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\chapter{\uppercase {Dark matter relic abundances in Scalar-Tensor Theories}\myfnt{Sections of this chapter are 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:DM}


A  popular framework to understand the origin of DM is the thermal relic scenario. In this scenario, at very early times when the universe was at a very high
temperature, thermal equilibrium was obtained and the number density of DM particles $\chi$ was roughly equal to the number density of photons.
During equilibrium, the dark matter number density decayed exponentially as $n_\chi^{eq} \sim e^{-m_\chi/T}$ for a non-relativistic DM candidate, where $m_\chi$ is the mass of the DM
particle $\chi$. As the universe cooled down as it expanded, DM interactions became less frequent and eventually,  the DM interaction rate dropped below
the expansion rate ($\Gamma_\chi < H$). At this point, the density number froze-out, and the universe was left with a ``relic" of DM particles.  
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Therefore, the dependence of the number density at the time of freeze-out is crucial to determine the DM relic abundance. The longer the DM particles remain in equilibrium, the lower its density will be at freeze-out and vice-versa. In the standard $\Lambda CDM$ scenario, particle freeze-out happens during the radiation era and DM species with weak scale interaction cross-section freeze-out with an abundance that matches the current observed value. The weakness of
the interactions is reflected in the predicted thermally-averaged annihilation cross section, $\langle\sigma v \rangle$, which is around $3.0\times10^{-26}cm^3s^{-1}$. Despite such a small value, the Fermi-LAT and Planck experiments have been exploring upper bounds on $\langle\sigma v \rangle$ (see  \cite{Fermi-LAT:2016uux,Ade:2015xua}). From observations, it appears that the annihilation cross-section can be smaller than the thermal average value for lower dark matter masses ($\leq 100$ GeV), whereas an annihilation cross-section larger than the thermal average value can still be allowed for larger DM mass.
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If, from future  measurements,  $\langle\sigma v \rangle\neq3.0\times10^{-26}cm^3s^{-1}$ is established, what can we say about the origin of the dark matter? Can it still be the thermal dark matter or do we need non-thermal origin of dark matter? In the case of non-thermal origin, the DM can arise from the decay of a heavy particle, e.g., moduli, and can satisfy the DM content with any value of $\langle\sigma v \rangle$.   The primary motivation of this chapter is to show that the DM content can still have a thermal origin with larger or smaller $\langle\sigma v \rangle$ by utilizing non-standard cosmology.

\section{Impact of the modified expansion rate on relic abundances }\label{Impact}


After presenting the ST models in chapters \ref{Ch:Conformal} and \ref{Ch:Disformal},  and how the presence of the scalar field changes the cosmic evolution of the universe, we are ready to discuss the impact of the modified expansion rates on the relic abundance of dark matter species. 
For a dark matter species $\chi$ with mass $m_\chi$ and annihilation cross-section $\langle \sigma v \rangle$, where $v$ is the relative velocity, the dark matter number density $n_\chi$ evolves according to the Boltzmann equation 
\be\label{Boltzn}
\frac{d n_\chi}{dt} = -3 \tilde H n_\chi - \langle \sigma v \rangle \left( n_\chi^2  - (n_\chi^{eq})^2 \right)\,,
\ee
where, as we have discussed above, the relevant  expansion rate  is the Jordan frame one, which can give interesting effects due to the presence of the scalar field. Further $n_\chi^{eq}$ is the equilibrium number density. We can  rewrite this equation in terms of $x=m_\chi/\tilde T$
\be\label{Boltzy}
\frac{d Y}{dx} = - \frac{\tilde s \langle \sigma v \rangle }{x \tilde H}  \left( Y^2  - Y_{eq}^2 \right) \,,
\ee
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where, $Y = \frac{n_\chi}{\tilde s}$, $\tilde s= \frac{2\pi}{45} g_s(\tilde T) \tilde T^3$.
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\section{Dark Matter relics in the pure conformal scenario}
In this section, we discuss the modifications to the DM relic abundances due a faster-than-usual expansion rate, which is caused by the presence of a scalar field coupled conformal and disformally to matter. In particular, we study the effect of the expansion rate, $\tilde H_{Conformal}$, discussed in chapter \ref{Ch:Conformal}, which is shown in Figure \ref{Hdisformal}. Also, at the end of this section, we comment the effect of  $\tilde H_{Disformal}$ presented in the same figure.
 
