# Non-Linear Matter Power Spectrum Modeling in Interacting Dark Energy
Cosmologies

Emanuelly Silva<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal do Rio Grande do Sul, 91501-970 Porto Alegre RS, Brazil  Ubaldo
Zúñiga-Bolaño<EMAIL_ADDRESS>Instituto de Física, Universidad de
Antioquia, Cl. 67 #53-108, Medellín, Colombia  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  Eleonora Di Valentino
<EMAIL_ADDRESS>School of Mathematics and Statistics, University
of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom

###### Abstract

Understanding the behavior of the matter power spectrum on non-linear scales
beyond the $\Lambda$CDM model is crucial for accurately predicting the large-
scale structure (LSS) of the Universe in non-standard cosmologies. In this
work, we present an analysis of the non-linear matter power spectrum within
the framework of interacting dark energy-dark matter cosmologies (IDE). We
employ N-body simulations and theoretical models to investigate the impact of
IDE on these non-linear scales. Beginning with N-body simulations
characterized by a fixed parameter space delineated by prior observational
research, we adeptly fit the simulated spectra with a simple parametric
function, achieving accuracy within 5%. Subsequently, we refine a modified
halo model tailored to the IDE cosmology, exhibiting exceptional precision in
fitting the simulations down to scales of approximately 1 h/Mpc. To assess the
model’s robustness, we conduct a forecast analysis for the Euclid survey,
employing our refined model. We find that the coupling parameter $\xi$ will be
constrained to $\sigma(\xi)=0.0046$. This marks a significant improvement by
an order of magnitude compared to any other current observational tests
documented in the literature. These primary findings pave the way for a novel
preliminary approach, enabling the utilization of IDE models for observational
constraints concerning LSS data on non-linear scales.

##  I Introduction

Dark energy (DE) is one of the most enigmatic and perplexing phenomena in the
field of astrophysics and cosmology. It may represent a new field that
permeates the universe, driving its current accelerated expansion Perlmutter
_et al._ (1999); Riess  _et al._ (1998). DE as a cosmological constant
($\Lambda$) represents our simplest understanding of the universe’s expansion
and evolution in the last two decades. The DE existence and influence on the
cosmos are supported by a wealth of observational evidence, Supernova Surveys,
Cosmic Microwave Background (CMB), Large-Scale Structure (LSS), Gravitational
Lensing, among others Huterer and Shafer (2017); Mortonson  _et al._ (2013).
The success of the $\Lambda$ term in explaining these observations has led to
the $\Lambda$CDM scenario being considered the standard model of cosmology.

However, recently, a few tensions and anomalies have emerged as statistically
significant when analyzing different datasets. The most noteworthy
disagreement lies in the value of the Hubble constant, $H_{0}$, which exists
between the Planck-CMB estimate Aghanim  _et al._ (2020), assuming the
standard $\Lambda$CDM model, and the direct local distance ladder measurements
conducted by the SH0ES team Riess  _et al._ (2022a, b); Murakami  _et al._
(2023), reaching a significance of more than 5$\sigma$. Furthermore, within
the $\Lambda$CDM framework, the CMB measurements from Planck and ACT-DR6
Aghanim  _et al._ (2020); et al (2023) provide values of
$S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$ in 2-3$\sigma$ statistical tension
with those inferred from various weak lensing, galaxy clustering, cluster
counts, and redshift-space distortion measurements Di Valentino  _et al._
(2021a); Nunes and Vagnozzi (2021); Amon  _et al._ (2022); Asgari  _et al._
(2021a). Various other anomalies and tensions have emerged within the
$\Lambda$CDM framework in recent years Perivolaropoulos and Skara (2022);
Abdalla  _et al._ (2022). Motivated by such discrepancies, which are unlikely
to disappear completely by introducing multiple systematic errors, there has
been widespread discussion in the literature regarding whether new physics
beyond the standard cosmological model can resolve these tensions Di Valentino
_et al._ (2021b); Knox and Millea (2020); Jedamzik  _et al._ (2021);
Kamionkowski and Riess (2023); Khalife  _et al._ (2023). Interacting Dark
Energy - Dark Matter (IDE) models have been extensively investigated as
potential solutions to the current cosmological tensions Kumar and Nunes
(2016); Murgia  _et al._ (2016); Kumar and Nunes (2017); Di Valentino  _et
al._ (2017); Yang  _et al._ (2019a, 2018, b); Kumar  _et al._ (2019); Pan  _et
al._ (2019a, b); Di Valentino  _et al._ (2020a, b); Di Valentino and Mena
(2020); Di Valentino  _et al._ (2021c); Gómez-Valent  _et al._ (2020); Yang
_et al._ (2020a); Lucca and Hooper (2020a); Martinelli  _et al._ (2019); Yang
_et al._ (2020b); Yao and Meng (2020); Pan  _et al._ (2020a); Di Valentino
(2020); Yao and Meng (2021); Amirhashchi  _et al._ (2022); Yang  _et al._
(2021a); Gao  _et al._ (2021); Lucca (2021a); de Cruz Perez and Sola Peracaula
(2024); Kumar (2021); Yang  _et al._ (2021b); Lucca (2021b); Halder and Pandey
(2021); Bernui  _et al._ (2023); Poulin  _et al._ (2023); Lucca (2021c); Zhai
_et al._ (2023); Gariazzo  _et al._ (2022); Valentino  _et al._ (2020); Mawas
_et al._ (2021); Yang  _et al._ (2021c); Lucca and Hooper (2020b); Pan  _et
al._ (2020b); von Marttens  _et al._ (2020); van der Westhuizen and Abebe
(2024); Goh  _et al._ (2023); Gómez-Valent  _et al._ (2022); Escamilla  _et
al._ (2023); Forconi  _et al._ (2023); Hoerning  _et al._ (2023); Wang  _et
al._ (2024a); Benisty  _et al._ (2024).

At the intersection of various cosmic phenomena, especially at the LSS
observation level, lies the matter power spectrum, an essential cosmological
tool for unraveling the interplay between cosmological parameters and the
emergence of LSS. The power spectrum provides a unique probe through which we
can peer into the universe’s past and present, revealing the traces of
primordial fluctuations, the gravitational dynamics of dark matter (DM), and
the elusive nature of DE. On the other hand, is well known that the non-linear
power spectrum plays a pivotal role in current LSS cosmological surveys,
offering crucial insights into the complex and dynamic behavior of cosmic
structures. The non-linear power spectrum is a crucial component in any
current LSS cosmological survey because it serves as a bridge between
theoretical predictions and observational data.

