# A comparison of effective field theory models of redshift space galaxy power
spectra for DESI 2024 and future surveys

M. Maus  0000-0002-9020-911X Y. Lai  H. E. Noriega  0000-0002-3397-3998 S.
Ramirez-Solano  A. Aviles  0000-0001-5998-3986 S. Chen  0000-0002-5762-6405 S.
Fromenteau  H. Gil-Marín  0000-0003-0265-6217 C. Howlett  0000-0002-1081-9410
M. Vargas-Magaña  0000-0003-3841-1836 M. White  0000-0001-9912-5070 P. Zarrouk
0000-0002-7305-9578 J. Aguilar  S. Ahlen  0000-0001-6098-7247 O. Alves  S.
Brieden  0000-0003-3896-9215 D. Brooks  E. Burtin  T. Claybaugh  S. Cole
0000-0002-5954-7903 K. Dawson  M. Icaza-Lizaola  0000-0002-2547-3184 A. de la
Macorra  0000-0002-1769-1640 A. de Mattia  P. Doel  S. Ferraro
0000-0003-4992-7854 N. Findlay  0009-0007-0716-3477 J. E. Forero-Romero
0000-0002-2890-3725 E. Gaztañaga  S. Gontcho A Gontcho  0000-0003-3142-233X C.
Hahn  0000-0003-1197-0902 K. Honscheid  M. Ishak  0000-0002-6024-466X A.
Kremin  0000-0001-6356-7424 M. Landriau  0000-0003-1838-8528 L. Le Guillou
0000-0001-7178-8868 M. Manera  0000-0003-4962-8934 R. Miquel  E. Mueller  S.
Nadathur  0000-0001-9070-3102 G. Niz  0000-0002-1544-8946 N. Palanque-
Delabrouille  0000-0003-3188-784X W. J. Percival  0000-0002-0644-5727 C.
Poppett  F. Prada  0000-0001-7145-8674 M. Rezaie  0000-0001-5589-7116 A.
Rocher  0000-0003-4349-6424 G. Rossi  E. Sanchez  0000-0002-9646-8198 D.
Schlegel  M. Schubnell  D. Sprayberry  G. Tarlé  0000-0003-1704-0781 S. Yuan
0000-0002-5992-7586 R. Zhao  0000-0002-7284-7265 R. Zhou  0000-0001-5381-4372
H. Zou  0000-0002-6684-3997

###### Abstract

In preparation for the next generation of galaxy redshift surveys, and in
particular the year-one data release from the Dark Energy Spectroscopic
Instrument (DESI), we investigate the consistency of a variety of effective
field theory models that describe the galaxy-galaxy power spectra in redshift
space into the quasi-linear regime using 1-loop perturbation theory. These
models are employed in the pipelines velocileptors, PyBird, and Folps$\nu$.
While these models have been validated independently, a detailed comparison
with consistent choices has not been attempted. After briefly discussing the
theoretical differences between the models we describe how to provide a more
apples-to-apples comparison between them. We present the results of fitting
mock spectra from the AbacusSummit suite of N-body simulations provided in
three redshift bins to mimic the types of dark time tracers targeted by the
DESI survey. We show that the theories behave similarly and give consistent
constraints in both the forward-modeling and ShapeFit compressed fitting
approaches. We additionally generate (noiseless) synthetic data from each
pipeline to be fit by the others, varying the scale cuts in order to show that
the models agree within the range of scales for which we expect 1-loop
perturbation theory to be applicable. This work lays the foundation of Full-
Shape analysis with DESI Y1 galaxy samples where in the tests we performed, we
found no systematic error associated with the modeling of the galaxy redshift
space power spectrum for this volume.

##  1 Introduction

The evolution of structures in the Universe is a powerful observational tool
that holds a wealth of information about its initial conditions and expansion
history. Large Scale Structure (LSS) can be probed with spectroscopic surveys
that use galaxies, quasars, the Ly$\alpha$ forest, etc. as tracers of the
underlying matter distributions. The completed Sloan Digital Sky Survey (SDSS;
[1]) consisting of the Baryon Oscillation Spectroscopic Survey (BOSS; [2, 3,
4, 5]) and the SDSS-IV extended Baryon Oscillation Spectroscopic Survey
(eBOSS; [6, 7]) is the largest, public galaxy redshift survey to date.
However, it has only scanned a small fraction of the observable Universe.
Therefore, much of the information on fundamental physics and the Universe’s
history we can potentially gain from LSS observations lies ahead of us with
upcoming surveys such as the Dark Energy Spectroscopic Instrument (DESI; [8,
9, 10, 11, 12]), Euclid [13], and beyond [14].

In parallel with the design of new instruments is also the development of more
sophisticated analysis pipelines to interpret the increasingly more
constraining data. In galaxy clustering analyses, the most common approach is
to employ methods based on cosmological perturbation theory. Such theories
have a history dating back decades (see e.g. ref. [15] for an early review and
references to the original literature) but have seen continual development to
the present day (see e.g. ref. [16] for a recent review). Modern perturbation
theories include contributions from non-linear dynamics [15], broadening of
features by long-wavelength fields111This frequently goes under the name “IR
resummation”, because it involves summing the contributions of long-wavelength
perturbations to higher order than other terms. A comparison of the Eulerian
and Lagrangian approaches can be found in Appendix A of ref. [17], and the
close numerical correspondence of the approaches can be found in Fig. 3 of
ref. [18]. [19, 20, 21, 22, 23], redshift-space distortions [24, 25], galaxy
bias [26, 27] and ‘counterterms’ to mitigate sensitivity to small-scale
physics [16]. Such models can be formulated in either a Lagrangian or Eulerian
framework and are able to self-consistently describe a wide range of
cosmological observables.

There are four main versions of these perturbative models that have been
implemented in efficient codes and are typically applied to data. These will
be introduced in some detail below. In principle, these codes all agree on a
‘unique’ prediction for the low-order statistics of biased tracers in redshift
space, for both configuration-space and Fourier-space statistics. However, in
practice, these models are normally applied to data or simulations with
different datasets or assumptions about window functions, nuisance parameters,
fitting range, priors and other details. This makes understanding the degree
of agreement between the models difficult, and can erroneously give the
impression that the models are inconsistent in some way. In order to dispel
this misconception, and in anticipation of the upcoming Y1 data release from
DESI [28, 29, 30, 31, 32, 33, 34, 35, 36] for which these models will be
employed, it is important to understand how differences in constraints between
modeling pipelines are driven by internal priors and to show that the
different codes perform similarly under the same assumptions. The purpose of
this paper is to compare the constraints on $\Lambda$CDM and compressed
parameters generated by three Fourier-space models that will be used on
upcoming DESI data analyses, and show that under consistent assumptions these
models do in fact agree in their constraints. This issue has been partially
investigated for the PyBird, class-pt, and CLASS-OneLoop models in Refs. [37,
38]. However, those models are based on the same theory and differ only in
their prior choices. The models we compare here (velocileptors, Folps$\nu$,
and PyBird) additionally differ in their theoretical frameworks and
resummation schemes. We also compare the behavior of our models within two
different fitting techniques, where one (“Full-modeling”) directly varies the
parameters within a given cosmological model (e.g. $\Lambda$CDM) while the
other (“ShapeFit”) uses a single reference linear power spectrum and then
varies compressed parameters that control the shape information and late-
universe dynamics. The comparison in this paper focuses on Fourier space
(power spectrum) models, but a similar comparison is underway that compares
the predictions of these models, transformed to configuration space, to those
of the purely configuration space model EFT-GSM [39].

The outline of the paper is as follows. In §2 we describe the mock galaxy
catalogs that are used in the bulk of this work to test the agreement of the
models on complex datasets for which the ‘true’ cosmology is known. In §3 we
first give an overview of the Eulerian and Lagrangian Perturbation Theory (EPT
and LPT) formalisms before introducing each model under consideration. We note
that detailed descriptions of these models can be found in their respective
individual papers and only a brief summary of their relevant features are
discussed here. We then describe two parametrization choices (minimal and
maximal freedom) for the bias terms that we will employ in our comparisons and
then conclude the section with a short summary of the key differences between
the models. Then, in §4 we briefly describe the Full-modeling and ShapeFit
fitting methods. In §5, we present results both on Abacus simulations data as
well as on noiseless theoretical mock power spectra created by the models
themselves. Finally, we present our conclusions in §6.

##  2 Data

In this paper, we present a comparison between theories using mock data
generated from N-body simulations and from the theories themselves. The N-body
data represent plausible and complex galaxy populations that the models should
all be able to fit with the advantage of knowing the ‘true’ cosmology from
which they were drawn. Data generated from the theories themselves is
essentially noiseless and thus allows us to compare the theories in great
detail.

