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The following document attempts to answer the role additional degrees of freedom have as to initial inflationary cosmology, i.e. the idea is to cut down on the number of independent variables to get as simple an emergent space time structure of entropy and its generation as possible. One parameter being initial degrees of freedom, the second the minimum allowed grid size in space time, and the final parameter being emergent space time temperature. In order to initiate this inquiry, a comparison is made to two representations of a scale evolutionary Friedman equation, with one of the equations based upon LQG, and another involving an initial Hubble expansion parameter with initial temperature
used as an input into T
^{4} times
*N*(
*T)*
. Initial assumptions as to the number of degrees of freedom have for
a maximum value of
*N*(
*T*) ~ 10
^{3}. Making that upper end approximation for the value of permissible degrees of freedom is dependent upon a minimum grid size length as of about
centimeters. Should the minimum uncertainty grid size for space time be higher than
centimeters, then top value degrees of freedom of phase space as given by a value
*N*(
*T*) ~ 10
^{3} drops. In addition, the issue of bits,
* i*.
*e*. information is shown to not only have temperature dependence, but to be affected by minimum “grid size” as well.

^{1}Papers on LCQ at the 12th Marcell Grossman Meeting in 2009 (http://www.icra.it/MG/mg12/en/).

Recently, a big bounce has been proposed^{1} as an alternative to singularity conditions that Hawkings, Ellis [

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. Equation (1.1) leads to, as given to

The author, Beckwith, wishes to determine inputs into n ν above, in terms of frequency, and also initial temperature. Doing so will, if one gets inputs into Equation (1.1) right lead to examining how the arrow of time initial configuration, of entropy, influences choices as to models of what to chose from in terms of inflation. The author is convinced an answer to the above which will be dependent upon the number of degrees of freedom present in early universe cosmology. In

the LQG version by [

Doing so leads to, first considering, a non standard Friedman equation which is written up as [

( a ˙ a ) 2 ≡ κ 6 ⋅ p ϕ 2 a 6 . (1.2)

This Equation (1.2) assumes that the conjugate dimension in this case has a quantum connection specified via an effective scalar field, ϕ obeying the relationship

ϕ ˙ = − ℏ i ⋅ ∂ ∂ ⋅ p ϕ . (1.3)

This inquiry explicitly assumes a Friedman equation dominated by temperature with N(T) a temperature dependent number of degrees of freedom present in a region of “phase space”, and a ⌣ a radiation constant, as given by Saunders (2005) [

H 2 = [ 4 π G ⋅ a ⌣ ⋅ T 4 N ( T ) / 3 c 2 ] . (1.4)

If we make the following minimum uncertainty value for momentum as given by Baez-Olson [

[ 4 π G ⋅ a ⌣ ⋅ T 4 N ( T ) / 3 c 2 ] ≈ κ 6 ⋅ [ p ϕ 2 = ( ℏ / l Planck ) 2 ] a 6 . (1.5)

The consequence of Equation (1.5) would be to set conditions for which the following could be true.

N ( T ) ~ 10 3 ≅ [ c 2 κ ⋅ ℏ 8 π G ⋅ a ⌣ ] ⋅ 1 T 4 a initial 6 ⋅ l Planck 2 . (1.6)

If we take a dimensional re scaling of Equation (1.6), with

N ( T ) ~ 10 3 ~ 1 [ T 4 ≈ T Planck 4 ] ⋅ [ a initial 6 = 10 6 ⋅ β ] ⋅ l Planck 2 . (1.7)

One can then obtain an algebraic equation to the effect that

76 − 4 δ + − 66 + 6 β ≈ − 3 ⇒ a initial ~ 1 . (1.8)

This above approximation would be assuming that T ~ 10 19 − δ + GeV i.e. close to the Planck temperature.

The other assumption is that the starting point for Planck expansion, has a initial = 1 with an enormous value for a in the present era as opposed to another scaling convention that a = [ 1 / 1 + z ] where one can have the red shift with values at the onset of inflation of the order of z initial ~ 10 25 at the start of inflation, and z CMBR ~ 1100 − 1000 at the moment of CMBR photon radiation “turn on” with z Today = 0 in the present era. Examining what happens if one substitutes in for l Planck l 1 / 3 l Planck 2 / 3 in Equation (1.7) would mean a substantially lower value for N ( T ) if the following holds, i.e. l ≫ l Planck making plausible even at the onset of inflation N ( T ) ~ 10 2 as reported by Kolb and Turner, 1991 [

A consequence of Verlinde’s [

n B i t = Δ S Δ x ⋅ c 2 π ⋅ k B 2 T ≈ 3 ⋅ ( 1.66 ) 2 g * [ Δ x ≅ l p ] ⋅ c 2 ⋅ T 2 π ⋅ k B 2 . (1.9)

This Equation (1.9) has a T^{2} temperature dependence for information bits, as opposed to [

S ~ 3 ⋅ [ 1.66 ⋅ g ˜ ∗ ] 2 T 3 . (1.10)

Should the Δ x ≅ l p order of magnitude minimum grid size hold, then conceivably when T ~ 10^{19} GeV

n Bit ≈ 3 ⋅ ( 1.66 ) 2 g * [ Δ x ≅ l p ] ⋅ c 2 ⋅ T 2 π ⋅ k B 2 ~ 3 ⋅ [ 1.66 ⋅ g ˜ ∗ ] 2 T 3 . (1.11)