The non-linear power spectrum is well modeled for the $\Lambda$CDM model and
some of its basic extensions Takahashi  _et al._ (2012); Matsubara (2008);
Hand  _et al._ (2017); Vlah  _et al._ (2016); Chen  _et al._ (2021); Baumann
_et al._ (2012); Cabass  _et al._ (2023). However, modeling non-linear regimes
in non-standard cosmologies poses a significant challenge. Various
methodologies have been employed to address this challenge, including the
Effective Field Theory of LSS D’Amico  _et al._ (2021a, b); Ivanov  _et al._
(2020); Colas  _et al._ (2020); D’Amico  _et al._ (2020); Bose  _et al._
(2018), the Halo models Bose  _et al._ (2023); Carrilho  _et al._ (2022); Bose
_et al._ (2020); Gupta  _et al._ (2023); Carrion  _et al._ (2024); Spurio
Mancini and Pourtsidou (2022), and emulators trained via N-body simulation
data Winther  _et al._ (2019); Fiorini  _et al._ (2023); Mauland  _et al._
(2023); Sáez-Casares  _et al._ (2024); Ruan  _et al._ (2024). These approaches
offer promising avenues for improving our understanding of non-linear regimes
in cosmology, but further research is needed to explore their capabilities and
limitations.

Given the imperative to anticipate model behavior on non-linear scales, it
becomes indispensable to employ phenomenological models for the interpretation
of current and forthcoming data derived from galaxy clustering and cosmic
shear. By doing so, we can better understand the perturbation dynamics at play
within these complex systems and extract meaningful insights from
observational data.

In this work, our main aim is to build a robust non-linear $P(k)$ model in the
presence of a DE-DM interaction. To this end, we carry out a series of N-body
gravity simulations and then we develop these perspectives in two ways. First,
we capture the ratio $R(k)=P_{\rm NL}(k)/P_{\rm linear}(k)$ up to non-linear
scales, and we find reliable analytical modeling for the function $R(k)$ up to
scales $k\leq 1$ h/Mpc. In a second modeling context, we modify and propose a
new $P(k)$ Halo model, which can fit the simulation data. Finally, we applied
our models in a forecast analysis from the perspective of the Euclid survey.
We demonstrate that an Euclid-like survey will possess strong sensitivity to
constrain DM-DE coupling, rendering it a crucial probe for this particular
class of cosmological models.

The paper is structured as follows: In Section II, we delve into the
fundamental physical aspects of the IDE model considered in this study.
Section III outlines the methodology and development of our N-body simulation
data. Moving forward, in Section IV, we introduce our adapted Halo model,
tailored to accurately fit the N-body simulations. Next, Section V presents
our forecast analysis, leveraging the sensitivity of an Euclid-like survey.
Finally, in Section VI, we offer our concluding remarks along with future
perspectives.

##  II Background expansion and linear perturbations

Let us start by examining the fundamental characteristics of IDE models.
Throughout our discussion, we will operate on the assumption of a spatially
flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric. We can introduce a
phenomenological parameterization for the DM-DE interaction within the
conservation equations such that the individual stress-energy tensors are no
longer conserved, although their sum remains so.111See Nunes  _et al._ (2022a)
for a more detailed description and Wang  _et al._ (2024b, 2016) for a general
review.  Typically, it is assumed that the covariant derivatives of the DM and
DE stress-energy tensors evolve according to:

$\displaystyle\nabla_{\nu}T^{\mu\nu}_{c}$ $\displaystyle=$
$\displaystyle\frac{Qu^{\mu}_{c}}{a}$ (1)
$\displaystyle\nabla_{\nu}T^{\mu\nu}_{x}$ $\displaystyle=$
$\displaystyle-\frac{Qu^{\mu}_{c}}{a},$ (2)

where $u^{\mu}_{c}$ denotes the DM 4-velocity vector, and $Q$ represents the
DM-DE interaction rate, measured in energy per volume per time.

At this point, we should decide on the functional form of $Q$, and we adopt a
widely used choice in the literature, that is $Q=\xi{\cal H}\rho_{x}$, where
${\cal H}$ signifies the conformal Hubble rate, $\rho_{x}$ represents the
energy density of DE, and $\xi$ serves as a dimensionless parameter governing
the strength of the DM-DE interaction. A positive $\xi$ indicates energy
transfer from DE to DM, while a negative $\xi$ indicates energy transfer from
DM to DE.

When accounting for this specified coupling, the continuity equations
dictating the evolution of the energy densities of DM and DE become:

$\displaystyle\dot{\rho}_{c}+3{\cal H}\rho_{c}$ $\displaystyle=$
$\displaystyle\xi{\cal H}\rho_{x}\,\quad$ (3)
$\displaystyle\dot{\rho}_{x}+3{\cal H}(1+w_{x})\rho_{x}$ $\displaystyle=$
$\displaystyle-\xi{\cal H}\rho_{x}\,$ (4)

where $w_{x}$ represents the equation of state (EoS) for DE. Equations (3) and
(4) can be analytically integrated, and in the case that both $w_{x}$ and
$\xi$ remain constant throughout cosmic time, they give:

$\displaystyle\rho_{c}$ $\displaystyle=$
$\displaystyle\frac{\rho_{c,0}}{a^{3}}+\frac{\rho_{x,0}}{a^{3}}\left[\frac{\xi}{3w_{x}+\xi}\left(1-a^{-3w_{x}-\xi}\right)\right]\,;$
(5) $\displaystyle\rho_{x}$ $\displaystyle=$
$\displaystyle\frac{\rho_{x,0}}{a^{3(1+w_{x})+\xi}}\,,$ (6)

where $\rho_{c,0}$ and $\rho_{x,0}$ represent the energy densities of DM and
DE today, respectively.

The expression for the Hubble parameter is then given by:

$\displaystyle\frac{H^{2}(a)}{H^{2}_{0}}=\Omega_{r,0}a^{-4}+\Omega_{b,0}a^{-3}+\Omega_{c,0}a^{-3}$
$\displaystyle+\frac{\Omega_{x,0}}{a^{3}}\left[\frac{\xi}{3w_{x}+\xi}\left(1-a^{-3w_{x}-\xi}\right)\right]+\frac{\Omega_{x,0}}{a^{3(1+w_{x})+\xi}},$
(7)

with the first two terms representing the contributions from radiation and
baryons, respectively. The relationship between the cosmic and conformal time
is defined by ${\cal H}=aH$.

