Journal of Modern Physics, 2011, 2, 730-751 doi: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: [email protected] Received April 12, 2011; revised June 1, 2011; accepted June 13, 2011
Abstract
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 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 Model
1. Introduction condensed matter application. The string theory metho- dology is merely extending much the same thinking up to Gravitons 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 entropy sition, 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 two graviton production with entropy, provided we have structures are related to each other in terms of kinks and sufficiently HFGW at the origin of the big bang. The anti-kinks. It is asserted that how they form and break up linkage 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 entropy kink-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 after In this vacuum the instantons and anti-instantons have a a second-order phase transition, another rapid increase in preferred direction of orientation”, and furthermore entropy. “Gravitons appear as the mode describing propagation of The supposition we are making here is that the value the gauge field which strongly interacts with the oriented of N so obtained is actually proportional to a numerical instantons” Furthermore, as given by Ivan Andrić, Larisa graviton density we will refer to as
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a that Dr. Jack Ng’s research into entropy [6] not only used Hm a 3 the Shannon entropy model, but also as part of his quan- tum infinite statistics lead to a quantum counting algo- where we assume 0 due to inflation rithm with entropy proportional to “emergent field” par-
2 ticles. If as an example a quantum graviton gas exists, as Hm5 ~~~Ht ~10 suggested by Glinka[4,5] if each quantum gas “particle” (1) MMPlanck Planck is equivalent to a graviton, and that graviton is an “emergent” from quantum vacuum entity, then we for- At the very onset of inflation, M Planck , and if 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 . 88 This is a counter part as to what Asakawa et al. [7] sug- ascertain where SNgravitons 10 comes from and plays a role as to the development of entropy in cosmo- gested for quark-gluongases, and the 2nd order phase logical evolution In the next Chapter , we will discuss if transition written up by Torrieri et al. [10] brought up at or not it is feasible/reasonable to have data compression the nuclear physics Erice school, in discussions with the of prior universe “information”. It suffices to say that if author. 5 Furthermore, finding out if or not it is either a drop in Sinitial ~10 is transferred from a prior universe to our own universe at the onset of inflation, at times less than viscosity [7,8,9], then Planck time t ~1044 seconds, that enough information P 1 , MAY exit for the preservation of the prior universe’s s 4π cosmological constants, i.e. , G, (fine structure constant) and the like. Confirmation of this hypothesis or a major increase in entropy density may tell us how depends 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 effective of 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 present 2. 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 singularity It is useful to state this convention for analyzing the re- theorems do not give precise answers as to information sulting entropy calculations, because it is a way to ex- flow from a prior to the present universe. Hawking’s plain how and why the number of instanton-anti in- singularity theorem is for the whole universe, and works stanton pairs, and their formulation and break up can be backwards-in-time: it guarantees that the big-bang has linked to the growth of entropy. If, as an example, there infinite density. This theorem is more restricted, it only is 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 that number 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 of worm 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 a tropy. Such an approach may permit asking how infor- fifth dimensional leakage of “information” from a prior mation 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 the lence between a quantum number, n, say of a quantum origin, then perhaps no information from a prior universe graviton 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
Copyright © 2011 SciRes. JMP 732 A. BECKWITH and the fine structure constant. arose, and may have is not justified analytically. i.e. it breaks down. Beckwith been 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], if proof of the singularity theorems is the Raychaudhuri 2 equation, which describes the divergence θ of a congru- ,1H t ence (family) of geodesics, which has a lot of assump- 16πG tions behind it, as stated by Naresh Dadhich [11]. As so that one has a scale factor behaving as indicated by Hawkings theorem, infinite density is its at() t1/ usual modus operandi, for a singularity, and this assump- (5) tion may have to be revisited. Natário [12] has more de- Then, if tails on the different type of singularities involved. The 2 supposition is that the value of N is proportional to a VG 4π (6) numerical DM density referred to as
Vg (4) 3. Linking Instaton-Anti Instaton Construction in both Entropy Generation De facto, what we come up with pre, and post and Gravitons Planckian space time regimes, when looking at consis- tency 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 tt 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 tt 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 peoples
Copyright © 2011 SciRes. JMP A. BECKWITH 733 visualization, an outrageous simplication, the issue of EMMTotal4 p j,0 p j ,0 squeezing 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 treating analogy done for initially non squeezed states, which the mass and the energy of the brane contribution along when squeezed have their own non quantum mechanical the lines of Mathur’s CQG article [15] where he has a flavor. 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 via stanton-anti instanton construction and go from there: nnLTnLTE11 22 1 , 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 a of a minimum Planck length for a grid of space-time of cycle of length L, which leads to an entropy defined in mT L Planck length, whereas this grid is changed to something terms of an energy value of mass of iP j (TP bigger is the tension of the ith brane, and Lj are spatial di- mensions of a complex torus structure). The toroidal lNl~1033 cm P Quantum Gravity threshold P . structure is to first approximation equivalent dimension- ally to the minimum effective length of Nl P ~ N So far, we this only covers a typical string gas model for times Planck length 1035 centimeters entropy. N is assigned as the as numerical density of brains and anti-branes. A brane-antibrane pair corre- EmnTotal 2 i i (11) sponds to solitons and anti-solitons in density wave phys- i ics. 