Clustering of LRGs in the DECaLS DR8 Footprint: Distance Constraints from Baryon Acoustic Oscillations Using Photometric Redshifts

The DESI Legacy Imaging Surveys will provide the target catalog for the upcoming DESI survey. One among the three imaging projects conducted for the Legacy Survey is DECaLS (Dey et al. 2019), which covers the South Galactic Cap region at DEC $\leqslant$ 34°. The data makes use of three optical bands (g, r, and z) to a depth of at least g = 24.0, r = 23.4, and z = 22.5, which is 1–2 magnitudes deeper than SDSS. We use the DECaLS data from the Legacy Surveys eighth data release (DR8), which is the first release to include images and catalogs from all three of the Legacy Surveys in a single release. The Legacy Surveys also processed some of the imaging data from the Dark Energy Survey (DES; Dark Energy Survey Collaboration 2005), and we include the DES imaging with DEC ≥  −30° in our analysis. In addition to the optical imaging, 4 yr of Wide-Field Infrared Survey Explorer (WISE; Wright et al. 2010; Meisner et al. 2017) data in the W1 and W2 bands are also included, which provide additional color information.

For the parent luminous red galaxy (LRG) sample in this study, we use a nonstellar cut of $(z-{\rm{W}}1)-0.8* (r-z)\,\gt -0.6$, a faint limit of z < 20.41, a color cut of $0.75\,\lt (r-z)\lt 2.45$, and a sliding magnitude–color cut of $(z-17.18)/2\lt (r-z)$. These selection cuts are motivated by the current DESI LRG target selection cuts (DESI Collaboration et al. 2016).

We also apply masks to get the final footprint for our parent sample using the "MASKBITS" column in the DR8 catalog. ¹⁰ Objects (and randoms) with the following bits are removed: 1 (Tycho-2 and GAIA bright stars), 8 (WISE W1 bright stars), 9 (WISE W2 bright stars), 11 (fainter GAIA stars), 12 (large galaxies), and 13 (globular clusters). Imaging data sets often suffer from systematic effects, and one such major contribution toward the systematic contamination comes from correlation with stellar density (Rezaie et al. 2020). A more detailed test of this effect on the large-scale structure correlation is explained in Appendix A.

After applying the magnitude cuts and masking scheme, we use random-forest-based (Breiman 2001) photo-z's from Zhou et al. (2020) to obtain the final photometric redshifts. The dispersion on the redshift is usually approximated by

$$\begin{eqnarray}&&{\sigma }_{z}\equiv {\sigma }_{0}\times (1+{z}_{\mathrm{true}}),\end{eqnarray} \tag{ 1 }$$

where σ₀ denotes the dispersion at redshift z = 0, and z_true is the true redshift or the spectroscopic redshift. The mean redshift uncertainty for the DECaLS DR8 sample within the range $0.3\lt {z}_{\mathrm{phot}}\lt 1.2$ is σ₀ = 0.0264. The angular distribution of the parent LRG sample after applying the masking and selection cuts is plotted in the left panel of Figure 1, with higher-density regions denoted by a darker shade. The redshift distribution of the parent sample within the range $0.3\lt {z}_{\mathrm{phot}}\lt 1.2$ is plotted as a blue filled histogram in the right panel of Figure 1. For comparison, we also overplot the forecasted dN/dz LRG redshift distribution from DESI for a sky coverage of 9000 deg² (given by the green solid line) and 14,000 deg² (given by the orange solid line). The DECaLS DR8 sample has a sky coverage of ≈9500 deg², which is more than two-thirds of the 14,000 deg² DESI footprint.

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In order to test and validate the results we obtain, we also need to analyze the clustering from a realistic mock catalog based on numerical simulations that calculate the nonlinear evolution of structure and predict the dependence of survey observables on cosmological parameters. We use the publicly available Dark Sky simulation set as described in Skillman et al. (2014) for this purpose. The simulation has been generated using particle numbers varying from 2048³ to 10,240³ and in a comoving cosmological volume varying from 100 h⁻¹ Mpc to 8 h⁻¹ Gpc on a side. The objects have been placed in the simulation using a simple (time-evolving) halo occupation distribution (HOD) assuming spherical, Navarro–Frenk–White (Navarro et al. 1996) halos for the satellite. To identify dark matter halos and substructures, the ROCKSTAR halo finder (Behroozi et al. 2013) has been used. The halo-finding approach is based on an adaptive hierarchical refinement of friends-of-friends groups in both position and velocity. From the set of simulations (with varying particle numbers), we make use of the ds14_a simulation, which has 10,240³ particles and a particle mass of $3.9\times {10}^{10}{h}^{-1}{M}_{\odot }$. These specific mocks contain only R.A., decl., and true redshift information for objects preidentified to be LRGs and do not contain color or luminosity information.

