# New tests of dark sector interactions from the full-shape galaxy power
spectrum

Rafael C. Nunes<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal do Rio Grande do Sul, 91501-970 Porto Alegre RS, Brazil  Divisão de
Astrofísica, Instituto Nacional de Pesquisas Espaciais, Avenida dos
Astronautas 1758, São José dos Campos, 12227-010, São Paulo, Brazil  Sunny
Vagnozzi<EMAIL_ADDRESS>Kavli Institute for Cosmology (KICC)
and Institute of Astronomy,
University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom
Suresh Kumar<EMAIL_ADDRESS>Department of Mathematics, Indira Gandhi
University, Meerpur, Haryana 122502, India  Eleonora Di Valentino
<EMAIL_ADDRESS>School of Mathematics and Statistics, University
of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom  Olga Mena
<EMAIL_ADDRESS>Instituto de Física Corpuscular (IFIC), University of
Valencia-CSIC, E-46980, Valencia, Spain

###### Abstract

We explore the role of redshift-space galaxy clustering data in constraining
non-gravitational interactions between dark energy (DE) and dark matter (DM),
for which state-of-the-art limits have so far been obtained from late-time
background measurements. We use the joint likelihood for pre-reconstruction
full-shape (FS) galaxy power spectrum and post-reconstruction Baryon Acoustic
Oscillation (BAO) measurements from the BOSS DR12 sample, alongside Cosmic
Microwave Background (CMB) data from Planck: from this dataset combination we
infer $H_{0}=68.02^{+0.49}_{-0.60}\,{\rm km}/{\rm s}/{\rm Mpc}$ and the
2$\sigma$ lower limit $\xi>-0.12$, among the strongest limits ever reported on
the DM-DE coupling strength $\xi$ for the particular model considered.
Contrary to what has been observed for the $\Lambda$CDM model and simple
extensions thereof, we find that the CMB+FS combination returns tighter
constraints compared to the CMB+BAO one, suggesting that there is valuable
additional information contained in the broadband of the power spectrum. We
test this finding by running additional CMB-free analyses and removing sound
horizon information, and discuss the important role of the equality scale in
setting constraints on DM-DE interactions. Our results reinforce the critical
role played by redshift-space galaxy clustering measurements in the epoch of
precision cosmology, particularly in relation to tests of non-minimal dark
sector extensions of the $\Lambda$CDM model.

##  I Introduction

Dark energy (DE) accounts for approximately $70\%$ of the Universe’s energy
budget, yet its nature remains puzzling. Within the standard six-parameter
$\Lambda$CDM cosmological model, DE takes the form of a smooth, time-
independent, spatially uniform vacuum energy component. This simple model is
able to provide an extremely good fit to a wide variety of cosmological and
astrophysical observations, including anisotropies in the Cosmic Microwave
Background (CMB), the clustering of the large-scale structure (LSS), the
magnitude-redshift relation of Type Ia Supernovae (SNeIa) in the Hubble flow,
the distortion of images of distant galaxies due to weak lensing from the
intervening LSS (cosmic shear), and the abundances of light elements Riess
_et al._ (1998); Perlmutter  _et al._ (1999); Aghanim  _et al._ (2020a); Aiola
_et al._ (2020); Alam  _et al._ (2021); Asgari  _et al._ (2021); Mossa  _et
al._ (2020); Dutcher  _et al._ (2021); Abbott  _et al._ (2022); Scolnic  _et
al._ (2021).

Of course, this very economical picture does not need to be the end of the
story as far as DE is concerned. One reason for this belief is the (in)famous
cosmological constant problem Weinberg (1989). In an attempt to construct more
realistic physical models for DE, significant effort has been placed into
going beyond the minimal scenario. Examples include models endowing DE with a
dynamical nature Wetterich (1988); Ratra and Peebles (1988); Caldwell  _et
al._ (1998), or featuring non-trivial interactions between DE and other
components of the Universe’s energy budget. Examples of the latter are
interacting DE (IDE) models, which exhibit interactions between dark matter
(DM) and DE, historically motivated by attempts to address the coincidence
problem Zlatev  _et al._ (1999); Huey and Wandelt (2006); Velten  _et al._
(2014).111However, the amount of energy exchange required to address this
problem is now excluded by observations. See in addition Refs. D’Amico  _et
al._ (2016); Marsh (2017) for further issues which have been raised in this
context owing to quantum corrections.

State-of-the-art constraints on IDE cosmologies arise primarily from
measurements of temperature and polarization anisotropies in the CMB, in
combination with measurements of the late-time background expansion history
from BAO and Hubble flow SNeIa. In combination, these probes have set
stringent constraints on the strength of the DM-DE interaction, typically
denoted by $\xi$: the constraints depend on the specific assumptions
concerning the form of the interaction, and on the datasets adopted, but
typically restrict $|\xi|\lesssim{\cal O}(0.1)$ (see e.g. Refs. Di Valentino
_et al._ (2017); Yang  _et al._ (2019a); Cheng  _et al._ (2020); Yang  _et
al._ (2020a, 2021a) for some of the latest constraints, see also Refs. Zhang
_et al._ (2019, 2018); Liu  _et al._ (2022)). Incidentally, these tight
constraints from BAO and SNeIa are the key reason why IDE scenarios fall short
of fully addressing the Hubble tension Bernal  _et al._ (2016); Addison  _et
al._ (2018); Mörtsell and Dhawan (2018); Lemos  _et al._ (2019); Aylor  _et
al._ (2019); Knox and Millea (2020); Arendse  _et al._ (2020); Zhang and Huang
(2021); Efstathiou (2021); Cai  _et al._ (2022), despite these models having
experienced somewhat of a revival in recent years in attempts to address
cosmological tensions (see e.g. Refs. Di Valentino  _et al._ (2021a, b);
Perivolaropoulos and Skara (2021); Schöneberg  _et al._ (2021); Abdalla  _et
al._ (2022) for reviews on the Hubble tension).

In recent years, significant efforts have gone into extracting LSS clustering
information beyond that contained within the (reconstructed) BAO peaks, using
information from the full-shape (FS) power spectrum of biased tracers of the
LSS. These efforts have been driven in part by significant advances in the so-
called Effective Field Theory of LSS (EFTofLSS) Baumann  _et al._ (2012),
which have allowed for concrete applications of this framework to real data
from galaxy redshift surveys: these advances motivate us to re-analyze IDE
models in this context. It is our major goal in the present work to explore
whether redshift-space galaxy clustering data can improve state-of-the-art
constraints on the DM-DE interaction strength.222We note that the same
interacting DE model that will be examined in this work had actually been
confronted against FS galaxy power spectrum measurements by one of us in 2009
in Ref. Gavela  _et al._ (2009). However, this early study was performed using
the FS power spectrum of luminous red galaxies from the Sloan Digital Sky
Survey (SDSS) Data Release 4 (DR4) sample, alongside CMB and SNeIa
measurements from WMAP and the Union dataset. All these datasets are now
significantly outdated, and the modeling of the FS galaxy power spectrum was
significantly simplified. Therefore it is extremely timely to re-examine this
issue nearly 15 years later, making use of state-of-the-art cosmological
measurements, in combination with a more robust theoretical modeling of the FS
galaxy power spectrum. See also the later Ref. Duniya  _et al._ (2015) for an
investigation of horizon-scale relativistic effects in the galaxy power
spectrum in the presence of interacting DE.

The rest of this paper is then organized as follows. In Sec. II, we review the
physics of IDE models, present the specific model we will study, before
briefly discussing the EFTofLSS and how our model fits within this framework.
In Sec. III we discuss the datasets and analysis methodology we make use of.
The results of this analysis are presented in Sec. IV. Finally, in Sec. V we
provide our concluding remarks.