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 circle anti-instanton structure. First, a similar pairing in both midway in the length of a cylinder. The structure the black hole models and models of the early universe is string wraps about is a compact object construct Dp examined, and a counting regime for the number of in- branes and anti-branes. Compactness is used to roughly stanton 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”. This and in early universe models is employed as a way to get leads to entropy expressed as a strict numerical count of a net entropy-information count value. One can observe different pairs of Dp brane-Dp anti-branes, which form a this 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’s which 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 stated lieu 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 SNgravitons ber of branes in an early universe configuration, while N M pj,0 is anti-brane number. i.e., there is a kink in the SAnTotal i (12) given i brane~ Mpj,0 CDWe Now, how do we make sense of the following entropy electron charge and for the corresponding anti-kink values? Note the following: As an example of present confusion, please consider the anti brane~ Mpj,0 CDWe following discussion where leading cosmologists, i.e. Sean Carroll [17] asserted that there is a distinct possibility that N positron charge. Here, in the bottom expression, is mega black holes in the center of spiral galaxies have the number of kink-anti-kink charge pairs, which is more entropy, in a calculated sense, i.e. up to 1090 in non analogous to the simpler CDW structure [17]. dimensional units. This has to be compared to Carroll’s 88 N [17] stated value of up to 10 in non dimensional units for ETotal SaTotal~ n M p j,0 M p j ,0 (9) observable non dimensional entropy units for the observ- 2 j1 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, each 1090 brane-anti-brane pairs) has a net energy value of ETotal with entropy , and then there is due to spiral galaxy 1069096 10 10 as expressed in Equation (9) above, where ETotal is entropy contributions entropy units to proportional to the cosmological vacuum energy pa- contend with, vs. 1088 entropy units to contend with for rameter; in string theory, ETotal is also defined via the observed universe. i.e. at least a ten to the eight order
Copyright © 2011 SciRes. JMP 734 A. BECKWITH difference in entropy magnitude to contend with. The 4. Different Senarios for Entropy Growth author is convinced after trial and error that the standard Depending Upon if or Not We Have Low which should be used is that of talking of information, in to High Frequency GW from the Big Bang the Shannon sense, for entropy, and to find ways to make a 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 the entropy as being written as S ~ ln[partition function]. big bang. To begin with let us look at frequency range of This 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 the What is chosen as a partition function will vary with our amplitude of a GW, as detected today chosen model of how to input energy into our present 56 18 Tg** universe. This idea as to an input of energy, and picking hc f 1.62 10 100 GeV 100 different models of how to do so leading to partition (14) 32 12 functions models is what motivated research in entropy 3 12 kfH0 ijij 2π f,2π f generation. From now on, there will be an effort made to 0.01 0.01 identify different procedural representations of the parti- The equation, as given by Kahniashvili [25] with a ton function, and the log of the partion function with frequency f given below in Equation (15) which is for both 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 synthesizing version 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 2nd tually determine if or not gravity/gravitational waves are order phase transition. If as an example T is many an 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. at its variance of density fluctuations may eventually be temperatures ~ 1032 Kelvin , then it is likely that f in linkable to Kolmogrov theory as far as structure forma- Equation (10) below is capable of approaching values of tion. 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 ul00, 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 of least as large, up to 106 time larger than Planck length l a GW would be, detected today, as opposed to what it Planck would be at the origin, say, of the big bang. The larger f 3 u 0 is, the more likely the amplitude is, of Equation (14) (13) l would be very large. In both Equation (14) above, and 0 g Equation (13) above can be linked to an eddy break Equation (15) below, is a degree of freedom for down process, which leads to energy dissipated by vis- spatial conditions factor, which has, according to Kolb cosity. 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 the through a “worm hole” from a prior to a present universe, modern era. i.e. the degrees of freedom radically dropped it can explain in the evolution of space time. 1）How there could be a break up of “encapsulating” 16 1 gT structure which may initially suppress additional entropy fHz1.55 103 ** * S ~105 beyond initial , in the onset of inflation k0 100 0.01 100 GeV 2) Provide a “release” mechanism for [6] (15) SN 1054 10 88 , with SN ~1021 gravitons gravitons 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 44 ttPlanck ~5 10 sec, rising to SNgravitons factor given by Kahniashvili [25] as 1054 10 88 for times up to 1000 seconds after the big 1 HXK,, dde3iK RX ,, ijkl4 ijkl bang. 2π Here is, in a nutshell the template for the Gravitons which will examine, and eventually link to Gravitational (16) waves, and entropy. Why? The factor Hijkl is due to complicated physics
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which gives a tensor/scalar ratio SN 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 RXijkl,, SXt ij , (17) w2 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 correlation functions, i.e. greens functions of Quantum field theory. 5. Minimum Amount of Information Needed See [24] Peskin’s QFT reference as to how such func- to Initiate Placing Values of Fundamental tional calculations are to show the degree of interaction Cosmological Parameters between SxtSxt,& , , with each individual ij,, kl A. K. Avessian’s [30] article (2009) about alleged time Sij, defined as part of a GR “stress tensor” contribution of variation of Planck’s constant from the early universe depends heavily upon initial starting points for t , as 1 k given below, where we pick our own values for the time SXtTXti, j,, ij ij TXt k , (18) 3 parameters, for reasons we will justify in this manuscript: This is where, commonly, we have a way to interpret tttHtt initial initial Planck exp macro ~ Planck hij, in terms of Sij, via (24) SXtXXij , hXtG,4d 3 X (19) The idea is that we are assuming a granular, discrete ij, XX 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- 2hXt,,16 hXt π GSXt , (20) ijt 2 ij ij tionship of what t is with respect to the other physical parameters, i.e. having the values of written What is above, is a way for making sense of GW as te 2 tc , as well as note how little the fine “density” as given by the formula structure constant actually varies. Note that if we assume
