#  Cosmology with the EFTofLSS and BOSS: dark energy constraints and a note on
priors

Pedro Carrilho  Chiara Moretti  and Alkistis Pourtsidou

###### Abstract

We analyse the BOSS DR12 multipoles of the galaxy power spectrum jointly with
measurements of the BAO scale for three different models of dark energy. We
use recent measurements performed with a windowless estimator, and an
independent and fast pipeline based on EFTofLSS modelling implemented via the
FAST-PT algorithm to compute the integrals of the redshift-space loop
corrections. We accelerate our analysis further by using the bacco linear
power spectrum emulator instead of a Boltzmann solver. We perform two sets of
analyses: one including $3\sigma$ Planck priors on $A_{s}$ and $n_{s}$, and
another that is fully CMB-free, i.e., letting the primordial parameters vary
freely. The first model we study is $\Lambda$CDM, within which we reproduce
previous results obtained with the same estimator. We find a low value of the
scalar amplitude in the CMB-free case, in agreement with many previous
analyses of the BOSS data. We then study $w$CDM, finding a lower value of the
amplitude in the CMB-free run, coupled with a preference for phantom dark
energy with $w=-1.17^{+0.12}_{-0.11}$, again in broad agreement with previous
results. Finally, we investigate the dark scattering model of interacting dark
energy, which we label $wA$CDM. In the CMB-free analysis, we find a large
degeneracy between the interaction strength $A$ and the amplitude $A_{s}$,
hampering measurements of those parameters. On the contrary, in our run with a
CMB prior, we are able to constrain the dark energy parameters to be
$w=-0.972^{+0.036}_{-0.029}$ and $A=3.9^{+3.2}_{-3.7}$, which show a 1$\sigma$
hint of interacting dark energy. This is the first measurement of this
parameter and demonstrates the ability of this model to alleviate the
$\sigma_{8}$ tension. Our analysis can be used as a guide for the analysis of
any model with scale-independent growth. Finally, we study the dependence of
the results on the priors imposed on the nuisance parameters and find these
priors to be informative, with their broadening generating shifts in the
contours. We argue for an in depth study of this issue, which can affect
current and forthcoming analyses of LSS data.

##  1 Introduction

The standard model of cosmology has enjoyed great success in describing the
observations of cosmological surveys, from the cosmic microwave background
(CMB) [1] to the large-scale structure (LSS) [2]. In spite of these successes,
the main ingredients of this $\Lambda$CDM model – cold dark matter (CDM) and a
cosmological constant ($\Lambda$) – are still poorly understood from the
fundamental point of view and it is possible that their replacement with
alternative components is necessary. Alternative theories for dark energy
include modified gravity, as well as quintessence, with the possibility of
additional interactions with dark matter [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

In addition to its theoretical issues, the $\Lambda$CDM model has recently
been challenged by a number of observed tensions between different sets of
cosmological data [13, 14, 15, 16, 17]. A particularly interesting discrepancy
is that between the amplitude of perturbations as observed at late times, when
compared to the estimate from CMB observations. This discordance arises in the
determination of $\sigma_{8}$, the variance of density fluctuations averaged
over spheres of $8~{}{\rm Mpc}/h$, which is measured to be lower than expected
from the Planck best-fit $\Lambda$CDM cosmology [1]. This has been seen in
multiple probes of the growth of structure, such as in measurements of weak
lensing and photometric clustering [18, 19, 20, 21, 22, 23, 24, 25],
spectroscopic clustering [26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and galaxy
clusters [36]. In particular, the values of the best-constrained quantity,
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{m}/0.3}$, are found to be up to 10% lower
($S_{8}\approx 0.77$) than the value inferred by Planck ($S_{8}\approx 0.83$).

A number of models have been proposed to alleviate this tension, with dark
sector interactions being a promising possibility [4, 37]. The main reason for
this is that the addition of the dark interaction can slow the growth of dark
matter density fluctuations, thus reducing their amplitude at late times.
Different flavours of these interacting dark energy (IDE) models have been
explored, most of which include energy exchange as a result of the
interaction. While these energy-transfer models have had some recent success
in fitting observations [38, 39, 40], they typically are heavily constrained
by CMB data [41, 42, 43, 44] due to the substantial modifications they
introduce to background evolution. Here we focus instead on models for which
only momentum is exchanged between dark matter and dark energy [45, 12, 46,
47]. This avoids any modification to background evolution beyond that
generated by dynamic dark energy, which allows for better fits to data [48,
49, 50, 51]. In this work we investigate the Dark Scattering model [45], for
which dark matter particles are hypothesised to scatter elastically with dark
energy, exchanging only momentum. This is a well-developed model, thoroughly
studied with N-body simulations [52, 53] and with robust nonlinear modelling
already developed [54, 55]. Similarly to other interacting models, it is able
to alleviate the $S_{8}$ tension by slowing down the clustering of dark matter
[56, 52, 53, 50]. Similar models which also have this property include Refs.
[57, 49, 58, 59, 60, 61, 62, 63, 64].

Current and forthcoming stage IV galaxy surveys, such as the Dark Energy
Spectroscopic Instrument (DESI) [65], ESA’s Euclid satellite mission [66, 67],
Rubin’s Legacy Survey of Space and Time (LSST) [68], and the Nancy Grace Roman
space telescope [69] are expected to pin down the details of the dark sector
to unprecedented accuracy and will allow for the distinction between many
alternatives to $\Lambda$CDM, including IDE models. To ensure the success of
these efforts, it is important that the theoretical predictions from these
non-standard cosmologies have the highest quality, as the volume of data and
its precision will be unprecedented, leaving no place for theoretical
systematics to hide [70, 71, 72, 73, 74, 75, 76, 23, 22]. This has led to the
development of new techniques extending the range of scales for which
predictions are trustworthy. A key recent development is the effective field
theory of large-scale structure (EFTofLSS) [77, 78]. This technique models the
unknown effects of very small scales on mildly nonlinear scales via the
introduction of effective stresses and adds corresponding time-dependent
parameters, improving the fits to simulations [74].

These new techniques are powerful enough to be useful already in the analysis
of current data. In particular, the EFTofLSS has been used by different groups
to re-analyse the summary statistics measured by BOSS, both in the context of
$\Lambda$CDM [32, 34, 35, 33, 79], as well as for extended models [80, 81,
38]. Our aim in this paper is exactly this – use the EFTofLSS to analyse the
BOSS data in order to constrain exotic dark energy and dark scattering
interactions. We use an independent implementation of the EFTofLSS-based
modelling validated in previous work [54] to predict the multipoles of the
power spectrum and run an MCMC analysis for all cosmological parameters on the
BOSS DR12 data, including also post-reconstructed BAO data in a broad redshift
range. We are thus able to constrain the strength of the interaction, in
parallel with the dark energy equation of state (EOS) parameter, for the first
time.

This paper is organised as follows. The data and the methods we use in our
analysis pipeline are described in Section 2. Our baseline results are shown
in Section 3, with full contour plots shown in Appendix B and a detailed
discussion on the information content of the priors of nuisance parameters and
their effects on the results in Appendix A. Finally, we detail our conclusions
in Section 4.

##  2 Data and Methods

###  2.1 Datasets

The main dataset we analyse is the Baryon Oscillation Spectroscopic Survey
(BOSS) Data Release 12 (DR12) measurement of the power spectrum multipoles
[82, 83, 84, 30]. This is based on the CMASS and LOWZ samples, containing in
excess of a million galaxies, observed in the Northern and Southern galactic
caps (NGC and SGC). These are separated into two redshift bins (0.2 < z < 0.5
and 0.5 < z < 0.75) with effective redshifts $z_{1}=0.38$ and $z_{3}=0.61$.
This results in four distinct sets of measurements with differing selection
criteria, number density and volume.

We use here the measurements of the windowless power spectrum multipoles, as
provided by Ref. [33]111The measured spectra and the code to generate them can
be found at github.com/oliverphilcox/BOSS-Without-Windows, while a likelihood
using them can be found at github.com/oliverphilcox/full_shape_likelihoods.
and based on the estimator of Ref. [85] (see also Ref. [86] for a similar
treatment for the bispectrum). This allows our theoretical predictions to be
directly compared to data, without the need for a convolution with the window
function.

Additionally, we include the post-reconstructed BAO measurements from the same
BOSS data release as provided in Ref. [33], giving 4 different measurements of
the parameters $\alpha_{\parallel}$, $\alpha_{\perp}$ at $z=0.38$ and
$z=0.61$. We also include low redshift measurements of the volume distance
from 6DF at $z=0.106$ [87] and from the SDSS DR7 MGS at $z=0.15$ [88], as well
as high redshift measurements of the Hubble and angular diameter distance from
Lyman-$\alpha$ forest auto-correlation and cross-correlation with quasars from
eBOSS DR14 measurements at $z=2.334$ [89].

The covariance matrix for the full shape measurements is also provided by Ref.
[33] and includes the cross-covariance between the BAO measurements and the
multipoles of the power spectrum. It was produced using the publicly available
suite of 2048 ‘MultiDark-Patchy’ mock catalogs [90, 91].

We include a Gaussian prior on the baryon density from Big Bang
Nucleosythesis, $\omega_{b}=0.02268\pm 0.00038$ [34, 92, 93, 94, 95]. For each
case, we perform two different analysis, one of which includes a 3-$\sigma$
Planck prior on the parameters of the primordial power spectrum, using $\log
A_{s}=3.044\pm 0.042$ and $n_{s}=0.9649\pm 0.012$ [1]. As discussed below in
more details, this is crucial for breaking degeneracies in the interacting
dark energy model and we consider that our baseline analysis in that case. Our
more general analysis is fully CMB-free, and therefore does not include any
priors on the primordial parameters.

###  2.2 Modelling

We consider three distinct dark energy models, starting with $\Lambda$CDM,
then moving to $w$CDM and finally investigating the Dark Scattering model.
Since these models are built as extensions of each other, we describe here the
most general case of Dark Scattering, from which the other models can be
easily obtained via appropriate limits. This is an interacting dark energy
model, in which cold dark matter exchanges momentum with dark energy via
elastic scattering. This model does not introduce an exchange of energy
between those two species and for that reason does not change the expansion
history with respect to the $w$CDM model. Here we briefly review the evolution
of the different species as well as the modelling used to construct the galaxy
power spectrum in redshift space.

####  2.2.1 Evolution equations

In the assumed flat Friedmann-Lemaître-Robertson-Walker spacetime, the
Friedmann equation at late times is given by

$H^{2}=H_{0}^{2}\left(\Omega_{m}a^{-3}+\Omega_{\rm{DE}}a^{-3(1+w)}\right)\,,$
(2.1)

where $H=\dot{a}/a$ is the Hubble rate, with $a$ the scale factor and $H_{0}$
its value at $z=0$; $\Omega_{m}$ is the matter density parameter at $z=0$,
including cold dark matter ($\Omega_{c}$), baryons ($\Omega_{b}$) and massive
neutrinos ($\Omega_{\nu}$); dark energy is assumed to be dynamic, with
constant equation of state parameter $w\equiv P_{\rm DE}/\rho_{\rm DE}$ and
density parameter $\Omega_{\rm{DE}}$.

We describe the matter fluctuations with perturbation theory [96] in the
Newtonian approximation [97]. Within the Dark Scattering model [45, 52, 53,
54, 55], the only equations that are modified relative to $w$CDM are the
momentum evolution equations for dark energy and dark matter. Under the
assumption that the dark energy sound speed is unity, we can neglect dark
energy fluctuations and their evolution. Given that, the evolution equations
for dark matter fluctuations are given by

$\displaystyle
a\partial_{a}\delta(\mathbf{k})+\Theta(\mathbf{k})=-\int{\frac{\rm{d}^{3}\mathbf{k_{1}}\rm{d}^{3}\mathbf{k_{2}}}{(2\pi)^{3}}\delta^{(3)}(\mathbf{k}-\mathbf{k_{1}}-\mathbf{k_{2}})}\alpha(\mathbf{k_{1}},\mathbf{k_{2}})\Theta(\mathbf{k_{1}})\delta(\mathbf{k_{2}})\,,$
(2.2) $\displaystyle
a\partial_{a}\Theta(\mathbf{k})+\left(2+(1+w)\xi\frac{\rho_{\rm
DE}}{H}+\frac{a\partial_{a}H}{H}\right)\Theta(\mathbf{k})-\left(\frac{k}{aH}\right)^{2}\phi(\mathbf{k})=$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\frac{1}{2}\int{\frac{\rm{d}^{3}\mathbf{k_{1}}\rm{d}^{3}\mathbf{k_{2}}}{(2\pi)^{3}}\delta^{(3)}(\mathbf{k}-\mathbf{k_{1}}-\mathbf{k_{2}})}\beta(\mathbf{k_{1}},\mathbf{k_{2}})\Theta(\mathbf{k_{1}})\Theta(\mathbf{k_{2}})\,,$
(2.3)

where $\delta$ is the density contrast, $\Theta$ is the normalized divergence
of the velocity field, time derivatives $\partial_{a}$ are written with
respect to the scale factor and $\alpha$ and $\beta$ are the usual nonlinear
couplings [98, 97, 54]. The key difference between this and the standard case
is the interaction term, which depends on the interaction cross section,
$\sigma_{\rm DS}$, and the dark matter mass, $m_{c}$, via $\xi=\sigma_{\rm
DS}/m_{c}$.

