BAOfit_wtheta: Angular BAO Fitting Pipeline

• BAOfit_wtheta is a template-based angular BAO pipeline that models the two-point angular correlation function w(θ) using a CAMB-derived physical template and tomographic projections.
• It integrates nonlinear BAO damping and smooth broadband nuisance terms with Gaussian covariance from CosmoLike, ensuring accurate BAO scale measurements.
• Validated with 1,952 ICE-COLA mocks, the code produces a DESI-independent BAO likelihood essential for robust cosmological inference in DES Y6 analyses.

BAOfit_wtheta is a template-based BAO fitting code for angular clustering, designed for the angular correlation function observable $w(\theta)$ and publicly released alongside a DESI-independent measurement of the angular BAO scale from the completed Dark Energy Survey dataset. In the released configuration, it is a lightly updated version of the pipeline used in the fiducial DES Y6 BAO ACF analysis. The code models tomographic measurements of the two-point angular correlation function with a physical BAO template derived from a linear-theory matter power spectrum, includes nonlinear BAO broadening and a smooth broadband nuisance term, and was used to construct a DES BAO likelihood that can be combined with DESI BAO without double-counting correlated survey volume (Mena-Fernández et al., 21 Jan 2026).

1. Definition, release context, and code organization

BAOfit_wtheta was introduced as the dedicated fitting pipeline for the angular correlation function BAO observable $w(\theta)$ used in the DES Y6 BAO analysis, and it was publicly released together with the DESI-independent reanalysis at:

$$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$

The release accompanies a measurement strategy in which the DES sky area overlapping DESI is removed so that the resulting DES BAO likelihood can be combined with DESI BAO without double-counting correlated volume (Mena-Fernández et al., 21 Jan 2026).

The appendix of the release paper states that the repository is organized into four main modules, with responsibilities tied directly to the angular BAO workflow.

Module	Role
utils_data.py	Loading redshift distributions and covariances
utils_cosmology.py	Defining cosmology
utils_template.py	Computing template $w(\theta)$
utils_baofit.py	Performing BAO fits

The same source also states that the release includes notebooks for computing and plotting $w(\theta)$ and for running fits (Mena-Fernández et al., 21 Jan 2026). In operational terms, BAOfit_wtheta is therefore not a generic cosmology package but a specialized angular BAO pipeline centered on tomographic ACF fitting.

2. Observable, survey configuration, and DESI-independent construction

The fitted observable is the angular two-point correlation function estimated with the Landy–Szalay estimator,

$$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$

where $DD$, $DR$, and $RR$ are normalized pair counts in bins centered at $\theta$ with width $w(\theta)$0 (Mena-Fernández et al., 21 Jan 2026). BAOfit_wtheta is therefore tied to a purely angular clustering observable rather than to a 3D comoving-separation estimator.

In the DES Y6 BAO application, the code is used on tomographic auto-correlations in six photometric-redshift bins of width $w(\theta)$1, over the range

$w(\theta)$2

with effective redshift

$w(\theta)$3

The default angular binning is

$w(\theta)$4

and the scale cuts are

$w(\theta)$5

The underlying DES Y6 BAO sample is optimized for BAO science using the red-galaxy selection

$w(\theta)$6

together with a flux cut

$w(\theta)$7

and a redshift-dependent magnitude limit

$w(\theta)$8

The full fiducial Y6 BAO footprint covers

$w(\theta)$9

and contains about

$$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$0

galaxies (Mena-Fernández et al., 21 Jan 2026).

For the DESI-independent likelihood, the DESI-overlap region is removed. Two footprint splits are defined. The DR1tiles split, designed for combination with DESI DR1 and DR2, uses the public DESI DR1 tiles mask and yields a DES no-DESI area of $$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$1 and a DES–DESI overlap of $$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$2. The Deccut split, designed for future DR3 and DR4 combinations, keeps only pixels with $$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$3, giving a DES no-DESI area of $$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$4 and an overlap of $$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$5. The cosmological inference uses the DR1tiles no-DESI sample as the main DES BAO likelihood because it has the larger area and is the appropriate independent companion for DESI DR1 and DR2. The redshift distribution $$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$6 is unchanged by the mask split, so the same fiducial $$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$7 as in the Y6 BAO analysis is used for templates and covariance (Mena-Fernández et al., 21 Jan 2026).

