# Dissecting kinetically coupled quintessence:
phenomenology and observational tests

Elsa M. Teixeira<EMAIL_ADDRESS>School of Mathematics and
Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United
Kingdom  Bruno J. Barros<EMAIL_ADDRESS>Cosmology and Gravity Group,
Department of Mathematics and Applied Mathematics, University of Cape Town,
Rondebosch 7700, Cape Town, South Africa  Vasco M. C. Ferreira
<EMAIL_ADDRESS>Instituto de Astrofísica e Ciências do Espaço,
Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
Noemi Frusciante<EMAIL_ADDRESS>Dipartimento di Fisica “E.
Pancini”, Università degli Studi di Napoli “Federico II”, Compl. Univ. di
Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

###### Abstract

We investigate an interacting dark energy model which allows for the kinetic
term of the scalar field to couple to dark matter via a power-law interaction.
The model is characterised by scaling solutions at early times, which are of
high interest to alleviate the coincidence problem, followed by a period of
accelerated expansion. We discuss the phenomenology of the background
evolution and of the linear scalar perturbations and we identify measurable
signatures of the coupling in the dark sector on the cosmic microwave
background, the lensing potential auto-correlation and the matter power
spectra. We also perform a parameter estimation analysis using data of cosmic
microwave background temperature, polarisation and lensing, baryonic acoustic
oscillations and supernovae. We find that the strength of the coupling between
the dark sectors, regulated by the parameter $\alpha$, is constrained to be of
order $10^{-4}$. A model selection analysis does not reveal a statistical
preference between $\Lambda$CDM and the Kinetic model.

###### Contents

  1. I Introduction
  2. II Theory
    1. II.1 The Kinetic model
    2. II.2 Background equations
    3. II.3 Linear cosmological perturbations
    4. II.4 The parameter space
  3. III Phenomenology of the Kinetic model
    1. III.1 Background evolution
    2. III.2 Cosmological observables
  4. IV Cosmological constraints and model selection analysis
    1. IV.1 Data sets
    2. IV.2 Cosmological bounds
  5. V Conclusions
  6. A Synchronous gauge

##  I Introduction

The existence of dark energy (DE) and dark matter (DM) is supported by
multiple cosmological observations, though their nature still remains unknown.
The former is postulated as a repulsive force acting on the largest scales,
needed to explain the late time cosmic acceleration, whereas the latter is a
non-baryonic matter component, responsible for the formation and evolution of
large scale structures in the Universe. The Standard Model of Cosmology, known
as $\Lambda$-cold dark matter ($\Lambda$CDM), is based on General Relativity
(GR) and includes a cosmological constant, $\Lambda$, as the simplest model of
DE and a cold dark matter (CDM) component as weakly interacting non-
relativistic particles. In this base scenario, it is assumed that the two dark
components do not directly couple to each other. Although it provides a fairly
accurate description of the Universe, there are some unexplained theoretical
and observational conundrums that indirectly affect the $\Lambda$CDM model
Weinberg (1989, 2000); Martin (2012). Such is the case of the Cosmological
Constant problem or the need for a primordial inflationary period.
Observational tensions, if not stemming from systematics, pose an additional
challenge Abdalla  _et al._ (2022) namely concerning the mismatch in the
estimation of the values of the Hubble constant, $H_{0}$ Aghanim  _et al._
(2020a); Riess  _et al._ (2019); Wong  _et al._ (2020); Riess  _et al._
(2021); Pesce  _et al._ (2020), and the amplitude of the matter power spectrum
at present time, $\sigma_{8}$ Heymans  _et al._ (2021); Di Valentino  _et al._
(2021); Asgari  _et al._ (2021), when using high- and low-redshift data from
different surveys. These shortcomings might signal the need to go beyond the
vanilla $\Lambda$CDM model Saridakis  _et al._ (2021).

Promoting DE to a dynamical scalar field is an enticing approach to extend
$\Lambda$CDM and still achieve the late-time accelerated expansion. Recent
experimental advancements in particle physics have lead to the detection of a
Higgs-like particle Aad  _et al._ (2012); Chatrchyan  _et al._ (2012) and
scalar fields also comprise the most promising proposal to solve the early
Universe trinity puzzle (i.e. the horizon, flatness, and magnetic-monopole
problems) Guth (1981); Linde (1982); Starobinsky (1982). The quintessence
model Wetterich (1995); Caldwell  _et al._ (1998); Chiba (1999) was the first
attempt to include a scalar degree of freedom, $\phi$, portraying a time-
varying DE component with dynamics assigned by the form of the potential,
$V(\phi)$, and its kinetic term,
${X=-\partial_{\mu}\phi\partial^{\mu}\phi}/2$.

In particular, it should resemble the cosmological constant at late times,
that is, its negative pressure must have a magnitude close to its energy
density, $p_{\phi}\approx-\rho_{\phi}$, while not revealing effective
clustering properties at small scales. One appealing feature of this theory
(or, more in general, of scalar tensor theories) lies in obtaining, under
particular conditions, scaling solutions Wetterich (1995); Copeland  _et al._
(1998); Ferreira and Joyce (1998); Liddle and Scherrer (1999); Barreiro  _et
al._ (2000); Amendola (2000); Guo  _et al._ (2003a, b); Chimento  _et al._
(2003); Tsujikawa and Sami (2004); Piazza and Tsujikawa (2004); Pettorino  _et
al._ (2005); Amendola  _et al._ (2006); Ohashi and Tsujikawa (2009); Gomes and
Amendola (2014); Chiba  _et al._ (2014); Amendola  _et al._ (2014);
Albuquerque  _et al._ (2018); Frusciante  _et al._ (2018); Amendola  _et al._
(2018); Barros (2019); Albuquerque  _et al._ (2022); Abdalla  _et al._ (2022);
Pace and Frusciante (2022). These are characterised by a constant ratio
between the energy density of the matter components and that of the scalar
field. In this case the DE contribution remains hidden throughout the
radiation and matter domination eras, despite allowing the energy density of
the scalar field to be of the same order of magnitude as these components.
This mechanism is relevant in addressing the cosmic coincidence problem Zlatev
_et al._ (1999); Velten  _et al._ (2014), namely why the magnitude of the
energy densities for matter and DE are comparable at present, while still
preserving compatibility with the energy scale associated with particle
physics. Accordingly, here we will focus on a specific model that already
revealed to feature scaling solutions Barros (2019).

In this work we are interested in exploring a setting in which the scaling
regime is achieved through a “fifth-force” acting on DM particles, induced by
a quintessence field. An effective field theory formulation of such
phenomenological interaction can be set at the level of the action and
provides a fully covariant way to construct theoretically viable models
Tamanini (2015), thus avoiding the propagation of unphysical modes on large
scales Valiviita  _et al._ (2008). One such approach consists of considering
the presence of a field dependent function $f\left(\phi\right)$ multiplying
the CDM Lagrangian, $\mathcal{L}_{c}$, in the total action, that is, a
coupling of the form $f(\phi)\mathcal{L}_{c}$ Koivisto (2005). Recently, this
formulation was generalised to accommodate interactions of the matter sector
with the kinetic term of the scalar as well, through a functional form
$f\left(\phi\,,X\right)\mathcal{L}_{c}$ Barros (2019). Lagrangian-based models
have been further explored in the context of the Schutz-Sorkin action Schutz
(1970); Schutz and Sorkin (1977); Brown (1993), allowing for the inclusion of
interaction terms depending on single derivatives of the scalar field in the
action for CDM Pourtsidou  _et al._ (2013); Boehmer  _et al._ (2015a, b).
Along similar lines, in Ref. Kase and Tsujikawa (2020) the energy exchange is
achieved via two terms of the form $f_{1}(\phi,X)\rho_{c}(n_{c})$ and
$f_{2}(n_{c},\phi,X)J_{c}^{\mu}\partial_{\mu}\phi$, where $\rho_{c}$ and
$n_{c}$ are the energy density and number density of CDM, respectively, and
$J_{c}^{\mu}$ is a vector field related to the CDM four-velocity Kase and
Tsujikawa (2020). In the presence of a $f(\phi)$-coupling, scaling solutions
have been shown to exist for quintessence with an exponential potential
Amendola (1999, 2000). Likewise, general forms of the Lagrangian allowing for
scaling behaviour given either a constant or field-dependent interaction, have
been derived for k-essence Piazza and Tsujikawa (2004); Tsujikawa and Sami
(2004); Tsujikawa (2006); Amendola  _et al._ (2006) and scalar-tensor theories
such as Horndeski Gomes and Amendola (2014, 2016); Amendola  _et al._ (2018);
Frusciante  _et al._ (2018) and quadratic-order degenerate higher-order
scalar-tensor theory Frusciante  _et al._ (2019a).

Setting cosmological constraints on the interaction between a scalar field and
DM has been the subject of many investigations Amendola and Quercellini
(2003); Pettorino and Baccigalupi (2008); Bean  _et al._ (2008); Pettorino
_et al._ (2012); Pettorino (2013); Xia (2013); Ade  _et al._ (2016); van de
Bruck  _et al._ (2017); Pourtsidou and Tram (2016); Van De Bruck and Mifsud
(2018); Barros  _et al._ (2019); Agrawal  _et al._ (2021); Gómez-Valent  _et
al._ (2020); Pan  _et al._ (2020); da Fonseca  _et al._ (2022); Archidiacono
_et al._ (2022). A well tested class of proposals is the coupled DE model in
which DM particles interact with the scalar field due to a $\phi$-dependent
mass, characterised by a constant coupling strength $\beta$. This parameter
has been constrained to be $\beta=0.036\pm 0.016$ (Planck13 + WMAP + baryon
acoustic oscillations (BAO)), deviating from the vanishing interaction case at
$2.2\sigma$, and $\beta=0.066\pm 0.018$ when including polarisation, with
increasing significance at 3.6$\sigma$ Pettorino (2013). Similar results were
reported lately by the Planck collaboration, also showing a tension at $\sim
2.5\sigma$ with $\Lambda$CDM when Planck15 + BAO + Supernovae Ia + $H_{0}$
data are considered Ade  _et al._ (2016), and in Refs. van de Bruck  _et al._
(2017); Barros  _et al._ (2019); Gómez-Valent  _et al._ (2020) resorting to
more recent data sets. Additionally it has been realised that such constant
coupling can remove the $\sigma_{8}$ tension if the background is assumed to
be identical to the $\Lambda$CDM one Barros  _et al._ (2019). Moreover in Ref.
Bean  _et al._ (2008) the authors provided cosmological bounds for a variety
of models, which differ from each other through the form of the nontrivial
coupling between the DM and the quintessence field. The strength of the
coupling was constrained to be less than 7% of the coupling to gravity.

Let us remark that non-minimal couplings of the DE field to other matter
components have also been explored, e.g. to massive neutrinos Afshordi  _et
al._ (2005); Brookfield  _et al._ (2006), to baryons Aviles and Cervantes-Cota
(2011) or to the electromagnetic field Carroll (1998); Chiba and Kohri (2002).
A universal coupling has also been investigated and its magnitude is tightly
constrained through Solar System experiments Hui  _et al._ (2009); Creminelli
_et al._ (2014). Therefore such couplings are often chosen to be minimal, i.e.
there is no additional coupling of the matter fields to the scalar curvature,
thus motivating the choice of a direct coupling between the dark species only.

In this work we explore the model presented in Ref. Barros (2019), in which a
purely kinetic coupling between the quintessence field and DM is considered.
This coupling is expressed in terms of a power law interaction function,
$f\propto X^{\alpha}$, with $\alpha$ being a constant parameter quantifying
the strength of the interaction. Hereafter this will be referred to as the
Kinetic model. At a more fundamental level, the low-energy limit of a scalar
field theory with a shift symmetry only allows for kinetic couplings to matter
Brax and Valageas (2017), where the scalar field is identified as the
Goldstone mode of the broken symmetry. Although in the literature it is much
more natural to consider a universal coupling, such as in dilaton gravity
Damour and Polyakov (1994), it is possible to construct a specific (non-
universal) interaction with an individual matter source Damour  _et al._
(1990) or it can even naturally emerge in an effective description of a
fundamental theory, such as Type II string theory Koivisto  _et al._ (2014).
The toy model considered in this present work also allows for scaling
solutions at early times Barros (2019), already found to be fruitful to tackle
the cosmic coincidence problem. The specific kinetic power law coupling here
assumed was employed in the literature to couple quintessence to
electromagnetism Barros and da Fonseca (2022) inducing a time variation on the
fine-structure constant. The authors showed that the theory encapsulates a
plethora of new analytical coupled solutions motivated by the dark energy
kinematics. We remark that kinetically coupled models have never been fully
explored in terms of theoretical predictions at linear order in perturbations
and, as such, cosmological bounds on the parameters are not present in
literature. In this work we present such kind of analysis for the first time,
by comparing the theoretical predictions to the $\Lambda$CDM model for the
temperature-temperature (TT) power spectrum, lensing potential auto-
correlation power spectrum and matter power spectrum. These are then used to
provide cosmological constraints by means of Markov Chain Monte Carlo (MCMC)
methods. For this purpose we resort to large sets of data including
measurements of the background expansion of the Universe, temperature
fluctuations power spectra and those of gravitational potentials.

The manuscript is organised as follows. We lay down the theoretical framework
in Sec. II: the Kinetic model is introduced in Sec. II.1 and the explicit
equations of motion for the background dynamics and linear scalar
perturbations in the Newtonian gauge are presented in Sec. II.2 and Sec. II.3,
respectively; in Sec. II.4 we discuss the parameter space of the model in
order to guarantee its theoretical viability. In Sec. III we focus on the
cosmological properties of the Kinetic model, exploring the signatures left by
the dark coupling on the background expansion in Sec. III.1, and on the
relevant cosmological observables in Sec. III.2. Finally, in Sec. IV we
present the observational constraints on the free cosmological and model
parameters along with a model selection analysis. Finally, we summarise our
findings in Sec. V. Appendix A provides the linear perturbation equations for
the Kinetic model in the Synchronous gauge as well.

##  II Theory

In this Section we will present the theoretical formulation of the kinetically
coupled dark energy model in consideration. We present the covariant
formulation and the corresponding equations in Section II.1, followed by the
background evolution and the framework for linear scalar perturbations in
Sections II.2 and II.3, respectively. We then discuss the parameter space in
Section II.4.

