# Full-Shape Galaxy Power Spectra and the Curvature Tension

Aaron Glanville 1 , Cullan Howlett 1, Tamara Davis 1
1The University of Queensland, School of Mathematics and Physics, QLD 4072,
Australia E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

With recent evidence for a possible “curvature tension" among early and late
universe cosmological probes, Effective Field Theories of Large Scale
Structure (EFTofLSS) have emerged as a promising new framework to generate
constraints on $\Omega_{k}$ that are independent of both CMB measurements, and
some of the assumptions of flatness that enter into other large-scale
structure analyses. In this work we use EFTofLSS to simultaneously constrain
measurements from the 6dFGS, BOSS, and eBOSS catalogues, representing the most
expansive full-shape investigation of curvature to date. Fitting the full-
shape data with a BBN prior on $\Omega_{b}h^{2}$ and fixed $n_{s}$, we measure
$\Omega_{k}=-0.089^{+0.049}_{-0.046}$, corresponding to a $\sim 2\sigma$
preference for curvature. We argue that this result cannot be biased towards
flatness by assumptions in the fitting methodology. Using the Bayesian
evidence ratio our full-shape data assigns betting odds of 2:1 in favour of
curvature, indicating present measurements remain broadly compatible with both
flat and curved cosmological models. When our full-shape sample is combined
with Planck 2018 CMB measurements, we break the geometric degeneracy and
recover a joint fit on $\Omega_{k}$ of $-0.0041^{+0.0026}_{-0.0021}$. Using
the suspiciousness statistic (built on the standard Bayes factor), we find
evidence for a moderate tension between Planck 2018 and our suite of full-
shape measurements, at a significance of $1.76^{+0.14}_{-0.11}\sigma$ ($p\sim
0.08\pm 0.02$). These results demonstrate the usefulness of full-shape
clustering measurements as a CMB independent probe of curvature in the ongoing
curvature tension debate.

###### keywords:

cosmology: cosmological parameters, large-scale structure of Universe, early
Universe

††pubyear: 2022††pagerange: Full-Shape Galaxy Power Spectra and the Curvature
Tension–Full-Shape Galaxy Power Spectra and the Curvature Tension

##  1 Introduction

Over the past decade, remarkable improvements in the precision and accuracy of
our constraints have exposed clear inconsistencies between cosmological
datasets. While much discussion has centred around the consistency of early
and late universe measurements in their constraints of $H_{0}$, recent
attention has been drawn to a potential tension among early universe curvature
constraints (Ooba et al., 2019; Handley, 2021; Park & Ratra, 2019; Di
Valentino et al., 2020; Efstathiou & Gratton, 2020; Vagnozzi et al., 2020).
Determining what is driving these inconsistencies, and how to ameliorate the
tensions which arise, remains a pressing challenge in modern cosmology. When
taken alone, constraints made using the fiducial Planck measurements and
likelihoods exhibit a strong preference for closed universes, in contrast to
other early universe probes of curvature (Aghanim et al., 2020; Di Valentino
et al., 2020). This apparent discrepancy has raised questions regarding the
internal consistency of data sets employed in joint constraints that use CMB
data, and the confidence which can be given to such joint fits (Handley, 2021;
Di Valentino et al., 2020; Efstathiou & Gratton, 2020). With recent advances
in modelling the full shape of the matter power spectrum, a powerful new
framework has emerged to generate competitve curvature constraints directly
from clustering measurements, independently of CMB information (Chudaykin et
al., 2021).

One of the most promising of these full-shape models is the state-of-the art
effective field theory of large scale structure (hereafter EFTofLSS), which
reliably models the power spectrum of biased matter-density tracers up to
mildly non-linear regimes ($k\sim 0.2$) (D’Amico et al., 2020; Chudaykin et
al., 2020). This is in contrast to methods which fit the imprinted BAO feature
exclusively (Beutler et al., 2011, 2017; eBOSS Collaboration et al., 2020), or
fit the full-shape of the power spectrum using template-based methods (where
the template is usually based on a fiducial flat-$\Lambda$CDM cosmology,
Tegmark et al. 2006; Sánchez & Cole 2008; Reid et al. 2010; Parkinson et al.
2012). While such mildly non-linear models provide a relatively modest
extension in k-space (from $k_{\rm{max}}\sim 0.1$ to $k_{\rm{max}}\sim 0.2$),
the cubic scaling of $n_{\rm{modes}}$ with $k_{\rm{max}}$ opens up a
substantial amount of information for cosmological model fitting.

In this work, we combine power spectrum measurements from the 6dFGS, BOSS, and
eBOSS surveys in the most comprehensive full-shape analysis of curvature to
date (Jones et al., 2009; Alam et al., 2017; eBOSS Collaboration et al.,
2020)111While drafting this manuscript, Neveux et al. (2022) also released a
full-shape analysis of the combined BOSS and eBOSS catalogues, although their
work focuses on flat cosmological models and uses a different theoretical
model and fitting methodology.. We use this combined sample to constrain both
flat and curved $\Lambda$CDM cosmological models, with minimal external
assumptions or priors. Using the Bayesian evidence ratio, we quantify the
statistical preference for curved/flat cosmological models exhibited by these
full-shape measurements in isolation. By strictly analysing the full-shape
data alone, we are able to disentangle the cosmological information provided
by clustering measurements in isolation, allowing us to understand the
statistical confidence of joint fits with external probes. We then determine
the consistency of these results with Planck CMB measurements, using the
suspiciousness statistic (Handley & Lemos, 2019a; Handley, 2021).

In order to meaningfully explore the significance of this possible tension, it
is also critical to quantify any aspects of our analysis that could
potentially bias our results towards flatness. In particular, the support for
flatness (within $\pm 10\%$) from early CMB experiments, along with the
success of the inflationary model (which predicts flatness at late times, Guth
1981; Linde 1982; Jaffe et al. 2001; Netterfield et al. 2002) have made the
assumption of flatness a convenient axiom in large scale structure physics.
Many cosmological analysis tools and techniques also assume flatness as a
matter of theoretical and computational simplicity. As such, it is imperative
that we guard against confirmation bias by rigorously investigating the impact
of these common assumptions in all steps of our analysis, from observation to
results. In this work, we systematically review the potential assumptions of
flatness which can arise in measurements of large scale structure, and
determine the biases these can introduce in the emerging curvature constraints
provided by EFTofLSS.

Our paper is structured as follows. Through Section 2, we expand on the
current state of the curvature tension, provide some background for the
EFTofLSS models used in this work, and justify our choice to fit the full-
shape directly by detailing some common points where flatness is assumed in
normal measurements of large scale structure. In Section 3, we detail the
methodology behind parameter constraints and model comparisons used in this
research, alongside the datasets employed in our work. In Section 4, we
validate the accuracy of our full-shape pipeline (and in particular, fits to
$\Omega_{k}$) using the publicly available Nseries mock catalogues. We then
examine the common assumption of flatness in the choice of fiducial cosmology,
to identify if this could introduce any substantial bias in fits to full-shape
clustering measurements. In Section 5 we use our pipeline to simultaneously
fit clustering measurements from the 6dFGS, BOSS, and eBOSS catalogues
providing the most robust full-shape only constraints of $\Omega_{k}$ to date.
Using the Bayesian evidence ratio, we then measure the statistical preference
for flatness/curvature exhibited by our suite of full-shape measurements in
isolation. Finally, we combine full-shape measurements with Planck 2018 CMB
information, first recovering their joint posterior, and then determining the
consistency of these datasets using the suspiciousness statistic.

##  2 Background

###  2.1 Curvature Tension

In conjunction with the axioms that form the standard $\Lambda$CDM model,
spatial flatness has been a common assumption in cosmological analysis for
some time. This preference is supported by the overwhelming body of evidence
(across early and late universe probes), which suggest the universe is either
spatially flat, or remarkably close to flat. Planck temperature and
polarization spectra, when combined with lensing reconstruction information,
yield fits which are consistent with flatness to within $\sim 1\%$
($\Omega_{k}=-0.0106\pm 0.0065$). These constraints are further sharpened with
the addition of BAO measurements, yielding $\Omega_{k}=0.0007\pm 0.0019$,
corresponding to a $1\sigma$ detection of flatness at an accuracy of $0.2\%$
(Alam et al., 2017; Aghanim et al., 2020). This preference for flatness is
also supported amongst joint fits with other late-universe cosmological probes
— constraints using CMB spectra in combination with type Ia supernovae from
Panetheon once again recover flatness to within $1\sigma$
($\Omega_{k}=-0.0061^{+0.0062}_{-0.0054}$).