##  II Dark sector interactions and the full-shape galaxy power spectrum

###  II.1 Interacting dark energy

We begin by reviewing the basic features of IDE models. In the following, we
shall work under the assumption of a spatially flat Friedmann-Lemaître-
Robertson-Walker metric. In the absence of DM-DE interactions the stress-
energy tensors of DM and DE, which we denote by $T^{\mu\nu}_{c}$ and
$T^{\mu\nu}_{x}$ respectively, are separately covariantly conserved. This
implies:

$\displaystyle\nabla_{\nu}T^{\mu\nu}_{c}=\nabla_{\nu}T^{\mu\nu}_{x}=0\,,$ (1)

where $\nabla_{\mu}$ denotes the covariant derivative. However, particularly
in the case where DE is described by a new light field (as in quintessence
models), couplings of DE to matter fields (typically with gravitational
strength) are somewhat unavoidable, unless protected by a specific symmetry
Carroll (1998). When introducing non-gravitational interactions between DM and
DE, there are essentially two ways to proceed. One can either work from first
principles in the context of a specific (possibly UV-complete) model, where
the interaction between the dark components is introduced at the level of the
action. Historically, this was in fact the approach first followed,
particularly in the context of so-called coupled quintessence models, where a
light quintessence DE field is coupled to the DM field Wetterich (1995);
Amendola (2000a, b); Mangano  _et al._ (2003); Farrar and Peebles (2004).
Alternatively, one can introduce a phenomenological parametrization for the
DM-DE interaction at the level of the conservation equations in Eq. (1), in
such a way that the two stress-energy tensors are separately not conserved,
but their sum is. In this work, we shall follow the second approach.333We note
that a separate but nonetheless associated possibility recently considered in
the literature is that where DE interacts either with baryons Vagnozzi  _et
al._ (2020); Jiménez  _et al._ (2020); Vagnozzi  _et al._ (2021a); Benisty and
Davis (2022); Ferlito  _et al._ (2022) or with electromagnetism Calabrese  _et
al._ (2014); Martins and Pinho (2015); Martins  _et al._ (2015, 2016);
Martinelli  _et al._ (2021). See also Refs. He and Zhang (2017); Zhang (2022)
for works discussing the related possibility of direct detection of DE.
Typically, one assumes that the covariant derivatives of the DM and DE stress-
energy tensors evolve as:

$\displaystyle\nabla_{\nu}T^{\mu\nu}_{c}$ $\displaystyle=$
$\displaystyle\frac{Qu^{\mu}_{c}}{a}\,;$ (2)
$\displaystyle\nabla_{\nu}T^{\mu\nu}_{x}$ $\displaystyle=$
$\displaystyle-\frac{Qu^{\mu}_{c}}{a}\,,$ (3)

where $u^{\mu}_{c}$ is the DM velocity 4-vector, and $Q$ is the DM-DE
interaction rate (with units of energy per volume per time). At this point,
one needs to make a (phenomenological) choice for the functional form of $Q$.
A choice commonly adopted in the literature is the following:

$\displaystyle Q=\xi{\cal H}\rho_{x}\,,$ (4)

where ${\cal H}$ is the conformal Hubble rate, $\rho_{x}$ is the DE energy
density and $\xi$ is a dimensionless parameter which controls the strength of
the DM-DE interaction: a value of $\xi>0$ ($\xi<0$) indicates energy transfer
from the DE (DM) to the DM (DE) sector.

A comment is in order regarding the appearance of ${\cal H}$ in the
interaction rate. This might appear puzzling at first, as it may suggest that
an interaction rate which should ultimately be determined by local
interactions is actually sensitive to a global quantity such as the expansion
rate, see e.g. Ref. Valiviita  _et al._ (2008). In reality, the appearance of
${\cal H}$ is ultimately a consequence of the first principle of
thermodynamics, a local law (independently of the context to which one applies
it, for example cosmology) which essentially states that changes in the
density respond to changes in the volume due to cosmic expansion. In other
words, the conservation equations do not explicitly know about the underlying
cosmology or theory of gravity, but only about the change in scale or volume.
In fact, one can eliminate ${\cal H}$ entirely and use the scale factor as
time variable (as in the non-interacting case).444We thank the anonymous
referee of one of our previous papers for drawing our attention to this issue,
whereas S.V. thanks Marco Bruni for sharing this illuminating explanation. In
addition, we note that Ref. Pan  _et al._ (2020) explicitly showed how
interaction rates featuring factors related to ${\cal H}$, including but not
limited to the case considered in Eq. (4), may naturally emerge from first
principles when considering well-motivated field theories for IDE scenarios.

In the presence of the coupling given by Eq. (4), the continuity equations for
the DM and DE energy densities $\rho_{c}$ and $\rho_{x}$ are modified to:

$\displaystyle\dot{\rho}_{c}+3{\cal H}\rho_{c}$ $\displaystyle=$
$\displaystyle\xi{\cal H}\rho_{x}\,;$ (5) $\displaystyle\dot{\rho}_{x}+3{\cal
H}(1+w_{x})\rho_{x}$ $\displaystyle=$ $\displaystyle-\xi{\cal H}\rho_{x}\,,$
(6)

where $w_{x}$ is the DE equation of state (EoS). Assuming that both $w_{x}$ ad
$\xi$ are constant in cosmic time, the above Eqs. (5,6) can be analytically
integrated to give:

$\displaystyle\rho_{c}$ $\displaystyle=$
$\displaystyle\frac{\rho_{c,0}}{a^{3}}+\frac{\rho_{x,0}}{a^{3}}\left[\frac{\xi}{3w_{x}+\xi}\left(1-a^{-3w_{x}-\xi}\right)\right]\,;$
(7) $\displaystyle\rho_{x}$ $\displaystyle=$
$\displaystyle\frac{\rho_{x,0}}{a^{3(1+w_{x})+\xi}}\,,$ (8)

where $\rho_{c,0}$ and $\rho_{x,0}$ are the present-day energy densities of DM
and DE. From Eq. (8), we see that in the presence of such an interaction the
DE component effectively behaves as a component with EoS $w_{x,{\rm
eff}}=w_{x}+\xi/3$.

The presence of the DM-DE coupling also modifies the evolution of
perturbations. Working in synchronous gauge, the coupled system of linear
Einstein-Boltzmann equations for the evolution of the DM and DE density
perturbations ($\delta_{c}$ and $\delta_{x}$) and velocity divergences
($\theta_{c}$ and $\theta_{x}$) is given by Valiviita  _et al._ (2008); Gavela
_et al._ (2010); Lopez Honorez  _et al._ (2010):

$\displaystyle\dot{\delta}_{c}$ $\displaystyle=$
$\displaystyle-\theta_{c}-\frac{1}{2}\dot{h}+\xi{\cal
H}\frac{\rho_{x}}{\rho_{c}}(\delta_{x}-\delta_{c})+\xi\frac{\rho_{x}}{\rho_{c}}\left(\frac{kv_{T}}{3}+\frac{\dot{h}}{6}\right)\,;$
(9) $\displaystyle\dot{\theta}_{c}$ $\displaystyle=$ $\displaystyle-{\cal
H}\theta_{c}\,;$ (10) $\displaystyle\dot{\delta}_{x}$ $\displaystyle=$
$\displaystyle-(1+w_{x})\left(\theta_{x}+\frac{\dot{h}}{2}\right)-\xi\left(\frac{kv_{T}}{3}+\frac{\dot{h}}{6}\right)$
(11) $\displaystyle-3{\cal H}(1-w_{x})\left[\delta_{x}+\frac{{\cal
H}\theta_{x}}{k^{2}}\left(3+3w_{x}+\xi\right)\right]\,;$
$\displaystyle\dot{\theta}_{x}$ $\displaystyle=$ $\displaystyle 2{\cal
H}\theta_{x}+\frac{k^{2}}{1+w_{x}}\delta_{x}+2{\cal
H}\frac{\xi}{1+w_{x}}\theta_{x}-\xi{\cal H}\frac{\theta_{c}}{1+w_{x}}\,,$ (12)

where we set the DE sound speed squared to $c_{s,x}^{2}=1$, and $h$ and
$v_{T}$ refer to the trace of the metric perturbation $h_{ij}$ in the
synchronous gauge and to the center of mass velocity for the total fluid
respectively, where the appearance of the latter is required by gauge
invariance arguments Gavela  _et al._ (2010). The initial conditions for the
DE density perturbation and velocity divergence are also modified, following
Ref. Gavela  _et al._ (2010).

