Applied Mathematics > Vol.4 No.7, July 2013

Andrew Walcott Beckwith

Physics Department, Chongqing University, Chongqing, China.

We once again reference Theorem6.1.2of the book by Ellis, Maartens, and MacCallum in order to argue that if there is a non zero initial scale factor, that there is a partial breakdown of the Fundamental Singularity theorem which is due to the Raychaudhuri equation. Afterwards, we review a construction of what could happen if we put in what Ellis, Maartens, and MacCallum call the measured effective cosmological constant and substitute Λ→Λ_effective in the Friedman equation. i.e. there are two ways to look at the problem, i.e. after Λ→Λ_effective, set Λ_Vac as equal to zero, and have the left over as scaled to background cosmological temperature, as was postulated by Park (2002) or else have Λ_Vac as proportional to Λ_Vac～10³⁸GeV² which then would imply using what we call a 5-dimensional contribution to Λ as proportional to Λ≈Λ_5D～-const/Tβ. We find that both these models do not work for generating an initial singularity. Λ removal as a non zero cosmological constant is most easily dealt with by a Bianchi I universe version of the generalized Friedman equation. The Bianchi I universe case almost allows for use of Theorem 6.1.2. But this Bianchi 1 Universe model almost in fidelity with Theorem 6.1.2 requires a constant non zero shear for initial fluid flow at the start of inflation which we think is highly unlikely.

Raychaudhuri Equation; Fundamental Singularity Theorem; Bianchi I Universe; Effective Cosmological Parameter

A. Beckwith, "How an Effective “Cosmological Constant” May Affect a Minimum Scale Factor, to Avoid a Cosmological Singularity (Breakdown of the First Singularity Theorem)," Applied Mathematics, Vol. 4 No. 7, 2013, pp. 1038-1042. doi: 10.4236/am.2013.47141.