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\chapter{\uppercase {Introduction}}\label{Ch:Intro}

Recent cosmological data support the phenomenological $\Lambda$-Cold Dark Matter ($\Lambda CDM$) model of cosmology, which describes the energy density content of the universe in terms of a cosmological constant, $\Lambda$, cold dark matter and baryonic matter. The first two components together make up  95\% of the universe's energy density budget, while the other $5\%$ is made of baryonic standard model (SM) particles. This phenomenological model is complemented by the inflationary mechanism, which is the most successful framework to account for the origin of structure in the universe. After inflation, the universe reheated, providing the conditions for the hot big bang.

 The predicted abundances of the light elements resulting from big bang nucleosynthesis (BBN) are in very good agreement with available observational data \cite{Cyburt:2015mya}, which strongly supports the $\Lambda CDM$ model and our understanding of the universe's evolution back to the first few seconds after the big bang.

However, the physics describing the universe's evolution from the end of inflation (reheating) to just before BBN ($t\lesssim 10$ s) remains relatively unconstrained.  During this period, the universe could have gone through a \say{non-standard} period of expansion, and still be compatible with BBN predictions.  If such modification happened at temperatures around few GeV, during the dark matter (DM) decoupling, the DM freeze-out may have been modified with measurable consequences for the relic DM abundances. Also, if this modification occured at a higher temperature, say few TeV, it may have had an impact on the scale of the leptogenesis mechanism that produces the baryon asymmetry of the universe. Indeed, one of our the main goals is to study the effect of a \say{non-standard} cosmological expansion on the thermal DM freeze-out picture and on the scale of leptogenesis.

Departures from the standard cosmology between reheating and BBN will mainly be a consequence of a modified expansion rate ($\tilde H$), which can be due to a modification of General Relativity (GR). Such modifications are well motivated by attempts to embed the $\Lambda CDM$ and inflation models into a fundamental theory of gravity and particle physics, such as theories with extra-dimensions, supergravity and string theory.

String theory approaches, to  SM  particle physics and inflation model building, generically predict the presence of several new ingredients, and in particular scalar fields with clear geometrical interpretations. For example, type II string theory models of particle physics introduce new ingredients such as D-branes, where matter (and DM) is to be localised\footnote{For a review on D-brane models of particle physics see \cite{IU}.}. In D-brane constructions, longitudinal string fluctuations are identified with the matter fields such as the SM and/or DM particles, while transverse fluctuations correspond to scalar fields, which may play a role during the cosmological evolution\footnote{For example, a coupled dark energy-dark matter D-brane scenario was proposed in\cite{KWZ}.}. These scalar fields couple conformally and disformally to the matter living on the brane \cite{KWZ} and thus may change the cosmological expansion rate felt by matter.  The gravitational theory arising from the coupling belongs to a class of theories of gravity called Scalar-Tensor (ST) theories.

In this work, we focus on  ST  theories where the gravitational interaction is mediated by both the metric and only one scalar field. In these type of theories, the cosmological evolution deviates from the standard expansion of the Universe at early times, but an attractor mechanism ~\cite{Damour:1992kf,Damour:1993id} relaxes the theory towards GR prior to the onset of BBN.
ST theories are often formulated in one of two frames of reference, namely, the Jordan or Einstein frames. As it is shown in \cite{Bekenstein:1992pj}, the most general transformation, physically consistent, between the metrics of these frames is given by \footnote{More generally, the functions, $C$ and $D$ can depend on $X=\frac12 (\partial \phi)^2$ as well. We do  not consider this case in the present paper. }
%%%%%%%%%%%%%
\be\label{Dmetric}
\tilde g_{\mu\nu} = C(\phi) g_{\mu\nu} + D(\phi) \partial_\mu \phi \partial_\nu \phi\,.
\ee
%%%%%%%%%%%%%
 where $\tilde g_{\mu\nu}$ is metric in the Jordan frame, $C(\phi)$  is the conformal coupling, $ g_{\mu\nu}$ is the metric in the Einstein frame  and $D(\phi)$ is the so-called disformal coupling.

 The first term in \eqref{Dmetric}  is the well-known conformal transformation which characterizes the Brans-Dicke class of scalar-tensor theories explored in \cite{Catena,Catena2,Gelmini,RG,WI,MW}. The second term is the disformal contribution, which is generic in extensions of general relativity. In particular, it arises naturally in D-brane models, as discussed in \cite{KWZ},  in a model of coupled dark matter and dark energy. 

