We ask the question if a formula for entropy, as given by with a usual value ascribed of initial entropy of the onset of inflation can allow an order of magnitude resolution of the question of if there could be a survival of a graviton from a prior to the present universe, using typical Planckian peak temperature values of . We obtain values consistent with up to 10 38 gravitons contributing to an energy value of if we assume a relic energy contribution based upon each graviton initially exhibiting a frequency spike of 10 10 Hz. The value of is picked from looking at the aftermath of what happens if there exists a quantum bounce with a peak density value of [1] in a regime of LQG bounce regime radii of the order of magnitude of meters. The author, in making estimates specifically avoids using , by setting the chemical potential for ultra high temperatures for reasons which will be brought up in the conclusion.
1Papers on LCQ at the 12th Marcell Grossman Meeting in 2009 (http://www.icra.it/MG/mg12/en/)
Recently, a big bounce has been proposed1 as an alternative to singularity conditions that Hawking’s, Ellis [
Modeling how much information may be carried by an individual graviton can be achieved by measuring the graviton via instrumentation. Normalized energy density of gravitational waves, as given by Maggiore [
Ω g w ≡ ρ g w ρ c ≡ ∫ ν = 0 ν = ∞ d ( log ν ) ⋅ Ω g w ( ν ) ⇒ h 0 2 Ω g w ( ν ) ≅ 3.6 ⋅ [ n ν 10 37 ] ⋅ ( ν 1 kHz ) 4 (1.1)
where n ν is a frequency-based count of gravitons per unit cell of phase space?
Is Equation (1.1) above fundamental physics? And what is the significance of the n ν and ν terms with regards to if gravitons could have been cycled from a prior to the present universe? The rest of the document will attempt to answer the question of what ultra high frequency inputs into the n ν as well as ν term are relevant to, assuming that the quantum bounce model of a “recycled” universe is in part, correct.
As suggested earlier by Beckwith [
q = − a ¨ a a ˙ 2 (1.2)
We wish next to consider what happens not a billion years ago, but at the onset of creation itself. If a correct understanding of initial graviton conditions is presented, it may add more credence to the idea of a small graviton mass, in a rest frame, which may give backing, in part, to Beckwith’s use of Equation (1.2) for re acceleration of the universe, in a manner usually associated with Dark Energy. Here, we are making use of refining the following estimates. In what follows, we will have even stricter bounds upon the energy value (as well as the mass) of the graviton based upon the geometry of the quantum bounce, with a radii of the quantum bounce on the order of l Planck ~ 10 − 35 meters [
m graviton | RELATIVISTIC < 4.4 × 10 − 22 h − 1 eV / c 2 ⇔ λ graviton ≡ ℏ m graviton ⋅ c < 2.8 × 10 − 8 meters (1.3)
For looking at the onset of creation, with a bounce; if we look at ρ max ∝ 2.07 ⋅ ρ planck for the quantum bounce with a value put in for when ρ planck ≈ 5.1 × 10 99 grams / meter 3 , where
E e f f ∝ 2.07 ⋅ l Planck 3 ⋅ ρ planck ~ 5 × 10 24 GeV (1.4)
Then, taking note of this, one is obtaining having a scaled entropy of S ≡ E / T ~ 10 5 when one has an initial Planck temperature T ≈ T Planck ~ 10 19 GeV . One needs, then to consider, if the energy per given graviton is, if a frequency ν ∝ 10 10 Hz and E graviton-effective ∝ 2 ⋅ h v ≈ 5 × 10 − 5 eV , then
S ≡ E e f f / T ~ [ 10 38 × E graviton-effective ( v ≈ 10 10 Hz ) ] / [ T ~ 10 19 GeV ] ≈ 10 5 (1.5)
Having said that, the E graviton-effective ∝ 2 ⋅ h v ≈ 5 × 10 − 5 eV is 1022 greater than the rest mass energy of a graviton if E ~ m graviton [ red-shift ~ 0.55 ] ~ ( 10 − 27 eV ) grams is taken when applied to Equation (1.2) above.
Note that J. Y. Ng uses the following. [
Mass-energy relationship:
m graviton ( energy − ν ≈ 10 10 Hz ) ≈ [ 100 ⋅ GeV ~ 10 11 eV − WIMP ] × 10 − 16 ~ 10 − 5 eV (1.6)
If one drops the effective energy contribution to ν ≈ 10 0 ~ 1 Hz , as has been suggested, then the relic graviton mass-energy relationship is:
m graviton ( energy − ν ≈ 10 0 Hz ) ≈ [ 100 ⋅ GeV ~ 10 11 eV − WIMP ] × 10 − 26 ~ 10 − 15 eV (1.7)
Finally, if one is looking at the mass of a graviton a billion years ago, with
m graviton ( red-shift-value ~ 0.55 ) ≈ [ 100 ⋅ GeV ~ 10 11 eV − WIMP ] × 10 − 38 ~ 10 − 27 eV (1.8)
i.e., if one is looking at the mass of a graviton, in terms of its possible value as of a billion years ago, one gets the factor of needing to multiply by 1038 in order to obtain WIMP level energy-mass values, congruent with Y. Jack Ng’s S ~ N counting algorithm, i.e., the equivalence relationship for entropy and “particle count” may work out well for the WIMP sized DM candidates, and may break down for the graviton mass-energy problem.
