DESI DR1 BAO Analysis

1. Introduction¶

In the hot plasma of the early Universe, sound waves launched by primordial overdensities propagated outward until recombination released the photons and the waves stalled. The distance each wave had travelled — the sound horizon at the drag epoch, $r_d$ — was thereby frozen into the clustering of matter as a preferred comoving separation, visible today as a localised peak in the galaxy two-point correlation function $\xi(s)$ near $100\,h^{-1}$ Mpc. Because $r_d$ is calibrated to a quarter of a percent by the cosmic microwave background, this baryon-acoustic-oscillation (BAO) feature is the most robust standard ruler in large-scale structure: measuring its apparent size across and along the line of sight yields the comoving distance $D_M(z)/r_d$ and the Hubble distance $D_H(z)/r_d$, mapping the expansion history of the Universe.

The ruler is not pristine. Over cosmic time, bulk flows and non-linear structure growth displace galaxies from their original positions, which blurs the acoustic peak (Eisenstein et al., 2006) and degrades the achievable distance precision by a factor of roughly three (Eisenstein et al., 2006). Density-field reconstruction was introduced to undo this damage: by estimating the large-scale displacement field from the observed density and moving galaxies back along it, reconstruction re-sharpens the acoustic peak (Eisenstein et al., 2006) and recovers roughly a factor of two in distance precision (Eisenstein et al., 2006).

This article reproduces the DESI DR1 BAO measurement in configuration space, fitting every tracer both before and after reconstruction so the gain is measured rather than assumed; Fourier-space $P(k)$ is out of scope. The headline result is twofold: the acoustic feature is BAO detected post-recon in every tracer after reconstruction, and reconstruction Reconstruction tightens α_iso — delivering ~1 % α_iso precision for the best-measured LRG bins.

2. Data¶

The measurement consumes three classes of input. The first is the raw DESI DR1 LSS clustering catalogs — galaxy positions plus eighteen random realisations per Galactic cap, which define the survey geometry and selection. The second is the tabulated DESI fiducial cosmology (Planck 2018 ΛCDM (Planck Collaboration et al., 2018)), which converts redshifts to comoving distances; every apparent BAO scale is measured relative to this fiducial expectation. The third is the set of RascalC semi-analytic covariance matrices for $\xi(s)$ — one pre- and one post-reconstruction file per tracer, taken from the published DESI release.

The covariances are ingested as published rather than recomputed, and this is the load-bearing data decision: because each RascalC matrix is calibrated against the fiducial DESI reconstruction and clustering configuration, adopting it fixes the $\xi(s)$ binning to the published linear grid and, as discussed below, pins several upstream pipeline choices to their fiducial settings.

3. Methods¶

The pipeline runs in three stages: Reconstruction produces shifted catalogs, Clustering measures correlation functions from them, and a template-fitting stage turns each correlation function into posterior constraints on the BAO scale — one MCMC chain per (tracer, reconstruction state), sixteen chains in total.

Reconstruction. The linear displacement field is estimated from the Gaussian-smoothed galaxy density and applied symmetrically to galaxies and randoms (the Reconstruction convention choice, RecSym), moving structure approximately back to its initial position and thereby sharpening the acoustic peak. The single load-bearing knob is the Reconstruction Gaussian smoothing scale (BGS/LRG/ELG): it sets the scale $\Sigma_\mathrm{sm}$ of the smoothing applied before the displacement solve, and because the same scale re-enters the BAO damping template downstream, the reconstruction stage inherits it from the root of the analysis rather than choosing independently — the two stages cannot drift apart.

Clustering. We measure the Landy–Szalay $\xi(s,\mu)$ multipoles on the raw catalogs (pre-reconstruction) and on the shifted catalogs (post-reconstruction), per tracer redshift slice. Because the covariance is pinned to the published RascalC grid (the Covariance matrix source decision), the $s$-binning, $\mu$-binning, and estimator are locked to that grid, and the non-fiducial smoothing options are declared incompatible with it — the baseline universe only validates at fiducial smoothing. The one remaining free choice here, Imaging-systematics weights, is null-tested by the same covariance at sub-$\sigma$ significance and left unlocked.

