New Conservation Law as to Hubble Parameter, Squared Divided by Time Derivative of Inflaton in Early and Late Universe, Compared with Discussion of HUP in Pre Planckian to Planckian Physics, and Relevance of Fifth Force Analysis to Gravitons and GW

Abstract

We use as given by Li and Koyama, as to using their idea of a fifth force. In doing so, we are assuming a force which is minus the spatial derivative of a scalar field. The scalar field we are using is one from Padmanabhan, and the problem is that the scalar field in the Padmanabhan representation is initially only dependent on time. The time component is stated to be in the Pre Planckian regime is proportional to a radial distance divided by the speed of light. The rephrasing of time as justified by stating that time in its initial configuration does not exist before the expansion of the universe and that we reintroduce time separately from a radial component divided by the speed of light upon entrance into Planckian space–time. In doing this we also refer to a new assumed conservation law which will give new structure as to inflationary expansion and its immediate aftermath. That of the Hubble “constant” divided by the ‘time derivative’ of the scalar field in the inflation regime and then a long time afterwards.

1. Introduction

Our idea is to regularize inflation and its aftermath by a Hubble parameter divided by the derivative of a scalar field, as being about the same ratio in Planckian space time and then say in the time frame within a billion years of the present. The benefits of such an interpretation is to regularize how we obtain GW frequency, in the initial phase of the universe expansion to near the present era. In addition we use a fifth force as to explain how we would have almost an infinite expansion rate in the beginning of inflation. The almost infinite expansion rate is due to the fifth force which triggers rapid expansion. We then conclude as to a regime of black hole physics, with a table as to a pre present universe well before our big bang super massive black hole, which would then be through a nonsingular start to the universe break up from pre big bang configuration into millions of micro sized black holes. The way to do this assumes a variant of the Penrose cyclic conformal cosmology model, with a pre universe giant sized black holes broken up into tiny black holes by the uncounted millions [1, 2].

2. How we will obtain scalar field behavior we want which yields input into a fifth force

Using

$$a\left(t\right)=a_{\mathit{\text{initial}}}{t}^{\nu }$$E1

Which will lead to

$$\frac{H^2}{\dot{\varphi }}\approx \sqrt{\frac{4\mathit{\pi G}}{\nu }}\cdot t\cdot T^4\cdot \frac{{\left(1.66\right)}^2\cdot g_{\ast }}{m_P^2}\approx {10}^{-5}$$E2

The Eq. (2) is a conservation law which is considered to be true in the initial expansion, Planck regime of space–time.

This of course makes uses of Eq. (3) for the Hubble parameter, the Padmabhan value of the scalar field due to Eq. (1) and this is all assuming a value of

$$H=1.66\sqrt{g_{\ast }}\cdot \frac{{T_{\mathit{\text{temperature}}}}^2}{m_P}$$E3

We will make the following calculation [3, 4].

$$V_0={\left(\frac{.022}{\sqrt{{\mathit{qN}}_{\mathit{\text{efolds}}}}}\right)}^4=\frac{\nu \left(\nu -1\right){\lambda }^2}{8\pi {\mathit{Gm}}_P^2}$$E4

$\lambda$ as a dimensionless parameter which we refer to later. From [3], page 17 we have ae\ Chamelon mechanism for fifth force as

$$F_{5\mathit{th}-\mathit{\text{force}}}=-\frac{\tilde{\beta }\cdot \left(\overrightarrow{\nabla }\varphi \right)}{m_P}$$E5

Eq. (5) equals zero of we have a scalar field solely dependent upon time. i.e. we need to have a re set of time as initially spatial divided by the speed of light.

3. Pre-Planckian to Planckian regime of space-time reset so Eq. (5) for fifth force does not vanish

To do this we will assume in an initial “bubble’ of space–time of which we have spatial r value and a speed of light given by c

$$t=\frac{r}{\mathit{\varpi c}}$$E6

The term of $\varpi$ is a dimensionless value and never negative.

If so, Eq. (5) will yield [3, 4].

$$F_{5\mathit{th}-\mathit{\text{force}}}=-\frac{\tilde{\beta }\cdot \left(\overrightarrow{\nabla }\varphi \right)}{m_P}\approx -\frac{\tilde{\beta }}{2m_{P}r}\cdot \sqrt{\frac{\nu }{\mathit{\pi G}}}$$E7

4. What is the power of production of gravitons due to fifth force?

The easiest way is to look at power expressions for GW and to make them linked to Eq. (7).

First Power = Force, time velocity.

$$P=\mathit{\text{Power}}=F\left(\mathit{\text{force}}\right)\times v\left(\mathit{\text{velocity}}\right)$$E8

Compare Eq. (8) for Power in terms of gravitational waves using [5, 6, 7].