A typical value and relationship between an inflation potential V [ ϕ ] , and a Hubble parameter value, H is
H 2 ~ V [ ϕ ] / m Planck 2 (1.9)
Also, if we look at the temperature T ∗ occurring about the time of the Electro weak transition, if T ≤ T ∗ when T ∗ = T c was a critical value, (of which we can write v ( T c ) / T c > 1 , where v ( T c ) denotes the Higgs vacuum expectation value at the critical temperature Tc., i.e. v ( T c ) / T c > 1 according to C. Balazc et al. (2005) [
H ~ 1.66 ⋅ [ g ˜ ∗ ] ⋅ [ T 2 / m Planck 2 ] (1.10)
Here, the factor put in, of g ˜ ∗ is the number of degrees of freedom. Kolb and Turner [
S ~ 3 m Plank 2 [ H = 1.66 ⋅ g ˜ ∗ ⋅ T 2 / m planck ] 2 T ~ 3 ⋅ [ 1.66 ⋅ g ˜ ∗ ] 2 T 3 (1.11)
Should the degrees of freedom hold, for temperatures much greater than T ∗ , and with g ˜ ∗ ≈ 1000 at the onset of inflation, for temperatures, rising up to, say T-1019 GeV, from initially a very low level, pre inflation, then this may be enough to explain how and why certain particle may arise in a nucleated state, without necessarily being transferred from a prior to a present universe.
I.e. the suggestion being presented is that a more standard thermodynamic dependence of entropy upon temperature, i.e. S ∝ T 3 for values of degrees of freedom may be envisioned if one has S ∝ T 3 when g ˜ ∗ ≈ 1000 or even higher even if T ~ 10 19 GeV ≫ T ∗ is envisioned, in place of S ≠ T 3 if T ~ 10 19 GeV ≫ T ∗ , and assuming that g ˜ ∗ ≠ 1000 , i.e. that an upper limit of g ˜ ∗ ≈ 100 - 110 in degrees of freedom is all that is permitted.
Furthermore, if one assumes that S ∝ T 3 [
q ¨ ( t ) + ω 0 2 q ( t ) = 0 , for t < 0 and t > T ⌣ ; q ¨ ( t ) − Ω 0 2 q ( t ) = 0 , for 0 < t < T ⌣ (1.12)
Given Ω 0 T ⌣ ≫ 1 , with a starting solution of q ( t ) ≡ q 1 sin ( ω 0 t ) if t < 0 , Mukhanov state that for t > T ⌣ ;
q 2 ≈ 1 2 1 + ω 0 2 Ω 0 2 ⋅ exp [ Ω 0 T ⌣ ] (1.13)
The Mukhanov et al. argument leads to an exercise which Mukhanov et.al. [
n = [ 1 / 2 ] ⋅ ( 1 + [ ω 0 2 / Ω 0 2 ] ) ⋅ sinh 2 [ Ω 0 T ⌣ ] (1.14)
I.e. for non zero [ Ω 0 T ⌣ ] , Equation (1.14) leads to exponential expansion of the numerical state. For sufficiently large [ Ω 0 T ⌣ ] , Equation (1.12) and Equation (1.14) are equivalent to placing of energy into a system, leading to vacuum nucleation. A further step in this direction is given by Mukhanov [
n ~ sinh 2 [ m 0 η 1 ] (1.15)
Equation (1.12) to Equation (1.14) are, for sufficiently large [ Ω 0 T ⌣ ] a way to quantify what happens if initial thermal energy are placed in a harmonic system, leading to vacuum particle “creation” Equation (1.15) is the formal Bogolyubov coefficient limit of particle creation. Note that q ¨ ( t ) − Ω 0 2 q ( t ) = 0 , for 0 < t < T ⌣ corresponds to a thermal flux of energy into a time interval 0 < t < T ⌣ . If T ⌣ ≈ [ t Planck ∝ 10 − 44 sec ] or some multiple of t Planck and if Ω 0 ∝ 10 10 Hz , then Equation (1.12), and Equation (1.14) plus its generalization as given in Equation (1.15) may be a way to imply either vacuum nucleation, or transport of gravitons from a prior to the present universe. Having said that, the problem of Heavy Gravity raises its ugly head in the following field theory example.