dGW 33 2 an unchanging Planck’s mass GW16π Gw THijij , (21) dln 19 mtcGtPlanck ~1.2 10 GeV Here, the temperature T for the onset of a phase this means that G has a time variance, too. This leads to transition, i.e. usually interpreted as a 2nd order phase us asking what can be done to get a starting value of transition plays a major role as to if or not the frequency, tt recycled from a prior universe, to f, for today is very low, or higher, and if or not energy initial initial Planck our present universe value. What is the initial value, and density is high, or low, as well as the attendant amplitude of a GW, as given by Equation (19) above is important. how does one insure its existence? We obtain a minimum Furthermore 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 as by Equation (17) is correctly done, allowing for a mini- 2 SHmax π (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 article ~1 lution is so subtle that no simple use of correlation func- [30] 2009 value of, where we pick tions will be sufficient to screen noise by typical field HmacroHH Hubble (26) theory derived methods. If temperature T for the onset of a phase transition, is very high, it is almost certain that i.e. a choice as to how t has an initial value, and 2 we are looking at HFGW, and relic gravitons which are entropy as scale valued by SHmax π gives us an severely energized, i.e. * would be enormous. If so, estimate as to compressed values of then for high T and enormous *, at the onset of infla- initialtt initial Planck which would be transferred from a 25 tion, we are looking at HFGW, and that [6] prior universe, to todays universe. If SHmax π ~10 , SN this would mean an incredibly small value for the INI- gravitons (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 before ticle-wave duality has large , for DM candidates the introduction vacuum energy, or emergent field en-
Copyright © 2011 SciRes. JMP 736 A. BECKWITH ergy from a prior universe, to our present universe. call the quadrupole moment, with tx, a density
Typically though, the value of the Hubble parameter, measurement. Now, the following value of the Qij as during inflation itself is HUGE, i.e. H is many times lar- given gives a luminosity function L , where R is the ger than 1, leading to initially very small entropy values. “characteristic size” of a gravitational wave source. Note This means that we have to assume, initially, for a that if M is the mass of the gravitating system minimum transfer of entropy/information from a prior 321 universe, that H is neligible. If we look at Hogan’s holo- Qdxxxij i j ij x tx, (31) 3 graphic model, this is consistent with a non finite event horizon [31] 3 35ij 2 1 GGMNNdQij dQ c 1 L 53 3 2 (32) rH0 (27) 5 cdtdtGN Rc This is tied in with a temperature as given by After certain considerations reported by Camp and 1 Cornish [32], one can recover a net GW amplitude Trblack hole 2π 0 (28) GMNN GM Nearly infinite temperatures are associated with tiny h ~2 (33) Rc22 rc event horizon values, which in turn are linked to huge Hubble parameters of expansion. Whereas initially This last equation requires that nearly zero values of temperature can be arguably linked to nearly non existent H values, which in term would be GMN 25 RRG 2 consistent with SHmax π ~10 as a starting point to c entropy. 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 sets can there be a transfer of entropy/information from a pre GMN inflation state to the present universe. Doing this will RRG 2 . require that we keep in mind, as Hogan writes, that the c number of distinguishable states is writable as [31] Note that when 2 GMN NH exp π RR~ G (29) c2 If, in this situation, that N is proportional to entropy, we are at an indeterminate boundary where one may pick i.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 that 5 tions, we will have opportunities for N ~ 10 2 21 15 Mpc M 90 km h 10 (34) 6. How the CMBR Permits, via Maximum rM2.8 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 2.8MRsolar mass (35) Camp and Cornish [32], as does Fangyu Li [33] use the typical transverse gravitational gauge hij with a typically As well as a mean time GW for half of gravitational wave potential energy to be radiated away as traceless value summed as 00 hhand off diago- nal elements of h on each side of the diagnonal to mix 3 x R GM N ~ with a value of GW 2 2π c Rc 2 TT GN 2 d 4 3 (36) hQij ij (30) R 2.8M 1 cdt42r solar mass sec retarded 90 kmM 2 This assumes r is the distance to the source of gravitational radiation, with the retarded designation on The assumption we make is that if we model d GMN the Equation (30) denoting replaced by a retarded RR~ G 2 , dt c d for a sufficiently well posed net mass M that the star time derivative , while TT means take the d trc 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 can
Copyright © 2011 SciRes. JMP A. BECKWITH 737 use the approximation 4 10 210Hz hij ~10 M 90 km m graviton 100 Hz 1/2 2.8MRsolar mass 7 M 5.9 10 T 13 2.8 M solar mass that we also have ~10 or higher, so, that at a TeV 107 Hz 5.9 M minimum we recover Grishchuck’s [26] value of (40) mMgraviton5.6 solar mass 310T f Peak 10 Hz ~ 10 Hz c 1 TeV This Equation (40) is in units where . 60 (37) If 10 grams per graviton, and 1 electron volt is in M 90 km 33 32 rest mass, so 1.6 10 grams gram 6.25 10 eV . MRsolar mass Then Equation (36) places, for a specified value of R, which 107 Hz can be done experimentally, an upper bound as far as far mgraviton as what a mass M would be. Can this be exploited to answer the question of if or not there is a minimum value 10715 Hz 6.582 10 eV s for the Graviton mass? 2 60 28 9 The key to the following discussion will be that 10grams 6.25 10 eV 2.99 10 meter / sec 22 M 90 km 8 10 13 10 , or larger. ~~109 2.8MRsolar mass 10 Then, exist 7. Inter Relationship between Graviton Mass 26 MM~10 solar mass m and the Problem of a Sufficient Num- (41) g 1.99 1033 26 1.99 10 7 grams ber of Bits of from a Prior Universe, to Preserve Continuity between Fundamen- If each photon, as stated above is 3.68 1048 grams tal Constants from a Prior to the Present per photon, [35] then
Universe 54 M ~5.44 10 initially transmitted photons. (42) P. Tinyakov [34] gives that there is, with regards to the Futhermore, if there are, today for a back ground halo of sub structures in the local Milky Way galaxy an CMBR temperature of 2.7 degrees Kelvin amplitude factor for gravitational waves of 5108 photons / cubic meter, with a wave length speci-
fied as max 1cm . This is for a numerical density of 210Hz 4 h ~1010 photons per cubic meter given by ij m (38) graviton T 4 n max (43) If we use LISA values for the Pulsar Gravitational photon hc 2 wave 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 33 M 90 km 8 place at a temperature T 10 Kelvin, then, if there 10 2.8MR was that temperature for a cubic meter of space, the nu- solar mass merical density would be , roughly 10132 times greater than what it is today. Forget it. So what we have to do is leads to looking at, if to consider a much smaller volume area. If the radii of the volume area is hhij ~~ 35 1/2 1/2 r4 10 meters lP Planck length 53015Mpc M (39) 10 10 then we have to work with a de facto initial volume r 2.8 M solar mass 64 10105 ~ 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 per light-years ~4300 Mpc or much greater, so then we have, unit volume area based upon Planck length, in stead of as an example meters, cubed is 1029 5 10 8 5 10 37 photons for a
