Original Paper From Astrophysicist Dr. Andrew Beckwith (Journal of Modern Physics 7/11) "Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves & How Squeezed States Affect Upper Limits To Graviton Mass"
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Original Paper From Astrophysicist Dr. Andrew Beckwith (Journal of Modern Physics 7/11) "Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves & How Squeezed States Affect Upper Limits To Graviton Mass"

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Original scientific paper on which September 3&9 presentations by Chongquing University's Dr. Andrew Beckwith is based, and published within the July 2011 edition of the Journal of Modern Physics. ...

Original scientific paper on which September 3&9 presentations by Chongquing University's Dr. Andrew Beckwith is based, and published within the July 2011 edition of the Journal of Modern Physics. (Scientific Research Publishing)

First part of abstract, i.e. part I•

Detailing Coherent, Minimum Uncertainty States of Gravitons, g , y , as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass /•

“We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW and also gravitons. A side ii f li GW, d l i id issue to consider is if gravitons can be configured as semi classical “particles”, which is akin to the Pilot particles model of Quantum Mechanics as embedded in a larger non linear “deterministic” background.” g g -Dr. Andrew Beckwith, Chongquing University, PRC

Links:

G999 Conference, Philadelphia, PA http://ggg999.org/

San Marino Workshop on Astrophysics and Cosmology For Matter and Antimatter, in San Marino, N. Italy.

http://www.workshops-hadronic-mechanics.org/

Slideshare: http://t.co/vEv2ci5



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Original Paper From Astrophysicist Dr. Andrew Beckwith (Journal of Modern Physics 7/11) "Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves & How Squeezed States Affect Upper Limits To Graviton Mass" Document Transcript

  • 1. Journal of Modern Physics, 2011, 2, 730-751doi:10.4236/jmp.2011.27086 Published Online July 2011 (http://www.SciRP.org/journal/jmp) Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass Andrew Beckwith1 Department of Physics, Chongqing University, Chongqing, China E-mail: abeckwith@uh.edu Received April 12, 2011; revised June 1, 2011; accepted June 13, 2011AbstractWe present what is relevant to squeezed states of initial space time and how that affects both the compositionof relic GW, and also gravitons. A side issue to consider is if gravitons can be configured as semi classical“particles”, which is akin to the Pilot model of Quantum Mechanics as embedded in a larger non linear “de-terministic” background.Keywords: Squeezed State, Graviton, GW, Pilot Model1. Introduction condensed matter application. The string theory metho- dology is merely extending much the same thinking up toGravitons may be de composed via an instanton-anti higher than four dimensional situations.instanton structure. i.e. that the structure of SO(4) gauge 1) Modeling of entropy, generally, as kink-anti-kinks theory is initially broken due to the introduction of pairs with N the number of the kink-anti-kink pairs.vacuum energy [1], so after a second-order phase tran-  This number, N is, initially in tandem with entropysition, the instanton-anti-instanton structure of relic production, as will be explained later,gravitons is reconstituted. This will be crucial to link 2) The tie in with entropy and gravitons is this: the twograviton production with entropy, provided we have structures are related to each other in terms of kinks andsufficiently HFGW at the origin of the big bang. The anti-kinks. It is asserted that how they form and break uplinkage to SO(4) gauge theory and gravitons was brought is due to the same phenomenon: a large insertion of vac-up by [1] Kuchiev, M. Yu, and we think it leads to a uum energy leads to an initial breakup of both entropykink-anti kink pair tie in for attendant gravitons. Note that levels and gravitons. When a second-order phase transi-Kuchiev [1] writes that “Conventional non-Abelian SO(4)gauge theory is able to describe gravity provided the tion occurs, there is a burst of relic gravitons. Similarly,gauge field possesses a specific polarized vacuum state. there is an initial breakup of net entropy levels, and afterIn this vacuum the instantons and anti-instantons have a a second-order phase transition, another rapid increase inpreferred direction of orientation”, and furthermore entropy.“Gravitons appear as the mode describing propagation of The supposition we are making here is that the valuethe gauge field which strongly interacts with the oriented of N so obtained is actually proportional to a numericalinstantons” Furthermore, as given by Ivan Andrić, Larisa graviton density we will refer to as <n>, provided thatJonke and Danijel Jurman [2], what is called an n-soliton there is a bias toward HFGW, which would mandate asolution is shown to have an equivalence with the very small value for V  RH   3 . Furthermore, struc- 3following, namely “semiclassical solutions corres- ture formation arguments, as given by Perkins [3] giveponding to giant gravitons described by matrix models ample evidence that if we use an energy scale, m , overobtained in the framework of AdS/CFT correspondence”. a Planck mass value M Planck , as well as contributionsSolitons have a kink-anti kink structure, even in low from field amplitude  , and using the contribution ofdimensions, as was worked out by Beckwith in a scale factor behaviorCopyright © 2011 SciRes. JMP
  • 2. A. BECKWITH 731  a  that Dr. Jack Ng’s research into entropy [6] not only used  H  m  the Shannon entropy model, but also as part of his quan- a 3   tum infinite statistics lead to a quantum counting algo-where we assume   0 due to inflation  rithm with entropy proportional to “emergent field” par- ticles. If as an example a quantum graviton gas exists, as  H2  m     5 ~ H t ~ ~   ~ 10 (1) suggested by Glinka[4,5] if each quantum gas “particle”    M Planck   M Planck   is equivalent to a graviton, and that graviton is an At the very onset of inflation,   M Planck , and if “emergent” from quantum vacuum entity, then we for- m (assuming   c  1 ) is due to inputs from a prior tuitously connect our research with gravitons with Shan-universe, we have a wide range of parameter space as to non entropy, as given by S ~ ln  partition  function  .ascertain where S  N gravitons  1088 comes from and This is a counter part as to what Asakawa et al. [7] sug-plays a role as to the development of entropy in cosmo- gested for quark-gluongases, and the 2nd order phaselogical evolution In the next Chapter , we will discuss if transition written up by Torrieri et al. [10] brought up ator not it is feasible/reasonable to have data compression the nuclear physics Erice school, in discussions with theof prior universe “information”. It suffices to say that if author. Sinitial ~ 105 is transferred from a prior universe to our Furthermore, finding out if or not it is either a drop inown universe at the onset of inflation, at times less than viscosity [7,8,9], thenPlanck time t P ~ 1044 seconds, that enough information  1MAY exit for the preservation of the prior universe’s   , s 4πcosmological constants, i.e. , G,  (fine structureconstant) and the like. Confirmation of this hypothesis or a major increase in entropy density may tell us howdepends upon models of how much “information” much information is, indeed, transferred from a prior , G,  actually require to be set in place, at the onset universe to our present. If it is s   , for all effectiveof our universe’s inflation, a topic which we currently purposes, at the moment after the pre big bang configu-have no experimental way of testing at this current time. ration , likely then there will be a high degree of “infor- mation” from a prior universe exchanged to our present2. Is Each “Particle Count Unit” as Brought universe. If on the other hand,   0 due to restriction up by Ng, Is Equivalent to a Brane-Anti of ‘information from four dimensional “geometry” to a Brane Unit in Brane Treatments of variable fifth dimension, so as to indicate almost infinite Entropy? How does This Tie in with collisions with a closure of a fourth dimensional “portal” String/Brane Theory Treatments of for information flow, then it is likely that significant data Entropy? compression has occurred. While stating this, it is note worthy to state that the Penrose-Hawking singularityIt is useful to state this convention for analyzing the re- theorems do not give precise answers as to informationsulting entropy calculations, because it is a way to ex- flow from a prior to the present universe. Hawking’splain how and why the number of instanton-anti in- singularity theorem is for the whole universe, and worksstanton pairs, and their formulation and break up can be backwards-in-time: it guarantees that the big-bang haslinked to the growth of entropy. If, as an example, there infinite density. This theorem is more restricted, it onlyis a linkage between quantum energy level components holds when matter obeys a stronger energy condition,of the quantum gas as brought up by Glinka [4,5] and a called the dominant energy condition, which means thatnumber of instanton-anti instanton pairs, then it is possi- the energy is bigger than the pressure. All ordinary mat-ble to ascertain a linkage between a Wheeler De Witt ter, with the exception of a vacuum expectation value ofworm hole introduction of vacuum energy from a prior a scalar field, obeys this condition.universe to our present universe, and the resulting This leaves open the question of if or not there is “in-brane-anti brane (instanton-anti instanton) units of en- finite” density of ordinary matter, or if or not there is atropy. Such an approach may permit asking how infor- fifth dimensional leakage of “information” from a priormation is transferred from a prior to the present uni- universe to our present. If there is merely infinite “den-verse .What would be ideal would be to make an equiva- sity”, and possibly infinite entropy density/disorder at thelence between a quantum number, n, say of a quantum origin, then perhaps no information from a prior universegraviton gas, as entering a worm hole, i.e. going back to is transferred to our present universe. On the other hand,the Energy (quantum gas)  n   , and the number <n> having   0 , or at least be very small may indicateof pairs of brane-anti brane pairs showing up in an en- that data compression is a de rigor way of treating howtropy count, and the growth of entropy. We are fortunate information for cosmological parameters, such as  , G,Copyright © 2011 SciRes. JMP
