# Cosmological constraints on early dark energy from the full shape analysis
of eBOSS DR16

Rafaela Gsponer,1 Ruiyang Zhao,2,3,1 Jamie Donald-McCann,1 David Bacon, 1
Kazuya Koyama,1 Robert Crittenden,1 Théo Simon,4 Eva-Maria Mueller5
1Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama
Building, Portsmouth, PO1 3FX, UK
2 National Astronomy Observatories, Chinese Academy of Sciences, Beijing,
100101, P.R. China
3University of Chinese Academy of Sciences, Beijing 100049, P.R.China
4 Laboratoire Univers & Particules de Montpellier (LUPM), CNRS & Université de
Montpellier (UMR-5299), F-34095 Montpellier Cedex 05, France
5 Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We evaluate the effectiveness of Early Dark Energy (EDE) in addressing the
Hubble tension using data from the completed eBOSS survey, focusing on
luminous red galaxies (LRGs), quasars (QSOs), and emission line galaxies
(ELGs). We perform cosmological parameter measurements based on full shape
analysis of the power spectrum of all three tracers. We conduct this full
shape analysis with the effective field theory of large-scale structure
(EFTofLSS). EDE is known to strongly suffer from volume projection effects,
which makes the interpretation of cosmological constraints challenging. To
quantify the volume projection effects within an EDE full shape analysis, we
explore the impact of different prior choices on the nuisance parameters of
EFTofLSS through an extensive mock study. We compare classical Gaussian priors
to the non-informative Jeffreys prior, known to mitigate volume projection
effects in $\Lambda$CDM. Our full shape analysis combines eBOSS and BOSS data
with Planck, external Baryon Acoustic Oscillation (BAO), PantheonPlus, and
SH0ES supernova data. EDE demonstrates to reduce the tension from $5.2\sigma$
to $3\sigma$ compared to $\Lambda$CDM. The derived values at a 68% credible
interval with Gaussian and Jeffreys priors are $H_{0}=71.73_{-0.86}^{+0.82}$
km/s/Mpc with $f_{\mathrm{EDE}}=0.1179_{-0.022}^{+0.025}$ and
$H_{0}=72.03_{-0.87}^{+0.82}$ km/s/Mpc with
$f_{\mathrm{EDE}}=0.1399_{-0.022}^{+0.023}$, respectively. Although the Hubble
tension is mitigated compared to $\Lambda$CDM, the inclusion of eBOSS data
amplifies the tension within EDE from $2\sigma$ to $3\sigma$, in contrast to
the full shape analysis of BOSS data with Planck, external BAO, PantheonPlus,
and SH0ES. This highlights the significance of incorporating additional large-
scale structure data in discussions concerning models aiming to resolve the
Hubble tension.

###### keywords:

large-scale structure of the Universe – methods: data analysis – cosmology:
cosmological parameters

††pubyear: 2023††pagerange: Cosmological constraints on early dark energy from
the full shape analysis of eBOSS DR16–A

##  1 Introduction

Over the last century $\Lambda$ Cold Dark Matter ($\Lambda$CDM) has been
established as the standard model in cosmology, but despite its successes, it
faces both observational and theoretical challenges. One observational
challenge which appears particularly persistent is the Hubble tension, which
describes the difference in value of the Hubble constant $H_{0}$ as inferred
from direct versus indirect measurements. The most significant discrepancy in
its value comes from the cosmic microwave background (CMB) as measured by the
Planck satellite, which implies $H_{0}=67.27\pm 0.60$ km/s/Mpc (Aghanim et
al., 2020a) in $\Lambda$CDM, and the direct measurements of Cepheid-calibrated
Type Ia supernovae by the SH0ES project, which reports $H_{0}=73.04\pm 1.04$
km/s/Mpc (Riess et al., 2022a, b). While each observational method presents
unique systematic challenges, it is unlikely that a single source of
systematic error can entirely account for the Hubble tension because the
significance of the tension remains relatively stable when any individual
measurement is excluded. Therefore, the tension may instead be indicating
limitations in the current standard model. For comprehensive reviews on the
Hubble tension and specific measurement systematics, readers are referred to
Di Valentino et al. (2021) and Abdalla et al. (2022), and references therein.

Due to the robustness of the tension on the observational side, many proposals
have been put forward to explain the Hubble tension by introducing new
physics. Early-time solutions (models which alter the expansion history prior
to recombination) as well as late-time solutions (models which alter the
expansion history after recombination) have been extensively been studied;
early-time solutions have demonstrated particular promise in alleviating the
tension considering the current low redshift data, which tightly constrains
the expansion history at $z\lesssim 2$ (Bernal et al., 2016; Aylor et al.,
2019; Knox & Millea, 2020; Camarena & Marra, 2021; Schöneberg et al., 2022). A
subclass of these early-time solutions that has gained much attention in
recent years is early dark energy (EDE) models (Karwal & Kamionkowski, 2016b;
Poulin et al., 2019b; Lin et al., 2019; Smith et al., 2020; Sakstein &
Trodden, 2020; Niedermann & Sloth, 2020, 2021; Ye & Piao, 2020; Karwal et al.,
2022; Seto & Toda, 2021; Rezazadeh et al., 2022; McDonough et al., 2022;
Agrawal et al., 2023; Poulin et al., 2023). EDE aims to alleviate the Hubble
tension by introducing a light scalar degree of freedom just before
recombination, which leads to a decreased sound horizon at last scattering, so
that the CMB is consistent with a larger Hubble constant $H_{0}$ at late
times.

EDE has shown promise in reconciling the Hubble tension when analysing Planck
CMB data, Baryon Acoustic Oscillation measurements, Pantheon supernova
observations, and data from SH0ES (Poulin et al., 2019b; Smith et al., 2020).
However, the inclusion of large-scale structure and weak lensing data imposes
stringent upper bounds on the overall contribution of EDE (Hill et al., 2020;
Ivanov et al., 2020b; D’Amico et al., 2021c; Simon et al., 2023b; Rebouças et
al., 2023; Goldstein et al., 2023; McDonough et al., 2023). This limitation
stems from the fact that EDE enhances clustering on small scales, exacerbating
another prominent issue in $\Lambda$CDM: the $S_{8}$ tension. The $S_{8}$
tension quantifies the mismatch of the observed clustering in our universe
when comparing Planck CMB data with weak lensing surveys. The parameter
$S_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.5}$, is a combination of the matter
fluctuations on a smoothing scale of $8$ Mpc/$h$ scale and the matter density
today $\Omega_{m}$; its value inferred from the CMB differs by 2-3$\sigma$
from measurements obtained by weak lensing surveys such as CFHTLenS (Heymans
et al., 2012), KiDS-1000 (Heymans et al., 2021), and DES (Abbott et al.,
2022). Discrepancies in the $S_{8}$ value have also been observed in large-
scale structure studies, including the full shape analysis of BOSS data (Zhang
et al., 2022; Philcox & Ivanov, 2022) \- implying a moderate tension of below
$2\sigma$. The cause of this tension is still under scrutiny, with recent
reanalyses of BOSS data suggesting that lower values of $S_{8}$ might be
attributed to projection effects (D’Amico et al., 2022a; Simon et al., 2023c;
Carrilho et al., 2023; Donald-McCann et al., 2023). Furthermore, a recent
combined analysis of cosmic shear of KiDS-1000 and DES suggests that the
tension may only be 1.7$\sigma$ (Abbott et al., 2023).

For EDE to effectively address the Hubble tension, a substantial energy
injection of $\sim 12\%$ of the critical density around a redshift of 3500 is
essential. The introduction of EDE leads to concurrent adjustments not only in
the sound horizon $r_{s}$, but also in the spectral index $n_{s}$, and CDM
energy density $\omega_{\mathrm{CDM}}$ to ensure a good fit to the CMB data
(Poulin et al., 2019b; Hill et al., 2020; Vagnozzi, 2021). These shifts lead
to a worsening of the $S_{8}$ tension of around $0.5\sigma$ (Hill et al.,
2020). The enhancement in power of small-scale clustering seen in EDE can be
directly probed by large-scale structure surveys such as BOSS (Beutler et al.,
2017) and eBOSS (Alam et al., 2021). Large-scale structure analyses typically
rely on a template fitting method, wherein the quasi-nonlinear galaxy power
spectrum in redshift space is deduced from a given linear power spectrum,
which is extended based on a perturbation theory model (e.g. Taruya et al.,
2010) and galaxy bias scheme (e.g. McDonald & Roy, 2009). The resulting galaxy
power spectrum is then fitted to data, enabling constraints on the Alcock-
Paczynski (AP) and redshift-space distortion (RSD) parameters. While the
template fitting method is computationally very efficient - the template needs
to be evaluated only once - the compression of the full power spectrum $P(k)$
to the AP and RSD parameters leads to some loss of information (Brieden et
al., 2021a, b; Chen et al., 2022; Maus et al., 2023). Recently, there has been
much effort to derive cosmological parameters from a direct fit to the power
spectrum. Extending the modelling into modes which are mildly affected by non-
linearities affords a higher $k_{\mathrm{max}}$, and therefore a higher number
of modes ($N_{\mathrm{modes}}\propto k_{\mathrm{max}}^{3}$). The development
of the effective field theory of large-scale structure (EFTofLSS; Baumann et
al., 2012a; Carrasco et al., 2012a; Senatore, 2015b; Ivanov, 2022) was a
crucial step towards this goal, and it is now possible to obtain constraints
on cosmological parameters coming from direct fits, utilizing modes from the
linear, as well as from the quasi-nonlinear regime (Ivanov et al., 2020a;
D’Amico et al., 2020; Kobayashi et al., 2022; Chen et al., 2022).

In this work, we present constraints on EDE coming from the full shape
analysis of the publicly available BOSS and eBOSS data. Previously, Bayesian
analyses of the EFTofLSS full shape fit suffered from projection effects
(Simon et al., 2023c), which originate from marginalization processes and lead
to biases in the resulting posteriors. It was suggested in Herold et al.
(2022); Herold & Ferreira (2023); Holm et al. (2023), that by using a profile
likelihood analysis, these projection effects are mitigated due to the prior
independence and reparametrization invariance of the frequentist approach. In
the context of Early Dark Energy (EDE), additional projection effects arise
due to the introduction of new EDE parameters. Herold et al. (2022) and Herold
& Ferreira (2023) argue that, when considering large-scale structure data, a
profile likelihood analysis favours a higher fractional amount of EDE compared
to Bayesian analysis. In this paper, we address how the issue of projection
effects in EDE can be tackled and mitigated within a Bayesian framework, and
how this impacts the constraints on EDE. To do this, we extend the findings of
Donald-McCann et al. (2023) to a beyond $\Lambda$CDM context by utilizing a
Jeffreys prior (Jeffreys, 1998a) on the marginalised nuisance parameters.

The paper is organised as follows. In section 2, we briefly review the axion-
like EDE model. In section 3, we further introduce the EFTofLSS and detail the
data sets that we consider. We lay special focus on the BOSS and eBOSS
measurements here, addressing the various systematics in the data sets. In
section 4, we present a series of mock analyses for EDE, as well as
$\Lambda$CDM, to test our inference pipeline and the effect of different prior
choices. In section 5, we briefly discuss results from the full shape analysis
of eBOSS in $\Lambda$CDM, before moving on to section 8, where we present
constraints on EDE coming from the EFTofLSS analysis of BOSS and eBOSS data
and their combination with other measurements such as Planck, BAO, Pantheon+
and SH0ES. We conclude in section 7.

##  2 Early Dark Energy Model

The quantity measured by the CMB is the angular acoustic scale $\theta^{*}$ at
recombination. It is constrained to a precision of $0.03\%$ by Planck 2018
(Aghanim et al., 2020a). This scale is a ratio of the sound horizon at
recombination $r_{s}^{*}$ and the associated angular diameter distance
$D_{A}^{*}$, where $r_{s}^{*}$ is defined as:

$r_{s}^{*}=\int_{z_{*}}^{\infty}\frac{dz}{H(z)}c_{s}(z),$ (1)

and $D_{A}^{*}$:

$D_{A}^{*}=\int_{0}^{z_{*}}\frac{dz}{H(z)},$ (2)

where $z_{*}\approx 1100$ denotes the redshift at last scattering, $c_{s}(z)$
is the sound speed in the baryon-photon plasma and $H(z)$ is the Hubble
parameter. Both integrals are dominated by the expansion history of their
lower bounds. Assuming an underlying model for the expansion of the universe
and determining the baryon and matter density from the CMB power spectra, it
is possible to calculate $r_{s}^{*}$ and infer $H_{0}$ from
$D_{A}^{*}=r_{s}^{*}/\theta^{*}$. EDE (Kamionkowski et al., 2014; Karwal &
Kamionkowski, 2016a) amends the assumptions of the underlying model with
respect to $\Lambda\mathrm{CDM}$ by introducing an extra degree of freedom at
early times. Most EDE models postulate the existence of a scalar field whose
background dynamics is described by the Klein-Gordon equation and undergo the
three following phases: i) The field starts out initially frozen in its
potential with an energy density constant in time, similarly to DE, ii) a
mechanism (i.e. the Hubble friction no longer holding the field in place or a
spontaneous trip change of the potential shape due to a phase transition)
triggers the field to become dynamical, iii) the field radiates away faster
than matter in order to keep the expansion history unchanged at late times.
The dilution ensures that the impact of EDE is localised in time. With the
field becoming dynamical around recombination, EDE affects the sound horizon
and decreases its value. In order to keep the value $\theta^{*}$ fixed, this
decrease in $r_{s}^{*}$ needs to be compensated by a decrease in $D_{A}$,
leading to a higher inferred $H_{0}$. To fully account for the Hubble tension,
most models need to achieve a maximal fractional contribution of EDE of
$\simeq 10\%$ (roughly the percentage difference between the estimate of
$H_{0}$ between Planck and SH0ES) around the matter-radiation equality scale.

EDE models come in various different flavours (see Poulin et al. (2023) for a
recent review). In this work, we adopt a canonical EDE model which introduces
a pseudo scalar field with an axion-like potential of the form (Poulin et al.,
2019a):

$V(\phi)=V_{0}(1-\cos(\phi/f))^{n},\quad V_{0}\equiv m^{2}f^{2},$ (3)

where $f$ is the decay constant of the axion-like field, $m$ denotes its mass
and $n$ is an integer value, which controls the decay rate of the potential.
For convenience, we define $\theta\equiv\phi/f$ as a renormalized field
variable such that $-\pi\leq\theta\leq\pi$ (without loss of generality, we set
$0\leq\theta\leq\pi$ ). The parameter $\theta_{i}$ represents the renormalized
initial field value of the frozen field and primarily controls the dynamics of
field perturbations through an effective sound speed $c_{s}^{2}$. At early
times, the field starts out in a slow-roll behaviour before the Hubble
friction drops below a critical value ($H\sim m)$ and the field starts
evolving to its potential minimum. Oscillating around its minimum, the field
dilutes with a time averaged approximate equation of state
$\omega_{\phi}=(n-1)/(n+1)$. To make sure that EDE dilutes at least as fast as
radiation, $n$ needs to be chosen $\geq 2$. In Smith et al. (2020), it was
found that the data are quite insensitive to this parameter in the range of
$2<n\leq 6$, where the best fit was found to be $n=3$. For our analysis, we
fix $n$ to this best fit value. In this configuration, the EDE final equation
of state is $\omega_{\phi}=1/2$, which is effectively higher than the
radiation equation of state $\omega_{\rm r}=1/3$. In principle, the background
dynamics can be characterised by the three theoretical parameters $m$, $f$ and
$\theta_{i}$. For a more direct interpretation in terms of observables, we are
mapping $m$ and $f$ to two phenomenological parameters $z_{c}$ and
$f_{\mathrm{EDE}}$ following the shooting mechanism described in (Hill et al.,
2020) where $z_{c}$ describes the critical redshift at which the field makes
its maximal fractional contribution
$f_{\mathrm{EDE}}(z_{c})\equiv\rho_{\mathrm{EDE}}/(3M_{P}^{2}H^{2})|_{z_{c}}$
to the energy budget of the universe. We therefore describe our cosmological
model in terms of three additional EDE parameters - $f_{\mathrm{EDE}}$,
$z_{c}$ and $\theta_{i}$ \- on top of the usual $\Lambda$CDM parameters.

