N-body simulations, halo mass functions, and halo density profile in
  $f(T)$ gravity
  We perform N-body simulations for $f(T)$ gravity using the ME-Gadget code, in
order to investigate for the first time the structure formation process in
detail. Focusing on the power-law model, and considering the model-parameter to
be consistent within 1$\sigma$ with all other cosmological datasets (such as
SNIa, BAO, CMB, CC), we show that there are clear observational differences
between $\Lambda$CDM cosmology and $f(T)$ gravity, due to the modifications
brought about the latter in the Hubble function evolution and the effective
$Newton\prime s$ constant. We extract the matter density distribution, matter
power spectrum, counts-in-cells, halo mass function and excess surface density
(ESD) around low density positions (LDPs) at present time. Concerning the
matter power spectrum we find a difference from $\Lambda$CDM scenario, which is
attributed to about 2/3 to the different expansion and to about 1/3 to the
effective gravitational constant. Additionally, we find a difference in the
cells, which is significantly larger than the Poisson error, which may be
distinguishable with weak-lensing reconstructed mass maps. Moreover, we show
that there are different massive halos with mass $M>10^{14}M_{\odot}/h$, which
may be distinguishable with statistical measurements of cluster number
counting, and we find that the ESD around LDPs is mildly different. In
conclusion, high-lighting possible smoking guns, we show that large scale
structure can indeed lead us to distinguish General Relativity and $\Lambda$CDM
cosmology from $f(T)$ gravity.
