Journal of Modern Physics, 2011, 2, 977991 doi:10.4236/jmp.2011.29118 Published Online September 2011 (http://www.SciRP.org/journal/jmp) Copyright © 2011 SciRes. JMP 977 What the Null Energy Condition (and When It May Be Violated) Tells Us about Gravitational Wave Frequencies in/for Relic Cosmology? Andrew Beckwith Department of Physi cs , Chongqing University, Chongqing, China Email: abeckwith@uh.edu Received April 13, 2011; revised June 12, 2011; accepted June 28, 2011 Abstract We introduce a criterion as to the range of HFGW generated by early universe conditions. The 1 to 10 Giga Hertz range is constructed initially starting with what Grupen writes as far as what to expect of GW frequen cies which can be detected assuming a sensitivity of 27 7~10h . From there we examine the implications of an earlier Hubble parameter at the start of inflation, and a phase transition treatment of pre to post Planckian inflation physics via use of inflatons. We close with an analysis of how gravitational constant G may vary with time, the tie in with the NEC condition and how to select a range of relic GW frequencies. The gravitational frequencies in turn may enable resolving a mis match between the datum that the entropy of the center of the galaxy black hole is greater than the entropy of the present four dimensional universe as we can infer and measure. Keywords: Null Energy Condition, Violation of Null Energy Condition, Cyclic Conformal Cosmology, Entropy, Multiverse 1. Introduction We begin looking at what to expect via the ratio of the energy of relic gravitational waves, over a fixed energy density as a way to quantify the allowed frequency range, and sensitivity allowed, i.e. 1 GHz. This permits, if we do it right at looking at a phenomenological treatment of acquisition of the data needed to understand the Hubble parameter, via experimental temperature inputs. Next, if that same Hubble parameter is proportional to the square root of the inflaton potential, the regime of potential change from say to as given by Beckwith’s [1] adaptation of Weinberg’s [2] discussion of scale factor and potentials may signify a phase transition. This out lined set of new results assumes that the inflaton keeps growing. The choice of is tantamount to the de facto decrease in the scalar field contribution to the scalar potential. It is Beckwith’s contention that rising and lowering temperatures as presented in [1,3] are im portant in determining a range of frequencies from 1 GHz to 10 GHz. The final tabulation of the frequency range is due to looking at uncertainty relations as far as energy and time versus Planck’s constant. To get the frequency range tabulated, this inquiry examines the con cept of the null energy condition [4]. Fidelity to the null energy condition as assume is combined with G ~ G(t). i.e. Is there a gravitiational “constant” parameter having a slight time variation, and a cosmological vacuum en ergy parameter changing with background temperature as part of what helps give a range of values as to the relic GW frequency? That is the important question asked. Part of this document will be a way to test for inputs into the spectral index, S. That objective is sought, by use of an article by Finelli, Cerioni, and Gruppuso, [5,6]. A case by case analysis of what can be ascertained via such inputs will be presented, with recommendations as to how to get these inputs set up experimentally. The spec tral inputs will also be a way to answer a question about comparing entropy as of the universe, and the center of major black holes in the center of galaxies. A mis match which needs resolution. n 2. Vacuum Energy, Sources and Commentary We begin first looking at different value of the cosmo
978 A. BECKWITH logical vacuum energy parameters, in four and five di mensions [7]. i.e. by looking at a five dimensional vac uum energy parameter written as in Equation (1) below. 5dim1 1cT (1) This Equation (1) is in contrast with the more tradi tional fourdimensional version of vacuum energy, minus the minus sign of the brane world theory version. The fivedimensional version is connected with Brane theory and higher dimensions, whereas the fourdimensional version is linked to more traditional De Sitter spacetime geometry, as given by Park [8]. Beckwith gives addi tional refinements [7] as presented in Equation (2) 4dim 2 cT (2) Right after gravitons are released, one sees a dropoff of temperature contributions to the cosmological con stant .Then one writes, for small time values 1 tt , and for temperatures sharply lower than , a drop off to the present low value of the cosmological constant, Beckwith, writes the inter relationship between the two version of the cosmological vacuum energy, if is an integer as having the given order of magnitude inter relationsip as given in Equation (3) below [7] 1 0 12 10 KeT 1 lvin n 4dim 5dim 1 1n (3) If there is order of magnitude equivalence between such representations, there is a quantum regime of grav ity that is consistent with fluctuations in energy and growth of entropy. An orderofmagnitude estimate will be used to present what the value of the vacuum energy should be in the neighborhood of Planck time in the ad vent of nucleation of a new universe. The significance of Equation (3) is that at very high temperatures, it reen forces what Beckwith brought up with Tigran Tchrakian, in Bremen, [9] August 29th, 2008. i.e., one would like to have a uniform value of the cosmological constant in the gravitating YangMills fields in quantum gravity in order to keep the gauges associated with instantons from changing. When one has, especially for times 12 ,tt Planck time t and 12 , with temperature vaues given so 12 Tt , then . i.e. in the regime of high temperatures, one has tt Tt 41 42 t t 12 TtTt for times Planck time 12 ,tt t12 tt and . The last set of conditions in the prior sentence is such that gauge invariance necessary for soliton (instanton) stability would be broken [10]. That breaking of instanton stabil ity due to changes of 41 42 t t will be where we move from an embedding of quantum mechanics in an