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%%                           SECTION IV
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\chapter{\uppercase{Conclusions} }\label{Ch:Conclusions}

Since it is difficult to probe the universe between inflation and the onset of BBN, the evolution of the universe is mostly unconstrained during this period. Origins of DM, baryon abundances, etc, crucially depend on the evolution history around that time. During this epoch, the expansion rate can be different in ST theories compared to the standard cosmology even though the universe is still radiation dominated. 

Scalar-tensor theories where the gravitational interaction is mediated by both the metric and scalar fields 
 arise commonly in modifications of GR.
The prototype example is the Brans-Dicke theory where the metric and a scalar field are related  via the conformal coupling as $\tilde g_{\mu\nu} =C(\phi) g_{\mu\nu}$.  However, the most general physically consistent relation between the two metrics  includes a disformal (or derivative) coupling \cite{Bekenstein}: $\tilde g_{\mu\nu} =C(\phi) g_{\mu\nu} + D(\phi) \partial_\mu\phi\partial_\nu\phi$.  

Both couplings $C$ and $D$ can give rise to a different expansion rate from the standard cosmological model in the early universe, and still be in agreement with current constraints from BBN and gravity. In particular, BBN imposes a strong constraint on the speed-up parameter, $\xi$, which needs to be very close to one before the onset of BBN.

In Chapter \ref{Ch:Conformal}, we explored an ST theory coupled both conformally and disformally to matter. For the pure conformal case, we investigated a conformal factor of the type $C=(1+b\,e^{-\beta\,\varphi})^2$. We found the modified expansion rate, $\tilde H$, during the radiation dominated. When comparing the expansion rate, $\tilde H$, to the standard expansion rate, $H_{GR}$, we found that the speed-up factor, $\xi=\frac{\tilde H}{H_{GR}}$, increases up to 200 and then become of order 1 prior to BBN (see Figure ~\ref{HubblesFig}). 

We also started to investigate the effects on the early evolution of a disformal factor in the metric \eqref{gtilde}. We noticed that in order to have a  consistent solution, i.e. a real positive $\tilde H$, the conformal and disformal factors need to satisfy a very specific relation, \eqref{ConditionD}.  We studied the effect of a disformal factor by turning on a small disformal contribution to the conformal case, given by $D(\varphi)=-4.9\times 10^{-14}\varphi^2$. We found, in Section \ref{Sec:CoD}, that when both, the conformal and disformal function, are turned on, $\tilde H$  has a very similar profile as for the pure conformal case, with an enhancement and a notch compared to the standard expansion rate. However, the position of the notch changes. The disformal factor is moving the notch to higher temperatures.

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In Chapter \ref{Ch:Disformal} we introduce the disformal D-brane coupling scenario. In this case, the functions $C$ and $D$ are closely related and are dictated by another theory, for example, type IIB string theory. Moreover, the scalar field has a geometrical origin in terms of the transverse fluctuations of the D-brane, while matter lives on the brane and it comes from the longitudinal fluctuations. In this chapter, we investigated the modification to the expansion rate due to the disformally coupled scalar, where $D=1/M^4 C$ (Figures ~\ref{HubbleC1},  \ref{HubbleScaleM} and \ref{HubbleCnon1}).  

We numerically solved the coupled equations for $H$ and $\varphi$ ( see \eqref{Hprime13} and \eqref{phiHeq13}) and used this to find the modified expansion rate. We noted that contrary to the purely conformal case, in the presence of the disformal term it is not possible to eliminate $H$ from the system to solve a single master equation, as in Section \ref{Sec:PureC}. So we needed to carefully take into account both equations as well as the initial conditions for $H$. This introduced a cubic equation for $H$ in terms of the other parameters $(\varphi, \tilde\rho)$ and  a lower bound for the scale $M$, given the initial conditions for $(\varphi_i, \tilde\rho_i)$ (see Section \ref{ICLowM}).

In section \ref{Ceq1} we presented for the first time the purely disformal case corresponding to $C=$ const. where the modification to the expansion rate is fully driven by the derivative coupling through $\gamma$ (see Eq.~\eqref{xiConst3}). For the disformal D-brane coupling scenario, the expansion rate is always faster-than-usual (Figure \ref{HubbleC1}). We found that the larger the value of the new scale $M$ the earlier the enhancement in the expansion rate appears (Figure \ref{HubbleScaleM}). Therefore, depending on the value of $M$, the modified expansion rate can occur at a different temperature in the early universe. We also found that the expansion rate can be up to $\sim 1000$ times bigger than the standard GR case.  