To validate our results on photometric redshifts, we generate a set of photo-z's using the true redshift information available. In reality, the statistical nature of the photo-z error is too complicated to be specified with any known distribution function, but it is assumed that the error propagation of photo-z uncertainty into cosmological information is mainly caused by the dispersion length (Arnalte-Mur et al. 2009). Thus a simple Gaussian function of statistical distribution is usually chosen to generate the photo-z uncertainty distribution, and the photo-z error dispersion σ_z as defined in Equation (1) (a more detailed description is given in Section 3.4) is applied. In reality, the precision is dependent on many factors such as magnitude and spectral type, but here only the redshift factor is counted in Equation (1), in which the coherent statistical property determined only by z is applied for all types of galaxies in the simulation.

In this paper, rather than assuming a true Gaussian distribution for the photo-z errors, we use the spectroscopic redshifts of galaxies from our training set along with their photo-z's and σ_z 's to get the $({z}_{\mathrm{phot}}-{z}_{\mathrm{spec}})/{\sigma }_{z}$ distribution. We use this distribution to create our photometric redshifts, which we believe is a more accurate representation than using a true Gaussian distribution. The detailed description of generating the photo-z's for our simulations is given in Section 2.3, and the comparison of a true Gaussian distribution and the $({z}_{\mathrm{phot}}-{z}_{\mathrm{spec}})/{\sigma }_{z}$ distribution is given in Appendix B and plotted in Figure A2.

Table 1 summarizes the number density information from both the DECaLS data and the Dark Sky data for the entire redshift range and for the two redshift cuts used in this paper. It can be seen that the number density in all three redshift ranges for the mock is smaller than in the DECaLS sample. Thus, the comparison in the clustering between the two samples should only be looked into as a consistency check rather than a precise validation of our results.

To compute the error on ${\xi }_{w}(s,{\mu }_{i})$, we make use of 100 EZmock (Chuang et al. 2015) simulations, all of which have the DESI expected sky coverage. To match the DECaLS DR8 footprint, we cut the EZmock samples within $-30^\circ \lt \mathrm{DEC}\lt$ 34°. The EZmock sample after the DEC cut has an area of ≈9300 deg², which is similar to the area of the DECaLS DR8 parent sample that we use in this paper. The mocks contain R.A., decl., z_cosmo, and dz_rsd information. Thus, we need to generate photometric redshifts for the mocks so that they can be used to obtain the covariance matrix. The methods of generating photometric redshifts for the EZmock samples are explained in detail in Section 2.3.

To generate the photo-z's for our simulations, we obtain ${\sigma }_{z}$'s randomly from the parent DECaLS sample, along with random points from the $({z}_{\mathrm{phot}}-{z}_{\mathrm{spec}})/{\sigma }_{z}$ distribution. To select the random σ_z 's, we restrict them to galaxies of similar redshifts that are within a redshift range of ±0.01 z_phot, which ensures that the dependence of errors on redshift is included. For example, for N number of EZmock galaxies that are within $0.60\lt {z}_{\mathrm{cosmo}}\lt 0.61$, N σ_z 's from the DECaLS data within 0.60 < z_phot < 0.61 are randomly selected. This process is repeated over the entire redshift range. Once we generate the photo-z's, they are diluted according to the DECaLS N(z) to make sure that they are consistent. The number of galaxies, volume within the redshift range, and the σ₀ for the two redshift cuts are listed in Table 1. Our detailed analysis comparing ${\xi }_{w}(s,\mu )$ between the EZmock sample and the DECaLS sample is explained in Appendix B and shown in Figure A1.

The excess probability of finding two objects relative to a Poisson distribution at volumes dV₁ and dV₂ separated by a vector distance  r is given by the two-point correlation function ξ(r) (Totsuji & Kihara 1969; Davis & Peebles 1983). The galaxy distribution seen in redshift space exhibits an anisotropic feature distorting ξ(r) into $\xi (\sigma ,\pi )$ along the line of sight (LOS), where σ and π denote the transverse and radial components of the separation vector  r . Acoustic fluctuations of the baryon–radiation plasma of the primordial universe leave a signature on the density perturbation of baryons. This standard ruler length scale, set by the acoustic wave, propagates until it is frozen at the decoupling epoch to remain in the large-scale structure of the universe. The threshold length scale of the acoustic wave is called the sound horizon, which is given by

$$\begin{eqnarray}&&{r}_{d}={\displaystyle \int }_{{z}_{\mathrm{drag}}}^{\infty }\displaystyle \frac{{c}_{s}(z)}{H(z)}{dz}\end{eqnarray} \tag{ 2 }$$

