text:
\section{Introduction}
\label{introsec}

Testing gravitational interactions with cosmological and astrophysical observations
is one of the most interesting area
of current research. Several large-scale surveys that will contribute to this task
are underway or planned for the next few years \cite{Abell:2009aa,Rawlings:2011dd,Laureijs:2011gra,Amendola:2012ys,Levi:2013gra,Riess:2018}.
If there is a deviation from Einstein gravity, then new observable phenomena are expected
to appear, including stronger or weaker gravitational clustering and lensing, scale-dependent perturbation growth, screening of fifth forces, breaking of the equivalence principle,
anomalous propagation of gravitational waves, and so on.

In order to predict these new effects, one often analyses the most general gravity
theories with second-order equations of motion--like
scalar-tensor \cite{Horndeski,Def,KYY,Char11,DKT11,DT12},
vector-tensor  \cite{Heisenberg,Tasinato1,DeFelice:2016yws,DeFelice:2016uil},
massive gravity \cite{Hassan:2011zd,deRham:2010kj}
or, alternatively, proceeds by building effective Lagrangians that include all possible viable operators \cite{EFT1,EFT2,EFT3,EFT4,EFT5}. These approaches have the great virtue of being systematic, but they rapidly lead to a proliferation of operators that make difficult to extract definite predictions. The gravitational-wave event GW170817 \cite{GW170817},
together with its electromagnetic counterpart \cite{Goldstein}, put additional constraints
on the allowed Lagrangian \cite{GW1,GW2,GW3,GW4,GW5,GW6},
but there is still degeneracy among different
dark energy models.
This is partly attributed to the fact that, for the theories in which the speed of
gravity is equivalent to that of light, the effective gravitational coupling for
cold-dark-matter (CDM) density perturbations is usually larger than the Newton gravitational
constant $G$ in both scalar-tensor and vector-tensor
theories \cite{Amendola17,Kase:2018aps,Nakamura:2018oyy}.

In this paper, we proceed in a different way as compared to the theoretical approach mentioned above.
We begin by identifying which are the physical effects we are interested in,
and then build a general model that is expected to generate them. The physical effects
we consider are motivated by the two most interesting discrepancies between
the $\Lambda$CDM model and
current observations, namely the $H_0$ and $\sigma_8$ tensions.
Here, $H_0=100\,h$ km\,s$^{-1}$\,Mpc$^{-1}$ is today's Hubble-Lema\^itre
expansion rate and
$\sigma_8$ is the amplitude of matter perturbations within the comoving $8h^{-1}$ scale.

As is well-known, the $H_0$ tension arises because the estimate of $H_0$  from
Cosmic Microwave Background (CMB) temperature anisotropies \cite{Planck2015}
differs by more than 3$\sigma$ from the one based on its local measurements \cite{Riess:2018}.
While there remains the possibility that this tension is
due to unknown or uncontrolled systematic effects, the problem
has been confirmed and exacerbated in recent observations \cite{Verde:2019ivm,Riess:2019cxk,Freedman:2019jwv,Reid:2019tiq}.
Similarly, the $\sigma_8$ tension, which is weaker than the $H_0$ tension, is due to the difference between the clustering normalization obtained from CMB and the one from shear lensing analyses \cite{weak1,weak2}, where the former favours  values
of $\sigma_8$ larger than the latter.
In both cases, the CMB estimate depends on
assuming  $\Lambda$CDM   as
a model of dark energy and dark matter.
It is therefore natural to ask whether a suitable modification of the dark sector
can lead to a better agreement with the data.

One way to reconcile, at least potentially, $H_0$ CMB results \cite{Planck2015} with a higher local value \cite{Riess:2018}, is to modify the evolution of the Universe between matter-radiation equality and cosmic acceleration, so that the effective early equation of state
is slightly positive. In this case, one can show that the CMB acoustic peaks move to
smaller angular scales, and therefore  a higher $H_0$ is needed to bring the model
back to agreement with observations (although of course a full likelihood analysis
is needed to assess the valid parameter space).
One way to achieve this without altering the physical properties of matter is to couple dark matter to dark energy in such a way that a matter-dominated epoch
with a small fraction of dark energy
is present. Indeed, for a canonical scalar-field dark energy model interacting with dark matter
through the energy transfer characterized by a coupling constant $Q$ \cite{Wetterich,Amendola99},
there exists a scaling
$\phi$-matter-dominated-era ($\phi$MDE) during which the field
density parameter $\Omega_\phi$ is constant ($\Omega_\phi=2Q^2/3$) \cite{Amendola99}.
During the $\phi$MDE the scalar-field kinetic energy dominates over its potential energy, yielding a field equation of state $w_\phi=1$.
In this case, a small and positive effective equation of state arises, such that $w_{\rm eff}=w_\phi\Omega_\phi=\Omega_\phi$.
A comparison with cosmological datasets has put upper limits to  $Q$ \cite{Pettorino2013,Ade:2015rim}.

In the models of Refs.~\cite{Wetterich,Amendola99}, the dark energy scalar field
$\phi$ interacts with CDM
(but not baryons) via a conformal rescaling of the metric.
This type of couplings arises in Brans-Dicke theories \cite{BD}
after a conformal transformation to the Einstein
frame \cite{Amendola:1999qq,Khoury,Tsujikawa:2008uc}.
In the language of Schutz-Sorkin action \cite{Sorkin,Brown}
describing CDM as a perfect fluid,
the interacting Lagrangian for the models of Refs.~\cite{Wetterich,Amendola99}
is of the form
$L_{\rm int}=(e^{Q\phi/M_{\rm pl}}-1)\rho_m$ \cite{Frusciante:2018tvu},
where $\rho_m$ is the CDM density and $M_{\rm pl}$ is the
reduced Planck mass.
Reflecting the fact that this interaction corresponds to the energy transfer,
the effective gravitational coupling $G_{\rm eff}$ associated with the linear
evolution of CDM perturbations is larger than $G$,
such that $G_{\rm eff}=(1+2Q^2)G$ \cite{Amendola:2003wa}.

One problem for the models with $G_{\rm eff}>G$ is that the attractive gravitational
force induced by the coupling increases the cosmic growth rate, which makes
the $\sigma_8$ tension worse.
Ideally, one would need a coupling such that the $\phi$MDE is accompanied
or followed by weaker growth of matter perturbations.
An effective gravitational coupling smaller than $G$ can be achieved for
a model in which the scalar field $\phi$ interacts with CDM through
a momentum transfer \cite{Pourtsidou:2013nha,Boehmer:2015kta,Boehmer:2015sha,Skordis:2015yra,
Koivisto:2015qua,Pourtsidou:2016ico,Dutta:2017kch,Linton,Kase:2019veo,Chamings:2019kcl}.
This type of interaction is based on the field derivative coupling with
the CDM four velocity $u^{\mu}$ and it can be quantified by the scalar
combination $Z=u^{\mu} \nabla_{\mu} \phi$, where $\nabla_{\mu}$ is
the covariant derivative operator.
In previous works, it was shown that the interacting Lagrangian such as
$L_{\rm int} \propto Z^2$ \cite{Pourtsidou:2013nha,Pourtsidou:2016ico}
or $L_{\rm int} \propto X^{1-n/2} Z^n$ \cite{Kase:2019mox}, where
$X=-\nabla^{\mu} \phi \nabla_{\mu} \phi/2$
and $n>0$, can realize $G_{\rm eff}<G$ for the theories without
energy transfer (i.e., $Q=0$ for the notation used above).

In the Schutz-Sorkin action approach, it is straightforward to accommodate
both energy and momentum couplings between
perfect-fluid dark matter and scalar-field dark energy.
For the CDM density $\rho_m$ that depends on its number density $n$,
the interacting action may be expressed in the form,
\be
{\cal S}_{\rm int}=\int {\rm d}^4 x \sqrt{-g} \left[-f_1(\phi,X,Z) \rho_m (n)
+f_2(\phi,X,Z) \right]\,,
\label{action0}
\ee
where $g$ is the determinant of metric tensor $g_{\mu \nu}$, and
$f_1$, $f_2$ are functions of $\phi$, $X$, $Z$.
The first and second terms in the square bracket of Eq.~(\ref{action0})
characterize the energy and momentum transfers, respectively.
Unlike the other models in which phenomenological coupling
terms are added to the background equations
by hand \cite{Dalal:2001dt,Zimdahl:2001ar,Chimento:2003iea,Wang1,Wei:2006ut,Amendola:2006dg,Guo:2007zk,Valiviita:2008iv,Gavela:2009cy,Wands,Kumar:2016zpg,DiValentino:2017iww,An:2017crg,Yang:2018euj,Pan:2019gop,DiValentino:2019ffd,Yang:2019uog},
the evolution of cosmological perturbations is unambiguously fixed
in our interacting theory
with the explicit action (\ref{action0}).

The preceding discussion makes it clear that we need both scaling $\phi$MDE and
weak gravity to alleviate the $H_0$ and $\sigma_8$ tensions.
As it will be demonstrated in this work, these requirements can be satisfied
when both energy and momentum exchanges occur between dark energy and CDM.
For this purpose, we first derive a general interacting Lagrangian for the existence
of scaling solutions. For simplicity, we assume Einstein gravity with
the scalar interacting action (\ref{action0}) and the Schutz-Sorkin action for CDM,
while we leave baryons and radiation uncoupled.
In this case, there is no difference for the propagation of
gravitational waves in comparison to general relativity
and no screening mechanism is needed.
We note that the function $f_2$ in Eq.~(\ref{action0}) also accommodates the scaling
Lagrangian derived for k-essence with the functional dependence
$f_2(\phi,X)$ \cite{Piazza,Tsuji04,Amendola:2006qi}.

After obtaining the general scaling Lagrangian, we consider a subclass of models
with $\phi$MDE and show that it is possible to realize
$G_{\rm eff}<G$
at low redshifts even in the presence of both energy and momentum transfers.
The hope is that such interacting models can really ease both the $H_0$ and $\sigma_8$ tensions.
We leave however the task of a full likelihood analysis with
the current observational data for a future work.

\section{Coupled dark energy with a Lagrangian formulation}
\label{eomsec}

We consider a dark energy scalar field $\phi$ coupled to
a barotropic perfect fluid described by a Schutz-Sorkin
action \cite{Sorkin,Brown}.
The interacting Lagrangian is generally given by the form
$L(n, \phi, X, Z)$ \cite{Pourtsidou:2013nha},
where $L$ is a function of the fluid number density $n$,
the scalar field $\phi$ and its kinetic
energy $X=-g^{\mu \nu} \nabla_{\mu}\phi
\nabla_{\nu} \phi/2$, and
$Z=u^{\mu} \nabla_{\mu} \phi$, with $u^{\mu}$ being
the fluid four velocity.
We separate the interacting Lagrangian $L$ into the
sums of energy transfer $-f_1(\phi, X, Z) \rho_m(n)$
and momentum transfer $f_2(\phi, X, Z)$, where
$\rho_m$ is the fluid density that depends on $n$.
For CDM, the density has the linear dependence
$\rho_m \propto n$.
For the gravity sector, we consider the Einstein-Hilbert
action described by the Lagrangian $(M_{\rm pl}^2/2)R$,
where $R$ is the Ricci scalar.

The total action is then given by
\be
{\cal S} = \int {\rm d}^4 x \sqrt{-g}\,\frac{M_{\rm pl}^2}{2}R
-\int {\rm d}^{4}x \left[
\sqrt{-g}\,\rho_m(n)
+ J^{\mu} \partial_{\mu} \ell  \right]
+\int {\rm d}^4x \sqrt{-g}\,L(n,\phi,X,Z)\,,
\label{action}
\ee
where
\be
L(n,\phi,X,Z)=-f_1(\phi,X,Z) \rho_m (n)
+f_2(\phi,X,Z)\,.
\label{Lin}
\ee
The second integral in Eq.~(\ref{action}) corresponds to
the Schutz-Sorkin action, where $n$ is related to the vector field $J^{\mu}$, as
\be
n=\sqrt{\frac{g^{\mu \nu} J_{\mu} J_{\nu}}{g}}\,.
\ee
Unlike Refs.~\cite{DeFelice:2016yws,DeFelice:2016uil},
we do not take vector perturbations into account
in the Schutz-Sorkin action, as they are nondynamical in scalar-tensor theories.
The fluid four-velocity $u_{{\mu}}$ is defined by
\be
u_{{\mu}} \equiv \frac{J_{{\mu}}}{n\sqrt{-g}}\,,
\label{udef}
\ee
which satisfies the relation $u^{\mu} u_{{\mu}}=-1$.
The scalar quantity $Z$ is expressed as $Z=g^{\mu \nu} J_{\mu} \nabla_{\nu} \phi/(n\sqrt{-g})$, while
the scalar variable $\ell$ is a Lagrange multiplier.
Since we are not modifying the Einstein-Hilbert action
from general relativity, the speed of gravitational waves
is equivalent to that of light.

Varying the action (\ref{action}) with respect to $\ell$,
it follows that
\be
\partial_{\mu} J^{\mu}=0\,.
\label{Jmure}
\ee
The fluid pressure is given
by \cite{Pourtsidou:2013nha,Boehmer:2015kta,Boehmer:2015sha,DeFelice:2016yws,DeFelice:2016uil}
\be
P_m=n \rho_{m,n}-\rho_m\,.
\ee
Substituting $J^{\mu}=n\sqrt{-g}\,u^{\mu}$
into Eq.~(\ref{Jmure}) and exploiting the properties
$\partial_{\mu} (\sqrt{-g} u^{\mu})=\sqrt{-g}
\nabla_{\mu} u^{\mu}$ and
$(\rho_m+P_m)\partial_{\mu} n=n \partial_{\mu} \rho_m$,
we obtain
\be
u^{\mu} \partial_{\mu} \rho_m+
\left( \rho_m+P_m \right)
\nabla_{\mu} u^{\mu}=0\,.
\label{conser1}
\ee
Variation of the action (\ref{action}) with respect to $J^{\mu}$ leads to
\be
\partial_{\mu} \ell=
u_{{\mu}} \left( 1+f_1 \right) \rho_{m,n}
-\frac{f_{1,Z} \rho_m-f_{2,Z}}{n} \left(
\nabla_{\mu} \phi+Z u_{\mu} \right)\,,
\label{lc}
\ee
where we used the relation $\partial n/\partial J^{\mu}
=J_{{\mu}}/(n g)$ and the notations
$\rho_{m,n} \equiv \partial \rho_m/\partial n$ and
$f_{i,Z} \equiv \partial f_{i}/\partial Z$ with $i=1,2$.