analog reality, to the quantum regime. Let us now look at different characterizations of the discontinuity, which is the boundary between analog reality, and Octonian grav ity [10,11]. Table 1 below is also using material from Barvinsky [12], and will be referred to later. For times today, a stable instanton is as sumed, along the lines brought up by t’Hooft [13], due to an asymptotic approach to a final, stable con stant value, as the temperature of the universe reaches a net value of P tt 3.2 KT 4dim . That constant for a four dimen sional vacuum energy is a very small value, roughly at the value of the cosmological constant given given today. The results given in Table 1 assume a radical dropoff of the cosmological constant after the electroweak transi tion. That drop off is in line with Kolb’s assertion of the net degrees of freedom in spacetime drop from about 100 to at most 1000 down to a low value of two, espe cially if today in the present era. The supposi tion is that the value of N is proportional to a numerical graviton density referred to as <n>, provided that there is a bias toward HFGW, which would mandate a very small value for P 3 H VR tt 3 . Furthermore, structure forma tion arguments, as given by Perkins [14] give evidence that if we use an energy scale, , over a Planck mass value m lanck M, as well as contributions from field am plitude , and using the contribution of scale factor behavior 3 a aHm , where we assume 0 due to inflation 2 5 ~~~~10 Planck Planck Hm Ht MM (4) At the very onset of inflation, lanck M , and if (assuming m 1c gravito SN ~1 initial S ) is due to inputs from a prior universe, we have a wide range of parameter space as to ascertain where ns [12] comes from. In the next section, we will discuss if it is feasible and reason able to have data compression of prior universe “infor mation”. If is transferred from a prior uni 88 10 5 0 Table 1 Time 0 tt Time 0 tt Time tt Time today P tt 5 undefined, T 32 10TK 4dim almost 5 , 4dim extremely large 32 12 10 10 TK 54di m , T much smaller than 12 10TK 5 huge, 4dim constant , 3.2TK Copyright © 2011 SciRes. JMP
A. BECKWITH Copyright © 2011 SciRes. JMP 979 verse to our own universe at the onset of inflation,, at times less than Planck time seconds, that enough information MAY exit for the preservation of the prior universe’s cosmological constants, i.e. 44 ~10 P t ,,G (fine structure constant) and the like. We do not have a reference for this and this supposition is being presented for the first time. Times after t = 10−44 are not less im portant. Issues raised in [1117] are important as to the research protocols 3. Consider Now What Could Happen with a Phenomenological Model Bases upon the Following Inflection Point i.e. Split Regime of Different Potential Behavior Vg (5) Given the above potential, as in Equation (5), two re gimes of space time behavior are examined. Manipulat ing formalism as given to use by Weinberg [2] we have [1], V For Lanck tt (6) Also, we would have 1V For Lanck tt (7) Equations (12) and (13) are predicated on the idea that increases, with V becoming smaller as Equation (7) approaches the present era. i.e. the potential system van ishes at or before one billion years ago. The switch between Equations (6) and (7) is not justi fied analytically. Beckwith [1] designated this divide in behavior as represented by Equations (6) and (7) as the boundary of a causal discontinuity. According to Wein berg [2], if 2 16πG , 1 t so that one has a scale factor behaving as [2] 1/ ()att (8) Then, if [14] 2 4πV G (9) there are no quantum gravity effects worth speaking of. i.e., if one uses an exponential potential a scalar field could take the value of, when there is a drop in a field from 1 to 2 for flat space geometry and times to [2] 1 t 2 t 22 18π ln 3 Gg t t (10) Then the scale factors, from Planckian time vary as given in reference [2] as written in Equation (11) below. 1/ 22 2 11 exp 2 at t at t 1 (11) The more 2 1 1 at at , then the less likely Equation (11) represents space time conditions requiring quantum gravity. Note those that the way this inflaton as given for a typical Equation (5) behavior Vg 1 0 TimePlanck Time potential is defined is for a flat, RobersonWalker geometry, and that if lanck tt then Equation (11) no longer applies. If Equation (11) no longer holds, then a physics observer would observationally finds that one is no longer having any connection with even an Octonionic Gravity regime. The details as to what may be expected via Octonionic gravity and its violation are given in Beckwith [1] as an adaptation of the argument given above. And linked to the next section which is that there is a way to link the phase shift involved in Equations (5)(11) with a degrees of freedom mapping as given in the next section. 4. Increase in Degrees of Freedom in the Sub Planckian Regime Starting with [18,19] 0 1 2 thermalB temperature EKT T (12) The assumption is that there would be an initial fixed entropy arising, with N as a nucleated structure arising in a short time interval as a temperature 0, temperature T arrives. One then obtains, dimensionally speaking [18,19] 19 10 GeV 52~~ Btempnet electricfield N kT qETSdist dist dist (13) The parameter, as given by will be one of the parameters used to define chaotic Gaussian mappings. Candidates as to the inflation potential would be in pow ers of the inflation, i.e. in terms of , with N = 4 ef fectively ruled out, and perhaps N = 2 an admissible can didate (chaotic inflation). For N = 2, one gets [11,18], Note that any such entropy as introduced into our uni verse would have to be consistent with a change given by (if ddVV , where V is an inflaton potential, and dist = distance of Planck length, or more) then Beckwith (2010) (see Equation (14)). which in the limit of typical chaotic inflation reduces to a more manageable behavior as (see Equation (15)). Note also in the limit of decline of inflation, Equations