where c_s is the sound speed of the plasma. This scale is imprinted on the correlation function as a peak and is imprinted on the matter power spectrum as a series of waves. Assuming standard matter and radiation content in the universe, the Planck Collaboration et al. (2020) measurements of the matter and baryon density determine the sound horizon to 0.2%. By measuring the BAO feature using an anisotropic analysis, one can separately measure D_A (z) and ${H}^{-1}(z)$. But adjustments to the cosmological parameters or changes to the prerecombination energy density can alter the value of r_d (Alam et al. 2017). So, the BAO measurements constrain the combinations ${D}_{A}(z)/{r}_{d}$ and ${H}^{-1}(z){r}_{d}$. The sound horizon for this fiducial model is ${r}_{d,\mathrm{fid}}=147.21$ Mpc as obtained from Planck Collaboration et al. (2020). The scalings of r_d with cosmological parameters can be found in detail in Aubourg et al. (2015). The distance constraints quoted in this paper are in units of megaparsecs and with a scaling factor, for example, ${D}_{A}(z)\times ({r}_{d,\mathrm{fid}}/{r}_{d})$, so that the numbers provided are independent of the fiducial cosmological parameters used.

The Landy & Szalay estimator (hereafter LS) in (s, μ) coordinates is best suited to calculate the two-point correlation function (Farrow et al. 2015; Sridhar et al. 2017) and extract BAO information from photometric redshift galaxy maps (Sridhar & Song 2019). The radius to shell s and the observed cosine of the angle the galaxy pair makes with respect to the LOS μ are given by ${s}^{2}={\sigma }^{2}+{\pi }^{2}$ and μ = π/s, respectively, where σ and π denote the transverse and radial separation between the galaxy pairs.

It is common practice to separate the random sample distributions into the angular and redshift components separately. We make use of the random catalog provided in the DR8 data release ¹¹ by the Legacy Survey, which gives us the angular component. These randoms have been downsampled to the surface density of 10,000/deg² and require +2 exposures in the g, r, and z bands. The number of objects is usually twice or more than the data catalog to avoid shot noise effects. In our case, the random catalog has five times more objects than the data catalog. For the redshift component, we extract redshifts randomly from the data catalog within the chosen redshift range (see Ross et al. 2012, 2017; Veropalumbo et al. 2016; Sridhar & Song 2019, for more info). The same number of exposure requirements, footprint cuts, and bright star masks are applied on the randoms as used in constructing the LRG sample. We use the publicly available KSTAT (KD-tree Statistics Package) code (Sabiu 2018) to calculate all of our correlation functions.

We pay attention to the usefulness of exploiting the wedge correlation function to separate the radial contamination from the BAO signal imprinted on perpendicular configuration pairs. The wedge correlation function ξ_w is given by

$$\begin{eqnarray}&&{\xi }_{w}(s,{\mu }_{i})={\displaystyle \int }_{{\mu }_{i}^{\min }}^{{\mu }_{i}^{\max }}d\mu ^{\prime} W(\mu ^{\prime} :{\mu }_{\mathrm{cut}}={\mu }_{i}^{\max })\xi (s,\mu ^{\prime} ),\end{eqnarray} \tag{ 3 }$$

where μ_i is the mean μ in each bin (we will refer to the mean value of the μ bin using $\bar{\mu }$ hereafter), ${\mu }_{i}^{\min }$ and ${\mu }_{i}^{\max }$ are the minimum and maximum values of μ, and W is a window function within the chosen minimum and maximum limits of μ. Using too many μ bins will complicate the covariance matrix, and by using very few μ bins we will not be able to separate the error propagation along the LOS clearly (Sabiu & Song 2016). Thus, we choose six bins in the μ direction with Δμ = 0.17 between μ = 0 and 1, with $\mu \to$ 0 corresponding to the transverse plane and $\mu \to$ 1 corresponding to the LOS plane.

In the case of photometric redshift samples, the noise on the pairs increases along the radial configuration and thus causes a smearing of the BAO peak (Estrada et al. 2009). This smearing not only increases with increasing photometric uncertainty but also increases along the LOS for a given σ₀. These noisy pairs can be removed using a cutoff $\bar{\mu }$. It has been shown in Sridhar & Song (2019) that using a cutoff $\bar{\mu }=0.42$ for photometric redshift samples can remove most of the contaminated pairs, so we use cutoff $\bar{\mu }=0.42$ in this paper. The empirical model that we use to fit the correlation function and obtain the BAO peak location is similar to the one proposed by Sánchez et al. (2011, 2012) and is given by

