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\chapter{\uppercase {LEPTOGENESIS In a faster-than-usual Universe}\myfnt{The contents of this chapter appear in "A cosmological pathway to testable leptogenesis'' by Bhaskar Dutta, Chee Sheng Fong, Esteban Jimenez and Enrico Nardi. arXiv: 1804.07676. This article was submitted to JCAP.}}\label{Ch:Leptogenesis}

The cosmological baryon asymmetry is very elegantly explained via the leptogenesis mechanism~\cite{Fukugita:1986hr} according to which an initial asymmetry is generated in lepton number and then partly converted in a baryon number asymmetry by $B+L$ violating sphaleron processes~\cite{Kuzmin:1985mm,Rubakov:1996vz} which, above the
temperature of the electroweak (EW) phase transition, proceed with in-equilibrium rates (for reviews on the leptogenesis mechanism see \cite{Davidson:2008bu,Fong:2013wr}).  A very attractive feature of the standard leptogenesis realization based on the type-I seesaw~\cite{Minkowski:1977sc,GellMann:1980vs,Yanagida:1979as, Mohapatra:1980yp} is that it provides a semi-quantitative relation connecting the out-of-equilibrium condition~\cite{Sakharov:1967dj} for the decays of the heavy right-handed (RH) neutrinos with the light neutrino mass scale. RH neutrino decays can be sufficiently out of equilibrium if $m_\nu \sim 10^{-2\pm 1}\,$eV, which is in beautiful agreement with neutrino oscillation data.  On the other hand, type-I seesaw leptogenesis has also an unpleasant facet.  A lepton asymmetry is preferably generated in the decay of the lightest RH neutrino $N_{1}$ since they generally occur at temperatures when the dynamics of the heavier $N_{2,3}$ neutrino is no more efficient. However, the CP asymmetry in $N_1$ decays is bounded by the following relation~\cite{Davidson:2002qv}:
%%%%%%%%%%%%%
\begin{equation}
  \label{eq:DIbound}
 \left|\epsilon_1\right| \leq \frac{3}{16\pi}\frac{M_1}{v^2}
\frac{\Delta m^2_\oplus}{m_1+m_3}  \,, 
\end{equation}
%%%%%%%%%%%%%
where $M_1$ is the $N_1$ mass, $v\sim 174\,$GeV is the SM electroweak (EW) breaking vacuum expectation value (VEV), $\Delta m^2_\oplus\sim 2.4 \times 10^{-3}\,{\rm eV}^2$ is the atmospheric neutrino mass square difference, and $m_{1,3}$ are the lightest and heaviest light neutrino masses, which are bounded by cosmological data to lie not much above $\sim 10^{-1}\,$eV. Since a
minimum CP asymmetry $\left|\epsilon_1\right| \gsim {\rm few}\times 10^{-7}$ is required to account quantitatively for the observed baryon asymmetry, the $N_1$ mass cannot lie much below $ 10^9\,$GeV.\footnote{The bound  \eqref{eq:DIbound} can be somewhat weakened if the RH neutrinos masses are not sufficiently hierarchical~\cite{Hambye:2005tk}, if $N_{2,3}$  decays also contribute to the generation of a lepton asymmetry \cite{Engelhard:2006yg}, or if flavor effects \cite{Abada:2006fw,Nardi:2006fx,Abada:2006ea} play a  relevant role \cite{Blanchet:2006be,Racker:2012vw}.  However, the main conclusion  regarding non-testability of the type-I seesaw leptogenesis model   does not change.}  The conclusion that the CP asymmetry is too small
to explain the baryon asymmetry if the leptogenesis scale is too low, implies that direct tests of the standard type-I seesaw leptogenesis are out of experimental reach. Since the argument does not involve any cosmological input, it holds regardless of the assumed cosmological model.

In more generic realization of leptogenesis~\eqref{eq:DIbound} does not necessarily hold: the simple relation between the CP asymmetries and the light neutrino masses is in fact quite specific of the type-I seesaw and is often lost in other models.  The most direct way to relax this bound is to rescale $v$ in \eqref{eq:DIbound} and this can be realized in a model where neutrinos only couple through a neutrinophilic Higgs which obtains a VEV $v_\nu \ll v$ ~\cite{Haba:2011ra,Clarke:2015hta}. Other examples are the inert scalar doublet model~\cite{Ma:2006km} complemented with heavy Majorana neutrinos~\cite{Racker:2013lua}, as well as many other models, see~\cite{Sierra:2014tqa,Sierra:2013kba,Racker:2012vw,Hambye:2012fh,AristizabalSierra:2009bh,
GonzalezGarcia:2009qd,AristizabalSierra:2007ur} for a sample list. Still, the vast majority of models that attempt to generate the baryon asymmetry from heavy particle decays are subject to an additional constraint which, although less tight than the one implied by \eqref{eq:DIbound}, is much more general.  This constraint stems from a general relation between the strength of the washout scatterings which tend to erase any lepton number asymmetry present in the thermal bath, and the CP asymmetries in the decays of the heavy states.  To our knowledge, in standard cosmologies only models which invoke a resonant enhancement of the CP asymmetries~\cite{Pilaftsis:2003gt,Pilaftsis:2004xx,Pilaftsis:2005rv,GonzalezGarcia:2009qd,Dev:2017wwc}
can evade the corresponding bound and bring leptogenesis from heavy particle decays down to a testable scale~\cite{Chun:2017spz}.

 In this chapter we point out that this conclusion can be avoided if, in the very early stages, the cosmological history of the Universe is described by an ST theory~\cite{jordan1955schwerkraft,Fierz:1956zz,Brans:1961sx} rather than by GR.  As we have seen,  the cosmic expansion rate is boosted in both the pure conformal case and the disformal D-brane coupling scenario. The magnitude of the enhancement depends on boundary conditions and it can be a few orders of magnitude larger compared to the standard GR expansion. We will make use of this enhancement in the expansion rate in our study of leptogenesis.

Differently, from the case of DM, for which the typical decoupling temperature falls in the few GeV range, the generation of a baryon asymmetry via leptogenesis must occur above the EW scale before the EW sphaleron processes get out of equilibrium. Since a leptogenesis scale up to a couple of TeV might still be within the reach of collider tests, we are interested in modifications of the standard cosmology at temperatures in the range $100\,{\rm GeV}-{\rm few}\,{\rm TeV}$.  Indeed, due to the larger scale in the game, we find that in the framework of conformally coupled ST theories the modified expansion does allow to lower the scale for successful leptogenesis down to $M_1 \lsim 1\,{\rm TeV}$. Hence, in our analysis, we will mainly focus on conformally coupled ST theories since this conclusion holds also for ST theories with disformal couplings.