The galaxy clustering simulations we use are from the AbacusSummit [40] suite,
which were created on the Summit supercomputer, located at the Oak Ridge
Leadership Computing Facility, using the Abacus [41] N-body code. We focus on
the 25 realizations with a fixed cosmology, each in a cubic box geometry of 8
[$h^{-3}$Gpc3] in volume. In order to reduce sample variance ‘noise’ in the
power spectra we average over the 25 realizations and work with the mean
$P_{\ell}(k)$, corresponding to a volume of 200 [$h^{-3}$Gpc3]. In this paper
we focus on HOD samples calibrated to reproduce the power spectra of the
luminous red galaxies (LRG) at a redshift of $z=0.8$, emission line galaxies
(ELGs) at $z=1.1$, and QSOs at $z=1.4$ [42, 43]. The mock data power spectra
are shown in the left panels of Fig. 1 for each of the three tracers. We only
use the monopole ($P_{0}(k)$) and quadrupole ($P_{2}(k)$) data, as we only
focus our comparison within the standard $\Lambda$CDM model for which the
hexadecapole does not significantly influence constraints[44, 45, 46]. In the
right panels we show comparisons of predictions from each of the three EFT
models with $\Lambda$CDM parameters fixed to the true input values of the
mocks. These will be discussed further in §5.3

In our fits, we assume a Gaussian likelihood, and so we need an estimate of
the covariance matrix. Our covariance matrix is obtained by Monte-Carlo using
1000 “effective Zel’dovich approximation” mocks (EZmocks; [47]). Since it is
numerically computed from mocks, we do not assume the covariance to be
Gaussian and thus there are nonzero off-diagonal terms. In principle we can
also compute analytic covariances for these samples following the methods of
Ref. [48], where assumptions about Gaussianity are relevant, but we do not do
so here. As we are fitting the combined volume of 25 Abacus realizations, the
covariance matrix for a single-box volume can be rescaled by a factor of up to
1/25 before we expect to see statistically significant shifts in the
parameters just from fluctuations in the (mean) data vector. While the implied
statistical precision in this case exceeds the convergence that has been
demonstrated for the N-body simulations themselves [49, 50], we currently see
no evidence of a bias that projects into the subspace of the cosmological
parameters. We discuss results from fitting data with both rescaled and
unrescaled covariance, and specify which is used with the labels $V_{25}$(200
[$h^{-3}$Gpc3]) and $V_{1}$(8 [$h^{-3}$Gpc3]) respectively.

We also employ the theoretical models themselves as noiseless “data”, to be
fit by the other modeling codes. This can aid in analyzing the behavior of
bias, stochastic, and counterterms of each model when the true values are
known. In addition, we have more freedom in scaling up/down the covariance
rather than being limited by the number of mocks and possible inaccuracies in
the simulations.

##  3 Theory and Models

###  3.1 Overview of EPT and LPT

As stated in the introduction, cosmological perturbation theory comes in two
flavors, Eulerian (EPT) and Lagrangian (LPT). In the Eulerian formulation, the
cold dark matter and baryons are treated as a perfect, pressure-less fluid
which obeys the continuity, Euler and Poisson equations. Defining the
overdensity, $\delta=\rho/\bar{\rho}-1$, and velocity, $\mathbf{v}$, fields
these equations become [15, 16, 51]:

$\displaystyle\partial_{\tau}\delta+\nabla\cdot[(1+\delta)\bm{v}]$
$\displaystyle=0$ (3.1)
$\displaystyle\partial_{\tau}\bm{v}+\mathcal{H}\bm{v}+\bm{v}\cdot\nabla\bm{v}$
$\displaystyle=-\nabla\Phi$ (3.2) $\displaystyle\nabla^{2}\Phi$
$\displaystyle=\frac{3}{2}\mathcal{H}^{2}\delta,$ (3.3)

where $\mathcal{H}$ is the conformal Hubble parameter. These equations can be
solved perturbatively by expanding $\delta$ and the velocity divergence,
$\theta=\nabla\cdot\bm{v}$, in powers of their lowest-order (linear)
solutions:

$\displaystyle\delta(\bm{k},z)=\sum_{n=1}^{\infty}a^{n}\delta^{(n)}(\bm{k}),\quad,\quad\theta(\bm{k},z)=\mathcal{H}\sum_{n=1}^{\infty}a^{n}\theta^{(n)}(\bm{k}),$
(3.4)

in which

$\displaystyle\delta^{(n)}(\bm{k})=\int\frac{d^{3}\bm{k}_{1}\cdots
d^{3}\bm{k}_{n}}{(2\pi)^{3n}}(2\pi)^{3}\delta^{D}(\sum\bm{k}_{i}-\bm{k})\delta^{(1)}(\bm{k}_{1})\cdots\delta^{(1)}(\bm{k}_{n})F_{n}(\bm{k}_{1},...,\bm{k}_{n},\bm{k}),$
(3.5)

and similarly for $\theta^{(n)}$. In these equations, we have followed the
standard practice of approximating the time-dependence of each term as $a^{n}$
and factoring this out of the $\delta^{(n)}$ and $\theta^{(n)}$. This will be
an excellent approximation for our purposes [52, 53].

By contrast, Lagrangian Perturbation Theory treats cold dark matter as
collisionless particles whose observed (Eulerian) coordinates, $\bm{x}$, are
related to their initial (Lagrangian) positions, $\bm{q}$, via the
displacement field such that $\bm{x}(\bm{q})=\bm{q}+\bm{\Psi}(\bm{q})$.
Nonlinear dynamics is then governed by a second order differential equation in
the displacement field:

$\displaystyle\partial_{\tau}^{2}\bm{\Psi}+\mathcal{H}\partial_{\tau}\bm{\Psi}=-\nabla_{\bm{x}}\Phi(\bm{q}+\bm{\Psi}(\bm{q}))\quad.$
(3.6)

where $\tau$ is the conformal time. The matter overdensity is then derived
from number conservation as

$\displaystyle 1+\delta(\bm{x})=\int d^{3}\bm{q}\ \delta^{\rm
D}(\bm{x}-\bm{q}-\bm{\Psi})\quad\Rightarrow\quad 1+\delta(\bm{k})=\int
d^{3}\bm{q}\ e^{i\bm{k}\cdot(\bm{q}+\Psi(\bm{q}))}.$ (3.7)

Expanding the above exponential as a Taylor series, as well as expanding the
displacement field
$\bm{\Psi}=\bm{\Psi}^{(1)}+\bm{\Psi}^{(2)}+\bm{\Psi}^{(3)}+\cdots$ gives
expressions for $\delta^{(n)}(\bm{k})$ that agree with the Eulerian case at
each order. That the Eulerian and Lagrangian approaches agree to all orders in
perturbation theory, while the behavior of the two systems that they model
disagree222In a collisional fluid when two fluid elements cross a shock
converts kinetic energy into thermal energy. In a collisionless system of
particles, the crossing of trajectories implies multi-streaming, and the
velocity field becomes multivalued at a given $\mathbf{x}$. when trajectories
of fluid elements or particles cross, is a hint that we will need to include
beyond-PT contributions to our model [54].

Since the galaxy redshift surveys target galaxies as tracers of the matter
density field we need a ‘bias model’ to connect the two [27]. We follow
standard practice and express the galaxy overdensity as a functional of the
matter field that can include all combinations allowed by the symmetries.
Specifically, we use a Taylor expansion, with bias coefficients that are free
parameters when modeling data. In the Eulerian prescription, the galaxy
biasing is a function of the non-linear matter field at $\mathbf{x}$:

$\displaystyle\delta_{\rm g}(\bm{x})=c_{1}\delta_{\rm
m}(\bm{x})+c_{2}(\delta_{\rm m}^{2}(\bm{x})-\left\langle\delta_{\rm
m}^{2}(\bm{x})\right\rangle)+c_{\rm s}(s^{2}(\bm{x})-\left\langle
s^{2}(\bm{x})\right\rangle)+...$ (3.8)

The Lagrangian case is expanded in the initial density field at Lagrangian
coordinate $\bm{q}$:

$\displaystyle 1+\delta_{\rm g}(\bm{k})=\int d^{3}\bm{q}\
e^{i\bm{k}\cdot(\bm{q}+\Psi)}F[\delta^{(1)}(\bm{q})],$ (3.9)

with the bias functional $F[\delta^{(1)}(\bm{q})]$, written as $F(\bm{q})$ for
convenience, given by

$\displaystyle
F(\bm{q})=1+b_{1}\delta_{0}+\frac{1}{2}b_{2}(\delta_{0}(\bm{q})^{2}-\left\langle\delta_{0}^{2}\right\rangle)+b_{s}(s_{0}^{2}(\bm{q})-\left\langle
s^{2}\right\rangle).$ (3.10)

Finally, the “effective” field theories employed in the models include
additive corrections that correspond to the sensitivity to small-scale physics
that can not be described perturbatively. Couplings of short wavelength modes
are included in the form of counterterms ($\sim k^{2}P_{\rm lin}$) and
stochastic terms within each model.

With these preliminaries understood, we now turn to the different models that
we shall compare in this paper.

###  3.2 Velocileptors

Full-Modeling | ShapeFit | LPT Bias | Stoch/Counter
---|---|---|---
|  | Min. F. | Max. F. |
H0 | $f\sigma_{8}$ | $(1+b_{1})\sigma_{8}$ | $\alpha_{0}$
$\mathcal{U}[55,79]$ | $\mathcal{U}[0,2]$ | $\mathcal{U}[0.5,3.0]$ | —
$\omega_{\mathrm{b}}$ | $\alpha_{\parallel}$ | $b_{2}$ | $\alpha_{2}$
$\mathcal{N}[0.02237,0.00037]$ | $\mathcal{U}[0.5,1.5]$ | $\mathcal{N}[0,10]$ | $\mathcal{U}[-15,15]$ | —
$\omega_{\mathrm{cdm}}$ | $\alpha_{\perp}$ | $b_{s}$ | SN0
$\mathcal{U}[0.08,0.16]$ | $\mathcal{U}[0.5,1.5]$ | 0 | $\mathcal{U}[-15,15]$ | —
$\log(10^{10}A_{\mathrm{s}})$ | $m$ | $b_{3}$ | SN2
$\mathcal{U}[2.03,4.03]$ | $\mathcal{U}[-3.0,3.0]$ | 0 | $\mathcal{U}[-15,15]$ | —
Table 1: Velocileptors LPT priors on parameters used in the Full-Modeling
($\Lambda$CDM) and ShapeFit fitting methods. The $\Lambda$CDM model involves
H0, $\Omega_{\mathrm{b}}$,$\omega_{\mathrm{cdm}}$,
$\log(10^{10}A_{\mathrm{s}})$ and all of the bias, stochastic, and
counterterms. The ShapeFit method fits $f\sigma_{8}$, $\alpha_{\parallel}$,
$\alpha_{\perp}$, $m$ as well as the same bias, stochastic and counterterms.
The entries $\mathcal{U}[{\rm min,max}]$ and $\mathcal{N}[\mu,\sigma]$ refer
to uniform and Gaussian normal distributions, respectively. For the bias terms
we show both minimal and maximal freedom cases, defined in § 3.5.

