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\chapter{\uppercase {Disformal D-brane Coupling Scenario}\myfnt{Reprinted with permission from "D-brane disformal coupling and thermal dark matter'' by Bhaskar Dutta, Esteban Jimenez, Ivonne Zavala, 2017. PhysRevD.96.103506, Copyright 2017 by The American Physical Society.}}\label{Ch:Disformal}

In this chapter, we introduce the second class of ST theories, which is Disformal D-Brane Coupling scenario. This scenario, as was pointed out in the introduction,  can arise from a post-string inflationary scenario. At this stage, the universe is already four-dimensional and moduli associated to the compactification have been properly stabilised\footnote{Though these fields might be displaced from their minima, giving rise to a matter dominated regime, with interesting consequences (see e.g. ~\cite{Allahverdi:2013noa}).}. However, the relevant parameters in the model will depend on the string theory quantities such as the string scale, string coupling and compactification volume as we will argue. 

One of the main differences between this scenario and the one studied in the previous chapter is that the disformal coupling enters not only enter through the metric (as in the previous scenario) but also through the action. We can see this clearly by comparing the action of the conformally coupled ST theory, given by \eqref{S1}, and the action we consider for the disformal D-brane coupling scenario, given below by \eqref{S1A3}.

The starting action we consider is given by 
%
\bea\label{S1A3}
S=S_{EH} + S_{brane}\,,
\eea
%
where: 
\bea
&& \hskip -0.9cm S_{EH}=\frac{1}{2\kappa^2} \!\!\int{\!d^4x\sqrt{-g}\,R}, \\
&& \hskip -0.9cm S_{brane} = - \!\!\int{\!d^4x\sqrt{- g} \left[ M^4 C ^2(\phi)\sqrt{1+\frac{D(\phi)}{C(\phi)} (\partial\phi)^2}+V(\phi)\right]} % \,,\\
 - \!\!\int{\!d^4x\sqrt{-\tilde g} \,{\cal L}_{M}(\tilde g_{\mu\nu}) } \,, \label{braneS3}%\nn% \\
\eea
%
where in a string setup, $\kappa^2=M_{P}^{-2}=8\pi G$ is related to the string coupling, scale and  overall compactification volume by 
$M_P^2 = \frac{2{\cal V}_6}{2\pi g_s^2 \alpha'} $, where $M_s^{-2}=\ell_s^2=\alpha'(2\pi)^2$  is the string scale,  ${\cal V}_6$ is the dimensionless six-dimensional (6D) volume in string units and $g_s$ is the string coupling. Note also that $G$ is not in general equal to Newton's constant as measured by e.g.~local experiments. 



In   \eqref{braneS3}  we describe the brane dynamics (of transverse and longitudinal fluctuations associated to the scalar and matter respectively) given by the Dirac-Born-Infeld (DBI) and Chern-Simons actions for a single D3-brane. The DBI part gives rise to the noncanonically normalized scalar field $\phi$, associated to the single overall position, $r^2=\sum_i^6 y_i^2$, of the brane in the internal 6D space\footnote{In general, a D3-brane can move in all six of the internal dimensions.}  with coordinates $y_i$.  In this case, the scale $M$ is dictated by the tension of a D3-brane as $M^4=T_3 =(g_s\alpha'^2(2\pi)^3)^{-1} = M_s^4(2\pi)g_s^{-1}=\frac{g_s^3}{8\pi {\cal V}_6^2}M_P^4$ and thus by the string scale and coupling.  In reality, one would most likely have a stack of branes moving in the internal space. However, to study the cosmological evolution after inflation, it is enough to model all matter living on the moving brane as in \eqref{braneS3} (see also \cite{KWZ,DWZ}) via the disformally coupled matter Lagrangian ${\cal L}_M$. 

In \eqref{braneS3},  the disformally coupled metric $\tilde g_{\mu\nu}$  is given by the induced metric on the brane, which for a brane moving along a single internal direction can be written as 
%
\be\label{gammaA3}
\tilde g_{\mu\nu} = C(\phi) g_{\mu\nu} + D(\phi) \partial_\mu \phi \partial_\nu \phi\,.
\ee
%
where the scalar field is related to the D-brane position by\footnote{For the D3-brane case,  one can also consider different dimensionalities, which will add extra factors due to the internal volumes wrapped by the brane in that case. } $\phi = \sqrt{T_3} \,r$, and while $C(\phi)$ is dimensionless, $D(\phi)$ has units of mass$^{-4}$. These functions are specified by the ten-dimensional (10D) compactification and therefore in general will be related to each other as we  see below (see also \cite{KWZ}). 
%


\section{The equations of motion}



Einstein's equations  obtained from \eqref{S1A3} are given by
%
\be
R_{\mu\nu} -\frac{1}{2}g_{\mu\nu} R = \kappa^2\left(T^\phi_{\mu\nu} + T_{\mu\nu}\right)\,,
\ee
where the energy-momentum tensors are defined with respect to the Einstein-frame metric $g_{\mu\nu}$ and are given by
%
\be\label{Tm3}
T_{\mu\nu}= P g_{\mu\nu} +(\rho + P) u_\mu u_\nu\,,
\ee
for matter, where $\rho$, $P$ are the energy density and pressure for matter with equation of state $P/\rho = \omega$. For the scalar field, the energy-momentum tensor takes the form:
\be\label{EMphiA3}
T_{\mu\nu}^{\phi} = - g_{\mu\nu} \left[M^4C^2 \gamma^{-1} + V \right] 
+ M^4 CD \,\gamma \partial_\mu\phi \, \partial_\nu \phi
\ee
%
where the energy density and pressure for the scalar field are identified as:
\bea\label{rhoPA3}
\rho_\phi =M^4C^2 \gamma + V  \,, \qquad P_\phi =  - M^4C^2\gamma^{-1} - V  \,,
\eea
%
 and the  ``Lorentz factor"  $\gamma$ introduced above is defined by
%
\be\label{LorentzA3}
\gamma \equiv \left(1+ \frac{D}{C}\, (\partial\phi)^2\right)^{-1/2}\,.
\ee
%
It will be  convenient to rewrite \eqref{rhoPA3} by introducing  ${\mathcal V} \equiv V + C^2 M^4$, as 
%
\be\label{rhoPA23}
\rho_\phi =- \frac{M^4C D\gamma^2}{\gamma+1} (\partial\phi)^2 + {\mathcal V}  \,, \qquad 
P_\phi = - \frac{M^4C D\,\gamma}{\gamma+1}(\partial\phi)^2 - {\mathcal V}  \,.
\ee
%
The equation of motion for the scalar field is:
%
\bea\label{EqphiA3}
&&\hskip-1cm - \nabla_\mu\!\left[M^4 DC\gamma\, \,\partial^\mu \phi\right ] \!+ \!\frac{\gamma^{-1}M^4 C^2}{2} \!\left[\!\frac{D_{,\phi}}{D} +3\frac{C_{,\phi}}{C} \right] \!+ \!\frac{\gamma \,M^4 C^2}{2} \!\left[\!\frac{C_{,\phi}}{C} -\frac{D_{,\phi}}{D} \right] \!+\! V_\phi
\nonumber \\&& \hskip3cm 
- \frac{T^{\mu\nu}}{2}\!\left[\frac{C_{,\phi}}{C} g_{\mu\nu}  +\frac{D_{,\phi}}{C}\partial_\mu\phi\partial_\nu\phi\right]
+\nabla_\mu \left[\frac{D}{C}T^{\mu\nu} \partial_\nu\phi \right] =0\,, %\nonumber \\
\eea
%
where $C_{,\phi}$ denotes derivative of $C$ with respect to $\phi$, and similarly for $D, V$.
Finally, the energy-momentum conservation  equation, $\nabla_\mu T^{\mu\nu}_{tot} = \nabla_\mu \left(T^{\mu\nu}_{\phi}+  T^{\mu\nu}\right)=0$, combined with the equation of motion for the scalar field allows us to define $Q$ as
%
\be
Q\equiv \nabla_\mu \left[\frac{D}{C} \,T^{\mu\lambda} \,\partial_\lambda \phi\right] - \frac{T^{\mu\nu} }{2} \left[\frac{C_{,\phi}}{C} g_{\mu\nu} +
\frac{D_{,\phi}}{C} \,\partial_\mu\phi \,\partial_\nu\phi\right]\,,
\ee
so that, 
$\nabla_\mu T^{\mu\nu}_{\phi}=- \nabla_\mu T^{\mu\nu}=  Q \partial^\nu \phi$.

