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Open Access
Issue
A&A
Volume 695, March 2025
Article Number A54
Number of page(s) 12
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/202452007
Published online 07 March 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model.

Open Access funding provided by Max Planck Society.

1. Introduction

Measuring the expansion history of the Universe is of paramount importance in the field of modern cosmology. It will allow us to improve our understanding of the nature of dark energy and gravity itself. It can be revealed by diverse cosmic distance measures in tomographic redshift space, such as cosmic parallax (Benedict et al. 1999), standard candles (Fernie 1969), and standard rulers (Eisenstein et al. 1998, 2005). The best constraints to date come from the distance–redshift relation and imply that the expansion rate has changed from a decelerating phase to an accelerated one (Riess et al. 1998; Perlmutter et al. 1999). While current observations largely affirm the Λ-cold dark matter (CDM) model, which incorporates the cosmological constant, the quest for a high-precision confirmation of this model, or the identification of any deviations, remains a pivotal observational goal. One of the most robust methods for measuring the distance–redshift relation is to use the baryon acoustic oscillation (BAO) feature that is observed as a bump in the two-point correlation function or as wiggles in the power spectrum. The tension between gravitational infall and radiative pressure caused by the baryon-photon fluid in the early Universe gave rise to an acoustic peak structure that was imprinted on the last-scattering surface (Peebles & Yu 1970). Acoustic features on density fluctuations are formed by a tension between the gravitational force from dark matter clustering and the radiation pressure from baryon-radiation plasma during the radiation-dominated epoch, and this length scale of the sound horizon, rd, is frozen as a standard ruler at the decoupling epoch to remain in the large-scale structure of the Universe. Notably, this scale manifests as a distinct peak in the isotropic two-point correlation function. However, by analyzing the anisotropic wedge correlation function, we can discern the perpendicular and parallel components of cosmic distance. The BAO peak positions convert to cosmic distances of DA(z)/rd and H−1(z)rd in perpendicular and parallel directions. The quoted rd in this manuscript is fixed to 147.21 Mpc, as obtained from Planck Collaboration XIII (2016, hereafter Planck 2015). The BAO feature has been measured through the correlation function (Eisenstein et al. 2005), and one of the most successful measurements in the clustering of large-scale structure at low redshifts was obtained using data from Sloan Digital Sky Survey (SDSS; Eisenstein et al. 2005; Estrada et al. 2009; Padmanabhan et al. 2012; Hong et al. 2012; Veropalumbo et al. 2014, 2016). The final results from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS; Ross et al. 2015; Alam et al. 2017) and the SDSS-IV Extended Baryon Oscillation Spectroscopic Survey (eBOSS; Bautista et al. 2021; Gil-Marín et al. 2020; Raichoor et al. 2021; de Mattia et al. 2021; Hou et al. 2021; Neveux et al. 2020; du Mas des Bourboux et al. 2020) traced the expansion of the Universe and showed an overall agreement with the Λ-CDM model. The Dark Energy Spectroscopic Instrument (DESI; Levi et al. 2013; DESI Collaboration 2016a,b, 2022; Silber et al. 2023; Miller et al. 2024) is the main instrument of an ongoing survey (Guy et al. 2023; Schlafly et al. 2023). The first early data of this survey have already been released to the public consisting of the data collected during its survey validation (DESI Collaboration 2024a,b). Furthermore, the first cosmological results based on BAO measurements (DESI Collaboration 2024d,e, 2025a, 2024f, 2025b, 2024c) of the first year of main survey operations have also been published and provide unprecedented precision; the full data release is still in preparation (DESI Collaboration et al, in preparation). While the results are still compatible with the Λ-CDM model, they show, when combined with other probes, a preference for a w0wa-CDM cosmology (Chevallier & Polarski 2001; Linder 2003; de Putter & Linder 2008). Furthermore, there is a tension between the DESI and eBOSS BAO data around an effective redshift of 0.7.

While the first year DESI data (DESI Y1) only cover a fraction of its final footprint, the full DESI photometric footprint has already been covered by the Legacy Imaging Surveys (Dey et al. 2019; Schlegel et al. 2021; Schlegel et al., in preparation). Photometric surveys provide more observed galaxies compared to spectroscopic surveys, even at deeper redshifts (Euclid Collaboration: Adam et al. 2019), but the uncertainty on the redshift obtained from photometric surveys is larger than the uncertainty on the redshift obtained from spectroscopic surveys. These photometric redshifts are measured with a much poorer resolution, which causes unpredictable damping of clustering at small scales and a smearing of the BAO peak (Estrada et al. 2009). However, possible BAO signatures that the photometric redshift uncertainty has not washed out might still be present in the data.

Thanks to recent advancements, we improved upon the methodology and formulation that was applied in Sridhar & Song (2019) to simulated photometric galaxy catalogs to get cosmological distance constraints. Chan et al. (2022b) show that using the anisotropic correlation function expressed in terms of the projected distance significantly improves the consistency of the data between different wedges. For this study, this allowed us to make improvements not only in terms of data but also in terms of methods compared to Sridhar et al. (2020).

We first discuss the observational and simulated data used for our work in Sect. 2. We then introduce the methods used in Sect. 3 and highlight the little-discussed systematic shift of the BAOs for photometric redshift data in Sect. 4. We then present our results in Sect. 5 and discuss them in detail in Sect. 6. We provide a brief summary and conclusions in Sect. 7. Furthermore, we present some additional figures in the Appendix.

2. Observed and mock catalogs

The DESI Legacy Imaging Survey Data Release 9 (DR9 hereafter) data are introduced in this section along with the simulations that are used to support data analysis, such as the EZmocks and the AbacusSummit simulation data as well as some additional dark matter simulations specifically created to test our methods.

2.1. Photometric catalogs

The DR9 (Dey et al. 2019; Schlegel et al. 2021) provides imaging data for the entire DESI footprint and beyond by including additional areas used by the Dark Energy Survey (DES; The Dark Energy Survey Collaboration 2005). Because of the large sky coverage of DR9, it was observed using three different telescopes:

  • The Bok 2.3 m telescope on Kitt Peak was used for the Beijing-Arizona Sky Survey (BASS; Zou et al. 2017), which observed at the sky on the North Galactic Cap (NGC) for declinations ≥32.375° using optical bands g and r to a depth of 24.0 mag and 23.4 mag, respectively.

  • The 4-meter Mayall telescope, which is also on Kitt Peak, was used to conduct the Mayall z-band Legacy Survey (MzLS) for the same footprint as Bok but using the z band up to a depth of 22.5 mag.

  • For the majority of the observations the Blanco 4m telescope at the Cerro Tololo Inter-American Observatory was used. This is the same instrument that also carried out the observations for DES (The Dark Energy Survey Collaboration 2005), but extended the footprint to incorporate areas of the NGC at declinations ≤32.375° and also areas of the South Galactic Cap (SGC) at declinations ≤ 34°, to create the Dark Energy Camera Legacy Survey (DECaLS). The depths of these observations were 24.0 mag, 23.4 mag, and 22.5 mag in the g, r, and z bands, respectively.

In addition to the optical imaging, near-infrared-bands W1 and W2 were added using four years of Wide-Field Infrared Survey Explorer (WISE, Wright et al. 2010; Meisner et al. 2017) data, which contributed to determine more precise photometric redshifts and target selection. In addition to DR9, we used the photometric redshifts based on DR9 data provided by Zhou et al. (2021), which were obtained using a random forest-based methodology (Breiman 2001).

