# Catalog-level blinding on the bispectrum for DESI-like galaxy surveys

Sergi Novell-Masot  Héctor Gil-Marín  Licia Verde  J. Aguilar  S. Ahlen  S.
Brieden  D. Brooks  T. Claybaugh  A. de la Macorra  J. E. Forero-Romero  E.
Gaztañaga  S. Gontcho A Gontcho  G. Gutierrez  K. Honscheid  C. Howlett  R.
Kehoe  T. Kisner  A. Lambert  M. E. Levi  M. Manera  A. Meisner  R. Miquel  G.
Niz  F. Prada  G. Rossi  E. Sanchez  M. Schubnell  H. Seo  D. Sprayberry  G.
Tarlé  B. A. Weaver

###### Abstract

We evaluate the performance of the catalog-level blind analysis technique
(blinding) presented in Brieden et al. (2020) in the context of a power
spectrum and bispectrum analysis. This blinding scheme, which is tailored for
galaxy redshift surveys similar to the Dark Energy Spectroscopic Instrument
(DESI), has two components: the so-called “AP blinding” (Alcock-Paczyński,
concerning the dilation parameters $\alpha_{\parallel},\alpha_{\bot}$) and
“RSD blinding” (redshift space distortions, affecting the growth rate
parameter $f$). Through extensive testing, including checks for the RSD part
in cubic boxes, the impact of AP blinding on mocks with realistic survey sky
coverage, and the implementation of a full AP+RSD blinding pipeline, our
analysis demonstrates the effectiveness of the technique in preserving the
integrity of cosmological parameter estimation when the analysis includes the
bispectrum statistic. We emphasize the critical role of sophisticated—and
difficult to accidentally unblind—blinding methods in precision cosmology.

##  1 Introduction

In recent decades, cosmology has undergone remarkable progress, entering the
era of precision cosmology, with the $\Lambda$CDM model as a cornerstone,
bridging theory and observations throughout all observable epochs of the
Universe. However, there are still several open questions at the heart of our
understanding of cosmology. We still do not know the fundamental physics of
dark matter and dark energy, which amount to roughly 95% of the Universe, and
some tensions arise from the $\Lambda$CDM model when applied to both early and
late-time physics. Current and forthcoming experiments—both early-time, such
as the Southern Pole Telescope (SPT) [1], the Atacama Cosmology Telescope
(ACT) [2], the Simons Observatory [3], as well as late-time, such as the Dark
Energy Spectroscopic Instrument (DESI) [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17], Euclid [18], Vera Rubin [19]—will play a pivotal role in these
unresolved issues, yielding complementary cosmological information with an
unprecedented level of precision.

In order to ensure the reliability of results and mitigate confirmation bias,
blind analyses, whereby the true result is hidden to the experimenter until
the full analysis is done and “frozen”, have been increasingly adopted in
cosmology. Blind analyses, already standard practice in fields like
experimental particle physics [20], have been extended to cosmology in recent
years [21, 22, 23, 24, 25, 26]. However, there is no one-size-fits-all
standard for making an analysis blind; hence there is a need to develop and
use a suitable method for each context. Hereafter, we refer to the action of
making an analysis blind as simply “blinding”, while we denote as “unblinding”
the procedure of recovering the original, non-blind results.

One can organize the different approaches to blinding relative to the stage of
the analysis in which they are implemented. Generally, in large-scale
structure surveys, the earlier it is implemented the more difficult it is to
accidentally unblind. For example, while adding a blinding scheme just at the
final analysis stages, e.g. shifting the recovered cosmological parameters by
an unknown amount, is certainly convenient, it is nevertheless relatively easy
to accidentally unblind. An intermediate approach involves adding a random
shift in the summary statistics or their covariance [27, 28, 29], e.g.
shifting the wave-vectors in the power spectrum. The strongest methods are
generally the ones that are implemented already at the phase of catalog
production, as done for example in [26].

In the framework of large-scale structure analyses, focusing on the power
spectrum baryon acoustic oscillations (BAO) feature and redshift-space
distortions, the method developed by [30] operates at catalog-level, which is
currently part of the official DESI pipeline [31]. While this method has been
extensively validated in the context of a power spectrum template-based
approach [31, 15], it still has not been tested on higher-order statistics
such as the bispectrum, which is what we set out to do. The purpose of this
work is to validate the blinding strategy of [30] for an analysis featuring
the combination of the power spectrum and bispectrum multipoles in DESI-like
periodic boxes and cutsky mocks.

In Section 2 we review the main aspects of this blinding procedure. In Section
3 we describe the simulations and the main ingredients of our analysis. In
Section 4 we present our results and we conclude in Section 5.

##  2 Blinding the data at the catalog level

The full blinding procedure consists of two separate steps, which we refer to
as respectively AP and RSD parts of the blinding, following the nomenclature
from [30]. It was shown in [30] that this combination results in a modular
effect in the cosmological parameters (with the AP and RSD parts affecting
different sets of cosmological parameters) while being very difficult to
accidentally unblind. In particular, the RSD blinding modifies the linear
growth rate parameter $f$, while the AP part of the blinding results in shifts
in the Alcock-Paczyński parameters $\\{\alpha_{\parallel},\alpha_{\bot}\\}$,
which are respectively the parallel and perpendicular to the line of sight
dilation scales with respect to the fiducial cosmology. At any redshift $z$,
given the Hubble parameter $H(z)$, the sound horizon scale at baryon drag
redshift $r_{\textrm{s}}(z_{\textrm{d}})$ and the angular distance parameter
$D_{\textrm{A}}(z)$, the Alcock-Paczyński parameters are defined as [32]

$\alpha_{\parallel}(z)=\frac{H^{\textrm{fid}}(z)r_{\textrm{s}}^{\textrm{fid}}(z_{\textrm{d}})}{H(z)r_{\textrm{s}}(z_{\textrm{d}})};\quad\alpha_{\bot}(z)=\frac{D_{\textrm{A}}(z)r_{\textrm{s}}^{\textrm{fid}}(z_{\textrm{d}})}{D_{\textrm{A}}^{\textrm{fid}}(z)r_{\textrm{s}}(z_{\textrm{d}})},$
(2.1)

with the “fid” superscript referring to the parameter corresponding to the
fiducial cosmology.

In this section, we provide an updated summary of the procedure of [30]. In a
nutshell, the AP component of the blinding acts only along the line of sight
by hiding (blinding) the reference cosmology used to convert the redshifts
($z$) into comoving distances. It does so by converting the original redshifts
(indicated by $z$) of the catalog to distances using a hidden (blind or
shifted in [30] nomenclature) cosmology and then transforming back the
distances into redshifts using a reference (known) cosmology, thus producing
an effectively blind redshift catalog. Blind redshifts are indicated by
$z^{\prime}$. It is easy to see that this transformation only affects the
dilation parameters $\alpha_{\parallel},\,\alpha_{\bot}$.