980 A. BECKWITH (14) and (15) imply that eventually one can work with 1/ frequency (16) If one makes the identification of later time physics, not necessarily in the initial space time regime one no longer has a vacuum energy and/or an inflaton contribu tion potential at all to contend with, namely 1/2 2 arg / LeTime T S distdistkn 2 (17) Furthermore, the entropy count is related to what Seth Lloyd (2002) gave in the number of operations as 3/4 7 /ln2# ~10 total B I Skoperations (18) as implying at least one operation per unit graviton, with gravitons being one unit of information per produced graviton. Note, Smoot (2007) gave initial values of the operations as 10 # initially operations~10 (19) What would be interesting to investigate would be a tie in to the number of operations, i.e. maybe 10 to the tenth power, and then the evolution of degrees of free dom which will be mentioned below. If the inputs into the inflation, as given by 2 becomes a random influx of thermal energy from temperature, we will see the par ticle count on the right hand side of Equations (14) and (15) above a partly random creation of articleCount which we claim has its counterpart in the following treatment of an increase in degrees of freedom. The way to introduce the expansion of the degrees of freedom from nearly zero, at the maximum point of contraction to having N(T) ~ 102 to 103 is to define the classical and quantum regimes of gravity in such a way as to minimize the point of the bifurcation diagram affected by quantum processes. [18]. The diagram, in a bifurcation sense would be an application of the Gauss mapping of [11,18]. n 2 1exp ii xx (20) In dynamical systems type terminology, one would achieve a diagram, with tree structure looking like what was given by Binous [19], using material written up by Lynch [20] .Now that we have a model as to what could be a change in space time geometry, let us consider what may happen during the Higgs mechanism break down, as given by Beckwith [1] and in very early universe geome try. 5. The Role Critical Density Plays in Analyzing the Frequency Produced in Relic GW Production We are now going to bring up what Grupen [21] brings up about the role of energy density, GW, and also of GW in terms of setting up frequency changes due to phase shifts in early universe cosmology. To do this, note first that 22 32π GW GW h G (21) This expression for gravitational wave (energy based) density leads to 22 2 12 GW GW h H (22) Frequently, if we assume that GW would be very high, we also wind up having that the Hubble parameter H is also very large. Otherwise, if the GW frequency is low, then Equation (22) is often immeasurably small, a datum which shows up in models of GW generation, in the early universe. As given in [21] having 27 ~10h ~ 1000Hz GW . We can now seriously consider candi dates as to the Hubble frequency, as far as phenomenol ogy and to use that to be part of an estimate as far as a permitted range of GW frequencies generated by relic early universe phase transitions. The current idea by [22] is that the Electro weak regime, as designated by Duerrer et al. [22] is by far a greater contributor to GW produc tion, and it is now time to revisit that assumption in de tail. As stated by Sarkar [23], page 481 of his reference, a good temperature based phenomenological treatment of a Hubble parameter would look like 2 1.66 L HgTTM (23) As stated by Beckwith [1] and re duplicated in Equation (20), the argument given is that there would be, if certain conditions are met, a starting low temperature, rapidly rising,with at about the Planck regime of space time a top 1/2 2 22 2 2 1633 /2 16π4π4π Planck Planck V T S distdistkMM VV V (14) 12 2 22 22 1612161 2~ 4π4π4π lanckpartick Count STk Mn (15) Copyright © 2011 SciRes. JMP
A. BECKWITH 981 degrees of freedom expression of about ~ Maximum gT , for the temperature reaching 1000 ~ L Maximum TT in value from an initially much lower value. This is also a datum, which if we reach ~1000 Maximum gT would be in sync with Sarkar’s [23] 2 ~8π3 L HVM (24) The matter to consider, would be, frankly, that looking at the following expression of energy flux being re for mulated for each universe. I.e. start with the Alcubierre’s [24] formalism about energy flux, assuming that there is a solid angle for energy distribution for the energy flux to travel through. [24,25] looking at a change of energy if 2 2 ' 4 dlimd d d16π t Er rt t (25) The expression 4 is a Weyl scalar which we will write in the form of [24,25] 22 4 22 12 4 2 4 ttrr xx ttrr hhh ihhh (26) Our assumptions are simple, that if the energy flux expression is to be evaluated properly, before the electro weak phase transition, that time dependence of both h and h is miniscule and that initially hh , so as to initiate a rewrite of Equation (21) above as [24] 2 4 11 4rh i (27) The upshot, is that the initial energy flux about the in flationary regime would lead to looking at [25] '2 4 1 d2 t r Planck thnt (28) This will lead to an initial energy flux at the onset of inflation which will be presented as [25] 222 2 d d64πr Planck Er hnt t (29) If we are talking about an initial energy flux, we then can approximate the above as [25] 223 2 64π initial fluxrPlanckeffective r Ehnt (30) Inputs into both the expression 2 rh , as well as effective will comprise the rest of this document, plus our conclusions. The derived value of effective as well as will be tied into a way to present energy per graviton, as a way of obtaining initial E flux n. The n value so obtained, will be used to make a relationship, using Y. J. Ng’s entropy [15] counting algorithm of roughly . We assert that in order to obtain from initial graviton production, as a way to quantify ~ entropy f S ~ entropy f Sn n n, that a small mass of the graviton can be assumed. For the sake of convenience, one can write [2427] 22 h~ rhk (31) So, then [25] 223 4 ~ initial flux 44 ~10 Planck t 64π r Ekhnt cm Planck rl effective (32) For our purposes, we shall call , , effective 34 ~10 Planck sec an effective cross sectional area as to the emission of gravitons, and defined as a physical wave vector. L. Crowell stated that GW would undergo massive red shifting, [28,29] Needless to state, the value of to consider would be for the GHz band of GW [26,27]. k k 2 2 1d d a 2 GW kk a 0 (33) Also, for the frequencies of [27,30] Hz, then 9 10 a 1 10 30 ~~10 rms hh 34 10 (34) Namely, if a net acceleration is such that accel 2πB kcT as mentioned by Verlinde [31,32] as an Unruh result, and that the number of “bits” is 2* 222 22 31.66 ππ Bit B p Sc c xkk xl gT n (35) This Equation (30) has a T2 temperature dependence for information bits, as opposed to [15,23,32,33] 23~ * ~ 31.66 S T g n (36) Should the l order of magnitude minimum grid size hold, then when T ~ 1019 GeV [31,32] 2*22 23 * 2 .66 ~ 3 πB p gcT ngT k xl 311.66 Bit (37) The situation for which one has [31,32] 13 23 lanck xll with ~ lanck corresponds to whereas if ll 2 Bit nT 3 T Bit n 13 23 lanck planck xll l . Here, we make ths assumption that either ~Bit nn or Bit per unit volume of phase space with the temperature T varying from a low value to up to 1034 Kelvin ( Planck temperature scale). 