While fits which combine CMB measurements with external probes exhibit a
consistent preference for flatness, CMB measurements in isolation exhibit a
stark preference for closed cosmologies (measured as
$\Omega_{k}=-0.044^{+0.018}_{-0.015}$ in Aghanim et al. 2020). This is broadly
driven by a preference for models with a higher than expected amount of
lensing, with the most recent measurements recovering a lensing amplitude
which exceeds theoretical forecasts by $\sim 2\sigma$. In the absence of any
external information (most notably lensing reconstruction), closed cosmologies
provide a natural explanation for this lensing excess via an increase in
$\Omega_{m}$. This preference for high lensing amplitudes and closed
cosmologies within CMB measurements has been extensively discussed in the
literature, with some evidence to suggest this effect may simply be a
statistical fluctuation (Ade et al., 2016; Aghanim et al., 2020; Efstathiou &
Gratton, 2021). In revisiting curvature constraints from Planck information,
Efstathiou & Gratton (2019) show the inclusion of $\Omega_{k}$ primarily
modulates the $\chi^{2}$ values from the $\mathcal{\ell}<30$ Commander
temperature likelihood. When removing the $\mathcal{\ell}<30$ multipoles,
Efstathiou & Gratton (2019) argue models with a free $\Omega_{k}$ do not
perform substantially better than flat models, as measured by their respective
$\chi^{2}$ values. Additionally, Planck measurements exhibit a strong
preference for inflationary models with a large number of e-foldings,
inconsistent with models of incomplete inflation which may naturally result in
closed cosmologies.

Several recent papers have argued that the significance of this preference for
closed universes, alongside the ubiquity of joint likelihoods which include
CMB information, warrant further investigation. Di Valentino et al. (2020)
provide evidence for a substantial discord between CMB measurements and a host
of late universe probes (including BAO, Supernovae, lensing, and cosmic shear
measurements) in $\Lambda$CDM models which permit curvature. This is supported
by Handley (2021), who determine Planck data is inconsistent with BAO and
lensing measurements at a significance of $\leq 2.5\sigma$, and consistent
with a closed universe (assigning Bayesian betting odds of 50:1 against
flatness). This comparison was further expanded in Vagnozzi et al. (2020), who
identify evidence of a substantial tension between full-shape clustering
measurements (evaluated up to $k_{\textrm{max}}=0.135$), and CMB spectra in
models which permit curvature. While the origin of this discord continues to
be discussed, any major inconsistency between the Planck measurements and our
suite of late-universe probes represent a substantial challenge for our
standard approach to fitting $\Omega_{k}$. Precise measurements of
$\Omega_{k}$ (as part of the standard $\Lambda$CDM model) almost always rely
on information from a combination of cosmic probes. As the earliest available
measurement of cosmic structure, CMB measurements provide an invaluable
“lever-arm” across cosmic history to break parameter degeneracies in late-
universe measurements, making CMB measurements a ubiquitous element of
curvature constraints. Implicit in the combination of multiple datasets is the
assumption that all measurements contained within the joint likelihood could
plausibly arise from the same universe. Evidence of a significant
incompatibility within the CMB data, regardless of its origin, opens serious
questions on the reliability of these joint fits.

As the potential significance of these apparent inconsistencies continues to
be discussed, it is more important than ever to generate a range of robust
curvature constraints that do not rely on CMB measurements. Measurements of
the BAO feature provide one such independent probe of cosmic curvature, with
the recent eBOSS survey recovering flatness to within $1\sigma$
($\Omega_{k}=0.078^{+0.086}_{-0.099};$ eBOSS Collaboration et al., 2020).
Additionally, advances in fitting the broadband power spectrum up to non-
linear scales have offered a valuable new perspective in this discussion,
which has already been capitalised on in recent work. Most notably, Chudaykin
et al. (2021) show that combining post-reconstruction BAO information with
full-shape measurements allows for competitive constraints on curvature
($\Omega_{k}=-0.043\pm 0.036$), which are wholly independent of CMB
information.

###  2.2 Flatness Assumptions in Large Scale Structure

With the emergence of the first reliable CMB anisotropy measurements two
decades ago, a flat universe (within $\pm 10\%$) has been consistently
preferred (Balbi et al., 2000; Jaffe et al., 2001; Masi et al., 2002). The
significant reduction in computational time afforded by fixing $\Omega_{k}=0$,
in conjunction with the strong body of observational and theoretical evidence
supporting a flat universe, has made flatness a very common “short-cut”
assumption in cosmological analyses. The ubiquity of even implicit assumptions
of flatness (as a default in many cosmological analysis tools or data
products), makes it important to guard against confirmation bias when
considering curvature constraints provided by new models such as EFTofLSS. As
such, we outline some of the more significant assumptions of flatness in large
scale structure measurements and analyses, in order to systematically evaluate
whether these introduce any significant bias in our EFTofLSS curvature
constraints.

####  2.2.1 Fiducial Cosmology

Measurements of cosmic structure begin with a galaxy redshift catalogue, which
details the angular sky position and redshift of hundreds of thousands of
individual galaxies. In order to measure clustering statistics from this
sample, a fiducial cosmology must be employed to convert observed redshift to
comoving distances. While measurements of cosmic structure (such as the BAO
feature) are broadly robust to systematics in redshift measurements themselves
(Glanville et al., 2021), offsets in fiducial cosmology ($\sim 0.5\%$ in
$\Omega_{\rm m}$) from the underlying sample truth are known to introduce
biases in anisotropic BAO fits of $\sim 0.1\%$ (Vargas-Magaña et al., 2018).
For most realistic settings this effect is firmly sub-dominant to statistical
errors, and does not substantially affect parameter constraints when
propagated through BAO analysis (Gil-Marín et al., 2020). Model independent
approaches to fitting the BAO feature have been proposed in the literature,
most notably by measuring the radial two-point correlation function (Sánchez
et al., 2010; Sánchez et al., 2013), however such techniques come at the
expense of a dampened BAO signal. While the increased scale of next generation
surveys such as the Dark Energy Spectroscopic Instrument (DESI; DESI
Collaboration et al., 2016) may allow for competitive model independent fits
of the BAO feature, at present most major investigations rely on fiducial
cosmologies which are overwhelmingly flat (Beutler et al., 2011; Parkinson et
al., 2012; Alam et al., 2017; Bautista et al., 2021; Gil-Marín et al., 2020;
Tamone et al., 2020; de Mattia et al., 2021; Hou et al., 2021; Neveux et al.,
2020). In this work, we systematically investigate the potential biases
introduced by this assumption by “stress-testing” our fits — that is, by
constraining clustering measurements made using non-flat fiducial cosmologies,
for comparison with a flat baseline measurement.

####  2.2.2 Reconstruction

In BAO analyses, it is common practice to use reconstruction techniques to
improve the clarity of the BAO feature. Broadly, the non-linear evolution of
galaxies arising from gravitational effects work to dampen the amplitude of
the BAO wiggles, weakening the observed signal. Reconstruction reduces this
signal loss by using the observed density field to infer the local velocity
field, partially reversing these non-linear motions to generate a
“reconstructed” sample. Importantly, reconstruction techniques must assume a
fiducial cosmology (when converting from observed redshifts to comoving
distances), a fiducial bias parameter in the modelling of the underlying
density field from the observed galaxy distribution, and a fiducial growth
rate of structure to account for redshift-space distortions.

Reconstruction significantly improves the precision of BAO distance
measurements, and the fiducial cosmology of reconstruction only has a small
impact at this scale (Padmanabhan et al., 2012; Burden et al., 2014; Kazin et
al., 2014; Carter et al., 2020). However, there is evidence to suggest that
the use of a fiducial reconstruction cosmology which deviates from the
underlying cosmology of a sample can bias the monopole shape and quadrupole
amplitude, at the $1\%$ and $5\%$ levels respectively (Sherwin & White, 2019).
As such, the use of reconstruction on observed data (where the underlying
sample cosmology is not known) represents a source of potential bias for fits
to the full shape of the power spectrum.

Additionally, it is common practice to fold in post-reconstruction BAO
information with other large scale structure measurements to improve joint
constraints. In the context of EFTofLSS, this approach was taken by Chudaykin
et al. (2021), who used post-reconstruction BAO information in combination
with full-shape data to constrain a range of extensions to the standard model
(such as oCDM, $w_{0}{\rm CDM}$ and $w_{0}w_{a}{\rm CDM}$ models). In these
fits, Chudaykin et al. (2021) correctly note this post-reconstruction BAO
information is covariant with the pre-reconstruction power spectrum. However,
it is important to first verify that pre-reconstruction full-shape and post-
reconstruction BAO measurement are internally consistent, before we can
confidently combine this information. At the cost of a reduction in our
constraining power, we therefore opt to disentangle the assumptions of
reconstruction from our analysis by fitting full-shape measurements
independently of post-reconstruction BAO information. This means that there
are no assumptions of flatness entering in our analysis due to reconstruction.

####  2.2.3 Covariance Matrix

When estimating the errors associated with binned clustering measurements, one
must compute the covariance matrix for this sample from a large number of mock
catalogues. Due to the high computational overhead associated with generating
these mocks, both the underlying cosmology of these catalogues, and the
fiducial cosmology used to compute comoving distances, are generally fixed to
$\Omega_{k}=0$ (Kitaura et al., 2016; Rodríguez-Torres et al., 2016, 2017;
Favole et al., 2016, 2017).222The recently completed Quijote N-body
simulations provide a suite of 11,000 mock power spectra generated from
universes which extend over a hypercube of parameters, including $\Omega_{k}$
Villaescusa-Navarro et al. (2020).

Even at linear order, the covariance matrix is proportional to the power
spectrum and shot-noise, and hence is dependent on cosmology. However, in
practice the clustering from simulations is tuned to match the amplitude of
the measured power spectrum (and often bispectrum) of the data because the
galaxy bias is unknown. As we demonstrate in Section 4 and Figure 3, changing
$\Omega_{k}$ mainly also changes the amplitude of the power spectrum (and is
very degenerate with the galaxy bias). Hence, we expect the covariance matrix
to be only very weakly dependent on the choice of $\Omega_{k}$ used to produce
the simulations.

