150, México. 22 Instituto de Física, Universidad Nacional Autónoma de México, Circuito de la Investigación CientíFca, Ciudad Universitaria, Cd. de México C. P. 04510, México 23 NSF NOIRLab, 950 N. Cherry Avenue, Tucson, AZ 85719, USA 24 Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10, EdiFcio Ip, CP 111711, Bogotá, Colombia 25 Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10, EdiFcio H, CP 111711 Bogotá, Colombia 26 Institut d’Estudis Espacials de Catalunya (IEEC), c/ Esteve Terradas 1, EdiFci RDIT, Campus PMT-UPC, 08860 Castelldefels, Spain 27 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK 28 Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA 29 The Ohio State University, Columbus, OH 43210, USA 30 Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA 31 Department of Physics, The University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USA 32 Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), FR-75005 Paris, France 33 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, EdiFci Cn, Campus UAB, 08193, Bellaterra (Barcelona), Spain 34 Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain 35 Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain 36 University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, USA 37 Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA 38 Instituto de Física Teórica (IFT) UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, E-28049, Madrid, Spain 39 Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain 40 Instituto de Astrofísica de Canarias, C/ Vía Láctea, s/n, E-38205 La Laguna, Tenerife, Spain 41 Departament de Física, EEBE, Universitat Politècnica de Catalunya, c/Eduard Maristany 10, 08930 Barcelona, Spain 42 Department of Physics and Astronomy, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 05006, Republic of Korea 43 CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain 44 Department of Physics & Astronomy, Ohio University, 139 University Terrace, Athens, OH 45701, USA 45 University of Michigan, 500 S. State Street, Ann Arbor, MI 48109, USA 46 Department of Physics and Center for Cosmology and Particle Physics, New York University, New York, NY 10003, USA 47 National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing, 100012, People’s Republic of China Received 2025 March 5; revised 2025 April 15; accepted 2025 May 2; published 2025 June 19 The Astrophysical Journal, 986:218 (18pp), 2025 June 20 https://doi.org/10.3847/1538-4357/add5df © 2025. The Author(s). Published by the American Astronomical Society. aaaaaaa Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1 https://orcid.org/0000-0002-5762-7571 https://orcid.org/0000-0003-3997-4606 https://orcid.org/0000-0002-4534-3125 https://orcid.org/0000-0002-2949-2155 https://orcid.org/0000-0002-2733-4559 https://orcid.org/0000-0002-5954-7903 https://orcid.org/0000-0002-2338-716X https://orcid.org/0000-0003-2644-135X https://orcid.org/0000-0002-8010-6715 https://orcid.org/0000-0002-7697-3306 https://orcid.org/0000-0003-3203-3299 https://orcid.org/0000-0001-9852-9954 https://orcid.org/0000-0003-0822-452X https://orcid.org/0000-0001-6098-7247 https://orcid.org/0000-0001-9712-0006 https://orcid.org/0000-0002-8458-5047 https://orcid.org/0000-0002-4928-4003 https://orcid.org/0000-0002-6397-4457 https://orcid.org/0000-0002-6550-2023 https://orcid.org/0000-0002-6024-466X https://orcid.org/0000-0003-1838-8528 https://orcid.org/0000-0001-7178-8868 https://orcid.org/0000-0002-6610-4836 https://orcid.org/0000-0001-9070-3102 https://orcid.org/0000-0003-0512-5489 https://orcid.org/0000-0001-6979-0125 https://orcid.org/0000-0002-9646-8198 https://orcid.org/0000-0002-5042-5088 https://orcid.org/0000-0002-6588-3508 https://orcid.org/0000-0002-3461-0320 https://orcid.org/0000-0001-7583-6441 https://orcid.org/0000-0003-1704-0781 https://orcid.org/0000-0002-6684-3997 mailto:wenting.wang@sjtu.edu.cn mailto:xyang@sjtu.edu.cn mailto:ypjing@sjtu.edu.cn https://doi.org/10.3847/1538-4357/add5df https://crossmark.crossref.org/dialog/?doi=10.3847/1538-4357/add5df&domain=pdf&date_stamp=2025-06-19 https://creativecommons.org/licenses/by/4.0/ Abstract We measure the luminosity functions (LFs) and stellar mass functions (SMFs) of photometric satellite galaxies around spectroscopically identiFed isolated central galaxies (ICGs). The photometric satellites are from the DESI Legacy Imaging Surveys (DR9), while the spectroscopic ICGs are selected from the DESI Year-1 BGS sample. We can measure satellite LFs down to r-band absolute magnitudes of Mr,sat ∼ −7, around ICGs as small as /< M Mlog 