Furthermore, one needs to avoid instabilities in the system of Eqs. (9–12).
Gravitational and non-adiabatic (early-time) instabilities can be avoided
provided that a) $w_{x}\neq-1$, and b) $\xi$ and $(1+w_{x})$ carry opposite
signs Gavela  _et al._ (2009, 2010). In this work, our goal will be that of
examining a scenario which is as close as possible to that of an interacting
vacuum, where $w_{x}=-1$. This is strictly speaking not possible, due to
gravitational instabilities. However, here we will opt for the
phenomenological choice of setting $1+w_{x}$ to a small but non-zero value.
Specifically, we fix $w_{x}=-0.999$, thus requiring $\xi<0$. The rationale
behind this approach is that, for $1+w_{x}$ sufficiently small, the effect of
the DE perturbations is negligible at the level of the coupled Einstein-
Boltzmann system, which are instead essentially only capturing the effect of
the DM-DE interaction governed by $\xi$. This was explicitly demonstrated by
one of us through simulated data in Ref. Di Valentino and Mena (2020).555In
principle, we could equally well have opted for the choice of setting
$w_{x}=-1.001$ and $\xi>0$. However, there are at least two good reasons to
not consider this possibility. The first is that quintessence-like DE
scenarios where $w_{x}>-1$ are actually theoretically more motivated (or at
least theoretically easier to come about) compared to phantom scenarios where
$w_{x}<-1$. The second is that a scenario where energy flows from the DM to
the DE ($\xi<0$) is somewhat theoretically more natural than the reverse case
($\xi>0$). Note, however, that the simplest quintessence models appear on
their own to be observationally disfavored, as they worsen the $H_{0}$ tension
Banerjee  _et al._ (2021). We note that such an approach has already been
followed in several previous works, see e.g. Refs. Di Valentino  _et al._
(2020a, b); Lucca and Hooper (2020). We also note that another approach
towards avoiding gravitational instabilities, investigated in Refs. Li  _et
al._ (2014a, b); Guo  _et al._ (2017); Zhang (2017); Guo  _et al._ (2018a), is
to extend the parameterized post-Friedmann approach to the IDE case.

In closing, it is worth noting that IDE models have received significant
attention in recent years. For a selection of important works on various
aspects of IDE models besides those already discussed, ranging from model-
building to structure formation simulations to observational constraints, we
refer the reader to Refs. Amendola and Quercellini (2003); Pettorino  _et al._
(2005a, b); Barrow and Clifton (2006); Bean  _et al._ (2008); He and Wang
(2008); Pettorino and Baccigalupi (2008); Baldi  _et al._ (2010); Majerotto
_et al._ (2010); Jamil  _et al._ (2010); Martinelli  _et al._ (2010); Baldi
and Pettorino (2011); De Bernardis  _et al._ (2011); Baldi (2012); Pettorino
_et al._ (2012); Carbone  _et al._ (2013); Piloyan  _et al._ (2013); Pettorino
(2013); Pourtsidou  _et al._ (2013); Piloyan  _et al._ (2014); Faraoni  _et
al._ (2014); Ferreira  _et al._ (2017); Ade  _et al._ (2016); Skordis  _et
al._ (2015); Tamanini (2015); Marra (2016); Murgia  _et al._ (2016);
Pourtsidou and Tram (2016); Nunes  _et al._ (2016); Kumar and Nunes (2016,
2017a); Benisty and Guendelman (2017); Yang  _et al._ (2018a); Mifsud and Van
De Bruck (2017); Yang  _et al._ (2017); Kumar and Nunes (2017b); Guo  _et al._
(2018b); Dutta  _et al._ (2018); Feng  _et al._ (2019); Benisty and Guendelman
(2018); Barros  _et al._ (2019); Costa  _et al._ (2018); Yang  _et al._
(2018b); von Marttens  _et al._ (2019a); Elizalde  _et al._ (2018); Guo  _et
al._ (2019); Yang  _et al._ (2018c); Li  _et al._ (2019); von Marttens  _et
al._ (2019b); Cárdenas  _et al._ (2019); Benisty  _et al._ (2019); Bonici and
Maggiore (2019); Martinelli  _et al._ (2019); Kumar  _et al._ (2019); Pan  _et
al._ (2019a); Li  _et al._ (2020); Yang  _et al._ (2019b); Pan  _et al._
(2019b); Landim (2019); Benetti  _et al._ (2019); von Marttens  _et al._
(2020); Kase and Tsujikawa (2020); Liu  _et al._ (2020); Yang  _et al._
(2020b); Chamings  _et al._ (2020); Sharma and Dubey (2022); Yang  _et al._
(2020c); Hogg  _et al._ (2020); Amendola and Tsujikawa (2020); Benisty  _et
al._ (2020); Gómez-Valent  _et al._ (2020); Aljaf  _et al._ (2021); Di
Valentino  _et al._ (2020c); Mukhopadhyay  _et al._ (2021); Di Valentino
(2021); Di Valentino  _et al._ (2021c); Yao and Meng (2021); Beltrán Jiménez
_et al._ (2021); Yang  _et al._ (2021b); Sinha (2021); Gao  _et al._ (2021);
Zhang  _et al._ (2021); Salzano  _et al._ (2021); Wang  _et al._ (2021);
Benetti  _et al._ (2021); Kumar (2021); Figueruelo  _et al._ (2021); Bora  _et
al._ (2021); Jiménez  _et al._ (2021); Carrilho  _et al._ (2021a); Lucca
(2021a); Bora  _et al._ (2022); Linton  _et al._ (2021); Nunes and Di
Valentino (2021); Anchordoqui  _et al._ (2021); Hogg and Bruni (2022); Guo
_et al._ (2021); Alestas  _et al._ (2021); Mancini and Pourtsidou (2022);
Gariazzo  _et al._ (2021); Carrilho  _et al._ (2021b); Sharma and Sur (2021);
Duniya (2022) and to the review of Ref. Wang  _et al._ (2016).

###  II.2 Interacting dark energy and the Effective Field Theory of Large-
Scale Structure

There is a wealth of information contained in the linear and mildly non-linear
clustering of tracers of the LSS, such as galaxies, on which our analysis will
focus on. In recent years, significant effort has been devoted to extract
information beyond that contained within the (reconstructed) BAO peaks
Eisenstein  _et al._ (2007); Sherwin and Zaldarriaga (2012), considering the
Full-Shape (broadband) redshift-space galaxy power spectrum. Nevertheless, in
order to fully exploit current and future FS redshift-space galaxy power
spectra measurements, a robust theoretical modeling is mandatory Bose  _et
al._ (2018a); Fonseca de la Bella  _et al._ (2020); Osato  _et al._ (2019);
Bose  _et al._ (2020, 2019). Various approaches towards such a robust modeling
exist in the literature. Here, we shall make use of the EFTofLSS approach
Baumann  _et al._ (2012), see Ref. Cabass  _et al._ (2022a) for a recent
comprehensive review. While, in the very beginning, one of the most popular
theoretical modeling approaches was Standard Perturbation Theory (SPT)
Scoccimarro and Frieman (1996), the EFTofLSS can be regarded as the final
product of a consistent perturbation theory approach to describe the mildly
non-linear clustering of tracers of the LSS: for an incomplete list of other
approaches developed throughout this evolution, see e.g. Refs. Zeldovich
(1970); Crocce and Scoccimarro (2006); Bernardeau  _et al._ (2008); Matsubara
(2008); Taruya  _et al._ (2010); Bernardeau  _et al._ (2012); Carlson  _et
al._ (2013); Vlah  _et al._ (2016); Hand  _et al._ (2017); Chen  _et al._
(2021).

The EFTofLSS is a LSS perturbation theory approach (where one can think of the
expansion variable as being the overdensity field smoothed over an appropriate
scale) which can be used to robustly model the mildly non-linear clustering of
LSS tracers Baumann  _et al._ (2012). In a nutshell, the EFTofLSS provides a
framework to characterize the back-reaction and impact of unknown or poorly-
known short-scale physics, such as the complex details of galaxy formation, on
long-wavelength modes: this is achieved through a set of additional counter-
terms, whose functional form is fully fixed once the symmetries obeyed by the
LSS field are specified, and whose amplitude is controlled by a set of free
coefficients of unknown magnitude which, lacking a precise knowledge of the
short-wavelength physics in question, have to be treated as nuisance
parameters to be fit to the data. When combined with an IR resummation
procedure to treat long-wavelength displacements (i.e. bulk motions, whose
effect is crucial to properly model the non-linear evolution of the BAO peak),
stochastic contributions, non-linear biasing terms, and Alcock-Paczynski
effects, one obtains the most general, symmetry-driven model for the mildly
non-linear clustering of biased LSS tracers such as galaxies, which integrates
out the complex and poorly-known details of short-scale (UV) physics.