2.2. Simulations

We used several different simulations to test and calibrate our methods. AbacusSummit (Maksimova et al. 2021), a comprehensive cosmological N-body simulation, is designed to support large-scale surveys like DESI. Using the Abacus N-body code (Garrison et al. 2018, 2019, 2021) as outlined in Metchnik 2009, this suite consists of 150 box simulations across various cosmologies. For our study, we utilized 25 simulations based on their base cosmology, informed by the Planck Collaboration VI (2020) results. These simulations feature 69123 particles in boxes measuring 2 Gpc/h on each side. Specifically, we focused on the simulation run at a redshift of 0.8. Based on this run, mock catalogs implemented the halo occupation distribution (HOD) model (Yuan et al. 2021) for DESI luminous red galaxies (LRGs) and adapted to a cut-sky geometry, to closely mirror observational conditions.

To obtain covariance matrices for our BAO fits, we used 1000 EZmocks (Chuang et al. 2015). They are relatively accurate mock catalogs that can be generated quickly using the Zel’dovich approximation (Zel’dovich 1970). We used EZmocks that are similar to AbacusSummit N-body simulations.

For the initial tests of our method, we created a set of 100 simulations, called G2P1. We used Gadget2 by Springel (2005), which is a code for cosmological N-body simulations. It is fed by initial conditions generated based on Lagrangian perturbation theory up to second order using 2LPTic by Crocce et al. (2006), given an input power spectrum from CAMB by Lewis et al. (2000). For the initial conditions, Np = 10243 number of particles are populated at z = 49 in the simulations box of one side, 1.89 Gpc/h with Nmesh = 1024 for the number of grids in the fast Fourier Transform. 1.89Gpc/h is corresponding to the shell volume at z = 0.9 with the redshift bin size Δz = 0.2, where the DESI survey is expected to have the largest target number density in Table 3 of Levi et al. (2013). After implementing the initial condition, the particles evolve with a mesh size of 2048 for TreePM (Bagla 2002) and a softening length of 92.28 kpc, which is 5% of the mean particle displacement, in a periodic boundary condition. We created 100 realizations using a cosmology based on Planck Collaboration XIII (2016, Table 3) with the following parameters: ΩM = 0.3132, ΩΛ = 0.6868, Ωb = 0.049, and H0 = 67.31 (km/s)/Mpc.

To generate the halo catalog, we applied the group finder ROCKSTAR Behroozi et al. (2013) to each simulation run. We found all halos with at least ten dark matter particles. The halo position is evaluated by averaging the particle locations for the inner subgroup that best minimizes the Poisson error. The halo velocity is calculated by using the average particle velocity within the innermost 10% of the halo radius. To fill these dark matter only simulations with a realistic representation of galaxies, we applied a HOD (Berlind & Weinberg 2002) model sample of LRGs that will be observed by DESI. For this model, we used the HOD parameters from Zhou et al. (2021) for a photometric redshift range between 0.61 and 0.72 and applied them to our simulations using the HALOTOOLS implementation of the HOD model from Zheng et al. (2007).

3. Methods

3.1. Sample selection of the photometric data

We selected a subset of the DESI LRG sample. We followed the official target selection outlined in Zhou et al. (2023), which consists of a fiber magnitude cut, a series of color cuts (to remove stars and select red galaxies in the right redshift range), and several quality checks (photometric information in all bands and masking of foreground objects). In addition to these selection criteria, we also applied masks for objects near bright stars2. We decided to limit the range of sample to photometric redshifts (Zhou et al. 2021) between 0.6 and 0.8 and thereby obtain an effective redshift of 0.701.

In the next step, we divided the dataset into four subsamples of comparable size (see Table 1). The data from the BASS/MzLS with declinations greater than 32.375 deg, to avoid overlap, form the NGC North sample. As they were collected by different telescopes, these data have distinct photometric characteristics that required minor target selection adjustments (see Zhou et al. 2023) and have a different error budget for the photometric redshifts. We further divided the data collected from DECaLS into one sample on the NGC and two samples on the SGC. The NGC South sample consists of the DECaLS data on the NGC between a declination of 32.375 deg (to avoid overlap with NGC North) and −10 deg (to avoid a few disconnected patches farther south). We split the data for the SGC along a declination of −17 deg into the SGC North and SGC South areas. The SGC North roughly corresponds to the area of the SGC that will be covered by the spectroscopic DESI survey. We provide an overview map of these samples in Fig. 1.

thumbnail Fig. 1.

Sky map of our different footprint subsamples. The NGC North footprint corresponds to the area covered by the Bok and MzLS surveys, while all other footprints were observed as part of DECaLS. The split between SGC North and SGC South is motivated by the planned reach of DESI spectroscopy.

Table 1.

Comparison of the different footprint subsamples.

3.2. The treatment of the random catalogs

As we aimed to calculate correlation functions for these data, we needed suitable random catalogs. The DESI Legacy Imaging Survey provides 20 random catalogs with a density of 2500 points per square degree covering its entire footprint. Additionally, these random catalogs contain the same mask and photometric pass information as the data. Hence, we simply applied the same selection for the mask bits and pass information as we did for the observational data. To match our four samples of different regions, we applied the same geometric cuts to the randoms as well. Furthermore, we used the number counts of the real data to define the density of each of the randoms in (photometric) redshift space. As the uncertainties and densities are slightly different based on the telescopes used to collect the imaging data, we treated the BASS/MzLS (essentially NGC North) and DECaLS (all the others) areas separately. We used the observed densities, applied a simple Gaussian smoothing, and used cubic splines for interpolation between the bin centers (see Fig. 2). We used this smoothed distribution to populate them with the suitable redshift distribution and adjusted the density accordingly. In the last step, we combined the 20 random catalogs of each footprint into one big random catalog for each footprint.

thumbnail Fig. 2.

Number density of LRGs as a function of photometric redshifts. First (leftmost) panel: Distribution for BASS/MzLS with the redshift slice used in our analysis highlighted. Dotted black lines show the smoothed interpolation function used for populating the redshift distribution in the random catalogs. Second panel: Same but for DECaLS within the NGC. Third panel: Same but for DECaLS within the northern part of the SGC. Forth panel: Same but for DECaLS within the southern part of the SGC.

3.3. Sample selection in simulations

We focused on the cut-sky simulations that were generated from the z = 0.8 snapshot with a LRG HOD applied to them. They were created specifically for the DESI to validate its pipeline on mock catalogs. As the photometric footprint is further extended than the spectroscopic footprint of DESI, we had to create suitable masks for it in the EZmocks and AbacusSummit simulations. To this end, we used the randoms from the DESI Legacy Imaging survey to create a HEALPix map (Górski et al. 2005) with nside of 256 to determine which parts of the sky are covered by our samples. We selected all HEALPix cells that have at least one random point in them to be the mask of the photometric footprint. An additional advantage that this method has is that it also retains the biggest hole created by the various MASKBITS (typically large galaxies and bright stars) as well and not just the overall shape of the distribution. Additional geometric cuts to recreate our four distinct samples were carried out after the mask was applied to the cut-sky simulations. In addition to the projected distribution on the sky, we also wanted to recover the redshift-space distribution correctly, and therefore, we need to understand the scatter created by the large uncertainties of the photometric redshifts.

3.4. Photometric redshift painting

As the photometric redshifts are the dominant effect that changes the shape of the correlation function, we needed to reproduce the scatter created by them in the redshift distribution as well as reasonably possible in the simulations. We analyzed the photometric redshifts, zphoto, within our LRG sample and our selected redshift range and compared them to the spectroscopic redshifts zspec of the training sample used by Zhou et al. (2021) to calibrate them. The distribution of (zphoto − zspec)/(1 + zspec), which are shown, for both the BASS/MzLS and the DECaLS separately, in Fig. 3 can be used to apply the correct scatter to the redshifts in the simulations. The distribution roughly corresponds to a Gaussian with a width σ0 of about 0.02. However, as there are deviations in its Gaussian shape, we decided to use a smoothed interpolation of the measured distribution instead of a Gaussian interpolation to reproduce the scatter created by the uncertainty of the photometric redshift estimates. We drew the photometric scatter Δz for the interpolated distribution and add it to the redshift zrsd obtained from the simulations using

thumbnail Fig. 3.