\section{Constraints on the leptogenesis scale}
\label{sec:constraint}
%
The quantum field theory conditions required in order that loop
diagrams can generate a lepton ($L$) number (or any other global quantum
number) violating CP asymmetry in the decays of an heavy state $X$
are: (i) complex couplings between $X$ and the particles running in
the loop (say $Y$ and $Z$); (ii) a CP even phase from the loop
factors, which only arises if the $Y,Z$ propagators inside the loops
can go on-shell; (iii) $L$ violation inside the loop.  Condition (ii)
then implies that $Y$ and $Z$ can also participate as external
asymptotic states in scattering processes, and condition (iii) implies
that these scatterings are necessarily $L$ violating. This means that
decay CP asymmetries unavoidably imply $L$ violating washout
scatterings~\cite{Racker:2013lua,Sierra:2013kba}. Since the same
couplings enter both in the expression for the CP asymmetries and for
the washout scattering rates, it is not surprising that a quantitative
relation between CP asymmetries and scattering rates can be worked
out. A general expression for this relation has been obtained
in~\cite{Sierra:2013kba} and reads:
%
\begin{equation}
  \label{eq:2}
\Gamma(YZ \leftrightarrow  \bar Y\bar Z) \approx \frac{64}{\pi}
\,T\,\left(\frac{T}{M_X}\right)^n \, \epsilon_X^2\,,  
\end{equation}
%
where $\Gamma$ is the rate of the $\Delta L =2$ washout scatterings,
$\epsilon_X$ is the CP asymmetry in $X$ decays, and $n=0$ for a scalar
$X$ decaying into two scalars $Y,Z$; $n=2$ for a fermion $X$ decaying
into a fermion and scalar pair, and $n=4$ for a scalar $X$ decaying into two fermions. 
Note that \eqref{eq:2} relates scattering washout
rates to CP asymmetries without any reference to the cosmological
model. In relation to successful leptogenesis, cosmology enters
through the requirement that at the relevant temperature $T\sim M_X$
the washout rates do not attain thermal equilibrium:
%
\begin{equation}
  \label{eq:3}
  \Gamma(YZ \leftrightarrow  \bar Y\bar Z) \lsim \tilde
  H(T)\big|_{T\sim M_X}\,, 
\end{equation}
%
where we parametrize the deviations of the expansion in terms of a
temperature dependent function $\xi(T)$ multiplying the canonical GR
expansion rate $H(T)= 1.66 \sqrt{g_*}\, T^2/M_P$ (with $g_\star$ the 
relativistic degrees of freedom and $M_P = 1.22 \times 10^{19}$ GeV), namely
$\tilde H(T)= \xi(T)H(T)$ (two examples of $\xi(T)$ are given
in Figure~\ref{fig:speedup})\,.

In the relevant temperature range $T\sim M_X$ the out-of-equilibrium
condition \eqref{eq:3} yields:
%
\begin{equation}
  \label{eq:generic_constraint}
  M_X \gsim 1.2 M_P \frac{\epsilon^2_X}{\xi(M_X)}\,. 
\end{equation}
%
Assuming as a benchmark value
$\epsilon_X \gsim 10^{-7}$ as the lowest possible CP asymmetry able to
explain $n_b/n_\gamma\sim 10^{-10}$, we see that standard cosmology
with $\xi(T) = 1$ yields the (conservative) limit
$M_X \gsim 1.4\cdot 10^{5}\,$GeV, so that in any generic model of
leptogenesis from heavy particle decays the relevant scale lies well
above experimental reach.  As an example, we see from the left plot of Figure
\ref{fig:speedup} that in modified ST cosmologies the function
$\xi(T)$ can remain of order $10^2$ in an interval centered at
$T \sim$ TeV and spanning about two orders of magnitude in $z={\rm TeV}/T$ . 
Because of the boosted expansion, in the relevant temperature range, the dynamical 
processes that govern leptogenesis, in particular, the $\Delta L = 2$ washout processes discussed above,
can more easily go out of equilibrium, rendering viable
scales as low as $M_X \lesssim\; $TeV for which direct tests can be foreseen.



\section{A few comments on the modified expansion rate and the general Boltzman equation for RH neutrinos}
\label{sec:expansion}


In Section \ref{Sec:PureC} we described the pure conformal scenario in ST theories and mentioned that the modified expansion rate depends on the initial conditions of the scalar field and initial temperature, the effective potential, and the particle content of the cosmic plasma (through the equation of state parameter $\tilde \omega$).  

In general, both the initial position and velocity of the scalar field can take any positive or negative values. The most interesting result arises when considering negative velocities. In this case, the field will start rolling-up the effective potential towards smaller values of the field, eventually turning back down and moving towards its final value. So, if the field starts at a positive value, and given a sufficiently negative initial velocity, it will move towards negative values until its velocity becomes zero and then positive again, as it rolls back down the effective potential. This change in sign for the scalar field will produce a peak in the conformal coupling, which will give rise to a non-trivial modification of the expansion rate $\xi\neq 1$ (see \eqref{xieq}). This particular behavior is shown again, for a new set of initial conditions, in Figure \ref{fig:speedup}.

The equation of state parameter $\tilde \omega$ plays an important role in locating the temperature at which the speedup factor, $\xi$, drops back to 1. Slight variations from the radiation dominated value, $\tilde\omega=1/3$, appear when particles become non-relativistic. So, to calculate $\tilde\omega$, as discussed in Section \eqref{Sec:EqState}, one has to take into account all the SM particles and, depending on the specific SM extensions one is dealing with,  one would add RH neutrinos, supersymmetric partners or other types of heavy species.  


Another interesting scenario yielding modified Hubble parameters is the disformal D-brane coupling scenario, which was studied extensively in Chapter \ref{Ch:Disformal}. In Figure \ref{fig:speedup} we also present the speedup factor $\xi$ in this scenario (thin red lines). Recall, $M$ plays the most important role in the location and shape of $\xi$. The maximum $\xi$ happens close to a temperature equal to $M$. It is interesting to notice that by rescaling $M$, $\xi$ moves to a higher (or lower) temperature without changing shape.