The first model we consider is the fully resummed, 1-loop, LPT pipeline
implemented in velocileptors333github.com/sfschen/velocileptors [55, 56]. The
redshift space power spectrum is computed using the biasing scheme of eq. 3.10
and the overdensity of eq. 3.9 such that:

$\displaystyle P_{\rm s}^{\rm LPT}(\bm{k})=\int d^{3}\bm{q}\left\langle
e^{i\bm{k}\cdot(\bm{q}+\Delta_{\rm
s})}F(\bm{q}_{1})F(\bm{q}_{2})\right\rangle_{\bm{q}=\bm{q}_{1}-\bm{q}_{2}},$
(3.11)

where $\Delta_{\rm s}=\Psi_{\rm s}(\bm{q}_{1})-\Psi_{s}(\bm{q}_{2})$ and the
“s” subscripts denote the redshift space displacement fields. We use the “Full
LPT” model in velocileptors which resums both the long-wavelength displacement
and velocity contributions.

The integrals in eq. 3.11 are done using FFTs (see Appendix D of [55]). The
model also includes stochastic contributions and counter terms. These terms
enter the power spectrum with $\sigma_{n}$ and $\alpha_{n}$ coefficients,
respectively, such that the final LPT power spectrum is:

$\displaystyle P_{\rm s}(\bm{k})=P_{\rm s}^{\rm
LPT}(\bm{k})+k^{2}(\alpha_{0}+\alpha_{2}\mu^{2}+\alpha_{4}\mu^{4})P_{\rm
s,Zel}(\bm{k})+R_{\rm
h}^{3}(1+\sigma_{2}k^{2}\mu^{2}+\sigma_{4}k^{4}\mu^{4}),$ (3.12)

where $P_{\rm s,Zel}(\bm{k})$ refers to the linear Zeldovich approximation and
$\mu$ is the cosine of the angle between $\bm{k}$ and the line-of-sight
(assumed fixed). For the stochastic contributions we use the parametrization
$\text{SN}_{0}=R_{\rm h}^{3}$, $\text{SN}_{2}=R_{\rm h}^{3}\sigma_{2}$, and
$\text{SN}_{4}=R_{h}^{3}\sigma_{4}$ where $R_{\rm h}$ is a characteristic
scale of halo formation. In Table 1 we report the priors used on parameters in
our velocileptors model that enter the monopole and quadrupole components of
the power spectra. We note that when excluding the hexadecapole from our
analysis, we set $\alpha_{4}=\text{SN}_{4}=0$.

velocileptors also includes a model based on EPT, with the terms collected in
a way informed by LPT but with resummation handled in a different way to the
LPT module. This makes it an interesting counterpoint to the LPT model and the
“natively EPT” models we discuss below. This EPT model is constructed from the
LPT expressions but with the IR scale, $k_{\rm IR}\rightarrow 0$ and bias
parameters remapped to the Eulerian prescription of ref. [57]. The IR
resummation scheme then involves applying a damping factor to the BAO signal
of the power spectrum prediction by separating it from the smooth part via a
wiggle/no-wiggle split using the sine transform method of Refs. [58, 59].

###  3.3 PyBird

The next model is based on 1-loop Eulerian perturbation theory.
PyBird444github.com/pierrexyz/pybird (The Python code for Biased tracers in
redshift space) employs an effective field theory formalism that is very
similar to CLASS-PT [60]. The derivation for the EFT in PyBird can be found in
ref. [61, 62], so we will not repeat it here. The redshift-space galaxy power
spectrum up to one-loop order is given by [62, 61]:

$\displaystyle P_{\rm g}(k,\mu)$
$\displaystyle=Z_{1}(\mu)^{2}P_{\mathrm{lin}}(k)+2\int\frac{d^{3}q}{(2\pi)^{3}}Z_{2}(\bm{q},\bm{k-q},\mu)^{2}P_{\mathrm{lin}}(|\bm{k-q}|)P_{\mathrm{lin}}(q)$
$\displaystyle+6Z_{1}(\mu)P_{\mathrm{lin}}(k)\int\frac{d^{3}q}{(2\pi)^{3}}Z_{3}(\bm{q},\bm{-q},\bm{k},\mu)P_{\mathrm{lin}}(q)+2Z_{1}(\mu)P_{\mathrm{lin}}(k)\times$
$\displaystyle\times\left(c_{\rm ct}\frac{k^{2}}{k_{\rm M}^{2}}+c_{\rm
r,1}\mu^{2}\frac{k^{2}}{k_{\rm M}^{2}}+c_{\rm r,2}\mu^{4}\frac{k^{2}}{k_{\rm
M}^{2}}\right)+\frac{1}{\overline{n}_{\rm
g}}\left(c_{\epsilon,1}+c_{\epsilon,2}\frac{k^{2}}{k_{\rm
M}^{2}}+c_{\epsilon,3}f\mu^{2}\frac{k^{2}}{k_{\rm M}^{2}}\right)$ (3.13)

where $P_{\mathrm{lin}}$ is the linear power spectrum, $c_{\rm ct},c_{\rm
r,1},c_{\rm r,2}$ are the counterterms,
$c_{\epsilon,1},c_{\epsilon,2},c_{\epsilon,3}$ are the stochastic terms, $f$
is the linear growth rate of structure, $\overline{n}_{\rm g}$ is the number
density of galaxies in the survey, and the redshift space kernels are given in
eq. (2.2) of ref. [62], following [15, 63]. By default the scale that
suppresses higher order derivatives in the bias expansion is set to $k_{\rm
M}=0.7\,h\,\mathrm{Mpc}^{-1}$. In addition to the counter terms and stochastic
terms, PyBird has 4 bias parameters: $b_{1},b_{2},b_{3},b_{4}$. This is the
bias parameterization described in Ref. [61], often referred to as the ‘West
Coast’ (WC) parameterization in the literature [64, 65, 37]. For the
stochastic terms, the main difference between PyBird and velocileptors is that
PyBird has an additional $k^{2}/k_{\rm M}^{2}$ term but does not include
$k^{4}\mu^{4}$. For the counter terms, velocileptors uses the Zeldovich
approximation while PyBird uses the redshift space linear power spectrum
$Z_{1}(\mu)P_{\mathrm{lin}}(k)$. When the difference between the two power
spectra is small, $c_{\rm ct}=\alpha_{0},c_{\rm r,1}=\alpha_{2}\ \mathrm{and}\
c_{\rm r,2}=\alpha_{4}$. For later use we define
$c_{\epsilon,\mathrm{mono}}=c_{\epsilon,1}+(2f/3)c_{\epsilon,2}$, and
$c_{\epsilon,\mathrm{quad}}=(4f/15)c_{\epsilon,3}.$.

Full-Modeling | ShapeFit | Bias | Stoch/Counter
---|---|---|---
H0 | $f\sigma_{8}$ | $b_{1}$ | $c_{\rm ct}$
$\mathcal{U}[55.36,79.36]$ | $\mathcal{U}[0,1]$ | $\mathcal{U}[0.0,4.0]$ | —
$\omega_{\mathrm{b}}$ | $\alpha_{\parallel}$ | $b_{2}$ | $c_{\rm r,1}$
$\mathcal{N}[0.02237,0.00037]$ | $\mathcal{U}[0.9,1.1]$ | $1$ | —
$\omega_{\mathrm{cdm}}$ | $\alpha_{\perp}$ | $b_{3}$ | $c_{\rm r,2}$
$\mathcal{U}[0.08,0.16]$ | $\mathcal{U}[0.9,1.1]$ | $1$ | $0$
$\log(10^{10}A_{\mathrm{s}})$ | $m$ | $b_{4}$ | $c_{\epsilon,1}$
$\mathcal{U}[2.0364,4.0364]$ | $\mathcal{U}[-1.0,1.0]$ | $\mathcal{U}[-15.0,15.0]$ | —
$w$ |  |  | $c_{\epsilon,\mathrm{mono}}$
$\mathcal{U}[-1.3,-0.7]$ |  |  | —
|  |  | $c_{\epsilon,\mathrm{quad}}$
|  |  | —
Table 2: The priors of Shapefit and Full-Model parameters for PyBirdẆe use
$\mathcal{U}$ ($\mathcal{N}$) to denote the uniform (Gaussian) prior. The
first number is the lower bound (mean), and the second number is the upper
bound (standard deviation). This configuration is the minimum freedom (“MinF")
configuration. We use this configuration to match with other pipelines in
DESI. The value for $b_{2},b_{3}$, and $b_{4}$ are obtained through the co-
evolution relation. The detailed derivation is given in section 3.5. For the
maximum freedom configuration, we put an infinite flat prior on $b_{3}$ and a
$\mathcal{U}[-10,10]$ prior on $b_{3}$ and $b_{4}$.