These equations are solved in conjunction with the Poisson equation for the
gravitational field order by order in perturbations, within the Einstein-de
Sitter (EdS) approximation, up to 1-loop order. Additionally, we evolve the
fluctuations of baryons and dark matter as a single species, using the
effective coupling approximation employed in Ref. [55]. This approximation
consists of using Eqs. (2.2) and (2.3) for the evolution of all cold species,
while taking into account the effect of non-interacting baryons by lowering
the effective coupling between dark matter and dark energy. In practice, this
is done by substituting the coupling $\xi$ with the effective coupling
$\bar{\xi}$ defined by

$\bar{\xi}=\frac{f_{c}}{1+C_{0}(1-f_{c})}\xi\,,$ (2.4)

in which $f_{c}\equiv\Omega_{c}/\Omega_{m}$ is the dark matter fraction and
$C_{0}=(1+w)\xi\rho_{\rm DE\,0}/H_{0}$ is the coupling term in Eq. (2.3)
evaluated at $z=0$. The effective strength of the interaction, and the
parameter we will ultimately probe is therefore the combination

$A\equiv(1+w)\bar{\xi}\,.$ (2.5)

We denote the Dark Scattering model in this work by $wA$CDM, to make it
explicit that this is a simple extension to $w$CDM with all distinguishing
effects parameterised by the interaction parameter $A$.

Finally, massive neutrinos are approximated to evolve linearly, with their
effect taken into account via the linear power spectrum that will later be
used in constructing the galaxy power spectrum.

####  2.2.2 EFTofLSS power spectrum

To construct the galaxy power spectrum in redshift space, we use the EFTofLSS
model of Refs. [34, 99] (see also Refs. [100, 101, 32, 102] for other similar
models), which was also used in the context of Dark Scattering in Ref. [54]
and shown to be more effective in describing the power spectrum of biased
tracers compared to alternatives such as the Taruya-Nishimichi-Saito (TNS)
model [103]. The model we adopt includes a number of ingredients which we now
briefly review.

The bias model used is based on a perturbative and gradient expansion of the
galaxy density field (see Refs. [104, 105, 106, 107] for details), given by

$\delta_{g}=b_{1}\delta_{cb}+\frac{b_{2}}{2}\delta_{cb}^{2}+b_{\mathcal{G}_{2}}\mathcal{G}_{2}+b_{\Gamma_{3}}\Gamma_{3}+\epsilon,$
(2.6)

where $b_{1}$ is the linear bias, $b_{2}$ is the quadratic bias and
$\mathcal{G}_{2}$ and $\Gamma_{3}$ are non-local functions of density, with
corresponding bias parameters. We neglect here a higher derivative term
proportional to $\nabla^{2}\delta_{cb}$ which is exactly degenerate with an
EFTofLSS counter-term. The stochastic contribution is given here by
$\epsilon$, which is taken to have a power spectrum of the form

$P_{\epsilon\epsilon}(\mbox{\boldmath$k$})=N+e_{0}k^{2}+e_{2}k^{2}\mu^{2}\,,$
(2.7)

where we assume the constant part, $N$, to include the Poisson shot noise and
possible deviations from it, and we add two scale-dependent terms with
different redshift-space dependence. Here and elsewhere,
$\mu\equiv\hat{}\mbox{\boldmath$k$}\cdot\hat{}\mbox{\boldmath$n$}$ is the
cosine of the angle between the wave-vector $k$ and the line-of-sight
direction $\hat{}\mbox{\boldmath$n$}$. As is clear from our bias modelling, we
follow the standard approximation for massive neutrinos in which the galaxy
field is assumed to depend only on the density of cold matter, excluding
neutrinos. This approach is also used in Refs. [99, 108, 80], and is based on
the simulation-based results of Refs. [109, 110].

We use the EFTofLSS to include the effects of small scales on mildly nonlinear
ones by adding counter-terms to the redshift-space power spectrum, which is
based on 1-loop perturbation theory [97]. This results in the following
expression for the galaxy power spectrum:

$\displaystyle P_{gg}(\mbox{\boldmath$k$})=$ $\displaystyle
Z_{1}^{2}(\mbox{\boldmath$k$})P_{L}(k)+2\int\mathrm{d}^{3}\mbox{\boldmath$q$}\,\left[Z_{2}(\mbox{\boldmath$q$},\mbox{\boldmath$k$}-\mbox{\boldmath$q$})\right]^{2}P_{L}(q)P_{L}(|\mbox{\boldmath$k$}-\mbox{\boldmath$q$}|)$
$\displaystyle+6Z_{1}(\mbox{\boldmath$k$})P_{L}(k)\int\mathrm{d}^{3}\mbox{\boldmath$q$}\,Z_{3}(\mbox{\boldmath$k$},\mbox{\boldmath$q$},-\mbox{\boldmath$q$})P_{L}(q)+P_{\rm
ctr}(\mbox{\boldmath$k$})+P_{\epsilon\epsilon}(\mbox{\boldmath$k$})\,,$ (2.8)

where the redshift-space kernels $Z_{1}(\mbox{\boldmath$k$})$,
$Z_{2}(\mbox{\boldmath$k$}_{1},\mbox{\boldmath$k$}_{2})$,
$Z_{3}(\mbox{\boldmath$k$}_{1},\mbox{\boldmath$k$}_{2},\mbox{\boldmath$k$}_{3})$
can be found in Eq. (2.14) of Ref. [34], which includes more details and a
derivation with the same bias basis used here (see also Ref. [111]). We denote
the linear power spectrum as $P_{L}(k)$ and have omitted the redshift
dependence for brevity. We use the EdS approximation when computing these
kernels, such that their redshift dependence is encoded in the linear growth
factor, $D$, and the linear growth rate, $f$. The contribution from counter-
terms is given by

$\displaystyle P_{\rm ctr}(k,\mu)=$
$\displaystyle-2\tilde{c}_{0}k^{2}P_{L}(k)-2\tilde{c}_{2}k^{2}f\mu^{2}P_{L}(k)-2\tilde{c}_{4}k^{2}f^{2}\mu^{4}P_{L}(k)$
$\displaystyle+c_{\nabla^{4}\delta}k^{4}f^{4}\mu^{4}(b_{1}+f\mu^{2})^{2}P_{L}(k)\,,$
(2.9)

where we define four counter-term parameters, resulting in eleven nuisance
parameters in total.

Infrared re-summation is performed by splitting the linear power spectrum in a
smooth, broadband component $P_{\rm nw}$ and a wiggle component $P_{\rm w}$.
We then evaluate the loop terms in Eq. (2.2.2) with both the usual linear
power spectra and $P_{\rm nw}$, and construct a wiggle version of the loop
terms by subtracting the two. This, together with the linear contribution
$P_{\rm w}$, constitutes $P_{gg}^{\rm w}$. We then apply the appropriate
damping factors to the 1-loop and linear terms and sum the results via

$P_{gg}^{\rm re-sum}=P_{gg}^{\rm nw}+e^{-k^{2}\Sigma_{\rm tot}^{2}}P_{gg}^{\rm
w}+e^{-k^{2}\Sigma_{\rm tot}^{2}}k^{2}\Sigma_{\rm
tot}^{2}(b_{1}+f\mu^{2})^{2}P_{L}^{\rm w}(k)\,,$ (2.10)

where $\Sigma_{\rm tot}$ is given by Eq. (2.30) of Ref. [99]. The wiggle-no-
wiggle decomposition is performed using an Eisenstein-Hu fitting function
[112].

Due to the assumption of a particular fiducial cosmology in the measurement of
the data, we apply the Alcock-Paczynski rescaling to our theoretical
prediction. This is done in the standard way, by modifying the scales and
angles via

$\displaystyle\bar{k}^{2}=k^{2}\left(\frac{H_{0}^{\rm
fid}}{H_{0}}\right)^{2}\left(\left(\frac{H}{H^{\rm
fid}}\right)^{2}\mu^{2}+\left(\frac{D_{A}^{\rm
fid}}{D_{A}}\right)^{2}(1-\mu^{2})\right)\,,$ (2.11)
$\displaystyle\bar{\mu}^{2}=\mu^{2}\frac{H}{H^{\rm fid}}\left(\frac{H_{0}^{\rm
fid}}{H_{0}}\right)^{-2}\left(\left(\frac{H}{H^{\rm
fid}}\right)^{2}\mu^{2}+\left(\frac{D_{A}^{\rm
fid}}{D_{A}}\right)^{2}(1-\mu^{2})\right)^{-1}\,,$ (2.12)

and applying a rescaling to the power spectrum of

$A_{\rm AP}=\left(\frac{H_{0}^{\rm fid}}{H_{0}}\right)^{3}\frac{H}{H^{\rm
fid}}\left(\frac{D_{A}^{\rm fid}}{D_{A}}\right)^{2}\,,$ (2.13)

where all quantities are evaluated at the effective redshift of the sample,
and fiducial quantities are evaluated in the fiducial cosmology, which is a
flat $\Lambda$CDM cosmology with $\Omega_{m}=0.31$, $H_{0}=0.676$,
$\Omega_{b}h^{2}=0.022$ and $M_{\nu}=0.06~{}{\rm eV}$ [30].

After the application of the AP distortion we compute the multipoles of the
power spectrum with:

$P_{\ell}(k)=\frac{2\ell+1}{2}\int_{-1}^{1}{\rm d}\mu
P(k,\mu)\mathcal{P}_{\ell}(\mu)\,,$ (2.14)

where $\mathcal{P}_{\ell}(\mu)$ is the Legendre polynomial of order $\ell$. We
follow Ref. [99] and re-define three of our counter-term parameters to be the
combinations arising for each multipole, resulting in

$c_{0}=\tilde{c}_{0}+\frac{f}{3}\tilde{c}_{2}+\frac{f^{2}}{5}\tilde{c}_{4}\,,\
c_{2}=\tilde{c}_{2}+\frac{6f}{7}\tilde{c}_{4}\,,\ c_{4}=\tilde{c}_{4}\,.$
(2.15)

The full set of 11 nuisance parameters per redshift bin that we use is
therefore

$\theta_{\rm
nuis}=\\{b_{1},b_{2},b_{\mathcal{G}_{2}},b_{\Gamma_{3}},N,e_{0},e_{2},c_{0},c_{2},c_{4},c_{\nabla^{4}\delta}\\}\,.$
(2.16)

To accelerate our analysis, the linear power spectrum is computed using the
bacco emulator [113, 114, 115]. This provides accurate evaluations of the
output of Boltzmann solvers for general cosmologies in a fraction of the time,
giving order 100 speed-up of the linear calculations. The calculation of the
growth is done via a fast numerical integration of the linear version of Eqs.
(2.2), (2.3). The integrals of the redshift-space loop corrections are
computed using a custom implementation based on the Fast-PT algorithm [116,
117], which also improves the speed of our theoretical predictions
substantially with respect to standard integration techniques. We note that
this is an independent pipeline from those publicly available, such as CLASS-
PT [99] and pybird [80]. The python-based theory code employed here is part of
a complete likelihood pipeline, which will be made available in a dedicated
paper [118]. In this work, we use only the theoretical prediction of the power
spectrum and a distinct likelihood constructed specifically to analyse this
data. However, this code includes also a full likelihood for the joint
analysis of the power spectrum and bispectrum, which has been used in Refs.
[119, 120].

To predict the BAO signal, we compute the Hubble rate and angular diameter
distance numerically with a simple code validated against CLASS [121] and CAMB
[122], and obtain the size of the sound horizon at the drag epoch using a
fitting formula from Ref. [123], given by

$r_{d}=\frac{55.154\exp\left[-72.3(\omega_{\nu}+0.0006)^{2}\right]}{\omega_{b}^{0.12807}(\omega_{c}+\omega_{b})^{0.25351}}\,,$
(2.17)

where we also approximate the neutrino density via
$\omega_{\nu}=0.0107~{}M_{\nu}/{\rm eV}$. We then compute the different
combinations of distances as required by the different datasets available.

####  2.2.3 Analysis set-up

We start by considering a Gaussian likelihood constructed with the power
spectrum multipoles, the BAO signal and including their cross covariance where
relevant. We then perform analytical marginalization over the 8 nuisance
parameters that appear linearly in the power spectrum, which are $b_{\Gamma
3},N,e_{0},e_{2},c_{0},c_{2},c_{4},c_{\nabla^{4}\delta}$. We follow the
procedure explained in Refs. [32, 80] and thus obtain a modified posterior
distribution which is no longer Gaussian with respect to the data vector, but
drastically reduces the dimension of the sampled parameter space, accelerating
the convergence of the MCMC analysis.

In our main analyses, we include multipole data from all 3 multipoles from 2
redshifts and 2 sky cuts up to a maximum wavenumber of $k_{\rm
max}=0.2~{}h/{\rm Mpc}$ as well as all BAO data, including that from the same
redshifts and sky cuts as the multipoles and the data from lower
($z=0.106,0.15$) and higher redshifts ($z=2.334$). We also perform simplified
analyses without BAO data to evaluate the impact of that dataset on the
constraints.