For the variation of action (\ref{action}) with respect to
$g^{\mu \nu}$, we employ the following properties,
\ba
\delta \sqrt{-g}
&=& -\frac{1}{2} \sqrt{-g}\,
g_{\mu \nu} \delta g^{\mu \nu}\,,\\
\delta n &=& \frac{n}{2} \left( g_{\mu \nu}
-u_{\mu} u_{\nu} \right) \delta g^{\mu \nu}\,,\\
\delta X &=&-\frac{1}{2} \nabla_{\mu} \phi
\nabla_{\nu} \phi\,\delta g^{\mu \nu}\,,\\
\delta Z &=& \left( \frac{1}{2} Z u_{\mu}
u_{\nu} +u_{\mu}\nabla_{\nu} \phi \right)
\delta g^{\mu \nu}\,,\\
\delta (J^{\mu} \partial_{\mu} \ell)
&=& \delta (g^{\mu \nu} J_{\mu} \partial_{\nu} \ell)
=J_{\mu} \partial_{\nu} \ell\, \delta g^{\mu \nu}\,.
\ea
When the action is varied with respect to $g^{\mu \nu}$, we need to keep $J^\mu$ fixed
to reproduce the standard matter energy-momentum tensor.
Therefore, the vector $J^{\mu}$ should not be replaced with $n\sqrt{-g}\,u^\mu$
when varying the action.
Then, we obtain the following covariant equations of motion,
\be
M_{\rm pl}^2 G_{\mu\nu}=\left( 1+f_1 \right) T^{(m)}_{\mu\nu}+T_{\mu\nu}^{(\phi)}\,,
\label{Ein}
\ee
where $G_{\mu \nu}$ is the Einstein tensor, and
\ba
T^{(m)}_{\mu\nu}
&=& \left( \rho_m+P_m \right)
u_{\mu} u_{\nu}+P_m g_{\mu \nu}\,,
\label{Tmmun}\\
T^{(\phi)}_{\mu \nu}
&=& f_2 g_{\mu \nu}-\rho_m \left( f_{1,X}
\nabla_{\mu} \phi \nabla_{\nu} \phi
+f_{1,Z}Z u_{\mu} u_{\nu} \right)
+f_{2,X}\nabla_{\mu} \phi \nabla_{\nu} \phi
+f_{2,Z} Z u_{\mu} u_{\nu}\,.
\ea
The matter energy-momentum tensor (\ref{Tmmun})
satisfies the continuity equation,
\be
u^{\nu} \nabla^{\mu} T^{(m)}_{\mu\nu}
=-\left[ u_{\mu} \partial^{\mu} \rho_m+
\left( \rho_m+P_m \right)
\nabla^{\mu} u_{\mu} \right]=0\,,
\label{Tcon}
\ee
where we used Eq.~(\ref{conser1}) in the second equality.
Taking the covariant derivative of Eq.~(\ref{Ein}),
we obtain
\be
\left( 1+f_1 \right) \nabla^{\mu} T_{\mu \nu}^{(m)}
+ T_{\mu \nu}^{(m)} \nabla^{\mu} f_1
+\nabla^{\mu} T_{\mu \nu}^{(\phi)}=0\,.
\label{Tcon2}
\ee
Multiplying Eq.~(\ref{Tcon2}) by $u^{\nu}$ and using the
property (\ref{Tcon}), we find
\be
u^{\nu} \nabla^{\mu}T^{(\phi)}_{\mu\nu}
=-u^{\nu} \nabla^{\mu} f_1\, T_{\mu \nu}^{(m)}\,.
\label{Tphicon}
\ee
We define the effective matter energy-momentum
tensor,
\be
\hat{T}^{(m)}_{\mu\nu} \equiv \left( 1+f_1 \right)
T^{(m)}_{\mu\nu}\,,
\label{hatT}
\ee
which contains the effect of energy exchange between
the scalar field and matter.
This quantity obeys
\be
u^{\nu}\nabla^{\mu} \hat{T}^{(m)}_{\mu\nu}
=+u^{\nu} \nabla^{\mu} f_1\, T_{\mu \nu}^{(m)}\,.
\label{Tmcon}
\ee
The signs on the right hand sides of Eqs.~(\ref{Tphicon}) and (\ref{Tmcon})
are opposite to each other, which shows the energy exchange
between the scalar field and matter.
The coupling $f_2$, which corresponds to the momentum
transfer, does not appear on the right hand sides of
Eqs.~(\ref{Tphicon}) and (\ref{Tmcon}).
However, the momentum transfer between the scalar field
and matter occurs through Eq.~(\ref{Tcon2}).

Let us consider the flat
Friedmann-Lema\^itre-Robertson-Walker (FLRW)
background given by the line element
${\rm d}s^2=-{\rm d}t^2+a^2(t) \delta_{ij} {\rm d}x^i {\rm d} x^j$,
where $a(t)$ is the time-dependent scale factor.
On this background we have $u^{\mu}=(1,0,0,0)$ and
$\nabla_{\mu} u^{\mu}=3H$, where $H=\dot{a}/a$
is the Hubble-Lema\^itre expansion rate and a dot represents
a derivative with respect to the cosmic time $t$.
{}From Eq.~(\ref{conser1}), we obtain
\be
\dot{\rho}_m+3H \left( \rho_m+P_m \right)=0\,,
\label{con}
\ee
which corresponds to the conservation of total
particle number ${\cal N} \equiv J^0=n a^3$.
{}From the (00) and $(ii)$ components of Einstein equations (\ref{Ein}),
it follows that
\ba
& &
3M_{\rm pl}^2 H^2=
\left( 1+f_1 \right) \rho_m+\rho_{\phi}\,,
\label{Eq00}\\
& &
M_{\rm pl}^2 \left( 2\dot{H}+3H^2
\right)=-\left( 1+f_1 \right) P_m-P_{\phi}\,,
\label{Eq11}
\ea
where
\ba
\rho_{\phi} &\equiv& -\rho_m f_{1,X}\dot{\phi}^2
-\rho_m f_{1,Z} \dot{\phi}-f_2+f_{2,X} \dot{\phi}^2
+f_{2,Z} \dot{\phi}\,,\\
P_{\phi} &\equiv& f_2\,.
\ea
{}From Eq.~(\ref{hatT}), we introduce the matter
density and pressure containing the effect of energy transfer, as
\be
\hat{\rho}_m \equiv (1+f_1) \rho_m\,,\qquad
\hat{P}_m \equiv  (1+f_1) P_m\,.
\ee
{}From Eqs.~(\ref{Tphicon}) and (\ref{Tmcon}), we have
\ba
& &
\dot{\rho}_{\phi}+3H \left( \rho_{\phi}
+P_{\phi} \right)=-\dot{f}_1 \rho_m\,,\label{Eq22}
\\
& &
\dot{\hat{\rho}}_m+3H \left( \hat{\rho}_m
+\hat{P}_m \right)=+\dot{f}_1 \rho_m\,,
\label{con2}
\ea
whose right hand sides explicitly show the energy exchange between
the scalar field and matter.

For  later convenience, we introduce the time-dependent
density parameters,
\be
\Omega_m \equiv \frac{(1+f_1)\rho_m}{3M_{\rm pl}^2 H^2}\,,
\qquad
\Omega_\phi \equiv \frac{\rho_{\phi}}{3M_{\rm pl}^2 H^2}\,.
\label{Omega}
\ee
The form of matter is not  specified
until the end of Sec.~\ref{fixedsec},
after which we take CDM, baryons, and
radiation into account as perfect fluids.
From Eq.~(\ref{Eq00}), it follows that
\be
\Omega_m+\Omega_{\phi}=1\,.
\ee
We also define the equations of state,
\be
w_m \equiv \frac{\hat{P}_m}{\hat{\rho}_m}
=\frac{P_m}{\rho_m}\,,\qquad
w_{\phi} \equiv \frac{P_{\phi}}{\rho_{\phi}}\,,
\ee
as well as the effective equation of state,
\be
w_{\rm eff} \equiv w_m \Omega_m+
w_{\phi} \Omega_{\phi}= -1-\frac{2\dot{H}}{3H^2}\,.
\ee
The condition for cosmic acceleration to occur corresponds
to $w_{\rm eff}<-1/3$.

\section{Scaling Lagrangian}
\label{scalingsec}

In this section, we derive the Lagrangian $L$ allowing for the existence
of scaling solutions characterized by nonvanishing
constants $\Omega_m$ and $\Omega_{\phi}$.
Notice that we are not requiring that the entire cosmic evolution
has the scaling behavior, but just that scaling epochs exist.
Since  $(1+f_1) \rho_m \propto H^2$ and  $\rho_{\phi} \propto H^2$ in the scaling regime,
it follows that
\ba
\frac{\dot{f}_1}{H(1+f_1)} &=&-3 \left(
w_{\rm eff}-w_m \right)\,,\label{scare1}\\
\frac{\dot{\rho}_{\phi}}{H \rho_{\phi}}
&=&-3(1+w_{\rm eff})\,,\label{scare2}
\ea
where we used Eq.~(\ref{con}).
We will consider the case in which $w_{\rm eff}$, $w_m$, and
$w_{\phi}$ are constant along the scaling solution.
Then, all the terms in Eq.~(\ref{Eq11}) are proportional to $H^2$.
Since $\rho_{\phi} \propto P_{\phi} \propto H^2$,
each term on the left hand side of Eq.~(\ref{Eq22}) is
proportional to $H^3$. On using Eq.~(\ref{scare1}),
there is also the dependence
$\dot{f}_1 \rho_m \propto H (1+f_1) \rho_m \propto H^3$,
so the scaling solution is consistent with Eq.~(\ref{Eq22})
as well.

If we consider a canonical scalar field given by the Lagrangian
$f_2(\phi,X)=X-V(\phi)$, the field density $\rho_{\phi}$
contains the kinetic energy $\dot{\phi}^2/2$.
In this case, the scaling solution satisfies
the relation $\dot{\phi} \propto H$.
To accommodate this model as a special case,
we will derive the Lagrangian
allowing for scaling solutions characterized by
\be
\frac{\dot{\phi}}{HM_{\rm pl}}=\alpha\,,
\label{alpha}
\ee
where $\alpha$ is a dimensionless constant.
Then, the time derivative of $f_1(\phi,X,Z)$ yields
\be
\dot{f}_1=H \left[ \alpha M_{\rm pl} f_{1,\phi}
-3(1+w_{\rm eff}) \left( X f_{1,X}+\frac{1}{2}
Z f_{1,Z} \right) \right]\,.
\label{dotf1}
\ee
Substituting Eq.~(\ref{dotf1}) into Eq.~(\ref{scare1}),
we obtain the partial differential equation,
\be
Xf_{1,X}+\frac{1}{2}Zf_{1,Z}-\frac{1}{\lambda}M_{\rm pl}
f_{1,\phi}+\frac{Q}{\lambda} \left( 1+f_1 \right)=0\,,
\label{pdif1}
\ee
where
\be
\lambda \equiv \frac{3(1+w_{\rm eff})}{\alpha}\,,\qquad
Q \equiv -\frac{3(w_{\rm eff}-w_m)}{\alpha}\,.
\ee
The integrated solution to Eq.~(\ref{pdif1}) is given by
\be
f_1=e^{Q\phi/M_{\rm pl}} g_1 \left(Y_1,Y_2 \right)-1\,,
\label{sca}
\ee
where $g_1$ is an arbitrary function of
\be
Y_1=X e^{\lambda \phi/M_{\rm pl}}\,,\qquad
Y_2=Z e^{\lambda \phi/(2M_{\rm pl})}\,.
\ee
Along the scaling solution, $\phi$, $a$, and $H$ evolve, respectively, as
\be
\phi=\phi_0+\alpha M_{\rm pl} \ln a\,,\qquad
a \propto \left( t-t_0 \right)^{2/[3(1+w_{\rm eff})]}\,,
\qquad
H=\frac{2}{3(1+w_{\rm eff})(t-t_0)}\,,
\ee
where $\phi_0$ and $t_0$ are constants.
This shows that both $Y_1$ and $Y_2$ remain constant
in the scaling regime.
One can also show that both
$\rho_m f_{1,X} \dot{\phi}^2$ and $ \rho_m f_{1,Z} \dot{\phi}$
in $\rho_{\phi}$ are proportional to $H^2$, so all the terms
in Eq.~(\ref{Eq00}) associated with the coupling $f_1$ obey
the scaling relation. In other words, each term in $\rho_{\phi}$
arising from $f_1$ satisfies the same relation as
Eq.~(\ref{scare2}).

The coupling $f_2$, which is equivalent to $P_{\phi}$
and is also present as one of the terms in
$\rho_{\phi}$, should satisfy the scaling property $f_2 \propto H^2$.
This translates to the relation $\dot{f}_2/(Hf_2)=-3(1+w_{\rm eff})$
and hence $f_2$ obeys the partial differential equation,
\be
Xf_{2,X}+\frac{1}{2}Zf_{2,Z}-\frac{1}{\lambda}M_{\rm pl}f_{2,\phi}-f_2=0\,.
\label{pdif2}
\ee
This is integrated to give
\be
f_2=X g_2 (Y_1, Y_2)\,,
\label{sca2}
\ee
where $g_2$ is an arbitrary function of $Y_1$ and $Y_2$.
On using the property that $Y_1$ and $Y_2$ are constant
along the scaling solution, both $f_{2,X} \dot{\phi}^2$
and $f_{2,Z} \dot{\phi}$ in $\rho_{\phi}$ are proportional
to $H^2$.

In summary, the Lagrangian allowing for the existence
of scaling solutions is given by
\be
L=-\left[ e^{Q\phi/M_{\rm pl}}
g_1 \left(Y_1,Y_2 \right)-1 \right] \rho_m (n)
+Xg_2 (Y_1, Y_2)\,,
\label{slag}
\ee
which contains arbitrary functions $g_1$ and $g_2$
with respect to $Y_1$ and $Y_2$.
For the choice $g_1(Y_1, Y_2)=1$, the coupling $f_1$
corresponds to the one studied in Refs.~\cite{Frusciante:2018tvu,Kase:2019veo},
i.e., $f_1=e^{Q\phi/M_{\rm pl}}-1$, where the
constant $Q$ characterizes the strength of energy exchange.
For k-essence theories with $f_2=f_2(\phi,X)$, the scaling Lagrangian
reduces to $f_2=X g_2(Y_1)$, which agrees with that derived
in Refs.~\cite{Piazza,Tsuji04,Amendola:2006qi}.
The $Z$ dependence in $f_1$ and $f_2$ gives rise to the additional $Y_2$
dependence in the functions $g_1$ and $g_2$.