The introduction of the DM-DE coupling further alters the evolution of
perturbations. Working in the Newtonian gauge, the coupled system of linear
Einstein-Boltzmann equations governing the evolution of dark matter density
perturbations, $\delta_{c}$, and velocity divergences, $\theta_{c}$, is
expressed as follows Väliviita  _et al._ (2008); Honorez  _et al._ (2010);
Gavela  _et al._ (2010):

$\displaystyle\dot{\delta}_{c}$ $\displaystyle=$
$\displaystyle-\left(\theta_{c}-3\dot{\Phi}\right)+\frac{Q}{\rho_{c}}\left(\frac{\delta
Q}{Q}-\delta_{c}+\Psi\right),$ (8) $\displaystyle\dot{\theta}_{c}$
$\displaystyle=$ $\displaystyle-{\mathcal{H}}\theta_{c}+k^{2}\Psi.$ (9)

It is worth noting that the Euler equation remains unaltered compared to
$\Lambda$CDM, indicating the absence of a fifth force in this model.
Additionally, we will not consider the potential presence of clustering DE in
this study. As argued in Dakin  _et al._ (2019), DE perturbations may only
have discernible effects on very large scales, necessitating a relativistic
treatment of simulations. In the regime of Newtonian N-body simulations that
we are focused on modeling, DE clustering does not significantly impact the
matter power spectrum Hassani  _et al._ (2019). Moreover, recent analyses
Dinda and Banerjee (2023) have demonstrated that observations favor a
homogeneous DE component over a clustering one. However, the presence of a
homogeneous DE component influences simulations in various ways, including
background evolution, growth function calculation, and initial conditions. To
provide a more precise quantification within the IDE context, Figure 1
illustrates the relative difference in $P(k,z=0)$ with and without DE
clustering effects. In constructing the figure, we adopted IDE-01 and IDE-02
baseline values, which will be defined in the subsequent section. It is
evident that on large scales, the observed effect is minimal, amounting to
less than 1%. As for scales where non-linear effects are expected to dominate
(i.e., $k>0.2h/\text{Mpc}$), the relative difference is approximately 1%.
Consequently, we can infer that the perturbations due to DE can be considered
negligible. The results were obtained using a customized version of the CLASS
code Blas  _et al._ (2011), which integrates the IDE model elaborated upon in
this study.

Figure 1: The relative difference in $P(k,z=0)$, accounting for DE clustering
effects, is evaluated within the IDE model discussed in this study.

On the other hand, within the framework of IDE, the principles of general
relativity remain applicable. Consequently, the Poisson equation retains its
conventional form:

$-k^{2}\Psi=4\pi Ga^{2}\rho_{c}\delta_{c}.$ (10)

It is important to emphasize that in equation (10), we have disregarded
contributions from DE clustering as argument above. Our focus is currently on
sub-horizon scales, where relativistic effects are negligible. Introducing the
linear growth function would prove beneficial in this context

$D(a)=\frac{\delta_{c}(a)}{\delta_{c}(1)},$ (11)

where $D$ at the present time is unity, $D(1)=1$.

Following a conventional non-relativistic perturbation-theory approach, which
originates from the Boltzmann equation for DM particles (eq. (9)), Poisson’s
equation (10), and the continuity equations for the energy densities of DM
(3), we derive the linear growth equation in the presence of interaction
between DM and DE species. In this scenario, we obtain:

$\begin{gathered}D^{\prime\prime}+D^{\prime}\left[\frac{\mathcal{H}^{\prime}}{\mathcal{H}}+\frac{r_{xc}\xi}{a}+3\right]\\\
-D\left[\frac{3}{2}\Omega_{c}-\frac{r_{xc}\xi\mathcal{H}^{\prime}}{\mathcal{H}}+\frac{r_{xc}\xi}{a^{2}}(\xi+3\omega_{x}+r_{xc}\xi-a)\right]=0,\end{gathered}$
(12)

where we define $r_{\rm xc}=\rho_{x}/\rho_{c}$. Here, a prime denotes
differentiation concerning the scale factor, denoted as $x^{\prime}=dx/dlna$.
It is noteworthy that when $\xi=0$, the prediction of the $\Lambda$CDM model
is recovered.

The equation above characterizes the first-order growth factor $D(a)$ and
serves as a fundamental equation for N-body gravity simulations. It ensures
the accurate implementation of the Zel’dovich approximation at each timestep,
utilizing the Zel’dovich equation of motion $x(a)=q+D(a)s$.

Figure 2: The growth factor is depicted as a function of redshift. The dashed
black curve represents the standard model, $\Lambda$CDM, whereas the solid
curves correspond to the interacting model utilized, showcasing variations of
the $\xi$ parameter.

Figure 2 illustrates the growth factor’s variation with redshift $z$. It is
apparent that as the value of $\xi$ changes, notable differences in the
curves’ amplitudes arise, particularly at lower redshifts. However, as
redshift values increase, the curves gradually converge asymptotically. This
trend indicates that within this framework, the interaction between DM and DE
significantly influences structures at lower redshifts, whereas at higher $z$
the effects diminish.

##  III N-body simulations

###  III.1 Methodology

The cosmic structure formation in a given theory is most accurately modeled
with N-body simulations, where the particles in the simulation are
incrementally displaced from their initial positions according to the Poisson
equation, the expansion rate of the universe and initial conditions. In this
work, we will use the pmwd code Li  _et al._ (2022), a differentiable
cosmological particle-mesh N-body library.

During each simulation, N-body particles discretize the uniform distribution
of matter at the beginning of cosmic history at their Lagrangian positions
$q$, from which they then evolve by displacements $s$ to their later positions
$\mathbf{x}=q+s(q)$. In what follows, we describe in a nutshell the main
physical quantities that are essential, and model-dependent, when running the
simulations.

The initial conditions of particles can be set perturbatively when the linear
approximation of the density fluctuation is much smaller than 1. We can
compute the initial displacements and momenta using the second-order
Lagrangian perturbation theory (2LPT) Bouchet  _et al._ (1994):

$\displaystyle\mathbf{s}=D_{1}\mathbf{s}^{(1)}+D_{2}\mathbf{s}^{(2)},$
$\displaystyle\mathbf{p}=a^{3}H[D_{1}^{\prime}\mathbf{s}^{(1)}+D_{2}^{\prime}\mathbf{s}^{(2)}],$
(13)

where we account for the cosmic background expansion $H$, modified by eq. (7),
allowing the physical position to grow alongside the expansion of the scale
factor. The amount $\mathbf{p}$ is the canonical momentum for the canonical
coordinate $\mathbf{x}$ and $D$ is the growth factor, with the index 1 and 2
representing first (linear) and second order calculations. The input linear
growth equation is given for the solution of our eq. (12).