$$\begin{eqnarray}&&{\xi }_{\mathrm{mod}}(s)=B+{\left(\displaystyle \frac{s}{{s}_{0}}\right)}^{-\gamma }+\displaystyle \frac{N}{\sqrt{2\pi {\sigma }^{2}}}\exp \left(-\displaystyle \frac{{\left(s-{s}_{m}\right)}^{2}}{2{\sigma }^{2}}\right),\end{eqnarray} \tag{ 4 }$$

where B takes into account a possible negative correlation at very large scales, s₀ is the correlation length (the scale at which the correlation function ≃1), and γ denotes the slope. The remaining three parameters, N, σ, and s_m , are the parameters of the Gaussian function that model the BAO feature, and, in particular, s_m represents the estimate of the BAO peak position. This empirical model can be used to accurately extract the BAO peak position (Sánchez et al. 2011; Veropalumbo et al. 2016; Sridhar & Song 2019) when the correlation function is provided.

The likelihood on s_m from previous BAO studies is either obtained by using a 1D grid on s_m where the χ² is minimized at each grid point or from the marginalized posterior from a Monte Carlo Markov Chain (MCMC) analysis. In this study, the fitting is performed by applying the MCMC technique (we make use of the emcee Python package, Foreman-Mackey et al. 2013), using the full covariance matrix obtained using Equation (17). The fitting parameter space is given by

$$\begin{eqnarray}&&{x}_{p}=(B,{s}_{0},\gamma ,N,{s}_{m},\sigma ),\end{eqnarray} \tag{ 5 }$$

and we place flat, wide priors on all six parameters. This five-parameter space is used for all of our μ bins when we perform the fitting. For the first three parameters, the ranges of the priors are $0.0\lt B\lt 1.0$, 0.0 < s₀ < 3.0, and 0.0 < γ < 3.0, and for the remaining three parameters of the Gaussian function, the ranges of the priors are 0.0 < N < 1.0, 85.0 < s_m  < 130.0, and $0.0\lt \sigma \lt 35.0$. Several variations of the range of these priors were tested, especially the range of the prior on s_m and σ. A smaller range for the s_m affects the posterior distribution, and we miss most of the information at high $s({h}^{-1}\,\mathrm{Mpc})$. A similar effect is seen when we use a smaller range for the σ prior, which eventually amplifies the BAO peak. We have also tested several ranges within which to perform the fit, and we fit the correlation function within the range $30.0\lt s({h}^{-1}\,\mathrm{Mpc})\,\lt 130.0$ after experimenting with other ranges. We find that there is a maximum shift of 1% in the BAO peak when we vary the range of the fit. This 1% shift is negligible compared to the error on the BAO peak point we obtain (as discussed in Section 3.4), and thus we believe it is subdominant. The constraints on the BAO peak s_m for the wedge correlation function are obtained after fully marginalizing all other parameters in x_p . We adopt a standard likelihood, ${\mathscr{L}}\,\propto \exp (-{\chi }^{2}/2)$, where the function χ² is defined as

$$\begin{eqnarray}&&\begin{array}{l}{\chi }_{{\mu }_{i}}^{2}({x}_{p})=\displaystyle \sum _{s,s^{\prime} }({\xi }_{\mathrm{mod}}(s)-{\xi }_{w}(s,{\mu }_{i})){C}_{s;s^{\prime} }^{-1}({\mu }_{i})\\ ({\xi }_{\mathrm{mod}}(s^{\prime} )-{\xi }_{w}(s,{\mu }_{i})),\end{array}\end{eqnarray} \tag{ 6 }$$

where ${\xi }_{\mathrm{mod}}(s)$ is the model correlation function as given by Equation (4), ${\xi }_{w}(s,{\mu }_{i})$ is the observed correlation function for the ith $\bar{\mu }$ bin, and ${C}^{-1}$ is the inverse covariance matrix.

We compute the theoretical correlation function ξ_th(s, μ) in redshift space, exploiting the improved power spectrum based upon the Taruya, Nishimichi, and Saito (TNS) model as

$$\begin{eqnarray}\begin{array}{rcl}{\xi }_{\mathrm{th}}(s,\mu ) & = & \displaystyle \int \displaystyle \frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}\tilde{P}(k,\mu ^{\prime} ){e}^{i{\boldsymbol{k}}\cdot {\boldsymbol{s}}}\\ & = & \displaystyle \sum _{{\ell }:\mathrm{even}}{\xi }_{{\ell }}(s){{ \mathcal P }}_{{\ell }}(\mu ),\end{array}\end{eqnarray} \tag{ 7 }$$

with ${ \mathcal P }$ being the Legendre polynomials. Here, we define $\mu =\pi /s$ and $s={\left({\sigma }^{2}+{\pi }^{2}\right)}^{1/2}$. The moments of the correlation function, ξ_ℓ (s), are defined by

$$\begin{eqnarray}&&{\xi }_{{\ell }}(s)={i}^{{\ell }}\int \displaystyle \frac{{k}^{2}{dk}}{2{\pi }^{2}}\,{\tilde{P}}_{{\ell }}(k){j}_{{\ell }}({ks}).\end{eqnarray} \tag{ 8 }$$