To expound further, consider if the true Universe had, e.g., $\Omega_{k}=0.1$.
Imagine we then generated two sets of simulations and two covariance matrices
from those simulations, one with $\Omega_{k}=0.1$ and the other assuming
flatness. These would both get tuned to match the clustering in the data, with
the result that the simulations with $\Omega_{k}=0.1$ would be tuned to have a
lower value for the galaxy bias. However, in both cases, the resulting
covariance matrices would be very similar.

We hence argue that the choice of cosmology used to generated the covariance
matrix is very unlikely to bias measurements from large-scale structure
towards flatness.

####  2.2.4 Summary

Galaxy clustering measurements, and the statistics derived from these, assume
spatial flatness at all stages of analysis — at the catalogue level (with the
use of reconstruction algorithms), through our analysis pipelines (by assuming
flat fiducial cosmologies), and in our final model fitting (through
assumptions in common cosmological codes, and model choices in computing the
covariance matrix). In order to produce results as independent of these
assumptions as possible, we purposefully choose to fit only pre-reconstruction
full-shape data, we fit directly for both flat and non-flat cosmological
models, and we use mock catalogues to quantify the impact of assuming a
consistent non-flat fiducial cosmology in our analysis pipeline.

###  2.3 Effective Field Theory of Large Scale Structure

####  2.3.1 Theoretical Overview

EFTofLSS comes from a rich theoretical background, initially developed to
model the back reactions from small scale inhomogeneities, to determine their
impact on spacetime at larger scales (Baumann et al., 2012). As improvements
in observation have pushed cosmology to a precision science, the EFTofLSS
model has evolved into a framework that can provide highly accurate
theoretical predictions of the power spectrum, which are directly comparable
to the clustering statistics derived from biased tracers of the matter field
(Carrasco et al., 2012; Porto et al., 2014; Senatore & Zaldarriaga, 2015).
While EFTofLSS initially only accommodated models of the dark matter density
field, the equations of motion associated with the baryon, massive neutrino,
and dark energy density fields have since been incorporated into a unified
formalism (Lewandowski et al., 2015; Senatore & Zaldarriaga, 2017; Lewandowski
et al., 2017). We use this section to provide a brief theoretical overview of
the EFTofLSS model used in this work, in order to motivate some model choices
in our analysis. For a more detailed description of the EFTofLSS model and its
theoretical derivation, we direct the reader to D’Amico et al. (2020) and
Chudaykin et al. (2020).

Broadly, EFTofLSS models use physically motivated perturbations of the linear
power spectrum to evaluate the power spectrum in the non-linear regimes. To
derive these non-linear perturbations, each component of the cosmic density
(e.g. dark matter, baryonic matter, neutrinos etc.) is modelled as a distinct,
fluid-like field. The clustering contributions from each component field may
be computed using perturbative expansions in powers of $k/k_{\rm{nl}}$, where
$k$ is the mode of interest, and $k_{\rm{nl}}$ is a non-linear cut-off scale.
The contribution of each perturbation to the overall field is weighted by a
series of bias parameters, which parameterise effects such as the linear
galaxy bias, shot-noise, and counter-terms which encode the effects of back-
reactions on large-scales from modes below the cut-off scale which have
otherwise been ignored. In order to avoid modelling complex and
computationally expensive small scale non-linear effects, EFTofLSS models
(such as D’Amico et al., 2020; Ivanov et al., 2020b) frequently restrict
themselves to the linear ($k<0.1$) and mildly non-linear ($0.1<k<0.2$)
regimes. Despite the relatively small extension in $k$ space associated with
fitting this mildly non-linear regime, the cubic scaling of $n_{\rm{modes}}$
with $k_{\rm{max}}$ ensures even small improvements in scale resolution open
up a large number of new density modes for analysis. Since most galaxy
clustering statistics already provide measurements at these mildly non-linear
scales, EFTofLSS models can massively extend the useful information that can
be extracted from these previously unconstrained modes.

A number of codes now exist that implement EFTofLSS theories to model
clustering statistics from an ensemble cosmological model, most notably the
PyBird and CLASS-PT codes (D’Amico et al., 2020; Chudaykin et al., 2020).
These codes have already been used to provide competitive, independent
constraints on a range of cosmological parameters directly from the BOSS DR12
power spectra, with blinded challenges verifying the robustness of these
results (D’Amico et al., 2020; Philcox et al., 2020; Nishimichi et al., 2020).
EFT models have also been used to place constraints on a number of common
extensions to the standard, flat $\Lambda$CDM model. Notably, Chudaykin et al.
(2021) used full-shape measurements in conjunction with post-reconstruction
BAO and supernova information to constrain curved $\Lambda$CDM models,
yielding results consistent with flatness at just over $1~{}\sigma$
($\Omega_{k}=-0.043\pm 0.036$). A number of investigations have also used
full-shape power spectrum measurements in conjunction with the EFTofLSS model
to measure the dark energy equation of state parameter, and the sum of
neutrino masses (Chudaykin et al., 2020; Ivanov et al., 2020a; D’Amico et al.,
2021; Zhang et al., 2021).

####  2.3.2 Model Parameterisation and Marginalisation

We use this section to provide a brief overview of the technical choices made
in this analysis which relate to our full-shape model. In this work we use the
EFTofLSS model incorporated in the PyBird code, which models power spectra up
to 1-loop order (i.e. up to $3^{\rm{rd}}$ order in perturbations), introducing
a total of 10 EFT bias parameters for each sample in our analysis. Four of
these terms correspond to the galaxy bias parameters $b_{\rm{i}}$, which arise
from the expansion of the galaxy density and velocity fields in terms of their
dark matter counterparts. The small scale (or UV) dependence of redshift space
distortions is cancelled through counter-terms parameterised as $c_{\rm{r,1}}$
and $c_{\rm{r,2}}$, and non-linearities in the dark matter density are
similarly accounted for by the $c_{\mathrm{ct}}$ counter-term, which models
the effective sound speed of dark matter. This leaves 3 remaining stochastic
bias parameters, $c_{\epsilon,\rm{i}}$, which are shot-noise terms that absorb
the difference between the observed noisy galaxy distribution and the
expectation values of the underlying fields.

Following the procedure of D’Amico et al. (2020), these bias parameters can be
separated into terms which appear at linear order in the power spectrum, and
higher-order terms. The 7 counter-terms which appear at linear order may be
analytically marginalised when computing our likelihood, leaving
{$b_{1},b_{2},b_{4}$} as free bias parameters. D’Amico et al. (2020) find that
$b_{2}$ and $b_{4}$ are almost completely anti-correlated in simulations and
data. As such, they perform a change of variables from {$b_{2}$, $b_{4}$} to
{$c_{2}$, $c_{4}$}, where:

$c_{2}=\dfrac{1}{\sqrt{2}}(b_{2}+b_{4})$ (1)
$c_{4}=\dfrac{1}{\sqrt{2}}(b_{2}-b_{4})$ (2)

In this work we follow this convention, performing a change of variables from
{$b_{2}$, $b_{4}$} to {$c_{2}$, $c_{4}$}, and marginalising over the 7 bias
terms which appear at linear order. D’Amico et al. (2020) determine that for
BOSS-like volumes, the functions that are scaled by $c_{4}$ are small enough
to be negligible, and as such they fix $c_{4}$ to 0 in their work. We do the
same, and as such, our standard fitting procedure consists of our chosen suite
of cosmological parameters (e.g {$\ln(10^{10}A_{s}),\ h,\
\Omega_{\textit{cdm}}h^{2},\ \Omega_{b}h^{2},\ n_{s},\ \Omega_{\rm k}$}), with
2 additional bias parameters {$b_{1}$, $c_{2}$} for each independent
clustering sample.

