# Hint of a long-range dark sector fifth force in the current cosmological
data?

Vidhya Ganesan<EMAIL_ADDRESS>Indian Institute of Astrophysics,
Bengaluru, Karnataka 560034, India  Amlan Chakraborty
<EMAIL_ADDRESS>Indian Institute of Astrophysics, Bengaluru,
Karnataka 560034, India  Tulip Ray<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
Subinoy Das<EMAIL_ADDRESS>Indian Institute of Astrophysics, Bengaluru,
Karnataka 560034, India  Arka Banerjee<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Science Education and Research, Homi Bhabha Road,
Pashan, Pune 411008, India

###### Abstract

Many previous studies have explored models of dynamically interacting dark
energy scenarios instead of a cosmological constant and cold dark matter. This
work aims to offer a comprehensive investigation of the Yukawa-type
interaction between dark energy and dark matter where dark matter mass is also
a function of a dynamical scalar field value. Unlike previous work, instead of
taking an approximate solution, we numerically solve the Klein-Gordon equation
that governs the intricate interplay between these two components of the dark
sector along with the corresponding perturbations, using a suitable shooting
algorithm to determine precise initial conditions. We have conducted a
thorough analysis of this interaction using the Markov Chain Monte Carlo
method, incorporating diverse datasets such as Planck, BAO, Pantheon+, and
SH0ES. Our results, for the first time, reveal a discernible detection of the
coupling constant, which encapsulates the strength of the dark sector
interaction. We find non-zero $\beta$ values of 0.0487, 0.0680, 0.0712, and
0.0822 ($68\%$ Confidence Limit) corresponding to the inclusion of each
dataset, respectively. Notably, the addition of SH0ES data led to a
significant increase in the prominence of $\beta$, suggesting the potential
for new physics in the dark sector. This study serves as a strategic road map
for future endeavours with non-linear cosmology, which will contribute to an
enhanced understanding and refinement of constraints on the dark sector
interaction.

††preprint: APS/123-QED

##  I Introduction

The standard six-parameter phenomenological $\Lambda$CDM model has been
successful in explaining the evolution of the Universe, both for the smooth
background, as well as for fluctuations at linear scales [1, 2, 3, 4, 5].
However, recent precision observations have revealed several anomalies, such
as the “Hubble tension” and the“$S_{8}$ tension” [6, 7]. On the theoretical
front, the smallness of cosmological constant ($\Lambda$) has remained a
challenge for fundamental particle physics [8]. While current observations are
mostly consistent with dark energy being a cosmological constant, cosmological
measurements have thus far been unable to offer any insights about its origin
or smallness.

The existence of dynamic dark energy has been explored as an alternative
possibility where the equation of state of dark energy is time dependant [9,
10], which results in a unique imprint on the growth of structures in the
universe. Such models can be probed through their imprints in existing and
upcoming cosmological observations. In this regard, an even more interesting
possibility could be interactions in the dark sector, for instance, between
dark energy and dark matter [11, 12], which could potentially leave signatures
on various cosmological observables. In fact, in the absence of any symmetry
forbidding such an interaction, the scalar field dark energy interacting with
dark matter fermion can occur in nature [11]. Consequently, several scalar-
tensor theories have been proposed in the literature to model interacting dark
energy and dark matter [13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43].

The Chameleon dark energy model [25], is one such prominent model that
exhibits an interesting feature of dark energy equation of state parameter
$w<-1$ without violating the weak energy principle or cosmic catastrophe [44,
45, 46]. This model is predominantly parameterised by two parameters, namely
the slope of the potential $\alpha$ and the interaction strength $\beta$
between dark matter and dark energy, which have their origin in string theory
of compactification [47]. This paper presents a detailed investigation of the
Chameleon dark energy model. Unlike the previous work in Boriero et. al.
[48]111In Boriero et. al. [48], an approximate analytical solution of the
modified Klein-Gordon equation (refer Eq. 1) was used for the analysis while
in the current work we use the exact solution obtained by numerically solving
it using the shooting method., here we have numerically solved the Klein-
Gordon equation (modified form for Chameleon model shown in Eq. 1) by
deploying a shooting algorithm to evaluate the initial condition of the scalar
field. The analysis demonstrates that this model is consistent with
cosmological data. Moreover, a peak feature is observed in $\beta$ for the
first time, indicating a mild detection of the fifth force. It is interesting
to note that the previous studies have only put an upper limit on the $\beta$
value. Perhaps the exact numerical solution adopted in this work and the use
of recent data sets for the analysis could have led to this slight detection
of fifth force in the dark sector. If this is true, the inclusion of non-
linear scales in our future analysis will open up new avenues to probe fifth
force. Specifically, its imprint on the highly precise measurements from
various stage IV cosmological surveys in the next decade will provide much
more insight into either ruling out or detecting such a fifth force with more
significance. We can also expect a unique signature of non-zero $\beta$ value
on the difference in infall between the dark matter and baryonic matter, into
the virialised halos [49, 50, 51]. These effects will be accurately studied
with N-body simulations in our future work.

The plan of the paper is as follows. In section II, we briefly describe the
chameleon model and its effect on background cosmology as well as
perturbations. In section III, we discuss the possible impact of dark sector
interaction on various cosmological probes. In section IV, we elaborate on the
numerical implementation of this model and present the result from our MCMC
analysis with different datasets. We conclude in section V.

##  II  The Chameleon Model

This work endeavours to thoroughly study the Yukawa-type interaction between
dark energy and dark matter. We adopt a popular Chameleon-type model, which
represents dark energy as a scalar field whose mass depends on local dark
matter density [52, 26, 53, 54, 55, 56]. In our case, to circumvent the
stringent constraints imposed by the solar system tests of gravity on
Chameleon models [57, 58], we assume that the dark energy only interacts with
dark matter and does not have any interaction with the visible sector [46,
59].

####  II.0.1 Background evolution

The dark energy is modeled as a scalar field, $\phi$, coupled to dark matter
as $f(\phi)\overline{\psi}\psi$, where $f(\phi)$ is an arbitrary function of
$\phi$ and $\psi$ is the dark matter Dirac spinor. The Klein-Gordan equation
governing the dynamics of this scalar field (obtained by minimizing the action
describing the Chameleon scalar field [53]) is given by

$\ddot{\phi}+3H\dot{\phi}=-V_{,\phi}(\phi)-\frac{\rho_{DM}^{(0)}}{a^{3}}\
\frac{f_{,\phi}(\phi)}{f(\phi_{0})}.$ (1)

The “dot” in the equation represents differentiation with respect to cosmic
time, while $\phi_{0}$ and $\rho_{DM}^{(0)}$ denote the present-day scalar
field and dark matter density, respectively. The scalar field experiences an
effective potential

$V_{\text{eff}}(\phi)=V(\phi)+\frac{\rho_{DM}^{(0)}}{a^{3}}e^{\beta(\phi-\phi_{0})/M_{Pl}},$
(2)

where the coupling function is taken as $f(\phi)=e^{\beta\phi/M_{Pl}}$, where
$M_{Pl}\equiv(8\pi G)^{-1/2}$ is the reduced Planck mass. Such exponential
form emerges from string theory models of compactification [47].

