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 [
.
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→Λeffective in the Friedman equation. i.e. there are two ways to look at the problem, i.e. after Λ→Λeffective, set ΛVac as equal to zero, and have the left over as scaled to background cosmological temperature, as was postulated by Park (2002) or else have ΛVac as proportional to ΛVac～1038GeV2 which then would imply using what we call a 5-dimensional contribution to Λ as proportional to Λ≈Λ5D～-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.
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 [
is in reference to a scale factor, as written by Ellis, Maartens, and MacCallum [
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 then next examine the consequences of
. i.e. what if 
for a FLRW cosmology?
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 [
is less than or equal to zero. In reality this does not happen, and we have
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 [
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.
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 [
which lead to a lower bound to the radius of the universe at the start of the Universe’s initial expansion, with manipulations. The term 
is defined via Equation (A2) afterwards. We start off with Kauffmann’s [
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.