The situation for which one has [

n B i t ≪ 3 ⋅ [ 1.66 ⋅ g ˜ ∗ ] 2 T 3 (1.12)

even if one has very high temperatures. Note that for WIMPS a situation as Y. J. Ng has it that [

S ≅ n Particle-Count . (1.13)

Note that Y. Jack Ng [

Next, if the additional degrees of freedom are warranted, comes the question of what are measurable protocols which may confirm/falsify this supposition. The following discussion will in part recap and extend a discussion which the author, Beckwith has presented in DICE 2010, in Italy [

The problem, though, is that there may be more than one graviton per information bit as given by Beckwith’s calculations for entropy, and also energy carried per graviton. As given by Beckwith, in DICE 2010, Beckwith has made the following estimate, i.e. [

Note that J. Y. Ng uses the following [

m graviton ( energy- ν ≈ 10 10 Hz ) ≈ [ 100 GeV ~ 10 11 eV-WIMP ] × 10 − 16 ~ 10 − 5 eV . (1.14)

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 . (1.15)

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 (1.16)

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 10^{38} in order to obtain WIMP level energy-mass values, congruent with Y. Jack Ng’s S counting algorithm [^{38} gravitons to form coherent clumps to obtain GW of sufficient semi classical initial conditions, to obtain conditions, initially to have the S ~ N counting algorithm work.

The author will later on attempt to prove that the 10^{38} factor so recorded is an artifact of Equation (1.9), i.e. that the scaling so implied in Equation (1.9) with the square of temperature, divided by grid size length means that for very light particles, the influence of high levels temperature will make the 10^{38} factor inevitable.

Still though, it would be important to come up with criteria as to how one can obtain a temperature and a mass of a “particle” regime for which S ~ N work may be solvable via making the Ng. “Entropy” linkable to particle count. AND bits of information at the same time. To do so may entail introducing a new concept, that of “configurational entropy”, as introduced below.

The author has been advised that Rubi et al., 2008 [

1 T ˜ ( x , v ) = ∂ S C ∂ e K i n . (1.17)

In the case that the graviton has a very slight rest mass, one can, if [

v graviton 2 = c 2 ⋅ [ 1 − ( m graviton 2 c 4 / E 2 ) ] . (1.18)

The net temperature may be considered to be a calculated function of a rise in temperature from almost nonexistent status, up to nearly Planck temperature, and the author is convinced, that one would have to, given different geometries, reconstruct the configurational entropy, once an idea of a minimum to the peak temperature, T, for Plank temperature values is obtained.

By doing so, the author hopes to obtain an evolution of S C with different values of the temperature, in order to come up with an emergent structure with S C ~ S ~ 3 ⋅ [ 1.66 ⋅ g ˜ ∗ ] 2 T 3 . This should be done while paying attention to t’ Hooft’s idea that an emergent structure would by necessity likely engage more than 100 dimensions, i.e. as Beckwith wrote about in [

Recently, the author has been fortunate enough to obtain Leff’s [

S = ( 4 / 3 ) b V T 3 . (1.19)

This should be compared with Beckwith’s derived “graviton clumping” entropy result [

What the author supposes, is that fine tuning the inter play between these two formulas, from the onset of inflation when there was likely coupling between gravitons, clumps of gravitons, and photons, may permit experimental measurements permitting investigation if there is an interplay between E & M and gravity, and also modifications of gravity theory along the lines brought up by Sidharth [

A μ = ℏ ⋅ Γ μ μ v (1.20)

where A μ can be identified with the electromagnetic four potential. The idea, as Beckwith sees it would be to determine if there could be coupling between E & M effects, and gravitation along the lines of employing the Quantum (coupled) oscillator frequency relationship for coherent “state” oscillation as given by Sidarth [

G ℏ ω max = c 5 . (1.21)

This would be to come up with a realistic way to talk about clumps of gravitons which may have coherent oscillatory behavior and to use this to make sense of the structure of up to 10^{38} coherent gravitons to form coherent clumps to obtain GW of sufficient semi classical initial conditions, to obtain conditions, initially to have the S ~ N counting algorithm work for gravitons as coherent clumps, allegedly in a structure defined by Equation (1.21).

Then, after employing Equation (1.21) to next examine the limits of, and interexchange of effects given in Equation (1.14) and Equation (1.15) to determine from there to what degree is Equation (1.16) is giving us joint linkage of E&M and gravitational waves in early universe conditions. Also , the author hopes that examining a potential inter play of Equation (1.10) to Equation (1.21) that the datum that the 10^{38} coherent gravitons [

n f = [ 1 / 4 ] ⋅ [ v ( a initial ) v ( a ) − v ( a ) v ( a final ) ] (1.22)

could be investigated as being part of the bridge between phenomenology of both photon gases, and their entropy, as well as a modified treatment of L. Glinka’s graviton gas [

It is well worth noting that tests concerning the alleged Graviton gas should be tested against the predictions given in [

Formula (1.22) as well would be useful in determining the would be existence of frequency response functions as far as a would be experimental datum for analysis, and also of falsifiable tests of alternatives to General Relativity as far as the foundation of gravity itself.

Finally, we should note as to the existence of positive identification of Gravitational waves. As seen in [

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

Beckwith, A.W. (2018) Supplying Conditions for Having up to 1000 Degrees of Freedom in the Onset of Inflation, Instead of 2 to 3 Degrees of Freedom, Today, in Space-Time. Journal of High Energy Physics, Gravitation and Cosmology, 4, 143-151. https://doi.org/10.4236/jhepgc.2018.41013