##  3 Methodology

| $\\{z_{\rm{min}}-z_{\rm{max}}\\}$ | $z_{\rm{eff}}$ | $N_{\rm{sources}}$ | $V_{\rm{eff}}\ (\rm{Gpc}^{3})$ | $k_{\rm{max}}\ \\{k_{0},k_{2},k_{4}\\}$
---|---|---|---|---|---
6dFGS DR3 | $\\{0.01-0.2\\}$ | $0.096$ | $75\ 117$ | $0.12$ | $\\{0.15,\ 0.15,\ 0.1\\}$
BOSS z1 NGC | $\\{0.2-0.5\\}$ | $0.38$ | $429\ 182$ | $2.6$ | $\\{0.2,\ 0.2,\ 0.15\\}$
BOSS z1 SGC | $\\{0.2-0.5\\}$ | $0.38$ | $174\ 819$ | $1.0$ | $\\{0.2,\ 0.2,\ 0.15\\}$
BOSS z2 NGC | $\\{0.4-0.6\\}$ | $0.51$ | $500\ 872$ | $3.1$ | $\\{0.2,\ 0.2,\ 0.15\\}$
BOSS z2 SGC | $\\{0.4-0.6\\}$ | $0.51$ | $185\ 498$ | $1.1$ | $\\{0.2,\ 0.2,\ 0.15\\}$
LRGpCMASS NGC | $\\{0.6-1.0\\}$ | $0.698$ | $255\ 741$ | $2.72\ (\rm{NGC+SGC})$ | $\\{0.2,\ 0.2,\ 0.15\\}$
LRGpCMASS SGC | $\\{0.6-1.0\\}$ | $0.698$ | $121\ 717$ | $2.72\ (\rm{NGC+SGC})$ | $\\{0.2,\ 0.2,\ 0.15\\}$
eBOSS QSO NGC | $\\{0.8-2.2\\}$ | $1.52$ | $218\ 209$ | $0.35$ | $\\{0.2,\ 0.2,\ 0.15\\}$
eBOSS QSO SGC | $\\{0.8-2.2\\}$ | $1.52$ | $125\ 499$ | $0.18$ | $\\{0.2,\ 0.2,\ 0.15\\}$
Nseries | $\\{0.43-0.7\\}$ | $0.56$ | $\sim 660\ 000$ | $3.67\ (\times 84)$ | $\\{0.2,\ 0.2,\ 0.15\\}$
Table 1: Full list of clustering measurements which constitute our baseline
full-shape data vector, and mock catalogues used for model verification.
Values like $z_{\rm{eff}}$ and $V_{\rm{eff}}$ have been taken directly from
the Fourier space analysis papers, and may be computed using different
techniques between surveys. Note that the values quoted for our Nseries sample
correspond to the properties of a single mock realisation. Figure 1: Redshift
distribution of catalogues used in this analysis, counted over bins of width
$\Delta z=0.005$. While the bulk of our redshifts are within the
$z=\\{0.2-0.8\\}$ regime, the inclusion of nearby (6dFGS) and very distant
(eBOSS QSO) sources help address degeneracies in the full-shape modelling. The
purple region corresponds to the overlapping samples of the BOSS $z_{1}$ and
$z_{2}$ catalogues— While this overlap does not introduce any additional
sources over total $z=\\{0.2-0.6\\}$ region, it does increase the number of
galaxy pairs available for modelling.

###  3.1 Power Spectrum Measurements

Throughout this work, we use the PyBird code to simultaneously constrain
clustering measurements from the 6dFGS, BOSS, and eBOSS catalogues (Jones et
al., 2009; Alam et al., 2017; eBOSS Collaboration et al., 2020). The BOSS and
eBOSS catalogues are composed of measurements made in two non-overlapping sky
patches, referred to as the Northern and Southern Galactic Caps (hereafter NGC
and SGC). Following the convention of Alam et al. (2017), we further divide
the BOSS sample into two overlapping redshift bins, $0.2<z_{1}<0.5$, and
$0.4<z_{2}<0.6$, for a total of four BOSS clustering measurements. Note that
the clustering statistics for these four BOSS samples were computed directly
from observational catalogues by us, using the Nbodykit toolkit (Hand et al.,
2018). In line with the approach of Gil-Marín et al. (2020), we combine the
high-$z$ BOSS CMASS and eBOSS LRG samples, yielding two “LRGpCMASS” samples at
$z_{\textrm{eff}}=0.698$. Finally, we include the NGC and SCG clustering
measurements from the eBOSS QSO catalogue, at $z_{\textrm{eff}}=1.52$. This
combination of cuts yields a total of 9 unique clustering measurements
(summarised in Table 1, and visualised in Figure 1), which form the joint data
vector for our full-shape analysis In order to verify the accuracy and
reliability of our analysis pipeline (across both flat and curved cosmological
models), we also constrain cosmology from the mean of 84 Nseries N-body mock
catalogues (hereafter referred to as the “Nseries” sample), once again
computed using Nbodykit.

When computing clustering statistics from these observational catalogues, a
fiducial cosmology must be assumed in the conversion of redshifts to comoving
distances. To test how this choice could impact full-shape curvature
constraints, we compute clustering statistics from the Nseries mock catalogues
using the three fiducial cosmologies given in Table 2. The power spectra used
in our mock validation runs (the fits of Section 4.1) are conducted using our
flat fiducial cosmology. We then repeat our analysis of the Nseries mock
catalogues in Section 4.2, using power spectra which employ closed
$\Omega_{k}=+0.1$) and open ($\Omega_{k}=-0.1$) fiducial cosmologies. These
spatially curved fiducial cosmologies are otherwise identical to our flat
fiducial cosmology, with $\Omega_{k}$ set by fixing $\Omega_{\Lambda}$.

| $\Omega_{m}$ | h | $\Omega_{b}$ | $\Omega_{\Lambda}$ | $\Omega_{k}$
---|---|---|---|---|---
Flat | 0.31 | 0.676 | 0.0481 | 0.693 | 0
Closed | 0.31 | 0.676 | 0.0481 | 0.593 | 0.1
Open | 0.31 | 0.676 | 0.0481 | 0.793 | -0.1
Table 2: Fiducial cosmologies used to analyse Nseries mocks. Our
$\Omega_{k}=0$ cosmology corresponds to the fiducial cosmology employed in
Alam et al. (2017), where we modify $\Omega_{k}$ by fixing $\Omega_{\Lambda}$
from this baseline. In principle, the offset between our flat fiducial
cosmology from the underlying cosmology of the Nseries sample
($\Omega_{m}=0.286$) exposes our analysis to the systematics described in
Section 2.2.1. In practice, the effect of this flat fiducial offset is known
to be negligible for standard analyses, whereas the same is not a priori known
for different choices of $\Omega_{k}$.

A number of investigations have already used full-shape measurements to place
constraints on curvature from different combinations of the BOSS data.
Notably, Chudaykin et al. (2021) recover $\Omega_{k}=-0.043\pm 0.036$ from
joint fits which include power spectrum measurements derived from the DR12
$z_{1}$ and $z_{3}$ bins, along with post reconstruction BAO and supernovae
information. Similarly, Vagnozzi et al. (2020) combine power spectrum
measurements from the DR12 CMASS sample ($0.43<z<0.7$) with Planck
information, recovering constraints consistent with flatness to $\sim 0.2\%$.
Our combination of 6dFGS, BOSS, and eBOSS data represents an almost
uninterrupted series of clustering measurements across a large fraction of
cosmic history, providing an unprecedented level of constraining power from
full-shape analyses. We find this remarkable completeness across expansion
history helps insulate EFT constraints against parameter degeneracies which
emerge in full-shape only constraints of extended cosmological models
(particularly the curved $\Lambda$CDM fits conducted in this work).

Importantly, this completeness does require the use of the overlapping $z_{1}$
and $z_{2}$ redshift bins from BOSS, introducing non-trivial correlations
across these samples. To account for this effect, we compute the full cross-
covariance matrix for our overlapping BOSS samples using the publically
available MultiDark Patchy mock catalogues, provided as part of BOSS DR12
(Kitaura et al., 2016; Rodríguez-Torres et al., 2016). All other sky-chunks in
our analysis are treated as independent, and combined through the simple
addition of their respective likelihoods. It should be noted that while there
is a very small overlap between our LRGpCMASS and QSO samples across
$z=\\{0.8-1\\}$ (as seen in Figure 1), this region does not significantly
contribute to the two sets of measurements. 333Stöcker et al. (2021) found a
cross-correlation coefficient between the DR14 eBOSS LRG and QSO BAO
measurements of 0.29, but this was found to be $<0.1$ for the DR16 LRGpCMASS
and QSO samples as the overlapping cosmological volume constitutes a far
smaller fraction of the total volume for these latter two samples (eBOSS
Collaboration et al., 2020). Finally, we de-bias our estimate of the numerical
inverse of the covariance matrix for all sky-chunks using the correction
proscribed in Hartlap et al. (2007).

###  3.2 Analysis Methodology

####  3.2.1 Summary Statistics

In cosmology, we are generally interested in measuring the posterior
distribution of some ensemble of variables, to update their constraints in
light of some new information or measurement. This is succinctly encoded in
Bayes theorem,

$P(\theta|D)=\dfrac{P(D|\theta)P(\theta)}{P(D)}=\dfrac{\mathcal{L}(\theta)\pi(\theta)}{\mathcal{Z}},$
(3)

for a posterior P($\theta$|D), likelihood $\mathcal{L}$, prior $\pi$, and
evidence $\mathcal{Z}$.