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

\section{Cosmological equations}

Let us now look at the cosmological evolution. We start with a FRW background  metric:
\be
ds^2= -dt^2 + a^2(t)dx_idx^i \,,
\ee
where $a(t)$ is the scale factor in the Einstein frame. With this metric, the equations of motion become
\bea
&& H^2 =\frac{\kappa^2}{3} \left[\rho_\phi +\rho\right]\,, \label{friedmann1A3}\\
&& \dot H + H^2 = -\frac{\kappa^2}{6}\left[ \rho_\phi+ 3P_\phi +\rho +3 P \right]\,,\label{friedmann2A3}\\
&& \ddot \phi +3H\dot\phi \,\gamma^{-2}  
+ \frac{C}{2D}\left( \frac{D_{,\phi}}{D}- \frac{C_{,\phi}}{C} + \gamma^{-2}\left[\frac{5 C_{,\phi}}{C}  - \frac{D_{,\phi}}{D}\right] 
  -4\gamma^{-3}\frac{C_{,\phi}}{C} \right)
 + \frac{1}{M^4CD\gamma^{3}}\, ({\mathcal V}_{,\phi}+Q_0) =0 \label{kgA3} \,,  \nonumber \\
\eea
%
where, $H= \frac{\dot a}{a}$, dots are derivatives with respect to $t$, $$\gamma= (1-D \,\dot\phi^2/C)^{-1/2},$$ and 
\bea
Q_0 = \rho \left[ \frac{D}{C} \,\ddot \phi + \frac{D}{C} \,\dot \phi \left(\!3H + \frac{\dot \rho}{\rho} \right) \!+ \!\left(\!\frac{D_{,\phi}}{2C}-\frac{D}{C}\frac{C_{,\phi}}{C}\!\right) \dot\phi^2 +\frac{C_{,\phi}}{2\,C} (1-3\,\omega)
\right], \nonumber \\
\eea
where we have used the equation of state for matter $P=\omega\rho$.
The continuity equations for the scalar field and matter are given by
\bea 
&&\dot\rho_\phi + 3H(\rho_{\phi}+P_{\phi}) = -Q_0\dot\phi\,, \label{contA3}\\
&&\dot\rho + 3H(\rho+P) = Q_0\,\dot\phi\,.\label{cont1A3}
\eea
%
Using \eqref{cont1A3}, we can rewrite this  as 
\be\label{Q0A3}
Q_0 = \rho\left( \frac{\dot \gamma}{\dot \phi\, \gamma} + \frac{C_{,\phi}}{2C}  (1-3\,\omega \,\gamma^2) -3H\omega\,\frac{(\gamma -1) }{\dot \phi}\right) \,.
\ee
%
Plugging this into  the  
(non)conservation equation for  matter \eqref{cont1A3} gives
\be\label{conservaDMA3}
\dot \rho + 3H (\rho + P\,\gamma^{2}) = \rho \left[\frac{\dot \gamma}{\gamma} + \frac{C_{,\phi} }{2C} \,\dot\phi\, (1-3\,\omega \gamma^2)\right]\,.
\ee
%


%
\section{Modified and standard expansion rates}

The modified expansion rate felt by matter $\tilde H$ (which will enter into the Boltzmann equation) is the Jordan-frame expansion rate, given in terms of Jordan (or disformal) frame quantities, and defined with respect to the disformal metric $\tilde g_{\mu\nu}$. 
In this frame, the  Hubble parameter is given by: 
%
\be\label{tildeH3}
\tilde H \equiv \frac{d \ln{\tilde a}}{d\tilde \tau} = \frac{\gamma}{C^{1/2}}\left[ H + \frac{C_{,\phi}}{2C}\dot \phi \right]\,.
\ee
%
and it is thus a function of the Einstein-frame rate $H$, the scalar field and its derivatives. The  proper time and the scale factors in the Jordan and Einstein  frames are related by
\be\label{tildea3}
\tilde a = C^{1/2} a   \,, \qquad \quad d\tilde \tau = C^{1/2} \gamma^{-1} d\tau\,.
\ee
%
Furthermore, the energy densities and pressures in the two frames are related by
\be\label{rhodisform3}
\tilde \rho = C^{-2} \gamma^{-1} \rho \,, \qquad \tilde P = C^{-2}\gamma P\,,
\ee
while the equation of state is given by 
\be
\tilde \omega = \omega \gamma^2\,.
\ee
%
One can check that in the Jordan frame, the continuity equation for  matter takes the standard form:
\be\label{tilderho23}
\frac{d{\tilde \rho}}{d\tilde\tau}  +  3 \tilde H( \tilde \rho +\tilde P)  =0 \,.
\ee
%

To proceed further,  we next  swap time derivatives with derivatives with respect to the number of efolds, $N=\ln a/a_0$, so $dN=Hdt$. We also define a dimensionless scalar field $\varphi = \kappa\phi$. In this case, \eqref{tildeH3} becomes: 
\be\label{tildeH23}
\tilde H = \frac{H\gamma}{C^{1/2}}\left[ 1 + \alpha(\varphi) \varphi' \right]\,,
\ee
where a prime  denotes  a derivatives with respect to $N$ and we have  defined 
\be\label{alphaeq3}
\alpha(\varphi) = \frac{d\ln C^{1/2}}{d\varphi}\,.
\ee
Note also that in terms of $\varphi$ and  $N$ derivatives, the Lorentz factor is now given by
\be\label{gammaH3}
\gamma^{-2} = 1-\frac{H^2}{\kappa^2}\frac{D}{C}\varphi'^2\,.
\ee