Following Li  _et al._ (2022), we need now to write the growth equation using
the suppression factors such that

$\displaystyle\mathrm{G}_{m}\equiv D_{m}/a^{m},$ (14)

where the index $m$ means the order of perturbative expansion in respective
quantities. Moreover, the derivatives are given by

$\displaystyle D_{m}=a^{m}\mathrm{G}_{m}$ (15) $\displaystyle
D_{m}^{\prime}=a^{m}\left(m\mathrm{G}_{m}+\mathrm{G}_{m}^{\prime}\right),$
(16) $\displaystyle
D_{m}^{\prime\prime}=a^{m}\left(m^{2}\mathrm{G}_{m}+2m\mathrm{G}_{m}^{\prime}+\mathrm{G}_{m}^{\prime\prime}\right).$
(17)

The first equation in the 2LPT approximation is then written as

$\displaystyle D_{1}=a\mathrm{G}_{1},$ (18) $\displaystyle
D_{1}^{\prime}=a(\mathrm{G}_{1}+\mathrm{G}_{1}^{\prime}),$ (19) $\displaystyle
D_{1}^{\prime\prime}=a(\mathrm{G}_{1}+2\mathrm{G}_{1}^{\prime}+\mathrm{G}_{1}^{\prime\prime}).$
(20)

Then we find that the $G$ equation takes the following form for the IDE
framework under consideration in this work:

$\begin{split}&\mathrm{G}_{1}^{\prime\prime}+\mathrm{G}_{1}^{\prime}\left[\frac{\mathcal{H}^{\prime}}{\mathcal{H}}+\frac{r_{xc}\xi}{a}+5\right]+\mathrm{G}_{1}\Bigg{[}\frac{\mathcal{H}^{\prime}}{\mathcal{H}}+\frac{r_{xc}\xi}{a}+4\\\
&-\Bigg{(}\frac{3}{2}\Omega_{c}-\frac{r_{xc}\xi\mathcal{H}^{\prime}}{\mathcal{H}}+\frac{r_{xc}\xi}{a^{2}}(\xi+3\omega_{x}+r_{xc}\xi-a)\Bigg{)}\Bigg{]}=0.\end{split}$
(21)

The second order follows as

$\displaystyle D_{2}=a^{2}\mathrm{G}_{2},$ (22) $\displaystyle
D_{2}^{\prime}=a^{2}(2\mathrm{G}_{2}+\mathrm{G}_{2}^{\prime}),$ (23)
$\displaystyle
D_{2}^{\prime\prime}=a^{2}(4\mathrm{G}_{2}+4\mathrm{G}_{2}^{\prime}+\mathrm{G}_{2}^{\prime\prime}),$
(24)

and similarly, we obtain

$\begin{gathered}\mathrm{G}_{2}^{\prime\prime}+\mathrm{G}_{2}^{\prime}\bigg{(}4+\left[\frac{\mathcal{H}^{\prime}}{\mathcal{H}}+\frac{r_{xc}\xi}{a}+3\right]\bigg{)}+\\\
\mathrm{G}_{2}\bigg{(}4+2\Bigg{[}\frac{\mathcal{H}^{\prime}}{\mathcal{H}}+\frac{r_{xc}\xi}{a}+3\Bigg{]}-\\\
\Bigg{[}\frac{3}{2}\Omega_{c}-\frac{r_{xc}\xi\mathcal{H}^{\prime}}{\mathcal{H}}+\frac{r_{xc}\xi}{a^{2}}(\xi+3\omega_{x}+r_{xc}\xi-a)\Bigg{]}\bigg{)}=\frac{3}{2}\Omega_{m}\mathrm{G}_{1}^{2},\end{gathered}$
(25)

where the $\mathrm{G}_{1}$ term is given by eq. (21). This forms a system of
coupled equations.

In the cosmological consensus, the overdensity field is a homogeneous and
isotropic Gaussian random field characterized by the linear matter power
spectrum, $P_{\mathrm{lin}}$,

$\displaystyle\langle\delta^{(1)}(k)\delta^{(1)}(k^{\prime})\rangle$
$\displaystyle=(2\pi)^{3}\delta^{D}(\mathbf{k}+\mathbf{k^{\prime}})P_{\mathrm{lin}}(k)$
$\displaystyle\simeq\delta^{K}(\mathbf{k}+\mathbf{k^{\prime}})VP_{\mathrm{lin}}(k),$
(26)

where $V$ is a periodic box.

We can calculate the linear contrast $\delta^{(1)}$ by sampling each Fourier
mode independently

$\delta^{(1)}(k)=\sqrt{VP_{\mathrm{lin}}}\omega(k),$ (27)

where $\omega(k)$ is the Fourier transforming of a real white noise field
$\omega(q)$.

The input cosmological linear power spectrum can be written as

$\frac{k^{3}}{2\pi^{2}}P_{\mathrm{lin}}(k)=\frac{4}{25}A_{s}\left(\frac{k}{k_{\mathrm{pivot}}}\right)^{n_{s}-1}T^{2}(k)\frac{c^{4}k^{4}}{\Omega_{m}^{2}H_{0}^{4}}\left(\frac{D^{2}(z)}{D^{2}(0)}\right),$
(28)

where $D$ is the linear growth equation, $T$ is the transfer function, $A_{s}$
is the amplitude of the primordial power spectrum defined at some fixed scale
$k_{\mathrm{pivot}}$, and $n_{s}$ describes the shape of the primordial power
spectrum. In summary, one needs to discretize the white noise modes $\omega$,
and solve the growth and transfer functions which both depend on a set of
cosmological parameters.

Finally, the core of gravitational N-body simulation is the gravity solver.
The gravitational potential sourced by matter density fluctuations follows the
Poisson equation, which can be solved efficiently in the Fourier space. As we
are not considering modifications in gravitational theory, the Poisson
equation, and the gravity solver core remain the same by default.

To test our framework for the IDE model in N-body simulations introduced
above, we shall consider three scenarios: the $\Lambda$CDM model from Planck-
CMB best-fit values, and, based on the recent LSS analysis Nunes  _et al._
(2022b) in the IDE context, two different input baselines for IDE. Therefore,
we will consider the following common fiducial values: $10^{9}A_{s}=2.1$,
$n_{s}=0.96$, and $\Omega_{b}=0.05$, while we will choose for:

  * •

$\Lambda$CDM: $\Omega_{m}$=0.31, $H_{0}$=67.36 km/s/Mpc,

  * •

IDE-01: $\Omega_{m}$= 0.29, $H_{0}$= 68.02 km/s/Mpc, $\xi$ = -0.1,

  * •

IDE-02: $\Omega_{m}$= 0.22, $H_{0}$= 71.6 km/s/Mpc, $\xi$ = -0.1.

For the simulations, we use $256^{3}$ for the number of particles with single
precision and a $512^{3}$ mesh in a $(256\,\,{\rm Mpc/h})^{3}$ box for 63-time
steps, from $a=1/64$ to $a=1$. These are relatively small simulations but
suffice for our purpose in this work. The box size mainly affects the power
spectrum at large scales caused by the cosmic variance, and we are interested
in modeling $P(k)$ on small scales. On the other hand, the resolution mainly
affects the power spectrum close to the Nyquist limit at small scales. For a
better comparison, we apply smoothing with a Gaussian filter to remove the
noise in our figures. Hereby a standard deviation of $\sigma=2.5$ is adopted
for the Gaussian kernels in all simulations. For the three scenarios, i.e,
$\Lambda$CDM, IDE-01 and IDE-02, we evaluate the non-linear power spectrum,
$P(k)$, at $z=0$. For other values of $z$, the form of $P(k)$ remains
unchanged; it merely undergoes a simple amplitude rescaling.

###  III.2 Discussion

To quantify our findings, we introduce the ratio

$\displaystyle R_{\rm NL}(k)=\frac{P_{\rm IDE}(k)}{P_{\rm\Lambda CDM}(k)},$
(29)

which quantifies deviations between simulations among the different $P_{\rm
NL}(k)$ model predictions.