The multipole power spectra ${\tilde{P}}_{{\ell }}(k)$ are explicitly given by

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{P}}_{0}(k) & = & {p}_{0}(k),\\ {\tilde{P}}_{2}(k) & = & \displaystyle \frac{5}{2}\left[3{p}_{1}(k)-{p}_{0}(k)\right],\\ {\tilde{P}}_{4}(k) & = & \displaystyle \frac{9}{8}\left[35{p}_{2}(k)-30{p}_{1}(k)+3{p}_{0}(k)\right],\end{array}\end{eqnarray} \tag{ 9 }$$

where we define the function p_m (k):

$$\begin{eqnarray}&&{p}_{m}(k)=\displaystyle \frac{1}{2}\displaystyle \sum _{n=0}^{4}\displaystyle \frac{\gamma (m+n+1/2,\kappa )}{{\kappa }^{m+n+1/2}}\,{Q}_{2n}(k)\end{eqnarray} \tag{ 10 }$$

with $\kappa ={k}^{2}{\sigma }_{p}^{2}$. The function γ is the incomplete gamma function of the first kind:

$$\begin{eqnarray}&&\gamma (n,\kappa )={\int }_{0}^{\kappa }{dt}\,{t}^{n-1}\,{e}^{-t}.\end{eqnarray} \tag{ 11 }$$

The Q_2n is explained below.

The observed power spectrum in redshift space $\tilde{P}(k,\mu )$ is written in the following form:

$$\begin{eqnarray}&&\tilde{P}(k,\mu )=\displaystyle \sum _{n=0}^{8}\,{Q}_{2n}(k){\mu }^{2n}\,{G}^{\mathrm{FoG}}(k\mu {\sigma }_{p}),\end{eqnarray} \tag{ 12 }$$

where the velocity dispersion σ_p is set to be a free parameter for the Finger of God (FoG) effect, and the function Q_2n is given by

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{0}(k) & = & {P}_{\delta \delta }(k),\\ {Q}_{2}(k) & = & 2{P}_{\delta {\rm{\Theta }}}(k)+{C}_{2}(k),\\ {Q}_{4}(k) & = & {P}_{{\rm{\Theta }}{\rm{\Theta }}}(k)+{C}_{4}(k),\end{array}\end{eqnarray} \tag{ 13 }$$

where C_n includes the higher-order polynomials caused by the correlation between density and velocity fluctuations, and P_XY (k) denotes the power spectrum in real space. The standard perturbation model exhibits the ill-behaved expansion leading to the bad UV behavior. In this manuscript, we use the resummed perturbation theory RegPT, which is regularized by introducing a UV cutoff (Taruya et al. 2012). The auto and cross spectra of P_XY (k) are computed up to first order, and higher-order polynomials are computed up to zeroth order, which are consistent in the perturbative order.

Although cosmic distances are estimated using the BAO at linear regimes, there are smearing effects at small scales that need to be computed. These small-scale corrections are included to make the final precise constraints on the BAO (Taruya et al. 2010). We make use of an improved model of the redshift-space power spectrum (Taruya et al. 2010), in which the coupling between the density and velocity fields associated with the Kaiser and the FoG effects is perturbatively incorporated into the power spectrum expression. The result includes nonlinear corrections consisting of higher-order polynomials (Taruya et al. 2010):

$$\begin{eqnarray}\begin{array}{rcl}\hat{P}(k,\mu ) & = & \left\{{P}_{\delta \delta }(k)+2{\mu }^{2}{P}_{\delta {\rm{\Theta }}}(k)+{\mu }^{4}{P}_{{\rm{\Theta }}{\rm{\Theta }}}(k)\right.\\ & & +\left.A(k,\mu )+B(k,\mu )\right\}{G}^{\mathrm{FoG}}.\end{array}\end{eqnarray} \tag{ 14 }$$

Here, the A(k, μ) and B(k, μ) terms are the nonlinear corrections and are expanded as a power series of μ. Those spectra are computed using the fiducial cosmological parameters. The FoG effect G^FoG is given by a simple Gaussian function that is written as

$$\begin{eqnarray}&&{G}^{\mathrm{FoG}}\equiv \exp \left[-{\left(k\mu {\sigma }_{p}\right)}^{2}\right]\end{eqnarray} \tag{ 15 }$$

where ${\sigma }_{p}$ denotes the one-dimensional velocity dispersion.

