_{1}

^{*}

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}～10^{38}GeV^{2} which then would imply using what we call a 5-dimensional contribution to Λ as proportional to Λ≈Λ_{5D}～-const/T^{}β. 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.

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 [

Again, we restate at what is given by Ellis, Maartens, and MacCallum [

Theorem 6.1.2 (Irrotational Geodestic singularities) If, , and in a fluid flow for which, and at some time, then a spacetime singularity, where either or, occurs at a finite proper time before.

As was brought up by Beckwith [

What was done by Beckwith [

We would argue that a given amount of mass, would be fixed in by initial conditions, at the start of the universe and that if energy, is equal to mass that in fact locking in a value of initial energy, according to the dimensional argument of that having a fixed initial energy of, with Planck’s constant fixed would be commensurate with, for very high frequencies, of having a non zero initial energy, thereby confirming in part Kauffmann [

We will start off with with an initial huge Hubble parameter

Equation (2) above becomes, with introduced will lead to

The equation to look at if we have put into Equation (3) is to go to, instead to looking at

Case 1 set, and [

This would change to , if the temperature T were about

The upshot, is that if we have Case 1, we will not have a singularity if we use Theorem 6.1 Case 2 set, and such that in the present era with T about 2.7 today The upshot, is that if we have Case 2, we will not have a singularity if we use Theorem 6.1 [

Case 3, set, and set for all eras. Such that

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 [

In this case, we have pressure as the negative quantity of density, and this will be enough to justify writing [

If, 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 such that

If we speak of a fluid approximation, this will lead to for Planck times looking at so we solve

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

Ellis, Maartens, and MacCallum [

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 given above by 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.

Case 1 if. But the cosmological parameter has a temperature dependence. Is the following true when the temperatures get enormous [2,5]?

Not necessarily, It could break down.

Case 2 set, and such that (cosmological constant). Then we have

Yes, but we have problems because the cosmological parameter, while still very small is not zero or negative. So Theorem 6.1.2 above 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 [

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

, And we also are assuming then, a thermal expression for the Hubble parameter given by EllisMaartens and Mac Callum [

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.

The situation when we look at effective cosmological “constants” is even worse. i.e. Case 1 to Case 3 in Section eight no where come even close to what we would want for satisfying the initial singularity theorem (Theorem 6.1).

We as a result of these results will in future work examine applying Penrose’s CCC cosmology [

This work is supported in part by National Nature Science Foundation of China grant No. 110752. Also thanks to my father for his lifetime commitment to making me engaged in science before his death in September 29, 2012, in Eagle, Idaho.

We follow the recent work of Kauffmann [

Kauffmann calls a “Planck force” which is relevant due to the fact we will employ Equation (A1) at the initial instant of the universe, in the Planckian regime of space-time. Also, we make full use of setting for small r, the following:

i.e. what we are doing is to make the expression in the integrand proportional to information leaked by a past universe into our present universe, with Ng style quantum infinite statistics use of

Then Equation (A1) will lead to

Here, , and

, and

where we set with, and

. Typically is about at the outset, when the universe is the most compact. The value of const is chosen based on common assumptions about contributions from all sources of early universe entropy, and will be more rigorously defined in a later paper.