Numerical solutions to \eqref{Boltzy}, with the expansion rate $\tilde H_{Conformal}$ shown in Figure \ref{Hdisformal},  were found for dark matter particles
with masses ranging from 5 GeV to 1000 GeV. For instance, we show solutions in Figures \ref{Y130Fig} and \ref{Y1000Fig} 
for two different masses. As we can see from \eqref{Boltzy}, the annihilation cross-section influences the evolution of the abundance $Y$. The current
value of $Y$ determines the present dark matter content of the universe. This can be seen clearly by recalling the current value of the energy density parameter
$\Omega_0=\frac{\rho_0}{\rho_{c,0}}=\frac{m\,Y_0\,s_0}{\rho_{c,0}}$, where $\rho_{c,0}$ and $s_0$ are the well-known current values of the critical energy
density and the entropy density of the universe, respectively. So, for every single mass, the thermally-averaged annihilation
cross section, $\langle\sigma v \rangle$, was chosen such as the current DM content of the universe is 27 \%, so $\Omega_0=0.27$. 

In Figure \ref{AnniFig} we show the annihilation cross-section, 
$\langle\sigma v \rangle_{Conformal}$, found for all masses and compare it to the annihilation cross sections for the standard cosmology model, 
$\langle\sigma v \rangle_{Standard}$. As it is shown, for large masses, $\langle\sigma v \rangle_{Conformal}$ is larger than $\langle\sigma v \rangle_{Standard}$,
up to a factor of four. As the mass decreases  $\langle\sigma v \rangle_{Conformal}$ decreases up to the point where it is smaller than $\langle\sigma v \rangle_{Standard}$. 
Then, for masses smaller than 100 GeV,  $\langle\sigma v \rangle_{Conformal} \approx \langle\sigma v\rangle_{Standard}$. 
Thus, we have found that the annihilation cross-sections can be larger or smaller than the thermal average cross-section.
Just to give an example of larger and smaller cross-section, in table \ref{examplecs} we compare the numerical values of $\langle\sigma v \rangle_{Conformal}$ and $\langle\sigma v \rangle_{Standard}$ for two dark matter masses, 1000 GeV and 130 GeV.

\begin{table}
\begin{center}
 \begin{tabular}{|c|c|c|}
 \hline
  Mass (GeV) & $\langle\sigma v \rangle_{Conformal} (\times 10^{-26}\rm{cm^3/s})$&$\langle\sigma v \rangle_{Standard} (\times 10^{-26}\rm{cm^3/s})$\\ \hline
   1000 &9.57 &2.23\\ \hline
    130 &1.68 &2.04\\ \hline
 \end{tabular}
\end{center}
\caption{Annihilation cross-section for masses of 1000 GeV and 130 GeV in the conformal and standard scenarios. }
\label{examplecs}
\end{table}

Figures \ref{Y130Fig} and \ref{Y1000Fig} show the evolution of the abundance $\tilde Y(x)$
for DM particles with masses 130 GeV and 1000 GeV, respectively. These figures also include the abundance $Y_{GR}(x)$ calculated in the standard 
cosmology model and the equilibrium abundance $Y_{Eq}(x)$.

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\begin{figure}[h!]
\centerline{
\includegraphics[width=.85\textwidth]{Y130plotV2.pdf}}
\caption{Evolution of the abundance as temperature changes for a DM particle of mass 130 GeV.}
\label{Y130Fig}
\end{figure}
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The temperature evolution of the abundance for a 130 GeV mass is not noticeably affected by the presence of the scalar field $\phi$. In this case, $\tilde Y$
and $Y_{GR}$ are almost indistinguishable from one another.
On the other hand, the scalar field $\phi$ has a prominent effect on the temperature evolution of the abundance for a 1000 GeV DM particle. First of all,
the freeze-out happens earlier than expected due to the enhancement of the expansion rate, $\tilde H$. Then, an unusual effect appears. As the temperature decreases, $\tilde H$ becomes smaller than the interaction rate\footnote{The interaction rate is defined as $\tilde\Gamma\equiv\langle\sigma v \rangle_{Conformal}\,\tilde s\,\tilde Y$.} 
 $\tilde \Gamma$ and a short period of annihilation starts again called ``re-annihilation''. The re-annihilation process reduces the abundance of dark matter until a second and final freeze-out happens.  After this final freeze-out, the abundance remains constant.

