Andrew Beckwith Chongqing University Department of Physics; Chongqing, PRC, Effective

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1 ow an effective cosmological constant may affect a minimum scale factor, to avoid a cosmological singularity.( Breakdown of the First Singularity heorem ) Andrew Beckwith Chongqing University Deartment of Physics; abeckwith@uh.edu Chongqing, PRC, We once again reference heorem 6.1. of the book by Ellis, aartens, and accallum in order to argue that if there is a non zero initial scale factor, that there is a artial breakdown of the Fundamental Singularity theorem which is due to the Raychaudhuri equation. Afterwards, we review a construction of what could haen if we ut in what Ellis, aartens, and accallum call the measured effective cosmological constant and substitute in the Friedman equation. I.e. there are two ways to look at the roblem, i.e. after, set as equal to zero, and have the left over as scaled to background cosmological temerature, as was ostulated by Park () or else have as roortional 8 to ~1 ev which then would imly using what we call a 5 dimensional contribution to as roortional to ~ const/ 5 D. 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. he Bianchi I universe case almost allows for use of heorem But this Bianchi 1 Universe model almost in fidelity with heorem requires a constant non zero shear for initial fluid flow at the start of inflation which we think is highly unlikely. Key words: Raychaudhuri equation, Fundamental Singularity theorem, Bianchi I universe, effective cosmological arameter 1. Introduction he resent document is to determine what may contribute to a nonzero initial radius, i.e. not just an initial nonzero energy value, as Kauffman s aer would imly, and how different models of contributing vacuum energy, initially may affect divergence from the first singularity theorem. he 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. he second choice will robably necessitate a tie in with Kaluza Klein geometries, leaving oen ossible string theory cosmology. In order to be self contained, this aer will give artial re roductions of Beckwith s (1) earlier aer, but the nd half of this document will be comletely different, ie. When considering an effective cosmological constant. With four different cases. he last case is unhysical, even if it has, via rescaling zero effective cosmological constant, due to an effective fluid mass. Looking at the First Singularity theorem and how it could fail Again, we restate at what is given by Ellis, aartens, and accallum. (1) as to how to state the fundamental singularity theorem heorem 6.1. (Irrotational eodestic singularities) If,, and in a fluid flow for which u, and at some time s, then a sacetime singularity, where or, occurs at a finite roer time before either As was brought u by Beckwith, (1), if there is a non zero initial energy for the universe, a suosition which is counter to AD theory as seen in Kolb and urner (1991), then the suosition by Kauffman (1) is suortable with evidence. I.e. then if there is a non zero initial energy, is this in s. eff

2 any way counter to heorem 6.1 above? We will review this question, keeing in mind that. is in reference to a scale factor, as written by Ellis, aartens, and accallum. (1), vanishing.. Looking at how to form for all scale factors. What was done by Beckwith (1) involved locking in the value of s constant initially. Doing that locking in of an initial s constant would be commensurate with some ower of the mass within the ubble arameter, namely, (1) 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 ( E = ) that in fact locking in a value of initial energy, according to the dimensional argument of E ~ that having a fixed initial energy of E ~, with s constant fixed would be commensurate with, for very high frequencies, of having a non zero initial energy, thereby confirming in art Kauffmann( 1), as discussed in Aendix 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 a(). for a FLRW cosmology? and what to look for in terms of the Raychaudhuri-Elders equation for a() We will start off with () the start of cosmological exansion in FLRW cosmology a a e with an initial huge ubble arameter initial at a / a a 8 a a const () Equation () above becomes, with () introduced will lead to a ainitial e ainitial const / 8 ainitial const / 8 () 5. Analyzing Eq. () for different candidate values of, with for three cases. he equation to look at if we have ut into Eq. () is to go to, instead to looking at () Case 1, set, and such that in the resent era with about.7 today start value 8 ~1 (today) (5) ev 19 his would change to, if the temerature were about 1 Kelvin ~1 ev 8 ~1 ( Plank era) (6) ev he ushot, is that if we have Case 1, we will not have a singularity if we use heorem 6.1 Case, set, and such that in the resent era with about.7 today start value he ushot, is that if we have Case, we will not have a singularity if we use heorem 6.1 Unless start value 8 ~1 ev is less than or equal to zero. In reality this does not haen, and we have (always) (7) 8 Case, set ~1 ev, and set ~ const/ 5 for all eras. Such that 8 8 ~1 ev 5D~ const/ ~1 ev ( today) D (8)

3 8 8 ~ 1 ev 5D ~ const/ ~ 1 ev ( era) (9) he only way to have any fidelity as to this theorem 6.1 would be to eliminate the cosmological constant entirely. here is, one model where we can, in a sense remove a cosmological constant, as given by Ellis, aartens, and accallum. (1), and that is the Bianchi I universe model, as given on age Bianchi I universe in the case of 1 const In this case, we have ressure as the negative quantity of density, and this will be enough to justify writing (1) If initial e, we can re write Eq.(1) 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) 6 In this situation, we are seaking of a cosmological constant and we will collect eff 6 eff If we seak of a fluid aroximation, this will lead to for times looking at 1/ eff 1/6 such that ~ initial so we solve he above equation no longer has an effective cosmological constant, i.e. if matter is the same as energy, in early inflation, Eq. (1) is a requirement that we have, effectively, for a finite but very large eff (1) 7. Use of hermal history of ubble arameter equation reresented by Eq.(1) Ellis, aartens, and accallum. (1) treatment of the thermal history will then be, if 1 g 1 (15) (1.7) g hen we have for Eq. (1),if the value of Eq.(15) is very large due to Plank temerature values initially (1.7) g (16) eff his assumes that there is an effective mass which is equal to adding both the ass and a cosmological constant together. In a fluid model of the early universe. his is of course highly unhysical. But it would lead to Eq. (1) having a non zero but almost infinitesimally small Eq. (1) value. he vanishing of a means that if we cosmological constant inside an effective (fluid) mass, as given above by eff treat Eq. I5 above as ALOS infinite in value, that we ALOS can satisfy heorem 6.1 as written above. he fact that 1 g 1, i.e. we do not have infinite degrees of freedom, means that we get out of having Eq. (15) become infinite, but it comes very close. 8. Use of hermal history of ubble arameter equation reresented by Eq.() and an effective cosmological arameter. Case 1, if. But the cosmological arameter has a temerature deendence. Is the following true when the temeratures get enormous? (11) (1) (1)

