Constraints, in Pre-Planckian Space-Time via Padmabhan’s Λ_{initial}·*H*^{-2}_{initial}≈*o*(1) Approximation Leading to Initial Inflaton Constraints and Its Relation to Early Universe Graviton Production

Author(s)
Andrew Walcott Beckwith

Affiliation(s)

Physics Department, College of Physics, Chongqing University Huxi Campus, Chongqing, China.

Physics Department, College of Physics, Chongqing University Huxi Campus, Chongqing, China.

ABSTRACT

We are looking at what if the initial cosmological constant is due to if we furthermore use as the variation of the time component of the metric tensor in Pre-Planckian Space-time up to the Planckian space-time initial values. This assumes as an initial inflaton value, as well as employing Non-Linear Electrodynamics to the scale factor in , and the upshot is an expression for as an initial inflaton value/squared which supports Corda’s assumptions in the Gravity’s breath Electronic Journal of theoretical physics article. We close with an idea to be worked in further detail as to density matrices and how it may relate to gravitons traversing from a Pre-Planckian to Planckian space-time regime. We will write up an idea in far greater detail in a future publication.

We are looking at what if the initial cosmological constant is due to if we furthermore use as the variation of the time component of the metric tensor in Pre-Planckian Space-time up to the Planckian space-time initial values. This assumes as an initial inflaton value, as well as employing Non-Linear Electrodynamics to the scale factor in , and the upshot is an expression for as an initial inflaton value/squared which supports Corda’s assumptions in the Gravity’s breath Electronic Journal of theoretical physics article. We close with an idea to be worked in further detail as to density matrices and how it may relate to gravitons traversing from a Pre-Planckian to Planckian space-time regime. We will write up an idea in far greater detail in a future publication.

1. Basic Idea, the Padmabhan Approximation of

To do this, we look at [1] which is of the form

(1)

Our objective is to use Equation (1) with [2]

(2)

and [2]

(3)

and [2] [3]

(4)

and [3] [4] [5]

(5)

and [4]

(6)

The next step will be to utilize [6]

(7)

where [6]

(8)

as well as use the Non-Linear Electrodynamic minimum value of the scale factor [7] which is in the spirit of [8] and which is avoiding using [9] .

2. Using the Section 1 Material to Isolate a Minimum Value of the Inflaton, beyond Equation (4)

From [4] we make the following approximation, i.e. simply put a relationship of the Lagrangian multiplier giving us the following: if

(9)

If the following is true, i.e. in a Pre-Plankian to Planckian regime of space- time

(10)

Here, −g is a constant, as assumed in [4] which means in the Pre-Planckian to Plackian regime we would have Equation (5) as a constant, so then we are looking at, if, an energy density as given by Zeldovich, as talked about with [10] setting a minimum energy density given by

(11)

And with the following substitution of

(12)

Then to first order we would be looking at Equation (11) re written as leading to

(13)

And if Equation (1) holds, we would have by [1]

(14)

So

(15)

And if is the square of Planck’s length, after some algebra, and assuming

(16)

We will examine the consequences of these assumptions as to what this says about the NLED approximation for the initial scale factor, as given in [7] .

3. Conclusions: Examining the Contribution of the Inflaton

In [11] Corda gives a very lucid introduction as to the physics of the inflaton. We urge the readers to look at it as it refers to Equation (17), second line. In particular, it gives the template for the possible range of values for in Equation (16).

The take away is that we are assuming a relatively large initial entropy (based upon a count of massive gravitons) being recycled from one universe to the next, which would influence the behavior of the first line of Equation (16) and tie into the behavior of the 2^{nd} line of the inflaton Equation (16) given above. The exact particulars of are being investigated.

Keep in mind the importance of the result from reference [12] below which forms the core of Equation (17) below

(17)

We have to adhere to this e fold business, and this will influence our choices as to how to model the inflaton.

Furthermore the constraints given in [13] [14] and [15] as to the influence of LIGO on our gravity models have to be looked into and not contravened.

This is a way of also showing if general relativity is the final theory of gravitation. i.e., if massive gravity is confirmed, as given in [16] , then GR is perhaps to be replaced by a scalar-tensor theory, as has been shown by Corda.