2 T~nn 3 T We will next reference as to conditions permitting re Copyright © 2011 SciRes. JMP
982 A. BECKWITH lease of or per unit vol ume of phase space, while also noting a way to also identify dimensional contributions to relic particle condi tions. Taking into account, as given by U. Sarkar [23] for relic Graviton production, as a function of extra dimen sions we can denote by 2 ~Bit nn T 3 ~Bit nn T 2 2 ()~ d relic gravitonproduction nTTT M (38) We can though, if we wish to reconcile Equation (38) with release of or look at temperature dependence of the scaled mass value, M, Furthermore, if a phase transition exists, as mentioned by Subir Sarkar the following change is revealing recall Subir Sarkar’s [34] 2001 investigation of a simple choice of variant of the standard chaotic inflationary potential given by 2 ~Bit nn T 3 ~Bit nn T 322 03 1 2 VVc (39) Sarkar treated the inflaton as having a varying effec tive mass, with an initial value of effective mass of 2 2 2 d d V m given a before and after phase transition value of [34] 2 3 2 3 6 6 phase transition Beforephase transition afterphase transition mc c (40) This is, when Hunt and Sarkar [34] did it, with 22 mM as a coupling term. This would also affect the spectral index value, and it also would be a way to consider an increase in inflation based entropy The value of M so given in Equation (38) we believe is connected with an appropriate choice of the details of the phase transition alluded to in Equation (40) above. There are two ways to reconcile information from Equation (40) as far as a temperature dependence affect ing M, as I see it, and connecting it with Equation (40) and—release of or First [23] 2 ~Bit nn T 3 ~Bit nn T 2 22 ~ d TTMTMT (41) This will as we will present below apparently imply ing 1323 lanck planck xll l It so happens that Equation (41) with a direct temperature dependence of a net mass M is equivalent to the production of gravitons/relic parti cles as dictated by an initially fixed starting temperature, i.e. making the 13 23 lanck xll with ~ lanck ll 2 T corre sponds to whereas Bit if 3 Bit nTn13 l 23 lanck planck ll. The minimum grid size being possibly of the form 13 23 lanckplanck implies a fixed set of initial space parameters, with temperature not affected by extra dimensions. Secondly, one can make the following approximation as obeying [23] xll l 2 23 ~ d TTMTMT , 1 (42) This may correspond to implying changing the mini mum length fluctuation to. 1323 lanck xll with ~l lanck l 6. Summary as to What Is Known, and Not Known about the Null Energy Condition in Cosmology, and Information Exchange between Prior to Present Universes As stated in [4], the NEC is linked to the following, i.e. look at the general null energy condition first. The null energy condition stipulates that for every futurepointing null vector field (for all of the GR) k 0 ab ab Tkk (43) With respect to a frame aligned with the motion of the matter particles, the components of the matter tensor take the diagonal form, in Euclidian space that ab 000 00 T000 000 P P P 0 (44) The simplest statement of the Null energy condition is that he null energy condition stipulates that density and momentum obey 0P (45) i.e. the equation of state to consider is, if 1w 1w , then if what [1] suggests is true, then there will be a reason to consider the relative import of Equations (43)(45) in terms of contributions. i.e. we do have problems with the idea of variance of the cosmological constant, G. We also will build upon the consequences of , We can generalize this idea to initial domain wall physics. Spherical geometry does not violate the NEC. Further domain wall physics may lead to a break down of the NEC [4]. We also refer to a treatment of the NEC if we look at an effective Friedman equation as given by [2], as seen by 22 2 2 8π 3243 213 aaG HG an nw e (46) The scaling done in this situation has [2], especially if e is a constant in Equation (46) leading to using, 31 w a (47) As stated in [16]. We expect that there will be flat Copyright © 2011 SciRes. JMP
A. BECKWITH 983 space geometry almost in the beginning of the early big bang. I.e. this will lead to Equation (47), if 1w im plying that if there is a viola tion of the NEC. As quoted from [18]. i.e. as seen in a colloquium presentation done by Dr. Smoot in Paris [35] (2007); he alluded to information theory as to how much is transferred between a prior to the present universe in terms of information “bits”’. 31 ~ w aa 0 1) Physically observable bits of information possibly in present Universe  180 10 . 2) Holographic principle allowed states in the evolu tion/development of the Universe  120 10 . 3) Initially available states given to us to work with at the onset of the inflationary era  10. 