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35 cubic area with sides rl 4 10 meters P at is how to obtain semi classical, minimal uncertainty T 1033 Kelvin However, M ~5.44 1054 wave states, in this case de rigor for coherent wave func- quantum effects initially transmitted photons! Either the minimum dis- tion states to form. Ford uses gravitons in a so called tance, i.e. the grid is larger, or T 1033 “squeezed vacuum state” as a natural template for relic quantum effects Kelvin 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 uncertainty also information with certain qualifications. Next, we principle is saturated: in QM coherence would be when will attempt to quantify the treatment of gravitons, as xp 2 . In the case of string theory it would have to given in Figure 1 above, with thin wall (box shape) be 2 treatment of quantum mechanics rendition of a Graviton. l 2 x pp S . When the thin wall approximation fails, we approach 22 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 for queezed states, with a break down of the purely quantum treating a template of how to nucleate gravitons into a mechanical modeling of gravitons. coherent state with respect to relic conditions [41]. Ford, in 1994, wrote a squeezed state operation S( ) via 8. 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 scalar and other bosonic quantum field theories, coherent states field , where were introduced by the work of [37] Roy J. Glauber in 222111ij hhhij T (45) 1963 Now, what is appropriate for presenting gravitons 30 15 180 Thermai bath as coherent states? Coherent states, to first approxima- Here, the value T has yet to be specified, and tion are retrievable as minimum uncertainty states. If one Thermai bath takes 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 for which can be written as given by Venziano [38] (1993), the expanding universe mid way, to the end of inflation, 33 which does not equal current even smaller string theory where llS10 Planck , with 0, and lPlanck 10 centimeters estimates as presented by Li et al. h 2 THERMAL BATH lS x p (44) 18 30 34 p 10 Hz hrms ~ 10 10 Hz 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 predicting minimize graviton position uncertainty, we will likely be almost infinitesimal GW fluctuations. It is not clear from out of luck if string theory is the only tool we have for Ford’s [41] treatmentof gravitons, and fluctuations, if he early universe conditions. Mainly, the momentum will is visualizing fluctuation of gravitons/GW, but if one not 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, xlS 10 l Planck sidering what would be the optimal way to obtain a way In 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 39 stated value of hrms ~10 Hz for solar plasma from gravitons are excitations of closed strings, then one will the sun as a graviton source, would be a way of obtaining have to look for conditions for which a coherent state of fluctuations 1059 10 times weaker, i.e. going to gravitons, as stated by [40] Mohaupt occurs. What Mo- hrms values so small that the requirement for a minimum haupt is referring to is a string theory way to re produce fluctuation , in line with not contradicting what Ford gave in 1995, i.e. conditions for how Gravi- 2 l 2 tons in a squeezed vacuum state, the natural result of x pp S , quantum creation in the early universe will introduce 22 metric fluctuations. Ford’s [41] treatment is to have a if we consider experimental conditions for obtaining 39 metric averaged retarded Green’s function for a massless xhrms ~10 / Hz. Note that this would put severe field becoming a Gaussian. The condition of Gaussianity restrictions upon the variations in momentum. A subject
Copyright © 2011 SciRes. JMP A. BECKWITH 739 which will be referenced in whether or not the Li-Baker relic conditions, which may not involve squeezing? Note detector can suitably obtain such small values of L. Grishchuk [44] wrote in “On the quantum state of 39 xhrms ~10 / Hz in detection capacity. To do so relic gravitons”, where he claimed in his abstract that “It will require an investigation into extreme sensitivity re- is shown that relic gravitons created from zero-point quirements, 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. 26 30 The authors have determined the parameters of the h ~10 10 Hz rms squeezed state generated in a simple cosmological model 5 would require up to 10 seconds in evaluative time for a which includes a stage of inflationary expansion. It is clean signal, for GW. What will be asked in further sec- pointed out that, in principle, these parameters can be 5 tions is if or not the 10 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 perturbation would have to do to distinguish if or not coherent states hnlm via the following argument. How we work with the of gravitons which merge to form GW may be measured argument will affect what is said about the necessity, or via 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 a and Suresh [43], built up a coherent state via use of a measure of curvature which is 1, 0 , a is a scale displacement operator Daa exp , ap- factor of a FRW metric, and na2π is a plied to a vacuum state , where is a complex number, way to scale a wavelength, , with n, and with a and aa, as annihilation, and creation operations l aa,1 , where one has hGxPlanck nlma nlm nlm (49) D 0 (46) 2 a nlm nK nlm 0 (50) However, what one sees in string theory, is a situation 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 equation this 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- ˆ i 3 use of a squeezing parameter r as a strength of squeezing tion (50) above, with Py , and Ma , interaction term, with 0 r , and also an angle of y 22 squeezing, ππ as used in nK , aa lPlanck , a r 22 Zr, exp exp( ia ) exp( ia ) , 2 and F an arbitrary function. yy . Also, we where combining the Z r, with (47) leads to a sin- have a finite volume Vgdx 3 3 . gle mode squeezed coherent state, as they define it via finite Then the Lagrangian for deriving Equation (50) is (and leads to a Hamiltonian which can be also derived (48) Zr,,0,0 Zr D Zr from the Wheeler De Witt equation), with 1 for 0 zero point subtraction of energy The right hand side of Equation (48) given above be- My2222 M ay comes a highly non classical operator, i.e. in the limit LaF 22a (51) that the super position of states 0Zr ,0 occurs, there is a many particle version of a “vacuum Pˆ 2 state” which has highly non classical properties. 111 ˆ y 22 HMy ˆ (52) Squeezed states, for what it is worth, are thought to occur ia 22 M 2 at the onset of vacuum nucleation, but what is noted for then there are two possible solutions to the S. E. Grush- 0Zr ,0 being a super position of vac- uum states, means that classical analog is extremely dif- chuk [45] created in 1989, one a non squeezed state, and ficult to recover in the case of squeezing, and general another a squeezed state. So in general we work with non classical behavior of squeezed states. Can one, in any case, faced with DZr 0,0 do a yCBy exp a (53) better job of constructing coherent graviton states, in