See [7],

$$P_{\mathit{GW}}\approx \frac{{\mathit{GM}}_{\mathit{\text{mass}}}{\omega }_{\mathit{gw}}^2}{c^2}$$E9

Usually, we want to look at GW quadrupoles, [6], page 312

$${\ddot{Q}}^2\approx {\left[\left(\frac{{\omega }_{\mathit{gw}}^2}{c^2}\right)\cdot ⟨r^2⟩\right]}^2$$E10

Keep in mind that we are using GW power which is given by

$$P_{\mathit{GW}}\approx \frac{\mathit{Gc}\cdot {\left(M_{\mathit{\text{mass}}}\right)}^2{\omega }_{\mathit{gw}}^6{⟨r^2⟩}^2}{c^6}$$E11

Eq. (7) times c (speed of light) will alter Eq. (11) to be read as

$$\begin{array}{l}{P}_{\mathit{GW}}\approx \frac{\mathit{Gc}\cdot {\left(M_{\mathit{\text{mass}}}\right)}^2{\omega }_{\mathit{gw}}^6{⟨r^2⟩}^2}{c^6}\\ \approx c\times \left|F_{5\mathit{th}-\mathit{\text{force}}}\right|=\left|-c\times \frac{\tilde{\beta }\cdot \left(\overrightarrow{\nabla }\varphi \right)}{m_P}\right|\approx c\times \frac{\tilde{\beta }}{2m_{P}r}\cdot \sqrt{\frac{\nu }{\mathit{\pi G}}}\end{array}$$E12

This leads to

$$\begin{array}{l}{\omega }_{\mathit{gw}}^6\approx c^7\times \frac{\tilde{\beta }}{2m_{P}r}\cdot \sqrt{\frac{\nu }{\mathit{\pi G}}}\times \frac{1}{\mathit{Gc}\cdot {\left(M_{\mathit{\text{mass}}}\right)}^2{⟨r^2⟩}^2}\\ \Rightarrow {\omega }_{\mathit{gw}}\approx {\left(\sqrt{\frac{\nu }{4\mathit{\pi G}}}\times \frac{\tilde{\beta }\cdot c^6}{G\cdot {\left(M_{\mathit{\text{mass}}}\right)}^2{m}_{P}r\cdot {⟨r^2⟩}^2}\right)}^{1/6}\end{array}$$E13

In terms of Planck Units

$$\begin{array}{l}{\omega }_{\mathit{gw}}^6\approx c^7\times \frac{\tilde{\beta }}{2m_{P}r}\cdot \sqrt{\frac{\nu }{\mathit{\pi G}}}\times \frac{1}{\mathit{Gc}\cdot {\left(M_{\mathit{\text{mass}}}\right)}^2{⟨r^2⟩}^2}\\ \Rightarrow {\omega }_{\mathit{gw}}\approx G,m_P,r\approx {\mathrm{\ell }}_p\underset{\mathit{\text{Planck}}-\mathit{\text{normalization}}}{\to }1\\ M_{\mathit{\text{mass}}}\approx \varsigma \cdot m_P\underset{\mathit{\text{Planck}}-\mathit{\text{normalization}}}{\to }\varsigma \\ {⟨r^2⟩}^2\approx {\mathrm{\ell }}_p^4\underset{\mathit{\text{Planck}}-\mathit{\text{normalization}}}{\to }1\\ \therefore {\omega }_{\mathit{gw}}\underset{\mathit{\text{Planck}}-\mathit{\text{normalization}}}{\to }{\left(\sqrt{\frac{\nu }{4\pi }}\times \frac{\tilde{\beta }}{{\left(\varsigma \right)}^2}\right)}^{1/6}\end{array}$$E14

We find then we have at the immediate beginning of inflation, an almost Planck frequency value of 1.855 times 10^43 Hertz, we would need $\nu$ be 10^502 which would be factored into Eq. (1) and the scale factor value for the term $\nu$. This would mean for the fifth force argument that we would have an almost infinitely quick expansion in the neighborhood of Planck length for the start of inflation.

What this means is that coefficient $\nu$ in the initial genesis of GW which will be in Planckian space–time to be

$$\nu \underset{\mathit{\text{Planck}}-\mathit{\text{normalization}}}{\to }4\pi \times {\left({\omega }_{\mathit{gw}}\right)}^{12}\times \frac{{\left(\varsigma \right)}^4}{{\tilde{\beta }}^2}$$E15

If we are looking at Planck time, in the Planck era, $\nu \propto {\left({\omega }_{\mathit{\text{Planck}}}\right)}^{12}$, meaning that the rate of expansion in the early universe is commensurate with inflation.

