_{1}

^{*}

The idea is to identify via ephemeris time as given by Barbour and an inflaton field as given by Padmanabhan, for scale factor proportional to time to the alpha power and a velocity given by Will for massive gravitons, an initial energy for a massive graviton in space-time. The spatial values for the graviton production could be from the Planckian to Electro weak regime, with a nod to using a worm hole from a prior to a present universe as a delivery font for gravitational energy, as an information carrying bridge from prior universe “information settings” to the present space-time. The number of Gravitons will be set as N, and the initial time, as a tie in with Barbour’s ephemeris time, a constant times Planck time. In setting up the positions, as input into the positions and distributions of gravitons in our model, we will compare results as could be generated by Racetrack inflation, for presumed position of relic gravitons when just produced in the universe, as compared with results given by an adaptation of an argument presented by Crowell, in a modification of the Wheeler de Witt equation he gave germane to worm hole physics. In addition, with this presentation we will discuss entropy generation via graviton production. And compare that with semi classical arguments, as well as Brane-anti brane combinations. The idea will be to in all of this to re set the particulars of massive gravity in such a way as to revisit the outstanding problem of massive gravity: Its predictions do not match those of general relativity in the limit when a massive graviton mass approaches zero In particular, while at small scales, Newton’s gravitational law is recovered, the bending of light is only three quarters of the result Albert Einstein obtained in general relativity

One of the inquiries as to graviton physics, is to ascertain how to gauge the real actual energy of a “massive” graviton. The reason for doing this, is due to the well known physics problem of how the bending of light by massive gravitons via the Planet Mercury is 3/4th that of the actual results seen in GR i.e. In the 1970s, van Dam and Veltman [

We will try to avoid using Yukawa style screening, and our start will be to ascertain an actual “rest energy” of a “massive” graviton, where we may be able to recover the limit behavior we want as m g → 0 . To do this, we will be using [

Afterwards, in 2^{nd} part of the manuscript we will briefly state some phenomenological consequences of what we have derived, and then detail those findings with possible consequences to the problem of early universe graviton generation and of an average energy, for a graviton, resulting from early universe production of gravitons.

The 3^{rd} part of the manuscript introduces in a general sense the problem of the position of gravitons, as assumed to be evaluated.

In the 4^{th} part of the manuscript, we will allude to racetrack inflation [

The 5^{th} part of this manuscript will be a discussion between different choices of entropy.

The 6^{th} part of this manuscript is a review of applications of non-standard treatments of the WdW equation which were written up by Crowell, in 2005 [

Our conclusion will be a wrap up of our findings plus a prospectus as we see it as to what to possibly expect next, and to ascertain what may be fruitful lines of inquiry. As to the originally stated problem of fixing massive gravitons, only ¾ of the angular deviation of light about the planet mercury is given.

From the use of [

δ t = ∑ i = 1 N m i ⋅ ( δ d ) i 2 2 ⋅ ( E − V ) (1)

m i refers to the mass of an ith body, ( δ d ) i is position of ith body, and ( E − V ) is kinetic energy of the system we are analyzing. We will use the construction given in [

a ≈ a min t γ ⇔ ϕ ≈ γ 4 π G ⋅ ln { 8 π G V 0 γ ⋅ ( 3 γ − 1 ) ⋅ t } ⇔ V ≈ V 0 ⋅ exp { − 16 π G γ ⋅ ϕ ( t ) } (2)

Using r = t ⋅ c

r = t ⋅ c = c 2 ⋅ ( 3 γ − 1 ) 8 π G V 0 ⋅ exp ( 4 π G γ ⋅ ϕ ) (3)

Hence, we get a kinetic energy value of

K . E . = 2 m ⋅ π G γ ⋅ ( d ϕ d t ) 2 ⋅ r 2 (4)

Thereby leading to

δ t = ∑ i = 1 N m i ⋅ ( δ d ) i 2 2 ⋅ ( E − V ) = ∑ i = 1 N m i ⋅ ( δ d ) i 2 ∑ i = 1 N m i ⋅ ( r i t ) 2 ⋅ ( γ π G ) = ∑ i = 1 N m i ⋅ ( δ d ) i 2 ∑ i = 1 N m i ⋅ ( v i ) 2 ⋅ ( γ π G ) (5)

Here is where we will use the reduced speed of the massive graviton. [

v graviton = c ⋅ ( 1 − m g 2 c 4 E 2 ) 1 / 2 (6)

Secondly set the m i → early-universe m g , in the early universe, with N the number of gravitons.