We  want to compare the Jordan-frame expansion rate with that  expected in GR, which is given by 
\be\label{HGR3}
H_{GR}^2 = \frac{\kappa_{GR}^2}{3} \tilde \rho\,.
\ee
%
We can write this in terms of $H$, $\varphi$ and its derivatives as follows. 
We first write \eqref{friedmann1A3} as  (see \cite{Catena}):
\be\label{H23}
H^2 = \frac{\kappa^2}{3} \frac{(1+\lambda)}{B}\rho= \frac{\kappa^2}{3} \frac{C^2\gamma(1+\lambda)}{B} \tilde \rho\,,
\ee
where $\lambda = {\cal V}/\rho\, (=\tilde {\cal V}/\tilde \rho)$, 
\be \label{B3}
B= 1-\frac{M^4CD\gamma^2}{3(\gamma+1)}\varphi'^2 \,,
\ee
and we have used \eqref{rhodisform3} in the second equality of \eqref{H23}.
%
By inserting  \eqref{H23} into \eqref{HGR3}, we can  write $H_{GR}$ entirely as a function of $H, \varphi, \varphi'$ as:
\be\label{HGR23}
H_{GR}^2 = \frac{\kappa_{GR}^2}{\kappa^2} \frac{C^{-2}B\,\gamma^{-1}H^2}{(1+\lambda)}\,.
\ee
%
Therefore, once we find a solution for $H$ and  $\varphi$, we can compare the expansion rates $\tilde H$ with $H_{GR}$ using \eqref{tildeH23} and \eqref{HGR23}. To measure  the departure from the standard expansion, we recall the speed-up factor parameter defined in \eqref{xieq}:  
\ben
\xi= \frac{\tilde H}{H_{GR}}\,.
\een
%
Notice that $\xi$ can be larger or smaller than one, indicating an enhancement or reduction of  $\tilde H$ with respect to $H_{GR}$. This means that $\tilde H$ can grow during the cosmological evolution. However notice that this does not imply a violation of the the NEC. This is because  the Einstein-frame expansion rate $H$ is dictated by the energy density $\rho$ and pressure $p$, which obey the NEC and therefore $\dot H<0$ during the whole evolution, as it should (see \cite{Creminelli}).

In the following section, we describe the procedure to solve the system of coupled equations for $H$ and $\varphi$ derived from \eqref{friedmann2A3} and \eqref{kgA3}.  



\subsection{Coupled equations for $\varphi$ and $H$}\label{Coupledeq3}

The field  equations \eqref{friedmann2A3} and \eqref{kgA3} can be written as  
\bea
&& H' = - H \left[ \frac{3B}{2(1+\lambda)}   (1+ \omega) + \frac{\varphi'^2 M^4CD\gamma}{2}\right], \label{Hprime3}\\ \nonumber \\
&&\!\!\varphi'' \left[1\!+\! \frac{3H^2\gamma^{-1} B}{M^4CD\kappa^2(1+\lambda)} \frac{D}{C} \right]  
+ 3\,\varphi'  \left[\gamma^{-2}-  \frac{ 3H^2\gamma^{-1} B \omega}{M^4CD\kappa^2(1+\lambda)} \frac{D}{C} \right]  \nonumber \\
 \nonumber \\
&&\hskip0.5cm +\frac{H'}{H} \varphi' \left[1+ \frac{3H^2\gamma^{-1} B}{M^4CD\kappa^2(1+\lambda)} \frac{D}{C} \right] + \frac{3B\gamma^{-3}}{M^4CD(1+\lambda)} \alpha(\varphi)(1-3 \,\omega\gamma^2) \nonumber \\ \nonumber\\
&& \hskip0.95cm 
  + \frac{3B\lambda\gamma^{-3}}{M^4CD(1+\lambda)} \frac{{\cal V}_{,\varphi}}{{\cal V}}  + \frac{3 H^2 \gamma^{-1} B}{M^4CD\kappa^2(1+\lambda)} \frac{D}{C}\left[ (\delta(\varphi) - \alpha(\varphi)) \,\varphi'^2 \right]   \nonumber \\
  \nonumber \\
  && \hskip1.5cm  +\frac{\kappa^2}{H^2}\frac{C}{D}\left[  \gamma^{-2}\left( 5 \alpha(\varphi) -\delta(\varphi)  \right)  +\delta(\varphi)-  \alpha(\varphi) \left(1+4\gamma^{-3}\right) \right]=0 \,, 
  \label{phiHeq3}
\eea
%
where 
\be
\delta(\varphi) = \frac{d \ln D^{1/2}}{d\varphi} \label{deltaeq3}\,.
\ee
%
We notice here that, contrary to the pure conformal case discussed in chapter \ref{Ch:Conformal}, we can not eliminate the equation for $H$, and we do not end up with a single master equation for the scalar field. Due to the disformal term, we need to consider the coupled  equations for $\varphi$ and $H$. 


\subsubsection*{The cubic equation for $H$}

Below we solve the equations numerically, for which we need the initial conditions for $H_i$ and $(\varphi_i, \varphi_i')$. Therefore, we need to find an expression for $H$ in terms of all other quantities and in particular $\tilde \rho$. We can obtain this from the Friedmann equation written in terms of $\tilde \rho$ in \eqref{H23}.
Recalling that $\gamma$ depends nontrivially on $H$  \eqref{gammaH3}, one obtains a cubic equation for $H^2$ given by\footnote{A similar equation was found in chapter \ref{Ch:Conformal}, see \eqref{eqforH2}, for the phenomenological disformal case. In that case, $A_2=-1$ and $A_3=0$. } : 
\be\label{cubic3}
A_1 H^6 + A_2 H^4 + A_3 H^2 + A_4 =0 
\ee
where 
\bea
&&   A_1= \frac{D\varphi'^2}{C\kappa^2} \label{A13}  \,,\\
&& A_2 =  \frac{2M^4CD\varphi'^2}{3}-1 \,,\\
&& A_3 =  \frac{M^4C^2\kappa^2}{3} \left(  \frac{M^4CD\varphi'^2}{3} -2  \right) \,,\\
&& A_4 =\left(\frac{M^4\kappa^2C^2}{3}\right)^2 \frac{(1+\lambda)\,\tilde\rho}{M^4} \left( \frac{(1+\lambda)\,\tilde\rho}{M^4} + 2 \right) \label{A43} \,.
\eea

One of the solutions to \eqref{cubic3} can be written as
%
\be\label{Hdis3}
H^2=\frac{1}{3 A_1}\left(-A_2+\left(A_2^2-3A_1A_3\right)\left(\frac{2}{\Delta}\right)^{1/3}+\left(\frac{\Delta}{2}\right)^{1/3}\right)\,,
\ee

