we refer to [67] for the full quantitative discussion comparing the mocks and
the data, here we note the excellent consistency between the overall
clustering amplitude of the Abacus-2 DR1 mocks and that of the data as a
function of scale within the fitting ranges pre and post reconstruction, as
shown in Figures 5 and 6.

Figures 5 and 6 also compare the pre-reconstruction versus post-reconstruction
clustering of DESI DR1 within the fitting ranges. As we adopted the RecSym
convention, which preserves the linear RSD boost along the line of sight
through the displacement of the random particles, the clustering before and
after reconstruction is quite consistent on large scales, as expected. On
smaller scales we see in Figure 6 that the post-reconstruction power spectrum
is reduced relative to the pre-reconstruction one—this is not unexpected as in
general we expect the reconstructed density field to have different nonlinear
power due to the coordinate remapping, and there is some analytic evidence for
the decrease of power specifically in the case of the post-reconstruction
matter power spectrum [106, 17, 154, 155]. From Figure 6, the change in
clustering before and after reconstruction of the data (colored solid and
dotted lines for monopole and quadrupole, respectively) appears highly
consistent with the expectation based on Abacus-2 DR1 mocks within the
$1\sigma$ dispersion expected from the RascalC covariance matrice (the black
solid and dashed lines with the gray shade for the $1\sigma$ dispersion).

The BAO feature appears moderately sharpened by reconstruction in the LRGs
redshift bins, while the improvement is less obvious for other tracers. One
can see that the Abacus-2 DR1 mocks replicate the observed level of the BAO
sharpening in the DESI DR1 data. Hence, we qualitatively find that the
reconstruction of the data is performing as expected given the survey
configuration and the reconstruction method we chose. In the next section,
where the results of the BAO fits are discussed, we make quantitative
comparisons on the aforementioned aspects and show that all tracers had
moderate gain from reconstruction at the level consistent with the mocks.

###  7.2 BAO measurements from the DESI DR1 galaxies

In this section, we present the BAO fitting of all galaxy and quasar tracers
using the default fitting method defined in Section 4.3. We focus on the
fitting results in configuration space for our discussion below, although we
get consistent results in Fourier space Appendix A.262626Additionally, since
our analytic and mock-based covariances have not fully converged in our
Fourier space analysis, we view the configuration space results as more
robust. All results except for ELG1 and LRG3 will be directly used for the
DESI DR1cosmology analysis of [43], with ELG1 and LRG3 incorporated into the
analysis via LRG3$+$ELG1. Based on the unblinding tests and the decisions
presented in Table 14, we performed two-dimensional BAO fits (${\alpha_{\rm
iso}}$ and ${\alpha_{\rm AP}}$) for all tracers other than BGS, ELG1, and QSO.
For the latter, we derive ${\alpha_{\rm iso}}$ using only the monopole data.

Table 15, Table 19, Figure 7 and Figure 8 summarize the resulting BAO
constraints and visualize the isolated BAO features. Table 15 shows that the
$\chi^{2}$ values of the 2-D fits to the correlation functions fall between
33.1 for LRG3$+$ELG1 and 58.6 for ELG2, all for 39 degrees of freedom. The
cases of 1-D fits show $\chi^{2}=17.3$ for BGS, $\chi^{2}=20.4$ for ELG1, and
$\chi^{2}=32.4$ for QSO for 19 degrees of freedom. The goodness-of-fit is
reasonable; the worst cases, ELG2 gives an associated p-value of 2.27% and the
QSO gives an associated p-value of 2.82%. Table 19 in appendix summarises the
corresponding fits using the power spectrum multiples and shows a reasonable
level of the goodness-of-fit with $P(k)$ as well (a smallest p-value of 4.35%
for QSO). The BAO measurement scales for the two conjugate spaces are very
consistent (less than 0.3% in $\alpha_{\rm iso}$ and $\alpha_{\rm AP}$ except
for QSO). This level of consistency between the BAO fits between $\xi(r)$ and
$P(k)$ agrees with the Abacus-2 DR1 mocks ([46]).

|
---|---
|
|
Figure 9: Pre- and post-reconstruction BAO measurements, comparing DESI DR1 measurements (stars) with the 25 Abacus-2 DR1 (the open points with the error bars are the means and the standard deviations around the means of the 25 mocks). Due to the discrepancy between the BAO scales imprinted in the mocks and the BAO scales recovered from the data, we focus on the relative comparison between the pre and the post reconstruction. All DESI DR1 fits are consistent with what is expected from the mocks. |  |
---|---|---
Figure 10: Similar to Figure 9, but showing results for BGS, ELG1, and QSO.
The DESI DR1 results are, again, consistent with the range covered by Abacus-2
DR1, except for QSO; the result of QSO shows the reconstruction of DESI DR1 is
less efficient than the worst case of the mocks.

Figure 7 displays the BAO features in the correlation function selected for
each tracer, compared to the best-fit $\xi$ models for pre-reconstruction
(dashed lines) and post-reconstruction data (solid lines). For the 2-D cases,
LRG1 shows a distinct BAO feature in the post-reconstruction quadrupole
measurement. This is consistent with the $2.29\sigma$ offset of ${\alpha_{\rm
AP}}-1$ measured in Table 15. The more the underlying ${\alpha_{\rm AP}}$
differs from the assumed ${\alpha_{\rm AP}}=1$ of the fiducial cosmology, the
greater extent to which the feature in the quadrupole would be shifted from
that of the monopole and also amplified, as the BAO feature leaks from the
monopole to an extent proportional to ${\alpha_{\rm AP}}-1$.272727From [11],
for a small change in $(\epsilon+1)^{3}\equiv{\alpha_{\rm AP}}$, the monopole
and quadrupole change as  $\displaystyle P_{0}$ $\displaystyle\rightarrow$
$\displaystyle P_{0}-\frac{2\epsilon}{5}\frac{dP_{2}}{d\ln
k}-\frac{6\epsilon}{5}P_{2},$ (7.1) $\displaystyle P_{2}$
$\displaystyle\rightarrow$
$\displaystyle\left(1-\frac{6\epsilon}{7}\right)P_{2}-\frac{4\epsilon}{7}\frac{dP_{2}}{d\ln
k}-2\epsilon\frac{dP_{0}}{d\ln k}.$ (7.2) This behaviour of the BAO feature in
the quadrupole is not evident in the mock data, as the mocks consistently
assume ${\alpha_{\rm AP}}=1$ when the they were analyzed. That is, the
discrepancy between the BAO signal in the mock and the data merely indicates a
measurement of ${\alpha_{\rm AP}}\neq 1$ with $2.29\sigma$ significance. A
similar quadrupole feature and the consistent ${\alpha_{\rm AP}}$ best fit is
derived from the power spectrum fit.

Figure 11: Detection significance of the BAO features in various DESI DR1
galaxy tracers. Solid lines show $\Delta\chi^{2}$ of the BAO fits as a
function of $\alpha_{\rm iso}$, i.e., $\chi^{2}$ offset from the minimum
$\chi^{2}$. The dashed lines correspond to $\Delta\chi^{2}$ of the noBAO fits,
i.e., using a model without the BAO wiggles, after subracting the minimum
$\chi^{2}$ of the BAO fit for each tracer. The detection level of the BAO
feature for each tracer is shown in the legend.

The reconstruction applied to the QSO sample is less effective than the other
tracers due to its high shot noise, as depicted in Figure 7 and detailed in
Tables 15 and 19. The mock analysis in [46] suggests $3-8\%$ gain in the
precision after reconstruction, though individual realizations will vary
considerably due to high shot noise. Additionally, [7] and [138] find that on
average the reconstruction will unbias the BAO location, even in this regime
of high shot noise. Given these findings, in Table 14, we opted to use the
reconstructed QSO result as our fiducial while recognizing its inefficiency
from the unblinding test. This decision ensures consistency across cases
rather than relying on the realization-dependent performance. In particular,
with the future DESI data releases, we anticipate an improved reconstruction
performance for the QSO sample, with a reduced fluctuation from realization to
realization.

Figures 9 and 10 compare the pre- and post-reconstruction BAO fits for the
data and the 25 Abacus-2 DR1 mocks. Due to the discrepancy between the BAO
location assumed in the mocks and its measurement in the data, we focus on the
changes in the best fits and precision. Except for QSO, the measurements from
the data (indicated by stars) lie well within the distribution of the 25
Abacus-2 DR1 mocks, which is the criteria we set as “consistent” given the
small number of the mocks. For QSO, the blinded data BAO fit was marginally
consistent with the upper boundary (the least effective reconstruction limit)
reached by the mocks, therefore passing the unblinding test ([46]). However,
the reconstruction of the unblinded DESI DR1 is slightly worse than the
blinded case and less efficient than in the worst case of the mocks. We
believe that such a slight change before and after unblinding is acceptable.

In detail, we observe the change in the best fit ${\alpha_{\rm iso}}$ with
reconstruction is the greatest for LRG2, giving an 1.6% shift ($0.87\sigma$
compared to its pre-reconstruction precision) in $\xi$. In terms of the
precision, LRGs show the biggest improvement from reconstruction, a factor of
1.3-1.6 higher precision in $\sigma({\alpha_{\rm iso}})$ and a factor of
1.5-1.8 in $\sigma({\alpha_{\rm AP}})$. For other tracers, we find an
improvement of 1.4 for BGS, a factor of 3 for ELG1282828For ELG1, the pre-
reconstruction constraint is overestimated as the posterior distribution
deviates from Gaussian distribution due to its low signal-to-noise., and 1.2
($\alpha_{\rm iso}$) and 1.4 ($\alpha_{\rm AP}$) for ELG2. Reconstruction of
these tracers is not as effective as LRGs. In particular, the reconstruction
for ELG2 is very likely inhibited due to its low completeness and irregular
footprint for DESI DR1. Again, with the future DESI data releases, we
anticipate an improved reconstruction performance for the ELGs sample. To
summarize, the BAO reconstruction fits lead to a substantial gain for the
LRGs, but only a moderate gain for the other tracers.

Figure 11 shows the detection level of the BAO feature. For this plot, one-
dimensional BAO fits were performed for all tracers including LRGs and ELG2.
Following the standard procedure [156], the fits with the template without the
BAO feature were also made (i.e., setting the second term in Eq. 4.1 to zero,
hereafter a ‘noBAO fit’). The square root of the difference in $\chi^{2}$
between the pair of the BAO and noBAO fits at the best fit $\alpha_{\rm iso}$
of the former is quoted as the detection significance.292929[156] presented
the detection significance estimation using the same number of free parameters
in the BAO fit and the noBAO fit, with the only degree of freedom being the
existence of the BAO. In our case, the BAO fit parameters that would have
changed the second term in Eq. 4.1 have no impact on the noBAO fit. Therefore,
we can effectively consider the BAO and noBAO fits have the same number of
free parameters. The flat likelihood of the noBAO fit is a trivial result of
the improved separation between the BAO and noBAO modeling in Eq. 4.1 that the
DESI DR1 analysis adopted, i.e., the noBAO modeling now has naturally no
dependence on $\alpha_{\rm iso}$, in contrast to previous configuration space
analyses (see, e.g., the right-hand panel of figure 12 in [112]). The
detection significance, in the increasing order, is 4.0$\sigma$ for BGS,
4.2$\sigma$ for QSO, 6.4$\sigma$ for LRG1, 6.8$\sigma$ for LRG2, 7.0$\sigma$
for ELG2, and 8.7$\sigma$ for LRG3, with the most significant case being
$9.1\sigma$ for LRG3+ELG1, closely correlated with the precision.