Let us refer to the vector of cosmological parameters that completely specify
a given cosmological model by ${\bf\Omega}$. For example, in a standard
$\Lambda$CDM model, ${\bf\Omega}$ includes parameters such as the matter
density parameter $\Omega_{m}$, the baryon density parameter $\Omega_{b}$, the
primordial power spectrum spectral slope $n_{s}$, the Hubble constant $H_{0}$,
the rms dark matter fluctuations filtered on $8$ Mpc/$h$ scales $\sigma_{8}$.
The blinding scheme that the DESI collaboration uses shifts a given reference
cosmology ${\bf\Omega}^{\rm ref}$ by a (hidden) shift $\Delta{\bf\Omega}$ such
that a blind cosmology is obtained:

${\bf\Omega}^{\prime}={\bf\Omega}^{\rm
ref}+\Delta{\bf\Omega}\equiv{\bf\Omega}^{\rm ref}+\left(\Delta f,\Delta
w_{0},\Delta w_{a}\right)$ (2.2)

where the linear growth rate is taken to be $f=\Omega_{m}^{\gamma}$,111Here,
the exponent $\gamma$ is chosen following the General Relativity prescription,
i.e. $\gamma\approx 6/11$.[33, 34] and $w_{0},w_{a}$ describe dynamical dark
energy with dark energy equation of state parameter being a function of the
scale factor: $w(a)=w_{0}+w_{a}(1-a)$. When the analysis on the blind data is
performed, a fiducial cosmology ${\bf\Omega}^{\rm fid}$ is used (hence
transforming equation 2.2 such that ${\bf\Omega}^{\rm
ref}\rightarrow{\bf\Omega}^{\rm fid}$). Since in this specific application we
do not use real data or an unknown true cosmology, the prime quantities are
the blinded ones, while the non prime quantities are the reference (and true)
ones.

The resulting shifts from Equation 2.2 in the Alcock-Paczyński parameters are:

$\frac{\alpha_{\parallel}^{\prime}(z^{\prime})}{\alpha_{\parallel}(z)}=\frac{H^{\prime}(z)}{H(z)};\quad\quad\frac{\alpha_{\bot}^{\prime}(z^{\prime})}{\alpha_{\bot}(z)}=\frac{D_{M}(z)}{D^{\prime}_{M}(z)},$
(2.3)

where $\alpha$ with no superscript refer to the reference, fiducial depending
on the specific application, with $H$ and $D_{M}$ being respectively the
Hubble expansion parameter and the angular comoving distance at redshift $z$,

$\displaystyle H(z)$
$\displaystyle=H_{0}\sqrt{\Omega_{m}(1+z)^{3}+(1-\Omega_{m})(1+z)^{3(1+w_{0}+w_{a}(1-a(z)))}}$
(2.4) $\displaystyle D_{M}(z)$
$\displaystyle=\int_{0}^{z}dz^{\prime}\frac{c}{H(z^{\prime})}.$ (2.5)

In other words, the dilation parameters expected to be measured from the
blinded catalog $\alpha^{\rm blind}_{\parallel,\perp}$ are related to the pre-
blinding ones $\alpha_{\parallel,\perp}$ by the same ratio as the
$\alpha^{\prime}$’s to the $\alpha^{\rm ref}$s. Of course, in tests applied to
simulations the reference quantities coincide with the fiducial and true
quantities, and the prime coincides with the blind, i.e. ${\bf\Omega}^{\rm
ref}={\bf\Omega}^{\rm fid}={\bf\Omega}^{\rm true}$ and
${\bf\Omega}^{\prime}={\bf\Omega}^{\rm blind}$.

The RSD part of the blinding operates in a different way. Built on the RSD
part of the reconstruction algorithm [35, 36, 37], this step of the blinding
procedure smooths the density field with a Gaussian filter and then uses the
reconstructed field to shift the redshift space positions r according to,

$\mathbf{r}^{\prime}=\mathbf{r}-f^{\textrm{ref}}(\Psi\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}+f^{\prime}\textrm{(}\Psi\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}.$
(2.6)

The quantity $\Psi\cdot\hat{\mathbf{r}}$, where $\Psi$ is the displacement
field, can be inferred from the reconstructed real-space positions, such that
the comoving distance is changed by

$d^{\prime}=d^{\textrm{rec}}+\frac{f^{\prime}}{f^{\textrm{ref}}}(d^{\textrm{ref}}-d^{\textrm{rec}}),$
(2.7)

with $d^{\textrm{rec}},d^{\textrm{ref}}$ being respectively the reconstructed
and reference comoving distances, and $f^{\prime},f^{\textrm{ref}}$ the
shifted and reference values of the growth factor $f$. In this way, the
redshifts of the tracers are modified such that the catalog has effectively
different RSD signal of amplitude given by $f^{\rm blind}=f^{\rm true}+\Delta
f$.

Both parts of the blinding procedure have been tested to be robust with a
template-based power spectrum analysis [30], and we set to quantify their
performance (both separately and in combination) for analyses involving the
bispectrum.

##  3 Test design and analysis set-up

###  3.1 Test design and current limitations

In this paper we perform the following sequence of tests for a joint power
spectrum and bispectrum analysis which employs cubic (periodic) boxes and more
realistic mock surveys which include partial sky coverage (cutsky
mocks):222While cubic mocks simulate the cosmic structure within a periodic
cubic volume, cutsky mocks apply observational features to these simulations
to mimic real sky survey data, accounting for survey geometry and selection
effects.

  1. 1.

Performance of the RSD part with cubic boxes for the combination of the power
spectrum and bispectrum redshift space multipoles.

  2. 2.

Performance of the AP part with cutsky mocks for the combination of redshift
space power spectrum multipoles and bispectrum monopole.

  3. 3.

Performance of the full AP+RSD blinding with cutsky mocks for the combination
of redshift space power spectrum multipoles and bispectrum monopole.

The reason why we first test the RSD part of the blinding algorithm in cubic
boxes is that the RSD part of the algorithm could be more sensitive than the
AP part to the specific analysis settings. Implementing the AP part of the
blinding (which was designed with cutsky data in mind) in a cubic box is
challenging both conceptually and practically. First of all, AP blinding
cannot be done correctly in plane-parallel approximation (unless in the limit
of large distances and small survey areas, which is not the case here).
Secondly, the advantage of using cubic boxes is their periodicity, but
periodicity is lost when applying the AP blinding. For more details see
Section 3.2. Given this, we separately test the AP part of the blinding in
step 2.

To perform step 1, we modify the blinding code to be used by DESI in order to
apply the RSD part of the blinding to cubic
boxes.333https://github.com/desihub/LSS/ For this test, we use the global
plane-parallel approximation by obtaining the redshifts of the galaxies from
the third direction of the box, $\hat{x}_{3}$. Then, the first two coordinates
$(x_{1},x_{2})$ are left equal, while the third coordinate $x_{3}$ is
transformed according to Equation 2.7444If the transformed value for $x_{3}$,
which we can denote as $\tilde{x}_{3}$, falls outside of the box, it is
reintroduced assuming periodic boundary conditions as $\tilde{x}_{3}$ modulus
$L_{\textrm{box}}$.

In this work, for steps 2 and 3 we limit ourselves to adding only the
bispectrum monopole to the power spectrum multipoles. The currently available
treatment of the bispectrum window function [38, 39, 40] is only applicable to
the bispectrum monopole $B_{0}$. We recognize that this bispectrum window
function treatment is known to suffer from some degree of inaccuracy,
especially in the squeezed configurations [41]; as we will show, these do not
interfere with the performance of the blinding (and unblinding) procedure.

We leave to future work to update steps 2 and 3 to include also the bispectrum
quadrupoles, subject to the availability of an improved window function
implementation suitable for the bispectrum multipoles.

###  3.2 Simulations and blinding parameters

We utilize 25 AbacusSummit LRG simulations at redshift $z=0.8$, which we may
refer to as simply as the AbacusSummit LRG mocks [42]. LRG stands for Luminous
Red Galaxies, which constitutes the highest signal-to-noise DESI sample at
$z\lesssim 0.8$ [12]. For the first test in this work (RSD blinding, as
presented in Section 2) we use the 25 available periodic boxes, while for the
second and third tests, we use the corresponding 25 cutsky mocks. The sky area
and completeness of the window that we use in cutsky mocks coincide with the
DESI Year 1 footprint, and we measure the power spectra and bispectra using
random catalogs (randoms) with 20 times the density of the data.