As given by M. Maggiore [
∂ μ ∂ ϖ h μ ν = 32 π G ⋅ ( T μ ν − 1 2 η μ v T μ μ ) (1.16)
When m graviton ≠ 0 , the above becomes
( ∂ ∂ μ ϖ − m graviton ) ⋅ h μ ν = [ 32 π G + δ + ] ⋅ ( T μ ν − 1 3 η μ v T μ μ + ∂ μ ∂ ν T μ μ 3 m graviton ) (1.17)
The mismatch between these two equations, when m graviton → 0 is largely due to m graviton h μ μ ≠ 0 as m graviton → 0 , which in turn is due to setting
m graviton ⋅ h μ μ = − [ 32 π G + δ + ] ⋅ T μ μ . The mismatch between these two expressions is one of several reasons for exploring what happens for semi-classical models when m graviton ≠ 0 , m graviton ~ 10 − 65 grams, noting that in QM, a spin 2 only two
m graviton ≠ 0 has five degrees of freedom, whereas the m graviton → 0 gram case has helicity states. Note that string theory treats gravitons as “excitations” of a closed string, as given by Keifer [
Δ x ≥ ℏ Δ p + l s 2 ℏ Δ p , where G ~ g 2 l s 2 . Kieffer [
m graviton ~ 10 − 65 g , it will be hard to measure as an individual “particle.” But, if m graviton ~ 10 − 65 g exists, as a macro effect one billion years ago, i.e. as a substitute for DE, it also would be potentially relevant toward information exchange between a prior to the present universe, provided that there was no cosmic singularity and that the LQG quantum bounce hypothesis has some validity., Note that the author has been informed by J. Dickau of research by [
A way to obtain traces of information exchange, from prior to present universe cycles is finding a linkage between information and entropy. If such a parameterization can be found and analyzed, then Seth Lloyd’s [
I = S total / k B ln 2 = [ # operations ] 3 / 4 = [ ρ ⋅ c 5 ⋅ t 4 / ℏ ] 3 / 4 (1.18)
could be utilized as a way to represent information which can be transferred from a prior to the present universe. The question to ask, if Equation (1.18) does permit a linkage of gravitons as information carriers, and there can be a linkage of information, in terms of the appearance of gravitons in the time interval of, say 0 < t < t Planck either by vacuum nucleation of gravitons/information packets along the lines of Equation (1.12) and Equation (1.14) or by reconciling the counting algorithm questions put up in Section 2.2.
The problem of reconciling the existence of a graviton mass with quantum mechanics, in spin two particles usually having zero mass appears to be resolvable, and may imply a linkage between DE and DM [
r U ≡ 1 H ⋅ | Ω − 1 | (1.19)
Specifically, the author is convinced that analyzing Equation (1.19) will be tied in, with appropriate analysis of the following
The relation between Ω g and the spectrum h ( v g , τ ) is often expressed as written by Grishchuk, [
Ω g ≈ π 2 3 ( v v H ) 2 h 2 ( v , τ ) , (1.20)
The importance of understanding the radius of the universe question, and making sense of Equation (1.19) lies in reconciling the conflicting estimates put in Section 2.2 above. If one can get an answer to reconciling the estimates put in Section 2.2, one has gone a long way toward answering, or laying the ground work to answering the question as to the classical nature of gravitons, or if they have a semi classical interpretation.
t’Hooft [
Beckwith has very deliberately set S ≡ [ E − μ N ] / T → E / T with μ ≠ 0 approaching zero. Note that L. Glinka [
If the author, Beckwith, is wrong, he will be quite happy to amend his work along the lines given by L. Glinka’s 2007 work [
Note that Appendix A below summarizes some of the methods used by the author in terms of counting of gravitons and initial entropy assumed in this document. The reader should also review [
In addition, in Appendix B, the author gives a summary as to some emerging trends in gravitational wave astronomy which are extremely important. References [
This work is supported in part by National Nature Science Foundation of China grant No. 11375279
Beckwith, A.W. (2017) Could Gravitons from a Prior Universe Survive a (LQG Inspired) “Quantum Bounce” to Re-Appear in Our Present Universe? Journal of High Energy Physics, Gravitation and Cosmology, 3, 624-634. https://doi.org/10.4236/jhepgc.2017.34047
The author brings up entropy development as given by [
Z N ~ ( 1 N ! ) ⋅ ( V λ 3 ) N (A.1)
This, according to Ng, leads to entropy of the limiting value of, if S = ( log [ Z N ] )
S ≈ N ⋅ ( log [ V / N λ 3 ] + 5 / 2 ) → Ng-infinite-Quantum-Statistics N ⋅ ( log [ V / λ 3 ] + 5 / 2 ) ≈ N (A.2)
But V ≈ R H 3 ≈ λ 3 , so unless N in Equation (A.2) above is about 1, S (entropy) would be < 0, which is a contradiction. Now Equation (A.2) is where [
Keep in mind as well that there has been recent confirmation by Abbot [