3. Physical template, nuisance sector, and BAO dilation parameter

BAOfit_wtheta implements a physical BAO template constructed from the linear matter power spectrum computed with CAMB. Nonlinear damping of the BAO feature is introduced by splitting the spectrum into wiggle and no-wiggle components,

$$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$8

Here $$\texttt{https://github.com/juanejo95/BAOfit\_wtheta}$$9 is the linear galaxy bias, $w(\theta)$0 is the linear growth rate, $w(\theta)$1 is the cosine to the line of sight, $w(\theta)$2 is a smooth no-wiggle spectrum, $w(\theta)$3 is the BAO wiggle component, and $w(\theta)$4 is the anisotropic damping scale. The no-wiggle spectrum is modeled with Gaussian smoothing in log-space, and the damping follows infrared resummation ideas from Senatore and Zaldarriaga and from Blas, Garny, and Ivanov, as summarized in the release paper (Mena-Fernández et al., 21 Jan 2026).

The code then projects the configuration-space correlation function over the redshift distributions to obtain the angular observable,

$w(\theta)$5

The fitted model is

$w(\theta)$6

with a broadband term

$w(\theta)$7

where $w(\theta)$8 by default. In this parameterization, $w(\theta)$9 is the physical BAO template, $w(\theta)$0 is an overall amplitude rescaling, $w(\theta)$1 is the BAO dilation or shift parameter, and $w(\theta)$2 absorbs smooth broadband structure (Mena-Fernández et al., 21 Jan 2026).

The release paper defines

$w(\theta)$3

so that $w(\theta)$4 measures the angular BAO shift relative to the fiducial template cosmology. In practice, $w(\theta)$5 is the parameter that maps the template BAO scale onto the measured one (Mena-Fernández et al., 21 Jan 2026).

This fitting structure is continuous with earlier DES angular BAO methodology. In the DES Y1 optimization study, the fitted template was written as

$w(\theta)$6

and the analysis compared the Maximum Likelihood Estimator, Profile Likelihood, and Markov Chain Monte Carlo. After removing extreme mocks with non-detected BAO signal, the MLE was found to be the least biased and to give $w(\theta)$7 errors closest to the Gaussian expectation (Chan et al., 2018). BAOfit_wtheta inherits the same general template-plus-broadband logic while operating in the DES Y6 ACF framework (Mena-Fernández et al., 21 Jan 2026).

4. Covariance model, mock validation, and systematic budget

The covariance treatment used with BAOfit_wtheta follows the DES Y3 and Y6 BAO analyses by employing CosmoLike Gaussian covariances. The release paper states that non-Gaussian terms such as the trispectrum and super-sample covariance were tested and do not materially change the results. The covariance inputs are the footprint area, the number of galaxies in each redshift bin, cosmological parameters, redshift distributions, and linear galaxy bias per bin. For the mocks, $w(\theta)$8 is the average of the 1952 simulations; for the data, the fiducial Y6 $w(\theta)$9 is used. Biases are estimated in the range

$$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$0

(Mena-Fernández et al., 21 Jan 2026).

Validation is performed with 1,952 COLA mocks produced with ICE-COLA and matched to the DES Y6 BAO sample properties. Each mock has $$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$1 particles, box size $$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$2, and is repeated 64 times to make a lightcone to $$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$3. The release paper explicitly states that these mocks are not used for covariance estimation because the periodic replications can induce spurious correlations; instead, they are used for validation and for interpreting statistical fluctuations. Applying all four masks to the 1,952 mocks produces

$$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$4

$$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$5 measurements for direct comparison with the data (Mena-Fernández et al., 21 Jan 2026).

The pre-unblinding tests compare the shift in best-fit $$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$6 and the ratio of errors $$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$7 between the split samples and the full footprint. The reported conclusion is that the shifts and uncertainty increases are consistent with mock statistics, with no evidence that removing the DESI overlap induces bias (Mena-Fernández et al., 21 Jan 2026).

The systematic uncertainty budget is inherited from the Y6 BAO analysis and added in quadrature to the statistical uncertainty. The release paper lists two contributions,

$$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$8

which combine to

$$w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)},$$9

This systematic is folded into the reported uncertainty on $DD$0 (Mena-Fernández et al., 21 Jan 2026).