Over the past decade, significant work has gone into developing the EFTofLSS
from its initial formulation, to the point of enabling practical applications
to real data from current galaxy surveys. While several of the intermediate
results are not, strictly speaking, required to analyze galaxy clustering
data, much as the “West Coast” EFTofLSS team we believe that it is only fair
to acknowledge key developments on the EFTofLSS theory McDonald and Roy
(2009); Carrasco  _et al._ (2012); Pajer and Zaldarriaga (2013); Carrasco  _et
al._ (2014a); Mercolli and Pajer (2014); Carrasco  _et al._ (2014b); Carroll
_et al._ (2014); Porto  _et al._ (2014); Senatore and Zaldarriaga (2015);
Baldauf  _et al._ (2015a); Angulo  _et al._ (2015a); Senatore (2015); Senatore
and Zaldarriaga (2014); Lewandowski  _et al._ (2015); Mirbabayi  _et al._
(2015); McQuinn and White (2016); Foreman and Senatore (2016); Angulo  _et
al._ (2015b); Baldauf  _et al._ (2015b); Assassi  _et al._ (2015a); Baldauf
_et al._ (2016a, b, 2015c); Foreman  _et al._ (2016); Abolhasani  _et al._
(2016); Assassi  _et al._ (2015b); Lewandowski  _et al._ (2018); Bertolini
_et al._ (2016a, b); Blas  _et al._ (2016); Cataneo  _et al._ (2017);
Bertolini and Solon (2016); Fujita  _et al._ (2020); Perko  _et al._ (2016);
Lewandowski  _et al._ (2017); Lewandowski and Senatore (2017); Senatore and
Zaldarriaga (2017); Simonović  _et al._ (2018); Senatore and Trevisan (2018);
Nadler  _et al._ (2018); Bose  _et al._ (2018b); de Belsunce and Senatore
(2019); Lewandowski and Senatore (2020); Konstandin  _et al._ (2019);
Nishimichi  _et al._ (2020); Donath and Senatore (2020); Steele and Baldauf
(2021a); Bragança  _et al._ (2021); Steele and Baldauf (2021b); Baldauf  _et
al._ (2021) and data D’Amico  _et al._ (2020a); Ivanov  _et al._ (2020a);
Colas  _et al._ (2020); Ivanov  _et al._ (2020b); Philcox  _et al._ (2020);
D’Amico  _et al._ (2021a); Chudaykin  _et al._ (2020); Ivanov  _et al._
(2020c); D’Amico  _et al._ (2021b); Wadekar  _et al._ (2020); Chudaykin  _et
al._ (2021a, b); D’Amico  _et al._ (2020b); Ivanov (2021); Ivanov  _et al._
(2022); D’Amico  _et al._ (2021c); Ivanov  _et al._ (2021); Philcox and Ivanov
(2022); Cabass  _et al._ (2022b); D’Amico  _et al._ (2022) sides in any work
which applies the EFTofLSS to real data.666For a more complete accounting of
the theoretical developments associated to Refs. McDonald and Roy (2009);
Carrasco  _et al._ (2012); Pajer and Zaldarriaga (2013); Carrasco  _et al._
(2014a); Mercolli and Pajer (2014); Carrasco  _et al._ (2014b); Carroll  _et
al._ (2014); Porto  _et al._ (2014); Senatore and Zaldarriaga (2015); Baldauf
_et al._ (2015a); Angulo  _et al._ (2015a); Senatore (2015); Senatore and
Zaldarriaga (2014); Lewandowski  _et al._ (2015); Mirbabayi  _et al._ (2015);
McQuinn and White (2016); Foreman and Senatore (2016); Angulo  _et al._
(2015b); Baldauf  _et al._ (2015b); Assassi  _et al._ (2015a); Baldauf  _et
al._ (2016a, b, 2015c); Foreman  _et al._ (2016); Abolhasani  _et al._ (2016);
Assassi  _et al._ (2015b); Lewandowski  _et al._ (2018); Bertolini  _et al._
(2016a, b); Blas  _et al._ (2016); Cataneo  _et al._ (2017); Bertolini and
Solon (2016); Fujita  _et al._ (2020); Perko  _et al._ (2016); Lewandowski
_et al._ (2017); Lewandowski and Senatore (2017); Senatore and Zaldarriaga
(2017); Simonović  _et al._ (2018); Senatore and Trevisan (2018); Nadler  _et
al._ (2018); Bose  _et al._ (2018b); de Belsunce and Senatore (2019);
Lewandowski and Senatore (2020); Konstandin  _et al._ (2019); Nishimichi  _et
al._ (2020); Donath and Senatore (2020); Steele and Baldauf (2021a); Bragança
_et al._ (2021); Steele and Baldauf (2021b); Baldauf  _et al._ (2021), we
refer the reader to what by now is a standard footnote in all the papers by
the “West Coast” EFTofLSS team, for instance footnote 1 in Ref. D’Amico  _et
al._ (2022).

In this work, we shall adopt the EFTofLSS to model the mildly non-linear full-
shape galaxy power spectrum in the presence of the interactions between DM and
DE previously described in Sec. II.1. As the complete set of equations for the
redshift-space galaxy power spectrum is rather cumbersome, we will not report
it here, but refer the reader to Ref. Chudaykin  _et al._ (2020), where the
model and its implementation in the CLASS-PT Boltzmann solver are described in
detail. In our case, the only ingredient of the underlying EFTofLSS model we
need to modify is the linear power spectrum $P_{\rm lin}(k,z)$, which we
compute as usual by evolving the linear Einstein-Boltzmann equations given in
Eqs. (9–12). In turn, $P_{\rm lin}(k,z)$ enters in the convolution integrals
required to compute the 1-loop SPT power spectrum $P_{\rm 1-loop,SPT}(k,z)$,
and is also required to compute the counter-term $P_{\rm ctr}(k,z)$, needed
for consistency of the 1-loop result, whose amplitude is modulated by the
effective sound speed parameter. We keep the functional form of the SPT
kernels, counter-terms, bias expansion, redshift-space kernels, and stochastic
contributions unchanged, while the implementation of IR resummation is also
left unchanged.

A comment is in order regarding the fact that we only need to modify the input
linear power spectrum, which is the only place where new physics due to DM-DE
interactions enters. While the default implementation of CLASS-PT is targeted
towards the $\Lambda$CDM model, one can essentially envisage extending the
underlying model to three classes of new fundamental physics scenarios with
increasing levels of complexity:

  1. 1.

New physics which only affects the expansion, thermal history, and linear
evolution of perturbations. In this case, no changes to the standard routines
are required, as these scenarios only modify the ingredients which are already
present, such as the linear power spectrum, growth factor, and so on.

  2. 2.

New physics which modifies the mode-coupling kernels while preserving the
equivalence principle. Several modified gravity models fall within this class,
in which case one needs to recompute the perturbation theory matrices
incorporating the new kernels, required to compute the 1-loop power spectrum.
The specific details will of course depend on the modified gravity model, see
e.g. Ref. Cusin  _et al._ (2018).

  3. 3.

New physics which modifies the mode-coupling kernels and violates the
equivalence principle, as also occurring in certain modified gravity models.
In this case, one needs to alter the IR resummation procedure accordingly
Creminelli  _et al._ (2014); Crisostomi  _et al._ (2020); Lewandowski (2020).

As we consider a purely phenomenological implementation of the DM-DE
interactions at the level of linear Einstein-Boltzmann equations, without
committing to any specific UV-complete underlying field theory model which
describes such interactions, the new physics explored here belongs to the
first level described above. Given the size of the uncertainties associated
with current galaxy clustering measurements, our approach can be viewed as a
conservative one. In closing, we also note that the same approach has been
adopted by several other works in the recent literature, where both early- and
late-time new physics scenarios have been constrained against FS galaxy
clustering measurements using the EFTofLSS implementation of either CLASS-PT
Chudaykin  _et al._ (2020) or PyBird D’Amico  _et al._ (2021a), with the only
modifications being the input linear matter power spectra (see e.g. Refs.
Niedermann and Sloth (2021); Chudaykin  _et al._ (2021c); Braglia  _et al._
(2021); Allali  _et al._ (2021); Laguë  _et al._ (2022); Jiang and Piao
(2021); Ye  _et al._ (2021a); Xu  _et al._ (2021); Herold  _et al._ (2021);
Chudaykin  _et al._ (2022)).

Figure 1: Left panel: measurements of the monopole ($\ell=0$) and quadrupole
($\ell=2$) of the full-shape power spectrum of the BOSS DR12 North galactic
cap galaxy field at the effective redshift $z_{\rm eff}=0.61$ correspond to
the black datapoints. The blue and red curves correspond to the theoretical
predictions for the monopole and quadrupole respectively, for the $\Lambda$CDM
(solid) and IDE (dashed) models. Cosmological parameters are fixed to their
best-fit values obtained from the CMB+BAO+FS dataset combination (see Sec. III
for a discussion of these datasets), with the DM-DE coupling strength $\xi$
instead being fixed to $\xi=-0.05$, whereas nuisance parameters are fixed to
their $\Lambda$CDM best-fit values. Note that for both the monopole and
quadrupole what is being plotted is not $P_{\ell}(k)$, but $kP_{\ell}(k)$.
Right panel: relative difference $\Delta P_{\ell}/P_{\ell}$ between the IDE
and $\Lambda$CDM monopole (blue) and quadrupole (red), with the same choice of
cosmological and nuisance parameters as in the left panel.