We run 3 different sets of analysis, one for each model of interest:
$\Lambda$CDM, $w$CDM, and $wA$CDM. We vary 5 standard cosmological parameters,
$\log(10^{10}A_{s})$, $n_{s}$, $\omega_{c}$, $\omega_{b}$ and $h$ in all
analyses, as well as the dark energy EOS parameter, $w$, and the interaction
strength $A$ as defined in Eq.(2.5), in the relevant cases. Throughout our
work, we adopt a BBN prior on $\omega_{b}$. Some of our main analyses include
a 3$\sigma$ Planck prior on the early Universe parameters $\log(10^{10}A_{s})$
and $n_{s}$, but we also run CMB-free analyses by relaxing this assumption.
Furthermore we fix the sum of the neutrino masses to $M_{\nu}=0.06~{}{\rm eV}$
and use broad flat priors for the remaining cosmological parameters.
Specifically, for $\omega_{c}$ and $h$ we impose flat priors matching the
ranges of parameters allowed by the bacco linear emulator, which are

$\Omega_{cb}\sim\mathcal{U}(0.06,0.7)\,,\ h\sim\mathcal{U}(0.5,0.9)\,,$ (2.18)

in which $\mathcal{U}(a,b)$ denotes a uniform distribution with edges $a,b$
and $\Omega_{cb}=\Omega_{c}+\Omega_{b}$. For numerical reasons, we also impose
flat priors on $w$ and $A$, given by

$w\sim\mathcal{U}(-3,-0.5)\,,\
\frac{A}{{\rm[b/GeV]}}\sim\mathcal{U}(-10^{3},10^{3})\,.$ (2.19)

Finally, these parameters also have to obey the physical condition that the
dark scattering cross-section is positive, giving the prior
$\xi\sim\mathcal{U}(0,\infty)$. This translates into a prior on $w$ and $A$
that forces ${\rm sign}(A)={\rm sign}(1+w)$, which gives rise to a
characteristic butterfly shape of the posteriors, as seen in Refs. [54, 120].

Our baseline priors for the nuisance parameters are the same as those in Ref.
[33] and are as follows:

$\displaystyle b_{1}\sim\mathcal{U}(0,4)\,,\ b_{2}\sim\mathcal{N}(0,1)\,,\
b_{\mathcal{G}_{2}}\sim\mathcal{N}(0,1)\,,\
b_{\Gamma_{3}}\sim\mathcal{N}\left(\frac{23}{42}(b_{1}-1),1\right)\,,$
$\displaystyle
N\sim\mathcal{N}\left(\frac{1}{\bar{n}},\frac{2}{\bar{n}}\right)\,,\
e_{0}\sim\mathcal{N}\left(0,\frac{2}{\bar{n}k_{\rm NL}^{2}}\right)\,,\
e_{2}\sim\mathcal{N}\left(0,\frac{2}{\bar{n}k_{\rm NL}^{2}}\right)\,,\
\frac{c_{0}}{[{\rm Mpc}/h]^{2}}\sim\mathcal{N}(0,30)\,,$ (2.20)
$\displaystyle\frac{c_{2}}{[{\rm Mpc}/h]^{2}}\sim\mathcal{N}(30,30)\,,\
\frac{c_{4}}{[{\rm Mpc}/h]^{2}}\sim\mathcal{N}(0,30)\,,\
\frac{c_{\nabla^{4}\delta}}{[{\rm Mpc}/h]^{4}}\sim\mathcal{N}(500,500)\,,$

in which $\mathcal{N}(\mu,\sigma)$ denotes a normal distribution with mean
$\mu$ and standard deviation $\sigma$. We use a nonlinear scale $k_{\rm
NL}=0.45~{}h/{\rm Mpc}$ and we assume an inverse number density of
$\bar{n}^{-1}=3500~{}({\rm Mpc}/h)^{3}$ for the $z_{1}=0.38$ bin and
$\bar{n}^{-1}=5000~{}({\rm Mpc}/h)^{3}$ for the $z_{3}=0.61$ bin.

These choices of priors are informed by the arguments of Refs. [99, 124],
which are based on the validity of the perturbative EFT modelling used here.
In particular, bias parameters are expected to be $O(1)$ and noise parameters
are expected to be the same order as the inverse number density and to scale
with the appropriate power of the nonlinear scale. The counter-terms are also
expected to be $O(1)$ in units of $k_{\rm NL}$, but it is argued in Ref. [99]
that for this effect the relevant nonlinear scale is larger, given the rather
large velocity dispersion values ($\sigma_{v}\sim 5~{}{\rm Mpc}/h$) found in
previous BOSS analyses [30]. This justifies using standard deviations of
$30~{}({\rm Mpc}/h)^{2}$ and $500~{}({\rm Mpc}/h)^{4}$ for the quadratic and
quartic counter-terms respectively. While some of these expectations for bias
parameters are supported by measurements from simulations [125, 119, 126, 127,
120], this is less so for the counter-term parameters, for which it is much
more difficult to get definite predictions. These are our baseline priors used
in most of our analyses. However, we also perform additional analyses in
Appendix A to test the relevance of these assumptions by enlarging the
variance of these priors by factors 3 and 10. This will also allow us to
include potential caveats to the arguments presented above due to the modified
cosmologies which we study here. As is shown there, at least for the BOSS
data, these priors appear to be informative and a further study of their
effects is essential for obtaining robust results.

We sample our posterior via an MCMC analysis using Goodman & Weare’s affine
invariant sampler as implemented in emcee [128]. We use GetDist [129] to plot
our posterior distributions.

##  3 Results

We now present our results for our main analyses for the three cosmological
models under study. We begin with $\Lambda$CDM, demonstrating that we
reproduce the results of Ref. [33]. Next we add dynamical dark energy with a
constant $w$ and finally we show our results for the dark scattering model,
which we denote $wA$CDM, where we show the first ever constraints on the
interaction strength $A$.

###  3.1 $\Lambda$CDM analysis

We present the results of our full-shape analysis for $\Lambda$CDM in Fig. 1,
showing the marginalised posteriors we obtain, compared to the means for the
results of Ref. [33] (which we denote as P+I), which performed the first
analysis with the data measured with the estimator used here. We also show a
summary of those results in Table 1. It is clear that our results are in
perfect agreement with those of Ref. [33], using an entirely independent
likelihood pipeline and theory code, with only the EFT and bias modelling
being the same. We consider this as a validation test of our theory and
likelihood code and its application to data. We will now show and discuss our
main results.

Figure 1:  Marginalized posteriors for the sampled cosmological parameters and $\Omega_{m}$ and $\sigma_{8}$ for $\Lambda$CDM for the analysis using only the full-shape power spectrum data. We use $k_{\rm max}=0.20~{}h/{\rm Mpc}$ for all 3 multipoles. This case mimics the analysis of Philcox and Ivanov (P+I) [33], and solid red lines show the mean of the parameters obtained there, shown in Table III of that reference, demonstrating perfect agreement. Parameter | FS only [this work] | P+I analysis [33]
---|---|---
$\omega_{c}$ | $0.134^{+0.011}_{-0.015}$ | $0.139^{+0.011}_{-0.015}$
$h$ | $0.695^{+0.011}_{-0.015}$ | $0.699^{+0.015}_{-0.017}$
$\log(10^{10}A_{s})$ | $2.67\pm 0.17$ | $2.63^{+0.15}_{-0.16}$
$n_{s}$ | $0.906\pm 0.078$ | $0.883^{+0.076}_{-0.072}$
$\Omega_{m}$ | $0.326^{+0.017}_{-0.022}$ | $0.333^{+0.019}_{-0.020}$
$\sigma_{8}$ | $0.707^{+0.042}_{-0.051}$ | $0.704^{+0.044}_{-0.049}$
Table 1: Parameter constraints for the $\Lambda$CDM analysis using full shape
data only compared to the results of Ref. [33] for the equivalent analysis. We
show the 5 sampled parameters as well as 2 derived parameters, $\Omega_{m}$,
and $\sigma_{8}$. The values shown are the means with the $1\sigma$ confidence
intervals.

Our main $\Lambda$CDM analyses are shown in Fig. 2, with the summary results
reported in Table 2. There we show both the case with a CMB prior on the
primordial parameters $A_{s}$ and $n_{s}$, as well as the CMB-free analysis,
similar to the one shown in Fig. 1. Both these analyses differ from the
previous one in the inclusion of the measurement of the BAO scale from the
post-reconstructed power spectrum at various redshifts ranging from 0.15 to
2.334. The addition of this information constrains the expansion history
better, and therefore provides a sharper measurement of $\omega_{c}$ and $h$.
For this reason, the degeneracy between $\omega_{c}$ and $n_{s}$ is somewhat
broken, thus moving the measurement of $n_{s}$ to larger values as
$\omega_{c}$ is lowered, placing it in agreement with the Planck measurement,
even in the CMB-free analysis.

Figure 2:  Marginalized posteriors for the sampled cosmological parameters for $\Lambda$CDM, showing both the analysis with (blue empty contours) and without (orange filled contours) a 3$\sigma$ Planck prior on the primordial parameters (as well as a BBN prior on $\omega_{b}$). We use $k_{\rm max}=0.20~{}h/{\rm Mpc}$ for all 3 power spectrum multipoles and use the full-shape data and the BAO measurement. Solid green lines mark the best fit values from the Planck $\Lambda$CDM analysis, while red lines show the $\sigma_{8}$ and $S_{8}$ measured in the fiducial DES analysis [22, 23]. Parameter |  $\Lambda$CDM CMB prior |  $\Lambda$CDM CMB-free
---|---|---
$\omega_{c}$ |  $0.118\pm 0.005$ ($0.120$) |  $0.121^{+0.008}_{-0.009}$ ($0.120$)
$100\omega_{b}$ |  $2.264\pm 0.037$ ($2.256$) |  $2.266\pm 0.038$ ($2.244$)
$h$ |  $0.678\pm 0.008$ ($0.681$) |  $0.681\pm 0.010$ ($0.678$)
$\log(10^{10}A_{s})$ |  $3.021\pm 0.040$ ($3.023$) |  $2.821\pm 0.158$ ($2.832$)
$n_{s}$ |  $0.969\pm 0.012$ ($0.974$) |  $0.975\pm 0.063$ ($0.972$)
$\Omega_{m}$ |  $0.306\pm 0.009$ ($0.308$) |  $0.311^{+0.013}_{-0.015}$ ($0.311$)
$\sigma_{8}$ |  $0.793\pm 0.025$ ($0.806$) |  $0.734^{+0.046}_{-0.053}$ ($0.731$)
$S_{8}$ |  $0.801\pm 0.033$ ($0.816$) |  $0.746^{+0.044}_{-0.049}$ ($0.744$)
Table 2: Parameter constraints for the two $\Lambda$CDM analyses using full
shape data as well as post-reconstructed measurements of the BAO scale
(FS+BAO). We show the 5 sampled parameters as well as 3 derived parameters,
$\Omega_{m}$, $\sigma_{8}$ and $S_{8}$. The values shown are the means with
the $1\sigma$ confidence intervals and the best-fit value in parenthesis.

The two analyses are broadly consistent, except for the measured value of the
scalar amplitude, which is substantially lower in the CMB-free case, with a
correspondingly low value of $\sigma_{8}$ being measured. The value measured
in the full-shape + BAO analysis (FS+BAO) is approximately $1\sigma$ larger
than the value measured in the FS only analysis discussed above
($\log(10^{10}A_{s})=2.67$ for FS vs $2.82$ for FS+BAO). This is a consequence
of the more precise measurement of $\omega_{c}$ whose degeneracy with $A_{s}$
raises the value of the amplitude. In spite of this, the scalar amplitude is
still over $1\sigma$ away from the value measured by Planck, with $\sigma_{8}$
agreeing more closely with the values measured by weak lensing experiments
such as DES [22, 23].

Low values of the scalar amplitude have been found in various previous
analyses of the BOSS data [33, 35, 81, 80, 34, 108, 32, 130]. Precise
comparisons are complicated by different choices of data, estimators and
priors used. However, our FS+BAO analysis is more similar to that conducted by
Ref. [81], which also use BAO data in a broad redshift range and find
$\Lambda$CDM results consistent with ours. Ref. [80] also uses similar BAO
data but focuses only on beyond-$\Lambda$CDM analyses, which we discuss below
in detail. Other analyses which set a Planck prior on $n_{s}$ [130, 33] but do
not use the same BAO data also obtain very similar results for the amplitude
as our FS+BAO CMB-free analysis. This is likely because adding the BAO data
effectively moves the values of $\omega_{c}$ and consequently of $n_{s}$ close
to the Planck value and thus results in similar values for all parameters. Our
results using only the full shape data are also in agreement with those of
Ref. [35], in spite of the different nonlinear modelling used there.

Our analysis with a Planck prior on $A_{s}$ and $n_{s}$, essentially
approximates the inclusion of CMB data and therefore results in all parameters
being in agreement with Planck constraints. However, because a lower scalar
amplitude is preferred by the BOSS data, the best-fit value of $\sigma_{8}$ is
still lower than in Planck, as is the value of the matter density. This
preference is shown even in $A_{s}$ which, while still dominated by its prior
($\log(10^{10}A_{s})=3.044\pm 0.042$) is $0.5\sigma$ lower with a best-fit
value of $\log(10^{10}A_{s})=3.023$.