We consider the self-interaction potential to be a simple power law runaway
type,

$V(\phi)=M_{\phi}^{4}\left(\frac{M_{Pl}}{\phi}\right)^{\alpha}.$ (3)

The slope of the potential is described by a positive constant, denoted by
$\alpha$.
$M_{\phi}\approx\left[\rho_{\phi}^{(0)}\left(\frac{\phi_{0}}{M_{Pl}}\right)^{\alpha}\right]^{1/4}$
is the scalar field mass.

In the limit of vanishing coupling parameter $\beta$ with constant potential
$V(\phi)$ and constant scalar field, the model reduces to the standard
$\Lambda$CDM model.

Furthermore, the dark matter and dark energy density for this model, are given
by

$\rho_{DM}=\frac{\rho_{DM}^{(0)}}{a^{3}}e^{\beta(\phi-\phi_{0})/M_{Pl}},\;\
\rho_{DE}=\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi).$ (4)

An interesting aspect of this model, as shown in the work [46], is that an
observer unaware of the existence of dark sector interaction would end up
estimating an equation of state (EoS) different from the actual value,
$w_{\phi}=P_{\phi}/\rho_{\phi}$. The observed EoS will become
$w_{obs}=\dfrac{w_{\phi}}{1-x},\ \ \text{where}\ \
x=-\dfrac{\rho^{(0)}_{DM}}{a^{3}\rho_{\phi}}\left[e^{\beta(\phi-\phi_{0})/M_{Pl}}-1\right]$.
Specifically, it was shown that even though the actual Chameleon dark energy
does not have any phantom behaviour, the observer ends up noting an apparent
phantom behaviour. This may explain the slight phantom EoS observed in several
independent probes such as [60, 61, 62], indirectly indicating the presence of
dark sector interaction.

####  II.0.2 Evolution of inhomogeneities

The coupling in the dark sector leads to modification in the evolution of
inhomogeneities in the Universe.

In the synchronous gauge, the equations of motion for the dark matter density
contrast $\delta_{DM}(k,t)$, velocity divergence $\theta_{DM}(k,t)$ and scalar
field perturbations [63] for a Fourier mode $k$ are given by

$\displaystyle\dot{\delta}_{DM}=-\left(\dfrac{\theta_{DM}}{a}+\dfrac{\dot{\tilde{h}}}{2}\right)+\dfrac{\beta}{M_{Pl}}\delta\dot{\phi},$
$\displaystyle\dot{\theta}_{DM}=-H\theta_{DM}+\dfrac{\beta}{M_{Pl}}\left(\dfrac{k^{2}}{a}\delta\phi-\dot{\phi}\theta_{DM}\right),$
$\displaystyle\ddot{\delta\phi}+3H\dot{\delta\phi}+\left(\frac{k^{2}}{a^{2}}+V_{,\phi\phi}\right)\delta\phi+\frac{1}{2}\dot{\tilde{h}}\dot{\phi}=-\dfrac{\beta}{M_{Pl}}\rho_{DM}\delta_{DM}$
(5)

where $\tilde{h}\equiv\tilde{h}_{i}^{i}$ is the trace of metric perturbation
$\tilde{h}_{ij}$, obtained from perturbed Einstein equation.

##  III Impact of Dark Sector Interaction on the Cosmological Probes

In the standard cosmological framework, the dark sector of the Universe can
only be detected and studied through their gravitational effects. Any
detection of $w_{\rm DE}\neq-1$ can be interpreted as a departure from the
standard cosmological constant, without any implications for dark matter. This
comes under $w$CDM paradigm, where the energy densities of dark energy and
dark matter are treated as background quantities with separate (uncoupled)
evolution equations.

When we consider models with dark sector interaction, it leads to degeneracy
between the features induced due to the presence of dark sector interaction
and those from the standard cosmological parameters. Specifically, the dark
sector coupling uniquely modifies the evolution of matter/radiation
perturbations and hence also the clustering properties of galaxies. In order
to test a dark sector interaction model, one needs to identify these unique
features on the evolving density perturbations in the linear and non-linear
phases using the latest data. Here in this subsection, we note some of the
consequences of Chameleon dark sector interaction on the cosmological probes.

The ratio of dark matter density in the Chameleon model to the corresponding
dark matter density in $\Lambda$CDM model has to be small otherwise it leads
to conflicts with the observational estimates of matter density at various
redshifts such as galaxy counts, weak lensing etc [64]. Hence in order for the
Chameleon dark matter to be consistent with these observations it needs to
behave as a normal dark matter for most of the cosmological history except in
the later times. This requires that the self interacting scalar field
potential in Eq. 3 to be sufficiently flat i.e. $\alpha\lesssim 0.2$ [46]. In
our current work, we set the value of $\alpha$ as $0.01$, satisfying this
condition as well as increasing the computational efficiency for the analysis
in the subsequent section.

The evolution equation for dark matter inhomogeneities in synchronous gauge
with slow roll approximation is given by [65]

$\ddot{\delta}_{DM}+2H\dot{\delta}_{DM}=\dfrac{3}{2}H^{2}\left(1+\dfrac{2\beta^{2}}{1+a^{2}V_{,\phi\phi}/k^{2}}\right)\delta_{DM}.$
(6)

This equation differs from the corresponding expression for $\Lambda$CDM by
the factor inside the brackets. The extra term proportional to $\beta^{2}$
arises due to the fifth force mediated by the scalar field. The comoving
finite range of this force, for the potential form in Eq. 3, is given by

$\lambda_{c}=a^{-1}V_{,\phi\phi}^{-1/2}\approx
a^{2}H_{0}^{-1}\sqrt{\dfrac{\Omega_{\phi}}{\Omega^{2}_{DM}}\dfrac{\alpha}{3\beta^{2}}}e^{-\beta(\phi-\phi_{0})/M_{Pl}},$
(7)

where $\alpha\ll 1$. The perturbations whose physical scale are much larger
than $\lambda_{c}$ i.e. $a/k\gg\lambda_{c}$ evolves similar to that in the
$\Lambda$CDM model. However, perturbations with smaller comoving scales
evolves such that the Newton’s constant is modified by a factor of
$(1+2\beta^{2})$. Thus the existence of dark sector interactions results in an
extra attractive force between the dark matter particles which leads to
enhancement of power on small scales and also enhances the growth of
perturbations. Hence these small scale perturbations becomes non-linear
earlier than those in the $\Lambda$CDM model.

