COSMIC INITIAL SINGULARITIES IN A SINGLE REPEATING UNIVERSE AS OPPOSED TO THEIR BEHAVIOR IN A MULTIVERSE

When initial radius of the universe in four dimensions and there is only ONE repeating universe then 0 initial R → or gets very close to zero if Stoica actually derived Einstein equations in a formalism which remove in four dimensions the big bang singularity pathology. So then the reason for Planck length no longer holds. This assumes a repeating single universe. We present entanglement entropy in the early universe with a shrinking scale factor, due to Muller and Lousto , and show that there are consequences due to initial entanged 2 2 .3 Entropy H S r a = for a time dependent horizon radius H r in cosmology, with (flat space conditions) H r η = for conformal time . Even if the 3 dimensional spatial length goes to zero. This construction preserves a minimum non zero Λ vacuum energy, and in doing so keep the bits, for computational bits cosmological evolution even if in four dimensions we have 0 initial R → . We also find that in the case of a multiverse, that such considerations will not hold and that cosmic singularities have a different characteristic in the multiverse setting than in the single universe repeated over and over again. i.e. using an argument borrowed and modified from Kauffman, the multiverse will not mandate ‘perfect’ singularities. The existence of a multiverse may allow for non zero singularities in lieu with the Kauffman argument cited at the end of the document, plus the lower pre big bang temperatures which may allow for the survivial of gravitons just before the onset of the cosmological expansion phase, if a multiverse exists embedding our present universe.

.3 Entropy H S r a = for a time dependent horizon radius H r in cosmology, with (flat space conditions) H r η = for conformal time . Even if the 3 dimensional spatial length goes to zero. This construction preserves a minimum non zero Λ vacuum energy, and in doing so keep the bits, for computational bits cosmological evolution even if in four dimensions we have 0 initial R → . We also find that in the case of a multiverse, that such considerations will not hold and that cosmic singularities have a different characteristic in the multiverse setting than in the single universe repeated over and over again. i.e. using an argument borrowed and modified from Kauffman, the multiverse will not mandate 'perfect' singularities. The existence of a multiverse may allow for non zero singularities in lieu with the Kauffman argument cited at the end of the document, plus the lower pre big bang temperatures which may allow for the survivial of gravitons just before the onset of the cosmological expansion phase, if a multiverse exists embedding our present universe.

Introduction
We first examine what is to be expected in the four dimensional case as to what happens if there is a single repeating universe. In such a situation, one can employ the following argument as to a singularity with the aforementioned behavior as given below. First before doing it, we investigate via simple arguments involving scaling arguments for the Friedman equations what to expect in the case that the cosmological "constant" is indeed a constant or has a temperature dependence as T to the beta power, according to formalism developed by Park et.al. [1]. In doing so, a case can be made using the Weinberg argument [2] that if there is a high initial background "viscosity" ( for graviton propagation) there is a high initial temperature. This high temperature would be consistent with the modus operandi of a single repeating universe, done again and again. To make the point of this, we also can refer to the Penrose CCC [3] hypothesis as yet another way to delineate this same repeating universe, with black holes in four dimensions. I.e. in the case of the single repeating universe, one may use the Stoica convention [4] as to non pathological singularities, or a near singularity at the beginning of space-time.
The situation changes if we have a multiverse. Here, the behavior of viscosity changes fundamentally in terms of its contribution to temperature, and this in turn has manifest implications as to possible advoidance of singularities in initial space time. To set the frame work for doing this, we will generalize the Penrose CCC hypothesis [3] in a way presented by the author in San Marino, Italy, and other places. First now, let us look at the single repeating universe case and comment upon it.
The final part will be a summary in section 13 below which states a generalized treatment as to how the singularity is avoided in the multiverse with reccomendations as to future research in section 14.