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


 \begin{figure}[h!]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=0.48\textwidth]{Examplecondis10_5_GeV.png}&\includegraphics[width=0.48\textwidth]{Examplecondis10_8_GeV.png}
\end{tabular}}
\caption{Two examples for the speedup factor $\xi(z)$ for conformal (thick blue lines) and disformal (thin red lines)  scenarios, as a function of $z=1\,\rm{TeV}/T$ with $T$ the temperature. Left plot: $\xi(z)$ in a conformal scenario with initial conditions $(\varphi_i,\varphi'_i)=(0.8,-1.83)$ and initial temperature  $100\,$TeV, and in a disformal scenario with $(\varphi_i,\varphi'_i)=(0.2,-2\times 10^{-6})$ and a mass scale $M=2.5\,$TeV. Right plot: $\xi(z)$ in a conformal and disformal scenario with  initial conditions as before, but initial temperature $T=10^5\,$TeV, and $M=2500\,$TeV for the disformal case.}
%\end{center}
\label{fig:speedup}
\end{figure}


Due to the fact that, as in GR, also in ST cosmologies the total entropy is conserved, adapting the BE for leptogenesis 
to ST cosmologies is rather straightforward. Considering just the BE for the evolution of the RH neutrinos density $n_N$ including only decays and inverse decays will suffice to illustrate this. Denoting 
with $a$ the scale factor, the BE reads:
%
\begin{equation}
  \label{eq:BE1}
  a^{-3}\frac{d(n_N a^3)}{d t} = \langle \Gamma_N\rangle (n_N^{\rm eq}-n_N)\,,
\end{equation}
%
where  $\langle \Gamma_N\rangle$ is the thermal averaged decay rate and $n_N^{\rm eq}$ the 
equilibrium density. We use the fiducial variable $z=M_N/T$ with $T$ the temperature, and write the time derivative as:
%
\begin{equation}
\label{eq:BE2} 
\frac{d}{d t} = z \left(\frac{1}{z}\frac{d z}{dt}\right) \frac{d}{dz}
= z  \left(\frac{1}{a}\frac{d a}{dt}\right)\frac{d}{dz} = z \tilde H \frac{d}{dz}\,.
\end{equation}
%
The second step relies on entropy conservation $d(s a^3)/dt = 0$
with $s\propto T^3$ the entropy density which implies the usual 
temperature-scale factor relation  $T\propto 1/a$,  
while $\tilde H$ is the physical Hubble parameter
defined as the rate of change of the physical length scale.
The rest is standard: denoting by $\gamma _N = n_N^{\rm eq}  \langle \Gamma_N\rangle$ 
the density of the reaction, and normalizing the particle number
densities to the entropic density as $Y_N = n_N/s$ we have:
%
\begin{equation}
  \label{eq:BE3}
  \frac{d Y_N }{d z} = 
\frac{1}{  s z \tilde H}  \left(1- \frac
  {Y_N}{Y_N^{\rm eq}}\right)\gamma_N=
\frac{1}{  s z H}  \left(1- \frac    {Y_N}{Y_N^{\rm eq}}\right)\frac{\gamma_N}{\xi(z)}
\,.
\end{equation}
%
In the second equation we have rewritten $\tilde H = H \, \xi(z)$
with $H= H_{GR}$  and 
$\xi (z)>1$ the $T$-dependent speedup factor, to put in evidence
how in the BE its effect is equivalent to ``slowing down'' the decay
and inverse decay reactions, favoring the enforcement of the
out-of-equilibrium condition.

In the next section, we will show how a simple (non-resonant) leptogenesis model, when embedded in a non-standard 
cosmology characterized by a boosted expansion rate allows getting around the constraint \eqref{eq:generic_constraint}. 
It is interesting to remark that if a particle physics model can be experimentally established as responsible for the 
cosmological baryon asymmetry via (non-resonant)  baryogenesis via decays of TeV scale particles, this would constitute a direct 
evidence of non-standard cosmology in a temperature range unreachable by all other cosmological probes 
(DM freeze-out, EW phase transition, etc.).


\section{A simple test model}  
\label{sec:model}

Besides the generic constraint in \eqref{eq:generic_constraint}, the type-I (non-resonant) leptogenesis is subject 
to \eqref{eq:DIbound} from neutrino mass. In order to get around the latter, a simple way is to assume that the VEV 
responsible for the neutrino masses is much smaller than the full EW breaking VEV: $v_\nu \ll v\sim 174\,$GeV.  
By requiring a sufficient CP
asymmetry $\epsilon_1 \gtrsim 10^{-7}$, the scale $M_1 \sim 1\,$TeV can be
reached for $v_\nu \lsim 0.2\,$GeV.  One has to introduce an
{\it ad hoc} Higgs field with $\langle H_\nu \rangle = v_\nu$ coupled
to RH neutrinos $N_j$ as $\lambda_{j\alpha } \bar N_j L_\alpha H_\nu$,
and forbid the couplings with the standard Higgs $H$ via some $\zb_2$
or $U(1)$ symmetry (a $U(1)$ softly broken might be preferable to
avoid domain wall problems with a spontaneously broken $\zb_2$).  Such
model exists, see for example~\cite{Haba:2011ra},
or~\cite{Clarke:2015hta} for various different possibilities (in the
last paper, Model Type I with $m^2_{12}>0$ and $\lambda_5=0$ is
probably the best option).  Although the `neutrinophilic' VEV model (we will denote it as $v_\nu$-model)
might not represent the most elegant possibility, its structure
remains very similar to the standard type-I see-saw model, with the advantage that it minimizes the differences with respect to the
standard leptogenesis case, rendering it suitable as a test model to
illustrate the effects of non-standard cosmologies.\footnote{Some
  $\Delta L=1$ $2 \leftrightarrow 2$ washouts involving the top quark,
  like $Q_{3L} L \leftrightarrow N t_R$ and
  $Q_{3L} \bar t_R \leftrightarrow N \bar L$ will be absent, since
  $H_\nu$ does not couple to the top-quark. This has no major impact
  in determining the viable leptogenesis scale.}
%
The usual seesaw formula still holds:
\begin{equation}
  \label{eq:seesaw}
m_\nu \simeq \lambda^T\frac{v^2_\nu}{M}\lambda,
\end{equation}
%
and so does the Casas
Ibarra parametrization of the Yukawa couplings:
%
\begin{equation}
  \label{eq:CI}
  \lambda_{j\alpha} = \frac{1}{v_\nu} \sqrt{M_j}R_{j\beta}
  \sqrt{m_\beta} (U^\dagger)_{\beta\alpha}\,, 
\end{equation}
%
with $M_j$ and $m_\beta$ the heavy and light neutrino mass eigenvalues, $U$ the
neutrino mixing matrix, and $R$ a generic complex orthogonal matrix ($RR^T=I$).