Tracer | Redshift | Recon | $\alpha_{\rm iso}$ | $\alpha_{\rm AP}$ | $r_{\rm off}$ | $\chi^{2}/{\rm dof}$
---|---|---|---|---|---|---
BGS | 0.1–0.4 | Post | $0.9829\pm 0.0185$ |  |  | 17.3/19
LRG1 | 0.4–0.6 | Post | $0.9792\pm 0.0113$ | $0.9148\pm 0.0372$ | -0.0558 | 41.8/39
LRG2 | 0.6–0.8 | Post | $0.9661\pm 0.0117$ | $1.0459\pm 0.0432$ | -0.0514 | 46.5/39
LRG3 | 0.8–1.1 | Post | $1.0042\pm 0.0090$ | $1.0027\pm 0.0299$ | -0.0597 | 46.2/39
LRG3+ELG1 | 0.8–1.1 | Post | $0.9983\pm 0.0081$ | $1.0262\pm 0.0278$ | -0.1019 | 33.1/39
ELG1 | 0.8–1.1 | Post | $0.9887\pm 0.0200$ |  |  | 20.4/19
ELG2 | 1.1–1.6 | Post | $0.9876\pm 0.0147$ | $0.9904\pm 0.0466$ | -0.3031 | 58.6/39
QSO | 0.8–2.1 | Post | $1.0015\pm 0.0256$ |  |  | 32.4/19
BGS | 0.1–0.4 | Pre | $0.9646\pm 0.0258$ |  |  | 14.3/19
LRG1 | 0.4–0.6 | Pre | $0.9760\pm 0.0171$ | $0.9258\pm 0.0590$ | 0.2674 | 25.8/39
LRG2 | 0.6–0.8 | Pre | $0.9505\pm 0.0191$ | $1.0332\pm 0.0769$ | 0.4130 | 31.1/39
LRG3 | 0.8–1.1 | Pre | $1.0085\pm 0.0120$ | $1.0034\pm 0.0446$ | 0.2737 | 48.4/39
LRG3+ELG1 | 0.8–1.1 | Pre | $1.0027\pm 0.0114$ | $1.0344\pm 0.0436$ | 0.2569 | 53.0/39
ELG1 | 0.8–1.1 | Pre | $0.9404\pm 0.0620$ |  |  | 29.2/19
ELG2 | 1.1–1.6 | Pre | $0.9882\pm 0.0176$ | $0.9913\pm 0.0640$ | 0.0484 | 42.7/39
QSO | 0.8–2.1 | Pre | $0.9967\pm 0.0229$ |  |  | 8.1/19
Table 15: Mean values and standard deviations from the marginalized posteriors of the BAO scaling parameters from fits to the unblinded DESI DR1 correlation functions in the $\alpha_{\rm iso}$-$\alpha_{\rm AP}$ basis. These do not include the systematic effect. $r_{\rm off}=C_{\alpha_{\rm iso},\alpha_{\rm AP}}/\sqrt{C_{\alpha_{\rm iso},\alpha_{\rm iso}}C_{\alpha_{\rm AP},\alpha_{\rm AP}}}$.  Tracer | Redshift | Recon | $\alpha_{\parallel}$ | $\alpha_{\perp}$ | $r_{\rm off}$ | $\chi^{2}/{\rm dof}$
---|---|---|---|---|---|---
LRG1 | 0.4–0.6 | Post | $0.9219\pm 0.0269$ | $1.0098\pm 0.0186$ | -0.4623 | 41.8/39
LRG2 | 0.6–0.8 | Post | $0.9955\pm 0.0296$ | $0.9527\pm 0.0179$ | -0.4386 | 46.5/39
LRG3 | 0.8–1.1 | Post | $1.0057\pm 0.0214$ | $1.0038\pm 0.0139$ | -0.4181 | 46.2/39
LRG3+ELG1 | 0.8–1.1 | Post | $1.0148\pm 0.0193$ | $0.9903\pm 0.0126$ | -0.4190 | 33.1/39
ELG2 | 1.1–1.6 | Post | $0.9797\pm 0.0297$ | $0.9922\pm 0.0245$ | -0.4545 | 58.6/39
LRG1 | 0.4–0.6 | Pre | $0.9242\pm 0.0485$ | $1.0035\pm 0.0243$ | -0.4434 | 25.8/39
LRG2 | 0.6–0.8 | Pre | $0.9766\pm 0.0642$ | $0.9402\pm 0.0248$ | -0.5015 | 31.1/39
LRG3 | 0.8–1.1 | Pre | $1.0098\pm 0.0353$ | $1.0080\pm 0.0165$ | -0.4360 | 48.4/39
LRG3+ELG1 | 0.8–1.1 | Pre | $1.0230\pm 0.0323$ | $0.9930\pm 0.0151$ | -0.4549 | 53.6/39
ELG2 | 1.1–1.6 | Pre | $0.9810\pm 0.0475$ | $0.9934\pm 0.0273$ | -0.4743 | 42.7/39
Table 16: Mean values and standard deviations from the marginalized posteriors
of the BAO scaling parameters from fits to the unblinded DESI DR1 correlation
functions in the $\alpha_{\parallel}$-$\alpha_{\perp}$ basis. These do not
include the systematic effect. $r_{\rm
off}=C_{\alpha_{\perp},\alpha_{\parallel}}/\sqrt{C_{\alpha_{\perp},\alpha_{\perp}}C_{\alpha_{\parallel},\alpha_{\parallel}}}$.

###  7.3 BAO constraints projected onto $\alpha_{\perp}$ and
$\alpha_{\parallel}$

The parameterisation of the BAO-scale fitting by ${\alpha_{\rm iso}}$ and
${\alpha_{\rm AP}}$ provides a natural separation of the isotropic and
anisotropic dilation of the BAO that can be constrained somewhat
independently. Table 15 (and Table 19 in Appendix for $P(k)$) displays the
cross-correlation coefficients, $r_{\rm off}=C_{\alpha_{\rm iso},\alpha_{\rm
AP}}/\sqrt{C_{\alpha_{\rm iso},\alpha_{\rm iso}}C_{\alpha_{\rm AP},\alpha_{\rm
AP}}}$. We find the absolute values of $r_{\rm off}$ are less than $\sim 0.10$
for the three LRGs after reconstruction, although it is larger for ELG2($\sim
0.3$). The low value of the cross-correlation allows us to incorporate
systematic errors ignoring the covariance between the two dilation parameters,
i.e., by adding the statistical covariance from Table 15 with the systematic
covariance in Eq. 5.1.

We can also decompose the BAO locations into transverse and line-of-sight
dilation parameters $\alpha_{\perp}$ and $\alpha_{\parallel}$ (Eq. 1.1) that
are more directly linked to the angular diameter distance and the Hubble
parameter at the given redshift. Table 16 (and Table 20 for $P(k)$) shows the
BAO constraints from re-fitting $\xi(r)$ into this parameterization rather
than transforming them using Eq. 5.2.303030Transforming using Eq. 5.2 gives
slightly different numbers due to the assumption of two-dimensional Gaussian
likelihood The cross-correlation coefficients in this case, $r_{\rm
off}=C_{\alpha_{\perp},\alpha_{\parallel}}/\sqrt{C_{\alpha_{\perp},\alpha_{\perp}}C_{\alpha_{\parallel},\alpha_{\parallel}}}$,
are in excellent agreement with the theoretical Fisher matrix prediction when
the BAO feature is singled out [157]. The covariance based on Table 16 will be
combined with the covariance for systematics (Eq. 5.4).

###  7.4 Test of the systematics using the unblinded data

We replicate the tests conducted on the blinded catalogs with the unblinded
catalogs to investigate any indications of systematics. Figure 12 illustrates
the BAO measurements when comparably optimal variations are introduced around
our fiducial setup. The figure compares the baseline fit (using $\xi(r)$)
versus the $P(k)$ fit, a different choice of the damping priors, a different
choice of the broadband modeling, and a different choice of the reconstruction
convention. These are the choices that we consider equally optimal (e.g., the
power spectrum fit) or close to being optimal, compared to the default setup.
In addition, we also include the pre-reconstruction results (second row).

Figure 12: Response of the constraints on the isotropic (top) or anisotropic
(bottom) BAO scaling parameters to changes in the data vector or the model
assumptions. The baseline configuration adopted for the BAO analysis is shown
in the top row, while the remaining rows show single variations around that
baseline (see the main text for a description of each case). The consistency
between the $\xi(r)$ and $P(k)$ fits and the consistency between the spline
broadband versus polynomial broadband are also very high. We identified the
Gaussian prior on the sampling parameters as a more robust choice, but even
the suboptimal flat priors return very consistent results. The difference
between NGC alone and the default (NGC + SGC) shows mild discrepancies for
${\alpha_{\rm AP}}$ of LRG1.

The comparison between the baseline fits versus the fits using $P(k)$ (third
whisker) shows high consistency in the derived BAO scales across all cases, as
discussed in Section 7.2. The BAO precision from the DESI DR1 $\xi(r)$ fit
reasonably agrees with these mock results, especially for ${\alpha_{\rm
iso}}$. In some cases, such as LRG2 and ELG2, the $P(k)$-BAO precision tends
to be 20-30% better than the $\xi(r)$-BAO error. The Abacus-2 DR1 mock test
shows 5-10% larger error with $\xi$; LRG2 lies somewhat beyond the range of
the mocks. Thus, opting for the $\xi-BAO$ fits is deemed a conservative
choice.

In the fourth whisker, we display the constraints obtained by only fitting
data from the NGC, which differs from the baseline case where both galactic
caps (NGC+SGC) are combined at the clustering level. In most cases, we only
see a small degradation in the parameter constraints compared to the baseline
method. The largest deviations appear in ${\alpha_{\rm AP}}$ of LRG1 and
LRG3$+$ELG1, but the comparison with mocks shows that this level of deviation
can occur for DESI DR1 [46].

The fifth whisker in Figure 12 shows the effect of using a flat, non-
informative prior on the BAO non-linear damping parameters. Based on [7], we
identified the Gaussian prior on the damping parameters as a more robust
choice than a flat prior, but even the sub-optimal, flat priors return very
consistent results.

Previous BAO analyses from BOSS [104] and eBOSS [4] adopted a different
parameterization for the broadband component of the power spectrum and
correlation function, using polynomials with varying degrees of freedom and
functional forms. The comparison between our default spline broadband analysis
and the polynomial-based broadband method (described in Section 4.3.3)
demonstrates excellent consistency (sixth whisker). We note that we have
performed this comparison in $\xi(r)$ and have only modified the parameters
that capture the broadband component in the modeling, while still integrating
all the other theoretical improvements in our analysis.

The second to last whisker shows the constraints using the RecIso convention
for reconstruction. This convention, which was the default in previous SDSS
BAO analyses [104, 158, 111] supresses higher-order multipoles by removing
linear RSD on large scales. To perform this test, we used the (e)BOSS
polynomial broadband parametrization for consistency with previous analysis.
Overall, we find excellent agreement in the BAO constraints compared to our
baseline case using the RecSym convention, in agreement with what was found
with mock galaxy catalogs [7] and the blinded DESI data [46].

Finally, in the last row, we show results using a smaller smoothing scale when
reconstructing the density field. This corresponds to $10h^{-1}\,{\rm Mpc}$
for BGS, LRGs, and ELGs, and $20h^{-1}\,{\rm Mpc}$ for QSO. We find that the
constraints on $\alpha_{\rm iso}$ are only slightly affected by this choice of
scale. For ${\alpha_{\rm AP}}$, the largest shift is of $1\sigma$ for LRG3. In
[46], tests on the full ensemble of Abacus-2 DR1 show that no statistically
significant shifts are observed when comparing these choices of smoothing
scales with the baseline. However, variations in individual mocks are expected
due to noise fluctuations and are consistent with what is observed in the
data.

In summary, the systematic tests conducted directly using unblinded DESI DR1
data demonstrate high consistency across variations of the fiducial setups and
data selections (e.g., NGC versus NGC+SGC). Hence, this test bolsters the
robustness of our measurements. As a caveat, these consistency tests may not
identify systematics that shift all the results together.