Figure 3: Behavior of the function, eq. (29), as predicted by N-body
simulation data. The upper and lower lines represent deviation limits for
IDE-02 (lower line) and IDE-01 (upper line) parameter inputs, respectively.

Figure 3 depicts the behavior of the ratio $R_{\rm NL}(k)$, providing insights
into the deviation of the IDE model from the $\Lambda$CDM model. The shaded
region indicates variations in parameter values between the IDE-01 and IDE-02
baselines. The upper and lower lines highlight the deviation limits predicted
for IDE-02 and IDE-01, respectively. As anticipated, IDE-02 demonstrates a
more pronounced deviation from $\Lambda$CDM compared to IDE-01. This
discrepancy can be attributed to the fact that IDE-01 has only a marginal
difference in the chosen parameters with respect to the $\Lambda$CDM case. On
the contrary, the important difference between IDE-01 and IDE-02 is not due to
a different coupling among DE and DM, but rather a different matter density
and Hubble constant. Between non-linear scales ranging from 0.15 to 0.40
h/Mpc, IDE models exhibit differences of approximately 20% to 70% compared to
the $\Lambda$CDM model. Moreover, across these scales, general variations in
the IDE baseline parameters may result in deviations of approximately 30% to
70% up to scales of about $\sim$ 1 h/Mpc. Thus, it is noteworthy that even
minimal variations in the IDE baseline values carry significant implications
for predicting $P(k)$ at the simulation level.

Let us also introduce the ratio:

$\displaystyle R_{\rm IDE}(k)=\frac{P_{\rm NL}(k)}{P_{\rm linear}(k)},$ (30)

which quantifies the deviations between the linear and non-linear power
spectra as predicted by IDE models themselves.

In our initial approach, we propose to fit the function

$\displaystyle R_{\rm IDE}(k)=A\cdot e^{-n(k-B)^{2}}+C.$ (31)

Figure 4: Comparison between predictions of the function with simulation data,
$R_{IDE}^{s}(k)$, and with the fitting function, $R_{IDE}(k)$. The comparison
for IDE-01 (IDE-02) are shown in red (blue), respectively.

In the following discussion, the superscript $s$ indicates quantities
evaluated differently from N-body simulation data. Figure 4 illustrates the
ratio between $R_{\text{IDE}}^{s}(k)$ and $R_{\text{IDE}}(k)$, parameterized
by equation (31). Our analysis reveals that the parameters
$A\in[23.23,24.24]$, $B\in[3.77,4.74]$, $C\in[-0.03,-0.04]$, and
$D\in[-9.30,-16.35]$ can effectively fit the non-linear spectra within a $<$5%
relative difference for the values within the range adopted in IDE-01 and
IDE-02 samples. It is noteworthy that any observed noise in Figure 4 on very
large scales is solely attributable to the box size of our simulations. Thus,
for the regime where non-linear effects are predominant, i.e., $k>0.2$ h/Mpc,
we note that our fit is very precise with accuracy $<$1%.

The fitting function, expressed as a parametric approximation like in equation
(31), has been widely employed to fit non-linear scales in phenomenological
models. In principle, this approach could suffice to generate valid $P(k)$
outputs within the confines of the simulations conducted. However, in the
subsequent section, we will deviate from the conventional methodology and
explore a modified Halo model capable of fitting the simulation data. We
emphasize that N-body simulations for interacting models, distinct from those
explored here, have been extensively investigated in the literature Baldi  _et
al._ (2010); Baldi (2012, 2023); Hashim  _et al._ (2018); Zhang  _et al._
(2018); Palma and Candlish (2023); Zhao  _et al._ (2023); Liu  _et al._
(2022); Ferlito  _et al._ (2022).

##  IV The Halo Model for IDE

The leading order moment of the cosmological matter distribution is the non-
linear matter power spectrum, $P_{\rm NL}(k)$. We follow the halo model (see
Asgari  _et al._ (2023) for a review) based approach of HMCode model Mead  _et
al._ (2016), in a target theory of cosmology to modeling $P_{\rm NL}(k)$ for
the IDE scenario under consideration in this work.

In the halo model, the matter power spectrum $P(k)$ is the contribution of two
parts. The first part is associated with the description of non-linearities
and is called the one-halo term, $P_{\rm 1H}$. The second part is called the
two-halo term and describes the linear part of the spectrum, associated with
small values of $k$, $P_{\rm 2H}$. Thus, we have that the total power spectrum
is given by

$\displaystyle P(k)=P_{\rm 1H}+P_{\rm 2H}.$ (32)

In many cases, it is also convenient to use dimensionless quantities
$\Delta^{2}(k)=4\pi V\left(\frac{k}{2\pi}\right)^{3}P(k)$. For the first term,
it is observed that the power spectrum presents the characteristics of a shot
noise, moderated by the halo density profile

$\displaystyle\Delta_{1\text{H}}^{2}(k)=\left(\frac{k}{2\pi}\right)^{3}\frac{4\pi}{\overline{\rho}^{2}}\int_{0}^{\infty}M^{2}W^{2}(k,M)F(M)\mathrm{d}M.$
(33)

In the above equation, the power spectrum is formulated as an integral over
halo masses, $M$, with the differential halo mass function denoted as $dM$,
determined by the halo mass function, $F(M)$. Additionally, this integral
incorporates the term $W(k,M)$, representing the normalized Fourier transform
of the halo density profile Asgari  _et al._ (2023). On larger spatial scales,
the distribution of haloes exhibits non-random patterns, and the displacements
between them necessitate the incorporation of a two-halo term into the power
spectrum analysis. This term accounts for the interactions between haloes and
is akin to the linear-theory power spectrum for the matter distribution,
$\Delta_{2\mathrm{H}}^{2}(k)=\Delta_{\mathrm{lin}}^{2}(k)$.