If there is a Single repeating universe, what can we expect in terms of entropy and Singularies ? Case written below.
This first part of the article is to investigate what happens physically if there is a non pathological singularity in terms of Einstein's equations at the start of space-time if there is a single repeating universe. This eliminates the necessity of having then put in the Planck length since then ther would be no reason to have a minimum non zero length. The reasons for such a proposal come from [4] by Stoica who may have removed the reason for the development of Planck's length as a minimum safety net to remove what appears to be unadvoidable pathologies at the start of applying the Einstein equations at a space-time singularity, and are commented upon in this article.
in particular is remarked upon. This is a counter part to Fjortoft theorem in Appendix I below. The idea is that entanglement entropy will help generate bits, due to the presence of a vacuum energy, as derived at the end of the article, and the presence of a vacuum energy non zero value, is necessary for comsological evolution. Before we get to that creation of what is a necessary creation of vacuum energy conditions we refer to constructions leading to extremely pathological problems which [4] could lead to minus the presence of initial non zero vacuum energy. [5]also adds more elaboration on this.
Note a change in entropy formula given by Lee [6] about the inter relationship between energy, entropy and temperature as given by 2 Lee's formula is crucial for what we will bring up in the latter part of this document. Namely that changes in initial energy could effectively vanish if [4]is right, i.e. Stoica removing the non pathological nature of a big bang singularity. That is, unless entanglement entropy is used.
If the mass m, i.e. for gravitons is set by acceleration (of the net universe) and a change in enthropy 38

10
Then we are really forced to look at Eq. (1) as a paring between gravitons (today) and gravitinos (electro weak) in the sense of preservation of information.
Having said this note by extention , t hen a is also altered i.e. goes to zero..
What will determine the answer to this question is if initial E ∆ goes to zero if 0 initial R → which happens if there is no minimum distance mandated to avoid the pathology of singularity behavior at the heart of the Einstein equations. In doing this, we avoid using the energy 0 E + → situation, i.e. of vanishing initial space-time energy, and instead refer to a nonzero energy, with initial E ∆ instead vanishing. In particular, the Entanglement entropy concept as presented by Muller and Lousto [7] is presented toward the end of this manuscript as a partial resolution of some of the pathologies brought up in this article before the entanglement entropy section. No matter how small the length gets, entropy S if it is entanglement entropy, will not go to zero. The requirement is that the smallest length of time, t, rescaled, does not go to zero. This preserves a minimum non zero Λ vacuum energy, and in doing so keep non zero amounts of initial bits, for computational bits cosmological evolution even if 0 initial R → Before doing that, we review Ng [8] and his quantum foam hypothesis to give conceptual underpinnings as to why we later even review the implications of entanglement.entropy. I.e. the concept of bits and computations is brought up because of applying energy uncertainty, as given by [8] and the Margolis theorem appears to indicate that the universe could not possibly evolve if [8] is applied, in a 4 dimensional closed universe. This bottle neck as indicated by Ng's [8] formalism is even more striking in the author's end of article proof of the necessity of using entanglement entropy in lieu of the conclusion involving entanglement entropy, which can be non zero, even if 0 initial R → provided therre is a minimum non zero time length.