###  7.5 BAO measurements from the combined tracers

There are several redshift ranges in which more than one DESI target class
substantially overlaps. Over $0.8<z<1.1$, LRGs, ELGs, and QSO overlap, and
over $1.1<z<1.6$, ELGs, and QSO overlap. In particular, over $0.8<z<1.1$, LRG3
and ELG1 are expected to have comparable densities for DESI Y5 and there is a
compensatory crossing of the two populations within this redshift range. As
both tracers probe the same volume, the two BAO constraints would be highly
covariant on large scales, providing a test for consistency in the tracer-
independent parameter, such as post-reconstructed BAO locations. Although we
have estimated the tracer-dependent systematics on BAO in Section 5.2 within a
reasonable (or limited, depending on a view) range of the HODs for each
tracer, these overlapping tracers provide an utmost test of the tracer
dependence.

When combining the two BAO constraints we must account for this cross-
covariance, either using a set of large mocks in which both tracers are
present or with an analytical method. Another approach is to construct a
combined catalog. A BAO constraint from such a catalog seamlessly combines the
information from two BAO measurements from auto-clustering statistics and one
from the cross-clustering between ELG1 and LRG3 (hereafter
‘LRG3$\times$ELG1’). The combined catalog, which we defined as LRG3$+$ELG1 in
Section 2, may have an additional gain: due to the higher number density, the
reconstruction can be more effective. [47] tested the optimal construction of
DESI DR1 LRG3 and ELG1. The work includes the test of systematics using auto-
clustering measurements and the cross-clustering between the two tracers.

For the unblinded DESI DR1 LRG3$+$ELG1, we find 11% precision-improvement in
$\alpha_{\rm iso}$ and 14% improvement in $\alpha_{\rm AP}$ relative to LRG3,
as shown in Figure 8 and Table 15. This is highly consistent with the $\sim
10\%$ improvement predicted by [47] based on the mocks.

We find an offset of 0.4% in $\alpha_{\rm iso}$ between LRG3 and
LRG3$\times$ELG1 ([47]) and 0.6% between LRG3 versus LRG3$+$ELG1 (Table 15).
In $\alpha_{\rm AP}$, we find an offset of 0.4% between LRG3 and
LRG3$\times$ELG1 ([47]) and 1.9% between LRG3$+$ELG1 (Table 15). These
differences are well within the range covered by the mocks. Therefore, we
detect no tracer-dependent bias among the DESI DR1 BAO measurements from LRG3,
ELG1, LRG3$\times$ELG1, and LRG3$+$ELG1. With Y3 and Y5, as the completeness
of the ELGs increases, we expect a greater improvement from the combined
tracer analysis in terms of reconstruction as well as the systematic test.
Note that we perform only the $\xi(r)$ fit to the combined tracer, as we have
only the RascalC covariance available for the combined tracer.

###  7.6 Comparison to previous analyses

Figure 13: A comparison of the impact of different pipelines on the derived distances, analyzing the published BOSS and eBOSS LRG data. There are three groups of points, one for each of the redshift bins, while the different panels show results for $D_{V}/r_{\mathrm{d}}$, $D_{M}/r_{\mathrm{d}}$, and $D_{H}/r_{\mathrm{d}}$. In each group, the leftmost point is the published BOSS/eBOSS result. The second point from the left uses the published correlation functions and covariance matrices, but refits these using the methodology presented in this paper, while the last two points show the results of reprocessing the BOSS and eBOSS catalogs and randoms through the full DESI pipeline for different fiducial cosmologies. To make the comparison more direct (and to leave the covariance matrices unchanged), we use the RecIso reconstruction scheme here. The fiducial cosmology used to normalize the distance scales on the $y$-axis is the DESI fiducial cosmology used in this paper. Tracer | $z$ | $D_{\rm M}/r_{\rm d}$ | $D_{\rm H}/r_{\rm d}$ | $r_{\rm off}$
---|---|---|---|---
SDSS published |  |  |  |
BOSS LOWZ+CMASS | 0.2 - 0.5 | $10.23\pm 0.17$ | $25.00\pm 0.76$ | -0.39
BOSS LOWZ+CMASS | 0.4 - 0.6 | $13.36\pm 0.21$ | $22.33\pm 0.58$ | -0.39
eBOSS LRG | 0.6 - 1.0 | $17.86\pm 0.33$ | $19.33\pm 0.53$ | -0.33
SDSS published/desilike refit |  |  |  |
BOSS LOWZ+CMASS | 0.2 - 0.5 | $10.18\pm 0.15$ | $25.1\pm 0.65$ | -0.425
BOSS LOWZ+CMASS | 0.4 - 0.6 | $13.35\pm 0.17$ | $22.27\pm 0.43$ | -0.404
eBOSS LRG | 0.6 - 1.0 | $17.9\pm 0.3$ | $19.16\pm 0.42$ | -0.416
DESI pipeline |  |  |  |
BOSS LOWZ+CMASS | 0.2 - 0.5 | $10.15\pm 0.15$ | $25.10\pm 0.66$ | -0.45
BOSS LOWZ+CMASS | 0.4 - 0.6 | $13.39\pm 0.17$ | $22.44\pm 0.58$ | -0.43
eBOSS LRG | 0.6 - 1.0 | $17.72\pm 0.30$ | $19.33\pm 0.41$ | -0.40
Table 17: Comparison between the SDSS published results and the reanalysis
using the DESI pipeline. Among several options we tested in Section 7.6, we
present the DESI reanalysis with IFFT RecIso under the BOSS fiducial
cosmology, to best match the clustering amplitude assumed in the published
covariance matrices. We did not add the SDSS systematic errors in the middle
group. We also did not add the DESI systematic errors from Eq. 5.4 to the
bottom group, but the difference is negligible.

Given the changes described above in our pipeline, we revisit the SDSS (BOSS
and eBOSS) data previously analyzed to quantify the impact of these changes on
previously published results [159]. We do so in two stages. First, we refit
the published correlation functions (using the published covariance matrices)
to measure the distance scale. Second, we rerun the reconstruction pipeline
using our new convention and then re-fit the updated correlation functions.
Figure 13 shows the results of these comparisons for the three redshift bins
used in the SDSS ([159]).

The red point of each group in Figure 13 (first from the left) is the
published BOSS/eBOSS result for $\alpha$, while the yellow point (second from
the left) uses the published correlation functions and covariance matrices,
but refits these using the baseline methodology described in this paper. For
the fits performed here, we use damping parameters with Gaussian priors to
match this analysis, while the published analyses used a fixed damping scale.
The differences between these points are due to two primary factors. The first
are changes in how the BAO features are damped. When computing the theoretical
correlation functions in the previous analyses, the power spectrum was damped
by an additional smoothing of $1h^{-1}\,{\rm Mpc}$ to accelerate the
convergence of the Hankel transform and reduce ringing. This effectively
changes the damping scales. Furthermore, previous analyses applied the small-
scale streaming velocities to both the BAO and the broadband shape, while our
current analysis only applies it to the broadband shape (Section 4.3.4).
Additionally, the current analysis has a new separation of the BAO and
broadband shape. The top and middle group of Table 17 shows the net change
from all of these effects.

We also explore the effect of different reconstruction conventions using the
full DESI pipeline and setup. We reprocess the BOSS and eBOSS catalogs and
randoms using pyrecon and pycorr under cosmodesi and fit using the fiducial
setup of Section 4.3 implemented in desilike. The two rightmost points in each
group of Figure 13 show the results of this reprocessing, and the bottom group
of Table 17 shows their impact. When comparing these numbers, we emphasize
that the exact details of how the data sets are assigned to grids and padded
will also affect the derived displacement fields. These different
implementations should be thought of as yielding different data sets, even
though they are derived from the same initial sets of galaxy positions. Given
that the remaining differences are well within the statistical precision of
these samples, we therefore conclude that our new analysis does yield
consistent results with published results. The companion cosmological
interpretation paper [43] will present a detailed consistency check at the
level of the underlying cosmological parameters.

##  8 The final distance measurements and the Hubble Diagram

|
---|---
Figure 14: The final BAO-only distance posteriors, shown as measurements of
$D_{\mathrm{M}}/r_{\mathrm{d}}$, $D_{\mathrm{H}}/r_{\mathrm{d}}$,
$D_{\mathrm{V}}/r_{\mathrm{d}}$, and $D_{\mathrm{H}}/D_{\mathrm{M}}$, compared
against the fiducial Planck 2018-$\Lambda$CDM distances. In the bottom panel,
the prediction at the effective redshift of each sample is shown by the black
dots along the dashed lines: $z_{\rm eff}=0.51$ (LRG1), 0.71 (LRG2), 0.93
(LRG3), and 1.32 (ELG2). The uncertainty intervals on the plot do not take
account of the systematic errors but doing so would make little visible
difference. Figure 15: Hubble diagram of the BAO distance scales measured from
the unblinded galaxy and quasar data, compared to those from earlier BAO
measurements by the 6 degree Field Galaxy Survey (6dFGS, [27]), WiggleZ [103],
the Sloan Digital Sky Survey (SDSS, [159]), and the Dark Energy Survey (DES
Y6, [160]), as labelled. From top to bottom, the panels show
$D_{\mathrm{M}}/r_{\mathrm{d}}$, $D_{\mathrm{H}}/r_{\mathrm{d}}$,
$D_{\mathrm{V}}/r_{\mathrm{d}}$ and $D_{\mathrm{M}}/D_{\mathrm{H}}$, all
relative to the respective quantities evaluated in the DESI fiducial cosmology
described in Section 1. For 6dFGS, WiggleZ and some redshift bins of SDSS and
DESI, only $D_{\mathrm{V}}/r_{\mathrm{d}}$ measurements were possible due to
the low signal-to-noise ratio, so these points are only shown in the third
panel. For the DESI and SDSS redshift bins where both
$D_{\mathrm{M}}/r_{\mathrm{d}}$ and $D_{\mathrm{H}}/r_{\mathrm{d}}$ were
measured, results for $D_{\mathrm{V}}/r_{\mathrm{d}}$ and
$D_{\mathrm{M}}/D_{\mathrm{H}}$ in the third and fourth panels are displayed
with open markers to indicate the repetition of information in the top two
panels in a different parametrisation. Note that a slight offset has been
applied to the effective redshifts of the SDSS results at $z_{\rm eff}=0.51$
and $0.70$ to avoid overlap and ensure visibility in this figure.

In this section, we present our final distance measurements, construct the
Hubble diagram, compare with the previous BAO measurements, and discuss any
implications. We adopt the configuration space measurements as our fiducial
results.

Tracer | $z_{\rm eff}$ | $D_{\rm M}/r_{\rm d}$ | $D_{\rm H}/r_{\rm d}$ | $D_{\rm V}/r_{\rm d}$ | $D_{\rm H}/D_{\rm M}$ | $r_{\rm off}$
---|---|---|---|---|---|---
BGS | 0.30 | — | — | $7.93\pm 0.15$ | — | —
LRG1 | 0.51 | $13.62\pm 0.25$ | $20.98\pm 0.61$ | $12.56\pm 0.15$ | $1.542\pm 0.063$ | $-0.445$
LRG2 | 0.71 | $16.85\pm 0.32$ | $20.08\pm 0.60$ | $15.90\pm 0.20$ | $1.193\pm 0.049$ | $-0.420$
LRG3 | 0.92 | $21.81\pm 0.31$ | $17.83\pm 0.38$ | $19.82\pm 0.18$ | $0.818\pm 0.025$ | $-0.393$
LRG3+ELG1 | 0.93 | $21.71\pm 0.28$ | $17.88\pm 0.35$ | $19.86\pm 0.17$ | $0.824\pm 0.022$ | $-0.389$
ELG1 | 0.95 | — | — | $20.01\pm 0.41$ | — | —
ELG2 | 1.32 | $27.79\pm 0.69$ | $13.82\pm 0.42$ | $24.13\pm 0.36$ | $0.498\pm 0.023$ | $-0.444$
QSO | 1.49 | — | — | $26.07\pm 0.67$ | — | —
Table 18: Final summary of the derived BAO distance scales from fits to the
post-reconstruction correlation function multipoles for each DESI target
sample, including the systematics from Eqs. 5.1 and 5.4. We display results in
terms of the mean values and standard deviations from the marginalized
posteriors of each parameter. $r_{\rm off}$ is the correlation coefficient
between $D_{\rm M}/r_{\rm d}$ and $D_{\rm H}/r_{\rm d}$. The results in this
table are used for the cosmology analysis in [43].