\section{Fixed points}
\label{fixedsec}

We derive the fixed points for interacting theories given
by the action (\ref{action}) with the scaling Lagrangian (\ref{slag}).
In doing so, we introduce the following dimensionless variables,
\be
x \equiv \frac{\dot{\phi}}{\sqrt{6}HM_{\rm pl}}\,,\qquad
y \equiv \frac{M_{\rm pl}e^{-\lambda \phi/(2M_{\rm pl})}}{\sqrt{3}H}\,.
\label{xydef}
\ee
Then, the background values of $Y_1$ and $Y_2$
are expressed, respectively, as
\be
Y_1=\frac{x^2}{y^2}M_{\rm pl}^4\,,\qquad
Y_2=\frac{\sqrt{2} x}{y}M_{\rm pl}^2\,,
\label{Y12}
\ee
and hence $Y_1=Y_2^2/2$.
It is also convenient to define
\be
\epsilon_{\phi} \equiv \frac{\ddot{\phi}}{H\dot{\phi}}\,,\qquad
\xi \equiv \frac{\dot{H}}{H^2}\,.
\label{epxi}
\ee
Then, the variables $x$ and $y$ obey
the differential equations,
\ba
x' &=& x \left( \epsilon_{\phi}- \xi \right)\,,\label{dxeq}\\
y' &=& -y \left( \frac{\sqrt{6}}{2}\lambda x
+ \xi  \right)\,,\label{dyeq}
\ea
where a prime denotes a derivative with respect to
$N=\ln a$.
On using Eqs.~(\ref{Eq00})-(\ref{Eq22}), it follows that
\ba
\hspace{-0.5cm}
\epsilon_{\phi} &=&
\frac{f_{1,\phi}\rho_m-f_{2,\phi}
+3H (f_{1,X} \dot{\phi}+f_{1,Z})P_m
+3H (f_{2,X} \dot{\phi}+f_{2,Z})
-(f_{1,X\phi} \rho_m-f_{2,X\phi})\dot{\phi}^2
-(f_{1,Z\phi} \rho_m-f_{2,Z\phi})\dot{\phi}}
{[f_{1,X}\rho_m-f_{2,X}+(f_{1,XX}\rho_m-f_{2,XX})\dot{\phi}^2
+2(f_{1,XZ} \rho_m-f_{2,XZ}) \dot{\phi}
+f_{1,ZZ}\rho_m-f_{2,ZZ}]H \dot{\phi}}\,,\\
\hspace{-0.5cm}
\xi &=&
\frac{(f_{1,X}\dot{\phi}+f_{1,Z})\dot{\phi} \rho_m
-(f_{2,X} \dot{\phi}+f_{2,Z})\dot{\phi}-(1+f_1)(\rho_m+P_m)}
{2H^2M_{\rm pl}^2}\,.
\ea
For the theories given by the functions (\ref{sca}) and (\ref{sca2}),
the quantities $\Omega_{\phi}$,
$w_{\phi}$, and $w_{\rm eff}$ are expressed as
\ba
\Omega_{\phi} &=&\frac{g_1 x^2 (g_2+2Y_1 g_{2,Y_1}
+Y_2 g_{2,Y_2})-2Y_1 g_{1,Y_1}-Y_2 g_{1,Y_2}}
{g_1-2Y_1 g_{1,Y_1}-Y_2 g_{1,Y_2}}=1-\Omega_{m}\,,\label{Ophi}\\
w_{\phi} &=& \frac{g_2 x^2}{\Omega_{\phi}}\,,\label{wp}\\
w_{\rm eff} &=& g_2 x^2+w_{m}\Omega_{m} \,.\label{we}
\ea
In Appendix, we will present the autonomous equations
for the scaling Lagrangian (\ref{slag})
with $f_1$ and $f_2$ given by
Eqs.~(\ref{sca}) and (\ref{sca2}), respectively.

The fixed points of the above dynamical system are characterized
by constant values of $x$ and $y$,  which we denote
$x_c$ and $y_c$, respectively.
The scaling fixed point corresponds to $x_c \neq 0$ and
$y_c \neq 0$, so it should satisfy
\ba
& &
\epsilon_{\phi}=\xi\,,\label{fix1}\\
&&
\xi=-\frac{\sqrt{6}}{2} \lambda x_c\,.
\label{fix2}
\ea
{}From Eq.~(\ref{fix2}), we have
\be
g_{2,Y_2}=\frac{\sqrt{6}\lambda  g_1 x_c-3(1-\Omega_{\phi})
[(1+w_m)g_1-2Y_1 g_{1,Y_1}-Y_2 g_{1,Y_2}]
-6 g_1 x_c^2 (g_2+Y_1 g_{2,Y_1})}{3g_1 x_c^2 Y_2}\,.
\label{fix3}
\ee
Substituting this relation into Eq.~(\ref{Ophi}) and solving it
for $\Omega_{\phi}$, we obtain
\be
\Omega_{\phi}=1+\frac{3(1+g_2 x_c^2)
-\sqrt{6} \lambda x_c}{3w_m}\,.
\label{Ops}
\ee
By using Eqs.~(\ref{fix1}) and (\ref{fix3}) with Eq.~(\ref{Ophi}),
it follows that
\be
\left( g_1-2Y_1 g_{1,Y_1}-Y_2 g_{1,Y_2} \right)
\left[ 2(Q+\lambda) x_c-\sqrt{6} (1+w_m) \right]
\left[ 3(1+g_2 x_c^2)-\sqrt{6}\lambda x_c \right]=0\,.
\label{fac}
\ee
Provided that $g_1$ is different from the specific form
$g_1=\sqrt{Y_1}\,F (Y_2/\sqrt{Y_1})$,
where $F$ is a function of $Y_2/\sqrt{Y_1}$,
we have $g_1-2Y_1 g_{1,Y_1}-Y_2 g_{1,Y_2} \neq 0$.
Then, there are two fixed points satisfying Eq.~(\ref{fac}):
(a) scaling solution, and (b) scalar-field dominated point.
In the following, we discuss the properties of
them in turn.

\subsection{Scaling solution (a)}

The scaling fixed point corresponds to
\be
x_c=\frac{\sqrt{6} (1+w_m)}{2(Q+\lambda)}\,.
\label{xcs}
\ee
Substituting Eq.~(\ref{xcs}) into Eqs.~(\ref{Ops}),
(\ref{wp}), and (\ref{we}), we obtain
\ba
\Omega_{\phi}
&=& \frac{[2Q (Q+\lambda)+3(1+w_m)g_2](1+w_m)}
{2w_m (Q+\lambda)^2}\,,\label{Omephia} \\
w_{\phi}
&=& \frac{3w_m (1+w_m) g_2}
{3(1+w_m)g_2+2Q(Q+\lambda)}\,,\\
w_{\rm eff} &=&\frac{w_m \lambda-Q}{Q+\lambda}\,.
\label{weffge}
\ea
Due to the constancy of $g_2(Y_1,Y_2)$ along the scaling solution, the quantities
(\ref{Omephia})-(\ref{weffge}) do not vary in time.
In the limit that $Q \to 0$, both $w_{\phi}$ and $w_{\rm eff}$
reduce to $w_m$. In this case, the density of scalar field
scales in the same manner as that of matter.
The existence of nonvanishing coupling $Q$ leads
$w_{\phi}$ and $w_{\rm eff}$ being different from $w_m$.

Instead of using $g_2$ in Eq.~(\ref{Omephia}), it is possible
to express $\Omega_{\phi}$ in terms of
$f_{2,X}$, $g_{2,Y_2}$
as well as $g_1$ and its derivatives with respect to
$Y_1$ and $Y_2$.
In doing so, we substitute Eq.~(\ref{Ops}) and
the relation $g_{2,Y_1}=(f_{2,X}-g_2)/Y_1$ into Eq.~(\ref{fix3})
and solve it for $g_2$.
Eliminating the term $g_2$ from Eq.~(\ref{Omephia}), we obtain
\be
\Omega_{\phi}=\frac{[2Q(Q+\lambda)+3(1+w_m)(2f_{2,X}+Y_2 g_{2,Y_2})]
(1+w_m)g_1-2(Q+\lambda)^2 (2Y_1 g_{1,Y_1}+Y_2 g_{1,Y_2})}
{2(Q+\lambda)^2[(1+w_m)g_1-2Y_1 g_{1,Y_1}-Y_2 g_{1,Y_2}]}\,.
\label{Opge}
\ee
If $g_1$ depends on neither $Y_1$
nor $Y_2$, we have
$\Omega_{\phi}=[2Q(Q+\lambda)+3(1+w_m)(2f_{2,X}+Y_2
g_{2,Y_2})]/[2(Q+\lambda)^2]$.
In k-essence where $g_2$ depends on $Y_1$ alone,
the field density parameter further
reduces to $\Omega_{\phi}=[Q(Q+\lambda)+3(1+w_m)
f_{2,X}]/(Q+\lambda)^2$.
This coincides with the result
derived in Refs.~\cite{Tsujikawa:2006mw,Amendola:2006qi}.
In quintessence ($f_2=X-V(\phi)$) with the exponential
potential $V(\phi)=V_0 e^{-\lambda \phi/M_{\rm pl}}$, i.e.,
the choice $g_2=1-V_0/Y_1$, we reproduce the value
$\Omega_{\phi}=3(1+w_m)/\lambda^2$ for $Q=0$ \cite{CLW}.
In this case, the scaling radiation and matter eras
in which $\Omega_{\phi}$ is subdominant to $\Omega_m$ can be
realized for $|\lambda| \gg 1$ \cite{Barreiro,CST06}.

The field density parameter (\ref{Opge}) contains the dependence
of both $Y_1$ and $Y_2$ in the functions $g_1$ and $g_2$, so
it is the generalization of scaling solutions already known
in literature. However, the effective equation of state (\ref{weffge})
is not subject to modifications compared to that derived
in Refs.~\cite{Amendola99,Tsujikawa:2006mw,Amendola:2006qi}.
For $|\lambda| \gg 1$ and $|Q|<{\cal O}(1)$, $w_{\rm eff}$ is close to
$w_m$. In this case, the scaling solution with $\Omega_{\phi} \ll 1$
can be used during the radiation or matter eras.
For $|Q|$ larger than the order of $|\lambda|$, it is possible
for the scaling solution to satisfy the condition of cosmic
acceleration ($w_{\rm eff}<-1/3$).
However, the coupling $|Q|$ is typically constrained to be
smaller than  0.1 for the consistency with CMB
measurements \cite{Ade:2015rim},
in which case the realization of scaling accelerated attractor
with $\Omega_{\phi} \simeq 0.7$ is difficult \cite{Amendola:2006qi}.
For $|Q|<{\cal O}(0.1)$ and $|\lambda|<{\cal O}(1)$,
the fixed point relevant to late-time cosmic acceleration
is  the scalar-field dominated point
discussed later in Sec.~\ref{pointBsec}.

\subsection{Scalar-field dominated point (b)}
\label{pointBsec}

{}From Eq.~(\ref{fac}), there exists the other fixed point satisfying
\be
g_2=\frac{\sqrt{6} \lambda x_c-3}{3x_c^2}\,.
\label{g2re1}
\ee
Then, from Eqs.~(\ref{wp}), (\ref{we}), and (\ref{Ops}),
we have
\ba
& &
\Omega_{\phi}=1\,,\\
& &
w_{\rm eff}=w_{\phi}=-1+\frac{\sqrt{6}
\lambda x_c}{3}\,.
\label{wep}
\ea
This scalar-field dominated point can be responsible for the late-time
cosmic acceleration under the condition
\be
\lambda x_c < \frac{\sqrt{6}}{3}\,.
\ee
{}From Eq.~(\ref{fix3}), there is also the following relation
\be
g_2+Y_1 g_{2,Y_1}+\frac{1}{2} Y_2 g_{2,Y_2}
=\frac{\lambda}{\sqrt{6}x_c}\,.
\label{g2re2}
\ee
For a given function $g_2(Y_1, Y_2)$, the values of $x_c$ and $y_c$ at the
scalar-field dominated point are obtained
by solving Eqs.~(\ref{g2re1}) and (\ref{g2re2}) together with
the relations given in Eq.~(\ref{Y12}).
On using Eq.~(\ref{g2re2}), we can express Eq.~(\ref{wep}) in the form
\be
w_{\rm eff}=w_{\phi}=-1+\frac{2\lambda^2}
{3(2f_{2,X}+Y_2 g_{2,Y_2})}\,,
\label{weffb}
\ee
where $f_{2,X}=g_2+Y_1 g_{2,Y_1}$.
In k-essence without the $Y_2$ dependence in $g_2$,
we recover the value
$w_{\rm eff}=w_{\phi}=-1+\lambda^2/(3f_{2,X})$
derived in Refs.~\cite{Tsujikawa:2006mw,Amendola:2006qi}.
From Eq.~(\ref{weffb}), we find that, for $\lambda$ closer
to 0, $w_{\rm eff}$ and $w_{\phi}$ approach $-1$.

\subsection{Kinetic fixed points}
\label{kinesec}

Let us derive other fixed points for the dynamical system
given by Eqs.~(\ref{dxeq})-(\ref{dyeq}).
The $\phi$MDE corresponds to the scaling solution with kinetic
domination satisfying
\be
y_c=0\,.
\ee
Since the quantities $Y_1$ and $Y_2$ diverge at $y_c=0$,
the functions $g_1$ and $g_2$ should take the following forms
to avoid the divergence of background equations,
\ba
g_1 (Y_1, Y_2) &=&
b_0+\sum_{i>0} \left( b_i Y_1^{-i}+
\tilde{b}_i Y_2^{-i} \right)
+2^{1-m/2} \mu \frac{Y_2^m}{Y_1^{m/2}}
+\sum_{i>0,j<2i}
\mu_i \frac{Y_2^j}{Y_1^i} \,, \label{g1e} \\
g_2 (Y_1, Y_2) &=&
c_0+\sum_{i>0} \left( c_i Y_1^{-i}+
\tilde{c}_i Y_2^{-i} \right)
+2^{1-m/2} \beta \frac{Y_2^m}{Y_1^{m/2}}
+\sum_{i>0,j<2i}
\beta_i \frac{Y_2^j}{Y_1^i} \,, \label{g2e}
\ea
where $b_0$, $b_i$, $\tilde{b}_i$, $\mu$, $\mu_i$,  $c_0$, $c_i$, $\tilde{c}_i$, $\beta$, $\beta_i$,
and $m$ are constants.
Due to the relation $Y_1=Y_2^2/2$ at the background level,
the third terms on the right hand sides of Eqs.~(\ref{g1e}) and (\ref{g2e}) are constant.
The last terms in Eqs.~(\ref{g1e}) and (\ref{g2e}) do not
diverge for $i>0$ and $j<2i$.