If we make the following approximation, i.e. δ t = 10 χ t P

E 2 = m g 2 c 4 ( 1 − π G N ⋅ c 2 ⋅ γ ⋅ ( 10 χ t P ) 2 ⋅ ∑ i = 1 N ( δ d i ) 2 ) (7)

This is the net energy associated with a graviton and we will spend the rest of our article analyzing the consequences of such for our questions as what is to known as the vDVZ discontinuity. And its possible resolution.

To do this we will examine what if we are working with a randomized set of value for the δ d i graviton positions, i.e. roughly like

δ d i → thermal-cavity 〈 d i 〉 semi-classical (8)

In the case of black body radiation, this would be for a random distribution of “gravitons” in a closed thermal box.

If true, i.e. that assumption. We would likely then be able to generate some version of Bose-Einstein statistics, here, for a graviton “gas” i.e. along the lines of N for the number of n. assumed gravitons, roughly

N ∝ 1 ( exp [ E k B T ] − 1 ) (9)

In this case, we would be revisiting the Solvay conference arguments as of 1927 with respect to [

n ( ω ) d ω = ω 2 d ω π 2 ⋅ [ exp ( 2 ⋅ π ⋅ ℏ ⋅ ω k ¯ T ) − 1 ] − 1 (10)

Note that both [

Hence, we will have to, if either [

To do that, we will consider, racetrack inflation, and also some of the ideas of what Crowell wrote up in [

Afterwards, in making some assumptions, as to this set of entries into terms δ d i in Equation (7) we will go to what we mentioned earlier, which is how to recast the problem of massive gravity in a way which may avoid the vDVZ discontinuity. [

This requires looking at Appendix A. And to comment upon what Appendix A has to say about positioning of the space-time domain of production of gravitons, in terms of space-time physics. [

Quote: From [

Once the slow-roll conditions break down, the scalar field switches from being overdamped to being underdamped and begins to move rapidly on the Hubble timescale, oscillating at the bottom of the potential. As it does so, it decays into conventional matter.

End of quote

I.e. this is well after the onset of inflation. [

h ( k , η i ) = A ⋅ ( k k H ) 2 + β coth [ k 2 T ] (11)

Specifically, if [ k 2 T ] is less than one, due to elevated temperatures, which is

what occurs in inflation. Hence, as by [

〈 a k † a k ′ 〉 = ( 1 exp ( k / T ) − 1 ) ⋅ δ 3 ( k − k ′ ) (12)

Notwithstanding what was said about [

In [

Quote

Hubble scale during inflation is bounded by the present value of the gravitino mass, i.e., H < m 3 / 2 . This relation, which ties the amplitude of primordial gravitational waves to the scale of supersymmetry breaking, appears to be rather generic.

End of quote

What this says, is that the racetrack though, in common with other string theory cosmology, has, at the point of symmetry breaking, of the racetrack, a regime where gravitinos, when produced, are giving bounding behavior to the Hubble scale, which in turn [

H 2 ~ V B / 3 ~ | V A d S | / 3 ~ m 3 / 2 2 & m 3 / 2 ~ 6 × 10 10 GeV & V B ∝ V racetrack (13)

The mass of a graviton, massive, is of the order of [

m g ~ 7.7 × 10 − 32 GeV / c m 3 / 2 ~ 6 × 10 10 GeV / c ~ 10 42 m g ⇒ Electro-weak ⇒ t electro-weak ~ 10 − 36 s ~ 10 7 t P ⇒ N ~ 10 42 (14)

Using this, we would have a numerical factor of N, and a time factor of δ t put in Equation (7)