Uncertainties of the photometric redshifts for the 0.6–0.8 redshift range. First (leftmost) panel: Distribution for BASS/MzLS. Dotted black lines show the smoothed interpolation function used for the photometric redshift painting. Second panel: Same but for DECaLS within the NGC. Third panel: Same but for DECaLS within the northern part of the SGC. Forth panel: Same but for DECaLS within the southern part of the SGC.

( 1 + z photo ) = ( 1 + Δ z ) · ( 1 + z rsd ) . $$ \begin{aligned} (1+z_{\text{ photo}})=(1+\Delta z)\cdot (1+z_{\text{ rsd}}). \end{aligned} $$(1)

The zrsd is already taken into account the redshift space distortion introduced by the peculiar motions of the galaxies. We applied this to all our datasets from the AbacusSummit cut-sky simulation and EZmocks. In the case of the box simulations, which we used only for complementary tests of our methods, we went for the simplified method using a simple Gaussian distribution with σ0 = 0.02 instead and applied it to the z-direction taking into account the periodic boundary conditions and scaling of the box given its cosmology. Based on the overall characteristics of the photometric redshifts (Zhou et al. 2021), we assumed the redshift evolution of the scatter of the photometric redshifts slice to be sufficiently small within our ultimate redshift.

3.5. Correlation function measurements

Two point correlation function ξ(r) (Totsuji & Kihara 1969; Davis & Peebles 1983) measures the excess probability of two galaxies separated by a cosmic distance r, relative to the Poisson expectation, which exhibits an anisotropic distortion along the line of sight (LOS) at redshift space. The two-point correlation function is typically estimated using the Landy & Szalay estimator (hereafter LS; Landy & Szalay 1993):

ξ LS = D D − 2 D R + R R RR , $$ \begin{aligned} \xi _{LS} = \frac{DD - 2 DR + RR}{RR} ,\end{aligned} $$(2)

where DD and RR are the normalized number of pairs in the data and randoms catalog, respectively within a given binning scheme. DR correspond to pairs between the data (D) and random catalog (R). Throughout this work, we used random catalogs of a density 20 times the data. We used the Planck 2015 cosmology for the redshift distance relation:

s 2 = s ⊥ 2 + s ‖ 2 μ = s ‖ / s . $$ \begin{aligned} s^2&=s_{\perp }^2+s_{\parallel }^2 \nonumber \\ \mu&=s_{\parallel }/s. \end{aligned} $$(3)

In this study we used the anisotropic correlation function, which is typically binned in (s, μ) coordinates, where s denotes the radius to the shell and μ denotes the observed cosine of the angle to the galaxy pair. The transverse (s⊥) and radial (s∥) separations are related to these coordinates by Eq. (3). While spectroscopic data with minimal scatter along the LOS enables a consistent recovery of BAO features in the ξ(s, μ) function, the increased scatter in photometric redshift measurements tends to wash out these features, especially along the LOS. As demonstrated in Sridhar & Song (2019) and Sridhar et al. (2020), sufficient information perpendicular to the LOS remains to discern the BAO peak. However, Chan et al. (2022b) show that in the case of data with large scatter, the correlation function between the various μ bins is more consistent in (s⊥, μ) coordinates than (s, μ), with s⊥ being the transverse separation (perpendicular to the LOS). As our own tests with simulated data confirm this, we consequently calculated all correlation functions that use photometric redshifts as functions of (s⊥, μ), while for data without large uncertainties (such as simulated spectroscopic data) we calculated the correlation function in (s, μ) coordinates.

Throughout this work, unless stated otherwise, we used the following binning scheme: in s or s⊥ we used 13 equal-sized bins of 7 Mpc/h width ranging between 42.5 and 133.5 Mpc/h, with bin centers at 46, 53,... 130 Mpc/h. We used five equal-size μ bins between 0 and 0.5 with bin centers at μ ¯ = 0.05 $ \bar{\mu}=0.05 $, 0.15,... 0.35; using too many μ bins would complicate the covariance matrix, and using too few μ bins would not allow us to separate the error propagation along the LOS clearly (Sabiu & Song 2016). The higher μ between 0.5 and 1 are not used as there is too little information retained in them. The large photo-z error destroys the radial BAO signal efficiently, while still retaining some angular BAO signal. Hence, with increasing μ, as we are probing more and more of the radial component, the pair counts decrease for the same separation. The calculations of the correlation functions were performed with the Python package corrfunc3 (Sinha & Garrison 2020; Sinha et al. 2019), as well as additional code, creates a wrapper around the ξ(s⊥, s∥)4 calculations of corrfunc to obtain ξ(s⊥, μ).

3.6. Covariance matrix

For the subsequent fits to the observational data (and simulations) suitable covariance matrices are required. To this end, we calculated the correlation functions for the 1000 EZmocks for each of the four subsamples as well as using the 100 G2P simulations for supplementary tests in Sect. 4. Consequently, we calculated the covariance matrix as

C ij ( ξ i , ξ j ) = ∑ n = 1 N mocks [ ξ n ( x i ) − ξ ¯ ( x i ) ] [ ξ n ( x j ) − ξ ¯ ( x j ) ] N mocks − 1 , $$ \begin{aligned} C^{ij}(\xi ^i,\xi ^j)=\frac{\sum ^{N_{\text{ mocks}}}_{n=1}[\xi ^n(\boldsymbol{x}_i)-\overline{\xi }(\boldsymbol{x}_i)][\xi ^n(\boldsymbol{x}_j)-\overline{\xi }(\boldsymbol{x}_j)]}{N_{\text{ mocks}}-1}, \end{aligned} $$(4)

where the total number of simulations is given by Nmocks. The ξn(xi) represents the value of the of the correlation function of ith bin of xi in the nth realization and ξ ¯ ( x i ) $ \overline{\xi}(\boldsymbol{x}_i) $ is the mean of all the realization. xi stands for each of the 65 combinations of the (s,μ) or (s⊥,μ) bins, depending on which type of data is used.

Additionally, we also accounted for the offset caused by the finite number of realization (Hartlap et al. 2007) as

C − 1 = N mocks − N bins − 2 N mocks − 1 C ̂ − 1 , $$ \begin{aligned} C^{-1}=\frac{N_{\text{ mocks}} - N_{\text{ bins}}-2}{N_{\text{ mocks}}-1}\ \hat{C}^{-1}, \end{aligned} $$(5)

where Nbins denotes the total number of i bins, which are 65 in our case. C ̂ − 1 $ \hat{C}^{-1} $ and C−1 are the inverted covariance matrix before and after the correction was applied.

Apart from the above correction factor, an additional correction to the inverse covariance matrix is proposed by Percival et al. (2014):

C − 1 = p cor C ̂ − 1 , p cor = 1 + B ( N bins − N para ) 1 + A + B ( N para + 1 ) , A = 2 ( N mocks − N bins − 1 ) ( N mocks − N bins − 4 ) , B = N mocks − N bins − 2 ( N mocks − N bins − 1 ) ( N mocks − N bins − 4 ) , $$ \begin{aligned} C^{-1}&=p_{\text{ cor}} \hat{C}^{-1} , \nonumber \\ p_{\text{ cor}}&= \sqrt{\frac{1+B (N_{\text{ bins}} - N_{\text{ para}})}{1+A+B (N_{\text{ para}} + 1)}},\\ A&=\frac{2}{(N_{\text{ mocks}}-N_{\text{ bins}}-1)(N_{\text{ mocks}}-N_{\text{ bins}}-4)},\nonumber \\ B&=\frac{N_{\text{ mocks}}-N_{\text{ bins}}-2}{(N_{\text{ mocks}}-N_{\text{ bins}}-1)(N_{\text{ mocks}}-N_{\text{ bins}}-4)}, \end{aligned} $$(6)

with Npara being the number of free parameters used in the fits.