##  3 Data and Methodology

###  3.1 Theory model: EFTofLSS

In recent years, substantial efforts have been invested into the description
of non linear modelling for large-scale structure data. Although modelling
modes up to a fully non-linear regime remains a very challenging task,
significant progress has been made in modelling modes which are just mildly
affected by non-linearities. In this paper, we follow an effective field
theory (EFT) approach, most often referred to as EFTofLSS (Baumann et al.,
2012b; Carrasco et al., 2012b)111The initial formulation of the EFTofLSS was
undertaken in Eulerian space in Carrasco et al. (2012a); Baumann et al.
(2012a). Porto et al. (2014) supplemented these efforts with a description of
EFTofLSS in Lagrangian space. Following the establishment of these theoretical
frameworks, various efforts were made to enhance their predictive
capabilities. These efforts included the understanding of renormalization
(Pajer & Zaldarriaga, 2013; Abolhasani et al., 2016), the IR-resummation of
the long displacement fields (Senatore & Zaldarriaga, 2014; Baldauf et al.,
2015; Senatore & Zaldarriaga, 2015; Senatore & Trevisan, 2018; Lewandowski &
Senatore, 2020; Blas et al., 2016), and calculating the two-loop matter power
spectrum (Carrasco et al., 2014a, b). Subsequently, this theory was further
developed within the context of biased tracers, such as galaxies and quasars,
as described in the references (Senatore, 2015a; Mirbabayi et al., 2015;
Angulo et al., 2015; Fujita et al., 2020; Perko et al., 2016; Nadler et al.,
2018).. Within this framework, we take into account small non-linear
corrections onto the long wavelength modes, while modelling the dark matter
field as an imperfect fluid. EFTofLSS incorporates a cutoff scale, acting as
an effective low pass filter and implying that the fluid equations are solved
in terms of long-wavelength overdensity and velocity fields. Additionally, an
effective stress energy tensor is introduced to account for the effects of
small-scale physics onto the larger scales. At a given order $n$, the effect
of these small scales and their backreactions can be captured by a finite
number of so called "counterterms" with coefficients $c_{i}$. These $c_{i}$
are free parameters and need to be either fitted to data or simulations.

PyBird (D’Amico et al., 2021b) generates predictions for the power spectrum of
biased tracers up to 1-loop order on the basis of EFTofLSS. Employing a
nonlinear bias scheme that correlates the underlying dark matter field with
observed galaxy densities, the expression for the one-loop redshift-space
galaxy power spectrum in Fourier space is derived as:

$\displaystyle P_{g}(k,\mu)$ $\displaystyle=Z_{1}(\mu)^{2}P_{11}(k)$
$\displaystyle+2\int\frac{d^{3}q}{(2\pi)^{3}}Z_{2}(\textbf{q},\textbf{k -
q},\mu)^{2}P_{11}(|\textbf{k - q}|)P_{11}(q)$
$\displaystyle+6Z_{1}(\mu)P_{11}(k)\int\frac{d^{3}q}{(2\pi)^{3}}Z_{3}(\textbf{q},\textbf{-q},\textbf{k
},\mu)P_{11}(q)$
$\displaystyle+2Z_{1}(\mu)P_{11}(k)\left(c_{ct}\frac{k^{2}}{k_{M}^{2}}+c_{r,1}\mu^{2}\frac{k^{2}}{k_{M}^{2}}+c_{r,2}\mu^{4}\frac{k^{2}}{k_{M}^{2}}\right)$
$\displaystyle+\frac{1}{\hat{n}_{g}}\left(c_{\epsilon,1}+c_{\text{mono}}\frac{k^{2}}{k^{2}_{M}}+\frac{3}{2}c_{\text{quad}}\left(\mu^{2}-\frac{1}{3}\right)\frac{k^{2}}{k^{2}_{M}}\right),$
(4)

where $k$ is the norm of the wavenumber k and $\mu\equiv\hat{k}\cdot\hat{z}$
is the cosine angle to the line of sight. The redshift-space galaxy kernels
$Z_{i}$ encapsulate the velocity and density characteristics of galaxies (for
their precise expressions, refer to D’Amico et al. (2020)). Additionally,
$\hat{n}_{g}$ denotes the mean galaxy density, and $k_{M}^{-1}$ serves as the
scale governing the expansion of spatial derivatives222We note that EFTofLSS
introduces in principle three different scales: $k_{NL},k_{M}$ and $k_{R}$,
where $k_{R}^{-1}$ controls the renormalization scale of the velocity products
which appear in the redshift-space expansion (Senatore & Zaldarriaga, 2014).
In this analysis, we explicitly set $k_{NL}=k_{M}=k_{R}=0.7~{}h/\mathrm{Mpc}$.
Any deviation from this equivalence will then be represented in a deviation of
$c_{r,1}$ and $c_{r,2}$ away from $\mathcal{O}(1)$ and by simple rescaling of
$c_{r,1,2}$ the true value of $k_{R}$ can be inferred..

EFTofLSS introduces a total of 10 nuisance parameters up to first-loop order,
denoted as:

$\\{b_{1},b_{2},b_{3},b_{4},c_{ct},c_{r,1},c_{r,2},c_{\epsilon,1},c_{\mathrm{mono}},c_{\mathrm{quad}}\\}.$
(5)

Among these, the four galaxy bias parameters ($b_{1-4}$) emerge in the
expansion of the galaxy density and velocity field with respect to the
underlying dark matter field and are found in the galaxy kernels $Z_{i}$. In
particular, $b_{1}$ corresponds to the linear galaxy bias parameter used in
the Kaiser formula (i.e., the first term in eq. (4), which is the tree-level
order of the EFTofLSS), while $b_{2-4}$ represent the non-linear galaxy bias
parameters. Due to being highly degenerate (D’Amico et al., 2020), $b_{2}$ and
$b_{4}$ are commonly reparameterized as follows:

$\displaystyle c_{2}=(b_{2}+b_{4})\ /\ \sqrt{2}\ ,$ $\displaystyle
c_{4}=(b_{2}-b_{4})\ /\ \sqrt{2}\ .$ (6)

The EFTofLSS model involves three stochastic parameters
($c_{\epsilon,1},c_{\text{mono}},c_{\text{quad}}$) that are introduced to
account for discrepancies between the actual observed galaxy field and its
expected value. The first term describes a constant shot noise, while the
other two terms correspond to the scale-dependant stochastic contributions of
the monopole and the quadrupole. Additionally, there are three counter-terms
used to incorporate the influence of UV physics: $c_{\text{ct}}$, which is a
linear combination of the dark matter sound speed Baumann et al. (2012a);
Carrasco et al. (2012a) and a higher-derivative bias Senatore (2015a), and
$c_{r,1}$ and $c_{r,2}$, which are the redshift-space counterterms that govern
the impact of small scales on redshift space distortions. The 2D power
spectrum in Eq. (4) can be decomposed into multipoles via

$P_{l}(k)=\frac{2l+1}{2}\int_{-1}^{1}P_{g}\left(k,\mu\right)\mathcal{L}_{l}\left(\mu\right)\mathrm{d}\mu\
,$ (7)

where $\mathcal{L}_{l}(\mu)$ are the Legendre polynomial of order $l$.

###  3.2 Data I: BOSS + eBOSS

The main results of this paper are based on the power spectrum analysis of the
completed (extended) Baryon Oscillation Spectroscopic Survey (eBOSS) data.
Here we give a brief overview of the data sets under consideration. For more
details see Alam et al. (2021) and references therein.

####  3.2.1 eBOSS

The eBOSS commenced full operations in July 2014. Its primary spectroscopic
targets were luminous red galaxies (LRGs), emission line galaxies (ELGs) and
quasars (QSOs). The survey concluded its operations on March 1, 2019 333All
eBOSS data products are publicly available at:
https://svn.sdss.org/public/data/eboss/DR16cosmo/tags/v1_0_1/dataveccov/lrg_elg_qso/.

  * •

LRGpCMASS: The eBOSS LRG sample consists of 174,816 redshift measurements over
the redshift interval $0.6<z<1$, covering an effective volume of
$V=2.7(\mathrm{Gpc}/h)^{3}$. In this work, we are supplementing the eBOSS LRG
sample with BOSS DR12 LRGs which were observed in the high redshift tail
$z>0.6$. The catalogue information can be found in Ross et al. (2020). The
multipole power spectrum measurements for the monopole and quadrupole come
from Gil-Marin et al. (2020). The effective redshift $z_{\mathrm{eff}}$ is
equal to $0.698$ for both galactic caps and the mean number density is set to
$\hat{n}_{g}=5\cdot 10^{-5}(h/\mathrm{Mpc})^{3}$. In agreement with Gil-Marin
et al. (2020), we are setting the following scale cuts for our analysis:
$k_{\mathrm{min}}=0.02~{}h/\mathrm{Mpc}$ and
$k_{\mathrm{max}}=0.2~{}h/\mathrm{Mpc}$.

  * •

ELG: eBOSS obtained redshift measurements for 173,736 ELGs in the redshift
range $0.6<z<1.1$ with an effective volume of $V=0.6(\mathrm{Gpc}/h)^{3}$. The
details of the ELG catalogs are described in Raichoor et al. (2020). We use
the full shape measurements of de Mattia et al. (2021) in a redshift range of
$0.7<z<1.1$ for our analysis of the multipoles, $l=0,2$, in the NGC and SGC
sky patches. The effective redshift for NGC and SGC is $z_{\mathrm{eff}}=0.86$
and $z_{\mathrm{eff}}=0.853$, respectively. We set the mean number density
$\hat{n}_{g}$ to $2.5\cdot 10^{-4}(h/\mathrm{Mpc})^{3}$ for both caps. To be
consistent with de Mattia et al. (2021), we impose the following scale cuts:
$k_{\mathrm{min}}=0.03~{}h/\mathrm{Mpc}$ and
$k_{\mathrm{max}}=0.2~{}h/\mathrm{Mpc}$.

  * •

QSO: This sample includes 343,708 quasars in the redshift range $0.8<z<2.2$.
The description of the QSO catalogs can be found in Ross et al. (2020). We use
the power spectrum multipoles, $l=0,2$, as measured from two different patches
of the sky (NGC and SGC) in Neveux et al. (2020). For our analysis, we use
$z_{\mathrm{eff}}=1.48$ as an effective redshift of the samples with an
effective volume of $V=0.6(h/\mathrm{Mpc})^{3}$ and a mean number density
$\hat{n}_{g}=1.5\cdot 10^{-5}(h/\mathrm{Mpc})^{3}$. In line with Neveux et al.
(2020), we impose the following scale cuts:
$k_{\mathrm{min}}=0.02~{}h/\mathrm{Mpc}$ and
$k_{\mathrm{max}}=0.2~{}h/\mathrm{Mpc}$.

####  3.2.2 BOSS

Between the years 2009 and 2014, the Baryon Oscillation Spectroscopic Survey
(BOSS) conducted large scale structure spectroscopy within the redshift range
of 0.2 to 0.75. The final galaxy catalogue used for clustering measurements
counts redshift data from a total of 1,372,737 galaxies. The sample was
divided into three distinct redshift bins: $0.2<z_{1}<0.5,0.4<z_{2}<0.6,$ and
$0.5<z_{3}<0.75$. The final galaxy catalogue is detailed in Reid et al.
(2016). We analyse the full shape of the BOSS power spectrum multipoles
$l=0,2$ in the redshift bins $z_{1}$ and $z_{3}$ measured in Beutler &
McDonald (2021)444Publicly available at
https://fbeutler.github.io/hub/deconv_paper.html. We deconvolve the data and
the associated covariance matrices with the window functions provided in
Beutler & McDonald (2021), such that one does not need to apply them to the
theoretical predictions. The effective redshift for $z_{1}$ is set to $0.38$
with an effective volume $V=3.7(\mathrm{Gpc/h})^{3}$ and mean number density
$\hat{n}_{g}=4\cdot 10^{-4}(h/\mathrm{Mpc})^{3}$. For $z_{3}$, the effective
redshift is $0.61$ with $V=4.1(\mathrm{Gpc}/h)^{3}$ and $\hat{n}_{g}=4.5\cdot
10^{-4}(h/\mathrm{Mpc})^{3}$. We set the following scale cuts for our
analysis: $k_{\mathrm{min}}=0.01~{}h/\mathrm{Mpc}$ and
$k_{\mathrm{max}}=0.2~{}h/\mathrm{Mpc}$.

####  3.2.3 BOSSz1+eBOSS

In the context of combining data from the BOSS and eBOSS surveys, we adopt a
simplified approach. Specifically, we include the low-z redshift bin ($z_{1}$)
in BOSS, while substituting $z_{3}$ with LRGpCMASS. This simplification leads
to the omission of BOSS galaxies falling within the redshift range of
$0.5<z<0.6$.

###  3.3 Systematic effects

####  3.3.1 AP effect

To convert measured celestial coordinates of galaxies and their redshifts into
comoving distances, a reference cosmology must be assumed for this
transformation. If the reference cosmology does not accurately represent the
true cosmology, this transformation introduces artificial distortions along
and parallel to the line of sight, which need to be accounted for. This
distortion is known as the Alcock-Paczynski (AP) effect (Alcock & Paczynski,
1979) and can be expressed using the parameters $q_{\parallel}$ and
$q_{\perp}$:

$\displaystyle
q_{\parallel}=\frac{D_{A}(z)H(z=0)}{D_{A}^{\mathrm{ref}}(z)H^{\mathrm{ref}}(z=0)}\
,$ $\displaystyle
q_{\bot}=\frac{H^{\mathrm{ref}}(z)H(z=0)}{H(z)H^{\mathrm{ref}}(z=0)}\ ,$ (8)

where $D_{A}(z)$ and $H(z)$ describe the angular-diameter distance and Hubble
parameter as a function of redshift, respectively. The true scales and angles
are then related to the reference ones by:

$\displaystyle
k=\frac{k^{\mathrm{ref}}}{q_{\bot}}\left[1+\left(\mu^{\mathrm{ref}}\right)^{2}\left(F^{-2}-1\right)\right]^{1/2}\
,$
$\displaystyle\mu=\frac{\mu^{\mathrm{ref}}}{F}\left[1+\left(\mu^{\mathrm{ref}}\right)^{2}\left(F^{-2}-1\right)\right]^{-1/2}\
$ (9)

with $F=q_{\parallel}\ /\ q_{\perp}$. PyBird gives predictions for the
multipole of the galaxy power spectrum rather than for the 2D power spectrum
(see Eq. (7)). In order to account for the AP effect, the 2D power spectrum
needs to be reconstructed from the theoretical multipoles via

$P_{g}(k,\mu)=\sum_{l=0}P_{l}(k)\mathcal{L}_{l}(\mu)\ .$ (10)

The multipole expansion of the power spectrum in the observed frame
$k^{\mathrm{ref}}$, can then be related to the theoretical model of the line-
of-sight power spectrum in terms of $k$ and $\mu$ by

$P^{\mathrm{AP}}_{l}(k^{\mathrm{ref}})=\frac{2l+1}{2q_{\parallel}q_{\bot}^{2}}\int_{-1}^{1}P_{g}\left(k,\mu\right)\mathcal{L}_{l}\left(\mu^{\mathrm{ref}}\right)\mathrm{d}\mu^{\mathrm{ref}}\
.$ (11)

Table 1: The relevant information about the fibre collision parameters for the different samples under consideration. $f_{s}$ describes the fraction of sky which is affected by fibre collision for each tracer and $D_{\mathrm{FC}}$ is the comoving distance of the 62” fibre collision angular scale. Since the loss of targets due to fibre collision is minimal in the LRGpCMASS sample, we are not applying any fibre collision correction.  |  $z_{1}/z_{3}$ BOSS (Hahn et al., 2017) | ELG eBOSS (de Mattia et al., 2021) | QSO eBOSS (Neveux et al., 2020) |
---|---|---|---|---
$f_{s}$ | 0.6 | 0.46 (NGC) / 0.38 (SGC) | 0.36 (NGC) / 0.45 (SGC) |
$D_{\mathrm{FC}}$ [Mpc/h] | 0.43 | 0.62 | 0.9 |

####  3.3.2 Fibre collision

To measure the spectroscopic redshift of a galaxy, a fibre needs to be placed
on the location in the sky where the galaxy is located. However, if two
galaxies are lying within the same angular scale that is covered by just one
fibre, only one redshift measurement is obtained. This leads to a systematic
bias in the clustering statistic because certain galaxies with close angular
proximity are missing from the sample. A common approach to correct for fibre
collisions is by the nearest neighbour (NN) method (Zehavi et al., 2002, 2005;
Anderson et al., 2014), where a statistical weight is assigned to the nearest
angular neighbour of the fibre-collided galaxy with no redshift information.
While this method provides a reasonable correction on scales much larger than
the fibre collision scale, it fails elsewhere. In this paper, we therefore
employ the effective window method as presented in Hahn et al. (2017), where
the fibre collision effect is modelled at the prediction level. It was shown
that the effective window method is able to mitigate fibre collision effects
up to very small scales ($k\gtrsim 0.3~{}h/$Mpc) (Hahn et al., 2017), much
smaller than those relevant to the scope of this paper.