  • 3. 732 A. BECKWITHand the fine structure constant.  arose, and may have is not justified analytically. i.e. it breaks down. Beckwithbeen recycled from a prior universe. Details about this et al [18] stated this as the boundary of a causal discon-have to be worked out, and this because that as of present tinuity.one of the few tools which is left to formulation and Now according to Weinberg [16], ifproof of the singularity theorems is the Raychaudhuri 2equation, which describes the divergence θ of a congru-  , H 1 tence (family) of geodesics, which has a lot of assump- 16πGtions behind it, as stated by Naresh Dadhich [11]. As so that one has a scale factor behaving asindicated by Hawkings theorem, infinite density is its a(t )  t1/ (5)usual modus operandi, for a singularity, and this assump-tion may have to be revisited. Natário [12] has more de- Then, iftails on the different type of singularities involved. The V     4πG  2supposition is that the value of N is proportional to a (6)numerical DM density referred to as <n> Dark  matter . there are no quantum gravity effects worth speaking of.HFGW would play a role if V  RH   has each  3 3 i.e., if one uses an exponential potential a scalar fieldof the order of being within an order of magnitude of the could take the value of, when there is a drop in a fieldPlanck length value, as implied by Beckwith [13]. Ex- from 1 to 2 for flat space geometry and times t1amined, and linked to modeling gravity as an effective to t 2 [16]theory, as well as giving credence to how to avoid dS/dt= ∞ at S = 0. If so, then one can look at the research re- 1  8πGg 2 t 2 sults of Mathur [14]. This is part of what has been de-  t   ln   (7)   3 veloped in the case of massless radiation, where for Dspace-time dimensions, and E, the general energy is Then the scale factors, from Planckian time scale as  D 1/ D  [16] S~E (2) a  t2        1/ This suggests that entropy scaling is proportional to a t   2   exp  2 1  (8)power of the vacuum energy, i.e., entropy ~ vacuum en- a  t1   t1   2 ergy, if E ~ Etotal is interpreted as a total net energyproportional to vacuum energy, as given below. Conven- a  t2  The more a t  1 then the less likely there is a tietional brane theory actually enables this instanton struc-  1ture analysis, as can be seen in the following. This is in with quantum gravity. Note those that the way thisadapted from a lecture given at the ICGC-07 conference potential is defined is for a flat, Roberson-Walker ge-by Beckwith [15] ometry, and that if and when t1  t Planck then what is  MaxV4  T 00V4    V4  Etotal (3) done in Equation (8) no longer applies, and that one is no 8 π G longer having any connection with even an octonionic The approximation we are making, in this treatment Gravity regime. If so, as indicated by Beckwith, et al.initially is that Etotal  V   where we are looking at a [15] one may have to tie in graviton production due topotential energy term [15]. photonic (“light”) inputs from a prior universe, i.e. a What we are paying attention to, here is the datum that causal discontinuity, with consequences which will showfor an exponential potential (potential energy) [16] in both GW and graviton production. V    g    (4) 3. Linking Instaton-Anti Instaton De facto, what we come up with pre, and post Construction in both Entropy GenerationPlanckian space time regimes, when looking at consis- and Gravitonstency of the emergent structure is the following. Namely, Here is a quick review of how to have an instaton-anti[18] instanton construction for entropy, and then proposing a V      for t  t PLanck (4a) similar construction for gravitons. Afterwards, we will analyze squeezed states. It is the authors conviction that Also, we would have semi classical treatment of Gravitons, if gravitons are in V    1   for t  t PLanck (4b) an instanton-anti instanton paring is equivalent to the break down of the “thin wall approximation” used in The switch between Equation (4a) and Equation (4b) density wave physics. In what may be by some peoplesCopyright © 2011 SciRes. JMP
  • 4. A. BECKWITH 733visualization, an outrageous simplication, the issue of ETotal  4  M p j ,0  M p j ,0squeezing of graviton states is similar to what happens (10)with the break down of the purely quantum mechanical Equation (10) can be changed and rescaled to treatinganalogy done for initially non squeezed states, which the mass and the energy of the brane contribution alongwhen squeezed have their own non quantum mechanical the lines of Mathur’s CQG article [15] where he has aflavor. string winding interpretation of energy: putting as much We will start first looking at entropy, as an in- energy E into string windings as possible viastanton-anti instanton construction and go from there:  n1  n1  LT   2n1  LT  E 2 , where there are n1 wrap- Traditionally, minimum length for space-time bench- pings of a string about a cycle of the torus , and n1 be-marking has been via the quantum gravity modification ing “wrappings the other way”, with the torus having aof a minimum Planck length for a grid of space-time of cycle of length L, which leads to an entropy defined inPlanck length, whereas this grid is changed to something terms of an energy value of mass of m  T  L ( TP i P jbigger is the tension of the ith brane, and L j are spatial di- mensions of a complex torus structure). The toroidal lP ~ 1033 cm  N   lP .  Quantum  Gravity  threshold structure is to first approximation equivalent dimension-  ally to the minimum effective length of N   lP ~ N  So far, we this only covers a typical string gas model for times Planck length  1035 centimetersentropy. N is assigned as the as numerical density ofbrains and anti-branes. A brane-antibrane pair corre- ETotal  2 mi ni (11)sponds to solitons and anti-solitons in density wave phys- iics. The branes are equivalent to instanton kinks in den- The windings of a string are given by Becker et al. [19],sity wave physics, whereas the antibranes are an as the number of times the strings wrap about a circleanti-instanton structure. First, a similar pairing in both midway in the length of a cylinder. The structure theblack hole models and models of the early universe is string wraps about is a compact object construct Dpexamined, and a counting regime for the number of in- branes and anti-branes. Compactness is used to roughlystanton and anti-instanton structures in both black holes represent early universe conditions, and the brane-anti brane pairs are equivalent to a bit of “information”. Thisand in early universe models is employed as a way to get leads to entropy expressed as a strict numerical count ofa net entropy-information count value. One can observe different pairs of Dp brane-Dp anti-branes, which form athis in the work of Gilad Lifschytz [17] in 2004. Lifsch- higher-dimensional equivalent to graviton production.yztz codified thermalization equations of the black hole, The tie in between Equation (12) below and Jack Ng’swhich were recovered from the model of branes and treatment of the growth of entropy is as follows: First, antibranes and a contribution to total vacuum energy. In look at the expression below, which has N as a statedlieu of assuming an antibrane is merely the charge number of pairs of Dp brane-antibrane pairs: The suf-conjugate of say a Dp brane. Here, M pj ,0 is the num-  fix N is in a 1-1 relationship with S  N gravitonsber of branes in an early universe configuration, while  N M pj ,0 is anti-brane number. i.e., there is a kink in the STotal  A   ni (12)given i brane ~ M pj ,0  CDWe  Now, how do we make sense of the following entropyelectron charge and for the corresponding anti-kink values? Note the following: As an example of present confusion, please consider the anti  brane ~ M pj ,0  CDWe  following discussion where leading cosmologists, i.e. Sean  Carroll [17] asserted that there is a distinct possibility thatpositron charge. Here, in the bottom expression, N is mega black holes in the center of spiral galaxies havethe number of kink-anti-kink charge pairs, which is more entropy, in a calculated sense, i.e. up to 1090 in nonanalogous to the simpler CDW structure [17]. dimensional units. This has to be compared to Carroll’s  E    N   [17] stated value of up to 1088 in non dimensional units for STotal ~ a   Total     M p j ,0  M p j ,0  n (9) observable non dimensional entropy units for the observ-  2  j 1   able universe. Assume that there are over one billion spiral This expression for entropy (based on the number of galaxies, with massive black holes in their center, eachbrane-anti-brane pairs) has a net energy value of ETotal with entropy 1090 , and then there is due to spiral galaxyas expressed in Equation (9) above, where ETotal is entropy contributions 106  1090  1096 entropy units toproportional to the cosmological vacuum energy pa- contend with, vs. 1088 entropy units to contend with forrameter; in string theory, ETotal is also defined via the observed universe. i.e. at least a ten to the eight orderCopyright © 2011 SciRes. JMP