 Again, we studied the effect of $M$ on the profiles of the expansion rates.  We considered in detail for concreteness only the conformal function used in Chapter \ref{Ch:Conformal}. For this function, the numerical analysis is relatively simple; however, there is in principle no obstruction to find similar effects for other functions. We looked at the function  $C= (b x^2 +c)^2$ (for  $b=4,8,15$, $c=1$), which can be a toy model for a smooth warp factor in a string theory setup. For this example, we found a relatively small enhancement with $\xi\sim 4$.
 We expect that a  wider search of parameters and conformal functions will give rise to a larger enhancement as well as a decrease in the modified expansion rate. 




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As we showed in Chapter \ref{Ch:DM}, the enhanced expansion rate has important consequences on the evolution of the abundance of dark matter particles. So, we also investigated the effect on the abundance of dark matter particles. In the pure conformal ST scenario, we observed that for dark matter particles of large mass, the particles undergo a second annihilation process and then freeze-out once and for all in (see Figure \ref{Y1000Fig}). Moreover, we found that for large masses the annihilation cross-section has to be up to four times larger than that of standard cosmology models in order to satisfy the dark matter content of the universe of 27 \%. On the other hand, for small masses, this re-annihilation process is not present, but we found that for masses around 130 GeV, the annihilation cross-section can be smaller than the annihilation cross-section for the standard cosmological model (see Figure \ref{AnniFig}).

 For the disformal D-brane coupling scenario, the modified expansion rate is always enhanced with respect to the standard one (Figure \ref{HubbleC1}), which implies an anticipated freeze-out and an enhancement of the cross section $\langle \sigma v\rangle$ (Figure \ref{Xsections}). For the  $C\ne$ const case (section \ref{Cne1}) we saw that it is possible to have an enhancement as well as a reduction of the expansion rate with respect to the standard case, that is $\xi>1$ and $\xi<1$ (Fig.~\ref{HubbleCnon1}). This diminution gives the possibility of a reannihilation process, as in the conformal and disformal cases studied in Chapter \ref{Ch:Conformal}. Thus the effect on the relic abundance and annihilation rate is analogous.  

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Finally, in Chapter \ref{Ch:Leptogenesis} we study leptogenesis in a faster-than-usual cosmic expansion. The scale of leptogenesis in the case of a typical type I seesaw model is very high and is out of reach for the ongoing experimental facilities. However, many models with a much lower leptogenesis scale exist where the RH neutrino masses arise due to new physics around multi-TeV scale. In these models, it is found that (if no resonant enhancement of CP asymmetries is assumed), there exists a lower bound on the scale of leptogenesis which is $\sim$ 10 TeV under the assumption of an initial thermal abundance for RH neutrinos along with no washout. The lower bound increases in the case of zero initial abundance. This conclusion changes in  ST theories with an enhanced expansion rate which helps the leptogenesis models to be probed in the ongoing experiments.


In the case of an enhanced expansion, the requirement of a larger washout scattering rate demands the scale of leptogenesis to be smaller since the scattering rate is inversely proportional to $M^n$ where the exact value of $n$ ($\geq 0$) depends on the details of the initial and final state particle properties. We used a toy model of leptogenesis to manifest the lowering of the leptogenesis scale due to an enhanced expansion rate.  In this model, the RH neutrinos do not couple to the SM Higgs, instead, they couple to a new Higgs. We found that the scale of leptogenesis can be lowered down to $\sim$ TeV for both zero and thermal initial abundances for the RH neutrinos for a wide range of model parameter space which allows these models to be probed at the ongoing experimental facilities. In some parameter space of the model, we showed that an enhancement of the expansion rate can lower the leptogenesis scale down to $\sim 400$ GeV. The existence of an enhanced expansion rate between 100 GeV to a few TeV due to the SM particle spectrum (plus the RH neutrinos) in the case of a conformal modification of the metric is crucial to lowering the scale of leptogenesis. If an enhancement happens at a higher scale, the scale of leptogenesis is not lowered and an enhancement at a smaller scale is also not helpful in lowering the leptogenesis scale with the correct amount of asymmetry since the EW sphaleron freezeout occurs at around 130 GeV. All of our findings for this model should apply to any leptogenesis model.