10 4) Observable bits of information present due to quan tum/statistical fluctuations  8 10 . Our guess is as follows. That the thermal flux accounts for perhaps bits of information. These could be transferred from a prior universe to our present, and that there could be minus bytes of information suppressed during the initial bozonification phase of matter right at the onset of the big bang itself. Beckwith [37] stated criteria as far as graviton production, and a toy model of the universe. If one has Equation (42) shut off due to , so then that 10 10 10 1w 120 10 10 31 ~0 w aa occurs, then the causal discontinuity so references in [18] by Beckwith et al, will have major consequences as far as a away to determine if gravitons have a small mass, and if there is a way to determine if a prior universe has contribution as to the information transferred as to the present universe. We will now assume, that the catastro phe given as s 31 w aa ~0 n 7. Determination If the NEC Is Valid Is Essential as Establishing a Necessary Condition for Transfer of Information from a Prior Universe to Today’s Cosmos How to do this? i.e. how to determine if, as an example there is a thermal, flux from a prior universe carrying prior universe information? We will briefly revisit a first principle introduction as to inflaton fluctuations in the beginning which may be part of how to obtain experi mental falsifiable criterion. From Weinberg [2], we can write, from page 192  193, if an inflaton potential 4 ~VM then, the inflaton potential has the fluctuation behavior given by [1] ~t (48) Then, this result for Equation (48) assumes The resulting contributions to the CMBR, if worked out, and also connections to gravitational wave astron omy can be used to pin point an eventual CMBR physics behavior as referred to by Beckwith [1] and its relation ship to falsifiable experimental tests of the NEC. 8. How to Calculate the Spectral Index n for a Dissipative Regime of the Inflaton? We are largely borrowing in this introduction from work done by Finelli, Cerion, and Gruppuso [2,3] and we will introduce the motivation behind their work as well as the actual Spectral index S. To begin with look at what Finelli et al [5,6] postulate as to the case of warm infla tion. i.e. as given by [1,5,6], if the equation of state n FF p is linked to 2 31 FF H so then we get the statement of 30 V H (50) We can count the term given as 3H as a damping term, as well as consider the slow roll value of 30 V H (51) The above dynamics, if 2 2 d d V V , and 0 4 0 C b F M , and 3 , 2 , 2 3 V (52) For the sake of convenience, we can use V ~ con stant, i.e. the quadratic scalar potential. But this is a spe cial case of what we will refer to later. If so, then the equations for perturbations, inflaton perturbations, ,Q Q as respectively the inflaton, and the fluid fluctua tions leads to initial conditions of 3 12 3 11 12 4 2 exp2 , exp 4 F i k g iF Fk Qik ak Qik ak (53) The upshot is that one gets the following as far as a running index [1,5,6] 2 * 41 1 232 13 6 11 S n (54) 2 61 1 .25 16 2 (49) Copyright © 2011 SciRes. JMP
984 A. BECKWITH Here, the * factor is for values of the parameters when the cosmological evolution crosses a radius defined by (k_ = a_ H_). In [2] there are two tables as far as inputs/ out puts into running index, which have to take into ac count several constraints. i.e. when one has, as was stated a situation for which [1] 4 0 0 C b Fconst M (55) Either b = C = 0, which is possible, or one could have, if , a situation for which one can have 0, 0bC 4 0 C b Fconst M (56) What if one had, 0 being a present day, very small value of a scalar field [1] / 04 Cb F M const (57) We can probably assume in all of this that M as a mass scale is fixed. When the author looks at Equation (57), it appears to be implying the relative value of density, i.e. varies with time. i.e. if one looked at the Octonian gravity formation regime we could look at variation of looking maybe like 2 ~observed G. The term about the relationship of [36], where a is a constant, and () T is the number of degrees of freedom, 24 ~4π()/3 F observed2 GaTgT c (58) There are two different scenarios as far as temperature build up and how it affects () T , and also initial tem peratures. 8.1. 1st, Version of Classical/Standard Cosmology Treatment of the Start of Inflation. i.e. the Ultra High Temperature Regime to Cooler Temperatures Here, as given by Kolb and Turner [37], () T has a peak of about 100  120 during the electro weak regime, and that there is allegedly little sense in terms of model ing of talking about () T before the electro weak re gime. What it means? In so many words, we would then have undefined before the electro weak regime. would then be undefined before the electro weak regime. It does mean that at the start of the electro weak regime, we would see an increasing . Which is the opposite of what we see. i.e. we need decreasing. Meaning that either () T is defined before the electro weak phase transition, or Equation (58) no longer holds. How to tie in the entropy with the growth of the scale function? Racetrack models of inflation, assuming far more detail than what is given in this simplistic treatment provide a power spectrum for the scalar field given by [38,39] 2 1 ~150π V P (59) This is very close to what Giovannini puts in, [40], and being the spectral index S n 1 4 8 3 S n PkaH V Pk k M (60) This Equation (60) result is assuming a slow roll pa rameter treatment with 1 , and for t. An in crease in scalar power, is then proportional to an increase in entropy via use of the following equation. t 2 33 150π ~ PP EP S ll (61) This Equation (61) result presumes that there exists awell defined V before the start of the Planck time interval. That is, if we want to make the equivalent state ment S ~ n for [15] a numerical relic count, as done by Ng [12] does not tell us where the relic particles came from, As we also note in [20] we can employ Sherrer k essence arguments [3] as to how to form relic particles without using a potential explicitly for times less than Planck time interval. 8.2. 1st, New Treatment of the Start of Inflation, i.e. First Low Temperature, Then Ultra High Temperature Regime to Cooler Temperatures (Low to High Then Low Temperature Evolution) This model of low temperature to higher temperatures involves using the initial analysis, except that one has g*(T) defined initially as of about 2 in pre Planckian space time, rising to about 100 to a peak possible value of 1000, as of Planck time, [18] and then from there de clining. The initial temperature would be low, which would rise to a peak temperature, i.e. Planck temperature value, and then subsequently moving to values seen to day. This scenario is outlined in [1], and has the advan tage of explaining at least before to about the Planck time interval, how Equation (54) could resort to a rising temperature. Now, having said, that, what is the advan tage toward having Equation (55) set as a constant with a rising inflaton value and with ? 0, 0bC 9. Comparing the Reacceleration of the Universe, via Deceleration Parameter, Initially and Finally Speaking The use of Equation (62) below to have reacceleration in Copyright © 2011 SciRes. JMP