In Appendix A, we discuss how these results depend on the choice of priors for
the EFTofLSS nuisance parameters. In the CMB-free analysis, shown in Fig. 9,
we find that a large broadening of the prior ranges ($10\times\sigma$) can
change the results substantially, particularly for $n_{s}$, but also for the
amplitude $A_{s}$. This is driven by the fact that the nuisance parameters are
drawn towards values considered extreme by most current literature. An
increase of the prior by a factor of 3 has a smaller impact, but does change
the primordial parameters somewhat. For the CMB-dependent cases shown in Fig.
10, shifts in parameters are largest in $\omega_{c}$ and $h$, and can be
significant, changing the results by $O(1)$ $\sigma$. And while all this is
fueled by rather large values of nonlinear bias parameters, this indicates
that the priors are informative and without knowledge of them, the inferred
values of parameters such as the scalar amplitude or the matter density are
highly dependent on the choice of prior.

We show the full contours including bias parameters in Fig. 15 of Appendix B,
with summary statistic shown in Table 9, where more details on our main
analyses can be seen.

###  3.2 $w$CDM analysis

We now turn to the $w$CDM analysis, whose main results are shown in Fig. 3 and
in Table 3, and are obtained using both the full shape data and the BAO
information. We show there both the CMB-free case and the one with a Planck
prior on the primordial parameters. This Planck prior has a considerably
larger impact here than in $\Lambda$CDM with many parameters having different
posteriors between the two cases. This is due to the fact that in the CMB-free
analysis a number of degeneracies are introduced when $w$ is varied,
particularly with $h$ and $A_{s}$ but also with $\omega_{c}$. The BAO data
plays a crucial role in reducing these degeneracies and without it, the
background parameters would have very large error bars [80, 81]. This can be
seen below in our study of $wA$CDM, where we also show results without BAO.

Figure 3:  Marginalized posteriors for the sampled cosmological parameters for
$w$CDM, showing both the analysis with (blue empty contours) and without
(orange filled contours) a 3$\sigma$ Planck prior on the primordial parameters
(as well as a BBN prior on $\omega_{b}$). We use $k_{\rm max}=0.20~{}h/{\rm
Mpc}$ for all 3 multipoles and use the full shape data and the BAO
measurement. Solid green lines mark the best fit values from the Planck
$\Lambda$CDM analysis while red lines show the $\sigma_{8}$ and $S_{8}$
measured in the fiducial DES analysis [22, 23].

The parameters measured in the CMB-free analysis show some deviation from the
Planck results, most of which is driven by the low amplitude preferred by the
BOSS data. The degeneracies mentioned above influence this too. We therefore
find a rather low amplitude with $\log(10^{10}A_{s})=2.62\pm 0.21$ and
$\sigma_{8}=0.709^{+0.046}_{-0.055}$ as well as a preference for phantom dark
energy, with $w=-1.17^{+0.12}_{-0.11}$. However, projection effects are
exacerbating this deviation, given that the best-fit values are less
discrepant than their means, resulting instead in $\log(10^{10}A_{s})=2.77$,
$\sigma_{8}=0.737$ and $w=-1.093$.

Comparison with previous works is not straightforward, since previous analyses
with the EFTofLSS of Refs. [80, 81] do not use the same estimator and it is
unclear whether the the power spectrum normalization is consistently applied
[131, 132, 133, 33]. Still, our results are broadly consistent with those of
Ref. [80], particularly in terms of the best-fit values. The results of Ref.
[81] are consistent at the $1\sigma$ level.

Parameter |  $w$CDM CMB prior |  $w$CDM CMB-free
---|---|---
$w$ |  $-1.002^{+0.081}_{-0.073}$ ($-1.023$) |  $-1.17^{+0.12}_{-0.11}$ ($-1.093$)
$\omega_{c}$ |  $0.118\pm 0.005$ ($0.118$) |  $0.127^{+0.009}_{-0.011}$ ($0.122$)
$100\omega_{b}$ |  $2.265\pm 0.038$ ($2.286$) |  $2.269\pm 0.038$ ($2.273$)
$h$ |  $0.679^{+0.014}_{-0.016}$ ($0.689$) |  $0.713\pm 0.024$ ($0.699$)
$\log(10^{10}A_{s})$ |  $3.021\pm 0.041$ ($3.027$) |  $2.62\pm 0.21$ ($2.77$)
$n_{s}$ |  $0.968\pm 0.012$ ($0.966$) |  $0.947\pm 0.064$ ($0.9721$)
$\Omega_{m}$ |  $0.306\pm 0.014$ ($0.298$) |  $0.297\pm 0.016$ ($0.297$)
$\sigma_{8}$ |  $0.793\pm 0.030$ ($0.802$) |  $0.709^{+0.046}_{-0.055}$ ($0.737$)
$S_{8}$ |  $0.800\pm 0.033$ ($0.799$) |  $0.705^{+0.050}_{-0.057}$ ($0.733$)
Table 3: Parameter constraints for the two $w$CDM analyses using full shape
data as well as post-reconstructed measurements of the BAO scale (FS+BAO). We
show the 6 sampled parameters as well as 3 derived parameters, $\Omega_{m}$,
$\sigma_{8}$ and $S_{8}$. The values shown are the means with the $1\sigma$
confidence intervals and the best-fit value in parenthesis.

The analysis with a CMB prior on $A_{s}$ and $n_{s}$ once again moves all
parameters closer to the values measured by Planck, as expected. In addition,
the measured value of $w$ is now in full agreement with $\Lambda$CDM, being
$w=-1.002^{+0.081}_{-0.073}$. Furthermore, the errors on all parameters are
substantially reduced relative to the CMB-free analysis, since degeneracies
are alleviated or broken. As we have already shown in the corresponding
$\Lambda$CDM analysis, the scalar amplitude is still pushed down somewhat,
resulting in a slightly lower value of $\sigma_{8}$ than inferred from Planck.

Full contours including bias parameters can be seen in Fig. 16 of Appendix B,
with summary statistics shown in Table 10.

We repeat the exercise done for $\Lambda$CDM of extending the prior range of
the nuisance parameters also for $w$CDM, showing the results in Figs. 11 and
12 and Table 7 of Appendix A. The conclusions are very similar to the case of
$\Lambda$CDM, but we find in addition that broader priors have a small effect
on the constraint on $w$. In the CMB-dependent cases, the uncertainties on $w$
grow only slightly and agree with the base case shown here. However, in the
CMB-free case, there is a small shift to $w=-1.20^{+0.13}_{-0.12}$ when the
priors are 3 times larger and a larger shift to $w=-1.11^{+0.19}_{-0.12}$ for
10 times broader prior ranges. While relatively small, these shifts imply that
the choice of priors directly affects the results of the analysis and one
should therefore be cautious in the interpretation of the constraints.

###  3.3 $wA$CDM analysis

Figure 4:  Marginalized posteriors for the sampled cosmological parameters in
the analysis with a Planck prior on the primordial parameters (as well as a
BBN prior on $\omega_{b}$). We use $k_{\rm max}=0.20~{}h/{\rm Mpc}$ for all 3
multipoles and show both the analysis with the full shape only (FS, blue empty
contours) and with both the full shape and the BAO measurement (FS+BAO, orange
filled contours). The shape of the $w-A$ contour is due to the physical prior
imposed on those parameters, i.e. ${\rm sign}(A)={\rm sign}(1+w)$. Solid green
lines mark the best fit values from Planck $\Lambda$CDM analysis.

Finally, we now describe our analyses for dark scattering, which we denote by
$wA$CDM. We begin by presenting our baseline analysis, allowing all
cosmological parameters to vary, but imposing a 3$\sigma$ Planck prior on
$A_{s}$ and $n_{s}$; and leave the CMB-free analysis, which suffers from
severe degeneracies, to the last part of this section.

The posteriors for all cosmological parameters sampled are shown in Figs. 4
and 5, with the means and best-fit parameters shown in Table 4. We show both
the case including only the full shape data (FS) as well as the full data set,
including also BAO data (FS+BAO).

From this analysis we find that there is a slight preference for a positive
value of $A$ at a significance of $\sim 1\sigma$, with our measurement giving

$A=3.9^{+3.2}_{-3.7}\ (5.4)\ {\rm b/GeV}\,,$ (3.1)

with the best-fit value shown in parenthesis. The origin of this preference
for $A>0$ is born out of the ability of the dark scattering interaction to
lower the value of $\sigma_{8}$ relative to that predicted from $\Lambda$CDM
or $w$CDM. This has the effect shown in Fig 6: increasing $A$ while
marginalizing over nuisance parameters mostly suppresses the quadrupole, while
only mildly affecting the monopole. This effect of lowering $\sigma_{8}$ on
the quadrupole had already been shown by Ref. [33], but here this is achieved
with a concrete model, which additionally also lowers the growth factor $f$.

Parameter |  $wA$CDM FS |  $wA$CDM FS+BAO
---|---|---
$w$ |  $-0.954^{+0.024}_{-0.046}$ ($-0.987$) |  $-0.972^{+0.036}_{-0.029}$ ($-0.994$)
$A$ |  $4.8^{+2.8}_{-3.8}$ ($3.7$) |  $3.9^{+3.2}_{-3.7}$ ($5.4$)
$\omega_{c}$ |  $0.1239^{+0.0062}_{-0.0069}$ ($0.1335$) |  $0.1201^{+0.0052}_{-0.0058}$ ($0.120$)
$100\omega_{b}$ |  $2.267\pm 0.038$ ($2.296$) |  $2.266\pm 0.038$ ($2.268$)
$h$ |  $0.680\pm 0.012$ ($0.6912$) |  $0.677\pm 0.011$ ($0.677$)
$\log(10^{10}A_{s})$ |  $3.038\pm 0.042$ ($3.060$) |  $3.039\pm 0.043$ ($3.033$)
$n_{s}$ |  $0.964\pm 0.012$ ($0.958$) |  $0.966\pm 0.012$ ($0.962$)
$\Omega_{m}$ |  $0.318\pm 0.014$ ($0.329$) |  $0.313^{+0.013}_{-0.011}$ ($0.313$)
$\sigma_{8}$ |  $0.773^{+0.028}_{-0.031}$ ($0.839$) |  $0.771^{+0.028}_{-0.034}$ ($0.762$)
$S_{8}$ |  $0.796\pm 0.037$ ($0.878$) |  $0.787\pm 0.034$ ($0.778$)
Table 4: Parameter constraints for the two baseline analyses with interacting
dark energy, here labelled $wA$CDM. We show results using full shape only (FS)
as well as those including the post-reconstructed measurements of the BAO
scale (FS+BAO). We show the 7 sampled parameters and 3 derived parameters,
$S_{8}$, $\sigma_{8}$ and $\Omega_{m}$. The values shown are the means with
the $1\sigma$ confidence intervals and the best-fit values in parenthesis.

Our hint of $A>0$ is therefore a translation of the preference of the BOSS
data for a low $\sigma_{8}$, already shown above in the analyses in
$\Lambda$CDM and $w$CDM. This is visible directly in Fig. 5, where we
additionally demonstrate that our measurement of $\sigma_{8}$ is in agreement
with that from weak lensing surveys such as DES [22, 23], and lower than that
inferred from the Planck measurements with $\Lambda$CDM.

Figure 5:  Marginalized posteriors for the dark energy parameters and the
derived parameters $S_{8}$, $\sigma_{8}$ and $\Omega_{m}$, for the same
analysis of Fig. 4. We also show the values of $S_{8}$ and $\sigma_{8}$
measured in the fiducial DES analysis [22, 23] with red solid lines,
demonstrating our good agreement with those experiments.

As in our previous analyses, the other cosmological parameters not fixed by
their priors are fully in agreement with the values measured by Planck. A
distinction with respect to the $\Lambda$CDM and $w$CDM analyses is that the
measurement of $A_{s}$ is not being pushed as much towards lower values as in
those cases, indicating that the preference of the data for a low $\sigma_{8}$
is being driven by the interaction, via $A$. Furthermore, given that this dark
sector interaction only affects the late-time evolution and does not modify
early-time physics, its effect on the CMB is negligible, as shown in Ref.
[50]. It is therefore justified that we use the Planck prior on the primordial
parameters, whose measurements would be unaffected by the inclusion of the
interaction. This indicates that this model of interacting dark energy can
alleviate the $\sigma_{8}$ tension, thus re-establishing the concordance
between the early- and late-Universe measurements of the clustering amplitude.

As discussed in the context of the $w$CDM analysis, including BAO data has a
considerable effect in breaking the degeneracy between $w$ and $h$, and
therefore allows for stronger constraints on those parameters as well as on
$\omega_{c}$ [80, 33, 134, 81]. Despite having a small impact on the
measurement of $A$, since it only probes the expansion history, it allows for
a tighter constraint on $w$, and therefore alleviates the degeneracy between
$w$ and $A$ [54], which moves the bounds on $A$ towards lower values.

Figure 6:  Comparison of the monopole (blue lines and points) and quadrupole
(orange lines and points) for the best-fit point in our baseline $wA$CDM case
(dashed lines), to the corresponding $\Lambda$CDM case (solid lines). The
multipoles are computed with all cosmological parameters matching except for
$w$ and $A$, but with marginalised nuisance parameters differing. It is this
compensation from the marginalised parameters that induces this variation to
affect the quadrupole substantially more than the monopole. Fixing all
nuisance parameters alters the amplitude of both multipoles close to
proportionally, as expected.