\ni with 
\bea
\Delta&=&-27A_1^2A_4+9A_1A_2A_3-2A_2^3+\sqrt{\left(-27A_1^2A_4+9A_1A_2A_3-2A_2^3\right)^2 
-4\left(A_2^2-3A_1A_3\right)^3} \nonumber \\
& \equiv& L+\sqrt{L^2-4\ell^3}.\label{Delta}
\eea 
The other two solutions can be  obtained by replacing 
$$ \left(\frac{2}{\Delta}\right)^{1/3} \to e^{2\pi i/3} \left(\frac{2}{\Delta}\right)^{1/3}
\qquad {\rm and } \qquad \left(\frac{2}{\Delta}\right)^{1/3} \to e^{4\pi i/3} \left(\frac{2}{\Delta}\right)^{1/3}\,.$$
%
We are interested in real positive solutions for $H^2$. These can be  identified by considering  a complex $\Delta$, that is, $4\ell^3>L^2$, which implies a condition on $\tilde \rho, \,\varphi'$, and $C$.  For this choice, the imaginary parts of $(\Delta/2)^{1/3}$ and  $\ell (\Delta/2)^{-1/3}$ cancel each other\footnote{In this case, we can write  $Z=\frac{\Delta}{2} =\frac{ L+i\sqrt{4\ell^3-L^2}}{2}$,  then  $Z\bar Z = \ell^3$ and $\frac{2}{\Delta} = \frac{\bar Z}{\ell^3} $ and thus the  imaginary parts in \eqref{Hdis3} cancel out. }. 
We will use the real positive solutions in our numerical implementations to find the initial condition for $H$. 



\section{D-brane Disformal Solutions}

 As we discussed at the beginning of the chapter, when considering a probe D3-brane moving in a warped 10D space, which is a solution to the 10D equations of motion, $C$ and $D$ are related and given in terms of the warp factor of the geometry \cite{KWZ}. In particular,  in the normalization where $\phi$ becomes canonically normalized once the DBI action is expanded,  $M^4CD=1$, (see Appendix \ref{App2}). Other normalizations are possible; however, the results will be equivalent. 
 Thus in this section, we study solutions for the D-brane conformally and disformally coupled matter with the choice above, which implies $\delta(\varphi)=-\alpha(\varphi)$. We start by presenting the equations of motion for this case, followed by a discussion on the constraints and initial conditions we use in our numerical analysis.  We first discuss in detail the numerical solutions for the $C=$ const or a pure disformal case, followed by the  $C\ne $ const case.  


\subsection{Equations of motion and Jordan frame }\label{Sec:EQM3}

%
We are interested in the radiation and matter dominated eras during which the potential energy of the scalar field is subdominant. Therefore in what follows we consider $\lambda\sim0$. Also, to solve the equations \eqref{Hprime3} and \eqref{phiHeq3}, we  need to write them in terms of Jordan-frame quantities $\tilde \omega = \omega \gamma^2$ and $\tilde \rho = C^{-2}\gamma^{-1} \rho$. After doing this, the coupled equations above become

\bea
&& H' = - H \left[ \frac{3 B}{2}   (1+\tilde \omega\gamma^{-2}) + \frac{\varphi'^2 }{2}\gamma\right], \label{Hprime13}\\ \nonumber \\
&&\!\!\varphi'' \left[1\!+\! \frac{3H^2\gamma^{-1} B}{ M^4 C^2\kappa^2} \right]  
+ 3\,\varphi' \gamma^{-2} \left[1-  \frac{ 3H^2\gamma^{-1} B}{M^4C^2\kappa^2}\tilde\omega\right] %+ \nonumber \\
% \nonumber \\
%&&
+\frac{H'}{H} \varphi' \left[1+\! \frac{3H^2\gamma^{-1} B}{ M^4 C^2\kappa^2} \right]\nonumber \\ \nonumber\\
 && \hskip0.5cm 
 - \frac{6H^2\gamma^{-1} B}{ M^4 C^2\kappa^2}\,\alpha(\varphi)\varphi'^2 %\nonumber \\ \nonumber\\
%&& \hskip0.5cm 
 +3B\gamma^{-3} \alpha(\varphi)(1-3 \,\tilde\omega)-\frac{2M^4C^2\kappa^2}{H^2}\left[ 2\gamma^{-3}-3 \gamma^{-2} +1 \right]\alpha(\varphi)=0 \,. \nonumber \\
  \label{phiHeq13}
\eea

Furthermore,  we also  convert derivatives with respect to  $N$  to derivatives with respect to  $\tilde N$, the  number of e-folds in the Jordan frame by following the procedure described at the end of section \ref{Sec:Mod2}. Therefore,  derivatives w.r.t.~$N$ transform to  derivatives w.r.t.~$\tilde N$  (assuming well behaved functions) according to \eqref{NtotildeN} as: 
\bean
%&&
\varphi' =  \frac{1}{\left(1-\alpha(\varphi) \frac{d\varphi}{d\tilde N}\right)} \frac{d\varphi}{d\tilde N} \,,\qquad %\\
%&& 
\varphi'' = \frac{1}{\left(1-\alpha(\varphi) \frac{d\varphi}{d\tilde N}\right)^3} \left( \frac{d^2\varphi}{d\tilde N^2} + \frac{d\alpha}{d\varphi}  \left(\frac{d\varphi}{d\tilde N}\right)^3\right)\,. \\ \nonumber 
\eean
%
To avoid clutter we  write down expressions with derivatives with respect to $N$, but it should be understood that all our numerical calculations are made using derivatives with respect to $\tilde N$. 

%
Let us start by discussing \eqref{phiHeq13} to understand the behavior of the solutions. Similarly to the  conformal case discussed in section \ref{Sec:PureC}, the derivative of $C(\varphi)$ acts as an effective potential, given by\footnote{Notice that the last term in \eqref{phiHeq13} proportional to $\alpha$ is not part of an effective potential, as it vanishes when taking the velocity terms,   $\varphi'$ to zero (so $B=1$ and $\gamma=1$).} 
\be\label{EffecV3}
V_{eff}\sim  3 (1-3 \,\tilde\omega)\ln C \,.
\ee
%
Deep in the radiation-dominated era, the equation of state is given by $\tilde\omega=1/3$ and the effective potential vanishes. As the temperature of the universe decreases, particle species in the cosmic soup become nonrelativistic. When the temperature of the universe drops below the rest mass of each of the particle types, nonzero contributions to $1-3\tilde \omega$  arise, activating the effective potential. On the other hand, during the matter-dominated era,  $\tilde\omega=0$, and the effective potential is active through it.
In section \ref{Sec:EqState} we showed how to calculate $\tilde\omega$ during the radiation-dominated era, and plotted $\tilde \omega$, for the SM particle spectrum, during that era in Figure \ref{plotomegaradiation}. 


\subsection{Initial conditions and parameter constraints }

Before we move on to solving the coupled equations  \eqref{Hprime13} and \eqref{phiHeq13}   to find the modified expansion rate, $\tilde H$, and compare it with the standard one, $H_{GR}$, we stop here to describe the constraints and initial conditions we use in our numerical analysis. 