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\begin{figure}[h!]
\centerline{
\includegraphics[width=.65\textwidth]{Y1000plotV2.pdf}}
\caption{Abundance for a mass of 1000 GeV.}
\label{Y1000Fig}
\end{figure}
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\begin{figure}[h!]
\centerline{
\includegraphics[width=.65\textwidth]{Annihilation.pdf}}
\caption{Annihilation cross section as function of mass. The presence of the scalar field enhances the $\langle\sigma v \rangle$ for large masses,
and diminishes $\langle\sigma v \rangle$ for masses around 130 GeV, while small mass the effect is almost negligible. }
\label{AnniFig}
\end{figure}
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\begin{figure}
\centerline{
\includegraphics[width=.85\textwidth]{plotsHgsV2.pdf}}
\caption{Expansion rate (as in Figure \ref{HubblesFig}) and interaction rate as function of temperature. The interaction rate, $\tilde \Gamma$, is given by $\langle\sigma v \rangle_{Conformal}\,\tilde s\,\tilde Y$. We use $\tilde Y$ from Figures \ref{Y130Fig} and \ref{Y1000Fig} and the values of $\langle\sigma v \rangle_{Conformal}$ presented previously for 130 GeV and 1000 GeV masses.  }
\label{plotsHgs} 
\end{figure}
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The re-annihilation phase can be described better by discussing the relation between the expansion rate $\tilde H$  and the interaction rate $\tilde\Gamma$.
The first freeze-out happens when $\tilde\Gamma$ becomes smaller than $\tilde H$  which can be seen in Figure \ref{plotsHgs} to happen around a temperature of 50 GeV
for a 1000 GeV particle. Then, near to 7 GeV, $\tilde H$  drops below $\tilde\Gamma$, and so the re-annihilation process starts and goes on until the second freeze-out occurs. Around 2 GeV $\tilde H$ becomes much larger than $\tilde\Gamma$ and so the abundance becomes almost constant. 
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Our analysis shows that, as found in \cite{Catena}, re-annihilation occurs for this particular choice of the conformal factor. However, we found that when fully integrating the master equation, the re-annihilation occurs only for very large masses of the dark matter particles (in \cite{Catena} it was found for $m=50$GeV). 
On the other hand, in \cite{MW}, no re-annihilation was found\footnote{Although \cite{MW} used a different conformal factor to \cite{Catena},  we expect the re-annihilation effect to be present also in that case.}, which was probably due to the initial conditions used and the values of the DM masses explored. 

 We have seen that the enhancement of $\tilde H$ allows bigger values of 
$\langle \sigma v \rangle$ for particles with masses within a certain range, and also, the notch of $\tilde H$ implies smaller $\langle \sigma v \rangle$ for particles with masses within a small interval  (see Figure \ref{AnniFig}). Thus, the location and shape of the notch determine for which masses the annihilation cross section is smaller.
In the disformal scenario (see $\tilde H_{Disformal}$ in Figure \ref{Hdisformal}), the notch has been moved to higher temperatures, which allows particles with higher masses to have smaller and larger annihilation cross sections for the observed DM content.

\section{Dark Matter relics in the disformal D-brane coupling scenario}

Since we computed the modified expansion rate in the disformal D-brane coupling scenario in Chapter \ref{Ch:Disformal}, we can now move on to discuss its impact and implications on the dark matter relic abundance and cross section. In this section, we focus on the case $C=$ const. since the $C\ne $ const. case gives similar results to those studied in the previous section, as we discuss below. 

As a  concrete example, we solve  \eqref{Boltzy} numerically, for the expansion rate corresponding to $M= 12$ GeV shown in the top left plot of Figure \ref{HubbleC1} and for dark matter particles with masses ranging from 10 GeV to 5000 GeV. Other choices of $M$ would give similar results. In Figure \ref{Y100} we show the solution for a DM particle of mass $m_\chi=100$ GeV. In this plot, we also include the abundance $Y_{GR}(x)$ calculated in the standard cosmology model and the abundance when dark matter particles are in thermal equilibrium, $Y_{Eq}(x)$. 