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The non squeezed state has a parameter 9. Necessary and Sufficient Conditions for String/Brane Theory Graviton Coherent BBbb 2 b States? where is an initial time, for which the Hamiltonian b 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 state erators 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 written rameter B for squeezing which when looking at a up by GXuv uv hX uv , with uv flat space time, squeeze parameter r, for which 0 r , then Equa- and the Polyakov action, is generalized as follows, the S Polyakov action is computed and compared with ex- tion (53) has, instead of Bbb 2 ponentiated values BB, b b 1 2 ab SdggGXXX uv a b (55) 4π M i a coshrir exp 2 sinh (54) becomes 22a coshrir exp 2 sinh expS Taking Grishchuck’s formalism [45] literally, a state for 1 2 ab expSgghXXXPuvab 1 d ... a graviton/GW is not affected by squeezing when we are 4π looking at an initial frequency, so that initially M b (56) corresponds to a non squeezed state which may have coherence, 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 hX4exp g ikX , and the uv c Suv i a action of S a coherent state of a graviton. Now the B , b bb b22a important question to ask, is if this coherent state of a graviton, as mentioned by Polochinski can hold up in A reasonable research task would be to determine, whe- relic, early universe conditions. R. Dick [47] argued as ther or not stating that the “graviton multiplet as one particular dark matter source in heterotic string theory. In particular, it is B, b b 2 pointed out that an appreciable fraction of dark matter would correspond to a vacuum state being initially formed from the graviton multiplet requires a mass generating 8 phase transition around Tc 10 GeV, where the symme- right after the point of nucleation, with b at time try partners of the graviton would evolve from an ultra- with an initial cosmological time some order of b hard fluid to pressureless dark matter. indicates m 10 44 magnitude of a Planck interval of time ttPlanck 10 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 by there could be a point of no squeezing, as Grishchuck im- Berkenstein [48] with regards to BPS states, and SHO 5 plied, for initial times, and initial frequencies, and an im- models for AdS5 S geometry. The upshot is that string mediate transition to times, and frequencies afterwards, theory appears to construct coherent graviton states, but it where 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- eezed point of departure, i.e. what to do with when into non classical configurations in relic conditions. DZr0,0 . 10. Does LQG Give us More Direct Having D0 with D a possible dis- Arguments as to Coherent States, placement operator, seems to be in common Squeezed States, and the Break Down of with Bbb 2 , whereas Zr ,0 which is Classical Behavior at the Onset of highly non classical seems to be in common with a solu- Inflation? tion for which Bbb 2 This leads us to the next section, i.e. does Bbb 2 when of time Carlo Rovelli [49], in a PRL article states that a vertex 44 ttPlanck 10 seconds, and then what are the initial amplitude that contributes to a coherent graviton state is conditions for forming “frequency” b ? the exponential of the Regge action: the other terms, that
Copyright © 2011 SciRes. JMP A. BECKWITH 741 have raised doubts on the physical viability of the model, referring to the existence of squeezed states, as either are suppressed by the phase of the vacuum state, and being necessarily, or NOT necessarily a consequence of Rovelli writes a coherent vacuum state as given by a the quantum bounce. As Bojowald [56] wrote it up, in
Gaussian peaked on parts of the boundary d of a four both his Equation (26) which has a quantum Hamiltonian dimensional sphere. VHˆ , with ˆ qqmns , j , (57) dV 0 (59) Rovelli [49] states that “bad” contributions to the be- d existence of un squeezed states 0 0 havior of Equation (57) are cancelled out by an appropri- ate (Gaussian?) vacuum wave functional which has “ap- and Vˆ is a “volume” operator where the “ volume” is propriately” chosen contributions from the boundary d set as V , Note also, that Bojowald has, in his initial of a four dimensional sphere. This is to avoid trouble Friedman equation, density values with “bad terms” from what is known as the Bar- Hmatter a ret-Crane vertex amplitude contributions, which are can , a3 be iminized by an appropriate choice of vacuum state so that when the Friedman equation is quantized, with an amplitude being picked. Rovelli [49] calculated some initial internal time given by , with becoming a components of the graviton two-point function and found more general evolution of state variable than “internal that the Barrett-Crane vertex yields a wrong time”. If so, Bojowald writes, when there are squeezed long-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 dVˆ N()0 value (60) not to work unless an infinite number of adjustments are d existence of squeezed states made. The criticism is harsh, but until one really knows 0 admissible early universe geometry one cannot rule out for his Equation (26), which is incidently when links to the Rovelli approach, or confirm it. In addition, Jakub classical behavior break down, and when the bounce Mielczarek [51] considered tensor gravitational perturba- from a universe contracting goes to an expanding present tions produced at a bounce phase in presence of the universe. Bojowald [56] also writes that if one is looking holonomy 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 classical What comes to the fore are corrections due to what is behavior, and a reduction in the strength of a quantum called quantum holonomy, l. A comment about the bounce. Bojowalds [56] states that “Especially the role of quantum bounce. i.e. what is given by Dah-Wei Chiou, squeezed states is highlighted. The presence of a bounce Li-Fang Li [52], is that there is a branch match up be- is proven for uncorrelated states, but as squeezing is a tween a prior to a present set of Wheeler De Witt equa- dynamical property and may change in time”. The upshot tions for a prior to present universe, as far as modeling is that although it is likely in a quantum bounce state that how the quantum bounce links the two Wheeler De Witt the states should be squeezed, it is not a pre requisite for solution branches, i.e. one Wheeler De Witt wave func- the states to always start off as being squeezed states. So tion for a prior univers, and another wave function for a a physics researcher can, look at if an embedding of the present universe. Furthermore, Abhay Ashtekar [53] present universe in a higher dimensional structure which wrote a simple treatment of the Bounce causing Wheeler could have lead to a worm hole from a prior universe to De Witt equation along the lines of, for our present for re introduction of inflationary growth.