The HMCode exhibits specific modifications about the halo model, involving
parameters fine-tuned through simulations and functions optimized to align
with them. Concerning the power spectrum, an adaptation is introduced to the
one-halo and two-halo terms. The one-halo term is enhanced with a function to
more effectively tailor the spectrum at these scales. Consequently, it is
redefined as follows:

$\displaystyle\bar{\Delta}_{2\mathrm{H}}^{2}(k)=\left[1-f\tanh^{2}(k\sigma_{\mathrm{V}}/\sqrt{f})\right]\Delta_{\mathrm{lin}}^{2}(k),$
(34)

where $f$ represents a free parameter, $\sigma_{\mathrm{V}}$ is the 1D linear-
theory displacement variance, and $\Delta_{\mathrm{lin}}^{2}(k)$ refers to the
adaptation of the linear term made by Crocce  _et al._ (2006) in the form

$\displaystyle\Delta_{\mathrm{lin}}^{2}(k)\to\mathrm{e}^{-k^{2}\sigma_{\mathrm{v}}^{2}}\Delta_{\mathrm{lin}}^{2}(k).$
(35)

The one-halo term has also undergone modifications; in this instance, it
exhibits a more rapid decay and is consequently redefined as:

$\displaystyle\bar{\Delta}_{1\text{H}}^{2}=[1-\mathrm{e}^{-(k/k_{*})^{2}}]\Delta_{1\text{H}}^{2},$
(36)

where $k_{*}$ represents a free parameter associated with the one-halo damping
wavenumber. The complete spectrum incorporates an additional $\alpha$ term,
markedly enhancing the transition between the respective terms. Its revised
definition is as follows:

$\displaystyle\Delta^{2}(k)=[(\bar{\Delta}_{2\text{H}}^{2})^{\alpha}+(\bar{\Delta}_{1\text{H}}^{2})^{\alpha}]^{1/\alpha}.$
(37)

Additionally, the halo model incorporates key elements crucial to its
formulation. One of these elements is the halo mass density profile
$\rho(r,M)$, which is selected using the Navarro-Frenk-White (NFW) profile
Navarro  _et al._ (1997).

Moreover, another significant component is the halo mass function (HMF),
denoted as $F(M)$, which can be universally represented by $f(\nu)$:

$\frac{M}{\overline{\rho}}F(M)\mathrm{d}M=f(\nu)\mathrm{d}\nu.$ (38)

Here, $\nu$ represents the peak threshold of a halo with mass $M$, defined by:

$\nu=\frac{\delta_{c}(z)}{\sigma_{8}(M,z)},$ (39)

where $\delta_{c}(z)$ denotes the critical threshold for spherical collapse
based on linear theory, and the variance $\sigma_{8}(M,z)$ is computed from
the linear matter power spectrum. In the HMCode, the improved formula
developed by Sheth and Tormen (1999) is applied to determine $f(\nu)$.

Now, it is crucial to understand the model dependency up to this point. By
default, any non-standard cosmology will automatically change the predictions
on $\sigma_{8}(M,z)$ and $\Delta_{\mathrm{lin}}^{2}(k)$. On the other hand,
$\delta_{c}(z)$ is also a quantity depending on the model. Therefore, we
changed the default prediction to $\delta_{c{\rm
IDE}}=({\texttt{HMCode}})^{\beta_{1}}$, where HMCode represents the default
term implemented in the code. The changes in $\sigma_{8}(M,z)$ and
$\Delta_{\mathrm{lin}}^{2}(k)$ are already automatically captured by the
modifications described in Section II.

Another essential quantity in the formulation of halo models, which depends on
the adopted cosmology, is the concentration-mass relationship $c(M,z)$. In
this context, Ref. Dolag  _et al._ (2004) underscores the necessity of
incorporating a correction factor into the value. This correction factor
involves multiplying the value by the ratio of the linear growth function
associated with the chosen cosmology to that of the standard model. This
adjustment accounts for exotic cosmological scenarios. For the IDE model, our
proposition involves:

$c(M,z)=A\frac{1+z_{\mathrm{f}}(M)}{1+z}\times\frac{g_{\rm
IDE}(z\to\infty)}{g_{\mathrm{\Lambda CDM}}(z\to\infty)},$ (40)

where $z_{f}$ represents the redshift at which halo formation occurs, a
quantity that depends on the mass and is further detailed in Bullock  _et al._
(2001). Additionally, $g_{\rm IDE}$ depends on the growth index, which is
described by

$\frac{\mathrm{d}\ln g}{\mathrm{d}\ln a}=\Omega_{\mathrm{m}}^{\gamma_{\rm
IDE}}(z).$ (41)

Notably, the growth index we employ deviates from standard values, as
elucidated in Marcondes  _et al._ (2016):

$\gamma_{\rm IDE}=\frac{3(1-\omega_{0}+5\xi)}{5-6\omega_{0}-6\xi}.$ (42)

We have also made a crucial modification in the term
$(\bar{\Delta}_{2\text{H}}^{2})$, redefining it for the IDE model as:

$\bar{\bar{\Delta}}_{2\mathrm{H}}^{2}(k)=\beta_{2}\bar{\Delta}_{2\mathrm{H}}^{2}\mathrm{e}^{k\beta_{3}},$
(43)

where $\beta_{2}$ and $\beta_{3}$ are free parameters to be calibrated by
simulation data.

Another aspect that we note in the IDE framework is the location and magnitude
of the transition that occurs in the one-halo and two-halo terms. To mitigate
these impacts, the value of $k_{*}$ at $\bar{\Delta}_{1\text{H}}^{2}$ is
modified to the new ones in the form $k_{*\rm
IDE}=({\texttt{HMCode}}){\beta_{4}}$, and the value of $\alpha$ is also
changed to $\alpha_{\rm IDE}=({\texttt{HMCode}}){\beta_{5}}$.

Taking these modifications into account, our complete matter power spectrum is
given by

$\displaystyle\Delta^{2}(k)=[(\bar{\bar{\Delta}}_{2\text{H}}^{2})^{\alpha_{\rm
IDE}}+(\bar{\Delta}_{1\text{H}}^{2})^{\alpha_{\rm IDE}}]^{1/\alpha_{\rm
IDE}}.$ (44)

In summary, our adaptation of the Halo Model for IDE introduces five new
fitting parameters, namely $\beta_{1}$, $\beta_{2}$, $\beta_{3}$, $\beta_{4}$,
and $\beta_{5}$. These parameters essentially quantify deviations that are
contingent upon cosmological models within the framework of the halo model and
need to be determined by N-body simulation data.

Our analysis indicates that $\beta_{1}\in[3.4,3.6]$,
$\beta_{2}\in[0.15,0.28]$, $\beta_{3}\in[0.15,1.31]$, $\beta_{4}\in[3.5,5.1]$
and $\beta_{5}\in[1.09,1.11]$ can fit with good accuracy the entire parameter-
space range between the IDE-01 and IDE-02 model baselines. The creation of
these parameters played a crucial role in our model’s ability to accurately
describe the power spectra of matter on non-linear scales. Figure 5
illustrates the ratio between the modified halo model and simulation data for
two scenarios (IDE-01 and IDE-02). Notably, modeling accuracy within a 5%
margin extends to values of $k\approx 1$. A relative difference of less than
5% compared to simulations is a promising outcome, suggesting that our model
may be robust enough to capture the subtleties of matter structures even in
different settings, as tested with IDE-01 and IDE-02.

Figure 5: Ratio between the non-linear matter power spectrum generated by
N-body simulation, $P(k)_{s}$, and our adapted model $P(k)_{m}$. The
comparison for IDE-01 is shown in red and for IDE-02 in blue.

In the following sections, we implement our halo model in a forecast analysis
based on the sensitivity of the Euclid survey.