The cosmology of all the mocks as well as our adopted reference and fiducial
cosmology are the same and compatible with the Planck best fit $\Lambda$CDM
model [43]: $\Omega_{m}=0.3138,\Omega_{b}=0.0493,$
$n_{\textrm{s}}=0.9649,\sigma_{8}=0.8114,h=0.6736,w=-1.$ The cubic boxes have
a size of $L_{\textrm{box}}=2000\,\textrm{Mpc}\,h^{-1}$, resulting in a total
physical volume for the 25 boxes of $V_{25}=200\,(\textrm{Gpc}/h)^{3}$. The
effective volume of a single cubic box—computed as in [44]—is $\sim
6.4\,(\textrm{Gpc}\,h^{-1})^{3}$, for a total effective volume of $V^{\rm
eff}_{25}\sim 160\,(\textrm{Gpc}\,h^{-1})^{3}$. The cutsky mocks are generated
by the Generate Survey Mocks code555https://github.com/Andrei-
EPFL/generate_survey_mocks [45, 46] and matched the DESI Year 1 survey
footprint with the mkfast-Y1
code666https://github.com/desihub/LSS/blob/main/scripts/mock_tools/mkfast_Y1.py.
The effective volume of each cutsky mock is estimated to be $\sim
1.6(\textrm{Gpc}\,h^{-1})^{3}$ resulting in the 25 cutsky mocks having a
cumulative effective volume777The effective volume of each cutsky mock is
calculated by multiplying the effective volume of its correspondent cubic mock
($\sim 6.4\,(\textrm{Gpc}\,h^{-1})^{3}$ [44]) by the ratio between the number
of particles in the cutsky mock and the cubic box. of $V^{\rm eff}_{\rm
25cutsky}\sim 40\,(\textrm{Gpc}\,h^{-1})^{3}$, of the order of the expected
DESI final volume (5 years of survey) for LRGs [12].

From the original AbacusSummit suite, we produce a total of 75 blinded mocks,
obtained by applying to each original AbacusSummit LRG simulation the
corresponding blinding as per Section 2 (25 blinded mocks for each of the 3
tests). We blind the cubic boxes by a different amount than the cutsky mocks.
The blinding is such that the expected shift in the growth rate $f$ is $\Delta
f=f^{\prime}-f^{\textrm{ref}}=-0.068$ for the cubic boxes (where the value of
$f^{\textrm{ref}}$ is set to 0.8), while for the cutsky mocks we change the
sign of the expected shift so that $\Delta
f=f^{\prime}-f^{\textrm{ref}}=0.060$. The AP part of the blinding in the
cutsky mocks has expected shifts of
$\Delta\alpha_{\parallel}=\Delta\alpha_{\bot}=-0.013$.888Note that in general
the $\alpha_{\parallel}$ and $\alpha_{\bot}$ parameters are not necessarily
shifted equally. The fact that in our case the shift is the same is just a
coincidence.

The values of the shifts were chosen in order to be close to the limit of the
DESI blinding pipeline, which enforces $|\Delta f|\leq 0.08$
$|\Delta\alpha_{\parallel}|\leq 0.03$, $|\Delta\alpha_{\bot}|\leq 0.03$ [31].
Compared to the expected standard deviation of the parameters recovered from
the total volume of the 25 mocks ($V_{25}$), these offsets represent a
blinding shift in $f$ for the cubic boxes of $\sim 3\sigma$, and for the
cutsky mocks of $\sim 1\textrm{--}2\sigma$ for all the parameters
$f,\alpha_{\parallel},\alpha_{\bot}$. We choose not to go beyond $3\sigma$ of
the expected errorbars in order to not produce a degradation in the recovered
constraints [30].

It should now be clearer why we do not implement the AP blinding on cubic
boxes. The size of the cubic boxes
($L_{\textrm{box}}=2000\,\textrm{Mpc}\,h^{-1}$) is of comparable magnitude to
the comoving distance from $z=0$ to $z=0.8$ where the tracers are located.
Hence, to introduce a significant AP shift at $z=0.8$ would require
arbitrarily setting the observer at a relatively short distance from the box.
This would break periodicity and the plane parallel approximation for RSD
would not apply. This limitation does not apply to the cutsky mocks.

For estimating the covariance matrices, we measure the power spectra and
bispectra from 1000 EZmocks, which generate the density field using the
effective Zel’dovich approximation [47]. Both the underlying cosmology and the
tracers’ clustering properties are designed to match the AbacusSummit LRG
mocks. Throughout, we measure the power spectrum and bispectrum of both cubic
and cutsky mocks using the publicly available Rustico
code999https://github.com/hectorgil/Rustico [48].

###  3.3 Modeling

Our theoretical modelling is the one adopted in our previous works [49, 50],
where the interested reader can find an in-depth description. The modelling of
the power spectrum follows the renormalized perturbation theory (RPT)
prescription [51, 52]. For the bispectrum we use the phenomenological GEO-FPT
model [49], as provided in the publicly available GEO-FPT
code101010https://github.com/serginovell/Geo-FPT. In particular for this
application, the phenomenological parameters that model the bispectrum shape
and scale dependence, $\\{f_{1},...,f_{5}\\}$, are obtained at $z=0.8$ by
interpolation from the tabulated values provided by GEO-FPT.

In all cases use the power spectrum monopole and quadrupole data-vector,
$P_{02}=\\{P_{0},P_{2}\\}$, while we consider different parts of the
bispectrum multipole expansion according to the analysis settings: for cubic
mocks (Section 4.1) we use the bispectrum monopole together with the first two
quadrupoles,111111We found in [49] that it is redundant to use all three
quadrupoles $B_{200},B_{020},B_{002}$. We follow the bispectrum multipole
expansion convention by [53]. and refer to our full bispectrum data-vector as
$B_{02}=\\{B_{0},B_{200},B_{020}\\}$; for cutsky mocks (Section 4.2) we only
consider the bispectrum monopole $B_{0}$.

As in our previous works [49, 50], we perform the cosmological parameter
inference via a Markov chain Monte Carlo sampling (MCMC), specifying broad
uniform priors in all parameters. We use the full shape of the power spectrum
and bispectrum, under the assumption of local Lagrangian bias [54, 55, 56].
The parameters of interest in this work are
$\\{f,\sigma_{8},\alpha_{\parallel},\alpha_{\bot}\\}$, and we marginalize over
the nuisance parameters
$\\{b_{1},b_{2},A_{\textrm{P}},A_{\textrm{B}},\sigma_{\textrm{P}},\sigma_{\textrm{B}}\\}$,
respectively: the linear and quadratic bias parameters, the shot noise
amplitude correction for the power spectrum and bispectrum, and the power
spectrum and bispectrum peculiar velocities rms.

The main difference with the analysis presented in previous works is that we
limit the bispectrum $k$-range to $0.02\,h{\rm Mpc}^{-1}<k<0.11\,h{\rm
Mpc}^{-1}$ (the bispectrum model was calibrated for
$k_{\textrm{max}}=0.12\,h{\rm Mpc}^{-1}$). The reason for this choice is that
the size of the data-vector is limited by the available number of simulations
(1000 EZmocks). In order to estimate the full covariance matrix (as
recommended in [50]) the number of simulations should be much bigger than the
size of the data-vector [57]. By adopting this $k_{\textrm{max}}$ and the
relatively large binning size of $\Delta k=0.01\,h\,\textrm{Mpc}^{-1}\approx
3.2k_{\textrm{f}}$,121212The fundamental wave-vector, $k_{\textrm{f}}$, is
defined as $k_{\textrm{f}}=2\pi/L_{\textrm{box}}$. the full $P_{02}$+$B_{02}$
data-vector size is of 342 elements, while the $P_{02}$+$B_{0}$ data-vector
used in the cutsky mocks has 152 elements. We further account for the errors
due to the finite number of simulations in the covariance matrix estimation by
employing the Sellentin-Heavens likelihood function [58].