We will investigate two classes of ST theories in the two following chapters. In Chapter \ref{Ch:Conformal}, we consider the general action consider in \cite{Catena,MW} and take into account the disformal coupling, entering only through the metric. We revisit the conformal case, that is $D=0$, and explore the expansion rate modification providing new interesting results with new boundary conditions. We further discuss the general modifications to the expansion rate due to a disformal coupling and present an explicit non-trivial example for the case in which the conformal term in \eqref{Dmetric} is a monomial. In such a case, the functions $C$ and $D$ are in principle independent functions, as long as they satisfy the causality constraints:  $C(\phi)>0$  and $C(\phi) +2D(\phi)X>0$ ($X=\frac12 (\partial \phi)^2$) \cite{Bekenstein}.

In Chapter \ref{Ch:Disformal}, we study in detail the effects on the expansion rate of the disformal coupling in \eqref{Dmetric},  which arises in the case of matter localized on D-branes.  In this case, the conformal and disformal terms are closely related and dictated by the underlying theory, such as type IIB flux compactifications in string theory. The picture we have in mind goes as follows. After string theory inflation, reheating took place, giving rise to a thermal universe. At this stage standard model particles and dark matter was produced. The SM would arise from stacks of D-branes at singularities or D-branes intersecting at suitable angles \cite{IU}, while DM particles could arise from the same or a different stack of D-branes, which may be moving towards their final stable positions in the internal six-dimensional space before the onset of BBN. From the end of inflation to BBN, a nonstandard cosmological evolution can take place without spoiling the predictions of BBN, in particular, a change in the expansion rate felt by the matter particles due to the D-brane conformal and disformal couplings between the scalar field and matter fields. As we will see, due to the coupling the expansion rate will generically be enhanced\footnote{It is interesting to notice that a phenomenological model with a faster-than-usual expansion at early times,  driven by a new cosmological species, has recently been discussed in \cite{DEramo}. }. Let us stress that a  realistic string theory scenario would be more complicated and may include nonuniversal couplings to baryonic and dark matter. However, it is very interesting that scalar couplings present in string theory can give interesting modifications of the post-inflationary evolution after string inflation. 

In this work, we show for the first time that the enhanced expansion rate can also happen due to a purely disformal contribution or a combination of conformal and disformal terms. The former case - a disformal enhancement - is particularly interesting as it can be interpreted in terms of an ``unwarped" compactification, which is typical of a large-volume compactification of string theory,  which is needed for perturbative control. When we identify the scale arising from the disformal coupling with the tension of a moving D3-brane, where matter is localized, this scale is determined by the string scale and the string coupling.  Interestingly, the modification of the expansion rate can take place at different temperature scales, depending on the value of the string scale. 

The enhancement of the expansion rate, due to conformal and disformal couplings, impact the early universe cosmology. In this thesis,  we study DM phenomenology and a simple model of leptogenesis as well. In Chapter \ref{Ch:DM}, we focus on the general thermal DM freeze-out picture as an example and study how it is modified due to the faster-than-usual expansion of the universe. We further investigate the correlation of the annihilation cross section with the dark matter content. In Chapter \ref{Ch:Leptogenesis}, we study a simple model of leptogenesis type-I seesaw and show that the enhanced expansion rate lowers the scale of leptogenesis to few TeV. 
 
In an attractive scenario of the cosmic history, the universe is radiation dominated prior to BBN and dark matter is produced from the thermal bath, which was created at the end of inflation. In this thermal picture,  the observed relic density is satisfied for  DM species with weak-scale interaction rate $\langle\sigma v \rangle$, which is around $3.0\times10^{-26}{\rm cm^3s}^{-1}$ (or $\sigma \sim 1$ picobarn), corresponding to weak interactions. Despite such a small value, the Fermi-LAT and Planck experiments have been exploring upper bounds on $\langle\sigma v \rangle$ \cite{Fermi-LAT:2016uux,Ade:2015xua}. In the future, HAWC~\cite{Abeysekara:2014ffg} and CTA~\cite{Carr:2015hta} will probe the annihilation rate for a wide range of dark matter masses.  
  It is thus worth establishing whether an annihilation rate, larger or smaller than the standard thermal prediction, could still have a thermal origin due a modified cosmological evolution before BBN. 