Let us consider \eqref{eq:seesaw}. In the usual seesaw with the SM VEV
$v\sim 174\,$GeV, to allow for a low value of $M$ while still ensuring
$m_\nu \lsim 0.1\,$eV, one has to take tiny Yukawa couplings $\lambda$, which in turn imply tiny CP asymmetries.\footnote{One could
  arrange for cancellations in the matrix multiplications to
  keep the coupling $\lambda$
  sizable~\cite{Hambye:2003rt,Raidal:2004vt}. However, 
  this requires exponential fine-tunings in the phases of the complex angles of the $R$ matrix in 
  \eqref{eq:CI}~\cite{Ibarra:2010xw,Shuve:2014zua} which, moreover, unless protected by some specific symmetry,
  are unstable under quantum corrections~\cite{AristizabalSierra:2011mn}. 
}
In the $v_\nu$-model instead, the couplings $\lambda$ can be large
since it is $v_\nu$ that is small, and thus the CP asymmetries can be
also large.  However, if the scale $M_1$ is low, leptogenesis will
occur when the Universe expansion is slower, and then the
$\Delta L =2$ washouts $L H \leftrightarrow \bar L \bar H$ or
$LL\leftrightarrow \bar H \bar H$ that are mediated by the same Yukawa
couplings can attain thermal equilibrium, realizing the situation in which
leptogenesis cannot be successful because of the constraint discussed
in sec.~\ref{sec:constraint}.

In order to illustrate these constraints in the GR, we show in Figure \ref{fig:GR_bound_M1} 
the bounds on the lightest RH neutrino mass $M_1$ from leptogenesis in the $v_\nu$-model as a function of \emph{washout} parameter defined as 
%
\be
K_1 = \left.\frac{\Gamma_{N1}}{H}\right|_{T = M_1}
\label{eq:K_1}
\ee
%
where $\Gamma_{N_1} = \frac{(\lambda\lambda^\dagger)_{11} M_1}{8\pi}$ is the total decay width of $N_1$. Outside these regions, one cannot generate sufficient baryon asymmetry.
Notice that for $K_1 = 1$, the \emph{effective} neutrino mass is
%
\be
{m_{\rm eff}} \equiv 
\frac{(\lambda\lambda^\dagger)_{11} v_\nu^2}{M_1} 
= 3.6\times 10^{-8} 
\sqrt{\frac{g_\star}{110.75}}
\left(\frac{v_\nu}{1\,{\rm GeV}}\right)^2\,{\rm eV}.
\label{eq:m_eff}
\ee
%
This implies that even in the strong washout regime $K_1 \gg 1$, 
the lightest light neutrino remains essentially massless i.e. $m_1 \approx 0$.

In Figure \ref{fig:GR_bound_M1}, the red/thick and blue/thin lines correspond respectively to values of $v_\nu = 1, 2$ GeV.\footnote{The results are obtained from solving eqs. \eqref{eq:BE_N1} and \eqref{eq:BE_Delta_alpha_sim} by setting $\xi = 1$ and the heaviest neutrino mass $m_3 = 0.05$ eV. For further details, refer to sec.~\ref{sec:boltzmann}.} 
The solid lines are for the case of vanishing initial $N_1$ abundance $Y_{N_1}(0)=0$ while dotted lines for thermal initial $N_1$ abundance $Y_{N_1}(0)=Y_{N_1}^{\rm eq}(0)$. 
The horizontal dashed lines refer to the absolute lower bounds obtained for the respective $v_\nu$ if leptogenesis proceeds with $Y_{N_1}(0)=Y_{N_1}^{\rm eq}(0)$ in the absence of washout (or in the weak washout regime $K_1 \ll 1$). 
The lower and upper bounds are respectively due to \eqref{eq:DIbound} and $\Delta L = 2$ washout scattering discussed in sec. \ref{sec:constraint}. Notice that one cannot lower the scale of $M_1$ indefinitely by lowing $v_\nu$, at some point, the washout will be too strong to generate sufficient baryon asymmetry. This is the case for $v_\nu = 1$ GeV where no solution exists for the case of $Y_{N_1}(0)=0$. In this case, one arrives at lower bound on $M_1$ of few times $10^5$ GeV in agreement with the estimation in sec. \ref{sec:constraint}.\footnote{Our results for $Y_{N_1}(0)=Y_{N_1}^{\rm eq}(0)$ are also consistent with the estimation in refs.~\cite{Haba:2011ra,Clarke:2015hta}.}   

\begin{figure}[t!!]
\begin{center}
\includegraphics[width=0.8 \textwidth]{Mbound_GR_2.pdf}
\end{center}
\caption{The bounds on $M_1$ for $v_\nu = 1, 2$ GeV (red/thick, blue/thin lines) as a function of $K_1$ defined in \eqref{eq:K_1}. Outside these closed regimes, one cannot obtain sufficient baryon asymmetry. The solid lines are for zero initial abundance of $N_1$ while dotted lines for thermal initial abundance of $N_1$. The horizontal dashed lines are the absolute lower bounds obtained for the respective $v_\nu$ which correspond to having thermal initial abundance of $N_1$ and no washout.}
\label{fig:GR_bound_M1}
\end{figure}