The effects of IDE on the linear matter power spectrum have been amply
discussed in Ref. Lucca (2021b) (to which we refer the reader for more in-
depth discussions), which however did not carry out the comparison against FS
measurements we will instead perform. Introducing DM-DE interactions leads to
two important effects. Notice from Eq. (7) that in the presence of DM-DE
interactions with strength $\xi<0$ (for a fixed value of $\rho_{c,0}$) the
amount of DM grows with respect to its $\Lambda$CDM counterpart when going
back in time. This on its own would result in a higher redshift of matter-
radiation equality, and therefore a larger equality wavenumber since $k_{\rm
eq}\propto\sqrt{\omega_{m}(1+z_{\rm eq})}$. However, as we shall see later,
IDE introduces correlations between parameters which result in parameter
shifts once $\xi\neq 0$, most notably a decrease in both $\omega_{c}$ and
$\Omega_{m}$. Accounting for these parameter shifts from a fit to the full
CMB+BAO+FS dataset as we will discuss later, we see that the overall effect is
to slightly decrease $z_{\rm eq}$, from the $\Lambda$CDM value $z_{\rm
eq}\approx 3387$ to $z_{\rm eq}\approx 3330$ (the shift is larger if we were
to consider the best-fit CMB+FS or CMB+BAO values, changing to $z_{\rm
eq}\approx 2984$ and $z_{\rm eq}\approx 2819$ respectively).

Recall that the equality wavenumber $k_{\rm eq}$ determines the turnaround
scale in the matter power spectrum (as the growth of modes with $k>k_{\rm
eq}$, which entered the horizon during radiation domination, was suppressed by
the large radiation pressure, which instead did not hinder the growth of modes
with $k<k_{\rm eq}$). Note that the shifts in $z_{\rm eq}$ and therefore
$k_{\rm eq}$ discussed previously result in slight changes in the horizon
scale. In addition, for $\xi<0$, the conformal time today $\eta_{0}$
increases, leading to a small overall increase in the large-scale (small-$k$)
amplitude of the power spectrum. Another feature will be that due the onset of
DE domination which, for $\xi<0$, occurs earlier, resulting in a small
suppression of the power spectrum on smaller scales (which however are beyond
the scales we will probe in our analysis).

The theoretical predictions for the monopole ($\ell=0$) and quadrupole
($\ell=2$) of the full-shape power spectrum within $\Lambda$CDM and the IDE
model considered here are shown in Fig. 1. We have fixed the cosmological
parameters to the best-fit values obtained from a combined analysis of CMB,
pre-reconstructed FS, and post-reconstruction BAO data (see details in Secs.
III and IV), whereas we have fixed the DM-DE coupling strength to $\xi=-0.05$.
Note that we have kept nuisance parameters fixed to their $\Lambda$CDM best-
fit values when moving across the two models, in order to focus exclusively on
the effects induced by the cosmological parameters. In reality, nuisance
parameters shift slightly when moving from one model to another, and as a
result absorb part of the differences shown in Fig. 1. However, we have
explicitly checked that the shifts in all nuisance parameters remain below the
$1\sigma$ level, and that the extent to which they will absorb part of the
signal of interest is rather limited. Notice from Fig. 1 that the cleanest
signature of IDE appears to be in the monopole. Already from these qualitative
considerations we can expect that broadband information in the FS galaxy power
spectrum, and in particular information related to the equality scale, will
play an important role in setting constraints on IDE. Our analysis will
confirm these expectations, as we will later discuss in Sec. IV.

##  III Datasets and methodology

In order to set constraints on the parameters of the IDE model we shall
consider various combinations of the following datasets:

  * •

Measurements of CMB temperature anisotropy and polarization power spectra, as
well as their cross-spectra, from the Planck 2018 legacy data release. We
consider the high-$\ell$ Plik likelihood for TT (in the multipole range
$30\leq\ell\leq 2508$) as well as TE and EE (in the multipole range
$30\leq\ell\leq 1996$), in combination with the low-$\ell$ TT-only
($2\leq\ell\leq 29$) likelihood based on the Commander component-separation
algorithm in pixel space, as well as the low-$\ell$ EE-only ($2\leq\ell\leq
29$) SimAll likelihood Aghanim  _et al._ (2020b). We also include measurements
of the CMB lensing power spectrum, as reconstructed from the temperature
4-point function Aghanim  _et al._ (2020c). We refer to this dataset
combination as CMB.

  * •

Post-reconstruction Baryon Acoustic Oscillation measurements from the BOSS
DR12 Alam  _et al._ (2017) survey. We refer to this dataset as BAO.

  * •

Measurements of the monopole and quadrupole ($\ell=0$ and $\ell=2$
respectively) of the full-shape power spectrum of the BOSS DR12 galaxies,
divided into four independent inputs: two distinct redshift bins at $z_{\rm
eff}=0.38$ and $z_{\rm eff}=0.61$, observed in the North and South galactic
caps (NGC and SGC respectively). We refer to this dataset combination as FS.

  * •

A Gaussian prior on the physical baryon density parameter $\omega_{b}$ arising
from Big Bang Nucleosynthesis (BBN) constraints on the abundance of light
elements: $100\omega_{b}=2.233\pm 0.036$ Mossa  _et al._ (2020). We refer to
this prior as BBN.

For more details regarding window function, post-reconstruction BAO
measurements extraction, BAO-FS covariance matrix, and implementation of the
joint FS+BAO likelihood, we refer the reader to Refs. Ivanov  _et al._
(2020a); Philcox  _et al._ (2020); Chudaykin  _et al._ (2020), which discusses
these issues in depth.

Model-wise, we consider a seven-parameter IDE model, where the strength of the
DM-DE interaction $\xi$ is allowed to vary alongside the six standard
$\Lambda$CDM parameters. We set a flat prior on $\xi\in[-1;0]$ in order to
avoid gravitational and early-time non-adiabatic instabilities (see Sec.
II.1), verifying a posteriori that our results are not affected by the choice
of lower prior boundary. We set the default, wide conservative priors on the
EFTofLSS nuisance parameters following Ref. Ivanov  _et al._ (2020a). As we
are including CMB measurements, we do not fix either the scalar spectral index
$n_{s}$ nor the baryon-to-dark-matter density ratio $\omega_{b}/\omega_{c}$,
given that both these parameters are extremely well constrained by CMB data.

Theoretical predictions for the relevant observables are obtained using the
Boltzmann solver CLASS-PT Chudaykin  _et al._ (2020), which is itself an
extension of the Boltzmann solver CLASS Blas  _et al._ (2011); Lesgourgues
(2011), and allows to compute the 1-loop auto- and cross-power spectra for
matter fields and biased tracers both in real and redshift space,
incorporating all the ingredients discussed in Sec. II.2 required for the
comparison to data. Our theoretical model for the FS measurements is based on
the EFTofLSS predictions at 1-loop order (corresponding to perturbative order
$n=3$), and we consider FS measurements in the wavenumber range
$k\in[0.01;0.2]\,h{\rm Mpc}^{-1}$ (slightly more conservative than the
wavenumber range adopted in other works which have used the very same
likelihood). As explained earlier, we only modify the input linear power
spectrum obtained by solving the system of coupled linear Einstein-Boltzmann
equations for the IDE model given in Eqs. (9–12), without altering the SPT
kernels, the structure of the counter-terms and stochastic contributions, the
IR resummation implementation, and so on.777Note that, besides the EFTofLSS, a
number of other theoretical modeling approaches have been adopted in analysis
and forecasts for galaxy clustering data. Some of these studies considered
simpler phenomenological and/or theory-motivated models for the FS power
spectrum and higher order correlators, whereas others considered compressed
versions thereof (see e.g. Refs. Escudero  _et al._ (2015); Cuesta  _et al._
(2016); Giusarma  _et al._ (2016); Grieb  _et al._ (2017); Sanchez  _et al._
(2017); Beutler  _et al._ (2017a, b); Doux  _et al._ (2018); Gualdi  _et al._
(2018); Sprenger  _et al._ (2019); Gil-Marín  _et al._ (2018); Giusarma  _et
al._ (2018); Vagnozzi  _et al._ (2018a); Loureiro  _et al._ (2019); Gualdi
_et al._ (2019); Tröster  _et al._ (2020); Chen  _et al._ (2020); Kobayashi
_et al._ (2020); Gil-Marin  _et al._ (2020); Gualdi  _et al._ (2021a); Cuceu
_et al._ (2021); Gualdi  _et al._ (2021b); Brieden  _et al._ (2021a, b); Chen
_et al._ (2022); Semenaite  _et al._ (2021); Neveux  _et al._ (2022); Gualdi
and Verde (2022); Brieden  _et al._ (2022); Gil-Marín (2022)).