Furthermore, it can be noted from the Eq. 7 that the fifth force range,
$\lambda_{c}$, today is comparable to the Hubble length today. While at
earlier times the Chameleon interaction range was suppressed due to the scale
factor. For example at the time of CMB decoupling, the Chameleon force range
was much smaller then the Hubble horizon at the time. Hence we can conclude
that the presence of Chameleon dark sector interaction will not effect the CMB
primary anisotropy.

##  IV Detecting and Constraining The Dark Sector Interaction Strength

Figure 1: The top panels show the power spectrum for the CMB temperature
(left) and E mode (middle), and also the matter power spectrum (right), for
$\Lambda$CDM model (red) and Chameleon model (blue) with
$(\alpha,\beta)=(0.01,0.1)$. The bottom panels shows the fractional difference
in these power spectra for the Chameleon model with respect to the
$\Lambda$CDM model. The Planck 2018 power spectrum error bars corresponding to
the CMB temperature and E mode are also shown in the respective panels.

###  IV.1 Numerical method: Solving the Background and Perturbation equations
using the Shooting Algorithm

We have modified the publicly available CLASS [66, 67] to solve the background
and perturbation equations of the Chameleon model (Eqs. 1, 5 together with the
Friedmann and perturbed Einstein equation). In order to get the correct dark
energy density today, $\rho^{(0)}_{DE}$, we use the shooting feature inbuilt
in CLASS to set the initial scalar field value, $\phi_{i}$, such that the
current dark energy density value matches its observational value today. This
is similar to what we do in the case of standard $\Lambda$CDM model, where we
set the value of cosmological constant, $\Lambda$, such that its value matches
the local dark energy density. Specifically, we first choose a $\phi_{i}$
value, evolve the background equations and then we estimate the current dark
energy density. We keep repeating this process by varying the $\phi_{i}$ value
until we find a value which finally gives the correct local dark energy
density.

The CMB power spectrum of temperature and E mode, and the matter power
spectrum obtained are shown in the top panels of the Fig. 1 respectively, for
the Chameleon model (blue) and $\Lambda$CDM model (red). The fractional
difference of the Chameleon model with respect to $\Lambda$CDM model
corresponding to each of these power spectrum are shown in the bottom panels.
The Planck 2018 power spectrum error-bars [68] for the CMB temperature and E
mode are also shown in the respective panels.

###  IV.2 Details of the MCMC Analysis

Dataset | Planck | Planck + BAO
---|---|---
Model |  $\Lambda$CDM | Chameleon |  $\Lambda$CDM | Chameleon
$100\omega_{b}$ | $2.2352(2.2489)^{+0.01525}_{-0.01552}$ | $2.2430(2.2500)^{+0.01655}_{-0.01736}$ | $2.2415(2.2442)^{+0.01399}_{-0.01371}$ | $2.2491(2.2570)^{+0.01505}_{-0.01590}$
$\omega_{dm}$ | $0.1202(0.1194)^{+0.00141}_{-0.00142}$ | $0.1230(0.1224)^{+0.00133}_{-0.00131}$ | $0.1193(0.1195)^{+0.00102}_{-0.00103}$ | $0.1222(0.1214)^{+0.00098}_{-0.00098}$
$100\theta_{s}$ | $1.0419(1.0419)^{+0.00030}_{-0.00030}$ | $1.0419(1.0418)^{+0.00030}_{-0.00029}$ | $1.0420(1.0421)^{+0.00028}_{-0.00028}$ | $1.0419(1.0419)^{+0.00029}_{-0.00028}$
$log(10^{10}A_{s})$ | $3.0455(3.0378)^{+0.01568}_{-0.01676}$ | $3.0429(3.0448)^{+0.01556}_{-0.01632}$ | $3.0462(3.0415)^{+0.01559}_{-0.01731}$ | $3.0431(3.0513)^{+0.01607}_{-0.01653}$
$n_{s}$ | $0.9647(0.9663)^{+0.00450}_{-0.00458}$ | $0.9649(0.9637)^{+0.00447}_{-0.00456}$ | $0.9669(0.9644)^{+0.00389}_{-0.00388}$ | $0.9669(0.9687)^{+0.00388}_{-0.00395}$
$\tau_{reio}$ | $0.0545(0.0521)^{+0.00763}_{-0.00820}$ | $0.0533(0.0548)^{+0.00758}_{-0.00796}$ | $0.0557(0.0524)^{+0.00740}_{-0.00829}$ | $0.0543(0.0595)^{+0.00760}_{-0.00799}$
$\beta$ | $-$ | $0.0512(0.0487)^{+0.03253}_{-0.02785}$ | $-$ | $0.0537(0.0680)^{+0.03372}_{-0.02712}$
$H_{0}$ | $67.3000(67.6870)^{+0.62469}_{-0.63540}$ | $67.6770(67.9310)^{+0.62567}_{-0.65097}$ | $67.6990(67.6710)^{+0.44644}_{-0.47054}$ | $68.0350(68.4270)^{+0.47362}_{-0.48113}$
$\sigma_{8}$ | $0.8114(0.8059)^{+0.00771}_{-0.00763}$ | $0.8233(0.8178)^{+0.00978}_{-0.01368}$ | $0.8092(0.8074)^{+0.00700}_{-0.00764}$ | $0.8222(0.8261)^{+0.01024}_{-0.01402}$
$\chi^{2}_{min}$ | $2771.7913$ | $2771.1117$ | $2777.0758$ | $2776.0405$
$\Delta\chi^{2}_{min}$ | $0$ | $-0.6796$ | $0$ | $-1.0353$
Table 1: The mean (bestfit) $\pm 1\sigma$ error for the cosmological parameters constrained with the datasets [Planck] and [Planck + BAO] for the $\Lambda$CDM model and Chameleon model are shown in the above table. The corresponding $\chi_{min}^{2}$ and $\Delta\chi_{min}^{2}$ values are also shown. Dataset | Planck + BAO + Pantheon+ | Planck + BAO + Pantheon+ + SH0ES
---|---|---
Model |  $\Lambda$CDM | Chameleon |  $\Lambda$CDM | Chameleon
$100\omega_{b}$ | $2.2385(2.2421)^{+0.01382}_{-0.01337}$ | $2.2452(2.2501)^{+0.01409}_{-0.01605}$ | $2.2606(2.2475)^{+0.01351}_{-0.01323}$ | $2.2699(2.2845)^{+0.01483}_{-0.01579}$
$\omega_{dm}$ | $0.1197(0.1188)^{+0.00095}_{-0.00097}$ | $0.1227(0.1231)^{+0.00091}_{-0.00094}$ | $0.1174(0.1171)^{+0.00088}_{-0.00086}$ | $0.1206(0.1202)^{+0.00083}_{-0.00086}$
$100\theta_{s}$ | $1.0419(1.0418)^{+0.00030}_{-0.00026}$ | $1.0419(1.0418)^{+0.00028}_{-0.00030}$ | $1.0422(1.0420)^{+0.00028}_{-0.00027}$ | $1.0421(1.0419)^{+0.00029}_{-0.00028}$
$log(10^{10}A_{s})$ | $3.0459(3.0354)^{+0.01548}_{-0.01607}$ | $3.0430(3.0566)^{+0.01575}_{-0.01695}$ | $3.0485(3.0451)^{+0.01642}_{-0.01804}$ | $3.0444(3.0423)^{+0.01621}_{-0.01730}$
$n_{s}$ | $0.9659(0.9650)^{+0.00363}_{-0.00379}$ | $0.9657(0.9661)^{+0.00397}_{-0.00368}$ | $0.9716(0.9726)^{+0.00374}_{-0.00364}$ | $0.9714(0.9742)^{+0.00379}_{-0.00373}$
$\tau_{reio}$ | $0.0552(0.0513)^{+0.00750}_{-0.00776}$ | $0.0538(0.0576)^{+0.00752}_{-0.00799}$ | $0.0589(0.0574)^{+0.00752}_{-0.00903}$ | $0.0568(0.0565)^{+0.00756}_{-0.00841}$
$\beta$ | $-$ | $0.0520(0.0712)^{+0.03204}_{-0.02781}$ | $-$ | $0.0660(0.0822)^{+0.03519}_{-0.02168}$
$H_{0}$ | $67.5210(67.8240)^{+0.42498}_{-0.43026}$ | $67.8220(67.6550)^{+0.43965}_{-0.46707}$ | $68.6160(68.5760)^{+0.38578}_{-0.39766}$ | $68.9440(69.1480)^{+0.42046}_{-0.41728}$
$\sigma_{8}$ | $0.8102(0.8026)^{+0.00726}_{-0.00689}$ | $0.8228(0.8358)^{+0.00966}_{-0.01354}$ | $0.8045(0.8027)^{+0.00727}_{-0.00778}$ | $0.8224(0.8258)^{+0.01147}_{-0.01598}$
$\chi^{2}_{min}$ | $4189.5447$ | $4189.3447$ | $4104.1945$ | $4101.2884$
$\Delta\chi^{2}_{min}$ | $0$ | $-0.2000$ | $0$ | $-2.9061$
Table 2: The mean (bestfit) $\pm 1\sigma$ error for the cosmological
parameters constrained with the datasets [Planck + BAO + Pantheon Plus] and
[Planck + BAO + Pantheon Plus + SH0ES] for the $\Lambda$CDM model and
Chameleon model are shown in the above table. The corresponding
$\chi_{min}^{2}$ and $\Delta\chi_{min}^{2}$ values are also shown.