We show the full contours with all sampled parameters in Fig. 17 and summary
results in Table 11 of Appendix B. It can be seen there that this result does
not come at the cost of large or unusual bias parameters, with the linear bias
being always $\sim 2$. There are some differences, however, with respect to
the bias parameters measured by Ref. [33], which detect larger values of
linear bias. However, it is plausible that these would be different in a
cosmology with a dark matter interaction, since this affects the dynamics of
collapse, as seen for example in Ref. [55].

Figure 7:  Marginalized posteriors for the sampled cosmological parameters in
the CMB-free analysis (with a BBN prior on $\omega_{b}$). We use $k_{\rm
max}=0.20~{}h/{\rm Mpc}$ for all 3 multipoles and show both the analysis with
the full shape only (FS) and with both the full shape and the BAO measurement
(FS+BAO). The shape of the $w-A$ contour is due to the physical prior imposed
on those parameters, i.e. ${\rm sign}(A)={\rm sign}(1+w)$.

We now explore the more general CMB-free analysis. We show the posteriors of
the sampled cosmological parameters in Fig. 7 and with derived parameters in
Fig. 8. The summary of the measured parameters can be found in Table 5. Full
contours and more details can be found in Fig. 18 and Table 12 of Appendix B.

We can see explicitly here that without prior early Universe information, the
effect of the BAO data is even stronger. Again in this case, one of the
important effects of the BAO is to break the degeneracy between $w$ and $h$.
Indeed, as seen in the $w$CDM analyses above and also found in Refs. [80, 81],
the BAO data from various different redshifts is essential for achieving a
better constraint on $w$.

Parameter |  $wA$CDM FS |  $wA$CDM FS+BAO
---|---|---
$w$ |  $-1.48\pm 0.20$ ($-1.28$) |  $-1.16^{+0.12}_{-0.069}$ ($-1.097$)
$A$ |  $-18.9^{+17}_{-7.7}$ ($-0.7631$) |  $-21^{+14}_{-11}$ ($-0.1077$)
$\omega_{c}$ |  $0.148^{+0.013}_{-0.018}$ ($0.138$) |  $0.1275^{+0.0082}_{-0.010}$ ($0.1249$)
$100\omega_{b}$ |  $2.269\pm 0.038$ ($2.266$) |  $2.268\pm 0.038$ ($2.253$)
$h$ |  $0.785^{+0.041}_{-0.046}$ ($0.746$) |  $0.711^{+0.017}_{-0.024}$ ($0.6992$)
$\log(10^{10}A_{s})$ |  $1.85^{+0.20}_{-0.28}$ ($2.37$) |  $2.07\pm 0.35$ ($2.725$)
$n_{s}$ |  $0.844\pm 0.080$ ($0.885$) |  $0.945\pm 0.062$ ($0.9553$)
$\Omega_{m}$ |  $0.279^{+0.020}_{-0.026}$ ($0.290$) |  $0.298\pm 0.015$ ($0.303$)
$\sigma_{8}$ |  $0.601^{+0.032}_{-0.049}$ ($0.665$) |  $0.642^{+0.044}_{-0.061}$ ($0.730$)
$S_{8}$ |  $0.579^{+0.040}_{-0.054}$ ($0.654$) |  $0.640^{+0.044}_{-0.060}$ ($0.733$)
Table 5: Parameter constraints for the two CMB-free analyses with interacting
dark energy, here labelled $wA$CDM. We show results using full shape only (FS)
as well those including the post-reconstructed measurements of the BAO scale
(FS+BAO). We show the 7 sampled parameters as well as 3 derived parameters,
$S_{8}$, $\sigma_{8}$ and $\Omega_{m}$. The values shown are the means with
the $1\sigma$ confidence intervals and the best-fit value in parenthesis.

Let us now comment on the results themselves. We can immediately see that the
data now appear to prefer a negative value of $A$, as well as a very low value
of the scalar amplitude, independently of the inclusion of BAO data. This is
explained by a very strong degeneracy between $A$ and $A_{s}$, which is also
clearly seen in the posterior plots. Physically, as $A$ grows more negative,
the amplitude of the linear power spectrum increases and $A_{s}$ decreases to
compensate, being even lower due to the preference of the data for a low
$\sigma_{8}$. This is compounded by the linear bias also being largely
degenerate with the amplitude. This is seen very clearly in the full contours
in Fig. 18 of Appendix B, where we can also see that the linear bias is very
large and is already dominated by its prior in some cases. Should that prior
be relaxed, this degeneracy would likely drive the bias to even larger values,
further reducing the scalar amplitude and increasing $A$. We can also see in
Fig. 8 that the preferred value of $\sigma_{8}$ and $S_{8}$ is equivalently
low, given this degeneracy with the bias.

In general, the BOSS full shape data is able to constrain two combinations of
these three amplitude parameters, essentially via the measurement of the
amplitudes of the monopole and quadrupole. Therefore, when all three are
allowed to vary this degeneracy is inevitable, whereas in the analysis with
the CMB prior on $A_{s}$, this problem is resolved and a fairly precise
measurement of $A$ and $b_{1}$ is possible. In principle, nonlinear effects
can help resolving this degeneracy, since they are sensitive to other
combinations of $A_{s}$, $A$ and $b_{1}$. This can be achieved by the enhanced
precision of future stage-IV surveys and/or by improved modelling.

Figure 8:  Marginalized posteriors for the dark energy parameters and the
derived parameters $S_{8}$, $\sigma_{8}$ and $\Omega_{m}$, for the same
analysis of Fig. 7. We also show the values of $S_{8}$ and $\sigma_{8}$
measured in the fiducial DES analysis [22, 23], now showing complete
disagreement due to the degeneracy detailed in the text.

In spite of the degeneracy found in these analyses, there still appears to be
a preference for a negative value of $A$, which remains to be explained. We
believe this preference is driven by the slight preference of LSS data for
$w<-1$. This is seen in the $w$CDM analyses in the literature (e.g. [80, 81])
as well as our own (CMB-free) analysis above, where the phantom model is
preferred. Planck also appears to find a preference for phantom dark energy
[1]. Due to the physical requirement of matching signs between $A$ and $1+w$,
this naturally implies that negative $A$ is preferred, since that is
determined by $w<-1$ and given that $A$ is otherwise unconstrainable due to
the strong degeneracies mentioned above.

Finally, we note once more that these results are somewhat dependent on the
priors imposed on the nuisance parameters. As seen in more detail in Figs. 13
and 14 and Table 8 of Appendix A, changes to these priors can alter
cosmological parameter constraints to some degree, including the value of $A$.
In the case with a CMB prior on the primordial parameters, the analyses with
larger priors reduce the significance of the above hint for $A>0$ to
approximately $0.5\sigma$, both because the contours are larger, but also due
to a shift of parameters towards their $\Lambda$CDM values. This is
complemented by a fairly unrealistic shift of $\omega_{c}$ to small values,
which would be in $2$ to $3\sigma$ tension with Planck. However, in such a
situation where the priors appear to be informative, it is important to be
conservative on any claimed hint or detection.

##  4 Conclusions

We have analysed the power spectrum multipoles from the BOSS DR12 data in
conjunction with BAO data from a broad range of redshifts, in the context of
three different cosmological models. We use data vectors generated from the
new windowless estimator for the power spectrum [85, 33] and follow state-of-
the-art modelling and analysis techniques using a fully independent likelihood
pipeline, with a theoretical prediction based on the linear emulator bacco
[113] and the perturbation theory code FAST-PT [116, 117]. For all
cosmological models, we perform two analyses, one including a Planck prior on
primordial parameters $A_{s}$ and $n_{s}$ and one free of CMB information.

In the context of $\Lambda$CDM we find a very low scalar amplitude when using
only the full-shape data, which is fully in agreement with Ref. [33], as can
be seen in Fig. 1. Adding the BAO data and in the CMB-free analysis, we
measure a larger value of the amplitude,

$\log(10^{10}A_{s})=2.821\pm 0.158\ (2.832)\,,$ (4.1)

which is still low relative to the Planck results. The measured value is,
however, in broad agreement with that found in many other analyses of the BOSS
data. In our CMB-dependent analysis, all parameters are broadly consistent
with Planck, with only the amplitude being slightly pushed to low values
within the prior we use, resulting in a lower $\sigma_{8}$ than expected from
Planck. See Fig. 2 and Table 2 for more details.

For $w$CDM, our CMB-free analysis shows a degeneracy between $w$ and the
amplitude, which brings the amplitude to lower values than in the $\Lambda$CDM
case and drives the EOS to the phantom regime:

$w=-1.17^{+0.12}_{-0.11}\ (-1.093)\,.$ (4.2)

The analysis with a CMB prior, on the other hand, moves the constraint on $w$
towards the $\Lambda$CDM value and all other parameters are consistent with
Planck, with the amplitude being the same as in the corresponding $\Lambda$CDM
case. All these results are shown in Fig. 3 and Table 3.

In the analysis for $wA$CDM, we find very large degeneracies between the
interaction strength $A$, the scalar amplitude $A_{s}$ and the linear bias, in
the context of the CMB-free case, which is shown in Figs. 7 and 8. This makes
it very difficult to obtain a constraint on $A$, with a negative value being
preferred, driven by the preference for $w<-1$ and the mentioned degeneracy.
However, in our main analysis including a CMB prior shown in Figs. 4 and 5,
the degeneracies are broken and we find a hint for a positive interaction
strength:

$w=-0.972^{+0.036}_{-0.029}\ (-0.994)\,,\quad A=3.9^{+3.2}_{-3.7}\ (5.4)\,.$
(4.3)

This leaves the scalar amplitude $A_{s}$ closer to the Planck value than in
the other corresponding analyses, while also lowering $\sigma_{8}$ to similar
values as seen in weak lensing experiments. This represents the first
measurement of this parameter and shows that the Dark Scattering model can be
successful in re-establishing concordance between early and late Universe
measurements of the scalar amplitude. It is therefore important to study this
model in more detail with different and future data, both spectroscopic and
photometric, to fully understand whether it can consistently explain all
observations.

We studied the dependence of all analyses on the priors of the nuisance
parameters in Appendix A. We find in all cases that these priors are
informative, as broadening them modifies the posterior distribution for
cosmological parameters. For the CMB-free cases, the most affected parameter
is the spectral index, $n_{s}$, although the amplitude $A_{s}$ can also be
substantially affected, with other parameters less so. When a CMB prior is
imposed, the effect is instead to change mostly $\omega_{c}$ as well as $h$ to
a smaller extent. The dark energy parameters $w$ and $A$ are also shifted by
fractions of their base uncertainties, typically in the direction of their
$\Lambda$CDM values. This means, for example, that our main result on the
strength of the interaction is less significant when the priors are allowed to
vary in a larger range.

Finally, let us comment on the overall conclusions of this work. While we find
interesting new constraints on dark energy, in all cases tested and for all
models under study, varying the nuisance parameter priors leads to significant
shifts in the cosmological parameter constraints, in addition to broadening of
the posteriors. This implies that the priors on the nuisance parameters
contain non-negligible information, which determines the possible constraints
on cosmology, at least at the level of precision of the BOSS data. The main
question is whether this information is correct and how it can be obtained. As
mentioned in the main text, these prior ranges were chosen based on several
theoretical arguments relating to the consistency of the bias model, and based
on the values typically seen in simulation studies. However, galaxies in the
real Universe could be different from simulations, particularly given the
difficulty of modelling galaxy formation robustly, as well as simply due to
selection effects on the galaxy samples used for measuring spectra. Moreover,
the theoretical arguments are typically order-of-magnitude statements, which
are difficult to implement exactly. Given these results and the uncertainty in
predicting values of nuisance parameters, we argue that it is crucial to study
this in more depth in the future. A better understanding of how nuisance
parameter priors affect constraints is crucial for confirming or ruling out
future apparent tensions or the detection of exotic new physics.

## Acknowledgments

We acknowledge use of the Cuillin computing cluster of the Royal Observatory,
University of Edinburgh. AP is a UK Research and Innovation Future Leaders
Fellow [grant MR/S016066/1]. PC and CM’s research is supported by a UK
Research and Innovation Future Leaders Fellowship [grant MR/S016066/1]. For
the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission.

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##  Appendix A Dependence on priors of nuisance parameters

In this appendix we study the prior dependence of our results. We focus on the
nuisance parameters, namely the bias parameters $b_{2}$, $b_{\mathcal{G}_{2}}$
and $b_{\Gamma_{3}}$, the counter-term parameters $c_{0}$, $c_{2}$, $c_{4}$
and $c_{\nabla^{4}\delta}$, and the shot-noise parameters $N$, $e_{0}$ and
$e_{2}$. We repeat the main FS+BAO analyses of the main text, but modify the
standard deviation of the Gaussian prior of these nuisance parameters222We do
not modify the prior on $b_{1}$, since the range chosen of $[0,4]$ is already
very broad. by factors of 3 and 10, denoting those cases by $3\times\sigma$
and $10\times\sigma$, respectively. The finding of different constraints on
the cosmological parameters in this exercise when compared to the baseline
cases implies that these priors are informative and therefore need to be
chosen very carefully.