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\begin{figure}[h]
\centerline{
\includegraphics[width=0.80\textwidth]{DBrane_Y100.png}}
\caption{Abundance $\tilde Y$ for a dark matter particle  with a mass of 100 GeV. }
\label{Y100}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the plot, we can see how the modification of the expansion rate gives rise to an earlier than standard freeze-out (see Figure \ref {Y100} around $x=20$). This is due to the enhancement of the expansion rate $\tilde H$. As the temperature decreases ($x$ increases), $\tilde H$ becomes comparable to the interaction rate\footnote{The interaction rate is defined as $\tilde\Gamma\equiv\langle\sigma v \rangle\,\tilde s\,\tilde Y$.}  $\tilde \Gamma$ and for a small period, between $x=20$ and $x=1000$,  the abundance decreases slowly until it becomes constant.
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It is interesting to notice that a similar behavior was found in \cite{DEramo}, where an extra scalar species drives a faster than usual expansion rate, giving rise to a similar behavior in the relic abundance. 
%%%
The comparison between $\tilde H$ (brown) and $\tilde \Gamma$ (purple) can be seen  in Figure \ref{HGamma}. 
Between around 5 GeV ($x=20$) and 0.1 GeV ($x=1000$), $\tilde H$ and $\tilde \Gamma$ are close to each other as temperature decreases. 

In  Figure \ref{HGamma}, we also show the interaction rate for two other DM particle masses, 600 GeV (green) and 2500 GeV (red). Notice that for the three masses shown, once the interaction rate becomes smaller than the expansion rate $\tilde H$ (brown), it always stays smaller than it. Therefore, there is no reannihilation effect. However, reannihilation can occur for the $C\ne $ const. case, where after the first freeze-out $\tilde \Gamma$ can overcome $\tilde H$ due to $\xi <1$,  and later become smaller again. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\centerline{
\includegraphics[width=0.80\textwidth]{DBrane_HGamma.png}}
\caption{Expansion rate corresponding to $M=12$ GeV shown in the top left plot of Figure \ref{HubbleC1}, and interaction rates of 100 GeV (purple), 600 GeV (green) and 2500 GeV (red) GeV DM particle masses as a function of temperature. The interaction rate $\tilde \Gamma$  is given by $ \langle \sigma v \rangle\tilde s \tilde Y$.  }
\label{HGamma}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Let us now turn our attention to the dark matter annihilation cross sections that we have used when solving the Boltzmann equation \eqref{Boltzy}. To determine $\langle\sigma v \rangle$ numerically, we required that the observed dark matter density today be $\Omega_0=0.27$. The present dark matter content of the universe is determined by the current value of the relic abundance. This  can be obtained from the current value of the energy density parameter
$\Omega_0=\frac{\rho_0}{\rho_{c,0}}=\frac{m\,Y_0\,s_0}{\rho_{c,0}}$. Here $\rho_{c,0}$ and $s_0$ are the well-known current values of the critical energy density and the entropy density of the universe, respectively. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\centerline{
\includegraphics[width=0.65\textwidth]{DBrane_Xsections.png}}
\caption{$ \langle  \sigma  v \rangle$ as function of dark matter particle mass. $\langle\sigma v \rangle_{GR}$ predicted by the standard cosmology model correspond to the black line, while the colored lines correspond to  $\langle\sigma v \rangle$ predicted by using the expansion rates (shown in Figure \ref{HubbleC1}), representing mass scales of $M= 12 $  (brown), 34  (red), 106  (green) and 333 GeV (blue).   }
\label{Xsections}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The resulting annihilation cross sections,  found in this way,  are shown in Figure \ref{Xsections}  for dark matter masses between 10 GeV and 5000 GeV. In this figure, we also show the effect that different values of $M$ have on $\langle\sigma v \rangle$. The values of $M$ chosen correspond to the expansion rates $\tilde H$ shown in Figure \ref{HubbleC1}. We compare this to the annihilation cross sections  $\langle\sigma v \rangle_{GR}$ predicted by the standard cosmology model (black line), which is around $2.1 \times 10^{-26} cm^3/s$. 
%

The behavior of the cross section $\langle\sigma v \rangle$ in Figure \ref{Xsections} shows an enhancement with respect to the standard case, with a maximum that moves towards larger DM masses as the scale $M$ increases.  Therefore, the smaller the scale $M$ the larger the annihilation cross section $\langle\sigma v \rangle$ for the DM masses shown in the figure. 

As was discussed, a modification to the expansion rate of the universe prior to BBN 
has tremendous consequences on the abundance, $\tilde Y$, of DM particles. This modification implies that the thermally-averaged annihilation cross section, $\langle \sigma v \rangle$,  differs significantly from the one predicted by the standard cosmological model, which is approximately $3.0 \times 10^{-26}\rm{cm^3/s}$.