###  3.4 FOLPS$\nu$

Our final EPT model, Folps$\nu$555https://github.com/henoriega/FOLPS-nu. With
JAX: https://github.com/cosmodesi/folpsax. [66], is a Python code based on a
perturbative model, with the added value of accounting for the presence of
massive neutrinos [67, 68, 66]. As for the previous models, Folps$\nu$ is
based on EPT up to one-loop. The redshift space power spectrum is given by:

$\displaystyle P^{\text{EFT}}_{\rm s}(k,\mu)$
$\displaystyle=P_{\delta\delta}(k)+2f_{0}\mu^{2}P_{\delta\theta}(k)+f_{0}^{2}\mu^{4}P_{\theta\theta}(k)+A^{\rm
TNS}(k,\mu)+D(k,\mu)$
$\displaystyle\quad+(\alpha_{0}+\alpha_{2}\mu^{2}+\alpha_{4}\mu^{4})k^{2}P_{\rm
lin}(k)$ $\displaystyle\quad+P_{\rm shot}\big{[}\alpha^{\rm
shot}_{0}+\alpha^{\rm shot}_{2}(k\mu)^{2}\big{]}.$ (3.14)

where each of these terms is defined in detail in ref. [66]. A key feature of
Folps$\nu$ is that it employs beyond-EdS kernels (called fk-kernels [69, 68])
to properly account for the effects of the free-streaming scale introduced by
massive neutrinos rather than making the $\omega_{\nu}\ll\omega_{\rm cdm}$
approximation made in the other codes. The first line on the right hand side
of eq. (3.4) gives the perturbative, non-linear, redshift-space power spectrum
as for the other models. The second and third lines are the counterterms and
stochastic terms, respectively. We employ the biasing scheme first introduced
in [57, 70], and generalized to cosmologies with additional scales (such as
the neutrino free streaming scale) in [67]. For simplicity, we adopt the
approximate bias treatment outlined in Appendix A.1 in ref. [71].

Folps$\nu$ also accounts for large scale bulk flows via the IR-resummations
technique of refs. [72, 73], and computes the multipoles using the IR-resummed
EFT power spectrum, as specified in equations (2.20) and (2.24) in Ref. [45].

In this work, we keep the total mass of neutrinos fixed at its fiducial value
$M_{\nu}=0.06\,\text{eV}$, for which differences in results between EdS- and
fk-kernels are small. Folps$\nu$ demonstrates its full potential in the
neutrino mass constraints, as explored in ref. [45] using synthetic data
mimicking DESI Year-1 and Year-5 tracers and error bars, revealing a
significant enhancement in the neutrino mass constraints of around 14% when
switching from the usual EdS kernels to the accurate fk-kernels.

Full-Modeling | ShapeFit | Bias | Stoch/Counter
---|---|---|---
|  | Min. F. | Max. F. |
H0 | $f\sigma_{8}$ | $b_{1}$ | $\alpha_{0}$
$\mathcal{U}[50,90]$ | $\mathcal{U}[0,1]$ | $\mathcal{U}[0,10]$ | —
$\omega_{\mathrm{b}}$ | $\alpha_{\parallel}$ | $b_{2}$ | $\alpha_{2}$
$\mathcal{N}[0.02237,0.00037]$ | $\mathcal{U}[0.8,1.4]$ | $\mathcal{U}[-50,50]$ | —
$\omega_{\mathrm{cdm}}$ | $\alpha_{\perp}$ | $b_{s^{2}}$ | $\alpha^{\rm shot}_{0}$
$\mathcal{U}[0.05,0.20]$ | $\mathcal{U}[0.8,1.4]$ | $=-\tfrac{4}{7}(b_{1}-1)$ | Uninformative | —
$\log(10^{10}A_{\mathrm{s}})$ | $m$ | $b_{\rm 3nl}$ | $\alpha^{\rm shot}_{2}$
$\mathcal{U}[2.0,4.0]$ | $\mathcal{U}[-3.0,3.0]$ | $=\tfrac{32}{315}(b_{1}-1)$ | Uninformative | —
Table 3: Folps$\nu$ priors in both Full-Modeling and ShapeFit analyses for
minimal (Min. F.) and maximal (Max. F.) freedom settings. We utilize
uninformative priors for the EFT counterterms $\alpha_{0}$ and $\alpha_{2}$,
as well as for the stochastic variables $\alpha^{\rm shot}_{0}$ and
$\alpha^{\rm shot}_{2}$. However, we analytically marginalize over these
parameters.

###  3.5 Minimal and Maximal freedom bias parametrizations

One of the difficulties in comparing the theories above is that they employ
different conventions for the bias parameters that relate fluctuations in the
matter density to galaxy overdensities. While the translation is known in
principle, in practice it is not easy to find direct mappings from the biasing
scheme of one model to that of all the others as one code can group terms in
ways that cannot be reproduced with simple transformations of bias parameters
in the other codes. This also makes it more difficult to be fully consistent
in the priors applied to bias parameters in each model. We can mitigate this
problem, however, by comparing our models within two opposing
parameterizations, which we call minimal and maximal freedom. For the maximal
freedom case, all four bias parameters of each model are allowed to vary
freely, with very wide uninformative priors. While this does not mean that all
models perform identically, it ensures that any differences are kept small.

We define the minimal freedom, or “coevolution” case by assuming that at
initial times the bias is purely given by the linear ($b_{1}$) and second-
order bias ($b_{2}$), and that non-local bias ($b_{s}$) as well as third-order
bias ($b_{3}$) terms emerge at later times only through non-linear evolution
due to gravity. Since the Lagrangian picture defines the biases at initial
coordinates $\bm{q}$, we assume specifically that the Lagrangian biases
$b_{s}^{\rm L}$ and $b_{3}^{\rm L}$ are fixed to zero, while $b_{1}^{\rm L}$
and $b_{2}^{\rm L}$ are allowed to vary. In the Eulerian picture, the bias
parameters are defined at the observed positions and time and thus depend on
the combination of initial (Lagrangian) values and the evolution of the
density fields. Therefore, when the Lagrangian $b_{s}^{\rm L}$ and $b_{3}^{\rm
L}$ biases are zero, the corresponding Eulerian biases are directly
proportional to $b_{1}^{\rm L}=b_{1}^{\rm E}-1$. While we can define a
“minimal freedom” parameterization within each model that involves only two
bias parameters, these are not the exact same parameters in each model but
rather linear combinations of each other. Nonetheless, the minimal freedom or
“coevolution” parameter choice allows us to reduce the number of terms
responsible for the different behavior between models.

Between the velocileptors LPT and EPT models, a linear mapping exists at one-
loop order to relate the two bias schemes [74]. These bias relations imply
that coevolution in the velocileptors EPT model involves setting

$b_{2}^{\rm E}=b_{2}^{\rm L}+\frac{8}{21}b_{1}^{\rm L}\quad,\quad b_{s}^{\rm
E}=-\frac{2}{7}(b_{1}^{\rm E}-1)\quad,\quad b_{3}^{\rm E}=b_{1}^{\rm E}-1.$
(3.15)

Meanwhile, in Folps$\nu$ one fixes the tidal $b_{s^{2}}$ and 3rd order $b_{\rm
3nl}$ biases to coevolution via [75, 76, 77]

$b_{s^{2}}=-\frac{4}{7}(b^{\rm E}_{1}-1)\quad\text{and}\quad b_{3\rm
nl}=\frac{32}{315}(b^{\rm E}_{1}-1).$ (3.16)

The connection with the PyBird parameters is slightly more complex, and is
most easily accomplished using the “monkey bias” formalism of [78]. Using
$b_{3}^{\rm L}=b_{s}^{\rm L}=0$ we have

$\displaystyle\tilde{b}_{1}$ $\displaystyle=b_{1}^{\rm v}+1$
$\displaystyle\tilde{b}_{2}$
$\displaystyle=\frac{7}{2}\left(\frac{2}{7}+b_{s}^{\rm L}\right)=1$
$\displaystyle\tilde{b}_{3}$
$\displaystyle=\frac{7(42-145b_{1}^{v}-21b_{3}^{v}+630b_{s}^{\rm
L})}{441}=\frac{294-1015(b_{1}-1)}{441}$ $\displaystyle\tilde{b}_{4}$
$\displaystyle=-\frac{7}{5}(b_{1}-1)-\frac{7}{10}b_{2}^{\rm v}$ (3.17)

After the conversion, the only two free parameters are $b_{1}$ and $b_{2}^{\rm
v}$. To simplify the equations further, instead of putting priors on
$b_{2}^{\rm v}$ and then finding $b_{4}$ we put a flat prior on $b_{4}$
directly. The range of this flat prior is determined by the prior on $b_{1}$
and $b_{2}^{\rm v}$ in velocileptors.

###  3.6 Overview of key model differences

While the models have been derived by different groups, starting from
different assumptions and proceeding in different ways, they arise from a
systematic procedure for capturing the effects of dynamics, redshift-space
distortions and biasing. The end result should thus be consistent predictions
for the redshift-space power spectrum of biased tracers once we account for
different conventions. Any remaining differences should arise only at 2-loop
order or be due to the addition of higher-order counterterms666Different
parameterizations can also correspond to different priors, but we try to
minimize this issure through the minimal and maximal freedom choices. With
this in mind, in this section, we try to summarize the key differences between
the models.

Perhaps the easiest differences to see are that the LPT flavor of
velocileptors has its counterterms of the form $k^{2}P_{\rm Zel}$777In Ref.
[44] a reparameterization of the counterterms is used that changes changes
slightly the definition of $\alpha_{n}$ parameters and additionally uses
$k^{2}P_{\rm lin}$. This gives identical results as it doesn’t change the
theory but slightly modifies how priors are chosen on $\alpha_{n}$. This is
not important here because we choose infinite priors on counterterm parameters
for all models, while the other models use $k^{2}P_{\rm lin}$. These differ at
high-$k$ by terms that are formally 2-loop order. PyBird includes an extra,
isotropic stochastic term of the form $k^{2}/k_{\rm M}^{2}$ that the other
models do not include.