As a first step, the correlation function $\xi_{l}$ is corrected using a
spherical 2D top hat window function $\mathcal{W}_{\mathrm{FC}}$ and the
fraction of the sky $f_{s}$ affected by fibre collision:

$\xi^{\mathrm{FC}}_{l}=\xi_{l}-\Delta\xi^{\mathrm{FC}}_{l},$ (12)

where

$\Delta\xi^{\mathrm{FC}}=\frac{2l+1}{2}\int_{-1}^{1}f_{s}\mathcal{W}_{\mathrm{FC}}\left(s_{\bot}\right)(1+\xi\left(s,\mu\right))\mathcal{L}_{l}\left(\mu\right)\mathrm{d}\mu\
.$ (13)

The step size of the top hat function corresponds to the comoving distance of
the 62” fibre collision angular scale, $D_{\mathrm{FC}}$ and
$s_{\bot}=s\sqrt{1-\mu^{2}}$ is the transverse part of the separation vector
$\bm{s}=|s|$. By Fourier transforming Eq. (13) we obtain the correction to the
theoretical power spectrum. It consists of two terms: The first term
corresponds to the Fourier transform of the top hat function, which accounts
for uncorrelated pair collision. The second term accounts for correlated pair
collision and is described by a convolution of the power spectrum with the top
hat function:

$\displaystyle\Delta P^{\mathrm{FC}}_{l}$ $\displaystyle=\Delta
P^{\mathrm{uncorr}}_{l}+\Delta P^{\mathrm{corr}}_{l}$ $\displaystyle=\Delta
P^{\mathrm{uncorr}}_{l}+\left.\Delta
P^{\mathrm{corr}}_{l}\right|_{q=0}^{q=\mathrm{k_{trust}}}+\left.\Delta
P^{\mathrm{corr}}_{l}\right|^{q=\infty}_{q=\mathrm{k_{trust}}}$
$\displaystyle=\Delta P^{\mathrm{uncorr}}_{l}+\left.\Delta
P^{\mathrm{corr}}_{l}\right|_{q=0}^{q=\mathrm{k_{trust}}}+\sum_{l\leq
n,n=0,2,4...}C_{l,n}k^{n}.$ (14)

The correlated part of the power spectrum correction involves an integration
from scales 0 to $\infty$. However, in practice, it is not possible to compute
the multipoles reliably up to an infinitely small distance. To address this,
we split the integral in Eq. (14) into a low k integral (from 0 to
$\mathrm{k_{trust}}$), which can be calculated numerically, while the high k
part (from $\mathrm{k_{trust}}$ to $\infty$) of the integral can be
approximated by a polynomial in k. The form of the polynomial in Eq. (14)
coincides with the stochastic contribution to the theoretical predictions of
the monopole and quadrupole (compare with Eq. (4)). As a consequence, we
incorporate the effect of the polynomial in the stochastic term governed by
the parameters $c_{\epsilon},c_{\mathrm{mono}}$ and $c_{\mathrm{quad}}$. By
marginalising over the stochastic parameters, the impact of fibre collision is
then fully accounted for 555In the case of the ELG, the fine-grained veto mask
introduces window function kernels which are scale dependent even on small
scales (see figure 1 in de Mattia et al. (2021)). The form of the stochastic
contribution and the fibre collision polynomial are therefore no longer
completely degenerate (Ivanov, 2021). Nevertheless we expect the remaining
difference to be small and safe to ignore due to the low constraining power of
the sample.. For our analysis, we set $k_{\mathrm{trust}}$ to be
$0.25~{}h/\mathrm{Mpc}$. Table 1 displays the corresponding values of
$D_{\mathrm{FC}}$ and $f_{s}$ for the BOSS and eBOSS data sets. We are not
applying any fibre collision to the LRGpCMASS sample since less than 4 percent
of targets were not observed due to the fibre collision (Bautista et al.,
2020).

Figure 1: Effect of the radial and angular integral constraint correction on
the monopole (upper panel) and quadrupole (lower panel) of the ELG NGC EZmock
sample. We show mocks without systematics (EZmock ref.) and with induced
radial and angular integral constraint (EZmock pix.). The different mocks are
described in Sec. 4.1. The integral constraints lead to a suppression of power
on large scales for the monopole and enhance power for the quadrupole. The
solid lines correspond to the best-fit model fitted to the pixelated mock,
where the integral constraints are modelled. The dashed lines correspond to
the pixelated best-fit model, where we did not account for the effect of the
angular and radial integral constraint in the theory model. The dashed lines
agree with the reference mocks on all scales.

####  3.3.3 Survey window function

For the eBOSS samples, to address the survey’s selection effects on the
observed galaxy density, we model the survey window effect on the prediction
level. This approach differs from the deconvolution method used for BOSS data
(see Sec. 3.2.2) and follows the procedure presented in Beutler et al. (2017);
Wilson et al. (2017); Beutler et al. (2019). The approach assumes a local
plane-parallel approximation and amounts to a multiplication of the true
correlation function with the multipole moments of the window mask $W_{n}^{2}$
in configuration space:

$\displaystyle\xi^{\mathrm{W}}_{l}(s)$
$\displaystyle=\sum_{p,q}A^{p}_{l,q}\frac{2l+1}{2p+1}\mathrm{W}^{2}_{p}(s)\xi_{q}(s)$
(15) $\displaystyle=Q_{l,l^{\prime}}(s)\xi_{l^{\prime}}(s),$ (16)

where $A^{p}_{l,q}$ arises from multiplying two Legendre polynomials together
and accounts for the weighted volume average (for its exact form see for
example (Wilson et al., 2017)). As the multipole moments of the configuration
space correlation function and the Fourier space power spectrum function form
a Hankel transform pair, we can easily evaluate Eq. (16) in Fourier space
using a 1D FFT algorithm (Hamilton, 2000). The convolved power spectrum can
then be expressed as:

$\displaystyle P_{l}^{\mathrm{W}}(k)$ $\displaystyle=4\pi(-i)^{l}\int
s^{2}ds\xi^{\mathrm{W}}_{l^{\prime}}(s)j_{l}(ks)$ (17)
$\displaystyle=\mathrm{W}(k,k^{\prime})_{l,l^{\prime}}P_{l^{\prime}}(k^{\prime}),$
(18)

where the window function for the power spectrum is given by:

$W(k,k^{\prime})_{l,l^{\prime}}=\frac{2}{\pi}(-i)^{l}(i)^{l^{\prime}}k^{\prime
2}\int dss^{2}j_{l}(ks)Q_{l,l^{\prime}}(s)j_{l^{\prime}}(k^{\prime}s).$ (19)

Here, $j_{l}(ks)$ denotes spherical Bessel functions. Following the approach
adopted in previous EFTofLSS full shape analyses (D’Amico et al., 2020;
Ivanov, 2021; Chudaykin & Ivanov, 2023), we restrict our analysis to multipole
moments up to the hexadecapole in $W_{n}^{2}$.

####  3.3.4 Integral constraints

To define a proper over-density, a mean density of the Universe needs to be
assumed. The most common assumption is to set the mean density of the Universe
equal to the mean density of the survey. However, this assumption is flawed,
particularly for the largest modes (corresponding to the size of the survey),
as the sample variance becomes significant. This leads to a phenomenon known
as the global integral constraint (IC), where the power spectrum near $k=0$ is
forced to be zero. Since the window function correlates various modes with
$k=0$, this effect propagates to other scales, which are important for
cosmological measurements. Similarly, the radial survey selection function is
estimated from the data, leading to a corresponding radial integral constraint
(RIC) which mainly suppresses power on large scales. While the RIC is
subdominant for the LRG and QSO data sets (Beutler et al., 2017; de Mattia &
Ruhlmann-Kleider, 2019; Gil-Marin et al., 2020; Neveux et al., 2020), it
represents one of the most significant systematics in the ELG eBOSS data set
(de Mattia et al., 2021). Moreover, the ELG data set exhibits remaining
angular photometric systematics. In order to mitigate these angular effects,
we remove contaminated modes by averaging the mean density fluctuations in
each HEALPix666http://healpix.jpl.nasa.gov/ (Górski et al., 2005) pixel to
zero, which removes angular modes larger than the pixel scale. This step
introduces an additional angular integral constraint.

We correct for the impact of integral constraints on the theoretical power
spectrum following the method described in (de Mattia & Ruhlmann-Kleider,
2019; de Mattia et al., 2021). All three integral constraints are modelled in
a similar manner, and the observed correlation function (after window function
and integral correction) is described by:

$\displaystyle\xi^{\mathrm{W+IC}}_{l}(s)$
$\displaystyle=\xi^{\mathrm{W}}_{l}(s)$
$\displaystyle-\sum_{l^{\prime},s^{\prime}}s^{\prime
2}ds^{\prime}\frac{4\pi}{2l^{\prime}+1}\mathrm{W}_{l,l^{\prime}}^{\mathrm{IC}}(s,s^{\prime})\xi_{l^{\prime}}(s^{\prime})-sn\cdot
W^{\mathrm{IC_{sn}}}_{l}(s),$ (20)

where $sn$ is the shot noise and
$\mathrm{W}_{l,l^{\prime}}^{\mathrm{IC}}(s,s^{\prime})$ is the integral
correction window function in configuration space, accounting for the global,
radial and angular integral constraints. To provide a complete description, we
add the correction to the shot noise contribution already in configuration
space (for details, refer to de Mattia & Ruhlmann-Kleider (2019)). Using the
Hankel transformation from Eq. (17), we derive the window-convolved, integral-
constraint-corrected power spectrum multipoles as:

$\displaystyle P^{\mathrm{W+IC}}_{l}(k)$ $\displaystyle=P^{\mathrm{W}}_{l}(k)$
$\displaystyle-\mathrm{W}^{\mathrm{IC}}_{l,l^{\prime}}(k,k^{\prime})P_{l^{\prime}}(k^{\prime})-sn\cdot\mathrm{W}_{l}^{\mathrm{IC_{sn}}}(k),$
(21)

where $\mathrm{W}^{\mathrm{IC}}_{l,l^{\prime}}(k,k^{\prime})$ is given by:

$\displaystyle\mathrm{W}^{\mathrm{IC}}_{l,l^{\prime}}(k,k^{\prime})$
$\displaystyle=\frac{8}{2l^{\prime}+1}(-i)^{l}i^{l^{\prime}}k^{\prime
2}\quad\times$ $\displaystyle\int s^{2}ds\int s^{\prime
2}ds^{\prime}\mathrm{W}_{l,l^{\prime}}^{\mathrm{IC}}(s,s^{\prime})j_{l}(ks)j_{l^{\prime}}(k^{\prime}s^{\prime}).$
(22)

To solve Eq. (22), we utilize a 2D FFTlog algorithm based on Fang et al.
(2020) and Umeh (2021). Fig. 1 shows the effect of the combined radial and
angular integral constraint correction on the monopole and quadrupole.

###  3.4 Data II: complementary data sets

We compare and combine the data sets mentioned in Sec. 3.2 with additional
external data sets. Here we give a brief description of the data sets under
consideration:

  * •

PlanckTTTEEE+lensing: We use the high $l$ TT,TE, EE and the low $l$ EE and TT
power spectra from the Planck PR3-2018 data release (Aghanim et al., 2020a).
Additionally, we include information from the gravitational lensing potential
reconstructed from the temperature and polarisation data of Planck 2018
(Aghanim et al., 2020b).

  * •

BAO: We include complementary BAO data sets that are not correlated with BOSS
z1,z3 and eBOSS LRG, ELG and QSO. These include the 6dFGS at $z=0.106$
(Beutler et al., 2011) and the SDSS DR7 MGS at $z=0.15$ (Ross et al., 2015).

  * •

PantheonPlus: We include the Pantheon+ SN Ia catalogue, which consists of 1701
light curves of 1550 distinct SNIa spanning a redshift range from $0.001$ to
$2.26$ (Brout et al., 2022).

  * •

SH0ES: We also consider SH0ES Cepheid host anchors (Riess et al., 2022a). When
using SH0ES alone, we impose a Gaussian likelihood for $H_{0}=73.04\pm
1.04\mathrm{km/s/Mpc}$. When combining it with Pantheon+, we use the
PantheonPlusSH0ES likelihood (unless otherwise specified), where the distance
calibration for SNIa in Cepheid host galaxies is provided by Cepheids,
offering an absolute calibration for the SNIa absolute magnitude $M_{B}$.

  * •

BBN: In the case where we present constraints without Planck, we impose a Big
Bang nucleosynthesis (BBN) prior on $\omega_{b}$ from Schöneberg et al.
(2019). This prior incorporates the theoretical prediction from Consiglio et
al. (2018), the experimental deuterium fraction of Cooke et al. (2018), and
the experimental helium fraction of Aver et al. (2015).

###  3.5 Likelihood

For BOSS and eBOSS data, we sample from a Gaussian likelihood of the form:

$\ln\left[\mathcal{L}\right]=-\frac{1}{2}\sum_{i,j}(P^{\mathrm{M}}_{l}(k_{i})-P^{\mathrm{D}}_{l}(k_{i}))\tilde{C}^{-1}_{ij}(P^{\mathrm{M}}_{l^{\prime}}(k_{j})-P^{\mathrm{D}}_{l^{\prime}}(k_{j})),$
(23)

where $P_{l}^{\mathrm{M}}$ represents the multipoles of the modelled power
spectrum, as described in Sec. 2, and $P_{l}^{\mathrm{D}}$ are the multipoles
obtained by the data. Due to the limited constraining power of the
hexadecapole, our analysis focuses on the monopole and quadrupole ($l=0,2$)
only777Although the hexadecapole may become important in the case of the ELGs,
where its power is increased due to the integral constraints (for a more
detailed discussion on this, see (de Mattia et al., 2021; Ivanov, 2021)), we
choose to exclude it. In order to model the hexadecapole, we would need to
have an additional free nuisance parameter $c_{r,2}$, which would result in
less overall constraining power of the ELGs. The ELGs are already the least
constraining data set under consideration.. The covariance matrices $C$ for
the different data sets have been estimated either from 1000 realisations of
the EZmocks for each galactic cap (NGC and SGC) (Zhao et al., 2021) in the
case of eBOSS or from 2048 realisations for NGC and SGC of the MultiDark-
Patchy mocks (Kitaura et al., 2016; Rodríguez-Torres et al., 2016) in the case
of BOSS, respectively. To account for the finite set of mocks, we use the
bias-corrected estimator $\tilde{C}$ (Hartlap et al., 2007) of the inverse
covariance matrix $C$ as:

$\tilde{C}^{-1}=\frac{N^{\mathrm{real}}-N^{\mathrm{data}}-2}{N^{\mathrm{real}}-1}C^{-1},$
(24)

where $N^{\mathrm{real}}$ is the total number of realisation of the mocks and
$N^{\mathrm{data}}$ is the number of data points considered in the analysis.

Each sample is treated as independent888 In principle, the eBOSS samples are
correlated and have overlapping volume, however the correlation was found to
be less than $0.1$ (Alam et al., 2021)., and the joint likelihood, denoted as
$\ln{\left.[\mathcal{L}_{\mathrm{joint}}(\theta|\phi_{\mathrm{joint}})\right]}$,
is defined as the sum of the individual likelihoods:
$\ln{\left.[\mathcal{L}_{\mathrm{joint}}(\theta|\phi_{\mathrm{joint}})\right]}=\sum_{i}\ln{\left.[\mathcal{L}(\theta|\phi_{i})\right]}$.
Here, $\theta$ represents the shared cosmological parameters,
$\phi_{\mathrm{joint}}$ is the comprehensive set of nuisance parameters
($\phi_{\mathrm{joint}}=[\phi_{1},\phi_{2},\ldots,\phi_{n}]$), and
$\ln{\left[\mathcal{L}(\theta|\phi_{i})\right]}$ is defined according to Eq.
(23). The parameter space can grow significantly large: for instance,
combining BOSS z1 and z3 for NGC and SGC leads to $4\times 10=40$ nuisance
parameters. Combining BOSS z1 with LRGpCMASS, QSO and ELG further increases
the number of nuisance parameters to 80. When also considering the Planck
likelihoods, the number of nuisance parameters exceeds 100. In order to reduce
the dimension of the parameter sampling space, it is worth noting that many of
the nuisance parameters in the EFTofLSS theory enter in the power spectrum
linearly (and in the likelihood quadratically). This allows for analytical
marginalisation over these parameters. Following the procedure in D’Amico et
al. (2020), we compute the likelihood with free parameters
$\\{b_{1},c_{2},c_{4}\\}$, while marginalising over
$\\{b_{3},c_{ct},c_{r,1},c_{r,2},c_{\epsilon,1},c_{\mathrm{mono}},c_{\mathrm{quad}}\\}$,
effectively reducing the number of nuisance parameters by more than a factor
of two.

To obtain theoretical predictions of the power spectrum multipoles, we combine
CLASS_EDE999https://github.com/mwt5345/class_ede (Hill et al., 2020), an
extension of the publicly available Einstein-Boltzmann code CLASS (Blas et
al., 2011) including early dark energy, with the EFTofLSS code
PyBird101010https://github.com/pierrexyz/pybird (D’Amico et al., 2021b). To
put constraints on cosmological parameters, we perform Markov chain Monte
Carlo (MCMC) analyses, where we sample from the posterior distribution via
Metropolis-Hastings; as implemented in
MontePython-v3.5111111https://github.com/brinckmann/montepython_public
(Brinckmann & Lesgourgues, 2019). As a convergence criterion we define a
Gelman-Rubin (Gelman & Rubin, 1992) value of $R-1<0.01$121212 We examine the
$R-1$ values for each individual sampling parameter using MontePython.. To
determine best fit values we use iMinuit131313https://github.com/scikit-
hep/iminuit (James & Roos, 1975). For post-processing chains, we use
GetDist141414https://github.com/cmbant/getdist (Lewis, 2019).