###  II.1 The Kinetic model

Let us start by considering a phenomenological theory minimally coupled to
gravity in the Einstein frame, where the dark energy source is portrayed by a
dynamical quintessence field, $\phi$, interacting with a dark matter component
via the action Barros (2019),

$\mathcal{S}=\int{\rm
d}^{4}x\sqrt{-g}\left[\frac{\text{M}_{\text{Pl}}^{2}}{2}R+X-V(\phi)+{f}(X)\tilde{\mathcal{L}}_{c}(\zeta,g_{\mu\nu})+\mathcal{L}_{\text{SM}}(\psi_{i},g_{\mu\nu})\right]\,,$
(1)

where $g$ denotes the determinant of the metric tensor, $g_{\mu\nu}$, $R$ is
the curvature scalar and ${\text{M}_{\text{Pl}}^{2}=(8\pi G)^{-1}}$ is the
Planck mass in units of $c=1$, with $G$ being the Newtonian constant. The
second and third terms in the action denote the scalar field Lagrangian, in
which ${X=-g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi}/2$ stands for the
kinetic term of $\phi$ and $V(\phi)$ is the scalar self-interacting potential.
In this work we extend the conventional quintessence formulation by taking a
purely kinetic function, $f(X)$ multiplying the Lagrangian of cold dark
matter, $\tilde{\mathcal{L}}_{c}$, which mediates a coupling of $\phi$ to the
dark matter field $\zeta$. Finally,
$\mathcal{L}_{\text{SM}}(\psi_{i},g_{\mu\nu})$ denotes a collective
representation of Lagrangians of the uncoupled standard model fields,
$\psi_{i}$.

Variation of the action in Eq. (1) with respect to the metric $g^{\mu\nu}$
yields the following field equations

$\text{M}_{\text{Pl}}^{2}G_{\mu\nu}=T^{(\phi)}_{\mu\nu}+T^{(c)}_{\mu\nu}+T^{(b)}_{\mu\nu}+T^{(r)}_{\mu\nu}\,,$
(2)

with $G_{\mu\nu}$ being the Einstein tensor and $T^{(i)}_{\mu\nu}$ the energy
momentum tensor for the $i$th species, defined as:

$T^{(i)}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}_{i}\right)}{\delta
g^{\mu\nu}}\,,$ (3)

where $i=\phi,c,b,r$ and $c$ denotes the cold dark matter, $b$ the baryons and
$r$ the radiation. Let us note that, for the previous definition to be valid
for all the fluids present in theory, we define an effective dark matter
Lagrangian as follows Barros (2019); Koivisto (2005); Kase and Tsujikawa
(2020)

${\mathcal{L}_{c}\equiv f(X)\tilde{\mathcal{L}}_{c}},$ (4)

incorporating the effect of the coupling. We follow to consider that all the
matter components in the theory can be modelled as perfect fluids, with energy
density $\rho_{i}$, pressure $p_{i}$, and equation of state (EoS) parameter
$w_{i}=p_{i}/\rho_{i}$. Therefore, the energy momentum tensor of each $i$th
species becomes fully defined in terms of the fluid variables:

$T^{(i)}_{\mu\nu}=\rho_{i}\left[\left(1+w_{i}\right)u^{(i)}_{\mu}u^{(i)}_{\nu}+w_{i}g_{\mu\nu}\right]\,,$
(5)

with $u^{(i)}_{\mu}$ being the 4-velocity vector associated with the $i$th
species, under the individual constraint
$g^{\mu\nu}{u^{(i)}_{\mu}u^{(i)}_{\nu}=-1}$. Regarding the EoS parameter, we
have, $w_{r}=1/3$ for radiation, and $w_{b}=w_{c}=0$ for baryons and cold dark
matter, respectively. In view of these considerations, the dark matter
Lagrangian takes the particular form Koivisto (2005); Avelino and Sousa
(2018),

$\mathcal{L}_{c}=-\rho_{c}\,.$ (6)

The scalar field admits a perfect fluid description as well Faraoni (2012),
provided that

$u^{(\phi)}_{\mu}=-\frac{\partial_{\mu}\phi}{\sqrt{2X}}\,,$ (7)

and $X>0$, where the energy density and pressure associated to the
quintessence field are given by:

$\displaystyle\rho_{\phi}$ $\displaystyle=$ $\displaystyle X+V\,,$ (8)
$\displaystyle p_{\phi}$ $\displaystyle=$ $\displaystyle X-V\,.$ (9)

The scalar field EoS parameter is

$w_{\phi}=p_{\phi}/\rho_{\phi}\,.$ (10)

The equation of motion for the quintessence field, or simply the Klein-Gordon
equation, is obtained through variation of the action in Eq. (1) with respect
to $\phi$ and reads:

$\square\phi-V_{,\phi}=-Q\,,$ (11)

with ${V_{,\phi}=\mathrm{d}V/\mathrm{d}\phi}$. The term on the right-hand side
of Eq. (11) includes the interaction in the dark sector in terms of $f(X)$
Barros (2019), and may be expressed as

$\displaystyle Q$ $\displaystyle=$
$\displaystyle-\mathcal{L}_{c}\left\\{\frac{{f}_{,X}}{{f}}\left[\square\phi+\partial^{\mu}\phi\left(\frac{\nabla_{\mu}\mathcal{L}_{c}}{\mathcal{L}_{c}}+\frac{{f}_{,X}}{{f}}\,\partial_{\alpha}\phi\nabla_{\mu}\partial^{\alpha}\phi\right)\right]-\frac{{f}_{,XX}}{{f}}\,\partial^{\mu}\phi\partial_{\alpha}\phi\left(\nabla_{\mu}\partial^{\alpha}\phi\right)\right\\}\,,$
(12)

where ${f_{,X}\equiv\mathrm{d}f/\mathrm{d}X}$ and
${f_{,XX}\equiv\mathrm{d}^{2}f/\mathrm{d}X^{2}}$. The uncoupled case ($Q=0$)
is naturally recovered when $f$ is a constant function. Let us note that Eq.
(11) could likewise be found through the contracted Bianchi identities,
yielding the following conservation relations,

$\nabla_{\mu}T^{(c)}{}^{\mu}{}_{\nu}=-\nabla_{\mu}T^{(\phi)}{}^{\mu}{}_{\nu}=Q\nabla_{\nu}\phi\,.$
(13)

These equations illustrate clearly the energy transfer between the scalar
field and DM when $f$ is not a constant, meaning that the dark components are
not individually conserved. However, since radiation and baryons remain non-
interacting, i.e.,

$\nabla_{\mu}T^{(r)}{}^{\mu}{}_{\nu}=\nabla_{\mu}T^{(b)}{}^{\mu}{}_{\nu}=0\,,$
(14)

then, consistently, the overall energy momentum tensor of the theory is
conserved, rendering the total action covariant.

In this work, we will focus on the case of a power-law interaction, motivated
in Barros (2019), and parameterised by the function

$f(X)=\left(\text{M}_{\text{Pl}}^{-4}\,X\right)^{\alpha}\,,$ (15)

where $\alpha$ is a dimensionless constant. Therefore Eq. (12) becomes

$Q=-\rho_{c}\frac{\alpha}{X}\left(\square\phi+\frac{\partial^{\mu}\phi\partial_{\nu}\phi\nabla_{\mu}\partial^{\nu}\phi}{X}+\partial^{\mu}\phi\frac{\partial_{\mu}\rho_{c}}{\rho_{c}}\right)\,.$
(16)

From Eq. (16) it is straightforward to conclude that the parameter $\alpha$
governs the strength of the coupling within the dark sector. Additionally, we
fully specify the model by considering the case of an exponential potential,
that is,

$V(\phi)=V_{0}\mathup{e}^{-\lambda\phi/\text{M}_{\text{Pl}}}\,,$ (17)

where $V_{0}$ is the energy scale of the potential (a constant with dimensions
of mass4), and $\lambda$ is a dimensionless parameter depicting the steepness
of the potential. The particular choices in Eqs. (15) and (17) are motivated
by the possibility of having a scaling regime at early times, which is then
followed by a period of accelerated expansion driven by $\phi$ Barros (2019).
In terms of a dynamical systems analysis, the kinetic coupling is indeed
responsible for the emergence of two novel critical points corresponding to
scaling solutions. Finally, the role of the exponential potential is to drive
the evolution of the system out of this scaling regime and towards the late
time attractor.

We conclude this section by remarking that the theory described by the action
Eq. (1) is mathematically equivalent (namely it reproduces the same field
equations and thus leads to equivalent cosmological dynamics) to that of the
following scalar-tensor theory in the Einstein frame Brax and Valageas (2017)

$\mathcal{S}=\int{\rm
d}^{4}x\sqrt{-g}\left[\frac{\text{M}_{\text{Pl}}^{2}}{2}R+X-V(\phi)+\mathcal{L}_{i}(\psi_{i},g_{\mu\nu})\right]+\mathcal{S}_{c}\left[\tilde{g}_{\mu\nu}(X),\zeta\right]\,,$
(18)

where $\tilde{g}_{\mu\nu}$ is the Jordan frame metric. However, it is worth
noting that their physical interpretation differs: while in the action (1) the
coupling is imposed directly through $f$, in the action (18) the metric
$\tilde{g}(X)$ defines a posteriori the coupling. In order for both theories
to give rise to the same cosmological physics, the two metrics must be
conformally related by the following Weyl scaling

$\tilde{g}_{\mu\nu}=f^{2}(X)g_{\mu\nu}\,,$ (19)

with conformal factor given by the square of $f$. Note that the square term
automatically guarantees the signature of the metric to be preserved in both
the Jordan and Einstein frames. These conformally coupled theories can also be
written in terms of a non-minimal coupling to matter in the Einstein frame
Pettorino and Baccigalupi (2008); Amendola (1999); Damour  _et al._ (1990);
nevertheless it is most common to assume a sole field dependence, i.e.,
${\tilde{g}_{\mu\nu}=\Omega(\phi)g_{\mu\nu}}$ Teixeira  _et al._ (2019);
Pettorino and Baccigalupi (2008); Barros  _et al._ (2019). The mapping between
the different formulations still applies as we have assumed in Eq. (6) that
the cold dark matter on-shell Lagrangian can be described by its trace Avelino
and Azevedo (2022); Ferreira  _et al._ (2020); Avelino and Azevedo (2018),
$T^{c}$, more generally,

$\mathcal{L}_{c}=T^{c}\equiv g^{\mu\nu}T^{c}_{\mu\nu}\,.$ (20)

If a different form for the nature of the cold dark matter Lagrangian had been
adopted, departing from the perfect fluid description, then the relation in
Eq. (20) might not hold, in which case the mapping between the theories would
break down. Notice that the same power law coupling, ${f\propto X^{\alpha}}$,
was considered in Ref. Barros and da Fonseca (2022) to couple a quintessence
field to Maxwell’s electromagnetism. It was found that the model dynamics were
mathematically equivalent to a disformally coupled theory. The reason for this
stems from the fact that radiation is conformally invariant (since it has a
vanishing energy-momentum trace ${T^{r}=0}$) thus one needs to consider a more
general Weyl scaling such as to induce an interaction at the level of the
field equations. This clearly shows that the correspondence between the
theories strongly depends on the nature of the matter fields one wishes to
couple the scalar source to.

###  II.2 Background equations

For what concerns the cosmological background dynamics, let us assume a flat
Friedmann–Lemaître–Robertson–Walker (FLRW) metric, expressed in terms of the
conformal time $\tau$, as

$\mathrm{d}s^{2}=a(\tau)^{2}\left(-\mathrm{d}\tau^{2}+\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}\right)\,,$
(21)

where $a\equiv a(\tau)$ is the scale factor of the Universe.

The equations governing the background evolution can be derived from Eq. (2),
more precisely the modified Friedmann equation and the conservation relations,
Eqs. (11), (13) and (14), which become,

$\displaystyle 3\text{M}_{\text{Pl}}^{2}\mathcal{H}^{2}$ $\displaystyle=$
$\displaystyle a^{2}(\rho_{c}+\rho_{b}+\rho_{r}+\rho_{\phi})\,,$ (22)
$\displaystyle\phi^{\prime\prime}+2\mathcal{H}\phi^{\prime}+a^{2}V_{,\phi}$
$\displaystyle=$ $\displaystyle a^{2}Q\,,$ (23)
$\displaystyle\rho_{c}^{\prime}+3\mathcal{H}\rho_{c}$ $\displaystyle=$
$\displaystyle-Q\phi^{\prime}\,,$ (24)
$\displaystyle\rho^{\prime}_{b}+3\mathcal{H}\rho_{b}$ $\displaystyle=$
$\displaystyle 0\,,$ (25)
$\displaystyle\rho^{\prime}_{r}+4\mathcal{H}\rho_{r}$ $\displaystyle=$
$\displaystyle 0\,,$ (26)

where a prime is used to refer to derivatives with respect to conformal time,
$\mathcal{H}=a^{\prime}/a$ is the Hubble rate in conformal time, and the
coupling term in Eq. (16) may now be written as:

$Q=2\alpha\rho_{c}\frac{3\mathcal{H}\phi^{\prime}+a^{2}V_{,\phi}}{2\alpha
a^{2}\rho_{c}+\left(1+2\alpha\right)\phi^{\prime 2}}\,.$ (27)

We can further define the energy density and pressure of the $\phi$ field at
the background level, through Eqs. (8) and (9), as

$\displaystyle\rho_{\phi}$ $\displaystyle=$ $\displaystyle\frac{\phi^{\prime
2}}{2a^{2}}+V\,,$ (28) $\displaystyle p_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\phi^{\prime 2}}{2a^{2}}-V\,,$ (29)

respectively. Therefore Eq. (23) can be written as:

$\rho_{\phi}^{\prime}+3\mathcal{H}(1+w_{\phi})\rho_{\phi}=Q\phi^{\prime}.$
(30)

Equations (24) and (30) imply that, when $Q\phi^{\prime}>0$, energy is being
transferred from the cold dark matter source to the scalar field, and,
accordingly, the opposite holds when $Q\phi^{\prime}<0$, and it is the
$\phi$-field granting energy to cold dark matter. At the classical level the
energy exchange in the dark sector may be interpreted as a mass variation for
dark matter particles, since $m_{c}=a^{3}\rho_{c}$, assuming conservation of
the number of particles, i.e. $N_{c}=N_{c}(\tau_{0})$, with $\tau_{0}$ being
the present conformal time. Integration of Eq. (24) yields an expression for
the total energy density of coupled dark matter,

$\rho_{c}=\rho_{c}(\tau_{0})a^{-3}\exp\left(2\alpha\int_{\tau_{0}}^{\tau}Q\frac{\phi^{\prime}}{\rho_{c}}{\rm
d}\tau\right)\,,$ (31)

that can be expressed equivalently in terms of the mass of the dark matter
particles:

$m_{c}(\tau)=m_{c}(\tau_{0})\exp\left(2\alpha\int_{\tau_{0}}^{\tau}Q\frac{\phi^{\prime}}{\rho_{c}}{\rm
d}\tau\right)\,.$ (32)

Finally let us note that the modified Friedmann equation, Eq. (22), can be
cast to the form of the well-known Friedmann constraint:

$1=\Omega_{\phi}+\Omega_{m}+\Omega_{r}\,,$ (33)

where we have defined a collective matter density
$\rho_{m}=\rho_{c}+\rho_{b}$, and the fractional density parameter of the
$ith$ species
$\Omega_{i}=\rho_{i}a^{2}/(3\text{M}_{\text{Pl}}^{2}\mathcal{H}^{2})$. Eq.
(33) can be rewritten in the form of a constraint on the present scalar field
fractional density, $\Omega_{\phi}^{0}=1-\Omega_{m}^{0}-\Omega_{r}^{0}$, where
“0” stands for quantities evaluated at the present time,
$\Omega_{i}^{0}=\rho_{i}^{0}/(3\text{M}_{\text{Pl}}^{2}H_{0}^{2})$, where
$H_{0}$ is the Hubble parameter. For numerical purposes, $V_{0}$, implicitly
entering the definition of $\Omega_{\phi}^{0}$, is used to perform a shooting
method that yields the fiducial value of $\Omega_{\phi}^{0}$ fulfilling the
constraint relation in Eq. (33), while simultaneously avoiding degeneracies.
As such, $V_{0}$ will no longer be considered a free parameter of the model,
leaving $\\{\lambda,\alpha\\}$ as the model free parameters.