The resummation of long-wavelength modes is handled quite differently in the
different codes. In velocileptors the long-wavelength components of the
displacements are exponentiated while the short-wavelength components are
treated perturbatively. The split is governed by a parameter $k_{\rm IR}$. For
reasonable volumes, the choice of $k_{\rm IR}$ has little effect, however, if
errors corresponding to the (physically unrealizable)
$200\,h^{3}\mathrm{Gpc}^{-3}$ volume are used we see differences. This is
expected, as the implied errors are small enough that 2-loop effects cannot be
ignored in any of the models. A similar procedure is done for PyBird. By
contrast, the EPT variant of velocileptors and the Folps$\nu$ model split the
linear theory spectrum into a “wiggle” and “no-wiggle” piece, and then damp
only the “wiggle” component. The results for the final sum are quite
insensitive to the precise decomposition. The dependence on the precise
wiggle/no-wiggle decomposition methods (e.g the methods described in Refs.
[79, 80, 59]) has been tested in Ref.[46], while tests on the choice of IR
cutoff scale are presented in Refs. [56, 46].

##  4 Fitting methods

There are two primary methods for fitting power spectra that we consider here.
The first, “ShapeFit” (SF; [81]), extends the standard template fitting used
in many previous galaxy clustering analyses. For the standard template fitting
it is assumed that the shape of the power spectrum is well-determined by early
Universe measurements (e.g. CMB anisotropies) and late-time effects do not
alter the shape of this spectrum. A fiducial cosmology, and thus a fiducial
linear power spectrum, $P_{\rm lin}$, is picked and held fixed in the fits.
The parameters being varied, corresponding to changes in “late time cosmology”
are the amplitude $f\sigma_{8}$ and the distance scalings,

$\displaystyle\alpha_{\parallel}=\frac{H^{\rm ref}(z)}{H(z)}\left(\frac{r_{\rm
d}^{\rm ref}}{r_{\rm d}}\right),\quad\quad\alpha_{\perp}=\frac{D_{\rm
A}(z)}{D^{\rm ref}_{\rm A}(z)}\left(\frac{r_{\rm d}^{\rm ref}}{r_{\rm
d}}\right),$ (4.1)

where $H(z)$, $D_{\rm A}(z)$, and $r_{\rm d}$ are the comoving Hubble
parameter, angular diameter distance, and sound horizon scale at drag epoch.
These distance scalings are driven by both the Alcock-Paczinsky effect
(AP;[82]), which is the anisotropic distortion when the true cosmology differs
from the fiducial in the conversion of angles and redshift to physical
distances, and the changes in $r_{\rm d}$ with varying cosmology. Varying
these parameters effectively changes the power spectrum without the need to
recompute $P_{\rm lin}$ in its entirety. One disadvantage of the template fit
is that one sacrifices information contained in the shape of $P_{\rm lin}$,
and so obtains sub-optimal constraints on parameters affecting its shape when
not combining with e.g. CMB measurements. The ShapeFit method aims to maintain
some of the simplicity of the standard template fit, while also capturing some
of the information from the shape of $P_{\rm lin}$ via two additional
compressed parameters $m$ and $n$, defined through

$P^{\prime}_{\rm lin}(\bm{k})=P_{\rm lin}(\bm{k})\
\exp\left\\{\frac{m}{a}\tanh\left[a\ln\left(\frac{k}{k_{\rm
p}}\right)\right]+n\ln\left(\frac{k}{k_{\rm p}}\right)\right\\}.$ (4.2)

The scale-dependent $m$ parameter above modulates the shape of the power
spectrum and aims to mimic the effect that different $\omega_{m}$ and
$\omega_{b}$ values have on the shape of $P_{\rm lin}(\bm{k})$. We use for the
amplitude $a=0.6$ and pivot scale $k_{\rm p}=0.03\ \,h{\rm Mpc}^{-1}$, as
suggested in Ref. [81]. These constants are chosen in order for changes in $m$
to best replicate how $\omega_{m}$ and $\omega_{b}$ influence $P_{\rm
lin}(\bm{k})$. The scale-independent $n$ parameter changes the slope in a way
that exactly reproduces the effect of different $n_{\rm s}$ when computing
$P_{\rm lin}(\bm{k})$, i.e. $n=n_{\rm s}-n_{\rm s}^{\rm ref}$. The inclusion
of the shape parameters therefore makes ShapeFit more constraining than the
standard template fit, which is particularly evident when only LSS data are
analysed, not in combination with CMB data or priors. It is one of the primary
methods that will be used in analyzing power spectra/correlation function data
in DESI, especially when studying the impact of systematic effects. All models
in this paper implement the ShapeFit method by sampling in the parameters
$(f\sigma_{8},\alpha_{\parallel},\alpha_{\perp},m)$888In this paper we keep
$n$ fixed to its fiducial value. This parameter is directly related to the
spectral tilt $n_{\rm s}$. In full-shape analyses, $n_{\rm s}$ is a weakly
constrained parameter and historically has been kept fixed to the e.g. CMB-
measured value. While the increased constraining power of DESI and beyond
allows us to vary $n_{\rm s}$ going forward, an informative prior is still
needed, and for DESI its width is chosen such that constraints on other
parameters are unaffected[32]. Since this paper focuses on the comparison
between different effective field theories, we do not find the inclusion of
such a prior-dominated parameter to provide any additional insights. However,
the effect of varying it is explored in companion DESI papers[44, 45, 39] and
the relevance of projection effects when varying $n$ (for SF) and $n_{\rm s}$
(for Full-Modeling) will be discussed in Ref. [32]. with the same $P_{\rm
lin}(\bm{k})$ in order to obtain the observable power spectrum multipole
predictions. In velocileptors a Taylor series emulator is used in order to
improve the evaluation of theory multipoles, while Folps$\nu$ and PyBird use
as default settings an approximation for the loop integrals [81]:

$\displaystyle\int d^{3}\bm{q}P^{\prime}_{\rm
lin}(q)F_{n}(...)\approx\left(\frac{P^{\prime}_{\rm lin}(k)}{P_{\rm
lin}(k)}\right)\int d^{3}\bm{q}P_{\rm lin}(q)F_{n}(...),$ (4.3)

with $F_{n}$ kernels defined in the beginning of §3. Under this approximation
the loop integrals only need to be computed once and then are simply rescaled
as $m$ is varied. This allows for fast evaluation of theory multipoles in lieu
of an emulator. The robustness of the emulator for velocileptors is shown in
Appendix E in Ref. [44] and the stability of the above approximation used for
PyBird and Folps$\nu$ is tested in Appendix B of Ref. [45]. Since both the
emulator and the approximation show neglibible differences in constraints on
Abacus mocks when compared to direct computations, we do not expect these
implementation choices to contribute to any disagreements between EFT models.

The second method used in our comparison is based on a more direct approach
and involves choosing a cosmological model, e.g. $\Lambda$CDM, and directly
varying the underlying parameters of this model, predicting the data and
comparing the prediction to the measurement. This direct-fitting approach,
hereafter referred to as Full-Modeling (FM), captures information from both
the early and late universe as the parameters of the model, ($H_{0}$,
$\omega_{\mathrm{b}}$, $\omega_{\mathrm{cdm}}$ and
$\log(10^{10}A_{\mathrm{s}})$ in our examples) control the shape of $P_{\rm
lin}$ as well as the late-time dynamics/geometry. While this method does
involve generating a new linear power spectrum at every step of a Markov Chain
Monte Carlo (MCMC), the use of emulators can speed up the computation enough
that differences in convergence time between FM and compressed parameters are
insignificant. The Full-modeling method also takes into account the AP effect
through the scaling parameters $q_{\parallel,\perp}$ are related to
$\alpha_{\parallel,\perp}$ but absent the factors of $r_{\rm d}^{\rm
ref}/r_{\rm d}$ because the changes in BAO scale are implicit when varying
cosmological parameters. The purely geometrical $q_{\parallel,\perp}$
parameters are not sampled in this modeling method but are derived and applied
to the power spectrum predictions as the cosmological parameters are varied.
The actual implementation of the AP distortion is consistent in all models and
involves re-scaling the power spectrum to observed coordinates using
$q_{\parallel,\perp}$ (for FM) or $\alpha_{\parallel,\perp}$(for FM) via:

$\displaystyle P^{\rm obs}_{s}(\bm{k}_{\rm
obs})=q_{\perp}^{-2}q_{\parallel}^{-1}P_{s}(\bm{k})\quad,\quad k^{\rm
obs}_{\parallel,\perp}=q_{\parallel,\perp}\ k_{\parallel,\perp}.\quad$ (4.4)

Finally, for all models and modeling methods, we compute the linear theory
power spectrum, $P_{\rm lin}$ from CLASS [83] or CAMB [84] before constructing
the 1-loop EFT power spectrum multipoles.

##  5 Results

###  5.1 Fixed-cosmology fits

Figure 1: Abacus LRG, ELG, and QSO mock data along with each Fourier space
model’s prediction with $\Lambda$CDM parameters fixed to truth but allowing
the nuisance parameters to vary to best fit the data. The error bars on the
data reflect the rescaled covariance with 200 ($h^{-1}$Gpc)3 volume, while the
shaded region shows the covariance corresponding to a single cubic box with 8
($h^{-1}$Gpc)3 volume, which is more similar to the constraining power of a
physical survey.

(a) Full-Modeling

(b) ShapeFit

Figure 2: Comparison between the three Fourier space models using the Full-
Modeling method (left) and ShapeFit method (right), fitting the mean of the
Abacus LRG ($z=0.8$) simulations. In both fits we show the constraints in the
“maximal” freedom parametrization where the four bias parameters are allowed
to vary independently in each model. The results shown use the full covariance
with the 1/25 rescaling from the single-box volume. In both cases we use scale
cuts of $0.02\leq k\leq 0.18\ \,h{\rm Mpc}^{-1}$ for all models.