###  3.6 Parameters and priors

####  3.6.1 Cosmological parameters

Here we discuss the selection of cosmological parameters for our analysis and
their corresponding prior. For runs without Planck data, we impose the
following uniform priors on the $\Lambda$CDM parameters,

$\displaystyle\omega_{cdm}\in\mathcal{U}[0.05,0.25],$ $\displaystyle
h\in\mathcal{U}[0.5,1.0],$
$\displaystyle\ln(10^{10}A_{s})\in\mathcal{U}[1.,4.].$ (25)

Since the constraining power of BOSS and eBOSS data is not sufficient to
tightly constrain the physical baryon density $\omega_{b}$, we impose a BBN
prior on $\omega_{b}$ (as discussed in Sec. 3.4). We fix the primordial
spectral tilt to its Planck best fit value (Aghanim et al., 2020a),

$n_{s}=0.965,$ (26)

in the case of $\Lambda$CDM. For EDE runs, we vary $n_{s}$ according to

$n_{s}\in\mathcal{U}[0.7,1.3].$ (27)

Additionally, on the level of the linear power spectrum, we include two
massless neutrinos and one massive neutrino with $m_{\nu}=0.06~{}\mathrm{eV}$
in our analysis. For runs including Planck data, we impose wide uniform priors
on all the above mentioned parameters. For easier comparison with literature,
we present cosmological results for the $\Lambda$CDM parameters in terms of
the reduced Hubble constant $h$ and two derived parameters, the fractional
matter abundance $\Omega_{m}$ and the clustering amplitude $\sigma_{8}$.

Regarding the EDE parameters, we adopt the following uniform priors,

$\displaystyle\mathrm{f}_{\mathrm{EDE}}\in\mathcal{U}[0.001,1.0],$
$\displaystyle\log_{10}\mathrm{z}_{\mathrm{c}}\in\mathcal{U}[3.1,4.0],$
$\displaystyle\theta_{i}\in\mathcal{U}[0.1,3.1].$ (28)

Throughout our analysis, we keep the index of the EDE potential fixed to
$n=3$, equal to the best fit found in Smith et al. (2020).

####  3.6.2 Priors on nuisance parameters

As discussed previously, the EFTofLSS algorithm incorporates 10 nuisance
parameters (see Eq. (5)). $c_{r,2}$, $c_{\mathrm{mono}}$ and $c_{4}$ are
commonly set to zero (D’Amico et al., 2021b), since the signal-to-noise ratios
of the data under consideration are too low to properly constrain the two
first parameters, while $b_{2}$ and $b_{4}$ are almost completely anti-
correlated implying that $c_{4}\sim 0$. Thus, we restrict our analysis on a
submodel of EFTofLSS with the following nuisance parameters:

$\\{b_{1},c_{2},b_{3},c_{ct},c_{r,1},c_{\epsilon,1},c_{\mathrm{quad}}\\}.$
(29)

The priors for $b_{1}$ and $c_{2}$ are respectively set to be $[0,4]$ and
$[-4,4]$. In order to maintain the perturbative nature of the EFTofLSS, the
nuisance parameters are expected to remain of the order of the linear bias
$b_{1}$ (D’Amico et al., 2020). It is therefore common practice to set tight
zero-centered Gaussian priors on the nuisance parameters which are
marginalised over. Table 2 shows the prior choices for the marginalised
nuisance parameters used in this work, which are in agreement with previous
BOSS and eBOSS analyses with PyBird (D’Amico et al., 2020; D’Amico et al.,
2021b; Simon et al., 2023c, a).

Recent works have examined different prior choices and their impact on the
cosmological parameter constraints (Carrilho et al., 2023; Simon et al.,
2023c; Donald-McCann et al., 2023; Holm et al., 2023). Particularly,
Hadzhiyska et al. (2023) and Donald-McCann et al. (2023) demonstrated that the
use of a (partial) Jeffreys prior (Jeffreys, 1998a) can mitigate prior volume
effects. Prior volume effects are a commonly seen feature in EFTofLSS where
the marginalised posteriors exhibit biases away from their maximum a
posteriori (MAP) estimates due to the marginalisation over nuisance
parameters. This is especially evident in $\Lambda$CDM when we look at the
parameter $\sigma_{8}$, which is degenerate with $b_{1}$ (and other nuisance
parameters). In this work, we therefore decide, besides presenting results
with a classical Gaussian prior choice, to explore the use of a Jeffreys prior
in an extended $\Lambda$CDM context. The Jeffreys prior is notable for its
non-informative nature, refraining from favouring any specific parameter
region a priori. The Jeffreys prior is defined as:

$J(\theta)=\sqrt{|F(\theta)|}\ ,$ (30)

where $F(\theta)$ is the Fisher information matrix. For a Gaussian likelihood
with covariance independent of model parameters $\theta$, this becomes:

$F_{ij}(\theta)=\frac{\partial
M(\theta)}{\partial\theta_{i}}C^{-1}\frac{\partial
M(\theta)}{\partial\theta_{j}}^{T}\ ,$ (31)

where $M(\theta)$ is the model and $C$ is the covariance matrix. Due to the
involvedness of the partial derivatives, we impose the Jeffreys prior only on
the parameters that appear linearly in the model, where the derivatives are
trivial. In this case, any volume effect attributed to the linear parameters
is mitigated, while volume effects caused by the remaining nuisance parameters
($b_{1}$ and $c_{2}$), as well as cosmological parameters, still remain. For
this reason, we present cosmological constraints by quoting the $68\%$
credible interval as well as the best-fit, i.e. the MAP. The best-fit value
is, by definition, not affected by volume effects (D’Amico et al., 2022a;
Simon et al., 2023c) and is therefore a self-diagnostic way to check for
remaining volume effects. For further details on the implementation of the
partial Jeffreys prior in the marginalised likelihood, we refer the reader to
Zhao et al. (2023).

Parameter |  LRG |  ELG |  QSO
---|---|---|---
$b_{3}$ |  $\mathcal{N}(0,2)$ |  $\mathcal{N}(0,2)$ |  $\mathcal{N}(0,2)$
$c_{ct}$ |  $\mathcal{N}(0,2)$ |  $\mathcal{N}(0,2)$ |  $\mathcal{N}(0,2)$
$c_{r,1}$ |  $\mathcal{N}(0,\textbf{8})$ |  $\mathcal{N}(0,\textbf{16})$ |  $\mathcal{N}(0,\textbf{16})$
$c_{\epsilon,1}$ |  $\mathcal{N}(0,2)$ |  $\mathcal{N}(0,2)$ |  $\mathcal{N}(0,2)$
$c_{\mathrm{quad.}}$ |  $\mathcal{N}(0,2)$ |  $\mathcal{N}(0,2)$ |  $\mathcal{N}(0,2)$
Table 2: Priors on the marginalised nuisance parameters of the PyBird EFTofLSS
model. $\mathcal{N}(\mu,\sigma)$ denotes a normal distribution with mean $\mu$
and standard deviation $\sigma$. Except for the counterterm $c_{r,1}$ the
priors are always chosen to be zero centered Gaussians with width 2. The width
of the prior on $c_{r,1}$ ensures that the best-fit value lies well within
1$\sigma$. While we find consistent priors with literature (D’Amico et al.,
2020, 2021a; Simon et al., 2023b, a) for the LRGs and QSOs, the absolute width
of the prior on $c_{r,1}$ is different. This is solely due to a redefinition
of the perturbative scale $k_{R}$ (see footnote 2). For the ELGs, we find the
same width as for QSOs.

##  4 Tests on Mock catalogues

In this section, we present results derived from a set of mock analyses
performed to validate the accuracy of our cosmological inference pipeline. Our
goal is to ensure unbiased parameter constraints for both $\Lambda$CDM and EDE
scenarios. We commence by analysing EZ- and PATCHY mocks within the context of
$\Lambda$CDM. Subsequently, we extend our mock analyses to cosmological models
with various different fractional contributions of EDE. In both cases, we
investigate the impact of different nuisance parameter priors (refer to Sec.
3.6.2) on the cosmological results.

Figure 2: Summary of the 1D marginalised posteriors and best-fit values for
the cosmological parameters of interest resulting from analyses of the
contaminated (pixelated) EZmocks, and PATCHY mocks as described in Sec. 4.1.
Coloured squares show the mean, thick horizontal coloured lines show the width
of the 68% credible interval, thin coloured lines with caps show the 95%
credible interval. The black crosses indicate the MAPs for each corresponding
tracer. The black dashed line corresponds to the fiducial cosmologies of the
EZmocks and PATCHY mocks.

###  4.1 LCDM

We fit the mean of 1000 EZmock realisations (Zhao et al., 2021) and the mean
of 2048 MultiDark PATCHY mock realisations (Kitaura et al., 2016) for each
corresponding tracer and redshift bin. The EZmock algorithm is based on the
Zel’dovich approximation (Zel’dovich, 1970) to model the nonlinear matter
density field and populate the field using an effective bias description. The
fiducial cosmology in this set of mocks is
$\Omega_{m}=0.307115,\,\Omega_{b}=0.048206,\,h=0.6777,\,\sigma_{8}=0.8225,\,n_{s}=0.9611$.
The EZmocks model the systematics and survey geometry in order to match the
redshift evolution and clustering properties of eBOSS DR16 LRG, ELG and QSO
samples. The 1000 realisations are also used to calculate the covariances
matrices used in the official eBOSS analyses (Gil-Marin et al., 2020; Bautista
et al., 2020; Tamone et al., 2020; de Mattia et al., 2021; Raichoor et al.,
2020; Neveux et al., 2020; Hou et al., 2020; Alam et al., 2021) and in this
work (see section 3.5). Two different sets of mocks are produced: the
reference mock (without any observational systematics) and the contaminated
mock (with systematics). The systematics included are redshift failure, fibre
collision, depth dependent radial density and angular systematics. The
contaminated mock adopts a so-called shuffled scheme, where the radial number
density distribution $n(z)$ is estimated from the mock galaxy catalogue. This
introduces a RIC effect which needs to be accounted for (see section 3.3.4,
for information on the integral constraints). For the ELG sample, the main
systematics are of an angular photometric nature (de Mattia et al., 2021). We
therefore work with a third set of mocks called pixelated mocks. In the
pixelated mock the measurements are obtained by rescaling weighted randoms in
HEALPix (Górski et al., 2005) pixels (nside = 64 ($\simeq
0.84~{}\mathrm{deg^{2}}$)) such that the mean density fluctuation in each
pixel is 0. This leads to an additional angular integral constraint effect,
which together with the radial integral constraint (ARIC) biases the
clustering measurements on large scales (see Fig. 1). The MultiDark PATCHY
mock is produced with approximated gravity solvers and analytical-statistical
biasing models. The mock is tuned to a reference catalogue from the high
resolution N-body simulation BigMultiDark (Klypin et al.,
2016)151515https://www.cosmosim.org/, assuming the following fiducial
cosmology:
$\Omega_{m}=0.307115,\,\Omega_{b}=0.048206,\,h=0.6777,\,\sigma_{8}=0.8288,\,n_{s}=0.9611$.
It has been calibrated to match the survey geometry, redshift distribution and
systematics of the BOSS DR12 galaxy data. As the EZmock for the eBOSS data
sets, the 2048 realisations of the MultiDark PATCHY are used to calculate the
covariance matrices for the BOSS data sets. For PATCHY as well as EZmocks
analyses, we enforce the same k-range as for their corresponding tracers (see
Sec. 3.2).

Fig. 2 shows the mean (coloured squares) and the 68% (95%) credible interval
(thick (thin) coloured lines) of the 1D marginalised posteriors on the
cosmological parameters $\Omega_{m},~{}h,~{}\sigma_{8}$ from the analysis of
the above described mock data for all tracers and redshift bins under
consideration. The black dashed line corresponds to the fiducial cosmologies
of the mocks and the best-fit values are indicated with black crosses. We are
comparing two different prior choices on the marginalised nuisance parameters
as described in Section 3.6.2. We observe that depending on the priors
applied, the mean of the marginalised posteriors are more or less shifted away
from the truth of the mocks. Meanwhile, the best fits are in agreement with
the truth. This suggests that these shifts are not due to inaccurate modelling
and rather are a form of volume effect. Depending on the volume in the
nuisance parameter space which is integrated over in the process of
marginalisation, these effects can be more or less severe (see Simon et al.
(2023c) for more details). This becomes obvious when we compare the shifts in
mocks from higher signal-to-noise ratio (SNR) data (BOSS) to lower SNR data
(eBOSS), where the impact of the volume effects worsens due to less
constraining power of the data. As a result, the best-fit values may not fall
within the 68% credible interval in the case of the EZmocks when zero-centered
Gaussian priors are applied. In order to quantify the shift due to the volume
effects, we introduce the following metric commonly used in EFTofLSS analyses
(see e.g. D’Amico et al. (2022a); Simon et al. (2023c); Piga et al. (2023):
($\mu-\mu_{\mathrm{truth}})/\sigma(\mu)$, where we quantify the shift of the
marginalised mean $\mu$ away from the truth ($\mu_{\mathrm{truth}}$) in number
of $\sigma$. In the case where zero-centered Gaussian priors are applied, we
observe that the mean of the marginalised 1D posteriors shift up to
$1.6\sigma$ away from the truth for the EZmocks and up to $0.8\sigma$ for the
PATCHY mocks.

While we acknowledge that the observed shifts in the considered data sets are
moderate, we aim to investigate strategies to alleviate these volume effects.
A promising approach, demonstrated in recent analyses (Donald-McCann et al.,
2023; Zhao et al., 2023), involves the application of a Jeffreys prior on the
nuisance parameters which enter linearly in the theory of EFTofLSS. In Fig. 2,
we illustrate the impact of implementing a Jeffreys prior on the marginalized
nuisance parameters, showcasing its influence on the 1D marginalized posterior
distributions. We can see that for all samples, the agreement with the
fiducial values has increased (where the maximum deviation for the EZmocks is
$0.6\sigma$ and $0.4\sigma$ for the PATCHY mocks). Although the Jeffreys
prior, as the less informative prior, leads to an inflation of the contours
(see Table 3) in comparison to the Gaussian prior, the improvement in
agreement is not just coming from bigger error bars. Rather, we find that the
means of the posteriors shift to their respective truth values and are more
consistently located around their best-fit values. Finding unbiased posteriors
when imposing a Jeffreys prior on the marginalised nuisance parameters, proves
that the observed shifts with the Gaussian priors are due to volume effects
and that our analysis pipeline as described in Sec. 3 is accurate.

We note that in the case of very low SNR as for the ELGs and QSOs, also the
Jeffreys prior fails to perfectly recover unbiased means for cosmological
parameters (especially in $\Omega_{m}$ and $h$). This suggests the presence of
residual volume effects arising from non-trivial degeneracies among nonlinear
nuisance parameters. The reasoning behind this is the following: The Jeffreys
prior facilitates exploration of an expanded parameter space for linear
nuisance parameters compared to Gaussian priors. The size of the expanded
parameter space strongly depends on the constraining power of the data.
Broadening the range that these parameters can explore inevitably results in a
degradation of the constraints on cosmological parameters. In scenarios
characterized by low SNR, this exacerbates the impact of volume effects
arising from nonlinear parameters. Nevertheless, imposing a Jeffreys prior on
the linear nuisance parameters allows for mean values consistent with the
truth within $1\sigma$.

$1-\sigma_{GP}/\sigma_{JP}$
---
Sample |  $\Omega_{m}$ |  $h$ |  $\sigma_{8}$
LRG |  0.49 |  0.32 |  0.31
QSO |  0.55 |  0.39 |  0.34
ELG |  0.68 |  0.36 |  0.51
z1 |  0.44 |  0.32 |  0.33
z3 |  0.40 |  0.18 |  0.30
Table 3: Fractional differences on the constraints of cosmological parameters
between using a Jeffreys prior or Gaussian priors on the marginalised nuisance
parameters in the inference process for different mock data sets. Utilizing
the Jeffreys prior leads to an inflation of the contours for all data sets,
while the degree of inflation depends on the constraining power of the data
set under consideration.

Figure 3: 1D and 2D posterior distributions on the cosmological parameters of
EDE resulting from the analyses of two different eBOSS-like PyMocks (defined
in Sec. 4.2). The upper panel corresponds to mocks with a maximal fractional
contribution of EDE of $\sim 12\%$, while the lower panel corresponds to mocks
with a minimal EDE contribution of $0.1\%$. The two contour levels in the off-
diagonal elements represent $68\%$ (inner) and $95\%$ (outer) credible
intervals and the best-fit values are indicated with crosses. The dashed lines
are the fiducial values of the cosmologies used to produce the mocks. We show
contours for two different prior choices on the marginalised nuisance
parameters in each subplot: Jeffreys prior (violet, dash-dotted line) vs.
zero-centered Gaussian priors (orange, solid line). Left: The introduction of
additional cosmological parameters leads to further projection effects with
both prior choices. The Jeffreys prior allows for stronger projection effects
than the Gaussian priors since the degeneracy axis $(f_{\mathrm{EDE}},h$) is
less constrained. Right: If the eBOSS full shape analysis is combined with
another data set, that has more statistical power or is able to break
parameter degeneracies (here mimicked by a tight prior on $H_{0}$), the
projection effects are reduced in both cases.