5. Measurement outputs and role in cosmological inference

For the DESI-independent sample used in cosmology, the fiducial BAO result obtained with the angular ACF pipeline is

$DD$1

equivalently

$DD$2

For comparison, the full DES Y6 BAO ACF result is

$DD$3

The no-DESI sample therefore has slightly larger uncertainty, as expected from the smaller footprint, but remains consistent with the full-footprint result and with the mock-based expectation. The BAO detection significance in the no-DESI samples remains between $DD$4 and $DD$5 (Mena-Fernández et al., 21 Jan 2026).

The same paper is explicit that BAOfit_wtheta follows the same methodology as the DES Y6 BAO analysis but uses only $DD$6, whereas the fiducial Y6 BAO analysis combined three clustering statistics: $DD$7, $DD$8, and $DD$9. The angular-only choice is motivated by the difficulty of consistently computing cross-covariances for irregular no-DESI footprints, especially when mixing angular and Fourier-space estimators. The quoted cost is about a $DR$0 increase in statistical uncertainty relative to the combined Y6 consensus analysis (Mena-Fernández et al., 21 Jan 2026).

The DESI-independent likelihood is then used in combinations with DES Y5 Type Ia supernovae or DES SN-Dovekie, Planck 2018 CMB, and DESI BAO DR1 or DR2. With DES Y5 SN + CMB + DESI DR1, the preference for $DR$1CDM over $DR$2CDM changes from

$DR$3

without DES BAO-noDESI to

$DR$4

with DES BAO-noDESI. With DES Y5 SN + CMB + DESI DR2, it changes from

$DR$5

to

$DR$6

Using the newer DES SN-Dovekie sample reduces the significance of the $DR$7 preference to about $DR$8 (Mena-Fernández et al., 21 Jan 2026).

The release therefore assigns BAOfit_wtheta a specific inferential role: it produces a DESI-independent angular BAO likelihood for joint DES + DESI work. The same source cautions that analyses using DES BAO data alone should still use the original fiducial Y6 BAO likelihood because it is more precise. The two DESI-independent DES BAO likelihoods are intended to be made available in CosmoSIS (Mena-Fernández et al., 21 Jan 2026).

6. Methodological lineage, contrast with earlier angular BAO analyses, and scope of the name

BAOfit_wtheta belongs to a broader line of angular BAO analyses based on $DR$9, but it is not identical to the earlier quasi model-independent 2PACF programs. In SDSS DR10 luminous red galaxies, the angular BAO scale was extracted from narrow redshift shells by identifying a bump in the two-point angular correlation function, fitting a power-law continuum plus a Gaussian bump, and correcting the fitted peak for projection effects. That analysis emphasized two sources of model dependence: the acoustic scale from CMB measurements and the correction in the $RR$0 position due to projection effects (Carvalho et al., 2015). A related SDSS DR12Q quasar analysis at $RR$1 used the same general strategy, including robustness checks under diverse bin-size choices and under small displacements of the quasars’ angular coordinates (Carvalho et al., 2017).

The DES optimization study for Year 1 moved toward the more survey-pipeline-oriented formulation that is closer to BAOfit_wtheta. It calibrated and optimized a full BAO fitting pipeline for tomographic angular clustering, studied template mismatch from cosmology and photo-$RR$2 errors, and showed that covariance mismatches between mocks and data can be corrected with Gaussian covariance corrections or, more effectively, with an eigenmode expansion of the mock covariance (Chan et al., 2018). BAOfit_wtheta is therefore best understood as the DES Y6 public realization of a template-based angular BAO pipeline rather than as a generic implementation of the earlier bump-finding 2PACF method (Mena-Fernández et al., 21 Jan 2026).

A recurrent source of ambiguity is the string “theta” in the package name. In BAOfit_wtheta, the notation refers to the angular separation in the correlation function $RR$3. It is unrelated to the lattice theta sums $RR$4 and $RR$5 studied in randomized approximation algorithms (Barvinok, 2022), and equally unrelated to Borwein or Ramanujan-type theta functions attached to positive definite binary quadratic forms, such as

$RR$6

which are studied in the theory of alternative modular bases (Bagis, 2015). The package name is thus domain-specific shorthand for BAO fitting in the angular correlation function.

Within that scope, BAOfit_wtheta is characterized by a precise combination of ingredients: a CAMB-based physical BAO template, nonlinear BAO damping, tomographic projection through $RR$7, Gaussian covariances from CosmoLike, mock-based validation with 1,952 ICE-COLA realizations, and direct inference of the dilation parameter $RR$8 for a DESI-independent angular BAO likelihood at $RR$9 (Mena-Fernández et al., 21 Jan 2026).