 
\section{Boltzmann equations in modified cosmology}
\label{sec:boltzmann}

In the following, we will describe a particle $X$ in term of abundance $Y_X \equiv n_X/s$ 
defined as its number density $n_X$ normalized over entropic density $s = \frac{2\pi^ 2}{45}g_\star T^3$. 
We will fix $g_\star = 110.75$ for the SM with an additional (neutrinophilic) Higgs doublet. 
Taking into account the modification due to speedup factor as discussed in ~\eqref{eq:BE3}, 
the BEs for $Y_{N_{1}}$ and $Y_{\Delta_{\alpha}}$ with $\Delta_{\alpha}\equiv\frac{B}{3}-L_{\alpha}$
can be written down as follows\footnote{To avoid double counting in the BE
for $Y_{\Delta_{\alpha}}$, we have subtracted off the CP-violating
$\Delta L=2$ scattering involving on-shell $N_{1}$ and ignored the
off-shell contribution \cite{Kolb:1979qa,Fry:1980ph}.}
%
\begin{eqnarray}
s\xi Hz\frac{dY_{N_{1}}}{dz} & = & -\gamma_{N_{1}}\left(\frac{Y_{N_{1}}}{Y_{N_{1}}^{{\rm eq}}}-1\right), 
\label{eq:BE_N1} \\
s\xi Hz\frac{dY_{\Delta_{\alpha}}}{dz} & = & -\epsilon_{1\alpha}\gamma_{N_{1}}\left(\frac{Y_{N_{1}}}{Y_{N_{1}}^{{\rm eq}}}-1\right)+\frac{1}{2}P_{1\alpha}\gamma_{N_{1}}\left(\frac{Y_{\Delta\ell_{\alpha}}}{Y_{f}}+\frac{Y_{\Delta H_{\nu}}}{Y_{b}}\right)\nonumber \\
 &  & +\gamma_{22}^{\alpha\alpha}\left(\frac{Y_{\Delta\ell_{\alpha}}}{Y_{f}}+\frac{Y_{\Delta H_{\nu}}}{Y_{b}}\right)+\sum_{\beta\neq\alpha}\gamma_{22}^{\alpha\beta}\left(\frac{Y_{\Delta\ell_{\alpha}}}{2Y_{f}}+\frac{Y_{\Delta\ell_{\beta}}}{2Y_{f}}+\frac{Y_{\Delta H_{\nu}}}{Y_{b}}\right),
\label{eq:BE_Delta_alpha}
\end{eqnarray}
%
where we have defined $z\equiv\frac{M_{1}}{T}$, $Y_{f}\equiv\frac{15}{8\pi^{2}g_{\star}}$
and $Y_{b}\equiv\frac{15}{4\pi^{2}g_{\star}}$. 
In the above, $Y_{\Delta\ell_{\alpha}}$
and $Y_{\Delta H_{\nu}}$ refer to abundances per gauge degrees of
freedom. Explicitly, the total thermal averaged decay reaction density $\gamma_{N_{1}}$ is given by
%
\be
\gamma_{N_1} =
\sum_{\alpha}\gamma_{N_{1}\to\ell_{\alpha}H_{\nu}} 
= n_{N_1}^{\rm eq} \Gamma_{N_1} \frac{{\cal K}_1(z)}{{\cal K}_2(z)},
\label{eq:gamma_N_1}
\ee
%
where ${\cal K}_n(z)$ refers to the modified Bessel function of second type of order $n$ and the branching ratio for $N_1$ decay to lepton of flavor $\alpha$ as $P_{1\alpha}\equiv\frac{\gamma_{N_{1}\to\ell_{\alpha}H_{\nu}}}{\gamma_{N_{1}}}$.
The $\Delta L=2$ washout mediated by off-shell $N_{i}$ is described by $\gamma_{22}^{\alpha\beta}$.\footnote{The $\Delta L=1$ scatterings 
involving gauge bosons are not considered since to consider them consistently, one also needs to consider CP violation in them which will result in a small net
effect \cite{Nardi:2007jp,Fong:2010bh}. As for flavor changing but $\Delta L=0$ scatterings, their rates go as $\frac{T^{5}}{M_{i}^{4}}$
for $T<M_{i}$ which are less important than that of $\Delta L=2$ reactions which go as $\frac{T^{3}}{M_{i}^{2}}$ for $T<M_{i}$. Furthermore, in the following, we will consider democratic flavor structure where
they are either not relevant or in thermal equilibrium and can be
dropped from the BEs. } In order to minimize the complication from flavor effects and focus solely on the effect of ST cosmology, we choose the democratic flavor structure as follows
%
\begin{eqnarray}
\epsilon_{1e} & = & \epsilon_{1\mu}=\epsilon_{1\tau}\equiv\frac{\epsilon_{1}}{3},\\
P_{1e} & = & P_{1\mu}=P_{1\tau}=\frac{1}{3},\\
\gamma_{22}^{\alpha\beta} & \equiv & \frac{1}{9}\gamma_{22},
\end{eqnarray}
%
where $\epsilon_{1}\equiv\sum_{\alpha}\epsilon_{1\alpha}$ and $\gamma_{22}\equiv\sum_{\alpha\beta}\gamma_{22}^{\alpha\beta}$.
With the above assumptions, the BE for $Y_{\Delta_{\alpha}}$
becomes
%
\begin{eqnarray}
s\xi Hz\frac{dY_{\Delta_{\alpha}}}{dz} & = & -\frac{1}{3}\gamma_{N_{1}}\epsilon_{1}\left(\frac{Y_{N_{1}}}{Y_{N_{1}}^{{\rm eq}}}-1\right)+\frac{1}{6}\gamma_{N_{1}}\left(\frac{Y_{\Delta\ell_{\alpha}}}{Y_{f}}+\frac{Y_{\Delta H_{\nu}}}{Y_{b}}\right)\nonumber \\
 &  & +\frac{1}{9}\gamma_{22}\left(\frac{Y_{\Delta\ell_{\alpha}}}{Y_{f}}+\frac{Y_{\Delta H_{\nu}}}{Y_{b}}\right)+\frac{1}{9}\gamma_{22}\left(\frac{Y_{\Delta\ell_{\alpha}}}{Y_{f}}+2\frac{Y_{\Delta H_{\nu}}}{Y_{b}}+\frac{1}{2}\sum_{\beta\neq\alpha}\frac{Y_{\Delta\ell_{\beta}}}{Y_{f}}\right).\label{eq:BE_Delta_alpha_sim}
\end{eqnarray}
%

