The Dark Energy Spectroscopic Instrument DESI Peculiar Velocity Survey aims to measure the peculiar velocities of early and late type galaxies within the DESI footprint using both the Fundamental Plane and TullyFisher relations. Direct measurements of peculiar velocities can significantly improve constraints on the growth rate of structure reducing uncertainty by a factor of approximately 2.5 at redshift 0.1 compared to the DESI Bright Galaxy Surveys redshift space distortion measurements alone. We assess the quality of stellar velocity dispersion measurements from DESI spectroscopic data. These measurements along with photometric data from the Legacy Survey establish the Fundamental Plane relation and determine distances and peculiar velocities of earlytype galaxies. During Survey Validation we obtain spectra for 6698 unique earlytype galaxies up to a photometric redshift of 0.15. 64 of observed galaxies 4267 have relative velocity dispersion errors below 10. This percentage increases to 75 if we restrict our sample to galaxies with spectroscopic redshifts below 0.1. We use the measured central velocity dispersion along with photometry from the DESI Legacy Imaging Surveys to fit the Fundamental Plane parameters using a 3D Gaussian maximum likelihood algorithm that accounts for measurement uncertainties and selection cuts. In addition we conduct zeropoint calibration using the absolute distance measurements to the Coma cluster leading to a value of the Hubble constant H0  76.05 pm 0.35statistical pm 0.49systematic FP pm 4.86statistical due to calibration mathrmkm  s1 Mpc1. This H0 value is within 2sigma of Planck Cosmic Microwave Background results and within 1sigma of other low redshift distance indicatorbased measurements.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies']
DESI Peculiar Velocity Survey  Fundamental Plane
{'DESI Peculiar Velocity Survey - Fundamental Plane': "Khaled Said , 1 ★ Cullan Howlett , 1 Tamara Davis , 1 John Lucey , 2 Christoph Saulder , 3 Kelly Douglass , 4 Alex G. Kim , 5 Anthony Kremin , 5 Caitlin Ross, 1 Greg Aldering, 5 Jessica Nicole Aguilar, 5 Steven Ahlen , 6 Segev BenZvi , 4 Davide Bianchi , 7 David Brooks, 8 Todd Claybaugh, 5 Kyle Dawson, 9 Axel de la Macorra , 10 Biprateep Dey , 11 Peter Doel, 8 Kevin Fanning , 12 , 13 Simone Ferraro , 5 , 14 Andreu Font-Ribera , 15 , 8 Jaime E. Forero-Romero , 16 , 17 Enrique Gaztañaga, 19 , 20 , 18 Satya Gontcho A Gontcho , 5 Julien Guy , 5 Klaus Honscheid, 23 , 21 , 22 Robert Kehoe, 24 Theodore Kisner , 5 Andrew Lambert, 5 Martin Landriau , 5 Laurent Le Guillou , 25 Marc Manera , 26 , 15 Aaron Meisner , 27 Ramon Miquel, 28 , 15 John Moustakas , 29 Andrea Muñoz-Gutiérrez, 10 Adam Myers, 30 Jundan Nie , 31 Nathalie Palanque-Delabrouille , 5 , 32 Will Percival , 34 , 33 , 35 Francisco Prada , 36 Graziano Rossi, 37 Eusebio Sanchez , 38 David Schlegel, 5 Michael Schubnell, 39 , 40 Joseph Harry Silber , 5 David Sprayberry, 27 Gregory Tarlé , 40 Mariana Vargas Magana , 10 Benjamin Alan Weaver, 27 Risa Wechsler , 41 , 12 , 13 Zhimin Zhou , 31 Hu Zou 31  \n- 1 School of Mathematics and Physics, University of Queensland, 4072, Australia\n- 2 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\n- 3 Max Planck Institute for Extraterrestrial Physics, Gießenbachstraße 1, 85748 Garching, Germany\n- 4 Department of Physics & Astronomy, University of Rochester, 206 Bausch and Lomb Hall, P.O. Box 270171, Rochester, NY 14627-0171, USA\n- 5 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA\n- 6 Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA\n- 7 Dipartimento di Fisica 'Aldo Pontremoli', Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy\n- 8 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK\n- 9 Department of Physics and Astronomy, The University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA\n- 10 Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México\n- 11 Department of Physics & Astronomy and Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh,\n- 3941 O'Hara Street, Pittsburgh, PA 15260, USA\n- 12 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Menlo Park, CA 94305, USA\n- 13 SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA\n- 14 University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, USA\n- 15 Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain\n- 16 Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio Ip, CP 111711, Bogotá, Colombia\n- 17 Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio H, CP 111711 Bogotá, Colombia\n- 18 Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain\n- 19 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK\n- 20 Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain\n- 21 Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA  \n22  \nDepartment of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA  \n- 23 The Ohio State University, Columbus, 43210 OH, USA\n- 24 Department of Physics, Southern Methodist University, 3215 Daniel Avenue, Dallas, TX 75275, USA\n- 25 Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), FR-75005 Paris, France\n- 26 Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain\n- 27 NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA\n- 28 Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain\n- 29 Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA\n- 30 Department of Physics & Astronomy, University of Wyoming, 1000 E. University, Dept. 3905, Laramie, WY 82071, USA\n- 31 National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Rd., Chaoyang District, Beĳing, 100012, P.R. China\n- 32 IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France\n- 33 Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada\n- 34 Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada\n- 35 Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada\n- 36 Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain\n- 37 Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea\n- 38 CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain\n- 39 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA\n- 40 University of Michigan, Ann Arbor, MI 48109, USA  \nMNRAS 000  \n, 1-18 (2024)", 'ABSTRACT': "The Dark Energy Spectroscopic Instrument (DESI) Peculiar Velocity Survey aims to measure the peculiar velocities of early and late type galaxies within the DESI footprint using both the Fundamental Plane and Tully-Fisher relations. Direct measurements of peculiar velocities can significantly improve constraints on the growth rate of structure, reducing uncertainty by a factor of approximately 2.5 at redshift 0.1 compared to the DESI Bright Galaxy Survey's redshift space distortion measurements alone. We assess the quality of stellar velocity dispersion measurements from DESI spectroscopic data. These measurements, along with photometric data from the Legacy Survey, establish the Fundamental Plane relation and determine distances and peculiar velocities of early-type galaxies. During Survey Validation, we obtain spectra for 6698 unique early-type galaxies, up to a photometric redshift of 0.15. 64% of observed galaxies (4267) have relative velocity dispersion errors below 10%. This percentage increases to 75% if we restrict our sample to galaxies with spectroscopic redshifts below 0.1. We use the measured central velocity dispersion, along with photometry from the DESI Legacy Imaging Surveys, to fit the Fundamental Plane parameters using a 3D Gaussian maximum likelihood algorithm that accounts for measurement uncertainties and selection cuts. In addition, we conduct zero-point calibration using the absolute distance measurements to the Coma cluster, leading to a value of the Hubble constant, 𝐻 0 = 76 . 05 ± 0 . 35(statistical) ± 0 . 49(systematic FP) ± 4 . 86(statistical due to calibration) km s -1 Mpc -1 . This 𝐻 0 value is within 2 𝜎 of Planck Cosmic Microwave Background results and within 1 𝜎 , of other low redshift distance indicator-based measurements.  \nKey words: galaxies: distances and redshifts - cosmology: observations - cosmology: cosmological parameters - cosmology: large-scale structure of Universe", '1 INTRODUCTION': "The Lambda Cold Dark Matter ( Λ CDM)modelstands as the prevailing cosmological framework to describe the Universe. It combines Einstein's cosmological constant, Λ , representing dark energy (Carroll 2001; Peebles & Ratra 2003), with nonbaryonic cold dark matter (CDM; Hut 1977; Lee & Weinberg 1977; Sato & Kobayashi 1977; Dicus et al. 1977; Vysotski ˇ i et al. 1977). Despite its consistency with a wide range of cosmological observations, the nature of its two main components, dark matter and dark energy, remains unknown to us (Peebles 2021). Moreover, tensions persist between measurements of the present-day expansion rate of the universe from the earlyuniverse (e.g., Planck Collaboration et al. 2020) and those derived from the late-universe (e.g., Riess et al. 2022). Additional tensions arise from measurements of the strength of clustering of matter in the universe between early- and late-universe observations (Hildebrandt et al. 2020). Furthermore, discrepancies emerge from measurements of the bulk flow, directly derived from late-universe observations, which do not align with the expected values from the Λ CDMmodel (Courtois et al. 2023b; Whitford et al. 2023; Watkins et al. 2023). In addition, current cosmological observations seem to favor a slightly higher growth index, 𝛾 than the value predicted by general relativity and the Λ CDM model, which might indicate weaker gravity or suppression of the growth of structure (Nguyen et al. 2023).  \nAddressing these tensions requires substantial efforts in acquiring additional data to precisely understand the underlying discrepancies. Nevertheless, examining the tension using independent techniques should be our utmost priority, as it involves overcoming potential sources of systematics that may be linked to new physics (Lahav & Silk 2021).  \nThe Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et al. 2022) is a ground-based spectroscopic survey that is targeting 40 million galaxies and quasars over a 14,000 square degrees footprint. It is over two years into its 5-year observing campaign. DESI's aims align very closely with the above suggestion  \nof unveiling the nature of dark energy by conducting the most precise measurement of the universe expansion history using the baryon acoustic oscillation (BAO) (Levi et al. 2013). DESI will also measure the growth rate of cosmic structure using redshift space distortions (RSD) (DESI Collaboration et al. 2016a).  \nThe DESI peculiar velocity survey serves as a secondary target program, designed to complement and enhance the primary goal of the DESI survey (Saulder et al. 2023). Peculiar velocities of galaxies can be measured through two primary methods: 'directly' using distance indicators, and 'indirectly' via velocity field reconstruction based on local density measurements. In this paper, we employ the direct method, specifically aiming to enrich the dataset by incorporating ∼ 180 , 000 directly measured distances using redshiftindependent distance indicators. These directly measured distances provide valuable independent information about galaxy velocities, helping to disentangle the effects of the smooth Hubble flow due to the cosmic expansion and the peculiar velocity due to gravitational interactions on galaxy motion. The peculiar velocity survey offers an independent and complementary approach to the main DESI surveys at higher redshift, allowing for a better understanding of large-scale structure through measurement of 𝑓 𝜎 8 below 𝑧 = 0 . 1 and the overall expansion of the universe through measurements of 𝐻 0 .  \nHistorically, large peculiar velocity surveys, from the Two Micron All Sky Survey (2MTF: Masters et al. 2008) and the 6dF Galaxy Survey (6dFGS: Springob et al. 2014) have relied on either Tully-Fisher relation (Tully & Fisher 1977) or the Fundamental Plane relation (Djorgovski & Davis 1987; Dressler et al. 1987) to obtain distance measurements. The DESI peculiar velocity survey will be able to provide observations for these two distance indicators in an unprecedented and innovative way, thanks to the DESI's substantial increase in the survey speed due to the large field of view and the densely populated focal plane with 5000 robotic fibre positioners (DESI Collaboration et al. 2016b).  \nThe Fundamental Plane relation utilizes two distance-independent observables of elliptical galaxies, the mean surface brightness ( 𝐼 𝑒 ) and the central velocity dispersion ( 𝜎 0 ) to infer the physical effective radius ( 𝑅 𝑒 ), which is distance dependent. Comparing the physical ra-  \nular effective radius ( 𝜃 𝑒 ) allows for the measurement of distances. Similarly, the Tully-Fisher relation uses the rotational velocity of galaxies as a distance-independent observable to predict the absolute magnitude. Then, by comparing the predicted absolute magnitude with the observed apparent magnitude, one can derive distance estimates for these galaxies. These directly measured distances and peculiar velocities have been instrumental in constraining crucial cosmological parameters.  \nThe 6dF Galaxy Survey (Jones et al. 2009) primary objective was to measure peculiar velocities for a sample of approximately 9,000 early-type galaxies up to redshift 𝑧 < 0 . 055 using the Fundamental Plane relation, aiming to constrain the growth rate of cosmic structure (6dFGSv: Magoulas et al. 2012). Qin et al. (2019) utilized the 6dFGSv data to estimate the Density-Momentum power spectrum and derived a value of 𝑓 𝜎 8 = 0 . 404 ± 0 . 082. Similarly, Adams & Blake (2020) employed the 6dFGSv data and applied the cross-covariance between galaxy redshift-space distortions and peculiar velocities to constrain the growth rate of structure, obtaining 𝑓 𝜎 8 = 0 . 384 ± 0 . 052(statistical) ± 0 . 061(systematic). Said et al. (2020) compared a sub-sample of the 6dFGSv Fundamental Plane galaxies along with a sample of SDSS galaxies to the velocity field reconstruction and reported a value of 𝑓 𝜎 8 = 0 . 338 ± 0 . 027. More recently, Turner et al. (2023) used the full 6dFGSv sample to measure galaxy-galaxy, galaxy-velocity, and velocity-velocity auto- and cross-correlation functions, finding a value of 𝑓 𝜎 8 = 0 . 358 ± 0 . 075. While all these measurements, using different methods, are consistent, they all favour a lower value than the predicted value from the Planck Λ CDMmodel (Planck Collaboration et al. 2020).  \nMore recently, Cosmicflows-4 has undertaken the ambitious task of combining almost all previous peculiar velocity surveys into a comprehensive dataset (Tully et al. 2023). This compilation incorporates distances and peculiar velocities for 55,877 galaxies, utilizing eight different methodologies. The largest number of galaxies were derived from two new datasets, with approximately 35,000 galaxies measured using the Fundamental Plane relation (Howlett et al. 2022) and about 10,000 galaxies using the Tully-Fisher relation (Kourkchi et al. 2020). The integration of these diverse datasets has resulted in a Hubble constant measurement of 𝐻 0 = 74 . 6 ± 0 . 8 (statistical) ± 3 . 0 (systematic) (Tully et al. 2023). The Cosmicflows-4 catalogue has proven invaluable for various cosmological measurements, including assessing the impact of our local environment on 𝐻 0 (Giani et al. 2024), measuring the growth rate of structure (Boubel et al. 2024b), and developing an improved method for determining the Hubble constant using the Tully-Fisher relation (Boubel et al. 2024a). While the statistical errors from these studies are relatively small, thanks to the considerable increase in the sample size, there remain concerns about potential large systematic errors due to the combination of different methods and calibrators.  \nThis reinforces the need for larger and more homogeneous surveys, like the DESI peculiar velocity survey, to further advance our understanding of cosmological parameters and address potential sources of systematic effects.  \nThe DESI peculiar velocity survey is expected to yield a remarkable number of directly measured distances and peculiar velocities. Specifically, it is projected to obtain 186,000 such measurements, with 133,000 of them using the Fundamental Plane relation and 53,000 using the Tully-Fisher relation (Saulder et al. 2023). These numbers represent a significant increase in the scale, being about four times all previous peculiar velocity surveys combined (Tully et al. 2023). These vast dataset will be significant in constraining cosmological parameters, such as the Hubble Constant ( 𝐻 0 ) and the growth rate of cosmic structure ( 𝑓 𝜎 8 ).  \nWith this extensive dataset, DESI will achieve a new level of precision in constraining cosmological parameters. The large number of directly measured distances and peculiar velocities will enable more robust and statistically significant results compared to all previous surveys.  \nThis work, accompanied by a parallel work conducted by Douglass et al. in prep, introduces the initial outcomes of the DESI peculiar velocity survey using Survey Validation data. In Douglass et al. in prep, the emphasis is on the Tully-Fisher relation, while this work focuses only on the Fundamental Plane relation.  \nDuring the Survey Validation (SV) phase, our approach involved conducting observations for a randomized subset of our designated targets. This selection addressed key questions: First, we evaluated the achievability of the Signal-to-Noise Ratio (SNR) needed for accurate velocity dispersion measurements across our magnitude and redshift range. Additionally, we examined potential sources of systematic errors in velocity dispersion measurements that might impact the precision of our distance and peculiar velocity determinations. Furthermore, we evaluated the efficiency of our photometric selection in accurately pinpointing genuine elliptical galaxies.  \nWhile the primary objective of this paper is to offer insights into these fundamental questions, we also conducted the fitting of the Fundamental Plane relation. By doing so, we produced the initial catalogue of distances and peculiar velocities from DESI. Moreover, we present our preliminary findings concerning the measurement of the Hubble constant.  \nThis paper is organized as follows: Section 2 provides a description of the Fundamental Plane relation. In Section 3, we offer an introduction to the dataset employed. Our sample selection process, encompassing photometric and spectroscopic aspects, is explained in Section 4. The derivation of Fundamental Plane parameters, inclusive of internal and external consistency checks, is presented in Section 5. We show the process of fitting the Fundamental Plane relation in Section 6. In Section 7, we discuss the zero-point calibration process and present absolute distance measurements. Section 8, unfolds our measurement of the Hubble constant. Summary and Conclusions are in Sections 9 and 10, respectively.  \nUnless otherwise stated, in this paper we assume a flat Λ CDM cosmological model with Ω 𝑚 = 0 . 31 and 𝐻 0 = 100 ℎ kms -1 Mpc -1 . All magnitudes are on the AB magnitude system. All uses of 'log' should be taken to mean logarithms taken to the base 10.", '2 FUNDAMENTAL PLANE': "Distance indicators operate on the premise of linking distanceindependent parameters such as kinematics, with distance-dependent characteristics like luminosity or size. This relationship allows for the estimation of galaxy distances based on known distance-independent parameters. The first distance-indicator relation for elliptical galaxies was the luminosity-stellar velocity dispersion correlation, known as the Faber-Jackson relation (FJ: Faber & Jackson 1976). It is worth noting that the foundation of this relation was first suggested by Minkowski (1962), although they regarded it as inadequate probably due to the inclusion of flattened galaxies with high rotational velocities. The FJ relation follows the form 𝐿 ∝ 𝜎 4 . Simultaneously, Kormendy (1977) identified a correlation between surface brightness and size of elliptical galaxies. Both FJ and Kormendy relations did not seem very promising as distance indicators due to their large scatter.  \nA significant advancement emerged approximately a decade later when it became evident that the FJ and Kormendy relations were  \nspecial instances of a more general relation known as the Fundamental Plane (Djorgovski & Davis 1987; Dressler et al. 1987). These works show that a galaxy's effective radius ( 𝑅 𝑒 ), surface brightness ( 𝐼 𝑒 ), and stellar velocity dispersion are related through a power-law relationship, expressed as 𝑅 𝑒 ∝ 𝜎 𝑎 𝐼 𝑏 𝑒 .  \nThe obvious explanation of the Fundamental Plane rested upon the virial equilibrium, linking a galaxy's mass ( 𝑀 ) to its velocity dispersion ( 𝜎 ) and effective radius ( 𝑅 𝑒 ) through the equation 𝑀 ∝ 𝜎 2 𝑅 𝑒 (Faber et al. 1987). However, the Fundamental Plane coefficients showed significant deviations from the predictions of the virial theorem (Hudson et al. 1997; Colless et al. 2001; Bernardi et al. 2003b; Magoulas et al. 2012; Said et al. 2020; D'Eugenio et al. 2021; Howlett et al. 2022). Varied contributors to this divergence were identified, including the fluctuation of the mass-to-light ratio (Faber et al. 1987), variations in the surface brightness profiles of early-type galaxies (Ciotti et al. 1996), and the proportion of dark matter within the kinematic observation region (Moster et al. 2010). Despite the deviations, the Fundamental Plane's applicability persisted across all early-type galaxies, maintaining a scatter of approximately 0.1 dex, which translates to an accuracy in distance estimation of around 23% (Lynden-Bell et al. 1988).  \nWhile the current work concentrates solely on distance and peculiar velocity measurements, the forthcoming data from DESI holds the promise of delving into sources of the Fundamental Plane's tilt (its deviation from the virial theorem). This potential for detailed analysis arises from the vast and comprehensive datasets that DESI will provide.", '3 DATA': "Constructing the Fundamental Plane relation involves utilizing two datasets: photometry and spectroscopy. The photometric data for this work are drawn from the DESI Legacy Imaging Surveys (Dey et al. 2019). The spectroscopic data for our analysis are from the DESI Survey Validation data (DESI Collaboration et al. 2023a).  \nThe DESI Legacy Imaging Surveys encompass the combination of three individual surveys: the Dark Energy Camera Legacy Survey (DECaLS), the Beĳing-Arizona Sky Survey (BASS), and the Mayall z-band Legacy Survey (MzLS). This combination is designed to capture imagery across the expansive 14,000 deg 2 footprint ( 𝛿 > -20 · and | 𝑏 | > 15 · ) of the DESI survey in three optical bands ( 𝑔, 𝑟, and 𝑧 ).  \nDECaLS,whichbegan observations in August 2014, uses the Dark Energy Camera (DECam; Flaugher et al. 2015) at the 4m Blanco telescope at the Cerro Tololo Inter-American Observatory. It covers approximately 9,350 deg 2 , including 3,580 deg 2 in the SGC and 5,770 deg 2 in the NGC, complementing the Dark Energy Survey's (The Dark Energy Survey Collaboration 2005) coverage of 1,130 deg 2 within the DESI footprint.  \nBASS (Zou et al. 2017), which started in spring 2015, images the 𝛿 > + 32 · region of the DESI NGC footprint (approximately 5,100 deg 2 ) in the 𝑔 and 𝑟 optical bands. It utilizes the 90Prime camera (Williams et al. 2004) at the prime focus of the University of Arizona's Bok 2.3m telescope on Kitt Peak.  \nMzLS complements BASS by imaging the same 𝛿 > + 32 · region of the NGC footprint in the z-band, covering approximately 5,100 deg 2 .  \nIn addition to optical bands, the Legacy Surveys incorporate midinfrared photometry from the Wide-field Infrared Survey Explorer (WISE) satellite. WISE conducted an all-sky survey in four bands  \ncentered at 3.4, 4.6, 12, and 22 𝜇 m(known as W1, W2, W3, and W4; Wright et al. 2010) during its mission.  \nThese surveys collectively contribute to the creation of a photometric catalogue covering the three optical bands ( 𝑔, 𝑟, and 𝑧 ) and incorporating the four WISE channels. For the purposes of this paper, our analysis is based on Data Release 9 (DR9) of the Legacy Surveys.  \nThe DESI Early Data Release (EDR) marks the initial public release of DESI spectroscopic data (DESI Collaboration et al. 2023b). It encompasses data from the Survey Validation (SV) phase, conducted between December 2020 and May 2021, prior to the start of the DESI Main Survey. A detailed description of the DESI pipeline can be found in Guy et al. (2023). For discussion of the survey operations, see Schlafly et al. (2023). The SV phase consisted of three stages:  \n- (i) Target Selection Validation (SV1): This phase refined and validated the selection of targets for the Milky Way Survey (MWS), Bright Galaxy Survey (BGS), Luminous Red Galaxies (LRG), Emission Line Galaxies (ELG), and Quasar (QSO) samples. It employed looser target selection cuts and higher signal-to-noise ratios than the Main Survey to optimize selection criteria and survey requirements.\n- (ii) Operations Development (SV2): A brief phase for operational refinements.  \n(iii) 1% Survey (SV3): This final SV stage further optimized observing procedures and produced high completeness samples over approximately 1% of the final DESI Main Survey area.  \nThe EDR demonstrates DESI's capabilities, and its advanced instrumentation at the 4m Mayall telescope at Kitt Peak National Observatory. The instrument's wide-field prime focus corrector enables a field of view just over 8 deg 2 , allowing for efficient large-scale observations. At the heart of DESI's design are 5,020 roboticallycontrolled fiber positioners (DESI Collaboration et al. 2016b; Silber et al. 2023; Miller et al. 2023), each directing light from individual targets. These fibers feed into ten spectrographs, each equipped with three cameras covering distinct wavelength ranges: B (3600-5800 Å), R (5760-7620 Å), and Z (7520-9824 Å). The spectrographs provide a resolving power that increases from approximately 2000 at 3600 Å to 5500 at 9800 Å (DESI Collaboration et al. 2023b).  \nIn addition to the primary target classes (MWS, BGS, LRG, ELG, and QSO), the EDR also includes observations from DESI's secondary programs, which serve as filler targets.", '4 SAMPLE SELECTION': 'As part of the DESI secondary target programs, we provided an initial set of targets to complement the main survey targets. As a result, our sample selection is made in two steps. The initial phase, which we refer to as the photometric selection, was exclusively reliant on the DESI Legacy Imaging Surveys DR9 (Dey et al. 2019), supplemented by photometric redshift catalogues from Zhou et al. (2021). Subsequently, the second step or the spectroscopic selection involves the integration of spectroscopic data from the DESI survey (DESI Collaboration et al. 2023b). Notably, most of our Fundamental Plane targets are also part of the DESI Bright Galaxy Survey (BGS: Ruiz-Macias et al. 2020; Hahn et al. 2023).  \nAcomprehensive description of the photometric sample selection process can be found in Saulder et al. (2023). For full details of our photometric selection, we refer the reader to that paper. Here, we only provide a brief summary of the photometric selection criteria. Additionally, we will describe how we conducted the spectroscopic  \nsample selection, which facilitated the construction of the peculiar velocity sample for our Fundamental Plane analysis.', '4.1 Photometric Selection': 'Our adopted selection criteria are outlined as follows: (1) An 𝑟 -band magnitude cut of 𝑟 < 18, which has been corrected for external galactic extinction, ensuring a robust signal-to-noise ratio that subsequently enhances the success rate for velocity dispersion measurements; (2) Three distinct colour cuts 1 of  \n𝑔 -𝑟 > 0 . 68 (1)  \n𝑔 -𝑟 > 1 . 3 ( 𝑟 -𝑧 ) -0 . 05 (2)  \n𝑔 -𝑟 < 2 ( 𝑟 -𝑧 ) -0 . 15 . (3)  \nThese colour cuts serve to eliminate galaxies located beneath the red sequence, dusty galaxies, and peculiar objects like galaxy mergers or galaxies displaying image artefacts, respectively; (3) a circularized radius, which is the half-light radius of the galaxy circularized using the axial ratio, condition of 𝑅 circ > 0; (4) An axial ratio requirement of 𝑏 / 𝑎 ≥ 0 . 3; (5) The application of either a de Vaucouleurs or Sérsic fit to the surface brightness profile, incorporating a Sérsic index of 𝑛 𝑠 > 2 . 5, measured collectively across 𝑟 , 𝑔 , and 𝑧 bands.  \nThese specific criteria for ETGs selection are established based on a combination of past experience in ETG identification from previous surveys like 6dFGSv and SDSS peculiar velocity surveys (Saulder et al. 2013; Said et al. 2020; Howlett et al. 2022), along with testing involving visual identifications sourced from the Siena Galaxy Atlas (SGA; Moustakas et al. 2023) and GalaxyZoo (Lintott et al. 2011)  \nOur photometric selection process, yielding a pool of over 400,000 galaxies deemed suitable for Fundamental Plane analysis. Since our selected sample is from the DESI Legacy Imaging Surveys DR9, it encompasses galaxies extending beyond the bounds of the DESI spectroscopic survey footprint. Refining our sample to adhere to the DESI spectroscopic survey footprint, spanning 14,000 deg 2 above a declination of -18 degree, yielded a final count of 373,533 galaxies eligible for Fundamental Plane analysis.', '4.2 Spectroscopic Selection': 'Our spectroscopic selection process depends on the data obtained from the DESI Survey Validation (SV; DESI Collaboration et al. 2023a). This validation survey unfolded in three phases: firstly, SV1, focused on target selection validation and the refinement of target selection algorithms (Myers et al. 2023); secondly, SV2, referred to as the Operation Development phase, serving as a practice run for the third stage; finally, SV3, also known as the 1% Survey, aimed to further validate both the survey operations procedures and the final target selection. The culmination of all three stages is encapsulated within the DESI Early Data Release (EDR; DESI Collaboration et al. 2023b), which has now been released and is publicly accessible. 2  \nFrom our photometrically selected Fundamental Plane sample, we identified the galaxies that were spectroscopically observed in the SV data. We then applied two key constraints to this subset: firstly, the warning bitmask ( ZWARN ) must be zero, indicating the absence of known issues with the data or the fit; secondly, the spectral classification ( SPECTYPE ) should be "GALAXY" (DESI Collaboration et al.  \n2023b). The outcome of this cross-matching process yielded a total count of 6698 distinct Fundamental Plane galaxies. Figure 1 shows the distribution of these galaxies in a Mollweide projection in equatorial coordinates. The open circles in the figure represent several clusters within the DESI footprint, which are key to the Fundamental Plane analysis, especially for setting the zero-point calibration.', '4.3 Visual inspection': 'In order to identify galaxies that were unsuitable for inclusion in our Fundamental Plane analysis, a visual inspection of all galaxies was conducted using 1 × 1 arcmin colour cutouts sourced from the PanSTARRS1 (Chambers et al. 2016) and the DESI Legacy Imaging Surveys (Dey et al. 2019) images. This visual inspection was carried out by John R. Lucey. Using deeper Legacy Survey images, particularly model residual images, significantly improved our discrimination capability compared to previous surveys. Our methodology followed established procedures from prior FP studies (Campbell et al. 2014; Said et al. 2020; Howlett et al. 2022).  \nDuring the visual inspection, the following categories of objects were identified:  \n- (i) Galaxies that are not bulge-dominated, including those with prominent spiral arms.\n- (ii) Galaxies for which the measurements of the Fundamental Plane photometric parameters, specifically total magnitude and effective radius, were likely to be unreliable due to the presence of overlapping sources, whether stars or other galaxies.  \n(iii) Galaxies with pronounced central asymmetries, including those with strong dust features, which are likely to bias the velocity dispersion measurements.  \nThe results of our visual inspection process, are summarized in table 1. While our primary objective for the visual inspection process was to select a clean sample of bulge-dominated systems for our Fundamental Plane fit (the first step of the two-step process), it is important to note that these flagged galaxies were reintroduced into our dataset for the purpose of measuring peculiar velocities and distances.', '5 FUNDAMENTAL PLANE PARAMETERS': "With both the essential photometric and spectroscopic data in hand, we possess the necessary components for deriving the Fundamental Plane parameters. The formulation of the FP relation employed in this study is characterized by its structure:  \nlog 𝑅 𝑒 = 𝑎 log 𝜎 0 + 𝑏 log 𝐼 𝑒 + 𝑐. (4)  \nIn this equation, 𝑅 𝑒 stands for the effective radius, measured in (kpc h -1 ), which serves as the parameter that allows us to measure distance. On the other side of the equation, 𝜎 0 signifies the central velocity dispersion, expressed in (km s -1 ), while 𝐼 𝑒 represents the meansurfacebrightness within the angular effective radius, presented in (L ⊙ pc -2 ). Notably, 𝜎 0 and 𝐼 𝑒 are both distance-independent parameters 3 . The coefficients of the FP relation are represented by 𝑎 , 𝑏 , and 𝑐 .  \nThe computation of the 𝑟 -band angular effective radius, 𝜃 𝑒 was  \nFigure 1. The distribution of Fundamental Plane galaxies within the DESI SV dataset, presented here in a Mollweide projection in equatorial coordinates. Regions obscured by Galactic extinction in the Milky Way are shown as shaded Reds. The orange dots indicate the Fundamental Plane data within the SV dataset. The blue circles encompass all the DESI tiles, which collectively define the DESI survey footprint. Notably, the open black circles pinpoint a few clusters within the DESI footprint, which have previously measured distances (Bell et al. 2023), making them invaluable for the zero-point calibration in the future. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->  \nderived from the 𝑟 -band half-light radius, 𝑟 , as well as the ellipticity components: 𝜖 1 and 𝜖 2 . This relationship is expressed through the following equations:  \n𝜃 𝑒 = 𝑟 √︁ 𝑏 / 𝑎 (5)  \n𝑏 / 𝑎 = 1 . 0 - | 𝜖 | 1 . 0 + | 𝜖 | (6)  \n| 𝜖 | = √︃ 𝜖 2 1 + 𝜖 2 2 . (7)  \nIn the above equations, 𝑟 , 𝜖 1 , and 𝜖 2 are all directly extracted from the DESI Legacy Imaging Surveys DR9 4 (Dey et al. 2019).  \nWe converted the angular effective radius in arcseconds to the physical effective radius in units of kpc h -1 using the angular diameter distance (Weinberg 1972). This conversion was performed with respect to the observed redshift in the CMB frame, following the standard Λ CDMcosmological model:  \nlog 𝑅 𝑒 = log ( 𝜃 𝑒 ) + log ( 𝑑 ( 𝑧 cmb )) -log ( 1 + 𝑧 helio ) + log GLYPH<18> 1000 𝜋 180 × 3600 GLYPH<19> . (8)  \nWhile the comoving distance calculation employed the redshift in the CMB frame, the heliocentric redshift was used for the conversion from comoving to angular diameter distance as recommended by Davis et al. (2019).  \nThe calculation of the second parameter in the Fundamental Plane relation, the effective surface brightness 𝐼 𝑒 in L ⊙ pc -2 , was based  \non the model flux in the 𝑟 -band, 𝑓 𝑟 , and Galactic transmission in the same band MW 𝑟 . This, along with the above calculated angular effective radius 𝜃 𝑒 , was employed in the following manner:  \nlog 𝐼 𝑒 = 0 . 