Figure 4: Same as Fig. 3 but for EZmock runs, as described in Sec. 4.1. The
different contours represent the two different prior choices on the linear
nuisance parameters of the model: the Jeffreys prior (blue, dash-dotted line)
vs. Gaussian priors (green, solid line). Solutions fully consistent with
$f_{\mathrm{EDE}}=0$ are found, if the mock data is combined with a tight
prior on $H_{0}$ (right side).

###  4.2 EDE

As we have seen in the previous section, it is possible to strongly mitigate
volume effects by utilizing a partial Jeffreys prior in a $\Lambda$CDM set-up.
In this section, we want to test if this is still the case when we extend to
beyond $\Lambda$CDM models where possible new degeneracies between parameters
can arise.

In the case of EDE, we introduce three additional parameters:
$f_{\mathrm{EDE}}$, $\log_{10}z_{c}$ and $\theta_{i}$. We want to highlight
again that imposing a partial Jeffreys prior as we have done in this analysis,
just removes volume effects assigned to parameters which appear linearly in
the likelihood. Therefore any volume effect due to non-linear parameters, as
for example from EDE parameters, will remain. In order to test the severity of
the projection effect coming from the introduction of these new cosmological
parameters in our pipeline, it is useful to have mocks where the cosmology, as
well as the nuisance parameters are known. For this purpose, we produce two
synthetic data vectors with two different $f_{\mathrm{EDE}}$ values using our
pipeline. We consider two limiting cases: 1) EDE entirely resolving the Hubble
tension - an EDE cosmology fixed to the best-fit value of Smith et al. (2020)
(with $n=3$), corresponding to a maximal EDE contribution of $\sim 12\%$ to
the energy density,

$\displaystyle f_{\mathrm{EDE}}=0.122,\log_{10}z_{c}=3.562,\theta_{i}=2.83,$
$\displaystyle\Omega_{m}=0.293836,\Omega_{b}=0.011741,h=0.7219,$
$\displaystyle\sigma_{8}=0.8606,n_{s}=0.9889,$ (32)

and 2) $f_{\mathrm{EDE}}\rightarrow 0$ \- the fiducial EZmock cosmology but
with a fractional contribution of EDE of $0.1\%$ (corresponding to the lower
bound of the $f_{\mathrm{EDE}}$ prior):

$\displaystyle f_{\mathrm{EDE}}=0.001,\log_{10}z_{c}=3.562,\theta_{i}=2.83,$
$\displaystyle\Omega_{m}=0.307115,\Omega_{b}=0.048206,h=0.6777,$
$\displaystyle\sigma_{8}=0.8225,n_{s}=0.9611,$ (33)

where we fixed $\log_{10}z_{c}$ and $\theta_{i}$ to their best-fit values from
Smith et al. (2020) as before. We fit the remaining EFT parameters to the mean
of 1000 comtaminated (or pixelated) eBOSS LRG, QSO and ELG EZmocks for both
NGC and SGC. This is done by finding the MAP estimate for the 7 EFT parameters
we vary, imposing the same uniform priors on $b_{1}$ and $c_{2}$ as discussed
in Sec. 3.6.2 and the Gaussian priors mentioned in Table 2 for the rest of the
nuisance parameters. We include the modelling of the systematics according to
their EZmock counterparts and use covariances rescaled by a factor of 10 to
find best-fit EFT parameters161616We rescale the covariances such that the
synthetic mocks have the same functional form as the simulated mocks on all
scales, as in Donald-McCann et al. (2023).. We refer to the resulting
multipoles as the "PyMocks" for LRG, QSO and ELG.

$(\mu-\mu_{\mathrm{truth}})/\sigma(\mu)$
---
Sample |  Prior | $\Omega_{m}$ | $h$ | $\sigma_{8}$
PyMocks: $f_{\mathrm{EDE}}=0.122$ |  JP | 0.83 | 0.23 | -0.25
GP | -1.51 | 0.13 | -2.23
PyMocks: $f_{\mathrm{EDE}}=0.001$ |  JP | 1.23 | -0.01 | -0.22
GP | \- 0.81 | 0.03 | -2.00
EZmocks |  JP | 1.52 | \- 0.06 | \- 0.23
GP | \- 0.62 | \- 0.02 | -2.07
Table 4: The difference of the marginalised mean away from the truth value of
the mocks in numbers of standard deviations $\sigma$ for the two different
prior choices on the linear nuisance parameters: Jeffreys prior (JP) and zero-
centered Gaussian priors (GP). The results shown are derived from runs where
an additional tight Gaussian prior on $H_{0}$ centered around or close to the
truth of the mocks is imposed. While the Jeffreys prior is able to reduce
projection effects in $\sigma_{8}$, the magnitude of the shift in $\Omega_{m}$
is on a similar level (while its direction is inversed), highlighting that in
the case of EDE additional projection effects appear due to newly introduced
degeneracies between cosmological parameters.

We analyse the combination of the LRG, QSO and ELG PyMocks for the two
different contributions of EDE. We fit the monopole and quadrupole of the NGC
and SGC synthetic galaxy power spectra for the same $k$-range as for the
corresponding data sets, using non-rescaled covariance matrices. We are
sampling 3 + 1 cosmological parameters, as well as 12 nuisance parameters,
fixing $\\{n_{s},\omega_{b},\log_{10}z_{c}$, $\theta_{i}\\}$ to their
corresponding truth values. The results for the two different configurations
of EDE are shown on the left side of Fig. 3. As for the $\Lambda$CDM case, we
consider two different prior choices on the linear nuisance parameters. Except
for $\sigma_{8}$ where the projection effect is mitigated in case of the
Jeffreys prior, the posteriors experience volume effects for both prior
choices. These shifts in the posteriors are present even in the case of the
partial Jeffreys prior, which indicates that the projection effects have their
origin in newly unlocked degeneracies between the cosmological parameters with
EDE parameters. One very obvious, and expected, degeneracy axis is between $h$
and $f_{\mathrm{EDE}}$. Indeed, the main purpose of introducing EDE is to
allow for higher $H_{0}$ values. It is important to notice that the volume
effect in case of Gaussian priors is less severe, since the sampled parameter
space of the linear nuisance parameters is restricted and very extreme
cosmologies (with $f_{\mathrm{EDE}}$ of $30\%$ and more), which are normally
accommodated by large linear nuisance parameters, are excluded171717Due to its
uninformative nature, the Jeffreys prior does not make any assumption of the
underlying theory model and these large nuisance parameters could, in
principle, indicate a breakdown of the model. Recent works (Bragança et al.,
2023; Donald-McCann et al., 2023) investigated additional theoretical priors
to ensure that the perturbative nature of the model is perserved as an
alternative (or extension) to classical Gaussian priors.. Furthermore, the
projection effect in $\Omega_{m}$ which is already slightly present in
$\Lambda$CDM for low SNR data is worsened in EDE, due to the new degeneracy
between $f_{\mathrm{EDE}}$ and $h$. In order to showcase the importance that
the degeneracy axis between $f_{\mathrm{EDE}}$ and $h$ holds, we show on the
right side of Fig. 3 mock runs where we additionally impose a prior on
$H_{0}$. For the PyMocks with $f_{\mathrm{EDE}}=0.122$, we impose a prior
centered on the truth of the mocks and with a SH0ES-like standard deviation:
$H_{0}\in\mathcal{N}(72.19,1.04)$. While for the PyMocks with
$f_{\mathrm{EDE}}=0.001$, the prior is $H_{0}\in\mathcal{N}(67.66,0.42)$ and
is motivated by TT,TE,EE+lowE+lensing+BAO constraints from Planck2018.
Imposing this additional prior clearly reduces the volume effects within $h$
and $f_{\mathrm{EDE}}$ for both cases and improves on the projection effect
visible in $\Omega_{m}$. The strong degeneracy of ($f_{\mathrm{EDE}}$, $h$)
makes it hard to determine the correct best-fit values. Without an additional
prior the best-fit values, especially of $f_{\mathrm{EDE}}$, are shifted away
from the truth up to $0.7\sigma$. Introducing the $H_{0}$ prior and
constraining this degeneracy axis, allows us to recover best-fit values
consistent with the truth (maximal deviation of $0.2\sigma$). As a last point,
we want to mention that only in the cases where this additional $H_{0}$ prior
is applied, the pipeline is able to clearly distinguish between a high ($\sim
12\%$) and a vanishing ($0.1\%$) contribution of EDE. This is true even in the
case of the Gaussian priors, where arguably the volume effects in
$f_{\mathrm{EDE}}$ are less severe.

In order to test the robustness of our EDE pipeline, we show results from mock
data which were produced independently of EFTofLSS. We fit the monopole and
quadrupole of the combined contaminated (pixelated) eBOSS LRG,QSO and ELG
EZmocks for both NGC and SGC, as already described in Sec. 4.1. The EZmocks
assume a flat $\Lambda$CDM cosmology, corresponding to $f_{\mathrm{EDE}}=0$.
The results are shown in Fig. 4 again for two different prior choices and with
(right) and without (left) imposing a tight $H_{0}$ prior. Consistent with the
synthetic mocks, the degeneracy axis between $h$ and $f_{\mathrm{EDE}}$ is
visible on the left side, leading to volume effects and the presence of a peak
in $f_{\mathrm{EDE}}$. Imposing an $H_{0}$ prior, as for the synthetic data,
resolves the volume effects in $h$ and $f_{\mathrm{EDE}}$ and leaves a reduced
projection effect in $\Omega_{m}$. Table 4 quantifies the shift of the mean in
regards to the truth for all EDE mock runs where an additional $H_{0}$ prior
is applied. The deviation in terms of $\sigma$ in $\Omega_{m}$ for all mock
sets (and all prior choices) is between $0.62\sigma$ and $1.51\sigma$,
highlighting the remaining projection effect in $\Omega_{m}$, while the
projection effect in $\sigma_{8}$ is clearly reduced in the Jeffreys prior
case (from $\leq 2.23\sigma$ (GP) to $\leq 0.23\sigma$ (JP)). The best-fit
values are shifted in regards to the truth by a maximum of $0.6\sigma$. While
this is slightly higher than the shift for the synthetic mocks ($0.2\sigma$),
it is unclear if this is due to an inaccuracy in the non-linear modelling of
EFTofLSS or the EZmocks.

We conclude that although the partial Jeffreys prior is able to resolve all
volume effects in $\Lambda$CDM, this is not necessarily true for extended
$\Lambda$CDM scenarios, where additional degeneracies between cosmological
parameters can arise. It is beyond the scope of this work to implement a
Jeffreys prior on the nuisance parameters, as well as on cosmological
parameters and we leave this for future work. We therefore express caution
about the use of a partial Jeffreys prior in an extended $\Lambda$CDM analysis
utilizing EFTofLSS, as long as the data set under consideration has no
significant statistical power or the data set is not combined with external
data sets (mimicked here by an additional tight $H_{0}$ prior), so that the
degeneracies between the parameters affected by volume effects can be broken.
Moving on to EDE data runs, we henceforth will not show EFTofLSS only
constraints and will always show combinations with either Planck or SH0ES
data181818 Since the constraining power on $z_{c}$ and $\theta_{i}$ from
current data sets is weak, we are aware that our analysis will have some
residual volume effects coming from degeneracies between the EDE parameters
which might not be entirely resolved by combining with external data sets
(Smith et al., 2021; Hill et al., 2020; Herold et al., 2022). .