  • 5. 734 A. BECKWITHdifference in entropy magnitude to contend with. The 4. Different Senarios for Entropy Growthauthor is convinced after trial and error that the standard Depending Upon if or Not We Have Lowwhich should be used is that of talking of information, in to High Frequency GW from the Big Bangthe Shannon sense, for entropy, and to find ways to makea relationship between quantum computing operations, As mentioned above, there is a question of what fre-and Shannon information. Making the identification of quency range of GW is dominant during the onset of theentropy as being written as S ~ ln[partition  function] . big bang. To begin with let us look at frequency range ofThis is Shannon information theory with regards to en- GW from relic conditions. As given by for a peak am-tropy, and the convention will be the core of this text. plitude as stated by Tina Kahniashvili [25]. Now for theWhat is chosen as a partition function will vary with our amplitude of a GW, as detected todaychosen model of how to input energy into our present 5 6  T*  g* universe. This idea as to an input of energy, and picking hc  f   1.62  1018 different models of how to do so leading to partition  100 GeV  100    32 12 (14)functions models is what motivated research in entropy        k fH ijij  2πf , 2πf   3 12generation. From now on, there will be an effort made to      0   0.01   0.01 identify different procedural representations of the parti- The equation, as given by Kahniashvili [25] with aton function, and the log of the partion function with frequency f given below in Equation (15) which is forboth string theory representations, i.e. the particle count todays detected GW frequency a detector would observe,algorithm of Y. Jack Ng [6], and the Wheeler De Witt whereas  is the frequency of a process synthesizingversion of the log of the partition function as presented GW during a 2nd order phase transition in the early uni-by Glinka [4]. Doing so may enable researchers to even- verse. Also, T is a mean temperature during that 2ndtually determine if or not gravity/gravitational waves are order phase transition. If as an example T is manyan emergent field phenomenon. times larger than 100 GeV, which is the case if GW nu- A further datum to consider is that Equation (8) with cleation occurred at the ORIGIN of the big bang, i.e. atits variance of density fluctuations may eventually be temperatures ~ 1032 Kelvin , then it is likely that f inlinkable to Kolmogrov theory as far as structure forma- Equation (10) below is capable of approaching values oftion. If we look at R. M. S. Rosa [22], and energy cas- the order of what was predicted by Grishkuk [26](2007),cades of the form of the “energy dissipation law”, as- i.e. approaching 10 Giga Hertz. Equation (8) and Equa-suming u0 , l0 are minimum velocity and length, with tion (9) above, would have either a small, or a huge T ,velocity less than the speed of light, and the length at which would pay a role as to how large the amplitude ofleast as large, up to 106 time larger than Planck length a GW would be, detected today, as opposed to what it lPlanck would be at the origin, say, of the big bang. The larger f 3 u0 is, the more likely the amplitude is, of Equation (14)  (13) l0 would be very large. In both Equation (14) above, and Equation (13) above can be linked to an eddy break Equation (15) below, g is a degree of freedom fordown process, which leads to energy dissipated by vis- spatial conditions factor, which has, according to Kolbcosity. If applied appropriately to structures transmitted and Turner [27] high values of the order of 100 right after the big bang, to values closer to 2 and/or 3 in thethrough a “worm hole” from a prior to a present universe, modern era. i.e. the degrees of freedom radically droppedit can explain in the evolution of space time. 1)How there could be a break up of “encapsulating” 16 1structure which may initially suppress additional entropy    g      3 T*  5 f  1.55  10 Hz  *   *     beyond Sinitial ~ 10 , in the onset of inflation  k0   100   0.01   100 GeV  2) Provide a “release” mechanism for [6] (15) S  N gravitons  1054  1088 , with S  N gravitons ~ 10 21 Here, in this choice of magnitude h of a GW today,perhaps a starting point for increase in entropy in and frequency f detected today, as presumed by using at  t Planck ~ 5  1044 sec , rising to S  N gravitons  factor given by Kahniashvili [25] as1054  1088 for times up to 1000 seconds after the big 1 H ijkl  X , K ,    i   K    d  d e 3 Rijkl  X ,  , bang.  2π  4 Here is, in a nutshell the template for the Gravitonswhich will examine, and eventually link to Gravitational (16)waves, and entropy. Why? The factor H ijkl is due to complicated physicsCopyright © 2011 SciRes. JMP
  • 6. A. BECKWITH 735which gives a tensor/scalar ratio S  N DM Candidates (23) As well as This graviton counting as given in Equation (22) will 1 next be connected to information counting which will be Rijkl  X ,  ,   2 Sij  X , t    (17) w a necessary and sufficient condition for information ex- Why? Equation (17) is a two correlation point function, changed from a prior to the present universe.much in the spirit of calculations of two point correlationfunctions, i.e. greens functions of Quantum field theory. 5. Minimum Amount of Information NeededSee [24] Peskin’s QFT reference as to how such func- to Initiate Placing Values of Fundamentaltional calculations are to show the degree of interaction Cosmological Parametersbetween Si , j  x, t  & Sk ,l  x, t    , with each individual Si , j defined as part of a GR “stress tensor” contribution A. K. Avessian’s [30] article (2009) about alleged timeof variation of Planck’s constant from the early universe depends heavily upon initial starting points for   t  , as 1 given below, where we pick our own values for the time Si , j  X , t   Tij  X , t    ij Tkk  X , t  (18) 3 parameters, for reasons we will justify in this manuscript: This is where, commonly, we have a way to interpret   t    initial tinitial  t Planck   exp   H macro   t ~ t Planck    hi , j in terms of Si , j via (24) Sij  X , t  X  X   The idea is that we are assuming a granular, discrete hi , j  X , t   4G  d X  3 (19) X  X nature of space time. Futhermore, after a time we will state as t ~ tPlanck there is a transition to a present value of As well as a wave equation we can write as space time, which is then probably going to be held con- 2 stant. It is easy to, in this situation, to get an inter rela-  2 hij  X , t   hij  X , t   16πGSij  X , t  (20) tionship of what   t  is with respect to the other t 2 physical parameters, i.e. having the values of  written What is above, is a way for making sense of GW as   t   e2   t   c , as well as note how little the fine“density” as given by the formula structure constant actually varies. Note that if we assume d GW an unchanging Planck’s mass  GW     16π3 3Gw2 TH ijij  ,   (21) d ln mPlanck    t  c G  t  ~ 1.2  1019 GeV Here, the temperature T for the onset of a phase this means that G has a time variance, too. This leads totransition, i.e. usually interpreted as a 2nd order phase us asking what can be done to get a starting value oftransition plays a major role as to if or not the frequency,  initial tinitial  t Planck  recycled from a prior universe, tof, for today is very low, or higher, and if or not energy our present universe value. What is the initial value, anddensity is high, or low, as well as the attendant amplitudeof a GW, as given by Equation (19) above is important. how does one insure its existence? We obtain a minimumFurthermore appropriate calculations of Equation (21) value as far as “information” via appealing to Hogans [31]very much depend upon the correlation function as given argument where we have a maximum entropy asby Equation (17) is correctly done, allowing for a mini- Smax  π H 2 (25)mization of sources of noise, of the sort alluded to by [29]Michelle Maggiore. Possibly though, cosmological evo- and this can be compared with A. K. Avessian’s articlelution is so subtle that no simple use of correlation func- [30] 2009 value of, where we pick  ~ 1tions will be sufficient to screen noise by typical field H macro     H Hubble  H  (26)theory derived methods. If temperature T for the onsetof a phase transition, is very high, it is almost certain that i.e. a choice as to how   t  has an initial value, andwe are looking at HFGW, and relic gravitons which are entropy as scale valued by Smax  π H 2 gives us anseverely energized, i.e. * would be enormous. If so, estimate as to compressed values ofthen for high T and enormous *, at the onset of infla-  initial tinitial  t Planck  which would be transferred from ation, we are looking at HFGW, and that [6] prior universe, to todays universe. If Smax  π H 2 ~ 105 , S  N gravitons this would mean an incredibly small value for the INI- (22) TIAL H parameter, i.e. in pre inflation, we would If the frequency is much lower, we will see, if the par- have practically NO increase in expansion, just beforeticle-wave duality has large  , for DM candidates the introduction vacuum energy, or emergent field en-Copyright © 2011 SciRes. JMP
  • 7. 736 A. BECKWITHergy from a prior universe, to our present universe. call the quadrupole moment, with   t , x  a densityTypically though, the value of the Hubble parameter, measurement. Now, the following value of the Qij asduring inflation itself is HUGE, i.e. H is many times lar- given gives a luminosity function L , where R is theger than 1, leading to initially very small entropy values. “characteristic size” of a gravitational wave source. NoteThis means that we have to assume, initially, for a that if M is the mass of the gravitating systemminimum transfer of entropy/information from a prior  1 universe, that H is neligible. If we look at Hogan’s holo- Qij   d 3 x  xi x j    ij  x 2     t , x  (31)  3 graphic model, this is consistent with a non finite eventhorizon [31] 3 1 GN d Qij d 3Q ij   c5 G M  2 L  5  3    N 2  (32) r0  H 1 (27) 5 c dt dt 3 GN  Rc  This is tied in with a temperature as given by After certain considerations reported by Camp and Cornish [32], one can recover a net GW amplitude Tblack  hole   2π  r0  1 (28) G  M  G  M  Nearly infinite temperatures are associated with tiny h ~ 2 N 2   N 2  (33)event horizon values, which in turn are linked to huge  Rc   r c Hubble parameters of expansion. Whereas initially This last equation requires thatnearly zero values of temperature can be arguably linkedto nearly non existent H values, which in term would be GN M R  RG consistent with Smax  π H 2 ~ 105 as a starting point to c2entropy. We next then must consider how the values of gravitational radius of a system, with a black hole result-initial entropy are linkable to other physical models. i.e. ing if one setscan there be a transfer of entropy/information from a pre G Minflation state to the present universe. Doing this will R  RG  N 2 .require that we keep in mind, as Hogan writes, that