\subsubsection*{Parameter Constraints}\label{ParCons}


As was discussed in section \ref{Sec:ParCons2}, in ST theories, deviations from GR can be parametrized in terms of the post-Newtonian parameters,  
$\gamma_{PN}$ and $\beta_{PN}$.  In the standard conformal case, these  parameters are given in terms of  $\alpha(\varphi_0)$ and  its derivative, $\alpha'_0= d\alpha/ d\varphi |_{\varphi_0}$  as  in \eqref{PostNewt}. Also, Solar System tests of gravity constrain  $\alpha_0$ to very small values of order $\alpha_0^2 \lesssim 10^{-5}$, while binary pulsar observations impose that $\alpha'_0\gtrsim -4.5$. Further, the relation between the bare gravitational constant and that measured by local experiments, for  conformally coupled theories, is given by \eqref{ratioG},  $\,\kappa_{GR}^2 = \kappa^2 C(\varphi_0) [1+\alpha^2(\varphi_0)]\,$. The strongest constraint applies to the speed-up factor $\xi$, which has to be of order 1 before the onset of BBN \cite{Coc}.

For the phenomenological disformal case, Solar System constraints and the ratio $\kappa_{GR}/\kappa$ have been studied for constant $D$ in Ref.~\cite{ISS}. In particular, they found  $\kappa_{GR}^2 = \kappa^2 (1+3\Upsilon/2)$, where $\Upsilon \propto \varphi'^2_0$. As we will see, all solutions we found have $\varphi'\approx 0$ at the onset of BBN. Therefore, for the constant disformal case, $\kappa_{GR}^2 = \kappa^2$. 
For the $C\ne $ const case, on the other hand, we will use the constraints on $\alpha$ above requiring that the standard expansion rate is recovered well before the onset of BBN. This is what we need to ensure that the predictions of the standard cosmological model are not modified.   

\subsubsection*{Initial conditions and the scale M}\label{ICLowM}

%
To find the numerical solutions, we need to fix the initial conditions for $H, \varphi, \varphi'$. Since $\varphi$ is given in Planck units, we take $\varphi_i, \varphi'_i \lesssim 1$.  To find the initial value for $H$,  based on the initial variables $\varphi_i, \varphi'_i $ and $\tilde \rho_i$, we need a real positive solution to \eqref{cubic3}, which can be found from \eqref{Hdis3}  for the  case $CDM^4=1$. In this case the coefficients $A_i$ simplify greatly. 

Writing \eqref{Delta} as
\be
\Delta = L+i \sqrt{\mathbb L} \,,\nonumber
\ee
where now  we have 
\bea
&&  L = 2+2\varphi'^2 -\frac{7}{3}\varphi'^4 + \frac{2}{27} \varphi'^6 -3\varphi'^4 R  \,, \\
&& \mathbb L= 4 \ell^3 - L^2 = -\frac{\varphi'^4}{9}(1+R)\left[81\varphi'^4 R -(3+4\varphi'^2)(\varphi'^2-6)^2\right]\,,\\
&& \ell= \left(1+\frac{\varphi'^2}{3}  \right)^2 \,,\\
&& R = \frac{\tilde \rho}{M^4} \left( \frac{\tilde \rho}{M^4} +2 \right) \,.
\eea
%
From here it is not hard to see that $L$ can be either positive or negative and we require that  $\mathbb L>0$ for $\Delta$ to be complex, as required to find real positive solutions.  
%
In terms of the initial values for $\varphi'_i$\footnote{Recall that in our numerical solutions we take derivatives w.r.t. $\tilde N$, so $\varphi'_i $ should be read as  $\frac{\varphi_i'}{1-\alpha(\varphi_i)}$.}, this requirement implies 
\be\label{bound3}
R\leq \frac{(3+4\varphi_i'^2)(\varphi_i'^2-6)^2}{81\varphi_i'^4}\,.
\ee
%
Recalling that during the radiation-dominated era the energy density is given by $\tilde\rho(\tilde T)=\frac{\pi}{30}g_{eff}(\tilde T)\tilde T^4$, once we fix $\varphi'_i$ and the initial temperature  $T_i$, the value of $M$ is bound via \eqref{bound3}. Indeed, \eqref{bound3}, is satisfied for $\tilde \rho_i/M^4$ in the interval 
$\left(0,- 1+\frac{2}{9\varphi_i'^2}\sqrt{(3+\varphi_i'^2)^3}\right)$. Or, in terms of $T_i$ and $\varphi_i'$, the value of $M$ lies in the interval:
\be\label{lowM3}
\left[\left(\frac{3\,\pi\,g_{eff}(\tilde T_i) \,\varphi_i'^2}{-90\,\varphi_i'^2+ 20 \sqrt{(3+\varphi_i'^2)^3}}\right)^{1/4} \tilde T_i  \,, + \infty \right]\,.
\ee


As an example, we show the lower bound for $M$ as a function of $(\varphi'_i)^2$ in Figure \ref{LowerM} for the initial temperature of  1.0 TeV. For simplicity, we take $C=1$, so that derivatives with respect to $N$ and $\tilde N$ are the same.  As can be seen from \eqref{lowM3} and  Fig.~\ref{LowerM}, for a given initial condition $T_i$,
 the closer $\varphi'_i$ goes to $\sqrt{6}$, the larger the values of $M$, and vice versa. Also, the larger the value of $T_i$, the larger also the lower bound of $M$. In the D-brane-like scenario, as was described at the beginning of the chapter, the scale $M$ is related to the string coupling and scale  (or the six-dimensional volume) as
  $M= M_s(2\pi g_s^{-1})^{1/4}$.  Therefore, we see that the scale decreases for small string scales (large compactification volumes) and small string couplings, which are needed for the string perturbative description to be valid. We will come back to this point below.  
%.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!]
\centerline{
\includegraphics[width=0.65\textwidth]{lowerM.png}}
\caption{Lower bound for  $M$ (see \eqref{lowM3}) as a function of  $(\varphi'_i)^2$ for  $C=1$. }
%\end{center}
\label{LowerM}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\subsection{Pure disformal case, $C=$ const.}\label{Ceq1}

We are now ready to discuss in detail the numerical solutions for $H$ and $\varphi$ and use them to compute the modified expansion rate. We start with the case $C(\varphi)=$ const which can be understood as a pure disformal case, which is presented here for the first time. Indeed, notice that in this case $\gamma\ne 1$, which precisely carries the disformal (or derivative) effect, while $\alpha=0$ (which carries the conformal effect)  
%

Without loss of generality we can take $C(\varphi)=1$ and therefore  $D(\varphi)=\frac{1}{M^4}$. Comparing with the phenomenological case studied in chapter \ref{Ch:Conformal}, one could think that an arbitrary choice of the function $D$ there (with $C=1$) would give different results. However,  we expect that the effects of an arbitrary function, in that case, can be encoded in the choice of the scale $M$ here, and therefore will give similar results to those presented here.