####  IV.2.1 Data sets

We use a combination of Cosmic Microwave Background [CMB], Baryon Acoustic
Oscillation [BAO], Pantheon+ and SH0ES datasets. The details of these datasets
are as follows:

  * $\blacktriangleright$

Planck 2018 measurements of the high-$\ell$ CMB TT, TE, EE, and low-$\ell$ CMB
TT, EE power spectra [1].

  * $\blacktriangleright$

The BAO measurements from SDSS-III Baryon Oscillation Spectroscopic Survey
DR12 galaxy sample at the redshifts $z=0.38,0.51$ and $0.61$ [69], 6dF Galaxy
Survey (6dFGS) at $z=0.106$ [70], and SDSS DR7 Main Galaxy Sample I at
$z=0.15$ [71].

  * $\blacktriangleright$

Pantheon+ data with Type Ia supernovae (SNe Ia) catalogue ranging in redshifts
from $z=0.001$ to $2.26$ [72].

  * $\blacktriangleright$

SH0ES data with Cepheids variables in the host galaxies with nearby or recent
SNe Ia discovered in the last four decades with $z\leq 0.01$ [73].

####  IV.2.2 Methodology

We consider the baseline cosmology to consist of six $\Lambda$CDM parameters
$[\omega_{b},\ \omega_{dm},\theta_{s},\ ln(10^{10}A_{s}),\ n_{s},\
\tau_{reio}]$ plus the two Chameleon model parameters $\alpha$ and $\beta$.

In this study, we perform a Markov Chain Monte Carlo (MCMC) analysis using the
publicly available package MontePython-v3 [74], which implements the
Metropolis Hasting algorithm. This is then interfaced with CLASS modified for
the Chameleon model with the datasets mentioned in the previous subsubsection.
We use Cholesky decomposition, which efficiently samples the fast and slow
parameters to cover the entire parameter space [75]. We consider the chains to
be converged when the Gelman-Rubin convergence criterion, $R-1<0.01$, is met
[76]. Finally, we obtain the minimized $\chi^{2}$ using the Sequential Least
Squares Programming algorithm built into the Montepython code.

Figure 2: Posterior distribution for the Chameleon model with different
datasets.

###  IV.3 Results

In this work, our main focus is on the dark sector interaction strength from
different cosmological datasets. In order to obtain an estimate for the
strength of Chameleon interaction, $\beta$, we fix $\alpha=0.01$ as discussed
in Sec.III. The mean and best fit values of the parameters obtained from the
MCMC runs are shown in Tables 1 and 2. The Table 1 shows the values for the
datasets Planck, and Planck + BAO. While the Table 2 shows for the datasets
Planck + BAO + Pantheon Plus, and Planck + BAO + Pantheon Plus + SH0ES. The
$\chi^{2}_{min}$ and $\Delta\chi^{2}_{min}$ values obtained for different
datasets are also shown in tables. The $\chi^{2}_{min}$ for each experiment
are reported in Appendix A. The plots of 1D and 2D posterior distribution of
$\beta,\omega_{dm},H_{0}$ and $\sigma_{8}$ are shown in the Fig. 2. This
figure shows the comparison of Chameleon model between different datasets
namely: Planck, Planck + BAO, Planck + BAO + Pantheon Plus, and Planck + BAO +
Pantheon Plus + SH0ES.