Review of Ng, [8] with comments.
First of all, Ng [8] refers to the Margolus-Levitin theorem with the rate of operations The key point as seen by Ng [8] and the author is in Assuming that the initial energy E of the universe is not set equal to zero, which the author views as impossible, the above equation says that the number of available bits goes down dramatically if one sets 1 # We summarize what may be the high lights of this inquiry leading to the present paper as follows.
4a. One could have the situation if 0 initial R → of an infinite point mass, if there is an initial nonzero energy in the case of four dimensions and no higher dimensional embedding even if [4] goes through verbatim. The author sees this as unlikely. The infinite point mass construction is verbatim if one assumes a closed universe, with no embedding superstructure and no entanglement entropy. Note this appears to nullify the parallel brane world construction used by Durrer [9] .The author, in lieu of the manuscript sees no reason as to what would perturb this infinite point structure, so as to be able to enter in a big bang era. In such a situation, one would not have vacuum energy unless entanglement entropy were used. That is unless one has a non zero entanglement entropy [7] present even if 0 initial R → . See [10]for a smilar argument.
4b. The most problematic scenario. 0 initial R → and no initial cosmological energy. I.e. this in a 4 dimensional closed universe. Then there would be no vacuum energy at all.initially. A literal completely empty initial state, which is not held to be viable by Volovik [7].
4c. If additional dimensions are involved in beginning cosmology, than just 4 dimensions will lead to physics which may give credence to other senarios. One scenario being the authors speculation as to initial degrees of freedom reaching up to 1000, and the nature of a phase transition from essentially very low degrees of freedom, to over 1000 as speculated by the author in 2010 [12].
4d. What the author would be particularly interested in knowing would be if actual semiclassical reasoning could be used to get to an initial prequantum cosmological state. This would be akin to using [13], but even more to the point, using [14] and [15], with both these last references relevant to forming Planck's constant from electromagnetic wave equations. The author points to the enormous Electromagnetic fields in the electroweak era as perhaps being part of the background necessary for such a semiclassical derivation, plus a possible Octonionic space-time regime, as before inflation flattens space-time, as forming a boundary condition for such constructions to occur [16] The relevant template for examinging such questions is given in the following table 1 as printed below.
4e. The meaning of Octonionic geometry prior to the introduction of quantum physics presupposes a form of embedding geometry and in many ways is similar to Penrose's cyclic conformal cosmology speculation.: 4f. It is striking how a semiclassical argument can be used to construct Table 1 below. In particular, we look at how Planck's constant is derived, as in the electroweak regime of space-time, for a total derivative [14] , [15] ( ) ( ) Similarly [10], [11] ( ) ( ) The A field so given would be part of the Maxwell's equations given by [9] as, when [ ] represents a D'Albertain operator, that in a vacuum, one would have for an A field [14], [15] [ ] 0 And for a scalar field φ Following this line of thought we then would have an energy density given by, if 0 ε is the early universe permeability [14] ( ) We integrate Eq. (10) over a specified E and M boundary, so that, then we can write the following condition namely [14], [15] .
(11) would be integrated over the boundary regime from the transition from the Octonionic regime of space time, to the non Octonionic regime, assuming an abrupt transition occurs, and we can write, the volume integral as representing [14] , [15] gravitational energy Then by applying [14], [15] we get  formed by semiclassical reasons In semi classical reasoning similar to [13] ( ) The question we can ask, is that can we have a prequantum regime commencing for Eq.
Eq. (11) and Eq (12)  In so many words, the formation period for  is our pre-quantum regime. This table 1 could even hold if 0 initial R → but that the 4 dimensional space-time exhibiting such behavior is embedded in a higher dimensional template. That due to 0 initial R → not removing entanglement entropy as is discussed near the end of this article.

