How an Effective “ Cosmological Constant ” May Affect a Minimum Scale Factor , to Avoid a Cosmological Singularity ( Breakdown of the First Singularity Theorem )

We once again reference Theorem 6.1.2 of 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 Effective  in the Friedman equation. i.e. there are two ways to look at the problem, i.e. after Effective  , set 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  Vac  as proportional to which then would imply using what we call a 5-dimensional contribution to 38 2 ~ 10 GeV Vac   as proportional to 5 ~ cons D t T      . We find that both these models do not work for generating an initial singularity. 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. 


Introduction
The present document is to determine what may contribute to a nonzero initial radius, i.e. not just an initial nonzero energy value, as Kauffman's paper [1] would imply, and how different models of contributing vacuum energy, initially may affect divergence from the first singularity theorem.The choices of what can be used for an effective cosmological constant will affect if we have a four dimensional universe in terms of effective contributions to vacuum energy, or if we have a five dimensional universe.The second choice will probably necessitate a tie in with Kaluza Klein geometries, leaving open possible string theory cosmology.In order to be self contained, this paper will give partial re productions of Beckwith's [2], but the 2nd half of this document will be completely different, i.e. when considering an effective cosmological constant.With four different cases, the last case is unphysical, even if it has, via rescaling zero effective cosmological constant, due to an effective "fluid mass" eff M .

Looking at the First Singularity Theorem and How It Could Fail
As was brought up by Beckwith [2], if there is a non zero initial energy for the universe, a supposition which is counter to ADM theory as seen in Kolb and Turner [4] (1991), then the supposition by Kauffman [1] is supportable with evidence, i.e. then if there is a non zero initial energy, is this in any way counter to Theorem 6.1 above?We will review this question, keeping in mind that.
is in reference to a scale factor, as written by Ellis, Maartens, and MacCallum [3], vanishing.
3. Looking at How to Form for All Scale Factors What was done by Beckwith [2] involved locking in the value of Planck's constant initially.Doing that locking in of an initial Planck's constant would be commensurate with some power of the mass within the Hubble parameter, namely 0 We would argue that a given amount of mass, 0 M would be fixed in by initial conditions, at the start of the universe and that if energy, is equal to mass   M E  that in fact locking in a value of initial energy, according to the dimensional argument of Ẽ that having a fixed initial energy of Ẽ    , with Planck's constant fixed would be commensurate with, for very high frequencies,  of having a non zero initial energy, there- by confirming in part Kauffmann [1], as discussed in Appendix A, for conditions for a non zero lower bound to the cosmological initial radius.If so then we always have .We will then next examine the consequences of .i.e. what if for a FLRW cosmology?

 a at the Start of Cosmological Expansion in FLRW Cosmology
We will start off with     initial e H a a Equation ( 2) above becomes, with

  
The upshot, is that if we have Case 2, we will not have a singularity if we use Theorem 6.1 [3] unless the expression start-value    is less than or equal to zero.In reality this does not happen, and we have Also, we have that The only way to have any fidelity as to this Theorem 6.1 would be to eliminate the cosmological constant entirely.There is, one model where we can, in a sense "remove" a cosmological constant, as given by Ellis, Maartens, and MacCallum [3], and that is the Bianchi I universe model, as given on page 459 [3].

Bianchi I Universe in the Case of 1 const      ω p ρ ρ
In this case, we have pressure as the negative quantity of density, and this will be enough to justify writing , we can re write Equation (10) as, if the sheer term in fluid flow, namely is a non zero constant term (i.e. at the onset of inflation, this is dubious) In this situation, we are speaking of a cosmological constant and we will collect  If we speak of a fluid approximation, this will lead to for Planck times looking at so we solve initial   The above equation no longer has an effective cosmological constant, i.e. if matter is the same as energy, in early inflation, Equation ( 13) is a requirement that we have, effectively, for a finite but very large 2

Use Thermal History of Hubble Parameter Equation Represented by Equation (14)
Ellis, Maartens, and MacCallum [3] treatment of the thermal history will then be, if Then we have for Equation ( 14), if the value of Equation (15) is very large due to Plank temperature values initially This assumes that there is an effective mass which is equal to adding both the Mass and a cosmological constant together.In a fluid model of the early universe.This is of course highly unphysical.But it would lead to Equation (13) having a non zero but almost infinitesimally small Equation (13) value.The vanishing of a cosmological constant inside an effective (fluid) mass, as means that if we treat Equation (15) above as ALMOST infinite in value, that we ALMOST can satisfy Theorem 6.1 as written above.The fact that , i.e. we do not have infinite degrees of freedom, means that we get out of having Equation (15) become infinite, but it comes very close.

Use of Thermal
Yes, but we have problems because the cosmological parameter, while still very small is not zero or negative.So Theorem 6.1.2above will not hold.But it can come close if the initial value of the cosmological constant is almost zero.
Case 3 when we can no longer use .Is the following true?When the Temperature is Planck temp?
Almost certainly not true.Our section eight is far from optimal in terms of fidelity to Theorem 6.1.
We are close to Theorem 6.1.2[3] on our Section seven.But this requires a demonstration of the constant value of the following term, in Section 7, namely in the Bianchi universe model, that the sheer term in fluid flow, namely  is a non zero constant term (i.e. at the onset of inflation, this is dubious).If it, , is not zero, then even close to Planck time, it is not likely we can make the assertion mentioned above in Section 7. , And we also are assuming then, a thermal expression for the Hubble parameter given by Ellis, Maartens and Mac Callum [3] as a     term which is almost infinite in initial value.Our conclusion is that we almost satisfy Theorem 6.1 if we assume an initially almost perfect fluid model to get results near fidelity with the initial singularity theorem (Theorem 6.1).This is dubious in that it is unlikely that , as a shear term is not zero, but constant over time, even initially.

For
Section 7 above we have almost an initial singularity, if we replace a cosmological constant with