Modifications to the  DM relic abundances in conformally coupled scalar-tensor theories (ST), such as generalizations of the Brans-Dicke theory, were first discussed in \cite{Catena}  by Catena et al. These authors showed a faster-than-usual expansion rate, $\tilde H$, which well before the onset of BBN, rapidly drops back to the standard GR expansion rate, $H_{GR}$. Due to this rapid relaxation of $\tilde H$ towards $H_{GR}$, they found a DM reannihilation effect:  after the initial particle decoupling, the dark matter species experienced a subsidiary period of annihilation as the expansion rate of the universe dropped below the interaction rate. Further  studies  on  conformally  coupled  ST  models  have  been  performed in recent  years in \cite{Catena2,Gelmini,RG,WI,MW} (see also\cite{LMN,Pallis,Salati,AM,IC,MW2,MW3}).

On the other hand, one of the most popular mechanisms to explain the cosmological baryon asymmetry is leptogenesis \cite{Fukugita:1986hr}, according to which an initial lepton asymmetry is generated and then partly converted to a baryon asymmetry by sphaleron processes \cite{Kuzmin:1985mm,Rubakov:1996vz} (for reviews on the leptogenesis mechanism see \cite{Davidson:2008bu,Fong:2013wr}).  In standard leptogenesis, based on type-I seesaw mechanism, this initial lepton asymmetry is produced in the decay of the lightest RH neutrino, $N_1$. In order to produce enough CP asymmetry to account for the observed baryon asymmetry, the mass of $N_1$  cannot be smaller than $\sim10^9$  GeV \cite{Davidson:2002qv}. Therefore, this implies that direct tests of standard type-I seesaw leptogenesis are out of experimental reach.

This constraint is quite specific of the type-I seesaw leptogenesis and is often lost in other models. The most direct way to go around this bound is in models where the neutrinos only couple through a neutrinophilic Higgs which obtains a vacuum expectation value (VEV) much smaller than the electroweak breaking  VEV, $v_\nu \ll 174$ GeV \cite{Haba:2011ra,Clarke:2015hta}. Other examples and their references are mention in Chapter \ref{Ch:Leptogenesis}. 

Most of the models that attempt to generate the baryon asymmetry from heavy particle decays are subject to an additional constraint.  This constraint originates 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 charge conjugation parity (CP) asymmetries in the decays of the heavy states. In this thesis, we show how this constraint can be avoided in the context of ST theories and how the scale of leptogenesis can be as low as few TeV.



The dissertation is organized as follows:

\bi
\item In Chapter \ref{Ch:Conformal}, we introduce the ST theory conformally and disformally coupled to matter. Then, we examine the formulation of this theory in the Einstein and Jordan frames, comment on their physical interpretation and derive the equations that describe the cosmological evolution of the Universe. Subsequently, we discuss the expansion rate modifications caused by the presence of the conformal and disformally coupled scalar field.

\item In Chapter \ref{Ch:Disformal}, we first briefly introduce the general D-brane-like setup. We then go directly to the cosmological equations and discuss how the expansion rate is modified in general. Then, we move on to the D-brane-like case, where the conformal and disformal functions are related. We start discussing the equations in the Jordan frame as well as the initial conditions and constraints that we use to numerically solve the full equations. We then discuss in detail the solutions for the unwarped case, that is,  a purely disformal effect (or $C=$const.). Next, we discuss the warped case, using for concreteness the same conformal function used in Chapter \ref{Ch:Conformal}. We also comment on the result of using other functions. 

\item In Chapter \ref{Ch:DM}, we briefly introduce the DM thermal freeze-out picture and the Boltzmann equation (BE). Then, we investigate in detail the impact of the modified expansion rates, described in both Chapters \ref{Ch:Conformal} and \ref{Ch:Disformal}, on the dark matter relic abundance and annihilation cross section.  

\item In Chapter \ref{Ch:Leptogenesis} we start by motivating leptogenesis in a faster-than-usual universe, discuss some constraints on the leptogenesis scale and then add few more comments on boosted expansion rates. Later, we present a simple benchmark model which will be used for the leptogenesis analysis and discuss the network of BE for leptogenesis in the modified cosmology. In our last two sections, we present our leptogenesis results and comment on the enhancement scales in the extensions of the SM, e.g., Minimal Supersymmetric Standard Model (MSSM).

\item In Chapter \ref{Ch:Conclusions},  we finally present our conclusions.

\ei