While most of our attention is generally focused on P($\theta$|D) for the
purposes of model fitting, the evidence plays a central role in model
comparison by quantifying the performance of a model in the light of the data
(Trotta, 2008). In particular, we can compare the compatibility of two models
with the same a priori likelihood through the ratio of their evidences. This
technique has already been employed to shed light on the possible curvature
tension, with Handley (2021) assigning 50:1 Bayesian betting odds against
flatness from Planck 2018 CMB information. The Bayesian evidence can also be
used to determine the congruity of multiple datasets under combination,
through the calculation of their Bayes factor,

$R=\dfrac{\mathcal{Z}_{AB}}{\mathcal{Z}_{A}\mathcal{Z}_{B}},$ (4)

where $\mathcal{Z}_{AB}$ denotes the evidence derived from the product of
likelihoods $\mathcal{L}_{A}$ and $\mathcal{L}_{B}$. While the Bayes factor is
intrinsically prior dependent, Handley & Lemos (2019b) demonstrate how this
sensitivity can be minimised using the change in Kullback-Leibler divergence
($\mathcal{D}$), yielding the “suspiciousness statistic”,

$S=\frac{R}{I},\ \
\textrm{log}(I)=\mathcal{D}_{1}+\mathcal{D}_{2}-\mathcal{D}_{12}.$ (5)

In order to more broadly understand how full-shape measurements inform
curvature constraints, we measure $\mathcal{Z}$ for a range of dataset/model
combinations, using the evidence ratio to determine whether full-shape
information alone exhibits a preference for the inclusion of curvature in
model evaluation. We then make use of the Bayes factor to determine the
compatibility of full-shape information with Planck CMB data. When we quantify
this internal consistency, we follow the approach of Handley (2021) and
convert the suspiciousness statistic into a “tension probability” ($p$), using
the survival function of the chi-squared distribution,

$p=\int^{\infty}_{d-2\textrm{log(S)}}\chi^{2}_{d}(x)dx,$ (6)
$d=d_{A}+d_{B}-d_{AB}.$ (7)

$d_{A}$, $d_{B}$, and $d_{AB}$ correspond to the Bayesian model
dimensionalities for our individual and joint datasets respectively, defined
as

$d=2*(\langle\textrm{log}(\mathcal{L})^{2}\rangle-\langle\textrm{log}(\mathcal{L})\rangle^{2})$
(8)

We also quote this tension probability as a sigma value, calibrated as

$\sigma=\sqrt{2}\textrm{Erfc}^{-1}(p).$ (9)

####  3.2.2 Sampling Strategy

In order to efficiently sample our posterior distribution, we optimize
parameter jumps using adaptive Metropolis-Hastings sampling through
MontePython (Brinckmann & Lesgourgues, 2018). When computing evidences for
model comparison, we use MontePython as an interface with the nested sampling
algorithm Polychord (Handley et al., 2015). For each step along our
cosmological parameter space, we use the Boltzmann code CLASS to compute
linear power spectra for each unique clustering signal, evaluated at their
respective $z_{\textrm{eff}}$. We then use PyBird to compute the one-loop non-
linear power spectrum from these linear signals, subject to our EFT bias
parameters. To replicate the survey geometry and selection effects present in
observational data, each model power spectrum must be convolved with their
corresponding window function. When fitting the full ensemble of observational
catalogues given in Table 1, this yields 9 model $P(k)$ which may be directly
compared to $P(k)$ from our observed catalogues.

For our full-shape only posterior evaluations, we directly sample up to 6
cosmological parameters {ln($10^{10}A_{s}$), $h$, $\Omega_{\textit{cdm}}$,
$\Omega_{b}$, $n_{s}$, $\Omega_{k}$}, with 2 additional free EFT bias
parameters for each clustering sample {$b_{1}$, $c_{2}$}. We begin by
employing broad, flat priors over almost all of our cosmological and bias
parameters,

$\begin{split}\rm{ln}(10^{10}A_{s})=\\{1,4\\}\hskip
56.9055pth=\\{0.5,0.85\\}\\\ \Omega_{\textit{cdm}}h^{2}=\\{0.05,0.25\\}\hskip
39.83368pt\Omega_{k}=\\{-0.25,0.25\\}\\\ b_{1}=\\{0,4\\}\hskip
65.44142ptc_{2}=\\{-4,4\\}\end{split}$ (10)

| $\rm{ln}(10^{10}A_{\textit{s}})$ | $h$ | $\Omega_{\textit{cdm}}h^{2}$ | $\Omega_{\textit{m}}$ | $\Omega_{k}$
---|---|---|---|---|---
Truth: | 3.066 | 0.7 | 0.117 | 0.286 | 0
Nseries, flat $\Lambda$CDM, std cov | $3.04^{+0.21}_{-0.22}\ (3.12)$ | $0.699^{+0.017}_{-0.017}\ (0.694)$ | $0.114^{+0.009}_{-0.009}\ (0.110)$ | $0.281^{+0.016}_{-0.015}\ (0.275)$ | -
Nseries, flat $\Lambda$CDM, red cov ($25$) | $3.05^{+0.05}_{-0.05}\ (3.05)$ | $0.700^{+0.006}_{-0.006}\ (0.701)$ | $0.116^{+0.003}_{-0.003}\ (0.116)$ | $0.284^{+0.004}_{-0.004}\ (0.284)$ | -
Nseries, curved $\Lambda$CDM, std cov | $2.96^{+0.39}_{-0.42}\ (3.17)$ | $0.702^{+0.031}_{-0.030}\ (0.693)$ | $0.114^{+0.008}_{-0.008}\ (0.111)$ | $0.275^{+0.026}_{-0.025}\ (0.279)$ | $-0.03^{+0.11}_{-0.11}\ (0.02)$
Nseries, curved $\Lambda$CDM, red cov (25) | $3.08^{+0.10}_{-0.11}\ (3.04)$ | $0.699^{+0.009}_{-0.010}\ (0.701)$ | $0.116^{+0.003}_{-0.003}\ (0.116)$ | $0.285^{+0.007}_{-0.006}\ (0.282)$ | $0.01^{+0.03}_{-0.03}\ (-0.01)$
Table 3: Constraints recovered through fitting our high quality mock mean data
against $\Lambda$CDM models with fixed curvature ($\Omega_{k}$ = 0), and
models which permit free variations in curvature. with the maximum a
posteriori cosmology for each model given in brackets. Our model is able to
robustly recover the truth cosmology of the Nseries reference sample, across
both standard and reduced covariance fits. While the inclusion of $\Omega_{k}$
as a free parameter does affect our constraining power on
$\rm{ln}(10^{10}A_{s})$ and $h$, this does not work to systematically shift
our model away from the underlying truth cosmology. This consistent preference
for our truth cosmology serves to emphasise the robustness of our EFT pipeline
when used to constrain $\Omega_{k}$.

While clustering measurements can weakly constrain $\Omega_{b}$ (through the
relative ratio of the power spectrum peaks), information from big bang
nucleosynthesis (BBN) measurements offer robust, CMB-independent constraints
on the baryonic matter density. We use the measurements provided in Cooke et
al. (2018), to place a Gaussian prior on our fits of
$100\times\Omega_{b}h^{2}=2.235\pm{0.049}$. While the free-streaming of
massive neutrinos does impact the shape of the matter power spectrum, for
realistic neutrino masses this effect is firmly subdominant to our parameters
of interest. For simplicity, we therefore assume a cosmological model with
three massless neutrino species ($N_{\textrm{eff}}=3.046$) across all full-
shape fits.

In principle, the full-shape of the power spectrum can be used to directly
constrain the tilt of the primordial power spectrum ($n_{s}$), however such
constraints are an order of magnitude weaker than CMB measurements (Chudaykin
et al., 2020). Across our full-shape only constraints, we therefore undertake
each analysis twice— once using a spectral tilt fixed to $n_{s}=0.965$ (taken
from Aghanim et al., 2020), and once with $n_{s}$ as a free parameter. The
only exception to this is our model verification, where $n_{s}$ is fixed to
the truth values of the Nseries catalogue ($n_{s}=0.96$). Finally, when
running fits to clustering measurements (across both mock and observed data-
sets), we choose to constrain the monopole, quadrupole, and hexadecapole
($P_{0}$, $P_{2}$, $P_{4}$) signals from $k_{\rm{min}}=\\{0.01,0.01,0.01\\}$
to $k_{\rm{max}}=\\{0.2,0.2,0.15\\}$, with the exception of the 6dFGS
measurements (which are fit from $k_{\rm{min}}=\\{0.01,0.01,0.01\\}$ to
$k_{\rm{max}}=\\{0.15,0.15,0.1\\})$.

##  4 Full-Shape Model Tests

###  4.1 Mock Validation

In order to test the reliability of our fitting pipeline, we begin by
constraining the mean of 84 pre-reconstruction Nseries mock catalogues. The
Nseries mocks are full N-body simulations which replicate the observational
pattern of the NGC $0.43<z<0.7$ sample from the BOSS DR12 catalogue, generated
using a fixed $\Lambda$CDM cosmology (defined with $\textrm{ln}10^{10}A_{s}$ =
3.066, $h$ = 0.7, $\Omega_{m}$ = 0.286, $\Omega_{b}$ = 0.047, $\Omega_{k}$ =
0.0). In this verification stage we constrain the mean of 84 individual
Nseries mock realisations (propagated using the flat fiducial cosmology of
Table 2), creating an effectively noiseless “reference sample” to establish
the accuracy of results recovered from our analysis scheme. In order to
rigorously examine the impact of potential systematic errors, we repeat our
analysis using a standard covariance matrix (designed to replicate the
expected errors from the real measurement in the same volume), and a
covariance matrix which has been reduced in amplitude by a factor of 25. To
compute the standard errors for this reference sample we follow the approach
of Gil-Marín et al. (2020), and use the variance in 2048 MultiDark Patchy Mock
catalogues over the above redshift range (with an additional $10\%$ rescaling
factor to account for veto effects applied to the MultiDark mocks, but not the
Nseries catalogues). We provide the confidence interval and best-fit from
constraints of this high fidelity sample in Table 3, with the full posterior
from fits to curved models provided in Figure 2.