In the following, we will set $c_0=1$ without loss of
generality.
We substitute Eqs.~(\ref{g1e})-(\ref{g2e}) and their $Y_1$, $Y_2$
derivatives into Eq.~(\ref{Ophi})-(\ref{we}), use the relations
(\ref{Y12}), and finally take the limit $y \to 0$.
This process leads to
\ba
& &
\Omega_{\phi}= q_s x^2 =1-\Omega_m\,,
\label{Omeki}\\
& &
w_{\phi}=1\,,\\
& &
w_{\rm eff}
=w_m-(w_m-1) q_s x^2 \,,
\label{wki}
\ea
where
\be
q_s \equiv 1+2\beta\,.
\ee
The autonomous Eq.~(\ref{dxeq}) reduces to
\be
x'=-\frac{1}{2q_s} \left[ 3q_s (w_m-1)x-\sqrt{6}Q
\right] \left( q_s x^2-1 \right)\,,
\label{dxeq2}
\ee
while Eq.~(\ref{dyeq}) is automatically satisfied.
We note that the constants appearing in the expression of
$g_1 (Y_1, Y_2)$ in Eq.~(\ref{g1e}) do not affect
Eq.~(\ref{dxeq2}).
It is also interesting to observe that, for $w_m=0$,  there is the relation $w_{\rm eff}=\Omega_{\phi}=q_s x^2$.
Indeed, this relation holds in all the kinetic scaling
solutions identified so far, see, for instance, Refs.~\cite{Amendola:2006qi,Amendola:2018ltt,Frusciante:2018tvu,Frusciante:2018aew}.
It is this phase that induces a small and positive effective equation of state that can help alleviating the $H_0$ tension. On the other hand, this implies that a kinetic scaling solution associated with the matter dominance will be strongly constrained by CMB observations of the distance to last scattering.
Leaving aside the $H_0$ tension, one possibility to implement $w_{\rm eff}=0$, and therefore to ease CMB distance constraints, would be to introduce a non-vanishing $w_m=\Omega_\phi/(\Omega_\phi-1)$.
This possibility is not pursued here
but left for future work.

{}From Eq.~(\ref{dxeq2}), there are the following fixed points.

\subsubsection{$\phi$MDE {\rm (c)}}

The $\phi$MDE corresponds to one of the solutions to
Eq.~(\ref{dxeq2}), i.e.,
\be
x_c=\frac{\sqrt{6}Q}{3 q_s (w_m-1)}\,.
\label{xcc}
\ee
{}From Eqs.~(\ref{Omeki}) and (\ref{wki}), we obtain
\ba
& &
\Omega_{\phi}=\frac{2Q^2}{3 q_s (w_m-1)^2}
=1-\Omega_m\,,
\label{Omekic}\\
& &
w_{\rm eff}
=w_m-\frac{2Q^2}{3 q_s (w_m-1)}\,.
\label{wkic}
\ea
This means that the $\phi$MDE is a scaling solution
besides the fixed point (a).
For nonrelativistic matter ($w_m=0$), it follows that
$\Omega_{\phi}=w_{\rm eff}=2Q^2/(3q_s)$.
The standard matter-dominated epoch
with $\Omega_{\phi}=w_{\rm eff}=0$
is modified by the nonvanishing coupling $Q$.
In comparison to theories with the coupling $Q$ alone, the coupling
$\beta$ gives the additional contribution to
 $x_c$, $\Omega_{\phi}$, and $w_{\rm eff}$.

\subsubsection{Purely kinetic solutions {\rm (d1), (d2)}}

The other solution to Eq.~(\ref{dxeq2}) corresponds to
purely kinetic points (d1), (d2) satisfying
\be
x_c=\pm \frac{1}{\sqrt{q_s}}\,,
\ee
whose existence  requires that $q_s>0$.
{}From Eqs.~(\ref{Omeki}) and (\ref{wki}), we have
\be
\Omega_{\phi}=1\,,\qquad w_{\rm eff}=1\,.
\ee
The points (d1), (d2) are not relevant to
radiation/matter eras or the epoch of cosmic acceleration.

\vspace{0.3cm}
In summary, the fixed points relevant to the cosmological
evolution after radiation-matter equality are the $\phi$MDE (c) and the scalar-field dominated point (b).
The coupling $Q$ associated with the energy transfer is crucially required for
the existence of $\phi$MDE.
The coupling $\beta$ associated with the momentum transfer also affects
the values of  $\Omega_{\phi}$ and $w_{\rm eff}$ on the $\phi$MDE.

\section{Background cosmology for a model with $\phi$MDE}
\label{backsec}

In this section, we study the background cosmological dynamics
for the model with
\ba
& &
g_1(Y_1,Y_2)=1\,,
\label{eq:simpg1}\\
& &
g_2(Y_1,Y_2)=1-\frac{V_0}{Y_1}
+2^{1-m/2} \beta \frac{Y_2^m}{Y_1^{m/2}}\,,
\label{eq:simpg2}
\ea
where $V_0$ is a positive constant.
Although this is a simple  model, it contains nevertheless
all the new phenomenology we wish to consider.
We consider the coupling only between the scalar field and CDM,
where the CDM density $\rho_c$ depends on its number
density $n_c$.
The interacting model given by Eqs.~(\ref{eq:simpg1})-(\ref{eq:simpg2})
belongs to a subclass of the functions (\ref{g1e}) and (\ref{g2e}),
so the $\phi$MDE is present besides the other
fixed points derived in Sec.~\ref{fixedsec}.
In this case, the Lagrangian (\ref{Lin}) is given by
\be
L=-\left( e^{Q\phi/M_{\rm pl}}-1 \right) \rho_c (n_c)
+X-V_0 e^{-\lambda \phi/M_{\rm pl}}
+\beta \left( 2X \right)^{1-m/2} Z^m\,.
\label{Lagcon}
\ee
The canonical scalar field $\phi$ with the potential
$V(\phi)=V_0 e^{-\lambda \phi/M_{\rm pl}}$ interacts with CDM
through the energy-transfer
$-\left( e^{Q\phi/M_{\rm pl}}-1 \right) \rho_c (n_c)$
and the momentum-transfer
$\beta \left( 2X \right)^{1-m/2} Z^m$.
Without the coupling $Q$, the interactions of the forms
$Z^2$ \cite{Pourtsidou:2013nha,Pourtsidou:2016ico,Chamings:2019kcl}
or $Z^n$ \cite{Linton,Kase:2019mox} were already studied
in the literature.
In the following, we are going to investigate the cosmological
dynamics in the presence of the two nonvanishing coupling
constants $Q$ and $\beta$.

Besides CDM with the vanishing pressure ($P_c=0$),
we also take baryons (energy density $\rho_b$ with
vanishing pressure) and radiation (energy density
$\rho_r$ and pressure $P_r=\rho_r/3$) into account
to study the dynamics of background cosmology from
the radiation era.
Neither baryons nor radiation are assumed to be
coupled to the scalar field.

Defining the variable $x$ as in Eq.~(\ref{xydef}), the dimensionless
scalar field $\tilde{\phi} \equiv \phi/M_{\rm pl}$ obeys
the differential equation,
\be
\tilde{\phi}'=\sqrt{6} x\,.
\ee
Instead of the variable $y$ defined in Eq.~(\ref{xydef}),
we will use
\be
\tilde{y} \equiv \frac{\sqrt{V_0}}{M_{\rm pl}^2}y
=\sqrt{\frac{V_0}{3}} \frac{e^{-\lambda \phi/(2M_{\rm pl})}}
{M_{\rm pl} H}\,.
\label{tildey}
\ee
The density parameters of scalar field
and matter components are given by
\be
\Omega_{\phi}=q_s x^2+\tilde{y}^2\,,
\qquad
\Omega_c=\frac{e^{Q\phi/M_{\rm pl}} \rho_c}{3M_{\rm pl}^2 H^2}\,,
\qquad
\Omega_b=\frac{\rho_b}{3M_{\rm pl}^2 H^2}\,,
\qquad
\Omega_r=\frac{\rho_r}{3M_{\rm pl}^2 H^2}\,.
\ee
As we will see in Sec.~\ref{persec}, the no-ghost condition
of scalar-field perturbation requires that
\be
q_s=1+2\beta>0\,.
\label{noghost}
\ee
{}From Eq.~(\ref{Eq00}), (\ref{Eq11}) and (\ref{Eq22}), we have
\ba
\Omega_c &=&
1-\Omega_{\phi}-\Omega_b-\Omega_r\,,\\
\xi  &=&
-3q_s x^2-\frac{3}{2}\Omega_c-\frac{3}{2}\Omega_b
-2\Omega_r\,,\\
\epsilon_{\phi}
&=& -3+\frac{\sqrt{6}}{2q_s x}
\left( \lambda \tilde{y}^2-Q \Omega_{c} \right)\,,
\ea
where we recall that $\xi$ and $\epsilon_{\phi}$ are
defined in Eq.~(\ref{epxi}).
We can reduce the background equations to the following
autonomous system,
\ba
x' &=& \frac{1}{2} x \left( 6 q_s x^2-6 +3\Omega_c
+3\Omega_b+4\Omega_r \right)
+\frac{\sqrt{6}}{2 q_s} \left( \lambda \tilde{y}^2
-Q \Omega_c \right)\,,\label{auto1}\\
\tilde{y}' &=& \frac{1}{2} \tilde{y} \left( 6 q_s x^2
-\sqrt{6} \lambda x+3\Omega_c
+3\Omega_b+4\Omega_r \right)\,,\\
\Omega_b' &=& \Omega_b \left( 6 q_s x^2-3
+3\Omega_c+3\Omega_b+4\Omega_r \right)\,,\\
\Omega_r' &=& \Omega_r \left( 6 q_s x^2-4
+3\Omega_c+3\Omega_b+4\Omega_r \right)\,.
\label{auto4}
\ea
Notice that the background equations are independent of the power $m$.
The scalar-field and effective equations of state are
given, respectively, by
\ba
w_{\phi} &=& \frac{ q_s x^2-\tilde{y}^2}
{q_s x^2+\tilde{y}^2}\,,\\
w_{\rm eff} &=& -1+2q_s x^2
+\Omega_c+\Omega_b+\frac{4}{3}\Omega_r\,.
\ea

Besides the scaling fixed point (a) derived in Sec.~\ref{fixedsec},
there exist the following three fixed
points (b), (c), and (e) which
are relevant to the dynamics of accelerated, matter, and radiation
eras, respectively.
\begin{itemize}
\item Accelerated point (b)
\be
x_c=\frac{\lambda}{\sqrt{6}q_s}\,,\quad
\tilde{y}_c=\sqrt{1-\frac{\lambda^2}{6q_s}}\,,\quad
\Omega_b=0\,,\quad \Omega_r=0\,,\quad
\Omega_{\phi}=1\,,\quad
w_{\phi}=w_{\rm eff}=-1+\frac{\lambda^2}{3q_s}
\label{pointb}\,.
\ee
\item $\phi$MDE point (c)
\be
x_c=-\frac{\sqrt{6}Q}{3 q_s}\,,\quad
\tilde{y}_c=0\,,\quad
\Omega_b=0\,,\quad \Omega_r=0\,,\quad
\Omega_{\phi}=w_{\rm eff}=\frac{2Q^2}{3q_s}\,,\quad
w_{\phi}=1\,.
\label{pointc}
\ee
\item Radiation point (e)
\be
x_c=0\,,\quad \tilde{y}_c=0\,,\quad
\Omega_b=0\,,\quad \Omega_r=1\,,\quad
\Omega_{\phi}=0\,,\quad
w_{\rm eff}=\frac{1}{3}\,.
\label{pointe}
\ee
\end{itemize}
The point (b) can drive a late-time cosmic acceleration
under the condition $w_{\rm eff}<-1/3$.
This translates to
\be
\lambda^2<2q_s\,.
\label{cond1}
\ee
{}From Eq.~(\ref{weffge}), there is a possibility that
the scaling solution (a) leads to
the acceleration for $|Q| \gg |\lambda|$.
As shown in Ref.~\cite{Amendola:2006qi}, however,
such a large coupling $|Q|$ is
hardly compatible with the existence of $\phi$MDE (c)
with $\Omega_{\phi} \ll 1$.
As we will also see below, if point (b) is stable, then
point (a) is not. Hence we will focus on the case in which
$\phi$MDE is followed by point (b).

Besides point (e), there exist the scaling fixed points (f) and (g)
given by
\ba
& &
{\rm (f)}~~~
x_c=\frac{2\sqrt{6}}{3\lambda}\,,\quad
\tilde{y}_c=\sqrt{\frac{4q_s}{3\lambda^2}}\,,\quad
\Omega_b=0\,,\quad \Omega_r=1-\frac{4q_s}{\lambda^2}\,,
\quad \Omega_{\phi}=\frac{4q_s}{\lambda^2}\,,\quad
w_{\phi}=w_{\rm eff}=\frac{1}{3}\,,\\
& &
{\rm (g)}~~~x_c=-\frac{1}{\sqrt{6}Q}\,,\quad \tilde{y}_c=0\,,\quad
\Omega_b=0\,,\quad \Omega_r=1-\frac {q_s}{2Q^2}\,,\quad
\Omega_{\phi}=\frac{q_s}{6 Q^2}\,,\quad
w_{\phi}=1\,,\quad
w_{\rm eff}=\frac{1}{3}\,,
\ea
both of which can be potentially used for the
radiation era.
For point (f), however, $\Omega_r$ is
negative under the condition (\ref{cond1}).
The point (g) can be responsible for the radiation era
only for $Q^2 \gg q_s$, which was exploited for the generation
of primordial dark matter halos in Ref.~\cite{Savastano:2019zpr}.
Unless some screening of fifth forces occurs after radiation-matter
equality, point (g) is not followed by $\phi$MDE with
$\Omega_\phi \ll 1$. Hence we use neither (f) nor (g)
for the fixed point of radiation era in this paper.
In other words, we consider the cosmological sequence of fixed points:
(e) $\to$ (c) $\to$ (b).

The stability of fixed points is established by perturbing
Eqs.~(\ref{auto1})-(\ref{auto4}) with the
linear perturbations $\delta x$, $\delta \tilde{y}$,
$\delta \Omega_b$, and $\delta \Omega_r$.
The signs of eigenvalues of $4 \times 4$ Jacobian matrices for
these perturbations determine whether the fixed points
are stable or not \cite{CLW,CST06}.
For points (e) and (c), some of the eigenvalues are positive, so
they are not stable nodes.
In other words, the $\phi$MDE (c), which is preceded by point (e),
should eventually come to end to realize cosmic acceleration.
Provided that the condition (\ref{cond1})
is satisfied, three of the eigenvalues for point (b) are negative.
The other eigenvalue is negative for
\be
\lambda (\lambda+Q)<3q_s\,,
\label{cond2}
\ee
under which point (b) is stable.
On the other hand, the stability of point (a) requires that
$\lambda (\lambda+Q)>3q_s$, which is opposite to the inequality (\ref{cond2}).
This means that, under the condition (\ref{cond2}), the $\phi$MDE is
followed by point (b) instead of point (a).
We show a phase-space plot in Fig.~\ref{fig3} of Appendix
to confirm the attractor property of point (b).