Finally, for the cases where we use cutsky mocks, we model the survey window
for the power spectrum and bispectrum exactly as in [38, 39]. In short, the
galaxy power spectrum model $P_{\textrm{gal}}$ is convolved with the window
function $W_{2}$ (obtained by performing pair counts in the random catalog, as
defined in [38, 39]) to obtain the windowed power spectra $P^{W}$ as

$P^{W}(\textbf{k})=\int\frac{d^{3}\textbf{k}^{\prime}}{(2\pi)^{3}}P_{\textrm{gal}}(\textbf{k}^{\prime})|W_{2}(\textbf{k}-\textbf{k}^{\prime})|^{2}+P_{\textrm{noise}}(\textbf{k}).$
(3.1)

where $P_{\rm noise}$ denotes the shot noise contribution to the power
spectrum.

Due to the computational challenge of operating with the analogous expression
of Equation 3.1 in the case of the bispectrum, in this paper we use the
approximation of [38, 39], which ignores the effect of the window on the
bispectum kernel. In fact, we can factorize the model of the galaxy bispectrum
as
$B_{\textrm{gal}}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})=P_{NL}(\textbf{k}_{1})P_{NL}(\textbf{k}_{2})\mathcal{Z}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})+\textrm{cyc.}$,
where $P_{NL}$ is the non-linear matter power spectrum, and the function
$\mathcal{Z}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})$ encompasses the
perturbation theory kernels, the geometrical correction and the bias
expansion. In our adopted approximation, the $\mathcal{Z}$ function is
unaffected by the window, resulting in a windowed bispectrum monopole theory
vector $B^{W}$ that reads

$B^{W}_{0}(k_{1},k_{2},k_{3})=\int_{-1}^{1}d\mu\int_{0}^{2\pi}d\phi\mathcal{Z}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})P_{NL}^{W}(\textbf{k}_{1})P_{NL}^{W}(\textbf{k}_{2})+\textrm{cyc}.$
(3.2)

In the above expression, $\mu_{i}$ is the cosine of the angle of the $i$-th
$k$-vector with respect to the line of sight, and $\phi$ is such that
$\mu_{2}=\mu_{1}\cos\theta_{12}-\sqrt{(1-\mu_{1}^{2})(1-\cos\theta_{12}^{2})}\cos\phi$,
with $\theta_{12}$ being the angle between $k_{1}$ and $k_{2}$. Since this
approximation cannot be easily generalized to the bispectrum quadrupoles, and
to date there is no suitable window modelling for the bispectrum quadrupoles
in this particular multipole expansion, in this work for cutsky tests we only
use the bispectrum monopole.131313As a consequence, in the cutsky cases, where
the bispectrum quadrupoles are not used, we do not consider variations from
Gaussian shot-noise in the bispectrum, as we do with the full joint power
spectrum and bispectrum multipoles analysis via the $A_{P}$ and $A_{B}$
parameters respectively. We find that letting $A_{B}$ vary when only using the
bispectrum monopole $B_{0}$ opens up an unnecessarily large and unconstrained
degeneracy direction where cosmological parameters take unphysical values,
e.g. $f>1$. For the purpose of testing the blinding, we find the current
approximation of Equation 3.2 to be sufficient. To show that the adopted model
for the data vector does not display obvious signature of systematics,
Appendix A shows the best fitting theoretical model for the data vector from
the blind and not blind catalogs, along with relevant $\chi^{2}$ values.

##  4 Results

In order to test the validity of the RSD part of the blinding scheme for the
addition of the bispectrum in the data-vector, we perform similar tests as in
[30], while using the set-up and analysis previously done in [49] and adapted
as mentioned in Section 3.

###  4.1 Cubic mocks: RSD blinding

We start by running the cosmological pipeline using only the power spectrum
monopole and quadrupole $P_{02}$ on the 25 original and RSD-blinded
AbacusSummit mocks. Additionally, we repeat the procedure for the mean of the
25 simulations, taking advantage of the fact that we have blinded all
realizations with the same shift $\Delta f$.

Figure 1: Recovered cosmological parameters
$\\{f,\sigma_{8},\alpha_{\parallel},\alpha_{\bot}\\}$ before and after
blinding for the power spectrum data-vector $P_{02}=\\{P_{0},P_{2}\\}$ from 25
AbacusSummit LRG cubic boxes. The scatter shows the relationship between the
maximum likelihood parameters before ($x$-axis) and after ($y$-axis) applying
RSD-type blinding. The blue dashed lines indicate the underlying/expected
value for each parameter, whilst the red dashed lines indicate the expected
relationship for the parameters between the original and blinded realizations.
The star points are obtained using the mean of the power spectra of the 25
boxes as data-vector, and the red ellipses represent the 1 and 2$\sigma$
regions of the scatter of the 25 realizations, computed with the eigenvalues
and eigenvectors of the covariance of the scatter.  Figure 2: Analogous to
Figure 1 but for the data-vector
$P_{02}$+$B_{02}=\\{P_{0},P_{2},B_{0},B_{200},B_{020}\\}$ as indicated in
Section 3. Original and blinded recovered parameters follow the expected
distribution. The inclusion of the bispectrum significantly reduces the
scatter and thus the errors.

The results are displayed in Figure 1, where we plot the scatter among the 25
boxes of the maximum likelihood of the cosmological parameters
$\\{f,\sigma_{8},\alpha_{\parallel},\alpha_{\bot}\\}$ before and after
blinding. In this figure, each of the filled blue circles corresponds to one
of the 25 boxes, and the coordinates represent the maximum likelihood
parameter values before ($x$ axis) and after ($y$ axis) blinding. The green
stars correspond to the maximum likelihood values for the mean signal from the
25 boxes. The expectation is that symbols should scatter around the line of
equation $y=x+\Delta\Omega$, where $\Delta\Omega$ is the (blinding) shift in
the given parameter.141414Note that $\Delta\Omega=0$ for parameters not
expected to be affected by blinding. In this test for RSD blinding in boxes,
the parameters with $\Delta\Omega=0$ are
$\\{\sigma_{8},\alpha_{\parallel},\alpha_{\bot}\\}$. This corresponds to the
red dashed line. The true underlying (and expected) values of the parameters
before (and after) blinding are indicated by the dashed vertical and
horizontal blue lines. This preliminary step shows that we can replicate the
results obtained in [30]. As expected, if only $P_{02}$ is used, the
$f$–$\sigma_{8}$ degeneracy is only weakly broken by the non-linear terms of
the power spectrum monopole and quadrupole. This degeneracy is not fully
disentangled unless the bispectrum monopole and quadrupoles are included in
the data-vector [49, 59].

Figure 3: Cubic boxes, RSD blinding only. Recovered 1-2D posteriors for the
cosmological parameters $\\{f,\sigma_{8},\alpha_{\parallel},\alpha_{\bot}\\}$
using the mean $P_{02}$+$B_{02}$ from the 25 AbacusSummit LRG boxes as the
data-vector. The statistical errors correspond to the full volume of the 25
boxes, $V_{25}=25\times 2^{3}\,(\textrm{Gpc}\,h^{-1})^{3}$. The blue and
orange contours are, respectively, the results from the original and RSD-
blinded simulations, while the green unblinded contours are obtained by
analytically shifting back the posteriors by the expected $\Delta f$. Even
though the effective volume ($\sim 160\,(\textrm{Gpc}\,h^{-1})^{3}$ [44]) is
approximately an order of magnitude larger than the expected DESI volume for
LRG’s, the unblinded case is almost identical to the original, while
recovering the expected values at $\sim 1\sigma$ level, thus providing
validation of the use of both the GEO-FPT bispectrum model and the RSD part of
the blinding scheme.