###  A.1 $\Lambda$CDM

Parameter |  $3\times\sigma$ CMB-free |  $10\times\sigma$ CMB-free |  $3\times\sigma$ CMB prior |  $10\times\sigma$ CMB prior
---|---|---|---|---
$\omega_{c}$ |  $0.124^{+0.009}_{-0.011}$ ($0.123$) |  $0.111^{+0.007}_{-0.010}$ ($0.111$) |  $0.108\pm 0.005$ ($0.107$) |  $0.100\pm 0.006$ ($0.096$)
$100\omega_{b}$ |  $2.267\pm 0.038$ ($2.275$) |  $2.267\pm 0.038$ ($2.279$) |  $2.263\pm 0.038$ ($2.245$) |  $2.264\pm 0.038$ ($2.246$)
$h$ |  $0.685\pm 0.010$ ($0.685$) |  $0.673^{+0.009}_{-0.011}$ ($0.676$) |  $0.671\pm 0.008$ ($0.672$) |  $0.665\pm 0.009$ ($0.661$)
$\log(10^{10}A_{s})$ |  $2.69\pm 0.18$ ($2.76$) |  $2.22^{+0.15}_{-0.24}$ ($2.14$) |  $3.029\pm 0.040$ ($3.034$) |  $3.028\pm 0.041$ ($3.048$)
$n_{s}$ |  $0.810\pm 0.083$ ($0.814$) |  $0.029^{+0.066}_{-0.19}$ ($-0.102$) |  $0.963\pm 0.012$ ($0.966$) |  $0.961\pm 0.012$ ($0.955$)
$b_{1}^{Nz1}$ |  $2.10\pm 0.24$ ($2.01$) |  $1.14^{+0.23}_{-0.28}$ ($1.02$) |  $1.91\pm 0.09$ ($1.91$) |  $1.89\pm 0.14$ ($1.847$)
$b_{2}^{Nz1}$ |  $-1.4^{+1.9}_{-3.0}$ ($-2.0$) |  $-4.6\pm 6.7$ ($-7.9$) |  $-1.1^{+1.5}_{-2.8}$ ($-1.8$) |  $-0.9\pm 4.6$ ($-2.4$)
$b_{\mathcal{G}_{2}}^{Nz1}$ |  $-1.3\pm 1.4$ ($-1.2$) |  $-9.5\pm 4.1$ ($-12.0$) |  $-0.5\pm 1.2$ ($-0.4$) |  $-0.4^{+3.3}_{-2.8}$ ($-1.5$)
$b_{1}^{Sz1}$ |  $2.17\pm 0.23$ ($2.14$) |  $1.21^{+0.24}_{-0.29}$ ($1.13$) |  $1.98^{+0.11}_{-0.10}$ ($2.01$) |  $1.90^{+0.20}_{-0.14}$ ($1.83$)
$b_{2}^{Sz1}$ |  $-0.4^{+2.4}_{-3.3}$ ($1.0$) |  $-1.2\pm 8.2$ ($1.6$) |  $0.0^{+2.1}_{-3.2}$ ($-1.2$) |  $0.2^{+9.1}_{-8.2}$ ($-5.4$)
$b_{\mathcal{G}_{2}}^{Sz1}$ |  $-0.2\pm 1.3$ ($0.5$) |  $-7.4\pm 4.7$ ($-7.6$) |  $0.2\pm 1.2$ ($0.1$) |  $0.9^{+3.2}_{-2.8}$ ($-0.1$)
$b_{1}^{Nz3}$ |  $2.00\pm 0.28$ ($1.86$) |  $0.99^{+0.24}_{-0.30}$ ($0.83$) |  $1.82^{+0.11}_{-0.10}$ ($1.83$) |  $1.77^{+0.16}_{-0.13}$ ($1.64$)
$b_{2}^{Nz3}$ |  $-5.9^{+1.8}_{-2.6}$ ($-6.4$) |  $-4.5\pm 7.4$ ($-3.3$) |  $-5.1^{+1.3}_{-1.9}$ ($-5.4$) |  $-5.2^{+1.4}_{-4.2}$ ($-7.2$)
$b_{\mathcal{G}_{2}}^{Nz3}$ |  $-2.1^{+1.7}_{-1.6}$ ($-2.7$) |  $-13.5^{+4.2}_{-4.9}$ ($-14.0$) |  $-0.9\pm 1.3$ ($-1.5$) |  $-1.3^{+3.3}_{-2.8}$ ($-1.7$)
$b_{1}^{Sz3}$ |  $2.40\pm 0.25$ ($2.27$) |  $1.40^{+0.26}_{-0.31}$ ($1.27$) |  $2.17^{+0.11}_{-0.10}$ ($2.16$) |  $2.15^{+0.18}_{-0.15}$ ($2.07$)
$b_{2}^{Sz3}$ |  $-0.5^{+2.7}_{-3.4}$ ($-3.0$) |  $-1.3\pm 9.1$ ($-5.5$) |  $-0.8^{+2.2}_{-3.3}$ ($-1.9$) |  $-3.5^{+2.1}_{-7.2}$ ($-7.3$)
$b_{\mathcal{G}_{2}}^{Sz3}$ |  $-1.0\pm 1.5$ ($-1.2$) |  $-8.1^{+5.0}_{-5.9}$ ($-13.1$) |  $0.0\pm 1.3$ ($0.2$) |  $1.6^{+3.3}_{-2.9}$ ($0.3$)
Table 6: Parameter constraints for all analyses with broader priors of
nuisance parameters for $\Lambda$CDM. We show results using full shape + BAO
data both for CMB-free analyses and those with a Planck prior. We show the 5
sampled cosmological parameters as well as 3 bias parameters per redshift bin
and sky cut. The values shown are the means with the $1\sigma$ confidence
intervals with the best-fit values in parenthesis.

We show the results of analysing the FS+BAO data using broader priors in Fig.
9 for the CMB-free case and in Fig. 10 for the case with a CMB prior on
$n_{s}$ and $A_{s}$. There are some effects of altering the prior range,
implying that the priors do include information. Allowing the nuisance
parameters to vary an order of magnitude more than our baseline case does have
a large effect, particularly on $n_{s}$ for the CMB-free case or on
$\omega_{c}$ for the case with a CMB prior. There appears to be sufficient
freedom in the BOSS data that the bias parameters can mimic the effects of
cosmological parameters that change the shape of power spectra. In addition to
this, there is also some change in the amplitude, with lower values found in
the analyses with larger prior ranges.

Figure 9:  Posteriors for the sampled cosmological parameters in the CMB-free
analysis in $\Lambda$CDM for three different options for the priors on
nuisance parameters: baseline case of Eq. (2.20) and the cases with standard
deviations of nuisance parameters, except for $b_{1}$, increased by factors of
$3$ and $10$. Dot-dashed lines show the best-fit points showing that those are
also shifted.

All of these changes are, however, dependent on allowing large values for the
nuisance parameters, particularly the nonlinear bias, $b_{2}$ and
$b_{\mathcal{G}_{2}}$, as can be seen in Table 6. In the CMB-free analysis and
for the $10\times\sigma$ case, the preferred values of $b_{\mathcal{G}_{2}}$
are all close to $-10$ and the values of $b_{2}$ approach $-5$. Additionally,
the counter-term parameters are larger than expected, with $c_{0}$ and $c_{2}$
being $O(-100)~{}\text{Mpc}^{2}h^{-2}$, and reaching
$-300~{}\text{Mpc}^{2}h^{-2}$ for the low redshift bin. The $3\times\sigma$
case is less severe, but such shifts in parameters also rely on large values
of the bias parameters, with $b^{Nz3}_{2}\sim-6$ and counter-term parameter
$c_{0}^{Nz3}\sim 100~{}\text{Mpc}^{2}h^{-2}$.

Figure 10:  Posteriors for the sampled cosmological parameters in the analysis
with a CMB prior in $\Lambda$CDM for three different options for the priors on
nuisance parameters: baseline case of Eq. (2.20) and the cases with standard
deviations of nuisance parameters, except for $b_{1}$, increased by factors of
$3$ and $10$. Dot-dashed lines show the best-fit points showing that those are
also shifted.

For the analysis with a CMB prior, the effects of enlarging the prior are
smaller and are transferred away from the primordial parameters into
$\omega_{c}$. Once more, the nuisance parameters are pushed to large absolute
values, with $b_{2}\sim-5$ and $c_{0}\sim 100~{}\text{Mpc}^{2}h^{-2}$ at the
$z=0.61$ bin, for the $10\times\sigma$ case. In the $3\times\sigma$ case only
the biases have large values, with $b^{Nz3}_{2}\sim-5$ and this is what drives
the cosmological parameters to shift considerably relative to the base case.

###  A.2 $w$CDM

Parameter |  $3\times\sigma$ CMB-free |  $10\times\sigma$ CMB-free |  $3\times\sigma$ CMB prior |  $10\times\sigma$ CMB prior
---|---|---|---|---
$w$ |  $-1.20^{+0.13}_{-0.12}$ ($-1.13$) |  $-1.11^{+0.19}_{-0.12}$ ($-0.98$) |  $-1.01^{+0.09}_{-0.08}$ ($-0.98$) |  $-1.03^{+0.10}_{-0.08}$ ($-1.02$)
$\omega_{c}$ |  $0.134\pm 0.012$ ($0.129$) |  $0.119^{+0.011}_{-0.018}$ ($0.103$) |  $0.108^{+0.005}_{-0.006}$ ($0.105$) |  $0.100^{+0.006}_{-0.007}$ ($0.099$)
$100\omega_{b}$ |  $2.266\pm 0.038$ ($2.245$) |  $2.266\pm 0.039$ ($2.267$) |  $2.263\pm 0.038$ ($2.242$) |  $2.261\pm 0.037$ ($2.249$)
$h$ |  $0.723\pm 0.026$ ($0.705$) |  $0.697^{+0.027}_{-0.042}$ ($0.662$) |  $0.674^{+0.015}_{-0.017}$ ($0.666$) |  $0.670^{+0.017}_{-0.019}$ ($0.668$)
$\log(10^{10}A_{s})$ |  $2.46\pm 0.22$ ($2.63$) |  $2.18^{+0.19}_{-0.23}$($2.23$) |  $3.030\pm 0.041$ ($3.042$) |  $3.025\pm 0.042$ ($3.020$)
$n_{s}$ |  $0.797\pm 0.078$ ($0.775$) |  $0.15^{+0.42}_{-0.33}$ ($-0.06$) |  $0.963\pm 0.012$ ($0.965$) |  $0.962\pm 0.012$ ($0.960$)
$b_{1}^{Nz1}$ |  $2.31\pm 0.27$ ($2.12$) |  $1.42^{+0.25}_{-0.68}$ ($1.01$) |  $1.90\pm 0.10$ ($1.88$) |  $1.88\pm 0.14$ ($1.82$)
$b_{2}^{Nz1}$ |  $-1.4^{+2.0}_{-2.8}$ ($-1.9$) |  $-4.1\pm 6.7$ ($-7.4$) |  $-1.3^{+1.4}_{-2.7}$ ($-2.4$) |  $0.9\pm 6.0$ ($-2.3$)
$b_{\mathcal{G}_{2}}^{Nz1}$ |  $-1.1\pm 1.4$ ($-1.1$) |  $-8.2^{+4.4}_{-5.0}$ ($-12.0$) |  $-0.5\pm 1.2$ ($-0.8$) |  $-0.3^{+3.3}_{-2.8}$ ($-0.1$)
$b_{1}^{Sz1}$ |  $2.36\pm 0.26$ ($2.16$) |  $1.47^{+0.28}_{-0.65}$ ($1.06$) |  $1.98\pm 0.11$ ($2.04$) |  $1.90^{+0.20}_{-0.16}$ ($1.89$)
$b_{2}^{Sz1}$ |  $-0.3^{+2.4}_{-3.0}$ ($-1.0$) |  $-2.2^{+8.1}_{-9.1}$ ($1.0$) |  $-0.1^{+2.1}_{-3.1}$ ($-0.4$) |  $0.2^{+9.1}_{-8.2}$ ($-5.2$)
$b_{\mathcal{G}_{2}}^{Sz1}$ |  $0.0\pm 1.4$ ($-0.9$) |  $-6.1\pm 5.2$ ($-7.7$) |  $0.2\pm 1.2$ ($0.2$) |  $1.1^{+3.2}_{-2.9}$ ($0.4$)
$b_{1}^{Nz3}$ |  $2.29\pm 0.34$ ($1.96$) |  $1.28^{+0.26}_{-0.66}$ ($0.77$) |  $1.82\pm 0.11$ ($1.78$) |  $1.77^{+0.16}_{-0.13}$ ($1.78$)
$b_{2}^{Nz3}$ |  $-5.6^{+2.0}_{-2.9}$ ($-6.8$) |  $-5.8^{+7.4}_{-9.2}$ ($-0.5$) |  $-5.2^{+1.3}_{-1.8}$ ($-6.1$) |  $-4.9^{+1.2}_{-4.7}$ ($-6.3$)
$b_{\mathcal{G}_{2}}^{Nz3}$ |  $-1.8\pm 1.6$ ($-2.4$) |  $-12.4^{+4.7}_{-5.2}$ ($-12.3$) |  $-0.9\pm 1.3$ ($-0.5$) |  $-1.0^{+3.2}_{-2.8}$ ($0.0$)
$b_{1}^{Sz3}$ |  $2.64\pm 0.30$ ($2.38$) |  $1.71^{+0.33}_{-0.73}$ ($1.24$) |  $2.17\pm 0.11$ ($2.20$) |  $2.14^{+0.18}_{-0.16}$ ($2.08$)
$b_{2}^{Sz3}$ |  $-0.3^{+2.8}_{-3.2}$ ($-0.3$) |  $-1.7\pm 9.2$ ($-2.1$) |  $-0.7^{+2.2}_{-3.4}$ ($-1.1$) |  $-3.7^{+2.1}_{-7.0}$ ($-7.1$)
$b_{\mathcal{G}_{2}}^{Sz3}$ |  $-0.8\pm 1.5$ ($-1.2$) |  $-7.1\pm 5.7$ ($-10.1$) |  $0.1\pm 1.3$ ($0.8$) |  $1.5^{+3.2}_{-2.9}$ ($1.3$)
Table 7: Parameter constraints for all analyses with broader priors of
nuisance parameters for $w$CDM. We show results using full shape + BAO data
both for CMB-free analyses and those with a Planck prior. We show the 6
sampled cosmological parameters as well as 3 bias parameters per redshift bin
and sky cut. The values shown are the means with the $1\sigma$ confidence
intervals with the best-fit values in parenthesis.