###  II.3 Linear cosmological perturbations

For the purpose of studying the background dynamics, we have assumed that the
Universe is homogeneous and isotropic on large scales. However, we know that
the global picture is far more complex and that, in particular, deviations to
the homogeneous model are needed in order to explain phenomena such as the
formation of structures in the Universe. For the purpose of this study, we
consider small inhomogeneities of the geometry (encoded in the metric) and the
matter fields, and investigate their synergy through the Einstein equations on
linear scales.

Let us consider the perturbed FLRW metric in the so called Newtonian gauge,
corresponding to a line element written as follows Ma and Bertschinger (1995):

$\mathrm{d}s^{2}=a^{2}(\tau)\left[-\left(1+2\Psi\right)\mathrm{d}\tau^{2}+\left(1-2\Phi\right)\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}\right]\,,$
(34)

where $\Psi(\vec{x},\tau)$ and $\Phi(\vec{x},\tau)$ are the Newtonian
potentials. We also consider linear perturbations around the relevant
background fluid variables:

$\displaystyle\phi(\vec{x},\tau)=\phi_{i}(\tau)+\delta\phi(\vec{x},\tau)\,,\quad\rho_{i}(\vec{x},\tau)=\rho_{i}(\tau)+\delta\rho_{i}(\vec{x},\tau)\,,\quad
p_{i}(\vec{x},\tau)=p_{i}(\tau)+\delta p_{i}(\vec{x},\tau)\,.$ (35)

In particular from Eqs. (8) and (9) we derive the perturbations for the energy
density and pressure of the scalar field:

$\displaystyle\delta\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\phi^{\prime}}{a^{2}}\delta\phi^{\prime}-\frac{\phi^{\prime
2}}{a^{2}}\Psi+V_{,\phi}\delta\phi\,,$ (36) $\displaystyle\delta p_{\phi}$
$\displaystyle=$
$\displaystyle\frac{\phi^{\prime}}{a^{2}}\delta\phi^{\prime}-\frac{\phi^{\prime
2}}{a^{2}}\Psi-V_{,\phi}\delta\phi\,.$ (37)

The perturbations of the energy-momentum tensor, Eq. (5), for each species and
at first order, read

$\delta T^{(i)}{}^{\mu}_{\nu}=(\delta\rho_{i}+\delta
p_{i})u^{(i)}{}^{\mu}u^{(i)}{}_{\nu}+\delta
p_{i}\delta^{\mu}_{\nu}+(\rho_{i}+p_{i})\left(\delta
u^{(i)}{}^{\mu}u^{(i)}{}_{\nu}+u^{(i)}{}^{\mu}\delta
u^{(i)}{}_{\nu}\right)\,,$ (38)

where $\delta u^{(i)}{}_{\mu}$ is the perturbation on the four velocity vector
of the $i$th-species, i.e. ${u^{(i)}{}_{\mu}=a(-1,v^{(i)}{}_{j})}$, with
$v_{j}$ being the peculiar velocity. In this study we will assume that there
is no anisotropic stress associated with the fluids under consideration.

Then we compute the linearised Einstein equations where we include the
modifications introduced by the coupling function. These are expressed in
terms of independent Fourier modes that characterise the evolution of the
perturbations for different scales:

$\displaystyle
k^{2}\Phi+3\mathcal{H}\left(\Phi^{\prime}+\mathcal{H}\Psi\right)$
$\displaystyle=$ $\displaystyle-4\pi Ga^{2}\sum_{i}\delta\rho_{i}\,,$ (39)
$\displaystyle k^{2}\left(\Phi^{\prime}+\mathcal{H}\Psi\right)$
$\displaystyle=$ $\displaystyle 4\pi
Ga^{2}\sum_{i}\rho_{i}(1+w_{i})\theta_{i}\,,$ (40)
$\displaystyle\Phi^{\prime\prime}+\mathcal{H}\left(\Psi^{\prime}+2\Phi^{\prime}\right)+\Psi\left(\mathcal{H}^{2}+2\mathcal{H}^{\prime}\right)+\frac{k^{2}}{3}\left(\Phi-\Psi\right)$
$\displaystyle=$ $\displaystyle 4\pi Ga^{2}\sum_{i}\delta p_{i}\,,$ (41)
$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\Psi\,.$ (42)

The first equation, corresponding to the time-time component, provides the
energy density constraint. Equation (40), computed from the time-space
components of the perturbed Einstein equations, gives the momentum constraint,
where we have adopted the definition of the velocity divergence
${\theta_{i}=\nabla\cdot v^{(i)}}$. The trace of the spatial components yields
Eq. (41) and, finally, Eq. (42) corresponds to the shear propagation for
vanishing anisotropic stress. This relation is expected due to the lack of a
non-minimal coupling in action (1).

The equations governing the evolution of each fluid’s perturbations can be
found through the conservation relations, Eqs. (13) and (14), perturbed at
first order. For the non-interacting species, i.e. baryons and radiation,
these are respectively

$\displaystyle\delta^{\prime}_{i}+3\mathcal{H}\left(\frac{\delta
p_{i}}{\delta\rho_{i}}-w_{i}\right)\delta_{i}+(1+w_{i})\left(\theta_{i}-3\Phi^{\prime}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ (43)
$\displaystyle\theta_{i}^{\prime}+\left[\mathcal{H}(1-3w_{i})+\frac{w_{i}^{\prime}}{1+w_{i}}\right]\theta_{i}-k^{2}\left(\Psi+\frac{\delta
p_{i}}{\delta\rho_{i}}\frac{\delta_{i}}{1+w_{i}}\right)$ $\displaystyle=$
$\displaystyle 0\,,$ (44)

where we have defined the dimensionless density contrast as
${\delta_{i}=\delta\rho_{i}/\rho_{i}}$. The dynamics for the coupled cold dark
matter is given by

$\delta_{c}^{\prime}+\theta_{c}-3\Phi^{\prime}=\frac{Q}{\rho_{c}}\left(\phi^{\prime}\delta_{c}-\delta\phi^{\prime}\right)-\frac{\phi^{\prime}}{\rho_{c}}\delta
Q\,,$ (45)

and the corresponding velocity divergence evolves according to

$\theta^{\prime}_{c}+\mathcal{H}\theta_{c}-k^{2}\Psi=\frac{Q}{\rho_{c}}\left(\phi^{\prime}\theta_{c}-k^{2}\delta\phi\right)\,,$
(46)

with the perturbed coupling term, obtained from Eq. (16), being defined as

$\displaystyle\delta Q$ $\displaystyle=$
$\displaystyle\frac{2\alpha\rho_{c}}{2\alpha
a^{2}\rho_{c}+(1+2\alpha)\phi^{\prime
2}}\left\\{-3\Phi^{\prime}\phi^{\prime}-\phi^{\prime}\theta_{c}+\left[3\mathcal{H}\phi^{\prime}+a^{2}(V_{,\phi}-Q)\right]\delta_{c}+\left(2k^{2}+a^{2}V_{,\phi\phi}\right)\delta\phi\right.$
(47) $\displaystyle\left.\hskip
108.12054pt-\left[3\mathcal{H}\phi^{\prime}+2a^{2}(V_{,\phi}-Q)\right]\frac{\delta\phi^{\prime}}{\phi^{\prime}}+2a^{2}\Psi\left(Q-V_{,\phi}\right)\right\\}\,,$

with $V_{,\phi\phi}=\mathrm{d}^{2}V/\mathrm{d}\phi^{2}$. One exceptional
feature of the Kinetic model can be readily identified at the level of the
perturbed coupling parameter, Eq. (47): it includes an explicit dependence on
$\theta_{c}$. This is not usual in other coupled dark energy models explored
so far, such as in Refs. van de Bruck  _et al._ (2017); van de Bruck and
Teixeira (2020), and it arises due to the $X$-dependence of the coupling, in
particular in relation to the term containing ${\nabla_{\mu}\mathcal{L}_{c}}$
in Eq. (16).

The evolution of the $\phi$-field perturbation is given by the linearisation
of Eq. (11):

$\delta\phi^{\prime\prime}+2\mathcal{H}\delta\phi^{\prime}+\left(a^{2}V_{,\phi\phi}+k^{2}\right)\delta\phi-\left(\Psi^{\prime}+3\Phi^{\prime}\right)\phi^{\prime}+2a^{2}\Psi
V_{,\phi}=a^{2}\delta Q+2a^{2}Q\Psi\,.$ (48)

For completeness we also provide the corresponding set of equations in the
synchronous gauge, see Appendix A.

In Sections III and IV we will evolve the full dynamics of the gravitational
potentials, scalar and matter fields. However in order to have a glimpse at
the phenomenology of the Kinetic model we resort to the so-called quasi-static
approximation (QSA) on sub-horizon scales Boisseau  _et al._ (2000); Tsujikawa
(2007); De Felice  _et al._ (2011). Under this approximation we find:

$\displaystyle
k^{2}\Psi\approx-\frac{3\mathcal{H}^{2}}{2}\big{(}\Omega_{c}\delta_{c}+\Omega_{b}\delta_{b}\big{)}\,,$
(49) $\displaystyle k^{2}\delta\phi\approx
a^{2}\Delta\bigg{\\{}\delta_{c}\Big{[}Q\phi^{\prime
2}+\rho_{c}\big{(}\phi^{\prime\prime}-\mathcal{H}\phi^{\prime}\big{)}\Big{]}-\rho_{c}\phi^{\prime}\delta_{c}^{\prime}\bigg{\\}}\,,$
(50)

where we have defined the following time and scale dependent functions:

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle-\frac{2\alpha\mathcal{M}^{2}}{\phi^{\prime
2}V_{\phi\phi}\left(1+\frac{a^{2}}{k^{2}}\mathcal{M}^{2}\right)}\,,$ (51)
$\displaystyle\mathcal{M}^{2}$ $\displaystyle=$
$\displaystyle\frac{\phi^{\prime 2}V_{\phi\phi}}{\phi^{\prime 2}-2\alpha
a^{2}\rho_{c}}\,.$ (52)

It follows that the equations for cold dark matter and baryonic fluids read

$\displaystyle\delta_{c}^{\prime\prime}+\mathcal{H}\big{(}1+\beta\big{)}\delta_{c}^{\prime}-\frac{3\mathcal{H}^{2}}{2G}\big{(}G_{cc}\Omega_{c}\delta_{c}+G_{cb}\Omega_{b}\delta_{b}\big{)}\approx
0\,,$ (53)
$\displaystyle\delta_{b}^{\prime\prime}+\mathcal{H}\delta^{\prime}_{b}-\frac{3\mathcal{H}^{2}}{2}\big{(}\Omega_{c}\delta_{c}+\Omega_{b}\delta_{b}\big{)}\approx
0\,,$ (54)

with

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{G_{cb}}{G\mathcal{H}\rho_{c}}\bigg{\\{}\big{(}1-a^{2}\rho_{c}\Delta\big{)}\Big{[}8\alpha\mathcal{H}\rho_{c}-Q\phi^{\prime}\big{(}1+2\alpha\big{)}\Big{]}-Q\phi^{\prime}\bigg{\\}}\,,$
(55) $\displaystyle G_{cc}$ $\displaystyle=$ $\displaystyle
G+\frac{2G}{3\mathcal{H}\rho_{c}\Omega_{c}\phi^{\prime
2}}\bigg{\\{}\phi^{\prime
2}\Big{[}\phi^{\prime}Q^{\prime}+Q\big{(}\phi^{\prime\prime}+4\mathcal{H}\phi^{\prime}\big{)}\Big{]}+\Big{[}Q\phi^{\prime
2}+\rho_{c}\big{(}\phi^{\prime\prime}-\mathcal{H}\phi^{\prime}\big{)}\Big{]}$
(57) $\displaystyle\times\Big{[}\phi^{\prime
2}Qa^{2}\Delta+2\alpha\big{(}\phi^{\prime\prime}+4\mathcal{H}\phi^{\prime}\big{)}\big{(}1-a^{2}\rho_{c}\Delta\big{)}\Big{]}\Bigg{\\}}\,,$
$\displaystyle G_{cb}$ $\displaystyle=$
$\displaystyle\frac{G}{1+2\alpha\big{(}1-a^{2}\rho_{c}\Delta\big{)}}\,.$ (58)

The cold dark matter perturbations are then modified by two effects emerging
from the coupling: a modified friction term, quantified by $\beta$, which
inevitably influences the growth rate of $\delta_{c}$; and a modified
effective gravitational potential encoded in

$\nabla^{2}\Psi^{\rm
eff}=4\pi\big{(}G_{cc}\rho_{c}\delta_{c}+G_{cb}\rho_{b}\delta_{b}\big{)}\,.$
(59)

The latter includes two effective gravitational couplings, $G_{cc}$ and
$G_{cb}$, defined in analogy to Kase and Tsujikawa (2020). We also find that
$G_{cb}$ is always an attractive contribution. These modifications clearly
show the emergence of a fifth force which is a standard signature of coupled
scalar field models. We expect that even a relative small value for the
coupling parameter $\alpha$ can lead to a significant effect on the
cosmological observables given the evolution equation for the cold dark matter
perturbations, which impacts the baryons dynamics and the gravitational
potentials. Among others we foresee a modification in the lensing angular
power spectrum due to a modified lensing potential
($\phi_{lens}=(\Phi+\Psi)/2=\Psi$). We will explore these signatures in more
detail in Section III. In the absence of the coupling, i.e. $\alpha=0$, we
recover $\beta=0$ and $G_{cc}=G_{cb}=G$, corresponding to the standard case of
quintessence.