For $C=1$, the system of coupled equations reduces to the following form,
%
\bea
&& H' = - H \left[ \frac{3}{2}   (1+\tilde \omega\gamma^{-2}) B+ \frac{\varphi'^2 }{2}\gamma\right]\label{HprimeC13},\\ \nonumber \\
&&\!\!\varphi'' \left[1\!+\! \frac{3H^2\gamma^{-1} B}{ M^4 \kappa^2} \right]  
+ 3\,\varphi' \gamma^{-2} \left[1-  \frac{ 3H^2\gamma^{-1} B}{M^4\kappa^2}\tilde\omega\right] + \frac{H'}{H} \varphi' \left[1+\! \frac{3H^2\gamma^{-1} B}{ M^4\kappa^2} \right]=0\,.   \label{phiHeqC13}
\eea
%
 As expected, the effective potential is flat, since $\alpha=0$ (see discussion above). 
We solve these equations numerically to find the dimensionless scalar field $\varphi$ and the Hubble parameter $H$, as functions of $\tilde N$. 
We have explored a wide range of initial conditions for $\varphi$ and $\varphi'$ and values of the scale $M$. To find the initial condition for $H$ ($H_{i}$),  we use the appropriate real positive solution of \eqref{cubic3}, as was pointed out earlier. 
%
We find that at most two of the solutions \eqref{cubic3} for $H_{i}$,  are real and positive.  For these  two $H_{i}$'s, the corresponding initial value of $\gamma$ ($\gamma_i$)  is obtained using \eqref{gammaH3} (setting $M^4CD=1$).  We find that one of these $ \gamma_i$'s is usually of order one while the other is 1 or 2 orders of magnitude larger (sometimes even larger). We find that  the solutions to \eqref{HprimeC13} and \eqref{phiHeqC13} that  obey the necessary constraints are those with $\gamma_i\sim 1$. 
%
Once we have found the solutions for $\varphi$ and $H$, we go back to \eqref{tildeH23} to obtain the expansion rate in the Jordan frame.  

%%%
Before looking into the full numerical solutions, let us take a closer look at the ratio between the modified expansion rate and the standard rate,  $\xi$. For $C=$ const this becomes 
\bea\label{xiConst3}
\xi = \frac{\gamma^{3/2}}{B^{1/2}} \,.
\eea
%
Since $\gamma, B \geq 1$, it is clear that $\xi\geq1$, that is,  $\tilde H \geq H_{GR}$. In other words, in this case, the expansion rate is always enhanced with respect to the standard evolution. Moreover, this enhancement is driven by $\gamma$ and $B$. As soon as $\gamma>1$, there will be a nontrivial disformal enhancement.
%%%

In Figure \ref{HubbleC1} we show the resulting modified expansion rate for different values of $\varphi'$ and the mass scale $M$. In these plots, we use $\varphi_i=0.2$, but any value in the interval $(0,1)$ and appropriate choices of $\varphi'_i$ and the mass scale $M$, will give similar results. 
As we can see in Figure \ref{HubbleC1}, $\tilde H$ (colored lines) is always enhanced with respect to the standard expansion rate (black line),  $H_{GR}$, as discussed above. From \eqref{xiConst3} and Figure \ref{Speedup}, it is clear that the ratio $\xi$  is always greater than or equal to 1. Moreover, as the temperature decreases, the ratio $\xi$ grows from a value close to 1 (recall that  $\gamma_i\sim 1$), reaches a maximum where $\gamma$ is maximal and eventually decreases towards one before BBN. The maximum value of the ratio increases and moves to lower temperatures as the mass scale $M$ becomes smaller. 
%

We can  understand this behavior by looking at the evolution of the factor $ f=\frac{3H^2\gamma^{-1} B}{ M^4 \kappa^2}$, inside the square brackets of \eqref{phiHeqC13}. We have seen numerically that this factor evolves as $f(\tilde T)\backsimeq \frac{3g_{eff}(\tilde T)}{10}\left(\frac{\tilde T}{M}\right)^4$ as temperature decreases (see Figure \ref{factorf}). For the  scale $M$ and temperatures plotted in Figure \ref{HubbleC1}, $f(\tilde T)$ starts  much bigger than one (up to $f(\tilde T_i)\backsimeq10^9$) and  decreases  as the temperature decreases. The bigger the  scale $M$, the earlier $f(\tilde T)$ becomes of order 1. While $f(\tilde T)$  is bigger than 1,  $\xi$ increases, the velocity of the scalar field $\varphi'$ increases slowly, and thus the scalar field increases very slowly too (see Figure \ref{Speedup} and \ref{phiplot}). As $f(\tilde T)$  approaches 1,  $\xi$ reaches a maximum and the scalar field starts  increasing faster. Then, as the temperature decreases further, $f(\tilde T)$ becomes smaller than 1. Meanwhile, $\tilde H$ starts converging  towards $H_{GR}$ (that is, $\xi$ starts decreasing) while  the scalar field keeps increasing. Finally, when $\tilde H$ becomes of order $H_{GR}$, $f(\tilde T)$ is much smaller than 1 and the scalar field  starts moving towards  a final constant value.



%%%%%%%%

\begin{figure}
%\centering
\centerline{
\begin{tabular}{cc}
\includegraphics[width=0.52\textwidth]{Dbranephip0_00002H.png}&\includegraphics[width=0.52\textwidth]{Dbranephip0_0002H.png}\\
\includegraphics[width=0.52\textwidth]{Dbranephip0_002H.png}&\includegraphics[width=0.52\textwidth]{Dbranephip0_02H.png}
\end{tabular}}
\caption{Modified expansion rate for the pure disformal case,  $C=1$. We show different boundary conditions and values of the scale parameter. The initial value of the scalar field for all the curves is $\varphi_i=0.2$. The black lines in all plots represent the standard expansion rate $H_{GR}$.}
\label{HubbleC1}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\centerline{
\includegraphics[width=0.60\textwidth]{DBranespeedup1.png}}
\caption{Speed-up factor, $\xi=\tilde H/H_{GR}$, as function of temperature for the expansion rates shown in the bottom left plot  in Figure \ref{HubbleC1}. The initial conditions chosen are $\varphi_i=0.2$ and $\varphi'_i=0.002$. }

\label{Speedup}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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

\begin{figure}
\centerline{
\includegraphics[width=0.65\textwidth]{Dbranephip0_002phi.png}}
\caption{Scalar field as a function of temperature. The initial conditions chosen are $\varphi_i=0.2$ and $\varphi'_i=0.002$. These solutions of Eqs.~\eqref{HprimeC13} and \eqref{phiHeqC13} correspond to the expansion rates shown in the bottom left plot in Figure \ref{HubbleC1}. }
\label{phiplot}
\end{figure}

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

We see the behavior described above in Figures \ref{Speedup} and \ref{phiplot}. 
For instance, for the  plot corresponding to  $M=106$GeV, $f(\tilde T)$ is approximately $ \frac{3g_{eff}(\tilde T)}{10}\left(\frac{\tilde T}{106 \,\text{GeV}}\right)^4$, which becomes 1 at around $\tilde T=50 \,\text{GeV}$. Between $1000$ GeV and $50$ GeV, $\tilde H$ differs from $H_{GR}$ and in this range the scalar field increases very slowly, looking almost constant. For lower temperatures, between $50$ GeV and $1$ GeV, $\tilde H$ converges towards $H_{GR}$ and the scalar field increases faster while for temperatures smaller than 1 GeV, $\tilde H\sim H_{GR}$ and the scalar field reaches its final value.