5. Interpreting force assumed in terms of Ehrenfests theorem

Gasiorowitz, [5] gives this Ehrenfests Theorem as

$$F=\frac{d⟨p⟩}{\mathit{dt}}=-{⟨\frac{\mathit{dV}}{\mathit{dr}}⟩}_t$$E16

We re write Eq. (16) as yielding in our procedure

$$⟨p⟩=-\frac{\tilde{\beta }}{2m_P}\cdot \sqrt{\frac{\nu }{\mathit{\pi G}}}\cdot \frac{lnt}{\mathit{\varpi c}}$$E17

For sufficiently small time step, t, use the ideas given in, [5] which leads to

$$\begin{array}{l}\mathit{\text{If}}⟨p⟩\approx \mathrm{\Delta }p\approx \frac{\tilde{\beta }}{2m_P}\cdot \sqrt{\frac{\nu }{\mathit{\pi G}}}\cdot \frac{\left|ln{\epsilon }^{+}\right|}{\mathit{\varpi c}}\\ \mathrm{\Delta }p\mathrm{\Delta }x\approx \mathrm{\hslash }\Rightarrow \mathrm{\Delta }x\approx \frac{\mathrm{\hslash }}{\frac{\tilde{\beta }}{2m_P}\cdot \sqrt{\frac{\nu }{\mathit{\pi G}}}\cdot \frac{\left|ln{\epsilon }^{+}\right|}{\mathit{\varpi c}}}\le l_P\end{array}$$E18

Meaning we will be obtaining to enormous energy values, for time smaller than Planck time.

6. Energy values, and the degrees of freedom initially

In an earlier paper, initial mass [8] is written as a huge value, namely

$$\begin{array}{l}M=\sqrt{\sqrt{g_{\ast }}\cdot \frac{1.66\mathrm{\hslash }}{64{\pi }^2{m}_P{G}^{2}k{}_B{}^2}}\cdot \sqrt{\frac{t}{\gamma }}\sqrt{N_{\mathit{\text{Gravitons}}}}\cdot m_{\mathit{\text{Planck}}}\\ \underset{\mathit{\text{Planck}}-\mathit{\text{Units}}}{\to }\approx \sqrt[4]{g_{\ast }}\cdot \sqrt{\frac{1.66}{64{\pi }^2}}\cdot m_{\mathit{\text{Planck}}}\approx \sqrt{N_{\mathit{\text{Gravitons}}}}\cdot m_{\mathit{\text{Planck}}}\\ \approx {10}^{60}\cdot m_{\mathit{\text{Planck}}}\end{array}$$E19

If so then

$$\sqrt[4]{g_{\ast }}\cdot \sqrt{\frac{1.66}{64{\pi }^2}}\approx {10}^{60}$$E20

i.e. the initial degrees of freedom, would be

$$g_{\ast }\approx {10}^{240}\cdot {\left(\frac{64{\pi }^2}{1.66}\right)}^2\approx {10}^{240}\times 144791\propto {10}^{245}$$E21

The magnitude of Eq. (21) is a ten to the 245 value, whereas degrees of freedom, is 10 to the 61st power.

Why we pay attention to this value. By [8].

$$\begin{array}{l}{m}_{\mathit{\text{graviton}}}\approx {10}^{-60}{m}_P\Rightarrow N_{\mathit{\text{Gravitons}}}\approx {10}^{120}\\ \Rightarrow N_{\mathit{\text{Gravitons}}}\approx {10}^{120}\approx S_{\mathit{\text{entropy}}}\\ \iff g_{\ast }\approx {10}^{240}\cdot {\left(\frac{64{\pi }^2}{1.66}\right)}^2\approx {10}^{240}\times 144791\propto {10}^{245}\end{array}$$E22

This is directly due, if assuming, an initial value of $t=\frac{r}{\mathit{\varpi c}}$ that Eq. (5) would be a large negative value, and

$$\varphi \left(\frac{r}{\mathit{\varpi c}}\right)=\sqrt{\frac{\nu }{4\mathit{\pi G}}}ln\left(\sqrt{\frac{8\pi {\mathit{GV}}_0}{\nu \left(\nu -1\right)}}\cdot \left(\frac{r}{\mathit{\varpi c}}\right)\right)$$E23

We will next discuss the fifth force in terms of the Dilaton model.

7. Dilaton model versus the general relativity theory. Can we extend general relativity?

In [3], page 17, the NON relativistic geodestic equation for a ‘test particle’

$$\overrightarrow{\ddot{x}}=-\overrightarrow{\nabla }\mathrm{\Psi }-\frac{\tilde{\beta }\cdot \left(\overrightarrow{\nabla }\varphi \right)}{m_P}$$E24

The first term is gravitational potential $\mathrm{\Psi }$. The second term is the fifth force. For Pre Planck to Plank

$$\overrightarrow{\ddot{x}}=-\overrightarrow{\nabla }\mathrm{\Psi }-\frac{\tilde{\beta }\cdot \left(\overrightarrow{\nabla }\varphi \right)}{m_P}\underset{Pre-\mathit{\text{Planck}}}{\to }\overrightarrow{\ddot{x}}=-\frac{\tilde{\beta }\cdot \left(\overrightarrow{\nabla }\varphi \right)}{m_P}$$E25