Template fitting. Each chain fits a damped-BAO template with desilike + emcee. The template is built in the fiducial cosmology and the fit measures how far the acoustic feature in the data is dilated away from it: $\alpha_\mathrm{iso}$ rescales the feature isotropically and traces $D_V/r_d$, while $\alpha_\mathrm{AP}$ warps it anisotropically and traces the Alcock–Paczyński ratio $D_M/D_H$. The five 2D tracers (the LRG bins, ELG2, and LRG3+ELG1) fit both parameters from the monopole and quadrupole; the three sparser 1D tracers (BGS, ELG1, QSO) fit $\alpha_\mathrm{iso}$ alone from the monopole. The smooth, BAO-free part of each multipole is absorbed by the Broadband model model — the spline form is fiducial and has been shown to give α consistent with the polynomial form (Chen et al., 2024), with the residual absorbed into the modelling-error budget. The non-linear smearing of the peak is modelled by damping parameters controlled jointly by the BAO damping parameter prior and Damping prior central values, anchored to the result that mis-centred damping priors bias α (Chen et al., 2024). Three further template-shape choices — FoG damping placement, Component that α dilates, and Wiggle / no-wiggle split — are pinned to fiducial and exposed to document the modelling-systematic budget they underlie; the fiducial dilation acts on the wiggle component only, the expected behaviour given that reconstruction reduces the non-linear BAO damping (Padmanabhan et al., 2008).

4. Results¶

4.1 Detection and peak sharpening¶

The most direct view of the measurement is the acoustic feature itself. Figure 1 isolates it by subtracting the smooth part of the best-fit model from each measured multipole: in every tracer the post-reconstruction peak is visibly narrower and better matched by the template than its pre-reconstruction counterpart — the peak BAO peak sharpens.

Provenance

Decisions (13)

• Broadband modelSpline (power, fiducial)
• BAO damping parameter priorGaussian (fiducial)
• Fit range in s[48, 152] Mpc/h (fiducial)
• Multipoles fit (LRG only; 1D tracers override)ℓ = 0, 2 (2D, LRG default)
• Damping prior central valuesFiducial (per tracer)
• BAO template cosmologyc000 -- Planck 2018 ΛCDM (fiducial)
• Reconstruction Gaussian smoothing scale (BGS/LRG/ELG)15 Mpc/h (fiducial)
• FoG damping placementSmooth-only (fiducial)
• Wiggle / no-wiggle splitPeak-average (Brieden+2022, fiducial)
• Component that α dilatesWiggle-only (fiducial)
• BAO fit methodSampling (emcee MCMC, fiducial)
• Imaging-systematics weightsvia clusteringOn (default)
• Reconstruction Gaussian smoothing scale (QSO only)30 Mpc/h (fiducial)

To quantify whether the feature is detected at all, each post-reconstruction correlation function is refit with the BAO wiggles removed, and Figure 2 profiles the $\chi^2$ difference between the two models as a function of $\alpha_\mathrm{iso}$. Every tracer develops a well-defined minimum near the fiducial scale — weakest for the sparse 1D tracers, strongest for the combined LRG3+ELG1 bin.

Provenance

Decisions (12)

• Broadband modelSpline (power, fiducial)
• BAO damping parameter priorGaussian (fiducial)
• Fit range in s[48, 152] Mpc/h (fiducial)
• Multipoles fit (LRG only; 1D tracers override)ℓ = 0, 2 (2D, LRG default)
• Damping prior central valuesFiducial (per tracer)
• BAO template cosmologyc000 -- Planck 2018 ΛCDM (fiducial)
• Reconstruction Gaussian smoothing scale (BGS/LRG/ELG)15 Mpc/h (fiducial)
• FoG damping placementSmooth-only (fiducial)
• Wiggle / no-wiggle splitPeak-average (Brieden+2022, fiducial)
• Component that α dilatesWiggle-only (fiducial)
• Imaging-systematics weightsvia clusteringOn (default)
• Reconstruction Gaussian smoothing scale (QSO only)30 Mpc/h (fiducial)