3.7. Template fits

In a different approach to the previously used method (Sánchez et al. 2011, 2012; Veropalumbo et al. 2016; Sridhar & Song 2019; Sridhar et al. 2020) of fitting an empirical function to the data, we implemented a simplified template fitter that takes advantage of the whole correlation function (within our bin-range). This approach is similar to the one used in Xu et al. (2012) and Ross et al. (2017) but is based on simulations that take into account the photometric redshift effects. To create the templates, we used the mean of the 100 G2P simulations with the photometric redshift painting applied to them and calculated ξ(s⊥, μ) at a resolution of 1 Mpc/h in s⊥ over a much-extended range between 20 and 170 Mpc/h and for μ bins of the 0.1 width. From these simulations, we obtained interpolation functions for each μ bin, as shown in Fig. 4. Following Moon et al. (2023), but with only one broadband term,

ξ fit ( s ⊥ , μ ) = B · ξ template ( s ⊥ α ⊥ , μ ) + A 0 , $$ \begin{aligned} \xi _{\text{ fit}}(s_{\perp },\mu )= B \cdot \xi _{\text{ template}}(s_{\perp } \alpha _{\perp },\mu ) + A_{0} , \end{aligned} $$(7)

thumbnail Fig. 4.

Basic templates for different μ bins derived from simulations assuming a photometric redshift uncertainty of σ0. They show the two-point correlation functions in terms of s⊥ for different μ bins.

we allowed this function to be stretched by α⊥ and shifted using the nuisance parameters B and A0. α⊥ in this case, is relative to the cosmology used to generate the simulations for the templates. For our application, we used a prior range of 0.8 to 1.2 on α⊥ and limited B to positive values. All other nuisance parameters were left unconstrained.

3.8. Joint fits of different μ bins

When fitting the different μ bins of ξ(s⊥, μ), we noticed that that data are still affected by cosmic variance. Chan et al. (2022b) show that for the level of photometric redshift uncertainty expected for the DESI Legacy Imaging Survey, we should expect all BAO peaks of different μ bins to align, if calculated as a function of s⊥ and thereby yield the same α-parameter. Therefore, we jointly fit the different μ bins with α⊥ being the only common parameter. We adopted a standard likelihood, ℒ ∝ exp(−χ2/2) where the function χ2 is defined as

χ 2 = ∑ s ⊥ , s ⊥ ′ , μ , μ ′ Δ ξ ( s ⊥ , μ ) C − 1 ( s ⊥ ; s ⊥ ′ ; μ ; μ ′ ) Δ ξ ( s ⊥ ′ , μ ′ ) $$ \begin{aligned} \chi ^{2}=\sum _{s_{\perp },s_{\perp }^{\prime },\mu ,\mu ^{\prime }} \Delta \xi (s_{\perp },\mu ) C^{-1}({s_{\perp };s_{\perp }^{\prime };\mu ;\mu ^{\prime }})\Delta \xi (s_{\perp }^{\prime },\mu ^{\prime })\end{aligned} $$(8)

Δ ξ ( s ⊥ , μ ) = ξ mod ( s ⊥ , μ ) − ξ data ( s ⊥ , μ ) . $$ \begin{aligned} \Delta \xi (s_{\perp },\mu ) = \xi _{\text{ mod}}(s_{\perp },\mu )-\xi _{\text{ data}}(s_{\perp },\mu ). \nonumber \end{aligned} $$(9)

This means that instead of several independent five-parameter fits for each μ bin, we have one 1 + 2 ⋅ nμ parameter fit for nμ jointly fitted μ bins since we had used the same α⊥ parameters for the joint fits to all the μ bins.

4. BAO peak shift

One of the reasons why we moved to template-based fits instead of an empirical function, was that we discovered that the location of BAO peak for data using photometric redshifts with their typical uncertainty is systematically shifted compared to the location of the BAO peak obtained from data with spectroscopic accuracy. This is the case when measuring the anisotropic two-point correlation function of the photometric data in terms of ξ(s⊥, μ). While Chan et al. (2022b) have already shown that the location of the BAO peak is consistent between the different μ bins, if expressed in terms of ξ(s⊥, μ), they did not report that the absolute location of it is shifted relative to the location of the BAO peak found using the same data, but with only redshift-space distortions (RSDs) or typical spectroscopic uncertainty accounted for. We found this effect to be present for all data within a redshift uncertainty σ0 of more than ∼0.01 (at the redshifts of our simulations of 0.7). The shift is also consistent between different μ bins and between different simulations that contain enough data to get beyond the cosmic variance threshold.

To better understand this phenomenon, we created a simple toy model that lets us explore how the larger uncertainties of the photometric redshifts affect the BAO feature. We used a s⊥-s∥ diagram, with s∥ being the separation parallel to the LOS and s⊥ being the separation orthogonal to the LOS, as the main tool in our toy model. We then populated this diagram with only a ring-like over-density with a Gaussian profile centered around a separation of 100 (in arbitrary units) and a width of 5 (again in arbitrary units). When further applied a redshift uncertainty σ0 = 0.001 along the LOS, which approximately corresponds to the magnitude of redshift space distortions. In the resulting map, which is shown in Fig. 5, we mark the different wedge-like μ bins used for the anisotropic correlation function as straight lines spreading from the origin of the coordinate system. Additionally, we mark circular bins representing the binning scheme of ξ(s, μ) and straight bins orthogonal to the LOS representing the binning scheme of ξ(s⊥, μ). Furthermore, Fig. 5 also shows the number counts within the bins, representing a non-normalized correlation function, in plots on the side for both binning schemes (s, μ) and (s⊥, μ). For practical reasons, we only show the first 5 μ bins. We see that our toy model, which represents the BAOs detected using spectroscopic redshifts, yields consistent locations for the center of the BAO peak in the (s, μ) bins and that it is located where we set it, at 100. However, for the ξ(s⊥, μ) the peak location is inconsistent, as the BAO feature for spectroscopic redshifts does not follow the straight s⊥ bins.

thumbnail Fig. 5.

Toy model for illustrations showing the anisotropic correlation function for spectroscopic redshifts. Left panel: Illustration of the BAOs after redshift space distortions are applied. Blue lines separate the μ bins. Dark green lines separate the s bins. Light green lines separate the s⊥ bins. Top-right panel: Counts within the s, μ bins. Bottom-right panel: Counts within the s⊥, μ bins.

For Fig. 6 we created the same plots representing photometric redshifts with a redshift uncertainty of σ0 = 0.02. The location of the BAO peak becomes inconsistent between different μ bins in ξ(s, μ), while it stays stable for the ξ(s⊥, μ) binning scheme. However, a closer look reveals that the center of the peak has shifted to lower values by a few percent. This is because the scatter introduced by the large uncertainty of photometric redshifts mixed parts of the BAO signal originally belonging to different μ bins together. As the BAO appears as a circular structure that gets washed out along the LOS, the bulk of the remaining feature appears at slightly lower values perpendicular to the LOS.

thumbnail Fig. 6.

Toy model for illustrations showing the anisotropic correlation function for photometric redshifts. Same as Fig. 5 but for the BAOs after applying a photometric redshift uncertainty of σ0 = 0.02.