the cnumber of distinguishable states is writable as [31] Note that when G M  N  exp πH 2  (29) R ~ RG  N 2 c If, in this situation, that N is proportional to entropy, we are at an indeterminate boundary where one may picki.e. N as ~ number of entropy states to consider, then as our system as having black hole properties.H drops in size, as would happen in pre inflation condi- Now for stars, Camp and Cornish [32] give us thattions, we will have opportunities for N ~ 105 2 21 15  Mpc   M   90 km  h  10     2.8M    (34)6. How the CMBR Permits, via Maximum  r   solar  mass   R  Frequency, and Maximum Wave Amplitude Values, an Upper Bound Value M 90 km f  frequency   100 Hz for Massive Graviton Mass mg 2.8M solar  mass R (35)Camp and Cornish [32], as does Fangyu Li [33] use thetypical transverse gravitational gauge hij with a typically As well as a mean time  GW for half of gravitationaltraceless value summed as 0  h  h  0 and off diago- wave potential energy to be radiated away asnal elements of hx on each side of the diagnonal to mix R  GN M  3with a value of  GW   ~ TT 2π  c  R  c 2    GN 2  d 2  3 (36) hij     2 Qij  (30)  R  4  2.8M solar  mass  1  4 c r  dt  retarded        sec   90 km   M  2  This assumes r is the distance to the source ofgravitational radiation, with the retarded designation on The assumption we make is that if we model d G Mthe Equation (30) denoting dt replaced by a retarded R ~ RG  N 2 , c d for a sufficiently well posed net mass M that the startime derivative d  t   r c   , while TT means take the   formulas roughly hold for early universe conditions, pro-transverse projections and substract the trace. Here, we vided that we can have a temperature T for which we canCopyright © 2011 SciRes. JMP
  • 8. A. BECKWITH 737use the approximation  2  10  4 Hz   hij  ~ 10  10    M 90 km  m graviton       100 Hz 1/ 2 2.8 M solar  mass R  M  5.9 107     T  13  2.8  M solar  mass that we also have   ~ 10 or higher, so, that at a  TeV  107 Hz   5.9  Mminimum we recover Grishchuck’s [26] value of    M (40)  mgraviton   5.6    solar  mass  T   f Peak  10 3 Hz     T eV  10  ~ 10 Hz This Equation (40) is in units where   c  1 . (37) If 1060 grams per graviton, and 1 electron volt is in M 90 km   rest mass, so 1.6  1033 grams  gram  6.25 1032 eV . M solar  mass R Then Equation (36) places, for a specified value of R, which 107 Hz can be done experimentally, an upper bound as far as far    mgraviton   as what a mass M would be. Can this be exploited toanswer the question of if or not there is a minimum value  107 Hz 6.582  1015 eV  s      for the Graviton mass?  1060 grams  6.25  1028 eV    2.99  109 meter / sec  2  The key to the following discussion will be that      M 90 km 1022   108 , or larger. ~ 9 ~ 1013 2.8M solar  mass R 10 Then, exist7. Inter Relationship between Graviton Mass M ~ 1026 M solar  mass  mg and the Problem of a Sufficient Num- (41) 1.99  1033 26  1.99  107 grams ber of Bits of  from a Prior Universe, to Preserve Continuity between Fundamen- If each photon, as stated above is 3.68  1048 grams tal Constants from a Prior to the Present per photon, [35] then Universe M ~ 5.44  1054 initially transmitted photons. (42)P. Tinyakov [34] gives that there is, with regards to the Futhermore, if there are, today for a back groundhalo of sub structures in the local Milky Way galaxy an CMBR temperature of 2.7 degrees Kelvinamplitude factor for gravitational waves of 5  108 photons / cubic  meter , with a wave length speci- fied as max  1  cm . This is for a numerical density of  2  10 4 Hz   hij  ~ 10 10    photons per cubic meter given by (38)  m graviton      T   max  4 n photon  (43) If we use LISA values for the Pulsar Gravitational h  c2wave frequencies, this may mean that the massive gravi- As a rough rule of thumb, if, as given by Weinberg [36]ton is ruled out. On the other hand that early quantum effects , for quantum gravity take M 90 km place at a temperature T  1033 Kelvin, then, if there   108 was that temperature for a cubic meter of space, the nu- 2.8M solar  mass R merical density would be , roughly 10132 times greater than what it is today. Forget it. So what we have to do isleads to looking at, if to consider a much smaller volume area. If the radii of  hij  ~ h ~ the volume area is 1/ 2 1/ 2 r  4  1035 meters  lP  Planck  length 515 Mpc   M  30 (39) 10       10 then we have to work with a de facto initial volume  r   2.8  M solar  mass   64  10105 ~ 10103 (meters)3 . i.e. the numerical value If the radius is of the order of r  10 billion for the number of photons at T  1033 , if we have a perlight-years ~4300 Mpc or much greater, so then we have, unit volume area based upon Planck length, in stead ofas an example   meters, cubed is 1029  5  108  5  1037 photons for aCopyright © 2011 SciRes. JMP
  • 9. 738 A. BECKWITHcubic area with sides r  4  1035 meters  lP at is how to obtain semi classical, minimal uncertaintyT quantum  effects  1033 Kelvin However, M ~ 5.44  1054 wave states, in this case de rigor for coherent wave func-initially transmitted photons! Either the minimum dis- tion states to form. Ford uses gravitons in a so calledtance, i.e. the grid is larger, or T quantum  effects  1033 “squeezed vacuum state” as a natural template for relicKelvin gravitons. i.e. the squeezed vacuum state (a squeezed We have, now, so far linked entropy, gravitons, and coherent state) is any state such that the uncertaintyalso information with certain qualifications. Next, we principle is saturated: in QM coherence would be whenwill attempt to quantify the treatment of gravitons, as xp   2 . In the case of string theory it would have togiven in Figure 1 above, with thin wall (box shape) betreatment of quantum mechanics rendition of a Graviton.  l2 xp   S   p  . 2When the thin wall approximation fails, we approach 2 2having a semi classical embedding for Gravitons. Corre-sponding to squeezed states, for gravitons, we will in- Putting it mildly, the string theory case is far more diffi-troduce coherent states of gravitons. cult. And that is the problem, with regards to string the- The next part of our discussion will be in linking se- ory, what is an appropriate vacuum expectation value forqueezed states, with a break down of the purely quantum treating a template of how to nucleate gravitons into amechanical modeling of gravitons. coherent state with respect to relic conditions [41]. Ford, in 1994, wrote a squeezed state operation S( ) via8. Issues about Coherent State of Gravitons   S    0 , Here, the operator. 0 is a ground (Linking Gravitons with GW) state, and frequently, as Ford did, in 1994, there is a definition of a root mean squared fluctuation of a gravi-In the quantum theory of light (quantum electrodynamics) ton/gravitational wave state via use of an average scalarand other bosonic quantum field theories, coherent states field  , wherewere introduced by the work of [37] Roy J. Glauber in 1 1 1 h2   hij hij    2  T 2 (45)1963 Now, what is appropriate for presenting gravitons 30 15 180 Thermai  bathas coherent states? Coherent states, to first approxima- Here, the value T Thermai bath has yet to be specified, andtion are retrievable as minimum uncertainty states. If onetakes string theory as a reference, the minimum value of that actually for energy values approximately of the or-uncertainty becomes part of a minimum uncertainty der of 1015 GeV which may be the mean temperature forwhich can be written as given by Venziano [38] (1993), the expanding universe mid way, to the end of inflation,where lS  10  lPlanck , with   0, and lPlanck  10 33 which does not equal current even smaller string theorycentimeters estimates as presented by Li et al. 2 h THERMAL  BATH   lS x     p  (44) 1018 Hz  hrms ~ 1030  1034 Hz  p  string theory values for inflationary Gravitational ampli- To put it mildly, if we are looking at a solution to tudes. i.e. the more modern treatments are predictingminimize graviton position uncertainty, we will likely be almost infinitesimal GW fluctuations. It is not clear fromout of luck if string theory is the only tool we have for Ford’s [41] treatmentof gravitons, and fluctuations, if heearly universe conditions. Mainly, the momentum will is visualizing fluctuation of gravitons/GW, but if onenot be small, and uncertainty in momentum will not be takes literally Equation (45) as a base line, and then con-small either. Either way, most likely, x  lS  10  lPlanck sidering what would be the optimal way to obtain a wayIn addition, it is likely, as Klaus Kieffer [39] in the book to obtain coherent states of gravitons, going to the Li“Quantum Gravity” on page 290 of that book that if stated value of hrms ~ 1039 Hz for solar plasma fromgravitons are excitations of closed strings, then one will the sun as a graviton source, would be a way of obtaininghave to look for conditions for which a coherent state of fluctuations 105  109 times weaker, i.e. going togravitons, as stated by [40] Mohaupt occurs. What Mo- hrms values so small that the requirement for a minimumhaupt is referring to is a string theory way to re produce fluctuation , in line with not contradictingwhat Ford gave in 1995, i.e. conditions for how Gravi- 2  lStons in a squeezed vacuum state, the natural result of   p  , 2 xp  quantum creation in the early universe will introduce 2 2metric fluctuations. Ford’s [41] treatment is to have a if we consider experimental conditions for obtainingmetric averaged retarded Green’s function for a massless x  hrms ~ 1039 / Hz . Note that this would put severefield becoming a Gaussian. The condition of Gaussianity restrictions upon the variations in momentum. A subjectCopyright © 2011 SciRes. JMP