7.110 ,ICG are already mostly observed given the Xux limit of r < 17.7, and thus, the fainter Xux limit of rdustcorr < 19.5 almost does not include more central galaxies with / >M Mlog 7.110 ,ICG . Over smaller stellar mass ranges of /< 40. Our parent BGS sample for selection is limited to the DESI Tier-1 BGS sample with rdustcorr < 19.5, which is a nearly Xux-limited sample (C. Hahn et al. 2023). Note, however, in the densest galaxy cluster or group environments, the completeness can be sometimes as low as ∼20% (e.g., A. Smith et al. 2019; R. Shi et al. 2024). The stellar masses of DESI BGS are estimated with the DESI FASTSPECFIT pipeline53 (J. Moustakas et al. 2023; J. Moustakas et al. 2025, in preparation) assuming a G. Chabrier (2003) initial mass function, and this would be the default choice of DESI based stellar mass for the majority of results in this paper. Moreover, based on the same initial mass function, DESI Year-1 data also has the CIGALE stellar mass measurements with modeling of active galactic nuclei (AGNs) through the combination of DESI and WISE photometry (M. Siudek et al. 2024), which we will use in a later section of this paper as a comparison to conFrm the results. We adopt the following selection criteria. Galaxies should be the brightest within the projected virial radius, R200, of their host dark matter haloes54 and within 3 times the virial velocity along the line of sight. To compensate for the incompleteness of the DESI BGS sample and avoid the situation that when a true companion is actually brighter but the galaxy mistakenly passes our selection because this companion does not have spectroscopic observations, we further discard galaxies that have a photometric companion satisfying the magnitude requirement, whose spectroscopic redshift information is not available but its photometric redshift (photo-z), zphot, satisFes the condition of ( )90% at / >M Mlog 11.510 ,ICG . The readers are referred to W. Wang et al. (2019) for additional details. However, the mock galaxy catalog does not incorporate the effect of incomplete spectro- scopic redshifts, and the readers may wonder whether the chosen criterion for photo-z above is appropriate, given the large relative photo-z uncertainties at low redshifts. To test this, we have tried to reject the central ICG, as long as it has a brighter companion galaxy projected within the virial radius, and does not have spectroscopic redshift measurement. The number of selected ICGs decreases at most by ∼20% in the most-massive bin, and for most of the other mass bins, the number of ICGs only decreases by < ∼10%, indicating our results are unlikely affected by the photo-z selection signiFcantly. In this study, we will calculate the LF for satellites projected within the halo virial radius of ICGs. To ensure consistencies with our previous studies (W. Wang et al. 2019; W. Wang et al. 2021a, 2021b; P. Alonso et al. 2023), here we choose the same R200 for all ICGs in the same bin of stellar mass. Historically, this R200 for each stellar mass bin was calculated based on ICGs selected from the mock galaxy catalog of Q. Guo et al. (2011b), and we maintain the choice in this paper. Note, however, the R200 here is different from the R200 we adopted above when selecting ICGs (which vary for each individual galaxy) and was calculated from the abundance matching formula of Q. Guo et al. (2010). So hereafter, we choose to denote this R200 (within which we calculate satellite number counts) explicitly as R200,mock. 52 The median redshift of SDSS Main galaxies is about z = 0.1. 53 https://fastspecFt.readthedocs.io/en/latest/ 54 R200 is deFned to be the radius within which the average matter density is 200 times the mean critical density of the Universe. The virial radius and velocity here are derived through the abundance matching formula between stellar mass and halo mass (Q. Guo et al. 2010) for each individual galaxy. In addition, based on mock catalogs, it was demonstrated that the choice of three times virial velocity along the line of sight is a safe criterion that identiFes all true companion galaxies. 