# Structure Formation in Various Dynamical Dark Energy Scenarios

Masoume Reyhani<EMAIL_ADDRESS>Department of Physics, K.N. Toosi
University of Technology, P.O. Box 15875-4416, Tehran, Iran  PDAT Laboratory,
Department of Physics, K.N. Toosi University of Technology, P.O. Box
15875-4416, Tehran, Iran  Mahdi Najafi<EMAIL_ADDRESS>Department
of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran,
Iran  PDAT Laboratory, Department of Physics, K.N. Toosi University of
Technology, P.O. Box 15875-4416, Tehran, Iran  Javad T. Firouzjaee
<EMAIL_ADDRESS>Department of Physics, K.N. Toosi University of
Technology, P.O. Box 15875-4416, Tehran, Iran  PDAT Laboratory, Department of
Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran,
Iran  School of Physics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395-5531, Tehran, Iran  Eleonora Di Valentino
<EMAIL_ADDRESS>School of Mathematics and Statistics, University
of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom

###### Abstract

This research investigates the impact of the nature of Dark Energy (DE) on
structure formation, focusing on the matter power spectrum and the Integrated
Sachs-Wolfe effect (ISW). By analyzing the matter power spectrum at redshifts
$z=0$ and $z=5$, as well as the ISW effect on the scale of $\ell=10-100$, the
study provides valuable insights into the influence of DE equations of state
(EoS) on structure formation. The findings reveal that dynamical DE models
exhibit a stronger matter power spectrum compared to constant DE models, with
the JBP model demonstrating the highest amplitude and the CPL model the
weakest. Additionally, the study delves into the ISW effect, highlighting the
time evolution of the ISW source term $\mathcal{F}(a)$ and its derivative
$d\mathcal{F}(a)/da$, and demonstrating that models with constant DE EoS
exhibit a stronger amplitude of $\mathcal{F}(a)$ overall, while dynamical
models such as CPL exhibit the highest amplitude among the dynamical models,
whereas JBP has the lowest. The study also explores the ISW auto-correlation
power spectrum and the ISW cross-correlation power spectrum, revealing that
dynamical DE models dominate over those with constant DE EoS across various
surveys. Moreover, it emphasizes the potential of studying the non-linear
matter power spectrum and incorporating datasets from the small scales to
further elucidate the dynamical nature of dark energy. This comprehensive
analysis underscores the significance of both the matter power spectrum and
the ISW signal in discerning the nature of dark energy, paving the way for
future research to explore the matter power spectrum at higher redshifts and
in the non-linear regime, providing deeper insights into the dynamical nature
of dark energy.

Dark Energy - The Integrated Sachs-Wolfe (ISW) effect - Structure Formation

##  I Introduction

It would not be an exaggeration to call the discovery of the expanding
universe, the triumph of modern cosmology Riess et al. (1998); Perlmutter et
al. (1999). This expansion could be explained through the inclusion of Dark
Energy (DE) component, which contributes to nearly 70% of the energy budget in
the standard $\Lambda$CDM model. This model posits the universe as homogeneous
and isotropic Di Valentino (2023); Peebles (1982). As the standard
cosmological model, the Lambda cold dark matter ($\Lambda$CDM) model has
proven highly successful in explicating a broad spectrum of cosmological
observations. These include the cosmic microwave background (CMB) Nolta et al.
(2004); Spergel et al. (2007); Ade et al. (2016); Aghanim et al. (2020a), type
Ia supernovae Riess et al. (1998); Perlmutter et al. (1999), galaxy surveys
Tegmark et al. (2006); Eisenstein et al. (2005), and weak lensing Hildebrandt
et al. (2017); Abbott et al. (2018).

However, in recent years, significant tensions have emerged in observational
data, particularly concerning the Hubble parameter $H_{0}$ Di Valentino et al.
(2021a); Krishnan et al. (2021); Guo et al. (2019); Abdalla et al. (2022); Di
Valentino et al. (2021b); Perivolaropoulos and Skara (2022) and the amplitude
of late-time matter clustering $\sigma_{8}-S_{8}$ Di Valentino et al. (2018);
Kazantzidis and Perivolaropoulos (2021); Di Valentino et al. (2021c, b). These
tensions suggest that the $\Lambda$CDM model may not be the complete picture.
The Hubble tension is a discrepancy between the direct and indirect
measurements of Hubble constant $H_{0}$ Bernui et al. (2023); Aghanim et al.
(2020a); Yang et al. (2023); Aiola et al. (2020); Dutcher et al. (2021); Verde
et al. (2019); Di Valentino et al. (2021a); McDonough et al. (2022);
Madhavacheril et al. (2023); Balkenhol et al. (2023); Huang et al. (2020);
Riess et al. (2022a, b); Murakami et al. (2023), and the $\sigma_{8}-S_{8}$
tension which represents the disagreement between the measurements of late-
time and early-time amount of matter clustering Perivolaropoulos and Skara
(2022); Asgari et al. (2021); McDonough et al. (2022); Abdalla et al. (2022);
Li et al. (2016); Gil-Marín et al. (2016); Aiola et al. (2020); Hu and Wang
(2023). Moreover, anomalies in CMB anisotropy Rassat et al. (2014); Akrami et
al. (2020); Schwarz et al. (2016); Perivolaropoulos (2014); Cayuso and Johnson
(2020) and CMB cold spots Vielva et al. (2004, 2004); Cruz et al. (2007);
Perivolaropoulos and Skara (2022) have spurred discussions among the
scientific community Hu and Wang (2023); Cayuso and Johnson (2020). Numerous
studies have been conducted to address these discrepancies Di Valentino et al.
(2021a), including models that challenge fundamental properties of dark
matter(DM) or dark energy (DE). These investigations encompass aspects such as
the mass-temperature relation Del Popolo et al. (2019); Naseri and Firouzjaee
(2020) and scaling relation Blanchard and Ilić (2021); Naseri and Firouzjaee
(2021) of galaxy clusters.

To address these tensions, one approach involves investigating potential
systematic errors. The consistency between independent direct and indirect
measurements, combined with the enhanced precision of measurement instruments,
underscores the need to explore alternative solutions to resolve these
cosmological crises. It is increasingly apparent that finding a resolution may
necessitate delving into new physics paradigms to accurately describe the
universe Di Valentino (2023); Natoli et al. (2018). One of the most promising
ways to address these tensions is to consider dynamical DE models Zhao et al.
(2017); Pan et al. (2019a); Yang et al. (2019), which allow the equation of
state of DE to vary with time. Dynamical DE models can be motivated by a
variety of theoretical considerations, such as the need to explain the late-
time acceleration of the Universe, the existence of a DE field that couples to
dark matter or other fields, or the possibility that DE is not a fundamental
constant but rather a dynamical quantity that evolves with time Yang et al.
(2021); Chevallier and Polarski (2001); Linder (2003); Efstathiou (1999);
Jassal et al. (2005); Barboza Jr and Alcaniz (2008). Some of the other
alternative DE models are Early DE Gómez-Valent et al. (2022); Escudero et al.
(2022); Karwal et al. (2022); Herold et al. (2022); Jiang and Piao (2022);
Smith et al. (2022); Reeves et al. (2023); Efstathiou et al. (2023);
Kamionkowski and Riess (2023); Poulin et al. (2023); Niedermann and Sloth
(2023); Eskilt et al. (2023); Gsponer et al. (2023); Goldstein et al. (2023),
Ginzburg-Landau Theory of DE Banihashemi et al. (2019); Banihashemi and
Khosravi (2022) and interacting Dark Energy-Dark Matter (IDMDE) Naseri and
Firouzjaee (2020); Farrar and Peebles (2004); Valiviita et al. (2008); Gavela
et al. (2009); Salvatelli et al. (2014); Di Valentino et al. (2017); Kumar and
Nunes (2017); Martinelli et al. (2019); Yang et al. (2020a); Di Valentino et
al. (2020); Pan et al. (2019b); Kumar et al. (2019); Kumar and Nunes (2016);
Murgia et al. (2016); Pourtsidou and Tram (2016); Lucca and Hooper (2020);
Gómez-Valent et al. (2020); Yang et al. (2020b); Nunes and Di Valentino
(2021); Nunes et al. (2022); Zhai et al. (2023); Escamilla et al. (2023).

An intriguing avenue for constraining DE models is to study the Integrated
Sachs-Wolfe (ISW) effect within observational data. This entails determining
the cross-correlation between the ISW signal and the distribution of galaxies.
The ISW effect is a cosmological phenomenon that occurs for the time
dependence of the gravitational potential as photons propagate through the
observable Universe. It is caused by the redshift or blueshift of photons as
they pass through regions of space where the gravitational potential undergoes
changes. In this respect, calculation of the ISW effect can be a convenient
measure for studying DE and can probe some modifications of the $\Lambda$CDM
model such as interacting dark sector models and modified gravity Kovács et
al. (2019); Yengejeh et al. (2023); Krolewski and Ferraro (2022); Ghodsi et
al. (2022); Kable et al. (2022); Seraille et al. (2024). The ISW effect is
especially sensitive to the time-varying nature of the DE equation of state,
making it a unique probe of dynamical DE models.