We sample the posterior distributions for the parameters of the IDE model
through Monte Carlo Markov Chain (MCMC) methods, using the cosmological
sampler MontePython Audren  _et al._ (2013); Brinckmann and Lesgourgues
(2019). We assess the convergence of the MCMC chains using the Gelman-Rubin
parameter $R-1$ Gelman and Rubin (1992), requiring $R-1<0.001$ for the chains
to be considered converged.

When considering analyses involving the BAO dataset but not the FS one (e.g.
when considering the CMB+BAO dataset combination), we make use of the
“standard” non-EFTofLSS likelihood which is publicly available at
github.com/brinckmann/montepython_public (in particular, we use the
bao_boss_dr12 likelihood and not the bao_fs_boss_dr12 one, with the latter
incorporating also $f\sigma_{8}$ measurements). When instead combining the BAO
and FS datasets, we make use of the joint BAO-FS likelihood which is publicly
available at github.com/oliverphilcox/full_shape_likelihoods, see Ref. Philcox
_et al._ (2020). This strategy will provide the key to isolate the advantages
of using the joint BAO-FS likelihood, and in particular the inclusion of FS
measurements in the context of IDE models. In fact, thanks to our choice, our
analyses involving the BAO dataset but not the FS one will be as close as
possible to pre-2020 analyses which preceded the public release of the joint
BAO-FS EFTofLSS likelihood we are using.888Recall that state-of-the-art
constraints on IDE models mostly arise from the combination of CMB and
geometrical information from the reconstructed BAO peaks, see for instance
Ref. Di Valentino  _et al._ (2020b).

Parameter |  CMB+BAO |  CMB+FS |  CMB+BAO+FS
---|---|---|---
$\omega_{c}$ | $0.094^{+0.022}_{-0.010}$ | $0.101^{+0.015}_{-0.009}$ | $0.115^{+0.005}_{-0.001}$
$\xi$ | $-0.22^{+0.18}_{-0.09}\,[>-0.48]$ | $>-0.35$ | $>-0.12$
$H_{0}\,[{\rm km}/{\rm s}/{\rm Mpc}]$ | $69.55^{+0.98}_{-1.60}$ | $69.04^{+0.84}_{-1.10}$ | $68.02^{+0.49}_{-0.60}$
$\Omega_{m}$ | $0.243^{+0.054}_{-0.030}$ | $0.261^{+0.038}_{-0.025}$ | $0.299^{+0.015}_{-0.007}$

Table 1: Constraints on selected parameters of the seven-parameter IDE model,
obtained from various dataset combinations as indicated in the upper section
of the Table, all involving the CMB dataset. For the DM-DE coupling strength
$\xi$ we report 95% C.L. lower limits, unless the 68% C.L. interval is
consistent with a detection, in which case we report the latter, alongside the
95% C.L. in square brackets (here this situation only occurs when considering
the CMB+BAO dataset combination), whereas for all other cosmological
parameters we report 68% C.L. intervals. The final two rows are separated from
the previous rows to highlight the fact that both $H_{0}$ and $\Omega_{m}$ are
derived parameters.

Parameter |  FS+BAO |  FS+BAO+BBN
---|---|---
$\omega_{c}$ | $0.100\pm 0.023$ | $0.104^{+0.021}_{-0.017}$
$\xi$ | $-0.31^{+0.20}_{-0.13}\,[>-0.62]$ | $>-0.62$
$H_{0}\,[{\rm km}/{\rm s}/{\rm Mpc}]$ | $70.7^{+5.5}_{-4.9}$ | $71.8^{+1.6}_{-1.9}$
$\Omega_{m}$ | $0.245^{+0.049}_{-0.036}$ | $0.257^{+0.044}_{-0.031}$

Table 2: As Tab. 1, but considering dataset combinations not involving the CMB
dataset. Note that we report the 68% C.L. interval on $\xi$ when considering
the FS+BAO dataset combination, as it is consistent with a detection, with the
95% C.L. lower limit given in square brackets.

##  IV Results

Figure 2: Triangular plot showing the 2D joint and 1D marginalized posterior
probability distributions for $\xi$, $\Omega_{m}$, and $H_{0}$, obtained from
the CMB+BAO (grey), CMB+FS (red), and CMB+BAO+FS (blue) dataset combinations
within the IDE model. The significant improvement in constraining power gained
by the introduction of the FS dataset is visually very clear. Note that
$H_{0}$ is in units of ${\rm km}/{\rm s}/{\rm Mpc}$.

Figure 3: Left panel: 1D marginalized posterior probability distributions for
$H_{0}$, obtained from various dataset combinations (see the color coding)
within the IDE model. The pink shaded band corresponds to the $1\sigma$
credible region for $H_{0}$ determined by the latest local distance ladder
measurements constructed out of Cepheid-calibrated SNeIa Riess  _et al._
(2021). Right panel: 1D marginalized posterior probability distributions for
$H_{0}$, obtained from the CMB+BAO and CMB+BAO+FS dataset combination within
the $\Lambda$CDM (green and red curves respectively) and IDE (blue and black
curves respectively) models. From this figure it is clear that there is a
significant gain in constraining power within the IDE model when including the
FS dataset, contrary to what is observed within $\Lambda$CDM. The pink shaded
band is equivalent to that depicted in the left panel. Figure 4: As Fig. 2,
but considering dataset combinations not involving the CMB dataset: FS+BAO
(red) and FS+BAO+BBN (blue). It is clear that the inclusion of the BBN prior
on $\omega_{b}$ helps enormously with the determination of $H_{0}$, given that
current galaxy clustering data mainly constrains $H_{0}$ through the sound
horizon rather than through equality scale information (with the prior on
$\omega_{b}$ sharpening the former), as discussed in the main text.

We begin by discussing the results obtained from dataset combinations
involving the CMB dataset: CMB+BAO, CMB+FS, and CMB+BAO+FS. Afterwards, to
better examine the role of shape versus geometrical information in setting
constraints on IDE, we consider CMB-free dataset combinations: FS+BAO and
FS+BAO+BBN. Our key results are reported in Tabs. 1 and 2, as well as Figs. 2,
3, and 4. Note that for $\xi$ we shall always report the 95% confidence level
(C.L.) lower bound when the 68% C.L. interval is not consistent with a
detection, whereas for all other cosmological parameters we report 68% C.L.
intervals.

In the context of the CMB+BAO dataset combination, i.e the one driving state-
of-the-art constraints on the DM-DE coupling strength $\xi$, we infer the
lower bound $\xi>-0.48$ at 95% C.L. (while the 68% C.L. interval is
$\xi=-0.22^{+0.18}_{-0.09}$). Note that this bound is slightly weaker than
those reported for instance in Refs. Di Valentino  _et al._ (2020a, b) because
we here deliberately do not include BAO measurements from the 6dFGS and SDSS-
MGS surveys: however, as will become clearer later, this particular CMB+BAO
dataset combination allows for a cleaner assessment of the constraining power
of shape and geometrical information in the IDE context. The reason is that
our pipeline does not include the 6dFGS and SDSS-MGS FS measurements.
Therefore, in this setting a full comparison of the constraining power of BAO
measurements from BOSS DR12+6dFGS+SDSS-MGS versus FS measurements from the
same combination is not possible. We thus make the choice of restricting our
LSS measurements to the one galaxy survey for which we have both a BAO and FS
pipeline readily available, i.e., the BOSS DR12 galaxies. In the following,
the limit $\xi>-0.48$ obtained from the CMB+BAO dataset combination will be
the reference limit to which we will compare and assess any improvement
brought about by the inclusion of the FS measurements.