We assume

$$\begin{array}{l}-\overrightarrow{\nabla }\mathrm{\Psi }\underset{Pre-\mathit{\text{Planck}}}{\to }0\\ -\frac{\tilde{\beta }\cdot \left(\overrightarrow{\nabla }\varphi \right)}{m_P}\underset{Pre-\mathit{\text{Planck}}}{\to }\mathit{\text{NOTzero}}\\ t\underset{Pre-\mathit{\text{Planck}}}{\to }t=\frac{r}{\mathit{\varpi c}}\\ t\underset{\mathit{\text{Planck}}}{\to }t\ne \frac{r}{\mathit{\varpi c}}\\ -\frac{\tilde{\beta }\cdot \left(\overrightarrow{\nabla }\varphi \right)}{m_P}\underset{\mathit{\text{Planck}}}{\to }\mathit{\text{Very}}-\mathit{\text{small}}-\mathit{\text{value}}\\ -\overrightarrow{\nabla }\mathrm{\Psi }\underset{\mathit{\text{Planck}}}{\to }\mathit{\text{Not}}-\mathit{\text{zero}}\end{array}$$E26

In addition [1, 2, 3, 4] we assume where we are using the values of an inflation potential given in [2] if we have Eq. (1) for the scale factor compared with another similar scalar potential

$$V\left(\varphi \right)=V_{0}exp\left(-\frac{\mathit{\lambda \varphi }}{m_P}\right)\leftrightarrow V_{0}exp\left(-\sqrt{\frac{16\mathit{\pi G}}{\nu }}\cdot \varphi \right)$$E27

In Pre Planck physics, Eq. (5) would be enormous, whereas the fifth force for Plank regime and beyond would be small. Also the Gravitational physics term due to a gravitational potential $\mathrm{\Psi }$ would be the largest term in Eq. (24).

8. Review of what our presumption of pre Planck to plank physics have gained us, before Gravimagnetism

We deviate from standard relativity and Newtonian physics by the existence of a fifth force in Pre Planckian to Planckian space–time physics.

We are considering what if Eq. (5) and Eq. (6) insert fifth force physics into cosmology What has to be determined are experimental verifications of Eq, (23) and Eq. (24). This is a test and a way to obtain falsifiable models. Furthermore we are presuming a nonsingular start to the universe. And these ideas need experimental verification.

Finally the model included in by use of references [9, 10] need to be seriously reviewed.

9. Gravimagnetism and an invariance model considered

We revisit Gravimagnetism and its links to this problem [11].

Note on page 48 of [11].

$$\begin{array}{l}\frac{d\stackrel{\rightharpoonup }{v}}{\mathit{dt}}\equiv -\mathit{\text{gradϕ}}+2\overrightarrow{\mathrm{\Omega }}\times \overrightarrow{v}\\ \leftrightarrow \mathit{\text{Lorentz}}-\mathit{\text{force}}\\ =\overrightarrow{K}=q\cdot \left(\overrightarrow{E}+\frac{\overrightarrow{v}}{c}\times \overrightarrow{B}\right)\end{array}$$E28

In Electromagnetic theory we note we have exact correspondences. However in GR the first line of Eq. (28) is approximate.

In the Pre Planck regime of space–time we use the following

$$\frac{d\stackrel{\rightharpoonup }{v}}{\mathit{dt}}\equiv -\mathit{\text{gradϕ}}+2\overrightarrow{\mathrm{\Omega }}\times \overrightarrow{v}$$E29

In the Pre Planckian regime we will have

$$\frac{d\stackrel{\rightharpoonup }{v}}{\mathit{dt}}\equiv -\mathit{\text{gradϕ}}+2\overrightarrow{\mathrm{\Omega }}\times \overrightarrow{v}\underset{Pre-\mathit{\text{Planckian}}}{\to }\frac{d\stackrel{\rightharpoonup }{v}}{\mathit{dt}}\equiv -\mathit{\text{gradϕ}}\approx -{\partial }_r\varphi$$E30

Whereas we use this substitution to obtain a nonzero fifth force in Pre Planckian physics

$$\varphi \left(t\right)\underset{Pre-\mathit{\text{Planckian}}}{\to }\varphi \left(\frac{r}{\mathit{\varpi c}}\right)$$E31

If this is done, then the Graviton condensate relationship as argued by the author before, should also be examined as far as experimental verification, It would be optimal if we find that the Pre Planckian to Planckian physics regime would have a lot of black holes, as given in [12].

$$\begin{array}{l}m\approx \frac{M_P}{\sqrt{N_{\mathit{\text{gravitons}}}}}\\ M_{\mathit{BH}}\approx \sqrt{N_{\mathit{\text{gravitons}}}}\cdot M_P\\ R_{\mathit{BH}}\approx \sqrt{N_{\mathit{\text{gravitons}}}}\cdot l_P\\ S_{\mathit{BH}}\approx k_B\cdot N_{\mathit{\text{gravitons}}}\\ T_{\mathit{BH}}\approx \frac{T_P}{\sqrt{N_{\mathit{\text{gravitons}}}}}\end{array}$$E32

Having a change in initial conditions from Pre Planckian physics to Planckian physics would be enough if we find that m in Eq. (32) is actually the mass of a graviton.