const18π G as a critical density, and the eignvalue 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 for bounce effect [54]. Initial un Squeezed States? Wheeler de 2 Witt Solution with Pseudo time aG 8π 1... H OT (58) Component Added in a 3 Furthermore, Bojowald [55,56] specified a criteria as This discussion is to present a not so well known but to how to use an updated version of and useful derivation of how instanton structure from a prior
const18π G in his GRG manuscript on what universe may be transferred from a prior to the present could constitute grounds for the existence of generalized universe. This discussion is partly rendered in [15], but is squeezed initial (graviton?) states. Bojowald [56] was reproduced here due to the relatively unknown feature of
Copyright © 2011 SciRes. JMP 742 A. BECKWITH a pseudo time component to the Wheeler de Witt equa- dr 2 dddSFrt22 2 tion Fr (65) 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. 2MQ2 2) Left unsaid is what embedding structure is assumed. Fr1 r r 2 3) A final exercise for the reader. Would a WKB style (66) solution as far as transfer of “material” from a prior to a 2 2 rrl 32 P present universe constitute procedural injection of non 33T10 Kelvin ~ compressed states from a prior to a present universe? This assumes that the cosmological vacuum energy Also if uncompressed, coherent states are possible, how parameter has a temperature dependence, leading to long would they last in introduction to a new universe? F This is the Wheeler-de-Witt equation with pseudo time ~2 rl P T rl P (67) component added. From Crowell [57] r 3 2 as a wave functional solution to a Wheeler-de-Witt equa- 11 3 rR r r (61) tion bridging two space-times, similar to two space-times rr22 rr with “instantaneous” transfer of thermal heat, as given by This has when we do it cost , and frequently Crowell [57] 3 R constant, so then we can consider 22 TACA 12 C (68) d aeik x a e ik x This has CC ,, tr as a pseudo cyclic and (62) 1 1 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 wave the solutions to Equation (61) above. function CC22 ,, tr , where C1 Equation (63) and C Equation (64) then we get that Equation (68) t 2 2 is a solution to the pseudo time WDM equation. CJr11 4 π 5 2 The question which will be investigated is if Equation 4 (68) is a way to present either a squeezed or un squeezed 5 sinrr cos r (63) state. A way forward is to note that Prado Mar- 15 6 tin-Moruno, Pedro F. Gonzalez-Diaz in July [58] wrote 55cosrSir 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 to Carco Cavaglià’s [59] treatment of use of conjugate 36 r ij CreCir2 1cos 4 (64) momentum π of h generalized momentum vari- 244 ij ables, also known as conjugate momenta This is where Si r and Ci r refer to inte ij grals of the form πˆ , ih ij x x sin x cosx leading to the sort of formalism as attributed to Luis J. dx and dx . x x Garay’s [60] article, of 3 ij Next, we should consider whether or not the instanton hxhijexp d π ij T (69) so formed is stable under evolution of space-time leading Now in the case of what can be done with the worm up 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 ij i πˆ i ˆ the Wheeler-De-Witt equation, with use of the Reinss- , π , gij 2rr ner-Nordstrom metric to help us obtain a solution that 1 passes through a thin shell separating two space-times. tt Fr πˆ i , The radius of the shell rt0 separating the two rr space-times is of length lP in approximate magnitude, and a kinetic energy value as given of the form leading to a domination of the time component for the ππˆˆtt ππ ˆˆ tt . The supposition which we have the Reissner-Nordstrom metric worm hole wave functional may be like, so, use the wave
Copyright © 2011 SciRes. JMP A. BECKWITH 743 functional looking like The question raised repeatedly in whether or not 1) if higher dimensions are necessary, and whether or not 2) 3 ij gxgijexp d π ij T mass gravitons are playing a role as far as the introduc- tion of DE speed up of cosmological expansion may lead where the g for the Weiner- Nordstrom metric will be ij to an improvement over what was specified for density 2 fluctuations and structure formation (the galaxy hierar- 22dr 2 dddSFrt chy problem) of density fluctuations given as Fr 410 2 ~10 10 (73) ij i j 2 22dr gxxdd rlP d t d 3 2 rlP 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 a tential of the form modified mass hierarchy to get suitable DM masses of the order of 100 GeV or more. Giving a suitable defini- m22 VV4 (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 end bare mass term of the field in de Sitter space, which dramatically improve the very crude estimate given by does not take into account quantum fluctuations) Equation (70) above which says next to nothing about 3H 3 how the problems illustrated by the break down of the mt22() M t t (71) galaxy mass formation/hierarchy can be fixed. Further- 4 0 more is considering the spectral index problem, where Specified non linearity of 2 at a time from the the spectral index is big bang, of the form 22 31VV 3H n 1 t (72) S (74) 1 2M 8π VV4π
Figure 1. Here, the left hand side corresponds to a soliton, the right hand side is an anti soliton [24].
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Usual experimental values of density fluctuations ex- should be compared with admissible values of perimentally are nS 12 which would closely correspond to 0 5 4 ~10 , instead of ~10 , and 00.2 . i.e. the precise values of this may help us out in determining how to unravel what is going on in and this is assuming that is extremely small. In addi- the galaxy formation i.e. how can we have earlier than tion, Linde [61] had expected galaxy formation?
d12 H a 2 Vm 13. Conclusions, as to How to Look at Early d 40 40 a Universe Topology and Later Flat Space inside a false vacuum bubble. If something other than the Klein Gordon relationship One of the aspects of early universe topology we need to consider is how to introduce a de facto break down of a 2 30Hm quantization in curved space time geometries, and this is a a problem which would permit a curved space treatment 3/2 occurs, then different models of how density fluctuation of ~/RReq . i.e. as R gets of the order of may have to be devised. A popular model of density Rl~ P , 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 3/2 3/2 3 3/2 Planck length, then how can we recover a field theory k k k 3 3/2 1/ 2π k quantization condition for ~/RReq in terms of Horizon 2π 2π path integrals. We claim that deformation quantization, if (75) applied successfully will eventually lead to a great re- finement of the above Wheeler De Witt wave functional where 10.2 , and 01ns and to first value, as well as allow a more through match up of a order, kHa . The values, typically of ns 1 If working with time independent solution of the Wheeler de Witt equa- tion, with the more subtle pseudo time dependent evolu- 2 aC 2 tion of the wave functional as Crowell wrote up. i.e. the H 2 22 4, linkage between time independent treatments of the wave a 336M 4 MaPlanck functional of the universe, with what Lawrence Crowell and with a density value [52] wrote will be made more explicit. This will, in addi-