In this paper, we study the effect of the nature of DE on structure formation
across a spectrum of DE models. We start with a review of the various DE
models and, subsequently, we conduct a comparative assessment of the
introduced models against the standard $\Lambda$CDM model. To gain insights
into the nature of Dark Energy, we adopt two distinct approaches. Firstly, our
exploration involves an examination of the linear matter power spectrum. This
analysis reveals that scrutinizing this spectrum is instrumental in
distinguishing between various DE models. In a second step, our focus shifts
to the calculation of the ISW signal amplitude within each model, compared
against the predictions of the standard $\Lambda$CDM model. For this purpose,
it has been suggested Schäfer (2008); Yengejeh et al. (2023); Ghodsi et al.
(2022) that ISW signal auto-correlation and cross-correlation with the galaxy
distribution is a powerful tool to differentiate proposed models from one
another.

Our results suggest the following: 1. Each DE model hints at the distinct
structure formation pattern in the matter power spectrum. 2. The ISW effect
could be a powerful tool for constraining DE models and distinguishing them
from the $\Lambda$CDM model. Future observations of the ISW effect, especially
when combined with other cosmological probes, could provide valuable insights
into the nature of DE and its evolution over time.

The outline of this work is presented as follows. In Sec. II, we illustrate
the theoretical framework and the DE models under the study. In Sec. III, we
briefly describe the methodology of the observational analysis tools and the
likelihoods of this study. Then in Sec. IV, we study the matter power spectrum
for these models. Furthermore, in Sec. V, we calculate the ISW effect within
the context of these DE models and compare our results with that of the
$\Lambda$CDM model. Finally, we scrutinize the results and summarize the
findings in Sec. VI.

##  II Dynamical Dark Energy

At very large scales ($\sim 100$ Mpc) the universe can be assumed homogeneous
and isotropic. By considering a flat expanding Friedmann-Lemaître-Robertson-
Walker (FLRW), the line element is described by

$ds^{2}=a^{2}(\tau)[-d\tau^{2}+d{\bf x}^{2}]$ (1)

where $a(\tau)$ is the expansion factor of the universe known as scale factor,
and $\tau$ is the conformal time defined as $d\tau=dt/da$. For this universe
the Einstein equations are given by Ma and Bertschinger (1995):

$\mathcal{H}^{2}(a)\equiv(\frac{{a^{\prime}}}{a})^{2}=\frac{8\pi
G}{3}a^{2}\rho_{\rm i}$ (2) $\frac{d}{d\tau}(\frac{a^{\prime}}{a})=\frac{-4\pi
G}{3}a^{2}(\rho_{\rm i}+3P_{\rm i})$ (3)

where the primes denote the derivation with respect to conformal time $\tau$,
$\mathcal{H}(a)=a^{\prime}/a$ is defined as the conformal Hubble parameter
which represents the expansion rate of the universe, $G$ is the Newtonian
constant, and $\rho_{\rm i}$ is the energy density of non-interacting
components of the universe (i.e. radiation represented by $\rho_{\rm r}$, DE
represented by $\rho_{\rm DE}$, dark matter represented by $\rho_{\rm c}$, and
baryonic matter represented by $\rho_{\rm b}$). The same notation applied to
$P_{\rm i}$ as well, which describes the pressure of each aforementioned
component. Also, the relation between density and pressure is described by
$w_{\rm i}=P_{\rm i}/\rho_{\rm i}$ which is called barotropic equation of
state. For the components studied in the paper we have, $w_{\rm r}=P_{\rm
r}/\rho_{\rm r}=1/3$, $w_{\rm c}=P_{\rm c}/\rho_{\rm c}=0$, $w_{\rm b}=P_{\rm
b}/\rho_{\rm b}=0$, $w_{\rm DE}=P_{\rm DE}/\rho_{\rm DE}=-1$. The cosmic fluid
can be presented as a perfect fluid with the following energy momentum :

$T^{\mu}_{\nu}=Pg^{\mu}_{\nu}+(\rho+P)U^{\mu}U_{\nu}$ (4)

Considering the FLRW metric, Eq. (4) leads to
$T^{\mu}_{\nu}=diag(-\rho,P,P,P)$. From Bianchi identity,
$\nabla_{\mu}G^{\mu}_{\nu}=0$ is obtained, which implies the conservation of
energy-momentum:

$\nabla_{\mu}T_{\nu}^{\mu}=0$ (5)

Considering $\nu=0$, the Eq. (5) will return the continuity equation which
simply depicts the time evolution of the components of the universe:

$\rho^{\prime}+3\mathcal{H}(\rho+P)=0,$ (6)

or

$\rho^{\prime}+3\mathcal{H}(1+w)\rho=0.$ (7)

We have ignored the subscript $i$, since this equation is true for every
component of the non-interacting universe.
With all that knowledge, the Eq. (2) can be written as below Yang et al.
(2021):

$\displaystyle\mathcal{H}^{2}(a)=\frac{8\pi G}{3}a^{2}[\rho_{\rm
r,0}a^{-4}+\rho_{\rm b,0}a^{-3}+\rho_{\rm c,0}a^{-3}$ $\displaystyle+\rho_{\rm
DE,0}(\frac{a}{a_{0}})^{-3}exp(-3\int_{\rm a_{0}}^{a}\frac{w_{\rm
DE}(a^{\prime})}{a^{\prime}}da^{\prime})]$ (8)

Where subscript $0$ indicates the current value of each component. Considering
$w_{\rm DE}=-1$, Eq. (II) simply returns the standard $\Lambda$CDM model.

By considering different scenarios for DE equation of state, we aim to
investigate the impact of the nature of DE equation of state on the matter
power spectra, the evolution of gravitational potential, and subsequently the
ISW effect. To this end, we will describe the DE models under study. A key
illustrative tool throughout this work will be the comparison of the
introduced models to the standard $\Lambda$CDM model and among themselves.

  * •

Model 1: In this model, we consider the DE equation of state to be constant
and $w_{\rm DE}=w_{0}\neq-1$. We will call this model $w$CDM.

  * •

Model 2: The first dynamical DE model under study is the well-known
Chevallier-Polarski-Linder parametrization, henceforth referred to as CPL
Chevallier and Polarski (2001); Linder (2003):

$w_{\rm DE}(a)=w_{0}+w_{\rm a}(1-a)$ (9)

  * •

Model 3: The second dynamical DE equation of state is proposed by Jassal-
Bagla-Padmanabhan and is referred to as JBP Jassal et al. (2005) in this
paper:

$w_{\rm DE}(a)=w_{0}+w_{\rm a}a(1-a)$ (10)

  * •

Model 4: The last model being under scrutiny in this article is proposed by
Barboza-Alcaniz, and it is known as BA Barboza Jr and Alcaniz (2008):

$w_{\rm DE}(a)=w_{0}+w_{\rm a}(\frac{1-a}{2a^{2}-2a+1})$ (11)

In order to study the time evolution of the potential, it is necessary to
consider perturbations to the FLRW metric. In synchronous gauge we can
reconsider the line element to be as Ma and Bertschinger (1995):

$ds^{2}=a^{2}(\tau)[-d\tau^{2}+(\delta_{\rm ij}+h_{\rm ij})dx^{i}dx^{j}]$ (12)

The metric perturbation $h_{\rm ij}$ can be decomposed into a trace part
$h\equiv h_{\rm ii}$ and a traceless part consisting of three pieces: $h_{\rm
ij}^{\parallel}$, $h_{\rm ij}^{\perp}$ and $h_{\rm ij}^{T}$. By moving to
Fourier space, the equations governing the dimensionless density perturbations
$\delta_{\rm i}=\delta\rho_{\rm i}/\rho_{\rm i}$ and velocity perturbations
$\theta_{\rm i}=\partial_{j}v_{i}^{j}$ are given by:

$\displaystyle\delta^{\prime}_{\rm DE}=-(1+w_{\rm DE})(\theta_{\rm
DE}+\frac{h^{\prime}}{2})-3\mathcal{H}(c^{2}_{\rm sDE}-w_{\rm DE})$
$\displaystyle\times[\delta_{\rm DE}+3\mathcal{H}(1+w_{\rm
DE})\frac{\theta_{\rm DE}}{k^{2}}]-3\mathcal{H}w^{\prime}_{\rm
DE}\frac{\theta_{\rm DE}}{k^{2}}$ (13) $\theta^{\prime}_{\rm
DE}=-\mathcal{H}(1-3c^{2}_{\rm sDE})\theta_{\rm DE}+\frac{c^{2}_{\rm
sDE}}{(1+w_{\rm DE})}k^{2}\delta_{\rm DE}$ (14) $\delta^{\prime}_{\rm
c}=-(\theta_{\rm c}+\frac{h^{\prime}}{2})$ (15) $\theta^{\prime}_{\rm
c}=-\mathcal{H}\theta_{\rm c}$ (16)

Where the $c^{2}_{\rm sDE}$ is the squared sound speed of DE component in rest
frame, $c^{2}_{\rm sDE}=\delta P_{\rm DE}/\delta\rho_{\rm DE}$, and $k$ is the
wavenumber in Fourier space. It is also worth to define adiabatic sound speed,
$c^{2}_{\rm aDE}=dP_{\rm DE}/d\rho_{\rm DE}=w_{\rm DE}+w^{\prime}_{\rm
DE}/(\rho^{\prime}_{\rm DE}/\rho_{\rm DE})$. Considering a barotropic equation
of state will lead to the $c^{2}_{\rm aDE}=c^{2}_{\rm sDE}=w_{\rm DE}$. By
considering DE as an adiabatic fluid, we will have $c^{2}_{\rm aDE}=c^{2}_{\rm
sDE}=w_{\rm DE}<0$ which as a consequence will lead to instabilities in DE
fluid. In order to solve this problem we will consider $c^{2}_{\rm sDE}=1$.