###  8.1 The final distance measurements with systematics

In Table 18, we combine the best-fit BAO measurements, based on Tables 15 and
16, with the total systematic error budget from Eqs. 5.1 and 5.4, and present
the final distance posteriors to be used for cosmology inference in [43]. With
the knowledge of the fiducial distance-to-redshift relation and the sound
horizon scale used in the analysis, we can convert the BAO measurements to the
distance observables: comoving angular diameter distance $D_{M}(z)$,

$\frac{D_{M}(z)}{r_{\mathrm{d}}}\equiv\frac{D_{A}(z)[1+z]}{r_{\mathrm{d}}}=\alpha_{\perp}\frac{D^{\rm
fid}_{M}(z)}{r^{\rm fid}_{d}},$ (8.1)

the Hubble distance $D_{H}(z)$,

$\frac{D_{H}(z)}{r_{\mathrm{d}}}\equiv\frac{c}{H(z)r_{\mathrm{d}}}=\alpha_{\parallel}\frac{D^{\rm
fid}_{H}(z)}{r^{\rm fid}_{d}},$ (8.2)

and the spherically-averaged distance $D_{V}(z)$,

$\frac{D_{V}(z)}{r_{\mathrm{d}}}\equiv\frac{[zD^{2}_{M}(z)D_{H}(z)]^{1/3}}{r_{\mathrm{d}}}=\alpha_{\rm
iso}\frac{D^{\rm fid}_{V}(z)}{r^{\rm fid}_{d}}.$ (8.3)

Figure 14 graphically presents our distance constraints of all tracers
relative to the fiducial Planck 2018-$\Lambda$CDM distances. The plot shows
offsets from the Planck 2018-$\Lambda$CDM predictions for $0.4<z<0.6$ (${\tt
LRG1}\ (z_{\rm eff}=0.51)$) and $0.6<z<0.8$. The deviation at ${\tt LRG1}\
(z_{\rm eff}=0.51)$ is mainly for $D_{H}/r_{\mathrm{d}}$. This means that the
line-of-sight BAO scale appears bigger than the transverse BAO scale, when the
observed space is mapped to the physical space using the metric of the Planck
2018-$\Lambda$CDM cosmology.

On the other hand, the deviation at ${\tt LRG2}\ (z_{\rm eff}=0.71)$ is mainly
in $D_{V}/r_{\mathrm{d}}$ in $D_{M}/r_{\mathrm{d}}$. That is, the transverse
BAO scale at this redshift appears much larger than the prediction of the
Planck 2018-$\Lambda$CDM, giving a smaller $D_{\mathrm{V}}/r_{\mathrm{d}}$ and
$D_{\mathrm{M}}/r_{\mathrm{d}}$ at this redshift. The cosmological implication
of these deviations is extensively tested in [43]. At $z>0.8$, our BAO
measurements are consistent with the expansion history predicted by the Planck
2018-$\Lambda$CDM.

We then construct a Hubble diagram using DESI DR1 BAO-only distance
measurements and also with previous spectroscopic BAO measurements overlaid
(Figure 15). Again, we defer the thorough investigation of the cosmological
implications of these measurements in combination with the DESI DR1
Ly-$\alpha$ BAO measurement ([42]) to [43].

###  8.2 Remarks on the discrepancy between SDSS and DESI over $0.6<z<0.8$

Figure 15 shows noticeable discrepancy between the DESI DR1 measurement at
$z_{\rm eff}=0.71$ (LRG2) and SDSS LRG (eBOSS DR16) at $z_{\rm eff}=0.7$
($0.6<z<0.85$). SDSS LRG measured $D_{M}(0.70)/r_{\mathrm{d}}=17.86\pm 0.33$
and $D_{H}(0.80)/r_{\mathrm{d}}=19.33\pm 0.53$, with $r_{\rm off}=-0.32$
[111], while this paper reports $D_{M}(0.71)/r_{\mathrm{d}}=16.85\pm 0.32$,
$D_{H}(0.71)/r_{\mathrm{d}}=20.08\pm 0.60$ with $r_{\rm off}=-0.420$ for LRG2.

An approximate estimate of the cross-correlation between the two tracers in
the power spectrum space is 0.21. Assuming that this directly propagates to
the cross-correlation of the BAO measurements, the discrepancy is close to
$\sim$3$\sigma$ for $D_{M}/r_{\mathrm{d}}$ and $D_{V}/r_{\mathrm{d}}$ when we
account for the difference between $z_{\rm eff}=0.7$ and $z_{\rm eff}=0.71$
using the Planck 2018-$\Lambda$CDM
cosmology.313131${D_{M}(0.71)/D_{M}(0.7)}]^{\rm fid}=1.011$.

An insight gleaned from Section 7.6 and numerous robustness tests conducted in
this paper is that an estimator derived from a particular dataset undergoing
different pipelines can be regarded as two distinct yet unbiased estimators of
the truth, provided each is individually demonstrated to be unbiased. Hence,
the cross-correlation coefficient could be much lower than the theoretical
estimate, suggesting that the observed 3$\sigma$ discrepancy between SDSS and
DESI is likely an upper limit. However, the lower limit, determined assuming
no correlation, is still 2.7$\sigma$. The DESI reanalysis in Table 17 yields a
slightly lower BAO measurement compared to the published SDSS measurements,
resulting in significances of 2.8$\sigma$ and 2.5$\sigma$, assuming cross-
correlations of 0.21 and 0, respectively.

While this 2.5$\sigma$–3$\sigma$ discrepancy warrants attention, we have not
identified any sources of non-statistical discrepancy from both sets of the
data. We reiterate the significant value of a blinded analysis of DESI DR1,
conducted and unblinded without being affected by prior knowledge of any
discrepancies compared to previous measurements. We anticipate gaining a
deeper understanding of its nature through the analysis of DESI Y3 and Y5
data.

##  9 Conclusion

In this paper, we presented the first BAO scale measurements323232Previously,
we reported the first detection of the DESI DR1 BAO [40] using the first two
months of the DESI data without presenting the BAO scales. from the DESI
galaxies and quasars since the start of its data collection in 2021. The
measurements from DESI DR1 include over 5.7 million galaxy and quasar
redshifts over the redshift range of $0.1<z<2.1$, with the total effective
survey volume of $\sim 18\ \text{Gpc}^{3}$.333333Computed using QSO only over
$1.1<z<1.6$. This marks the largest volume and the most redshifts used for any
spectroscopic BAO measurements. We summarize the final distance constraints
for six redshift bins, detailed in Table 18, as:

  * •

Distance precision (i.e., $D_{\mathrm{V}}/r_{\mathrm{d}}$) ranges of 1.9% (BGS
at $z_{\rm eff}=0.3$), 1.2% (LRG1 at $z_{\rm eff}=0.51$), 1.2% (LRG2 at
$z_{\rm eff}=0.71$), 0.8% (LRG3$+$ELG1 at $z_{\rm eff}\sim 0.95$), 1.5% (ELG2
at $z_{\rm eff}=1.32$), and 2.6% (QSO at $z_{\rm eff}=1.49$). The aggregate
precision of all measurements is 0.52% on the BAO scale.

  * •

The anisotropic distortion of the BAO (i.e., $D_{\mathrm{H}}/D_{\mathrm{M}}$)
is measured with the precision of 4.1% (LRG1), 4.1% (LRG2), 2.7%
(LRG3$+$ELG1), and 4.6% (QSO).

Our measurement marks the highest precision on the cosmological distances for
$z>0.8$, compared to the previous survey, and also collectively marks the
highest aggregate precision. Comparing this with the SDSS BAO measurements
([159, 161]), which report an aggregate precision of 0.64% on
$D_{V}/r_{\mathrm{d}}$ for all galaxy and quasar tracers combined, the
aggregate precision of DESI DR1 in this paper returns 0.52%. The detection
levels range from the highest 9.1$\sigma$ for LRG3$+$ELG1 to the lowest
4.0$\sigma$ for BGS. We produced the first high-significance BAO detection
using galaxies $z>1$, i.e., ELG2 at the significance of 7.0$\sigma$. Proving
the robust nature of the BAO for ELGs, despite its substantial imaging
systematics, is an important result for upcoming Stage-V BAO surveys.

Alongside the state-of-the-art dataset, the DESI DR1 BAO analysis incorporated
several significant novel elements. Our analysis is one of the first to
rigorously implement blinding in the BAO analysis (c.f. [160]) and the first
catalog-level blinded BAO analysis (Section 2.3). This approach was adopted to
mitigate confirmation bias. The true clustering was unveiled only once the
blinded catalogs successfully passed a series of unblinding tests (refer to
Table 14). Anticipating the unprecedented precision of DESI BAO beyond DESI
DR1, our analysis incorporated physically motivated enhancements to the BAO
fitting and reconstruction methods from previous surveys. This included the
adoption of a new reconstruction method (Section 4.2) and improved treatment
for separating BAO from broadband signals (Section 4.3). We introduced a
spline-based broadband model to reduce parameterization dependence on data
signal-to-noise. We constructed and analyzed a combined tracer catalog for
LRGs and ELGs spanning $0.8<z<1.1$, merging information from both tracers and
facilitating an additional systematic test (refer to Section 7.5). This
approach yielded an improvement of approximately 10%, with even greater
improvements anticipated for future DESI analyses.

We employed a unified BAO analysis pipeline and methodology across all galaxy
and quasar tracers, as well as between configuration space and Fourier space,
marking a first in BAO surveys. The algorithms utilized in this study are
incorporated into a publicly accessible repository developed by the
collaboration,343434https://github.com/cosmodesi with some currently available
and others planned for future release. We meticulously coordinated a
systematic study across all tracers and components. Our investigation covered
theoretical modeling effects (Section 4.4), tracer-specific systematics
(Section 5.2), assumptions regarding fiducial cosmology (Section 5.4),
observational systemtaics (Section 5.3), reconstruction algorithm choices
(Section 4.2), and covariance matrix choice (Section 4.5). Summary of these
findings is provided in Table 13, yielding conservative systematics estimates
of 0.245% for the isotropic BAO scale and 0.300% for the anisotropic
distortion of the BAO feature (i.e., the Alcock-Paczynski effect). Further
details are available in supporting papers listed in Table 1.

To quantify the impact of the changes we made in the DESI pipeline, we
revisited the SDSS BAO measurements using the DESI pipelines. We conclude that
the two measurement pipelines yield sufficiently similar results for these
datasets (refer to Section 7.6), although we anticipate that the new method
will be advantageous for the higher-precision dataset expected in the future.

From the Hubble diagram constructed from our results, we found that our BAO
measurements indicate systematically larger observed BAO scales than the
prediction of Planck 2018-$\Lambda$CDM at $z<0.8$, and therefore lower
distances. For $z_{\rm eff}=0.51$, this discrepancy is sourced from the line-
of-sight BAO: it appears bigger than the transverse BAO scale, when the
observed space is mapped to the physical space using the metric of the Planck
2018-$\Lambda$CDM cosmology. For $z_{\rm eff}=0.71$, the discrepancy is
sourced from the overall size of the BAO (and more transverse than the line-
of-sight). The transverse BAO scale at this redshift appears much larger than
the prediction of the Planck 2018-$\Lambda$CDM, implying a smaller
$D_{\mathrm{V}}/r_{\mathrm{d}}$ and $D_{\mathrm{M}}/r_{\mathrm{d}}$ at this
redshift. Accordingly, we found a large discrepancy between the DESI BAO
measurement at $z_{\rm eff}=0.71$ and the SDSS measurement at $z_{\rm
eff}=0.7$. The level of discrepancy is $\sim$3$\sigma$ on
$D_{\mathrm{M}}/r_{\mathrm{d}}$. At $z>0.8$, our BAO measurements are
consistent with that of the Planck 2018-$\Lambda$CDM. We emphasize the value
of our blinded analysis in achieving the DESI DR1 BAO measurements without
prior knowledge of comparisons with previous measurements. An extensive
investigation on the cosmological implication of these discrepancies from
Planck 2018-$\Lambda$CDM, in combination with the BAO measurement from the
Lyman-$\alpha$ forest ([42]), is presented in the DESI DR1 cosmology paper,
[43].

