The effective gravitation theory at large scale with Lorentz violation

Abstract

The Doopler effect of CMB (cosmic microwave background) may indicates the Lorentz boost violation at the cosmological scale. It is revealed that there are some anomalies in the low-l multipole expansion of the CMB spectrum in the obeservation of Planck, which are not consistent with LCDM model, the standard model of cosmology. A famous one is the existence of the axis of evil. Combining the the anomalies exhibited by the low-l multipole expansion of the CMB observation, we propose that the Lorentz symmetry may start to be violated partly from the scale of galaxy and the boost invariance is violated eventrully at the cosmological scale. We take the symmetry groups of very special relativity as the examples to illustrate the Lorentz violation effect on the large scale gravity by constructing the corresponding gauge theories as the effective large scale gravitation theories. The estimated symmetry groups for very special relativity are Sim(2), Hom(2) and E(2) which are all constructed by adding generators to the T(2) group, a 2-parameter subgroup of Lorentz group. We first review the gauge principle approach to the construction of gravitation theory based on the equivalence principle which indicates the local Lorentz invariance. The choice of dynamics of the spacetime gauge theory based on the gauge principle and the Newtonian limit at the weak coupling limit reveals that the local Lorentz invariant theory is a theory with torsion in general case. However in the case of scalar matter source, the theory is exactly the theory of general relativity with Leci-Civita connection and zero torsion. We investigate the gauge approaches for all the very special relativity cases, Sim(2), Hom(2) and E(2) as well as the case of T(2). We find that these gravitation theories share a common feather, the non-trivial distribution of space-time torsion and the contortion, which contribute an effective enegy momentum dirtribution addition to the ordinary matter souce distribution even for the case of scalar matter source, while general relativity predicts the zero torsion and contortion and space-time connection as the Levi-Civita connection in this case. To restrict the theory with a local Lorentz subgroup, one need to postpone constrain conditions on the Lorentz gauge potentials which is similar to the gauge fixing condition in the ordinary internal gauge theory. In the Sim(2) case, there needs 8 constrain conditions. The equations of motion for the gauge fields finally reduce the independent Kontortion components from 24 to 8, which is consistent with the 8 constrain equations in addition to the equation of motion for the gravitation field. The numbers of independent Kontortion components is 12 for the cases of Hom(2) and E(2) and 16 for the case of T(2), while there are 12 and 16 constrain equations in addition to the gravitation field equations respectively. The non-zero distribution of contortion may be regarded as the dark matter effect in the static case at the galaxy or the galaxy cluster scale and both the dark matter and dark energy effect in the case of expanding Robertron-Walker like solution at the cosmoligical sacle. We investigate the self consistence of the gauge theories in the Sim(2) case by examining the closure of Maurer-Cartan equation within the Sim(2) algebra and the first and second Bianchi identities induced by Jacobi identity of the Sim(2) gauge theory.