Thus the theoretical correlation function ${\xi }_{\mathrm{th}}(s,\mu )$ is parameterized by

$$\begin{eqnarray}&&{x}_{\mathrm{th}}=({D}_{A},{H}^{-1},{G}_{b},{G}_{{\rm{\Theta }}},{\sigma }_{p})\end{eqnarray} \tag{ 16 }$$

wherein G_b and G_Θ are the normalized density and coherent motion growth functions. When working with photo-z samples, the effect of the photo-z error on the correlation function is incoherent. Thus, an extra parameter is needed for the theoretical template to model ξ_th(s, μ) as a function of the photo-z error, but it is not well understood. So, we use Equation (4) instead to fit our observed ${\xi }_{w}(s,\mu )$. This functional form only assumes a power law at small scales and a Gaussian function to fit the BAO peak at large scales and seems to model ${\xi }_{w}(s,\mu )$ quite well, as we can see from Figure 2.

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In our previous work (Sridhar & Song 2019), we have verified that the BAO feature from the theoretical correlation function is weakly dependent on the growth functions and ${\sigma }_{p}$. We have verified that changing the value of σ_p by ±10% does alter the location of the BAO peak, but only by less than 0.1% for samples at $\bar{\mu }$ close to 0 and less than 1% for samples at $\bar{\mu }$ close to 1, which is negligible. Thus, when we fit the cosmic distances, we fix ${\sigma }_{p}$. To find the best-fit σ_p , we vary it within $3.0\lt {\sigma }_{p}({\text{}}{h}^{-1}\ \mathrm{Mpc})\lt 6.0$ (by fixing D_A and ${H}^{-1}$ to their fiducial values), compute the theoretical correlation function ${\xi }_{\mathrm{th}}(s,\mu )$, and use the ${\sigma }_{p}$ for which ${\xi }_{\mathrm{th}}(s,\mu )$ best matches our measured ${\xi }_{w}(s,\mu )$. The best-fit σ_p used in this work is ${\sigma }_{p}=4.8\,{\text{}}{h}^{-1}\ \mathrm{Mpc}$, and we fix it for both of the redshift ranges.

Note that we apply the TNS model to compute the theoretical ${\xi }_{w}(s,\mu )$, from which we extract the BAO peak information using the fitting procedure mentioned in Section 3.1. The theoretical BAO peaks in the fiducial cosmology can also be transformed into a new cosmology according to simple coordinate projections wherein we can vary the D_A and H⁻¹ from the fiducial values by a small fraction. But it has been shown in Sridhar & Song (2019; see Figure 6) that the location shift of the BAO peak with varying D_A and H⁻¹ is not completely consistent with this coordinate transformation. The difference is exceeding the detectability limit by about 5%, and thus the TNS model is adopted to determine the theoretical BAO points.

A random catalog that is approximately 20 times the mock data is provided for the EZmock samples, but it contains only the angular component (R.A., decl.). Since the density of the DECaLS randoms is only five times the data catalog, we dilute the EZmock randoms to the same density to ensure consistency in our results. For the redshift component, we follow the same method of randomly extracting σ_z from the data catalog.

We calculate the covariance matrix, which is given by

$$\begin{eqnarray}&&{C}_{w}^{{ij}}({\xi }_{w}^{i},{\xi }_{w}^{j})=\displaystyle \frac{1}{N-1}\displaystyle \sum _{n\,=\,1}^{N}[{\xi }_{w}^{n}({\vec{x}}_{i})-{\overline{\xi }}_{w}({\vec{x}}_{i})][{\xi }_{w}^{n}({\vec{x}}_{j})-{\overline{\xi }}_{w}({\vec{x}}_{j})],\end{eqnarray} \tag{ 17 }$$

where the total number of simulations is given by N. The ${\xi }_{w}^{n}({\vec{x}}_{i})$ represents the value of the wedge correlation function of the ith bin of ${\vec{x}}_{i}$ in the nth realization, and ${\overline{\xi }}_{w}({\vec{x}}_{i})$ is the mean value of ${\xi }_{w}^{n}({\vec{x}}_{i})$ over all realizations. Due to the limited number of mock samples (100) that we have, we use 12 bins in s. We obtain the correlation matrix as

$$\begin{eqnarray}&&{C}_{{ij}}=\displaystyle \frac{\mathrm{Cov}({\xi }_{i},{\xi }_{j})}{\sqrt{\mathrm{Cov}({\xi }_{i},{\xi }_{i})\mathrm{Cov}({\xi }_{j},{\xi }_{j})}},\end{eqnarray} \tag{ 18 }$$

which is plotted in Figure 3.