If 0
initial R → then if there is an isolated, closed universe, there is a disaster unless one uses entanglement entropy.
One does not have initial entropy, and the number of bits initially disappears. That is if one is not using entanglement entropy, as will be examined at the end of this article.
Abandoning the idea of a completely empty universe, this unperturbed point of matter-energy appears to be a recipede for a static point with no perturbation, as may be the end result of applying Fjortoft theorem [17] to the thermodynamic potential as given in [18], i.e. the non definitive anwer for fufillment of criteria of instability by applying Fjortoft's theorem [17] to the potential [18]leading to no instability as given by the potential given in [18] may lead to a point of space-time with no change, i.e. a singular point with 'infinite' mass which does not change at all. This issue will be reviewed in [19] a different procedure, i.e. a so called nonsingular universe construction. To get there we will first of all review an issue leading up to implimentation of [19]. 6. Can an alternative to a minimum length be put in? Consider the example of Planck time as the minimal component, not Planck length. From J. Dickau, [20] the following was given to the author, as a counter point to 0 initial R → leading to a disaster. "If we examine the Mandelbrot Set along the Real axis, it informs us about behaviors that also pertain in the Quaternion and Octonic case-because the real axis is invariant over the number types. If numbers larger than .25 are squared and summed recursively ( i.e. -z = z^2 +c ) the result will blow up, but numbers below this threshold never get to infinity, no matter how many times they are iterated. But once space-like dimensions are added-i.e. an imaginary compoent-the equation blows up exponentially, faser than when iterated" Dickau concludes: "Anyhow there may be a minimum (space-time length) involved but it is probably in the time direction". This is a counter pose to the idea of minimum length, looking at a beginning situation with a crucial parameter initial R even if the initial time step is "put in by hand". First of all, look at [7], if E is M, due to setting c = 1, then Everything depends upon the parameter initial R which can go to zero. We have to look at what Eq. (14) tells us, even if we have an initial time step for which time is initially indeterminate, as given by a redoing of Mitra's 00 g formula Eq.
[10] which we put in to establish the indeterminacy of the initial time step if quantum processes hold.  potentially going to zero, is less important than a minimum time length. The instability issue is reviewed in Appendix II. for those who are interested in the author's views as to lack proof of instability. It uses [18] which the author views as THE reference as far as thermodynamic potentials and the early universe.

Muller and Lousto Early universe entanglement entropy, and its implications.
Solving the spatial length issue, provided a minimum time step is preserved in the cosmos, in line with Dickau's suggestion. i.e.
So, then one has No matter how small the length gets, entropy S if it is entanglement entropy, will not go to zero. The requirement is that the smallest length of time, t, re scaled does not go to zero. This preserves a minimum non zero Λ vacuum energy, and in doing so keep the non zero initial bits, for computational bits contributions to evolving space time behavior even if 0 initial R → 8. Reviewing a suggestion as to how to quantify the shrinkage of the scale factor and its connections with entanglement entropy.
We are given by [19] if there is a non singular universe, a template as to how to evaluate scale factor a against time scaled over Planck time, with the following results. 6 The main thing we can take from this, is to look at the inter-relationship of how to pin down an actual initial Hubble "constant" expansion parameter, where we look at: Then, to first order, one is looking at Initial entropy to get a non zero but definite vacuum energy as leading to an entanglement entropy of about ( just before the electro weak regime) In line with Stoica [4] shrinking the minimum length and referring to both Eq.(29) and Eq. (27), the idea is to use a surface area treatment as to getting the initial entropy values as given in Eq. (29). To do so, the author looks at the following diagram: Figure 1, from [9] Brane world dynamics in the case of a single, repeating universe, as opposed to a multiverse. Now for a review of Figure 1 above. The two branes given at b y and s y refer to the two Brane world states, especially in line with [24], [25]. The first one, namely b y is the brane where our physical universe lives in, and is embedded in. If one uses this construction, with higher dimensions than just 4 dimensions, then it is possible to have a single point in 4 dimensional space as a starting point to a tangential sheet which is part of an embedding in more than 4 dimensions. Along the lines of having a 4 dimensional cusp with its valley (lowest) point in a more than 4 dimensional tangential surface. The second brane is about 30 10 − centimeters away from the brane our physical world lives in, and moves closer to our own brane in the future, leading to a slapping of the two branes together about a trillion years ahead in our future [24], [25]. The geometry we are referring to with regards to embedding is in the first brane b y . [9] uses this geometry to have graviton production which the author has used to model Dark Energy [21]