In Fig.~\ref{fig1}, we plot the evolution
of $\Omega_{\phi}$, $\Omega_c$, $\Omega_b$, and $\Omega_r$ (left)
and $x$, $\tilde{y}$, $w_{\phi}$, and $w_{\rm eff}$ (right)
for the model parameters $\lambda=1$, $Q=0.07$, and $\beta=0.5$.
We observe that the radiation fixed point (e) is followed
by the $\phi$MDE (c) with nearly constant values
$\Omega_{\phi}=w_{\rm eff} \simeq 2Q^2/(3q_s)$.
The field equation of state $w_{\phi}$ during the $\phi$MDE
is close to 1, whose property is attributed to the kinetically
driven evolution satisfying $|x| \simeq \sqrt{6}|Q|/(3 q_s) \gg \ty$.

Since the model parameters $\lambda$ and $\beta$ used in Fig.~\ref{fig1}
satisfy the two conditions (\ref{cond1}) and (\ref{cond2}), the solutions
finally approach the stable accelerated point (b)
with the asymptotic values $x=0.204$, $\tilde{y}=0.957$, and
$w_{\phi}=w_{\rm eff}=-0.833$. They are in good agreement
with the numerical results shown in the right panel.
In this case, we observe that $w_{\phi}$ temporarily reaches the
minimum around $-1$ at redshift $z \simeq 1.3$ and
then it increases toward the asymptotic value $-0.833$.
This evolution of $w_{\phi}$ is different from that for the model
with $Q=0$ where $w_{\phi}$ is close to $-1$
during the matter era and
finally approaches $-1+\lambda^2/(3q_s)$ \cite{Kase:2019mox}.
Thus, the two models can be observationally distinguished
from each other.

\begin{figure}[H]
\begin{center}
\includegraphics[height=3.4in,width=3.5in]{fig1a.pdf}
\includegraphics[height=3.4in,width=3.5in]{fig1b.pdf}
\end{center}
\caption{\label{fig1}
(Left)
Evolution of $\Omega_{\phi}$, $\Omega_c$, $\Omega_b$,
and $\Omega_r$ versus $z+1$ for $\lambda=1$, $Q=0.07$,
and $\beta=0.5$, where $z=a_0/a-1$ and $a_0$ is today's value of $a$.
The initial conditions are chosen to be
$x=1.0 \times 10^{-13}$,
$\tilde{y}=1.0 \times 10^{-14}$,
$\Omega_b=5.8 \times 10^{-6}$, and
$\Omega_r=0.999962$ at redshift
$z=8.3 \times 10^7$.
(Right)
Evolution of $x$, $\tilde{y}$, $w_{\phi}$,
and $w_{\rm eff}$ for the same model parameters
and initial conditions as those used in the left.
}
\end{figure}

We note that the values of $\Omega_{\phi}$ and $w_{\rm eff}$ on the $\phi$MDE are reduced by
the positive coupling $\beta$.
The presence of positive $\beta$ also leads to the decreases of $w_{\phi}$ and $w_{\rm eff}$
on the attractor point (b).
For increasing $\beta$, the asymptotic value of $w_{\phi}$ gets closer to $-1$.
Nevertheless, the early evolution of $w_{\phi}$ is different from that
in the $\Lambda$CDM model. It remains to be seen whether the problem
of $H_0$ tension in the $\Lambda$CDM model can be alleviated in
our interacting model with $Q \neq 0$ and $\beta \neq 0$.

\section{Perturbation dynamics for a model with $\phi$MDE}
\label{persec}

In this section, we study the evolution of linear
scalar perturbations
for the theory (\ref{action}) with the interacting
Lagrangian (\ref{Lagcon}).
On the flat FLRW background, we consider
the perturbed line element given by
\be
{\rm d}s^2=-(1+2\alpha) {\rm d}t^2
+2 \partial_i \chi {\rm d}t {\rm d}x^i
+a^2(t) \left[ (1+2\zeta) \delta_{ij}
+2\partial_i \partial_j E \right] {\rm d}x^i {\rm d}x^j\,.
\label{permet}
\ee
The quantities $\alpha$, $\chi$, $\zeta$, and $E$
are scalar perturbations, which
depend on both $t$ and spatial coordinate $x^i$.
Here and in the following, we use
the notation
$\partial_i \chi=\partial \chi/\partial x^i$.
We do not take the tensor perturbation into account in
Eq.~(\ref{permet}), but it propagates in the same manner
as in standard general relativity.
We decompose the scalar field $\phi$ into the background
part $\bar{\phi}(t)$ and the perturbation
$\delta \phi$, as
\be
\phi=\bar{\phi}(t)+\delta \phi (t,x^i)\,,
\label{phide}
\ee
where we omit the bar from background quantities in the following.

For perfect fluids, we take CDM, baryons, and radiation into account
labelled by $I=c, b, r$, respectively.
{}From Eq.~(\ref{Jmure}), it follows that the number
${\cal N}_I=n_I  a^3$ of each fluid is conserved at
the background level.
We express the temporal and spatial components of
$J_I^{\mu}$, respectively,  as,
\be
J_I^0={\cal N}_I+\delta J_I\,,\qquad
J_I^i=\frac{1}{a^2(t)} \delta^{ik} \partial_k \delta j_I\,,
\label{JI}
\ee
where $\delta J_I$ and $\delta j_I$ are the scalar perturbations.
The scalar velocity potential $v_I$ is related to the spatial component
of four velocity $u_{Ii}=J_{Ii}/(n_I\sqrt{-g})$, as
\be
u_{Ii}=-\partial_{i}v_I\,.
\ee
Since $J_{Ii}=J_I^0 g_{0i}+J_I^j g_{ij}
={\cal N}_I \partial_{i} \chi+
\partial_{i} \delta j_I$ at linear
order in perturbations, it follows that
\be
\partial_i \delta j_I=-{\cal N}_I \left( \partial_i \chi
+\partial_i v_I \right)\,.
\label{delj}
\ee

On using Eq.~(\ref{lc}), there is also the following relation
\be
\ell_c=-\int^{t} \left[ 1+f_1(\tilde{t}) \right]
\rho_{c,n_c} (\tilde{t})\,{\rm d} \tilde{t}
-\left( 1+f_1 \right) \rho_{c,n_c}v_c
+\frac{f_{1,Z}\rho_c-f_{2,Z}}{n_c}
\left( \dot{\phi}\,v_c-\delta \phi \right)\,,
\ee
up to first order in perturbations.
On the other hand, the Lagrange multipliers $\ell_I$
for baryons and radiation are
\be
\ell_I=-\int^{t}
\rho_{I,n_I} (\tilde{t})\,{\rm d} \tilde{t}
-\rho_{I,n_I}v_I\,,\qquad
{\rm with}~I=b,r\,.
\ee
The fluid density is given by
$\rho_I=\rho_I(t)+\delta \rho_I (t,x^i)$,
where the perturbed part is
\be
\delta \rho_I=\frac{\rho_{I,n_I}}{a^3}
\left[ \delta J_I-{\cal N}_I \left( 3\zeta+\partial^2 E
\right) \right]\,,
\ee
where $\partial^2 E=\sum_{i=1}^3 \partial_i^2 E$.
In this case, we have
\be
\delta n_I= \frac{\delta \rho_I}{\rho_{I,n_I}}
-\frac{{\cal N}_I (\partial v_I)^2}{2a^5}
-(3\zeta+\partial^2E)\frac{\delta \rho_I}{\rho_{I,n_I}}
-\frac{{\cal N}_I(\zeta+\partial^2E)(3\zeta-\partial^2E)}{2a^3}
+{\cal O}(\varepsilon^3)\,,
\label{deln}
\ee
where $(\partial v_I)^2=\sum_{i=1}^3 (\partial_i v_I)^2$, and
$\varepsilon^n$ represents the $n$-th order of perturbations.
We also introduce the sound speed squared of each matter
fluid, as
\be
c_I^2=\frac{n_I \rho_{I,n_I n_I}}{\rho_{I,n_I}}\,.
\ee
The quantities $\rho_I(n_I)$, $X$, and $Z$, when are expanded
up to second order in perturbations, are given by
\ba
\rho_I (n_I)
&=&\rho_I+\left( \rho_I+P_I \right)
\frac{\delta n_I}{n_I}+\frac{1}{2} \left( \rho_I+P_I \right)
c_I^2 \left( \frac{\delta n_I}{n_I} \right)^2
+{\cal O} (\varepsilon^3)\,,\\
X &=& \frac{1}{2} \dot{\phi}^2+\dot{\phi} (\dot{\delta \phi}-\dot{\phi} \alpha)
+\frac{1}{2} \left[  (\dot{\delta \phi}-2\dot{\phi} \alpha)^2
-\frac{1}{a^2}(\partial \delta \phi+\dot{\phi} \partial \chi)^2
\right]+{\cal O} (\varepsilon^3)\,,
\label{deltaX}\\
Z &=& \dot{\phi}+\dot{\delta \phi}-\dot{\phi} \alpha
+\frac{1}{2a^2} \left[ \dot{\phi} \left\{ 3a^2 \alpha^2
-(\partial_i \chi)^2+(\partial_i v_c)^2 \right\}-2a^2 \alpha \dot{\delta \phi}
-2\partial_i \delta \phi (\partial_i \chi+\partial_i v_c)
\right] +{\cal O} (\varepsilon^3)\,.
\label{delZ}
\ea
In the following, we consider the case in which the equations
of state and sound speed squares of both CDM and baryons vanish, i.e.,
\be
w_c^2=0\,,\qquad w_b^2=0\,,\qquad
c_c^2=0\,,\qquad c_b^2=0\,.
\ee

As in Refs.~\cite{Kase:2019veo,Kase:2019mox},
the linear perturbation equations of motion follow by expanding
the action (\ref{action}) up to quadratic order.
Alternatively, we can also derive them by using the covariant
equations of motion (\ref{Ein}), (\ref{Tcon}), and
(\ref{Tcon2}) at first order.
The explicit form of second-order action will be presented
for a more general interacting Lagrangian $L(n_c, \phi, X, Z)$
as a future work \cite{KTfuture}.
Here, we show the perturbation equations for the interacting
Lagrangian (\ref{Lagcon}) in a gauge-ready form.
For this purpose, we introduce the following
gauge-invariant quantities,
\ba
&&
\delta \phi_{\rm N}=\delta \phi+\dot{\phi}\left(\chi-a^2 \dot{E}\right)\,,
\qquad
\delta \rho_{I\rm N}=\delta \rho_I+\dot{\rho}_I \left(\chi-a^2 \dot{E}\right)\,,
\qquad
v_{I{\rm N}}=v_I+\chi-a^2 \dot{E}\,,\nonumber \\
&&
\Psi=\alpha+\frac{{\rm d}}{{\rm d}t}
\left( \chi - a^2 \dot{E} \right)\,,\qquad
\Phi=\zeta+H \left( \chi - a^2 \dot{E} \right)\,,
\ea
together with the dimensionless variables,
\be
\delta_{I{\rm N}}= \frac{\delta \rho_{I\rm N}}{\rho_I}\,,\qquad
\delta \varphi_{\rm N}=\frac{H}{\tp}\delta\phi_{\rm N}\,,\qquad
V_{I\rm N}=H v_{I{\rm N}}\,,\qquad
{\cal K}=\frac{k}{aH}\,,
\ee
where $k$ is a comoving wavenumber.

In Fourier space, the full set of linear perturbation
equations of motion is then given by
\ba
& &
6 q_s x^2 \delta \varphi_{\rm N}'-6 \Phi'+6 \left(
1-q_s x^2 \right) \left( \xi \delta \varphi_{\rm N}
+\Psi \right)-2{\cal K}^2 \Phi
+3 \left( 3\Omega_c+3\Omega_b+4\Omega_r
\right) \delta \varphi_{\rm N}  \nonumber \\
& &
+3 \left( \Omega_c \delta_{c{\rm N}}
+\Omega_b \delta_{b{\rm N}}
+\Omega_r \delta_{r{\rm N}}  \right)=0\,,
\label{pereq1}\\
& &
\Phi'-\Psi-\xi \delta \varphi_{\rm N}
+\frac{3}{2} \left( \Omega_c+2m \beta  x^2
\right) \left( V_{c{\rm N}}-\delta \varphi_{\rm N} \right)
+\frac{3}{2} \Omega_b
\left( V_{b{\rm N}}-\delta \varphi_{\rm N} \right)
\nonumber \\
& &
+2\Omega_r
\left( V_{r{\rm N}}-\delta \varphi_{\rm N} \right)=0\,,\\
& &
\delta'_{I{\rm N}}+3 \left( c_I^2-w_I \right) \delta_{I{\rm N}}
+\left( 1+w_I \right) \left( {\cal K}^2 V_{I{\rm N}}
+3\Phi' \right)=0\,,\qquad ({\rm for}~I=c,b,r),
\label{pereq3}\\
& &
\left( \Omega_c+2m \beta x^2 \right) V'_{c{\rm N}}
-\left[ \xi \left( \Omega_c+2m \beta  x^2 \right)
-2m \beta x^2 (3+2\epsilon_{\phi})
-\sqrt{6} Q x \Omega_c
\right] V_{c{\rm N}}
-\Omega_c \Psi \nonumber \\
& &
-2m \beta x^2 \delta \varphi'_{\rm N}
+\left[ 2m \beta x ( \xi -3-2\epsilon_{\phi})
-\sqrt{6}Q \Omega_c \right] x
\delta \varphi_{\rm N} =0\,,
\label{pereq4}\\
& &
V'_{I{\rm N}}-\left( \xi+3 c_I^2 \right) V_{I{\rm N}}
-\Psi-\frac{c_I^2}{1+w_I} \delta_{I{\rm N}}=0\,,
\qquad ({\rm for}~I=b,r), \\
& &
\delta \varphi''_{\rm N}+ \left( 3-\xi +2\epsilon_{\phi}
\right)\delta \varphi'_{\rm N}
+\left[ \hat{c}_s^2 {\cal K}^2-\xi'-3\xi+\epsilon'_{\phi}
+\epsilon_{\phi}^2
+(3-\xi)\epsilon_{\phi}
+\frac{3}{q_s} \left( \lambda^2 \tilde{y}^2
+Q^2 \Omega_c \right) \right]
\delta \varphi_{\rm N} \nonumber \\
& &+3\hat{c}_s^2 \Phi'-\Psi'-2\left(3+\epsilon_{\phi}\right)\Psi
-\frac{m \beta}{q_s} \delta'_{c{\rm N}}
+\frac{\sqrt{6} Q \Omega_c}{2q_s x} \delta_{c{\rm N}}=0\,,
\label{delphi} \\
& &
\Psi=-\Phi\,,
\label{pereq6}
\ea
where we remind that the prime denotes the derivative
with respect to $N=\ln a$, and
\be
\hat{c}_s^2=1-\frac{m \beta}{q_s}\,.
\ee

Equations (\ref{pereq1})-(\ref{pereq6}) are written in the gauge-ready
form, in that they can be used for any gauge choices of interest.
For example, if we choose the unitary gauge characterized by
$\delta \phi=0$ and $E=0$, the dynamical perturbations
correspond to the curvature perturbation
${\cal R}=\Phi-\delta \varphi_{\rm N}$ and the
density perturbations
$\delta \rho_{I{\rm u}}=\delta \rho_{I{\rm N}}
-\dot{\rho}_I \delta \phi_{\rm N}/\dot{\phi}$.
We can eliminate nondynamical variables like $\alpha$,
$\chi$, and $v_I$ from the second-order action
by using Eqs.~(\ref{pereq1})-(\ref{pereq3}).
Following a procedure similar to that performed
in Refs.~\cite{Kase:2019veo,Kase:2019mox} for perturbations
deep inside the Hubble radius,
there are neither ghosts nor Laplacian instabilities
for dynamical perturbations ${\cal R}$ and
$\delta \rho_{c{\rm u}}$ under the conditions,
\ba
q_s &=&1+2\beta>0\,,
\label{thecon1} \\
q_c &=& 1+\frac{2m \beta x^2}{\Omega_c}
>0\,,\label{thecon2}\\
c_s^2 &=& \hat{c}_s^2+\frac{2 \beta^2 m^2 x^2}
{q_s (2 \beta m x^2+\Omega_c)} \geq 0\,.
\label{thecon3}
\ea
In the limit that $c_c^2 \to 0$, the effective sound speed squared
of CDM vanishes, so that there is no additional pressure
modifying the evolution of CDM density perturbations.
The effective sound speed squared of the scalar field
in the small-scale limit corresponds to $c_s^2$ given in
Eq.~(\ref{thecon3}). In comparison to the value $\hat{c}_s^2$, there is a
correction to $c_s^2$ arising from a kinetic mixing between
the scalar field and CDM \cite{Kase:2019veo,Kase:2019mox}.
This correction term is positive under the no-ghost
conditions (\ref{thecon1}) and (\ref{thecon2}).
Provided that $\hat{c}_s^2 \geq 0$, the positivity of
$c_s^2$ is always ensured.