Including FS measurements (CMB+BAO+FS) by means of the joint BAO-FS EFTofLSS
likelihood) tightens the limit on $\xi$ to $\xi>-0.12$ at 95% C.L.,
representing a significant improvement. There is, in fact, significant
information contained in the BOSS DR12 FS measurements which helps
disentangling IDE from $\Lambda$CDM. This is clear from the previous arguments
in Sec. II.2, where we discussed the important role of the equality wavenumber
in constraining IDE, as well as from Fig. 1.

To further examine the impact of the shape information, we now consider the
CMB+FS dataset combination. The limit we obtain is $\xi>-0.35$ at 95% C.L.,
tighter than that obtained from the CMB+BAO dataset combination, implying that
BOSS full-shape information carries more constraining power than the purely
geometrical information from the reconstructed BAO wiggles after combining
with CMB data. The inferred values of selected cosmological parameters from
the three dataset combinations (CMB+BAO, CMB+FS and CMB+BAO+FS) are reported
in Tab. 1. In addition, the $\xi$-$\Omega_{m}$-$H_{0}$ triangular plot in Fig.
2 clearly shows the increased constraining power of the CMB+FS dataset
combination with respect to the CMB+BAO one, as well as the overall
significant improvement when considering the CMB+FS+BAO dataset combination,
which constitutes one of the novel results of this work.

At first glance, the above results might not appear to be surprising: after
all, FS information encodes geometrical BAO information in the form of wiggles
in $P(k)$, alongside other additional information contained in the broadband,
non-wiggly part of $P(k)$. However, this conclusion should nonetheless
surprise us, as a number of recent works have argued that shape and
geometrical information carry similar constraining power in light of the
precision of current data Ivanov  _et al._ (2020b). The EFTofLSS-based
analysis of Ref. Ivanov  _et al._ (2020b) argued that the comparable
constraining power of shape and geometrical information in current BOSS data
is purely a coincidence given the BOSS volume and redshift coverage, in
combination with the efficiency of BAO reconstruction algorithms. However,
Ref. Ivanov  _et al._ (2020b) also argued that FS information is expected to
dramatically supersede geometrical information in the context of future galaxy
redshift surveys whose coverage will expand to significantly wider volumes and
redshift ranges. Similar conclusions, albeit obtained from simpler (non-
EFTofLSS) analyses of FS information, have been reached in at least two
earlier works Hamann  _et al._ (2010); Vagnozzi  _et al._ (2017). These works
also discussed the possibility of such a conclusion being reversed in
extensions to $\Lambda$CDM where shape information might play a crucial role,
for instance in the context of beyond-$\Lambda$ DE models. Our results indeed
provide a clear example in this sense: this further highlights the extremely
important role of full-shape galaxy clustering information when probing non-
standard extensions of $\Lambda$CDM, particularly in the DE sector, such as
IDE models. In fact, as we have already argued earlier in Sec. II.2,
information on the equality wavenumber $k_{\rm eq}$, not present in post-
reconstruction BAO measurements, plays an important role in constraining IDE.
A visual indication of the gain in constraining power when including the FS
dataset (in particular when moving from the CMB+BAO dataset combination to the
CMB+BAO+FS one) is given in the right panel of Fig. 3, where it is clear that
there is virtually no improvement within the $\Lambda$CDM scenario (green
versus red curves), while the improvement is rather significant for the IDE
model (blue versus black curves).

To further understand the role of geometrical and equality information in our
results, we will closely follow the rationale of Refs. Philcox  _et al._
(2021); Farren  _et al._ (2022), and consider additional dataset combinations
not including CMB measurements. Full-shape measurements contain the imprint of
two “standard rulers”: the first is connected to the scale of the sound
horizon at baryon drag $r_{d}$, and leaves its imprint in the position of the
BAO wiggles, whereas the second is connected to the horizon wavenumber at
matter-radiation equality $k_{\rm eq}$, and governs the position of the
turnaround in the power spectrum, see Sec. II.2. Information from both scales
can be used to constrain $H_{0}$ or, for that matter, any other parameter
which affects $r_{d}$ and $k_{\rm eq}$ either directly or indirectly.999An
example of a parameter indirectly influencing $r_{d}$ or $k_{\rm eq}$ is one
that does so through the induced shifts in other parameters. For instance,
besides their direct impact on $\rho_{c}(z)$ clear from Eq. (7), which
obviously affects $z_{\rm eq}$, IDE models typically lead to shifts in
$\omega_{c}$, which translate into additional shifts in $r_{d}$ and in the
redshift of matter-radiation equality $z_{\rm eq}$, and therefore in $k_{\rm
eq}$.

One way of assessing the extent to which constraints involving FS data are
driven by geometrical BAO information is to artificially limit or even remove
any information on $r_{d}$ whatsoever. To zeroth order, this can be achieved
by removing any external calibration on $r_{d}$, which is tantamount to
removing any external informative prior on the physical baryon density
$\omega_{b}$: FS analyses not including CMB data typically include a BBN prior
on $\omega_{b}$ (see e.g. Ref. Philcox  _et al._ (2020)), given that knowledge
of $\omega_{b}$ (which controls the pre-recombination baryon sound speed) is
required to compute $r_{d}$ and therefore calibrate the geometrical BAO
information. Notice that this procedure is not exactly equivalent to
marginalizing over $r_{d}$, which is the proper procedure to follow to remove
any geometrical information and extract only information from the broadband
part of FS measurements. Nevertheless, given the precision of current galaxy
clustering data, this procedure works (as argued in Ref. Philcox  _et al._
(2021)), whereas for future, more precise FS data, the BAO features and the
small-scale $\omega_{b}$-induced Jeans suppression in FS measurements could in
principle cross-calibrate each other, effectively generating an $\omega_{b}$
or $r_{d}$ prior. The later work of Ref. Farren  _et al._ (2022) constructs an
improved method to remove geometrical information in galaxy clustering
measurements in future surveys. For the purposes of the present work however,
given the precision of BOSS FS measurements, removing any external informative
prior on $\omega_{b}$ is sufficient to ensure that the sound horizon
information is essentially completely (or mostly) removed, and we will
therefore follow the methodology of Ref. Philcox  _et al._ (2021).

In light of the above discussion, we shall now consider the FS+BAO+BBN and
FS+BAO dataset combinations: the former allows for the exploitation of sound
horizon information through the BBN prior on $\omega_{b}$, which calibrates
$r_{d}$, whereas such information is absent in the latter.101010Note that any
dataset combination involving the CMB dataset contains information on
$\omega_{b}$ and therefore $r_{d}$, as $\omega_{b}$ is determined by the
relative height of the odd versus even peaks. When considering FS+BAO+BBN we
infer $\xi>-0.62$ at 95% C.L., whereas removing the BBN prior on $\omega_{b}$
(thus considering the FS+BAO dataset combination) surprisingly returns exactly
the same lower limit on $\xi$, although in this case the 68% C.L. interval is
consistent with a $\simeq 1.5\sigma$ detection: $\xi=-0.31^{+0.20}_{-0.13}$.
This further confirms the fact that the constraints on $\xi$ are driven to an
important amount by broadband information rather than by just geometrical
information: if the latter were true, the constraints on $\xi$ should have
become significantly weaker when removing the BBN dataset, contrary to what we
observe. The inferred values of selected cosmological parameters from the two
CMB-free dataset combinations discussed above are reported in Tab. 2, whereas
in Fig. 4 we present the $\xi$-$\Omega_{m}$-$H_{0}$ triangular plot obtained
from these same two dataset combinations.