###  II.4 The parameter space

In order for a model to be theoretically viable, there are specific stability
requirements that need to be satisfied. We will present and examine them below
for the Kinetic model. Let us stress that the identification of a physically
motivated parameter space plays an important role when testing particular
gravity models with cosmological data Raveri  _et al._ (2014); Frusciante  _et
al._ (2016); Salvatelli  _et al._ (2016); Frusciante  _et al._ (2019b, 2020);
Frusciante and Perenon (2020); Albuquerque  _et al._ (2022).

According to the results in Ref. Barros (2019) the solutions with
$\lambda^{2}<2$ guarantee that the future dark energy attractor is a stable
fixed point of the system and describes an accelerated expanding Universe.
Albeit necessary at the attractor, this condition can be somewhat relaxed,
while still generating an accelerating behaviour at transient times. By
allowing the attractor to lie outside, but close to the accelerated region,
with say ${\lambda^{2}=2+\epsilon}$, the solution may still feature an
accelerated expanding scenario at present time, that is, with
${w_{\phi}(a_{0})<-1/3}$. We further discuss this point in Section III.1 and
we show some examples in the right panel of Fig. 3. Let us note that under
such condition, instead of accelerating forever, there should be a turning
point in the future when the expansion changes from accelerated to
decelerated, i.e. ${w_{\phi}(a_{x})=-1/3}$ at the crossover $a_{x}$, and
${w_{\phi}(a)>-1/3}$ thereafter, for $a<a_{x}$, as the attractor is
approached. From the critical points analysis conducted in Ref. Barros (2019)
we know that at the attractor ${w^{\star}_{\rm eff}=\lambda^{2}/3-1}$, which
using the following general identity

$\frac{1}{H}\frac{\mathrm{d}H}{\mathrm{d}\ln a}=-\frac{3}{2}\left(1+w_{\rm
eff}\right)\,,$ (60)

we find for the Hubble rate at the attractor:

$\frac{\mathrm{d}H^{\star}}{\mathrm{d}\ln
a}=-\frac{1}{2}H^{\star}\lambda^{2}\,,$ (61)

from which the accelerated condition is derived Barros (2019). Here we define
a star superscript denoting quantities evaluated at the attractor, and
$H=\mathcal{H}/a$ is the Hubble function in cosmic time, $t$. The relation
above corresponds to a cosmological expanding behaviour described as,

$H^{\star}=H_{0}a^{-\lambda^{2}/2}\quad\text{and therefore}\quad
a^{\star}=\left(1+H_{0}\frac{\lambda^{2}}{2}t\right)^{2/\lambda^{2}}\,.$ (62)

Indeed, the explicit time dependence of the scale factor in Eq (62), $a\propto
t^{2/\lambda^{2}}$, reveals that $\lambda^{2}=2$ is an inflection point of
$a(t)$, i.e. $\ddot{a}=0$ (with dots referring to derivatives with respect to
cosmic time), laying out the fine limit between an accelerated or decelerated
setting. Following the above discussion we will then consider $\lambda>0$.
Additionally, according to the power-law role of $\alpha$ in Eq. (15), we
choose to consider cases with $\alpha\geqslant 0$ only Barros (2019).

Furthermore, we take into account theoretical stability conditions to
guarantee the absence of ghost and gradient instabilities in the scalar sector
Sbisà (2015); Kase and Tsujikawa (2020). The first demands for positive
kinetic terms of the scalar field and cold dark matter perturbations
($q_{s}>0$ and $q_{c}>0$, respectively), and the second for their positive
speeds of propagation ($c_{s}^{2}\geqslant 0$ and $c_{c}^{2}\geqslant 0$). It
is possible to show that a very general way to write an action with an extra
scalar field and one matter component up to second order in perturbations, is
the following Kase and Tsujikawa (2020):

$\mathcal{S}^{(2)}=\int{}dtdk^{3}a^{3}\left[\dot{\vec{\chi}}^{t}{\bf
K}\dot{\vec{\chi}}-\frac{k^{2}}{a^{2}}\vec{\chi}^{t}{\bf
G}\vec{\chi}-\vec{\chi}^{t}{\bf M}\vec{\chi}-\frac{k}{a}\vec{\chi}^{t}{\bf
B}\dot{\vec{\chi}}\right]$ (63)

with $\vec{\chi}^{t}=(\delta\phi,\delta\rho_{c}/k)$ being $\delta\phi$ and
$\delta\rho_{c}$ the perturbations of the scalar field and cold dark matter
component respectively, the $2\times 2$ matrices are defined in terms of
background quantities and their general forms can be found in ref. Kase and
Tsujikawa (2020). For the action (1), with $f(X)$ and $V(\phi)$ defined in
Eqs. (15) and (17) respectively we have Kase and Tsujikawa (2020):

$\displaystyle q_{s}$ $\displaystyle=$ $\displaystyle
K_{11}=2\text{M}_{\text{Pl}}^{2}\left[1-\frac{\alpha(2\alpha-1)}{X}\rho_{c}\right]\,,$
(64) $\displaystyle q_{c}$ $\displaystyle=$ $\displaystyle
K_{22}=\left(\text{M}_{\text{Pl}}^{-4}X\right)^{\alpha}\,,$ (65)
$\displaystyle c_{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{G_{11}}{K_{11}}+\frac{B_{12}^{2}}{K_{11}K_{22}}=\frac{4\text{M}_{\text{Pl}}^{2}}{q_{s}}-1\,,$
(66) $\displaystyle c_{c}^{2}$ $\displaystyle=$
$\displaystyle\frac{G_{22}}{K_{22}}=0\,.$ (67)

For dark matter the conditions are trivial. The stability conditions for the
scalar field are more involved and need to be verified throughout the entire
expansion history. We find that both conditions are verified as long as:

$-1<\alpha(2\alpha-1)\frac{\rho_{c}}{X}<1\,,$ (68)

where the first inequality accounts for the no-ghost condition and the second
one for the positive (square) speed of propagation. This constraint then
selects the viable range for the parameter $\alpha$. The initial condition
$\phi^{\prime}_{i}$ plays a role in securing the stability toward the
cosmological evolution.

Let us now discuss the initial conditions (ICs) for the scalar field,
$\phi_{i}$ and its first derivative, $\phi^{\prime}_{i}$ which must be
specified, deep in the radiation dominated epoch, namely around redshift
$z_{i}\approx 10^{14}$ in order to solve the system of equations (23)-(26).

From the numerical study we concluded that for non-trivial ICs, the system
rapidly enters in the scaling regime. According to this feature we found that
the choice of values for $\phi_{i}$ and $\phi^{\prime}_{i}$ has a negligible
impact on the cosmological evolution111We have numerically verified that the
phenomenology of the cosmological observables, as discussed in Section III, is
not affected by the choice of ICs and neither are the cosmological
constraints.. Hence, without loss of generality, we set
$\phi(z_{i})=10^{-2}\,\text{M}_{\rm Pl}$ .

The IC for $\phi^{\prime}$ is chosen such as to avoid instabilities according
to Eq. (68). Moreover, it should be noted that when $\phi^{\prime}_{i}$ is
chosen to be positive, the condition $\lambda>0$ must hold for the
accelerating attractor solution to exist Barros (2019).

Finally, we recall that $V_{0}$ is not considered an extra parameter of the
model as discussed in Section II.2.

##  III Phenomenology of the Kinetic model

In this Section we shall explore the signatures left by the Kinetic model on
the background expansion and on some cosmological observables such as the
cosmic microwave background (CMB), the lensing potential auto-correlation and
the matter power spectra in Sections II.2 and II.3, respectively. We use our
own modification of the public version of the Einstein Boltzmann solver CLASS
Lesgourgues (2011a); Blas  _et al._ (2011); Lesgourgues (2011b).

(a)  (b)

Figure 1:  Left upper panel: Evolution of the energy densities $\rho_{i}$ with
redshift, $1+z$, of the scalar field (pink), matter (black) and radiation
(blue) for the uncoupled case (solid line), ${\alpha=0.01}$ (dotted line) and
${\alpha=0.03}$ (dashed line). Left lower panel: Ratio of the energy densities
of cold dark matter and dark energy, for ${\alpha=0.01}$ (solid line) and
${\alpha=0.03}$ (dashed line). Right panel: Differences relative to the
uncoupled case, ${\alpha=0}$, for ${\alpha=0.01}$ (solid line) and
${\alpha=0.03}$ (dashed line), on the quantities (from top to bottom): the
coupling strength parameter; the fractional deviation of the energy density of
matter, i.e. ${\Delta\rho_{m}/\tilde{\rho}_{m}=\rho_{m}/\tilde{\rho}_{m}-1}$,
where a tilde denotes variables in the uncoupled scenario, such that
${\tilde{\rho}_{m}=\rho^{0}_{m}a^{-3}}$; and the fractional deviation of the
Hubble rate of expansion.

###  III.1 Background evolution

We start by reviewing the background evolution in the kinetic coupled dark
sector scenario. A similar study has been previously presented in Ref. Barros
(2019) by means of a dynamical systems analysis, with a particular focus on
the late time dynamics (i.e. cosmological redshift $z\lesssim 40$), neglecting
the radiation and baryonic contribution. In this work we shall examine the
cosmological evolution starting from the early stages, deep into the radiation
dominated epoch ($z_{i}\approx 10^{14}$) up to present time ($z=0$). For the
numerical investigation in this work we fix the following cosmological
parameters to be Aghanim  _et al._ (2020a): ${H_{0}=67.56}$ km/s/Mpc,
${\Omega_{b}h^{2}=0.022}$ and ${\Omega_{c}h^{2}=0.12}$, with ${h\equiv
H_{0}/100}$. We also select some exemplifying values for the parameter
$\alpha$ controlling the coupling, namely ${\alpha=0.01}$ and ${\alpha=0.03}$,
and we fix the slope of the potential as ${\lambda=0.2}$, with the aim of
singling out the main phenomenology associated to the coupling function.
Moreover, for comparison purposes, we also include the case with $\alpha=0$,
which corresponds to an uncoupled scenario. It should be noted that the choice
for the values of the parameters associated with the scalar field are purely
illustrative, but nevertheless still satisfy the requirements discussed in
Section II.4. They are chosen in such a way that the overall effect of the
coupling can be grasped, and therefore are not necessarily realistic. This
will be assessed in Section IV, in which case these parameters are left to
vary when performing a parameter estimation according to cosmological data.

In the left panel of Fig. 1 we show the evolution with redshift, $1+z$, of the
energy densities for each species, $\rho_{i}$. We notice that the introduction
of the coupling results in the emergence of an early scaling regime, in direct
contrast with the uncoupled case, for which this behaviour can never be
achieved. The onset of this scaling behaviour takes place during the radiation
dominated epoch, with energy density of the scalar field proportional to the
dark matter one, approximately according to the relation
${\rho_{c}/\rho_{\phi}=1/\alpha}$, as shown in the left lower panel of Fig. 1.
Eventually the field will exit this scaling regime and head towards the future
attractor solution, in which case its energy density will remain forever
diluting as ${\rho_{\phi}\propto a^{-\lambda^{2}}}$.

In the upper right panel of Fig. 1 we show the evolution of the coupling
strength, expressed as $Q\phi^{\prime}/\rho_{c}$, as a function of the
redshift. The sign of this quantity is relevant to assess the direction of the
energy flow between cold dark matter and the scalar field.

We can notice that the interaction term is positive at all redshifts,
establishing the direction of the energy transfer from the dark matter fluid
to the scalar field. This is consistent with the fact that the dynamics of the
scalar field follows the relation ${\phi^{\prime}>\lambda
V/\left(3\text{M}_{\text{Pl}}^{2}\mathcal{H}\right)\Leftrightarrow Q>0}$ (see
Eq. (27)). Let us note that because we fixed the present day values of the
fluid densities this results in a larger value for the cold dark matter energy
density at early times because it is the CDM component granting energy to the
scalar field at later times, with this feature being more prominent for higher
values of $\alpha$. This effect is compensated as the matter energy density
decreases throughout time, while additional energy is being transferred for
the scalar field, when compared with the uncoupled case. We illustrate this
behaviour in the middle right panel of Fig. 1, where we report on the
deviations from the uncoupled case, denoted by a tilde. As consequence there
is a shift of the matter-radiation equality towards earlier times for
increasing values of $\alpha$, as shown in the left panel of Fig. 2. From the
same Figure, we can notice that because the $\phi$ field is acquiring energy
at a rate that is proportional to its energy density (see Eq. (27)), then the
matter-dark energy equality is achieved earlier.

Additionally, in the lower right panel of Fig. 1, we show the deviations in
the Hubble rate for the Kinetic model when compared with the uncoupled case,
i.e.
${\Delta\mathcal{H}/\tilde{\mathcal{H}}=\mathcal{H}/\tilde{\mathcal{H}}-1}$.
No significant deviations on $\mathcal{H}$ are observed during the radiation
dominated epoch, since any interactions between the dark and radiation sectors
have been excluded. However, when the matter contribution becomes non-
negligible, around ${z\approx 10^{6}}$, the Kinetic models show an enhanced
value of $\mathcal{H}$ with respect to the uncoupled case, with this effect
being larger for the higher values of $\alpha$.

(a)  (b)

Figure 2:  Left panel: Evolution of the relative energy densities $\Omega_{i}$
with redshift, $1+z$, of the scalar field (pink), matter (black) and radiation
(blue). Right panel: Equation of state parameters, $w_{\rm eff}$ (pink) and
${w_{\phi}}$ (black), along redshift. In accordance with Fig. 1, we present
the uncoupled case (solid line), ${\alpha=0.01}$ (dotted line) and
${\alpha=0.03}$ (dot-dashed line).

Finally, it is also worth analysing the evolution of two fluid-related
quantities: the equation of state parameters for the scalar field, $w_{\phi}$,
and for the total effective budget, $w_{\rm eff}$. These characterise the
nature of the dark energy fluid description and the overall effective
dominating fluid contribution in the Universe, and are defined according to
Eq. (10) and

$w_{\rm eff}=\frac{\sum_{i}p_{i}}{\sum_{i}\rho_{i}}\,,$ (69)

respectively. Their evolution with redshift is depicted in the right panel of
Fig. 2. We observe that during the scaling regime the field behaves as a stiff
fluid, with ${w_{\phi}=1}$, since ${V\ll\phi^{\prime 2}}$, and in agreement
with the findings of Ref. Barros (2019). As the field exits the scaling
regime, the Universe approaches the attractor scenario, for which
${w_{\phi}=-1+\lambda^{2}/3}$. During radiation domination, the effective
equation of state remains at a plateau with ${w_{\rm eff}\approx w_{r}=1/3}$.
At matter domination, and during the scaling regime, when radiation may be
neglected and under the limit ${V\ll\phi^{\prime 2}}$, the equation of state
follows

$w_{\rm
eff}\approx\frac{\alpha}{1+\alpha\left(1+\frac{\rho_{b}}{\rho_{\phi}}\right)}\,.$
(70)

Note that, in Ref. Barros (2019), a similar approximation was presented,
though stated as ${w_{\rm eff}\approx\alpha/(1+\alpha)}$. That is because the
contribution of radiation and baryons was not taken into account in that
study, which focused mainly on the late time dynamics, for which it still
stands as a good approximation. By neglecting the baryonic contribution we may
resort to the dynamical system analysis employed in Ref. Barros (2019) to find
the behaviour of the Hubble rate and coupled DM at matter domination during
the scaling:

$\rho_{c}\propto H^{2}\propto a^{-3\frac{1+2\alpha}{1+\alpha}}\,,$ (71)

which we numerically verified to be a good approximation. We remark that the
transition towards an accelerating state occurs later for increasingly larger
values of $\alpha$, owing to the fact that, for a stronger interaction, the
field remains frozen in the scaling regime for longer, with ${w_{\phi}=1}$. As
a direct outcome, when the accelerating stage finally starts (that is, when
$w_{\rm eff}<-1/3$), it will take place at a slower rate. This behaviour is
illustrated in the right panel of Fig. 2. Alternatively, this trend could be
intuitively understood by inspection of the deceleration parameter
${q=(1+3w_{\rm eff})/2}$, that scales linearly with the total equation of
state parameter of the Universe.