All the cases shown in Figure \ref{HubbleC1} satisfy the constraints discussed in section \ref{ParCons}.  In particular, $\varphi'_{BBN} = 0$ (so $\Upsilon=0$)  
and the speed-up factor $\xi$ is equal to 1 prior to BBN as shown in Figure \ref{Speedup}. For  scales $M$ smaller than 10 GeV  the last condition is not satisfied, that is $\xi >1$ by the onset of BBN. Therefore,  scales  $M$ smaller than 10 GeV are discarded.

As we have mentioned,  if we consider larger values of  $M$  than the ones presented in Figure \ref{HubbleC1}, 
the enhancement of the expansion rate will occur earlier, at higher temperatures, as long as $M$ is smaller than $\tilde T_i$.  We illustrate this in Figure \ref{HubbleScaleM}. Here we show a series of plots were the mass scale takes values up to order EeV. This figure shows that the speed-up factor $\xi$,  has the same behavior as long as the ratio $\tilde T_i/M$ does not change. For instance, see all the green lines $\tilde T_i/M$=58.8. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
%\centering
\centerline{
\begin{tabular}{cc}
\includegraphics[width=0.52\textwidth]{Dbranephip0_00002H10_12.png}&\includegraphics[width=0.52\textwidth]{Dbranephip0_00002H10_9.png}\\
\includegraphics[width=0.54\textwidth]{Dbranephip0_00002H10_6.png}&\includegraphics[width=0.54\textwidth]{Dbranephip0_00002H10_3.png}
\end{tabular}}
\caption{Modified expansion rate for the pure disformal case, $C=1$, for larger values of $M$ as compared to Fig.~\ref{HubbleC1}. For these  plots, $\varphi_i=0.2$ and $\varphi'_i=2\times10^{-5}$.}
\label{HubbleScaleM}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{figure}[h]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.5\textwidth]{Dbranefactorf.png}&\includegraphics[width=0.5\textwidth]{DbranefactorfC.png}
\end{tabular}
\caption{Evolution of the factor $f$  as a function of temperature for $C=$ const case (left) and $C\ne$ const. ($f_C$, right). The initial conditions chosen in the left plot are shown in Figure \ref{HubbleC1}, while in the right plot $\varphi_i=0.2$ and $\varphi'_i=-0.004$.}
\label{factorf}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%




\subsection{Conformal and disformal case $C\neq $ const } \label{Cne1}

We now move  to the  case where the conformal coupling is not constant, so both conformal and disformal effects are turned on.  For concreteness we  consider the same conformal coupling as that studied in chapter \ref{Ch:Conformal}, which is given by \eqref{Cfactor} as
\be\label{Cfactor3}
C(\varphi) = (1+ b \,e^{-\beta\, \varphi})^2\,,
\ee
with  the values $b=0.1$, $\beta=8$. 
%
We have also analysed other functions such as $C=(b\,\varphi^2+c)^2$ with $b=4,8,15$, $c=1$. However it is harder to find numerical solutions for this and other functions, which satisfy the phenomenological constraints. In those cases, the effect on the expansion rate  $\tilde H$ was smaller with respect to the case where $C$ is given by \eqref{Cfactor3}. 
% 

As  mentioned in section \ref{Sec:EQM3}, the conformal term acts as an effective potential, or force, in  \eqref{phiHeq13}, given by Eq.~\eqref{EffecV3}. This effective force can be neglected when the factor $f_C=\frac{3H^2\gamma^{-1} B}{ M^4 C^2\kappa^2}$ is much larger than 1, as can be seen from \eqref{phiHeq13}. In this regime, the evolution of the scalar field is given by a flat effective potential,  and the scalar field stays approximately constant. 
When $f_C$ is becomes of order 1 or smaller and $\tilde\omega\neq 1/3 $, the evolution of the scalar field is driven by the effective potential \eqref{EffecV3} and by the Hubble friction term. 

For the conformal coupling  \eqref{Cfactor3}, the effective potential allows for an interesting behavior, according to the choice of initial conditions chosen \ref{ERM}. That is, for negative initial velocities, $\varphi'_i<0$,  the scalar field will start rolling up the effective potential towards smaller values. After reaching a maximum point, it will turn back down the effective potential, eventually reaching its final value. 
This behavior in the scalar field sources a nontrivial behavior in  $C$  and (importantly) its derivative $\alpha$, and therefore in the modified expansion rate $\tilde H$. 
Indeed, when $C\ne $ const., the speed-up factor, $\xi$, becomes 
%
\be\label{xiC}
\xi = \frac{\kappa}{\kappa_{GR}} \frac{C^{1/2}\gamma^{3/2} }{B^{1/2}}\left[ 1+\alpha(\varphi) \varphi'  \right]  \,.
\ee
It is not hard to see that for the initial conditions above, due to the factor inside the parentheses, $\xi$ can become less than one during the evolution.  Recalling that   $\xi=\tilde H/H_{GR}$, $\xi<1$ implies that $\tilde H<H_{GR}$,  as shown in the explicit solutions below. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{
\includegraphics[width=0.65\textwidth]{Dbranephip-0004phi1+Exp.png}}
\caption{Scalar field as a function of temperature for different values of $M$. The conformal coupling is $(1+ 0.1 \,e^{-8\, \varphi})^2$ and the initial conditions chosen are $\varphi_i=0.2$ and $\varphi'_i=-0.004$. These solutions of Eqs.~\eqref{Hprime13} and \eqref{phiHeq13} correspond to the expansion rates shown in the right plot of Figure \ref{HubbleCnon1}. }
\label{phiplotC}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Let us now take a closer look at the evolution of $f_C$ with temperature. Numerically, we  found that when $f_C \gtrsim 1$  it behaves  as  $f_C(\tilde T)\backsimeq \frac{3g_{eff}(\tilde T)}{10}\left(\frac{\tilde T}{M}\right)^4$. But when  $f_C <1$ then it evolves as  $f_C(\tilde T)\backsimeq h(\tilde T) \frac{3g_{eff}(\tilde T)}{10}\left(\frac{\tilde T}{M}\right)^4$, where $h(\tilde T)$ is a function that measures the enhancement of $\tilde H$,  which is larger than 1 and depends on the scale $M$ (see right plot in Figure \ref{factorf}). 
When $f_C\gg1$, the effective force is negligible and the scalar field stays roughly constant. As $f_C$ decreases and approaches and/or becomes smaller than 1, the effective force takes over the evolution of the scalar field. The velocity of the scalar field starts decreasing (we use small negative velocities), and for suitable values, the scalar field goes up the effective potential and comes back down again as described above. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.50\textwidth]{Dbranephip-0004H1+Exp.png}&\includegraphics[width=0.50\textwidth]{Dbranephip-04H1+Exp.png}
\end{tabular}
\caption{Modified expansion rate for the case $C=1+0.1e^{-8\varphi}$.  The initial value of the scalar field for all the curves is $\varphi_i=0.2$. Also, $\varphi'_i=-0.004$ for the plot on the left and $\varphi'_i=-0.4$ for the plot on the right.}
\label{HubbleCnon1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In Figure \ref{phiplotC} we  plot the full numerical solution for  the scalar field for $\varphi_i=0.2$ and an initial velocity $\varphi'=-0.004$.  
The red, green and blue curves (scale masses smaller than  $\tilde T_i=1000$ GeV)  show the scalar field going up the effective potential toward smaller values of the field, and then rolling down  its terminal value., while for the brown curve ($M=$1000 GeV), the scalar field  stays almost constant because for this value of $M$ its initial velocity is not negative enough to move the field up the effective potential.