It is important to explore the impact of the inclusion of FS measurements on
the inference of parameters other than the coupling $\xi$. There are two key
parameters in IDE cosmologies: the matter density parameter $\Omega_{m}$ and
the Hubble constant $H_{0}$. The correlation between $\xi$ and $\Omega_{m}$
can be understood in terms of the direction of energy flow between DM and DE.
Since $\xi<0$, energy flows from DM to DE, and therefore the DM density today
(and correspondingly $\Omega_{m}$) will be smaller Di Valentino  _et al._
(2020a, b). This is also clear from Eq. (7), where the evolution of $\rho_{c}$
carries an extra term proportional to $\xi$ in addition to the standard
$\Lambda$CDM term $\rho_{c}\propto(1+z)^{3}$. However, as $\Omega_{m}h^{2}$ is
tightly constrained by CMB data, the decrease in $\Omega_{m}$ requires an
increase in $H_{0}$. This explains the mutual $\xi$-$\Omega_{m}$-$H_{0}$
degeneracies we observe: a direct correlation between $\xi$ and $\Omega_{m}$,
and an inverse correlation between $\xi$ and $H_{0}$, with a significantly
weaker inverse correlation between $\Omega_{m}$ and $H_{0}$. Focusing on
$\Omega_{m}$ and $H_{0}$, we find that their determination is significantly
improved by the inclusion of the FS dataset on top of the CMB+BAO dataset
combination. In particular, the uncertainties on $\Omega_{m}$ and $H_{0}$ are
decreased by factors of $\sim$3 and $\sim$2 respectively. This improvement
(particularly for $\Omega_{m}$) is due to the important role of the equality
scale in FS measurements. In fact, equality information in FS measurements
directly constrains the so-called “shape parameter” $\Gamma\equiv\Omega_{m}h$
Efstathiou  _et al._ (1992, 2002). In combination with CMB measurements
constraining $\Omega_{m}h^{2}$ through the early integrated Sachs-Wolfe Hou
_et al._ (2013); Cabass  _et al._ (2015); Kable  _et al._ (2020); Vagnozzi
(2021) and lensing effects Baxter and Sherwin (2021), and geometrical BAO
information, this helps pinning down $\Omega_{m}$, particularly in the context
of extended models (see for instance the discussion in Section 2 of Ref.
Vagnozzi  _et al._ (2021b) in the context of spatial curvature). In Sec. II.2
we already emphasized the important role of broadband information (and in
particular information related to $k_{\rm eq}$) in the context of IDE models,
which is anyhow clear from the monopole curves in Fig. 1.

Moving to the CMB-free dataset combinations discussed earlier (FS+BAO and
FS+BAO+BBN), we note that the parameter most significantly affected by the
addition of the BBN prior on $\omega_{b}$ is the Hubble constant $H_{0}$,
whose uncertainty decreases by a factor of $\sim 3$ once the BBN dataset is
included. This is not surprising, given that current galaxy clustering data
mostly constrain $H_{0}$ through the sound horizon rather than from equality
scale information Philcox  _et al._ (2021). The uncertainty we infer on
$H_{0}$ of order $5\,{\rm km}/{\rm s}/{\rm Mpc}$ is in fact comparable to that
obtained in Ref. Philcox  _et al._ (2021) for the $\Lambda$CDM model (of
course, our uncertainties are slightly larger because of the extra parameter
$\xi$ and its correlation with $H_{0}$). To conclude this section, it is clear
that full-shape information is highly precious when dealing with non-trivial
extensions of the $\Lambda$CDM model, particularly within IDE cosmologies. The
addition of FS information further limits the available parameter space for
the DM-DE coupling strength, even in the absence of CMB data.

##  V Conclusions

Driven by important theoretical advances, in the last couple of years
significant efforts have gone into extracting Large-Scale Structure (LSS)
clustering information beyond geometrical information contained in the
reconstructed BAO peaks, by exploiting the information content of the full-
shape (FS) power spectrum of LSS tracers. The power of FS measurements and
their ability to not only sharpen constraints on the parameters of the
baseline $\Lambda$CDM model, but perhaps more importantly test extensions
thereof, is by now established. In the wake of these important developments,
the present work is the first to test interacting dark energy (IDE)
cosmologies in light of state-of-the-art redshift-space galaxy clustering
data, with a robust theoretical modeling of the underlying mildly non-linear
power spectrum.

IDE cosmologies feature non-gravitational interactions between dark matter
(DM) and dark energy (DE) and have experienced nothing short of a resurgence
in recent years. In the context of our work, IDE models provide a unique
phenomenological stage to test the power and advantages of certain
cosmological observations with respect to others. Specifically, our goal is to
assess the improvement gained by considering FS measurements when constraining
the DM-DE coupling strength $\xi$, taking as baseline state-of-the-art
constraints which only consider geometrical BAO measurements (see e.g. Ref. Di
Valentino  _et al._ (2020b)). Focusing on an IDE model [see Eq. (4)] which is
well-studied, mostly due to its simplicity and connection to coupled
quintessence, we have demonstrated how the inclusion of FS measurements
significantly sharpens constraints on $\xi$, which is now safely constrained
to $|\xi|\lesssim 0.1$ for the model considered.

Our limits on $\xi$ imply that DM and DE cannot exchange energy at a rate
greater than ${\cal O}(10^{-48})\,{\rm g}/{\rm s}/{\rm cm}^{3}$, or
equivalently that they cannot exchange more than $\approx 7\%$ of the critical
energy density over the course of a Hubble time, a quantity which is way too
small to have any bearings on the coincidence problem (as was already known).
Moreover, these tight constraints further limit the ability of IDE cosmologies
to play an important role in the context of the Hubble tension. Crucially, the
same limits also constrain the DE effective equation of state (EoS) $w_{x,{\rm
eff}}=w_{x}+\xi/3\gtrsim-1.04$, consistent with independent constraints on the
DE EoS from various probes, which strongly limit the ability to move deep into
the phantom region (see e.g. Refs. Zhao  _et al._ (2017); Vagnozzi  _et al._
(2018b); Gerardi  _et al._ (2019); Vagnozzi (2020); Visinelli  _et al._
(2019); Bonilla  _et al._ (2021); Colgáin  _et al._ (2021); Teng  _et al._
(2021); Raveri  _et al._ (2021); Sharma  _et al._ (2022)). 111111Our results,
and particularly the inferred values of $H_{0}$, can also be compared to those
of Ref. Krishnan  _et al._ (2021), given the background equivalence between
IDE and dynamical DE models.

Our results highlight the extremely important role of FS measurements in the
era of precision cosmology, particularly in further sharpening constraints on
models beyond $\Lambda$CDM: for instance, a recently well-documented example
in this context is that of the early dark energy model Hill  _et al._ (2020);
Ivanov  _et al._ (2020c); D’Amico  _et al._ (2021b); Niedermann and Sloth
(2020); Smith  _et al._ (2021); Ye  _et al._ (2021b); Poulin  _et al._ (2021);
La Posta  _et al._ (2021); Smith  _et al._ (2022); Jiang and Piao (2022).
121212See also Ref. Rudelius (2022) for related discussions on theoretical
difficulties early dark energy faces. Furthermore, contrary to what occurs
within the $\Lambda$CDM model and simple extensions thereof, we have
demonstrated that constraints obtained by combining CMB and FS data are
tighter than those obtained by combining CMB and BAO measurements, suggesting
that in the context of IDE models there is significant information contained
in the broadband of the power spectrum. We have extensively tested these
findings, discussing the important role played by the equality scale $k_{\rm
eq}$ in constraining IDE cosmologies.

This work is the first to robustly test DM-DE interactions from state-of-the-
art redshift-space galaxy clustering measurements, and the constraints we
report on the coupling strength $\xi$ are among the strongest ever presented
in the literature. At the same time, there are many interesting follow-up
directions one could envisage. While here we have considered FS measurements
from the BOSS DR12 sample, it would be very interesting to repeat a similar
analysis using the much more recent eBOSS DR16 measurements Alam  _et al._
(2021). On the other hand, we expect the role of FS galaxy clustering
information to become even more important in this context with the advent of
future LSS surveys, such as Euclid Amendola  _et al._ (2018) and DESI
Aghamousa  _et al._ (2016). It would be interesting to forecast the ability of
these surveys to constrain DM-DE interactions when combined with future CMB
data, for instance from Simons Observatory Ade  _et al._ (2019); Abitbol  _et
al._ (2019) or CMB-S4 Abazajian  _et al._ (2016). Finally, it would be very
interesting to test other well-motivated beyond-$\Lambda$CDM models in light
of the same FS measurements adopted here. Most of these ideas are the subject
of work in progress, on which we hope to report soon.

###### Acknowledgements.

S.V. thanks Misha Ivanov for useful discussions. R.C.N. acknowledges financial
support from the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP,
São Paulo Research Foundation) under the project No. 2018/18036-5. S.V. is
supported by the Isaac Newton Trust and the Kavli Foundation through a Newton-
Kavli Fellowship, and by a grant from the Foundation Blanceflor Boncompagni
Ludovisi, née Bildt. S.V. acknowledges a College Research Associateship at
Homerton College, University of Cambridge. S.K. gratefully acknowledges
support from the Science and Engineering Research Board (SERB), Govt. of India
(File No. CRG/2021/004658). E.D.V. is supported by a Royal Society Dorothy
Hodgkin Research Fellowship. O.M. is supported by the Spanish grants
PID2020-113644GB-I00, PROMETEO/2019/083 and by the European ITN project HIDDeN
(H2020-MSCA-ITN-2019//860881-HIDDeN).

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