We find our analysis pipeline is able to recover the underlying cosmology of
the Nseries sample to a high degree of fidelity, across models with both fixed
and varying curvature. As the errors on this reference sample are reduced, our
model distinctly converges around the underlying Nseries cosmology (most
reassuringly, recovering $\Omega_{k,\textrm{true}}=0$). A notable degeneracy
between $\Omega_{k}$ and the scalar amplitude parameter $A_{s}$ can be
identified in our full posterior distribution, significantly weakening our
constraining power for curved models. Despite the presence of this degeneracy,
the fidelity of our posterior and best-fit across all validation fits strongly
indicates the full-shape model employed in this work provides unbiased
measurements of flat cosmologies, and can be applied to observation.

Figure 2: Posteriors recovered from analysis of our Nseries mock reference
sample in a curved $\Lambda$CDM framework, using our standard and reduced
amplitude covariance matrix (with the maximum a posterior cosmology for our
reduced covariance fit given in red). Notably, the inclusion of curvature as a
free parameter does not shift our posterior mean or a posteriori point away
from the truth value of the Nseries mock catalogues. As the errors assigned to
our reference sample are reduced our model remains unbiased, with both the
posterior mean and maximum a posteriori cosmology lining up with the truth
value of our reference catalogue. Figure 3: The Monopole (upper), Quadrupole
(middle) and Hexadecapole (lower) signals generated from propagating the
Nseries mocks using the three fiducial cosmologies given in Table 2. Note that
we also include the standard $P_{0}$ errors on our $\Omega_{\textrm{k,
fid}}=0$ sample for reference. Non-flat ($\Omega_{\textrm{k, fid}}=\pm 0.1$)
cosmologies are defined by fixing $\Omega_{m}$, $\Omega_{b}$, and $h$ to the
default values of our flat fiducial cosmology, where $\Omega_{\Lambda}$ is
varied to achieve the desired $\Omega_{k}$. The impact of our non-flat
fiducial cosmology is to uniformly dilate the matter power spectrum across
$k$, resulting in a visible dilation of the BAO feature. The modification of
our redshift-distance relationship also visibly modulates the overall
amplitude of the Monopole signal, well in excess of the measurement errors.
Figure 4: Reduced amplitude covariance fits to the mean of 84 Nseries mock
catalogues using the three fiducial cosmologies of Table 2, with a dotted line
corresponding to the posterior mean of our flat fiducial sample for reference.
We find the use of a fiducial cosmology which substantially deviates from the
underlying cosmology of our sample results in a small bias on $\Omega_{k}$
fits from our flat baseline, but which is subdominant to the statistical
variance of a standard measurement ($\sim 0.2\sigma$).

###  4.2 Impact of Fiducial Cosmology

With the accuracy of our full-shape pipeline validated against an effectively
noiseless mock-mean reference catalogue, we now consider the role that our
choice of fiducial cosmology plays in full-shape constraints of $\Omega_{k}$,
in line with Section 2.2.1. We fully recompute the clustering statistics for
our suite of 84 Nseries mock catalogues using two extremely curved fiducial
cosmologies (with $\Omega_{k\rm{,fid}}=\pm 0.1$, given in Table 2). It should
be noted that the offset between the underlying cosmology of our sample, and
the chosen fiducial cosmology of propagation, is much larger than the expected
offset in realistic fits. Indeed, this test was designed to be as sensitive as
possible to any residual effects which may arise from the use of a non-flat
fiducial cosmology in analysis, which can be traced back to model assumptions
or approximations. In line with this philosophy, we also use the reduced
amplitude covariance matrix of Section 4.1 throughout fitting, reducing the
final width of our measurement errors by a factor of 5. When recomputing
clustering measurements from this sample, we also recompute the number
density, FKP weights, and Poisson shot noises assigned to these samples,
yielding the clustering signals in Figure 3. Additionally, we fully recompute
the window function applied to our model power spectra, using random
catalogues which have been propagated using the same (curved) fiducial
cosmology as our mock data. Using these clustering signals to constrain our
curved $\Lambda$CDM model model (broadly repeating the technique of Section
4.1), we recover the posterior distributions provided in Figure 4.

After fully propagating our modified fiducial cosmology across all stages of
analysis, we recover a modest scatter in parameters which modulate the
amplitude of our clustering signal. Notably, the use of a fiducial cosmology
which is offset from our underlying sample by $\Omega_{\textrm{k, fid}}=+0.1$
yields a small offset in $\Omega_{k}$— Importantly, this effect is sub-
dominant to the statistical error assigned by our unmodulated covariance
matrix ($\sim 0.2\sigma$). Fiducial cosmological assumptions are therefore
unlikely to represent a major source of error in present day full-shape
constraints of curvature. As future galaxy redshift surveys provide
unprecedented improvements in scale and accuracy, the constraints offered by
full-shape measurements are likely to grow in importance as stand alone
measurements. While outside the scope of this work, it is important to
consider whether assumptions in fiducial cosmology (and indeed assumptions of
flatness in general) will continue to remain negligible with increasing
precision.

##  5 EFTofLSS and the "Curvature Tension"

Figure 5: Posteriors recovered from joint 6dFGS+BOSS+eBOSS constraints in both flat and curved $\Lambda$CDM models, when $n_{s}$ is fixed to the Planck 2018 value (left) and when $n_{s}$ is allowed to vary (right). Consistent with the findings of our mock verification, the inclusion of $\Omega_{k}$ substantially weakens our constraints on $\rm{ln}(10^{10}A_{s})$ and $h$, stretching these posteriors along their degeneracy direction. When $n_{s}$ is allowed to vary in addition to $\Omega_{k}$, the reduced constraining power on $\Omega_{\textit{cdm}}h^{2}$ further compounds this degeneracy, offsetting marginalised parameter constraints from flat models. Interestingly, across all tests our maximum a posterori cosmologies are consistently skewed along this degeneracy direction, yielding best-fits which are substantially closer to flatness than the posterior mean. | $\rm{ln}(10^{10}A_{\textit{s}})$ | $h$ | $\Omega_{\textit{cdm}}h^{2}$ | $\Omega_{\textit{m}}$ | $\Omega_{k}$ | $n_{s}$ | $2*\rm{log}(\mathcal{L})$
---|---|---|---|---|---|---|---
Flat, fixed $n_{s}$ | $2.85^{+0.11}_{-0.12}\ (3.03)$ | $0.667^{+0.011}_{-0.011}\ (0.672)$ | $0.114^{+0.005}_{-0.004}\ (0.115)$ | $0.307^{+0.010}_{-0.011}\ (0.304)$ | - | - | $367.2$
Curved, fixed $n_{s}$ | $2.55^{+0.21}_{-0.22}\ (2.77)$ | $0.686^{+0.015}_{-0.016}\ (0.665)$ | $0.115^{+0.004}_{-0.005}\ (0.111)$ | $0.291^{+0.014}_{-0.013}\ (0.302)$ | $-0.089^{+0.049}_{-0.046}\ (-0.042)$ | - | $366.3$
Flat, varying $n_{s}$ | $2.80^{+0.14}_{-0.13}\ (2.97)$ | $0.669^{+0.012}_{-0.011}\ (0.668)$ | $0.117^{+0.009}_{-0.008}\ (0.114)$ | $0.312^{+0.017}_{-0.014}\ (0.304)$ | - | $0.950^{+0.04}_{-0.051}\ (0.972)$ | $367.1$
Curved, varying $n_{s}$ | $2.19^{+0.29}_{-0.28}\ (2.62)$ | $0.707^{+0.021}_{-0.021}\ (0.686)$ | $0.127^{+0.011}_{-0.009}\ (0.116)$ | $0.300^{+0.016}_{-0.014}\ (0.295)$ | $-0.152^{+0.059}_{-0.053}(-0.089)$ | $0.878^{+0.053}_{-0.055}\ (0.932)$ | $364.8$
Table 4: $68\%$ confidence interval and best fit provided from analysis of our
combined 6dFGS+BOSS+eBOSS sample, across both fixed and varying
$n_{s}$/$\Omega_{k}$. When analysed in isolation, our suite of pre-
reconstruction full-shape measurements exhibit a $\sim 2\sigma$ preference for
curvature across all fits. Interestingly, while our curved models generally
perform better than their flat counterparts (as measured by
$\rm{log}(\mathcal{L})$), the maximum a posteriori point in both fits is
skewed from the posterior mean by $\sim 1\sigma$.

###  5.1 6dFGS+BOSS+eBOSS Fits

We now turn our attention to employing this validated pipeline to constrain
cosmological models using our joint 6dFGS+BOSS+eBOSS sample (following the
approach of Section 3.2.2). We provide the $68\%$ confidence interval across
all 4 full-shape only tests in Table 4, with the full posterior distribution
shown in Figure 5. Additionally, we provide the best-fitting model power
spectra (across all 9 sub-samples) from our curved, varying $n_{s}$ run in
Figure 6.