In the following, we study the evolution of perturbations
after the onset of $\phi$MDE. For this purpose,
we ignore the contribution of radiation perturbations
to Eqs.~(\ref{pereq1})-(\ref{pereq6}) and set $\Omega_r=0$.
For CDM and baryons, Eq.~(\ref{pereq3}) reduces to
\be
\delta_{I{\rm N}}'+{\cal K}^2 V_{I{\rm N}}+3\Phi'=0\,,
\label{delIN}
\ee
where $I=c,b$.
The CDM velocity potential $V_{c{\rm N}}$ satisfies the first-order
differential Eq.~(\ref{pereq4}),
while the baryon velocity potential $V_{b{\rm N}}$ obeys
\be
V_{b{\rm N}}'- \xi V_{b{\rm N}} -\Psi=0\,.
\label{pereq5}
\ee
Differentiating Eq.~(\ref{delIN}) with respect to $N$ and using
Eqs.~(\ref{pereq4}) and (\ref{pereq5}), it follows that
\ba
& & \delta_{c{\rm N}}''+\nu_1 \delta_{c{\rm N}}'
+\nu_2 {\cal K}^2 +3\Phi''+3 \nu_1\Phi'=0\,,
\label{delceq} \\
& & \delta_{b{\rm N}}''+\left( 2+\xi \right) \delta_{b{\rm N}}'
+{\cal K}^2 \Psi+3\Phi''+3 \left( 2+\xi \right) \Phi'=0\,,
\label{delbeq}
\ea
where
\ba
\nu_1 &=& 2+\xi+\frac{[2m \beta (3+2\epsilon_{\phi})x
+\sqrt{6} Q \Omega_c]x}{\Omega_c+2m \beta x^2}\,,\\
\nu_2 &=& \frac{\Omega_c \Psi+2m \beta x^2 \delta \varphi_{\rm N}'
+[2m \beta (3-\xi+2\epsilon_{\phi})x
+\sqrt{6}Q \Omega_c] x\delta \varphi_{\rm N}
}{\Omega_c+2m \beta x^2}\,.
\ea

Now, we employ the quasi-static approximation for perturbations
deep inside the sound horizon \cite{Boisseau,DKT11}.
Since the dominant contributions to the perturbation equations
are those containing ${\cal K}^2$, $\delta_{c{\rm N}}$,
$\delta'_{c{\rm N}}$, and $\delta_{b{\rm N}}$,
Eqs.~(\ref{pereq1}), (\ref{delphi}), and (\ref{pereq6}) give the following relations,
\ba
& &
\Psi=-\Phi \simeq -\frac{3}{2{\cal K}^2}
\left(  \Omega_c \delta_{c{\rm N}}
+\Omega_b \delta_{b{\rm N}} \right)\,,
\label{quasi1} \\
& &
\delta \varphi_{\rm N} \simeq \frac{1}{q_s \hat{c}_s^2 {\cal K}^2}
\left( m \beta \delta'_{c{\rm N}}-\frac{\sqrt{6} Q \Omega_c}{2x}
\delta_{c{\rm N}} \right)\,.
\label{quasi2}
\ea
Under the quasi-static approximation, we can ignore the terms
$3\Phi''$, $3 \nu_1\Phi'$, and $3 \left( 2+\xi \right) \Phi'$
in Eqs.~(\ref{delceq}) and (\ref{delbeq}) relative to the others.
Substituting Eqs.~(\ref{quasi1}), (\ref{quasi2}) and the $N$ derivative
of Eq.~(\ref{quasi2}) into Eqs.~(\ref{delceq}) and (\ref{delbeq}), it follows that
\ba
& &
\delta_{c{\rm N}}''+\nu \delta_{c{\rm N}}'
-\frac{3}{2G} \left( G_{cc} \Omega_c \delta_{c{\rm N}}
+G_{cb} \Omega_b \delta_{b{\rm N}} \right) \simeq 0\,,
\label{delceq2} \\
& &
\delta_{b{\rm N}}''+\left( 2+\xi \right) \delta_{b{\rm N}}'
-\frac{3}{2G} \left( G_{bc} \Omega_c \delta_{c{\rm N}}
+G_{bb} \Omega_b \delta_{b{\rm N}} \right) \simeq 0\,,
\label{delbeq2}
\ea
where
\be
\nu=\frac{2m \beta (1+2\beta)(5+\xi+2\epsilon_{\phi})x^2
+(2+\xi+\sqrt{6}Qx)[1+(2-m)\beta]\Omega_c}
{2m \beta (1+2\beta) x^2+\Omega_c
[1+(2-m)\beta]}\,,
\ee
and
\ba
& &
G_{cc} = \frac{1+r_1}{1+r_2}G\,,\qquad
G_{cb} = \frac{1}{1+r_2}G\,,\label{GCDM} \\
& &
G_{bc} = G_{bb}=G\,,\label{Gba}
\ea
with
\ba
r_1 &=& \frac{2Q[3Q \Omega_c+\sqrt{6} m \beta x
(2+\epsilon_{\phi}+\sqrt{6}Q x)]}
{3\Omega_c [1+(2-m)\beta]}\,,\\
r_2 &=& \frac{2m \beta (1+2\beta)x^2}
{\Omega_c [1+(2-m)\beta]}\,.
\ea
In the limit that $\beta \to 0$, we have $r_1=2Q^2$ and $r_2=0$,
so that $G_{cc}=(1+2Q^2)G$ and $G_{cb}=G_{bc}=G_{bb}=G$.
Since $G_{cc}$ is larger than $G$, the growth rate
of $\delta_{c{\rm N}}$
is larger than that for $Q=0$.
In the other limit $Q \to 0$, we have $r_1=0$ and hence
$G_{cc}=G_{cb}=G/(1+r_2)$ and $G_{bc}=G_{bb}=G$.
In this case, under the conditions (\ref{thecon1})-(\ref{thecon3}),
both $G_{cc}$ and $G_{cb}$ are smaller than $G$
for $m \beta>0$.

For the theories with $\beta \neq 0$ and $Q \neq0$,
$G_{cc}$ can be either smaller or larger than $G$
depending on the model parameters.
We note that $G_{cb}$ is equivalent to $G_{cc}$ in the limit $Q \to 0$.

During the $\phi$MDE, $G_{cc}$ and $G_{cb}$ are given, respectively, by
\ba
G_{cc}
&=&\left( 1+\frac{2Q^2}{1+2\beta} \right)G\,,
\label{Gcc1}\\
G_{cb}
&=& \left[ 1-\frac{4m \beta Q^2}{3-2Q^2+6(2-m)\beta^2
+\{ 12-3m+(6m -4)Q^2 \}\beta} \right]G\,.
\label{Gcb}
\ea
Under the condition (\ref{thecon1}), $G_{cc}$ is larger than $G$.

A simple approximation valid at the present time for the small
couplings $|\beta| \ll 1$ and $|Q|\ll 1$ is given by
\begin{equation}
G_{cc} \simeq \left( 1+2Q^2 -\frac{2\beta m x^2}{\Omega_c} \right)G\,,
\end{equation}
where we ignored the terms which are the products of $\beta$
and $Q$.
This shows how $G_{cc}$ can be larger or smaller than $G$ depending on the coupling parameters.

On the fixed point (b) given by Eq.~(\ref{pointb}),
$G_{cc}$ reduces to
\be
G_{cc}=\frac{4(1+2\beta)+\lambda(2Q-\lambda)}
{1+2\beta} \frac{Q}{\lambda} G\,,
\label{Gcc2}
\ee
whereas $G_{cb}$ vanishes.
{}From Eq.~(\ref{Gcc2}), we have $G_{cc} \to 0$ for $Q \to 0$.
If we impose the condition $G_{cc}>0$ on the future attractor
point (b), we require that
$[4(1+2\beta)+\lambda(2Q-\lambda)]Q/\lambda>0$.

Since both $G_{cc}$ and $G_{cb}$ are different from $G$,
this affects the evolution of $\delta_{c{\rm N}}$ through Eq.~(\ref{delceq2}).
While both $G_{bc}$ and $G_{bb}$ are equivalent to $G$,
the modified evolution of $\delta_{c{\rm N}}$ affects
the growth of $\delta_{b{\rm N}}$ through Eq.~(\ref{delbeq2}).
To study the evolution of total matter perturbations,
we introduce the effective
CDM background density $\hat{\rho}_c=(1+f_1) \rho_c=e^{Q \phi/M_{\rm pl}}\rho_c$
and the perturbed gauge-invariant density
$\hat{\delta \rho}_{c{\rm N}}=e^{Q\phi/M_{\rm pl}} (\delta \rho_{c{\rm N}}+Q\rho_c
\delta \phi_{\rm N}/M_{\rm pl})$.
The total matter density contrast is given by
\be
\delta_m=\frac{\hat{\delta \rho}_{c{\rm N}}+\delta \rho_{b {\rm N}}}
{\hat{\rho}_c+\rho_b}
=\left( \delta_{c{\rm N}}+\sqrt{6}Q x \delta \varphi_{\rm N} \right)
\frac{\Omega_c}{\Omega_m}+\delta_{b{\rm N}}
\frac{\Omega_b}{\Omega_m}\,,
\label{delmes}
\ee
where $\Omega_m=\Omega_c+\Omega_b$.
For the perturbations deep inside the Hubble radius (${\cal K} \gg 1$),
Eq.~(\ref{quasi2}) shows that
the term $\sqrt{6}Q x \delta \varphi_{\rm N}$
in Eq.~(\ref{delmes})
is negligibly small relative to $\delta_{c{\rm N}}$, so  Eq.~(\ref{delmes})
reduces to
$\delta_m \simeq \delta_{c{\rm N}} \Omega_c/\Omega_m
+\delta_{b{\rm N}} \Omega_b/\Omega_m$.
The quantity related to the measurement of redshift-space distortions
is $f (z) \sigma_8 (z)$, where $f=\delta_m'/\delta_m$
is the growth rate of
matter perturbations that depends on the redshift $z$.
On the critical points, $f$ is constant and can be obtained  analytically,
since all the coefficients of Eqs.~(\ref{delceq2}) and
(\ref{delbeq2}) are constant.
If we neglect the contribution of baryons to Eq.~(\ref{delceq}),
the value of $f$ corresponding to
the growing-mode solution is given by
\begin{equation}
    f= \frac{1}{2} \left( \sqrt{\nu^2+6\frac{G_{cc}}{G} \Omega_c}
     -\nu\right)\,.
\end{equation}
For instance, $f=1+2Q^2/q_s$ during the $\phi$MDE.
The matter growth rate is always larger than
in standard general relativity ($f=1$),
but, for $\beta>0$, it is smaller than in a pure
energy-exchange model with $\beta=0$ and
$Q \neq 0$. In general, however, we need the numerical integration to know the
precise evolution of $\delta_m$ at low redshifts, which we describe next.

\begin{figure}[H]
\begin{center}
\includegraphics[height=3.4in,width=3.5in]{fig2a.pdf}
\includegraphics[height=3.4in,width=3.5in]{fig2b.pdf}
\end{center}
\caption{\label{fig2}
(Left)
Evolution of $G_{cc}/G$ versus the redshift $z$
for $\lambda=1$ in three different cases:
(i) $Q=0.04$, $\beta=0$, $m=2$,
(ii) $Q=0.04$, $\beta=1$, $m=2$, and
(iii) $Q=0.02$, $\beta=0.5$, $m=3$.
The background initial conditions are chosen
to realize $\Omega_{\phi} \simeq 0.68$,
$\Omega_b \simeq 0.05$, and $\Omega_r \simeq 10^{-4}$ today.
(Right)
Evolution of $f \sigma_8$ versus the redshift $z$
for the three cases shown in the left panel.
Today's values of $\sigma_8$ and ${\cal K}$ are chosen
to be $\sigma_8 (z=0)=0.811$ and
${\cal K}(z=0)=300$,
respectively.
}
\end{figure}

In Fig.~\ref{fig2}, we plot the evolution of $G_{cc}/G$ (left)
and $f \sigma_8$ (right) versus the redshift $z$ for $\lambda=1$ in three different cases:
(i) $Q=0.04$, $\beta=0$, $m=2$,
(ii) $Q=0.04$, $\beta=1$, $m=2$, and
(iii) $Q=0.02$, $\beta=0.5$, $m=3$.
In these cases, the background cosmology corresponds to the $\phi$MDE
followed by the accelerated fixed point (b).
In case (i), all the terms associated with the coupling
$\beta$ disappear from the background and
perturbation equations, so their dynamics is
the same as that of standard coupled quintessence
with the exponential potential \cite{Amendola99,Amendola:2003wa}.
Indeed, $G_{cc}$ is equivalent to $(1+2Q^2)G=1.0032G$
in the simulation of Fig.~\ref{fig2}, with $G_{cb}=G$.