  • 10. A. BECKWITH 739which will be referenced in whether or not the Li-Baker relic conditions, which may not involve squeezing? Notedetector can suitably obtain such small values of L. Grishchuk [44] wrote in “On the quantum state ofx  hrms ~ 1039 / Hz in detection capacity. To do so relic gravitons”, where he claimed in his abstract that “Itwill require an investigation into extreme sensitivity re- is shown that relic gravitons created from zero-pointquirements, for this very low value of hrms . Fanguy Li. quantum fluctuations in the course of cosmological ex-et al. [42] reports in their PRD document pansion should now exist in the squeezed quantum state. hrms ~ 1026  1030 Hz The authors have determined the parameters of the squeezed state generated in a simple cosmological model 5would require up to 10 seconds in evaluative time for a which includes a stage of inflationary expansion. It isclean signal, for GW. What will be asked in further sec- pointed out that, in principle, these parameters can betions is if or not the 105 seconds in evaluative time for a measured experimentally”. Grishchuk [45], et al., (1989)clean signal can evaluate additional data. i.e. what if one reference their version of a cosmological perturbationwould have to do to distinguish if or not coherent states hnlm via the following argument. How we work with theof gravitons which merge to form GW may be measured argument will affect what is said about the necessity, orvia the protocols brought up by Li et al. [42] for relic lack of, of squeezed states in early universe cosmology.GW. Now what could be said about forming states close From [44], where hnlm has a component  nlm  to classical representations of gravitons? Venkatartnam, obeying a parametric oscillator equation, where K is aand Suresh [43], built up a coherent state via use of a measure of curvature which is  1, 0 , a   is a scale displacement operator D    exp   a      a , ap-  factor of a FRW metric, and n  2π   a     is a  plied to a vacuum state , where  is a complex number, way to scale a wavelength,  , with n, and with a  and a, a  as annihilation, and creation operations l  a, a    1 , where one has hnlm  Planck  nlm    Gnlm  x  (49)   a     D    0 (46)  a   However, what one sees in string theory, is a situation nlm     n 2  K     nlm    0 (50)  awhere a vacuum state as a template for graviton nuclea-tion is built out of an initial vacuum state, 0 . To do If y       is picked, and a Schrodinger equationthis though, as Venkatartnam, and Suresh [43] did, in- a  volved using a squeezing operator Z  r ,   defining via is made out of the Lagrangian used to formulate Equa- iuse of a squeezing parameter r as a strength of squeezing ˆ tion (50) above, with Py  , and M  a 3   ,interaction term, with 0  r   , and also an angle of ysqueezing,  π    π as used in n2  K 2   , a   a   lPlanck    ,   r  a     Z  r ,    exp    exp(i )   a 2   exp(i )   a  2  , 2  and F   an arbitrary function. y   y  . Also, wewhere combining the Z  r ,   with (47) leads to a sin- have a finite volume V finite    3 gd 3 x .gle mode squeezed coherent state, as they define it via Then the Lagrangian for deriving Equation (50) is   (and leads to a Hamiltonian which can be also derived (48) from the Wheeler De Witt equation), with   1 for Z  r ,     Z  r ,   D    0  Z  r ,    0  0  zero point subtraction of energy The right hand side of Equation (48) given above be- M  y 2 M 2   2 a  y 2comes a highly non classical operator, i.e. in the limit L   a  F   (51)that the super position of states   Z  r ,    0  2a   2  0occurs, there is a many particle version of a “vacuum 1  ˆ  P2 1 1 state” which has highly non classical properties.  ˆ  H   y   M 2 y 2        (52) ˆSqueezed states, for what it is worth, are thought to occur i a    2M 2  2  at the onset of vacuum nucleation, but what is noted for   Z  r ,    0 being a super position of vac-  0  then there are two possible solutions to the S. E. Grush-uum states, means that classical analog is extremely dif- chuk [45] created in 1989, one a non squeezed state, andficult to recover in the case of squeezing, and general another a squeezed state. So in general we work withnon classical behavior of squeezed states. Can one, in   any case, faced with   D    0  Z  r ,    0 do a y     C    exp   B  y  (53) a  better job of constructing coherent graviton states, inCopyright © 2011 SciRes. JMP
  • 11. 740 A. BECKWITH The non squeezed state has a parameter 9. Necessary and Sufficient Conditions for String/Brane Theory Graviton Coherent B   B b   b 2    b States?where b is an initial time, for which the Hamiltonian A curved spacetime is a coherent background of gravi-given in Equation (52) in terms of raising/lowering op- tons, and therefore in string theory is a coherent stateerators is “diagnonal”, and then the rest of the time for Joseph Gerard Polchinski [46] starting with the typical  b , the squeezed state for y   is given via a pa- small deviation from flat space times as can be writtenrameter B for squeezing which when looking at a up by Guv  X   uv  huv  X  , with uv flat space time,squeeze parameter r, for which 0  r   , then Equa- and the Polyakov action, is generalized as follows, thetion (53) has, instead of B b   b 2 S Polyakov action is computed and compared with ex- ponentiated values B   B  ,  b    1  b S    d 2  g  g ab  Guv  X    a X   b X  (55)  4π  M i   a     cosh r  exp  2i    sinh r   (54)    becomes 2   a    2 cosh r   exp  2i    sinh r   exp   S  Taking Grishchuck’s formalism [45] literally, a state for  1  exp   S P    1  4π    d   g  g  huv  X    a X  b X  ...  a graviton/GW is not affected by squeezing when we are   2 ab  looking at an initial frequency, so that   b initially M (56)corresponds to a non squeezed state which may havecoherence, but then right afterwards, if   b which Polochinski writes that the term of order h in Equa-appears to occur whenever the time evolution, tion (56) is the vertex operator for the graviton state of the  string, with huv  X   4 g c  exp  ikX Suv  , and the   i   a    b action of S a coherent state of a graviton. Now the  b    b  B  ,  b     2   a    2 important question to ask, is if this coherent state of a graviton, as mentioned by Polochinski can hold up inA reasonable research task would be to determine, whe- relic, early universe conditions. R. Dick [47] argued asther or not stating that the “graviton multiplet as one particular dark b matter source in heterotic string theory. In particular, it is B   ,   b   pointed out that an appreciable fraction of dark matter 2would correspond to a vacuum state being initially formed from the graviton multiplet requires a mass generating phase transition around Tc  108 GeV, where the symme-right after the point of nucleation, with   b at time try partners of the graviton would evolve from an ultra-  b with an initial cosmological time some order of hard fluid to pressureless dark matter. indicates m  10magnitude of a Planck interval of time t  t Planck  1044 MeV for the massive components of the graviton multi-seconds The next section will be to answer whether or not plet”. This has a counter part in a presentation made bythere could be a point of no squeezing, as Grishchuck im- Berkenstein [48] with regards to BPS states, and SHOplied, for initial times, and initial frequencies, and an im- models for AdS5  S 5 geometry. The upshot is that stringmediate transition to times, and frequencies afterwards, theory appears to construct coherent graviton states, but itwhere squeezing was mandatory. Note that Grischchuk has no answer to the problem that Ford [41], wrote on if[45] further extended his analysis, with respect to the same the existing graviton coherent states would be squ- eezedpoint of departure, i.e. what to do with when into non classical configurations in relic conditions.   D    0  Z  r ,    0 . 10. Does LQG Give us More DirectHaving   D    0 with D   a possible dis- Arguments as to Coherent States,placement operator, seems to be in common Squeezed States, and the Break Down ofwith B b   b 2 , whereas   Z  r ,    0 which is Classical Behavior at the Onset ofhighly non classical seems to be in common with a solu- Inflation?tion for which B b   b 2  This leads us to thenext section, i.e. does B b   b 2 when of time Carlo Rovelli [49], in a PRL article states that a vertex t  t Planck  1044 seconds, and then what are the initial amplitude that contributes to a coherent graviton state isconditions for forming “frequency”   b ? the exponential of the Regge action: the other terms, thatCopyright © 2011 SciRes. JMP
  • 12. A. BECKWITH 741have raised doubts on the physical viability of the model, referring to the existence of squeezed states, as eitherare suppressed by the phase of the vacuum state, and being necessarily, or NOT necessarily a consequence ofRovelli writes a coherent vacuum state as given by a the quantum bounce. As Bojowald [56] wrote it up, inGaussian peaked on parts of the boundary  d of a four both his Equation (26) which has a quantum Hamiltoniandimensional sphere. ˆ V  H , with  q  s    q  , jm, n  (57) d Vˆ  0  Rovelli [49] states that “bad” contributions to the be- d existence  of  un  squeezed  states    0 (59)   0havior of Equation (57) are cancelled out by an appropri-ate (Gaussian?) vacuum wave functional which has “ap- ˆ and V is a “volume” operator where the “ volume” ispropriately” chosen contributions from the boundary  d set as V , Note also, that Bojowald has, in his initialof a four dimensional sphere. This is to avoid trouble Friedman equation, density valueswith “bad terms” from what is known as the Bar- H aret-Crane vertex amplitude contributions, which are can   matter 3 , abe iminized by an appropriate choice of vacuum state so that when the Friedman equation is quantized, with anamplitude being picked. Rovelli [49] calculated some initial internal time given by  , with  becoming acomponents of the graviton two-point function and found more general evolution of state variable than “internalthat the Barrett-Crane vertex yields a wrong time”. If so, Bojowald writes, when there are squeezedlong-distance limit. A problem, as stated by Lubos Motel states [51](2007) [50], that there are infinitely many other compo-nents of the correlators in the LQG that are guaranteed d V ˆnot to work unless an infinite number of adjustments are  N (value)  0  (60) d existence  of  squeezed  statesmade. The criticism is harsh, but until one really knows  0admissible early universe geometry one cannot rule out for his Equation (26), which is incidently when links tothe Rovelli approach, or confirm it. In addition, Jakub classical behavior break down, and when the bounceMielczarek [51] considered tensor gravitational perturba- from a universe contracting goes to an expanding presenttions produced at a bounce phase in presence of the