At this point, there is a subtlety that should be noted. Although the ICs for
the scalar field do not have any influence on the parameter constraints, there
is a link between the initial values of the velocity of the field and the dark
energy density, as expressed in Eq. (8), which will have a subtle impact on
the early behaviour of the quintessence. Increasing the initial density of the
field inevitably leads to an earlier onset for the scaling regime by taking
higher values of $\phi^{\prime}_{i}$. On the other hand, the value for the
initial velocity is completely negligible when it comes to setting the time
for which the field exits the scaling and starts evolving towards the
accelerating attractor. This implies that the duration of the period in which
the energy density of dark energy scales with matter is extended for
increasing values of $\phi^{\prime}_{i}$. This trend is illustrated in the
left panel of Fig. 3. Nonetheless this does not mean that $\phi^{\prime}_{i}$
can take any arbitrary value, as the conditions in Eq. (68) still have to be
verified, in order to avoid instabilities in the theory. On the other hand,
the initial value for the field per se has no influence over the dynamics.
Indeed $\phi_{i}$ only appears in the exponential term of the potential, Eq.
(17), which can be equivalently absorbed by the shooting parameter $V_{0}$.

Finally, we conclude by providing some concrete examples to support the
argument in Section II.4, namely that values of $\lambda^{2}>2$ can still give
rise to present time accelerated expansion under exceptional conditions. In
the right panel of Fig. 3 we illustrate the behaviour of the effective
equation of state parameter close to the present epoch and up to some time in
the future for different values of $\lambda$, and for a fixed coupling
parameter, ${\alpha=0.03}$. Indeed we notice that transient acceleration
phases around the present time are achieved for ${\lambda^{2}>2}$, before
crossing the boundary given by $w_{\rm eff}<-1/3$, and exiting this region at
some point in the future. Accordingly, these solutions may still stand as
cosmologically valid, and such values for $\lambda$ need to be taken into
account in the statistical analysis of Sec. IV.

(a)  (b)

Figure 3:  Left Panel: Evolution of the energy densities of matter (black) and
scalar field (pink) for different ICs for the field’s velocity,
$\phi^{\prime}_{i}$, with fixed $\alpha=0.03$ and $\lambda=0.2$. Right Panel:
Effective equation of state, $w_{\rm eff}$, for different values of $\lambda$,
namely $\lambda=1.4$ (solid line), $\lambda=1.6$ (dashed line) and
$\lambda=1.8$ (dotted line), with fixed $\alpha=0.03$. The shaded green area
corresponds to the region where the Universe features accelerated expansion,
i.e. ${w_{\rm eff}<-1/3}$.  (a)  Figure 4:  Upper panel: The matter power
spectrum as function of $k$, for the uncoupled case (dashed line),
${\alpha=0.001}$ (dot-dashed line), ${\alpha=0.002}$ (dotted line) and
$\Lambda$CDM (pink solid line). Lower panel: Percentage deviations of the
matter power spectrum of the Kinetic model and the uncoupled case from the
$\Lambda$CDM model. (a)  Figure 5:  Evolution of the density contrast of cold
dark matter for the Kinetic model relative to the $\Lambda$CDM case, that is
$\delta_{c}/\delta^{\Lambda CDM}_{c}$, as a function of the Fourier scale $k$
and the redshift $z$ for $\alpha=0.001$ and $\alpha=0.002$.

###  III.2 Cosmological observables

In this section we discuss the effect of the coupling on some relevant
cosmological observables such as the matter power spectrum and the CMB
temperature-temperature (TT) and lensing angular power spectra. We assume
adiabatic perturbative initial conditions with an amplitude of curvature
fluctuations of $A_{s}=2.215\times 10^{-9}$, at the pivot scale $k_{\rm
piv}=0.05$ Mpc-1, and with the spectral index set to $n_{s}=0.962$ Aghanim
_et al._ (2020a). The remaining cosmological parameters and $\lambda$ are the
same as used in the previous Section. We adopt a different set of values for
$\alpha$, which are one order of magnitude smaller than the ones used in the
numerical analysis of the background quantities, with the reason being that
the latter would lead to drastic effects on the cosmological observables. On
the contrary, the values we will use to highlight the features on cosmological
observables do not produce any significant effects on the background
quantities. As a consequence the features we will show in this Section are
attributed solely to the modifications to the linear perturbation equations
presented in Section II.3. Then, for illustrative purposes, we set $\alpha$ to
be $1\times 10^{-3}$ and $2\times 10^{-3}$. Moreover, and without loss of
generality, we assume vanishing ICs for the scalar field perturbation and its
velocity, that is, $\delta\phi(z_{i})=\delta\phi^{\prime}(z_{i})=0$,
respectively.

In the upper panel of Fig. 4 we present the linear matter power spectrum at
present time up to the scale $k_{\rm max}=0.1h$ Mpc-1, above which the linear
perturbative approximation is expected to break down due to non-linear
effects, dominant at smaller scales. In the lower panel we also plot the
fractional differences between the coupled scenarios and the $\Lambda$CDM one.
We note that the matter power spectrum of the Kinetic model is significantly
suppressed at intermediate scales, $10^{-3}h$ Mpc${}^{-1}\lesssim k\lesssim
3\times 10^{-2}h$ Mpc-1, with respect to $\Lambda$CDM, and enhanced at the
smaller scales. These signatures emerge as a combination of the effects
produced by the changes in the evolution of the background and the cold dark
matter perturbations due to the positive exchange of energy that flows from
cold dark matter to dark energy. Because the radiation-matter equality era is
shifted towards earlier times, when compared with the uncoupled case (see left
panel of Fig. 2), the turnover in the matter power spectrum is shifted to
higher $k$. The growth of the matter perturbations is suppressed at
intermediate scales, with deviations from $\Lambda$CDM of $\sim 7\%$ and $\sim
14\%$ for $\alpha=0.001$ and $\alpha=0.002$, respectively, and enhanced at the
smaller scales, with deviations that can reach $\sim 45\%$ for $\alpha=0.002$.
This is illustrated in Fig. 5, where we can clearly see that the largest
deviations occur for scales $0.01h<k<0.1h$ Mpc-1 and at large redshift, with
some milder modifications close to present time as well for $k\sim 0.1h$
Mpc-1. At larger scales $k\sim 0.01h$ Mpc-1 the deviations are more
accentuated at intermediate redshifts ($z\sim 10$). The plots also show that
as expected the largest deviations are present for the higher values of
$\alpha$. As a consequence, the value of the amplitude of the matter power
spectrum at present time and scale of 8 $h^{-1}$Mpc, denoted by $\sigma_{8}$,
is expected to be larger for the Kinetic model.

(a)  (b)

Figure 6:  Left panel: (Top) Evolution of the sum of the gravitational
potentials as a function of the redshift at $k=0.01$ Mpc-1 for the cases:
uncoupled model (dashed line), $\alpha=0.001$ (dot-dashed line),
$\alpha=0.002$ (dotted line), and $\Lambda$CDM (pink solid line). (Bottom)
Relative percentage difference of $\Psi+\Phi$ computed with respect to
$\Lambda$CDM. Right panel: (Top) Evolution of the time derivative of the sum
of the gravitational potentials as a function of the redshift (Bottom)
Relative percentage difference of $\Psi^{\prime}+\Phi^{\prime}$ computed with
respect to $\Lambda$CDM. (a)  Figure 7:  Upper panel: Lensing angular power
spectra for $\Lambda$CDM (solid pink line), $\alpha=0.001$ (dot-dashed line),
$\alpha=0.002$ (dotted line) and the uncoupled case (dashed line). Lower
panel: Relative difference between the lensing power spectra of each model and
that of $\Lambda$CDM. (a)  Figure 8:  Upper panel: TT power spectrum as
function of the angular scale $\ell$, for the uncoupled case (dashed line),
${\alpha=0.001}$ (dot-dashed line), ${\alpha=0.002}$ (dotted line) and
$\Lambda$CDM (pink solid line) for reference. Lower panel: Percentage
deviations of the TT power spectra for the coupled and uncoupled cases with
respect to $\Lambda$CDM.

In Fig. 6 we show the sum of the gravitational potentials $\Phi+\Psi$ (left
panel) and their time derivative (right panel), as a function of the redshift,
for a fixed scale, $k=0.01$ Mpc-1. The evolution of the potentials is
regulated according to the Poisson equation. We can infer that the value of
the lensing potential, given by $\phi_{\text{lens}}=(\Psi+\Phi)/2$, is lower
in the Kinetic model when compared to the standard cosmological scenario,
resulting in a suppression of the lensing power spectrum, as shown in Fig. 7.
This effect becomes increasingly evident for larger value of $\alpha$. The
quantity $\Psi^{\prime}+\Phi^{\prime}$ instead is directly connected with the
integrated Sachs-Wolfe effect (ISW). The latter affects the shape of the TT
power spectrum as it enters in the radiation transfer function. The total ISW
effect is divided into: an early time contribution, produced during the
transition from radiation to matter dominated epochs, which in the Kinetic
model is shifted towards earlier times when compared to the standard scenario;
and a late time contribution, related with the presence of the dark energy
component. The impact of the ISW effect on the TT power spectrum is
illustrated in Fig. 8, as a function of the angular multipole $\ell$,
exhibiting an overall enhancement with respect to the reference case for
$\ell\lesssim 300$. While milder differences are identified around the plateau
at $\ell<10$, significant deviations can be appreciated around $10<\ell<200$,
in particular for $\ell\sim 50$, being as large as $\sim 40\%$ for
$\alpha=0.002$. Moreover, there is a clear increase in the amplitude of the
first peak, accompanied by a broadening of its shape. Likewise, the presence
of the coupling and the modifications to the background expansion also induce
small differences between the peaks and troughs at the higher multipoles.

These effects can be measured using cosmological data from background and
large-scale structure.

##  IV Cosmological constraints and model selection analysis

In this Section we present the constraints on the cosmological and model
parameters of the Kinetic model for different combinations of data sets. We
perform a Bayesian Monte Carlo Markov Chain (MCMC) analysis using the
Metropolis-Hastings algorithm implemented in the Monte
Python222https://github.com/brinckmann/montepython_public sampler Audren  _et
al._ (2013); Brinckmann and Lesgourgues (2019) interfaced with our personal
modified version of CLASS333https://github.com/lesgourg/class_public
Lesgourgues (2011a); Blas  _et al._ (2011); Lesgourgues (2011b). The general
aim is to estimate the sample posteriors that maximise the likelihood
associated to each data set, therefore minimising the statistical error
distribution. Subsequently, we analyse the MCMC chains and produce the results
reported in Tables 2 and 3, and in Figures 9, 10, 11, and 12, resorting to the
GetDist444https://github.com/cmbant/getdist Python package Lewis (2019). For
comparison purposes we also report on the constraints derived for the standard
cosmological scenario. Finally we examine whether the Kinetic model is
supported by the data over $\Lambda$CDM.

Parameter | Prior
---|---
$\Omega_{b}h^{2}$ | $[0.005,0.1]$
$\Omega_{c}h^{2}$ | $[0.001,0.99]$
$100\theta_{s}$ | $[0.5,10]$
$z_{reio}$ | $[0.,20.]$
$n_{s}$ | $[0.7,1.3]$
$\log\left(10^{10}A_{s}\right)$ | $[1.7,5.0]$
$\lambda$ | $[0,2]$
$\alpha$ | $[0,1]$
Table 1: Flat priors on the cosmological and model parameters sampled in this
work. Kinetic Model
---
Parameter | Plk18 | Plk18+BAO+SN | Plk18+BAO+SN+len
$S^{0}_{8}$ | $0.793^{+0.110}_{-0.064}$ | $0.875^{+0.037}_{-0.043}$ | $0.863^{+0.030}_{-0.039}$
$\Omega^{0}_{m}$ | $0.257^{+0.045}_{-0.025}$ | $0.2988^{+0.0072}_{-0.0036}$ | $0.2982^{+0.0070}_{-0.0035}$
$H_{0}$ | $64.0^{+3.3}_{-1.8}$ | $67.14\pm 0.62$ | $66.94^{+0.60}_{-0.54}$
$10^{-9}A_{s}$ | $2.088\pm 0.035$ | $2.096\pm 0.035$ | $2.111\pm 0.031$
$n_{s}$ | $0.9667\pm 0.0047$ | $0.9669\pm 0.0044$ | $0.9655\pm 0.0041$
$\lambda$ | $1.11\pm 0.48$ | $0.42^{+0.18}_{-0.21}$ | $0.41^{+0.17}_{-0.22}$
$10^{4}\alpha$ | $1.88\pm 0.95$ | $1.37^{+0.67}_{-1.00}$ | $1.05^{+0.51}_{-0.87}$
Table 2: $68\%$ C.L. bounds on the cosmological and model parameters for the
Kinetic model for the three different combinations of data sets: Planck,
Planck combined with BAO and SN, and their full combination with CMB lensing.
$\Lambda$CDM Model
---
Parameter | Plk18 | Plk18+BAO+SN | Plk18+BAO+SN+len
$S^{0}_{8}$ | $0.833\pm 0.016$ | $0.831^{+0.013}_{-0.015}$ | $0.834\pm 0.013$
$\Omega^{0}_{m}$ | $0.3163\pm 0.0085$ | $0.3151^{+0.0060}_{-0.0075}$ | $0.3162\pm 0.0073$
$H_{0}$ | $67.31\pm 0.61$ | $67.39^{+0.53}_{-0.45}$ | $67.32\pm 0.53$
$10^{-9}A_{s}$ | $2.102\pm 0.034$ | $2.102\pm 0.034$ | $2.105^{+0.028}_{-0.032}$
$n_{s}$ | $0.9652\pm 0.0044$ | $0.9656\pm 0.0039$ | $0.9651\pm 0.0041$
Table 3: $68\%$ C.L. bounds on the cosmological parameters for the
$\Lambda$CDM model for the three different combinations of data sets: Planck
2018, Planck 2018 combined with BAO and SN, and and their full combination
with CMB lensing.