A. BECKWITH 985 this formulation is dependent upon “heavy gravity” as the rest mass of gravitons in four dimensions has a small mass term. This equation below is developed by Beckwith [40 42] 2 aa qa (62) We wish next to consider what happens not a billion years ago, but at the onset of creation itself. If a correct understanding of initial graviton conditions is presented, it may add more credence to the idea of a small graviton mass, in a rest frame, Here, we are making use of refin ing the following estimates. In what follows, we will have even stricter bounds upon the energy value (as well as the mass) of the graviton based upon the geometry of the quantum bounce, with a radii of the quantum bounce on the order of meters [43,44]. So then the mass of a graviton implies a wavelength to the gravi ton as can be written as given in Equation (63) below. 35 ~10 Planck l 22 12 8 4.410eV / 2.8 10meters graviton RELATIVISTIC graviton graviton mh mc c (63) For looking at the onset of creation, with a LQG bounce; if we look at max 2.07 lanck for the LQG quantum bounce with a value put in for when grams/meter3 , where the effective energy is 99 5.1 10 planck 32 2.07~ 510GeV effPlanck planck El 4 (64) Then, taking note of this, one is obtaining having scaled entropy of 5 ~10SET when one has an ini tial Planck temperature . One then needs to consider, if the energy per given graviton is, if a frequency and , then there is a minimum entropy value we can write as. 19 ~10 GeV Planck TT graviton effective 10 10 Hz 25Eh v 5 10 eV 381019 5 ~ 1010~ 1010 eff graviton effective SET EvHzTGeV (65) Having said that, the 5 2510 graviton effective Ehv is greater than the rest mass energy of a gravi ton if eV 22 10 27 ~.55~10eVshift ~ graviton redE m grams is taken. 10. Now, for Permitted Frequency Ranges for the Relic Graviton As given by Hambler [45] for the effective Friedman equation, on his books pages 318  319. In the procedure which will be written up, we can set 0 a with as defined by what is known as the running gravitational coupling in the vicinity of the ultra violet fixed point as given in equation 9.1 of Hambler [45]. 0 a 1t Gt GC (66) The time varying value of G does lead to an effective density as given by () effective Gt t G t (67) If one is making an analysis of the effective energy, as given by an analysis in part given by Ng [15] and Beckwith [46] 3~ 1 relic gravitionrelic graviton P v relic graviton ESn l t C (68) The relic graviton frequency so described would be from 5 2510eV graviton effective Ehv which is greater than the rest mass energy of a graviton, taking 22 10 ~ relic gravitongraviton effective E , with raviton effective E is over times greater than the rest mass energy of a graviton. The spread in the fre quencies would be given by the factor 5 2510eV 22 10hv 1t C . Let us for the sake of completeness analyze where this came from. The Friedman equation, as given by Hambler [46] with k the curvature factor, and 2 2 2 8π1 1 33 at kGt Ct at at (69) In short, we get, a variance in the Friedman equation. The variance in the Friedman equation appears to be linked to a density variance as given below. 42 ~4π()/3 FaTgTc (70) As mentioned earlier, we have, in Equation (68) and also in Equation (69) a duration of time for which there is a build up of temperature, of the magnitude T just before the inflationary era, and that the time factor is tied into 1t C of Equation (68) It means that in the context of relic graviton production, that the frequency range as of GW production is, indeed nearly a delta func Copyright © 2011 SciRes. JMP
986 A. BECKWITH tion. Why is this delta function behavior significant? If one looks at a frequency for relic GW in terms of an up per bound as to GW frequency, i.e. if frequency as given by Buoanno [47], then the bound to Equation (71) follows. i.e. 9 4.4 10Hzff 2 29 4.8 10hff 0GW a (71) One gets a bias toward low frequencies, and this is accentuated by an estimate which needs to be looked at and questioned, namely, if there is, according to Buo nanno, [47] purely adiabatic evolution of the universe, Here, we have that 0 is todays value of the cosmo logical constant, whereas are initial scale factor and frequency values for the production of GWs as given by Buonnanno [47] , af 0 faa a (72) If the bias toward low frequency values is removed and we look at generation of say having for nearly relic conditions, one gets astonishingly high initial values for , i.e. 25 0 ~10a f 25 35 0~1010 Hz Today ffaa f (73) Note that next to Equation (73) we calculated de facto times greater than the rest mass energy of a gravi ton for relativistic graviton energy. i.e. what was being predicted by the adiabatic approximation has a value of, already about times larger for the frequency. 22 10 25 10 Of course, though, an adiabatic approximation is non sense for the initial phases of the universe, but it is still indicative as to what could be the starting point to a le gitimate inquiry We should note that researchers as of China and the United States have project work on answering the feasi bility of this sort of measurement. [48] Should there be a way to make such a measurement, some of the issues so referred to, i.e. the feasibility of semi adiabiatic ap proximations can be considered. Secondly, and most importantly, if the genesis of initial GW production is within the Planck regime as so mentioned above, for the initial value for frequency will be congruent with extremely tiny starting geometries. 