The strength of dark sector interaction, $\beta$, indicated by the mean value
of its posterior distribution from MCMC runs using different dataset
combinations of Planck, BAO, Pantheon Plus, and SH0ES are found to be about
$0.0512,0.0537,0.0520,0.0660$ in tandem with the inclusion of each dataset
respectively. The corresponding bestfit values are
$0.0487,0.0680,0.0712,0.0822$ respectively. Further, the corresponding
$\Delta\chi^{2}_{min}$ for the Chameleon model with respect to the
$\Lambda$CDM model comes out to be about $-0.6796,-1.0353,-0.2000,-2.9061$
respectively. The lowest $\Delta\chi^{2}$ corresponds to the Planck + BAO +
Pantheon Plus + SH0ES dataset while the highest $\Delta\chi^{2}$ corresponds
to the Planck + BAO + Pantheon Plus dataset. Even though these
$\Delta\chi^{2}$ values does not indicate a very significant preference but
still all of the different dataset combinations show a slight preference to
the Chameleon model over the standard $\Lambda$CDM model.

Further, we note that the Chameleon model results in higher mean values of
$H_{0}$, $\sigma_{8}$ and $\omega_{dm}$ in comparison to the $\Lambda$CDM
model as shown in the Tables 1 and 2.

Here, we list the interesting features noted in the 1D posterior distribution
of $\beta$ in Fig. 2, which are as follows:

  * $\blacktriangleright$

The posterior distribution corresponding to all of the different dataset
combinations shows a peak with upper bound but no lower bound.

  * $\blacktriangleright$

The addition of BAO dataset to the Planck dataset leads to a slight decrease
in the likelihood value at $\beta=0$.

  * $\blacktriangleright$

Further, the addition of Pantheon Plus dataset leads to a smoother drop in the
likelihood value near $\beta=0$ other than this it does not significantly
alter the shape in comparison to the previous case.

  * $\blacktriangleright$

Finally with the addition of SH0ES dataset, we see a dramatic improvement as
the likelihood value at $\beta=0$ decreases significantly. The mean value also
shifts significantly in comparison to the above cases.

  * $\blacktriangleright$

The consecutive addition of BAO, Pantheon Plus and SH0ES datasets to the
Planck dataset, decreases the likelihood value at $\beta=0$.

  * $\blacktriangleright$

All of the different dataset combinations show a higher likelihood value at
the non-zero mean $\beta$ value in comparison to the likelihood value at
$\beta=0$.

The last point implies that all of the different dataset combinations
consistently show a preference for non-zero $\beta$ value rather than for
$\beta=0$ (which refers to non-interacting dark sector). This clearly implies
that the cosmological datasets are hinting at a more versatile interacting
dark sector.

The 2D posterior distribution of $\beta$ with other parameters show features
similar to those of 1D posterior distribution and hence a similar conclusion
can be drawn.

Figure 3: Posterior distribution for the Chameleon model with Planck + BAO +
Pantheon Plus + SH0ES dataset for the $\alpha$ values $0.005,0.05,$ and $0.1$.

The above MCMC analysis was also implemented with different $\alpha$ values
such as $0.005,0.05$ and $0.1$. The 1D and 2D posterior distributions obtained
with the Planck + BAO + Pantheon Plus + SH0ES dataset corresponding to each of
these $\alpha$ values are shown in Fig. 3. The inferences that can be drawn
from the posterior distributions of $\beta$ are listed as follows:

  * $\blacktriangleright$

The posterior distributions corresponding to all of the different $\alpha$
value shows a peak with an upper bound but no lower bound.

  * $\blacktriangleright$

Posterior distributions corresponding to all of the different $\alpha$ values
consistently show a higher likelihood value at a non-zero mean $\beta$ value
in comparison to the likelihood value at $\beta=0$. This indicates a
preference for a non-zero $\beta$ value rather than for $\beta=0$ case.

  * $\blacktriangleright$

The peak position seems to remain robust to the variation of $\alpha$ value.
This can be attributed to the following reason: The $\alpha$ and $\beta$
encapsulates different aspects of dark energy component namely its self
interacting potential and its coupling with dark matter respectively, and
hence are independent of each other.

The above points further supports the discussion in the previous paragraphs
which hinted towards the possible existence of a fifth force within the dark
sector. Further, we can note that the mean values of $H_{0}$, $\sigma_{8}$ and
$\omega_{dm}$ seems to increase with the $\alpha$ value.

We can note that since the Chameleon interaction has no effect on the CMB
primary anisotropies (as described in Sec.III), the constraint obtained on the
parameter $\beta$ from the Planck dataset can be entirely credited to be
coming from the CMB secondary anisotropies only, which originates at later
times after the recombination era.

The BAO, Pantheon Plus, and SH0ES datasets capture information about the
physics in the late-time evolution of the Universe i.e. at specific low
redshift or at a range of low redshifts. And further the Chameleon model also
starts to significantly alter the evolution at later times. Hence adding these
late-time observational datasets to Planck dataset leads to a better and
better shaped posterior distributions of $\beta$. Further, as noted earlier
the likelihood value at $\beta=0$ decreases with the consecutive addition of
each of these datasets which in turn implies that the non-interacting dark
energy scenario is becoming less and less probable in these consecutive cases
respectively even though it cannot be ruled out entirely.

The analysis and discussions so far point towards a mild detection of fifth
force within the dark sector, which could have resulted due to the following
significant improvements in the current work: i) The use of exact solution
obtained by numerically solving the modified Klein-Gordan equation in Eq. 1
using the shooting method, which was then solved together with the background
and perturbation equations of Chameleon model with the CLASS package. ii) The
use of latest cosmological datasets for the MCMC analysis.

In Desmond et. al. [77], they constrain the relative strength of a fifth force
arising from a Yukawa type Chameleon scalar field, for screened and un-
screened cases, using the Alfalfa survey of neutral atomic hydrogen. They
quantify the relative strength as $\Delta G/G_{N}$, where $G_{N}$ is the
Newtonian gravitational constant, $\Delta G$ is the strength of new
interaction relative to gravitational force. They found an evidence of
$6.6\sigma$ for $\Delta G>0$ at $\lambda_{c}=2$Mpc with $\Delta G/G_{N}=0.025$
for the case with screening. While for the no-screening case, they found no
increase in the likelihood over the case with $\Delta G=0$ for any
$\lambda_{C}$. They mention that they could not rule out the possibility that
this detection could have originated from other effects such as the galaxy
formation physics. Hence they show a more conservative results which yields
only upper limits on $\Delta G/G_{N}$ ranging from $\sim 10^{-1}$ for
$\lambda_{c}\simeq 0.5$Mpc to $\sim$ few $\times 10^{-4}$ at
$\lambda_{c}\simeq 50$Mpc.