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The inverse of C_ij is well defined and thus does not require any denoising procedures such as singular value decomposition. Additionally, we also count the offset caused by the finite number of realizations (Hartlap et al. 2007) as

$$\begin{eqnarray}&&{C}^{-1}=\displaystyle \frac{{N}_{\mathrm{mocks}}-{N}_{\mathrm{bins}}-2}{{N}_{\mathrm{mocks}}-1}\ {\hat{C}}^{-1},\end{eqnarray} \tag{ 19 }$$

where N_bins denotes the total number of i bins. As mentioned in Section 3.1, we fit the correlation function within the range $30.0\lt s({h}^{-1}\mathrm{Mpc})\lt 130.0$. Thus, the number of s bins in this range is nine. As the final three μ bins along the LOS do not contain any BAO information, we perform the fitting by ignoring them. Thus, the shape of the matrix is 27 × 27, and for 100 mock realizations, the factor in Equation (19) becomes 0.71. Apart from the above correction factor, an additional correction to the inverse covariance matrix is proposed by Percival et al. (2014), which is given by

$$\begin{eqnarray}&&{m}_{1}=\displaystyle \frac{1+B({n}_{b}-{n}_{p})}{1+A+B({n}_{p}+1)}\end{eqnarray} \tag{ 20 }$$

where n_b is the number of bins used for the two-point correlation measurements, n_p is the number of parameters measured, and the A and B terms are given by

$$\begin{eqnarray}&&A=\displaystyle \frac{2}{({n}_{s}-{n}_{b}-1)({n}_{s}-{n}_{b}-4)}\end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray}&&B=\displaystyle \frac{({n}_{s}-{n}_{b}-2)}{({n}_{s}-{n}_{b}-1)({n}_{s}-{n}_{b}-4)}\end{eqnarray} \tag{ 21b }$$

where n_s is the number of simulations used for the covariance matrix calculations. Applying the square root of this expression to the measured standard deviation should take care of the extra correction. For our binning scheme as mentioned above (with 100 mock realizations), we get $\sqrt{{m}_{1}}=0.89$, which is significantly smaller than the correction factor we already apply. The BAO peak position s_m of the wedge correlation function is found by fitting the phenomenological model by considering the full covariance using C_ij .

From the parent DECaLS DR8 sample within the redshift range $0.3\lt {z}_{\mathrm{phot}}\lt 1.2$, we choose two redshift cuts between $0.6\lt {z}_{\mathrm{phot}}\lt 0.8$ and 0.8 < z_phot < 1.0 for our analysis. The reason for choosing these two redshift ranges is that the redshift distribution of the sample peaks at ${z}_{\mathrm{phot}}\approx 0.7$, as can be seen from Figure 1. Thus, we expect to have the maximum number of galaxies around this redshift range. The redshift uncertainty scales with redshift, so we quote the mean values for our two redshift cut samples in Table 1. The wedge correlation functions are calculated using Equation (3) by using six μ bins of thickness Δμ = 0.17. The ${\xi }_{w}(s,{\mu }_{i})$ calculated from the first three μ bins for the two redshift samples is shown in Figure 2. We fit ${\xi }_{w}(s,\mu )$ using Equation (4) by following the MCMC procedure described in Section 3.1, and the values of the BAO peak s_m from the fit are provided in Table 2.

Table 2. Results of Fitting the Correlation Function (Plotted Using the Dotted Lines in Figure 2) for the Two Redshift Samples and in the Three $\bar{\mu }$ Bins using Equation (4)

Redshift Range	$\bar{\mu }$ Bin	sm (h−1 Mpc)
	0.08	${111.6}_{-3.2}^{+3.0}$
$0.6\lt {z}_{\mathrm{phot}}\lt 0.8$	0.25	${106.8}_{-5.6}^{+4.8}$
	0.42	${111.9}_{-8.0}^{+5.7}$
	0.08	${114.2}_{-5.1}^{+4.7}$
$0.8\lt {z}_{\mathrm{phot}}\lt 1.0$	0.25	${109.4}_{-10.0}^{+10.9}$
	0.42	${110.5}_{-6.0}^{+5.5}$

Note. The s_m is the BAO peak point obtained from the fit, and the units are in h⁻¹ Mpc.

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One can observe that for both redshift samples, the BAO signal is diluted as μ increases and is also more clearly visible for the first redshift sample than for the second redshift sample. This is because the photometric redshift errors scale proportional to ${\sigma }_{0}\times (1+z)$, and thus the isotropy along the LOS is destroyed more strongly for the high-redshift sample than for the low-redshift one. The errors on s_m gradually increase with increasing $\bar{\mu }$, as expected.