4 The Astrophysical Journal, 986:218 (18pp), 2025 June 20 Wang et al. https://fastspecfit.readthedocs.io/en/latest/ The values of R200,mock in these different bins 55 are provided in Table 1. For the three least-massive bins, we simply Fx their values of R200,mock to the bin of /< 30 kpc for / >M Mlog 10.210 ,ICG and >10 kpc for /M Mlog 10.210 ,ICG . Table 1 also provides the total number of ICGs in each bin under different conditions. The third column shows the total number of ICGs selected from Tier-1 BGS in DESI Year-1 observation with rdustcorr < 19.5 (NICG,DESI). For DESI, the ICG numbers are a weighted sum of 1/pobs, to account for Fber incompleteness in DESI BGS. The readers are referred to Section 3.2 to review how 1/pobs is deFned. We will discuss the other columns later in this work. 2.3. Isolated Central Galaxies in SDSS In this paper we will perform detailed comparisons of satellite LF measurements around ICGs selected from both DESI BGS and from SDSS spectroscopic Main galaxies, and in this subsection, we introduce how ICGs are selected from SDSS. The parent sample used for selection is the NYU Value Added Galaxy Catalogue (NYU-VAGC; M. R. Blanton et al. 2005b), which is based on the spectroscopic Main galaxy sample from the seventh data release of SDSS (DR7; K. N. Abazajian et al. 2009). The sample is Xux limited down to an apparent magnitude of ∼17.7 in the SDSS r band, with most of the objects below redshift z = 0.25. Stellar masses in NYU-VAGC were estimated from the K-corrected galaxy colors by Ftting the stellar population synthesis model (M. R. Blanton & S. Roweis 2007) assuming a G. Chabrier (2003) initial mass function. In the previous study of W. Wang et al. (2021a), we selected ICGs from SDSS, with almost the same isolation criteria adopted in this paper, except for the compensating selection criteria adopted to photometric companions when their spectro- scopic redshift measurements are not available due to Fber collisions. For such cases, W. Wang et al. (2021a) adopted the photo-z probability distribution from C. E. Cunha et al. (2009) for ICG selections, whereas the selection criteria we adopted above do not require a full photo-z probatility distribution, but instead rely on the 1σ error, σp, of photo-z measurements ( ( ) 18 deg in Galactic coordinates (A. Dey et al. 2019). The surveys are composed of three imaging projects, the Beijing-Arizona Sky Survey (H. Zou et al. 2017), the Mayall z- band Legacy Survey, and the Dark Energy Camera Legacy Survey. The survey footprints are observed at least once, while most Felds are observed twice or more times. In fact, the depth varies over the sky. Data used in this study are downloaded from the ninth data release (DR9) of the DESI Legacy surveys (Schlegel et al. 2025, in preparation). All data from the Legacy Surveys are Frst processed through the NOAO Community Pipelines (A. Dey et al. 2019). The photometric source product is constructed by TRACTOR.56TRACTOR (D. Lang et al. 2016) generates an inference-based model of the sky that best Fts the real data. The sources are detected using the sky subtracted and point-spread function (PSF) convolved stacked images with a threshold of 6σ. For each detected source, TRACTOR models its pipeline-reduced images from different exposures and in multiple bands simultaneously. This is achieved by Ftting parametric proFles including a delta function (for point Table 1 Average Halo Virial Radii (R200,mock) for Our ICGs Grouped by Stellar Mass, which Are Estimated Using ICGs Selected from a Mock Galaxy Catalog of Q. Guo et al. (2011b) /M Mlog ,ICG R200,mock NICG,DESI NICG,SDSS NICG,DESI (kpc) M* corr (resel +rebin) (r < 17.7) 11.4–11.7 758.65 112585 19375 29968 11.1–11.4 459.08 468736 54718 92448 10.8–11.1 288.16 892661 85557 130368 10.5–10.8 214.80 955531 82512 109634 10.2–10.5 173.18 723499 60217 69635 9.9–10.2 142.85 488155 36340 41541 9.2–9.9 114.64 586346 41229 46936 8.5–9.2 82.68 180217 13582 15168 7.8–8.5 61.42 41515 3219 3394 7.1–7.8 61.42 7305 593 768 6.4–7.1 61.42 1383 102 211 5.7–6.4 61.42 228 11 48 Note. In each stellar mass bin, the numbers of ICGs selected under different conditions are provided. The third column shows the total number of ICGs selected from Tier-1 BGS of DESI Year-1 data (NICG,DESI), with Xux limit of rdustcorr < 19.5. Here, rdustcorr is Galactic extinction corrected. The numbers are calculated using 1/pobs as weights to account for Fber incompleteness in DESI (see Section 3.2). The fourth column gives the numbers of SDSS ICGs in these stellar mass bins, after median stellar mass corrections to match the stellar mass of DESI (see Figure 1) and with Fber incompleteness corrections (see Section 3.2). The last column is similar to the third column, but we further includes a Xux cut of r < 17.7 to DESI ICGs, with the other selections identical to the third column, to have a more fair comparison with SDSS. 55 The bins are deFned in log stellar mass following W. Wang & S. D. M. White (2012), with the sizes tuned to ensure a sufFcient