Assuming the same sample definitions (which are conservative for DESI DR1),
the final DESI dataset will be more than a factor of 3 in terms of the
effective volume (4-5 for ELGs, $\sim 3$ for LRGs, and 2-3 for BGS and QSO)
compared to DESI DR1, using more than 20 million unique redshifts. Through the
analysis of the DESI Y3 and Y5 data, we anticipate to gain a deeper
understanding of the nature of the intriguing measurements we observed.

##  10 Data Availability

The data used in this analysis will be made public along the Data Release 1
(details in https://data.desi.lbl.gov/doc/releases/).

## Acknowledgments

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 and Innovation of Spain (MICINN), 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 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.

## References

  * [1] D.H. Weinberg, M.J. Mortonson, D.J. Eisenstein, C. Hirata, A.G. Riess and E. Rozo, _Observational probes of cosmic acceleration_ , _Phys. Rep._ 530 (2013) 87 [1201.2434].
  * [2] E. Abdalla, G.F. Abellán, A. Aboubrahim, A. Agnello, Ö. Akarsu, Y. Akrami et al., _Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies_ , _Journal of High Energy Astrophysics_ 34 (2022) 49 [2203.06142].
  * [3] A.G. Riess, W. Yuan, L.M. Macri, D. Scolnic, D. Brout, S. Casertano et al., _A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s -1 Mpc-1 Uncertainty from the Hubble Space Telescope and the SH0ES Team_, _ApJ_ 934 (2022) L7 [2112.04510].
  * [4] eBOSS Collaboration, S. Alam, M. Aubert, S. Avila, C. Balland, J.E. Bautista et al., _Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory_ , _Phys. Rev. D_ 103 (2021) 083533 [2007.08991].
  * [5] D.J. Eisenstein, _Dark energy and cosmic sound [review article]_ , _New A Rev._ 49 (2005) 360.
  * [6] B. Bassett and R. Hlozek, _Baryon acoustic oscillations_ , in _Dark Energy: Observational and Theoretical Approaches_ , P. Ruiz-Lapuente, ed., p. 246 (2010), DOI.
  * [7] S.-F. Chen, C. Howlett, M. White, P. McDonald, A.J. Ross, H.-J. Seo et al., _Baryon Acoustic Oscillation Theory and Modelling Systematics for the DESI 2024 results_ , _arXiv e-prints_ (2024) arXiv:2402.14070 [2402.14070].
  * [8] P.J.E. Peebles and J.T. Yu, _Primeval Adiabatic Perturbation in an Expanding Universe_ , _ApJ_ 162 (1970) 815.
  * [9] R.A. Sunyaev and Y.B. Zeldovich, _Small-Scale Fluctuations of Relic Radiation_ , _Ap &SS_ 7 (1970) 3.
  * [10] C. Alcock and B. Paczynski, _An evolution free test for non-zero cosmological constant_ , _Nature_ 281 (1979) 358.
  * [11] N. Padmanabhan and M. White, _Constraining anisotropic baryon oscillations_ , _Phys. Rev. D_ 77 (2008) 123540 [0804.0799].
  * [12] D.J. Eisenstein, H.-J. Seo, E. Sirko and D.N. Spergel, _Improving Cosmological Distance Measurements by Reconstruction of the Baryon Acoustic Peak_ , _ApJ_ 664 (2007) 675 [astro-ph/0604362].
  * [13] H.-J. Seo, E.R. Siegel, D.J. Eisenstein and M. White, _Nonlinear Structure Formation and the Acoustic Scale_ , _ApJ_ 686 (2008) 13 [0805.0117].
  * [14] N. Padmanabhan, M. White and J.D. Cohn, _Reconstructing baryon oscillations: A Lagrangian theory perspective_ , _Phys. Rev. D_ 79 (2009) 063523 [0812.2905].
  * [15] H.-J. Seo, J. Eckel, D.J. Eisenstein, K. Mehta, M. Metchnik, N. Padmanabhan et al., _High-precision Predictions for the Acoustic Scale in the Nonlinear Regime_ , _ApJ_ 720 (2010) 1650 [0910.5005].
  * [16] Y. Noh, M. White and N. Padmanabhan, _Reconstructing baryon oscillations_ , _Phys. Rev. D_ 80 (2009) 123501 [0909.1802].
  * [17] M. White, _Reconstruction within the Zeldovich approximation_ , _MNRAS_ 450 (2015) 3822 [1504.03677].
  * [18] C. Hikage, K. Koyama and R. Takahashi, _Perturbation theory for the redshift-space matter power spectra after reconstruction_ , _Phys. Rev. D_ 101 (2020) 043510 [1911.06461].
  * [19] L. Anderson, E. Aubourg, S. Bailey, D. Bizyaev, M. Blanton, A.S. Bolton et al., _The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Data Release 9 spectroscopic galaxy sample_ , _MNRAS_ 427 (2012) 3435 [1203.6594].
  * [20] A.J. Ross, F. Beutler, C.-H. Chuang, M. Pellejero-Ibanez, H.-J. Seo, M. Vargas-Magaña et al., _The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: observational systematics and baryon acoustic oscillations in the correlation function_ , _MNRAS_ 464 (2017) 1168 [1607.03145].
  * [21] M. Ata, F. Baumgarten, J. Bautista, F. Beutler, D. Bizyaev, M.R. Blanton et al., _The clustering of the SDSS-IV extended Baryon Oscillation Spectroscopic Survey DR14 quasar sample: first measurement of baryon acoustic oscillations between redshift 0.8 and 2.2_ , _MNRAS_ 473 (2018) 4773 [1705.06373].
  * [22] G. Merz, M. Rezaie, H.-J. Seo, R. Neveux, A.J. Ross, F. Beutler et al., _The clustering of the SDSS-IV extended Baryon Oscillation Spectroscopic Survey quasar sample: testing observational systematics on the Baryon Acoustic Oscillation measurement_ , _MNRAS_ 506 (2021) 2503 [2105.10463].
  * [23] Rosado et al._, in preparation_ (2024) .
  * [24] Yu et al._, in preparation_ (2024) .
  * [25] D.J. Eisenstein, I. Zehavi, D.W. Hogg, R. Scoccimarro, M.R. Blanton, R.C. Nichol et al., _Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies_ , _ApJ_ 633 (2005) 560 [astro-ph/0501171].
  * [26] S. Cole, W.J. Percival, J.A. Peacock, P. Norberg, C.M. Baugh, C.S. Frenk et al., _The 2dF Galaxy Redshift Survey: power-spectrum analysis of the final data set and cosmological implications_ , _MNRAS_ 362 (2005) 505 [astro-ph/0501174].
  * [27] F. Beutler, C. Blake, M. Colless, D.H. Jones, L. Staveley-Smith, L. Campbell et al., _The 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant_ , _MNRAS_ 416 (2011) 3017 [1106.3366].
  * [28] S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J.A. Blazek et al., _The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample_ , _MNRAS_ 470 (2017) 2617 [1607.03155].
  * [29] C. Blake, S. Brough, M. Colless, C. Contreras, W. Couch, S. Croom et al., _The WiggleZ Dark Energy Survey: joint measurements of the expansion and growth history at z ¡ 1_ , _MNRAS_ 425 (2012) 405 [1204.3674].
  * [30] H. du Mas des Bourboux, J. Rich, A. Font-Ribera, V. de Sainte Agathe, J. Farr, T. Etourneau et al., _The Completed SDSS-IV Extended Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations with Ly $\alpha$ Forests_, _ApJ_ 901 (2020) 153 [2007.08995].
  * [31] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al., _Planck 2018 results. VI. Cosmological parameters_ , _A &A_ 641 (2020) A6 [1807.06209].
  * [32] G.E. Addison, G. Hinshaw and M. Halpern, _Cosmological constraints from baryon acoustic oscillations and clustering of large-scale structure_ , _MNRAS_ 436 (2013) 1674 [1304.6984].
  * [33] DESI Collaboration, A. Aghamousa, J. Aguilar, S. Ahlen, S. Alam, L.E. Allen et al., _The DESI Experiment Part I: Science,Targeting, and Survey Design_ , _arXiv e-prints_ (2016) arXiv:1611.00036 [1611.00036].
  * [34] DESI Collaboration, A. Aghamousa, J. Aguilar, S. Ahlen, S. Alam, L.E. Allen et al., _The DESI Experiment Part II: Instrument Design_ , _arXiv e-prints_ (2016) arXiv:1611.00037 [1611.00037].
  * [35] DESI Collaboration, B. Abareshi, J. Aguilar, S. Ahlen, S. Alam, D.M. Alexander et al., _Overview of the Instrumentation for the Dark Energy Spectroscopic Instrument_ , _AJ_ 164 (2022) 207 [2205.10939].
  * [36] A.D. Myers et al., _The Target Selection Pipeline for the Dark Energy Spectroscopic Instrument_ , 2208.08518.
  * [37] J. Guy et al., _The Spectroscopic Data Processing Pipeline for the Dark Energy Spectroscopic Instrument_ , _Astron. J._ 165 (2023) 144 [2209.14482].
  * [38] DESI Collaboration, A.G. Adame, J. Aguilar, S. Ahlen, S. Alam, G. Aldering et al., _Validation of the Scientific Program for the Dark Energy Spectroscopic Instrument_ , _AJ_ 167 (2024) 62 [2306.06307].
  * [39] DESI Collaboration, A.G. Adame, J. Aguilar, S. Ahlen, S. Alam, G. Aldering et al., _The Early Data Release of the Dark Energy Spectroscopic Instrument_ , _arXiv e-prints_ (2023) arXiv:2306.06308 [2306.06308].
  * [40] J. Moon, D. Valcin, M. Rashkovetskyi, C. Saulder, J.N. Aguilar, S. Ahlen et al., _First detection of the BAO signal from early DESI data_ , _MNRAS_ 525 (2023) 5406 [2304.08427].
  * [41] C. Gordon, A. Cuceu, J. Chaves-Montero, A. Font-Ribera, A.X. González-Morales, J. Aguilar et al., _3D correlations in the Lyman- $\alpha$ forest from early DESI data_, _J. Cosmology Astropart. Phys_ 2023 (2023) 045 [2308.10950].
  * [42] DESI Collaboration, _DESI 2024 IV: Baryon Acoustic Oscillations from the Lyman Alpha Forest_ , _in preparation_ (2024) .
  * [43] DESI Collaboration, _DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations_ , _in preparation_ (2024) .
  * [44] M. Rashkovetskyi, D.J. Eisenstein, J.N. Aguilar, D. Brooks, T. Claybaugh, S. Cole et al., _Validation of semi-analytical, semi-empirical covariance matrices for two-point correlation function for early DESI data_ , _MNRAS_ 524 (2023) 3894 [2306.06320].
  * [45] Chen et al._, in preparation_ (2024) .
  * [46] Paillas et al._, in preparation_ (2024) .
  * [47] Valcin et al._, in preparation_ (2024) .
  * [48] Forero-Sanchez et al._, in preparation_ (2024) .
  * [49] Rashkovetskyi et al._, in preparation_ (2024) .
  * [50] Alves et al._, in preparation_ (2024) .
  * [51] Perez-Fernandez et al._, in preparation_ (2024) .
  * [52] Mena-Fernandez et al._, in preparation_ (2024) .
  * [53] Garcia-Quintero et al._, in preparation_ (2024) .
  * [54] Andrade et al._, in preparation_ (2024) .
  * [55] N.A. Maksimova, L.H. Garrison, D.J. Eisenstein, B. Hadzhiyska, S. Bose and T.P. Satterthwaite, _ABACUSSUMMIT: a massive set of high-accuracy, high-resolution N-body simulations_ , _MNRAS_ 508 (2021) 4017 [2110.11398].