δ t ∝ t electro-weak ~ 10 − 36 s ~ 10 7 t P N ~ 10 42 (15)

Due to the uncertainty of the exact commencement of the relative distance of the radii of the universe in the electroweak era, [

δ d i → thermal-cavity 〈 d i 〉 semi-classical (16)

An e fold of 65 in inflation [^{28} magnification of an initial radius, and so if we consider an electro weak magnification at the end of inflation, for a radii of 10^{−35} meters start to a magnified initial radius of about 1 meter at the very end of inflation, tops, with an initial radii of say 10^{−7} meters at the start of the electro weak era, to about 1 meter at the close of the electro weak era.

Meaning N ~ 10 42 gravitons, in a spatial regime of say a ring in between a distance of 10^{−7} meters to 1 meter from the ‘center’ of inflation in a time regime of roughly δ t ∝ t electro-weak ~ 10 − 36 s ~ 10 7 t P .

This would be, if we use the idea of racetrack inflation, and of 1 gravitino roughly equivalent to N ~ 10 42 gravitons, input into Equation (7).

Keep in mind, that Guth, on page 135, of [^{75} in volume for inflation, i.e. this is then giving us the following inputs, put into Equation (7)

δ t ∝ t electro-weak ~ 10 − 36 s ~ 10 7 t P N ~ 10 42 10 − 7 meters < δ d i < 1 meter (17)

For the sake of convenience, in this first approximation model we will be initially assuming the rest mass of a graviton is about 10^{−}^{65} grams, in line with [

We next will, if we assume that there is a correlation between entropy, due to S - N with the number N = (count of particles) [

This is reviewing the substance of Appendix B and Appendix C below. Here are some first impressions. The given models, do not answer the question of if there is at least one unit of Entropy at the start of the inflationary era. To do that, one can look at what the author did in [

If there are no units of entropy, at the start of expansion of space time, we will choose the methodology of the Racetrack which implies that entropy production and graviton production, and gravity waves would have to await at a minimum, going to the electro weak regime of space time, I.e. That space time expands 10 million times past an initial starting point.

I.e. both the semi classical picture and brane picture tend to support the idea of graviton production starting at the electro-weak era, but if the graviton is a carrier of entropy, and if the radii of the initial configuration of the universe, is not zero, then we will be reason to bring up some of the issues the author raised in [

Note that Beckwith, in [

[

Ψ ( T ) ∝ − A ⋅ { η 2 ⋅ C 1 } + A ⋅ η ⋅ ω 2 ⋅ C 2 (18)

Appendix F, gives a statement largely based upon Mukhanov [

Appendix G and Appendix H give qualitative descriptions as to the behavior of the scalar field, presumably like an inflaton, which may be zero in the initial phases of entry into the “bubble” before a presumed causal barrier at H = 0, and Appendix G gives an interpretation of the largeness of a presumed energy flux which would go out of the cosmological “bubble” of initial space time.

Note that the end effect of all this is to argue for very different dynamics, of space time, i.e. for the entropy being generated just past the H = 0 barrier of space time, with a radii of say 10^{−}^{35} meters, and all that, the answers we will get out of Equation (7) will look profoundly different than say, entropy and gravitons, and GW produced at 10^{−7} meters to 1 meter in radii “distance” from the start of presumed space-time.

Either choice will have profound implications for interpreting Equation (7) of our text. What is given below is for what we would have for Equation (7) inputs if we have entropy produced well “before” the electro weak regime

10 − 35 meters < δ d i < 10 − 33 meters & δ t ∝ t P N ~ 1 - 10 10 (19)

In [

θ ≈ 4 G M b + 15 G 2 M 2 π 4 b 2 + ϑ ( G ℏ M b 3 = l p 2 r s b 3 ) (20)

The impact parameter, b of the “photons”, i.e. light ray, with the sun, and the Schwartzshield radius r s of the sun.