Having outlined the framework within which we will operate, it would be
illustrative to delineate the methodology that will assist us in constraining
the cosmological parameters.

##  III Methodology

In the following, we briefly describe the observational analysis tools and
likelihoods used in this study. Using a modified version of the publicly
available cosmological code CAMB Lewis et al. (2000); Howlett et al. (2012),
and by means of the publicly available Monte-Carlo Markov Chain Cobaya, we
successfully obtained the best fit to the data Powell et al. (2009); Cartis et
al. (2018, 2019); Torrado and Lewis (2021).
For the base cosmological model, $\Lambda$CDM, we have used the following 6
parameter space:

$\mathcal{P}=\\{\Omega_{\rm b}h^{2},\Omega_{\rm c}h^{2},100\theta_{\rm
MC},\tau,n_{\rm s},\ln[10^{10}A_{\rm s}]\\}$ (17)

For $w$CDM model, we have added an extra degree of freedom in the form of
$w_{0}$:

$\mathcal{P}=\\{\Omega_{\rm b}h^{2},\Omega_{\rm c}h^{2},100\theta_{\rm
MC},\tau,n_{\rm s},\ln[10^{10}A_{\rm s}],w_{0}\\}$ (18)

And for the dynamical parameterization [Eqs. (9)-(11)], we have added two
extra degrees of freedom, $w_{\rm a}$ and $w_{0}$ to the standard, 6 parameter
space:

$\mathcal{P}=\\{\Omega_{\rm b}h^{2},\Omega_{\rm c}h^{2},100\theta_{\rm
MC},\tau,n_{\rm s},\ln[10^{10}A_{\rm s}],w_{0},w_{\rm a}\\}$ (19)

The baseline likelihoods we used include Aghanim et al. (2020b, c):

  * •

Commander likelihood which provides low multipoles TT data in the range
($2\leq\ell\leq 29$).

  * •

SimAll likelihood which provides low multipoles EE data in the range
($2\leq\ell\leq 29$).

  * •

Plik TT,TE,EE likelihood which provides the high multipoles TT, TE, and EE
data in the range ($30\leq\ell\lesssim 2500$ for TT and $30\leq\ell\lesssim
2000$ for TE and EE).

  * •

lensing reconstruction likelihood which is obtained with a trispectrum
analysis and can provide complementary information to the Planck CMB power
spectra.

Figure 1: Comparison of the CMB temperature power spectrum of each model under
the study (BA: Red, JBP: Green, CPL: Blue, $\Lambda$CDM: Black, $w$CDM:
Yellow) based on the best-fit values obtained from the cosmological code
Cobaya assuming the baseline likelihoods with Planck 2018 (blue data points
with error bars).

Table 1 shows the flat priors used to obtain best-fit of the data. Table 2
shows the best-fit values for parameters used in each DE parameterization. It
has to be emphasized that in $w$CDM case, our best-fit analysis relies on
Planck baseline likelihoods listed above, and the value for $w_{0}$ resides in
the phantom region. We can see in Fig. 1 the temperature power spectrum of
each model obtained with the best-fit results, and it depicts the agreement of
these best-fits for each DE parameterization used in this study.

By understanding the methodology and datasets, one would be able to study the
DE equation of state effects on structure formation. To do so, we will start
by studying the effect of the impact of DE equation of state on the matter
power spectrum.

Parameter | Prior
---|---
$\Omega_{\rm b}h^{2}$ | $[0.005\,,\,0.1]$
$\Omega_{\rm c}h^{2}$ | $[0.001\,,\,0.99]$
$\tau$ | $[0.01,0.8]$
$100\,\theta_{\rm MC}$ | $[0.5\,,\,10]$
$\log(10^{10}A_{\rm S})$ | $[1.61\,,\,3.91]$
$n_{\rm s}$ | $[0.8\,,\,1.2]$
$w_{0}$ | $[-3\,,\,1]$
$w_{\rm a}$ | $[-3\,,\,2]$
Table 1: Ranges for the flat prior distributions on cosmological parameters in the study. For $w$CDM, we used [-3 , -0.333] to impose an expanding universe. Table 2: The best-fit dataset from our baseline dataset used to study each DE model. Model | $\Lambda CDM$ | $wCDM$ | CPL | JBP | BA
---|---|---|---|---|---
$\Omega_{\rm b}h^{2}$ | $0.022340$ | $0.02241$ | $0.02242$ | $0.02246$ | $0.02244$
$\Omega_{\rm c}h^{2}$ | $0.11984$ | $0.11902$ | $0.11863$ | $0.11858$ | $0.11860$
$H_{0}$ | $67.36$ | $81.31$ | $82.42$ | $99.79$ | $96.51$
$10^{9}A_{\rm S}$ | $2.111$ | $2.103$ | $2.104$ | $2.095$ | $2.103$
$n_{\rm s}$ | $0.9636$ | $0.9663$ | $0.9662$ | $0.9673$ | $0.9677$
$\tau$ | $0.059$ | $0.058$ | $0.058$ | $0.058$ | $0.057$
100$\theta_{\rm MC}$ | $1.04092$ | $1.04101$ | $1.04104$ | $1.04106$ | $1.04105$
$w_{0}$ | $-1$ | $-1.42$ | $-0.92$ | $-1.85$ | $-1.31$
$w_{\rm a}$ | $-$ | $-$ | $-2.53$ | $-0.38$ | $-1.69$

##  IV Matter power spectrum

In order to illustrate the effect of dynamical DE on structure formation over
time, we will study how the dynamical nature of DE affects the amplitude of
the matter power spectrum. By analyzing the matter power spectrum, we can also
discern the effect of DE across various scales. Here, we will focus on the
linear evolution of the matter power spectrum in different redshifts, but
studying nonlinear correction to the matter power spectrum could unveil
interesting results.

###  IV.1 Linear matter power spectrum

To study the statistical correlations, the linear matter power spectrum is a
powerful tool for this purpose Dodelson and Schmidt (2020); Orjuela-Quintana
et al. (2023). The matter power spectrum for any given redshift can be written
as a function of redshift $z$ and wavenumber $k$:

$P_{l}(k,z)=\frac{8\pi^{2}}{25}\frac{\mathcal{A}_{\rm s}}{\Omega^{2}_{\rm
m}}D_{+}^{2}(z)T^{2}(k)\frac{k^{n_{\rm s}}}{H_{0}^{4}k_{\rm p}^{n_{\rm s}-1}}$
(20)

Where $\mathcal{A}_{\rm s}$ and $n_{\rm s}$ represent the primordial amplitude
and spectral index respectively. $T(k)$ is the transfer function, where on
large-scale $T(k)=1$. $k_{\rm p}$ is an arbitrary pivot scalar, and
$H(a)=\dot{a}/a$ is the Hubble parameter, where the dot denotes the derivative
with respect to physical time $t$. As before, $H_{0}$ denotes the present
value of the Hubble parameter, known as the Hubble constant.

To scrutinize the evolution of structure formation, it is descriptive to
consider the matter power spectrum at different redshifts.

Figure 2: The matter power spectrum as a function of comoving wavenumber $k$
for 3 dynamical DE equation of state models (Top: BA, Middle: JBP, Bottom:
CPL), compared with 2 constant DE equation of state model ($\Lambda$CDM and
$w$CDM), at redshift $z=0$ (solid lines) and redshift $z=5$ (dashed lines).
The shaded area in each panel represents the range within which the matter
power spectrum of redshifts between $0<z<5$ is contained.

In Fig. 2, we present a comparison of the matter power spectrum as a function
of comoving wavenumber $k$ among dynamical dark energy (DE) equation of state
parameterizations –i.e., the BA, JBP, and CPL models – and the constant dark
energy equation of state model $w$CDM, along with the standard $\Lambda$CDM
model. This comparison is based on the best fit of these scenarios to our
baseline likelihoods. All three panels demonstrate that regardless of the
chosen model, the general amplitude of the matter power spectrum is suppressed
as we move towards higher redshifts (dotted lines). This observation arises
because structure formation is stronger at lower redshifts, closer to the
current age of the universe. It can also be concluded that in all three cases
of dynamical dark energy, regardless of redshift, the general amplitude of the
matter power spectrum has higher values compared to those with a constant dark
energy equation of state. One possible explanation for this behavior lies in
the dynamical nature of dark energy; considering dark energy to be dynamical
leads to a stronger effect from the matter component in the evolution history
of the universe. As indicated in Fig. 2, $w$CDM has a stronger general
amplitude compared to the $\Lambda$CDM model for the same redshifts. In the
top panel of Fig. 2, one can also observe a slight shift in the Baryon Wiggles
peaks towards lower wavenumbers $k$s (larger scales) in the BA model compared
to constant DE EoS scenarios. The middle panel of Fig. 2 shows the same
behaviour for the JBP parameterisation. However, the bottom panel of Fig. 2
shows an interesting attribute. The matter power spectrum of the CPL model,
though slightly higher in values in general, behaves similarly to the $w$CDM
model, i.e. the shift in Baryon Wiggles peaks is not as noticeable as in two
other cases. Also, it can be seen that distinguishing between the CPL and
$w$CDM models is challenging. Furthermore, one can observe a shift in $k_{eq}$
(the wavenumber representing the transition from a radiation-dominated to a
matter-dominated universe) towards lower $k$ values for BA and JBP compared to
$w$CDM and $\Lambda$CDM. Comparing $w$CDM with $\Lambda$CDM, a clear shift of
$k_{eq}$ towards lower $k$ values is noticeable in the case of $w$CDM. An
interesting point to note is that distinguishing the shift in $k_{eq}$ between
CPL and $w$CDM remains challenging.