##  5 Constraints on $\Lambda$CDM

Figure 5: Summary of the 1D marginalised posteriors on cosmological parameters resulting from the EFTBOSS and EFTeBOSS analyses, as well as their combined analysis. Coloured squares show the mean, thick horizontal coloured lines show the width of the 68% credible interval, thin coloured lines with caps show the 95% credible interval. In addition to the results from this work, we show results from the PlanckTTTEEE+lowl+lowE+lens analysis and results from Simon et al. (2023a); Zhao et al. (2023); Holm et al. (2023). The grey band, reflecting the $95\%$ credible interval of the Planck results, has been plotted to aid comparison. For Simon et al. (2023a) (triangle,blue) and Zhao et al. (2023) (circle, green), we only indicate the 68% credible intervals. Holm et al. (2023) (diamond, yellow) refers to the profile likelihood results with reconstructed 68% confidence intervals using the Neyman construction.  | BOSS | eBOSS | BOSSz1+eBOSS | Planck
---|---|---|---|---
| GP | JP | GP | JP | GP | JP |
$\Omega_{m}$ | $0.3069$ | $0.3548$ | $0.3240$ | $0.3158$
$0.301_{-0.0095}^{+0.0095}$ | $0.3125_{-0.016}^{+0.016}$ | $0.3043_{-0.015}^{+0.015}$ | $0.3758_{-0.035}^{+0.029}$ | $0.3012_{-0.01}^{+0.0097}$ | $0.3302_{-0.018}^{+0.016}$ | $0.3153\pm 0.0073$
$h$ | $0.6799$ | $0.6625$ | $0.6707$ | $0.6732$
$0.684_{-0.012}^{+0.011}$ | $0.681_{-0.015}^{+0.014}$ | $0.6865_{-0.016}^{+0.016}$ | $0.6609_{-0.02}^{+0.022}$ | $0.6793_{-0.011}^{+0.01}$ | $0.6733_{-0.013}^{+0.013}$ | $0.6736\pm 0.0054$
$\sigma_{8}$ | $0.8062$ | $0.8746$ | $0.8544$ | $0.8120$
$0.746_{-0.047}^{+0.042}$ | $0.7685_{-0.071}^{+0.065}$ | $0.7112_{-0.049}^{+0.044}$ | $0.8254_{-0.067}^{+0.06}$ | $0.735_{-0.041}^{+0.036}$ | $0.7938_{-0.053}^{+0.055}$ | $0.8111\pm 0.0060$
$\omega_{cdm}$ | $0.1185$ | $0.1324$ | $0.1224$ | $0.12011$
$0.1175_{-0.0053}^{+0.005}$ | $0.1217_{-0.011}^{+0.0092}$ | $0.12_{-0.006}^{+0.0063}$ | $0.1405_{-0.011}^{+0.011}$ | $0.1156_{-0.0048}^{+0.0047}$ | $0.1264_{-0.0089}^{+0.0087}$ | $0.1200\pm 0.0012$
$10^{2}\omega_{b}$ | $2.271$ | $2.269$ | $2.272$ | $2.2383$
$2.27_{-0.04}^{+0.036}$ | $2.269_{-0.04}^{+0.035}$ | $2.269_{-0.04}^{+0.036}$ | $2.268_{-0.039}^{+0.037}$ | $2.27_{-0.039}^{+0.037}$ | $2.268_{-0.041}^{+0.036}$ | $2.237\pm 0.015$
$ln(10^{10}A_{s})$ | $3.051$ | $3.088$ | $3.133$ | $3.0448$
$2.903_{-0.13}^{+0.13}$ | $2.919_{-0.18}^{+0.17}$ | $2.779_{-0.15}^{+0.15}$ | $2.903_{-0.17}^{+0.16}$ | $2.898_{-0.12}^{+0.11}$ | $2.943_{-0.14}^{+0.14}$ | $3.044\pm 0.014$
$N_{\mathrm{data}}$ | 153 | 213 | 289 | 2352
$N_{\mathrm{param}}$ | 4 + 8 | 4 + 12 | 4 + 16 | 6 + 21
$\chi^{2}$ | $110$ | $158$ | $219$ | 2775
Table 5:  Best-fit and mean $\pm$ $1\sigma$ bounds for the cosmological
parameters of $\Lambda$CDM measured from the full shape analysis of BOSS,
eBOSS and the combined BOSSz1+eBOSS data (always in combination with a BBN
prior). We also present parameter constraints from the
PlanckTTTEEE+lowl+lowE+lens analysis. We present the mean and $1\sigma$ bounds
of the full shape analyses for the two different prior choices on the linear
nuisance parameters: Jeffreys prior (JP) and zero-centered Gaussian priors
(GP). $N_{\mathrm{data}}$ indicates the number of fitted data points for each
data set combination. We consider $N_{\mathrm{BOSSz1}}=N_{\mathrm{BOSSz3}}=76$
(Alam et al., 2017), $N_{\mathrm{eBOSS}}=212$ (Alam et al., 2021),
$N_{\mathrm{BBN}}=1$ (Schöneberg et al., 2019) and $N_{\mathrm{Planck}}=2352$
(Aghanim et al., 2020a, b). $N_{\mathrm{param}}$ is the combination of the
number of varied cosmological and nuisance parameters. The $\chi^{2}$ values
are calculated at the best-fit points.
$N_{\sigma}=|\mu_{Planck}-\mu_{EFT}|/\sqrt{\sigma_{Planck}^{2}+\sigma_{EFT}^{2}}$
---
Sample |  Prior |  $\Omega_{m}$ |  $h$ |  $\sigma_{8}$
BOSS |  JP |  0.16 |  0.46 |  0.65
GP |  1.19 |  0.79 |  1.53
eBOSS |  JP |  1.69 |  0.56 |  0.21
GP |  0.66 |  0.76 |  2.25
BOSSz1+eBOSS |  JP |  0.77 |  0.02 |  0.31
GP |  1.16 |  0.47 |  2.09
Table 6: The average level of agreement between the 1D marginalised posteriors resulting from the full shape analyses of BOSS, eBOSS, and the combined BOSSz1+eBOSS and the Planck 2018 results. For each data set, we show results obtained with two prior choices on the linear nuisance parameters of the full shape analyses: Jeffreys prior (JP) and zero-centered Gaussian priors (GP). | BOSS + Baseline | BOSSz1 + eBOSS + Baseline | BOSSz1 + eBOSS + SH0ES + Baseline | Baseline
---|---|---|---|---
| GP | JP | GP | JP | GP | JP |
$\Omega_{m}$ | $0.3144$ | $0.3137$ | $0.3061$ | $0.3219$
$0.3137_{-0.0058}^{+0.006}$ | $0.3169_{-0.0062}^{+0.0061}$ | $0.3139_{-0.0058}^{+0.0058}$ | $0.3202_{-0.0061}^{+0.006}$ | $0.3_{-0.0052}^{+0.0051}$ | $0.3039_{-0.0055}^{+0.0052}$ | $0.3184_{-0.0067}^{+0.0067}$
$h$ | $0.6745$ | $0.6741$ | $0.6810$ | $0.6689$
$0.6749_{-0.0046}^{+0.0041}$ | $0.6726_{-0.0045}^{+0.0044}$ | $0.6747_{-0.0044}^{+0.0041}$ | $0.6702_{-0.0044}^{+0.0043}$ | $0.6855_{-0.004}^{+0.004}$ | $0.6825_{-0.0041}^{+0.0041}$ | $0.6715_{-0.0049}^{+0.0048}$
$\sigma_{8}$ | $0.8095$ | $0.8064$ | $0.8099$ | $0.8094$
$0.8104_{-0.006}^{+0.0062}$ | $0.8116_{-0.0059}^{+0.0059}$ | $0.8097_{-0.0058}^{+0.0058}$ | $0.8125_{-0.0058}^{+0.006}$ | $0.8074_{-0.0064}^{+0.006}$ | $0.8095_{-0.0063}^{+0.0061}$ | $0.8124_{-0.0059}^{+0.0061}$
$\omega_{cdm}$ | $0.12$ | $0.1196$ | $0.1187$ | $0.1211$
$0.1198_{-0.00095}^{+0.00099}$ | $0.1203_{-0.00098}^{+0.001}$ | $0.1198_{-0.00092}^{+0.00099}$ | $0.1208_{-0.001}^{+0.00094}$ | $0.1177_{-0.00089}^{+0.00087}$ | $0.1183_{-0.0009}^{+0.0009}$ | $0.1206_{-0.0011}^{+0.0011}$
$10^{2}\omega_{b}$ | $2.241$ | $2.231$ | $2.257$ | $2.229$
$2.24_{-0.013}^{+0.013}$ | $2.236_{-0.014}^{+0.013}$ | $2.24_{-0.014}^{+0.013}$ | $2.232_{-0.014}^{+0.014}$ | $2.262_{-0.014}^{+0.013}$ | $2.257_{-0.013}^{+0.014}$ | $2.234_{-0.014}^{+0.014}$
$ln(10^{10}A_{s})$ | $3.040$ | $3.033$ | $3.050$ | $3.032$
$3.044_{-0.015}^{+0.014}$ | $3.043_{-0.014}^{+0.014}$ | $3.042_{-0.014}^{+0.014}$ | $3.042_{-0.014}^{+0.014}$ | $3.052_{-0.016}^{+0.014}$ | $3.053_{-0.016}^{+0.014}$ | $3.043_{-0.015}^{+0.014}$
$n_{s}$ | $0.9663$ | $0.9684$ | $0.9697$ | $0.9635$
$0.9659_{-0.0039}^{+0.0037}$ | $0.9648_{-0.0039}^{+0.0038}$ | $0.9657_{-0.0036}^{+0.0038}$ | $0.9637_{-0.0039}^{+0.0038}$ | $0.9714_{-0.0038}^{+0.0037}$ | $0.9699_{-0.0037}^{+0.0036}$ | $0.9644_{-0.004}^{+0.004}$
$\tau_{\mathrm{reio}}$ | $0.0516$ | $0.0487$ | $0.0525$ | $0.04675$
$0.05405_{-0.0072}^{+0.0073}$ | $0.05321_{-0.0073}^{+0.0071}$ | $0.05315_{-0.0068}^{+0.0071}$ | $0.05192_{-0.0069}^{+0.007}$ | $0.06005_{-0.008}^{+0.007}$ | $0.05931_{-0.0081}^{+0.0069}$ | $0.05289_{-0.0074}^{+0.0072}$
$N_{\mathrm{data}}$ | 4086 | 4222 | 4090 | 3934
$N_{\mathrm{param}}$ | 6 + 30 | 6 + 38 | 6 + 38 | 6 + 22
$\chi^{2}$ | 4298 | 4409 | 4326 | 4187
AIC | 4370 | 4497 | 4414 | 4243
Table 7:  Best-fit and mean $\pm$ $1\sigma$ bounds for the cosmological
parameters of $\Lambda$CDM measured from the combined full shape analysis of
BOSS and BOSSz1+eBOSS with the baseline, as well as with the full
PantheonPlusSH0ES likelihood. The cosmological results from the baseline come
from the PlanckTTTEEE+lowl+lowE+lens+ext.BAO+PantheonPlus analysis. We present
the mean and $1\sigma$ bounds of the full shape analyses for the two different
prior choices on the linear nuisance parameters: Jeffreys prior (JP) and zero-
centered Gaussian priors (GP). $N_{\mathrm{data}}$ indicates the number of
fitted data points for each data set combination. We consider
$N_{\mathrm{BOSSz1}}=N_{\mathrm{BOSSz3}}=76$ (Alam et al., 2017),
$N_{\mathrm{eBOSS}}=212$ (Alam et al., 2021), $N_{\mathrm{BAO}}=2$ (Beutler et
al., 2011; Ross et al., 2015), $N_{\mathrm{PantheonPlus}}=1580$ (Brout et al.,
2022), $N_{\mathrm{PantheonPlusSH0ES}}=1448$ (Brout et al., 2022; Riess et
al., 2022a) and $N_{\mathrm{Planck}}=2352$ (Aghanim et al., 2020a, b).
$N_{\mathrm{param}}$ is the combination of the number of varied cosmological
and nuisance parameters. The corresponding Akaike Information Criterion (AIC)
and $\chi^{2}$ values are calculated at the best-fit points.  Figure 6: 1D and
2D posterior distributions on the cosmological parameters of the combined full
shape analysis of BOSSz1+eBOSS in combination with the baseline. The baseline
corresponds to PlanckTTTEEE+lowl+lowE+ext.BAO+PantheonPlus (green, solid
line). The two contour levels in the off-diagonal elements represent $68\%$
(inner) and $95\%$ (outer) credible intervals and the best-fit values are
indicated with corresponding markers. We compare two different prior choices
of the BOSSz1+eBOSS analysis: Jeffreys prior (JP) (violet, dash-dotted line)
and zero-centered Gaussian priors (GP) (orange, dashed line).

In this section, we present cosmological constraints for the $\Lambda$CDM
model coming from the full shape analysis of the public eBOSS data. Fig. 5 and
Table 5 summarizes our main findings. For the EFT only analysis of eBOSS and
BOSS, we fix $n_{s}$ to its Planck best-fit value and impose a BBN prior on
$\omega_{b}$, as described in Sec. 3.6.1. Furthermore, we perform combined
analyses of full shape with other LSS surveys such as the full shape of BOSS,
external BAO data, PantheonPlus, as well as with Planck data (see Sec. 3.2
$\&$ 3.4 for details). The results of the combined analysis are shown in Fig.
6 and Table 7.

###  5.1 Full Shape analysis of eBOSS and BOSS

We start by presenting full shape data constraints only. We show 1D
marginalized constraints on cosmological parameters for the full shape
analysis of BOSS and eBOSS individually, as well as the combined BOSSz1+eBOSS
analysis in Fig 5. The coloured squares correspond to the mean of the
posteriors and the thick (thin) coloured lines indicate the 68% (95%) credible
intervals. For comparison, we also show results from the CMB
PlanckTTTEE+lowl+lowE+lens analysis in black. As for the mock runs, we show
full shape results with two different priors on the linear nuisance
parameters: the Jeffreys prior in violet and Gaussian priors in orange. Table
5 summarises our findings.

We quantify the level of agreement between different configurations and
different data sets by the number of $\sigma$ between the peak values of two
marginalised distributions:

$N_{\sigma}=\frac{|\mu_{1}-\mu_{2}|}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}},$
(34)

where $\mu_{i}$ and $\sigma_{i}$ are the mean and $1\sigma$ errors calculated
from the 1D marginalised posteriors. When the distributions are asymmetric, we
use the $1\sigma$ errors between the corresponding peaks. Comparing the level
of agreement between Planck and the full shape analyses, as presented in Table
6, we find that the analyses assuming a Jeffreys prior show an overall better
agreement with Planck than the analyses with Gaussian priors. The agreement
for $\sigma_{8}$ with Planck is especially improved for all three data set
combinations if a Jeffreys prior is applied, resulting in no notable
disagreement between Planck and large-scale structure data. This suggests that
certain minor discrepancies that have been seen between previous EFTofLSS
analysis and Planck could be due to different prior choices of EFTofLSS
nuisance parameters. For a more thorough study of different analysis choices
of EFTofLSS and their influence on parameter constraints, we refer the reader
to Donald-McCann et al. (2023). The improved agreement with Planck in the case
where a Jeffreys prior is assumed is in general two fold. On one hand, the
peak of the posterior shifts towards the Planck values. On the other hand, the
widths of the 68% confidence intervals are enlarged. For BOSS, this broadening
can be as large as $\sim 70\%$, while the agreement for all parameters is
improved by at least a factor of $\gtrsim 1.7$. The only case where the
Jeffreys prior decreases the agreement with Planck compared to the Gaussian
priors is in the parameter $\Omega_{m}$ for the eBOSS full shape analysis,
where the posterior of $\Omega_{m}$ is shifted to high values. As already
discussed in Sec. 4.1, this is most likely due to residual volume effects
coming from degeneracies between EFT parameters that enter non-linearly in the
one-loop power spectrum, which can become important in the case of low SNR
data. From eBOSS alone, we constrain $\Omega_{m}$ to a precision of $5\%$
(9%), $h$ to 2% (3%) and $\sigma_{8}$ to 7% (8%) at $1\sigma$ level for the
runs with Gaussian priors (Jeffreys prior). For the constraints from the
combined BOSSz1+eBOSS data, we reconstruct $\Omega_{m}$, $h$, and $\sigma_{8}$
to 3% (5%), 2% (2%), and 6% (7%) precision at 68% credible interval.

We compare our constraints on $\Lambda$CDM from the full shape analyses of
BOSS, eBOSS and their combination with constraints from previous EFTofLSS
analyses, e.g. Simon et al. (2023a); Zhao et al. (2023); Holm et al. (2023).
The results of these comparisons are shown in Fig 5, together with the
comparison to Planck, and should be understood with the caveat that there are
some differences in the data sets included, modelling choices and priors on
the cosmological parameters $n_{s}$ and $\omega_{b}$. Simon et al. (2023a)
derives full shape constraints using BOSS multipoles, eBOSS QSO multipoles and
the combination of the two. The assumption of the priors on the linear
nuisance parameters corresponds to the Gaussian priors used in this work. Holm
et al. (2023) performs a profile likelihood analysis of EFTofLSS applied to
the BOSS and eBOSS QSO data set. Since the Jeffreys prior has the effect of at
least partially cancelling the contribution of the Laplace term in the full
shape likelihoods, we expect comparable results between these two analyses.
The analysis of Holm et al. (2023) further differs in that the spectral index
$n_{s}$ is a free parameter, while it is fixed to its Planck best-fit value in
our analysis and in Simon et al. (2023a). Zhao et al. (2023) performs a full
shape analysis of the eBOSS LRG data assuming a Jeffreys prior. The analysis
of Zhao et al. (2023) also differs in that the hexadecapole $P_{4}(k)$ is
included in addition to the monopole and quadrupole. In order to fit
$P_{4}(k)$ an additional nuisance parameter $c_{r,2}$ is varied. Each of the
three full shape analyses presented in Fig. 5 can be approximated by one of
the above described EFTofLSS literature works in the sense that similar data
combinations and analysis assumptions were made. It is important to notice
that this is the first work where all three different tracers (LRG, QSO and
ELG) of the eBOSS survey are analysed simultaneously within EFTofLSS, while
Simon et al. (2023a); Zhao et al. (2023); Holm et al. (2023) use sub-sets of
these.

We note the overall good agreement between the full shape analyses presented
in this work and their respective literature results. The full shape analyses
of BOSS and the combination of BOSS and eBOSS agree with the literature within
$\lesssim 1\sigma$ under the assumption of zero-centered Gaussian priors;
similar agreement is seen comparing the profile likelihood analysis with the
Jeffreys prior analysis. The results of the eBOSS analysis are slightly less
in agreement, mainly because the most constraining eBOSS data set (LRG) was
not taken into account in Simon et al. (2023a); Holm et al. (2023). For a more
thorough comparison with Holm et al. (2023) it would be interesting to see how
the consistency level behaves if $n_{s}$ is freed in an eBOSS QSO analysis
where the Jeffreys prior is imposed. For the results of the eBOSS analysis,
assuming a Jeffreys prior, we find good agreement with the eBOSS LRG analysis
of Zhao et al. (2023), with results within $\lesssim 1\sigma$.

###  5.2 Combination with external Data Sets

In a last step, we present EFTofLSS analysis of eBOSS and BOSS data in
combination with Planck, external BAO data and PantheonPlus in Fig. 6 and
table 7. We show results for the analysis assuming a Jeffreys prior, as well
as with classical zero-centered Gaussian priors on the linear nuisance
parameters and indicate the best fits. Including BOSS and eBOSS full shape
analysis improves constraints coming from Planck+BAO+PantheonPlus on
$\Omega_{m}$, $h$, $\sigma_{8}$ by 14% (10%), 16% (9%) and 5% (2%) for the
analysis imposing Gaussian priors (Jeffreys prior).