In case (ii), the analytic estimations (\ref{Gcc1})-(\ref{Gcc2})
show that $G_{cc}=1.0011G$ and $G_{cb}=0.9986G$
during the $\phi$MDE and $G_{cc}=0.1477G$
on the accelerated point (b).
In comparison to case (i), the existence of positive coupling
$\beta$ leads to the smaller value of $G_{cc}$ in $\phi$MDE,
which is also confirmed numerically.
At low redshifts, there is a transient period during which
$G_{cc}$ gets larger than that
in case (i). However, $G_{cc}$ starts to be smaller than $G$
around the redshift $z \lesssim 0.37$ and it reaches
today's value $0.90G$.
Thus, even for $Q \neq 0$, the presence of positive
coupling $\beta$ can realize $G_{cc}$ smaller than $G$ by today.
In case (iii), the decrease of $G_{cc}$ at low redshifts
is even more significant relative to case (ii).
In this case, today's values of CDM gravitational couplings
are $G_{cc}=0.72G$ and $G_{cb}=0.65G$
with the future asymptotic value $G_{cc}=0.070G$.

As we observe in the right panel of Fig.~\ref{fig2}, the decrease of
$G_{cc}$ less than $G$, which occurs in cases (ii) and (iii) at low redshifts,
leads to  values of $f \sigma_8$ smaller than that in case (i).
This suppression of growth of $\delta_m$ is the consequence of momentum transfer
associated with the positive coupling $\beta$.
The evolution of $f \sigma_8$ depends on the couplings
$Q$, $\beta$ as well as the model parameters $m$, $\lambda$.
For $Q$ close to 0, $G_{cc}$ approaches $G_{cb}=G/(1+r_2)$,
which is in the range $G_{cb}<G$ for $m \beta>0$.
In such cases, it is easy to realize a CDM gravitational interaction
smaller than the Newton gravitational constant.
The nonvanishing coupling $Q$ works to enhance $G_{cc}$,
but the energy transfer is required for
the existence of $\phi$MDE.

We note that our interacting model with the $\phi$MDE
is different from the early dark energy recently studied in Ref.~\cite{Hill},
in that the latter only modifies the background dynamics with
the standard growth of matter perturbations.
In our model, the weak cosmic growth rate is realized by
the momentum transfer, with the modified
early background dynamics by the energy transfer.
Hence there is the possibility that the observational tensions of
both $\sigma_8$ and $H_0$ are relaxed. However, clearly, we need detailed
Markov chain Monte Carlo simulations with the recent observational data
to see whether this is really the case or not.

\section{Conclusions}
\label{consec}

In this paper, we studied the cosmological dynamics of interacting
theories of dark energy and dark matter by paying particular attention to
the existence of a scaling $\phi$MDE.
Our goal is not only to alleviate the $H_0$ tension problem by realizing
the $\phi$MDE but also to ease the $\sigma_8$ tension problem
by achieving a weak gravitational interaction on scales relevant to
the growth of large-scale structures.
It is possible to satisfy these two demands by considering
the interacting action (\ref{action0}) containing
both energy and momentum transfers.
The dependence of $Z$ in the Lagrangian $f_2(\phi,X,Z)$,
where $Z=u^{\mu} \nabla_{\mu} \phi$ quantifies
the interaction between the CDM four velocity $u^{\mu}$
and the scalar derivative $\nabla_{\mu} \phi$, plays a crucial role
for the realization of weak gravity.

In Sec.~\ref{scalingsec}, we derived the interacting Lagrangian
allowing for the existence of scaling solutions which obey
the relation $\dot{\phi}/H={\rm constant}$.
We showed that the corresponding Lagrangian is of the form
(\ref{slag}), which contains two arbitrary functions $g_1$
and $g_2$ with respect to $Y_1=X e^{\lambda \phi/M_{\rm pl}}$
and $Y_2=Z e^{\lambda \phi/(2M_{\rm pl})}$.
In Sec.~\ref{backsec}, we identified the scaling fixed point
(a) as well as the other point (b) relevant to late-time cosmic
acceleration, without fixing concrete forms of $g_1$ and $g_2$.
We also found that, for models given by the functions
(\ref{g1e}) and (\ref{g2e}), there exists the scaling
$\phi$MDE satisfying the relation
$w_{\rm eff}=\Omega_{\phi}=2Q^2/(3q_s)$.
Thus, for the existence of $\phi$MDE, we require a nonvanishing
coupling constant $Q$ associated with the energy transfer.

In Sec.~\ref{backsec}, we studied the background cosmology for
a concrete model given by the interacting Lagrangian (\ref{Lagcon}).
We showed the existence of $\phi$MDE preceded by
the radiation fixed point (\ref{pointe}).
As long as the two conditions (\ref{cond1}) and (\ref{cond2})
are satisfied, the $\phi$MDE is followed by the stable fixed
point (\ref{pointb}) with cosmic acceleration.
As we observe in Fig.~\ref{fig1}, the field equation of
state $w_{\phi}$ is close to 1 in the deep matter era
and it approaches the asymptotic value
$w_{\phi}=-1+\lambda^2/(3q_s)$ after the
temporary approach to $-1$ at low redshifts.
This background dynamics is distinguished from
the coupled dark energy scenario with the momentum
transfer alone \cite{Kase:2019mox}.

In Sec.~\ref{persec}, we explored the dynamics of cosmological
perturbations for the same interacting model studied in Sec.~\ref{backsec}.
We derived the full linear perturbation equations of motion and
applied the quasi-static approximation to the modes deep
inside the sound horizon.
Under this approximation scheme, the effective gravitational couplings
for CDM and baryon density contrasts are given by
Eqs.~(\ref{GCDM}) and (\ref{Gba}), respectively.
We showed that, depending on model parameters,
both $G_{cc}$ and $G_{cb}$ can be smaller than
$G$ at low redshifts, while satisfying conditions for the
absence of ghosts and Laplacian instabilities.
The weak gravitational interaction of CDM leads to
the suppression of growth rate
of total matter density contrast $\delta_m$.
As we observe in Fig.~\ref{fig2}, this property is attributed
to the momentum transfer arising from the
coupling $\beta \left( 2X \right)^{1-m/2} Z^m$.

We thus showed that the coupled dark energy and dark matter
scenario with both energy and momentum exchanges offers an
interesting possibility for realizing the $\phi$MDE as well as
the weak gravitational interaction at low redshifts.
The next step is to investigate whether the concrete interacting
model proposed in this paper can alleviate the problems
of $H_0$ and $\sigma_8$ tensions present in the $\Lambda$CDM
model.
 Since the Lagrangian (5.3) contains additional
model parameters with respect to those in  $\Lambda$CDM,
one might expect that our model is hardly better than the
$\Lambda$CDM from the Bayesian statistical point of view.
However, it is known that there are dynamical dark energy models
that are comparable with  $\Lambda$CDM from the point of view of Bayesian statistics,
even with more than three  additional free parameters  \cite{Nakamura:2018oyy}.
The detailed observational constraint
on our interacting dark energy model
is left for a future publication.

\section*{Acknowledgements}

We thank Antonio De Felice and Ryotaro Kase for useful discussions.
ST is supported by the Grant-in-Aid for Scientific Research Fund of the JSPS No.\,19K03854 and
MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas
``Cosmic Acceleration'' (No.\,15H05890).

\section*{Appendix: Phase space for a simplified case}

In this Appendix, we express the dynamical Eqs.~(\ref{dxeq})-(\ref{dyeq})
in terms of the dimensionless variables $x$, $y$, and $\Omega_m$
for the scaling Lagrangian (\ref{slag}) containing
the functions (\ref{sca}) and (\ref{sca2}).
In doing so, we use the unit $M_{\rm pl}=1$ and notations
$g_{i,j}=\partial g_i/\partial Y_j$ and
$g_{i,jk}=\partial^{2}g_{i}/\partial Y_{j}\partial Y_{k}$,
where $i=1,2$.
Then, the autonomous equations are given by
\ba
x' &=&
\frac{\sqrt{6}g_{1}\left(x^{2}\Gamma_{2}+\sqrt{6}g_{2}xy^{4}
+Qy^{4}\Omega_{m}\right)-\sqrt{6}\Omega_{m}\Gamma_{1}}{2\Omega_{m}
\left(\bar{\Gamma}_{1}
+y^{2}g_{1,1}\right)-2g_{1}\left(x^{2}\bar{\Gamma}_{2}
+5x^{2}y^{2}g_{2,1}
+2\sqrt{2}x y^{3}g_{2,2}+g_{2}y^{4} \right)}
+\frac{3x\Theta}{2g_{1}y^{2}}\,,\label{autoge1} \\
y' &=&
\frac{3\Theta-\sqrt{6}g_{1}\lambda xy^{2}}{2g_{1}y}\,,
\label{autoge2}
\ea
where
\ba
\Gamma_{i} &=& \lambda x^2 (\bar\Gamma_i+y^{2}g_{i,1} )
+A_iy^{2}(yg_{i,2}+\sqrt{2} xg_{i,1}) \,,\\
\bar{\Gamma}_{i} &=&
2x^{2}g_{i,11}+2\sqrt{2}xyg_{i,12}+y^{2}g_{i,22}\,,\\
A_1 &=& \sqrt{2}Qx-\sqrt{3}w_{m}\,,\\
A_2 &=& \sqrt{2}\lambda x+\sqrt{3}\,,\\
\Theta &=& g_{1}\left[ \sqrt{2}x^{3}yg_{2,2}+2g_{2}x^{2}y^2
+y^2\left(w_{m}+1\right)\Omega_{m}+2x^{4}g_{2,1}\right]
-x\,\Omega_{m}\left(2xg_{1,1}+\sqrt{2}yg_{1,2}\right)\,,
\ea
and
\be
\Omega_{m} =\frac{g_{1}\left(2x^{4}g_{2,1}
+\sqrt{2}x^{3}yg_{2,2}+g_{2}x^{2}y^{2}-y^{2}\right)}
{x\left(2xg_{1,1}+\sqrt{2}yg_{1,2}\right)
-g_{1}y^{2}}\,.
\ee
Let us consider the model given by the functions
(\ref{eq:simpg1}) and (\ref{eq:simpg2}).
For the perfect fluid, we only take into account CDM with $w_m=0$.
Then, the autonomous Eqs.~(\ref{autoge1}) and (\ref{autoge2})
reduce to
\ba
x' &=& \frac{3}{2} x \left( 2 q_s x^2-2 +\Omega_m \right)
+\frac{\sqrt{6}}{2 q_s} \left( \lambda \tilde{y}^2
-Q \Omega_m \right)\,,\label{autom1}\\
\tilde{y}' &=& \frac{1}{2} \tilde{y} \left( 6 q_s x^2
-\sqrt{6} \lambda x+3\Omega_m \right)\,,\label{autom2}
\ea
where $\tilde{y}$ is defined by Eq.~(\ref{tildey}),
and $\Omega_m=1-q_s x^2-\tilde{y}^2$.
The dynamical system (\ref{autom1})-(\ref{autom2})
contains the scaling solution (a)
characterized by fixed point $(x_c,y_c)=(\sqrt{6}/[2(Q+\lambda)],
\sqrt{[2Q(Q+\lambda)+3q_s]/[2(Q+\lambda)^2]})$.
Besides this, there are also the fixed point
(b) and the $\phi$MDE (c) whose values of $x_c$, $y_c$,
$\Omega_{\phi}$, $w_{\rm eff}$ are the same
as those given in Eqs.~(\ref{pointb}) and (\ref{pointc}),
respectively. We also have the kinetic points (d1) and (d2),
but they are relevant to neither matter-dominated nor accelerated
epochs. As long as the two conditions (\ref{cond1}) and
(\ref{cond2}) are satisfied, the $\phi$MDE is followed
by point (b) with cosmic acceleration
[instead of the scaling solution (a)].
In Fig.~\ref{fig3}, we can confirm that,
for the model parameters used in Fig.~\ref{fig1}, the solutions
converge to the attractor point (b).

\begin{figure}[t]
\begin{center}
\includegraphics[width=3.5in]{fig3.pdf}
\end{center}
\caption{
Phase-space analysis for the same model parameters as those used
in Fig.~\ref{fig1}, but with neither baryons nor radiation
($\Omega_{b}=0$ and $\Omega_{r}=0$).
The $\phi$MDE saddle point and the final accelerated attractor
are denoted as c and b, respectively.
\label{fig3}}
\end{figure}

redpajama_set_name: RedPajamaArXiv


text:
Accidents & Traffic
UPDATE: 22-Year-Old Man Succumbs to Injuries from Crash on Trinity Church Road
By Charles County Sheriff's Office
Update: ALLENS FRESH, Md. - The driver at fault in this crash, Matthew Robert Eppley, 22, of Charlotte Hall, MD, was pronounced deceased yesterday afternoon as a result of the injuries he sustained in the crash. The woman who was driving the Pacifica is still being treated at a hospital.
ALLENS FRESH, Md. - On December 16 at 10:23 a.m., officers responded to a motor vehicle crash involving three cars on Trinity Church Road north of Old Sycamore Road in Allens Fresh, MD. A preliminary investigation revealed the driver of a Ford Mustang was following a Chevrolet Silverado which was traveling north on Trinity Church Road. The initial investigation revealed the driver of the Mustang attempted to pass the Silverado on a stretch of roadway marked by double-yellow lines, a no passing-zone, near the crest of a hill. The driver swerved to avoid striking a Chrysler Pacifica, which was southbound on Trinity Church Road, and struck the Silverado, sending it into a spin. The Mustang re-entered the southbound lane and struck the Pacifica head-on. The driver of the Mustang was trapped and the vehicle caught fire. Cpl. Stephen Miller, an off-duty Charles County Sheriff's officer, was nearby and used a fire extinguisher to suppress the fire in an effort to protect the driver until emergency fire crews arrived. The driver of the Mustang, a 22-year-old male of Charlotte Hall, was extricated from his vehicle and flown to a hospital in critical condition. The driver of the Pacifica, a 36-year-old woman from Mechanicsville, suffered critical injuries. The driver of the Silverado, a 35-year-old male from Waldorf suffered injuries which are not life threatening. Anyone with additional information about this case is asked to call Cpl. T. Hemsley of the Traffic Operations Unit at (301) 609-5903. The investigation is ongoing.
ALLENS FRESH, Md. -Two former Mechanicsville Volunteer Firefighters fought to save a young man trapped in a burning car this morning in Charlotte Hall.
The accident happened at 10:26 am on Trinity Church Road. Casey Sheckels, 34, of Mechanicsville, and his friend David Harding were riding in a vehicle together when they spotted the accident. "We started coming up the hill on Trinity Church and David said, there's fire up there.'"
As luck would have it, both Sheckels and Harding are former members of the Mechanicsville VFD. When they spotted the three-car accident, they immediately, jumped out to offer assistance. Casey says two other drivers did the same.
That's when the father-of-four said his first responder training kicked in. "You learn to go for the worst vehicle first and help the person most in need. I could see the truck was upright and not damaged. So, I went to the SUV. The woman said she couldn't move or get out of car. The steering wheel had her pinned. She was alert and talking, so I told a bystander to stay with her."
The Charles County Sheriff's Office says the accident happened when the 22-year-old driver of a Ford Mustang attempted to pass a truck in a no-passing zone and hit the truck before crashing head on into a SUV.
Incredibly, the driver of the truck turned out to be Casey's cousin. Fortunately, he was not seriously injured, even though his vehicle rolled and landed in an upright position.
The driver of the Mustang. was not so lucky. "He had a very faint pulse. He was breathing but severely pinned into the vehicle. The dashboard was where the backseat was." The severely damaged car was on fire. Casey said the flames were getting intense.
Unable to reach the young man in the driver's seat due to the condition of the vehicle, a bystander busted the passenger seat window out with a tire iron and Casey climbed inside to attempt to render aid. When he realized how close the flames were to the victim, he called for water. "The flames were four or five feet high."
That's when the two men, assisted by bystander Paul Brady and two nearby homeowners formed a bucket brigade. They ran back and forth from a nearby home throwing water on the engine in an attempt to keep the fire away from the trapped passenger.
"That worked for awhile, but they started coming underneath the dash. Two sheriff's deputies arrived with fire extinguishers. We used those as much as we could. We did everything we could to keep the fire away from him. We did the best that we could. I kept talking to him and I prayed at the same time."
According to the CCSO, off-duty officer Cpl. Stephen Miller was one of the officers with a fire extinguisher.
Shortly afterward, the fire department showed up and got the flames under control. They extricated the driver from the vehicle and transported him to a hospital. Both the driver of the Mustang and the SUV sustained critical injuries.
Casey said, "I've seen it all. knowing and seeing what I've seen. Tt's never something you can be prepared for when you get there. To see someone in desperate need. and even though I had the training, I couldn't do anything for him because he was stuck in the car. Thankfully, we had enough men there to make a 300 yard dash to a house with buckets until the fire department arrived."
The former first responder commended local police and fire for their work on the scene. "The fire department, EMS, and the police did an amazing job. Everybody's responsiveness and training was top-notch. Tt's good that a community can know that they are in good hands."
Casey said he texted his mother from the scene to let her know his cousin had been in an accident but was okay. "Mom texted me, 'God puts you in these places for a reason.' All I can do is pray and hope he makes it. That all of them are safe."
The Charles County Sheriff's Office says the accident is still under investigation. Anyone with additional information about this case is asked to call Cpl. T. Hemsley of the Traffic Operations Unit at (301) 609-5903.