3 6 42 tion allow us to understand better how graviton produc- a mcg aa21 0 tion in relic conditions may add to entropy, as well as 0 2 a 8πG 14 5 2 how to link the number of gravitons, say 1012 gravitons 65 per photon, as information as a way to preserve the con- where mg 10 grams, and 0.2 is usually picked to avoid over production of black holes, a very complex tinuity of values from a prior universe to the present picture emerges. Furthermore, if working with 0.2 universe. and 0 14. The Author Claims that in Order to do H 2 1/ 2π k 345 ~1010 This Rigorously, That Use of the (76) Horizon Material in [63] Gutt, and Waldmann Will be Necessary, Especially to The above equation gives inter relationships between Investigate if Quantization of Severely the time evolution of a pop up inflaton field , and a Curved Space Time Conditions Is Hubble expansion parameter H, and a wave length pa- Possible. We Claim that It Is Not [15] rameter 2π kat for a mode given as . k 3/2 What should be considered is the inter relation ship of Resolution of which add more detail to ~/RR . the constituent components of Equation (76) and eq 1 Having said this, it is now important to consider what H . What the author thinks is of particular import can be said about how relic gravitons/ information can is to look at whether or not the more general expression, pass through minimum vales of Rl~ P . as given by the below equation also holds [62]. We shall reference what A. W. Beckwith presented
nS 1 [64], which we think still has current validity for reasons 2 45 Ak 10 10 (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: when
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En() eff eff Sk ln (81) mL24 6 0 Ppower()2graviton net 5 (78) If the phase spaces can be quantified, as a starting 45cG point of say llminlength string10 Planck , with l Planck be- ing part of how to form the “dimensions” of , and This expression of power should be compared with the 0 lminlength string part of how to form the dimensions of , one presented by Giovannini on averaging of the en- ergy-momentum pseudo tensor to get his version of a and 10 being, for a given 0 , and in certain cases gravitational 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 2 4 3 27 HH obtaining 10 naturally, so this sort of procedure is non ,1H 2 GW 0 24 (79) controversial, and linked to falsifiable experimental 256 π M M measurement protocol, so quantum gravity becomes a de Giovannini states that should the mass scale be picked facto experimental science. This will mean looking at such that Mm~ Planck m graviton , that there are doubts Appendix B, fully. Appendix C, and Appendix D give that 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 future the approximation that development of this project. am H 15. Acknowledgements a 6 as stated by [68] by Linde, where we are following This work is supported in part by National Nature Sci- m 23 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 the tection schemes to get quantifiable experimental meas- physics of GW and Graviton physics, and also has a debt urements so we can start predicting relic gravitational of gratitude to Stuart Allen, CEO for his efforts to permit waves. This is especially true if we make use of the fol- the author to do physics work lowing formula for gravitational radiation, as given by. Kofman [69], with M V 1/4 as the energy scale, with a 16. References stated initial inflationary potential V. This leads to an initial approximation of the emission frequency, using [1] M. Y. Kuchiev, “Can Gravity Appear Due to Polarization present-day gravitational wave detectors. of Instantons in SO(4) Gauge Theory?” Classical and Quantum Gravity, Vol. 15, No. 7, 1998, pp. 1895-1913. ()MV 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 What we would like to do for future development of in Matrix Models,” Progress of Physics, Vol. 56, No. 4-5, 2008, pp. 324-329. entropy would be to consider a way to ascertain if or not the following is really true, and to quantify it by an im- [3] D. Perkins, “Particle Astrophysics,” Oxford University press, Oxford, 2003. provement of a supposition advanced by [70] Kiefer, Polarski, and Starobinsky. i.e. the author, Beckwith, has [4] L. Glinka, “Preliminaries in Many-Particle Quantum Gravity. Einstein-FriedmannSpacetime,” in this document presented a general question of how to http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.1380v4.pd avoid having dS/dt = ∞ at S = 0, f; 1) Removes any chance that early universe nucleation [5] L. Glinka, “Quantum Information from Graviton-Matter is 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/e10040441 idea of presenting the evolution of relic entropy via the [7] M. Asakawa, T. Hatsuda and Y. Nakahara, “Maximum evolution 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.
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Appendix A: ment accepts that there is a small graviton mass, which 60 Bounds upon Graviton Mass, and Making the author has estimated to be on the order of 10 Use of the Difference between Graviton kilograms. Small enough so the following approximation is valid. Here, v is the speed of graviton “propaga- Propagation Speed and HFGW Transit g tion”, g is the Compton wavelength of a graviton with Speed to Observe Post Newtonian Correc- 10 gg hm c, and f 10 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 fulfilling 2 The author presents a post Newtonian approximation hf mg c so that [71] based upon an earlier argument/paper by Clifford Will as 2 2 vc1 c f (A1) to Yukawa revisions of gravitational potentials in part gg initiated by gravitons with explicit mass dependence in But Equation (1) above is an approximation of a much their Compton wave length. more general result which may be rendered as
2 2 vc1 mcE2 Introduction gg (A2)
The terms mg and E refers to the graviton rest mass Post Newtonian approximations to General relativity and energy, respectively. Now specifically in line with have given physicists a view as to how and why infla- applying the Li Baker detector, [42] physics researchers tionary dynamics can be measured via deviation from can ascertain what E is, with experimental data from the simple gravitational potentials. One of the simplest de- Li Baker detector, and then the next question needs to be viations 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 source gravitational potentials. So happens that the mass of a 17 200Mpc t graviton would factor directly into the Yukawa exponen- 1510vcg (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 upon detector as a way to obtain more realistic upper bounds ttae (1 Zt ) , with where ta and te are the as to the mass of a graviton and to use it as a template to differences in arrival time and emission time of the two investigate 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 for Giving an Upper Bound to the Mass of a early universe conditions, that Z 1100 , in fact for very Graviton 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 not investigate 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 problem transit from a “source” to the detector. Visser’s [72] of the limit of small graviton mass does not coincide mass of a graviton paper [72] presents a theory which with pure GR, and the predicted perihelion advance, for passes the equivalence test, but which has problem with example, violates experiment. A way forward would be depending upon a non-dynamical background metric. I.e. to configure data sets so in the case of early universe gravitons are assumed by both him, and also Will’s [71] conditions that one is examining appropriate Z 1100 write up of experimental G.R. to have mass. This docu- but with extremely small te times, which would re-