This is the first item to discuss, and the last term is the one which should be minimized, whereas the first two terms are in sync with [^{nd} term is a post Newtonian contribution and the third term is a quantum correction largely based upon the Born approximation and can be seen in [

V = − G M m r ⋅ ( 1 + 3 G ⋅ ( M + m ) c 2 r + 41 G ℏ 10 c 3 r 2 ) (21)

If m is the mass of a graviton, almost 10^{−}^{65} grams, whereas M is the mass of a planet, say Mercury, and that Equation (21) has a quantum correction to the

tune of 41 G ℏ 10 c 3 r 2 put in, so that

V quantum-correction ≈ − G M m r . ( 41 G ℏ 10 c 3 r 2 ) ⇔ θ quantum-correction ≈ ϑ ( G ℏ M b 3 = l p 2 r s b 3 ) ∝ ε + ≈ 0 (22)

Our task would be to look at a total energy, say making this deduction, of

E graviton 2 = m g 2 c 4 ( 1 − π G N ⋅ c 2 ⋅ γ ⋅ ( 10 χ t P ) 2 ⋅ ∑ i = 1 N ( δ d i ) 2 ) ⇔ V graviton-Sun = − G M m r ⋅ ( 1 + 3 G ⋅ ( M + m ) c 2 r + 41 G ℏ 10 c 3 r 2 ) ⇔ V quantum-correction ≈ − G M m r . ( 41 G ℏ 10 c 3 r 2 ) ⇔ θ quantum-correction ≈ ϑ ( G ℏ M b 3 = l p 2 r s b 3 ) ∝ ε + ≈ 0 (23)

It would be a lot of work, but it would also be more direct than what De Rahm and other tried in [

What we have done, is to find a basis for a different way to address the issue of if we have relic gravitational waves at just the electro weak regime, as quantified in this paper, or if we have earlier based processes and/or the influence of recycled earlier universes, which may influence the transmission of gravitons, and possibly pre universe information to our present universe.

Do we have a repeating universe, with shared from the prior cosmos information? The logical extension of the inquiry so presented may allow for answering this question. In the meantime, the touch of using Barbour’s version of time, initially was put in to ascertain, a working benchmark for the twinning of a definite time step, with graviton production, and also then, if graviton production, i.e. the number of gravitons, is proportional to entropy, what has been done is in essence vetting the start of times arrow, via entropy production in the universe.

Equation (7) is by necessity very preliminary and we expect to revisit it with greater precision later on.

Finally we have presented a different way to start an inquiry as to working to a solution to the vDVZ discontinuity.

See Appendix I, as to the remarks made as to the foundations of gravitational astronomy. The document so presented is expected to be in fidelity with respect to these observations and guidelines.

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

Beckwith, A.W. (2017) Using Barbour’s Ephemeris Time, and Padmanabhan’s Inflaton Value, plus Will’s Massive Graviton Velocity to Isolate Rest Energy of Massive Graviton as Compared to Racetrack Inflation Results of Graviton Physics and Modified Wheeler de Witt Results of Wormhole Physics. Journal of High Energy Physics, Gravitation and Cosmology, 3, 754-775. https://doi.org/10.4236/jhepgc.2017.34056

P. Brax, A. Davis et al. [

V = V 0 + V 1 cos ( a Y ) + V 2 cos ( b Y ) + V 3 cos ( | a − b | ⋅ Y ) (a1)

This has scalar fields X , ϕ as relatively constant and we can look at an effective kinetic energy term along the lines of

ℑ Kinetic = 3 ⋅ ( ∂ Y ) 2 / 4 ( ∂ X ) 2 (a2)

This ultra simple version of the race track potential is chosen so that the following conditions may be applied

1) Exist a minimum at Y = Y 0 ; i.e. we have V ′ ( Y 0 ) = 0 , and V ′ ( Y 0 ) > 0 , when we are not considering scalar fields X , ϕ

2) We set a cosmological constant equal to zero with V ( Y 0 ) = 0

3) We have a flat saddle at Y ≈ 0 ; I.e. V ″ ( 0 ) = 0

4) We re-scale the potential via V → λ V so as to get the observed power spectra P = 4 × 10 − 10