###  IV.1 Data sets

For the present analysis we resort to the CMB Planck 2018 Aghanim  _et al._
(2020b) data for large angular scales $\ell=[2,29]$ and a joint of TT, TE and
EE likelihoods for the small angular scales. In detail, for the latter case,
$\ell=[30,2508]$ for the TT power spectrum and $\ell=[30,1996]$ for the TE
cross-correlation and EE power spectra. This will be our baseline data set and
we will refer to it as “Plk18” in what follows. Subsequently, we examine the
changes when adding to the Plk18 data set a compilation of BAO distance and
expansion rate measurements from the Sloan Digital Sky Survey (SDSS) DR7 Main
Galaxy Sample Ross  _et al._ (2015), SDSS DR12 consensus release Beutler  _et
al._ (2017) and the 6dF Galaxy Survey Beutler  _et al._ (2011) (see text and
Figure 11 in Ref. Aghanim  _et al._ (2020a) for more details), and distance
moduli measurements of type Ia Supernova (SN) data from Pantheon Scolnic  _et
al._ (2018), hereafter simply “Plk18+BAO+SN”. Finally we consider the
combination of “Plk18+BAO+SN” with the addition of the CMB lensing potential
data from Planck 2018 Aghanim  _et al._ (2020b, c), referenced as
“Plk18+BAO+SN+len” from now on. We note that both the CMB Planck 2018
temperature and polarisation angular power spectra data used corresponds to
the standard reference likelihood from the 2018 release
555http://pla.esac.esa.int/pla used in the Planck analysis. In particular this
is given by the product of the Commander, SimALL, and PlikTT,TE,EE likelihoods
Aghanim  _et al._ (2020b).

Our set of free parameters consists of the baseline $\Lambda$CDM cosmological
parameters, namely: ${\Omega_{b}h^{2}}$, ${\Omega_{c}h^{2}}$, $A_{\rm s}$ and
$n_{s}$, the angular size of the sound horizon at recombination $\theta_{s}$
and the reionisation redshift $z_{reio}$; moreover we add the two free
parameters associated to the Kinetic model, $\alpha$ and $\lambda$. We impose
flat priors for all the parameters sampled and these are specified in Table
1666We use a linear sampling for $\alpha$ but we have run chains using also a
logarithmic sampling. Comparing the results we concluded that the C.L. bounds
and the marginalised posterior distributions found are in agreement at
$1\sigma$ level. This supports the robustness of the results reported on with
a flat prior.. We will then provide derived constraints on $H_{0}$, and
$S^{0}_{8}=\sigma_{8}^{0}\sqrt{\Omega_{m}^{0}/0.3}$.

###  IV.2 Cosmological bounds

(a)  Figure 9:  $68\%$ and $95\%$ C.L. contours obtained in the Kinetic model
under consideration for the Planck 2018 data (grey), the Planck 2018, BAO and
SN combination (yellow), and their combination with CMB lensing (red). (a)
Figure 10:  Comparison between the $\Lambda$CDM (dashed lines) and Kinetic
model (solid lines) marginalised likelihood of the cosmological parameters for
the Planck 2018 data (grey), the Planck 2018, BAO and SN combination (yellow)
and their combination with CMB lensing (red).

We show the constraints on the model and cosmological parameters for the
Kinetic model in Table 2 and the corresponding contour plots in Figure 9, for
all the data set combinations considered. For comparison purposes, we include
the results for the $\Lambda$CDM model in Table 3 and in Figure 10.

We find that the parameter $\alpha$ is constrained to be of the order of
$10^{-4}$, regardless of the combination of data sets considered. The Planck
data alone prefer the higher mean value of $\alpha$, mainly as this allows to
better accommodate the TT likelihood; on the other hand, the inclusion of the
BAO and SN data results in a slight decrease of the mean value of $\alpha$; at
last, adding the CMB lensing data leads to a shift of the peak of the
posterior distribution for the $\alpha$ parameter to an even lower central
value. This feature is connected to the lensing excess reported by the Planck
collaboration Ade  _et al._ (2014); Adam  _et al._ (2016); Aghanim  _et al._
(2020a). As discussed in the previous section, the lensing power spectrum is
always suppressed in the Kinetic model, when compared to the $\Lambda$CDM one,
with higher values of $\alpha$ corresponding to lower amplitudes of the
lensing power spectrum (see Figure 7). Therefore, in order to better
accommodate the CMB lensing data, a lower mean value for $\alpha$ is
preferred.

Although the constraints on the cosmological parameters of the Kinetic model
are compatible with the $\Lambda$CDM ones within the errors, the cosmological
standard model yields higher mean values for $H_{0}$ and $\Omega_{m}^{0}$,
when compared to the Kinetic model. The latter is characterised by a positive-
correlation between $\Omega_{m}^{0}$ and $H_{0}$, contrary to the anti-
correlation that characterises the $\Lambda$CDM model, as shown in Figure 11.
In other words, a preference for lower values of $\Omega_{m}^{0}$ results in
lower values for $H_{0}$ alike. This characteristic correlation is persistent
through all the three data combinations considered. This trait can be ascribed
to the presence of a non-vanishing value for the $\alpha$ parameter,
associated with an enhancement of the TT power spectrum (see Fig 8).

Furthermore, in Figure 11 we depict the contour plots for the constraints in
the $S_{8}^{0}-\Omega_{m}^{0}$ plane. The parameters are positively correlated
for both the $\Lambda$CDM model and the Kinetic model. For the latter we find
$S_{8}^{0}=0.793^{+0.110}_{-0.064}$ at 68% C.L. with Plk18 data only, thus
alleviating the discordance with cosmic shear measurements Heymans  _et al._
(2021); Di Valentino  _et al._ (2021); Abdalla  _et al._ (2022) present in the
standard model, for which we report ${S_{8}^{0}=0.833\pm 0.016}$. However, as
seen in Table 2 when the other data sets are also taken into account the
discrepancy arises again, reflecting a tension between BAO and/or SN data
under this framework. A similar situation has also been reported in a Galileon
model Frusciante  _et al._ (2020). This contingency requires further
investigation since it has been suggested that there might be a bias towards
$\Lambda$CDM-like models enclosed in the BAO data Carter  _et al._ (2020).

The inclusion of BAO and SN data leads to narrower constraints on
$\Omega_{m}^{0}$, which in turn results in tighter constraints on other
parameters, such as $H_{0}$, $S_{8}^{0}$, and $\lambda$. The latter is
directly connected to the anti-correlation shown in Fig. 12 in the
$\Omega_{m}^{0}$-$\lambda$ plane, i.e. higher values of $\Omega_{m}^{0}$
select lower values for $\lambda$. This negative correlation is justified by
considering that the late time accelerated expansion, expressed in terms of
$w_{\phi}\approx 1-2V/3H^{2}$, is mainly regulated by two parameters, namely
$\Omega_{\phi}^{0}$ and $\lambda$. The former is given by the Friedmann
constraint $\Omega_{\phi}^{0}\approx 1-\Omega_{m}^{0}$, meaning that, in turn,
higher values of $\Omega_{m}^{0}$ are associated with lower values of
$\Omega_{\phi}^{0}$. Therefore, and in order to have a cosmological constant-
like scenario for the scalar field at present times, $w_{\phi}^{0}\approx-1$,
the mean value of $\lambda$ is pushed towards smaller values, explaining the
identified anti-correlation between $\Omega_{m}^{0}$ and $\lambda$.

(a)

(b)

Figure 11:  68% and 95% C.L. 2D contours obtained for the parameters $H_{0}$ and $\Omega_{m}^{0}$ (left panels) and $S_{8}^{0}$ and $\Omega_{m}^{0}$ (right panels) in the Kinetic model (upper panels) and $\Lambda$CDM model (lower panels) for the Planck 2018 data (grey), the Planck 2018, BAO and SN combination (yellow), and their combination with CMB lensing (red). (a)  Figure 12:  68% and 95% C.L. 2D contours obtained for the parameters $\lambda$ and $\Omega_{m}^{0}$ for the Kinetic model considering the Planck 2018 data (grey), the Planck 2018, BAO and SN combination (yellow), and their combination with CMB lensing (red). |  Plk18 | Plk18+BAO+SN | Plk18+BAO+SN+len
---|---|---|---
$\Delta\chi^{2}_{\rm eff}$ | $-0.9$ | $0.7$ | $1.0$
$\Delta$DIC | $-0.3$ | $0.8$ | $1.6$
Table 4: Results for the $\Delta\chi^{2}_{\rm eff}$ and $\Delta\text{DIC}$
obtained as the difference between the Kinetic and $\Lambda$CDM scenarios.

Finally, we wish to examine whether the Kinetic model is supported over the
$\Lambda$CDM case resorting to statistical indicators: the effective
$\chi^{2}$ corresponding to the maximum likelihood, namely
$\chi_{\text{eff}}^{2}$, and the Deviance Information Criterion (DIC)
Spiegelhalter  _et al._ (2014). The former will enable us to assess whether
the Kinetic model is preferred by the data against $\Lambda$CDM, by computing
$\Delta\chi_{\rm eff}^{2}=\chi^{2}_{\rm eff,Kinetic}-\chi^{2}_{\rm eff,\Lambda
CDM}$, with a negative outcome standing for a support for it, while a positive
result indicates no preference. The DIC will complement this analysis as a
tool for quantifying this preference, and it is defined as

$\text{DIC}:=\chi_{\text{eff}}^{2}+2p_{\text{D}},$ (72)

where ${p_{\text{D}}=\overline{\chi}_{\text{eff}}^{2}-\chi_{\text{eff}}^{2}}$,
with the upper bar denoting the average of the posterior distribution.
According to this definition, the DIC accounts for both the reliability of the
fit, through the $\chi_{\text{eff}}^{2}$ term, and for the Bayesian complexity
of the model, encoded in $p_{\text{D}}$. Hence, more complex models are
disfavoured, in line with a quantitative Occam’s razor criteria. Hence,
cosmological models with smaller DIC should be preferred over models with
larger DIC Liddle (2009); Peirone  _et al._ (2019a, b); Frusciante  _et al._
(2020); Frusciante and Benetti (2021); Anagnostopoulos  _et al._ (2021);
Rezaei and Malekjani (2021); Albuquerque  _et al._ (2022); Atayde and
Frusciante (2021). Finally, the quantity

$\Delta\text{DIC}=\text{DIC}_{\text{Kinetic}}-\text{DIC}_{\text{$\Lambda$CDM}}\,,$
(73)

will indicate support for the Kinetic model over the $\Lambda$CDM scenario
provided that $\Delta\text{DIC}<0$. In Tab. 4 we present the values for both
the $\Delta\chi_{\rm eff}^{2}$ and the $\Delta\text{DIC}$. We gather that, by
taking the Plk18 data alone, a better fit to the data for the Kinetic model is
suggested, compared to the $\Lambda$CDM case, since $\Delta\chi^{2}=-0.9$.
However, when the other data sets are included, this preference is no longer
present. This is linked to the fact that the BAO and SN data spoil the fit to
the TT likelihood which, after the inclusion of the CMB lensing data, becomes
worsened as a result of the Kinetic model predicting a suppressed lensing
amplitude, while the CMB lensing data actually shows an excess of power.
However, it should be noted that the support of the Kinetic model by the
Planck data over the standard cosmological scenario is not overly significant
($\Delta{\rm DIC}=-0.3$) and the remaining data combinations indicate a slight
preference for the $\Lambda$CDM model. Therefore, we conclude that there is no
statistical evidence in support for either of the two models in this analysis.

##  V Conclusions

In this work we have thoroughly explored the evolution of the background and
linear perturbations of the Kinetic model, a coupled quintessence theory
characterised by a power-law kinetic interaction, with strength characterised
by the parameter $\alpha$. We studied the impact of the coupling between the
scalar field and the dark matter fluid on the cosmological observables and we
have provided cosmological constraints on the parameters of the theory using
CMB, CMB lensing, BAO and SN data.

We have derived the background and linear scalar perturbation equations and we
have modified the public Einstein Boltzmann code CLASS. For our study we have
identified the theoretically viable parameter space by enforcing stability
requirements such as the absence of ghosts and gradient instabilities. These
mostly define the range of viability of the parameter $\alpha$. The other
additional free parameter of this model is the steepness of the potential
function, $\lambda$, which has a crucial role in regulating the late time
accelerated expansion. We employed an extended viable range for $\lambda$
compared to what had previously been presented Barros (2019), as we allowed
for transient accelerated regimes at the present time and not at the future
attractor only.

In Section III we studied in detail the phenomenology of the Kinetic model. At
the background level we found that a non-vanishing value of $\alpha$ allows
for the presence of a scaling regime at early times, during the radiation
dominated epoch, according to which the ratio of the densities of the cold
dark matter and the scalar field approximately scales with $\alpha$. The
initial condition for the velocity of the scalar field sets how long the
quintessence field stays in the scaling regime, hence quantifying the
deviations from a cosmological constant behaviour. Furthermore we found that,
due to the coupling in the dark sector, energy is being transferred from the
dark matter field to the scalar field. We also highlighted the presence of a
shift of the radiation matter equality towards earlier times. These two
features have a direct impact on the matter power spectrum: the latter leads
to a shift in the position of its peak towards higher $k$ modes, generating in
turn a suppression for scales $k\lesssim 3\times 10^{-2}h$ Mpc-1, when
compared to the $\Lambda$CDM case; the former affects the growth of the matter
perturbations on larger $k$, resulting in an enhancement with respect to the
standard scenario. Likewise, the differences in the growth of the matter
perturbations influence the evolution of the gravitational potentials through
the Poisson equation. Consequently we found an overall suppression of the
lensing potential (and lensing power spectrum), with respect to $\Lambda$CDM,
along with a modified ISW effect which alters the shape of the TT power
spectrum for large angular scales.

These theoretical predictions are then used to provide constraints on the
model through a Monte Carlo code for cosmological parameter extraction. We
found that the $S_{8}$ tension is alleviated since
$S_{8}^{0}=0.793^{+0.110}_{-0.064}$ at 68% with Planck data, while the $H_{0}$
tension is still present. Regardless of the combination of data considered,
the parameter $\alpha$ is consistently constrained to be of the order
$10^{-4}$. We also reported on the bounds for the other parameter of the
model, $\lambda$, for which the strongest constraints are for the two
combinations including BAO and SN data. This is attributed to the strong
constraining power of BAO data on $\Omega_{m}^{0}$, which indirectly impact
the bounds on $\lambda$. Finally we performed a model selection analysis based
on the effective $\chi_{\rm eff}^{2}$ and Deviance Information Criterion, but
we were not able to clearly identify the statistically favoured model between
$\Lambda$CDM and the Kinetic model. We want to stress that the purpose of our
work is not to make any claim on the class of models characterised by a
kinetic coupling with cold dark matter but to provide constraints on the
parameters of the specific model analysed. The latter being the first tested
model in such class of theories. Actually the present analysis can be
considered a starting point to construct and test new kinetic coupling models
with interesting cosmological signatures.