  * [56] DESI Collaboration, _DESI 2024 I: Data Release 1 of the Dark Energy Spectroscopic Instrument_ , _in preparation_ (2024) .
  * [57] B. Abareshi, J. Aguilar, S. Ahlen, S. Alam, D.M. Alexander, R. Alfarsy et al., _Overview of the Instrumentation for the Dark Energy Spectroscopic Instrument_ , _AJ_ 164 (2022) 207 [2205.10939].
  * [58] DESI Collaboration, A.G. Adame, J. Aguilar, S. Ahlen, S. Alam, G. Aldering et al., _Validation of the Scientific Program for the Dark Energy Spectroscopic Instrument_ , _arXiv e-prints_ (2023) arXiv:2306.06307 [2306.06307].
  * [59] J.H. Silber, P. Fagrelius, K. Fanning, M. Schubnell, J.N. Aguilar, S. Ahlen et al., _The robotic multiobject focal plane system of the dark energy spectroscopic instrument (DESI)_ , _The Astronomical Journal_ 165 (2022) 9.
  * [60] Raichoor et al. in, _Dark Energy Spectroscopic Instrument Fiber Assignment_ , xxxx.xxxxx.
  * [61] Schlafly et al. in, _Dark Energy Spectroscopic Instrument Survey Operations_ , xxxx.xxxxx.
  * [62] C. Hahn et al., _DESI Bright Galaxy Survey: Final Target Selection, Design, and Validation_ , 2208.08512.
  * [63] R. Zhou et al., _Target Selection and Validation of DESI Luminous Red Galaxies_ , 2208.08515.
  * [64] E. Chaussidon et al., _Target Selection and Validation of DESI Quasars_ , 2208.08511.
  * [65] A. Raichoor et al., _Target Selection and Validation of DESI Emission Line Galaxies_ , 2208.08513.
  * [66] A.J. Ross et al., _Construction of Large-scale Structure Catalogs for Data from the Dark Energy Spectroscopic Instrument_ , _in prep._ (2024) [xxxx.xxxxx].
  * [67] DESI Collaboration, _DESI 2024 II: Two Point Clustering Measurements and Validation_ , _in preparation_ (2024) .
  * [68] A. Dey, D.J. Schlegel, D. Lang, R. Blum, K. Burleigh, X. Fan et al., _Overview of the DESI Legacy Imaging Surveys_ , _AJ_ 157 (2019) 168 [1804.08657].
  * [69] Zhou et al._, in preparation_ (2024) .
  * [70] Chaussidon et al._, in preparation_ (2024) .
  * [71] Krolewski et al._, in preparation_ (2024) .
  * [72] Lasker et al._, in preparation_ (2024) .
  * [73] Bianchi et al._, in preparation_ (2024) .
  * [74] Pinon et al._, in preparation_ (2024) .
  * [75] H.A. Feldman, N. Kaiser and J.A. Peacock, _Power spectrum analysis of three-dimensional redshift surveys_ , _Astrophys. J._ 426 (1994) 23 [astro-ph/9304022].
  * [76] A. Font-Ribera, P. McDonald, N. Mostek, B.A. Reid, H.-J. Seo and A. Slosar, _DESI and other dark energy experiments in the era of neutrino mass measurements_ , _JCAP_ 05 (2014) 023 [1308.4164].
  * [77] C. Hahn, M.J. Wilson, O. Ruiz-Macias, S. Cole, D.H. Weinberg, J. Moustakas et al., _The DESI Bright Galaxy Survey: Final Target Selection, Design, and Validation_ , _AJ_ 165 (2023) 253 [2208.08512].
  * [78] J. Moustakas et al., _fastspecfit_ , _in prep._ (2024) [xxxx.xxxxx].
  * [79] J. Moustakas, D. Scholte, B. Dey and A. Khederlarian, “FastSpecFit: Fast spectral synthesis and emission-line fitting of DESI spectra.” Astrophysics Source Code Library, record ascl:2308.005, Aug., 2023.
  * [80] S. Yuan, H. Zhang, A.J. Ross, J. Donald-McCann, B. Hadzhiyska, R.H. Wechsler et al., _The DESI One-Percent Survey: Exploring the Halo Occupation Distribution of Luminous Red Galaxies and Quasi-Stellar Objects with AbacusSummit_ , _arXiv e-prints_ (2023) arXiv:2306.06314 [2306.06314].
  * [81] S. Brieden, H. Gil-Marín, L. Verde and J.L. Bernal, _Blind Observers of the Sky_ , _J. Cosmology Astropart. Phys_ 2020 (2020) 052 [2006.10857].
  * [82] B. Hadzhiyska, M.J. White, X. Chen, L.H. Garrison, J. DeRose, N. Padmanabhan et al., _Mitigating the noise of DESI mocks using analytic control variates_ , _The Open Journal of Astrophysics_ 6 (2023) 38 [2308.12343].
  * [83] L.H. Garrison, D.J. Eisenstein and P.A. Pinto, _A high-fidelity realization of the Euclid code comparison N-body simulation with ABACUS_ , _MNRAS_ 485 (2019) 3370 [1810.02916].
  * [84] B. Hadzhiyska, D. Eisenstein, S. Bose, L.H. Garrison and N. Maksimova, _COMPASO: A new halo finder for competitive assignment to spherical overdensities_ , _MNRAS_ 509 (2022) 501 [2110.11408].
  * [85] S. Bose, D.J. Eisenstein, B. Hadzhiyska, L.H. Garrison and S. Yuan, _Constructing high-fidelity halo merger trees in ABACUSSUMMIT_ , _MNRAS_ 512 (2022) 837 [2110.11409].
  * [86] S. Yuan, L.H. Garrison, B. Hadzhiyska, S. Bose and D.J. Eisenstein, _ABACUSHOD: A highly efficient extended multi-tracer HOD framework and its application to BOSS and eBOSS data_ , _MNRAS_ 510 (2021) 3301 [2110.11412].
  * [87] S. Yuan, L.H. Garrison, B. Hadzhiyska, S. Bose and D.J. Eisenstein, _ABACUSHOD: a highly efficient extended multitracer HOD framework and its application to BOSS and eBOSS data_ , _MNRAS_ 510 (2022) 3301 [2110.11412].
  * [88] S. Alam, J.A. Peacock, K. Kraljic, A.J. Ross and J. Comparat, _Multitracer extension of the halo model: probing quenching and conformity in eBOSS_ , _MNRAS_ 497 (2020) 581 [1910.05095].
  * [89] A. Rocher, V. Ruhlmann-Kleider, E. Burtin, S. Yuan, A. de Mattia, A.J. Ross et al., _The DESI One-Percent survey: exploring the Halo Occupation Distribution of Emission Line Galaxies with ABACUSSUMMIT simulations_ , _J. Cosmology Astropart. Phys_ 2023 (2023) 016 [2306.06319].
  * [90] A. Smith, C. Grove, S. Cole, P. Norberg, P. Zarrouk, S. Yuan et al., _Generating mock galaxy catalogues for flux-limited samples like the DESI Bright Galaxy Survey_ , _arXiv e-prints_ (2023) arXiv:2312.08792 [2312.08792].
  * [91] C.-H. Chuang, F.-S. Kitaura, F. Prada, C. Zhao and G. Yepes, _EZmocks: extending the Zel’dovich approximation to generate mock galaxy catalogues with accurate clustering statistics_ , _MNRAS_ 446 (2015) 2621 [1409.1124].
  * [92] C. Zhao, C.-H. Chuang, J. Bautista, A. de Mattia, A. Raichoor, A.J. Ross et al., _The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: 1000 multi-tracer mock catalogues with redshift evolution and systematics for galaxies and quasars of the final data release_ , _MNRAS_ 503 (2021) 1149 [2007.08997].
  * [93] P. Zarrouk, M. Rezaie, A. Raichoor, A.J. Ross, S. Alam, R. Blum et al., _Baryon acoustic oscillations in the projected cross-correlation function between the eBOSS DR16 quasars and photometric galaxies from the DESI Legacy Imaging Surveys_ , _MNRAS_ 503 (2021) 2562 [2009.02308].
  * [94] Y. Zel’dovich, _Gravitational instability: An Approximate theory for large density perturbations_ , _å_ 5 (1970) 84.
  * [95] Zhao et al._, in preparation_ (2024) .
  * [96] DESI Collaboration, _The Dark Energy Survey Instrument Year 1 Results: 2-Point Clustering Measurements and their Validation_ , _in preparation_ (2024) .
  * [97] A. de Mattia and V. Ruhlmann-Kleider, _Integral constraints in spectroscopic surveys_ , _JCAP_ 08 (2019) 036 [1904.08851].
  * [98] S.D. Landy and A.S. Szalay, _Bias and Variance of Angular Correlation Functions_ , _ApJ_ 412 (1993) 64.
  * [99] N. Padmanabhan, X. Xu, D.J. Eisenstein, R. Scalzo, A.J. Cuesta, K.T. Mehta et al., _A 2 per cent distance to z = 0.35 by reconstructing baryon acoustic oscillations - I. Methods and application to the Sloan Digital Sky Survey_ , _MNRAS_ 427 (2012) 2132 [1202.0090].
  * [100] K. Yamamoto, M. Nakamichi, A. Kamino, B.A. Bassett and H. Nishioka, _A Measurement of the Quadrupole Power Spectrum in the Clustering of the 2dF QSO Survey_ , _Publications of the Astronomical Society of Japan_ 58 (2006) 93.
  * [101] N. Hand, Y. Feng, F. Beutler, Y. Li, C. Modi, U. Seljak et al., _nbodykit: An Open-source, Massively Parallel Toolkit for Large-scale Structure_ , _AJ_ 156 (2018) 160 [1712.05834].
  * [102] R. Tojeiro, A.J. Ross, A. Burden, L. Samushia, M. Manera, W.J. Percival et al., _The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: galaxy clustering measurements in the low-redshift sample of Data Release 11_ , _MNRAS_ 440 (2014) 2222 [1401.1768].
  * [103] E.A. Kazin, J. Koda, C. Blake, N. Padmanabhan, S. Brough, M. Colless et al., _The WiggleZ Dark Energy Survey: improved distance measurements to z = 1 with reconstruction of the baryonic acoustic feature_ , _MNRAS_ 441 (2014) 3524 [1401.0358].
  * [104] S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J.A. Blazek et al., _The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample_ , _MNRAS_ 470 (2017) 2617 [1607.03155].
  * [105] H. Gil-Marín, J.E. Bautista, R. Paviot, M. Vargas-Magaña, S. de la Torre, S. Fromenteau et al., _The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: measurement of the BAO and growth rate of structure of the luminous red galaxy sample from the anisotropic power spectrum between redshifts 0.6 and 1.0_ , _MNRAS_ 498 (2020) 2492 [2007.08994].
  * [106] H.-J. Seo, J. Eckel, D.J. Eisenstein, K. Mehta, M. Metchnik, N. Padmanabhan et al., _High-precision Predictions for the Acoustic Scale in the Nonlinear Regime_ , _ApJ_ 720 (2010) 1650 [0910.5005].
  * [107] M. Schmittfull, T. Baldauf and M. Zaldarriaga, _Iterative initial condition reconstruction_ , _Phys. Rev. D_ 96 (2017) 023505 [1704.06634].
  * [108] R. Hada and D.J. Eisenstein, _An iterative reconstruction of cosmological initial density fields_ , _MNRAS_ 478 (2018) 1866 [1804.04738].
  * [109] H.-J. Seo, F. Beutler, A.J. Ross and S. Saito, _Modeling the reconstructed BAO in Fourier space_ , _MNRAS_ 460 (2016) 2453 [1511.00663].
  * [110] A. Burden, W.J. Percival and C. Howlett, _Reconstruction in Fourier space_ , _MNRAS_ 453 (2015) 456 [1504.02591].
  * [111] J.E. Bautista, R. Paviot, M. Vargas Magaña, S. de la Torre, S. Fromenteau, H. Gil-Marín et al., _The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: measurement of the BAO and growth rate of structure of the luminous red galaxy sample from the anisotropic correlation function between redshifts 0.6 and 1_ , _MNRAS_ 500 (2021) 736 [2007.08993].