If so, by Novello [13] we then scale mass m as given in Eq. (32) to the mass of a graviton, as in Eq. (33)

$$m_g=\frac{\mathrm{\hslash }\cdot \sqrt{\mathrm{\Lambda }}}{c}$$E33

In a word, the next step to ascertain would be how Eq. (31), as given breaks down, and we have then application of Eq. (32) with m set with m becoming the mass of a graviton as given in Eq. (33).

Confirming these details should be the object of future research as can also be seen in [13].

In addition we have the [14] as to the choice of the Starobinsky potential as well use of radial acceleration as a way of confirming the cosmological constant.

The way indicated in [14] may fix the value of m, after determining M, as an input into Eq. (32).

Then use the right hand side of Eq.(33) whereas in [8], we will be determining the right hand side of Eq. (33), namely $\mathrm{\Lambda }$ and then after doing that, assuming Eq. (33) to work backwards into the M of Eq. (32).

That is how to reconcile the [8] and [14] references whereas we will be using this current document to ascertain the existences of a Fifth force which would be a bridge between Pre Planckian to Planckian physics,. Finally though what is implicitly assumed is [15] which is an application of Klauder enhanced quantization.

Finally is the imponderable, i.e. the generalization of Penrose CCC theory which is in [8] which is a generalization of what is in Penrose single universe recycling of universes which may be seen in [16].

All these steps need to be combined and rationalized. Also remember [17].

10. Now for the invariance model

Also the Hawking argument as to the probability of finding a universe with $\mathrm{\Lambda }$ being a given value [18, 19].

$$P\left(\mathrm{\Lambda }\right)\sim exp\left(-2S_E\left(\mathrm{\Lambda }\right)\right)\approx exp\left(\frac{3\pi M_P^2}{\mathrm{\Lambda }}\right)$$E34

We get combining Eq. (32), Eq. (33) and Eq. (34) to realize having

$$\begin{array}{l}P\left(\mathrm{\Lambda }\right)\underset{\mathit{Eq}\left(1\right),\mathit{Eq}.\left(2\right),\mathit{Eq}.\left(3\right)}{\to }exp\left(\frac{3\pi c^2{N}_{\mathit{\text{graviton}}}}{{\mathrm{\hslash }}^2}\right)\\ \underset{\mathit{Eq}\left(1\right),\mathit{Eq}.\left(2\right),\mathit{Eq}.\left(3\right),\mathrm{\hslash }=c\to 1}{\to }exp\left(3\pi N_{\mathit{\text{graviton}}}\right)\end{array}$$E35

11. Now putting in the detail about the universe being treated as a giant black hole, of sorts

First sign in the mass m in Eq. (32) as being the same as the mass of a graviton, in Eq. (33).

We then would have

$$m\to m_g\approx \frac{M_P}{\sqrt{N_{\mathit{\text{graviton}}}}}\Rightarrow N_{\mathit{\text{graviton}}}\approx {10}^{122}$$E36

In addition the radius of the “particle” would be of the form given by

$$R\to R_{\mathit{\text{universe}}}\approx \sqrt{N_{\mathit{\text{graviton}}}}\cdot {\mathrm{\ell }}_P\approx {10}^{61}\cdot {\mathrm{\ell }}_P$$E37

Also the overall mass M would scale as

$$M\to M_{\mathit{\text{universe}}}\approx \sqrt{N_{\mathit{\text{graviton}}}}\cdot M_P\approx {10}^{61}\cdot M_P$$E38

Whereas the entropy

$$S\to S_{\mathit{\text{universe}}}\left(\mathit{\text{gravitons}}\right)\approx k_B\cdot {10}^{122}\underset{k_B\to 1}{\to }{10}^{122}$$E39

And the final temperature

$$T\to T_{\mathit{\text{universe}}}\left(\mathit{\text{gravitons}}\right)\approx \frac{T_P}{\sqrt{N_{\mathit{\text{graviton}}}}}\approx {10}^{-61}\cdot T_P$$E40

In this case, we have that the mass of the graviton, allowing for this scaling is given by [5, 6, 20, 21].

12. Consequences. We have a starting point determined by the following

From Eq. (1) and Eq. (2) of this manuscript we have the DNA for the working out of the coefficient of the scale factor, and this is in the end what we end up with.