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text:
What If Board Games Were Politically Correct
Expansion of Sections
Incorporate Statistics
Talk About Impact
Focus on POV
Examples and Experiences
If board games were politically correct, their themes and storylines would be tailored to suit modern sensibilities. Instead of featuring characters who fit into traditional gender stereotypes or exploring outdated concepts like conquest and wealth accumulation, these games might focus on themes that challenge the status quo and encourage players to have an open mind. This could include topics such as creativity, collaboration, and inclusivity that are seen in cooperative board games such as Pandemic or social deduction games like Secret Hitler. Players will likely be rewarded for making thoughtful contributions to their team while also learning about important current issues like environmentalism or the importance of cultural understanding.
In comparison to traditional board games, politically-correct ones could look quite different overall " from the types of materials used for pieces and cards to the length of playtime. The 'rules' may even be more fluid; multiple endings might be possible depending on how each round turns out " inculcating a sense of unpredictability which adds to the overall game experience. Additionally, sophisticated tools like card creation tools and dynamic game boards can provide added complexity and customization options for competitive play " encouraging players to explore deeper strategies instead of relying on luck.
Ultimately, if political correctness moves towards becoming an integral factor in game design then we may see an influx of new titles that feature meaningful lessons about values beyond sheer entertainment " potentially rewarding both existing lovers and establishing a whole new generation of gamers with valuable knowledge.
Player Expectations
Gone are the days where board games involve confrontational competition. According to a survey by the Pew Research Centre, 78% of people now prefer playing cooperative or team-based games as opposed to head-to-head competition. People also appreciate when game developers introduce elements that promote diversity and inclusion. A Nielsen survey found that 44% of adult gamers prefer games that feature diverse characters, settings and storylines, gaining further traction with millennials at 54%.
The Backlash
Developers have come under fire for creating board games that can be considered politically incorrect. As of December 2020, a quick review of BGG " BoardGameGeek " reveals over 500+ posts mentioning words like 'harassment' and 'problematic' in the context of the games on sale or new releases. The trend isn't likely to diminish any time soon considering how highly sensitive topics are being discussed in today's society, sometimes even dividing communities or friends within them. Discussions regarding rebranding certain games as well as introducing content warnings to alert players prior to purchasing violently-themed titles have picked up notably since 2018 suggesting potential industry reform is necessary if companies are going to keep up with changing player trends and preferences.
Bugs In The Kitchen Board Game Instructions
Having board games that are politically correct could have a major impact on the gaming community and industry. People from all backgrounds and cultures who previously may have felt excluded or not able to engage in certain board games would now be able to. This could lead to an increase in consumer spending for the gaming industry as people of all backgrounds may feel more included when playing these games.
The reception of politically correct board games may also be quite positive within the gaming society, as people will appreciate the effort being taken to adjust the game to make it accessible for everyone. Different perspectives and sensitivities that come with different cultures, can help shape and refine decisions made in creating politically correct board games so players from all backgrounds can enjoy their experiences.
Politically correct board games also lead to more inclusion amongst communities as they become commonplace and part of popular culture, prompting these messages even closer together, removing any parts of a game that were exclusive or idea-biased due to one's environmental circumstances and making these titles available for players around the world.
If board games were politically correct, they would focus on an inclusion of all voices. Not only would the games be designed for players of all backgrounds but also for players expressing different points of view. Intersectionality could be explored in these games by including narratives across multiple identities and experiences. This could be represented through characters that reflect those varied perspectives and perhaps allow the players to take direct action and influence those stories as they play through. Rulesets and objectives could be reframed so they support a shared narrative that emphasizes understanding, respect, and open mindedness rather than one-sided competition or predetermined outcomes. Furthermore, diverse game boards could create pathways to explore stories away from traditional "western" settings, allowing players to consider different worldviews and teachings. By doing so, the game can become an immersive learning experience while still being fun and engaging. Ultimately, these politically correct board games could provide a space for constructive dialogue around differences in culture, identity, race, ability, sexuality and more.
How To Create A Board Game Based On A Novel
What if board games were politically correct? One way often used to address this is by producing games that feature diverse content and inclusivity. Some examples of these titles include No Outsiders from Green Couch Games, an abstract game that celebrates diversity through objective card mechanics, and Spectrum from Renegade Game Studios, which features the colors of the rainbow to represent the five goals of the LGBTTQIA+ movements. There are additional games out on the market with similarly inspiring attitudes towards inclusivity such as Too Many Bones and Origins from Chip Theory Games, which each feature fantastic art and gender-neutral storytelling.
Personal experiences also contribute to this level of political correctness, as board gamers around the world have come together to ensure a safe space for all players. At gaming conventions and friendly meetups, stories of friendly competition but acceptance abound; regardless of one's background or beliefs all are welcomed into these gatherings to test their skills against those of others.
Careers Board Game Online
Is The Dark Souls Board Game Good


text:
Q: Core of divergence form operator with unbounded coefficient Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that $a(x) > 0$ for all $x$ and $a(x)$ is differentiable.  However, I make no assumptions on the boundedness of $a$.
Does this operator have a core?  If so, can it be identified explicitly?
Thanks.

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text:
600 Hewitson Street Thunder Bay , ON, P7B 6E4
Northwest Connector
Baakaakonaanan Ishkwaandemonan
Local Employment Planning Council
Bert Johnson
Home AboutBoard Bert Johnson
Bert Johnson is currently serving as Councillor for the Township of Terrace Bay and is in his sixth year of public service as Councillor. As part of that role, Bert sits on the Accessibility, Supportive Housing, Thunder Bay District Municipal League and Hand-Transit Committees for the Township.
Mr. Johnson is retired from Bowater (Great Lakes Paper, etc.) in 2006 following a successful career as Scaler, which included two short stints as Cut Foreman, in the bush for 30+ years. Bert is no stranger to public service as while employed by Bowater, he served sixteen (16) years as Chair of the Upsala District School Area Board. During that time, he also represented all of the isolate School Boards in Ontario and all of the Public School Boards west of Thunder at the Northwestern Ontario School Trustees' Association (NOSTA). While serving in that capacity, Bert was once awarded recognition as the Outstanding Trustee in Northern Ontario.
In addition to his passion for representing Northwestern Ontario constituents at the district school board and municipal levels, he is also an avid outdoorsman. Bert has a passion for introducing and mentoring others into the joys of the outdoors and worked for many years at the D&R Sporting Goods store. Bert also served in a volunteer capacity with the Northwestern Ontario Sportsmen's Alliance and was awarded "Outstanding Sportsman" in 2011.
Throughout his career, Bert has been dedicated to giving back to his community in the form of volunteering on various committees and boards. Some of those additional volunteer positions include:
Upsala District School Area Board
Northern Ontario School Trustees' Association during tenure with the school board
Northwestern Ontario Sportsmen's Alliance
Thunder Bay Forest Planning
Dog River Mattawan Forest
Thunder Bay District Stewardship Council and currently sits on
Terrace Bay Ducks Unlimited Committee.
Bert is retired and living in Terrace Bay with his best friend and wife Gail of forty-five (45) years.
Doris Rossi
John DeGiacomo
John Hatton
Roger Shott


Atlanta | How Does BOTOX® Cosmetic Work?
Home | How Does BOTOX® Cosmetic Work?
The years can take their toll on your body, and your face is especially vulnerable to age's worst effects. Crow's feet, frown lines, wrinkles, these all serve to make you look older and fatigued. Fortunately, board-certified Atlanta plastic surgeon Dr. Mark Crispin can restore youth and vitality to your face with BOTOX® Cosmetic, giving you the youthful appearance you desire.
The active ingredient in BOTOX® Cosmetic is a sterile form of the botulinum toxin. This contains no bacteria and presents you with little health risk.
Dr. Crispin will inject the BOTOX® Cosmetic into the area of your face that you are most concerned with. One possible example is your forehead and the unsightly frown lines that can develop there. BOTOX® Cosmetic effectively serves as a shield between your muscles and the surrounding nerves. It blocks signals the nerves send to your muscles, preventing them from contracting.
The muscles then relax and soften, which causes your face to look smoother and younger. Your BOTOX® Cosmetic injection should last you about four months, and you will need to return to Dr. Crispin for further treatments to retain your new look.
If you live in the Atlanta area and you're considering BOTOX® Cosmetic for a younger look, please contact Mark E. Crispin, MD, PC today to schedule a consultation.


text:
Supporting strands
Namrata ZakariaNamrata Zakaria / Updated: May 26, 2019, 06:00 IST
PIC: DEEPAK TURBHEKAR
Monica Shah and Karishma Swali — embroiderers to some of the biggest couture companies in the world, and fashion designers in their own right — take us around their new passion project, a girls-only embroidery school
HISTORICALLY, Girangaon is where Mumbai found its glory. The city's textile hub is what had made the city, then beloved as Bombay, an international port. The fabric produced here was exported to the UK, and then to the rest of the world, making India the largest exporter of cotton globally.
Today, two women are making Girangaon — as Parel, Byculla, Dadar, Lalbaug are collectively known — internationally famous again.
Monica Shah and Karishma Swali, who run the rather successful embroidery house Chanakya, which they inherited from Karishma's father and Monica's father-in-law Vinod Shah, are the chief embroiderers for several international labels like Gucci, Dior, Fendi, Giambattista Valli, Valentino and others. Like many other embroiderers and exporters, they are housed in Byculla.
Their formidable repertoire had renowned international fashion critic Suzy Menkes visit them in 2017, and Maria Grazia Chiuri, the chief designer of Dior (she has been working with Chanakya for over two decades, since her days as Fendi's accessories head) attend their tenth anniversary in October 2018.
The ladies count Chiuri, whose feminist bent at Dior has been iconoclastic, as their mentor. It was on Chiuri's suggestion that they decided to ensure artisanal Indian embroideries continue for coming generations.
This altruism birthed The Chanakya School of Embroidery & Fine Crafts. Also in Byculla, the school is less than six months old and invites women from lesser privileged homes to learn basic and sophisticated embroidery techniques over the two three-month courses. The school runs in a rather modernist space — a large airconditioned white shed that resembles a giant art gallery, replete with designer foldable chairs, swanky embroidery-ring style desks and an AV room.
"Embroidery is traditionally a maledominated field. But fewer men want to do this work now, despite it being a generational skill," Monica says. "We wanted to bring women to the workforce, and thus also empower them," Karishma adds.
There are 100 women who come here daily. They are each paid Rs 2,500 per month to cover their conveyance. Even if they come here to escape Mumbai's brutal summer, and to collect a little pocket money, they go home learning something new, and keep coming back for more.
One of the students says she wants to open her own fashion boutique; another wants to be a fashion designer. Their dreams don't seem impossible, as they learn couture-style embroideries and surface techniques; even the au courant three-dimensional embroidery.
As the ladies show me around, the classes are underway. Monica and Karishma are greeted with grateful smiles. But courtesies are brief as the students resume their work almost immediately. The fineness of their handiwork is surprisingly good.
"The teachers are professionals from local fashion institutes like Nirmala Niketan. For the advanced course, our own master craftsmen take over," Karishma says. The women are given an in-depth understanding of the art, with monthly museum visits and orientation in history. The study modules also deal with the use of technology, marketing strategies and business acumen. The programmes at The Chanakya are certified by the National Institute of Fashion Technology as well as the Export Council of Handicrafts.
Once the nine-month to one-year programme is over, the women are given a choice to either look for employment, or turn entrepreneurs. They are also offered a month-long internship at the ladies' embroidery firm.
Monica and Karishma seem satisfied with their passion project: their two-fold goal of empowering women and rejuvenating thread arts has met. The women didn't divulge the economics of the school, how much it cost to build it or their monthly expenses, saying they would prefer not to say.
The school is nestled right next to the Bhau Daji Lad Museum, a Byculla landmark, and also Mumbai's museum of textile arts and costume history. It's almost as if the opportune location also lends itself to the arts facility. The students have already visited a textile show, 'Connecting Threads: Textiles in Contemporary Practice', at the museum. They were shown around by the creativity teacher, commercial artist Radhika Chopra, as well as the museum's guides.
Monica and Karishma hail from privileged backgrounds and feel they are fortunate to be able to do this sort of social entrepreneurship. Their embroidery house already does a little bit of CSR by collaborating with NGOs, like Shrujan in the Kutch. But The Chanakya School is truly a fullfledged commitment.
By 2020, they hope to have 1,000 students in their fold. As Monica sums it up, "We want to create leaders."
Embroidery is traditionally a male-dominated field. But we wanted to bring women to the workforce, and thus also empower them
–Monica Shah
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