We begin by showing in Fig. 1 the LRG ($z=0.8$), ELG ($z=1.1$), and QSO
($z=1.4$) simulated data that we are fitting with our models. In the left
panels we show the monopole and quadrupole data for each tracer type along
with theoretical models generated with each pipeline with $\Lambda$CDM
parameters fixed to truth and nuisance parameters varied to best match the
data up to $k_{\rm max}=0.25h^{-1}$Mpc. We allow all four bias parameters to
vary, and use the covariance rescaled to the 200 ($h^{-1}$Gpc)3 volume of the
25 mock realizations. The error bars of the data reflect this covariance, but
we also show shaded regions corresponding to the error bars of the data from a
single cubic box volume of 8 ($h^{-1}$Gpc)3 that more closely resembles the
constraining power of a typical survey. In the right-hand panels of Fig. 1 we
show the difference between the data and the models at each $k$-bin. It is
instructive to see how the nonlinear terms in the models behave when the
linear power spectrum and cosmology dependence are fixed. We observe similar
trends from each of the models, most noticeably in the LRG quadrupole, where
all three of them show the same bump in $\Delta P_{2}^{(\mathrm{data-model})}$
at $k\sim 0.1h^{-1}$Mpc. Additionally, all three models seem to diverge from
the data the same way near $k\sim 0.25h^{-1}$Mpc, which sets the range of
validity of these models. For both of the LRG multipoles as well as the
monopole ELG curves, we see that the velocileptors and PyBird predictions
overlap pretty closely, whereas the Folps$\nu$ model is slightly more offset
from the data for some $k$ ranges.

###  5.2 Tests on Mock data

We show in Fig. 2 a comparison between each model’s posteriors fitting to the
LRG mocks with free parameters and priors described in Tables 1-3, with the
“Maximal freedom” choice of bias parameters. While all models give constraints
within around 1$\sigma$ of the true values, there are noticeable disagreements
between the models for both the ShapeFit and Full-modeling methods. These
shifts between models can be driven by differences in the handling of counter
or stochastic terms, bias parametrization, IR-resummation schemes, or even
numerical effects in the code that may become important in extremely
constraining data sets. Indeed, we note that the covariance used here reflects
a survey volume of 200 ($h^{-1}$Gpc)3, which far exceeds that of any
physically achievable survey on our past lightcone to these redshifts. At such
a volume, the data has a statistical error on the order of a third of a
percent at each point in the $k$ data vector, a level of precision at which we
could well be dominated by systematic uncertainties in the N-body simulations.
On the other hand, our goal in performing tests of our models on the Abacus
simulations is to detect systematic uncertainties in our models that will be
relevant in the DESI-Y5 data; and given the size of the Y5 footprint this is
not achievable without a very large simulation volume. We therefore present
the “full volume” results with the understanding that some hard-to-quantify
level of discrepancy between the models and true values is expected, while
focusing on ensuring that our models agree at the level of precision
attainable at volumes relevant to a physical survey. For this reason, we use
Fig. 2 simply to demonstrate the level of agreement we currently have at the
full volume, but for the remainder of this paper will restrict our attention
to fits using the covariance of a single box (V = 8 ($h^{-1}$Gpc)3), which is
more comparable to the volumes spanned by a realistic survey. However, the
data vectors we fit to remain the ones computed from the average of 25 abacus
boxes in order to reduce the noise.

Figure 3: Comparison between Folps$\nu$, velocileptors (LPT and EPT), and
PyBird full-modeling (top row) and ShapeFit (bottom row) constraints on the
mean of the mock LRG data, respectively. In the left panels we show the
“minimal” freedom parametrization, in which Eulerian $b_{s2}$ and $b_{3\rm
nl}$ parameters are fixed to coevolution and the corresponding Lagrangian
parameters fixed to zero. In the right panels we employ the “maximal” freedom
parameter choice for which all bias parameters vary independently with uniform
priors. These fits used the single-box covariance volume without any
rescaling. We use a scale cut of $0.02\leq k\leq 0.18\ \,h{\rm Mpc}^{-1}$ for
all fits shown.

As discussed in § 3.5, the bias terms and how they are defined can differ
between theories, making it difficult to convert between biasing schemes and
by extension use consistent priors. Choices made by the user about which
parameters to vary and which to fix also affect the volume of the parameter
space and can narrow/widen constraints purely from a degrees-of-freedom
standpoint. This can greatly complicate the comparisons as the baseline
settings preferred for each model as described in Refs. [44, 45, 46] are quite
different: the Folps$\nu$ model is often used with a coevolution
parametrization in which the tidal and third order parameters, b${}_{s^{2}}$
and b3nl, are determined by $b_{1}$, so only two bias parameters are varying
freely. Meanwhile, the velocileptors results often have Lagrangian b3 fixed
with the remaining three free. For PyBird it is common to fix $b_{4}$ to
anticorrelate with $b_{2}$. We thus compare the constraints between Fourier
space models while taking more care to standardize the amount of freedom
afforded to the bias terms. We show in Fig. 3 comparisons in the Full-Modeling
and ShapeFit constraints on LRG data, respectively, with both “maximal” (Max.
F) and “minimal” (Min. F) freedom settings, which are defined in § 3.5. We see
that in both cases there is very good agreement between the two velocileptors
models, Folps$\nu$, and PyBird in the ShapeFit and Full-Modeling methods. We
find that the models generally agree better in the minimal freedom case, and
this is a more well-defined parametrization that can be applied consistently
to each theory. In the maximal freedom case, we observe slight differences in
the widths of the $\log A_{\rm s}$ posteriors in Full-modelling. In ShapeFit
the PyBird model has a wider posterior in the $m$ parameter, and explores more
negative m values than the other theories.

We next show the results for the ELG and QSO tracers for Full-modeling and
ShapeFit methods in Figs. 4 and 5, respectively. For minimal freedom, we
observe consistent constraints on the QSO data between pipelines for both
modeling methods. However, when fitting the ELG data, the PyBird model
constrains the power spectrum shape slightly differently from the other
models. We observed this through a small shift in $\Omega_{\rm m}$ and $m$
constraints for Full-modeling and ShapeFit respectively. In the maximal
freedom case, we see better consistency between theories in Full-Modeling than
in ShapeFit. Similarly to the LRG tracer, we find that the PyBird model has
more freedom in exploring larger negative values of the $m$ parameter. These
extreme values for $m$ correspond to nonphysical $\Lambda$CDM models and are
therefore disfavored in the Full-modeling fit. We find that in PyBird these
large negative values of $m$ are accompanied by large negative $b_{3}$ (around
$-100$), resulting in canceling effects to the power spectrum such that a
moderately high likelihood is still obtained. In order to prevent this effect
we introduce a Gaussian prior on $b_{3}$ centered on zero with a width of 10,
and show the result in the left hand plot of Fig. 6. We find that with a more
informative prior on $b_{3}$, the tail of the $m$ posterior distribution goes
away and the PyBird model agrees more closely with the others. We still find a
slightly wider posterior in the $m$ parameter for velocileptors LPT compared
to the other models. This can also be attributed to the amount of freedom
allowed in the bias parameters, in this case both Lagrangian $b_{s}$ and
$b_{3}$. If we include Gaussian priors $\mathcal{N}[0,5]$ on both $b_{s}^{\rm
L}$ and $b_{3}^{\rm L}$, the constraints on ShapeFit parameters tighten enough
that velocileptors LPT matches the constraining power of the other models, as
shown in the right-hand side of Fig. 6. One can make the case that this is no
longer a “maximal” freedom scenario as we have added several informative
priors in order to get the best agreement between all models. We note however
that models with bias parameters in the tens or hundreds are not theoretically
self-consistent, and we would not be able to trust our results or theory if
the posteriors peaked at such values for a given data set. In our case, we do
not observe any preference (i.e. shifts in the posterior peaks) towards
extreme values when allowing more freedom in the bias parameters, but these
regions are still allowed by the likelihood. We are therefore justified in
cutting out those regions via Gaussian priors. If such priors turn out to be
informative, then this should be noted, i.e. the data alone are not able to
fully constrain the model without additional information. Our priors should be
as broad as possible, and consistent with the model and the data. We refer
readers to the appendix of Ref. [44] for a detailed discussion of projection
effects and priors relevant to DESI fullshape modeling, along with additional
references cited therein that encounter similar issues in other areas of
cosmology.

(a) ELG ($z=1.1$)

(b) ELG ($z=1.1$)

(c) QSO ($z=1.4$)

(d) QSO ($z=1.4$)

Figure 4: Comparison between Folps$\nu$ PyBird and velocileptors (LPT and EPT)
full-modeling constraints on the mean of the mock ELG and QSO data,
respectively. In the left column we show the “minimal” freedom parametrization
and the right column shows the “maximal” freedom. These fits use the single-
box covariance volume without any rescaling. We use a scale cut of $0.02\leq
k\leq 0.18\ \,h{\rm Mpc}^{-1}$ for all fits shown.

(a) ELG ($z=1.1$)

(b) ELG ($z=1.1$)

(c) QSO ($z=1.4$)

(d) QSO ($z=1.4$)

Figure 5: Comparison between Folps$\nu$ PyBird and velocileptors (LPT and EPT)
ShapeFit constraints on the mean of the mock ELG and QSO data, respectively.
In the left columns we show the “minimal” freedom parametrization and on the
right we show the “maximal” freedom case. These fits use the single-box
covariance volume without any rescaling. We use a scale cut of $0.02\leq k\leq
0.18\ \,h{\rm Mpc}^{-1}$ for all fits shown.