Because the toy model, albeit an excellent tool for illustrating the processes that shift the BAO peak, is a highly simplified approach that does not take the full picture of large-scale clustering into account, we needed to repeat the measurements of the BAO peak shift with data from proper numerical simulations. To this end, we carried out the basic tests using the G2P simulations produced by our group. We applied the photometric redshift painting (see Sect. 3.4) and calculated the anisotropic correlation function for both the photometric redshifts and spectroscopic redshifts. Afterward, we fit the empirical six-parameter model (see Sánchez et al. 2011, 2012; Sridhar et al. 2020) to the mean of the correlation functions of the G2P simulations. By comparing the peak locations, which is one of the free parameters, obtained by these fits, we find that the center of the BAO peak is systematically shifted to lower values for the photometric redshifts, as shown in Fig. 7. The shift is slightly larger than what we see in the toy model. We obtained the shift by taking the ratio between the photometric BAO peak locations of all μ bins and the spectroscopic BAO peak location of all μ bin. This is possible because the location of the peak is consistent in both datasets when using the respective binning schemes and it allows us to measure the value of the shift parameter with high precision. We find a shift parameter δsm of 0.93 ± 0.01 for the mean of the G2P simulations. We repeated the same method with the cut-sky AbacusSummit simulations and obtained values consistent with our simulations.

thumbnail Fig. 7.

Comparison of the BAO peak locations in spectroscopic (upper panel) and photometric (lower panel) redshift data. The vertical lines in various colors correspond to the location of the BAO peak found by fitting a six-parameter model to various μ bins. The vertical black line is the average and its error bar for all μ bins combined. There is a systematic shift that has to be accounted for by either a correction factor or dedicated templates for photometric redshifts.

This shift has to be accounted for in all methods that aim to measure cosmology using the location of the BAO peak. The empirical fits yield a notably different BAO scale for data based on photometric redshifts than for the same data based on spectroscopic redshifts. In the case of template fitting, it is also important to design the templates used specifically for photometric redshift data and not to use ones designed for spectroscopic data. Hence, we created special templates using simulations with photometric redshift uncertainty as shown in Sect. 3.7. While testing different levels of photometric redshift uncertainty, we found that the magnitude of the shift does not depend on the exact value of the photometric redshift uncertainty σ0, given that it is sufficiently large (σ0 > ∼0.01). This corresponds to the scale found by Chan et al. (2022b) at which the ξ(s⊥, μ) binning scheme starts to yield a consistent BAO peak location across all μ bins.

5. Results

5.1. Correlation function measurements

We analyzed the anisotropic correlation functions for the four subsamples independently and compared them to the predictions obtained from the 1000 EZmocks cut to the same geometry. In Fig. 8 we compare the observed correlation function to the expected one from the EZmocks for the first μ bin. For most subsamples, there is a reasonable agreement between the expected correlation function and the observed ones. Only the observations in SGC South strongly deviate from the expectations. Results are similar for the other μ bins, which are shown in Figs. A.1 to A.4.

thumbnail Fig. 8.

Correlation functions of the μ ¯ = 0.05 $ \bar{\mu}=0.05 $ bin for the four subsamples compared to the EZmocks correlation functions.

5.2. Template fits

We applied the template fits to the three subsamples (NGC North, NGC South, and SGC North) for which we obtained reasonable correlation function measurements. We excluded the fourth subsample (SGC South) from this part of our analysis because the shape of the correlation function is heavily distorted and we would not be able to trust the fits. The results of the fits for the three more reliable subsamples are provided in Table 2. The α⊥-parameter measurements were obtained using a joint fit across multiple μ bins (see Sect. 3.8) starting from zero to μmax as the upper limit of the range μ considered for the fit. Based on the reduced χ2, we determined that a μmax of 0.3 is the optimal choice to gain the most reliable results from our data.

Table 2.

Results of the template fits.

6. Discussion

6.1. Comparison of different footprints

As we divided the DR9 footprints into four different areas (see Fig. 1), we compared the peculiarities of the measured correlation functions within them intending to better understand the systematic differences between these footprints.

The data from NGC North shows a notably exaggerated BAO peak. This is surprising as it is the region with very marginally the lowest quality imaging data and the largest uncertainty on the photometric redshifts. However, aside from the strong BAO peak, the rest of the correlation function for this footprint aligns well with the expected shape from EZmocks and the AbacusSummit simulations. Such a sharp BAO peak is not seen in any of the other footprints and the fact the data skews toward a strong BAO in all μ bins indicates that it is more than just sample variance. The correlation function of NGC South pretty much aligns perfectly with one from the EZmocks. However, the BAO peak is washed out in some of the μ bins, most likely due to unfavorable cosmic variance. The shape of the correlation function in SGC North is slightly above the expected correlation function from the EZmocks, but the peak is recovered well. The data in the SGC South area is problematic. The upturn in the correlation function at larger separations is indicative of a mask issue, which becomes especially bad below a declination of −30°. This implies that the removal of stars and artifacts from foreground sources was not done well for this region, which also lies outside the planned spectroscopic footprint of DESI. We find a similar, albeit less extreme, behavior for the correlation functions in all footprints (see Fig. A.5 and compare it to Fig. 8), when the official LRG target selection (Zhou et al. 2023) was applied instead of our slightly modified version from Sect. 3.1. We do not think that this difference will seriously affect the spectroscopic DESI survey, as the spectroscopic redshift data will enable a better final cleaning of the sample than just the photometric redshifts used in our analysis.

6.2. Comparison of different μ bins

When comparing the correlation functions measured in the different μ bins for each sample (see Figs. 8 and A.1 to A.4), we found that the BAO feature becomes increasingly more difficult to detect with increasing μ, which is expected and consistent with previous works (Sridhar & Song 2019). Using the results presented in Table 2, we found that constraints on α⊥ start to become weaker after including the fourth and fifth μ bins. Hence, we recommend using the α⊥ values with a μmax of 0.3 and disregarding the values that were obtained using data from higher μ bins as well. We also found the same cutoff point in μmax when using the AbacusSummit cut-sky simulations. With this in mind, we took a closer look at the results of the first three μ bins for the different footprints. In the case of NGC North, the results for the α⊥ values stay consistent at 1.012 with error bars that are consistent with the fiducial cosmology within 1σ. The template fit yields values are compatible (within 1 to 2σ) with the fiducial cosmology for NGC South for all μ bins, albeit with larger uncertainties than the other footprints. For the first three μ bins the results for α⊥ for SGC North, we find a moderate tension with the fiducial cosmology of 2 to 3σ. Overall, we see that our results tend to be compatible with our fiducial cosmology used to create templates, the Planck 2015 cosmology (Planck Collaboration XIII 2016).

