A family of solutions representing an accelerating Universe

Abstract

U.S. Rahubedda, Department of Physics, Nalin de Silva

Since the observations of Perlmutter and others, it is now an established fact that the universe expands with an acceleration in the present epoch. In order to explain this phenomenon Hemantha and de Silva modified the Einstein’s field equations T G g ,0 T 0 : constant G T g ; = ; − Λ ; = ⇒ ; = Λ − = + Λ υ µυ υ µυ υ µυ υ µυ µυ µυ µυ κ κ with the following variations. g T 0 G 0 ( g T ) 0 Λ :-variable ; ; ; ; Λ + = = ⇒ Λ + = υ µυ µυ υ υ µυ µυ υ µυ κ κ Essentially they made Λ, the “Cosmological Constant”, a variable and replaced the energy conservation by the conservation of the energy of the Λ field, matter and radiation. According to the modified equations the energy of the Λ field and that of matter and radiation are together conserved instead of the energy of the matter and radiation only. This enables matter and radiation to be created from the Λ field during the expansion of the universe during certain epochs. Conversely in certain other epochs the matter and radiation contributes to the Λ field.

The modification leads to the following equations with respect to the Robertson - Walker metric. 2 2 2 2 2 2 2 2 2 3 3 2 R c R R k c R R R R R kc p & && & = + Λ + + + + = Λ κρ κ 67 3 0 2 2 = Λ  + +      + R c R c p κ ρ ρ & & & where dot ( & ) denotes differentiation with respect to cosmic time t. The modified equations give rise to solutions representing acceleration during certain epochs. We have found that with k=1, p =0, the modified equations are satisfied by the solution R = a + b1 cos ωt + b2 cos 2 ωt. It is believed by Perlmutter and others that the present acceleration epoch commenced at a redshift of 1.6. Taking that and minimum of R=0, as boundary conditions, we find that 2 1 3( 10) 4 3 b = + b , a = 2 2 2 1 8 b b b + . These conditions give a family of solutions representing an accelerating universe, with b1 as the parameter. As an example, we give below a schematic representation of the solution with b1=5x1030 .