Dynamically equivalent Λ CDM equations with underlying Bianchi Type geometry

Abstract

Solutions have been found for gravity coupled to electromagnetic field and a set of charged and uncharged perfect fluids for Bianchi Types VI(−1), VIII, IX. It has been assumed that the anisotropy is “frozen”, γμν=α(t)2mμν, where γμν and mμν are the spatial metric and some constant matrix respectively. This, according to previous works, results in the existence of a conformal Killing vector field proportional to the fluid velocity of the comoving matter, which guarantees the absence of parallax effects and the independence of the temperature (assuming black body spectrum) from the direction of observation. The electromagnetic field “absorbs” the “frozen” anisotropy and the remaining equations are dynamically equivalent with the equations of Λ CDM. There are solutions with flat, negative and positive effective spatial curvature corresponding to the three FLRW classes. Three equations of state for the charged perfect fluid were studied: non-relativistic w=0, relativistic w=1/3 and dark energy-like w=−1. For the first two cases, maximum values exist for the scale factor, in order for the weak energy conditions to be respected, which depend upon the geometric and charged fluid parameters. A minimum value for the scale factor exists (for the solutions to be valid) in all the cases and Types, indicating the absence of initial spacetime singularity (big bang). This minimum value depends upon the geometric and electromagnetic parameters. The number of essential constants in the final form of each metric is the minimum without loss of generality due to the use of the constant Automorphism's group. A known solution, with the anisotropy absorbed via one free scalar field is reproduced with our method and contains the minimum possible number of parameters.