We can compare our results with Desmond et. al. [77], since the parameter
$\beta$ is related to the relative strength as $\Delta G/G_{N}=2\beta^{2}$
(from Eq.6). The relative strength can then be estimated for the Chameleon
model corresponding to the different dataset combinations of Planck, BAO,
Pantheon Plus, and SH0ES, in tandem with the inclusion of each dataset, to be
$0.0052,0.0057,0.0054,0.0087$ respectively (using the mean value of $\beta$
posterior distribution). Further, the corresponding fifth force range now is
approximately equal to the current size of the observable Universe (from Eq.7)
implying that the fifth force in the Chameleon model becomes a long range
force at the later times. We can note that our estimates of fifth force
relative strength are consistent with their conservative estimate at
$\lambda_{c}\simeq 0.5$Mpc while it is about an order of magnitude higher in
comparison to their conservative estimate at $\lambda_{c}\simeq 50$Mpc. This
difference can be attributed to the different approaches adopted which are
listed as follows: i) They consider the fifth force to originate from the
coupling of screened scalar field with the stars, gas and dark matter while in
our work we consider the fifth force to originate from the dark energy which
is assumed to couple with only the dark matter. ii) Our estimates use
cosmological datasets while their analysis is based on searching for
displacements between galaxies stellar and gas mass centroids in galaxy
surveys. iii) While our study focuses on the impact of fifth force at
cosmological scales, their study focus on scales of $0.4-50$Mpc. In conclusion
irrespective of the different approaches adopted in these works, both offer
intriguing and complementary hints towards the existence of a fifth force.

##  V Conclusion

We have investigated the Yukawa-type interaction between the dark energy and
dark matter arising in the Chameleon model. The associated background and
perturbation equations were numerically solved using the shooting algorithm. A
thorough analysis was then conducted using the MCMC method. The main findings
of this analysis are as follows:

  * $\blacktriangleright$

The strength of dark sector interaction, obtained from the peak of posterior
distribution of $\beta$, for different dataset combinations of Planck, BAO,
Pantheon Plus, and SH0ES in tandem with the inclusion of each dataset are
$0.0512,0.0537,0.0520,0.0660$ respectively. The corresponding bestfit values
are $0.0487,0.0680,0.0712,0.0822$ respectively.

  * $\blacktriangleright$

The $\Delta\chi^{2}_{min}$ values obtained for the Chameleon model, with the
additional parameter $\beta$, relative to the $\Lambda$CDM model does not
indicate a very significant preference but still all of the different dataset
combinations consistently shows a slight but definite preference.

  * $\blacktriangleright$

The different dataset combinations show higher likelihood for non-zero $\beta$
value in comparison to that of $\beta=0$ (which refers to non-interacting dark
sector). This indicates that the cosmological datasets are hinting at an
interacting dark sector.

  * $\blacktriangleright$

The strength of the fifth force with respect to the gravitational force,
$\Delta G/G_{N}$, due to the Chameleon dark energy was estimated corresponding
to the different dataset combinations of Planck, BAO, Pantheon Plus, and SH0ES
in tandem with the inclusion of each dataset as $0.0052,0.0057,0.0054,0.0087$
respectively.

This mild detection of fifth force within the dark sector could have resulted
because we use the exact solution, which was obtained by numerically solving
the modified Klein-Gordan equation using the shooting method, and the use of
latest cosmological datasets for the MCMC analysis.

The hints of a non-zero value for $\beta$ being favored by the analysis of
linear scales motivate extending the analysis to smaller, more nonlinear
scales. Differences in the linear $P(k)$ on scales of $k>10^{-1}h/{\rm Mpc}$
at $z=0$, as seen in Fig. 1, should also translate to differences in the
observable, nonlinear $P(k)$ which will be measured with great precision in
the various Stage IV cosmological surveys in the next decade. Given the
statistical power on smaller scales (due to the number of independent modes),
we expect to be able to either detect a non-zero $\beta$ with much greater
significance or constrain $\beta$ much more stringently. There will also be
unique signatures of a non-zero $\beta$ value arising from the difference in
the infall of Dark Matter and baryonic matter onto virialized halos [49, 50,
51]. Numerical $N$-body simulations will be needed to quantify these effects
accurately, where the usual $N$-body methods will need to be updated to
correctly account for the Chameleon interactions, as well as differences in
the forces on Dark Matter vs baryons. This will be part of a future study.

###### Acknowledgements.

SD and VG acknowledge the DST-SERB Government of India grant CRG/2019/006147
for supporting the project. We acknowledge the use of NOVA, the high
performance computing cluster situated at Indian Institute of Astrophysics,
Bangalore where all of the numerical calculations in this paper was performed.