For $M_{2,3}\gg M_{1}$, the total CP parameter is given by \cite{Roulet:1998}
%
\begin{eqnarray}
\epsilon_{1} & \simeq & -\frac{3}{16\pi}\sum_{j>1}\frac{{\rm Im}\left[\left(\lambda\lambda^{\dagger}\right)_{1j}^{2}\right]}{\left(\lambda\lambda^{\dagger}\right)_{11}}\frac{M_{1}}{M_{j}},
\end{eqnarray}
%
and using \eqref{eq:CI}, one can derive the Davidson-Ibarra bound \cite{Davidson:2002qv}
%
\begin{eqnarray}
\left|\epsilon_{1}\right| & \leq & \frac{3}{16\pi}\frac{M_{1}}{v_{\nu}^{2}}\left(m_{3}-m_{1}\right)\equiv\epsilon_{1}^{{\rm max}},\label{eq:DIbound_vnu}
\end{eqnarray} 
%
as introduced in \eqref{eq:DIbound} but with $v \to v_\nu$.
We further parametrize the off-shell $\Delta L=2$ washout mediated by $N_{i}$ valid for $T<M_{i}$ as follows
\begin{eqnarray}
\gamma_{22} & \equiv & \frac{n}{\pi^{3}}\frac{{\rm Tr}\left[m_{\nu}m_{\nu}^{\dagger}\right]}{v_{\nu}^{4}}T^{3}
%=\frac{n}{\pi^{3}}\frac{\sum_{i}m_{i}^{2}}{v_{\nu}^{4}}T^{3}
=\frac{nM_{1}^{3}}{\pi^{3}z^{3}}\frac{\sum_{i}m_{i}^{2}}{v_{\nu}^{4}},
\label{eq:eff_22_washout}
\end{eqnarray}
where $n=\frac{2}{\pi^{2}}T^3$. As shown in \eqref{eq:m_eff}, the lightest light neutrino mass $m_1$ can be neglected and we can rewrite the above in term of \eqref{eq:DIbound_vnu} as follows
%
\begin{eqnarray}
\gamma_{22} & \simeq & \frac{256 n}{9\pi} \frac{M_1}{z^{3}}\epsilon_{1}^{{\rm max},2}.
\label{eq:eff_22_washout2}
\end{eqnarray}
%\begin{eqnarray}
%\gamma_{22} & = & \frac{nM_{1}^{3}}{\pi^{3}z^{3}}\frac{m_{3}^{2}}{v_{\nu}^{4}}=\frac{256}{9}\frac{nM_{1}}{\pi z^{3}}\epsilon_{1}^{{\rm max},2}.
%\label{eq:eff_22_washout2}
%\end{eqnarray}
%
From the above, we see that the $\Delta L=2$ washout is indeed proportional to $\epsilon_{1}^{{\rm max},2}$ as argued in \eqref{eq:2}, so that $M_1$ remains bounded from below by the general lower limit given in  \eqref{eq:generic_constraint}. 
As in the standard type-I seesaw, also in the present case an upper bound on $M_1$ exists, which follows from the requirement 
that $\Delta L=2$ washout scatterings will not become too strong to erase the asymmetry.
\eqref{eq:eff_22_washout} shows that once the neutrino mass scale $m_i$ is fixed, for each value of  $v_\nu$ there is a limiting upper value of $M_1$ for which $\gamma_{22}$ remains sufficiently out of equilibrium. However, 
while in the standard case this hints to a loose upper limit of order $\sim 10^{14}\,$GeV, due to the large 
hierarchy $v_\nu/v\lsim 10^{-2}$ and to the quartic dependence on the VEV values, in the neutrinophilic VEV model 
the corresponding constraint is much stronger.

For the spectator effects \cite{Buchmuller:2001sr,Nardi:2005hs}, we consider the temperature regime $T\lesssim 10^{5}$
GeV where all Yukawa interactions are in chemical equilibrium. We further assume that $H_{\nu}$ does not carry a conserved charge\footnote{This can be due to fast interactions induced by $\lambda_{5}\left(H^{\dagger}H_{\nu}\right)^{2}$
in the scalar potential.} and we have
%
\begin{eqnarray}
\left(\begin{array}{c}
Y_{\Delta\ell_{e}}\\
Y_{\Delta\ell_{\mu}}\\
Y_{\Delta\ell_{\tau}}
\end{array}\right) & = & \frac{1}{207}\left(\begin{array}{ccc}
-64 & 5 & 5\\
5 & -64 & 5\\
5 & 5 & -64
\end{array}\right)\left(\begin{array}{c}
Y_{\Delta_{e}}\\
Y_{\Delta_{\mu}}\\
Y_{\Delta_{\tau}}
\end{array}\right),\\
Y_{\Delta H_{\nu}} & = & -\frac{2}{23}\left(Y_{\Delta_{e}}+Y_{\Delta_{\mu}}+Y_{\Delta_{\tau}}\right).
\end{eqnarray}
%

Substituting the result above into (\ref{eq:BE_Delta_alpha_sim})
and summing over $\alpha$ on both sides, the BE becomes
%
\begin{eqnarray}
s\xi Hz\frac{dY_{\Delta}}{dz} & = & -\gamma_{N_{1}}\epsilon_{1}\left(\frac{Y_{N_{1}}}{Y_{N_{1}}^{{\rm eq}}}-1\right)+\frac{1}{6}\gamma_{N_{1}}\left(-\frac{6}{23}\frac{Y_{\Delta}}{Y_{f}}-\frac{6}{23}\frac{Y_{\Delta}}{Y_{b}}\right)\nonumber \\
 &  & +\frac{1}{9}\gamma_{22}\left(-\frac{6}{23}\frac{Y_{\Delta}}{Y_{f}}-\frac{6}{23}\frac{Y_{\Delta}}{Y_{b}}\right)+\frac{2}{9}\gamma_{22}\left(-\frac{6}{23}\frac{Y_{\Delta}}{Y_{f}}-\frac{6}{23}\frac{Y_{\Delta}}{Y_{b}}\right)\nonumber \\
 & = & -\gamma_{N_{1}}\epsilon_{1}\left(\frac{Y_{N_{1}}}{Y_{N_{1}}^{{\rm eq}}}-1\right)-\frac{3}{23}\left(\gamma_{N_{1}}+2\gamma_{22}\right)\frac{Y_{\Delta}}{Y_{f}},
\label{eq:BE_Delta_alpha_sim2}
\end{eqnarray}
%
where we have defined $Y_{\Delta}\equiv Y_{\Delta_{e}}+Y_{\Delta_{\mu}}+Y_{\Delta_{\tau}}$.
In ref. \cite{DOnofrio:2014rug}, assuming the SM, it was obtained that the EW sphaleron processes freeze out at $T_{\rm EWSp}=132$ GeV after the EW symmetry breaking at $T_c=159$ GeV. Assuming the EW symmetry breaking also happens before $T_{{\rm EWSp}}$ in $v_\nu$-model, we have \cite{Harvey:1990qw,Inui:1993wv}
\begin{eqnarray}
Y_{\Delta B} & = & \frac{30}{97}Y_{\Delta},
\label{eq:Delta_to_B}
\end{eqnarray}
excluding the contributions from heavy charged (neutrinophilic) Higgs and top quark.