Fig. 3 depicts the evolution of these three dynamical DE models at redshifts
$z=0$ and $z=5$. As observed, when we consider DE behavior as described by the
CPL model, the amplitude of the matter power spectrum is the lowest compared
to the BA and JBP parameterizations at the mentioned redshifts. However, the
JBP and BA parameterizations exhibit an interesting similarity in the given
redshifts. As depicted, the JBP and BA models are nearly indistinguishable. It
should be noted that the shaded area in each panel of Fig. 2, 3 represents the
range within which the matter power spectrum for redshifts between $0<z<5$ is
expected.

It is crucial to note that considering different dataset combinations may
yield diverse behaviors in the matter power spectrum. Particularly,
incorporating datasets that observe redshifts deep into the matter-dominated
era could provide valuable insights into the dynamical nature of the BA, JBP,
and CPL models.

Gaining insight into the impact of the nature of DE on the matter power
spectrum enables a more intuitive exploration of the Integrated Sachs-Wolfe
(ISW) signal.

Figure 3: Comparison of the matter power spectra as a function of comoving
wavenumber $k$ among 3 dynamical DE models in redshift $z=0$ (solid lines) and
redshift $z=5$ (dashed lines). The shaded area represents the range within
which the matter power spectrum for redshifts between $0<z<5$ is expected.

##  V Measuring Integrated Sachs-Wolfe effect

Here, we will focus on two changes in the wavelength of Cosmic Microwave
Background (CMB) radiation. The first is based on the fact that due to the
inhomogeneity in the universe, the photons will encounter potential wells
resulting in an increase in their energy or a decrease in their wavelength.
The second change is due to the expansion of the universe which will cause
photons to lose their energy. In other words, photons traveling toward us from
the last scattering surface will experience a change in their wavelength to a
larger value. Based on what we have discussed, when a photon enters a
potential well, it will experience a blueshift in its wavelength. However, due
to the expansion of the universe, we will observe only a slight difference
between the original and final wavelength of the photon.
To calculate the ISW effect for each model, it is necessary to obtain the
matter power spectrum (see Subsec. IV.1). Accordingly we have used the
modified version of CAMB, discussed in Sec. III.

The appearance of temperature anisotropy $\Theta_{\rm ISW}$ on the CMB map,
stemming from the evolving gravitational potential $\phi$ over time, can be
articulated as follows:

$\Theta_{\rm ISW}=\frac{\Delta T}{T_{\rm CMB}}\equiv\frac{2}{c^{2}}\int_{{\rm
a}_{\rm dec}}^{1}\frac{\partial\phi}{\partial
a}da=-\frac{2}{c^{3}}\int_{0}^{\chi_{\rm
H}}a^{2}H(a)\frac{\partial\phi}{\partial a}d\chi$ (21)

where $T_{\rm CMB}$ represents the CMB temperature which is equal to 2.725
$K$, $c$ parameterizes the velocity of light, $a_{\rm dec}$ denotes the scale
factor of matter-radiation equality, and $\chi$ is the comoving distance
defined as:

$\chi(a)=\int_{0}^{1}\frac{c}{a^{2}H(a)}da$ (22)

Assuming perturbations exist within the horizon, the potential can be related
to the matter fluctuation field, using Poisson’s equation in Fourier space,
leading to:

$k^{2}\phi(k,a)=-\frac{3}{2}a^{2}H^{2}(a)\Omega_{\rm m}(a)\delta_{\rm
m}(k,a),$ (23)

where the matter density parameter $\Omega_{\rm m}(a)$ is defined as
$\Omega_{\rm m}=\rho_{\rm m}/\rho_{\rm crit}$, where ${\rho_{\rm crit}}$
represents the critical density given by $\rho_{\rm crit}=3H^{2}(a)/8\pi G$
and $\delta_{\rm m}(k,a)$ denotes the matter perturbation.
Using Eqs. (21) and (23) will result in:

$\Theta_{\rm ISW}=\frac{3}{c^{3}}\int_{0}^{\chi_{\rm
H}}a^{2}H(a)\frac{\partial\mathcal{F}(a)}{\partial a}\frac{\delta_{\rm
m}(k,a=1)}{k^{2}}d\chi,$ (24)

where the $\mathcal{F}(a)$ is defined as:

$\mathcal{F}(a)=a^{2}H^{2}\Omega_{\rm m}(a)D_{+}(a)$ (25)

Here, $D_{+}(a)$ is the linear growth factor which is defined as:
$D_{+}(a)=\delta_{\rm m}(k,a)/\delta_{\rm m}(k,a=1)$. As long as the dark
energy fluid does not influence local structure formation, the equation
governing the evolution of $D_{+}(a)$ with respect of scale factor is as
follows Linder and Jenkins (2003); Peter and Uzan (2009); Schäfer (2008):

$\frac{d^{2}D_{+}}{da^{2}}+\frac{1}{a}\left(3+\frac{d\ln H}{d\ln
a}\right)\frac{dD_{+}}{da}=\frac{3}{2a^{2}}\Omega_{m}(a)D_{+}(a).$ (26)

Fig. 4 depicts the time evolution of the ISW source term $\mathcal{F}(a)$
(top) and its derivative with respect to the scale factor $d\mathcal{F}(a)/da$
(bottom) as a function of scale factor $a$. The top panel indicates the
amplitude of time evolution of the ISW source term as a function of scale
factor $a$. Here, it is clear to see the effect of the nature of DE EoS, as
the amplitude for those that are dynamical in nature is suppressed in this
measurement. Among the constant parameterization of DE EoS, the $\Lambda$CDM
model has a higher amplitude compared to the $w$CDM model. As for dynamical DE
models, the CPL has the largest amplitude among the dynamical models. As can
be seen, the amplitude of the time evolution of the ISW source for the $w$CDM
and CPL, as well as the BA and JBP parameterization, are very similar, with
the $w$CDM and BA dominating the CPL and JBP, respectively.

Figure 4: The time evolution of the ISW source term $\mathcal{F}(a)$ (Top) and
its derivative $d\mathcal{F}(a)/da$ (Bottom) as functions of scale factor $a$.

However, the impressive feature of the evolution of the ISW source term is
that in each model, as we move towards the scales where the DE becomes the
dominant force of expansion, a rapid decrease in amplitude can be seen. This
derivation with respect to scale factor $a$ is shown in the bottom panel of
Fig. 4. As expected, the most extreme case of the change in time evolution
belongs to $\Lambda$CDM. At scale factors, $a\approx 0.6$ and $a\approx 0.65$,
a change in the strength of the time evolution of the ISW source term can be
observed for the JBP and BA models, as well as the CPL and $w$CDM models,
respectively. It should be mentioned that around $a=1$, the values of
$d\mathcal{F}/da$ become close to each other once again.

Additionally, it is important to note that in a non-interacting dark sector,
the following equation holds for the density parameter :

$\frac{\Omega_{\rm m}(a)}{\Omega_{\rm m}}=\frac{H^{2}_{0}}{a^{3}H^{2}(a)}$
(27)

In order to study the potential wells encountered by photons traveling from
the last scattering surface to us, it is imperative to evaluate the
correlation between the ISW temperature and galaxy density. Acknowledging
that, the ISW effect is relatively minor on smaller scales but significantly
impactful on larger scales, galaxies serve as a suitable marker for large-
scale structures during later cosmic epochs.

Consequently, the line-of-sight integral for galaxy density is:

$\delta_{\rm g}=\int\frac{H(a)}{c}f(z)D_{+}(a)\delta_{\rm m}(k,a=1)d\chi$ (28)

The redshift distribution, denoted as $f(z)$, is formulated as
$f(z)=b(z)dN/dz$. To understand the vastness of galaxy clusters, the
utilization of large-scale surveys becomes necessary in order to compute the
abundance of galaxy clusters as a function of redshift $z$.
Using the CMB map and galaxy distribution, one can define the angular auto-
correlation and cross-correlation as:

$C^{TT}=<\Theta_{\rm ISW}\Theta_{\rm ISW}>$ (29)

$C^{Tg}=<\Theta_{\rm ISW}\delta_{\rm g}>$ (30)

Defining weight functions for photons and galaxies as follows:

$W_{\rm T}(\chi)=\frac{3}{c}a^{2}H(a)\frac{d\mathcal{F}(a)}{da}$ (31) $W_{\rm
g}(\chi)=\frac{H(a)}{c}f(z)D_{+}(a)$ (32)

this allows us to write the angular auto-correlation and cross-correlation in
a compact notation:

$C^{TT}_{\rm ISW}(\ell)=\int_{0}^{\chi_{\rm H}}\frac{W^{2}_{\rm
T}(\chi)}{\chi^{2}}~{}\frac{H^{4}_{0}}{k^{4}}~{}P(k=\frac{\ell+\frac{1}{2}}{\chi})~{}d\chi$
(33)

$C^{Tg}_{\rm ISW}(\ell)=\int_{0}^{\chi_{\rm H}}\frac{W_{\rm T}(\chi)W_{\rm
g}(\chi)}{\chi^{2}}~{}\frac{H^{4}_{0}}{k^{4}}~{}P(k=\frac{\ell+\frac{1}{2}}{\chi})~{}d\chi$
(34)

In this context, P(k) represents the current matter power spectrum and
$k=(\ell+1/2)/\chi$ is derived using the Limber approximation for larger value
of $\ell$ Afshordi et al. (2004); Stölzner et al. (2018)

Figure 5: The ISW auto-power spectrum $C^{TT}_{\rm ISW}$ as a function of the
multipole $\ell$, for BA (red solid line), JBP( green dotted line) and CPL
(blue dotted line) as 3 dynamical DE models, as well as $\Lambda$CDM (black
solid line) and $w$CDM (dotted-dashed line) as two models of DE with constant
equation of state.