6.3. Comparison to other photometric and spectroscopic BAO measurements

The main advantages of photometric surveys and photometric redshifts are that they are an efficient way to cover a large area quickly, while spectroscopic redshift surveys can reach much higher levels of precision at the cost of significantly more observing time and the requirement of preexisting imaging surveys for targeting. However, the differences between BAO measurements obtained from spectroscopic redshifts and ones obtained from photometric redshifts are more than just data quality. There are various systematic biases, and properly addressing them requires the use of different methods. Chan et al. (2022b) showed that for redshift uncertainties that are typical for photometric redshifts from broadband imaging surveys, such as the DESI Legacy Imaging Survey DR9, the binning scheme used to obtain the anisotropic correlation function from spectroscopic survey data yields inconsistent BAO peak locations between the different μ bins. The suggested alternative binning strategy, calculating ξ(s⊥, μ) instead of the usual ξ(s, μ), yields consistent BAO peak locations, and we can hereby confirm the results of Chan et al. (2022b) in this regard. However, on top of this issue, we discovered that the BAO peak location is systematically shifted in the case of data with typical photometric redshift uncertainty when compared to data with spectroscopic redshift quality as discussed in great detail in Sect. 4. This shift needs to be considered when comparing observations to theoretical predictions or templates from simulations. Both discussed effects were not considered in previous works on DESI Legacy Imaging Survey data such as in Sridhar et al. (2020). The samples that we considered are listed in Table 3 and illustrated in Fig. 9. The DES Y3 and Y6 footprint (Sevilla-Noarbe et al. 2021; Mena-Fernández et al. 2024) is located in the SGC and is within our SGC North and SGC South footprints. Chan et al. (2022a) and Abbott et al. (2024) found, using a very similar method, results comparable to ours, but at a higher effective redshift. The combined BOSS low-z and CMASS samples cover a lower redshift range than our sample. However, the eBOSS LRG sample (Ross et al. 2020) has an effective redshift of 0.698 (Gil-Marín et al. 2020) almost the same as our samples. Although the eBOSS LRG sample extends into all three footprints for which we have measured the BAO scale, it is dominated and the most complete in the areas belonging to NGC North that contains ∼40% of the galaxies and to a lesser extent in SGC North (∼32% of the galaxies) and also NGC South (∼28% of the galaxies). In the case of the DESI Y1 data (DESI Collaboration 2024e), most of it (∼53%) can be found in NGC South, which the LRG sample within our redshift range shows a similar low value in our photometric data as it does for the spectroscopic Y1 data. Less of the DESI Y1 data are distributed in SGC North (∼31% ) and NGC North (∼17%). Therefore, our BAO measurements using photometric redshifts support the assumption that the apparent difference between final eBOSS and DESI Y1 BAO measurements is caused by cosmic variance due to the footprints covered by both surveys.

Table 3.

Comparison of various recent BAO measurements of the angular distance scale within the intermediate redshift range.

thumbnail Fig. 9.

Comparison of various recent BAO measurements of the angular distance scale within the intermediate redshift range. The dashed black line shows the fiducial cosmology of our paper, pink circles the BOSS data (Alam et al. 2017), the olive diamond the eBOSS data (Bautista et al. 2021), the gray plus-shaped point the DES Y6 data (Abbott et al. 2024), lime-green crosses the DESI Y1 data (DESI Collaboration 2024e), and squares the data presented in this paper for NGC North (red), NGC South (blue), and SGC North (dark green).

7. Summary and conclusions

We carried out measurements of the correlation functions for four distinct footprints within the DESI Legacy Imaging Survey DR9 and discussed the differences between them. To this end, we used the DESI LRG target selection with additional masks to better clean the sample, and we further limited our sample to a photometric redshift range between 0.6 and 0.8. While we find that SGC South suffers from serious masking issues, we managed to obtain BAO measurements in all other footprints. We also discussed the systematic difference between BAO measurements that rely on photometric redshifts with their relatively large uncertainties and high-quality spectroscopic redshifts. In addition to the known issues in measuring the correlation function and the necessary modifications, we also find that the location of BAO peaks is systematically shifted in the case of photometric redshifts. We considered that shift when constructing fitting templates from simulations.

The key results of this paper are that our most robust measurements (μmax = 0.3) provide a possible explanation for the apparent tension between the eBOSS LRG (Bautista et al. 2021) and DESI Y1 LRG samples (DESI Collaboration 2024e) within the same redshift range. As we have measurements for the comoving angular distance in the three distinct footprints that make up the survey area for the final DESI release and therefore also contain the eBOSS footprint, we can study the variance of the BAO measurements at the critical redshift range for the tension within these areas. The value we obtain in NGC South, which covers the area in which most of the spectroscopic DESI Y1 data can be found, yields a value very similar to that in DESI Y1, while the areas in which most of the eBOSS LRG sample is located yield values comparable to the eBOSS LRG values (see Fig. 9). Given the nature of measuring BAOs with photometric redshifts, we can only probe the comoving angular distance and not the Hubble rate. However, given the strong agreement of our data with eBOSS and DESI in their respective footprints, we argue that the apparent tension is mainly caused by sample variance across the sky. Therefore, we expect that for the final DESI release, which will cover the three footprints discussed in this paper, the BAO measurements will yield a value compatible with the average of all of them.

Data availability

All data points shown in the figures are available in a machine-readable form at https://doi.org/10.5281/zenodo.14733076, as is the appendix.

2

MASKBITS 8 and 11 according to the bit masks of the DESI Legacy Imaging Survey https://www.legacysurvey.org/dr9/bitmasks/

4

In the literature, such as the corrfunc documentation, s⊥ is called σ and s∥ is called π.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MIST), Grant No. RS-2021-NR058702. YZ acknowledges the support from the National Natural Science Foundation of China (NFSC) through grant 12203107, the Guangdong Basic and Applied Basic Research Foundation with No.2019A1515111098, and the science research grants from the China Manned Space Project with NO.CMS-CSST-2021-A02. Data analysis was performed using the high-performance computing cluster Seondeok at the Korea Astronomy and Space Science Institute. This material is based upon work supported by the U.S. Department of Energy (DOE), Office of Science, Office of High-Energy Physics, under Contract No. DE–AC02–05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Humanities, Science and Technology of Mexico (CONAHCYT); the Ministry of Science, Innovation and Universities of Spain (MICIU/AEI/10.13039/501100011033), and by the DESI Member Institutions: https://www.desi.lbl.gov/collaborating-institutions. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U. S. National Science Foundation, the U. S. Department of Energy, or any of the listed funding agencies. The 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 Near-Earth 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/. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U. S. National Science Foundation, the U. S. Department of Energy, or any of the listed funding agencies. The authors are honored to be permitted to conduct scientific research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation. This research made use of the following python packages: ASTROPY(Astropy Collaboration 2013, 2018, 2022), NUMPY (Harris et al. 2020), CORRFUNC (Sinha & Garrison 2020; Sinha et al. 2019), SCIPY (Virtanen et al. 2020), EMCEE (Foreman-Mackey et al. 2013), and MATPLOTLIB (Hunter 2007).