Combining clustering measurements from 6dFGS, BOSS, and eBOSS, allows us to
measure $\Omega_{k}$ to a precision of $5\%$ using full-shape information
alone, a result which is broadly competitive with other CMB-independent fits
in the literature. Relaxing this model and allowing $n_{s}$ to vary in
conjunction with $\Omega_{k}$ introduces substantial degeneracies in our fits,
notably weakening our constraints on $\textrm{ln}(10^{10}A_{s})$ and $h$
(parameters which both help modulate the amplitude of our broadband clustering
signal). Across both models, our suite of pre-reconstruction full-shape
measurements exhibit a $\sim 2\sigma$ preference for closed cosmologies. This
result is notably more significant than the $\sim 1\sigma$ preference obtained
from analysis in Chudaykin et al. (2020), where BOSS full-shape measurements
were combined with post-reconstruction BAO information. It is unclear whether
this difference in marginalised fits to $\Omega_{k}$ arises due to the impact
of post-reconstruction BAO information (which overwhelmingly supports
flatness), or from our inclusion of high-redshift clustering information.
While beyond the scope of this paper, these results highlight the value of
investigating the role of reconstruction in measurements of curvature which
utilise BAO information. Finally, it is interesting to note the maximum a
posteriori model across our curved fits consistently prefers cosmologies which
are $1\sigma$ closer to flatness than the posterior mean. In addition, we find
our goodness of fit (as measured by $\rm{log}(\mathcal{L})$) remains broadly
consistent across analyses— this suggests the inclusion of $\Omega_{k}$ and
$n_{s}$ in our full-shape models does not substantially improve the quality of
our fits.

Figure 6: Summary of model spectra generated for the 9 survey chunks of our
joint 6dFGS+BOSS+eBOSS sample, corresponding to the maximum a posteriori
cosmology from our curved run (where $n_{s}$ is allowed to vary). The model
power spectrum for each chunk is derived from our best-fitting ensemble
cosmology (given in Table 4), with independent bias parameters to model the
distinct selection/observational effects of each sub-sample. This scheme
provides robust fits across the redshift range of our sample, recovering a
reduced $\chi^{2}$ of 1.09, despite the substantial variations in population
and selection effects across our suite of measurements.

####  5.1.1 Evidence Ratio

With the $\textrm{log}(\mathcal{L})$ of our full-shape constraints providing
some insight into the relative performance of curved and flat cosmological
models, we now turn our attention to developing this model comparison using
the evidence ratio. We use nested sampling to compute the Bayesian evidence
given by our full-shape measurements when used to constrain flat and curved
cosmologies (using a fixed $n_{s}$). Using our posterior evaluation to refine
our nested sampling bounds, we apply the following hard priors for our full-
shape model comparison:

$\begin{split}\textrm{ln}(10^{10}A_{s})=\\{1.5,3.5\\}\hskip
56.9055pth=\\{0.6,0.75\\}\\\ \Omega_{\textit{cdm}}h^{2}=\\{0.09,0.16\\}\hskip
31.2982pt\Omega_{b}h^{2}=\\{0.02,0.024\\}\\\ \Omega_{k}=\\{-0.25,0.15\\}\hskip
110.96556pt\end{split}$ (11)

While only $\ln(\mathcal{Z})$ is required for this comparison, we additionally
quote the Kullback-Leibler divergence ($\mathcal{D}$) and Bayesian model
dimensionality ($d$) for each model. The summary of our results is provided in
Table 5.

| $\textrm{ln}(\mathcal{Z})$ | $\mathcal{D}$ | $d$
---|---|---|---
Flat | $-224.51\pm 0.24$ | $31.39\pm 0.23$ | $22.68\pm 0.68$
Curved | $-223.87\pm 0.22$ | $30.80\pm 0.22$ | $25.35\pm 0.86$
Table 5: Summary of results from nested sampling analysis of our full-shape data in flat and curved models, with derived statistics. | $\rm{ln}(10^{10}A_{\textit{s}})$ | $h$ | $\Omega_{\textit{cdm}}h^{2}$ | $\Omega_{\textit{m}}$ | $\Omega_{k}$ | $n_{s}$ | $2*\rm{log}(\mathcal{L})$
---|---|---|---|---|---|---|---
Planck, flat | $3.045^{+0.017}_{-0.016}\ (3.045)$ | $0.679^{+0.007}_{-0.007}\ (0.678)$ | $0.120^{+0.001}_{-0.002}\ (0.120)$ | $0.309^{+0.009}_{-0.009}\ (0.310)$ | - | $0.966^{+0.005}_{-0.004}\ (0.965)$ | $1003.2$
Planck, curved | $3.030^{+0.018}_{-0.019}\ (3.010)$ | $0.542^{+0.043}_{-0.042}\ (0.529)$ | $0.118^{+0.002}_{-0.002}\ (0.118)$ | $0.470^{+0.076}_{-0.069}\ (0.501)$ | $-0.042^{+0.019}_{-0.021}\ (-0.051)$ | $0.971^{+0.005}_{-0.005}\ (0.972)$ | $994.2$
Planck + FS, flat | $3.038^{+0.017}_{-0.016}\ (3.041)$ | $0.679^{+0.005}_{-0.005}\ (0.679)$ | $0.120^{+0.001}_{-0.001}\ (0.120)$ | $0.309^{+0.007}_{-0.007}\ (0.308)$ | - | $0.966^{+0.004}_{-0.004}\ (0.968)$ | $1377.7$
Planck + FS, curved | $3.037^{+0.017}_{-0.016}\ (3.035)$ | $0.667^{+0.008}_{-0.009}\ (0.687)$ | $0.118^{+0.001}_{-0.002}\ (0.117)$ | $0.317^{+0.009}_{-0.008}\ (0.317)$ | $-0.0041^{+0.0026}_{-0.0021}\ (-0.0052)$ | $0.969^{+0.005}_{-0.005}\ (0.968)$ | $1373.5$
Table 6: $68\%$ confidence interval and best fit provided from full-shape
analysis of the Planck TTTEEE lite likelihoods (i.e. with foreground
parameters marginalised), both individually and under combination with our
full-shape sample via Metropolis-Hastings sampling. Note that during our
analysis of the Planck likelihoods, we assume a cosmology with 3 massless
Neutrino species (in contrast to the baseline Planck cosmology, which assumes
a single massive species of $\Sigma m_{\nu}=0.06\ \textrm{eV}$). Our
assumption of three massless Neutrino species does not introduce any
significant deviation from the standard Planck results, and allows for a more
reliable comparison with our full-shape models (which similarly assume a
cosmology populated with massless Neutrinos).

We find the full-shape data alone exhibits a very marginal preference for open
cosmological models, assigning Bayesian betting odds of $\sim 2:1$ in favour
of curvature. This level of support broadly indicates our full-shape
measurements are equally compatible with either model, a result supported by
the weak improvement in the quality of curved fits as measured by
log($\mathcal{L}$).

###  5.2 Planck + Full-Shape Analysis

We now turn our attention to exploring the reliability of cosmological
constraints which combine full-shape and Planck information. We make use of
the nuisance-marginalised “Planck-lite” likelihood for TTTEEE spectra beyond
$\ell=30$, in conjunction with the standard low-$\ell$ TT and EE spectra below
this cut-off. This “lite” likelihood marginalises over the standard CMB
foreground parameters, encoding these into a single bias parameter
$A_{\textrm{Planck}}$. Posterior evaluations which include Planck data follow
the default MontePython configuration, which includes a hard prior on
$\tau_{\rm{reio}}>0.004$. All remaining cosmological parameters (including
$\Omega_{k}$) are not defined with explicit priors, allowing our runs to
extend over the entire parameter space supported by Class. As previously, our
full-shape likelihood includes a Gaussian prior of
$100\times\Omega_{b}h^{2}=2.235\pm 0.049$ (informed from Cooke et al. (2018))—
while this is included in runs which combine full-shape and Planck
information, the effect of this prior is marginal in comparison to the
$\Omega_{b}h^{2}$ fits provided by the Planck likelihood. As outlined in
Section 3.2.2 our full-shape cosmological model assumes three massless
Neutrino species, in contrast to the single massive Neutrino species (with
$\Sigma m_{\nu}=0.06\ \textrm{eV}$) used in the baseline Planck model. To
ensure our comparison with the full-shape information remains robust, we
therefore opt to compute all chains which include Planck information with our
three massless neutrino configuration (rather than using the publicly
available Planck chains). The full $68\%$ confidence intervals and best fits
from fits to the Planck data are provided in Table 6.

Reassuringly we find our use of 3 massless neutrino species do not
substantially affect our CMB constraints, recovering fits across flat and
curved $\Lambda$CDM models which are consistent with the standard Planck
analysis. While the posterior distribution from CMB measurements alone are
skewed to closed universes at a significance of $\sim 3\sigma$, including
full-shape information yields constraints which are consistent with flatness
to within $0.5\%$. Breaking the geometric degeneracy also brings our
measurements of $h$ and $\Omega_{m}$ into closer alignment with standard
cosmological fits.