###  V.1 Redshift Distribution

In order to study the correlation between CMB photons and large-scale
structure, it is crucial to use information obtained from galaxy redshift
distribution surveys. These surveys typically employ spectroscopic and$/$or
multi-band photometric calibration survey of small patches of the sky Sánchez
et al. (2020). In this paper, we have used the redshift distribution of four
distinct surveys. Hence, in the following, we briefly discuss galaxies
redshift distribution surveys under investigation. In Sec 4.1 (see Fig. 4) of
Ref Yengejeh et al. (2023), the photometric normalized redshift distribution
for the surveys under study are discussed.

I. Dark Universe Explorer (DUNE): The main goal of this survey is to
investigate potential candidates for weak gravitational lensing and explore
the ISW effect. The redshift distribution of the survey is as follows:

$f_{\rm DUNE}(z)=b_{\rm
eff}[\frac{z_{*}}{a_{*}}\Gamma(\frac{3}{a_{*}})]^{-1}(\frac{z}{z_{*}})^{2}exp[-(\frac{z}{z_{*}})^{a_{*}}]$
(35)

II. The National Radio Astronomy Observatory (NRAO) Very Large Array (VLA) Sky
Survey (NVSS): The redshift distribution function in this survey, covering
about 82% of the sky, is as follows:

$f_{\rm NVSS}(z)=b_{\rm
eff}\frac{a_{*}^{a_{*}+1}}{\Gamma(a_{*})}\frac{z^{a_{*}}}{z_{*}^{a_{*}+1}}exp(-\frac{a_{*}z}{z_{*}})$
(36)

III. The Sloan Digital Sky Survey (SDSS): This survey gathers images and
spectroscopic data of galaxies, quasars, and stars, with the following
redshift distribution:

$f_{\rm SDSS}(z)=b_{\rm
eff}\frac{a_{*}}{\Gamma(\frac{m+1}{a_{*}})}\frac{z^{m}}{z_{*}^{m+1}}exp[-(\frac{z}{z_{*}})^{a_{*}}]$
(37)

IV. Euclid-like: The Euclid mission intends to explore the relationship
between the distance and redshift of galaxies up to a redshift of z $\approx$
2\. In this survey, the distribution of redshift is characterized by the
following Weaverdyck et al. (2018); Martinet et al. (2015):

$f_{\rm Euclid-like}(z)=b_{\rm
eff}\frac{3}{2z^{3}_{*}}z^{2}exp[-(\frac{z}{z_{*}})^{\frac{3}{2}}]$ (38)

In the aforementioned surveys, $b_{\rm eff}$, $z_{*}$, $a_{*}$ and $m$ are
free parameters that need to be determined. $\Gamma(x)$ represents the Gamma
function for all surveys. The values for the free parameters are presented in
Table 3.

Figure 6: The ISW-cross power spectrum $C_{\rm Tg}$ as a function of the
multipole $\ell$ in the range of $10$ to $100$, for BA (red solid line), JBP
(green dashed line), and CPL (blue dotted line) as 3 dynamical DE models, as
well as $\Lambda$CDM (black solid line) and $w$CDM (orange dot-dashed line) as
two models of DE with constant EoS, for DUNE (Top left), Euclid-like (Top
right), NVSS (Bottom left) and SDSS (Bottom right).

Figure 7: Corresponding variation with respect to $\Lambda$CDM for BA (red solid line), JBP (green dashed line), and CPL (blue dotted line) as 3 dynamical DE models, as well as $\Lambda$CDM (black solid line) and $w$CDM (orange dot-dashed line) as two models of DE with constant EoS, for DUNE (Top left), Euclid-like (Top right), NVSS (Bottom left) and SDSS (Bottom right).  Survey | $b_{\rm eff}$ | $z_{*}$ | $a_{*}$ | $m$
---|---|---|---|---
DUNE | 1.00 | 0.640 | 1.500 | -
NVSS | 1.98 | 0.790 | 1.180 | -
SDSS | 1.00 | 0.113 | 1.197 | 3.457
Euclid-Like | 1.00 | 0.700 | - | -
Table 3:  Redshift distribution parameters for DUNE, NVSS, SDSS and Euclid-
like surveys.

###  V.2 The ISW auto-power spectrum

The ISW auto-power spectrum $C^{TT}_{\rm ISW}$ is shown in Fig. 5 as a
function of multipole $\ell(=10-100)$ (inverse angular scale). As depicted,
the amplitude of the ISW auto-power spectrum for the JBP and BA models is
higher than the other three models, with JBP having the highest amplitude
compared to the others. As $\ell$ increases (corresponding to smaller scales),
the amplitude of the auto-power spectrum decreases. An intriguing result seen
in this figure is that the CPL auto-correlation spectrum amplitude is lower
than the $w$CDM model at any given $\ell$. At $\ell\sim 25$ the amplitude of
the $\Lambda$CDM auto-correlation spectrum surpasses that of the CPL model,
and at $\ell\sim 90$ the amplitudes for $\Lambda$CDM and CPL models start to
become indistinguishable from each other. By focusing on the comparison
between the CPL and $w$CDM, one can observe that as $\ell$ increases, the
difference in the auto-power spectrum amplitude becomes noticeable, indicating
a footprint of the dynamical nature of DE. Moreover, at $\ell\sim 20-80$ we
can observe a slight curve in the amplitude of the auto-power spectrum,
implying that at these scales, the effects of dynamical DE are most
pronounced.

###  V.3 The ISW-cross power spectrum

In Fig. 6, we have depicted the ISW-cross power spectrum $C^{Tg}_{\rm ISW}$
amplitude as a function of the multipoles $\ell\sim 10-100$. The top left
panel represents the ISW-cross power spectrum for the DUNE survey. We can see
that the amplitude of the ISW-cross power spectrum amplitude for the JBP
models is higher than that of any other models at any multipole $\ell$. The BA
model exhibits the next highest amplitude up until $\ell\sim 90$; it can be
observed that as we move toward higher $\ell$ values, the rate at which the BA
amplitude weakens is the most pronounced compared to other models. Thus, this
effect can be seen in the form of the BA model, with the ISW-cross power
spectrum amplitude approaching that of the $w$CDM model. At $\ell\sim 90$,
this observed property is best demonstrated as they become indistinguishable
at this scale. As depicted in the panel, the CPL model has a higher cross-
power spectrum amplitude than $w$CDM at lower $\ell$ values, but as we pass
$\ell\sim 20-30$, they converge, and then the $w$CDM model rises in amplitude
over the CPL model. It is worth mentioning that the amplitude of $\Lambda$CDM
model is lower than that of any other models studied at any $\ell$.

The top right panel of Fig. 6, illustrates the ISW-cross power spectrum for
Euclid-like survey. The general behavior observed in the DUNE survey holds
true for this survey as well, with some exceptions. The first important point
is the overall suppression of amplitude compared to the DUNE survey. The
second striking feature is the rate at which the BA amplitude is changing with
respect to the multipole $\ell$. This feature is so drastic that the BA
amplitude crosses below that of the $w$CDM model towards lower amplitudes at
$\ell\sim 70$. In this survey, $\Lambda$CDM model also exhibits the weakest
ISW-cross power spectrum amplitude.

The bottom left panel Fig. 6 shows the NVSS results for the ISW-cross-power
spectrum. The change in the amplitude of the ISW-cross power spectrum is an
easily observable feature of this panel compared to two other surveys
discussed previously. In this panel the prominent features are very similar to
those observed in the Euclid-like survey. Here, the crossing point for the CPL
and $w$CDM models, as well as the BA and $w$CDM models, is shifted towards
higher $\ell$ values ($\ell\sim 25$ and $\ell\sim 80$, respectively).
Additionally, the $\Lambda$CDM ISW-cross power spectrum amplitude remains the
weakest, similarly to the other two surveys.

However, the bottom right panel of Fig. 6, representing the SDSS survey, shows
drastically different characteristics compared to the other three surveys. In
this survey, the overall strength of the ISW-cross power spectrum is higher
compared to the other three surveys. The strength of the ISW-cross power
spectrum for the BA and JBP models is nearly indistinguishable up to $\ell\sim
60$. From that point onward, the JBP model exhibits the highest amplitude
compared to other models. Within the compared range of multipoles $\ell$, the
CPL model outperforms the $w$CDM and $\Lambda$CDM in the strength of the ISW-
cross power spectrum, with $\Lambda$CDM, similar to the other surveys, having
the least amplitude among all compared models.
Fig. 7 demonstrates the corresponding variation with respect to $\Lambda$CDM
model the for BA, JBP, and CPL as 3 dynamical DE models, as well as
$\Lambda$CDM and $w$CDM as two models of DE with constant EoS. This sheds
better light on the points discussed earlier.