  * [112] L. Anderson, É. Aubourg, S. Bailey, F. Beutler, V. Bhardwaj, M. Blanton et al., _The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Data Releases 10 and 11 Galaxy samples_ , _MNRAS_ 441 (2014) 24 [1312.4877].
  * [113] M. Vargas-Magaña, S. Ho, X. Xu, A.G. Sánchez, R. O’Connell, D.J. Eisenstein et al., _The clustering of Galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: potential systematics in fitting of baryon acoustic feature_ , _MNRAS_ 445 (2014) 2.
  * [114] M. Vargas-Magaña, S. Ho, A.J. Cuesta, R. O’Connell, A.J. Ross, D.J. Eisenstein et al., _The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: theoretical systematics and Baryon Acoustic Oscillations in the galaxy correlation function_ , _MNRAS_ 477 (2018) 1153 [1610.03506].
  * [115] F. Beutler, H.-J. Seo, A.J. Ross, P. McDonald, S. Saito, A.S. Bolton et al., _The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Fourier space_ , _MNRAS_ 464 (2017) 3409 [1607.03149].
  * [116] S. Brieden, H. Gil-Marín and L. Verde, _Model-agnostic interpretation of 10 billion years of cosmic evolution traced by BOSS and eBOSS data_ , _J. Cosmology Astropart. Phys_ 2022 (2022) 024 [2204.11868].
  * [117] H.-J. Seo, F. Beutler, A.J. Ross and S. Saito, _Modeling the reconstructed BAO in Fourier space_ , _MNRAS_ 460 (2016) 2453 [1511.00663].
  * [118] J.C. Jackson, _A critique of Rees’s theory of primordial gravitational radiation_ , _MNRAS_ 156 (1972) 1P [0810.3908].
  * [119] C. Park, M.S. Vogeley, M.J. Geller and J.P. Huchra, _Power Spectrum, Correlation Function, and Tests for Luminosity Bias in the CfA Redshift Survey_ , _ApJ_ 431 (1994) 569.
  * [120] N. Kaiser, _Clustering in real space and in redshift space_ , _Mon. Not. Roy. Astron. Soc._ 227 (1987) 1.
  * [121] N. Sugiyama, _Developing a Theoretical Model for the Resummation of Infrared Effects in the Post-Reconstruction Power Spectrum_ , _arXiv e-prints_ (2024) arXiv:2402.06142 [2402.06142].
  * [122] A.K. Chaniotis and D. Poulikakos, _High order interpolation and differentiation using B-splines_ , _Journal of Computational Physics_ 197 (2004) 253.
  * [123] E. Sefusatti, M. Crocce, R. Scoccimarro and H.M.P. Couchman, _Accurate estimators of correlation functions in Fourier space_ , _MNRAS_ 460 (2016) 3624 [1512.07295].
  * [124] A.J. Ross, W.J. Percival, A. Carnero, G.-b. Zhao, M. Manera, A. Raccanelli et al., _The clustering of galaxies in the SDSS-III DR9 Baryon Oscillation Spectroscopic Survey: constraints on primordial non-Gaussianity_ , _MNRAS_ 428 (2013) 1116 [1208.1491].
  * [125] F. Beutler and P. McDonald, _Unified galaxy power spectrum measurements from 6dFGS, BOSS, and eBOSS_ , _J. Cosmology Astropart. Phys_ 2021 (2021) 031 [2106.06324].
  * [126] J. Bradbury, R. Frostig, P. Hawkins, M.J. Johnson, C. Leary, D. Maclaurin et al., _JAX: composable transformations of Python+NumPy programs_ , 2018.
  * [127] D. Foreman-Mackey, D.W. Hogg, D. Lang and J. Goodman, _emcee: The mcmc hammer_ , _PASP_ 125 (2013) 306 [1202.3665].
  * [128] F. James and M. Roos, _Minuit - a system for function minimization and analysis of the parameter errors and correlations_ , _Computer Physics Communications_ 10 (1975) 343.
  * [129] S.R. Hinton, C. Howlett and T.M. Davis, _BARRY and the BAO model comparison_ , _MNRAS_ 493 (2020) 4078 [1912.01175].
  * [130] R. O’Connell, D. Eisenstein, M. Vargas, S. Ho and N. Padmanabhan, _Large covariance matrices: smooth models from the two-point correlation function_ , _MNRAS_ 462 (2016) 2681 [1510.01740].
  * [131] R. O’Connell and D.J. Eisenstein, _Large covariance matrices: accurate models without mocks_ , _MNRAS_ 487 (2019) 2701 [1808.05978].
  * [132] O.H.E. Philcox, D.J. Eisenstein, R. O’Connell and A. Wiegand, _RASCALC: a jackknife approach to estimating single- and multitracer galaxy covariance matrices_ , _MNRAS_ 491 (2020) 3290 [1904.11070].
  * [133] O.H.E. Philcox and D.J. Eisenstein, _Estimating covariance matrices for two- and three-point correlation function moments in Arbitrary Survey Geometries_ , _MNRAS_ 490 (2019) 5931 [1910.04764].
  * [134] M. Rashkovetskyi, D.J. Eisenstein, J.N. Aguilar, D. Brooks, T. Claybaugh, S. Cole et al., _Validation of semi-analytical, semi-empirical covariance matrices for two-point correlation function for early DESI data_ , _MNRAS_ 524 (2023) 3894 [2306.06320].
  * [135] D. Wadekar and R. Scoccimarro, _Galaxy power spectrum multipoles covariance in perturbation theory_ , _Phys. Rev. D_ 102 (2020) 123517 [1910.02914].
  * [136] D.J. Eisenstein, H.-J. Seo and M. White, _On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter_ , _ApJ_ 664 (2007) 660 [astro-ph/0604361].
  * [137] M. Crocce and R. Scoccimarro, _Nonlinear evolution of baryon acoustic oscillations_ , _Phys. Rev. D_ 77 (2008) 023533 [0704.2783].
  * [138] H.-J. Seo, E.R. Siegel, D.J. Eisenstein and M. White, _Nonlinear Structure Formation and the Acoustic Scale_ , _ApJ_ 686 (2008) 13 [0805.0117].
  * [139] N. Padmanabhan and M. White, _Calibrating the baryon oscillation ruler for matter and halos_ , _Phys. Rev. D_ 80 (2009) 063508 [0906.1198].
  * [140] B.D. Sherwin and M. Zaldarriaga, _Shift of the baryon acoustic oscillation scale: A simple physical picture_ , _Phys. Rev. D_ 85 (2012) 103523 [1202.3998].
  * [141] G. Rossi et al., _The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: N-body mock challenge for galaxy clustering measurements_ , _Mon. Not. Roy. Astron. Soc._ 505 (2021) 377 [2007.09002].
  * [142] BOSS collaboration, _The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: Observational systematics and baryon acoustic oscillations in the correlation function_ , _Mon. Not. Roy. Astron. Soc._ 464 (2017) 1168 [1607.03145].
  * [143] J. Yu, C. Zhao, V. Gonzalez-Perez, C.-H. Chuang, A. Brodzeller, A. de Mattia et al., _The DESI One-Percent Survey: exploring a generalized SHAM for multiple tracers with the UNIT simulation_ , _MNRAS_ 527 (2024) 6950 [2306.06313].
  * [144] A. Raichoor, J. Moustakas, J.A. Newman, T. Karim, S. Ahlen, S. Alam et al., _Target Selection and Validation of DESI Emission Line Galaxies_ , _AJ_ 165 (2023) 126 [2208.08513].
  * [145] W.E. Ballinger, J.A. Peacock and A.F. Heavens, _Measuring the cosmological constant with redshift surveys_ , _MNRAS_ 282 (1996) 877 [astro-ph/9605017].
  * [146] H.-J. Seo and D.J. Eisenstein, _Probing Dark Energy with Baryonic Acoustic Oscillations from Future Large Galaxy Redshift Surveys_ , _ApJ_ 598 (2003) 720 [astro-ph/0307460].
  * [147] K. Thepsuriya and A. Lewis, _Accuracy of cosmological parameters using the baryon acoustic scale_ , _Journal of Cosmology and Astroparticle Physics_ 2015 (2015) 034–034.
  * [148] P. Carter, F. Beutler, W.J. Percival, J. DeRose, R.H. Wechsler and C. Zhao, _The impact of the fiducial cosmology assumption on BAO distance scale measurements_ , _MNRAS_ 494 (2020) 2076 [1906.03035].
  * [149] J.L. Bernal, T.L. Smith, K.K. Boddy and M. Kamionkowski, _Robustness of baryon acoustic oscillation constraints for early-Universe modifications of $\Lambda$ CDM cosmology_, _Phys. Rev. D_ 102 (2020) 123515 [2004.07263].
  * [150] J. Pan, D. Huterer, F. Andrade-Oliveira and C. Avestruz, _Compressed baryon acoustic oscillation analysis is robust to modified-gravity models_ , 2023\.
  * [151] S. Sanz-Wuhl, H. Gil-Marín, A.J. Cuesta and L. Verde, _Bao cosmology in non-spatially flat background geometry with application to boss and eboss_ , 2024\.
  * [152] N.A. Maksimova, L.H. Garrison, D.J. Eisenstein, B. Hadzhiyska, S. Bose and T.P. Satterthwaite, _ABACUSSUMMIT: a massive set of high-accuracy, high-resolution N-body simulations_ , _MNRAS_ 508 (2021) 4017 [2110.11398].
  * [153] D. Baumann, D. Green and M. Zaldarriaga, _Phases of new physics in the bao spectrum_ , _Journal of Cosmology and Astroparticle Physics_ 2017 (2017) 007–007.
  * [154] C. Hikage, K. Koyama and A. Heavens, _Perturbation theory for BAO reconstructed fields: One-loop results in the real-space matter density field_ , _Phys. Rev. D_ 96 (2017) 043513 [1703.07878].
  * [155] S.-F. Chen, Z. Vlah and M. White, _The reconstructed power spectrum in the Zeldovich approximation_ , _J. Cosmology Astropart. Phys_ 2019 (2019) 017 [1907.00043].
  * [156] D.J. Eisenstein, I. Zehavi, D.W. Hogg, R. Scoccimarro, M.R. Blanton, R.C. Nichol et al., _Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies_ , _ApJ_ 633 (2005) 560 [astro-ph/0501171].
  * [157] H.-J. Seo and D.J. Eisenstein, _Improved Forecasts for the Baryon Acoustic Oscillations and Cosmological Distance Scale_ , _ApJ_ 665 (2007) 14 [astro-ph/0701079].
  * [158] A. Tamone, A. Raichoor, C. Zhao, A. de Mattia, C. Gorgoni, E. Burtin et al., _The completed SDSS-IV extended baryon oscillation spectroscopic survey: growth rate of structure measurement from anisotropic clustering analysis in configuration space between redshift 0.6 and 1.1 for the emission-line galaxy sample_ , _MNRAS_ 499 (2020) 5527 [2007.09009].
  * [159] S. Alam, M. Aubert, S. Avila, C. Balland, J.E. Bautista, M.A. Bershady et al., _Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory_ , _Phys. Rev. D_ 103 (2021) 083533 [2007.08991].
  * [160] DES Collaboration, T.M.C. Abbott, M. Adamow, M. Aguena, S. Allam, O. Alves et al., _Dark Energy Survey: A 2.1% measurement of the angular Baryonic Acoustic Oscillation scale at redshift $z_{\rm eff}$=0.85 from the final dataset_, _arXiv e-prints_ (2024) arXiv:2402.10696 [2402.10696].
  * [161] R. Neveux, E. Burtin, A. de Mattia, A. Smith, A.J. Ross, J. Hou et al., _The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: BAO and RSD measurements from the anisotropic power spectrum of the quasar sample between redshift 0.8 and 2.2_ , _MNRAS_ 499 (2020) 210 [2007.08999].

