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%%                           APPENDIX A 
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\phantomsection

\chapter{\uppercase{General Disformal Set-Up}} \label{App1}


The general scalar-tensor  action coupled to matter, which can include a realisation in string theory compactifications is given by: 
\bea\label{S1A}
S=S_{EH} + S_\phi + S_{m}\,,
\eea
%
where: 
\bea
&& S_{EH}=\frac{1}{2\kappa^2} \!\!\int{\!d^4x\sqrt{-g}\,R}, \\
&&  S_\phi = - \!\!\int{\!d^4x\sqrt{- g} \left[\frac{b}{2} (\partial\phi)^2+  M^4 \c1 ^2(\phi)\sqrt{1+\frac{\d1(\phi)}{\c1(\phi)} (\partial\phi)^2}+V(\phi)\right]}  \,,\\
&& S_{m} = - \!\!\int{\!d^4x\sqrt{-\tilde g} \,{\cal L}_{DM}(\tilde g_{\mu\nu}) } \,,
\eea
%
and the disformally coupled metric is given by
%
\be\label{gammaA}
\tilde g_{\mu\nu} = C_2(\phi) g_{\mu\nu} + D_2(\phi) \partial_\mu \phi \partial_\nu \phi\,.
\ee
%
%
 $b$ is a constant equal to $1$ or $0$, depending on the model one wants to consider; $C_i(\phi), D_i(\phi)$ are functions of $\phi$, which can be identified as  conformal and disformal couplings of the scalar to the  metric, respectively. 
  Finally,  we have introduced the mass scale $M$  to keep units right (remember that the conformal coupling is dimensionless, whereas the disformal  has units of $Mass^{-4}$.)

The connection of the general action \eqref{S1A} to the different models in the literature can be obtained as follows:  the case $C_1 = D_2$, $D_1=D_2$, $b=0$ arises when considering a D-brane moving along an extra dimension. This case was  studied in \cite{KWZ} as a model of a coupled dark matter dark energy sector scenario, where scaling solutions arise naturally.  
Note that in this case, the kinetic term for the scalar field, identified with dark energy for example, is automatically  non-canonical and  dictated by the DBI action (see \cite{KWZ}). 
On the other hand, phenomenological models considering a disformal coupling between matter and a scalar field, usually consider a canonical kinetic term, and therefore, in that case, $C_1=D_1=0$ and $b=1$ ($C_1$ can be taken to be non-zero and will be part of the scalar potential). 
Furthermore, the widely studied case of a conformal coupling is obtained for $b=1, C_1=D_1=D_2=0$ or, as in the case of a D-brane for example\footnote{For the system corresponding to a D-brane moving in a typically warped compactification in string theory, the functions $C(\phi)$ and $D(\phi)$ are identified with powers of the so-called warp factor, usually denoted as $h(\phi)$. In this approach,  the longitudinal and transverse fluctuations of the D-brane are identified with the dark matter and dark energy fluids respectively \cite{KWZ}.}, simply considering small velocities with   $b=0, C_1=C_2$ and $D_1=D_2$, and normalising canonically the scalar field (see Appendix \ref{App2}). 
%\bigskip 

Finally, let us clarify further our nomenclature on frames.  The action in \eqref{S1A} is written in the Einstein frame, which  in string theory, is usually related to the frame in which the dilaton and the graviton degrees of freedom are decoupled. From this point of view, the dilaton field as well as all other moduli fields not relevant for the cosmological discussion are considered as stabilised, massive, and are therefore decoupled from the low energy effective theory. 
In the literature of  scalar-tensor theories however (including conformal and disformal couplings), the Einstein and Jordan frames are identified with respect to the (usually single) scalar field to which gravity is coupled. In this paper, we follow this and call  ``Jordan" or ``disformal frame"  the frame in which dark matter is coupled only to the metric $\tilde g_{\mu\nu}$, rather than to the metric $g_{\mu\nu} $ and a scalar field $\phi$. 