The corresponding results of the $P_{02}$+$B_{02}$ joint analysis are shown in
Figure 2 using the same conventions. As with the case of the power spectrum,
we find excellent agreement between original and blinded results, given the
expected shift in $f$. Note the different scales in the figure: the errorbars
on the parameters are significantly reduced by including the bispectrum
monopole and quadrupoles in the analysis.

The cosmological constraints obtained with the mean signal of the 25
simulations and the corresponding covariance (i.e., for a physical volume of
$V_{25}=25\times 2^{3}\,(\textrm{Gpc}\,h^{-1})^{3}$) are shown in Figure 3.
The original (blue), blinded (orange) and unblinded (green) posteriors for all
the possible pairs of the
$\\{f,\sigma_{8},\alpha_{\parallel},\alpha_{\bot}\\}$ parameters are shown. By
“unblinded” we refer to the posteriors of the blinded case numerically shifted
back by the designed shift i.e. by $-\Delta f=0.068$. As we mentioned in
Section 3, the shift in $f$ produced by the blinding procedure is equivalent
to $\sim 3\sigma$ of the parameter error in the $P_{02}$+$B_{02}$ analysis in
the joint volume covered by the 25 simulations.

We can see that the original analysis, including the bispectrum multipoles,
recovers the underlying cosmological parameters within 1$\sigma$, even for a
volume (and thus statistical precision) that far exceeds the total DESI
volume. The overlap of the original (blue) and unblinded (green) posterior
shows that the RSD blinding component shifts the recovered posteriors exactly
as predicted.

###  4.2 Cutsky mocks

The AP component of the blinding scheme, by acting simply on the fiducial
cosmology adopted in the redshift-distance relation, by design shifts the
dilation parameters exactly as modeled in all the established BAO analyses,
e.g. [60, 61].

Figure 4: Analogous to Figure 3, using the same conventions but for cutsky
boxes, and AP blinding only. Recovered 1-2D posteriors for the cosmological
parameters $\\{f\sigma_{8},\alpha_{\parallel},\alpha_{\bot}\\}$ using the mean
$P_{02}$+$B_{0}$ from the 25 AbacusSummit LRG cutsky mocks at $z=0.8$. The
single realization covariance is rescaled by 25 so that it matches the total
volume of the 25 mocks, $V^{\textrm{eff}}_{\rm 25cutsky}\sim
40\,(\textrm{Gpc}\,h^{-1})^{3}$. We model the effect of the window function
according to the approximation in Section 3. Similarly to Figure 3, the AP
blinding operation shifts the posteriors exactly as expected. In this case,
the blinding procedure shifts the parameters as
$\Delta\alpha_{\parallel}=\Delta\alpha_{\bot}=-0.013$, which corresponds to
$1\sigma$ of the parameters errorbars. In the 1D distribution for
$f\sigma_{8}$ the orange curve is identical to the green one and thus
invisible.

Nevertheless, for the sake of systematically validating the blinding pipeline,
we first test the behaviour of the AP blinding in the presence of a survey
window, in isolation of the RSD part, which corresponds to our second step as
mentioned in Section 2. In Figure 4 we show the posterior distribution for the
main cosmological parameters of our analysis,
$\\{f\sigma_{8},\alpha_{\parallel},\alpha_{\bot}\\}$ for both the unblinded
and AP blinded cutsky mocks, together with the shifted (“unblinded”, in green)
posteriors. The statistical errors correspond to the cosmological volume
spanned by the 25 individual mocks, which is $\sim
40\,(\textrm{Gpc}\,h^{-1})^{3}$ as explained in Section 3.

In this application to cutsky mocks, we report constraints on the product
$f\sigma_{8}$ instead of the separate parameters $f$ and $\sigma_{8}$ since
the bispectrum monopole cannot fully resolve the well-known $f$–$\sigma_{8}$
degeneracy [49]. The inclusion of the bispectrum quadrupoles (once a suitable
modeling of the window becomes available) is expected to resolve this.
However, at this time, this is beyond the scope of this work, centered on
validating the blinding scheme.

The figure shows that the posterior distributions for the parameters of
interest are all within 1$\sigma$ of the true values, which is a validation of
both the power spectrum and bispectrum models and bias expansions, together
with the window approximation we use for $B_{0}$. The close similarity of the
blue (before blinding) and green (after blinding, but numerically shifted
back) posterior contours indicates that for the adopted magnitude of the
blinding shift of $\alpha_{\parallel}=\Delta\alpha_{\bot}=-0.013$
(corresponding to $1$–$2\sigma$ of our errorbars for the total of 25 mocks),
the blinding procedure performs as expected.

Figure 5: Analogous plot as Figure 4, in this case performing the full
blinding procedure, which also includes the RSD shift as described in Section
2. Both AP and RSD parts of the blinding are seen to perform exactly as
expected.

Finally, Figure 5 shows the results of the test of the full blinding
pipeline—AP and RSD—applied to the cutsky mocks, with shifts of $\Delta
f=0.06,\,\Delta\alpha_{\parallel}=\Delta\alpha_{\bot}=-0.013$. The
$\alpha_{\parallel},\alpha_{\bot}$ parameters are (correctly) shifted as in
the case without the RSD part of the blinding: even with the addition of the
bispectrum, and in the presence of a sky cut, the two parts of the blinding do
not interfere significantly. As for the redshift-space distortions
parametrization $f\sigma_{8}$, the RSD part of the blinding we applied is
designed to shift $f$ while not affecting $\sigma_{8}$, and the recovered
posterior distribution for $f\sigma_{8}$ is exactly as predicted.

The original and shifted (unblinded) posteriors in Figures 4 and 5 indicates
that estimated blind parameter constraints are slightly tighter than the
original ones. This is not unexpected, as for this application we do not
transform the covariance matrix under the blinding operation. The clustering
properties of the EZmocks used to compute the covariance are designed to match
those of the (original) before blinding catalogs. Blinding changes slightly
the clustering properties making the covariance matrix slightly mis-
calibrated. While this effect can be corrected—albeit in a somewhat time-
consuming way, by applying the blinding transformation to the 1000
realizations of EZmocks—for this application it is small enough that can be
left uncorrected.

##  5 Conclusions

In the era of large stage IV galaxy surveys, blind analyses have become
essential to safeguard against confirmation bias in cosmology, where
researchers may unconsciously influence their analyses to conform to their
prior beliefs. By masking certain aspects of the data or results until the
analysis is complete, blinding adds to the integrity of findings from present
and upcoming galaxy surveys [30, 27].

The blinding technique of [30] has been validated for two-point statistics and
currently adopted by and implemented in the official DESI pipeline. This
blinding scheme modifies the distance-redshift relation and thus the tracer’s
redshifts in two complementary parts. The first part, named AP blinding in
this work, is equivalent to modifying the cosmology used in transforming
redshifts into distances, by changing the default $\Lambda$CDM values of the
$w_{0},w_{a}$ parameters in the dark energy equation of state parameterization
given by $w(z)=w_{0}+(1-a(z))w_{a}$. This generates a (theoretically)
predictable shift in the recovered values of
$\alpha_{\parallel},\alpha_{\bot}$. The second part of blinding, the RSD
blinding, combines the observed density field with a suitably reconstructed
density field in order to modify the effective value of the linear growth rate
parameter $f$.

In this paper we have explored the effectiveness and performance of the
catalog-level blinding technique proposed by [30] within DESI-like galaxy
spectroscopic surveys, for a data-vector which combines the power spectrum and
bispectrum summary statistics.

We have confirmed that such a blinding scheme performs as expected when
including the bispectrum data-vector, in the three sequential tests below:

  1. 1.