##  V Forecast analysis from the Euclid Survey perspective

Building upon the development of the matter power spectrum model that captures
the effects of dark sector interactions on non-linear scales, as outlined in
the previous section, we now turn our attention to a forecast analysis. In
this section, we aim to leverage the sensitivity of galaxy clustering and
cosmic shear measurements from the Euclid mission. This analysis will
primarily focus on quantifying the accuracy and validation of our developed
model. Our forecast analysis methodology is based on the approach presented in
Sprenger  _et al._ (2019).

We know that the arrangement of galaxies in space provides a good tracer of
the hidden distribution of dark matter. The relationship between the power
spectrum of galaxies, $P_{g}$, and the power spectrum of matter, $P_{m}$, is
given by:

$\begin{gathered}P_{g}(k,\mu,z)=f_{\text{AP}}(z)\times
f_{\text{res}}(k,\mu,z)\times\\\ f_{\text{RSD}}(k,\mu,z)\times b^{2}(z)\times
P_{m}(k,z),\end{gathered}$ (45)

where $f_{\text{AP}}(z)$ is the term associated with the Alcock-Paczynski
effect, $f_{\text{res}}(k,\mu,z)$ is a suppression factor, $b(z)$ is the bias
function (for more details see Sprenger  _et al._ (2019)) and $f_{\text{RSD}}$
incapsule the effects of the redshift space distortions. The inclusion of the
last term mentioned is crucial for the accuracy of the analysis.

It turns out that the redshift information we receive is not solely generated
by the expansion of the universe. The Doppler effect, arising from the
peculiar random velocities of galaxies, also contributes, creating a distorted
appearance in the redshift pattern known as the ‘fingers of God’ Jackson
(1972). Here, we describe these distortions using an exponential suppression
factor, as outlined in Bull  _et al._ (2015), so that the third term of
equation 45 is given by

$f_{\mathrm{RSD}}(k,\mu,z)=\left(1+\eta(k,z)\mu^{2}\right)^{2}e^{-k^{2}\mu^{2}\sigma_{\mathrm{NL}}^{2}},$
(46)

where the first term in parentheses represents the Kaiser formula Kaiser
(1987), while the exponential term addresses the ‘fingers of God’ phenomenon.
Specifically, $\sigma_{\mathrm{NL}}$ is initially set at 7 Mpc, with a range
of 4-10 Mpc considered in our predictions. Meanwhile, $\eta$ represents the
corrected growth rate, potentially influenced by the galaxy bias:

$\eta(k,z)=-\frac{1+z}{2b(z)}\cdot\frac{\mathrm{d}\ln
P_{m}(k,z)}{\mathrm{d}z}.$ (47)

Continuing with the discussion of eq. (45), while it provides a comprehensive
definition by incorporating various factors to ensure accurate measurements,
it is crucial to account for the impact of experimental noise. As a result, we
partition our surveys into bins of width $\Delta z=0.1$, centered around the
mean redshift $\bar{z}$. Within each bin, we compute correlation functions to
investigate the power spectrum at a specific fixed redshift $\bar{z}$. The
volume of each redshift bin is determined as follows:

$V_{r}(\bar{z})=\frac{4\pi}{3}f_{\mathrm{sky}}\cdot\left[r^{3}\left(\bar{z}+\frac{\Delta
z}{2}\right)-r^{3}\left(\bar{z}-\frac{\Delta z}{2}\right)\right],$ (48)

where $f_{\text{sky}}$ represents the fraction of the survey’s coverage over
the sky. Subsequently, to account for this experimental noise, the observed
quantity in each bin is adjusted as:

$P_{\text{obs}}(k,\mu,\bar{z})=P_{g}(k,\mu,\bar{z})+P_{N}(\bar{z}),$ (49)

where $P_{N}(\bar{z})$ denotes the volume expressed in eq. (48) divided by the
number of galaxies in the respective bin.

We will focus on redshifts ranging from $0.7$ to approximately $2$, with a sky
coverage of 0.3636. In this analysis, the spectroscopic redshift measurement
accuracy, $\sigma_{z}$, is impressively low at $0.001(1+z)$. This small error
is crucial because it translates into a minimal radial distance uncertainty,
$\sigma_{||}$, via the formula $\sigma_{||}=c\sigma_{z}/H$.

Furthermore, the bias factor for Euclid-detected galaxies is approximated by a
simple linear relation, $b(z)=1+z$. To address potential deviations from this
relation, let us also consider two nuisance parameters, each with a mean value
of 1:

$b(z)=b_{0}(1+z)^{0.5b_{1}},$ (50)

where in our analysis we will impose a stringent 5% precision, 2$\sigma$ CL,
constraint on these $b$-factors to ensure accurate modeling.

In our forecast analysis, we will incorporate the cosmic shear survey, which
captures the alignments of galaxies induced by weak gravitational lensing
resulting from large-scale structures along the line of sight. We will
evaluate this phenomenon in terms of the angular power spectrum, as detailed
in Sprenger  _et al._ (2019) for the Euclid survey. In this step, we integrate
our three-dimensional matter power spectrum model into these calculations to
enhance the accuracy of our predictions. To define the $\chi^{2}$, we adopt an
expression akin to that presented in Hilbert  _et al._ (2017), given by

$\begin{gathered}\chi^{2}=\sum_{n}\int_{k_{\min}}^{k_{\max}}\mathrm{d}k\cdot
k^{2}\int_{-1}^{1}\frac{V_{r}(\bar{z}_{n})}{2(2\pi)^{2}}\\\
\times\left[\frac{(\Delta
P_{g}(k,\mu,\bar{z}_{n}))^{2}}{\sigma_{\mathrm{obs}}^{2}(k,\mu,\bar{z}_{n})+\sigma_{\mathrm{th}}^{2}(k,\mu,\bar{z}_{n})}\right]\mathrm{d}\mu,\end{gathered}$
(51)

where $\Delta P_{g}(k,\mu,\bar{z}_{n})$ denotes the difference between the
predicted and observed galaxy power spectrum and to bolster the analytical
rigor, we also take into account the term denoted as $\sigma_{th}$ (of which
take the role of correlation lengths $\Delta k$, $\Delta\mu$, $\Delta z$),
given by

$\sigma_{\mathrm{th}}(k,\mu,z)=\left[\frac{V_{r}(z)}{(2\pi)^{2}}k^{2}\Delta
k\frac{\Delta z}{\Delta\bar{z}}\right]^{1/2}\alpha(k,\mu,z)P_{g}(k,\mu,z).$
(52)

For the correlation length in wavenumber space, we set $\Delta
k=0.05h/\text{Mpc}$. This choice aligns closely with the BAO scale, which
represents the smallest inherent scale in the matter power spectrum, thus
providing a conservative estimate for the correlation length in k-space. The
purpose of the relative error envelope function $\alpha(k,\mu,z)$ is to
primarily address uncertainties in two types of non-linear corrections: the
prediction of the matter power spectrum itself and the bias (see discussion in
Sprenger  _et al._ (2019)).