%%%%%%%%%%%%%%%%%%%%%%%%%

The equations of motion obtained from  (\ref{S1A}) are \eqref{EM}: 
%
\be
R_{\mu\nu} -\frac{1}{2}g_{\mu\nu} R = \kappa^2\left(T^\phi_{\mu\nu} + T^{DM}_{\mu\nu}\right)\,,
\ee
where in the  frame  relative to $g_{\mu\nu}$ the energy momentum tensors are defined in \eqref{EM1} and \eqref{EM2}. 
 The energy-momentum  tensor for the scalar field in the general case  is modified from \eqref{EMphi} to:
%
\be\label{EMphiA}
T_{\mu\nu}^{\phi} = - g_{\mu\nu} \left[M^4C_1^2 \gamma_1^{-1} + \frac{b}{2} (\partial\phi)^2+ V \right] 
+ \left(M^4 C_1D_1 \,\gamma_1+ b\right) \partial_\mu\phi \, \partial_\nu \phi\,,
\ee
%
where now the energy density and pressure are given by:
\bea\label{rhoPA}
\rho_\phi =- \frac{b}{2}(\partial\phi)^2 + M^4C_1^2 \gamma_1 + V  \,, \qquad P_\phi = - \frac{b}{2}(\partial\phi)^2 - M^4C_1^2\gamma_1^{-1} - V  \,,
\eea
%
 and the  ``Lorentz factor"  $\gamma_1$ introduced above is defined by
%
\be\label{LorentzA}
\gamma_1 \equiv \left(1+ \frac{D_1}{C_1}\, (\partial\phi)^2\right)^{-1/2}\,.
\ee
%
We can rewrite \eqref{rhoPA} in a more succinct way, by defining ${\mathcal V} \equiv V + C_1^2 M^4$ 
%
\be\label{rhoPA2}
\rho_\phi =- \left [\frac{b}{2}+  \frac{M^4C_1 D_1\gamma_1}{\gamma+1}\right] (\partial\phi)^2 + {\mathcal V}  \,, \qquad 
P_\phi = - \left [\frac{b}{2}+  \frac{M^4C_1 D_1\gamma_1^{-1}}{\gamma+1}\right] (\partial\phi)^2 - {\mathcal V}  \,.
\ee

The equation of motion for the scalar field becomes (compare with \eqref{Eqphi})
%
\bea\label{EqphiA}
&&\hskip-1cm - \nabla_\mu\!\left[(M^4 D_1C_1\gamma_1\, +b)\,\partial^\mu \phi\right ] \!+ \!\frac{\gamma_1^{-1}M^4 C_1^2}{2} \!\left[\!\frac{D_1'}{D_1} +3\frac{C_1'}{C_1} \right] \!+ \!\frac{\gamma_1 \,M^4 C_1^2}{2} \!\left[\!\frac{C_1'}{C_1} -\frac{D_1'}{D_1} \right] \!+\! V' 
\nonumber \\&& \hskip3cm 
- \frac{T^{\mu\nu}}{2}\!\left[\frac{C_2'}{C_2} g_{\mu\nu}  +\frac{D_2'}{C_2}\partial_\mu\phi\partial_\nu\phi\right]
+\nabla_\mu \left[\frac{D_2}{C_2}T^{\mu\nu} \partial_\nu\phi \right] =0\,. \nonumber \\
\eea
%

Finally, the energy-momentum conservation equation gives rise to \eqref{conserva}, where $Q$ now is given in terms of $C_2, D_2$:
%
\be
Q\equiv \nabla_\mu \left[\frac{D_2}{C_2} \,T^{\mu\lambda} \,\partial_\lambda \phi\right] - \frac{T^{\mu\nu} }{2} \left[\frac{C_2'}{C_2} g_{\mu\nu} +
\frac{D_2'}{C_2} \,\partial_\mu\phi \,\partial_\nu\phi\right]\,.
\ee



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\section*{General cosmological equations}