35 10 Hz 11. 1st Part of Conclusions, What to Make of PrePlanckian Physics, in Terms of What to Measure via a GW Detector We will initially quote part of the conclusion as of [1] here, and add more to it. Finelli et al [6] claims that does not match observations, with 0.01 3 . We gave arguments in the prior session as to the feasibility of having as a constant, which often appears to create serious difficul ties. If one has as a constant, with rising inflaton value, up to Planck time interval we have a natural reason for 4dim varying, and also Fcon s t , as suming that with rising inflaton value, up to Planck time interval? 1st we have a natural reason for 4dim varying, and also F const varies with T varying from 2 to 1000 before the electro weak era, and Fconst hav ing 23 ~ 31.66S g T increasing in a net tempera ture increase up to at least 105 from nearly zero, initially. Having said that, we should also revisit what was brought up in [18] namely in how likely we are to be able to get such measurements. Doing so, asks the ques tion of if gravitons have a small rest mass, and that leads to the second real issue to consider. From [18] we wrote for how to isolate the effects of a 4 dimensional graviton with rest mass. If one looks at if a four dimensional graviton with a very small rest mass included [18] we can write how a graviton would interact with a magnetic field within a GW detector. 0 1 g veffective vggg FJJ x (74) where for 0 but very small ,~ v F (75) The claim which A. Beckwith made [18] is that 4ivecountD Gravtion Jnm effect (76) As stated by Beckwith, in [18], 65 4~10 DGravtion m rams , while is the number of gravitons which may be in the detector sample. What would be needed to do would be to try to isolate out an appropriate stress energy tensor contribution due to the interaction of gravitons with a static magenetic field assuming a non zero graviton rest mass. count n 1 uv T The details of the count would be affected by the de gree of the graviton mass, the frequency range and a whole lot of other parameters, but the key point would be in finding a specified frequency range, which the author claims for relic gravitons is almost a spike, as well as their energy level. From there, using some of the details brought up in this document would be relevant, in a pro gram of action as to how to get necessary experimental confirmation. We hope to do so, as soon as circum stances permit. We also seek to find ways to confirm what t’Hooft brought up in [13]. n 12. 2nd Part of Conclusions, the Future Game Plan 12.1. 2nd Part of Conclusions: Information Theory Considerations, and Solving the Copyright © 2011 SciRes. JMP
A. BECKWITH 987 Problem of a Black Hole in the Center of the Galaxy Having More Entropy than the Entire Universe 1st: If Entropy has for a single black hole, say versus a value of for the entire universe. An ex ample of such is given in a NASA news service [49] and is noteworthy, since we will claim that the black holes in such galaxies do have more than four dimensions. This is crucial. Note that these numbers are given by Carroll, as in reference [50]. The problem though is that no amount of conformal rescaling of space time geometry [51] will itself help us reconcile has for a single black hole, say versus a value of for the entire universe. It is useful though tor review the suggestion of what is im plied by conformal cyclic cosmology. 100 10 88 10 100 10 88 10 The heart of the hypothesis is in what Penrose called conformal re scaling, namely looking at what Paul Tod wrote for a spatial metric, to re scale almost infinite ge ometry back to a new big bang [51] 2 ab ab g (77) In so many words, after a near infinite expansion of the universe, re scale the “infinite expansion” via a con formal re scaling back to a new big bang. Also, Tod [51] writes exp t (78) The interesting addition to this hypothesis is given by Tod, [52] and Penrose to read as having a positive pa rameter = 3H2 (79) As stated by Tod [51], Penrose writes, namely that Quote: In (Penrose, [52] 2008), Penrose presents a picture of the very remote future with positive—as a physical worldin which propertime plays no role. He remarks that all stars will have completed their evolution and either collapsed to form black holes or been swallowed up by the massive black holes at the centres of galaxies. Black holes themselves will eventually decay by the Hawking process and the content of the universe will very largely be just electromagnetic and gravitational radiation, both of them massless felds. To complete the picture of a world from which propertime has vanished, Penrose hypothesises that all massive particles eventu ally either decay to radiation or lose their mass in some unspecified way. Still though there is no way that conformal mappings or conformal re scaling can make the following mapping, using what was presented by Lloyd, [33] in terms of in formation theory 100 cos log 3/4 7 10 Value /ln2# ~10 conformal mappingmoy total B I Skoperations (80) This is to be compared to Entropy of black holes in the center of galaxies, e.g. our own, can be greater than the entropy associated with the entire four dimensional ob servational universe, as given by [50] writes that the en tropy of the central black hole of the galaxy is 2 90  6  ~10 10   Blackhole center ofgalaxy solar mass ofsun M SM entropy ofobserveduniverse (81) Equation (81) is for a single black hole at the center of the galaxy. If there are over galaxies, the question is what happens to ~1099 units of entropy per Galaxy? Of a single black hole as opposed to to for the general universe. 6 10 88 10 90 10 2nd: The difference between units of entropy, versus for the entire universe (Carroll, 2004) [50] can only be resolved if Black holes are 5 dimensional (or higher dimensional objects). 106 10 88 10 12.2. 2nd Part of Conclusions: Arrow of Time/and 5 Dimensional Black Holes What Beckwith became convinced of, due to these ar guments is that Black holes, and other information col lection portals have to be considered in higher dimen sions. To do this, look at Re define a general entropy which may exist in five dimensions, so that if the starting point is to look at (Penrose, 2011) [52] in terms of a temperature value, the vacuum energy, and also entropy, directly. From the book (Penrose, 2011) [52], if G = 1 and is a vacuum energy and from Penrose, 2011 [52] 4dim 44 12π3π1 42π3 entropyblack holegeneral entropy A SS T (82) Then, using Table 1, we can have, the two different limiting values for entropy in four and five dimensions. The five dimensional entropy would initially be enor mous, whereas there is a different interpretation for the magnitude of four dimensional entropy. 5dim 5 dim 4dim 4 dim 12 44 12 44  general entropy general entropy A S A S tiny value (83) Copyright © 2011 SciRes. JMP