1 st Conclusion when looking at ONE repeating universe. Making computational bits, via (19).
As stated by Ng [8]. the idea would be to have to give imputs into (3) i.e.
Here in this case, even if the spatial contribution, due to [4] , even if the spatial componet goes to zero, according to [4]. We suggest an update as to what was written by Lloyd [26] [ ] when [27] energy vacuum While doing this, a good thing to do, would be to keep in mind the four dimensional version of vacuum energy as given by Park, [1] namely As well as the transition given by a combination of [1] , with [28], Barvinskey et. al.
[ ] Quantifying the above, and giving it experimental proof, via detector technology may allow us to investigate an old suggestion by the author as to four dimension and five dimensional vacuum energy which was given for small time values In particular, the author is interested in investigating if the following is true, i.e.
Look at an argument provided by Padmanabhan [29] and [30] (39) .Question. Do we always have this value of Eq.(39)? At the onset of Inflation? When we are not that far away from a volume of space characterized by 3 P l , or at most 100 or so times larger ? Contemporary big bang theories imply this. I.e. a very high level of thermal energy. We need to ask if this is something which could be transferred from a prior universe, i.e. could there be a pop up nucleation effect , i.e. emergent space time? This question is what should be investigated throughly. Appendix III and Appendix IV give suggestions which the author has thought of which may contribute to, if anything, models of how instantons from a prior universe may be transmitted to our present universe, i.e. Appendix V which is based in part on what Wesson formulated as to five dimensional universe constructions, and instantons [33]. The very interesting topic of vacuum fluctuations in such space-time has also been reviewed briefly in Appendix VI, and Appendix VII 11. What if we have a multiverse? Argue then that the above methodology should be modified. I.e. consider the following scaling of quantities from the Friedman equation. Using the formalism given in Peacock [34], page 80 we can make the following scaling argument which will prove useful as to the divergence of the multiverse case, and how there is no longer a convenient singularity to refer to, via Stoica [4] or anyone else Peacock, has it that one can write as follows, namely for a scale factor, with 0 a infinitesimally small and representing the intial scale factor in pre Planckian space time ` ` Here, the main evolving factor to consider is, Peacock [34] , page 80 with cos const − Λ from Eq. (32) above. If so then we have to look at one datum which will be important. Mainly, how the temperature changes. If the term beta in the coefficient of temperature, T , is zero, we have merely the Einstein vanilla Cosmological constant . If the term beta in the coefficient of temperature, T, is not zero, then we should take a good look at what is done in Weinberg [ 2] where there is a concerted effort to mix in background viscosity =. But first take note of the Grischuck [35] expression as to GW frequency in the hign end, with Temperature approaching Planck Tempure values, with behavior due to space-time temperature dependence as given by:" M = mass of 'universe' initially, and R = radius of initial dimensions, with Planck temperature values giving for a single universe, and four dimensions ( ) 3 10 10 Hz~10 Hz eV

km
Peak solar mass Then in doing this, we will also consider the case where the temperature is low, which we claim is due to a multiverse, and this will be leading to, for ultra low initial temperatures, for the "pre big bang, namely if there is a multiverse having ( ) 3 18 10 10 Hz~10 10 Hz eV Furthermore, we can write from Page 163 of Penrose [3] , that, if N ~ S (entropy) as given by Ng [8] for a regular single repeating universe structure, that we will have, for a cyclical universe the following power law relationship. Note that in Eq. (44) the abbreviation of F.T. stands for field theory. I.e that the cosmological constant is set by the initial release of, maybe, gravitons. i.e. and also note that if one does this that one is able to state that initially low entropy will lead to a tendency, later to a scaled cosmological constant which is todays value.
The question in reaching Eq (44) is , if entropy is commensurate with graviton production as brought up by both Beckwith [21] and Giovannini [36] that then one is looking at thermally induced GW production which may indeed impact the cosmological constant. The question to ask is the following reasonable? I.e. as given by Giovannini [36] ( ) ( ) Using this Eq.(46), if the viscosity drops, the temperature drops as well. We posit that this will then have immediate consequenes for the problem of a multiverse versus a single universe