References

  1. Abbott, T. M. C., Adamow, M., Aguena, M., et al. 2024, Phys. Rev. D, 110, 063515 [NASA ADS] [CrossRef] [Google Scholar]
  2. Alam, S., Ata, M., Bailey, S., et al. 2017, MNRAS, 470, 2617 [Google Scholar]
  3. Astropy Collaboration (Robitaille, T. P., et al.) 2013, A&A, 558, A33 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  4. Astropy Collaboration (Price-Whelan, A. M., et al.) 2018, AJ, 156, 123 [Google Scholar]
  5. Astropy Collaboration (Price-Whelan, A. M., et al.) 2022, ApJ, 935, 167 [NASA ADS] [CrossRef] [Google Scholar]
  6. Bagla, J. S. 2002, JA&A, 23, 185 [NASA ADS] [Google Scholar]
  7. Bautista, J. E., Paviot, R., Vargas Magaña, M., et al. 2021, MNRAS, 500, 736 [Google Scholar]
  8. Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109 [NASA ADS] [CrossRef] [Google Scholar]
  9. Benedict, G. F., McArthur, B., Chappell, D. W., et al. 1999, AJ, 118, 1086 [NASA ADS] [CrossRef] [Google Scholar]
  10. Berlind, A. A., & Weinberg, D. H. 2002, ApJ, 575, 587 [Google Scholar]
  11. Breiman, L. 2001, Mach. Learn., 45, 5 [Google Scholar]
  12. Chan, K. C., Avila, S., Carnero Rosell, A., et al. 2022a, Phys. Rev. D, 106, 123502 [NASA ADS] [CrossRef] [Google Scholar]
  13. Chan, K. C., Ferrero, I., Avila, S., et al. 2022b, MNRAS, 511, 3965 [NASA ADS] [CrossRef] [Google Scholar]
  14. Chevallier, M., & Polarski, D. 2001, Int. J. Mod. Phys. D, 10, 213 [Google Scholar]
  15. Chuang, C.-H., Kitaura, F.-S., Prada, F., Zhao, C., & Yepes, G. 2015, MNRAS, 446, 2621 [NASA ADS] [CrossRef] [Google Scholar]
  16. Crocce, M., Pueblas, S., & Scoccimarro, R. 2006, MNRAS, 373, 369 [NASA ADS] [CrossRef] [Google Scholar]
  17. Davis, M., & Peebles, P. J. E. 1983, ApJ, 267, 465 [Google Scholar]
  18. de Mattia, A., Ruhlmann-Kleider, V., Raichoor, A., et al. 2021, MNRAS, 501, 5616 [NASA ADS] [Google Scholar]
  19. de Putter, R., & Linder, E. V. 2008, JCAP, 2008, 042 [CrossRef] [Google Scholar]
  20. DESI Collaboration (Aghamousa, A., et al.) 2016a, arXiv e-prints [arXiv:1611.00036] [Google Scholar]
  21. DESI Collaboration (Aghamousa, A., et al.) 2016b, arXiv e-prints [arXiv:1611.00037] [Google Scholar]
  22. DESI Collaboration (Abareshi, B., et al.) 2022, AJ, 164, 207 [NASA ADS] [CrossRef] [Google Scholar]
  23. DESI Collaboration (Adame, A. G., et al.) 2024a, AJ, 167, 62 [NASA ADS] [CrossRef] [Google Scholar]
  24. DESI Collaboration (Adame, A. G., et al.) 2024b, AJ, 168, 58 [NASA ADS] [CrossRef] [Google Scholar]
  25. DESI Collaboration (Adame, A. G., et al.) 2024c, JCAP, submitted [arXiv:2411.12022] [Google Scholar]
  26. DESI Collaboration (Adame, A. G., et al.) 2024d, JCAP, submitted [arXiv:2411.12020] [Google Scholar]
  27. DESI Collaboration (Adame, A. G., et al.) 2024e, JCAP, accepted [arXiv:2404.03000] [Google Scholar]
  28. DESI Collaboration (Adame, A. G., et al.) 2024f, JCAP, submitted [arXiv:2411.12021] [Google Scholar]
  29. DESI Collaboration (Adame, A. G., et al.) 2025a, JCAP, 2025, 021 [Google Scholar]
  30. DESI Collaboration (Adame, A. G., et al.) 2025b, JCAP, 2025, 124 [CrossRef] [Google Scholar]
  31. Dey, A., Schlegel, D. J., Lang, D., et al. 2019, AJ, 157, 168 [Google Scholar]
  32. du Mas des Bourboux, H., Rich, J., Font-Ribera, A., et al. 2020, ApJ, 901, 153 [CrossRef] [Google Scholar]
  33. Eisenstein, D. J., Hu, W., & Tegmark, M. 1998, ApJ, 504, L57 [NASA ADS] [CrossRef] [Google Scholar]
  34. Eisenstein, D. J., Zehavi, I., Hogg, D. W., et al. 2005, ApJ, 633, 560 [Google Scholar]
  35. Estrada, J., Sefusatti, E., & Frieman, J. A. 2009, ApJ, 692, 265 [NASA ADS] [CrossRef] [Google Scholar]
  36. Euclid Collaboration (Adam, R., et al.) 2019, A&A, 627, A23 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  37. Fernie, J. D. 1969, PASP, 81, 707 [NASA ADS] [CrossRef] [Google Scholar]
  38. Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306 [Google Scholar]
  39. Garrison, L. H., Eisenstein, D. J., Ferrer, D., et al. 2018, ApJS, 236, 43 [NASA ADS] [CrossRef] [Google Scholar]
  40. Garrison, L. H., Eisenstein, D. J., & Pinto, P. A. 2019, MNRAS, 485, 3370 [CrossRef] [Google Scholar]
  41. Garrison, L. H., Eisenstein, D. J., Ferrer, D., Maksimova, N. A., & Pinto, P. A. 2021, MNRAS, 508, 575 [NASA ADS] [CrossRef] [Google Scholar]
  42. Gil-Marín, H., Bautista, J. E., Paviot, R., et al. 2020, MNRAS, 498, 2492 [Google Scholar]
  43. Górski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759 [Google Scholar]
  44. Guy, J., Bailey, S., Kremin, A., et al. 2023, AJ, 165, 144 [NASA ADS] [CrossRef] [Google Scholar]
  45. Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357 [NASA ADS] [CrossRef] [Google Scholar]
  46. Hartlap, J., Simon, P., & Schneider, P. 2007, A&A, 464, 399 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  47. Hong, T., Han, J. L., Wen, Z. L., Sun, L., & Zhan, H. 2012, ApJ, 749, 81 [NASA ADS] [CrossRef] [Google Scholar]
  48. Hou, J., Sánchez, A. G., Ross, A. J., et al. 2021, MNRAS, 500, 1201 [Google Scholar]
  49. Hunter, J. D. 2007, CiSE, 9, 90 [Google Scholar]
  50. Landy, S. D., & Szalay, A. S. 1993, ApJ, 412, 64 [Google Scholar]
  51. Levi, M., Bebek, C., Beers, T., et al. 2013, arXiv e-prints [arXiv:1308.0847] [Google Scholar]
  52. Lewis, A., Challinor, A., & Lasenby, A. 2000, ApJ, 538, 473 [Google Scholar]
  53. Linder, E. V. 2003, Phys. Rev. Lett., 90, 091301 [Google Scholar]
  54. Maksimova, N. A., Garrison, L. H., Eisenstein, D. J., et al. 2021, MNRAS, 508, 4017 [NASA ADS] [CrossRef] [Google Scholar]
  55. Meisner, A. M., Lang, D., & Schlegel, D. J. 2017, AJ, 153, 38 [Google Scholar]
  56. Mena-Fernández, J., Rodríguez-Monroy, M., Avila, S., et al. 2024, Phys. Rev. D, 110, 063514 [CrossRef] [Google Scholar]
  57. Metchnik, M. V. L. 2009, Ph.D. Thesis, The University of Arizona, USA [Google Scholar]
  58. Miller, T. N., Doel, P., Gutierrez, G., et al. 2024, AJ, 168, 95 [NASA ADS] [CrossRef] [Google Scholar]
  59. Moon, J., Valcin, D., Rashkovetskyi, M., et al. 2023, MNRAS, 525, 5406 [NASA ADS] [CrossRef] [Google Scholar]
  60. Neveux, R., Burtin, E., de Mattia, A., et al. 2020, MNRAS, 499, 210 [NASA ADS] [CrossRef] [Google Scholar]
  61. Padmanabhan, N., Xu, X., Eisenstein, D. J., et al. 2012, MNRAS, 427, 2132 [Google Scholar]
  62. Peebles, P. J. E., & Yu, J. T. 1970, ApJ, 162, 815 [NASA ADS] [CrossRef] [Google Scholar]
  63. Percival, W. J., Ross, A. J., Sánchez, A. G., et al. 2014, MNRAS, 439, 2531 [NASA ADS] [CrossRef] [Google Scholar]
  64. Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, ApJ, 517, 565 [Google Scholar]
  65. Planck Collaboration VI. 2020, A&A, 641, A6 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  66. Planck Collaboration XIII. 2016, A&A, 594, A13 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  67. Raichoor, A., de Mattia, A., Ross, A. J., et al. 2021, MNRAS, 500, 3254 [Google Scholar]
  68. Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, AJ, 116, 1009 [Google Scholar]
  69. Ross, A. J., Samushia, L., Howlett, C., et al. 2015, MNRAS, 449, 835 [NASA ADS] [CrossRef] [Google Scholar]
  70. Ross, A. J., Beutler, F., Chuang, C.-H., et al. 2017, MNRAS, 464, 1168 [Google Scholar]
  71. Ross, A. J., Bautista, J., Tojeiro, R., et al. 2020, MNRAS, 498, 2354 [NASA ADS] [CrossRef] [Google Scholar]
  72. Sabiu, C. G., & Song, Y. S. 2016, arXiv e-prints [arXiv:1603.02389] [Google Scholar]
  73. Sánchez, E., Carnero, A., García-Bellido, J., et al. 2011, MNRAS, 411, 277 [CrossRef] [Google Scholar]
  74. Sánchez, A. G., Scóccola, C. G., Ross, A. J., et al. 2012, MNRAS, 425, 415 [CrossRef] [Google Scholar]
  75. Schlafly, E. F., Kirkby, D., Schlegel, D. J., et al. 2023, AJ, 166, 259 [NASA ADS] [CrossRef] [Google Scholar]
  76. Schlegel, D., Dey, A., Herrera, D., et al. 2021, Am. Astron. Soc. Meeting Abstr., 237, 23503 [Google Scholar]
  77. Sevilla-Noarbe, I., Bechtol, K., Carrasco Kind, M., et al. 2021, ApJS, 254, 24 [NASA ADS] [CrossRef] [Google Scholar]
  78. Silber, J. H., Fagrelius, P., Fanning, K., et al. 2023, AJ, 165, 9 [NASA ADS] [CrossRef] [Google Scholar]
  79. Sinha, M., & Garrison, L. 2019, in Software Challenges to Exascale Computing, eds. A. Majumdar, & R. Arora (Singapore: Springer Singapore), 3 [NASA ADS] [CrossRef] [Google Scholar]
  80. Sinha, M., & Garrison, L. H. 2020, MNRAS, 491, 3022 [Google Scholar]
  81. Springel, V. 2005, MNRAS, 364, 1105 [Google Scholar]
  82. Sridhar, S., & Song, Y.-S. 2019, MNRAS, 488, 295 [NASA ADS] [CrossRef] [Google Scholar]
  83. Sridhar, S., Song, Y.-S., Ross, A. J., et al. 2020, ApJ, 904, 69 [NASA ADS] [CrossRef] [Google Scholar]
  84. The Dark Energy Survey Collaboration 2005, arXiv e-prints [arXiv:astro-ph/0510346] [Google Scholar]
  85. Totsuji, H., & Kihara, T. 1969, PASJ, 21, 221 [NASA ADS] [Google Scholar]
  86. Veropalumbo, A., Marulli, F., Moscardini, L., Moresco, M., & Cimatti, A. 2014, MNRAS, 442, 3275 [NASA ADS] [CrossRef] [Google Scholar]
  87. Veropalumbo, A., Marulli, F., Moscardini, L., Moresco, M., & Cimatti, A. 2016, MNRAS, 458, 1909 [NASA ADS] [CrossRef] [Google Scholar]
  88. Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nat. Methods, 17, 261 [Google Scholar]
  89. Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868 [Google Scholar]
  90. Xu, X., Padmanabhan, N., Eisenstein, D. J., Mehta, K. T., & Cuesta, A. J. 2012, MNRAS, 427, 2146 [NASA ADS] [CrossRef] [Google Scholar]
  91. Yuan, S., Garrison, L. H., Hadzhiyska, B., Bose, S., & Eisenstein, D. J. 2021, AbacusHOD: A highly efficient extended multi-tracer HOD framework and its application to BOSS and eBOSS data [Google Scholar]
  92. Zel’dovich, Y. B. 1970, A&A, 5, 84 [NASA ADS] [Google Scholar]
  93. Zheng, Z., Coil, A. L., & Zehavi, I. 2007, ApJ, 667, 760 [Google Scholar]
  94. Zhou, R., Newman, J. A., Mao, Y.-Y., et al. 2021, MNRAS, 501, 3309 [NASA ADS] [CrossRef] [Google Scholar]
  95. Zhou, R., Dey, B., Newman, J. A., et al. 2023, AJ, 165, 58 [NASA ADS] [CrossRef] [Google Scholar]
  96. Zou, H., Zhou, X., Fan, X., et al. 2017, PASP, 129, 064101 [CrossRef] [Google Scholar]