RSD-only blinding on cubic boxes. For a blinding shift of $\Delta f=-0.068$
(which corresponds to $\sim 3\sigma$ of the recovered errorbars151515In all
cases, both cubic and cutsky mocks, the total volume of both the signal and
covariance corresponds to the sum of the 25 available mocks. This yields an
effective volume of $\sim 160\,(\textrm{Gpc}\,h^{-1})^{3}$ in the case of
cubic mocks and of $\sim 40\,(\textrm{Gpc}\,h^{-1})^{3}$ in the cutsky mocks.
in $f$) and with a data-vector consisting of power spectrum and bispectrum
monopole and quadrupoles $\\{P_{0},P_{2},B_{0},B_{200},B_{020}\\}$, we found
perfect agreement between the predicted blinded and recovered blinded
parameters.

  2. 2.

AP-only blinding on cutsky mocks, with a data-vector composed of power
spectrum multipoles and bispectrum monopole $\\{P_{0},P_{2},B_{0}\\}$. The
modeling of the window function on the bispectrum monopole follows the
approximation proposed by [38, 39]. As expected, given that this component is
directly modifying the cosmology—which the $\alpha_{\parallel},\alpha_{\bot}$
parameters naturally quantify—the parameters recovered in the blinded mocks
closely follow the predictions. In this case, we shifted the $w_{0},w_{a}$
parameters such that the expected shifts are
$\Delta\alpha_{\parallel}=\Delta\alpha_{\bot}=-0.013$, about 1–2$\sigma$ of
errorbars obtained in $\alpha_{\parallel},\alpha_{\bot}$.

  3. 3.

Full blinding pipeline, including both AP and RSD parts on cutsky mocks for
$\\{P_{0},P_{2},B_{0}\\}$ data vector. The AP part is implemented as above,
and the RSD part yields a theoretical shift of $\Delta f=0.060$. The AP part
behaves exactly as expected. For this data vector, the RSD part can only be
reliably tested by considering the parameter combination $f\sigma_{8}$: the
addition of the bispectrum monopole is not sufficient to break the well-known
degeneracy between these two parameters (see [49] for a recent discussion).

We conclude that these results provide sufficient validation to the blinding
pipeline for present-day analyses involving the galaxy bispectrum. To date,
there is no sufficiently precise modeling of the survey window function on the
bispectrum multipoles. Should one become available, steps 2 and 3 above could
be extended to the full data vector including bispectrum multipoles. We leave
this to future work.

The work presented in this paper represents the first step to validate our
pipeline for the joint power spectrum and bispectrum analysis of the DESI
survey. Forthcoming work will include improvements of the survey window
approximation for the bispectrum and the treatment of the imaging and
systematic weights on the bispectrum.

## Data Availability

All data from the tables figures are available in machine-readable format at
10.5281/zenodo.11984896 in compliance with the DESI data management plan.

## Acknowledgements

SNM, HGM and LV thank Santiago Ávila, Mike Shengbo Wang and Benjamin Weaver
for the helpful discussion and suggestions. SNM acknowledges funding from the
official doctoral program of the University of Barcelona for the development
of a research project under the PREDOCS-UB grant. HGM acknowledges support
through the program Ramón y Cajal (RYC-2021-034104) of the Spanish Ministry of
Science and Innovation. LV and HGM acknowledge the support of the European
Union’s Horizon 2020 research and innovation program ERC (BePreSySe, grant
agreement 725327).

Funding for this work was partially provided by the Spanish MINECO under
project PGC2018-098866-B-I00MCIN/AEI/10.13039/501100011033 y FEDER “Una manera
de hacer Europa”, and the “Center of Excellence Maria de Maeztu 2020-2023”
award to the ICCUB (CEX2019-000918-M funded by
MCIN/AEI/10.13039/501100011033).

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.

This work has made use of the following publicly available codes: GEO-FPT
[49], Rustico [48], Brass [48], Emcee [62], GSL [63], SciPy [64], NumPy [65],
GetDist [66], Astropy [67], Matplotlib [68]. We are grateful to the developers
who made these codes public.

##  Appendix A Additional blinding considerations

We show in Figure 6 the best fitting theoretical model for the joint data-
vector $\\{P_{0},P_{2},B_{0}\\}$ of the cutsky mocks in the original and
blinded cases (both AP-only and AP+RSD). There is no significant difference in
the $\chi_{H}^{2}$ values (where the $H$ subscript stands for Hartlap-
corrected covariance matrix) when the data-vector is blinded. Furthermore, the
values for $\chi^{2}_{H}$ are close to the number of degrees of freedom, which
is equal to the number of elements of the full data-vector (141) minus the
number of free parameters (9). Even in this case, where the errorbars
correspond to an effective volume of $\sim 40\,(\textrm{Gpc}\,h^{-1})^{3}$,
the values for $\chi_{H}^{2}$ are all within $\sim 10\%$ of the number of
degrees of freedom. This (along with the results shown in the main text) is an
indication that the adopted models for the power spectrum and bispectrum do
not show evidence for large systematic errors given the expected statistical
error of present and upcoming surveys.

Figure 6: Best fits for the cutsky mocks for the three components of the data-
vector, $\\{P_{0},P_{2},B_{0}\\}$, in the cases where the mocks were not
blinded, AP-blinded, and fully blinded (AP+RSD). The quantity that is shown is
the discrepancy between the best-fitting model and the data (eg.
$P_{0}-\hat{P}_{0}$) divided by the corresponding error $\sigma$. The shaded
regions delimit the 1$\sigma$ region. The $x$-axis in the lower plot refers to
the triangle configuration index.

