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With this huge range of models available it’s important to place constraints on each one of those using observational data. The usual approach is to compare the $\Lambda \text{CDM}$ and different models predictions on the basis of a bayesian analysis.\\  On the largest scales observations of type Ia Supernovae provide information on the cosmic expansion around the redshift $z \lesssim 2$ [ ] and while those are the observables that first helped establish the $\Lambda \text{CDM}$ model there are some statistically significant deviations in the brightness of SnIa Supernovae Ia  data at z>1 $z > 1$  from the $\Lambda \text{CDM}$ prediction as stated by Kowalsky et al [ ].\\ Another large scale probe is the measurement of Cosmic Microwave Background Radiation. This is the radiation coming from the last scattering surface around z =~ 1090 $z \simeq 1090$  when photons ceased to be tightly coupled with baryons and started to stream freely across the universe. This radiation presents anisotropies that are directly related to the matter perturbations originated from inflation. The presence of dark energy Dark Energy  directly affects those anisotropies, for example by shifting the position of the acoustic peaks [68 book ] or by changing the entity of the Integrated sachs wolfe Sachs-Wolfe  effect due to a variation of the gravitational potential [118 book]. book].\\  But while LCDM $\Lambda \text{CDM}$  remains a really good fit for the above probes there are other observational challenges that might require some more drastic modifications in order to be resolved. resolved.\\  Some of those involve galactic scale phenomena such as the density distributions inside dark matter halos, observed to have a central density core[48 core [48  (2) ] whereas LCDM $\Lambda \text{CDM}$  predicts a steeper inner cusp [22 (2)], or the high density of cluster halos compared to the shallow profiles predicted by LCDM $\Lambda \text{CDM}$  model [46, 55 , 56 (2)] and their baryon fraction measured to be systematically lower than expected [ baldi pres] . Another problem involves the amount and properties of satellites of milky way sized halos [ ref (eg. the  Missing Satell, Satellite Probelm [ ] and the  Too Big To to  Fail ]. Problem []).\\  On larger scales instead it possible to measure the bulk flow corresponding to the CMB CMBR  dipole. This has been done in a number of large scale velocity surveys [27 (2) ] and has been observed that the galaxy velocities on scales larger than 100 mpc/h Mpc/h  are always bigger than expected [ Baldi pres ]. ].\\  Another LCDM prediction concerns the predicted amount of galaxies that should reside in the smaller underdense regions of the universe called voids. In fact LCDM $\Lambda \text{CDM}$  predicts the presence of many small dark matter halos that should host dwarf galaxies [19, 20, 21 (2) ] . While for the largest scale it’s possible to make predictions for every dark energy model simply by ignoring non linear effects whose contribution is negligible, for all the other observations presented above it is necessary, even for the simple LCDM $\Lambda \text{CDM}$  model, to resort to Numerical Simulations. In the last decade the advances in gravitational n-body simulations, computational fluid dynamics and accessibility of computational power allowed them to be really precise and readily available[ 1 baldi] . Simulations.\\  N-Body simulations follow the evolution and interaction of the matter species in the universe, starting from the tiny density fluctuations generated in the early universe by inflation, to the formation of large scale structures down the the galaxies in the present epoch.  This epoch and in the last decade the advances in gravitational n-body algorithms, computational fluid dynamics and accessibility of computational power allowed them to be really precise and readily available[ 1 baldi]. The wide dynamical range of modern Cosmological Simulation  has been made  possible thanks to the continued effort spent in improving the level of detail achieved by including the effects of baryonic physics [ 33 35 baldi] and a wide range of astrophysical processes as gas cooling, star formation, supernovae and agn feedback [36 - 41 baldi]. But only baldi].\\  Only  recently thoug  a bigger effort is being spent in adapting the algorithms of N-Body simulations to different dark energy scenarios, allowing cosmologist cosmologists  to study their effects on the evolution of density perturbations in the non-linear regime. regime.\\  Those were pioneered in 2003 by Klypin et al [penzo] with a dark-matter only simulation with an evolving equation of state parameter that found no major differences at redshift z=0 in the power spectrum and halo mass function but noted how the differences became more significant at higher redshift. redshift.\\  Subsequently multiple groups investigated the properties of dark matter structures in DE Dark Energy  cosmologies [note 1 penzo] looking at halo concentrations, velocity dispersions and abundance relations in quintessence and early dark energy models. Along the same lines Baldi [] investigated the evolution of coupled dark energy models that present an interesting signature due to new phenomena not present in the LCDM scenario.On the largest scales $\Lambda \text{CDM}$ scenario. The large scale implications of such models instead has been investigated by Jenkins [ ] that studied  the BAO peaks and the redshift space distortions have been investigate by Jenkins [ ]. in a variety of Quintessence and Early Dark Energy.\\  Recently new algorithms have been published that allow to simulate in great detail the evolution of Modified Gravity theories like f(R)-Gravity and DGP [ ] a feat that was not possible with the previously available codes. While most of those simulations concentrate on the dark matter properties additional effort has been spent recently to simulate the observable part of the universe by including baryonic physics and following it’s impact on the formation of the smallest structures structures;  this has been accomplished in Maio et al [125 baldi] and Penzo et al[ have print] that studied galaxy properties in hydrodynamical simulations of quintessence models, and by Fontanot et al.  [ two pap] that studied the statistical properties of galaxies applying a semi analytical model to Early dark energy Dark Energy  and f(R) gravity simulations. simulations.\\  Both those kind of simulations allow us to quantify the deviations from the LCDM behavior on the medium to large scale using the matter distribution and halo properties and then constraint the new models by comparing them against observations and also to observe a virtual realization of the small smaller  scale scenario regions  to understand how galactic properties are affected and if those modifications are able to tackle the challenges that persist in the LCDM $\Lambda \text{CDM}$  scenario.