deletions | additions       diff --git a/section_Simulations_For_the_analysis__.tex b/section_Simulations_For_the_analysis__.tex  index 1600606..3470ce2 100644  --- a/section_Simulations_For_the_analysis__.tex  +++ b/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 suite consists of 4 simulations that span the natural interval for the $\nu$ parameter and one control simulation that uses the standard $\Lambda CDM$ cosmology.  The  present day cosmological parameters used in this are the same between the  simulations used and  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  
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\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. code,  this procedure was repeated for 4 values of the $\nu$ parameter and also a mock table was created to allow the use of the same code for the $\Lambda CDM$ simulation.  The only remaining part of the simulations that needed careful consideration was the generation of the initial conditions. These where obtained by perturbing a glass particle distribution according to the 2LPT prescription described by \cite{Crocce_2006} along with a freely available implementation. This implementation needed some minor tweaks to take into account the modified evolution of the cosmological quantities, but used only two values for each of them, one at the starting redshift, chosen to be $z=49$ and one at the present time. These values where calculated using the lookup tables computed before.  The last inputs needed to generate the initial conditions are the shape and amplitude of the linear power spectrum, according to the discussion of the previous paragraph the matter power spectrum for the "running FLRW" model retains the same shape as that of the $\Lambda CDM$ model and so we can use the powerful CAMB code \cite{Lewis:2002ah} to obtain an accurate power spectrum shape given the current cosmological parameters.  The amplitude of the power spectrum is then set according to the $\sigma_8$ value given in [TABLE 1], scaled back to the starting redshift using the standard $\Lambda CDM$ formula for the growth factor and then scaled forward to the redshift $z=0$ using the numerical solution of equation \ref{scalefactorsecond}.  This choice is equivalent to normalize the power spectrum for every realization to the same $\sigma_8$ at the starting redshift even though careful tests show that no appreciable difference is carried over in the non-linear regime if we choose to normalize to the same $\sigma_8$ at redshift $z=0$.