## References

  * [1] B. A. Benson, P. Ade, Z. Ahmed, S. Allen, K. Arnold, J. Austermann, A. Bender, L. Bleem, J. Carlstrom, C. Chang, et al., “Spt-3g: a next-generation cosmic microwave background polarization experiment on the south pole telescope,” in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII, vol. 9153, pp. 552–572, SPIE, 2014.
  * [2] S. Aiola, E. Calabrese, L. Maurin, S. Naess, B. L. Schmitt, M. H. Abitbol, G. E. Addison, P. A. Ade, D. Alonso, M. Amiri, et al., “The atacama cosmology telescope: Dr4 maps and cosmological parameters,” Journal of Cosmology and Astroparticle Physics, vol. 2020, no. 12, p. 047, 2020.
  * [3] N. Galitzki, A. Ali, K. S. Arnold, P. C. Ashton, J. E. Austermann, C. Baccigalupi, T. Baildon, D. Barron, J. A. Beall, S. Beckman, et al., “The simons observatory: instrument overview,” in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy IX, vol. 10708, pp. 40–52, SPIE, 2018.
  * [4] M. Levi, C. Bebek, T. Beers, R. Blum, R. Cahn, D. Eisenstein, B. Flaugher, K. Honscheid, R. Kron, O. Lahav, P. McDonald, N. Roe, D. Schlegel, and representing the DESI collaboration, “The DESI Experiment, a whitepaper for Snowmass 2013,” arXiv e-prints, p. arXiv:1308.0847, Aug. 2013.
  * [5] DESI Collaboration, “The DESI Experiment Part I: Science,Targeting, and Survey Design,” arXiv e-prints, p. arXiv:1611.00036, Oct. 2016.
  * [6] DESI Collaboration, “The DESI Experiment Part II: Instrument Design,” arXiv e-prints, p. arXiv:1611.00037, Oct. 2016.
  * [7] DESI Collaboration, “Overview of the Instrumentation for the Dark Energy Spectroscopic Instrument,” AJ, vol. 164, p. 207, Nov. 2022.
  * [8] J. H. Silber, P. Fagrelius, K. Fanning, M. Schubnell, J. N. Aguilar, S. Ahlen, J. Ameel, O. Ballester, et al., “The Robotic Multiobject Focal Plane System of the Dark Energy Spectroscopic Instrument (DESI),” AJ, vol. 165, p. 9, Jan. 2023.
  * [9] T. N. Miller, P. Doel, G. Gutierrez, R. Besuner, Brooks, et al., “The Optical Corrector for the Dark Energy Spectroscopic Instrument,” arXiv e-prints, p. arXiv:2306.06310, June 2023.
  * [10] J. Guy, S. Bailey, A. Kremin, S. Alam, D. M. Alexander, C. Allende Prieto, S. BenZvi, A. S. Bolton, D. Brooks, E. Chaussidon, A. P. Cooper, K. Dawson, de la Macorra, et al., “The Spectroscopic Data Processing Pipeline for the Dark Energy Spectroscopic Instrument,” AJ, vol. 165, p. 144, Apr. 2023.
  * [11] E. F. Schlafly, D. Kirkby, D. J. Schlegel, A. D. Myers, A. Raichoor, K. Dawson, J. Aguilar, C. Allende Prieto, S. Bailey, S. BenZvi, et al., “Survey Operations for the Dark Energy Spectroscopic Instrument,” AJ, vol. 166, p. 259, Dec. 2023.
  * [12] R. Zhou, B. Dey, J. A. Newman, D. J. Eisenstein, K. Dawson, S. Bailey, A. Berti, J. Guy, T.-W. Lan, H. Zou, et al., “Target Selection and Validation of DESI Luminous Red Galaxies,” AJ, vol. 165, p. 58, Feb. 2023.
  * [13] DESI Collaboration, “Validation of the Scientific Program for the Dark Energy Spectroscopic Instrument,” AJ, vol. 167, p. 62, Feb. 2024.
  * [14] DESI Collaboration, “The Early Data Release of the Dark Energy Spectroscopic Instrument,” arXiv e-prints, p. arXiv:2306.06308, June 2023\.
  * [15] DESI Collaboration, “DESI 2024 III: Baryon Acoustic Oscillations from Galaxies and Quasars,” arXiv e-prints, p. arXiv:2404.03000, Apr. 2024\.
  * [16] DESI Collaboration, “DESI 2024 IV: Baryon Acoustic Oscillations from the Lyman Alpha Forest,” arXiv e-prints, p. arXiv:2404.03001, Apr. 2024.
  * [17] DESI Collaboration, “DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations,” arXiv e-prints, p. arXiv:2404.03002, Apr. 2024.
  * [18] R. Laureijs, J. Amiaux, S. Arduini, J.-L. Augueres, J. Brinchmann, R. Cole, M. Cropper, C. Dabin, L. Duvet, A. Ealet, et al., “Euclid definition study report,” arXiv preprint arXiv:1110.3193, 2011.
  * [19] Ž. Ivezić, S. M. Kahn, J. A. Tyson, B. Abel, E. Acosta, R. Allsman, D. Alonso, Y. AlSayyad, S. F. Anderson, J. Andrew, et al., “Lsst: from science drivers to reference design and anticipated data products,” The Astrophysical Journal, vol. 873, no. 2, p. 111, 2019.
  * [20] J. R. Klein and A. Roodman, “Blind analysis in nuclear and particle physics,” Annu. Rev. Nucl. Part. Sci., vol. 55, pp. 141–163, 2005.
  * [21] A. Conley, G. Goldhaber, L. Wang, G. Aldering, R. Amanullah, E. Commins, V. Fadeyev, G. Folatelli, G. Garavini, R. Gibbons, et al., “Measurement of $\omega_{m}$, $\omega_{\lambda}$ from a blind analysis of type ia supernovae with cmagic: Using color information to verify the acceleration of the universe,” The Astrophysical Journal, vol. 644, no. 1, p. 1, 2006\.
  * [22] R. MacCoun and S. Perlmutter, “Blind analysis: Hide results to seek the truth,” Nature, vol. 526, no. 7572, pp. 187–189, 2015.
  * [23] K. Kuijken, C. Heymans, H. Hildebrandt, R. Nakajima, T. Erben, J. T. de Jong, M. Viola, A. Choi, H. Hoekstra, L. Miller, et al., “Gravitational lensing analysis of the kilo-degree survey,” Monthly Notices of the Royal Astronomical Society, vol. 454, no. 4, pp. 3500–3532, 2015.
  * [24] B. R. Zhang, M. J. Childress, T. M. Davis, N. V. Karpenka, C. Lidman, B. P. Schmidt, and M. Smith, “A blinded determination of h 0 from low-redshift type ia supernovae, calibrated by cepheid variables,” Monthly Notices of the Royal Astronomical Society, vol. 471, no. 2, pp. 2254–2285, 2017.
  * [25] T. M. Abbott, F. B. Abdalla, A. Alarcon, J. Aleksić, S. Allam, S. Allen, A. Amara, J. Annis, J. Asorey, S. Avila, et al., “Dark energy survey year 1 results: Cosmological constraints from galaxy clustering and weak lensing,” Physical Review D, vol. 98, no. 4, p. 043526, 2018.
  * [26] M. Asgari, T. Tröster, C. Heymans, H. Hildebrandt, J. L. van den Busch, A. H. Wright, A. Choi, T. Erben, B. Joachimi, S. Joudaki, et al., “Kids+ viking-450 and des-y1 combined: Mitigating baryon feedback uncertainty with cosebis,” Astronomy & Astrophysics, vol. 634, p. A127, 2020.
  * [27] J. Muir, G. M. Bernstein, D. Huterer, F. Elsner, E. Krause, A. Roodman, S. Allam, J. Annis, S. Avila, K. Bechtol, et al., “Blinding multiprobe cosmological experiments,” Monthly Notices of the Royal Astronomical Society, vol. 494, no. 3, pp. 4454–4470, 2020.
  * [28] E. Sellentin, “A blinding solution for inference from astronomical data,” Monthly Notices of the Royal Astronomical Society, vol. 492, no. 3, pp. 3396–3407, 2020.
  * [29] A. Font-Ribera et al., “In prep.,” 2024.
  * [30] S. Brieden, H. Gil-Marín, L. Verde, and J. L. Bernal, “Blind observers of the sky,” Journal of Cosmology and Astroparticle Physics, vol. 2020, no. 09, p. 052, 2020.
  * [31] U. Andrade, J. Mena-Fernández, H. Awan, A. Ross, S. Brieden, J. Pan, A. de Mattia, J. Aguilar, S. Ahlen, O. Alves, et al., “Validating the galaxy and quasar catalog-level blinding scheme for the desi 2024 analysis,” arXiv preprint arXiv:2404.07282, 2024.
  * [32] C. Alcock and B. Paczynski, “An evolution free test for non-zero cosmological constant,” Nature, vol. 281, pp. 358–359, 1979.
  * [33] E. V. Linder, “Cosmic growth history and expansion history,” Physical Review D—Particles, Fields, Gravitation, and Cosmology, vol. 72, no. 4, p. 043529, 2005.