Copyright © 2011 SciRes. JMP A. BECKWITH 749 flect upon generation of HFGW before the electro weak 1/2 1/2 c 12 DHz100 1 transition, and after the INITIAL onset of inflation. i.e. mkmg 310 h 200Mpc f f t the Li-Baker detector system should be employed [42] as a 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 the ence in times, t . i.e. graviton mass mg , and the resulting Compton wave- length would be important to get values of the fol- ttae(1 Zt ) tt ] (A4) g aa lowing formula, namely The closer the emission times for production of the HFGW and Gravitons are to the time of the initial nu- MG Vr exp rg (A10) cleation of vacuum energy of the big bang, the closer we gravity r can be to experimentally using Equation (4) above as to Clifford Will gave for values of frequency f 100 give experimental criteria for stating to very high accu- Hertz enormous values for the Compton wave length, i.e. racy the following. 19 values like g 610 kilometers. Such enormous values for the Compton wave length make experimental 17 200Mpc ta 1510vcg (A5) tests of Equation (A10) practically infeasible. Values of D 1sec 5 g 10 centimeters or less for very high HFGW data More exactly this will lead to the following relation- makes investigation of Equation (A10) above far more ship which will be used to ascertain a value for the mass tractable. of a graviton. By necessity, this will push the speed of graviton propagation very close to the speed of light. In Application to Gravitational Synchrotron this, we are assuming an enormous value for D Radiation, in Accelerator Physics
17 200Mpc ta Eric Davis, quoting Pisen Chen’s article [74] estimates vcg 1510 (A6) D 1sec that a typical storage ring for an accelerator will be able to give approximately 1063 10 gravitons per second. This Equation (A6) relationship should be placed into Quoting Pisen Chen’s [75] 1994 article, the following for g hmc/ g with a way to relate this above value of graviton emission values for a circular accelerator system,
2 2 2 with m the mass of a graviton, and M P being Planck vcgg1 mcE, 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 which Baker detector [42] and field theory calculations, as well for the LHC is set by Pisen Chen [74] as of the value as to make the following argument rigorous, namely 2800, and N for the LHC is about 1011 . And, for the 2 2 LHC Pisen Chen sets as 88 10 , with 2 2 17 200Mpc ta mcg 1510 1 m 4300 . Here, mm~ graviton acts as a mass (A7) DE1sec charge. 24 22mc A suitable numerical treatment of this above equation, NnNGSR~5.6 b 2 (A11) M P with data sets could lead to a range of bounds for mg , as a refinement of the result given by Will [71] for graviton The immediate consequence of the prior discussion Compton wavelength bounded behavior for a lower would be to obtain a more realistic set of bounds for the bound to the graviton mass, assuming that h is the graviton mass, which could considerably refine the esti- 11 Planck’s constant. mate of 10 gravitons produced per year at the LHC, with realistically 365 × 86400 seconds = 31536000 sec- 1/2 1/2 3 hDHz12 100 1 onds in a year, leading to 3.171 10 gravitons pro- g 310km duced per second. Refining an actual permitted value of mcg 200 Mpc f f t bounds for the accepted graviton mass, m, as given 1/2 1/2 19 2 12 DHz100 1 above, while keeping M p 1.2209 × 10 GeV/c would 310km allow for a more precise set of gravitons per second 200Mpc f f ta 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 place mass mg as given by a detector for actually capturing/detecting a graviton.
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Conclusion, Falsifiable Tests for the Graviton achievable precision given by Are Closer than the Physics Community Thinks 2 C 214 Texp 22 l fel b The physics community now has an opportunity to ex- sky Clll21 f C perimentally infer the existence of gravitons as a know- sky able and verifiable experimental datum with the onset of (B3) the LHC as an operating system. Even if the LHC is not used, Pisen Chens parameterization of inputs from his fsky 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 is nity to make progress as to detection of Gravitons for, the width of a beam, while we have a minimum value of say the Brookhaven site circular ring accelerator system. lmin 1 which is one over the fluctuation of the Tony 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 from wave-particle duality induces so much uncertainty as to noise. The end result is the mass of the purported graviton, that worst case model 22 2 building and extraordinarily robust parameters for a 4π fCsky l exp l T (B4) Rothman style graviton detector have to be put in place. The Li-Baker detector [42] can help with bracketing a Appendix C: Cosmological Perturbation range of masses for the graviton, as a physical entity subject to measurements. Such an effort requires obtain- Theory and Tensor Fluctuations (Gravity ing rigorous verification of the approximation used to the Waves) effect that ttae(1 Zt ) ttaa is a defensible approximation. Furthermore, obtaining realis- Durrer [77] reviews how to interpret Cl in the region tic inputs for distance D for inputs into Equation (A9) where we have 2 l 100 , roughly in the region of the above is essential. The expected pay offs of making such Sachs-Wolf contributions due to gravity waves. We an 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 value tions. The author asserts that Equation (10) can only be of the Hubble expansion parameter realistically be tested and vetted for sub atomic systems, 2 3 2 H and that with the massive Compton wavelength specified k i (C1) by Clifford Will cannot be done with low frequency M P gravitational waves.Furthermore, a realistic bounding of and the graviton mass would permit a far more precise cali- bration of Equation (A11) as given by Pisen Chen in his 2 kAk3211nn 1994 article [74]. 0 (C2)
Here we are interpreting A amplitude of metric Appendix B. Basic Physics of Achieving perturbations at horizon scale, and we set k 1/ , Minimum ll10 Precision 0 min length string Planck where is the conformal time, according to in CMBR Power Spectra Measurements dt ad physical time, where we have a as the
scale factor. Then for 2 l 100 , and 33n , and Begin first of all looking at a pure power law given by T 2 32nn aYlm l, m , (B1) Hk,1/ k k AkTT T lm, T 0 (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 Callm , (B2) 2 T AT 1 Samtleben et al. [76] consider then what the experi- Cl (C4) mental variance in this power spectrum, to the tune of an ll3215
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Appendix D: Linking the Thin Shell Ap- a 32/3 xZxx1/6 ~ (D2) proximation, Weyl Quantization, and the 3 Wheeler de Witt Equation Similarly, x 1 leads to
This is a re capitulation of what is written by S. Ca- a 3 xZx1/6 (D3) poziello, et al. [78] for physical review A, which is as- 32 suming a generally spherically symmetric line element. The upshot is that we obtain a dynamical evolution equa- Also, when x 1 3 tion, similar in part to the Wheeler De Witt equation 2 8 3/2 21ax Z3/4 ax 1 (D4) which can be quantified as H 0 3 Which in turn will lead to, with qualifications, for thin shell approximations x 1, Realistically, in terms of applications, we will be consid- ering very small x values, consistent with conditions ax24 0 (D1) near a singularity/worm hole bridge between a prior to so that Z1/6 is a spherical Bessel equation for which we our present universe. This is for xRR equilibrium . can write
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