All Tables

Table 1.

Comparison of the different footprint subsamples.

In the text
Table 2.

Results of the template fits.

In the text
Table 3.

Comparison of various recent BAO measurements of the angular distance scale within the intermediate redshift range.

In the text

All Figures

thumbnail Fig. 1.

Sky map of our different footprint subsamples. The NGC North footprint corresponds to the area covered by the Bok and MzLS surveys, while all other footprints were observed as part of DECaLS. The split between SGC North and SGC South is motivated by the planned reach of DESI spectroscopy.
In the text
thumbnail Fig. 2.

Number density of LRGs as a function of photometric redshifts. First (leftmost) panel: Distribution for BASS/MzLS with the redshift slice used in our analysis highlighted. Dotted black lines show the smoothed interpolation function used for populating the redshift distribution in the random catalogs. Second panel: Same but for DECaLS within the NGC. Third panel: Same but for DECaLS within the northern part of the SGC. Forth panel: Same but for DECaLS within the southern part of the SGC.
In the text
thumbnail Fig. 3.

Uncertainties of the photometric redshifts for the 0.6–0.8 redshift range. First (leftmost) panel: Distribution for BASS/MzLS. Dotted black lines show the smoothed interpolation function used for the photometric redshift painting. Second panel: Same but for DECaLS within the NGC. Third panel: Same but for DECaLS within the northern part of the SGC. Forth panel: Same but for DECaLS within the southern part of the SGC.
In the text
thumbnail Fig. 4.

Basic templates for different μ bins derived from simulations assuming a photometric redshift uncertainty of σ0. They show the two-point correlation functions in terms of s⊥ for different μ bins.
In the text
thumbnail Fig. 5.

Toy model for illustrations showing the anisotropic correlation function for spectroscopic redshifts. Left panel: Illustration of the BAOs after redshift space distortions are applied. Blue lines separate the μ bins. Dark green lines separate the s bins. Light green lines separate the s⊥ bins. Top-right panel: Counts within the s, μ bins. Bottom-right panel: Counts within the s⊥, μ bins.
In the text
thumbnail Fig. 6.

Toy model for illustrations showing the anisotropic correlation function for photometric redshifts. Same as Fig. 5 but for the BAOs after applying a photometric redshift uncertainty of σ0 = 0.02.
In the text
thumbnail Fig. 7.

Comparison of the BAO peak locations in spectroscopic (upper panel) and photometric (lower panel) redshift data. The vertical lines in various colors correspond to the location of the BAO peak found by fitting a six-parameter model to various μ bins. The vertical black line is the average and its error bar for all μ bins combined. There is a systematic shift that has to be accounted for by either a correction factor or dedicated templates for photometric redshifts.
In the text
thumbnail Fig. 8.

Correlation functions of the μ ¯ = 0.05 $ \bar{\mu}=0.05 $ bin for the four subsamples compared to the EZmocks correlation functions.
In the text
thumbnail Fig. 9.

Comparison of various recent BAO measurements of the angular distance scale within the intermediate redshift range. The dashed black line shows the fiducial cosmology of our paper, pink circles the BOSS data (Alam et al. 2017), the olive diamond the eBOSS data (Bautista et al. 2021), the gray plus-shaped point the DES Y6 data (Abbott et al. 2024), lime-green crosses the DESI Y1 data (DESI Collaboration 2024e), and squares the data presented in this paper for NGC North (red), NGC South (blue), and SGC North (dark green).
In the text

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