universe. Bojowald [56] also writes that if one is lookingholonomy corrections. Here bounce phase and holonomy at an isotropic universe, that as the large matter “H” in-corrections originate from Loop Quantum Cosmology. creases, that in certain cases, one observes more classicalWhat comes to the fore are corrections due to what is behavior, and a reduction in the strength of a quantumcalled quantum holonomy, l. A comment about the bounce. Bojowalds [56] states that “Especially the role ofquantum bounce. i.e. what is given by Dah-Wei Chiou, squeezed states is highlighted. The presence of a bounceLi-Fang Li [52], is that there is a branch match up be- is proven for uncorrelated states, but as squeezing is atween a prior to a present set of Wheeler De Witt equa- dynamical property and may change in time”. The upshottions for a prior to present universe, as far as modeling is that although it is likely in a quantum bounce state thathow the quantum bounce links the two Wheeler De Witt the states should be squeezed, it is not a pre requisite forsolution branches, i.e. one Wheeler De Witt wave func- the states to always start off as being squeezed states. Sotion for a prior univers, and another wave function for a a physics researcher can, look at if an embedding of thepresent universe. Furthermore, Abhay Ashtekar [53] present universe in a higher dimensional structure whichwrote a simple treatment of the Bounce causing Wheeler could have lead to a worm hole from a prior universe toDe Witt equation along the lines of, for our present for re introduction of inflationary growth.   const  1 8πG   as a critical density, and  theeignvalue of a minimum area operator. Small values of 11. Other Models. Do Worm Hole Bridges  imply that gravity is a repulsive force, leading to a between Different Universes Allow forbounce effect [54]. Initial un Squeezed States? Wheeler de  2 Witt Solution with Pseudo time a 8πG       1        H .O.T . (58) Component Added in  a 3 Furthermore, Bojowald [55,56] specified a criteria as This discussion is to present a not so well known butto how to use an updated version of  and useful derivation of how instanton structure from a prior   const  1 8πG   in his GRG manuscript on what universe may be transferred from a prior to the presentcould constitute grounds for the existence of generalized universe. This discussion is partly rendered in [15], but issqueezed initial (graviton?) states. Bojowald [56] was reproduced here due to the relatively unknown feature ofCopyright © 2011 SciRes. JMP
  • 13. 742 A. BECKWITHa pseudo time component to the Wheeler de Witt equa- dr 2tion dS 2   F  r   dt 2   d 2 (65) F r  1) The solution as taken from L. Crowell’s [57] book,and re produced has many similarities with the WKB This has:method. i.e. it is semi classical. 2M Q 2 2) Left unsaid is what embedding structure is assumed. F r   1  2 3) A final exercise for the reader. Would a WKB style r r (66)  2 solution as far as transfer of “material” from a prior to a  r     r  lP  2  T 1032 Kelvin ~  present universe constitute procedural injection of non 3 3compressed states from a prior to a present universe? This assumes that the cosmological vacuum energyAlso if uncompressed, coherent states are possible, how parameter has a temperature dependence, leading tolong would they last in introduction to a new universe? F  This is the Wheeler-de-Witt equation with pseudo time ~ 2    r  lP    T    r  lP  (67)component added. From Crowell [57] r 3 1 2 1  as a wave functional solution to a Wheeler-de-Witt equa-  rR      r  r    3   2  (61) tion bridging two space-times, similar to two space-times  r r 2  r r with “instantaneous” transfer of thermal heat, as given by This has when we do it   cos   t  , and frequently Crowell [57]R    constant, so then we can consider   3  T    A   2  C1  A    2  C2 (68)     d  a    eik x  a     eik x    (62) This has C1  C 1  , t , r  as a pseudo cyclic and   0 evolving function in terms of frequency, time, and spatial In order to do this, we can write out the following for function. This also applies to the second cyclical wavethe solutions to Equation (61) above. function C2  C2  , t , r  , where C1  Equation (63) and C2  Equation (64) then we get that Equation (68)  t C1   2   4  π  5  J1   r   is a solution to the pseudo time WDM equation.  2 The question which will be investigated is if Equation 4  (68) is a way to present either a squeezed or un squeezed  sin   r     r   cos   r   (63) 5  state. A way forward is to note that Prado Mar- 15 6 tin-Moruno, Pedro F. Gonzalez-Diaz in July [58] wrote  5 cos   r   5 Si   r  up about thermal phantom-like radiation process coming   from the wormhole throat. Note that the Crowell con-and struction of a worm hole bridge is in some ways similar 3 6 to Carco Cavaglià’s [59] treatment of use of conjugate 4 C2   1  cos   r    4e  r  4  Ci   r  (64) momentum π ij of hij generalized momentum vari- 2   ables, also known as conjugate momenta This is where Si   r  and Ci   r  refer to inte  grals of the form πij   ˆ i   hij , x sin  x   x cos  x   leading to the sort of formalism as attributed to Luis J.  x dx and  x dx  . Garay’s [60] article, of   Next, we should consider whether or not the instanton   hij   exp  d 3 x  πij  hij    (69) Tso formed is stable under evolution of space-time leading Now in the case of what can be done with the wormup to inflation. To model this, we use results from Cro-well [57] on quantum fluctuations in space-time, which hole used by Crowell [57], with, if   1 ,gives a model from a pseudo time component version of  i  πij  i ˆ π   ˆthe Wheeler-De-Witt equation, with use of the Reinss-   gij , 2r   r ,ner-Nordstrom metric to help us obtain a solution that 1passes through a thin shell separating two space-times.  F  r    π  i   tt ˆ   ,The radius of the shell r0  t  separating the two  r    rspace-times is of length lP in approximate magnitude, and a kinetic energy value as given of the formleading to a domination of the time component for the π πtt  πtt π . The supposition which we have the ˆ ˆ ˆ ˆReissner-Nordstrom metric worm hole wave functional may be like, so, use the waveCopyright © 2011 SciRes. JMP
  • 14. A. BECKWITH 743functional looking like The question raised repeatedly in whether or not 1) if higher dimensions are necessary, and whether or not 2)   gij   exp  d x   π   gij   3   ij T mass gravitons are playing a role as far as the introduc- tion of DE speed up of cosmological expansion may leadwhere the gij for the Weiner- Nordstrom metric will be to an improvement over what was specified for density dr 2 fluctuations and structure formation (the galaxy hierar- dS 2   F  r   dt 2   d 2  chy problem) of density fluctuations given as F r    dr 2 ~ 10 4    1010 (73)   r  lP   dt 2   2 g ij dxi dx j   d 2 3    r  lP  2 Equation (73) is for four space, a defining moment as 3 to what sort of model would lead to density fluctuations. It totally fails as to give useful information as to the gal-12. Unanswered Questions, and What This axy hierarchy problem as given in Figure 1, above. Suggests for Future Research Endeavors Secondly, to what degree is the relative speed up of the q(z ) function is impacted by various inter plays between ,As far back as 1982, Linde [61], when analyzing a po- say a modified version of, say a KK DM model, using atential of the form modified mass hierarchy to get suitable DM masses of the order of 100 GeV or more. Giving a suitable defini- m 2 2 V      4  V (0) (70) tion as to q(z) as well as the inter play between DM val- 2 ues, 4 Dim Graviton mass issues, and/or what really con- This is when the “mass” has the form, (here M is the tributes to the speed up of the universe will in the endbare mass term of the field  in de Sitter space, which dramatically improve the very crude estimate given bydoes not take into account quantum fluctuations) Equation (70) above which says next to nothing about 3 H 3 how the problems illustrated by the break down of the m 2 (t )  M 2    t  t0  (71) galaxy mass formation/hierarchy can be fixed. Further- 4 more is considering the spectral index problem, where Specified non linearity of   2  at a time from the the spectral index isbig bang, of the form 2 2 3H 3  V  1  V  t1  (72) nS  1         (74) 2M 8π  V  4π  V  Figure 1. Here, the left hand side corresponds to a soliton, the right hand side is an anti soliton [24].Copyright © 2011 SciRes. JMP
  • 15. 744 A. BECKWITH Usual experimental values of density fluctuations ex- should be compared with admissible values ofperimentally are  nS  1 2  which would closely correspond to   0   and 0    0.2 . i.e. the precise values of this may help ~ 10 5 , instead of ~ 10 4 ,   us out in determining how to unravel what is going on inand this is assuming that  is extremely small. In addi- the galaxy formation i.e. how can we have earlier thantion, Linde [61] had expected galaxy formation? d H 1 a  V  m2    13. Conclusions, as to How to Look at Early d 2 40 40 a Universe Topology and Later Flat Spaceinside a false vacuum bubble. If something other than theKlein Gordon relationship One of the aspects of early universe topology we need to  a consider is how to introduce a de facto break down of    3H   m 2  0   quantization in curved space time geometries, and this is a a problem which 3/ 2 would permit a curved space treatmentoccurs, then different models of how density fluctuation of  ~  R / Req  . i.e. as R gets of the order of  may have to be devised. A popular model of density R ~   lP  , say that the spatial geometry of early uni-fluctuations with regards to the horizon is verse expansion is within a few orders of magnitude of Planck length, then how can we recover 3/a field theory    k 3/ 2  k k  3/ 2   3 3/ 2      Horizon  2π  2π    1/ 2π  k 3   2 quantization condition for  ~  R / Req  in terms of path integrals. We claim that deformation quantization, if (75) applied successfully will eventually lead to a great re-where 1    0.2 , and   0  ns  1 and to first finement of the above Wheeler De Witt wave functionalorder, k  Ha . The values, typically of ns  1 If value, as well as allow a more through match up of aworking with time independent solution of the Wheeler de Witt equa- tion, with the more subtle pseudo time dependent