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

\section{Results}
\label{sec:results}

The asymmetry $Y_\Delta$ can be parametrized in term of \emph{efficiency} factor $\eta = \eta(K_1,v_\nu,m_3,M_1)$ as follows
%
\begin{eqnarray}
Y_{\Delta} & = & \epsilon_{1}Y_{N_{1}}^{{\rm eq},0}\eta,
\end{eqnarray}
%
where $Y_{N_{1}}^{{\rm eq},0} \equiv Y_{N_{1}}^{{\rm eq}}\left(0\right)=\frac{45}{\pi^4 g_\star}$. 
The above parametrization is convenient because once we substitute it into \eqref{eq:BE_Delta_alpha_sim2}, for temperature-independent $\epsilon_1$, the BE becomes independent of $\epsilon_1$. The final asymmetry is obtained by evaluating the final efficiency $\eta = \eta(z\to\infty)$. In the case with an initial thermal abundance of $N_1$, one saturates to the maximal efficiency $\eta = 1$ in the limit of weak washout $K_1 \ll 1$ and small $\Delta L = 2$ washout. As we will see in more detail later, as $M_1$ gets close to the EW sphaleron freezeout temperature $T_{\rm EWSp}$, one might not be able to saturate the efficiency factor because the baryon asymmetry will be frozen before all $N_1$ can decay.

Using \eqref{eq:DIbound_vnu} and \eqref{eq:Delta_to_B}, the maximal asymmetry is given by
%
\begin{eqnarray}
Y_{\Delta B}^{{\rm max}} & = & \frac{30}{97}\frac{3}{16\pi}\frac{M_{1}m_{3}}{v_{\nu}^{2}}Y_{N_{1}}^{{\rm eq},0}\eta.
\end{eqnarray}
%
Setting $Y_{\Delta B}^{{\rm max}} = Y_{\Delta B}^{{\rm obs}} = 8.7 \times 10^{-11}$, we can derive both upper and lower bounds on $M_{1}$. Starting from a very small $M_{1}$ while keeping the $\Delta L = 2$ washout under control, the CP parameter might be too small and we need to increase $M_{1}$ until $Y_{\Delta B}^{{\rm max}} = Y_{\Delta B}^{{\rm obs}}$, which gives us the lower bound on $M_{1}$. As we continue to increase $M_{1}$, eventually the $\Delta L=2$ washout \eqref{eq:eff_22_washout} will become too strong until which we are no longer able to obtain sufficient baryon asymmetry and this gives us an upper bound on $M_1$. As we explain below \eqref{eq:eff_22_washout2}, this upper bound is specific to the model we have chosen due to neutrino mass constraint. In other words, from the following equation
%
\begin{eqnarray}
M_{1}\eta\left(K_{1},v_{\nu},m_3,M_{1}\right) & = & \frac{97}{30}\frac{16\pi}{3}\frac{v_{\nu}^{2}}{m_{3}}\frac{Y_{\Delta B}^{{\rm obs}}}{Y_{N_{1}}^{{\rm eq},0}},
\end{eqnarray}
%
for a given $v_{\nu}$ and $m_{3}$, we can have no solution, one solution,
or two solutions for $M_{1}$. The two solutions will correspond to 
upper and lower bounds on $M_{1}$. Notice that as we go to smaller $M_1$, the EW sphaleron freezeout temperature becomes relevant and we fix this to be $T_{\rm EWSp} =132$ GeV after which the value of baryon asymmetry will be frozen.

\begin{figure}[t!!]
\begin{center}
\includegraphics[width=0.8 \textwidth]{Mbound_conf3_2.pdf}
\end{center}
\caption{The bounds on $M_1$ for $v_\nu = 0.1, 1$ GeV (red/thick, blue/thin lines) as a function of $K_1$ defined in \eqref{eq:K_1}. 
Outside the closed regimes, one cannot obtain sufficient baryon asymmetry.
The solid lines are for zero initial abundance of $N_1$ while dotted lines for thermal initial abundance. 
The horizontal dashed lines represent the absolute lower bounds on $M_1$ for the respective $v_\nu$ 
obtained with initial thermal abundance of $N_1$ and no washout.}
\label{fig:ST_bound_M1}
\end{figure}

It is important to note the temperature where speedup happens is crucial for leptogenesis. 
As discussed in Section \ref{sec:expansion}, while the regime of speedup for the conformal case depends on initial temperature and $\varphi$ field configurations, the regime of speedup for the disformal case depends on a new mass scale $M_D$. 
Besides this point, the qualitative effect of the speedup for both scenarios on leptogenesis remains the same. Hence we will only illustrate the result for speedup factor for the conformal case as shown in the left plot of Figure \ref{fig:speedup}. 
In this example where speedup happens in the range $10\,{\rm GeV} < T < 10^5\,{\rm GeV}$, it will affect leptogenesis with $M_1$ which falls in the relevant mass range.

Our main results are presented in Figure \ref{fig:ST_bound_M1} in the $K_1 - M_1$ plane for a fixed $m_3 = 0.05$ eV and $v_\nu = 0.1, 1$ GeV (red/thick, blue/thin lines) where outside these closed regime, one cannot obtain sufficient baryon asymmetry.
The solid lines are for $Y_{N_1}(0)=0$ while dotted lines for $Y_{N_1}(0)=Y_{N_1}^{\rm eq}(0)$. 
The horizontal dashed lines are the absolute lower bounds obtained for the respective $v_\nu$ which correspond to having $Y_{N_1}(0)=Y_{N_1}^{\rm eq}(0)$ and no washout.
For $Y_{N_1}(0)=0$ and small $K_1$, due to the speedup in the Hubble expansion, the inverse decay is not efficient in populating $N_1$. Less $N_1$ results in less asymmetry being produced and hence $M_1$ needs to increase correspondingly to enhance the CP violation. As one goes to larger $K_1$, $N_1$ is more efficiently populated and one is allowed to have smaller $M_1$. Crucially, in all cases, $\Delta L = 2$ washout is suppressed sufficiently due to the speedup factor $\xi$ as evidence from the fact that one is able to obtain successful leptogenesis for $M_1$ much below $10^5$ GeV (cf. Figure \ref{fig:GR_bound_M1}). In fact, a smaller speedup in the early times (see Figure \ref{fig:speedup}) allows an efficient washout of an initial `wrong' sign asymmetry (generated during the production of $N_1$) by $\Delta L = 2$ scattering and one ends up enhancing the final asymmetry. Instead of a curse, $\Delta L = 2$ becomes a blessing. Numerically, we found the lowest $M_1$ to be around 350 GeV which corresponds to $v_\nu \sim 0.03$ GeV and $K_1 \sim 3000$. 