##  VI Conclusion

In this work, we approached the effect of the nature of DE EoS in two aspects:
first, we studied the impact of the DE nature on structure formation via the
matter power spectrum at two different redshifts, $z=0$ and $z=5$. Second, we
observed the ISW effect on the scale of $\ell=10-100$ by analyzing the ISW
auto-correlation power spectrum $C^{TT}$ and the ISW cross-correlation power
spectrum $C^{Tg}$.

In the first section of this study, with the best-fit values used in this
work, one can observe that regardless of the model, as the redshift increases,
the amplitude of the matter power spectrum is suppressed, confirming that the
strength of structure formation is highest in each model at the present day.
Moreover, from Fig. 2 and 3, one can infer that the strength of the matter
power spectrum for models with a dynamical nature in their EoS is stronger
compared to those with a constant nature. Among those with a dynamical EoS,
the JBP model exhibits the highest amplitude of the matter power spectrum,
while the CPL model has the weakest; to the extent that it is
indistinguishable from $w$CDM model (Bottom panel of Fig. 2).

In the second part of this work, we focused on the ISW effect within the
framework of dynamical DE in comparison with models featuring constant DE
equations of state. First, in Fig. 4 we have shown the time evolution of the
ISW source term $\mathcal{F}(a)$ and its derivative $d\mathcal{F}(a)/da$ as
functions of the scale factor $a$, which reveals a stronger amplitude of
$\mathcal{F}(a)$ for models with constant DE EoS. Here, $\Lambda$CDM exhibits
the highest amplitude among the models with constant DE equations of state
overall, while CPL exhibits the highest amplitude among the dynamical models,
whereas JBP has the lowest. Additionally, as we move towards the scales where
DE becomes the dominant force, one can observe a decrease in the amplitude of
$\mathcal{F}(a)$. This effect can be seen in the right panel of Fig. 4, which
depicts the time evolution of the ISW source term with respect to the scale
factor $a$. Fig. 4 also indicates that in all cases studied, DE becomes the
dominant force in the late-time universe, but the scale at which the effect of
DE on $\mathcal{F}(a)$ becomes noticeable is highly dependent on the
characterization of the DE equation of state. Together, Fig. 4 in combination
with Fig. 2 and 3, can help one observe the dynamical effect of DE on the
essence of structure formation.

It is known that the ISW effect can be represented by the auto-correlation
power spectrum and the cross-correlation power spectrum between the CMB
temperature and large-scale structures such as galaxies. However, the
contributions from auto-correlation are subdominant compared to the primordial
contributions, hence they cannot be detected by observational data. On the
other hand, the cross-correlation is large enough to be detected in various
studies Enander et al. (2015).

In this study, Fig. 5 helps to observe the strength of the ISW auto-power
spectrum, where the JBP and BA models rank the highest (with JBP being the
highest). One interesting outcome was that in this dataset of choice, the
$w$CDM model favors higher amplitude over the CPL parameterization in the
range of $\ell$s under study. In Fig. 5, one can even observe that after
passing $\ell\sim 25-30$, $\Lambda$CDM surpasses the CPL model in amplitude.

At the end, we have analyzed the ISW-cross power spectrum for 4 different
surveys (DUNE, Euclid-like, NVSS and SDSS). The three panels of Fig. 6 that
show DUNE, Euclid-like, and NVSS surveys indicate that the highest values for
the ISW-cross power spectrum belong to the dynamical DE models, specifically
the JBP and BA models. However, one can expect a shift between the dominance
of the CPL and $w$CDM models. Using the SDSS redshift distribution will reveal
a different characteristic. First the amplitude of the ISW-cross power
spectrum is higher compared to other surveys. Second, the dynamical DE models
always dominates over DE model with a constant DE equation of state. Thus, it
is difficult to discriminate between the BA and JBP parameterizations,
especially at lower $\ell$s. It should be noted that in all surveys, the
$\Lambda$CDM model is subdominant in terms of the strength of the ISW-cross
power spectrum. This study has shown that both the matter power spectrum and
the ISW signal provide valuable information for distinguishing the nature of
DE. Fig. 7 illustrates the corresponding variation with respect to
$\Lambda$CDM, confirming the earlier discussion.

It is worth mentioning that in this study, we have focused on the linear
approximation of the matter power spectrum. However, studying the non-linear
matter power spectrum could provide key information regarding the dynamical
nature of dark energy, especially on smaller scales (larger $k$).
Additionally, it would be informative to study the matter power spectrum at
higher redshifts by incorporating datasets that contain information from
redshifts deep into the matter-dominated era. An intriguing idea that could
spark new discussions is to investigate the matter power spectrum in the
future, as at that point, the dynamical nature of dark energy could leave its
footprint more prominently over time.

## ACKNOWLEDGMENTS

MR and MN gratefully thank Saeed Fakhry and Mina Ghodsi for the useful
discussion and constructive comments.
EDV acknowledges support from the Royal Society through a Royal Society
Dorothy Hodgkin Research Fellowship. This publication 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).