Doing all this though frequently leads to the odd situation that | a − b | must be small so that X ≫ 1 in a race track potential system when we analyze how to fit Equation (1) for flat potential behavior modeling inflation. This assumes that we are working with a spectra index of the form so that if the scalar field power spectrum is

P = V 150 π 2 ε (a3)

Then the spectral index of the inflaton is consistent with WMAP data. I.e. if we have the number of e foldings N > N ∗ ≈ 0.55

n s = 1 − d ln P d N ≈ 0.95 ± 0.02 (a4)

These sort of restrictions on the spectral index will start to help us retrieve information as to possible inflation models which may be congruent with at least one layer of WMAP data. This model says nothing about if or not the model starts to fit in the data issues Subir Sarkar [

Kolb and Turner [

s Density = 2 ⋅ π 2 45 ⋅ g ∗ ⋅ T 3 (b1)

This pre supposes when we do it that we are able to state a total entropy as the entropy density times space time volume V 4

S total ≡ s Density ⋅ V 4 (b2)

In this situation we are writing for initial conditions with a temperature T ≈ 10 32 K for the initiation of quantum effects for quantum gravity as given by Weinberg (1972) [

T ≈ 10 32 K ≈ 1.3 × 10 28 eV ~ 1.3 × 10 19 GeV . This gives us the option of comparing what we get in entropy with Seth Lloyds [

I = S total / k B ln 2 = [ # operations ] 3 / 4 = [ ρ ⋅ c 5 ⋅ t 4 / ℏ ] 3 / 4 (b3)

We will examine if or not the following is actually true in terms of time, i.e. can we write I = ( t / t P ) 2 ? This is assuming that the density ρ ≡ T 00 ~ Λ vacuum-energy which is initially enormous, and which will be due in terms of a transfer of energy density from a prior universe to our present universe, which will be elaborated upon later in this document.

We can if we take the absolute value of Equation (b3) and (b2) above get for small volume values good estimates as to the relative volume of the phase space in early universe cosmology where Equation (b2) and Equation (b3) are congruent with each other. For our purposes, we will take time as greater than (or equal) to a Planck time interval, in line with the temperature dependence of entropy density mentioned in Equation (b1) above.

We can compare this with Thanu Padamanadan’s [

exp ( S t o t a l ) = g ( E ) = A N ! ⋅ ∫ d 3 N x ⋅ [ E − 1 2 ⋅ ∑ i ≠ j U ( x i , x j ) ] 3 N 2 ≤ A N ! ⋅ ∫ d 3 N x ⋅ [ E ] 3 N 2 ≈ [ A N ! ⋅ ∫ d 3 N x ] ⋅ [ Λ M a x V 4 8 ⋅ π ⋅ G ] 3 N 2 (b4)

If A ~ O ( 1 ) , i.e. we re scale it as being of order unity, and N ~ 10 87 particles, and we re scale ∫ d 3 N x ~ V 4 N where we choose V 4 , and where we assume Equation (b2) and Equation (b3) are equivalent and we assume that there is grounds

for writing Λ M a x V 4 8 ⋅ π ⋅ G ~ T 00 V 4 ≡ ρ ⋅ V 4 ≫ 1 2 ⋅ ∑ i ≠ j U ( x i , x j ) , we can shed light on if or

not it is still feasible to treat entropy, with N ~ 10 87 as a micro canonical ensemble phenomena, which we claim has implications for the formation of an instanton in early universe cosmology. Frankly we would want, in early universe

cosmology that we have ρ ⋅ V 4 ≠ 1 2 ⋅ ∑ i ≠ j U ( x i , x j ) , but not by too much, so we can

form an instanton.