As a final remark, notice that the behaviors of $M_1$ lower bounds for $Y_{N_1}(0)=Y_{N_1}^{\rm eq}(0)$ for small $K_1$ are different for the case of $v_\nu = 0.1,1$ GeV. In the small $K_1$ regime, we expect them to approach the absolute lower bounds (the horizontal dashed lines). While this happens for the case of $v_\nu = 1$ GeV, the lower bound actually moves away from the horizontal line for the case of $v_\nu = 0.1$ GeV. The reason is that for $M_1 \sim$ TeV and small $K_1$, the decays happen very late close to the EW sphaleron freezeout temperature $T_{\rm EWSp}$. When we reach this temperature, the baryon asymmetry will be frozen before all $N_1$ can decay, resulting in smaller final asymmetry.

\section{Minimal Supersymmetric Standard Model and Right-handed Neutrinos}
\label{sec:mssm}

In section~\ref{Ceq1}, we mentioned that the scale for the enhanced expansion rate can be moved around as a function of the new scale associated with  $D(\varphi)$ term in the disformal D-brane coupling scenario. In the conformal case, an extension of the SM can change the enhancement scale. In this section, we discuss the scale for enhancement in the cases of MSSM and SM with 3 RH neutrinos.

In the left plot of Figure \ref{fig:speedup}, we show the speedup factor, in the conformal scenario, for one set of values for $\varphi$ and $\varphi'$ at an initial temperature of $100$ TeV considering only the SM particle spectrum. If we add three 10 TeV RH neutrinos, the speedup factor and its slope at around 1 TeV are the same as in the SM case, but $\xi$ drops to 1 slightly earlier. 

We now add the RH neutrinos to the spectrum of the SM and MSSM. In Figure \ref{fig:speedup1}, we show the enhancements  for various values of $\varphi'$ and initial temperatures. We add three RH neutrinos at $\sim$ 10 TeV in the particle spectrum along with the SM (solid lines). In the bottom panel of the figure, we show  $\tilde\omega$ as a function of $z$ (blue solid line) and we find a new small trough at around 10 TeV due to the new RH neutrinos. The other dips in $\tilde\omega$ are due to the SM particles. The expansion rate increases when  $\varphi'$ increases, however, if $\varphi'$ is too large, the speedup factor does not get reduced to 1 before the BBN.  As mentioned before, a sudden drop of the enhancement factor to the standard GR value occurs due to the troughs in $\tilde\omega$, which create an attractive effective potential when $\tilde\omega\neq1/3$. For large initial values of $\varphi'$, the scalar field overcomes this attractive potential and the enhancement factor never reduces to one.

In Figure \ref{fig:speedup1}, left plot, we consider an initial temperature of 100 TeV. The expansion rate can be enhanced by a factor of 100, or more, for temperatures between 100 GeV and 1 TeV. If, however, we increase the initial temperature to $10^5$ TeV or higher, the enhancement scale moves to a higher temperature and enhancements bigger than 100 can occur for temperatures between 1 TeV and  $10^4$ TeV. At higher temperature, the Hubble friction slows down the scalar field faster, and since the attractive effective potential kicks in early, due to the trough in $\tilde\omega$ caused by the RH neutrinos (at around 10 TeV), the enhancement factor drops to one at a higher temperature. 

Using dotted lines in Figure \ref{fig:speedup1}, we show the speedup factors due to the MSSM particles, where we keep the SUSY partners of the SM particles at around 1 TeV and, for illustration, three RH neutrinos (and their SUSY partners) at $\sim$ 10 TeV. 
In the panel below the figure, we show  $\tilde\omega$ (red dotted line) for MSSM + 3 RH neutrinos and we find a new deep trough in $\tilde\omega $ at around 1 TeV due to the SUSY particles (we put all of them together). The other dips are due to the SM particles. Like before, we find that the enhancement and its slope does not change much compared to the SM case but the speedup factor reduces to one for higher temperature.

It is also important to mention, no matter what particle spectrum we consider, that the initial value of the scalar field does not play a relevant roll in the shape and slope of the speedup factor, as long as this value is positive and order one.

We find that an enhancement of the expansion rate with an initial temperature at $10^{2-3}$ TeV is most effective in producing a successful leptogenesis, with the enhancement scale around TeV, caused by the SM particle spectrum. Now, the initial temperature is set by the inflation scale. In the case of the MSSM, it is shown that the thermal leptogenesis constraint from the type I seesaw is $T_R>10^6$ TeV~\cite{Buchmuller:2004nz,Giudice:2003jh}. This bound conflicts with the cosmological gravitino bound for unstable gravitinos. For a gravitino mass closer to a 100 GeV DM mass,  BBN implies stringent upper bound on reheating temperature $\leq O(1)\times 10^2$ TeV~\cite{Kawasaki:2008qe}. Based on our analysis, this low reheating scale is very helpful to have the leptogenesis scale to be around 1 TeV.



\begin{figure}[h!]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=0.48\textwidth]{Example10_5_GeV.png}&\includegraphics[width=0.48\textwidth]{Example10_8_GeV.png}\\
\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{Omega_paper.png}}
\end{tabular}}
\caption{Speedup factor $\xi(z)$ as a function of $z=1\,\rm{TeV}/T$ for SM+3 RH neutrinos (solid lines) and MSSM+3 RH neutrinos (dotted lines) for various values of $\varphi'$. The RH neutrinos mass value is $10\,$ TeV,  and the initial temperatures are $10^5$ GeV (left plot) and $10^8$ GeV (right plot). The bottom figure shows the equation of state parameter $\tilde\omega$ for the two cases.} 
%\end{center}
\label{fig:speedup1}
\end{figure}