##  6 Constraints on EDE

| BOSS + Baseline | BOSSz1 + eBOSS + Baseline | BOSSz1 + eBOSS + SH0ES + Baseline | Baseline
---|---|---|---|---
| GP | JP | GP | JP | GP | JP |
$\Omega_{m}$ | $0.3071$ | $0.3156$ | $0.3038$ | $0.3193$
$0.3126_{-0.0059}^{+0.0059}$ | $0.3156_{-0.006}^{+0.0066}$ | $0.3129_{-0.0059}^{+0.006}$ | $0.3185_{-0.0062}^{+0.0061}$ | $0.3005_{-0.0053}^{+0.0051}$ | $0.3061_{-0.0058}^{+0.0054}$ | $0.3172_{-0.0069}^{+0.0068}$
$h$ | $0.6857$ | $0.6759$ | $0.7319$ | $0.6817$
$0.6812_{-0.008}^{+0.0053}$ | $0.6794_{-0.0085}^{+0.0055}$ | $0.6803_{-0.0078}^{+0.0052}$ | $0.6796_{-0.011}^{+0.0061}$ | $0.7173_{-0.0086}^{+0.0082}$ | $0.7203_{-0.0087}^{+0.0082}$ | $0.6789_{-0.0095}^{+0.0062}$
$\sigma_{8}$ | $0.8161$ | $0.8207$ | $0.8528$ | $0.8220$
$0.8146_{-0.0082}^{+0.007}$ | $0.8165_{-0.0089}^{+0.0072}$ | $0.8136_{-0.0082}^{+0.0066}$ | $0.8199_{-0.011}^{+0.0075}$ | $0.8379_{-0.0098}^{+0.0094}$ | $0.8475_{-0.0094}^{+0.009}$ | $0.8182_{-0.0097}^{+0.007}$
$f_{\mathrm{EDE}}$ | $0.0256$ | $0.0087$ | $0.1702$ | $0.0415$
$<0.0575$ | $<0.0614$ | $<0.0584$ | $<0.0752$ | $0.1179_{-0.022}^{+0.025}$ | $0.1399_{-0.022}^{+0.023}$ | $<0.0635$
$\log_{10}z_{c}$ | $3.6232$ | $3.4968$ | $3.5550$ | $3.6027$
unconstr. | $>3.27$ | unconstr. | $>3.30$ | $3.621_{-0.11}^{+0.017}$ | $3.575_{-0.038}^{+0.031}$ | unconstr.
$\theta_{i}$ | $2.5916$ | $2.9075$ | $2.7653$ | $2.5876$
unconstr. | unconstr. | unconstr. | unconstr. | $2.747_{-0.11}^{+0.12}$ | $2.729_{-0.062}^{+0.089}$ | unconstr.
$\omega_{cdm}$ | $0.1212$ | $0.1211$ | $0.1394$ | $0.1252$
$0.1218_{-0.0024}^{+0.0012}$ | $0.1225_{-0.0027}^{+0.0014}$ | $0.1217_{-0.0023}^{+0.0011}$ | $0.124_{-0.0036}^{+0.0016}$ | $0.1311_{-0.0035}^{+0.0031}$ | $0.1354_{-0.0035}^{+0.0033}$ | $0.123_{-0.0029}^{+0.0016}$
$10^{2}\omega_{b}$ | $2.2522$ | $2.2420$ | $2.2669$ | $2.2529$
$2.25_{-0.019}^{+0.017}$ | $2.246_{-0.019}^{+0.017}$ | $2.249_{-0.018}^{+0.016}$ | $2.246_{-0.023}^{+0.016}$ | $2.286_{-0.022}^{+0.022}$ | $2.276_{-0.02}^{+0.021}$ | $2.246_{-0.021}^{+0.016}$
$ln(10^{10}A_{s})$ | $3.0535$ | $3.0615$ | $3.0679$ | $3.0463$
$3.047_{-0.015}^{+0.014}$ | $3.046_{-0.015}^{+0.014}$ | $3.045_{-0.015}^{+0.014}$ | $3.047_{-0.015}^{+0.015}$ | $3.066_{-0.015}^{+0.015}$ | $3.069_{-0.014}^{+0.014}$ | $3.047_{-0.016}^{+0.014}$
$n_{s}$ | $0.9735$ | $0.9676$ | $0.9926$ | $0.9708$
$0.9689_{-0.006}^{+0.0047}$ | $0.9685_{-0.0063}^{+0.0046}$ | $0.9684_{-0.0059}^{+0.0045}$ | $0.9686_{-0.0074}^{+0.0049}$ | $0.9894_{-0.0064}^{+0.0059}$ | $0.9903_{-0.0061}^{+0.006}$ | $0.9685_{-0.0067}^{+0.0049}$
$\tau_{\mathrm{reio}}$ | $0.0580$ | $0.0588$ | $0.0497$ | $0.0504$
$0.05414_{-0.0075}^{+0.007}$ | $0.05308_{-0.0073}^{+0.0071}$ | $0.05351_{-0.007}^{+0.0071}$ | $0.05175_{-0.007}^{+0.0073}$ | $0.05649_{-0.0075}^{+0.0074}$ | $0.05405_{-0.0072}^{+0.0069}$ | $0.05305_{-0.0076}^{+0.0071}$
$N_{\mathrm{data}}$ | 4086 | 4222 | 4090 | 3934
$N_{\mathrm{param}}$ | 9 + 30 | 9 + 38 | 9 + 38 | 9 + 22
$\chi^{2}$ | 4297 | 4407 | 4291 | 4185
AIC | 4375 | 4501 | 4385 | 4247
Table 8: Best-fit and mean $\pm$ $1\sigma$ bounds for the cosmological
parameters of EDE measured from the combined full shape analysis of BOSS and
BOSSz1+eBOSS with the baseline, as well as with the full PantheonPlusSH0ES
likelihood. The cosmological results from the baseline come from the
PlanckTTTEEE+lowl+lowE+lens+ext.BAO+PantheonPlus analysis. We present the mean
and $1\sigma$ bounds of the full shape analyses for the two different prior
choices on the linear nuisance parameters: Jeffreys prior (JP) and zero-
centered Gaussian priors (GP). $N_{\mathrm{data}}$ indicates the number of
fitted data points for each data set combination. We consider
$N_{\mathrm{BOSSz1}}=N_{\mathrm{BOSSz3}}=76$ (Alam et al., 2017),
$N_{\mathrm{eBOSS}}=212$ (Alam et al., 2021), $N_{\mathrm{BAO}}=2$ (Beutler et
al., 2011; Ross et al., 2015), $N_{\mathrm{PantheonPlus}}=1580$ (Brout et al.,
2022), $N_{\mathrm{PantheonPlusSH0ES}}=1448$ (Brout et al., 2022; Riess et
al., 2022a) and $N_{\mathrm{Planck}}=2352$ (Aghanim et al., 2020a, b).
$N_{\mathrm{param}}$ is the combination of the number of varied cosmological
and nuisance parameters. The corresponding Akaike Information Criterion (AIC)
and $\chi^{2}$ values are calculated at the best-fit points.

We present constraints on cosmological parameters within EDE through a full
shape analysis of eBOSS and BOSS data, incorporating various external data
sets. We begin with constraints from LSS data combined with Planck, probing
whether EDE can yield high values of $H_{0}$ without relying on information
from the local distance ladder. Subsequently, we integrate SH0ES data and
assess their impact on EDE constraints. The primary outcomes of this study are
summarized in Fig.7 and detailed in Table 8. To gauge the potential of EDE in
addressing the $H_{0}$ tension, we evaluate its effectiveness after the
inclusion of eBOSS data and examine its preference over $\Lambda$CDM in Table
9. Looking into the near future, we present the potential of Planck-free
constraints on EDE from eBOSS and BOSS in Fig. 8 and Table 10. While we
acknowledge the presence of projection effects at the current stage, analyses
of Stage-4 LSS survey have the potential to strongly mitigate these effects.

###  6.1 Combination with external Data Sets

First, we investigate whether large-scale structure data in combination with
Planck supports an EDE energy fraction capable of entirely resolving the
Hubble tension. In Table 8 we summarise our results of a joint analysis of the
full shape likelihood of BOSS and eBOSS data with Planck, external BAO and
PantheonPlus data. We refer to the combination of
PlanckTTTEEE+lens+ext.BAO+PantheonPlus data as our baseline data set. Fig. 7
presents the marginalised posteriors on
$\\{h,\Omega_{m},\sigma_{8},f_{\mathrm{EDE}}\\}$ as well the corresponding
best-fit values and $\chi^{2}$ statistics. The contours of all parameters can
be found in Fig. 9 in Appendix A. As in the previous section, we consider both
a Gaussian prior and a Jeffreys prior in our analysis (Section 3.6.2 for
details on the different prior choices). Assuming Gaussian priors we find that
while EDE predicts a higher $H_{0}$ value than $\Lambda$CDM, the increase is
small. The combination of baseline+BOSSz1+eBOSS gives an upper bound on
$f_{\mathrm{EDE}}<0.0584$ (95%CL), while the baseline+BOSS leads to
$f_{\mathrm{EDE}}<0.0575$. Both these analyses lead to tighter upper bounds on
$f_{\mathrm{EDE}}$ compared to the baseline ($f_{\mathrm{EDE}}<0.0635$),
indicating the importance of including the full shape analysis of large-scale
structure data in the EDE discussion. Our results indicate that the analysis
of large-scale structure data in combination with Planck does not favour an
EDE energy fraction able to entirely resolve the Hubble tension. These
findings agree with recent literature (Hill et al., 2020; Ivanov et al.,
2020a; D’Amico et al., 2021c; Simon et al., 2023b; Simon, 2023).

Figure 7: 1D and 2D posterior distributions on the cosmological parameters of
the baseline (PlanckTTTEEE+lens+ext.BAO+PantheonPlus) (dot-dashed),
baseline+EFTBOSS (dotted), baseline+EFTBOSS+EFTeBOSS (dashed) and
baseline+EFTBOSS+EFTeBOSS+SH0ES (solid), where we included the full likelihood
of PantheonPlusSH0ES. The two contour levels in the off-diagonal elements
represent $68\%$ (inner) and $95\%$ (outer) credible intervals and the best-
fit values are indicated with corresponding markers. The grey bands correspond
to the $1\sigma$ and $2\sigma$ bounds coming from the latest SH0ES analysis
(Riess et al., 2022a). Left: zero-centerd Gaussian priors on the linear
nuisance parameters. Right: Jeffreys prior imposed on the linear nuisance
parameters of the EFTofLSS analyses.

Next, we examine the impact of adopting a Jeffreys prior instead of Gaussian
priors on our results. While Jeffreys priors have been effective in mitigating
projection effects that pose challenges to the robustness of the EFTofLSS
analysis within the framework of LCDM, the scenario is more nuanced when
considering the EDE model. As was pointed out in various previous works
(Schöneberg et al., 2022; Smith et al., 2021; Murgia et al., 2021; Herold et
al., 2022), posteriors in EDE analyses are highly non-Gaussian and volume
effects can appear upon marginalisation. In the limit of
$f_{\mathrm{EDE}}\rightarrow 0$, the model becomes equivalent to $\Lambda$CDM
but with two additional redundant parameters ($z_{c}$ and $\theta_{i}$). This
can lead to an artificial preference for $f_{\mathrm{EDE}}=0$ in the
marginalised posteriors. The previous results should therefore be interpreted
with care. Recent literature (Herold et al., 2022; Herold & Ferreira, 2023;
Reeves et al., 2023; Cruz et al., 2023) complements Bayesian analyses of (new)
EDE with a profile likelihood approach, which are free of volume projection
effects. However, profile likelihoods introduce increased numerical
complexity, particularly with a large number of parameters.

When employing the Jeffreys prior in the context of EDE, we can effectively
mitigate projection effects arising from the linear nuisance parameters of the
EFTofLSS. However, it is important to note that projection effects stemming
from newly introduced EDE parameters persist. Nevertheless, as detailed in
Section 4.2, the joint application of full shape analyses on both BOSS and
eBOSS data sets, assuming a Jeffreys prior, is anticipated to result in fewer
volume projection effects compared to analyses employing Gaussian priors, as
long as we include external data sets possessing sufficient constraining power
to address degeneracies among EDE parameters. The 2D posteriors coming from
the full shape analysis of BOSS and eBOSS with a Jeffreys prior in combination
with the baseline are shown on the right of Fig. 7. As for the analysis with
the Gaussian priors, the full contours can be found in in Fig. 10 in Appendix
A. The corresponding reconstructed posteriors and best fit values are given in
Table 8. Employing a Jeffreys prior, we find an upper bound on
$f_{\mathrm{EDE}}<0.0614$ (95% CL) when baseline+BOSS is analysed and
$f_{\mathrm{EDE}}<0.0752$ (95% CL) when baseline+BOSSz1+eBOSS is analysed. In
both cases, we infer consistently higher upper bounds on $f_{\mathrm{EDE}}$ as
for analyses with Gaussian priors.

|  $\Lambda$CDM | EDE
---|---|---
Planck high-l TTTEEE |  - |  2347.6 |  2346.8 |  2349.4 |  2348.0 |  - |  2345.6 |  2346.1 |  2346.1 |  2352.3
Planck low-l EE |  - |  395.8 |  395.8 |  395.8 |  397.2 |  - |  395.7 |  396.8 |  397.2 |  395.9
Planck low-l TT |  - |  23.5 |  23.1 |  22.5 |  22.7 |  - |  22.8 |  22.2 |  23.3 |  21.3
Planck lensing |  - |  9.0 |  8.9 |  9.2 |  8.9 |  - |  9.6 |  9.0 |  9.5 |  10.6
BAO small-z |  - |  0.8 |  1.0 |  1.1 |  1.5 |  - |  0.8 |  1.4 |  1.0 |  1.6
BOSS z1 |  60.4 |  - |  59.0 |  59.2 |  59.3 |  59.4 |  - |  59.4 |  59.1 |  60.4
BOSS z3 |  - |  - |  52.8 |  - |  - |  - |  - |  50.3 |  - |  -
eBOSS LRG |  54.8 |  - |  - |  54.8 |  55.3 |  53.9 |  - |  - |  54.7 |  54.3
eBOSS QSO |  46.4 |  - |  - |  41.0 |  41.2 |  40.5 |  - |  - |  40.6 |  41.1
eBOSS ELG |  64.6 |  - |  - |  64.8 |  65.9 |  64\. 0 |  - |  - |  64.8 |  64.4
Pantheon+ |  - |  1410.1 |  1410.9 |  1410.9 |  - |  - |  1410.4 |  1411.9 |  1410.7 |  -
PantheonPlusSH0ES |  - |  - |  - |  - |  1326.1 |  |  - |  - |  - |  1289.2
SH0ES |  3.3 |  - |  - |  - |  - |  0.1 |  - |  - |  - |  -
$\omega_{b}$ |  0.5 |  - |  - |  - |  - |  0.2 |  - |  - |  - |  -
$\chi^{2}_{\mathrm{EDE}}-\chi^{2}_{\Lambda\mathrm{CDM}}$ |  |  |  |  |  |  -12 |  -2 |  -1 |  -2 |  -35
$\mathrm{AIC}_{\mathrm{EDE}}-\mathrm{AIC}_{\Lambda\mathrm{CDM}}$ |  |  |  |  |  |  -6 |  +4 |  +5 |  +4 |  -29
Table 9: Table of best-fit $\chi^{2}$ contributions from individual
experiments for $\Lambda$CDM and EDE when fitted to different data set
combinations: BOSSz1+eBOSS+SH0ES, Baseline+BOSS, Baseline+BOSSz1+eBOSS,
Baseline+BOSSz1+eBOSS+SH0ES. The individual total $\chi^{2}$ can be found in
Table 7 for $\Lambda$CDM constrained by baseline+EFT, Table 8 for EDE
constrained by baseline+EFT and Table 10 for EFT+SH0ES constraints on
$\Lambda$CDM and EDE. We report the difference in $\chi^{2}$ between EDE and
$\Lambda$CDM, as well as the difference in AIC.

With the inclusion of SH0ES data, we reconstruct
$f_{\mathrm{EDE}}=0.1179_{-0.022}^{+0.025}$ with
$h=0.7173_{-0.0086}^{+0.0082}$ km/s/Mpc and
$f_{\mathrm{EDE}}=0.1399_{-0.022}^{+0.023}$ with
$h=0.7203_{-0.0087}^{+0.0082}$ km/s/Mpc for analyses with the assumption of
Gaussian prior and Jeffreys prior, respectively. Both correspond to a more
than $5\sigma$ detection of a non-zero $f_{\mathrm{EDE}}$. The consistency of
$h$ with the SH0ES constraint $h=0.7304\pm 0.0104$ km/s/Mpc (Riess et al.,
2022a) is at the $1\sigma$ level for the analysis with Gaussian priors and the
$0.8\sigma$ level for the analysis with the Jeffreys prior. In comparison, for
$\Lambda$CDM, the consistency is much poorer ($4.1\sigma$ and $4.3\sigma$,
respectively). In line with previous work, the inclusion of SH0ES data not
only results in a higher value of $h$ and, consequently, $f_{\mathrm{EDE}}$,
but also induces a shift in the spectral tilt $n_{s}$ within EDE relative to
$\Lambda$CDM. For both choices of priors, we observe a value within $2\sigma$
of scale-invariant value of $n_{s}=1$. The resolution of the Hubble tension
through EDE would thus carry important implications for models seeking to
describe the primordial Universe (D’Amico et al., 2022b; Kallosh & Linde,
2022; Ye et al., 2021; Smith et al., 2022; Jiang et al., 2022).

###  6.2 The Hubble tension and EDE

Finally, we quantify how well EDE is able to resolve the tension and discuss
improvements in the overall fit to the data by examining the change in
$\chi^{2}$ assuming EDE and $\Lambda$CDM, respectively (see Table 9). Naively,
one would expect an improvement in $\chi^{2}$ equal to the number of
additional parameters. A common way to present data comparison is therefore
with the reduced $\chi^{2}$. But while the number of degrees of freedom can be
estimated for models where the fitting parameters enter in a linear way, the
effective number of degrees of freedom is unknown for models with non-linear
fitting parameters (Andrae et al., 2010). As a way to look beyond the regular
$\chi^{2}$ statistic and attempt a fairer model comparison of EDE with
$\Lambda$CDM, we, therefore, turn to the Akaike Information Criterion (AIC)
(Trotta, 2008), which penalises models with additional degrees of freedom:

$\mathrm{AIC}\equiv-2\ln\mathcal{L}_{\mathrm{max}}+2k,$ (35)

where $k$ is the number of fitted parameters and $\mathcal{L}_{\mathrm{max}}$
is the maximum likelihood value. We present $\Delta$AIC relative to
$\Lambda$CDM for the different data set combinations in Table 9, where the
model with minimum AIC is preferred. While the improvement in $\chi^{2}$ is
close to what is expected $\Delta\chi^{2}\simeq 3$, the AIC shows that
$\Lambda$CDM is preferred over EDE for all data combinations where no
additional information from SH0ES is added. Only when considering SH0ES data,
the AIC favors EDE over $\Lambda$CDM, with a substantial
$\Delta\mathrm{AIC}=-29$. The improvement of $\Delta\chi^{2}=-35$ is mainly
coming from the fit to the PantheonPlusSH0ES likelihood, where there is an
improvement of $\Delta\chi^{2}_{\mathrm{PantheonPlusSH0ES}}=-36.9$, while the
overall fit to Planck is just slightly worsened by
$\Delta\chi^{2}_{\mathrm{Planck}}=+3.3$. This demonstrates the overall
attraction of EDE: the fit to Planck is maintained, while $H_{0}$ is
compatible with SH0ES.