This is adapted from a lecture given at the ICGC-07 conference by Samir Mathur [

I.e. look at the case, first of massless radiation, and then we obtain for D space time dimensions, and E the general energy

S ~ E ( D − 1 / D ) (c1)

This has

Λ M a x V 4 8 ⋅ π ⋅ G ~ T 00 V 4 ≡ ρ ⋅ V = E total (c2)

The question now becomes how do we go about defining what the necessary volume is re scaled via a quantum gravity changing of how to measure gravitational lengths which are for the threshold of quantum gravity. Traditionally the bench marking has been via the Planck length l P ~ 10 − 33 cm → Quantum-Gravity-threshold N ˜ α ⋅ l P . This re scaling of the minimum length needed for the importance of quantum gravity effects showing up in a grid of space time resolves, as information paradox of black hole physics. So far we have merely been working with a typical string gas model for entropy. Now, let us add in a supposition for N ⌣ branes and anti branes to put in an instanton structure as to how we look at the entropy. Gilad Lifschytz [

S Total ~ a ⌣ ⋅ [ E Total 2 n ] λ ⋅ ∏ j = 1 N ( M p j , 0 + M p ¯ j , 0 ) (c3)

This has when we do it E Total as in Equation(c1) above, and proportional to the cosmological vacuum energy parameter. Of course, in string theory, the energy is also defined via

E Total = 4 λ ⋅ M p j , 0 ⋅ M p ¯ j , 0 (c4)

Furthermore, the values of M p j , 0 , and M p ¯ j , 0 refer to the mass of p branes and p anti branes, as Gilad Lifschytz refers to it. This can be changed and rescaled to treating the mass and the energy of the brane contribution along the lines of Mathur’s^{2} [

E Total = 2 ∑ i m i n i (c5)

This leads to entropy

S Total = A ⋅ ∏ i N n i (c6)

Our claim is that this very specific value of entropy for Equation (c6) above will in Planck interval of time at about the onset of inflation lead to

| [ S Total = A ⋅ ∏ i N n i ] / k B ln 2 | ≈ [ # operations ] 3 / 4 ≈ 10 8 (c7)

Furthermore we also claim that the interaction of the branes and anti-branes will form an instanton structure, which is implicit in the treatment outlined in Equation (c4), and that the numerical counting given in Equation (c6) merely reflects that branes and anti-branes, even if charge conjugates of each other have the same “wrapping number” n i .

We begin with a temperature estimate of T ≈ 10 45 K > T QG-Threshold ~ 10 32 K . Then, Equation (b4) above modified when we take the absolute value will lead to, if we look at when N ≈ 10 86 :

| S Total | ~ | k B ⋅ ln 2 | ⋅ [ # operations ] 3 / 4 ≈ N | − log N 10 + log V 4 3 + log E 3 / 2 | ~ 10 8 (d1)

Leading to solving for E as follows ⇒ ρ ⋅ V 4 ≠ 1 2 ⋅ ∑ i ≠ j U ( x i , x j ) , and also that

V 4 3 ⋅ E 3 / 2 ~ 10 85 ⇒ E ~ 10 57 V 4 2 ≡ Λ Max ⋅ V 4 8 ⋅ π ⋅ G (d2)

We can and will reference what we can say about Λ Max ~ c 2 ⋅ T β ˜ , as given by Park [

V 4 | Threshold-volume-for-quantum-effects ~ 10 − 4 cm (d3)

This is way too large, but it indicates that the interaction of material within

the region of space being considered does not obey ρ ⋅ V 4 ≫ 1 2 ⋅ ∑ i ≠ j U ( x i , x j ) . If

this is what we have, we can then begin to look at if the instanton picture is true or not. We will first review what can be said about different variants of vacuum energy. I.e. where the vacuum energy models of four and five dimensions could conceivably overlap. But to do this we will look at what these models are.

We will fill in the details inherent in Equation (18) above in the main text.. This will be to show some things about the worm hole we assert the instanton traverses en route to our present universe. Equation (18) of the main text actually comes from the following version of the Wheeler De Witt equation with a pseudo time component added. From Crowell [^{ }

− 1 η r ∂ 2 Ψ ∂ r 2 + 1 η r 2 ⋅ ∂ Ψ ∂ r + r R ( 3 ) Ψ = ( r η ϕ − r ϕ ¨ ) ⋅ Ψ (e1)

This has when we do it ϕ ≈ cos ( ω ⋅ t ) , and frequently R ( 3 ) ≈ constant , so then we can consider

ϕ ≅ ∫ 0 ∞ d ω [ a ( ω ) ⋅ e i k ϖ x μ − a + ( ω ) ⋅ e − i k ϖ x μ ] (e2)

In order to do this, we can write out the following with regards to the solutions to Equation (e1) put up above.