Figure 6: ShapeFit constraints on the QSO data in the maximal freedom case,
but introducing priors on some of the bias parameters for PyBird and
velocileptors LPT. In the left plot a Gaussian prior $\mathcal{N}[0.,10.]$ is
applied to the $b_{3}$ parameter of the PyBird model only, while the other
models still have uniform priors on all parameters. In the right plot we
additionally introduce Gaussian priors $\mathcal{N}[0.,5.]$ on both Lagrangian
$b_{s}$ and $b_{3}$ parameters in the velocileptors LPT model. We use a scale
cut of $0.02\leq k\leq 0.18\ \,h{\rm Mpc}^{-1}$ for all fits shown.

###  5.3 Tests on noiseless theoretical model

We next turn to fits of data that are generated from our theoretical models.
This “data” is shown in Fig. 1 and is generated by fixing the $\Lambda$CDM
parameters to their true Abacus values and then varying the nuisance
parameters to best fit the mock data up to $k_{\rm max}=0.25h^{-1}$Mpc. These
data vectors are noiseless, which allows for cleaner comparisons between the
theories without being obscured by cosmic variance or systematics of the
simulations. While the previous section compared the performance of the
different models on simulations, i.e. their ability to constrain parameters to
their truth, the purpose of this section is to quantify the agreement of
theories. To that end, we cycle through each of the theoretical data sets and
perform fits using the two pipelines that were not used to produce said data.
We perform these tests as a function of $k_{\rm max}$ in order to observe how
the predictions of the models diverge as higher $k$-modes are included in the
fits. We generally expect the models to agree better at lower $k\lesssim
0.1\,h{\rm Mpc}^{-1}$ (larger scales) as the physics at these scales is well-
described by linear theory, which is the same in all of the models. The models
differ in their handling of higher-order nonlinear terms, which are more
important at higher $k\gtrsim 0.1\,h{\rm Mpc}^{-1}$. For these tests, we use
the single-box covariance that was also used for the Abacus mocks in the
previous section, and we restrict our attention to the minimal freedom
parameterization. These tests have been repeated with maximal freedom but the
results show the same behavior, so we restrict our discussion to the minimal
freedom case for simplicity.

In Fig. 7 we show comparisons of constraints between the LPT and EPT
velocileptors and Folps$\nu$ models while fitting to the theoretical spectra
generated by PyBird to imitate LRG mocks. We present both the Full-Modeling
and ShapeFit results. For both methods, the models agree up to a $k_{\rm
max}\sim 0.24h^{-1}$Mpc (both with each other and with the PyBird data). For
Full-Modeling, see Folps$\nu$ shift upwards by $\gtrsim 1\sigma$ from truth in
$\log A_{\rm s}$ at $k_{\rm max}\sim 0.26h^{-1}$Mpc, while in the ShapeFit
method the velocileptors LPT and Folps$\nu$ models disagree with each other in
$\alpha_{\parallel}$ by $\gtrsim 1\sigma$ only at $k_{\rm max}\sim
0.28h^{-1}$Mpc and above. However, both velocileptors EPT and Folps$\nu$
already deviate from the true PyBird value by $\sim 1\sigma$ at $k_{\rm
max}\sim 0.26h^{-1}$Mpc. We also note that the velocileptors EPT model follows
the trends of Folps$\nu$ in $\alpha_{\parallel,\perp}$ at higher $k_{\rm max}$
rather than agreeing more closely with velocileptors LPT, which does not
deviate from the truth by more than $1\sigma$ for any Full-Modeling or
ShapeFit parameter until $k_{\rm max}\sim 0.3h^{-1}$Mpc.

We next repeat this test with noiseless data generated by the Folps$\nu$ model
in Fig. 8, focusing only on Full-modeling for simplicity. In this case, the
velocileptors LPT constraint on $\log A_{\rm s}$ begins to deviate from the
truth by 1$\sigma$ at $k_{\rm max}\gtrsim 0.26\,h{\rm Mpc}^{-1}$, while the
EPT model remains unbiased for the whole range of $k_{\rm max}$. For the
PyBird model, we observe similar offsets in $\log A_{\rm s}$ in the same
direction as the velocileptors LPT results. Finally, we show in Fig. 9 the
comparison of Full-Modeling constraints from fitting to the data vector
generated by the velocileptors LPT model. For the $k_{\rm max}\leq 0.28\,h{\rm
Mpc}^{-1}$ fits the PyBird model is consistent with the velocileptors LPT data
whereas the Folps$\nu$ model begins to deviate by 1$\sigma$ in $\log A_{\rm
s}$ for $k_{\rm max}\gtrsim 0.26\,h{\rm Mpc}^{-1}$. From Figs. 7-9 we notice a
trend that at lower $k$ all models behave similarly while at higher $k$ the
velocileptors LPT and PyBird models are more consistent with each other,
whereas Folps$\nu$ agrees better with velocileptors EPT. We can also see this
to some extent in the power spectrum predictions shown in the right-hand
panels of Fig. 1, where the velocileptors LPT and PyBird curves match each
other more closely than Folps$\nu$. This may be due to the differences in IR-
resummation schemes between the models. As noted in § 3.6, PyBird and
velocileptors LPT use a similar resummation procedure of exponentiating long-
wavelength displacements while perturbatively expanding the short-wavelength
modes, while velocileptors EPT and Folps$\nu$ both use the wiggle-no wiggle
split procedure. Our results suggest that the similarities between
velocileptors LPT and PyBird (and Folps$\nu$ with velocileptors EPT) at high
$k$ are related to the IR-resummation methods employed by the models (see ref.
[56] for further discussion).

In Appendix A we perform similar tests as those described here for theoretical
data vectors produced from fits to the ELG and QSO Abacus mocks. Our
conclusions mimic those above, and show that our models agree with one another
in the range of scales999The dependence on scale cuts of each model using
Abacus mocks is presented and discussed in Refs. [44, 45, 46, 39] appropriate
for Full-Shape analyses using 1-loop perturbation theory.

Finally, we find the maximum likelihood values from velocileptors LPT and EPT
when fit to the theoretical data sets generated by Folps$\nu$ and PyBird to
$k_{\rm max}=0.18\,\,h{\rm Mpc}^{-1}$. For each FM and SF parameter, the
average shift between best-fit and truth provides a rough estimate for the
theoretical-systematic error of the models that can be compared to the
statistical errors, $\sigma$. For FM we obtain systematic errors of $\lesssim
0.02\sigma$ in $\Omega_{\rm m}$, $H_{0}$ and $\log(10^{10}A_{\rm s})$. For SF
we find systematic errors of $\lesssim 0.05\sigma$ for $f\sigma_{s8}$,
$\alpha_{\parallel}$ and $\alpha_{\perp}$, while in $m$ we get a systematic
error of $0.17\sigma$.

Figure 7: Comparison of 1D constraints of Full-Modeling and ShapeFit fits from
velocileptors (EPT and LPT) and Folps$\nu$ to the theoretical model generated
by PyBird. The theoretical model has $\Lambda$CDM fixed to the true abacus
cosmology and nuisance parameters shifted to best-fit the LRG mock data.
Figure 8: Comparison of 1D constraints of Full-Modeling fits from
velocileptors (EPT and LPT) and PyBird to the theoretical model generated by
Folps$\nu$. The theoretical model has $\Lambda$CDM fixed to the true abacus
cosmology and nuisance parameters shifted to best-fit the LRG mock data.
Figure 9: Comparison of 1D constraints of Full-Modeling from PyBird and
Folps$\nu$ to the theoretical model generated by velocileptors LPT. The
theoretical model has $\Lambda$CDM fixed to the true abacus cosmology and
nuisance parameters shifted to best-fit the LRG mock data.

##  6 Conclusion

Analyses of two-point statistics from previous galaxy redshift surveys (e.g.
BOSS and eBOSS) using effective field theories (EFT) of Large Scale Structure
have often obtained constraints on cosmological parameters, such as
$\sigma_{8}$, that are in mild tension with one another. These differences can
usually be attributed to the settings of pipelines used and modeling choices
(see e.g. [85, 65, 37, 38]) but could have given the erroneous impression that
different effective field theories are not well understood and inconsistent.
In anticipation of the Y1 data release of the Dark Energy Spectroscopic
Instrument, this paper aims to show that various EFT models/pipelines used for
fitting galaxy power spectra in redshift space are actually consistent at the
level of precision of a physical survey– as long as one handles with care the
freedom afforded to bias/counter/stochastic terms in the models and is
consistent with choice of scale cuts. We focus our attention on three such
models: velocileptors, Folps$\nu$, and PyBird and show that, despite relying
on different theoretical frameworks (Eulerian vs Lagrangian perturbation
theories, EFT corrections, IR resummation schemes), the predictions from these
models are in close agreement with one another up to scales of $k_{\rm
max}\simeq 0.25\,h{\rm Mpc}^{-1}$ and at the level of precision of 8
($h^{-1}$Gpc)3 survey volumes. Compared to the full DESI footprint the
simulation volume used in this comparison is larger than those of the BGS
($0.1<z<0.4$) and two of the LRG ($0.4<z<0.6$,$0.6<z<0.8$) redshift bins,
whose volumes are rougbly 2,3,and 5 ($h^{-1}$Gpc)3 respectively. The third
DESI LRG bin ($0.8<z<1.1$), both ELG bins ($0.8<z<1.1$, $1.1<z<1.6$), and the
QSO ($0.8<z<2.1$) are all larger with volumes of 10, 20, and 60
($h^{-1}$Gpc)3, and the total comoving volume up to $z=2.1$ is around 70
($h^{-1}$Gpc)3[86].