The effect of broader priors for the $w$CDM analysis is shown in Fig. 11 for
the CMB-free case and in Fig. 12 for the case with the prior on $A_{s}$ and
$n_{s}$. Table 7 includes the constraints on all sampled parameters for all 4
different analyses, including 3 bias parameters. The conclusions of this
exercise are very similar to those of the $\Lambda$CDM case, with large
effects in the CMB-free and $10\times\sigma$ case, specially for $n_{s}$, but
also for $A_{s}$; and smaller effects in the $3\times\sigma$ scenario. This is
again driven by large values of the nonlinear bias and counter-term
parameters, with the former reaching values of $O(-10)$ and the latter of
$O(-300)$ in the $10\times\sigma$ case.

Figure 11:  Posteriors for the sampled cosmological parameters in the CMB-free
analysis in $w$CDM for three different options for the priors on nuisance
parameters: baseline case of Eq. (2.20) and the cases with standard deviations
of nuisance parameters, except for $b_{1}$, increased by factors of $3$ and
$10$. Dot-dashed lines show the best-fit points showing that those are also
shifted.

In these analyses with varying $w$, that parameter is also shifted slightly
when priors are modified. This small change occurs mostly in the CMB-free case
when the priors are the broadest, moving $w$ towards $-1$ and increasing the
size of its posterior. In the CMB-dependent analyses, only a minor broadening
of the posterior is seen, with no significant shift in $w$. As in the
$\Lambda$CDM case, the main effect of broadening the priors in the scenario
with CMB priors is to shift the matter density measurement towards lower
values.

Figure 12:  Posteriors for the sampled cosmological parameters in the analysis
with a CMB prior in $w$CDM for three different options for the priors on
nuisance parameters: baseline case of Eq. (2.20) and the cases with standard
deviations of nuisance parameters, except for $b_{1}$, increased by factors of
$3$ and $10$. Dot-dashed lines show the best-fit points showing that those are
also shifted.

###  A.3 $wA$CDM

The effect of larger priors in $wA$CDM is again similar to the two cases
above, with a few exceptions which we will discuss. This is shown in Fig. 13
for the CMB-free case and in Fig. 14 for the CMB-dependent case, with a
summary of constraints in Table 8.

Figure 13:  Posteriors for the sampled cosmological parameters in the CMB-free
analysis in $wA$CDM for three different options for the priors on nuisance
parameters: baseline case of Eq. (2.20) and the cases with standard deviations
of nuisance parameters, except for $b_{1}$, increased by factors of $3$ and
$10$. Dot-dashed lines show the best-fit points showing that those are also
shifted.

Starting with the CMB-free case, we can see that the effects of the
$10\times\sigma$ case are large, with an extreme bias on $n_{s}$ and smaller
shifts on most other parameters. The $3\times\sigma$ case shows much milder
shifts, but still substantial fractions of the parameter uncertainties.
Interestingly, the preferred value of $A_{s}$ is not shifted, always remaining
very low, even if its contours are broadened somewhat towards higher values.
The consistently low value of this parameter is likely due to the degeneracy
with the interaction parameter $A$, which is instead shifted towards zero as
the priors are broadened. So, while $A_{s}$ remains fixed, a change in $A$
towards less negative values reduces the overall amplitude of the matter power
spectrum, which is the same effect seen when enlarging the priors in
$\Lambda$CDM and $w$CDM.

Figure 14:  Posteriors for the sampled cosmological parameters in the analysis
with a CMB prior in $wA$CDM for three different options for the priors on
nuisance parameters: baseline case of Eq. (2.20) and the cases with standard
deviations of nuisance parameters, except for $b_{1}$, increased by factors of
$3$ and $10$. Dot-dashed lines show the best-fit points showing that those are
also shifted.

The CMB-dependent case is also similar to the corresponding $\Lambda$CDM and
$w$CDM analyses. The biggest shifts are once more in the value of
$\omega_{c}$, but there are also some changes in $w$ and $A$. For both dark
energy parameters, the effect is to increase the contours and to move them
towards the $\Lambda$CDM values of $w=-1$, $A=0$. The two cases with broader
priors are mostly consistent with each other, even when considering the bias
parameters, as can be seen in Table 8. Looking at the dark energy parameters
only, these analyses are also consistent with the baseline case shown in the
main text, but reduce the preference for interacting dark energy ($A\neq 0$)
to $0.5\sigma$.

Parameter |  $3\times\sigma$ CMB-free |  $10\times\sigma$ CMB-free |  $3\times\sigma$ CMB prior |  $10\times\sigma$ CMB prior
---|---|---|---|---
$w$ |  $-1.21\pm 0.14$ ($-1.16$) |  $-1.07^{+0.19}_{-0.10}$ ($-1.00$) |  $-0.985^{+0.081}_{-0.038}$ ($-0.982$) |  $-0.994^{+0.083}_{-0.046}$ ($-0.991$)
$A$ |  $-15^{+14}_{-13}$ ($-1$) |  $1\pm 16$ ($5$) |  $2.7^{+3.9}_{-4.5}$ ($3.4$) |  $3.0^{+4.9}_{-5.9}$ ($2.2$)
$\omega_{c}$ |  $0.133^{+0.011}_{-0.012}$ ($0.124$) |  $0.117^{+0.010}_{-0.018}$ ($0.104$) |  $0.110^{+0.006}_{-0.007}$ ($0.111$) |  $0.103^{+0.007}_{-0.008}$ ($0.098$)
$100\omega_{b}$ |  $2.265\pm 0.039$ ($2.256$) |  $2.269\pm 0.038$ ($2.263$) |  $2.261\pm 0.039$ ($2.261$) |  $2.262\pm 0.038$ ($2.264$)
$h$ |  $0.724\pm 0.028$ ($0.711$) |  $0.688^{+0.022}_{-0.044}$ ($0.665$) |  $0.671^{+0.011}_{-0.013}$ ($0.672$) |  $0.666^{+0.012}_{-0.015}$ ($0.664$)
$\log(10^{10}A_{s})$ |  $2.15^{+0.29}_{-0.49}$ ($2.61$) |  $2.31^{+0.37}_{-0.63}$ ($2.31$) |  $3.037\pm 0.042$ ($3.037$) |  $3.030\pm 0.043$ ($3.072$)
$n_{s}$ |  $0.824^{+0.098}_{-0.070}$ ($0.875$) |  $0.126^{+0.059}_{-0.32}$ ($-0.049$) |  $0.963\pm 0.012$ ($0.962$) |  $0.961\pm 0.012$ ($0.963$)
$b_{1}^{Nz1}$ |  $2.58^{+0.50}_{-0.32}$ ($2.19$) |  $1.32^{+0.16}_{-0.69}$ ($1.04$) |  $1.97\pm 0.14$ ($1.96$) |  $1.98\pm 0.21$ ($1.89$)
$b_{2}^{Nz1}$ |  $-1.2^{+2.2}_{-2.9}$ ($-0.9$) |  $-4.7^{+5.9}_{-6.7}$ ($0.3$) |  $-1.0^{+1.6}_{-2.8}$ ($-1.4$) |  $-1.1^{+3.2}_{-5.9}$ ($-3.1$)
$b_{\mathcal{G}_{2}}^{Nz1}$ |  $-0.9\pm 1.5$ ($-0.7$) |  $-8.1^{+4.2}_{-4.8}$ ($-7.4$) |  $-0.5\pm 1.2$ ($-0.7$) |  $-0.2\pm 3.5$ ($-2.6$)
$b_{1}^{Sz1}$ |  $2.63^{+0.48}_{-0.32}$ ($2.28$) |  $1.39^{+0.15}_{-0.72}$ ($1.05$) |  $2.04\pm 0.14$ ($2.09$) |  $1.99^{+0.25}_{-0.20}$ ($1.75$)
$b_{2}^{Sz1}$ |  $-0.1^{+2.6}_{-3.0}$ ($-0.2$) |  $-1.3\pm 8.0$ ($2.91$) |  $0.2^{+2.3}_{-3.0}$ ($-0.6$) |  $1.1\pm 5.9$ ($-6.9$)
$b_{\mathcal{G}_{2}}^{Sz1}$ |  $0.4\pm 1.5$ ($0.0$) |  $-6.1\pm 5.0$ ($-6.8$) |  $0.3\pm 1.2$ ($0.7$) |  $1.0^{+3.2}_{-2.8}$ ($-2.3$)
$b_{1}^{Nz3}$ |  $2.71^{+0.69}_{-0.45}$ ($2.19$) |  $1.21^{+0.13}_{-0.73}$ ($0.77$) |  $1.88\pm 0.14$ ($1.90$) |  $1.85^{+0.22}_{-0.19}$ ($1.70$)
$b_{2}^{Nz3}$ |  $-4.7^{+2.6}_{-3.5}$ ($-5.0$) |  $-6.0\pm 7.8$ ($-2.7$) |  $-4.8^{+1.4}_{-2.0}$ ($-4.1$) |  $-4.9^{+1.7}_{-4.7}$ ($-7.4$)
$b_{\mathcal{G}_{2}}^{Nz3}$ |  $-1.4\pm 1.8$ ($-0.8$) |  $-11.8\pm 5.5$ ($-12.9$) |  $-0.9\pm 1.3$ ($-1.3$) |  $-0.9^{+3.2}_{-2.8}$ ($-2.5$)
$b_{1}^{Sz3}$ |  $3.02^{+0.61}_{-0.42}$ ($2.54$) |  $1.61^{+0.21}_{-0.85}$ ($1.29$) |  $2.24\pm 0.13$ ($2.29$) |  $2.23\pm 0.22$ ($2.24$)
$b_{2}^{Sz3}$ |  $-0.2\pm 2.9$ ($0.0$) |  $-2.1^{+8.8}_{-9.9}$ ($-1.7$) |  $-0.4^{+2.3}_{-3.3}$ ($-1.1$) |  $-2.8^{+2.7}_{-7.7}$ ($-6.6$)
$b_{\mathcal{G}_{2}}^{Sz3}$ |  $-0.8\pm 1.7$ ($0.8$) |  $-7.0\pm 5.8$ ($-10.5$) |  $0.0\pm 1.3$ ($-0.5$) |  $1.6^{+3.4}_{-3.0}$ ($3.4$)
Table 8: Parameter constraints for all analyses with broader priors of
nuisance parameters for $wA$CDM. We show results using full shape + BAO data
both for CMB-free analyses and those with a Planck prior (labeled CMB). We
show the 7 sampled cosmological parameters as well as 3 bias parameters per
redshift bin and sky cut. The values shown are the means with the $1\sigma$
confidence intervals with the best-fit values in parenthesis.

As seen previously, when the priors are broadened, large second-order biases
are measured, particularly in the higher redshift bin, where $O(-5)$ or larger
values are preferred for $b^{Nz3}_{2}$ in all cases. In the CMB-free case with
a factor of 10 increase of the priors, even larger values are seen for various
bias parameters, similar to those measured in the $\Lambda$CDM and $w$CDM
versions of this exercise.

##  Appendix B Full contours

In this appendix, we show the contours for the full set of sampled parameters
in each analysis, including all nuisance parameters that were not marginalised
over analytically.

