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Antonio Bibiano added section_Simulations_For_the_analysis__.tex
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\section{Simulations}
For the analysis of structure formation in this model we performed a suite of dark matter only N-Body simulations.
These simulations follow the evolution of $1024^3$ cold dark matter particles each of mass $10^10 M_{sun}/h$ in a periodic cosmological box of 1024 Mpc/h on a side. The present day cosmological parameters used in this simulations reflect the latest PLANCK\cite{Ade:2015xua} determination for $\Omega_m$, $\Omega_\Lambda$, $\Omega_b$, $h$, $n$ and $\sigma_8$, these values are reported in [TABLE].
The simulations where carried out using a modified version of the parallel TreePM N-Body code Gadget \cite{Springel_2005}. The modified version of the code keeps the basic algorithms to follow the evolution of the dark matter particles but interpolates the cosmological quantities $H(z)$, $G(z)$, $\Omega_m(z)$, $\Omega_\Lambda (z)$ using look-up tables to avoid a performance hit for every time step.
The look up tables it is necessary to rewrite the equation \ref{syslambda} as
\begin{equation}
\Omega_\Lambda(z) = \frac{\Omega_\Lambda^0 + \nu (\Omega_m g(z) - 1)}{1- \nu g(z)}
\end{equation}
and combine it with \ref{syshubble} to obtain
\begin{equation}
\Omega_m + \Omega_\Lambda(z) = \frac{\Omega_m(z) + \Omega_\Lambda^0 - \nu}{1- \nu g(z)}.
\end{equation}
Then we differentiate the first one to obtain $d\Omega_\lambda(z)$ and substitute both in \ref{sysbianchi} to obtain
\begin{equation}
(\Omega_m(z) + \Omega_\Lambda^0 - \nu) dg + \nu (1 - \nu g) g^2 d\Omega_m(z) = 0
\end{equation}
that can be integrated by quadrature to obtain an g(z) as an implicit function of redshift:
\begin{equation}
\frac{1}{g(z)} - 1 + \nu \text{ ln } \left [ \frac{1}{g(z)} - \nu \right] = \nu \text{ ln } [\Omega_m(z) + \Omega_\Lambda^0 - \nu].
\end{equation}
It is now straightforward to solve this numerically for $g(z)$ and use the previous equations along with \ref{syshubble} to obtain the cosmological quantities needed by the code.
The only remaining part of the simulations that needed careful consideration was the generation of the initial conditions.