## References

  * Aghanim and et. al. [2020] N. Aghanim and et. al., Astronomy & Astrophysics 641, A6 (2020).
  * Semenaite  _et al._ [2022] A. Semenaite, A. G. Sánchez, A. Pezzotta, J. Hou, R. Scoccimarro, A. Eggemeier, M. Crocce, C.-H. Chuang, A. Smith, C. Zhao, J. R. Brownstein, G. Rossi, and D. P. Schneider, Monthly Notices of the Royal Astronomical Society 512, 5657 (2022), arXiv:2111.03156 [astro-ph.CO] .
  * Sevilla-Noarbe and et. al. [2021] I. Sevilla-Noarbe and et. al., The Astrophysical Journal Supplement Series 254, 24 (2021).
  * Dalal  _et al._ [2023] R. Dalal _et al._ , (2023), arXiv:2304.00701 [astro-ph.CO] .
  * Li  _et al._ [2023] X. Li, T. Zhang, S. Sugiyama, R. Dalal, and et. al., (2023), arXiv:2304.00702 [astro-ph.CO] .
  * Freedman [2017] W. L. Freedman, Nature Astronomy 1, 0121 (2017).
  * Heymans  _et al._ [2013] C. Heymans, E. Grocutt, A. Heavens, M. Kilbinger, and et. al., Monthly Notices of the Royal Astronomical Society 432, 2433 (2013).
  * Weinberg [1989] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
  * Caldwell  _et al._ [1998] R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998).
  * Liddle and Scherrer [1998] A. R. Liddle and R. J. Scherrer, Phys. Rev. D 59, 023509 (1998).
  * Wetterich [1988] C. Wetterich, Nuclear Physics B 302, 668 (1988).
  * Wetterich [1994] C. Wetterich, (1994), arXiv:hep-th/9408025 [hep-th] .
  * Damour  _et al._ [1990] T. Damour, G. W. Gibbons, and C. Gundlach, Phys. Rev. Lett. 64, 123 (1990).
  * Casas  _et al._ [1992] J. A. Casas, J. Garcia-Bellido, and M. Quiros, Classical and Quantum Gravity 9, 1371–1384 (1992).
  * Anderson and Carroll [1998] G. W. Anderson and S. M. Carroll, in  _COSMO-97_ (WORLD SCIENTIFIC, 1998).
  * Amendola [2000] L. Amendola, Phys. Rev. D 62, 043511 (2000).
  * Amendola and Tocchini-Valentini [2001] L. Amendola and D. Tocchini-Valentini, Phys. Rev. D 64, 043509 (2001).
  * Bean [2001] R. Bean, Phys. Rev. D 64, 123516 (2001).
  * Damour  _et al._ [2002] T. Damour, F. Piazza, and G. Veneziano, Phys. Rev. Lett. 89, 081601 (2002).
  * Comelli  _et al._ [2003] D. Comelli, M. Pietroni, and A. Riotto, Physics Letters B 571, 115–120 (2003).
  * Amendola  _et al._ [2003] L. Amendola, C. Quercellini, D. Tocchini-Valentini, and A. Pasqui, The Astrophysical Journal 583, L53 (2003).
  * Chimento  _et al._ [2003] L. P. Chimento, A. S. Jakubi, D. Pavón, and W. Zimdahl, Phys. Rev. D 67, 083513 (2003).
  * Amendola and Quercellini [2003] L. Amendola and C. Quercellini, Phys. Rev. D 68, 023514 (2003).
  * França and Rosenfeld [2004] U. França and R. Rosenfeld, Phys. Rev. D 69, 063517 (2004).
  * Khoury and Weltman [2004a] J. Khoury and A. Weltman, Physical Review Letters 93, 10.1103/physrevlett.93.171104 (2004a).
  * Khoury and Weltman [2004b] J. Khoury and A. Weltman, Phys. Rev. D 69, 044026 (2004b).
  * Gubser and Khoury [2004] S. S. Gubser and J. Khoury, Physical Review D 70, 10.1103/physrevd.70.104001 (2004).
  * Brax  _et al._ [2004a] P. Brax, C. van de Bruck, A.-C. Davis, J. Khoury, and A. Weltman, Physical Review D 70, 10.1103/physrevd.70.123518 (2004a).
  * Farrar and Peebles [2004] G. R. Farrar and P. J. E. Peebles, The Astrophysical Journal 604, 1–11 (2004).
  * Kaplan  _et al._ [2004] D. B. Kaplan, A. E. Nelson, and N. Weiner, Phys. Rev. Lett. 93, 091801 (2004).
  * Olivares  _et al._ [2005] G. Olivares, F. Atrio-Barandela, and D. Pavón, Physical Review D 71, 10.1103/physrevd.71.063523 (2005).
  * Wei and Cai [2005] H. Wei and R.-G. Cai, Phys. Rev. D 71, 043504 (2005).
  * Peccei [2005] R. D. Peccei, Physical Review D 71, 10.1103/physrevd.71.023527 (2005).
  * Fardon  _et al._ [2006] R. Fardon, A. E. Nelson, and N. Weiner, Journal of High Energy Physics 2006, 042–042 (2006).
  * von Marttens  _et al._ [2020] R. von Marttens, L. Lombriser, M. Kunz, V. Marra, L. Casarini, and J. Alcaniz, Physics of the Dark Universe 28, 100490 (2020).
  * Baker  _et al._ [2022] O. Baker, A. Afanasev, T. Lagouri, J. Pan, and C. Weber, Symmetry 14, 2238 (2022).
  * Zhang [2022] H.-C. Zhang, (2022), arXiv:2210.10204 [gr-qc] .
  * Piedipalumbo  _et al._ [2023] E. Piedipalumbo, S. Vignolo, P. Feola, and S. Capozziello, Phys. Dark Univ. 42, 101274 (2023), arXiv:2307.02355 [gr-qc] .
  * Brax  _et al._ [2024] P. Brax, C. P. Burgess, and F. Quevedo, JCAP 03, 015, arXiv:2310.02092 [hep-th] .
  * Käding  _et al._ [2023] C. Käding, M. Pitschmann, and C. Voith, Eur. Phys. J. C 83, 767 (2023), arXiv:2306.10896 [hep-ph] .
  * Singh  _et al._ [2023] A. Singh, A. Pradhan, and A. Beesham, New Astron. 100, 101995 (2023).
  * Burgess and Quevedo [2022] C. Burgess and F. Quevedo, Journal of Cosmology and Astroparticle Physics 2022 (04), 007.
  * Brax  _et al._ [2023] P. Brax, C. P. Burgess, and F. Quevedo, JCAP 08, 011, arXiv:2212.14870 [hep-ph] .
  * Huey and Wandelt [2006] G. Huey and B. D. Wandelt, Physical Review D 74, 10.1103/physrevd.74.023519 (2006).
  * Štefančić [2004] H. Štefančić, The European Physical Journal C 36, 523–527 (2004).
  * Das  _et al._ [2006] S. Das, P. S. Corasaniti, and J. Khoury, Phys. Rev. D 73, 083509 (2006).
  * Damour and Polyakov [1994] T. Damour and A. Polyakov, Nuclear Physics B 423, 532–558 (1994).
  * Boriero  _et al._ [2015] D. Boriero, S. Das, and Y. Y. Wong, Journal of Cosmology and Astroparticle Physics 2015 (07), 033.
  * Kesden and Kamionkowski [2006] M. Kesden and M. Kamionkowski, Physical Review D 74, 10.1103/physrevd.74.083007 (2006).
  * Keselman  _et al._ [2009] J. A. Keselman, A. Nusser, and P. J. E. Peebles, Physical Review D 80, 10.1103/physrevd.80.063517 (2009).
  * Secco  _et al._ [2018] L. F. Secco, A. Farah, B. Jain, S. Adhikari, A. Banerjee, and N. Dalal, The Astrophysical Journal 860, 32 (2018).
  * Khoury and Weltman [2004c] J. Khoury and A. Weltman, Phys. Rev. Lett. 93, 171104 (2004c).
  * Brax  _et al._ [2004b] P. Brax, C. van de Bruck, A.-C. Davis, J. Khoury, and A. Weltman, Phys. Rev. D 70, 123518 (2004b).
  * Brax  _et al._ [2006] P. Brax, C. van de Bruck, A. Davis, and A. Green, Physics Letters B 633, 441 (2006).
  * Copeland  _et al._ [2006] E. J. Copeland, M. Sami, and S. Tsujikawa, International Journal of Modern Physics D 15, 1753–1935 (2006).
  * Wang  _et al._ [2012a] J. Wang, L. Hui, and J. Khoury, Physical Review Letters 109, 10.1103/physrevlett.109.241301 (2012a).
  * Will [2001] C. M. Will, Living Reviews in Relativity 4, 10.12942/lrr-2001-4 (2001).
  * Kapner and et. al. [2007] D. J. Kapner and et. al., Phys. Rev. Lett. 98, 021101 (2007).
  * Wang  _et al._ [2012b] J. Wang, L. Hui, and J. Khoury, Phys. Rev. Lett. 109, 241301 (2012b).
  * Alam and et. al. [2021] S. Alam and et. al., Phys. Rev. D 103, 083533 (2021).
  * D’Amico  _et al._ [2021] G. D’Amico, L. Senatore, and P. Zhang, Journal of Cosmology and Astroparticle Physics 2021 (01), 006.
  * Abbott  _et al._ [2019] T. M. C. Abbott, F. B. Abdalla, and et. al. (DES Collaboration), Phys. Rev. D 99, 123505 (2019).
  * Corasaniti [2008] P. S. Corasaniti, Phys. Rev. D 78, 083538 (2008).
  * Abbott and et. al. [2018] T. Abbott and et. al., Physical Review D 98, 10.1103/physrevd.98.043526 (2018).
  * Brax  _et al._ [2004c] P. Brax, C. van de Bruck, A. C. Davis, J. Khoury, and A. Weltman, AIP Conference Proceedings 736, 105 (2004c).
  * Lesgourgues [2011] J. Lesgourgues, (2011), arXiv:1104.2932 [astro-ph.IM] .
  * Blas  _et al._ [2011] D. Blas, J. Lesgourgues, and T. Tram, Journal of Cosmology and Astroparticle Physics 2011 (07), 034.
  * Planck Collaboration  _et al._ [2020] Planck Collaboration, Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., and et. al., A&A 641, A5 (2020).
  * Alam and et. al. [2017] S. Alam and et. al., Monthly Notices of the Royal Astronomical Society 470, 2617–2652 (2017).
  * Beutler and et. al. [2011] F. Beutler and et. al., Monthly Notices of the Royal Astronomical Society 416, 3017–3032 (2011).
  * Ross and et. al. [2015] A. J. Ross and et. al., Monthly Notices of the Royal Astronomical Society 449, 835–847 (2015).
  * Brout and et. al. [2022] D. Brout and et. al., The Astrophysical Journal 938, 110 (2022).
  * Riess and et. al. [2022] A. G. Riess and et. al., The Astrophysical Journal Letters 934, L7 (2022).
  * Brinckmann and Lesgourgues [2018] T. Brinckmann and J. Lesgourgues, (2018), arXiv:1804.07261 [astro-ph.CO] .
  * Lewis  _et al._ [2000] A. Lewis, A. Challinor, and A. Lasenby, The Astrophysical Journal 538, 473 (2000).
  * Gelman and Rubin [1992] A. Gelman and D. B. Rubin, Statistical Science 7, 457 (1992).
  * Desmond  _et al._ [2018] H. Desmond, P. G. Ferreira, G. Lavaux, and J. Jasche, Phys. Rev. D 98, 064015 (2018).