Parameter | $\sigma({\rm Euclid})$
---|---
$\Omega_{\rm cdm}$ | 0.0013
$n_{s}$ | 0.0043
$H_{0}$ | 0.55
$\xi$ | 0.0046
$\sigma_{\rm NL}$ | 0.054
$b_{0}$ | 0.0034
$b_{1}$ | 0.0060
Table 1: Summary of the expected 1$\sigma$ sensitivity of Euclid to the
cosmological parameters. The parameters $b_{0}$ and $b_{1}$ represent the
galaxy bias parameters, and $\sigma_{\rm NL}$ accounts for the effects of RSD.
Figure 6: Marginalized 1$\sigma$ and 2$\sigma$ contours and one-dimensional
posteriors in the parameter space, showing the expected sensitivity of Euclid
to the cosmological parameters for the IDE-01 scenario baseline values. The
analysis is performed following the conservative approach for the description
of the theoretical error.

In what follows in our main results, we use a modified version of the
CLASS$+$MontePython code Blas  _et al._ (2011); Audren  _et al._ (2013);
Brinckmann and Lesgourgues (2019), adapting the likelihoods euclid_pk and
euclid_lensing as well as incorporating all background dynamics, linear
perturbations, and the modified halo models in CLASS, as discussed in the
previous sections. In our forecast analyses, we utilize a maximum scale of
$k=1$ h/Mpc, within which our model demonstrates good predictive capability in
fitting the simulations.

Table 1 presents a statistical summary detailing the anticipated sensitivity
of the Euclid mission to cosmological parameters at $1\sigma$ confidence level
(CL). These results provide invaluable insights into the precision with which
Euclid can probe the IDE framework adopted in this work. Throughout our
analysis, we have adopted the IDE-01 scenario baseline values, ensuring
consistency and comparability across our findings. This choice serves as a
robust foundation for our discussions, enabling us to explore the nuanced
implications of Euclid’s observations. Importantly, our sensitivity
constraints remain robust and independent of specific input values for the
parameters baseline. This inherent flexibility underscores the generality of
our results, allowing for broad applicability and facilitating meaningful
interpretations across this IDE framework.

Of particular significance is the examination of sensitivity towards the
coupling parameter $\xi$. According to the analysis of Euclid forecasts, the
coupling parameter can be constrained with a precision of
$\sigma(\xi)=0.0046$. This represents a substantial improvement by an order of
magnitude compared to any other current observational tests documented in the
literature. For example, in a recent study Nunes  _et al._ (2022b), which
employed the full-shape galaxy power spectrum from the BOSS DR12 sample in
conjunction with CMB data from Planck, a constraint of $\xi>-0.12$ was
established. Consequently, the forthcoming data from Euclid has the potential
to enhance these constraints by up to two orders of magnitude. Therefore, the
accuracy of the constraints generated by the Euclid mission sensitivity will
be crucial for the IDE models. The joint accuracy with $\sigma(\Omega_{\rm
cdm})$ and $\sigma(H_{0})$, two parameters highly sensitive to IDE dynamics,
will also be constrained with remarkable precision. Consequently, Euclid data
alone will possess the capability to constrain the dynamics of IDE models,
encompassing both background and perturbative levels, with unprecedented
accuracy.

Figure 6 show the marginalized 1$\sigma$ and 2$\sigma$ contours and one
dimensional posteriors in the parametric space: $\Omega_{\rm cdm}$, $n_{s}$,
$H_{0}$, $\xi$, $\sigma_{\rm NL}$, $b_{0}$, $b_{1}$. Let us refocus our
attention on the coupling parameter. As previously noted, the anticipated high
sensitivity forecast $\sigma(\xi)=0.0046$ suggests that with the current
observed input value of $\xi=-0.1$, there could potentially be an indication
of interaction within the dark sector. To generalize and repeat the analysis
with a different input value of $\xi=0$, we find $\xi>-0.0045$ at 1$\sigma$
CL. Therefore, it will be crucial to integrate forthcoming real data from
Euclid with existing data from the literature for a real comprehensive
analysis.

There are other noteworthy points to consider. In the forecast analysis from
the perspective of Euclid, it is observed that the well-established
correlation between $\xi$ and $H_{0}$ persists, indicating a significant
negative correlation. This correlation has played a crucial role in arguing
that this particular class of models may offer a resolution to the $H_{0}$
tension Zhai  _et al._ (2023); Bernui  _et al._ (2023); Gariazzo  _et al._
(2022). Additionally, it is intriguing to observe that $\xi$ may exhibit a
notable new correlation with the galaxy bias parameters $b_{0}$ and $b_{1}$
(see Figure 6). Furthermore, the correlation in the $\xi$-$H_{0}$ plane within
the IDE framework implies that the galaxy bias parameters will also exhibit
correlations with $H_{0}$. Consequently, this reveals a previously
unrecognized cosmological characteristic for this class of models, which will
be essential to comprehend for future analyses with real observational data.

##  VI Final Remarks

In conclusion, our study represents a significant step forward in the
utilization of IDE models for observational constraints within the realm of
LSS data analysis on non-linear scales. By leveraging N-body simulations and
refining a modified halo model tailored to IDE cosmologies, we have
demonstrated the potential for achieving remarkable precision in fitting
simulated spectra down to scales of approximately $\sim$ 1 h/Mpc. These
findings not only enhance our understanding of the dynamics of dark energy
interactions but also underscore the viability of IDE models as powerful tools
for cosmological analysis. Moreover, our work opens new avenues for future
research, with implications for upcoming observational surveys such as the
Euclid mission. The robustness of our model, as demonstrated through forecast
analyses, suggests its potential utility in informing and refining
cosmological parameter estimates from forthcoming observational data.

In anticipation of future developments, our next steps involve the application
and refinement of our models on real observational data, including datasets
such as the Dark Energy Survey et al. (2022); Survey  _et al._ (2023) and the
Kilo-Degree Survey (KIDS-1000) Asgari  _et al._ (2021b). Specifically, we aim
to investigate the robustness of our model in addressing the current tension
in the $S_{8}$ parameter. We are actively working on analyzing the results in
this regard, and we look forward to sharing our findings in future
communications.

###### Acknowledgements.

We thank Carsten van de Bruck and Yuejia Zhai for useful discussions. E. S.
thanks CAPES for partial financial support. R.C.N. thanks the financial
support from the Conselho Nacional de Desenvolvimento Científico e Tecnologico
(CNPq, National Council for Scientific and Technological Development) under
the project No. 304306/2022-3, and the Fundação de Amparo à Pesquisa do Estado
do RS (FAPERGS, Research Support Foundation of the State of RS) for partial
financial support under the project No. 23/2551-0000848-3. E.D.V. is supported
by a Royal Society Dorothy Hodgkin Research Fellowship. This article is based
upon work from COST Action CA21136 Addressing observational tensions in
cosmology with systematics and fundamental physics (CosmoVerse) supported by
COST (European Cooperation in Science and Technology).

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