In conclusion, we remark that it would be of interest to consider the Kinetic
model for future investigations when new probes from upcoming surveys will be
available. This progress will help in shedding light on the tensions and the
high accuracy data we expect to collect will allow us to set a definite
preference of one model over the other.

###### Acknowledgements.

We thank Eleonora di Valentino for useful comments on the results. E.M.T. is
supported by the grant SFRH/BD/143231/2019 from Fundação para a Ciência e a
Tecnologia (FCT). B.J.B. is supported by the South African NRF Grants No.
120390, reference: BSFP190416431035; No. 120396, reference: CSRP190405427545.
N.F. is supported by the Italian Ministry of University and Research (MUR)
through the Rita Levi Montalcini project “Tests of gravity on cosmic scales”
with reference PGR19ILFGP. B.J.B., E.M.T. and N.F. also acknowledge the FCT
project with ref. number PTDC/FIS-AST/0054/2021. The results of this work were
possible thanks to The University of Sheffield’s High Performance Computing
(HPC) clusters Bessemer and ShARC.

##  Appendix A Synchronous gauge

In this Appendix we write the linear perturbations equations of the Kinetic
model in synchronous gauge.

We use the following metric to describe perturbations in synchronous gauge

$\mathrm{d}s^{2}=a^{2}(\tau)\left[-\mathrm{d}\tau^{2}+\left(\delta_{ij}+h_{ij}\right)\mathrm{d}x^{i}\mathrm{d}x^{j}\right]\,,$
(74)

where the scalar modes of the perturbation components $h_{ij}$ are
parameterised in Fourier space,

$h_{ij}(\vec{x},\tau)=\int
d^{3}k\,\mathup{e}^{\mathup{i}\vec{k}\cdot\vec{x}}\left[\hat{\vec{k}}_{i}\cdot\hat{\vec{k}}_{j}\,h(\vec{k},\tau)+\left(\hat{\vec{k}}_{i}\cdot\hat{\vec{k}}_{j}-\frac{1}{3}\delta_{ij}\right)6\eta(\vec{k},\tau)\right]\,,$
(75)

with $\vec{k}=k\hat{\vec{k}}$. The perturbations in Newtonian gauge are
related with the scalar quantities $\eta$ and $h$ as follows Ma and
Bertschinger (1995):

$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\frac{1}{2k^{2}}\left[h^{\prime\prime}+6\eta^{\prime\prime}+\mathcal{H}\left(h^{\prime}+6\eta^{\prime}\right)\right]\,,$
(76) $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\eta-\frac{\mathcal{H}}{2k^{2}}\left(h^{\prime}+6\eta^{\prime}\right)\,,$
(77)

where a prime denotes derivatives with respect to the conformal time $\tau$.
We can the write the system of equations (39)-(42) in synchronous gauge as
follows:

$\displaystyle k^{2}\eta-\frac{1}{2}\mathcal{H}h^{\prime}$ $\displaystyle=$
$\displaystyle-4\pi Ga^{2}\sum_{i}\delta\rho_{i}\,,$ (78) $\displaystyle
k^{2}\eta^{\prime}$ $\displaystyle=$ $\displaystyle 4\pi
Ga^{2}\sum_{i}\rho_{i}(1+w_{i})\theta_{i}\,,$ (79) $\displaystyle
h^{\prime\prime}+2\mathcal{H}h^{\prime}-2k^{2}\eta$ $\displaystyle=$
$\displaystyle-24\pi Ga^{2}\sum_{i}\delta p_{i}\,,$ (80) $\displaystyle
h^{\prime\prime}+6\eta^{\prime\prime}+2\mathcal{H}\left(h^{\prime}+6\eta^{\prime}\right)-2k^{2}\eta$
$\displaystyle=$ $\displaystyle 0\,.$ (81)

Similarly one can find the equivalent of the linear perturbation equation for
the matter density perturbations and velocity:

$\displaystyle\delta^{\prime}_{i}+3\mathcal{H}\left(\frac{\delta
p_{i}}{\delta\rho_{i}}-w_{i}\right)\delta_{i}+(1+w_{i})\left(\theta_{i}+\frac{h^{\prime}}{2}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ (82)
$\displaystyle\theta_{i}^{\prime}+\left[\mathcal{H}(1-3w_{i})+\frac{w_{i}^{\prime}}{1+w_{i}}\right]\theta_{i}-\frac{\delta
p_{i}}{\delta\rho_{i}}\frac{k^{2}}{1+w_{i}}\delta_{i}$ $\displaystyle=$
$\displaystyle 0\,,$ (83)

and for the cold dark matter density and velocity perturbations:

$\delta_{c}^{\prime}+\theta_{c}+\frac{h^{\prime}}{2}=\frac{Q}{\rho_{c}}\left(\phi^{\prime}\delta_{c}-\delta\phi^{\prime}\right)-\frac{\phi^{\prime}}{\rho_{c}}\delta
Q\,,$ (84)
$\theta^{\prime}_{c}+\mathcal{H}\theta_{c}=\frac{Q}{\rho_{c}}\left(\phi^{\prime}\theta_{c}-k^{2}\delta\phi\right)\,,$
(85)

where

$\displaystyle\delta Q$ $\displaystyle=$
$\displaystyle\frac{2\alpha\rho_{c}}{2\alpha
a^{2}\rho_{c}+(1+2\alpha)\phi^{\prime
2}}\left\\{\frac{h^{\prime}}{2}\phi^{\prime}-\phi^{\prime}\theta_{c}+\left[3\mathcal{H}\phi^{\prime}+a^{2}(V_{,\phi}-Q)\right]\delta_{c}+\left(2k^{2}+a^{2}V_{,\phi\phi}\right)\delta\phi\right.$
(86) $\displaystyle\left.\hskip
108.12054pt-\left[3\mathcal{H}\phi^{\prime}+2a^{2}(V_{,\phi}-Q)\right]\frac{\delta\phi^{\prime}}{\phi^{\prime}}\right\\}\,.$

It is worth noting that the synchronous gauge defines a frame which is always
comoving with cold dark matter. That is, in the absence of a coupling, $Q=0$,
and for an initial condition $\theta_{c}(z_{i})=0$, the the velocity
divergence of CDM remains zero throughout time, as dictated by Eq. (85).

Finally we write the equation for the scalar field perturbation:

$\delta\phi^{\prime\prime}+2\mathcal{H}\delta\phi^{\prime}+\left(a^{2}V_{,\phi\phi}+k^{2}\right)\delta\phi+\frac{h^{\prime}}{2}\phi^{\prime}=a^{2}\delta
Q\,.$ (87)