number of ICGs in bins in both the most- and least-massive ends. 56 https://github.com/dstndstn/tractor 5 The Astrophysical Journal, 986:218 (18pp), 2025 June 20 Wang et al. https://github.com/dstndstn/tractor sources), a de Vaucouleurs law, an exponential model or de Vaucouleurs plus exponential to each image simultaneously. The model is assumed to be the same for all images and is convolved with the corresponding PSF in different exposures and bands before Ftting to each image. TRACTOR also outputs the quantity that can be used to distinguish extended sources (galaxies) from point sources (stars). Starting from the DESI Legacy Survey database sweep Fles, we remove all sources with TYPE classiFed as “PSF,” and require BITMASK not containing any of the following: BRIGHT, SATUR_G (saturated), SATUR_R, ALLMASK_G,57 and ALLMASK_R. In W. Wang et al. (2021a), we discussed that the average number counts of sources keep rising to r ∼ 23. However, given the variation of the depth over the sky, we only use the regions with depth deeper than 23 in r. This is achieved by at Frst selecting bricks with at least three exposures in both the g and r bands, and we also require the r-band GALdepth of the bricks to be deeper than 23. We only include galaxies within these selected bricks. Finally, when using the Legacy survey photometric sources to calculate companion LFs, we further make the Xux limit 0.5 magnitude brighter, i.e., we adopt a Xux cut of r < 22.5 to deFne a safe Xux-limited photometric sample throughout our analysis in this study. 3. Methodology 3.1. Satellite Counting and Background Subtraction Our method of counting photometric satellites around spectroscopically identiFed ICGs and computing the intrinsic luminosities of satellites follows W. Wang & S. D. M. White (2012). For each ICG in a given stellar mass bin, we, at Frst, count all of its photometric companions down to the Xux limit of r = 22.5 and projected within the halo virial radius, R200,mock (see Table 1). The physical scale is calculated based on the spectroscopic redshift and angular diameter distance of the ICG. However, without redshift information and accurate distance measurements for photometric companions, the companion counts can only be counted as a function of apparent magnitude and observed-frame color, and the total counts not only include true satellites, but also contamination by fore/background sources. We will discuss the subtraction of foreground and background sources shortly. We derive the intrinsic luminosities and rest-frame colors using the following method. For each companion, we employ the empirical K correction of E. Westra et al. (2010) to estimate its rest-frame color using the observed color and also assuming that the companion is at the same redshift as the ICG. The distance modulus is also calculated from the redshift of the ICG, in combination with the K correction, to infer the intrinsic luminosity or absolute magnitude. This is a reason- able approximation, because physically associated satellite galaxies are expected to share very similar redshifts as the ICG. In this paper, we mainly present measurements of satellite LFs, but we will also show our measurements of satellite SMFs. To obtain the stellar mass of satellites, W. Wang & S. D. M. White (2012) derived a relation between the r-band stellar mass-to-light ratio and the g − r galaxy color from SDSS spectroscopic Main galaxies. In this paper, we update the relation using DESI Year-1 BGS as M*/Lr = 2.0012×0.1 (g − r) −0.3133. Here, the upper index of 0.1 means all galaxies are K-corrected to z = 0.1. Note that for all r-band absolute magnitudes presented in this paper, the K correction is always done to z = 0.1, but we do not show the upper index of 0.1 to make the symbols short and easy to read. To ensure the completeness of satellite number counts in different luminosity bins, for each ICG, we convert the photometric Xux limit (r < 22.5; see Section 2.4) to a K-corrected absolute magnitude, Mr,lim, using the redshift of the ICG and a color chosen to be on the red envelope of the intrinsic color distribution for galaxies at that redshift. Here, Mr,lim can also be converted to a limit in stellar mass, * M ,lim, based on the same color on the red envelope. Then for a given luminosity or stellar mass bin, ICGs are allowed to contribute to the Fnal averaged companion counts only if Mr,lim (or * M ,lim) is fainter (smaller) than the fainter (smaller) bin boundary. As a result, the number of actual ICGs contributing to different luminosity or stellar-mass bins for satellites