##  Appendix A BAO measurements in Fourier space

Tracer | Redshift | Recon | $\alpha_{\rm iso}$ | $\alpha_{\rm AP}$ | $r_{\rm off}$ | $\chi^{2}/{\rm dof}$
---|---|---|---|---|---|---
BGS | 0.1–0.4 | Post | $0.9848\pm 0.0242$ |  |  | 27.4/41
LRG1 | 0.4–0.6 | Post | $0.9799\pm 0.0118$ | $0.9231\pm 0.0371$ | -0.1413 | 85.9/87
LRG2 | 0.6–0.8 | Post | $0.9624\pm 0.0098$ | $1.0431\pm 0.0356$ | -0.0373 | 82.1/87
LRG3 | 0.8–1.1 | Post | $1.0005\pm 0.0081$ | $1.0057\pm 0.0261$ | -0.0991 | 100.8/87
ELG1 | 0.8–1.1 | Post | $0.9913\pm 0.0249$ |  |  | 47.6/41
ELG2 | 1.1–1.6 | Post | $0.9827\pm 0.0112$ | $0.9986\pm 0.0352$ | -0.2217 | 103.8/87
QSO | 0.8–2.1 | Post | $1.0094\pm 0.0212$ |  |  | 57.7/41
BGS | 0.1–0.4 | Pre | $0.9657\pm 0.0364$ |  |  | 50.0/41
LRG1 | 0.4–0.6 | Pre | $0.9774\pm 0.0179$ | $0.9316\pm 0.0595$ | 0.2927 | 87.1/87
LRG2 | 0.6–0.8 | Pre | $0.9487\pm 0.0148$ | $1.0324\pm 0.0610$ | 0.4315 | 87.5/87
LRG3 | 0.8–1.1 | Pre | $1.0022\pm 0.0107$ | $1.0132\pm 0.0387$ | 0.2259 | 108.0/87
ELG1 | 0.8–1.1 | Pre | $0.9819\pm 0.0619$ |  |  | 51.2/41
ELG2 | 1.1–1.6 | Pre | $0.9873\pm 0.0152$ | $1.0191\pm 0.0558$ | 0.1395 | 95.6/87
QSO | 0.8–2.1 | Pre | $1.0006\pm 0.0206$ |  |  | 49.3/41
Table 19: Mean values and standard deviations from the marginalized posteriors of the BAO scaling parameters from fits to the unblinded DESI DR1 power spectra in the $\alpha_{\rm iso}$-$\alpha_{\rm AP}$ basis. Systematic effects are not included. The $P(k)$ fits do not include LRG3$+$ELG1, as we do not have a covariance available for the combined tracer in $P(k)$. $r_{\rm off}=C_{\alpha_{\rm iso},\alpha_{\rm AP}}/\sqrt{C_{\alpha_{\rm iso},\alpha_{\rm iso}}C_{\alpha_{\rm AP},\alpha_{\rm AP}}}$. Tracer | Redshift | Recon | $\alpha_{\parallel}$ | $\alpha_{\perp}$ | $r_{\rm off}$ | $\chi^{2}/{\rm dof}$
---|---|---|---|---|---|---
LRG1 | 0.4-0.6 | Post | $0.928\pm 0.023$ | $1.007\pm 0.018$ | -0.429 | 85.9/87
LRG2 | 0.6-0.8 | Post | $0.987\pm 0.022$ | $0.949\pm 0.014$ | -0.419 | 82.1/87
LRG3 | 0.8-1.1 | Post | $1.004\pm 0.018$ | $0.999\pm 0.012$ | -0.401 | 100.8/87
ELG2 | 1.1-1.6 | Post | $0.979\pm 0.022$ | $0.984\pm 0.018$ | -0.416 | 103.8/87
LRG1 | 0.4-0.6 | Pre | $0.924\pm 0.040$ | $1.003\pm 0.021$ | -0.399 | 87.1/87
LRG2 | 0.6-0.8 | Pre | $0.975\pm 0.044$ | $0.937\pm 0.017$ | -0.440 | 87.5/87
LRG3 | 0.8-1.1 | Pre | $1.012\pm 0.029$ | $0.997\pm 0.014$ | -0.406 | 108.0/87
ELG2 | 1.1-1.6 | Pre | $1.004\pm 0.041$ | $0.979\pm 0.020$ | -0.419 | 95.6/87
Table 20: Mean values and standard deviations from the marginalized posteriors
of the BAO scaling parameters from fits to the unblinded DESI DR1 power
spectra in the $\alpha_{\parallel}$-$\alpha_{\perp}$ basis. Systematic effects
are not included. $r_{\rm
off}=C_{\alpha_{\perp},\alpha_{\parallel}}/\sqrt{C_{\alpha_{\perp},\alpha_{\perp}}C_{\alpha_{\parallel},\alpha_{\parallel}}}$.

Tables 19 and 20 shows the BAO measurements using the power spectrum
measurements that should be directly compared with the measurements in Tables
15 and 16. As noted in the main section, our analytical and mock-based
covariances have not fully converged in our Fourier space analysis, thus we
consider the power spectrum results less robust. Nevertheless, the power
spectrum results serve as a consistency check of our results based on the
correlation function as the two are in good agreement, in line with
expectations derived from mock tests.

##  Appendix B Author Affiliations

1Instituto de Física Teórica (IFT) UAM/CSIC, Universidad Autónoma de Madrid,
Cantoblanco, E-28049, Madrid, Spain

2Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA

3Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215,
USA

4Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005,
India

5Centre for Extragalactic Astronomy, Department of Physics, Durham University,
South Road, Durham, DH1 3LE, UK

6Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK

7Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

8Leinweber Center for Theoretical Physics, University of Michigan, 450 Church
Street, Ann Arbor, Michigan 48109-1040, USA

9IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

10Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain

11Instituto Avanzado de Cosmología A. C., San Marcos 11 - Atenas 202.
Magdalena Contreras, 10720. Ciudad de México, México

12Instituto de Ciencias Físicas, Universidad Autónoma de México, Cuernavaca,
Morelos, 62210, (México)

13Physics Department, Yale University, P.O. Box 208120, New Haven, CT 06511,
USA

14Department of Physics and Astronomy, University of California, Irvine,
92697, USA

15Department of Physics, Kansas State University, 116 Cardwell Hall,
Manhattan, KS 66506, USA

16Department of Physics & Astronomy, University of Rochester, 206 Bausch and
Lomb Hall, P.O. Box 270171, Rochester, NY 14627-0171, USA

17Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK

18Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano,
Via Celoria 16, I-20133 Milano, Italy

19Centre for Astrophysics & Supercomputing, Swinburne University of
Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia

20NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA

21Department of Physics and Astronomy, The University of Utah, 115 South 1400
East, Salt Lake City, UT 84112, USA

22Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK

23Department of Astronomy and Astrophysics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL 60637, USA

24Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA

25Korea Astronomy and Space Science Institute, 776, Daedeokdae-ro, Yuseong-gu,
Daejeon 34055, Republic of Korea

26Institute of Cosmology and Gravitation, University of Portsmouth, Dennis
Sciama Building, Portsmouth, PO1 3FX, UK

27Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206, La
Laguna, Tenerife, Spain

28Instituto de Astrofísica de Canarias, C/ Vía Láctea, s/n, E-38205 La Laguna,
Tenerife, Spain

29Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
U.K

30Departamento de Física, Instituto Nacional de Investigaciones Nucleares,
Carreterra México-Toluca S/N, La Marquesa, Ocoyoacac, Edo. de México C.P.
52750, México

31Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

32Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, USA

33NASA Einstein Fellow

34School of Mathematics and Physics, University of Queensland, 4072, Australia

35Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510, México

36CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain

37Department of Physics & Astronomy and Pittsburgh Particle Physics,
Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh, 3941
O’Hara Street, Pittsburgh, PA 15260, USA

38Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China

39Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way,
Berkeley, CA 94720, USA

40University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA
94720, USA

41Universities Space Research Association, NASA Ames Research Centre

42Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA

43Department of Physics, The Ohio State University, 191 West Woodruff Avenue,
Columbus, OH 43210, USA

44The Ohio State University, Columbus, 43210 OH, USA

45Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Menlo Park, CA 94305, USA

46SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA

47Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía,
s/n, E-18008 Granada, Spain

48Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

49Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio Ip, CP 111711, Bogotá, Colombia

50Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio H, CP 111711 Bogotá, Colombia

51Department of Physics, The University of Texas at Dallas, Richardson, TX
75080, USA

52Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

53Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans
s/n, 08913 Bellaterra, Barcelona, Spain

54Departament de Física Quàntica i Astrofísica, Universitat de Barcelona,
Martí i Franquès 1, E08028 Barcelona, Spain

55Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), c.
Martí i Franquès, 1, 08028 Barcelona, Spain.

56Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582. Colonia
Crédito Constructor, Del. Benito Juárez C.P. 03940, México D.F. México

57Departamento de Física, Universidad de Guanajuato - DCI, C.P. 37150, Leon,
Guanajuato, México

58Centro de Investigación Avanzada en Física Fundamental (CIAFF), Facultad de
Ciencias, Universidad Autónoma de Madrid, ES-28049 Madrid, Spain

59Excellence Cluster ORIGINS, Boltzmannstrasse 2, D-85748 Garching, Germany

60University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81677 München, Germany

61Department of Astrophysical Sciences, Princeton University, Princeton NJ
08544, USA

62Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK

63Department of Astronomy, The Ohio State University, 4055 McPherson
Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA

64Department of Physics, Southern Methodist University, 3215 Daniel Avenue,
Dallas, TX 75275, USA

65Department of Physics and Astronomy, University of Waterloo, 200 University
Ave W, Waterloo, ON N2L 3G1, Canada

66Perimeter Institute for Theoretical Physics, 31 Caroline St. North,
Waterloo, ON N2L 2Y5, Canada

67Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave
W, Waterloo, ON N2L 3G1, Canada

68Graduate Institute of Astrophysics and Department of Physics, National
Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan

69Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), FR-75005 Paris, France

70Department of Astronomy and Astrophysics, UCO/Lick Observatory, University
of California, 1156 High Street, Santa Cruz, CA 95064, USA

71Department of Astronomy and Astrophysics, University of California, Santa
Cruz, 1156 High Street, Santa Cruz, CA 95065, USA

72Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON
M5S 3H4, Canada

73University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon
34113, Republic of Korea

74Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain

75Laboratoire de Physique Subatomique et de Cosmologie, 53 Avenue des Martyrs,
38000 Grenoble, France

76Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís
Companys, 23, 08010 Barcelona, Spain

77Max Planck Institute for Extraterrestrial Physics, Gießenbachstraße 1, 85748
Garching, Germany

78Department of Physics and Astronomy, Siena College, 515 Loudon Road,
Loudonville, NY 12211, USA

79Department of Physics & Astronomy, University of Wyoming, 1000 E.
University, Dept. 3905, Laramie, WY 82071, USA

80National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Rd., Chaoyang District, Beijing, 100012, P.R. China

81Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France

82Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical
Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum,
Germany

83Departament de Física, EEBE, Universitat Politècnica de Catalunya, c/Eduard
Maristany 10, 08930 Barcelona, Spain

84University of California Observatories, 1156 High Street, Sana Cruz, CA
95065, USA

85Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA

86Department of Physics and Astronomy, Sejong University, Seoul, 143-747,
Korea

87Abastumani Astrophysical Observatory, Tbilisi, GE-0179, Georgia

88Faculty of Natural Sciences and Medicine, Ilia State University, 0194
Tbilisi, Georgia

89Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218, USA

90Centre for Advanced Instrumentation, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK

91Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

92Beihang University, Beijing 100191, China

93Department of Astronomy, Tsinghua University, 30 Shuangqing Road, Haidian
District, Beijing, China, 100190

94Physics Department, Stanford University, Stanford, CA 93405, USA

95Department of Physics, University of California, Berkeley, 366 LeConte Hall
MC 7300, Berkeley, CA 94720-7300, USA