%The Ricci scalar is
%Sayan: Commented out Eqs. (5), (6), and (7) and moved them to the first appendix
%Sayan
%AB: the next 2 equations may now be too compressed.
%AB: I don't think this is what Tina meant.
%TK: Yes, agree with Axel. I eman have a less technical stuff, if we can keep clarity without detailed eqs. this will be fine. If not, than probably the previous form is better.
%Sayan: Commenting the next three lines
%$R=\frac{6}{N^2}\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^2-\frac{\dot{a}\dot{N}}{aN}+\dot{\sigma}^2\right]$,
%and the Maxwell Lagrangian is
%$-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{e^{4\sigma}}{2a^4}(E^2-B^2)$.
%Sayan
%AB: do we really need the 3 lines above
%Sayan: I think I commented them, but may have missed.
%Sayan: Done it again; also the next five lines.
%The coupling term $\mathcal{L}_C\equiv\xi\tilde{F}^{\mu\nu}\tilde{F}^{\rho\sigma}R_{\mu\nu\rho\sigma}$ becomes,
%\begin{eqnarray}\label{eCoupTerm}
%\mathcal{L}_C&=&\frac{4\xi e^{4\sigma}}{N^2a^6}\bigg[\left(\frac{a\dot{a}\dot{N}}{N}-a\ddot{a}+\frac{2a^2\dot{N}\dot{\sigma}}{N}-2a^2\ddot{\sigma}\right)B^2 \\
%&+&\dot{a}^2E^2-a\dot{a}\dot{\sigma}\left(4B^2-E^2\right)-2a^2\dot{\sigma}^2\left(E^2+2B^2\right)\bigg]. \nonumber
%\end{eqnarray}

%TK: Yes, agree with Axel. I eman have a less technical stuff, if we can keep clarity without detailed eqs. this will be fine. If not, than probably the previous form is better.
%AB: Its not about the length, but about how easily it can be digested.
%Sayan: I see what you mean. But only eqs. (1), (3) and (4) are repeated
%Sayan: from Mukohyama 2016.
%Sayan: I am reverting to the previous form for now.
%Sayan: Now I think (5), (6), and (7) can be relegated to the first appendix.
%Sayan: We can use that Appendix to list the explicit general form of the various terms.
%Sayan:
%Sayan: Moving the following to the appendix as mentioned above.
%\begin{equation}\label{eRicScal}
%R=\frac{6}{N^2}\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^2-\frac{\dot{a}\dot{N}}{aN}+\dot{\sigma}^2\right],
%\end{equation}
%and the Maxwell Lagrangian is (see Appendix \ref{AppMax})
%\begin{equation}
%-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{e^{4\sigma}}{2a^4}(E^2-B^2).
%\end{equation}
%The coupling term $\mathcal{L}_C\equiv\xi\tilde{F}^{\mu\nu}\tilde{F}^{\rho\sigma}R_{\mu\nu\rho\sigma}$ becomes,
%\begin{eqnarray}\label{eCoupTerm}
%\mathcal{L}_C&=&\frac{4\xi e^{4\sigma}}{N^2a^6}\bigg[\left(\frac{a\dot{a}\dot{N}}{N}-a\ddot{a}+\frac{2a^2\dot{N}\dot{\sigma}}{N}-2a^2\ddot{\sigma}\right)B^2 \\
%&+&\dot{a}^2E^2-a\dot{a}\dot{\sigma}\left(4B^2-E^2\right)-2a^2\dot{\sigma}^2\left(E^2+2B^2\right)\bigg]. \nonumber
%\end{eqnarray}
%Sayan

%Sayan: I don't think this appendix is very relevant to the rest of the paper.
%Sayan: I am thus commenting this out.
%\subsection{``Slow-roll" $k$-inflation}
%\label{kInfl}
%%Sayan: I now think this subsection is unnecessary
%%Sayan: k-inflation is not relevant to the rest of the paper
%%Sayan: I had just done the calculations as a curiosity
%%Sayan: But this section may end up being confusing
%A model for the generation of homogeneous magnetic fields was presented in
%\citep{Mukohyama:2016npi}, in the context of a scalar-tensor theory with the Horndeski
%non-minimal coupling term, as well as the shift-symmetric Horndeski terms (see \cite{Mukohyama:2016npi}).
%In 2017, the LIGO and VIRGO collaborations jointly detected a
%kilonova \cite{TheLIGOScientific:2017qsa,GBM:2017lvd,Monitor:2017mdv}.
%The simultaneous multimessenger detections put stringent constraints
%on various theories of modified gravity \citep{Baker:2017hug,Gong:2017kim};
%in particular, the Horndeski scalar-tensor theory of gravity was strongly disfavored.
%However these constraint do not apply to the case discussed in \citep{Mukohyama:2016npi}
%since in this scenario, the scalar field attains a large mass after inflation and is constant at
%the present, whereas for the constraint to apply, the scalar field has to have a
%non-vanishing derivative at the present time.
%
%We briefly argue below, for methodological purposes, that $k$-inflation is unable to
%sustain a constant magnetic field.
%During $k$-inflation, the scalar field in the Lagrangian in Eq.~\eqref{eKEss} drives
%inflation through its dynamics.
%Following Armendariz-Picon et al. \cite{ArmendarizPicon:1999rj} and analyze a
%specific form of the Lagrangian that leads to exponential expansion,
%\begin{equation}\label{eKInfl}
%\mathcal{L}_\varphi=K(\varphi)X+L(\varphi)X^2
%\end{equation}
%Following the definitions and notation from \cite{Mukohyama:2016npi}, we set,
%\begin{equation}\label{eDefsKinfl}
%\begin{gathered}
%N=1, \quad\quad a=e^{H_0t},\quad\quad \sigma=sH_0t, \\
%E=eH_0, \quad\quad B=bH_0
%\end{gathered}
%\end{equation}
%for constant $e,b,s,H_0$, and look at the equations of motions that arise from the
%action of the theory (see Appendices \ref{AppEOM-kInfl} and \ref{AppEB-Eom}
%for the list of equations (Eqs.~\eqref{eKInflFullEomPhi} -- \eqref{eKInflFullEomSig} and \eqref{eKInflFullEomE});
%also see these sections for details on how to obtain the full equations \tcr{Should we get rid of these appendices too?}).
%Unfortunately it was pointed out by Barrow et al. \cite{Barrow:2012ay} that the equations of
%motion coming from this scenario have no meaningful solution.
%Indeed, as shown in \citep{Mukohyama:2016npi,Mukohyama:2018obj}, the shift-symmetric
%Horndeski terms are necessary to consistently generate such a magnetic field during inflation.
%It is assumed, however, that after inflation, the massless Einstein and Maxwell
%theories are recovered; the non-minimal coupling survives.