## References

  * Riess et al. (1998) A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, et al., The astronomical journal 116, 1009 (1998).
  * Perlmutter et al. (1999) S. Perlmutter, G. Aldering, G. Goldhaber, R. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, et al., The Astrophysical Journal 517, 565 (1999).
  * Di Valentino (2023) E. Di Valentino, in _The Sixteenth Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories: Proceedings of the MG16 Meeting on General Relativity; 5–10 July 2021_ (World Scientific, 2023), pp. 1770–1782.
  * Peebles (1982) P. Peebles, _Large-scale background temperature and mass fluctuations due to scale-invariant primeval perturbations_ (1982).
  * Nolta et al. (2004) M. R. Nolta, E. Wright, L. Page, C. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. Meyer, et al., The Astrophysical Journal 608, 10 (2004).
  * Spergel et al. (2007) D. N. Spergel, R. Bean, O. Doré, M. Nolta, C. Bennett, J. Dunkley, G. Hinshaw, N. e. Jarosik, E. Komatsu, L. Page, et al., The astrophysical journal supplement series 170, 377 (2007).
  * Ade et al. (2016) P. A. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. Banday, R. Barreiro, J. Bartlett, N. Bartolo, et al., Astronomy & Astrophysics 594, A13 (2016).
  * Aghanim et al. (2020a) N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. Banday, R. Barreiro, N. Bartolo, S. Basak, et al., Astronomy & Astrophysics 641, A6 (2020a).
  * Tegmark et al. (2006) M. Tegmark, D. J. Eisenstein, M. A. Strauss, D. H. Weinberg, M. R. Blanton, J. A. Frieman, M. Fukugita, J. E. Gunn, A. J. Hamilton, G. R. Knapp, et al., Physical Review D 74, 123507 (2006).
  * Eisenstein et al. (2005) D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, M. R. Blanton, R. C. Nichol, R. Scranton, H.-J. Seo, M. Tegmark, Z. Zheng, et al., The Astrophysical Journal 633, 560 (2005).
  * Hildebrandt et al. (2017) H. Hildebrandt, M. Viola, C. Heymans, S. Joudaki, K. Kuijken, C. Blake, T. Erben, B. Joachimi, D. Klaes, L. t. Miller, et al., Monthly Notices of the Royal Astronomical Society 465, 1454 (2017).
  * Abbott et al. (2018) T. M. Abbott, F. B. Abdalla, A. Alarcon, J. Aleksić, S. Allam, S. Allen, A. Amara, J. Annis, J. Asorey, S. Avila, et al., Physical Review D 98, 043526 (2018).
  * Di Valentino et al. (2021a) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Classical and Quantum Gravity 38, 153001 (2021a).
  * Krishnan et al. (2021) C. Krishnan, E. Ó. Colgáin, M. Sheikh-Jabbari, and T. Yang, Physical Review D 103, 103509 (2021).
  * Guo et al. (2019) R.-Y. Guo, J.-F. Zhang, and X. Zhang, Journal of Cosmology and Astroparticle Physics 2019, 054 (2019).
  * Abdalla et al. (2022) E. Abdalla, G. F. Abellán, A. Aboubrahim, A. Agnello, Ö. Akarsu, Y. Akrami, G. Alestas, D. Aloni, L. Amendola, L. A. Anchordoqui, et al., Journal of High Energy Astrophysics 34, 49 (2022).
  * Di Valentino et al. (2021b) E. Di Valentino, L. A. Anchordoqui, Ö. Akarsu, Y. Ali-Haimoud, L. Amendola, N. Arendse, M. Asgari, M. Ballardini, S. Basilakos, E. Battistelli, et al., Astroparticle Physics 131, 102605 (2021b).
  * Perivolaropoulos and Skara (2022) L. Perivolaropoulos and F. Skara, New Astronomy Reviews 95, 101659 (2022).
  * Di Valentino et al. (2018) E. Di Valentino, C. Bœhm, E. Hivon, and F. R. Bouchet, Physical Review D 97, 043513 (2018).
  * Kazantzidis and Perivolaropoulos (2021) L. Kazantzidis and L. Perivolaropoulos, in _Modified Gravity and Cosmology: An Update by the CANTATA Network_ (Springer, 2021), pp. 507–537.
  * Di Valentino et al. (2021c) E. Di Valentino, L. A. Anchordoqui, Ö. Akarsu, Y. Ali-Haimoud, L. Amendola, N. Arendse, M. Asgari, M. Ballardini, S. Basilakos, E. Battistelli, et al., Astroparticle Physics 131, 102604 (2021c).
  * Bernui et al. (2023) A. Bernui, E. Di Valentino, W. Giarè, S. Kumar, and R. C. Nunes, arXiv preprint arXiv:2301.06097 (2023).
  * Yang et al. (2023) W. Yang, S. Pan, O. Mena, and E. Di Valentino, Journal of High Energy Astrophysics 40, 19 (2023).
  * Aiola et al. (2020) S. Aiola, E. Calabrese, L. Maurin, S. Naess, B. L. Schmitt, M. H. Abitbol, G. E. Addison, P. A. Ade, D. Alonso, M. Amiri, et al., Journal of Cosmology and Astroparticle Physics 2020, 047 (2020).
  * Dutcher et al. (2021) D. Dutcher, L. Balkenhol, P. Ade, Z. Ahmed, E. Anderes, A. Anderson, M. Archipley, J. Avva, K. Aylor, P. Barry, et al., Physical Review D 104, 022003 (2021).
  * Verde et al. (2019) L. Verde, T. Treu, and A. G. Riess, Nature Astronomy 3, 891 (2019).
  * McDonough et al. (2022) E. McDonough, M.-X. Lin, J. C. Hill, W. Hu, and S. Zhou, Physical Review D 106, 043525 (2022).
  * Madhavacheril et al. (2023) M. S. Madhavacheril, F. J. Qu, B. D. Sherwin, N. MacCrann, Y. Li, I. Abril-Cabezas, P. A. Ade, S. Aiola, T. Alford, M. Amiri, et al., arXiv preprint arXiv:2304.05203 (2023).
  * Balkenhol et al. (2023) L. Balkenhol, D. Dutcher, A. S. Mancini, A. Doussot, K. Benabed, S. Galli, P. Ade, A. Anderson, B. Ansarinejad, M. Archipley, et al., Physical Review D 108, 023510 (2023).
  * Huang et al. (2020) C. D. Huang, A. G. Riess, W. Yuan, L. M. Macri, N. L. Zakamska, S. Casertano, P. A. Whitelock, S. L. Hoffmann, A. V. Filippenko, and D. Scolnic, The Astrophysical Journal 889, 5 (2020).
  * Riess et al. (2022a) A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, G. S. Anand, L. Breuval, et al., The Astrophysical journal letters 934, L7 (2022a).
  * Riess et al. (2022b) A. G. Riess, L. Breuval, W. Yuan, S. Casertano, L. M. Macri, J. B. Bowers, D. Scolnic, T. Cantat-Gaudin, R. I. Anderson, and M. C. Reyes, The Astrophysical Journal 938, 36 (2022b).
  * Murakami et al. (2023) Y. S. Murakami, A. G. Riess, B. E. Stahl, W. Kenworthy, D.-M. A. Pluck, A. Macoretta, D. Brout, D. O. Jones, D. M. Scolnic, and A. V. Filippenko, arXiv preprint arXiv:2306.00070 (2023).
  * Asgari et al. (2021) M. Asgari, C.-A. Lin, B. Joachimi, B. Giblin, C. Heymans, H. Hildebrandt, A. Kannawadi, B. Stölzner, T. Tröster, J. L. van den Busch, et al., Astronomy & Astrophysics 645, A104 (2021).
  * Li et al. (2016) Z. Li, Y. Jing, P. Zhang, and D. Cheng, The Astrophysical Journal 833, 287 (2016).
  * Gil-Marín et al. (2016) H. Gil-Marín, W. J. Percival, L. Verde, J. R. Brownstein, C.-H. Chuang, F.-S. Kitaura, S. A. Rodríguez-Torres, and M. D. Olmstead, Monthly Notices of the Royal Astronomical Society p. stw2679 (2016).
  * Hu and Wang (2023) J.-P. Hu and F.-Y. Wang, Universe 9, 94 (2023).
  * Rassat et al. (2014) A. Rassat, J.-L. Starck, P. Paykari, F. Sureau, and J. Bobin, Journal of Cosmology and Astroparticle Physics 2014, 006 (2014).
  * Akrami et al. (2020) Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. Barreiro, N. Bartolo, S. Basak, K. Benabed, et al., Astronomy & Astrophysics 641, A7 (2020).
  * Schwarz et al. (2016) D. J. Schwarz, C. J. Copi, D. Huterer, and G. D. Starkman, Classical and Quantum Gravity 33, 184001 (2016).
  * Perivolaropoulos (2014) L. Perivolaropoulos, Galaxies 2, 22 (2014).
  * Cayuso and Johnson (2020) J. I. Cayuso and M. C. Johnson, Physical Review D 101, 123508 (2020).
  * Vielva et al. (2004) P. Vielva, E. Martinez-Gonzalez, R. Barreiro, J. Sanz, and L. Cayón, The Astrophysical Journal 609, 22 (2004).
  * Cruz et al. (2007) M. Cruz, L. Cayon, E. Martinez-Gonzalez, P. Vielva, and J. Jin, The Astrophysical Journal 655, 11 (2007).
  * Del Popolo et al. (2019) A. Del Popolo, F. Pace, and D. F. Mota, Physical Review D 100, 024013 (2019).
  * Naseri and Firouzjaee (2020) M. Naseri and J. T. Firouzjaee, Physical Review D 102, 123503 (2020).
  * Blanchard and Ilić (2021) A. Blanchard and S. Ilić, Astronomy & Astrophysics 656, A75 (2021).
  * Naseri and Firouzjaee (2021) M. Naseri and J. T. Firouzjaee, Physics of the Dark Universe 34, 100888 (2021).
  * Natoli et al. (2018) P. Natoli, M. Ashdown, R. Banerji, J. Borrill, A. Buzzelli, G. De Gasperis, J. Delabrouille, E. Hivon, D. Molinari, G. Patanchon, et al., Journal of Cosmology and Astroparticle Physics 2018, 022 (2018).
  * Zhao et al. (2017) G.-B. Zhao, M. Raveri, L. Pogosian, Y. Wang, R. G. Crittenden, W. J. Handley, W. J. Percival, F. Beutler, J. Brinkmann, C.-H. Chuang, et al., Nature Astronomy 1, 627 (2017).
  * Pan et al. (2019a) S. Pan, W. Yang, E. Di Valentino, E. N. Saridakis, and S. Chakraborty, Physical Review D 100, 103520 (2019a).
  * Yang et al. (2019) W. Yang, S. Pan, E. Di Valentino, E. N. Saridakis, and S. Chakraborty, Physical Review D 99, 043543 (2019).
  * Yang et al. (2021) W. Yang, E. Di Valentino, S. Pan, Y. Wu, and J. Lu, Monthly Notices of the Royal Astronomical Society 501, 5845 (2021).