C 1 = η 2 ⋅ ( 4 ⋅ π ⋅ t 2 ω 5 ⋅ J 1 ( ω ⋅ r ) + 4 ω 5 ⋅ sin ( ω ⋅ r ) + ( ω ⋅ r ) ⋅ cos ( ω ⋅ r ) ) + 15 ω 5 cos ( ω ⋅ r ) − 6 ω 5 S i ( ω ⋅ r ) (e3)

And

C 2 = 3 2 ⋅ ω 4 ⋅ ( 1 − cos ( ω ⋅ r ) ) − 4 e − ω ⋅ r + 6 ω 4 ⋅ C i ( ω ⋅ r ) (e4)

This is where S i ( ω ⋅ r ) and C i ( ω ⋅ r ) refer to integrals of the form

∫ − ∞ x sin ( x ′ ) x ′ d x ′ and ∫ − ∞ x cos ( x ′ ) x ′ d x ′ . It so happens that this is for forming the

wave functional permitting an instanton forming, while we next should consider if or not the instanton so farmed is stable under evolution of space time leading up to inflation. We argue here that we are forming an instanton whose thermal energy is focused into a wave functional which is in the throat of the worm hole up to a thermal discontinuity barrier at the onset, and beginning of the inflationary era.

We argue that the existence of the worm hole and an instanton formation in the throat of the worm hole will lead to a constant energy flux. Note that we are assuming a constant energy flux through the worm hole. This is equivalent to work with an expression given by Mukhanov [

∂ 2 δ ρ ( x ) ∂ t 2 − c s 2 Δ ⋅ δ ρ ( x ) − 4 π ⋅ G ρ 0 ⋅ δ ρ ( x ) = σ ⋅ Δ δ S ( x ) (f1)

This is Fourier transformed into being

∂ 2 δ ρ ( k ) ∂ t 2 + k 2 c s 2 ⋅ δ ρ ( k ) − 4 π ⋅ G ρ 0 ⋅ δ ρ ( k ) = − σ ⋅ k 2 δ S ( k ) (f2)

This has a time independent solution of the form given by, assuming small spatial dimensions

δ ρ ( k ) ≡ − σ k 2 δ S ( k ) ( k 2 c s 2 − 4 π G ρ 0 ) → k → big − σ δ S ( k ) ( c s 2 ) (f3)

This may be Fourier transformed, assuming near constant values of k and position x, to be in x position space

δ ρ ( x ) ≅ − 8 σ c s 2 δ S ( x ) (f4)

Here, c s 2 is the square of the speed of sound which is in early universe conditions close to unity. We also have that σ ≡ ( ∂ p / ∂ S ) ρ . Then we can state that when we have δ ρ ( x ) ∝ Λ initial → Λ max due to increasing temperature

| δ ρ ( x ) | ≅ 8 σ ⋅ δ S ( x ) (f5)

We claim that the increase in entropy, is connected with a breaking of the instanton structure of a packet of energy transferred from a prior space time to our own.

Look at an argument provided by Thanu Padmanabhan [

ρ V A C ~ Λ observed 8 π G ~ ρ U V ⋅ ρ I R ~ l Planck − 4 ⋅ l H − 4 ~ l Planck − 2 ⋅ H observed 2 (g1)

Δ ρ ≈ adarkenergydensity ~ H observed 2 / G (g2)

We can replace Λ observed , H observed 2 by Λ initial , H initial 2 . In addition we may look at inputs from the initial value of the Hubble parameter to get the necessary e folding needed for inflation, according to

E -foldings = H initial ⋅ ( t Endofinf − t beginningofinf ) ≡ N ≥ 100 ⇒ H initial ≥ 10 39 − 10 43 (g3)