Penrose Comology supposition and its tie in to temperature and viscosity as used in
The single universe CCC has, then, when black holes ( up to a million of them) take in space time from an expanding universe, a 'reverse' conformal re set which Penrose [ 3] sets as Eq.(53) with Eq.(52) are in reality a motivation of Ergodic mixing , i.e. a way to have continual re sets of the initial data, for the purpose of having the same initial conditions again and again, i.e. this is a way of a re set of Planck's constant, and of the fine structure constant without the annoyance of the anthropic principle.
The governing principle behind it would be the Ergodic theorem, i.e. see Dye [39] , as an averaging process again and again. In doing so, Eq. (52) and Eq.(53) would through mixing lead to the same re set again and again, but with one major different consequence than in the single repeating universe case, i.e there would be a low temperature pre big bang, using Eq. (43), instead of Eq. (42). This would be a way of having a very different answer as to the character of early universe GW generation, with the multiverse having very low GW frequency, even before red shifting of GW, instead of the high frequency GW predicted by Eq. (42). Next then, we will examine the consequences of the initial starting point of the multiverse recycled starting point. Note the multiverse would likely entail an initially low viscosity, which says low temperature.

Re examining the question of a 'near singularity' in a multiverse
We follow the recent work of Steven Kenneth Kauffmann [40], which sets an upper bound to concentrations of energy, in terms of how he formulated the following equation put in below as Eq. (54). Eq. (54) specifies an inter-relationship between an initial radius R for an expanding universe, and a "gravitationally based energy" expression we will call ( ) G T r 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 ( ) G T r is defined via Eq.(55) afterwards.We start off with Kauffmann's expression [40] ( ) a "Planck force" which is relevant due to the fact we will employ Eq. (54) 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:  10 Planck l ⋅ 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 We argue that the above methodology, giving a non zero initial starting point is made especially tendable if one is using a low temperature start, allowing for the existence of prior recycling universes gravitons to play a role, i.e. that in the single universe repeated again and again, there would be real issues as to the survival of the graviton allowing for the conclusion as to Eq. (57).
14. Fork the road. Nonzero radii of start of inflation may be linkable to low temperature pre universe due to multiverse. How to confirm it ?
The author's supposition and argument as to the results of Eq. (57) which would be key to identifying if a zero point starting point to inflation were mandatory is akin to the question of could gravity and gravitons exist prior to inflation. If there is a multiverse, i.e. repeatedly, as mentioned in section 12, then the answer is likely yes. Hence, Eq. (57) would indicate that a 'perfect' singularity is not mandatory. The problem though is that then likely initial relic GW would then be enormously long, ie as given in Eq. (43) above. This independent of red shifting.
If one has ultra high gravity waves as created at the big bang, then it is likely that a singularity is mandatory. We also then have referred to Stoica's work as to how this could be squeezed down below the Planck length limit, but itself would not be pathological for the reasons stated above.I.e. confirmation of the two alternatives likely hinges upon determination if ultra high GW are indeed part of the pre inflation to inflationary universe's signal heritage.
Note that Figure 1 above is pertinent to a single, repeating universe. The author knows of no viable counter part to Figure 1 in the case of a multiverse. Hence, as for higher dimensions, the author, in lieu of his generalization of the Penrose cosmology conjecture [3] has referenced Appendix V below which is really akin to the material given in Wesson [33] as to Appendix VIII below, as to a different geometry than Braneworlds as to the multiverse hypothesis. Suitable inquiry should be in terms of if Braneworlds as of Figure 1 can suitably be modified for the multiverse. The author doubts this is possible. I.e. if low GW are confirmed as to relic conditions for our particular universe, then if we choose the multiverse, disternment of if Figure  1 has a real generalization to the multiverse hypothesis is mandatory. The author believes that this will be futile.
Note also, if a single repeating universe is confirmed via relic HFGW, then Figure 1 is probably legitimate and brane world 'vibrations' may be necessary for HFGW. If so, then one can as an intellectual inquiry inquire if the generalization to Appendix VIII is then really necessary. It so happens that this is for forming the wave functional permitting an instanton forming , while we next should consider if or not the instanton so farmed is stable under evolution of space time leading up to inflation. We argue here that we are forming an instanton whose thermal energy is focused into a wave functional which is in the throat of the worm hole up to a thermal discontinuity barrier at the onset , and beginning of the inflationary era.