  * Chevallier and Polarski (2001) M. Chevallier and D. Polarski, International Journal of Modern Physics D 10, 213 (2001).
  * Linder (2003) E. V. Linder, Physical review letters 90, 091301 (2003).
  * Efstathiou (1999) G. Efstathiou, Monthly Notices of the Royal Astronomical Society 310, 842 (1999).
  * Jassal et al. (2005) H. K. Jassal, J. Bagla, and T. Padmanabhan, Physical Review D 72, 103503 (2005).
  * Barboza Jr and Alcaniz (2008) E. Barboza Jr and J. Alcaniz, Physics Letters B 666, 415 (2008).
  * Gómez-Valent et al. (2022) A. Gómez-Valent, Z. Zheng, L. Amendola, C. Wetterich, and V. Pettorino, Physical Review D 106, 103522 (2022).
  * Escudero et al. (2022) H. G. Escudero, J.-L. Kuo, R. E. Keeley, and K. N. Abazajian, arXiv preprint arXiv:2208.14435 (2022).
  * Karwal et al. (2022) T. Karwal, M. Raveri, B. Jain, J. Khoury, and M. Trodden, Physical Review D 105, 063535 (2022).
  * Herold et al. (2022) L. Herold, E. G. Ferreira, and E. Komatsu, The Astrophysical Journal Letters 929, L16 (2022).
  * Jiang and Piao (2022) J.-Q. Jiang and Y.-S. Piao, Physical Review D 105, 103514 (2022).
  * Smith et al. (2022) T. L. Smith, M. Lucca, V. Poulin, G. F. Abellan, L. Balkenhol, K. Benabed, S. Galli, and R. Murgia, Physical Review D 106, 043526 (2022).
  * Reeves et al. (2023) A. Reeves, L. Herold, S. Vagnozzi, B. D. Sherwin, and E. G. Ferreira, Monthly Notices of the Royal Astronomical Society 520, 3688 (2023).
  * Efstathiou et al. (2023) G. Efstathiou, E. Rosenberg, and V. Poulin (2023), eprint 2311.00524.
  * Kamionkowski and Riess (2023) M. Kamionkowski and A. G. Riess, Ann. Rev. Nucl. Part. Sci. 73, 153 (2023), eprint 2211.04492.
  * Poulin et al. (2023) V. Poulin, T. L. Smith, and T. Karwal, Phys. Dark Univ. 42, 101348 (2023), eprint 2302.09032.
  * Niedermann and Sloth (2023) F. Niedermann and M. S. Sloth (2023), eprint 2307.03481.
  * Eskilt et al. (2023) J. R. Eskilt, L. Herold, E. Komatsu, K. Murai, T. Namikawa, and F. Naokawa, Phys. Rev. Lett. 131, 121001 (2023), eprint 2303.15369.
  * Gsponer et al. (2023) R. Gsponer, R. Zhao, J. Donald-McCann, D. Bacon, K. Koyama, R. Crittenden, T. Simon, and E.-M. Mueller (2023), eprint 2312.01977.
  * Goldstein et al. (2023) S. Goldstein, J. C. Hill, V. Iršič, and B. D. Sherwin, Phys. Rev. Lett. 131, 201001 (2023), eprint 2303.00746.
  * Banihashemi et al. (2019) A. Banihashemi, N. Khosravi, and A. H. Shirazi, Physical Review D 99, 083509 (2019).
  * Banihashemi and Khosravi (2022) A. Banihashemi and N. Khosravi, The Astrophysical Journal 931, 148 (2022).
  * Farrar and Peebles (2004) G. R. Farrar and P. J. E. Peebles, The Astrophysical Journal 604, 1 (2004).
  * Valiviita et al. (2008) J. Valiviita, E. Majerotto, and R. Maartens, JCAP 07, 020 (2008), eprint 0804.0232.
  * Gavela et al. (2009) M. B. Gavela, D. Hernandez, L. Lopez Honorez, O. Mena, and S. Rigolin, JCAP 07, 034 (2009), [Erratum: JCAP 05, E01 (2010)], eprint 0901.1611.
  * Salvatelli et al. (2014) V. Salvatelli, N. Said, M. Bruni, A. Melchiorri, and D. Wands, Phys. Rev. Lett. 113, 181301 (2014), eprint 1406.7297.
  * Di Valentino et al. (2017) E. Di Valentino, A. Melchiorri, and O. Mena, Phys. Rev. D 96, 043503 (2017), eprint 1704.08342.
  * Kumar and Nunes (2017) S. Kumar and R. C. Nunes, Phys. Rev. D 96, 103511 (2017), eprint 1702.02143.
  * Martinelli et al. (2019) M. Martinelli, N. B. Hogg, S. Peirone, M. Bruni, and D. Wands, Mon. Not. Roy. Astron. Soc. 488, 3423 (2019), eprint 1902.10694.
  * Yang et al. (2020a) W. Yang, S. Pan, R. C. Nunes, and D. F. Mota, JCAP 04, 008 (2020a), eprint 1910.08821.
  * Di Valentino et al. (2020) E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Dark Univ. 30, 100666 (2020), eprint 1908.04281.
  * Pan et al. (2019b) S. Pan, W. Yang, C. Singha, and E. N. Saridakis, Phys. Rev. D 100, 083539 (2019b), eprint 1903.10969.
  * Kumar et al. (2019) S. Kumar, R. C. Nunes, and S. K. Yadav, Eur. Phys. J. C 79, 576 (2019), eprint 1903.04865.
  * Kumar and Nunes (2016) S. Kumar and R. C. Nunes, Phys. Rev. D 94, 123511 (2016), eprint 1608.02454.
  * Murgia et al. (2016) R. Murgia, S. Gariazzo, and N. Fornengo, JCAP 04, 014 (2016), eprint 1602.01765.
  * Pourtsidou and Tram (2016) A. Pourtsidou and T. Tram, Phys. Rev. D 94, 043518 (2016), eprint 1604.04222.
  * Lucca and Hooper (2020) M. Lucca and D. C. Hooper, Phys. Rev. D 102, 123502 (2020), eprint 2002.06127.
  * Gómez-Valent et al. (2020) A. Gómez-Valent, V. Pettorino, and L. Amendola, Phys. Rev. D 101, 123513 (2020), eprint 2004.00610.
  * Yang et al. (2020b) W. Yang, E. Di Valentino, O. Mena, S. Pan, and R. C. Nunes, Phys. Rev. D 101, 083509 (2020b), eprint 2001.10852.
  * Nunes and Di Valentino (2021) R. C. Nunes and E. Di Valentino, Phys. Rev. D 104, 063529 (2021), eprint 2107.09151.
  * Nunes et al. (2022) R. C. Nunes, S. Vagnozzi, S. Kumar, E. Di Valentino, and O. Mena, Phys. Rev. D 105, 123506 (2022), eprint 2203.08093.
  * Zhai et al. (2023) Y. Zhai, W. Giarè, C. van de Bruck, E. Di Valentino, O. Mena, and R. C. Nunes, JCAP 07, 032 (2023), eprint 2303.08201.
  * Escamilla et al. (2023) L. A. Escamilla, O. Akarsu, E. Di Valentino, and J. A. Vazquez, JCAP 11, 051 (2023), eprint 2305.16290.
  * Kovács et al. (2019) A. Kovács, C. Sánchez, J. García-Bellido, J. Elvin-Poole, N. Hamaus, V. Miranda, S. Nadathur, T. Abbott, F. B. Abdalla, J. Annis, et al., Monthly Notices of the Royal Astronomical Society 484, 5267 (2019).
  * Yengejeh et al. (2023) M. G. Yengejeh, S. Fakhry, J. T. Firouzjaee, and H. Fathi, Physics of the Dark Universe 39, 101144 (2023).
  * Krolewski and Ferraro (2022) A. Krolewski and S. Ferraro, Journal of Cosmology and Astroparticle Physics 2022, 033 (2022).
  * Ghodsi et al. (2022) M. Ghodsi, A. Behnamfard, S. Fakhry, and J. T. Firouzjaee, Physics of the Dark Universe 35, 100918 (2022).
  * Kable et al. (2022) J. A. Kable, G. Benevento, N. Frusciante, A. De Felice, and S. Tsujikawa, Journal of Cosmology and Astroparticle Physics 2022, 002 (2022).
  * Seraille et al. (2024) E. Seraille, J. Noller, and B. D. Sherwin (2024), eprint 2401.06221.
  * Schäfer (2008) B. M. Schäfer, Monthly Notices of the Royal Astronomical Society 388, 1403 (2008).
  * Ma and Bertschinger (1995) C.-P. Ma and E. Bertschinger, arXiv preprint astro-ph/9506072 (1995).
  * Lewis et al. (2000) A. Lewis, A. Challinor, and A. Lasenby, The Astrophysical Journal 538, 473 (2000).
  * Howlett et al. (2012) C. Howlett, A. Lewis, A. Hall, and A. Challinor, Journal of Cosmology and Astroparticle Physics 2012, 027 (2012).
  * Powell et al. (2009) M. J. Powell et al., Cambridge NA Report NA2009/06, University of Cambridge, Cambridge 26 (2009).
  * Cartis et al. (2018) C. Cartis, L. Roberts, and O. Sheridan-Methven, arXiv preprint arXiv:1812.11343 (2018).
  * Cartis et al. (2019) C. Cartis, J. Fiala, B. Marteau, and L. Roberts, ACM Transactions on Mathematical Software (TOMS) 45, 1 (2019).
  * Torrado and Lewis (2021) J. Torrado and A. Lewis, Journal of Cosmology and Astroparticle Physics 2021, 057 (2021).
  * Aghanim et al. (2020b) N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. Barreiro, N. Bartolo, S. Basak, et al., Astronomy & Astrophysics 641, A5 (2020b).
  * Aghanim et al. (2020c) N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. Barreiro, N. Bartolo, S. Basak, et al., Astronomy & Astrophysics 641, A8 (2020c).
  * Dodelson and Schmidt (2020) S. Dodelson and F. Schmidt, _Modern cosmology_ (Academic press, 2020).
  * Orjuela-Quintana et al. (2023) J. B. Orjuela-Quintana, S. Nesseris, and D. Sapone, arXiv preprint arXiv:2307.03643 (2023).
  * Linder and Jenkins (2003) E. V. Linder and A. Jenkins, Monthly Notices of the Royal Astronomical Society 346, 573 (2003).
  * Peter and Uzan (2009) P. Peter and J.-P. Uzan, _Primordial cosmology_ (Oxford University Press, 2009).
  * Afshordi et al. (2004) N. Afshordi, Y.-S. Loh, and M. A. Strauss, Physical review D 69, 083524 (2004).
  * Stölzner et al. (2018) B. Stölzner, A. Cuoco, J. Lesgourgues, and M. Bilicki, Physical Review D 97, 063506 (2018).
  * Sánchez et al. (2020) C. Sánchez, M. Raveri, A. Alarcon, and G. M. Bernstein, Monthly Notices of the Royal Astronomical Society 498, 2984 (2020).
  * Weaverdyck et al. (2018) N. Weaverdyck, J. Muir, and D. Huterer, Physical Review D 97, 043515 (2018).
  * Martinet et al. (2015) N. Martinet, J. G. Bartlett, A. Kiessling, and B. Sartoris, Astronomy & Astrophysics 581, A101 (2015).
  * Enander et al. (2015) J. Enander, Y. Akrami, E. Mörtsell, M. Renneby, and A. R. Solomon, Physical Review D 91, 084046 (2015).