If we are looking at Planck time, and assuming we have Plank frequency, this means in the Planck era

$$\nu \propto {\left({\omega }_{\mathit{\text{Planck}}}\right)}^{12},$$E41

This enormous initial coefficient to the scale factor time coefficient, would be put in initially in the last part of Eq. (41) which would subsequently, be invariant, namely from the beginning of inflation, to its near present day conditions, the following would be invariant, so the following would be approximately a constant

$${\left.\frac{H^2}{\dot{\varphi }}\approx {10}^{-5}\right\}}_{\mathit{\text{initial}}-\mathit{\text{conditions}}}\underset{\mathit{\text{Evolution}}-\mathit{\text{to}}-\mathit{\text{near}}-\mathit{\text{present}}}{\to }{\left.\frac{H^2}{\dot{\varphi }}\approx {10}^{-5}\right\}}_{\mathit{\text{present}}-\mathit{\text{conditions}}}$$E42

This would somehow have to be confirmed via data sets but the coefficients in the initial conditions to final, in ratio would be similar, in ratio value, but the magnitude of the H term, and the magnitude of the derivative of the scalar field would be vastly different, just their ratios would likely have a similar value [22, 23, 24, 25]. And this leads us to the final question to raise. What would it take to come up with the initial frequency as given in Eq. (41) leading to an initially extremely high rate of expansion of the scale factor? To see this let us conclude about energy initially,

13. So do we have something (space time) from nothing? Conclusion with speculations as to answer

To answer this, we look at the following. Namely the crazy geometry in the Pre Planckian regime of space time.

Let us first recall the Shalyt-Margolin and Tregubovich (2004, p.73) [26].

$$\begin{array}{l}\mathrm{\Delta }t\ge \frac{\mathrm{\hslash }}{\mathrm{\Delta }E}+\gamma t_P^2\frac{\mathrm{\Delta }E}{\mathrm{\hslash }}\Rightarrow {\left(\mathrm{\Delta }E\right)}^2-\frac{\mathrm{\hslash \Delta }t}{\gamma t_P^2}{\left(\mathrm{\Delta }E\right)}^1+\frac{{\mathrm{\hslash }}^2}{\gamma t_P^2}=0\\ \Rightarrow \mathrm{\Delta }E=\frac{\mathrm{\hslash \Delta }t}{2\gamma t_P^2}\cdot \left(1+\sqrt{1-\frac{4{\mathrm{\hslash }}^2}{\gamma t_P^2\cdot {\left(\frac{\mathrm{\hslash \Delta }t}{2\gamma t_P^2}\right)}^2}}\right)=\frac{\mathrm{\hslash \Delta }t}{2\gamma t_P^2}\cdot \left(1\pm \sqrt{1-\frac{16{\mathrm{\hslash }}^2\gamma t_P^2}{{\left(\mathrm{\hslash \Delta }t\right)}^2}}\right)\end{array}$$E43

For sufficiently small $\gamma$.

$$\begin{array}{l}\mathrm{\Delta }E\approx \frac{\mathrm{\hslash \Delta }t}{2\gamma t_P^2}\cdot \left(1\pm \left(1-\frac{8{\mathrm{\hslash }}^2\gamma t_P^2}{{\left(\mathrm{\hslash \Delta }t\right)}^2}\right)\right)\\ \Rightarrow \mathrm{\Delta }E\approx \mathit{\text{either}}\frac{\mathrm{\hslash \Delta }t}{2\gamma t_P^2}\cdot \frac{8{\mathrm{\hslash }}^2\gamma t_P^2}{{\left(\mathrm{\hslash \Delta }t\right)}^2},\mathit{\text{or}}\frac{\mathrm{\hslash \Delta }t}{2\gamma t_P^2}\cdot \left(2-\frac{8{\mathrm{\hslash }}^2\gamma t_P^2}{{\left(\mathrm{\hslash \Delta }t\right)}^2}\right)\end{array}$$E44

would lead to a minimal relationship between change in E and change in time as

$$\mathrm{\Delta }E\approx \frac{\mathrm{\hslash \Delta }t}{2\gamma t_P^2}\cdot \frac{8{\mathrm{\hslash }}^2\gamma t_P^2}{{\left(\mathrm{\hslash \Delta }t\right)}^2}\equiv \frac{4\mathrm{\hslash }}{\mathrm{\Delta }t}$$E45

Or

$$\mathrm{\Delta }E\mathrm{\Delta }t\approx 4\mathrm{\hslash }$$E46

In doing so, we will refer to Eq. (46) as the pre inflaton state of energy being delivered due to a non conserved interjection of energy into the new universe.

Doing so would be a way to have the frequency so alluded to given in [27, 28, 29] and this is what we conjecture as to the evolution of the change in energy if we have the inflaton included which would be in Planckian space–time

$$⟨{\left(\delta g_{\mathit{uv}}\right)}^2{\left({\widehat{T}}_{\mathit{uv}}\right)}^2⟩\ge \frac{{\mathrm{\hslash }}^2}{V_{\mathit{\text{Volume}}}^2}\underset{\mathit{uv}\to \mathit{tt}}{\to }⟨{\left(\delta g_{\mathit{tt}}\right)}^2{\left({\widehat{T}}_{\mathit{tt}}\right)}^2⟩\ge \frac{{\mathrm{\hslash }}^2}{V_{\mathit{\text{Volume}}}^2}\&\delta g_{\mathit{rr}}\sim \delta g_{\mathit{\theta \theta }}\sim \delta g_{\mathit{\varphi \varphi }}\sim 0^{+}$$E47