4 ( 𝑀 𝑟 ⊙ -𝑚 𝑟 -0 . 85 𝑧 cmb + 𝑘 𝑟 ) -log ( 2 𝜋𝜃 2 𝑒 ) + 4 log ( 1 + 𝑧 helio ) + 2 log ( 206265 / 10 ) , (9)  \nwhere,  \n𝑚 𝑟 = 22 . 5 -2 . 5 log 𝑓 MW 𝑟 . (10)  \nHere, 𝑀 𝑟 ⊙ = 4 . 65 signifies the 𝑟 -band absolute magnitude of the Sun (Willmer 2018), while 𝑚 𝑟 is the extinction-corrected magnitude. The term 0 . 85 𝑧 accounts for evolution correction (Bernardi et al. 2003a), 4 log ( 1 + 𝑧 helio ) represents surface brightness dimming correction, 𝑘 𝑟 approximates the 𝐾 -correction in the 𝑟 -band given by Chilingarian et al. (2010), and 𝑓 and MW 𝑟 are directly extracted from the Legacy Imaging Surveys DR9.  \nCompleting the set of parameters for the Fundamental Plane, the third parameter is the distance-independent central velocity dispersion, 𝜎 0 . In our analysis, we employed the Penalized Pixel-Fitting (pPXF) software to measure velocity dispersion and its associated uncertainty from DESI spectra. As the name suggests, pPXF utilizes the maximum penalized likelihood method for deriving stellar kinematics from absorption-line spectra of galaxies. The development of this method was initiated by Cappellari & Emsellem (2004) and has been refined in subsequent works by Cappellari (2017, 2022). Our stellar templates were drawn from the Indo-U.S. Coudé Feed Spectral Library (Valdes et al. 2004), which encompasses 1273 stars, though typically only about 10-20 are selected by pPXF for precise fits. This spectral library spans the range from 3460 to 9464Å at a resolution of 1.35Å, corresponding to 𝜎 = 30 km s -1 .  \nThe DESI spectrograph's resolving power ( 𝜆 / Δ 𝜆 ) varies as a function of wavelength: [2000, 3500] in the blue, [3300, 5000] in the red, and [3500, 5200] in the z band (see Fig. 33 in DESI Collaboration et al. 2022). This translates to resolutions for measureing velocity dispersion of 46, 31, and 29 km s -1 in the blue, red, and z arms respectively. In this paper, we use only the blue arm data. The stellar template library we are using here has a higher resolution than the data, which is necessary for accurate fitting. Additionally, the instrumental resolution in pPXF was handled by using the full resolution DESI data matrix, then output a 1D array with the RMS per pixel.  \nDESI spectra are categorized into two main groups: full-depth and per-tile spectra (DESI Collaboration et al. 2023b). To derive the final central velocity dispersion, we primarily utilized the full-depth spectra. These spectra merge exposures for targets positioned on a given sky pixel and also aggregate data across tiles when the same target is observed on multiple tiles. On the other hand, per-tile spectra do not combine data from various tiles, even if the same target was observed on multiple tiles. Per-tile spectra find utility in subsequent subsections, where they are used for internal consistency checks.  \nSubsequently, the measured velocity dispersion was transformed into central velocity dispersion using the formula introduced by Jorgensen et al. (1995):  \n𝜎 0 𝜎 = GLYPH<18> 𝜃 𝑒 / 8 𝜃 ap GLYPH<19> -0 . 04 . (11)  \nHere, 𝜃 𝑒 / 8 correspond to the standard aperture size (one-eighth of the optical effective radius), and 𝜃 ap = 0 . 75 represents the DESI fibre radius in arcseconds (DESI Collaboration et al. 2022).  \nFollowing the computation of the three Fundamental Plane parameters, 𝑅 𝑒 , 𝐼 𝑒 , and 𝜎 0 , we also calculated the corresponding uncertainties, 𝜖 𝑟 , 𝜖 𝑖 , and 𝜖 𝑠 , respectively. These uncertainties are pivotal in accounting for the overall scatter observed in the Fundamental Plane relation. Among these sources of scatter, the intrinsic scatter of the relation itself holds the primary contribution. Nonetheless, the velocity dispersion error also plays a substantial role, ranking as the second most influential factor. If the velocity dispersion error exceeds 20%, it begins to dominate the scatter, surpassing the intrinsic component.  \nFigure 2 illustrates the Signal-to-Noise Ratio (SNR) of the observed Fundamental Plane galaxies in the blue band against the relative error in velocity dispersion measurements derived from pPXF. In this figure, galaxies are colour-coded according to their redshift. The S/N ratio calculated is per-pixel, representing the median signal divided by the standard deviation of the residuals. Notably, the figure showcases that approximately 75% of the observed galaxies exhibit relative errors of less than 10%, rendering them suitable candidates for Fundamental Plane.", '5.1 Internal consistency': 'To perform an internal consistency assessment of our DESI Fundamental Plane sample, we will employ the entire set of 6698 galaxies, encompassing the full Fundamental Plane sample prior to implementing the redshift and velocity dispersion constraints. Within this subsection, we will systematically conduct an in-depth internal consistency examination of the Fundamental Plane parameters, both photometric and spectroscopic.  \nIn our Fundamental Plane analysis, we utilized data from the DESI Legacy Imaging Surveys (LS; Dey et al. 2019) to derive all our photometric parameters.  \nFigure 2. Observed Fundamental Plane galaxies SNR as a function of the relative error in the velocity dispersion measurements from pPXF. Galaxies are color-coded by their redshift. 75% of the observed galaxies have relative errors less than 10% which makes them suitable for FP. The fitted curve suggests that even with SNR as low as 4, one can still get a velocity dispersion measurement with less than 10% relative error. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->  \nIt is important to note that due to the utilization of various telescopes, cameras, and filters combinations in the BASS and DECam LS, systematic variations in the zero-point calibration between these two photometric systems are observed (as detailed in Dey et al. 2019, section 7.2).  \nTo investigate any systematic difference, we conducted a comparison of the 5 arcsec aperture magnitudes for galaxies in our FP sample that were observed in both the BASS and DECaLS surveys.  \nAmong the 1660 galaxies shared between the two surveys, we observed a median BASS -DECaLSdifferenceof + 0 . 0234mag,with root mean square (RMS) deviation of 0 . 02 mag. In light of this, we incorporate this correction into our analysis by adjusting the northern 𝑟 -band magnitudes, by 0 . 0234 mag. This offset will be investigated further with year 1 data, which will include approximately 100,000 galaxies, providing a more robust calibration for our Fundamental Plane data  \nFor the velocity dispersion measurements, we used data from the DESI spectroscopic survey early data release (DESI Collaboration et al. 2023b). In contrast to the employment of full-depth spectra for the final determination of central velocity dispersion, we utilize per-tile spectra for our spectroscopic data in this context. Per-tile spectra prove to be particularly valuable for the internal consistency assessment of our velocity dispersion measurements. These spectra combine observations across multiple exposures within the a single tile, but not across different tiles. This allows us to treat per-tile spectra as repeated observations of the same targets. By comparing velocity dispersion from the same target observed on different tiles, we can identify potential tile-to-tile offsets due to varying observing conditions.  \nFor each of our Fundamental Plane galaxies, our initial step involved determining whether they had been observed on different tiles more than once. Subsequently, we categorized these observations pairs as primary or secondary based on the signal-to-noise, with observations having the higher SNR being considered the primary one. After this categorization, we proceeded to measure the velocity  \ndispersion for all pairs, employing the same methodology applied to the full-depth spectra, as explained earlier.  \nIn order to assess the consistency of the velocity dispersion measurements and explore the possibility of systematic offsets between observations or tiles within the DESI data, we employed the relative error between pairs of observations. This relative error encompasses both the measurements of the velocity dispersion itself and its associated error. The assessment was conducted using the following formula:  \n𝜖 = 𝜎 𝑝 -𝜎 𝑠 ( 𝛿𝜎 2 𝑝 + 𝛿𝜎 2 𝑠 ) 1 2 . (12)  \nHere, 𝜎 𝑝 , 𝜎 𝑠 , 𝛿𝜎 𝑝 , and 𝛿𝜎 𝑠 represent the velocity dispersion measurements from primary and secondary tiles, accompanied by their respective error estimates. In the context of consistent and unbiased velocity dispersion measurements, with accurately estimated errors, this evaluation should yield a Gaussian distribution with a mean of zero and a standard deviation of unity.  \nIn Figure 3, we present an assessment of the velocity dispersion measurements conducted using pPXF and the Indo-US. stellar library. This evaluation is carried out on a subset of 4644 pairs from a total of 1420 unique galaxies in our DESI FP sample, utilizing per-tile DESI spectra. The left panel of the figure illustrates the one-to-one comparison between primary and secondary tiles. Reassuringly, the observations exhibit agreement within the uncertainties for the majority of tiles.  \nFor enhanced clarity regarding any differences, we plot the difference between the measurements of each pair as a function of their mean in the middle panel. To provide a quantitative analysis of these differences, we construct a histogram in the right panel to visualize the distribution of pairwise relative errors in velocity dispersion measurements. The solid curve in this panel represents a Gaussian distribution with a mean of zero and a standard deviation of one. Notably, this figure affirms that the measurements are consistent and unbiased, reinforcing the reliability of our velocity dispersion measurements.', '5.2 External consistency': 'With a successful internal assessment of our velocity dispersion measurements across different tiles and observing conditions, we now embark on a more extensive evaluation by comparing our measurements to those obtained by other surveys and telescopes. Our objective is to ensure that our measurements align closely with those from well-vetted sources. To achieve this, we perform a comparison between our velocity dispersion measurements derived from the fulldepth spectra and those from the Sloan Digital Sky Survey (SDSS).  \nIn this endeavor, we cross-match the entire FP sample of 6698 galaxies, prior to any spectroscopic cuts, with the SDSS Data Release 14 (Abolfathi et al. 2018). Our cross-match yields 4221 galaxies that are present in both the DESI FP sample and the SDSS survey. To assess the agreement, we employ the same methodology of relative error comparison that was used for the internal consistency check.  \nHowever, it is important to note that SDSS provides two distinct velocity dispersion values: firstly, from the base table containing all spectroscopic information, known as veldisp (referred to as the pipeline value); secondly, from the emissionLinesPort catalogue, from the Portsmouth group, which utilizes the pPXF method and the MILES stellar library (Sánchez-Blázquez et al. 2006) for stellar kinematics measurements, including velocity dispersion, denoted as sigmastar in the SDSS dataset. While there are other SDSS catalogs  \navailable for velocity dispersion, comparing two measurements from the same survey is sufficient for our purposes of external consistency checking.  \nFigure 4 presents the outcomes of our comparison. In the top panel, wecompare our results to the SDSS pipeline measurements. Notably, there exists an overall agreement between the two measurements. However, a small yet significant offset of 0.17 is observed, along with a standard deviation of approximately 1.3. This offset is around 7 times the standard error in the mean, and the deviation from unity of the standard deviation is 32 times the uncertainty in the standard deviation, highlighting its high significance.  \nGiven these findings, we extend our comparison to the SDSS pPXF measurements, as displayed in the middle panel. Interestingly, a comparable offset is observed, but with the opposite sign of -0.18. Importantly, the deviation from unity in the standard deviation is reduced to 1.17.  \nAs part of a comprehensive approach, to assess the potential systematic effects introduced by this offset/tilt on our measured cosmological parameters, we will calibrate our DESI velocity dispersion measurements to the SDSS pPXF measurements using the fitted red line in Figure 4. We will then evaluate the systematic bias and incorporate it into the total error budget, as discussed in Section 8.  \nAs part of a comprehensive approach to assess potential systematic effects, we calibrated our DESI velocity dispersion measurements against the SDSS pPXF measurements using the fitted red line in Figure 4. This calibration was done only to evaluate the impact of any systematic offset or tilt if incorrect velocity dispersions were used. Our fiducial cosmological analysis relies on the DESI measurements. The calibration with SDSS was included solely to identify and account for this potential source of systematic bias, which is incorporated into the total error budget, as discussed in Section 8.', '6 FUNDAMENTAL PLANE FITS': 'To derive distances and peculiar velocities, we employed the Maximum Likelihood method to fit the Fundamental Plane using a 3D Gaussian model. This approach, initially formulated by Saglia et al. (2001) and Colless et al. (2001), has since been refined and adapted by subsequent studies such as Magoulas et al. (2012); Springob et al. (2014); Said et al. (2020); Howlett et al. (2022). Here, we provide a brief summary of the method, but for a comprehensive review, see the aforementioned studies.  \nIn this study, we employ the conventional two-step approach. First, we perform a Fundamental Plane fitting without considering peculiar velocities. Then, the deviation from the optimal fit of the Fundamental Plane is attributed to the peculiar velocities of individual galaxies.  \nWeinitiate by establishing the core parameters of the Fundamental Plane as 𝑟 = log 𝑅 𝑒 , 𝑠 = log 𝜎 0 , and 𝑖 = log 𝐼 𝑒 . Following the approach proposed by Colless et al. (2001), we describe the threedimensional probability distribution in the ( 𝑟, 𝑠, 𝑖 ) space as follows:  \n𝑃 ( 𝑥 𝑛 ) = exp GLYPH<2> -0 . 5 x 𝑇 𝑛 ( V + E 𝑛 ) -1 x 𝑛 GLYPH<3> ( 2 𝜋 ) 3 / 2 | V + E 𝑛 | 1 / 2 𝑓 𝑛 . (13)  \nHere, x 𝑛 = ( 𝑟 -¯ 𝑟, 𝑠 -¯ 𝑠, 𝑖 -¯ 𝚤 ) signifies the position of galaxy 𝑛 within the Fundamental Plane domain. The matrix V encapsulates the intrinsic scatter of the Fundamental Plane relation, while E 𝑛 accounts for the measurement uncertainties associated with the Fundamental Plane parameters (see equation 14 by Said et al. (2020) for the full description of the error matrix). The normalization factor 𝑓 𝑛 ensures the distribution integrates to unity, accounting for selection criteria.  \nFigure 5 illustrates the forward projection of the Fundamental Plane relation, accompanied by the best fit of the Fundamental Plane parameters for the DESI 𝑟 -band sample as derived through this procedure. The Fundamental Plane parameters fitted under different conditions are listed in Table 2.  \n<!-- image -->  \n<!-- image -->  \nFigure 3. Pairwise comparison of velocity dispersion measured using pPXF in this work between repeat observations for a subset of DESI galaxies selected for Fundamental Plane relation. Left Panel: one to one comparison between primary and secondary observations where primary is defined as the observations with the highest SNR. The red line shows the linear fit result. Middle Panel: the difference between the two measurements as a function of the mean of the two measurements. Right Panel: distribution of pairwise relative errors in velocity dispersion measurements. The distribution is in a good agreement with a Gaussian with mean of zero and standard deviation of unity shown as a solid curve. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->  \n<!-- image -->  \n<!-- image -->  \n<!-- image -->  \n<!-- image -->  \nDESI  \n<!-- image -->  \n2  \nFigure 4. Velocity dispersion measured with pPXF in this work for Fundamental Plane galaxies in DESI SV data in comparison with SDSS measurements. The top panel shows a comparison with the SDSS pipeline velocity dispersion 𝜎 pipeline SDSS . The bottom panel present a comparison with the Portsmouth group (emissionLinesPort) velocity dispersion using pPXF 𝜎 pPXF SDSS . The agreement between DESI velocity dispersion and SDSS in general is better than the agreement between SDSS pipeline velocity dispersion measurements and the SDSS pPXF ones. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->  \nSubsequently, the likelihood can be expressed as:  \nL = 𝑁 𝑔 GLYPH<214> 𝑛 = 1 𝑃 ( 𝑥 𝑛 ) 1 / 𝑆 𝑛 . (14)  \nIn this context, 𝑆 𝑛 signifies the 1 / 𝑉 max weighting factor, which accommodates for galaxies that might be absent due to the selection function (Said et al. 2020). The objective is to determine the optimal parameters ( 𝑎 , 𝑏 , ¯ 𝑟 , ¯ 𝑠 , ¯ 𝚤 , V ) of the Fundamental Plane that best describe the data, with V encompassing the scatter intrinsic to each orthogonal direction, namely 𝜎 1 , 𝜎 2 , and 𝜎 3 .  \nIn our fitting procedure, the parameters 𝑓 𝑛 and 𝑆 𝑛 accommodate for three specific selection criteria: (1) a range of lower and upper  \nlimits on the 𝑟 -band magnitude, set at 10 ≤ 𝑚 𝑟 ≤ 18 . ; (2) lower and upper boundaries on redshift, restricted to 0 . 003 ≤ 𝑧 ≤ 0 . 1; (3) velocity dispersion, constrained by the instrumental resolution of the DESI spectrographs, adhering to 50 ≤ 𝜎 ≤ 420 km s -1 . Table 1 shows the number of remaining galaxies after each successive selection criterion.  \nWith the complete set of Fundamental Plane parameters at our hands, we are positioned to compare them with findings from previous works. Notably, as the Fundamental Plane parameters are con-  \nN  \nTable 1. Summary of the DESI FP peculiar velocity Selection criteria  \nFigure 5. The projected Fundamental Plane of DESI SV data. The data shows the measured effective radii against the predicted radii from the 3D Gaussian fit of the Fundamental Plane for DESI SV data. The solid black line shows the one-to-to line. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->  \ntingent upon wavelength, a straightforward method for comparison and assessment against earlier studies involves the computation of the root mean square (RMS) scatter of the Fundamental Plane in the 𝑟 direction. This measure should offer valuable insights into the actual distance error. However, it is worth emphasizing that the true distance error encompasses supplementary elements, including the correction for the selection function and the underlying distribution of galaxies within the Fundamental Plane, and the full covariance between parameters such as 𝐼 𝑒 and 𝜃 𝑒 . While we present a simplified estimate here, our full analysis incorporates these complex relationships through the use of complete covariance matrices.  \nThe comprehensive RMS scatter in the 𝑟 direction can be quantified through the expression:  \n𝜎 𝑟 = h ( 𝑎𝜖 𝑠 ) 2 + 𝜖 2 phot + 𝜎 2 𝑟, int i 1 / 2 . (15)  \nIn this equation 𝜖 𝑠 denotes the mean error in log 𝜎 , while 𝜖 phot is the total photometric error arising from both 𝜖 𝑟 and 𝜖 𝑖 , determined as 𝜖 phot = GLYPH<2> 𝜖 2 𝑟 + 𝑏𝜖 2 𝑖 GLYPH<3> 1 / 2 . Additionally, 𝜎 𝑟, int signifies the intrinsic scatter within the Fundamental Plane itself. Upon substituting these values into equation 15, the total RMS scatter in the 𝑟 direction is established as 23%. This outcome marks a notable improvement compared to the 6dFGSv reported value of 31% (Magoulas et al. 2012). It is worth noting that while this value is similar to the total scatter evident in the SDSS Fundamental Plane (Said et al. 2020), the DESI number of Fundamental Plane galaxies is projected to be at least five-fold greater than the SDSS Fundamental Plane sample, which itself stands as the most expansive peculiar velocity survey undertaken thus far.  \nThat was essentially the initial step in the traditional two-step Fundamental Plane approach. Up to this point, we have worked under  \nthe assumption of negligible peculiar velocities. However, the next phase involves calculating these peculiar velocities by gauging the deviations of the data from the best-fit Fundamental Plane parameters. By comparing the physical effective radius, 𝑟 , derived from equation 8, with the true effective radius, 𝑟 𝑡 , inferred from the bestfit Fundamental Plane parameters, we can derive the log-distance ratio as:  \n𝑟 -𝑟 𝑡 = 𝜂. (16)  \nThe log-distance ratio is the main derived value in our dataset.', '7 ZERO-POINT CALIBRATION AND ABSOLUTE DISTANCES': 'In the Fundamental Plane equation 4, the coefficient 𝑐 sets the zeropoint of the relation. The determination of this zero-point has a direct impact on the calculation of the actual effective radius of galaxies, subsequently influencing the derived distances and peculiar velocities. Throughout the process of fitting the Fundamental Plane, we have assumed that the net radial peculiar velocity of all galaxies is zero. This assumption implies the absence of a monopole term in the velocity field. While this assumption may hold for a homogeneous allsky galaxy sample, it requires special consideration for hemispheric surveys such as 6dFGSv, SDSS, and DESI surveys, essentially all ground-based galaxy surveys.  \nVarious surveys have adopted different approaches to establish the zero-point. For instance, Springob et al. (2014) utilized a sub-sample of the 6dFGSv near the celestial equator, defining a great circle sample to re-fit the Fundamental Plane and adjust the zero-point for the entire sample. This effectively treats the sample as a fullsphere, being degenerate only in the monopole term. Unfortunately, this approach is not feasible for the DESI early data release due to the sky coverage limitations. However, it holds promise for the full DESI data release.  \nAnother example can be found in the SDSS peculiar velocity survey by Howlett et al. (2022). In this study, they performed a crossmatch between the SDSS sample and the Cosmicflows-III catalogue (CF3; Tully et al. 2016), which contains distance measurements from alternative methods.  \nWe adopted an approach similar to SDSS\'s method by crossmatching our DESI FP sample with the SDSS peculiar velocity catalogue. Specifically, we utilized their calibrated log distance ratios, which were originally calibrated using the CF3 data sets. This dataset itself was calibrated through a distance ladder approach encompassing various standard candles, such as Cepheid variables, Tip of the Red Giant Branch, and Type Ia supernovae. We identified 896 galaxies common to both samples, providing a substantial number of galaxies for zero-point calibration.  \nFigure 6 shows a comparison of the log distance ratios for these common objects between SDSS and DESI peculiar velocity catalogue prior to zero-point calibration.  \nAone-to-one line (black) and a Hyper Fit line (red) are overlaid to show the difference between 𝜂 DESI and 𝜂 SDSS  \nHowever, this approach is not the optimal choice if the desired scientific goal is to measure the present-day expansion rate 𝐻 0 , as this value is already set to 75 ± 2 km s -1 Mpc -1 for the CF3 catalogue. To address this, we employed an alternative method to calibrate the zero-point (which is degenerate with the Hubble constant 𝐻 0 ) using measured distances to clusters in our sample, leveraging primary distance indicators and offering absolute distance measurements. This  \nTable 2. Fundamental Plane parameters for the DESI SV sample under different conditions: Fiducial (original), including non-elliptical galaxies in the fitting procedure, and calibrated DESI velocity dispersion measurements to the SDSS measurements from the Portsmouth group using pPXF. SDSS and 6dFGSv FP parameters are also included for comparison.  \nFigure 6. Comparison between the measured log-distance ratio for DESI SV galaxies and their counterpart in the SDSS PV. The solid line shows the oneto-one line. The red line shows the Hyper Fit result (y = 1.01x - 0.01). Mean uncertainties for 𝜂 DESI (x-axis) and 𝜂 SDSS (y-axis) are displayed in the upper left corner. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->  \napproach offers two key advantages: First, it grants us greater control over the selection of primary distance indicators, enabling us to choose and exclude indicators as needed; second, by averaging over multiple distances within the same cluster, we can mitigate the overall scatter in the calibration process, in contrast to using individual galaxy distances.  \nAlthough there is not a group/cluster catalogue available for the DESIearly data release yet, we know that this release has already covered the Coma cluster to a greater depth than any survey before. The Coma cluster, being massive, relatively nearby, and extensively studied, serves as an ideal candidate for setting the Fundamental Plane zero-point. To make use of additional galaxies in the Coma cluster that had not previously been observed before DESI, we adapted a modified friends-of-friends (FoF) algorithm based on the methodology introduced by Press & Davis (1982). We combined redshift data from the DESI early data release and supplemented it with SDSS redshifts for galaxies not observed by DESI. Our initial dataset encompassed all DESI and SDSS redshifts within a generous region around the Coma cluster, spanning a 10 degree radius centered on the Coma cluster RA=12h59\'48.7", Dec=27 ° 58\'50" and redshift range of 0.02 relative to the mean redshift of the Coma cluster ( 𝑧 = 0 . 0231, Abell et al. 1989). We applied a modified FoF algorithm, implementing separate linking lengths for the angular and radial directions, as outlined in Eke et al. (2004); Duarte & Mamon (2014, 2015),  \nand optimized these lengths using a cost function (Robotham et al. 2011) along with SDSS-like mock catalogues. Specifically, we employed linking lengths of 600 km s -1 in the radial direction and 0.3 ° in the angular direction. This yielded a catalogue of 1731 galaxies potentially belonging to the Coma cluster.  \nSubsequently, we employed the method of Jaffé et al. (2015) to remove galaxies that could not be kinematically bound to the Coma cluster, and we excluded galaxies located beyond the cluster\'s turnaround radius, following Korkidis et al. (2020). This process yielded a final catalogue of 1696 identified Coma cluster members. To validate our method and the chosen linking lengths, we applied the same process to DESI-only and SDSS-only datasets. The results were consistent with each other, differing by only a few galaxies. A more detailed description of this method and the associated code for cluster member identification will be provided in Saulder et al., (in preparation).  \nCross-matching our Fundamental Plane sample with this newly created Coma catalogue, we identified 226 galaxies in common. We then calculated the uncalibrated distance of the Coma cluster using the weighted mean of distances for these 226 galaxies. To perform zero-point calibration and obtain absolute distances for our full sample, we compared this calculated Coma distance to the most recent absolute distance measurement of the massive NGC 4874 galaxy at the core of the Coma cluster, determined using the Surface Brightness Fluctuation method (SBF; Jensen et al. 2021), which yielded 99 . 1 ± 5 . 8 Mpc.  \nWe applied this correction between the uncalibrated distance of Coma obtained from the Fundamental Plane and the measurement of Coma\'s distance from the surface brightness fluctuation method to obtain absolute distances for all galaxies in our sample.  \nAs with the previous methods, this approach also comes with its own set of limitations. Firstly, due to the absence of a dedicated full group catalogue for DESI data at present, we are confined to utilizing the Coma cluster alone. While this provides a valuable calibration point, it could potentially introduce bias compared to using multiple clusters and groups that span a broader region of the sky. However, it\'s worth noting that this limitation will be significantly alleviated in future DESI data releases and its associated group catalogue, which will encompass numerous clusters and groups with known distances, facilitating a more comprehensive zero-point calibration process. In Figure 1, we display a selection of these clusters and groups, each possessing known absolute distances, that are well-suited for the calibration procedure.  \nThe second limitation arises from the observation by Howlett et al. (2022) of a correlation between group richness and one of the Fundamental Plane parameters, specifically the mean surface brightness  \n( 𝐼 𝑒 ). Consequently, relying solely on the Coma cluster for zeropoint calibration might introduce bias into our results. However, this concern is mitigated by the potential approach of employing multiple Fundamental Plane fits as a function of group/cluster richness, as suggested by Howlett et al. (2022). In general, while the zeropoint calibration can be executed through various methodologies, the choice is contingent on the data available for analysis.  \nIn summary, the process of establishing the zero-point for the Fundamental Plane involves several considerations, each with its own merits and limitations. The methods applied depend on the available data and the specific scientific objectives of the study. With forthcoming DESI data, these challenges will be further addressed and the calibration process can be refined, resulting in even more accurate distance measurements and peculiar velocities for a much larger sample of galaxies.', '8 MEASURING THE HUBBLE CONSTANT': "The recession velocity-distance relation, often referred to as the Hubble-Lemaître law, establishes the connection between a galaxy's recession velocity and its distance. It is expressed as:  \n𝑣 = 𝐻 0 𝐷. (17)  \nIn this equation, the constant of proportionality, denoted as 𝐻 0 , represents the present rate of expansion of the universe, Hubble's constant. Our method of measuring the Hubble constant here involves the construction of a Hubble diagram, which necessitates a dataset presenting the distance modulus versus redshift.  \nHowever, our approach to creating the Hubble diagram and performing the fitting to estimate the Hubble constant differs from the traditional Hubble diagram. Instead of using apparent magnitude and absolute magnitude as the observable and measured quantities, we employ a different set of variables in the Fundamental Plane analysis. Specifically, our observable quantity is the angular effective radius, while the measured quantity is the true physical radius. Consequently, rather than plotting the distance modulus on the y-axis of the Hubble diagram, our y-axis represents the difference between the logarithm of the angular effective radius and the logarithm of the true physical radius. It is mathematically expressed as:  \n𝜇 𝐴 = 𝑟 𝑡 -𝑟 𝜃 (18)  \n= 𝑟 -𝜂 -𝑟 𝜃 -log ( 1000 𝜋 180 × 3600 ) (19)  \n= log 𝑑 𝐴 (20)  \nwhere 𝑑 𝐴 is the angular diameter distance parameterized as follows:  \n𝑑 𝐴 = 𝑑 𝐿 ( 1 + 𝑧 ) 2 (21)  \n= 𝑐 ( 1 + 𝑧 ) ∫ 𝑧 0 𝑑𝑧 ' 𝐻 ( 𝑧 ' ) (22)  \nwhere 𝑑 𝐿 is the luminosity distance. We then express the angular distance modulus as a power series of the form 5 :  \n𝜇 model 𝐴 = log 𝑐𝑧 GLYPH<18> 1 + 1 2 [ 1 -𝑞 0 ] 𝑧 -1 6 [ 1 -𝑞 0 -3 𝑞 2 0 + 𝑗 0 ] 𝑧 2 GLYPH<19> -log 𝐻 0 -2 log ( 1 + 𝑧 ) . (23)  \nOne can then use this formulation to fit for the Hubble constant 𝐻 0 , the deceleration parameter 𝑞 0 , and the jerk parameter 𝑗 0 by measuring the angular distance modulus as a function of redshift up to terms of order 𝑧 3 .  \nDESI FP Hubble diagram comprising 4191 Fundamental Plane galaxies within the redshift range of 0.01 to 0.1 is shown in the upper panel of Figure 7.  \nDue to the limited redshift range covered by our Fundamental Plane data ( 𝑧 < 0 . 1), it is not feasible to perform a concurrent fit for all three cosmological parameters in equation 23. Instead, we adopt 𝑞 0 = -0 . 55 and 𝑗 0 = 1, in line with the expectations for a flat Λ CDM cosmology with Ω 𝑚 = 0 . 3 and Ω Λ = 0 . 7. We solely perform a fit for 𝐻 0 under these conditions.  \nAdditionally, following the approach by Riess et al. (2022), we implement an additional redshift cut of 𝑧 > 0 . 023, to limit the effect of peculiar velocities, which results in a sample of 4063 galaxies. After applying these criteria, the derived value for the Hubble constant is 𝐻 0 = 76 . 05 ± 0 . 35 km s -1 Mpc -1 . This uncertainty exclusively encompasses statistical uncertainties.  \nFigure 7 illustrates our best-fit model for 𝐻 0 using a flat Λ CDM cosmology with Ω 𝑚 = 0 . 3 and Ω Λ = 0 . 7 (black line), alongside a curve representing our best-fit 𝐻 0 using Planck values for Ω 𝑚 = 0 . 315 and Ω Λ = 0 . 685 (red dashed line). The negligible difference between these curves (0 . 06% in 𝐻 0 ) demonstrates the insensitivity of our 𝐻 0 estimate to reasonable variations in Ω 𝑚 and Ω Λ at these low redshifts.  \nWe also plot the curve corresponding to Planck Collaboration et al. (2020) values for 𝐻 0 , Ω 𝑚 and Ω Λ (blue line), which shows an offset from our best-fit model. Quantitatively, an average change of 0.0522 in 𝜇 𝐴 over our fitting range ( 𝑧 > 0 . 023) would be required to reconcile our measurements with the Planck 𝐻 0 value of 67.4 km s -1 Mpc -1  \nThe lower panel of Fig. 7 illustrates the residuals in comparison to the best fit results for a flat Λ CDMcosmology, as detailed in equation 23. The observed larger 𝜇 𝐴 at 𝑧 < 0 . 023 can be attributed to the effect of LSS. This trend can be well explained by a single attractor model, which shows infall toward the attractor's centre and backside infall. This trend is also evident in velocity field reconstruction models. We have included three pv reconstructions (2M++ from Said et al. (2020), 2MRS from Lilow & Nusser (2021) and 2M++ from Carrick et al. (2015)), which all show the same signature as the DESI FP data. The smoother curves in these models results from their use of linear theory for reconstruction and smoothing to a scale of approximately 4 Mpc.  \nTo address potential systematic errors, we conducted an analysis, revisiting our calculations while considering several sources of systematic bias. Figure 8 presents the posterior distributions of the Hubble constant obtained from the DESI Fundamental Plane data after applying various sources of systematics to it. In Figure 8, the black probability density distribution represents our fiducial measurement. Although other potential sources of systematics may exist, the following are the most evident and notable.  \n- (i) Inclusion of Spiral Galaxies: Despite implementing selection criteria to isolate pure elliptical galaxies during our sample selection process, a number of non-ellipticals remained in our dataset. We endeavored to identify and exclude them via visual inspection, a task undertaken by JRL. This inspection identified 1081 galaxies out of 4191 as non-ellipticals, representing approximately 26% of our sam-  \nl  \ne  \nd  \no  \nm  \nFigure 7. The DESI Hubble diagram features 4191 Fundamental Plane galaxies with redshifts falling within the range of 0.01 to 0.1. In the upper panel, the plot displays the log of the angular diameter distances, denoted as 𝜇 𝐴 , as a function of redshift, 𝑧 . The solid black curve corresponds to the best-fit 𝐻 0 for a Λ CDM cosmology with Ω 𝑚 = 0 . 