evolu-  a 2   2  C tion of the wave functional as Crowell wrote up. i.e. the H 2        4  , a 2 2  3M 4 36 M Planck  a  linkage between time independent treatments of the wave   functional of the universe, with what Lawrence Crowelland with a density value [52] wrote will be made more explicit. This will, in addi- tion allow us to understand better how graviton produc-  a0   mg c   a 4 2a 2 1  3 6   0      2      tion in relic conditions may add to entropy, as well as  a   8πG    14   5 2 how to link the number of gravitons, say 1012 gravitonswhere mg  1065 grams, and   0.2 is usually picked per photon, as information as a way to preserve the con-to avoid over production of black holes, a very complex tinuity of  values from a prior universe to the presentpicture emerges. Furthermore, if working with   0.2 universe.and   0 14. The Author Claims that in Order to do    H2      Horizon    1/ 2π  k 3 ~   10 4 105 (76) This Rigorously, That Use of the Material in [63] Gutt, and Waldmann Will be Necessary, Especially to The above equation gives inter relationships between Investigate if Quantization of Severelythe time evolution of a pop up inflaton field  , and a Curved Space Time Conditions IsHubble expansion parameter H, and a wave length pa- Possible. We Claim that It Is Not [15]rameter    2π k   a  t  for a mode given as  k .What should be considered is the inter relation ship of 3/ 2 Resolution of which add more detail to  ~  R / Req  .the constituent components of Equation (76) and   Having said this, it is now important to consider what   H 1 . What the author thinks is of particular import can be said about how relic gravitons/ information canis to look at whether or not the more general expression, pass through minimum vales of R ~   lP  .as given by the below equation also holds [62]. We shall reference what A. W. Beckwith presented      n S 1   [64], which we think still has current validity for reasons    Ak  2   10  4  10 5 (77) we will elucidate upon in this document. We use a power    law relationship first presented by Fontana [65], who To first order, variations of   0.2 and   0 , used Park’s earlier [66] derivation: whenCopyright © 2011 SciRes. JMP
  • 16. A. BECKWITH 745Eeff  n( )    eff  S  k   ln (81)  0 mgraviton 2  L 4 net 6 P ( power )  2  If the phase spaces can be quantified, as a starting  45  c 5  G  (78) point of say lmin length  string  10  lPlanck , with l Planck be- This expression of power should be compared with the ing part of how to form the “dimensions” of  0 , andone presented by Giovannini on averaging of the en- lmin length  string part of how to form the dimensions of  ,ergy-momentum pseudo tensor to get his version of a and 10 being, for a given   0 , and in certain casesgravitational power energy density expression, namely   0 , then avoiding having dS/dt = ∞ at S = 0 will be[67] straight forward We hope to come up with an emergent structure for gravitational fields which is congruent with H   H 4  obtaining 10 naturally, so this sort of procedure is non 2 27 GW  , 0    3 H 2     1     4   (79) 256  π 2 M    M  controversial, and linked to falsifiable experimental   measurement protocol, so quantum gravity becomes a de Giovannini states that should the mass scale be picked facto experimental science. This will mean looking atsuch that M ~ mPlanck  mgraviton , that there are doubts Appendix B, fully. Appendix C, and Appendix D givethat we could even have inflation. However, it is clear further issues we describe later on. In future publications.that gravitational wave density is faint, even if we make We give them as pertinent information for the futurethe approximation that development of this project. a m  H  15. Acknowledgements a 6as stated by [68] by Linde, where we are following This work is supported in part by National Nature Sci-   m 2 3 in evolution, so we have to use different  ence Foundation of China grant No. 11075224. The au-procedures to come up with relic gravitational wave de- thor thanks Dr. Fangyu Li for conversations as to thetection schemes to get quantifiable experimental meas- physics of GW and Graviton physics, and also has a debturements so we can start predicting relic gravitational of gratitude to Stuart Allen, CEO for his efforts to permitwaves. This is especially true if we make use of the fol- the author to do physics worklowing formula for gravitational radiation, as given by.Kofman [69], with M  V 1/ 4 as the energy scale, with a 16. Referencesstated initial inflationary potential V. This leads to aninitial approximation of the emission frequency, using [1] M. Y. Kuchiev, “Can Gravity Appear Due to Polarizationpresent-day gravitational wave detectors. of Instantons in SO(4) Gauge Theory?” Classical and Quantum Gravity, Vol. 15, No. 7, 1998, pp. 1895-1913. ( M  V 1/ 4 ) doi:10.1088/0264-9381/15/7/008 f  Hz (80) 107 GeV [2] I. Andrić, L. Jonke and D. Jurman, “Solitons and Giants in Matrix Models,” Progress of Physics, Vol. 56, No. 4-5, What we would like to do for future development of 2008, pp. 324-329.entropy would be to consider a way to ascertain if or not [3] D. Perkins, “Particle Astrophysics,” Oxford Universitythe following is really true, and to quantify it by an im- press, Oxford, 2003.provement of a supposition advanced by [70] Kiefer, [4] L. Glinka, “Preliminaries in Many-Particle QuantumPolarski, and Starobinsky. i.e. the author, Beckwith, has Gravity. Einstein-FriedmannSpacetime,”in this document presented a general question of how to http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.1380v4.pdavoid having dS/dt = ∞ at S = 0, f; 1) Removes any chance that early universe nucleation [5] L. Glinka, “Quantum Information from Graviton-Matteris a quantum based emergent field phenomena. Gas,” Symmetry, Integrability and Geometry: Methods 2) Goldstone gravitons would arise in the beginning and Applications, Vol. 3, No. 087, 2007, pp. 1-13.due to a violation of Lorentz invariance. i.e. we have a [6] Y. J. Ng, “Spacetime Foam: From Entropy and Hologra-causal break, and merely having the above condition phy to Infinite Statistics and Nonlocality,” Entropy, Vol.does not qualify for a Lorentz invariance breakdown 10, No. 4, 2008, pp. 441-461. Kiefer, Polarski, and Starobinsky [70] presented the doi:10.3390/e10040441idea of presenting the evolution of relic entropy via the [7] M. Asakawa, T. Hatsuda and Y. Nakahara, “Maximumevolution of phase spaces, with   0 being the ratio of entropy analysis of the spectral functions in lattice QCD,”“final (future)”/“initial” phase space volume, for k modes Progress in Particle and Nuclear Physics, Vol. 46, No. 2,of secondary GW background. 2001, pp. 459-508.Copyright © 2011 SciRes. JMP
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  • 19. 748 A. BECKWITH[74] P. Chen, “Resonant Phton-Graviton Conversion in EM and Particle Science, Vol. 57, 2007, pp. 245-283. Fields: From Earth to Heaven,” 1994. doi:10.1146/annurev.nucl.54.070103.181232 http://www.slac.stanford.edu/cgi-gedoc/slac-pub-6666.pd [77] R. Durrer, “Cosmological Perturbation Theory,” In: E. f Papantonopoulous, Ed., Physics of the Early Universe,[75] T. Rothman and S. Boughn, “Can Gravitons be De- Lecture Notes in Physics 653, Springer-Verlag, Berlin, tected?” Foundations of Physics, Vol. 36, No. 12, 2006, 2004. pp. 1801-1825. doi:10.1007/s10701-006-9081-9 [78] S. Capozziello, A. Feoli, et al., “Thin Shell Quantization[76] D. Samtleben, S. Staggs and B. Witnstein, “The Cosmic in Weyl Spacetime,” Physics Letters A, Vol. 273, No. 1, Microwave Background for Pedestrians, a review for Par- 2000, pp. 25-30. doi:10.1016/S0375-9601(00)00478-3 ticle and Nuclear Physicists,” Annual Review of NuclearAppendix A: ment accepts that there is a small graviton mass, whichBounds upon Graviton Mass, and Making the author has estimated to be on the order of 1060 kilograms. Small enough so the following approximationUse of the Difference between Graviton is valid. Here, vg is the speed of graviton “propaga-Propagation Speed and HFGW Transit tion”, g is the Compton wavelength of a graviton withSpeed to Observe Post Newtonian Correc- g  h mg c , and f  1010 Hertz in line with L. Gris-tions to Gravitational Potential Fields chuck’s treatment of relic HFGW’s [26]. In addition, the high value of relic HFGW’s leads to naturally fulfillingThe author presents a post Newtonian approximation hf  mg c 2 so that [71]based upon an earlier argument/paper by Clifford Will as v c   2 2 g  1  c g f (A1)to Yukawa revisions of gravitational potentials in partinitiated by gravitons with explicit mass dependence in But Equation (1) above is an approximation of a muchtheir Compton wave length. more general result which may be rendered as v c   2 2Introduction g  1  mg c 2 E (A2) The terms mg and E refers to the graviton rest massPost Newtonian approximations to General relativity and energy, respectively. Now specifically in line withhave given physicists a view as to how and why infla- applying the Li Baker detector, [42] physics researcherstionary dynamics can be measured via deviation from can ascertain what E is, with experimental data from thesimple gravitational potentials. One of the simplest de- Li Baker detector, and then the next question needs to beviations from the Newtonian inverse power law gravita- addressed, namely if D is the distance between a detector,tional potential being a Yukawa potential modification of and the source of a HFGW/Graviton emitter sourcegravitational potentials. So happens that the mass of a  200Mpc   t graviton would factor directly into the Yukawa exponen- 1  vg c  5 1017      (A3)tial term modification of gravity. This appendix indi-  D   1sec cates how a smart experimentalist could use the Li-Baker The above formula depends upondetector as a way to obtain more realistic upper bounds t  ta  (1  Z )te , with where ta and te are theas to the mass of a graviton and to use it as a template to differences in arrival time and emission time of the twoinvestigate modifications of gravity along the lines of a signals (HFGW and Graviton propagation ), respectively,Yukawa potential modification as given by Will [71]. and Z is the redshift of the source. Z is meant to be the red shift. Specifically, the situation for HFGW is that forGiving an Upper Bound to the Mass of a early universe conditions, that Z  1100 , in fact for veryGraviton early universe conditions in the first few mili seconds after the big bang, that Z ~ 1025 . An enormous number.The easiest way to ascertain the mass of a graviton is to The first question which needs to be asked is, if or notinvestigate if or not there is a slight difference in the the Visser [72] non-dynamical background metric correct,speed of graviton ‘particle’ propagation and of HFGW in for early universe conditions so as to avoid the problemtransit from a “source” to the detector. Visser’s [72] of the limit of small graviton mass does not coincidemass of a graviton paper [72] presents a theory which with pure GR, and the predicted perihelion advance, forpasses the equivalence test, but which has problem with example, violates experiment. A way forward would bedepending upon a non-dynamical background metric. I.e. to configure data sets so in the case of early universegravitons are assumed by both him, and also Will’s [71] conditions that one is examining appropriate Z  1100write up of experimental G.R. to have mass. This docu- but with extremely small te times, which would re-Copyright © 2011 SciRes. JMP