$$\begin{array}{l}\mathit{\delta t}\mathrm{\Delta }E\ge \frac{\mathrm{\hslash }}{\delta g_{\mathit{tt}}}\ne \frac{\mathrm{\hslash }}{2}\\ \mathit{\text{Unless}} \delta g_{\mathit{tt}}\sim O\left(1\right)\end{array}$$E48

$$\delta g_{\mathit{tt}}\sim a^2\left(t\right)-\varphi <<1$$E49

This version of uncertainty would be for inclusion of energy once we are in the specific Planckian regime of space time and may be what is needed for sufficient energy imput from the fifth dimension, leading to a fifth force argument as given by [30] which may be from the work given by Wesson in [27]. This fifth force, in addition to fitting in the HUP in the Pre Planckian to Planckian physics regime would be encouraging us for an unbelievably high initial change in energy, as stated in Eq. (48), whereas once we are in the Planckian regime of the present universe we would be using Eq. (46) so as to specify a very high initial initial frequency, and this would be in tandeum with [27] being directly employed

$$\begin{array}{l}\int p_{\alpha }{\mathit{dx}}^{\alpha }=\pm \frac{h}{c}\cdot \frac{L}{\mathrm{\ell }}=\pm \frac{h}{c}\cdot \sqrt{\frac{3}{\mathrm{\Lambda }}}\frac{m_{\mathit{\text{particle}}}}{h}\\ =\frac{r}{m_{\mathit{pl}}}\cdot \left(1-log\left[\frac{r}{\varpi \cdot c}\right]\right)\\ \iff r\approx {\epsilon }^{+}\end{array}$$E50

This is linkable to z, as to red shift showing up in [27, 28, 29] and it shows how to obtain a very small radial value namely in a tiny scale factor due to an enormous z red shift as given in [29].

Quote.

$\mathit{\text{If}}z\approx {10}^{55}\mathit{\text{then}}a\approx {10}^{-55}$ so we do not have a space – time singularity.

End of quote.

The Eq. (46) would be for specifying, via the frequency being the inverse of change in time, after the Planckian regime of space time, whereas Eq. (47) would be the Pre Planckian to Planckian uncertainty principle used when Eq. (50) would be considered.

The application of the fifth force to the geometry of space–time in the beginning of expansion of the universe would employ Eq. (50) and Eq. (48) in the Planckian regime, whereas Eq. (47) would be just before Planckian space time.

All these details need to be worked out and given more foundation in future research.

14. Linkage to GW and their importance as to GW astronomy by making an analogy to black holes explicit for GW generation, and this has to be confirmed experimentally

We will for the sake of linkages to GW treat this problem as related to black holes, and gravitons and subsequent GW generation.

The Eq. (42) so mentioned, is an invariance procedure as far as space–time and its scaling may lead to black hole production’.

Assuming our BEC condensate argument leads to scaling as far as black hole production, we will make the following assumption, namely the following grouping leads to

$$\begin{array}{l}M\to M_{\mathit{\text{universe}}}\approx \sqrt{N_{\mathit{\text{graviton}}}}\cdot M_P\approx {10}^{61}\cdot M_P\\ M\to \mathit{\text{several}}\tilde{m}=\frac{8\mathit{\pi R}{\left(\mathit{\text{radius}}-\mathit{\text{of}}-\tilde{m}\right)}^3\tilde{\rho }}{3}\\ {\left.\frac{H^2}{\dot{\varphi }}\approx {10}^{-5}\right\}}_{\mathit{\text{initial}}-\mathit{\text{conditions}}}\underset{\mathit{\text{Evolution}}-\mathit{\text{to}}-\mathit{\text{near}}-\mathit{\text{present}}}{\to }{\left.\frac{H^2}{\dot{\varphi }}\approx {10}^{-5}\right\}}_{\mathit{\text{present}}-\mathit{\text{conditions}}}\end{array}$$E51

I estimate that this together leads to about 10^20 to 10^21 effective Planck mass s sized mini black holes in the beginning of the cosmos at the cosmos. Making use of page 46 of [31] we have that 1/1000 of a 3 + 1 dimensional mini black hole would, if not considered rotating contribute to graviton emission.

Using that rule, we could assume 10^`122 gravitons, as actually being generated from primordial conditions with say of this number, say at most about 10^21 Planck sized black holes being formed.

Then

$$\begin{array}{l}{E}_{\mathit{BEC}-\mathit{\text{Graviton}}}\approx \frac{k_B{T}_{\mathit{BH}}}{2}\approx \frac{k_B\times {10}^{-5}\times T_P}{2}\\ \Rightarrow {\omega }_{\mathit{BEC}-\mathit{\text{Graviton}}}\propto {10}^{-5}\times {10}^{43}\mathit{Hz}\approx {10}^{38}\mathit{Hz}\\ \Rightarrow {\omega }_{\mathit{BEC}-\mathit{\text{Graviton}}-\mathit{\text{to}}-\mathit{\text{CMBR}}}\approx {10}^{38}\times {10}^{-3}\mathit{Hz}\end{array}$$E52

We could see Primordial black holes as about $z\approx {10}^{25}$. Leading to present Gravitational wave signals from the primordial black holes today of about 1 Hertz, by massive red shifting.