can vary. The fainter or smaller the bin, the fewer number of ICGs can contribute to the satellite counts, and their redshifts are lower. In the end, the total companion counts are divided by the total number of ICGs, which actually contribute to the satellite counts in each bin. This provides the complete and average companion LF or SMF per ICG. To subtract fore/background contamination, we use a sample of random points, which are assigned the same redshift and stellar mass distributions as true central primaries, but their coordinates have been randomized within the survey footprint. The averaged companion counts per random point are calculated in exactly the same way as around real ICGs above, and then be subtracted off from the averaged counts around real primaries, to obtain the background-subtracted averaged satellite counts per host ICG. A red-end cut of ( ) /< +g r M M0.065log 0.350.1 10 is applied to the photometric companions to reduce the number of background sources that are too red to be at the same redshift of the ICG, and hence increase the S/N. The color cut is drawn from the color distribution of SDSS spectroscopic Main galaxies in W. Wang & S. D. M. White (2012). Again, we emphasize that the upper index of 0.1 means all galaxies are K-corrected to z = 0.1, and for all r-band absolute magnitudes in this paper, the K correction is always done to z = 0.1. 3.2. Correction for Fiber Incompleteness Though the DESI Tier-1 BGS sample is a nearly Xux- limited sample, its completeness fraction in fact varies signiFcantly with the density of the local environment. In dense galaxy cluster or group environments, the worse completeness fraction can be as low as ∼20% (e.g., A. Smith et al. 2019; D. Bianchi et al. 2025; R. Shi et al. 2024). In order to account for the incompleteness of the BGS sample due to Fber assignments, we weight each ICG by the inverse of pobs (the PROB_OBS column) from the DESI Year-1 large scale structure catalog (A. J. Ross et al. 2025). The Fnal averaged satellite LF per host ICG is in fact the weighted average, with 1/pobs as weights. Here, pobs is deFned as pobs = Nassign/129, with Nassign being the number of DESI mock Fber assignment realizations in which the target is assigned. The readers are 57 ALLMASK_X denotes a source that touches a pixel with problems in an entire set of overlapping X-band images. Explicitly, such pixels include BADPIX, SATUR (saturated), INTERP (interpolated), CR (hit by cosmic rays), or EDGE (edge pixels). 6 The Astrophysical Journal, 986:218 (18pp), 2025 June 20 Wang et al. referred to J. Lasker et al. (2025) for additional details. There are a total of 128 mock DESI Fber assignment simulations generated with different initial random seeds, and this is why the value in the denominator is 129=128+1. Adopting such a weighting strategy will increase the contribution by satellite counts around ICGs with lower Fber assignment completeness fractions. For SDSS spectroscopic Main galaxies, we use the Fber collision corrected Fles provided by the NYU-VAGC website58 to account for possible effects due to Fber assign- ment incompleteness. The collision correction is done simply by assigning the galaxy that does not have a spectroscopic redshift observation due to Fber collisions with the redshift of the nearest object in angular separation. 3.3. Correcting Incomplete Projected Area Due to the survey boundary and photometric masks, satellite counts in a projected circular region around each ICG may be masked or outside the survey footprint. We should estimate the completeness of the projected area around primaries. This is achieved by using the DESI Legacy survey photometric random samples provided in the database. We apply exactly the same selection and masks to random points. The completeness of the projected area is estimated as ( ) ( ) = × f number of actual random points area surface density of random points . 2 complete Our actual companion counts around both real and random primaries are divided by fcomplete for incompleteness corrections. 