For both redshift ranges and all three $\bar{\mu }$ bins used, we fit the correlation function within the range $30.0\lt s({h}^{-1}\mathrm{Mpc})\,\lt 130.0$. When using log binning instead of linear binning, we note that the BAO peak results slightly shift to higher values. However, the effect is of the order of 1%, which is well below the estimated accuracy of the BAO peak position, which is between 4% and 10% for our samples. We use the ${\chi }^{2}/{dof}$ goodness-of-fit indicator to validate the performance of our empirical model to fit the correlation function. We have nine data points within the range $30.0\lt s({h}^{-1}\mathrm{Mpc})\lt 130.0$, and we fit for six parameters in each of the three $\bar{\mu }$ bins. The overall fit to the $0.6\lt {z}_{\mathrm{phot}}\lt 0.8$ sample yields a χ²/dof = 10/9, including all cross-covariances between $\bar{\mu }$ bins. We obtain a χ²/dof = 8/9 for the $0.8\lt {z}_{\mathrm{phot}}\lt 1.0$ sample.

It has been shown from previous studies (Ross et al. 2012, 2017) that data from the different μ bins are expected to be correlated and that the results from splitting the clustering by μ show a slight decrease in the BAO information content with increasing μ. Thus, we perform the fit using the full data vector including all of the μ bins. By performing the fit using the full covariance matrix, we make sure that the correlations between the different μ bins that exist are taken into account. The one- and two-dimensional projections of the posterior probability distribution of the s_m parameter from the MCMC chains for the two redshift samples are shown in Figure 4. The marginalized distribution for each s_m value from each $\bar{\mu }$ bin is shown independently in the histograms along the diagonal and the marginalized two-dimensional distributions in the other panels. We find that the correlation between the s_m values obtained at the different μ bins for both redshift samples is minimal.

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To validate our clustering results obtained on the DECaLS data, we use the LRGs from the Dark Sky mock catalog. To mimic the photometric redshifts from the DECaLS data, we follow the methodology as explained in Section 2.3. To compare the correlation function results obtained from the two redshift samples of the DECaLS catalog, we compute ${\xi }_{w}(s,\mu )$ from the Dark Sky photometric redshift catalogs with the same redshift cuts and use the same binning scheme. We find that by using the true values of ${\xi }_{w}(s,\mu )$ for the two photometric redshift samples from the Dark Sky mocks, the amplitudes of ${\xi }_{w}(s,\mu )$ do not match with the ${\xi }_{w}(s,\mu )$ from the DECaLS data. There are a few reasons for this behavior, and one of them is the poor estimation of the survey volume (due to systematics). Apart from systematics, there can also be other potential sources for this behavior. If the photo-z's were overestimated, then when applied to the mock catalogs they can cause a lower amplitude of the correlation function. The mocks could also have a lower galaxy bias compared to the observed galaxies, which can also result in a lower amplitude of the correlation function. However, by adding a constant value of 0.0005 and 0.0010 to the ${\xi }_{w}(s,\mu )$ obtained from the Dark Sky mocks for the 0.6 < z_phot < 0.8 and 0.8 < z_phot < 1.0 samples, we see that the amplitudes match well. The results are presented in Figure 5.

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We are interested in two key features when comparing the data and the mock catalog. One is the amplitude of ${\xi }_{w}(s,\mu )$, and the other is the location of the BAO peak. We find that the amplitude of the clustering measurements from the Dark Sky mock catalog (given by the solid blue line) matches the amplitude of the clustering measurements from the DECaLS sample (given by the red scatter points) for both redshift samples in all three $\bar{\mu }$ bins after adding the constant values to our redshift samples as described above. We repeat the MCMC procedure to obtain the BAO peak for the Dark Sky sample and find that the location of the BAO peak from the Dark Sky samples for both redshift ranges agrees with the DECaLS sample, at least within 1σ. A similar result has been obtained by Ross et al. (2017) by doing a comparison between mock samples and model curves using mock photometric data.

We also verify the internal consistency of the BAO peaks obtained from the Dark Sky photometric catalog by comparing them with the BAO peaks obtained from the true redshift (z_pec, cosmological redshift with peculiar velocity added) catalog for the same $\bar{\mu }$ bins. The wedge correlation functions from the three $\bar{\mu }$ bins are calculated using Equation (3), and the s_m values obtained from the two samples are plotted in Figure 6. For all three $\bar{\mu }$ bins, it can be seen that the s_m values from the photometric samples are within 1σ compared to the s_m values from the z_pec sample, with the 1σ errors on s_m being larger for the z_phot sample.

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