While combinations of CMB and full-shape information provides incredibly
precise constraints on $\Omega_{k}$, in order to confidently combine these
measurements we must be convinced they are internally consistent. As with our
earlier full-shape only analysis, we evaluate this using the Bayesian
evidence, this time estimating the congruity of the Planck and full-shape data
via the suspicousness statistic. In order to include Planck data in our Nested
Sampling runs, we are forced to reduce our prior width on $\Omega_{k}$ to
$\\{-0.15,0.1\\}$, and perform a change of variables (substituting
$100\times\theta_{s}$ for $h$), with the full prior range given in Equation
LABEL:eq:PlanckxFSPrior. As noted in Handley (2021), unphysical cosmological
combinations at the edges of our prior volume (where the nested sampling run
begins) can cause standard cosmology calculations to fail. The full summary of
our analysis over this prior range is provided in Table 7, with the posteriors
from these nested sampling runs plotted in Figure 7

$\begin{split}\ln(10^{10}A_{s})=\\{1.5,3.5\\}\hskip
19.91684pt100\times\theta_{s}=\\{1.025,1.055\\}\\\
\Omega_{\textit{cdm}}h^{2}=\\{0.09,0.16\\}\hskip
34.14322pt\Omega_{b}h^{2}=\\{0.02,0.024\\}\\\ n_{s}=\\{0.78,1.08\\}\hskip
46.94687pt\Omega_{k}=\\{-0.15,0.1\\}\\\ \tau_{\rm{reio}}=\\{0.004,0.1\\}\hskip
113.81102pt\\\ \end{split}$ (12)

| $\textrm{ln}(\mathcal{Z})$ | $\mathcal{D}$ | $d$ |  Tension ($\sigma$)
---|---|---|---|---
FS | $-227.41\pm 0.20$ | $34.26\pm 0.21$ | $23.45\pm 0.67$ | -
Planck | $-519.97\pm 0.15$ | $19.01\pm 0.14$ | $8.35\pm 0.24$ | -
Planck + FS | $-747.41\pm 0.25$ | $50.72\pm 0.26$ | $26.28\pm 0.80$ | $1.76^{+0.14}_{-0.11}\sigma$
Table 7: Nested sampling analysis of our full-shape data and Planck
information, individually and in combination. Figure 7: Posterior distribution
provided from joint full-shape (6dFGS+BOSS+eBOSS) and CMB (Planck 2018) nested
sampling fits to curved $\Lambda$CDM models, in contrast with the individual
posteriors from both probes. In addition, we include our full-shape posterior
evaluated over our wide $h$ prior for reference (Note this is identical to the
curved posterior from Figure 5, right). While $\theta_{s}$ and $\Omega_{k}$
are directly sampled as part of our evidence calculations, $\Omega_{m}$ and
$h$ are included as derived parameters for clarity. The combination of full-
shape and CMB information provides remarkably tight constraints on
$\Omega_{k}$, arising due to the intersection of their perpendicular
degeneracy directions in $\Omega_{k}$/$\theta_{s}$ space. While full-shape
measurements can accomodate curved cosmologies (as visible through evaluation
over our wide $h$ prior), this comes at the expense of a fit to
$\theta_{s}$/$h$ which is strongly excluded by CMB measurements.

We find evidence for a moderate inconsistency between Planck 2018 and our
suite of full-shape measurements, corresponding to a tension probability of
$p\sim 0.08\pm 0.02$, or $1.76^{+0.14}_{-0.11}\sigma$. This inconsistency is
identifiable in the joint contours of Figure 7, where we find joint fits to
$\Omega_{k}$ and $100\times\theta_{s}$ are broadly incompatible with our
Planck only posterior. While full-shape measurements can accommodate closed
cosmologies (as evidenced by our Metropolis-Hastings analysis), this comes at
the expense of a $\theta_{s}$ which is strongly disfavoured by Planck.
Similarly, while Planck alone prefers a low value of $\Omega_{k}$, this comes
at the cost of a low $h$ and high $\Omega_{m}$ which is not supported by the
full-shape measurements.

As with any Bayesian statistic, it is essential to consider the choice of
prior in our analysis. While our full-shape models are strongly modified by
$\Omega_{m}$ and $h$, the extrapolation of these parameters from $\theta_{s}$
is highly sensitive to $\Omega_{k}$. Indeed, despite employing a very broad
prior of $100\times\theta_{s}=\\{1.025-1.055\\}$ (roughly 100 times the
statistical error from Planck), this low-redshift sensitivity enforces an
effective $\Omega_{k}$ prior on our full-shape measurements. Importantly, when
our full-shape measurements are combined with Planck information, the
resulting joint fits from our nested sampling runs remain wholly consistent
with the Metropolis-Hastings analysis provided in Table 6. While the
traditional Bayes ratio is explicitly prior dependent, our use of the
suspiciousness statistic ensures our model comparison remains more robust to
this choice of prior.

Interestingly, the tension measured in our analysis is much less significant
than tensions between Planck and other late-universe measurements available in
the literature, most notably the “decisive tension” between full-shape
information (from the BOSS $0.43<z<0.7$ signal) and Planck data measured in
Vagnozzi et al. (2020). While this result is surprising, there are a number of
novelties in our analysis which may explain this deviation. Firstly, our full-
shape analysis uses the widest range of clustering measurements to date, with
redshift measurements from our QSO sample extending out to $z=2.2$. The
inclusion of this (relatively) high redshift information from QSO clustering
may impact the resulting $\Omega_{k}/\theta_{s}$ degeneracy in full-shape
information which is so critical to our joint-fits. Indeed, in earlier tests
of this work fitting the individual BOSS DR12 redshift bins, we found that the
‘z3 NGC’ sample alone, which is similar to that used in Vagnozzi et al.
(2020), prefers a slightly lower $\Omega_{m}$ than our complete set of full-
shape data which could exacerbate the tension somewhat. In addition, we are
able to model these clustering statistics to much smaller scales than previous
studies, offering many more density modes for analysis. Finally, the impact of
our choice of prior (most notably the effect of our $\theta_{s}$ prior on
full-shape measurements in isolation) must be taken into account when
considering the significance of this measured tension. We stress that these
results should not be interpreted as conclusive evidence against the CMB/late
universe tension discussed in the literature. Indeed, until this inconsistency
is better understood, joint fits which combine CMB and late universe
information (including full-shape clustering measurements) should continue to
be treated with caution. Given the novel elements of our analysis however
(most notably our inclusion of high-redshift and small-scale clustering
information), these results represent an interesting contribution to the
continuing curvature tension debate.

##  6 Discussion and Conclusions

As improvements in measurement precision continue to expose inconsistencies
between cosmological datasets, the EFTofLSS provides a powerful new framework
to independently revisit measurements across a range of cosmological models.
In this paper we investigate the role of full-shape clustering measurements in
the emerging curvature tension debate, and explore how common analysis
assumptions could affect full-shape curvature constraints. Galaxy clustering
statistics frequently assume flatness through the use of reconstruction
algorithms, and in the choice of fiducial cosmology used to convert redshifts
to distances. We fully disentangle the assumptions of reconstruction from our
work by fitting the full-shape of the power spectrum in isolation (i.e.
without the inclusion of post-reconstruction BAO information). While our
analysis does rely on some choice of fiducial cosmology, we quantify the
impact of assuming a consistent, non-flat fiducial cosmology across all stages
of analysis. Using two extremely curved fiducial cosmologies
($\Omega_{k\textrm{,fid}}=\pm 0.1$), we identify a small scatter with respect
to our baseline (i.e. $\Omega_{k,\textrm{fid}}=0$) validation fits. The
resulting offsets are sub-dominant to the expected statistical error on
realistic measurements (at a significance of $0.2\sigma$), and as such are
unlikely to bias current full-shape constraints of $\Omega_{k}$.

With our pipeline validated against mock catalogues, we constrain flat and
curved $\Lambda$CDM cosmologies using a combination of measurements from the
6dFGS, BOSS, and eBOSS samples. Using these full-shape power spectrum
measurements alone, we recover robust constraints on flat $\Lambda$CDM
cosmologies which are consistent with the Planck results to within $1\sigma$.
When we extend our analysis to include curvature as a free parameter, we
recover fits of $\Omega_{k}$ of $-0.089^{+0.049}_{-0.046}$ (when $n_{s}$ is
fixed) and $\Omega_{k}=-0.152^{+0.059}_{-0.053}$ (when $n_{s}$ is allowed to
vary). Comparing the relative performance of these models using the Bayesian
evidence ratio yields betting odds of 2:1 in favour of curvature, indicating
our full-shape data remains broadly compatible with both cosmological models.

When our suite of full-shape measurements are combined with Planck CMB
information to jointly constrain curved $\Lambda$CDM models, we recover
results which are consistent with flatness to within $0.5\%$
($\Omega_{k}=-0.0041^{+0.0026}_{-0.0021}$). Using the suspiciousness statistic
we identify a moderate tension between CMB and full-shape measurements in
curved $\Lambda$CDM models, with a tension probability of $p\sim 0.08\pm
0.02$, or $1.76^{+0.14}_{-0.11}\sigma$. Like any Bayesian measurement, this
should be understood as the tension recovered given our chosen model
parameterisation and prior— as such, our results should not be understood as
broad evidence against the strong CMB/late universe tension recorded in the
literature. However, given the novel elements of our analysis (most notably,
our use of very high-$z$ and small-scale clustering information), we argue
this is an interesting result in the emerging curvature tension debate. It
will thus be interesting to explore further with upcoming data from surveys
such as DESI (DESI Collaboration et al., 2016), which our methodology and
pipeline enables us to do.

## Acknowledgements

The authors would like to thank Will Handley and David Parkinson for very
useful discussions and correspondence. This research was supported by the
Australian Government through the Australian Research Council’s Laureate
Fellowship funding scheme (project FL180100168). AG is the recipient of an
Australian Government Research Training Program (RTP) Scholarship.

## Data Availability

The PyBird code used in this analysis is available here, and the MontePython
likelihoods built on this model are available here. The Nseries mock
catalogues used in this work, alongside the LRGpCMASS samples are available
here. The remaining BOSS and eBOSS measurements used in this analysis are
available from the following url, with the 6dFGS catalogues available here.
Any additional data products will be made available upon reasonable request to
the corresponding author.

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