4. Results 4.1. Redshift Distribution, Limiting Magnitude, and Expectations Before presenting the satellite LF and SMF measurements, we Frst compare ICGs selected from SDSS spectroscopic Main galaxies and from DESI Tier-1 BGS sample in the Year- 1 observation, and use the comparison to discuss the expected improvements in measured satellite LFs and SMFs, around the deeper DESI BGS sample. First, Figure 1 shows a comparison between the SDSS and DESI FASTSPECFIT stellar mass measurements, for a sub- sample of DESI BGS matched to SDSS. In general, the black dots go through the red solid diagonal line. The bottom panel of Figure 1 shows the scatter in SDSS stellar mass at Fxed DESI stellar mass. The scatters are mostly ∼0.2 dex. The magenta solid and green dashed curves mark the mean and median of log SDSS stellar mass, at Fxed stellar mass in DESI. As we can see, the mean and median values indicate lower SDSS stellar masses at the most-massive end and higher SDSS stellar masses at lower masses, as compared with DESI FASTSPECFIT stellar masses. Though here we compare DESI FASTSPECFIT and SDSS stellar masses, similar signs and amounts of median deviations are found, if we compare SDSS stellar masses to DESI CIGALE stellar masses. We do not know exactly which stellar mass is closer to the ground truth, but we take the median bias and correct for the bias of SDSS stellar mass, to force them to agree with DESI FASTSPECFIT stellar mass on average.59 This is for fair comparisons between DESI and SDSS. After the correction, SDSS ICGs are binned into 12 different stellar mass bins, with their redshift distributions shown by the red hatched histograms in different panels of Figure 2. Here, the redshift distributions are all plotted after Fber incompleteness correc- tions. In the fourth column of Table 1, we provide the number for ICGs selected from SDSS Main galaxies after this stellar mass correction. Due to some modiFcation in selection criteria (see Section 2.3) and the stellar mass correction, the numbers differ from those of W. Wang et al. (2021a). The redshift distributions for ICGs selected from the DESI Tier-1 BGS sample in the Year-1 observation are shown by the green plain histograms in Figure 2. Here, to plot the histograms, we adopt 1/pobs as weights (see Section 3.2). With a deeper Xux limit of rdustcorr < 19.5, the green plain histograms of DESI Year-1 BGS (Tier-1) can extend to much higher redshifts. In particular, when the numbers of SDSS ICGs (red hatched histograms) already start to drop with the increase in redshifts, showing signiFcant incompleteness, the numbers of DESI ICGs still keep rising with the increase in redshifts, which start to drop at much higher redshifts. 6 7 8 9 10 11 12 log10M , DESI/M 0.0 0.2 0.4 0.6 (lo g 1 0M ,S DS S/M ) 7 8 9 10 11 12 lo g 1 0M ,S DS S/M 1:1 mean log10M , SDSS median log10M , SDSS Figure 1. Top panel: DESI FASTSPECFIT stellar masses vs. SDSS NYU- VAGC stellar masses. Black dots show a sample of DESI Year-1 BGS matched to SDSS. The red solid line marks y = x to guide the eye, while the magenta solid and green dashed curves mark the mean and median SDSS stellar mass at Fxed DESI stellar mass. Bottom panel: the scatter in log SDSS stellar mass, as a function of DESI FASTSPECFIT stellar mass. 58 http://sdss.physics.nyu.edu/vagc/ 59 Our selection of ICGs depend on stellar mass to infer the virial radius (see Section 2). We have tried to either reselect or not reselect SDSS ICGs after the median stellar mass correction, and Fnd with the reselection, the total numbers of SDSS ICGs become slightly smaller, but the redshift distributions and measured satellite LFs/SMFs are very similar. 7 The Astrophysical Journal, 986:218 (18pp), 2025 June 20 Wang et al. http://sdss.physics.nyu.edu/vagc/ We also choose a subset of ICGs from DESI BGS with the same Xux limit as SDSS, i.e., r < 17.7, which is shown as the black plain histogram. With the same Xux limit, the SDSS and DESI histograms are more similar.60 Note our corrections to the SDSS stellar mass (Figure 1) have redistributed SDSS ICGs in a few intermediate-mass bins to the most- and least- massive ends, bringing in much better agreement between the DESI and SDSS ICG redshift distributions under the same Xux limit of r < 17.7. Without the median stellar correction, the black plain and red hatched histograms show more signiFcant differences in the few most- and least-massive panels. However, in the two least-massive panels, the black plain histogram of DESI is still higher than the red hatched histogram of SDSS after the median stellar mass correction and with the same Xux limit. This is at least partially related to the accuracy in our median stellar mass correction, as the number of low-mass galaxies adopted to deduce the median bias is very limited in the two least-massive panels. The median correction shown in Figure 1 is very noisy at the low- mass end. Moreover, the red hatched histograms for SDSS in the three most-massive panels of Figure 2 are still narrower than the black plain histograms for DESI. This can be due to the remaining