  * [34] E. V. Linder and R. N. Cahn, “Parameterized beyond-einstein growth,” Astroparticle Physics, vol. 28, no. 4-5, pp. 481–488, 2007.
  * [35] D. J. Eisenstein, H.-J. Seo, and M. White, “On the robustness of the acoustic scale in the low-redshift clustering of matter,” The Astrophysical Journal, vol. 664, no. 2, p. 660, 2007.
  * [36] N. Padmanabhan, M. White, and J. Cohn, “Reconstructing baryon oscillations: A lagrangian theory perspective,” Physical Review D, vol. 79, no. 6, p. 063523, 2009.
  * [37] A. Burden, W. J. Percival, M. Manera, A. J. Cuesta, M. Vargas Magana, and S. Ho, “Efficient reconstruction of linear baryon acoustic oscillations in galaxy surveys,” Monthly Notices of the Royal Astronomical Society, vol. 445, no. 3, pp. 3152–3168, 2014.
  * [38] H. Gil-Marín, J. Noreña, L. Verde, W. J. Percival, C. Wagner, M. Manera, and D. P. Schneider, “The power spectrum and bispectrum of SDSS DR11 BOSS galaxies – I. Bias and gravity,” Mon. Not. Roy. Astron. Soc., vol. 451, no. 1, pp. 539–580, 2015.
  * [39] H. Gil-Marín, W. J. Percival, L. Verde, J. R. Brownstein, C.-H. Chuang, F.-S. Kitaura, S. A. Rodríguez-Torres, and M. D. Olmstead, “The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: RSD measurement from the power spectrum and bispectrum of the DR12 BOSS galaxies,” Mon. Not. Roy. Astron. Soc., vol. 465, no. 2, pp. 1757–1788, 2017.
  * [40] H. Gil-Marín, W. J. Percival, L. Verde, J. R. Brownstein, C.-H. Chuang, F.-S. Kitaura, S. A. Rodríguez-Torres, and M. D. Olmstead, “The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: RSD measurement from the power spectrum and bispectrum of the DR12 BOSS galaxies,” Monthly Notices of the Royal Astronomical Society, vol. 465, pp. 1757–1788, Feb. 2017.
  * [41] K. Pardede, F. Rizzo, M. Biagetti, E. Castorina, E. Sefusatti, and P. Monaco, “Bispectrum-window convolution via hankel transform,” Journal of Cosmology and Astroparticle Physics, vol. 2022, no. 10, p. 066, 2022.
  * [42] 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,” Monthly Notices of the Royal Astronomical Society, vol. 508, no. 3, pp. 4017–4037, 2021.
  * [43] P. Collaboration, “Planck 2018 results. vi. cosmological parameters,” 2018.
  * [44] M. Tegmark, “Measuring cosmological parameters with galaxy surveys,” Physical Review Letters, vol. 79, no. 20, p. 3806, 1997.
  * [45] A. Variu et al., “In prep.,” 2024.
  * [46] M. White, J. L. Tinker, and C. K. McBride, “Mock galaxy catalogues using the quick particle mesh method,” Monthly Notices of the Royal Astronomical Society, vol. 437, no. 3, pp. 2594–2606, 2014.
  * [47] 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,” Monthly Notices of the Royal Astronomical Society, vol. 446, no. 3, pp. 2621–2628, 2015.
  * [48] H. Gil-Marín, J. E. Bautista, R. Paviot, M. Vargas-Magaña, S. de la Torre, S. Fromenteau, S. Alam, S. Ávila, E. Burtin, C.-H. Chuang, K. S. Dawson, J. Hou, A. de Mattia, F. G. Mohammad, E.-M. Müller, S. Nadathur, R. Neveux, W. J. Percival, A. Raichoor, M. Rezaie, A. J. Ross, G. Rossi, V. Ruhlmann-Kleider, A. Smith, A. Tamone, J. L. Tinker, R. Tojeiro, Y. Wang, G.-B. Zhao, C. Zhao, J. Brinkmann, J. R. Brownstein, P. D. Choi, S. Escoffier, A. de la Macorra, J. Moon, J. A. Newman, D. P. Schneider, H.-J. Seo, and M. Vivek, “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, vol. 498, pp. 2492–2531, Oct. 2020.
  * [49] S. Novell-Masot, D. Gualdi, H. Gil-Marín, and L. Verde, “Geo-fpt: a model of the galaxy bispectrum at mildly non-linear scales,” Journal of Cosmology and Astroparticle Physics, vol. 2023, no. 11, p. 044, 2023.
  * [50] S. Novell-Masot, H. Gil-Marín, and L. Verde, “On approximations of the redshift-space bispectrum and power spectrum multipoles covariance matrix,” arXiv preprint arXiv:2306.03137, 2023.
  * [51] M. Crocce and R. Scoccimarro, “Renormalized cosmological perturbation theory,” Physical Review D, vol. 73, mar 2006.
  * [52] H. Gil-Marín, J. Noreña, L. Verde, W. J. Percival, C. Wagner, M. Manera, and D. P. Schneider, “The power spectrum and bispectrum of SDSS DR11 BOSS galaxies – I. Bias and gravity,” Monthly Notices of the Royal Astronomical Society, vol. 451, pp. 539–580, July 2015.
  * [53] R. Scoccimarro, H. M. P. Couchman, and J. A. Frieman, “The Bispectrum as a Signature of Gravitational Instability in Redshift-Space,” Astrophys. J., vol. 517, pp. 531–540, 1999.
  * [54] T. Baldauf, U. Seljak, V. Desjacques, and P. McDonald, “Evidence for quadratic tidal tensor bias from the halo bispectrum,” Physical Review D, vol. 86, no. 8, p. 083540, 2012.
  * [55] S. Saito, T. Baldauf, Z. Vlah, U. Seljak, T. Okumura, and P. McDonald, “Understanding higher-order nonlocal halo bias at large scales by combining the power spectrum with the bispectrum,” Physical Review D, vol. 90, no. 12, p. 123522, 2014.
  * [56] S. Brieden, H. Gil-Marín, and L. Verde, “PT challenge: validation of ShapeFit on large-volume, high-resolution mocks,” J. Cosmology Astropart. Phys., vol. 2022, p. 005, June 2022.
  * [57] J. Hartlap, P. Simon, and P. Schneider, “Why your model parameter confidences might be too optimistic: Unbiased estimation of the inverse covariance matrix,” Astron. Astrophys., vol. 464, p. 399, 2007.
  * [58] E. Sellentin and A. F. Heavens, “Parameter inference with estimated covariance matrices,” Mon. Not. Roy. Astron. Soc., vol. 456, no. 1, pp. L132–L136, 2016.
  * [59] D. Gualdi and L. Verde, “Galaxy redshift-space bispectrum: the Importance of Being Anisotropic,” JCAP, vol. 06, p. 041, 2020.
  * [60] D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, M. R. Blanton, R. C. Nichol, R. Scranton, H.-J. Seo, M. Tegmark, Z. Zheng, et al., “Detection of the baryon acoustic peak in the large-scale correlation function of sdss luminous red galaxies,” The Astrophysical Journal, vol. 633, no. 2, p. 560, 2005.
  * [61] J. E. Bautista, R. Paviot, M. Vargas Magana, S. de La Torre, S. Fromenteau, H. Gil-Marín, A. J. Ross, E. Burtin, K. S. Dawson, J. Hou, 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,” Monthly Notices of the Royal Astronomical Society, vol. 500, no. 1, pp. 736–762, 2021.
  * [62] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, “emcee: The MCMC hammer,” Publications of the Astronomical Society of the Pacific, vol. 125, pp. 306–312, mar 2013.
  * [63] M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi, and R. Ulerich, GNU scientific library.  Network Theory Limited Godalming, 2002.
  * [64] SciPy 1.0 Contributors, “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,” Nature Methods, vol. 17, pp. 261–272, 2020.
  * [65] C. R. Harris et al., “Array programming with NumPy,” Nature, vol. 585, pp. 357–362, Sept. 2020.
  * [66] A. Lewis, “Getdist: a python package for analysing monte carlo samples,” arXiv preprint arXiv:1910.13970, 2019.
  * [67] Astropy Collaboration, “The Astropy Project: Sustaining and Growing a Community-oriented Open-source Project and the Latest Major Release (v5.0) of the Core Package,” ApJ, vol. 935, p. 167, Aug. 2022.
  * [68] J. D. Hunter, “Matplotlib: A 2d graphics environment,” Computing in Science & Engineering, vol. 9, no. 3, pp. 90–95, 2007.