The equations of motion for the general system in an FRW background  become:
\bea
&& H^2 =\frac{\kappa^2}{3} \left[\rho_\phi +\rho\right]\,, \label{friedmann1A}\\
&& \dot H + H^2 = -\frac{\kappa^2}{6}\left[ \rho_\phi+ 3P_\phi +\rho +3 P \right]\,,\label{friedmann2A}\\
&& \ddot \phi \left[1+ \frac{b}{M^4C_1D_1\gamma_1^3}\right]+3H\dot\phi \,\gamma_1^{-2}\left[\frac{b}{M^4C_1D_1\gamma_1}+ 1\right]  \nonumber \\
&&\hskip0.5cm + \frac{C_1}{2D_1}\left(\gamma_1^{-2}\left[\frac{5 C_1'}{C_1}  - \frac{D_1'}{D_1}\right] 
 + \frac{D_1'}{D_1}- \frac{C_1'}{C_1} -4\gamma^{-3}_1 \frac{C'}{C}  \right)
 + \frac{1}{M^4C_1D_1\gamma_1^{3}}\, ({\mathcal V}'+Q_0) =0 \label{kgA} \,,  \nonumber \\
\eea
%
where, $H= \frac{\dot a}{a}$, dots are derivatives with respect to $t$, $'=d/d\phi$ and  $$\gamma_1= (1-D_1 \,\dot\phi^2/C_1)^{-1/2}.$$
We also have the continuity equations for the scalar field and matter given by
\bea 
&&\dot\rho_\phi + 3H(\rho_{\phi}+P_{\phi}) = -Q_0\dot\phi\,, \label{contA}\\
&&\dot\rho + 3H(\rho+P) = Q_0\,\dot\phi\,.\label{cont1A}
\eea
where $Q_0$ is given by
%
%
\bea
Q_0 = \rho \left[ \frac{D_2}{C_2} \,\ddot \phi + \frac{D_2}{C_2} \,\dot \phi \left(\!3H + \frac{\dot \rho}{\rho} \right) \!+ \!\left(\!\frac{D_2'}{2C_2}-\frac{D_2}{C_2}\frac{C_2'}{C_2}\!\right) \dot\phi^2 +\frac{C_2'}{2\,C_2} (1-3\,\omega)
\right]. \nonumber \\
\eea
%
Using \eqref{cont1A} we can rewrite this in a more compact  and useful form as 
\be\label{Q0A}
Q_0 = \rho\left( \frac{\dot \gamma_2}{\dot \phi\, \gamma_2} + \frac{C_2'}{2C_2}  (1-3\,\omega \,\gamma_2^2) -3H\omega\,\frac{(\gamma_2 -1) }{\dot \phi}\right) \,,
\ee
where $$\gamma_2= (1-D_2 \,\dot\phi^2/C_2)^{-1/2}.$$


Plugging this into  the  
(non-)conservation equation for dark matter \eqref{cont1A},  gives:
\be\label{conservaDMA}
\dot \rho + 3H (\rho + P\,\gamma_2^{2}) = \rho \left[\frac{\dot \gamma_2}{\gamma_2} + \frac{C_2' }{2C_2} \,\dot\phi\, (1-3\,\omega \gamma_2^2)\right]\,.
\ee
%

The energy densities and pressures in the Einstein and Jordan frames are now related similarly to \eqref{eqrhos}, replacing $\gamma \to \gamma_2$:
\be\label{eqrhosA}
\tilde \rho = C_2^{-2} \gamma_2^{-1} \rho \,, \qquad \tilde P = C_2^{-2}\gamma_2\, P,
\ee
and therefore the equation of states in both frames are related by $\tilde \omega = \omega \gamma_2^2$.
%
Similarly  the physical proper time and the scale factors in the two frames are related via $\gamma_2$:
\be\label{tildeaA}
\tilde a = C_2^{1/2} a   \,, \qquad \quad d\tilde \tau = C_2^{1/2} \gamma_2^{-1} d\tau \,.
\ee
Defining  the disformal frame Hubble parameter $\tilde H \equiv \frac{d \ln{\tilde a}}{d\tilde \tau}$, gives:
%
\be\label{tildeHA}
\tilde H = \frac{\gamma_2}{C_2^{1/2}}\left[ H + \frac{C_2'}{2C_2}\dot \phi \right]\,.
\ee
%
To  solve the equations of motion one now can proceed as in section \ref{ME} to write the equations in terms of derivatives w.r.t.~the number of e-folds $N$ and consider different cases by choosing appropriately the parameters $b,C_i,D_i$. 