3 and Ω Λ = 0 . 7, as determined by employing equation 23. The highlighted regions indicate the redshift ranges used for zero-point calibration and for fitting cosmological parameters. The red dashed line shows our best-fit 𝐻 0 using Planck values for Ω 𝑚 and Ω Λ . The blue curve represent Planck's values for 𝐻 0, Ω 𝑚 and Ω Λ . In the lower panel, the plot reveals the residuals relative to the best-fit model, calculated as 𝜇 𝐴 -𝜇 model 𝐴 . Additionally, in the bottom panel, a black line represents the average trend of the residuals. The observed larger 𝜇 𝐴 at redshift 𝑧 < 0 . 023 is due to LSS and can be explained by a single attractor model showing infall and backside infall toward the attractor's centre. The plotted peculiar velocity field reconstructions 2M++ from Said et al. (2020) and Carrick et al. (2015) and 2MRS from Lilow & Nusser (2021) also show this trend. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->  \nple. The inclusion of these galaxies could potentially influence the Fundamental Plane parameters, as illustrated in Table 2. Rather than outright removal, we opted to adopt the approach outlined by Howlett et al. (2022), wherein we excluded them from the Fundamental Plane fit but retained these galaxies during peculiar velocity measurements. This strategy ensures that these galaxies do not introduce bias into our Fundamental Plane fit but still allows us to calculate their peculiar velocities. As a systematic test, we conducted the Fundamental Plane fit with these galaxies included in our sample. This test assessed the impact of not applying visual inspection procedures, revealing Hubble constant value of 𝐻 0 = 76 . 24 ± 0 . 39 km s -1 Mpc -1 (represented by the red probability density) which is a slightly higher value than our fiducial value.  \nContinuing our exploration, we further examined the impact of  \nspirals by conducting an alternative test where all identified spiral galaxies were removed from the entire process, spanning from the Fundamental Plane fit to the zero-point calibration and subsequent cosmology fitting. The resulting probability density for this test is shown in blue, revealing a shift in the mean value of 𝐻 0 of about 1 . 6 km s -1 Mpc -1 . This significant alteration is attributed to the exclusion of these galaxies, introducing a selection bias during the Hubble constant fitting process. It suggests that visual inspection tends to eliminate more nearby and brighter galaxies while sparing those at higher redshifts or fainter. Despite this noticeable shift, we refrain from incorporating this posterior into our systematic error budget. Nevertheless, presenting it here serves as a point of interest for further investigation, particularly with the anticipated DESI year  \nFigure 8. The posterior distribution for the Hubble constant 𝐻 0 derived from our fiducial DESI Fundamental Plane analysis represented by the black probability density. Additional probability density distributions, displayed in various colours, illustrate the impact of different sources of systematics on the 𝐻 0 measurement process. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->  \n1 data, which promises a substantially larger sample for a more indepth analysis.  \n(ii) Correcting DESI 𝜎 to SDSS values: A key component of the Fundamental Plane parameters is the stellar velocity dispersion 𝜎 . An important consideration for DESI data is whether the signal-to-noise ratio is sufficient to obtain accurate measurements of velocity dispersion. To address this concern, we perform a comprehensive internal and external consistency check analysis in this paper (refer to Figs. 3 and 4). When we compare DESI velocity dispersion measurements to those of SDSS, we observe a slight difference, as shown in Figure 4. To assess this potential source of systematic bias, we employ linear fitting to rectify all velocity dispersion measurements across our dataset, encompassing not only the overlapping DESI and SDSS measurements but all DESI data. We then repeat the entire process of fitting the Fundamental Plane, perform re-calibration, and re-fit the Hubble diagram using these corrected velocity dispersion values. This results in a Hubble constant measurement of 𝐻 0 = 75 . 97 ± 0 . 34 km s -1 Mpc -1 , which we identify as the least influential source of systematic bias among the five we have identified. The posterior distribution for this measurement is illustrated in Figure 8 as the green probability density.  \n(iii) Higher Redshift Cut ( 𝑧 > 0 . 034): In the process of constructing the Hubble diagram and fitting for the Hubble constant, we adopted the practice recommended by Riess et al. (2022) of implementing a redshift cut to exclude low-redshift galaxies below redshift, 𝑧 < 0 . 023 to mitigate the impact of peculiar velocities. In this systematic analysis, we examine the consequences of applying a more stringent redshift cut of 𝑧 < 0 . 034. This specific redshift limit was chosen to exclude any data used in the Zero-point calibration process. The resulting probability density for this cut is represented in purple in Figure 8. Applying this stricter redshift cut yields a Hubble constant measurement of 𝐻 0 = 76 . 54 ± 0 . 36 km s -1 Mpc -1 , which is slightly higher than our fiducial value.  \n(iv) No Redshift Cut Applied: In this systematic assessment, we conducted the Hubble constant measurement without implementing any redshift cut, including all the Fundamental Plane data, even the low-redshift galaxies. The resulting probability density is shown in cyan, and it provides a Hubble constant value of 𝐻 0 = 75 . 58 ± 0 . 32 km s -1 Mpc -1 . This value is slightly lower than our fiducial Hubble constant measurement.  \n(v) Applying the peculiar velocity correction: Throughout our  \ncosmological fitting, we utilized the redshift in the CMB frame, 𝑧 cmb . However, peculiar velocities can introduce systematic effects. To assess this, we corrected the redshift from the CMB frame, which accounts for our own motion, to the cosmological redshift, incorporating peculiar velocities. This correction utilized the default option of the pvhub 6 velocity field maps developed by Carr et al. (2022). Implementing this correction resulted in a slightly higher value for the Hubble constant: 𝐻 0 = 76 . 36 ± 0 . 33 km s -1 Mpc -1 .  \n(vi) Fundamental Plane Analysis with TRGB Calibration: In our primary cosmology fitting, we utilized the Fundamental Plane data calibrated with the absolute distance to the Coma cluster, as measured using the Surface Brightness Fluctuation method. We introduced an alternative distance calibration using the Tip of the Red Giant Branch (TRGB) method. Using the TRGB method to measure the distance to Leo I group, and based on the relative distance between Leo I and Coma cluster, Sakai et al. (1997) derived a distance modulus of 𝜇 = 35 . 03 ± 0 . 37 for the Coma cluster. Comparing this absolute distance obtained via the TRGB calibration to our Fundamental Plane distances, we constructed a new catalogue of absolute distances for our sample. Subsequently, we created a Hubble diagram and remeasured the Hubble constant, yielding a value of 𝐻 0 = 74 . 33 ± 0 . 31 km s -1 Mpc -1 . This systematic shift represents the most substantial bias among all the potential sources of systematics we have identified. The probability density for this specific analysis is visualized in grey dotted line.  \nConcluding our investigation into potential systematic biases, we proceeded to evaluate the systematic error linked to our Hubble constant value. This analysis considered all MCMC chains, excluding the chain that involved the exclusion of non-ellipticals through visual inspection and the one utilizing the TRGB distance calibration to the Comacluster. The exclusion of the TRGB-calibrated chain is justified by its representation of a systematic bias intrinsic to the calibration method, rather than an issue within the Fundamental Plane fitting process. Moreover, the statistical error arising from the zero-point calibration is anticipated to address this aspect as well.  \nTo assess statistical uncertainties associated with the zero-point calibration process, we generated 1000 distances to the Coma cluster based on the measured distance and its associated uncertainties  \nderived from surface brightness fluctuation (Jensen et al. 2021). For each of these 1000 distances, we repeated the zero-point calibration process, generating a new catalogue used to fit the Hubble constant. The individual probability densities are depicted in the top panel of Figure 9 as dotted grey lines, while the combined chain is represented by the solid black line, serving to quantify the statistical errors linked to the SBF calibration process, resulting in a value of ± 4 . 86 km s -1 Mpc -1 . A similar process was conducted for the Tip of the Red Giant Branch calibration, yielding a statistical error of ± 1 . 87 km s -1 Mpc -1 , as shown in the bottom panel of Figure 9. Despite the lower statistical uncertainties associated with the TRGB calibration, we adopted the SBF calibration results as the main findings in this paper. This decision is based on the fact that the TRGB distance to Coma is not a direct measurement but relies on a measured distance to the Leo I group and assumes a known relative distance between Leo I and Coma cluster (Sakai et al. 1997).  \nNotably, this analysis reveals that the statistical uncertainty associated with the zero-point calibration process dominates the error budget. However, this result underscores the robustness of the Fundamental Plane analysis. The systematic biases arising from potential sources of systematic within the Fundamental Plane fitting process itself are notably smaller than the statistical errors introduced by the calibration process. Therefore, after accounting for all other sources of systematics, including the inclusion of spiral galaxies, the correction of DESI velocity dispersion to SDSS values, applying both high and low redshift cuts, and statistical uncertainties due to the calibration, our final estimate for the Hubble constant is 𝐻 0 = 76 . 05 ± 0 . 35(statistical) ± 0 . 49(systematic FP) ± 4 . 86(statistical due to calibration) km s -1 Mpc -1 .", '9 SUMMARY': 'The DESI peculiar velocity survey will be approximately four times larger than the combined size of all previous peculiar velocity surveys. For the science verification sample, we adopted a similar approach to the 6dFGSv and SDSS peculiar velocity surveys (Magoulas et al. 2012; Howlett et al. 2022) in order to select a clean and reliable sample of elliptical galaxies. This approach involved implementing various photometric cuts, including magnitude and colour cuts. Following the implementation of the aforementioned cuts, our selection process resulted in a sample of 6698 unique galaxies. This sample size is comparable to that of the complete 6dFGSv peculiar velocity sample which had been the largest peculiar velocity survey for a decade and helped refine our understanding of the growth rate of structure (Adams & Blake 2017; Qin et al. 2019; Adams & Blake 2020; Said et al. 2020).  \nWe apply only one spectroscopic cut: redshift. At this stage, we refrained from applying any H-alpha cuts, as this aspect will be investigated more extensively using data from the first year of the survey. Our final peculiar velocity sample includes 4191 elliptical galaxies.  \nDuring the process of fitting the Fundamental Plane, we implemented an additional visual inspection cut to further refine the selection of galaxies. It is important to note that this visual inspection cut was only applied during the initial step, to determine the cleanest sample with which to fit the Fundamental Plane parameters. Subsequently, these galaxies were re-introduced into the sample to calculate their peculiar velocities. This approach ensures that the determination of Fundamental Plane parameters remains unbiased, preserving the integrity of the other galaxies within the sample. Simultaneously, it allows for the calculation of peculiar velocities for all galaxies,  \naffording the flexibility to decide at a later stage whether to include or exclude these peculiar velocities in any subsequent cosmological analysis.  \nTo construct the Fundamental Plane, we constructed photometric parameters such as the angular effective radius and the mean surface brightness. These parameters were obtained from the Ninth Data Release (DR9) of the DESI Legacy Imaging Surveys (Dey et al. 2019). Thephotometric error was calculated as 𝜖 phot = [( 𝜖 𝑟 ) 2 +( 𝑏𝜖 𝑖 ) 2 ] 1 / 2 = 0 . 002 dex (<1%). This error is one order of magnitude smaller than the total photometric error observed in the SDSS peculiar velocity survey.  \nVelocity dispersion serves as the third component in constructing the Fundamental Plane. In our study, we employed the pPXF algorithm (Cappellari 2017) along with the Indo-U.S. Coudé Feed Spectral Library (Valdes et al. 2004) to measure velocity dispersion from DESI spectra.  \nIn order to test the internal (per tile) and external (per survey) consistency of velocity dispersion measurements and avoid any potential systematic offsets within our data, we used the pairwise relative error. We showed that the relative error distribution, both internally and externally, followed a Gaussian distribution centered at zero with a standard deviation of one. This showed that our velocity dispersion measurements are consistent and unbiased.  \nTo ensure the reliability of our results, we assessed the relative velocity dispersion error 𝛿𝜎 𝜎 . Our examination revealed that 75% of our sample exhibited a relative error of less than 10%.  \nWhile intrinsic scatter remained the largest source of uncertainty, this velocity dispersion error constituted the second largest component of uncertainty in the Fundamental Plane, specifically in the 𝑟 -direction.  \nThe fitting of a 3D Gaussian Fundamental Plane to our sample yielded results that were comparable to those obtained from the SDSS survey (Said et al. 2020; Howlett et al. 2022) in terms of scatter in the 𝑟 -direction of the FP. However, a significant improvement was observed when compared to the analysis based on the 6dFGSv survey.  \nIndependent of the Fundamental Plane, we defined Coma cluster membership. Using this criterion, we identified 226 galaxies belonging to the Coma cluster. These galaxies were then used in our Zeropoint calibration for the Fundamental Plane. This calibration utilized the absolute distance to the Coma cluster, which was determined by Jensen et al. (2021) through the Surface Brightness Fluctuation method.  \nAfter calibrating our sample, we proceeded to construct the Hubble diagram and estimate the Hubble constant. Our final result for the Hubble constant is 𝐻 0 = 76 . 05 ± 0 . 35 (statistical) ± 0 . 49 (systematic FP) ± 4 . 86 (statistical due to calibration) km s -1 Mpc -1 .  \nWhile our measured value of the Hubble constant, 𝐻 0 , is within 2 𝜎 of the measurement derived from cosmic microwave background (CMB) anisotropies (e.g., Planck Collaboration et al. 2020), it notably aligns well within 1 𝜎 with local Hubble constant determinations based on different distance indicators (e.g., Riess et al. 2022). This alignment includes includes the most recent findings from CosmicFlows-4 (Kourkchi et al. 2020, 2022) as discussed in Said (2023), where a comprehensive review of Hubble constant measurements from the Tully-Fisher relation is provided. Additionally, our measurement is consistent with the first standard siren measurement from the GW170817 analysis performed with DESI data (Ballard et al. 2023).  \nH 0 [kms 1 Mpc 1 ] Systematics due to using TRGB as Calibrator  \n<!-- image -->  \nFigure 9. MCMCsampling of the posterior for 𝐻 0 to assess systematics arising from the zero-point calibration. The top panel displays 1000 MCMC chains of the 𝐻 0 posterior as grey dotted lines, derived using randomly sampled distances to the Coma cluster, given its measured distance and associated error from SBF. The solid line represents the combined chain used to quantify systematic errors introduced by the calibration process when using SBF. The bottom panel mirrors the top one but employs the distance to Coma cluster derived from the TRGB method. For the Python code and data used to reproduce this plot, see this link.  \n<!-- image -->', '10 CONCLUSIONS AND FUTURE DIRECTIONS': 'This paper underscores the capability of the Dark Energy Spectroscopic Instrument (DESI) to deliver reliable velocity dispersion measurements, thereby facilitating the application of the Fundamental Plane analysis and the subsequent constraint of critical cosmological parameters, including the Hubble constant and the growth rate of cosmic structure.  \nIn this paper, we present an analysis of the Hubble constant ( 𝐻 0 ) based on Fundamental Plane measurements derived from a sample of 4191 galaxies within the redshift range of 0.01 to 0.1. Systematic uncertainties are explored, including potential biases introduced by spiral galaxies, velocity dispersion calibration, and redshift cuts. The final result, 𝐻 0 = 76 . 05 ± 0 . 35(statistical) ± 0 . 49(systematic FP) ± 4 . 86(statistical due to calibration) km s -1 Mpc -1 , demonstrates an agreement with previous Hubble constant measurements from other distance indicators.  \nSeveral avenues for expanding our analysis are on the horizon. Foremost is the utilization of the forthcoming DESI year 1 Fundamental Plane data (Ross et al. in preparation), expected to be substantially larger ( ∼ 100k elliptical galaxies), providing not only enhanced statistical power but also greater control over systematic uncertainties, which currently represent the primary source of uncertainty, particularly in the zero-point calibration process. Currently, our calibration relies on a single cluster, Coma, and a solitary source within the cluster, NGC 4874. However, we anticipate a substantial improvement in precision with the inclusion of multiple clusters and numerous sources within each of these clusters. This advancement will become feasible with the availability of the group catalogue from the full DESI dataset.  \nFuture research could explore the potential of using stellar population to enhance the precision and accuracy of the FP as a distance indicator. Recent work by (D\'Eugenio et al. 2024) discuss the concept  \nof a "hyperplane" for early-type galaxies, which incorporates stellar population observables alongside the traditional FP parameters.  \nFuture work can also focus on extending our peculiar velocity survey to higher redshifts using Brightest Cluster Galaxies (BCGs). As the most luminous galaxies in the Universe, BCGs offer the potential to probe peculiar velocities out to much greater distances than possible with typical spiral and elliptical galaxies. We plan to utilize both the Fundamental Plane relation and the Metric Plane (Lauer et al. 2014) for BCGs, with the latter offering intrinsically less scatter and thus more precise distance measurements. The DESI survey, particularly the combination of Bright Galaxy Survey (BGS) and Luminous Red Galaxy (LRG) data, is expected to provide a volume-limited sample of BCG-like galaxies. Moreover, BCGs are less affected by selection biases that impact normal ellipticals. This extension will enable us to investigate deviations from normal kinematic Hubble flow expansion out to 𝑧 ∼ 0 . 15, potentially shedding light on the existence and effects of large-scale structures such as a large cosmic voids on cosmological parameters like 𝐻 0 .  \nExtending our distance measurements to higher redshifts will enable simultaneous fitting of cosmological parameters like the Hubble constant, deceleration, and jerk parameters. Combining our distance measurementswithotherDESIdatasets,suchastheTully-Fisher relation, can further mitigate systematics. Additionally, incorporating our data with external datasets like SN Ia measurements holds promise. However, a key part of making these advancements is putting more effort into improving the zero-point calibration, which this paper shows is a major source of uncertainties.  \nExploring other scientific domains, like the measurements of the amplitude and directions of the Bulk Flow, necessitates complete sky coverage. While DESI encompasses 14,000 square degrees in the northern hemisphere, upcoming surveys such as WALLABY (Courtois et al. 2023a), which employs the Australian SKA Pathfinder (ASKAP), and the 4MOST Hemisphere Survey (Taylor et al. 2023), are set to encompass the southern hemisphere. This expanded coverage promises numerous scientific possibilities that would be unattainable with just half of the sky. It is noteworthy that surveys like WALLABY will operate in entirely different wavelengths. Such diversity is invaluable for testing phenomena like galaxy bias across varying wavelengths, and might unravel sources of systematics that we have not thought about yet.', 'ACKNOWLEDGEMENTS': "KS, CH, and TMD acknowledge support from the Australian Government through the Australian Research Council's Laureate Fellowship funding scheme (project FL180100168) and the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), through project number CE230100016.  \nKS would like to thank Francesco D'Eugenio for many useful discussions realted to pPXF and the associated stellar library templates. This research used data obtained with the Dark Energy Spectroscopic Instrument (DESI). DESI construction and operations is managed by the Lawrence Berkeley National Laboratory. This material is based upon work supported by the U.S. Department of Energy, 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;  \nthe 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 Science and Technology of Mexico (CONACYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: www.desi.lbl.gov/collaborating-institutions. The DESI collaboration is 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. 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.  \nThe DESI Legacy Imaging Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS), the Beĳing-Arizona Sky Survey (BASS), and the Mayall z-band Legacy Survey (MzLS). DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory. Legacy Surveys also uses data products from the NearEarth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Legacy Surveys was supported by: the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy; the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility; the U.S. National Science Foundation, Division of Astronomical Sciences; the National Astronomical Observatories of China, the Chinese Academy of Sciences and the Chinese National Natural Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. The complete acknowledgments can be found at https://www.legacysurvey.org/.", 'DATA AVAILABILITY': 'All data and codes necessary to reproduce the analyses presented in this paper are publicly available. The repository containing both the final catalogue and the analysis codes can be accessed at https://github.com/KSaid-1/DESI\\_fuji\\_FP . All data points shown in the figures are available in a machine-readable form on https://doi.org/10.5281/zenodo.13363598.', 'REFERENCES': '```\nAbell G. O., Corwin Harold G. J., Olowin R. P., 1989, ApJS, 70, 1 Abolfathi B., et al., 2018, ApJS, 235, 42 Adams C., Blake C., 2017, MNRAS, 471, 839 Adams C., Blake C., 2020, MNRAS, 494, 3275 Ballard W., et al., 2023, Research Notes of the American Astronomical Society, 7, 250 Bell R., Said K., Davis T., Jarrett T. 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