Leading to

a ( Endofinf ) / a ( Beginningofinf ) ≡ exp ( N ) (g4)

If we set Λ initial ~ c 1 ⋅ [ T ~ 10 32 Kelvin ] implying a very large initial cosmological constant value, we get in line with what Park [

of nucleation of a vacuum state

Λ initial ~ [ 10 156 ] ⋅ 8 π G ≈ hugenumber (g5)

Question. Do we always have this value of Equation (g5)? At the onset of Inflation? When we are not that far away from a volume of space characterized by l P 3 , or at most 100 or so times larger? Contemporary big bang theories imply this. I.e. a very high level of thermal energy. We need to ask if this is something which could be transferred from a prior universe, i.e. could there be a pop up nucleation effect, I.e. emergent space time? Appendix H gives a way for this to occur. We will now examine a mechanism which would allow for this to happen. It involves transfer of energy from a prior to the present universe.

We begin with the D’Albertain operator as part of an equation of motion for an emergent scalar field. We refer to the Penrose potential (with an initial assumption of Euclidian flat space for computational simplicity) to account for, in a high temperature regime an emergent non-zero value for the scalar field ϕ due to a zero effective mass, at high temperatures [

When the mass approaches far lower values, it, a non-zero scalar field re appears.

Leading to ϕ → T → 2.7 Kelvin ε + ≈ 0 + as a vanishingly small contribution to cosmological evolution

Let us now begin to initiate how to model the Penrose quintessence scalar field evolution equation. To begin, look at the flat space version of the evolution equation

ϕ ¨ − ∇ 2 ϕ + ∂ V ∂ ϕ = 0 (h1)

This is, in the Friedman?Walker metric using the following as a potential system to work with, namely:

V ( ϕ ) ~ − [ 1 2 ⋅ ( M ( T ) + ℜ 6 ) ϕ 2 + a ˜ 4 ϕ 4 ] ≡ − [ 1 2 ⋅ ( M ( T ) + κ 6 a 2 ( t ) ) ϕ 2 + a ˜ 4 ϕ 4 ] (h2)

This is pre supposing κ ≡ ± 1 , 0 , that one is picking a curvature signature which is compatible with an open universe.

That means κ = − 1 , 0 as possibilities. So we will look at the κ = − 1 , 0 values. We begin with.

ϕ ¨ − ∇ 2 ϕ + ∂ V ∂ ϕ = 0 ⇒ ϕ 2 = 1 a ˜ ⋅ { c 1 2 − [ α 2 + κ 6 a 2 ( t ) + M ( T ) ] } ⇔ ϕ ≡ e − α ⋅ r exp ( c 1 t ) (h3)

We find the following as far as basic phenomenology, namely

ϕ 2 = 1 a ˜ ⋅ { c 1 2 − [ α 2 + κ 6 a 2 ( t ) + ( M ( T ) ≈ ε + ) ] } → M ( T ~high ) → 0 ϕ 2 ≠ 0 (h4)

ϕ 2 = 1 a ˜ ⋅ { c 1 2 − [ α 2 + κ 6 a 2 ( t ) + ( M ( T ) ≠ ε + ) ] } → M ( T ~Low ) ≠ 0 ϕ 2 ≈ 0 (h5)

The difference is due to the behavior of M ( T ) . We use M ( T ) ~axion mass m a ( T ) in asymptotic limits with

m a ( T ) ≅ 0.1 ⋅ m a ( T = 0 ) ⋅ ( Λ Q C D / T ) 3.7 (h6).

The experimental gravity considerations are covered in [

Reference [

Finally, in lieu of [

Quote

In essence, for making a consistent cosmology, our results argue in favor of a string theory style embedding of the start of inflation and what we have argued so far is indicating how typical four-dimensional cosmologies have serious mathematical measure theoretic problems. This quantum measure theoretic problem are unphysical especially in light of the Stoica findings.

End of quote

This is a fairly consistent edorsement to the idea of what was presented for our model, starting with H = 0 (no initial expansion) to the jump toward massive expansion, We urge the readers to review it, as well as to review what Corda brought up in [

As to [