  • 20. A. BECKWITH 749flect upon generation of HFGW before the electro weak  100 Hz  1/ 2  1  1/ 2 c Dtransition, and after the INITIAL onset of inflation. i.e. mg    3  1012 km      the Li-Baker detector system should be employed [42] as  h  200 Mpc f   f  ta to pin point experimental conditions so to high accuracy, (A9)the following is an adequate presentation of the differ- Needless to say that an estimation of the bound for theence in times, t . i.e. graviton mass mg , and the resulting Compton wave- t  ta  (1  Z )te  ta     ta ] (A4) length g would be important to get values of the fol- lowing formula, namely The closer the emission times for production of the MGHFGW and Gravitons are to the time of the initial nu- V  r  gravity  exp  r g  (A10)cleation of vacuum energy of the big bang, the closer we rcan be to experimentally using Equation (4) above as to Clifford Will gave for values of frequency f  100give experimental criteria for stating to very high accu- Hertz enormous values for the Compton wave length, i.e.racy the following. values like g  6  1019 kilometers. Such enormous  200 Mpc   ta  values for the Compton wave length make experimental 1  vg c  5  1017      (A5) tests of Equation (A10) practically infeasible. Values of  D   1sec  g  105 centimeters or less for very high HFGW data More exactly this will lead to the following relation- makes investigation of Equation (A10) above far moreship which will be used to ascertain a value for the mass tractable.of a graviton. By necessity, this will push the speed ofgraviton propagation very close to the speed of light. In Application to Gravitational Synchrotronthis, we are assuming an enormous value for D Radiation, in Accelerator Physics  200 Mpc   ta  Eric Davis, quoting Pisen Chen’s article [74] estimates vg c  1  5  1017      (A6)  D   1sec  that a typical storage ring for an accelerator will be able to give approximately 106  103 gravitons per second. This Equation (A6) relationship should be placed into Quoting Pisen Chen’s [75] 1994 article, the following forg  h / mg c with a way to relate this above value of graviton emission values for a circular accelerator system, with m the mass of a graviton, and M P being Planck v c   2 2 g  1  mg c 2 E , mass. N as mentioned below is the number of ‘particles’ in a ring for an accelerator system, and nb is an accel- with an estimated value of E coming from the Li- erator physics parameter for bunches of particles whichBaker detector [42] and field theory calculations, as well for the LHC is set by Pisen Chen [74] as of the valueas to make the following argument rigorous, namely 2800, and N for the LHC is about 1011 . And, for the 2 LHC Pisen Chen sets  as 88  102 , with  200 Mpc ta  2  mg c 2  1  5  1017    1   (A7)   m  4300 . Here, m ~ mgraviton acts as a mass  D 1sec   E      charge. m2 c   4 A suitable numerical treatment of this above equation, N GSR ~ 5.6  nb  N 2  2  2 (A11)with data sets could lead to a range of bounds for mg , as MP a refinement of the result given by Will [71] for graviton The immediate consequence of the prior discussionCompton wavelength bounded behavior for a lower would be to obtain a more realistic set of bounds for thebound to the graviton mass, assuming that h is the graviton mass, which could considerably refine the esti-Planck’s constant. mate of 1011 gravitons produced per year at the LHC, with realistically 365 × 86400 seconds = 31536000 sec- 1/ 2 1/ 2 h  D 100 Hz   1  onds in a year, leading to 3.171  103 gravitons pro- g   3  1012 km       duced per second. Refining an actual permitted value of mg c  200 Mpc f   f t  bounds for the accepted graviton mass, m, as given 1/ 2 1/ 2  D 100 Hz   1  above, while keeping M p  1.2209 × 1019 GeV/c2 would  3  1012 km       allow for a more precise set of gravitons per second  200 Mpc f   f t a  which would significantly enhance the chance of actual (A8) detection, since right now for the LHC there is too much The above Equation (A8) gives an upper bound to the general uncertainty as to the likelihood of where to placemass mg as given by a detector for actually capturing/detecting a graviton.Copyright © 2011 SciRes. JMP
  • 21. 750 A. BECKWITHConclusion, Falsifiable Tests for the Graviton achievable precision given byAre Closer than the Physics Community Thinks  4   Texp   2 Cl 2  1 f sky  el  b  2 2    The physics community now has an opportunity to ex- Cl 2l  1  f sky Cl perimentally infer the existence of gravitons as a know-  able and verifiable experimental datum with the onset of (B3)the LHC as an operating system. Even if the LHC is notused, Pisen Chens parameterization of inputs from his f sky is the fraction of the sky covered in the meas-table [74] right after his Equation (8) as inputs into urement , and Texp is a measurement of the total ex-Equation (A11) above will permit the physics commu- perimental sensitivity of the apparatus used. Also  b isnity to make progress as to detection of Gravitons for, the width of a beam, while we have a minimum value ofsay the Brookhaven site circular ring accelerator system. lmin  1   which is one over the fluctuation of theTony Rothman’s [75] predictions as to needing a detec- angular extent of the experimental survey.tor the size of Jupiter to obtain a single experimentally i.e. contributions to Cl uncertainty from sample vari-falsifiable set of procedures is defensible only if the ance is equal to contributions to Cl uncertainty fromwave-particle duality induces so much uncertainty as to noise. The end result isthe mass of the purported graviton, that worst case modelbuilding and extraordinarily robust parameters for a  4π  f sky  Cl  exp  l 2 2      T  2 (B4)Rothman style graviton detector have to be put in place. The Li-Baker detector [42] can help with bracketing a Appendix C: Cosmological Perturbationrange of masses for the graviton, as a physical entitysubject to measurements. Such an effort requires obtain- Theory and Tensor Fluctuations (Gravitying rigorous verification of the approximation used to the Waves)effect that t  ta  (1  Z )te  ta     ta is adefensible approximation. Furthermore, obtaining realis- Durrer [77] reviews how to interpret Cl in the regiontic inputs for distance D for inputs into Equation (A9) where we have 2  l  100 , roughly in the region of theabove is essential. The expected pay offs of making such Sachs-Wolf contributions due to gravity waves. Wean investment would be to determine the range of valid- begin first of all by looking at an initial perturbation,ity of Equation (A10), i.e. to what degree is gravitation using a scalar field treatment of the “Bardeen potential”as a force is amendable to post Newtonian approxima-  This can lead us to put up, if H i is the initial valuetions. The author asserts that Equation (10) can only be of the Hubble expansion parameterrealistically be tested and vetted for sub atomic systems, 2 2 H and that with the massive Compton wavelength specified k3    i  (C1)by Clifford Will cannot be done with low frequency  MP gravitational waves.Furthermore, a realistic bounding of andthe graviton mass would permit a far more precise cali-bration of Equation (A11) as given by Pisen Chen in his  2  k 3  A 2k n 1 0 1 n1994 article [74]. (C2) Here we are interpreting A  amplitude of metricAppendix B. Basic Physics of Achieving perturbations at horizon scale, and we set k  1 / 0 ,Minimum lmin length  string  10  lPlanck Precision where  is the conformal time, according toin CMBR Power Spectra Measurements dt  ad  physical time, where we have a as the scale factor. Then for 2  l  100 , and 3  n  3 , andBegin first of all looking at a pure power law given by T   almYl , m  ,   (B1) H  k ,  1 / k  2   k 3  A T2k nT 0 n T T l ,m (C3)This leads to consider what to do with We get for tensor fluctuation, i.e. gravity waves and a 2 scale invariant spectrum with nT  0 Cl  al , m (B2) 2 AT 1 Cl T Samtleben et al. [76] consider then what the experi-   (C4)mental variance in this power spectrum, to the tune of an  l  3   l  2  15Copyright © 2011 SciRes. JMP
  • 22. A. BECKWITH 751Appendix D: Linking the Thin Shell Ap- a    xZ1/6  x3  ~ x 2/3 (D2)proximation, Weyl Quantization, and the 3 Wheeler de Witt Equation Similarly, x  1 leads toThis is a re capitulation of what is written by S. Ca-  a    xZ1/6  x3  (D3)poziello, et al. [78] for physical review A, which is as-  3 2 suming a generally spherically symmetric line element.The upshot is that we obtain a dynamical evolution equa- Also, when x  1tion, similar in part to the Wheeler De Witt equation 3 8 3/ 2 which can be quantified as H   0    2a 2   x  1  Z 3/ 4   a   x  1    (D4)   3  Which in turn will lead to, with qualifications, for thinshell approximations x  1 , Realistically, in terms of applications, we will be consid- ering very small x values, consistent with conditions    a 2 x 4   0 (D1) near a singularity/worm hole bridge between a prior toso that Z1/6 is a spherical Bessel equation for which we our present universe. This is for x  R Requilibrium .can writeCopyright © 2011 SciRes. JMP