We compare our models within two bias parametrization choices, “minimal” and
“maximal” freedom, where the first assumes tidal and third-order biases to be
fixed to coevolution and the latter allows all four bias parameters to vary
independently with wide, uninformative priors. We show that when fitting
Abacus simulations provided in 8 ($h^{-1}$Gpc)3 cubic boxes with three types
of tracers (LRG, $z=0.8$; ELG, $z=1.1$; QSO, $z=1.4$), the cosmological
constraints are consistent between models in both minimal and maximal freedom
cases. We perform this comparison using both parameter compression
(“ShapeFit”) and direct fitting (“Full-Modeling”) methods. Despite finding
close agreement between the models, we note a degeneracy between the third-
order bias and the ShapeFit parameter, m, in the PyBird model that results in
a tail in the posterior distribution of $m$ for the fits to the quasar sample.
By including a Gaussian prior on the $b_{3}$ parameter, we can minimize this
effect without biasing the cosmological constraints. Our conclusions on model
consistency are in agreement with those of previous work [37, 38]. In
particular, Ref. [37] performed a comparison between PyBird and CLASS-PT using
BOSS data and found that the primary effect driving the differences between
the two models is the different bias parameterizations, and that controlling
these differences results in a significant improvement in model consistency.
In this paper we come to a similar conclusion, but given the more substantial
theoretical differences between theories considered here, do not have a direct
mapping of bias parameters between all models available to us. However, the
minimal and maximal freedom choices were sufficient in controlling the effect
of having different parameterizations between models. The effect of applying
Gaussian vs flat priors on $b_{3}$ in PyBird is not directly explored as the
comparison used Guassian priors on higher order biases by default. In Ref.
[38] comparisons between CLASS-ONELOOP and CLASS-PT were only made at the
level of power spectrum predictions and not constraints on a data set.
However, their comparison of $P_{\ell}(k)$ predictions show similar levels of
agreement as the model predictions shown in Fig. 1 in this paper.101010To be
precise, the two models compared in Ref. [38] employed the exact same bias
expansions and, by setting counterterms and stochastic terms to zero,
$P_{\ell}(k)$ predictions were compared under the exact same parameter values.
In our case the model predictions used the same $\Lambda$CDM parameters but
all other parameters values were independently chosen by fitting to data. So
our theoretical spectra differ slightly more than in their tests, but
differences are $\mathcal{O}(1\%)$ for $k\leq 0.25\ \,h{\rm Mpc}^{-1}$ in both
cases.

In addition to fitting cubic mock samples, we generate noiseless data using
best-fitting predictions from each model. For each data vector generated by
one model, we run fits using the other models for a variety of $k_{\rm max}$
values. We show that significant deviations of model predictions only occur
when raising the scale cuts to $k_{\rm max}\sim 0.26\,h{\rm Mpc}^{-1}$ or
higher. Since the physics at such small scales (high $k$) is already known to
be sensitive to two-loop effects that are not included in our models, we
conclude that the stronger restriction on $k_{\rm max}$ comes from the
limitation of one-loop perturbation theory and not from disagreements of the
theories themselves. For the range of scales, i.e. $0.02<k\lesssim 0.2\,h{\rm
Mpc}^{-1}$, that will be the fiducial setting for the DESI Y1 analysis, the
EFT models all give consistent cosmological constraints.

While the settings (priors, bias parametrization, etc.) chosen in this paper
allow for better comparison between models, they are not necessarily the
optimal settings with which one may use the pipelines in future analyses. Our
aim in this work is simply to demonstrate that these theories agree when
consistent assumptions are made and pave the way towards the study of modeling
systematics for Full-Shape analysis with DESI. A similar comparison of these
models along with EFT-GSM is also being performed in configuration space[39].
We have thus far focused our analysis on $\Lambda$CDM models only with $n_{\rm
s}$ fixed. Additional tests have been performed that open up the parameter
space beyond $\Lambda$CDM and study/mitigate the parameter projection effects
that arise in more exotic models. These will be presented in future
publications([44, 45, 46, 39]).

##  7 Data availability

Chains from the plots in this paper are available on Zenodo as part of DESI’s
Data Management Plan (DOI: https://doi.org/10.5281/zenodo.10823205). The data
used in this analysis will be made public along the Data Release 1 (details in
https://data.desi.lbl.gov/doc/releases/)

## Acknowledgements

We thank Arnaud de Mattia for implementing our findings into the DESI pipeline
and also thank the members of the Galaxy and Quasar Clustering working group
within DESI for helpful discussions pertaining to this work.

MM is supported by the U.S. Department of Energy (DOE). YL is supported by the
Australian government through the Research Training Program scholarship,
Australian Research Council’s Laureate Fellowship (project FL180100168) and
Discovery Project (project DP20220101395) funding schemes. HN and SR are
supported by Investigacion in Ciencia Basica CONAHCYT grant No. A1-S-13051 and
PAPIIT IN108321 and IN116024, and Proyecto PIFF. HN is supported by Ciencia de
Frontera grant No. 319359 and PAPIIT IG102123. 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.

##  Appendix A Tests on noiseless data for ELG and QSO redshift bins

We repeat the above tests with noiseless data mimicking the ELG and QSO mock
samples in Figs. 10-12, this time with fewer $k_{\rm max}$ points just to see
if any new behavior shows up at higher redshifts. We observe that all models
are more consistent with each other up through $k_{\rm max}=0.26\,h{\rm
Mpc}^{-1}$ than for the LRG case, and that the maximum $k_{\rm max}$ for which
our models agree is not as restricted for ELG and QSO tracers as the LRGs.
That being said, even for the LRG tests the models behave consistently for
$k_{\rm max}\lesssim 0.24\,h{\rm Mpc}^{-1}$, which is a looser bound on
$k_{\rm max}$ than what we observe when fitting each model to the actual
abacus mock data sets (see individual papers [44, 45, 46, 39]). We therefore
conclude from this section that the EFT models discussed in this paper are
consistent for the range of scales that will be used in the analysis of real
DESI data, and that any significant deviations between models only occur at
scales where real data is already known to not be well-described by one-loop
theories.

Figure 10: Comparison of 1D constraints of Full-Modeling fits from
velocileptors (EPT and LPT) and PyBird to the theoretical model generated by
Folps$\nu$. The theoretical model has $\Lambda$CDM fixed to the true abacus
cosmology and nuisance parameters shifted to best-fit the ELG (top) and QSO
(bottom) mock data.

Figure 11: Comparison of 1D constraints of Full-Modeling fits from
velocileptors (EPT and LPT) and Folps$\nu$ to the theoretical model generated
by PyBird. The theoretical model has $\Lambda$CDM fixed to the true abacus
cosmology and nuisance parameters shifted to best-fit the ELG (top) and QSO
(bottom) mock data.

Figure 12: Comparison of 1D constraints of Full-Modeling fits from PyBird and
Folps$\nu$ to the theoretical model generated by velocileptors LPT. The
theoretical model has $\Lambda$CDM fixed to the true abacus cosmology and
nuisance parameters shifted to best-fit the ELG (top) and QSO (bottom) mock
data.

##  Appendix B Author Affiliations

.5cm1

1Department of Physics, University of California, Berkeley, CA 94720, USA

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

3School of Mathematics and Physics, University of Queensland, 4072, Australia

4Instituto de Ciencias Físicas, Universidad Autónoma de México, Cuernavaca,
Morelos, 62210, (México)

5Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510, México

6Instituto Avanzado de Cosmología A. C., San Marcos 11 - Atenas 202. Magdalena
Contreras, 10720. Ciudad de México, México

7Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

8Departament de Física Quàntica i Astrofísica, Universitat de Barcelona, Martí
i Franquès 1, E08028 Barcelona, Spain

9Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

10Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), c.
Martí i Franquès, 1, 08028 Barcelona, Spain.

11Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), FR-75005 Paris, France

12

13Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215,
USA

14University of Michigan, Ann Arbor, MI 48109, USA

15Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK

16Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK

17IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

18Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK

19Department of Physics and Astronomy, The University of Utah, 115 South 1400
East, Salt Lake City, UT 84112, USA

20Korea Astronomy and Space Science Institute, 776, Daedeokdae-ro, Yuseong-gu,
Daejeon 34055, Republic of Korea

21University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA
94720, USA

22Institute of Cosmology and Gravitation, University of Portsmouth, Dennis
Sciama Building, Portsmouth, PO1 3FX, UK

23Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio Ip, CP 111711, Bogotá, Colombia

24Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio H, CP 111711 Bogotá, Colombia

25Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans
s/n, 08913 Bellaterra, Barcelona, Spain

26Department of Astrophysical Sciences, Princeton University, Princeton NJ
08544, USA

27Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, USA

28Department of Physics, The Ohio State University, 191 West Woodruff Avenue,
Columbus, OH 43210, USA

29The Ohio State University, Columbus, 43210 OH, USA

30Department of Physics, The University of Texas at Dallas, Richardson, TX
75080, USA

31Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain

32Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain

33Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís
Companys, 23, 08010 Barcelona, Spain

34Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
U.K

35Departamento de Física, Universidad de Guanajuato - DCI, C.P. 37150, Leon,
Guanajuato, México

36Department of Physics and Astronomy, University of Waterloo, 200 University
Ave W, Waterloo, ON N2L 3G1, Canada

37Perimeter Institute for Theoretical Physics, 31 Caroline St. North,
Waterloo, ON N2L 2Y5, Canada

38Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave
W, Waterloo, ON N2L 3G1, Canada

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

40Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía,
s/n, E-18008 Granada, Spain

41Department of Physics, Kansas State University, 116 Cardwell Hall,
Manhattan, KS 66506, USA

42Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

43Department of Physics and Astronomy, Sejong University, Seoul, 143-747,
Korea

44CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain

45Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

46NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA

47SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA

48National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Rd., Chaoyang District, Beijing, 100012, P.R. China

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