scatter between the DESI and SDSS stellar masses, which is impossible to be easily corrected. With Figure 2 and Table 1 showing the redshift distribution and number of ICGs in different stellar mass bins, now we move on to talk about the completeness of satellite counts, and what is the expected improvement of measuring satellite LFs around ICGs selected from DESI BGS, compared with SDSS. Figure 3 shows the limiting absolute magnitude, Mr,lim, above which the photometric satellite counts are complete above the Xux limit of our photometric sample, and as a function of redshift. Here, Mr,lim is estimated with an r-band Xux limit of r = 22.5, and readers are referred to Section 3 for details about how we calculate Mr,lim. With the decrease in redshifts, Mr,lim gets fainter and fainter. This means that satellite LFs at fainter magnitudes are only contributed to by more nearby ICGs, because satellites around more distant ICGs become incom- plete. In particular, if we want to push fainter than Mr,sat = −10 for satellite LF measurement, only ICGs with redshifts lower than z = 0.01 can contribute complete satellite counts. Despite the fact that DESI bright galaxies can extend to much higher redshifts with a deeper Xux limit of rdustcorr < 19.5 in Figure 2, as compared with SDSS Main galaxies with r < 17.7, stellar mass bins more massive than / =M Mlog 7.810 ,ICG in Figure 2 show similar number 0.00 0.01 0.02 0.03 0.04 0.05 0.06 z 0 10 20 30 5.7-6.4 DESI BGS DESI BGS r<17.7 SDSS M corr 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 z 0 25 50 75 100 125 150 6.4-7.1 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 z 0 200 400 600 N 7.1-7.8 0.00 0.02 0.04 0.06 0.08 0.10 0 500 1000 1500 2000 2500 7.8-8.5 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 1000 2000 3000 4000 5000 8.5-9.2 0.0 0.1 0.2 0.3 0.4 0 2000 4000 6000 8000 10000 N 9.2-9.9 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 2000 4000 6000 9.9-10.2 0.0 0.1 0.2 0.3 0.4 0 2000 4000 6000 8000 10.2-10.5 0.0 0.1 0.2 0.3 0.4 0 2000 4000 6000 8000 10000 N 10.5-10.8 0.0 0.1 0.2 0.3 0.4 0.5 0 5000 10000 15000 10.8-11.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 2000 4000 6000 8000 11.1-11.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 N 11.4-11.7 Figure 2. Redshift distribution for ICGs selected from DESI Year-1 data (green plain histogram) with rdustcorr < 19.5 (extinction corrected according to DESI criteria) and SDSS spectroscopic Main galaxies (red hatched histogram) with r < 17.7 (no extinction correction according to SDSS NYU-VAGC documents), in 12 stellar mass bins, as indicated by the text in each panel ( /M Mlog10 ,ICG ). Here, SDSS ICGs have been corrected for the median bias of stellar mass from DESI (see Figure 1). Black histograms are based on a subset of DESI BGS but with Xux cut of r < 17.7 (no extinction correction), which is the Xux limit for SDSS Main galaxies. Here, the green and black plain histograms are all plotted with 1/pobs as weights, to account for Fber incompleteness in DESI (see Section 3.2). Red hatched histograms are based on the Fber collision corrected catalog from NYU-VAGC. 60 The total footprints of the SDSS Spectroscopic Main galaxy from NYU- VAGC and of DESI Year-1 BGS are 7818 deg2 (M. R. Blanton et al. 2005b) and 7473 deg2 (DESI Collaboration et al. 2024f). Assuming the same number density, we expect the SDSS and DESI ICG histograms to be close in amplitudes. 8 The Astrophysical Journal, 986:218 (18pp), 2025 June 20 Wang et al. distributions of DESI bright galaxies and SDSS Main galaxies at z < 0.01. This is because for galaxies more massive than / =M Mlog 7.810 ,ICG , they are already well observed at z < 0.01 with the shallower Xux limit of r < 17.7. Only in the two least-massive bins of /< M Mlog 11.110 ,ICG , however, the magenta squares are higher than black dots at fainter magnitudes. The difference is not large, but is signiFcant compared with the small error bars there. The same discrepancy also exists at the faint end in the third, fourth, and Ffth most-massive panels covering /< M Mlog 7.110 ,ICG , reaching Mr,sat ∼ −7 at the faint end of satellite LF. In Figures 2 and 3 above, we have also discussed that only in the two least-massive stellar mass bins ( /< M Mlog 11.110 ,ICG , there are some disagreements between SDSS and DESI at fainter magnitudes of satellites, and the discrepancy is likely due to the difference in SDSS and DESI stellar mass measurements. With ICGs selected from DESI BGS, we can measure their satellite LFs around ICGs spanning a wide range in stellar mass of /<