Parameter |  $\Lambda$CDM CMB prior |  $\Lambda$CDM CMB-free
---|---|---
$\omega_{c}$ |  $0.118\pm 0.005$ ($0.120$) |  $0.121^{+0.008}_{-0.009}$ ($0.120$)
$100\omega_{b}$ |  $2.264\pm 0.037$ ($2.256$) |  $2.266\pm 0.038$ ($2.244$)
$h$ |  $0.678\pm 0.008$ ($0.681$) |  $0.681\pm 0.010$ ($0.678$)
$\log(10^{10}A_{s})$ |  $3.021\pm 0.040$ ($3.023$) |  $2.821\pm 0.158$ ($2.832$)
$n_{s}$ |  $0.969\pm 0.012$ ($0.974$) |  $0.975\pm 0.063$ ($0.972$)
$b_{1}^{Nz1}$ |  $1.941\pm 0.068$ ($1.908$) |  $2.17\pm 0.15$ ($2.14$)
$b_{2}^{Nz1}$ |  $-0.71^{+0.77}_{-0.89}$ ($-1.10$) |  $-0.35\pm 0.88$ ($-0.62$)
$b_{\mathcal{G}_{2}}^{Nz1}$ |  $-0.21\pm 0.39$ ($-0.2539$) |  $-0.26\pm 0.43$ ($-0.32$)
$b_{1}^{Sz1}$ |  $1.972\pm 0.076$ ($1.96$) |  $2.19\pm 0.15$ ($2.16$)
$b_{2}^{Sz1}$ |  $-0.26^{+0.83}_{-0.94}$ ($0.04$) |  $-0.03\pm 0.91$ ($-0.36$)
$b_{\mathcal{G}_{2}}^{Sz1}$ |  $0.16\pm 0.40$ ($0.27$) |  $0.22\pm 0.44$ ($0.00$)
$b_{1}^{Nz3}$ |  $1.988\pm 0.074$ ($1.986$) |  $2.25\pm 0.17$ ($2.23$)
$b_{2}^{Nz3}$ |  $-1.90^{+0.88}_{-0.99}$ ($-1.73$) |  $-1.3\pm 1.0$ ($-1.3$)
$b_{\mathcal{G}_{2}}^{Nz3}$ |  $-0.09\pm 0.44$ ($-0.03$) |  $-0.04\pm 0.50$ ($-0.11$)
$b_{1}^{Sz3}$ |  $2.181\pm 0.079$ ($2.186$) |  $2.43\pm 0.17$ ($2.41$)
$b_{2}^{Sz3}$ |  $-0.17\pm 0.93$ ($-0.21$) |  $0.01\pm 0.95$ ($-0.01$)
$b_{\mathcal{G}_{2}}^{Sz3}$ |  $-0.11\pm 0.45$ ($-0.20$) |  $-0.23\pm 0.52$ ($-0.30$)
Table 9: Parameter constraints for the two $\Lambda$CDM analyses using full shape data as well as post-reconstructed measurements of the BAO scale (FS+BAO). We show the 5 sampled cosmological parameters as well as 3 bias parameters per redshift bin and sky cut. The values shown are the means with the $1\sigma$ confidence intervals and the best-fit value in parenthesis. Figure 15:  Posteriors for all sampled parameters in the baseline analyses in $\Lambda$CDM, showing both the case with a CMB prior and without. Parameter |  $w$CDM CMB prior |  $w$CDM CMB-free
---|---|---
$w$ |  $-1.002^{+0.081}_{-0.073}$ ($-1.023$) |  $-1.17^{+0.12}_{-0.11}$ ($-1.093$)
$\omega_{c}$ |  $0.118\pm 0.005$ ($0.118$) |  $0.128^{+0.009}_{-0.011}$ ($0.122$)
$100\omega_{b}$ |  $2.265\pm 0.038$ ($2.286$) |  $2.269\pm 0.038$ ($2.273$)
$h$ |  $0.679^{+0.014}_{-0.016}$ ($0.689$) |  $0.713\pm 0.024$ ($0.699$)
$\log(10^{10}A_{s})$ |  $3.021\pm 0.041$ ($3.027$) |  $2.62\pm 0.21$ ($2.77$)
$n_{s}$ |  $0.968\pm 0.012$ ($0.966$) |  $0.947\pm 0.064$ ($0.9721$)
$b_{1}^{Nz1}$ |  $1.942\pm 0.073$ ($1.91$) |  $2.31\pm 0.18$ ($2.19$)
$b_{2}^{Nz1}$ |  $-0.72^{+0.73}_{-0.89}$ ($-0.84$) |  $-0.39\pm 0.90$ ($-0.40$)
$b_{\mathcal{G}_{2}}^{Nz1}$ |  $-0.19\pm 0.39$ ($-0.2539$) |  $-0.22\pm 0.44$ ($-0.20$)
$b_{1}^{Sz1}$ |  $1.974\pm 0.080$ ($1.96$) |  $2.32\pm 0.18$ ($2.23$)
$b_{2}^{Sz1}$ |  $-0.22\pm 0.89$ ($-0.11$) |  $-0.11\pm 0.91$ ($-0.34$)
$b_{\mathcal{G}_{2}}^{Sz1}$ |  $0.17\pm 0.40$ ($0.00$) |  $0.32\pm 0.46$ ($0.00$)
$b_{1}^{Nz3}$ |  $1.987\pm 0.076$ ($1.960$) |  $2.43\pm 0.22$ ($2.297$)
$b_{2}^{Nz3}$ |  $-1.91^{+0.86}_{-1.0}$ ($-1.88$) |  $-1.2\pm 1.0$ ($-1.5$)
$b_{\mathcal{G}_{2}}^{Nz3}$ |  $-0.10\pm 0.44$ ($-0.07$) |  $0.14\pm 0.53$ ($-0.15$)
$b_{1}^{Sz3}$ |  $2.179\pm 0.082$ ($2.168$) |  $2.60\pm 0.22$ ($2.47$)
$b_{2}^{Sz3}$ |  $-0.17\pm 0.973$ ($-0.01$) |  $0.00\pm 0.95$ ($-0.01$)
$b_{\mathcal{G}_{2}}^{Sz3}$ |  $-0.12\pm 0.46$ ($-0.07$) |  $-0.16\pm 0.55$ ($-0.09$)
Table 10: Parameter constraints for the two $w$CDM analyses using full shape data as well as post-reconstructed measurements of the BAO scale (FS+BAO). We show the 6 sampled cosmological parameters as well as 3 bias parameters per redshift bin and sky cut. The values shown are the means with the $1\sigma$ confidence intervals and the best-fit value in parenthesis. Figure 16:  Posteriors for all sampled parameters in the baseline analyses in $w$CDM, showing both the case with a CMB prior and without. Figure 17:  Posteriors for all sampled parameters in the CMB-dependent analyses in $wA$CDM, showing both results with FS data only and those with FS+BAO. Parameter |  $wA$CDM FS |  $wA$CDM FS+BAO
---|---|---
$w$ |  $-0.954^{+0.024}_{-0.046}$ ($-0.987$) |  $-0.972^{+0.036}_{-0.029}$ ($-0.994$)
$A$ |  $4.8^{+2.8}_{-3.8}$ ($3.7$) |  $3.9^{+3.2}_{-3.7}$ ($5.4$)
$\omega_{c}$ |  $0.1239^{+0.0062}_{-0.0069}$ ($0.1335$) |  $0.1201^{+0.0052}_{-0.0058}$ ($0.1200$)
$100\omega_{b}$ |  $2.267\pm 0.038$ ($2.296$) |  $2.266\pm 0.038$ ($2.268$)
$h$ |  $0.680\pm 0.012$ ($0.6912$) |  $0.677\pm 0.011$ ($0.677$)
$\log(10^{10}A_{s})$ |  $3.038\pm 0.042$ ($3.060$) |  $3.039\pm 0.043$ ($3.033$)
$n_{s}$ |  $0.964\pm 0.012$ ($0.958$) |  $0.966\pm 0.012$ ($0.962$)
$b_{1}^{Nz1}$ |  $2.020\pm 0.085$ ($1.953$) |  $2.011\pm 0.090$ ($2.030$)
$b_{2}^{Nz1}$ |  $-0.36\pm 0.84$ ($-0.55$) |  $-0.53^{+0.79}_{-0.88}$ ($-0.48$)
$b_{\mathcal{G}_{2}}^{Nz1}$ |  $-0.28\pm 0.40$ ($-0.486$) |  $-0.28\pm 0.39$ ($-0.29$)
$b_{1}^{Sz1}$ |  $2.047\pm 0.090$ ($1.974$) |  $2.033\pm 0.092$ ($2.064$)
$b_{2}^{Sz1}$ |  $-0.04\pm 0.85$ ($-0.52$) |  $-0.07^{+0.86}_{-0.95}$ ($-0.38$)
$b_{\mathcal{G}_{2}}^{Sz1}$ |  $0.14\pm 0.41$ ($-0.32$) |  $0.16\pm 0.41$ ($0.00$)
$b_{1}^{Nz3}$ |  $2.059\pm 0.085$ ($1.963$) |  $2.045\pm 0.090$ ($2.065$)
$b_{2}^{Nz3}$ |  $-1.55\pm 0.97$ ($-2.10$) |  $-1.62^{+0.90}_{-1.0}$ ($-1.81$)
$b_{\mathcal{G}_{2}}^{Nz3}$ |  $-0.12\pm 0.45$ ($-0.08$) |  $-0.16\pm 0.45$ ($-0.29$)
$b_{1}^{Sz3}$ |  $2.229\pm 0.090$ ($2.104$) |  $2.232\pm 0.093$ ($2.232$)
$b_{2}^{Sz3}$ |  $0.04\pm 0.93$ ($0.83$) |  $-0.01\pm 0.97$ ($0.00$)
$b_{\mathcal{G}_{2}}^{Sz3}$ |  $-0.20\pm 0.45$ ($0.20$) |  $-0.19\pm 0.46$ ($-0.25$)
Table 11: Parameter constraints for the two CMB-dependent analyses for $wA$CDM. We show results using full shape only (FS) as well those including the post-reconstructed measurements of the BAO scale (FS+BAO). We show the 7 sampled cosmological parameters as well as 3 bias parameters per redshift bin and sky cut. The values shown are the means with the $1\sigma$ confidence intervals and the best-fit value in parenthesis. Figure 18:  Posteriors for all sampled parameters in the CMB-free analyses in $wA$CDM, showing both results with FS data only and those with FS+BAO. Parameter |  $wA$CDM FS |  $wA$CDM FS+BAO
---|---|---
$w$ |  $-1.48\pm 0.20$ ($-1.28$) |  $-1.16^{+0.12}_{-0.069}$ ($-1.097$)
$A$ |  $-18.9^{+17}_{-7.7}$ ($-0.7631$) |  $-21^{+14}_{-11}$ ($-0.1077$)
$\omega_{c}$ |  $0.148^{+0.013}_{-0.018}$ ($0.138$) |  $0.1275^{+0.0082}_{-0.010}$ ($0.1249$)
$100\omega_{b}$ |  $2.269\pm 0.038$ ($2.266$) |  $2.268\pm 0.038$ ($2.253$)
$h$ |  $0.785^{+0.041}_{-0.046}$ ($0.746$) |  $0.711^{+0.017}_{-0.024}$ ($0.6992$)
$\log(10^{10}A_{s})$ |  $1.85^{+0.20}_{-0.28}$ ($2.37$) |  $2.07\pm 0.35$ ($2.725$)
$n_{s}$ |  $0.844\pm 0.080$ ($0.885$) |  $0.945\pm 0.062$ ($0.9553$)
$b_{1}^{Nz1}$ |  $2.90^{+0.27}_{-0.21}$ ($2.45$) |  $2.69\pm 0.28$ ($2.215$)
$b_{2}^{Nz1}$ |  $-0.40\pm 0.97$ ($-0.63$) |  $-0.34\pm 0.96$ ($-0.66$)
$b_{\mathcal{G}_{2}}^{Nz1}$ |  $-0.09\pm 0.57$ ($-0.33$) |  $-0.05\pm 0.52$ ($-0.25$)
$b_{1}^{Sz1}$ |  $2.94^{+0.28}_{-0.22}$ ($2.47$) |  $2.71\pm 0.29$ ($2.245$)
$b_{2}^{Sz1}$ |  $-0.27\pm 0.95$ ($-0.39$) |  $-0.21\pm 0.92$ ($-0.28$)
$b_{\mathcal{G}_{2}}^{Sz1}$ |  $0.83\pm 0.62$ ($0.00$) |  $0.64^{+0.55}_{-0.62}$ ($0.00$)
$b_{1}^{Nz3}$ |  $3.26^{+0.36}_{-0.28}$ ($2.65$) |  $2.97^{+0.36}_{-0.42}$ ($2.32$)
$b_{2}^{Nz3}$ |  $-0.7\pm 1.0$ ($-1.2$) |  $-0.7\pm 1.0$ ($-1.6$)
$b_{\mathcal{G}_{2}}^{Nz3}$ |  $0.51\pm 0.71$ ($-0.23$) |  $0.40\pm 0.65$ ($-0.15$)
$b_{1}^{Sz3}$ |  $3.42^{+0.37}_{-0.28}$ ($2.77$) |  $3.14^{+0.36}_{-0.43}$ ($2.51$)
$b_{2}^{Sz3}$ |  $0.03\pm 0.99$ ($0.00$) |  $0.03\pm 0.97$ ($0.00$)
$b_{\mathcal{G}_{2}}^{Sz3}$ |  $-0.20\pm 0.73$ ($-0.25$) |  $-0.19\pm 0.68$ ($-0.28$)
Table 12: Parameter constraints for the two CMB-free analyses for $wA$CDM. We
show results using full shape only (FS) as well those including the post-
reconstructed measurements of the BAO scale (FS+BAO). We show the 7 sampled
cosmological parameters as well as 3 bias parameters per redshift bin and sky
cut. The values shown are the means with the $1\sigma$ confidence intervals
and the best-fit value in parenthesis.