988 A. BECKWITH So, how does one justify this result? Doing it implies coming up with a multi verse for containment of the four dimensional universe. i.e. demoting the present universe we are in as one out of perhaps billions of (Tegmark 2003) [53] level four universes contributing to entropy. In a manner which is still being worked out, Beckwith is attempting to take the Penrose suggestion of conformal recycling, but to do it in a way which avoids the problem as implied by Equations (80) and (81). Note that the five dimensional representation of black holes would be similar to what is presented in reference [54]. Equation (83) would probably point to a large degree of entropy dumped from four dimensional universes by black holes, extending from each 4 dimensional universe into a fifth dimension. 13. 3rd Part of Conclusions, What to Look for in Term of Observations and Information Transfer? Confirming or denying the importance of the multiverse would be crucial. The idea that black holes may have a 5 dimensional embedding space, as also part of their rep resentation also means taking into consideration the fol lowing as given by Penrose (2011) [52], i.e. Penrose claims that right at the time of the CMBR, that the en tropy per baryon is to . Similarly, Penrose claims that the entropy per baryon is about today. What the entropy per ‘particle’ before the turn on of CMBR, closer to the big bang would be is not stated by Penrose, but it probably would be far lower than . 9 10 10 10 21 10 9 10 Secondly, Penrose’s cyclic conformal cosmology is for times up to about seconds is allegedly imply ing (Penrose, 2011) in the last stated reference for [52] that the product of a [distance measure] times a [mo mentum measure] is an invariant quality. A reduction/ increase in information present in a distance measure increase/reduction in information present in a dis tance measure i.e. the key point being Equation (80) and then Equation (81) would still have to be explained. Even if the entropy per “particle” or clumping of “infor mation” were dramatically lower than , what infor mation may be transferred from prior universe embed ding of our present universe, should be reconciled with 1 to 10 GHz relic GW being generated initially. 31 10 9 10 The information content implied by Equation (63) to Equation (65), in terms of multiverses would need to be verified experimentally. Beckwith’s guess is that viola tion of the Null energy condition will be important as well as a slowly time varying G(t) value. This behavior would start off by an initial energy step being propor tional to the inverse of a varying initial time step as given in Equation (84) below. 1 1/ ~2 thermalB temperature EtE kTT (84) Beckwith submits that the smallness of the initial time step t , as given of the order of Planck Time, reflecting the variance in temperature is a consequence of the Null Energy condition. In turn, Equation (84) suggests a necessary re do of the Penrose cyclic conformal cosmol ogy suggestion [54], which will eventually lead to an indirect proof of Tegmarks [53] multiverse (level four) hypothesis. The transition given by Equation (84), i.e. a phase transition, from a pre quantum regime, perhaps represented by a multi verse [53] to a regime of space time given by Octonionic geometry, as given by Appen dix A below. This transition, and the information transfer as alluded to in the document would be where a violation of the Null Energy condition would be of paramount importance. The violation of the null energy condition would be in the transfer from Pre Octonionic to Oc tonionic geometry, with Octonionic geometry, signifying the initial regime of quantum gravity as given in appen dix A below [56]. t 14. Acknowledgements This work is supported in part by National Nature Sci ence Foundation of China grant No. 11075224 The au thor wishes to thank Dr. Fangyu Li for his repeated hos pitality in Chongquing, PRC, as well as Stuart Allen, of international media associates whom freed the author to think about physics, and get back to his work. 15. References [1] A. W. Beckwith, “What Violations of the Null Energy Condition Tell Us about Information Exchange between Prior to Present Universes? How to Obtain Spectral Index Confirmation?” http://vixra.org/abs/1102.0030 [2] S. Weinberg, “Cosmology,” Oxford University Press, New York, 2008 [3] A. W. 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A. BECKWITH 991 Negi Leads to a generalization for the Gell Mann Matri ces symmetry from SU(2) to SU(3) we replace three Pauli spin matrices by eight Gellmann i _matrices. Then Equation (A4) will be built up as [_ j, _k] = 2Fjkl_ l (8 j, k, l = 1, 2, 3, 4, 5, 6, 7, 8) (A5) The generalization as to expanding Equation (1) above, if ,, ABA B ee As according to Pushpa, Bisht, Li, and Negi would lead to associator structure (x, y, z) = (xy)z − x(yz), for any x, y, z (A6) Implying, depending upon the build up of entries into space and momentum [26] ,/ iPlanckijk k pllT x (A7) Here, in doing so, the scaling factor, for Planck energy term lanck E 1Planck EE (A8) whereas the ijk T is a structure term in some respects acting similar to the basis one used for the Gell Mann matrices, in part dependent upon how the momentum and spatial matrix entries are built up. Copyright © 2011 SciRes. JMP