The effect of the scalar field on the modified expansion rate is shown in Figure \ref{HubbleCnon1} (the black straight line is $H_{GR}$). The left plot shows $\tilde H$ corresponding to the scalar field solutions in Figure \ref{phiplotC}. For these solutions, the factor $f_C$ is initially much bigger than 1 and as the temperature decreases it passes one (around 200 GeV) and keeps decreasing to very small values. For some values of $M$,  the scalar field goes up and down the effective potential, producing the enhancement and the little notch in $\tilde H$ (blue), where $\xi<1$ as explained above. 
On the other hand, in the right plots, $f_C$ is initially of order 1 and then decreases to negligible values. The initial velocity used ($\varphi'=-0.4$) is sufficiently negative to produce the enhancement and notch in $\tilde H$ for some of the $M$ values (green and blue). 

Let us mention another point about the right plot in Figure \ref{HubbleCnon1}. For the brown curve corresponding to $M=5000\, \text{GeV}$, the enhancement is very small, and since the factor $f_C$ decreases as the mass scale $M$ increases, choosing larger values of $M$  would give a similar result, for the same choice of initial conditions. 
Indeed, as $M\to \infty$, $f_C=\frac{3H^2\gamma^{-1} B}{ M^4 C^2\kappa^2}\to 0$ and we recover the pure conformal case in \eqref{phiHeq13}. Notice that  the last term in this equation vanishes when $M$ increases,  since $\gamma\to 1$  as $M$ increases. So, by dropping all terms proportional to $f_C$ and the last term in \eqref{phiHeq13}  ones recovers the master conformal case equation, \eqref{masterConf}, studied in section \ref{Sec:PureC}.
Thus, for a very large value of $M$, we will recover the results of section \ref{Sec:PureC}, by suitably changing the initial conditions for $\varphi_i$ and $\varphi'_i$.

%
Let us finally comment on the differences between the present case (conformal plus disformal) and the pure disformal and pure conformal cases, where there is no derivative interaction.  We saw in the previous subsection that in the pure disformal case the enhancement in the expansion rate can be produced at any temperature (see Figure \ref{HubbleScaleM}), by suitably changing the value of the scale $M$. However, in the present $C\neq 1$ case, this does not happen at any scale since $\omega\ne1/3$ is needed and we get $\omega\ne1/3$ when SM particles become nonrelativistic.   % 
This is due to the last term in \eqref{phiHeq13}, which makes the evolution of $\varphi'$ go to zero very fast, effectively making $\gamma\sim 1$ throughout the evolution and thus an ineffective disformal enhancement. 
%
However, new physics at a higher scale causing a change in $\tilde \omega$ can introduce an enhancement at that scale. This will be similar to the case with the additional $M$ scale associated with the D-brane models.
%
The conformal  enhancement is effective so long as the effective potential \eqref{EffecV3} is active, that is, whenever $\omega\ne1/3$ (see Figure \ref{plotomegaradiation}). 
%

\section{Post-inflationary string cosmology}

The period after the end of inflation,  from reheating up to the onset of BBN remains largely unconstrained. 
 Let us now connect our results with a post-inflationary toy model of string cosmology and discuss the implications in terms of the parameters of the theory.  
% 

As described in the introduction of this chapter, we imagine a toy model where matter is coupled conformally and disformally to a scalar field, associated to an overall position of a (stack of) D-brane(s) in the internal six-dimensional compact space in a warped type IIB string compactification. In this case, we can relate the scale $M$ to the tension of a D3-brane  $T_3$ (for example), and thus to the string coupling $g_s$ and the string scale $M_s$ (or the six-dimensional volume ${\cal V}_6$) as $M=T_3^{1/4}= M_s (2\pi g_s^{-1})^{1/4}$.
%
The pure disformal case $C=1$ is very interesting and would correspond to a large-volume compactification, where the warping due to the presence of fluxes can be ignored \cite{LV1,LV2}. 
In this case, the lowest value we used for $M$ that is relevant for the DM relic abundance (as we will see in the next chapter) was $\sim 10$ GeV and the largest (with a large effect) was $M\sim 300$ GeV. For string couplings of order $g_s\sim 10^{-4}$, these give string scales $\sim 1-20$ GeV and thus exponentially large volumes ${\cal V}_6 \sim 10^{25}-10^{27}$,   that is,  a very low string scale and very weakly coupled compactification. 
%,  
%
On the other hand, for larger values of $M$, or larger values of the string scale, the enhancement on the expansion rate will occur earlier in the universe's evolution (see Fig.~\ref{HubbleScaleM}). For example,  the largest value we used, $M\sim 10^{10}$  GeV, for $g_s\sim 10^{-4}$ would give $M_s\sim 10^{9}$ GeV, volumes of order ${\cal V}_6\sim 10^{10}$ and the expansion enhancement occurs at around $\tilde T\sim 10^{10}$ GeV. 
Therefore, depending on the string scale, coupling and compactification, we may expect the early pre-BBN cosmology to be affected at different epochs with interesting consequences for the post-inflationary string cosmological evolution. This will also be connected to string inflation, which usually requires large string scales (see Ref.~\cite{BM} for a review).

%
Of course a more  realistic model may involve,  for example, other parameters of the theory in the scale $M$ (due for example to higher-dimensional D-branes wrapping the internal space), nonuniversal couplings among the scalar (or scalars) to SM matter and DM as briefly discussed in Ref.~\cite{MW} for the pure conformal case, etc. However, we find it very interesting that scalar couplings present in string theory can give important predictions for the post-inflationary evolution after string inflation. 

%
Let us finally stress that the cosmological implications of conformal and disformal couplings in scalar-tensor theories are in any case very interesting from a more phenomenological point of view. Here we have taken a further step in making progress to address these implications and have presented the disformal effects for the first time.