##  Appendix A $\chi^{2}$ per experiment

We report the bestfit $\chi^{2}$ values per experiment for the $\Lambda$CDM
and Chameleon model corresponding to different dataset combinations in Table
3.

We note from the table that the bestfit $\chi^{2}$ values of Planck
high-$\ell$, Pantheon Plus and SH0ES datasets seems to show a preference to
Chameleon model. While the Planck low-$\ell$ TT, Planck low-$\ell$ EE, BAO
BOSS DR12 and BAO low z datasets either does not show preference or shows a
comparatively smaller preference to the Chameleon model. The reason for the
behaviour in the case of Planck low-$\ell$ datasets can be interpreted as
follows: The Planck low-$\ell$ TT and EE datasets corresponds to large Cosmic
Microwave Background perturbation scales ($\ell<30$) that originated at early
times but has mostly remained unaltered due to the later evolution. Hence it
supports the standard $\Lambda$CDM model over the Chameleon model, which as
previously noted significantly alters the evolution only at later times.

Dataset Combination $\rightarrow$ | D1 | D2 | D3 | D4
---|---|---|---|---
Experiment $\downarrow$ Model $\rightarrow$ |  $\Lambda$CDM | Chameleon |  $\Lambda$CDM | Chameleon |  $\Lambda$CDM | Chameleon |  $\Lambda$CDM | Chameleon
Planck high-$\ell$ TTTEEE | $2351.5205$ | $2348.4028$ | $2353.1730$ | $2348.7670$ | $2353.4624$ | $2350.9443$ | $2357.8959$ | $2355.9107$
Planck low-$\ell$ EE | $396.7873$ | $397.3647$ | $395.7295$ | $397.2702$ | $395.7135$ | $395.7490$ | $396.4642$ | $396.1857$
Planck low-$\ell$ TT | $23.4834$ | $25.3441$ | $22.7220$ | $24.8302$ | $23.3171$ | $24.6166$ | $21.9501$ | $24.4752$
BAO BOSS DR12 | $-$ | $-$ | $4.1515$ | $3.5268$ | $4.0200$ | $6.7074$ | $3.4827$ | $3.6519$
BAO low z | $-$ | $-$ | $1.2996$ | $1.6461$ | $1.3488$ | $0.8624$ | $2.2277$ | $2.3994$
Pantheon Plus | $-$ | $-$ | $-$ | $-$ | $1411.6825$ | $1410.4646$ | $-$ | $-$
Pantheon Plus+SH0ES | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $1322.1737$ | $1318.6652$
Total $\chi_{min}^{2}$ | $2771.7913$ | $2771.1117$ | $2777.0758$ | $2776.0405$ | $4189.5447$ | $4189.3447$ | $4104.1945$ | $4101.2884$
$\Delta\chi_{min}^{2}$(Chameleon - $\Lambda$CDM) | $0$ | $-0.6796$ | $0$ | $-1.0353$ | $0$ | $-0.2000$ | $0$ | $-2.9061$
Table 3: The bestfit $\chi^{2}$ per experiment for $\Lambda$CDM and Chameleon
model corresponding to different dataset combinations namely D1, D2, D3, and
D4 (where D1 $\equiv$ Planck, D2 $\equiv$ D1 + BAO, D3 $\equiv$ D2 + Pantheon
Plus and D4 $\equiv$ D2 + Pantheon Plus + SH0ES). The total $\chi_{min}^{2}$
and $\Delta\chi_{min}^{2}$ are shown in the last rows.