In order to address the question of how the inclusion of SH0ES data is
impacting the fit of a certain model $\mathcal{M}$ to a given data set
$\mathcal{D}$, we compute the tension metric $Q_{\mathrm{DMAP}}$ as discussed
in Raveri & Hu (2019); Schöneberg et al. (2022):

$Q_{\mathrm{DMAP}}\equiv\sqrt{\chi^{2}_{\mathcal{M}}(\mathcal{D}+\mathrm{SH0ES})-\chi^{2}_{\mathcal{M}}(\mathcal{D})},$
(36)

where we compare the difference of the maximum a posterior (DMAP) values upon
the addition of a $H_{0}$ prior (Riess et al., 2022a)191919It was demonstrated
in Simon (2023) that incorporating the SH0ES prior onto the Pantheon+
likelihood yields constraints on EDE models equivalent to those from the full
PantheonPlusSH0ES likelihood (Riess et al., 2022a). For simplicity, we
calculate $Q_{\mathrm{DMAP}}$ by substituting the PantheonPlusSH0ES likelihood
with PantheonPlus along with the Gaussian prior on $H_{0}$ from Riess et al.
(2022a) in the corresponding minimization process of the chains.. In terms of
the $Q_{\mathrm{DMAP}}$ metric, the tension between SH0ES and the
baseline+BOSSz1+eBOSS is reduced from $5.2\sigma$ in $\Lambda$CDM to
$3.0\sigma$ in EDE. For baseline+BOSS alone, the tension is lowered to
$2.0\sigma$ in the case of EDE. The difference in $Q_{\mathrm{DMAP}}$ for EDE
between baseline+BOSSz1+eBOSS and baseline+BOSS is primarily attributed to a
poorer fit to the PlanckTTTEEE likelihood. Schöneberg et al. (2022)
established criteria for classifying the success of a given model in
addressing the $H_{0}$ tension. Models were categorized based on three
individual tests: Achieving a good fit to all data with a minimal
$Q_{\mathrm{DMAP}}$, significantly improving the fit over $\Lambda$CDM
according to the AIC and the ability of allowing for high values of $H_{0}$
without incorporating a local distance ladder prior. As per Schöneberg et al.
(2022), passing the $Q_{\mathrm{DMAP}}$ criterion required reducing the
tension to $\leq 3\sigma$, and the AIC criterion must suggest more than a weak
preference over $\Lambda$CDM following the Jeffreys scale (Jeffreys, 1998b;
Nesseris & Garcia-Bellido, 2013), specifically with
$\Delta\mathrm{AIC}\leq-6.91$. Although each criterion has its advantages and
drawbacks, they are computationally efficient and thought to complement one
another. A more detailed Bayes factor study (Kass & Raftery, 1995) is,
therefore, deferred to future work, and the question of whether EDE is a
suitable model to explain the tension is addressed here using the
aforementioned criteria. While EDE is incapable of accommodating sufficiently
high $H_{0}$ values to resolve the tension without SH0ES data, as discussed
earlier, it passed the $Q_{\mathrm{DMAP}}$ and AIC criterion in previous works
(Schöneberg et al., 2022; Simon, 2023). Our analysis, incorporating the full
shape analysis of BOSSz1+eBOSS, suggests that EDE clearly passes the AIC test
(with $\Delta\mathrm{AIC}=-29$), while the $Q_{\mathrm{DMAP}}$ test with
$Q_{\mathrm{DMAP}}=3\sigma$ represents the upper limit for passing the test.
Although eBOSS data currently does not decisively exclude EDE as a model to
address the $H_{0}$ tension, it reveals a tendency that the inclusion of
large-scale structure data exerts increasing pressure on EDE as a solution to
the tension. Consequently, it will be crucial to assess the implications of
Stage-4 large-scale structure data, such as from DESI (Aghamousa et al., 2016)
or Euclid Amendola et al. (2018), on our understanding of EDE.

###  6.3 Full Shape analysis in combination with SH0ES

In a final step, we discuss the constraining power of a Planck-free analysis
of EDE with a full shape analysis of eBOSS and BOSS, including a BBN prior as
defined in Sec. 3.4. For a discussion about the impact of different BBN priors
on the Hubble tension and EDE constraints, we refer the reader to Takahashi &
Toda (2023). As we have discussed in Sec. 4.2, we will not show full shape
only constraints due to possible strong volume projection effects. In order to
minimize potential volume effects, we are combining the full shape analysis of
BOSS and eBOSS with SH0ES. The results are presented in Table 10 and the
relevant $\chi^{2}$ statistics can be found in Table 9. We show the associated
reconstructed 2D posteriors in Fig. 8 and compare our results with the
posteriors reconstructed from PlanckTTTEE+lowl+lowE+lens+SH0ES. Best-fit
values are indicated by cross markers for both analyses. Upon incorporating
SH0ES data, we derive the following EDE parameter constraints:
$f_{\mathrm{EDE}}=0.2539_{-0.13}^{+0.089}$ with $h=0.7289_{-0.011}^{+0.011}$
km/s/Mpc and $f_{\mathrm{EDE}}=0.1794_{-0.06}^{+0.085}$ with
$h=0.7328_{-0.01}^{+0.011}$ km/s/Mpc for analyses assuming Gaussian and
Jeffreys priors, respectively. The consistency of $h$ with the SH0ES
constraint $h=0.7304\pm 0.0104$ km/s/Mpc (Riess et al., 2022a) is within
$<0.1\sigma$ for the analysis with Gaussian priors and $<0.2\sigma$ for the
analysis with the Jeffreys prior. In comparison, for $\Lambda$CDM, the
consistency is within $<1.5\sigma$ and $<0.8\sigma$, respectively. In the case
of Planck combined with SH0ES, we obtain
$f_{\mathrm{EDE}}=0.1084_{-0.024}^{+0.031}$ with $h=0.7157_{-0.01}^{+0.0091}$
km/s/Mpc. When contrasting the difference in $\chi^{2}$ between $\Lambda$CDM
and EDE, we observe an improvement of $\Delta\chi^{2}=-12$ for the full shape
only constraints within EDE. This improvement primarily stems from the fit to
the SH0ES prior, where there is a specific enhancement of
$\Delta\chi^{2}_{\mathrm{SH0ES}}=-3.2$. Furthermore, we note an enhancement in
the eBOSS QSO data set of $\Delta\chi^{2}_{\mathrm{QSO}}=-5.9$, as the QSOs
favor a higher $\Omega_{m}$ value, which is more likely accommodated within
the EDE model. The full shape analysis of BOSSz1+eBOSS+SH0ES hints at a mild
preference for EDE over $\Lambda$CDM with $\Delta$AIC = -6.

However, it is important to exercise caution in interpreting these results at
this stage. As highlighted in the earlier mock analysis (Sec. 4.2), eBOSS
faces substantial challenges related to projection effects when considering
EDE, both with Gaussian priors and a Jeffreys prior on the linear nuisance
parameters. These effects arise due to the increasing significance of
nonlinear degeneracies. Comparing the best-fit values with the mean of the
posteriors in Fig. 8, it becomes evident that imposing an $H_{0}$ prior is
insufficient to fully account for all projection effects when varying all EDE
parameters. Consequently, we conclude that while, in principle, we can extend
constraints beyond $\Lambda$CDM models with BOSS and eBOSS full shape analyses
alone, the observed projection effects need to be treated cautiously. To
achieve full shape constraints comparable to Planck, we must await forthcoming
large-scale structure surveys such as DESI or Euclid, where these projection
effects are expected to play a subdominant role due to the enhanced
constraining power of the data sets.

Figure 8: 1D and 2D posterior distributions on the cosmological parameters of the combined full shape analysis of BOSSz1 and eBOSS in combination with SH0ES (imposing a BBN prior (Schöneberg et al., 2019)) for the two different prior choices: Jeffreys prior (dark blue, dot-dashed line) and zero-centered Gaussian priors (light blue, dashed line). The two contour levels in the off-diagonal elements represent $68\%$ (inner) and $95\%$ (outer) credible intervals and the best-fit values are indicated with corresponding markers. We also show results from PlanckTTTEEE+lowl+lowE+lens in combination with SH0ES (red, solid line) for comparison. The grey bands correspond to $1\sigma$ and $2\sigma$ bounds of the latest SH0ES analysis (Riess et al., 2022a). | BOSSz1+eBOSS+SH0ES | BOSSz1+eBOSS+SH0ES | PlanckTTTEEE+lens+SH0ES
---|---|---|---
|  $\Lambda$CDM | EDE | EDE
| GP | JP | GP | JP |
$\Omega_{m}$ | 0.3417 | $0.3138$ | $0.2976$
$0.3092_{-0.018}^{+0.014}$ | $0.3845_{-0.027}^{+0.026}$ | $0.2909_{-0.017}^{+0.014}$ | $0.3574_{-0.033}^{+0.03}$ | $0.2992_{-0.0067}^{+0.0063}$
$h$ | $0.7116$ | $0.7330$ | $0.7240$
$0.7107_{-0.009}^{+0.0088}$ | $0.7197_{-0.0095}^{+0.0093}$ | $0.7289_{-0.011}^{+0.011}$ | $0.7328_{-0.01}^{+0.011}$ | $0.7157_{-0.0091}^{+0.01}$
$\sigma_{8}$ | 0.7874 | $0.8558$ | $0.8450$
$0.7162_{-0.041}^{+0.039}$ | $0.7208_{-0.053}^{+0.045}$ | $0.7331_{-0.045}^{+0.039}$ | $0.742_{-0.069}^{+0.059}$ | $0.8372_{-0.01}^{+0.01}$
$f_{\mathrm{EDE}}$ | $-$ | $0.1897$ | $0.1319$
$-$ | $-$ | $0.2539_{-0.13}^{+0.089}$ | $0.1794_{-0.085}^{+0.06}$ | $0.1084_{-0.024}^{+0.031}$
$\log_{10}z_{c}$ | $-$ | $3.4216$ | $3.5765$
$-$ | $-$ | $3.83^{+0.48}_{-0.65}$ | $3.59^{+0.57}_{-0.46}$ | $3.614_{-0.11}^{+0.032}$
$\theta_{i}$ | $-$ | $0.9013$ | $2.748$
$-$ | $-$ | $1.8^{+1.2}_{-1.4}$ | $1.75^{+0.89}_{-0.89}$ | $2.684_{-0.068}^{+0.22}$
$\omega_{cdm}$ | $0.1495$ | $0.1453$ | $0.1326$
$0.1326_{-0.01}^{+0.0084}$ | $0.1758_{-0.016}^{+0.016}$ | $0.1311_{-0.0093}^{+0.0074}$ | $0.1686_{-0.019}^{+0.016}$ | $0.1298_{-0.0036}^{+0.0038}$
$10^{2}\omega_{b}$ | $2.2899$ | $2.2667$ | $2.2825$
$2.296_{-0.039}^{+0.038}$ | $2.283_{-0.04}^{+0.036}$ | $2.274_{-0.039}^{+0.037}$ | $2.267_{-0.04}^{+0.036}$ | $2.284_{-0.021}^{+0.022}$
$ln(10^{10}A_{s})$ | $2.8076$ | $3.0780$ | $3.0762$
$2.714_{-0.14}^{+0.13}$ | $2.506_{-0.16}^{+0.13}$ | $2.906_{-0.15}^{+0.14}$ | $2.702_{-0.23}^{+0.19}$ | $3.07_{-0.017}^{+0.015}$
$n_{s}$ | $0.8310$ | $0.9818$ | $0.9931$
$0.8945_{-0.05}^{+0.05}$ | $0.7606_{-0.061}^{+0.015}$ | $0.9908_{-0.059}^{+0.062}$ | $0.8519_{-0.11}^{+0.071}$ | $0.9891_{-0.0065}^{+0.007}$
$N_{\mathrm{data}}$ | 288 | 288 | 2352
$N_{\mathrm{param}}$ | 5+16 | 8 + 16 | 8 + 21
$\chi^{2}$ | $230$ | $218$ | 2775
AIC | 272 | 266 | 2833
Table 10: Best-fit and mean $\pm$ $1\sigma$ bounds for the cosmological
parameters of $\Lambda$CDM and EDE measured from the combined full shape
analysis of BOSSz1+eBOSS+SH0ES, with a BBN prior on $\omega_{b}$, confronted
with the EDE analysis of Planck+SH0ES. We present the mean and $1\sigma$
bounds of the full shape analyses for the two different prior choices on the
linear nuisance parameters: Jeffreys prior (JP) and zero-centered Gaussian
priors (GP). We consider $N_{\mathrm{BOSSz1}}=N_{\mathrm{BOSSz3}}=76$ (Alam et
al., 2017), $N_{\mathrm{eBOSS}}=212$ (Alam et al., 2021),
$N_{\mathrm{H_{0}}}=1$ (Riess et al., 2022a), $N_{\mathrm{BBN}}=1$ (Schöneberg
et al., 2019) and $N_{\mathrm{Planck}}=2352$ (Aghanim et al., 2020a, b). The
corresponding Akaike Information Criterion (AIC) and $\chi^{2}$ values are
calculated at the best-fit points.

##  7 Conclusions

In this paper, we present constraints on EDE derived from a full shape
analysis applied to the complete eBOSS DR16 data sets, encompassing LRG, QSO,
and ELG data. For the full shape analysis of eBOSS and BOSS data in
combination with Planck, external BAO measurements, PantheonPlus (baseline),
and SH0ES data, we derive values of $H_{0}=71.73_{-0.86}^{+0.82}$ km/s/Mpc and
$H_{0}=72.03_{-0.87}^{+0.82}$ km/s/Mpc for analyses assuming Gaussian priors
and a Jeffreys prior on the linear nuisance parameters, respectively.
Irrespective of the priors applied, consistency with SH0ES is found at a level
of less than $1\sigma$ for EDE, contrasting with the $>4\sigma$ consistency
for $\Lambda$CDM. Utilizing the Akaike Information Criterion (AIC) to assess
the preference for EDE over $\Lambda$CDM, accounting for the additional
parameters introduced by EDE, reveals a mild preference for $\Lambda$CDM in
scenarios involving the baseline dataset, baseline+BOSS, and
baseline+BOSSz1+eBOSS. Only with the inclusion of additional information from
SH0ES does the AIC distinctly favor EDE over $\Lambda$CDM, indicating a
significant preference with $\Delta$AIC = -29. For an evaluation of the
residual Hubble tension using the $Q_{\mathrm{DMAP}}$ metric, we observe a
reduction of the tension to $3\sigma$ in EDE, in contrast to $5.2\sigma$ in
$\Lambda$CDM, considering the baseline+BOSSz1+eBOSS dataset. The introduction
of eBOSS data, however, elevates the tension in EDE from $2\sigma$ to
$3\sigma$. Although eBOSS does not currently exclude EDE, its inclusion
indicates a trend that additional large-scale structure data increases the
pressure of EDE of simultaneously fitting various data sets.

We have also demonstrated that using additional large-scale structure data in
EDE analyses could significantly limit the model as a potential resolution to
the Hubble tension. In particular, full shape analyses exhibit the ability to
impose stringent constraints on EDE parameters. Using the full shape analysis
of eBOSS and BOSS data, while incorporating a BBN prior and SH0ES data, yields
$H_{0}$ values of $72.9_{-1.1}^{+1.1}$ km/s/Mpc and $73.3_{-1.0}^{+1.1}$
km/s/Mpc, depending on the application of Gaussian or Jeffreys priors. The
analysis suggests a mild preference for EDE over $\Lambda$CDM with $\Delta$AIC
= -6. While we are capable of placing constraints on EDE parameters through
full shape-only analyses, we showed that it is crucial to understand the
impact of possible remaining projection effects on the cosmological
constraints.

While the projection effects coming from the marginalisation of linear
nuisance parameters are expected to be reduced in Stage-IV surveys, we have
proposed methods to address them within EDE for eBOSS data by exploring
different prior choices. Building on recent successes in addressing projection
effects in $\Lambda$CDM through the application of a Jeffreys prior to linear
nuisance parameters, we explored its effectiveness in mitigating projection
effects in EDE and compared it to zero-centered Gaussian priors which are
widely used in literature. Our comprehensive mock study, covering $\Lambda$CDM
and EDE, suggests that while the Jeffreys prior is able to mitigate projection
effect within $\Lambda$CDM, in the case of EDE - where the introduction of
additional parameters leads to degeneracies between non-linear cosmological
parameters - significant projection effects persist with both prior choices.
Full shape analysis in the context of beyond $\Lambda$CDM achieves unbiased
posteriors only when combined with data sets capable of breaking these newly
introduced parameter degeneracies.

We defer to future work the examination of the impact of a Jeffreys prior on
non-linear parameters within models beyond $\Lambda$CDM in EFTofLSS, as well
as the direct comparison with a profile likelihood analysis.

## Acknowledgements

We would like to thank Seshadri Nadathur and Arnaud de Mattia for useful
discussions regarding the eBOSS data sets. We also would like to thank Harry
Desmond for insights about the Jeffreys prior and Pierre Zhang regarding
questions about Pybird. RG and JD-M were supported by STFC studentships. RZ is
supported by NSFC grants 11925303 and 11890691, and is also supported by the
Chinese Scholarship Council (CSC) and the University of Portsmouth. KK, DB and
RC are supported by STFC grant ST/W001225/1. TS is supported by the European
Research Council (ERC) under the European Union’s HORIZON-ERC-2022 (Grant
agreement No. 101076865). For the purpose of open access, the author(s) has
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising.

## Data Availability

Supporting research data are available on request from the corresponding
author.

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