4 g (17) (1.7) 8 initial start Value Not necessarily,. It could break down. Case, set, and such that (cosmological constant). hen we have start value (1.7) g 8 initial (18) start Value Yes, but we have roblems because the cosmological arameter, while still very small is not zero or negative. So theorem 6.1. above will not hold. But it can come close if the initial value of the cosmological constant is almost zero. Case, when we can no longer use. Is the following true? When the emerature is tem? (1.7) g >> 8 8 ev 5D ev ~ 1 ~ const/ ~ 1 ( era) (19) Almost certainly not true. Our section eight is far from otimal in terms of fidelity to heorem 6.1. We are close to heorem 6.1. 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 time, it is not likely we can make the assertion mentioned above. In Section Conclusion: Non singular solutions to cosmological evolution require new thinking. For section 7 above we have almost an initial singularity, if we relace a cosmological constant with, And we also are assuming then, a thermal exression for the ubble arameter given by eff Ellis, aartens and ac Callum as a (1.7) g term which is almost infinite in initial value. Our conclusion is that we almost satisfy heorem 6.1 if we assume an initially almost erfect fluid model to get results near fidelity with the initial singularity theorem (heorem 6.1). his is dubious in that it is unlikely that, as a shear term is not zero, but constant over time, even initially. he situation when we look at effective cosmological constants is even worse. I.e. Case 1 to Case 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 alying Penrose s CCC cosmology to get about roblems we run into due to the singularity theorem cosmology as reresented by heorem 6.1 above. Aendix A: Indirect suort for a massive graviton We follow the recent work of Steven Kenneth Kauffmann, which sets an uer bound to concentrations of energy, in terms of how he formulated the following equation ut in below as Eq. (A1). Equation (A1) secifies an inter-relationshi between an initial radius R for an exanding universe, and a gravitationally based energy exression we will call r which lead to a lower bound to the radius of the universe at the start of the Universe s initial exansion, with maniulations. he term r is defined via Eq.(A) afterwards. We start off with Kauffmann s c R r r d r (A1) r R

5 Kauffmann calls c a force which is relevant due to the fact we will emloy Eq. (A1) at the initial instant of the universe, in the ian regime of sace-time. Also, we make full use of setting for small r, the following: r r r const ~ V( r) ~ mraviton ninitial entroy c (A) I.e. what we are doing is to make the exression in the integrand roortional to information leaked by a ast universe into our resent universe, with Ng style quantum infinite statistics use of n ~ S (A) raviton count entroy Initial entroy hen Eq. (A1) will lead to c R r r d r const m raviton ninitialentroy ~ S ravitoncountentroy r R c R const mraviton ninitialentroy ~ S ravitoncountentroy 1 c R const mraviton n 5 ere, ninitial entroy ~ S raviton count entroy ~ 1, Initialentroy ~ Sravitoncountentroy 1 length = l = meters mraviton ~ 1 6 grams, and (A) where we set l with R~ l 1, and. yically R~ l 1 is about 1 l at c the outset, when the universe is the most comact. he value of const is chosen based on common assumtions about contributions from all sources of early universe entroy, and will be more rigorously defined in a later aer. Acknowledgements his work is suorted in art by National Nature Science Foundation of China grant No1175. he author thanks Jonathan Dickau for his review in 1. Bibliograhy Andrew Beckwith, ow assive ravitons (and ravitinos) may Affect and odify he Fundamental Singularity heorem (Irrotational eodestic Singularities from the Raychaudhuri Equation); Aril 1, htt://vixra.org/abs/1.17. Ellis, R. aartens, and.a.. accallum; Relativistic Cosmology, Cambridge University ress,1, Cambridge, U.K. Steven Kauffmann, A Self ravitational Uer bound On Localized Energy Including that of Virtual Particles and Quantum Fields, which Yield a Passable Dark Energy Density Estimate, htt://arxiv.org/abs/11.6 Edward Kolb, ichael urner, he Early Universe, Westview Press (February 1, 199) D. K. Park,. Kim, and S. amarayan, Nonvanishing Cosmological Constant of Flat Universe in Brane world Scenarios, Phys.Lett. B55 (). 5-1 R. Penrose, Cycles of ime, he Bodley ead, 1, London, UK

Efficiencies. Damian Vogt Course MJ2429. Nomenclature. Symbol Denotation Unit c Flow speed m/s c p. pressure c v. Specific heat at constant J/kgK

Efficiencies. Damian Vogt Course MJ2429. Nomenclature. Symbol Denotation Unit c Flow speed m/s c p. pressure c v. Specific heat at constant J/kgK Turbomachinery Lecture Notes 1 7-9-1 Efficiencies Damian Vogt Course MJ49 Nomenclature Subscrits Symbol Denotation Unit c Flow seed m/s c Secific heat at constant J/kgK ressure c v Secific heat at constant