4.2 The dilation parameters¶

Condensing the chains, the post-reconstruction isotropic dilation is consistent with unity across the suite — the acoustic scale in DESI DR1 sits where the fiducial cosmology predicts. The strongest 2D bins reach the sub-percent regime: $\alpha_\mathrm{iso} =$ 0.9995 ± 0.0088 for the combined LRG3+ELG1 bin and 1.003 ± 0.0097 for LRG3. The gain from reconstruction is largest where the pre-reconstruction feature is most degraded — for ELG1, $\sigma(\alpha_\mathrm{iso})$ contracts from 0.0696 to 0.0207, and for LRG2 from 0.0195 to 0.0114. The full set of fits is collected in Table 1:

The fits are statistically well-behaved — $\chi^2/\mathrm{dof}$ χ²/dof near dof. The combined LRG3+ELG1 bin returns $\chi^2 =$ 29.12 for 39 degrees of freedom; QSO is the least well-behaved, at $\chi^2 =$ 33.49 for 19.

4.3 Cosmological distances¶

The dilation parameters convert directly into distances: each chain carries $D_M/r_d$, $D_H/r_d$, and $D_V/r_d$ as derived parameters, so the constraints in Table 2 inherit the full non-Gaussian shape of the posteriors rather than a linearised propagation. For the combined LRG3+ELG1 bin at $z_\mathrm{eff} =$ 0.93 we measure $D_M/r_d =$ 21.76 ± 0.29, $D_H/r_d =$ 17.85 ± 0.36, and $D_V/r_d =$ 19.88 ± 0.17. The 1D tracers contribute $D_V/r_d$ alone: 7.961 ± 0.17 at $z_\mathrm{eff} =$ 0.3 (BGS), 19.92 ± 0.42 at 0.95 (ELG1), and 26.1 ± 0.59 at 1.49 (QSO).

Placed on a BAO Hubble diagram against 6dFGS, WiggleZ, SDSS DR16, and DES Y6 (Figure 3), the DESI DR1 distances trace the Planck 2018 $\Lambda$CDM prediction across the full redshift range, with the standard-ruler anchor set by the Planck 2018 sound horizon (Planck Collaboration et al., 2018).

Provenance

Decisions (14)

• Reported error includes modelling systematicInclude modelling budget (headline)
• Broadband modelSpline (power, fiducial)
• BAO damping parameter priorGaussian (fiducial)
• Fit range in s[48, 152] Mpc/h (fiducial)
• Multipoles fit (LRG only; 1D tracers override)ℓ = 0, 2 (2D, LRG default)
• Damping prior central valuesFiducial (per tracer)
• BAO template cosmologyc000 -- Planck 2018 ΛCDM (fiducial)
• Reconstruction Gaussian smoothing scale (BGS/LRG/ELG)15 Mpc/h (fiducial)
• FoG damping placementSmooth-only (fiducial)
• Wiggle / no-wiggle splitPeak-average (Brieden+2022, fiducial)
• Component that α dilatesWiggle-only (fiducial)
• BAO fit methodSampling (emcee MCMC, fiducial)
• Imaging-systematics weightsvia clusteringOn (default)
• Reconstruction Gaussian smoothing scale (QSO only)30 Mpc/h (fiducial)

5. Systematics and robustness¶

The reported uncertainties are not statistical-only: a combined modelling + HOD + fiducial-cosmology systematic budget is folded into the quoted errors at the aggregation step, toggled by the Reported error includes modelling systematic decision as a table rebuild rather than a refit — the chains themselves are untouched, and the statistical-only errors are kept alongside for comparison. The budget is anchored to the DESI companion modelling result that sets the combined systematic budget (Chen et al., 2024), dominated by the wiggle/no-wiggle split, FoG placement, and template-dilation choices catalogued in the decision register. Fiducial-cosmology dependence is swept through BAO template cosmology (a set of AbacusSummit grids), and the remaining Fit range in s, Multipoles fit (LRG only; 1D tracers override), and BAO fit method decisions exist for sensitivity tests; none moves the baseline result beyond its quoted budget.

6. Conclusions¶

A configuration-space BAO analysis of DESI DR1 detects the acoustic feature in all eight tracer bins post-reconstruction, reaches sub-percent isotropic precision in the strongest LRG bins, and yields a self-consistent set of $D_M/r_d$, $D_H/r_d$, $D_V/r_d$ distances at the percent level — reproducing the DESI DR1 BAO distance ladder. Every numerical claim above is traceable to a registered ASTRA finding, the decision that configured it, and the materialised output product it summarises.