Whereas we can consider what would be gained by looking at the contribution near the CMBR, i.e. z ∼ 1100 or so for the CMBR,

whereas this would mean roughly that we would be looking in the regime of the CMBR generated processes

$${\omega }_{\mathit{\text{signal}}-\mathit{\text{from}}-\mathit{\text{Planck}}-\mathit{\text{to}}-\mathit{\text{CMBR}}}\propto {\left(\frac{3}{2}\right)}^{1/\gamma }\cdot {10}^{\left(25/\gamma \right)}\times {10}^{-3}\mathit{Hz}$$E53

Also

$$\mathrm{\Delta }E\mathrm{\Delta }t\approx \mathrm{\hslash }\equiv \mathrm{\hslash }\omega \mathrm{\Delta }t\approx \mathrm{\hslash }\omega \cdot {\left(\frac{2}{3a_{\min }}\right)}^{1/\gamma }\Rightarrow \omega \approx {\mathrm{\hslash }}^{-1}\cdot {\left(\frac{2}{3a_{\min }}\right)}^{-1/\gamma }$$E54

We claim that if we take the energy as consistent with a change in value as given by Eq. (45) and Eq. (48) that this will lead to a frequency which may, if $a{}_{min}\approx {10}^{-25}-{10}^{-20}$ (range from 10^-25 to 10^-20) lead to initial primordial production of Frequencies as to emergence from a near singularity along the lines of an initial value of

$$\omega \approx {\mathrm{\hslash }}^{-1}\cdot {\left(\frac{3}{2}\right)}^{1/\gamma }\cdot {10}^{25/\gamma }\propto {\left(\frac{3}{2}\right)}^{1/\gamma }\cdot {10}^{25/\gamma }\mathit{Hz}$$E55

Whereas as from [32] we assume the following table as given in that publication for say a huge number of initial primordial black holes.

This would lead to about an energy release initially of the order of say

$$\begin{array}{l}\dot{E}=\mathit{GW}-\mathit{\text{change}}-\mathit{\text{in}}-\mathit{\text{energy}}=\frac{32{\left(M_{1}M{}_2\right)}^2\left(M_1+M{}_2\right)}{5\cdot R^2{M}_{\mathit{\text{Planck}}}^5}\\ \underset{M_1=M{}_2=M_{\mathit{\text{Planck}}}}{\to }\frac{64}{5\cdot R^2\left(\mathit{\text{Planck}}-\mathit{\text{length}}\right)}\\ \equiv \mathit{\text{Change}}-\mathit{\text{in}}-\mathit{\text{power}}-\mathit{\text{from}}-\mathit{\text{Rotating}}-\mathit{\text{binary}}-\mathit{\text{blackholes}}\end{array}$$E56

We of course would be wanting to compare this with the change in energy as given in Eq. (45) and Eq. (48).

Table 1 from [32] assuming Penrose recycling of the Universe as stated in that document.

End of Prior Universe time frame	Mass (black hole): super massive end of time BH; 1.98910^+41 to about 10^44 grams	Number (black holes); 10^6 to 10^9 of them usually from center of galaxies
Planck era Black hole formation; Assuming start of merging of micro black hole pairs	Mass (black hole); 10^-5 to 10^-4 grams	Number (black holes); 10^40 to about 10^45,
Post Planck era black holes	Mass (black hole); Up to 10^6 grams per black hole	Number (black holes); 10^20 to at most 10^25

Table 1.

Black hole production per cosmological era.

Now for the sake of primordial black holes.

The formula in page 16 of reference [32] that two black holes emit GW with a wave frequency 2 times the rotation frequency of the orbit of the two black holes to each other.

If we assume that we are still using this approximation above, from [33].

$$\begin{array}{l}R\left(\mathit{\text{separation}}\right)\simeq r_g^{\mathit{eff}}=\frac{\left(M_1+M{}_2\right)}{{\left(M_{\mathit{\text{Planck}}}\right)}^2}\\ \underset{M_1=M{}_2=M_{\mathit{\text{Planck}}}\underset{M_{\mathit{\text{Planck}}}=1}{\to }}{\to }1\equiv R\left(\mathit{\text{Planck}}-\mathit{\text{length}}\right)\end{array}$$E57

I.e. this means that the primordial black holes, presumably of Planck size would be separated about 1 Planck length from each other, that their recombination would be quick and that the frequency range would likely be of the magnitude of about 10^25 Hz in terms of GRAVITATIONAL waves which would then be massively red shifted downward to about `1 Hz in an Earth bound detector system.

I.e. a huge downward red shifting from 10^25 Hz to about a 1 Hz value in Earth orbit.

Acknowledgments

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