## References

  * Weinberg (1989) S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
  * Weinberg (2000) S. Weinberg,  _4th International Symposium on Sources and Detection of Dark Matter in the Universe (DM 2000)_ , , 18 (2000), arXiv:astro-ph/0005265 .
  * Martin (2012) J. Martin, Comptes Rendus Physique 13, 566 (2012), arXiv:1205.3365 [astro-ph.CO] .
  * Abdalla  _et al._ (2022) E. Abdalla _et al._ , JHEAp 34, 49 (2022), arXiv:2203.06142 [astro-ph.CO] .
  * Aghanim  _et al._ (2020a) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020a), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Riess  _et al._ (2019) A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Astrophys. J. 876, 85 (2019), arXiv:1903.07603 [astro-ph.CO] .
  * Wong  _et al._ (2020) K. C. Wong _et al._ , Mon. Not. Roy. Astron. Soc. 498, 1420 (2020), arXiv:1907.04869 [astro-ph.CO] .
  * Riess  _et al._ (2021) A. G. Riess, S. Casertano, W. Yuan, J. B. Bowers, L. Macri, J. C. Zinn, and D. Scolnic, Astrophys. J. Lett. 908, L6 (2021), arXiv:2012.08534 [astro-ph.CO] .
  * Pesce  _et al._ (2020) D. W. Pesce _et al._ , Astrophys. J. Lett. 891, L1 (2020), arXiv:2001.09213 [astro-ph.CO] .
  * Heymans  _et al._ (2021) C. Heymans _et al._ , Astron. Astrophys. 646, A140 (2021), arXiv:2007.15632 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021) E. Di Valentino _et al._ , Astropart. Phys. 131, 102604 (2021), arXiv:2008.11285 [astro-ph.CO] .
  * Asgari  _et al._ (2021) M. Asgari _et al._ (KiDS), Astron. Astrophys. 645, A104 (2021), arXiv:2007.15633 [astro-ph.CO] .
  * Saridakis  _et al._ (2021) E. N. Saridakis _et al._ (CANTATA), (2021), arXiv:2105.12582 [gr-qc] .
  * Aad  _et al._ (2012) G. Aad _et al._ (ATLAS), Phys. Lett. B 716, 1 (2012), arXiv:1207.7214 [hep-ex] .
  * Chatrchyan  _et al._ (2012) S. Chatrchyan _et al._ (CMS), Phys. Lett. B 716, 30 (2012), arXiv:1207.7235 [hep-ex] .
  * Guth (1981) A. H. Guth, Phys. Rev. D 23, 347 (1981).
  * Linde (1982) A. D. Linde, Phys. Lett. B 108, 389 (1982).
  * Starobinsky (1982) A. A. Starobinsky, Phys. Lett. B 117, 175 (1982).
  * Wetterich (1995) C. Wetterich, Astron. Astrophys. 301, 321 (1995), arXiv:hep-th/9408025 .
  * Caldwell  _et al._ (1998) R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998), arXiv:astro-ph/9708069 .
  * Chiba (1999) T. Chiba, Phys. Rev. D 60, 083508 (1999), arXiv:gr-qc/9903094 .
  * Copeland  _et al._ (1998) E. J. Copeland, A. R. Liddle, and D. Wands, Phys. Rev. D 57, 4686 (1998), arXiv:gr-qc/9711068 .
  * Ferreira and Joyce (1998) P. G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503 (1998), arXiv:astro-ph/9711102 .
  * Liddle and Scherrer (1999) A. R. Liddle and R. J. Scherrer, Phys. Rev. D 59, 023509 (1999), arXiv:astro-ph/9809272 .
  * Barreiro  _et al._ (2000) T. Barreiro, E. J. Copeland, and N. J. Nunes, Phys. Rev. D 61, 127301 (2000), arXiv:astro-ph/9910214 .
  * Amendola (2000) L. Amendola, Phys. Rev. D 62, 043511 (2000), arXiv:astro-ph/9908023 .
  * Guo  _et al._ (2003a) Z.-K. Guo, Y.-S. Piao, R.-G. Cai, and Y.-Z. Zhang, Phys. Lett. B 576, 12 (2003a), arXiv:hep-th/0306245 .
  * Guo  _et al._ (2003b) Z. K. Guo, Y.-S. Piao, and Y.-Z. Zhang, Phys. Lett. B 568, 1 (2003b), arXiv:hep-th/0304048 .
  * Chimento  _et al._ (2003) L. P. Chimento, A. S. Jakubi, D. Pavon, and W. Zimdahl, Phys. Rev. D 67, 083513 (2003), arXiv:astro-ph/0303145 .
  * Tsujikawa and Sami (2004) S. Tsujikawa and M. Sami, Phys. Lett. B 603, 113 (2004), arXiv:hep-th/0409212 .
  * Piazza and Tsujikawa (2004) F. Piazza and S. Tsujikawa, JCAP 07, 004 (2004), arXiv:hep-th/0405054 .
  * Pettorino  _et al._ (2005) V. Pettorino, C. Baccigalupi, and F. Perrotta, JCAP 12, 003 (2005), arXiv:astro-ph/0508586 .
  * Amendola  _et al._ (2006) L. Amendola, M. Quartin, S. Tsujikawa, and I. Waga, Phys. Rev. D 74, 023525 (2006), arXiv:astro-ph/0605488 .
  * Ohashi and Tsujikawa (2009) J. Ohashi and S. Tsujikawa, Phys. Rev. D 80, 103513 (2009), arXiv:0909.3924 [gr-qc] .
  * Gomes and Amendola (2014) A. R. Gomes and L. Amendola, JCAP 03, 041 (2014), arXiv:1306.3593 [astro-ph.CO] .
  * Chiba  _et al._ (2014) T. Chiba, A. De Felice, and S. Tsujikawa, Phys. Rev. D 90, 023516 (2014), arXiv:1403.7604 [gr-qc] .
  * Amendola  _et al._ (2014) L. Amendola, T. Barreiro, and N. J. Nunes, Phys. Rev. D 90, 083508 (2014), arXiv:1407.2156 [astro-ph.CO] .
  * Albuquerque  _et al._ (2018) I. S. Albuquerque, N. Frusciante, N. J. Nunes, and S. Tsujikawa, Phys. Rev. D 98, 064038 (2018), arXiv:1807.09800 [gr-qc] .
  * Frusciante  _et al._ (2018) N. Frusciante, R. Kase, N. J. Nunes, and S. Tsujikawa, Phys. Rev. D 98, 123517 (2018), arXiv:1810.07957 [gr-qc] .
  * Amendola  _et al._ (2018) L. Amendola, D. Bettoni, G. Domènech, and A. R. Gomes, JCAP 06, 029 (2018), arXiv:1803.06368 [gr-qc] .
  * Barros (2019) B. J. Barros, Phys. Rev. D 99, 064051 (2019), arXiv:1901.03972 [gr-qc] .
  * Albuquerque  _et al._ (2022) I. S. Albuquerque, N. Frusciante, and M. Martinelli, Phys. Rev. D 105, 044056 (2022), arXiv:2112.06892 [astro-ph.CO] .
  * Pace and Frusciante (2022) F. Pace and N. Frusciante, Universe 8, 145 (2022), arXiv:2204.06420 [gr-qc] .
  * Zlatev  _et al._ (1999) I. Zlatev, L.-M. Wang, and P. J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999), arXiv:astro-ph/9807002 .
  * Velten  _et al._ (2014) H. E. S. Velten, R. F. vom Marttens, and W. Zimdahl, Eur. Phys. J. C 74, 3160 (2014), arXiv:1410.2509 [astro-ph.CO] .
  * Tamanini (2015) N. Tamanini, Phys. Rev. D 92, 043524 (2015), arXiv:1504.07397 [gr-qc] .
  * Valiviita  _et al._ (2008) J. Valiviita, E. Majerotto, and R. Maartens, JCAP 07, 020 (2008), arXiv:0804.0232 [astro-ph] .
  * Koivisto (2005) T. Koivisto, Phys. Rev. D 72, 043516 (2005), arXiv:astro-ph/0504571 .
  * Schutz (1970) B. F. Schutz, Phys. Rev. D 2, 2762 (1970).
  * Schutz and Sorkin (1977) B. F. Schutz and R. Sorkin, Annals Phys. 107, 1 (1977).
  * Brown (1993) J. D. Brown, Class. Quant. Grav. 10, 1579 (1993), arXiv:gr-qc/9304026 .
  * Pourtsidou  _et al._ (2013) A. Pourtsidou, C. Skordis, and E. J. Copeland, Phys. Rev. D 88, 083505 (2013), arXiv:1307.0458 [astro-ph.CO] .
  * Boehmer  _et al._ (2015a) C. G. Boehmer, N. Tamanini, and M. Wright, Phys. Rev. D 91, 123002 (2015a), arXiv:1501.06540 [gr-qc] .
  * Boehmer  _et al._ (2015b) C. G. Boehmer, N. Tamanini, and M. Wright, Phys. Rev. D 91, 123003 (2015b), arXiv:1502.04030 [gr-qc] .
  * Kase and Tsujikawa (2020) R. Kase and S. Tsujikawa, Phys. Rev. D 101, 063511 (2020), arXiv:1910.02699 [gr-qc] .
  * Amendola (1999) L. Amendola, Phys. Rev. D 60, 043501 (1999), arXiv:astro-ph/9904120 .
  * Tsujikawa (2006) S. Tsujikawa, Phys. Rev. D 73, 103504 (2006), arXiv:hep-th/0601178 .
  * Gomes and Amendola (2016) A. R. Gomes and L. Amendola, JCAP 02, 035 (2016), arXiv:1511.01004 [gr-qc] .
  * Frusciante  _et al._ (2019a) N. Frusciante, R. Kase, K. Koyama, S. Tsujikawa, and D. Vernieri, Phys. Lett. B 790, 167 (2019a), arXiv:1812.05204 [gr-qc] .
  * Amendola and Quercellini (2003) L. Amendola and C. Quercellini, Phys. Rev. D 68, 023514 (2003), arXiv:astro-ph/0303228 .
  * Pettorino and Baccigalupi (2008) V. Pettorino and C. Baccigalupi, Phys. Rev. D 77, 103003 (2008), arXiv:0802.1086 [astro-ph] .
  * Bean  _et al._ (2008) R. Bean, E. E. Flanagan, I. Laszlo, and M. Trodden, Phys. Rev. D 78, 123514 (2008), arXiv:0808.1105 [astro-ph] .
  * Pettorino  _et al._ (2012) V. Pettorino, L. Amendola, C. Baccigalupi, and C. Quercellini, Phys. Rev. D 86, 103507 (2012), arXiv:1207.3293 [astro-ph.CO] .
  * Pettorino (2013) V. Pettorino, Phys. Rev. D 88, 063519 (2013), arXiv:1305.7457 [astro-ph.CO] .
  * Xia (2013) J.-Q. Xia, JCAP 11, 022 (2013), arXiv:1311.2131 [astro-ph.CO] .
  * Ade  _et al._ (2016) P. A. R. Ade _et al._ (Planck), Astron. Astrophys. 594, A14 (2016), arXiv:1502.01590 [astro-ph.CO] .
  * van de Bruck  _et al._ (2017) C. van de Bruck, J. Mifsud, and J. Morrice, Phys. Rev. D 95, 043513 (2017), arXiv:1609.09855 [astro-ph.CO] .
  * Pourtsidou and Tram (2016) A. Pourtsidou and T. Tram, Phys. Rev. D 94, 043518 (2016), arXiv:1604.04222 [astro-ph.CO] .
  * Van De Bruck and Mifsud (2018) C. Van De Bruck and J. Mifsud, Phys. Rev. D 97, 023506 (2018), arXiv:1709.04882 [astro-ph.CO] .
  * Barros  _et al._ (2019) B. J. Barros, L. Amendola, T. Barreiro, and N. J. Nunes, JCAP 01, 007 (2019), arXiv:1802.09216 [astro-ph.CO] .
  * Agrawal  _et al._ (2021) P. Agrawal, G. Obied, and C. Vafa, Phys. Rev. D 103, 043523 (2021), arXiv:1906.08261 [astro-ph.CO] .
  * Gómez-Valent  _et al._ (2020) A. Gómez-Valent, V. Pettorino, and L. Amendola, Phys. Rev. D 101, 123513 (2020), arXiv:2004.00610 [astro-ph.CO] .
  * Pan  _et al._ (2020) S. Pan, G. S. Sharov, and W. Yang, Phys. Rev. D 101, 103533 (2020), arXiv:2001.03120 [astro-ph.CO] .
  * da Fonseca  _et al._ (2022) V. da Fonseca, T. Barreiro, and N. J. Nunes, Phys. Dark Univ. 35, 100940 (2022), arXiv:2104.14889 [astro-ph.CO] .
  * Archidiacono  _et al._ (2022) M. Archidiacono, E. Castorina, D. Redigolo, and E. Salvioni, (2022), arXiv:2204.08484 [astro-ph.CO] .
  * Afshordi  _et al._ (2005) N. Afshordi, M. Zaldarriaga, and K. Kohri, Phys. Rev. D 72, 065024 (2005), arXiv:astro-ph/0506663 .
  * Brookfield  _et al._ (2006) A. W. Brookfield, C. van de Bruck, D. F. Mota, and D. Tocchini-Valentini, Phys. Rev. Lett. 96, 061301 (2006), arXiv:astro-ph/0503349 .
  * Aviles and Cervantes-Cota (2011) A. Aviles and J. L. Cervantes-Cota, Phys. Rev. D 83, 023510 (2011), arXiv:1012.3203 [astro-ph.CO] .
  * Carroll (1998) S. M. Carroll, Phys. Rev. Lett. 81, 3067 (1998), arXiv:astro-ph/9806099 .
  * Chiba and Kohri (2002) T. Chiba and K. Kohri, Prog. Theor. Phys. 107, 631 (2002), arXiv:hep-ph/0111086 .
  * Hui  _et al._ (2009) L. Hui, A. Nicolis, and C. Stubbs, Phys. Rev. D 80, 104002 (2009), arXiv:0905.2966 [astro-ph.CO] .
  * Creminelli  _et al._ (2014) P. Creminelli, J. Gleyzes, L. Hui, M. Simonović, and F. Vernizzi, JCAP 06, 009 (2014), arXiv:1312.6074 [astro-ph.CO] .
  * Brax and Valageas (2017) P. Brax and P. Valageas, Phys. Rev. D 95, 043515 (2017), arXiv:1611.08279 [astro-ph.CO] .
  * Damour and Polyakov (1994) T. Damour and A. M. Polyakov, Nucl. Phys. B 423, 532 (1994), arXiv:hep-th/9401069 .
  * Damour  _et al._ (1990) T. Damour, G. W. Gibbons, and C. Gundlach, Phys. Rev. Lett. 64, 123 (1990).
  * Koivisto  _et al._ (2014) T. Koivisto, D. Wills, and I. Zavala, JCAP 06, 036 (2014), arXiv:1312.2597 [hep-th] .
  * Barros and da Fonseca (2022) B. J. Barros and V. da Fonseca, (2022), arXiv:2209.12189 [astro-ph.CO] .
  * Avelino and Sousa (2018) P. P. Avelino and L. Sousa, Phys. Rev. D 97, 064019 (2018), arXiv:1802.03961 [gr-qc] .
  * Faraoni (2012) V. Faraoni, Phys. Rev. D 85, 024040 (2012), arXiv:1201.1448 [gr-qc] .
  * Teixeira  _et al._ (2019) E. M. Teixeira, A. Nunes, and N. J. Nunes, Phys. Rev. D 100, 043539 (2019), arXiv:1903.06028 [gr-qc] .
  * Avelino and Azevedo (2022) P. P. Avelino and R. P. L. Azevedo, Phys. Rev. D 105, 104005 (2022), arXiv:2203.04022 [gr-qc] .
  * Ferreira  _et al._ (2020) V. M. C. Ferreira, P. P. Avelino, and R. P. L. Azevedo, Phys. Rev. D 102, 063525 (2020), arXiv:2005.07739 [astro-ph.CO] .
  * Avelino and Azevedo (2018) P. P. Avelino and R. P. L. Azevedo, Phys. Rev. D 97, 064018 (2018), arXiv:1802.04760 [gr-qc] .
  * Ma and Bertschinger (1995) C.-P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995), arXiv:astro-ph/9506072 .
  * van de Bruck and Teixeira (2020) C. van de Bruck and E. M. Teixeira, Phys. Rev. D 102, 103503 (2020), arXiv:2007.15414 [gr-qc] .
  * Boisseau  _et al._ (2000) B. Boisseau, G. Esposito-Farese, D. Polarski, and A. A. Starobinsky, Phys. Rev. Lett. 85, 2236 (2000), arXiv:gr-qc/0001066 .
  * Tsujikawa (2007) S. Tsujikawa, Phys. Rev. D 76, 023514 (2007), arXiv:0705.1032 [astro-ph] .
  * De Felice  _et al._ (2011) A. De Felice, T. Kobayashi, and S. Tsujikawa, Phys. Lett. B 706, 123 (2011), arXiv:1108.4242 [gr-qc] .
  * Raveri  _et al._ (2014) M. Raveri, B. Hu, N. Frusciante, and A. Silvestri, Phys. Rev. D 90, 043513 (2014), arXiv:1405.1022 [astro-ph.CO] .
  * Frusciante  _et al._ (2016) N. Frusciante, M. Raveri, D. Vernieri, B. Hu, and A. Silvestri, Phys. Dark Univ. 13, 7 (2016), arXiv:1508.01787 [astro-ph.CO] .
  * Salvatelli  _et al._ (2016) V. Salvatelli, F. Piazza, and C. Marinoni, JCAP 09, 027 (2016), arXiv:1602.08283 [astro-ph.CO] .
  * Frusciante  _et al._ (2019b) N. Frusciante, G. Papadomanolakis, S. Peirone, and A. Silvestri, JCAP 02, 029 (2019b), arXiv:1810.03461 [gr-qc] .
  * Frusciante  _et al._ (2020) N. Frusciante, S. Peirone, L. Atayde, and A. De Felice, Phys. Rev. D 101, 064001 (2020), arXiv:1912.07586 [astro-ph.CO] .
  * Frusciante and Perenon (2020) N. Frusciante and L. Perenon, Phys. Rept. 857, 1 (2020), arXiv:1907.03150 [astro-ph.CO] .
  * Sbisà (2015) F. Sbisà, Eur. J. Phys. 36, 015009 (2015), arXiv:1406.4550 [hep-th] .
  * Lesgourgues (2011a) J. Lesgourgues, (2011a), arXiv:1104.2932 [astro-ph.IM] .
  * Blas  _et al._ (2011) D. Blas, J. Lesgourgues, and T. Tram, Journal of Cosmology and Astroparticle Physics 2011, 034–034 (2011).
  * Lesgourgues (2011b) J. Lesgourgues, (2011b), arXiv:1104.2934 [astro-ph.CO] .
  * Audren  _et al._ (2013) B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, Journal of Cosmology and Astroparticle Physics 2013, 001 (2013).
  * Brinckmann and Lesgourgues (2019) T. Brinckmann and J. Lesgourgues, Phys. Dark Univ. 24, 100260 (2019), arXiv:1804.07261 [astro-ph.CO] .
  * Lewis (2019) A. Lewis,  (2019), arXiv:1910.13970 [astro-ph.IM] .
  * Aghanim  _et al._ (2020b) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A5 (2020b), arXiv:1907.12875 [astro-ph.CO] .
  * Ross  _et al._ (2015) A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden, and M. Manera, Mon. Not. Roy. Astron. Soc. 449, 835 (2015), arXiv:1409.3242 [astro-ph.CO] .
  * Beutler  _et al._ (2017) F. Beutler _et al._ (BOSS), Mon. Not. Roy. Astron. Soc. 464, 3409 (2017), arXiv:1607.03149 [astro-ph.CO] .
  * Beutler  _et al._ (2011) F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson, Mon. Not. Roy. Astron. Soc. 416, 3017 (2011), arXiv:1106.3366 [astro-ph.CO] .
  * Scolnic  _et al._ (2018) D. M. Scolnic _et al._ (Pan-STARRS1), Astrophys. J. 859, 101 (2018), arXiv:1710.00845 [astro-ph.CO] .
  * Aghanim  _et al._ (2020c) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A8 (2020c), arXiv:1807.06210 [astro-ph.CO] .
  * Ade  _et al._ (2014) P. A. R. Ade _et al._ (Planck), Astron. Astrophys. 571, A16 (2014), arXiv:1303.5076 [astro-ph.CO] .
  * Adam  _et al._ (2016) R. Adam _et al._ (Planck), Astron. Astrophys. 594, A1 (2016), arXiv:1502.01582 [astro-ph.CO] .
  * Carter  _et al._ (2020) P. Carter, F. Beutler, W. J. Percival, J. DeRose, R. H. Wechsler, and C. Zhao, Mon. Not. Roy. Astron. Soc. 494, 2076 (2020), arXiv:1906.03035 [astro-ph.CO] .
  * Spiegelhalter  _et al._ (2014) D. J. Spiegelhalter, N. G. Best, B. P. Carlin, and A. van der Linde, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76, 485 (2014).
  * Liddle (2009) A. R. Liddle, Ann. Rev. Nucl. Part. Sci. 59, 95 (2009), arXiv:0903.4210 [hep-th] .
  * Peirone  _et al._ (2019a) S. Peirone, G. Benevento, N. Frusciante, and S. Tsujikawa, Phys. Rev. D 100, 063540 (2019a), arXiv:1905.05166 [astro-ph.CO] .
  * Peirone  _et al._ (2019b) S. Peirone, G. Benevento, N. Frusciante, and S. Tsujikawa, Phys. Rev. D 100, 063509 (2019b), arXiv:1905.11364 [astro-ph.CO] .
  * Frusciante and Benetti (2021) N. Frusciante and M. Benetti, Phys. Rev. D 103, 104060 (2021), arXiv:2005.14705 [astro-ph.CO] .
  * Anagnostopoulos  _et al._ (2021) F. K. Anagnostopoulos, S. Basilakos, and E. N. Saridakis, Phys. Lett. B 822, 136634 (2021), arXiv:2104.15123 [gr-qc] .
  * Rezaei and Malekjani (2021) M. Rezaei and M. Malekjani, Eur. Phys. J. Plus 136, 219 (2021), arXiv:2102.10671 [astro-ph.CO] .
  * Atayde and Frusciante (2021) L. Atayde and N. Frusciante, Phys. Rev. D 104, 064052 (2021), arXiv:2108.10832 [astro-ph.CO] .