Modified Regge calculus as an explanation of dark energy

Abstract

Using Regge calculus, we construct a Regge differential equation for the time evolution of the scale factor $a(t)$ in the Einstein-de Sitter cosmology model (EdS). We propose two modifications to the Regge calculus approach: 1) we allow the graphical links on spatial hypersurfaces to be large, as in direct particle interaction when the interacting particles reside in different galaxies, and 2) we assume luminosity distance $D_L$ is related to graphical proper distance $D_p$ by the equation $D_L = (1+z)\sqrt{\overrightarrow{D_p}\cdot \overrightarrow{D_p}}$, where the inner product can differ from its usual trivial form. The modified Regge calculus model (MORC), EdS and $Λ$CDM are compared using the data from the Union2 Compilation, i.e., distance moduli and redshifts for type Ia supernovae. We find that a best fit line through $\displaystyle \log{(\frac{D_L}{Gpc})}$ versus $\log{z}$ gives a correlation of 0.9955 and a sum of squares error (SSE) of 1.95. By comparison, the best fit $Λ$CDM gives SSE = 1.79 using $H_o$ = 69.2 km/s/Mpc, $Ω_{M}$ = 0.29 and $Ω_Λ$ = 0.71. The best fit EdS gives SSE = 2.68 using $H_o$ = 60.9 km/s/Mpc. The best fit MORC gives SSE = 1.77 and $H_o$ = 73.9 km/s/Mpc using $R = A^{-1}$ = 8.38 Gcy and $m = 1.71\times 10^{52}$ kg, where $R$ is the current graphical proper distance between nodes, $A^{-1}$ is the scaling factor from our non-trival inner product, and $m$ is the nodal mass. Thus, MORC improves EdS as well as $Λ$CDM in accounting for distance moduli and redshifts for type Ia supernovae without having to invoke accelerated expansion, i.e., there is no dark energy and the universe is always decelerating.