Finally is a re-do of what was brought up in [17] by Tang. In a density equation of stated with a relaxation procedure, between different physical states, Tank writes if m, and n are different quantum level states, then, if is the “Atomic coherence time”

(18)

We will here, in our work assign the same sort of physical state which would in place have if in which then the solution to this problem would be given by Equation (11). The idea would be as follows. If model the density of states as having the flavor of gravitons preserving the essential quantum “state”, and not changing if we go from the Pre-Planckian to Planckian state.

There would be then the matter of identifying, and the time, if. In our review we would put likely as the Pre-Planck- ian to Planckian transition time.

Note that in the m = n time if our “density of states” was referring to gravitons, keeping the same states as if m = n is picked, that the second part of Equation (17) is in referral to quantum states of a graviton having a non-planar character which would not have a planar wave character.

In the case of we are then referring to changes in the states of presumed gravitons as information carriers, and the density equation, has in Eq. (17) as a wave with explicit damped by time evolution wave component times a planar wave component.

We presume here that the frequency term, would be in the high gigahertz range.

In any case, the details of this sketchy idea should be from the Pre-Planckian to Planckian regime of space-time given far more structure in a future document.

We should note that the removal of initial singularities is due to Non-Linear Electrodynamics, as seen in [7] , by Camara et al., which is also in tandem with [18] [19] [20] which give also frequency specifications, which could also affect, i.e. a tie in, with Gravitons, and Nonlinear Electrodynamics, which should be developed further.

Acknowledgements

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

Cite this paper

Beckwith, A. (2017) Constraints, in Pre-Planckian Space-Time via Padmabhan’s Λ_{initial}·*H*^{-2}_{initial}≈*o*(1) Approximation Leading to Initial Inflaton Constraints and Its Relation to Early Universe Graviton Production. *Journal of High Energy Physics, Gravitation and Cosmology*, **3**, 322-327. doi: 10.4236/jhepgc.2017.32027.

Beckwith, A. (2017) Constraints, in Pre-Planckian Space-Time via Padmabhan’s Λ

References

[1] https://ned.ipac.caltech.edu/level5/Sept02/Padmanabhan/Pad7.html

[2] Padmanabhan, T. (2005) Understanding Our Universe: Current Status and Open Issues. In: Ashtekar, A., Ed., 100 Years of Relativity, Space-Time, Structure: Einstein and Beyond, World Scientific, Singapore, 175-204. http://arxiv.org/abs/gr-qc/0503107

[3] Beckwith, A. (2016) La Grangian Multiplier Equation in Pre Planckian Space-Time Early Universe and the Cosmological Constant. http://vixra.org/abs/1612.0333

[4] Beckwith, A. (2016) Gedanken Experiment for Refining the Unruh Metric Tensor Uncertainty Principle via Schwarzschild Geometry and Planckian Space-Time with Initial Nonzero Entropy and Applying the Riemannian-Penrose Inequality and Initial Kinetic Energy for a Lower Bound to Graviton Mass (Massive Gravity). Journal of High Energy Physics, Gravitation and Cosmology, 2, 106-124. https://doi.org/10.4236/jhepgc.2016.21012

[5] Giovannini, M. (2008) A Primer on the Physics of the Cosmic Microwave Background. World Press Scientific, Hackensack. https://doi.org/10.1142/6730

[6] Padmanabhan, T. (2006) An Invitation to Astrophysics. World Scientific Series in Astronomy and Astrophysics, Vol. 8, World Press Scientific, PTE. Ltd., Singapore, Republic of Singapore.

[7] Camara, C.S., de Garcia Maia, M.R., Carvalho, J.C. and Lima, J.A.S. (2004) Nonsingular FRW Cosmology and Non Linear Dynamics. Physical Review D, 69, Article ID: 123504.

[8] Rovelli, C. and Vidotto, F. (2015) Covariant Loop Quantum Gravity. Cambridge University Press, Cambridge.

[9] Penrose, R. (1965) Gravitational Collapse and Space-Time Singularities. Physical Review Letters, 14, 57. https://doi.org/10.1103/PhysRevLett.14.57

[10] https://ned.ipac.caltech.edu/level5/Sept02/Padmanabhan/Pad1_2.html

[11] Corda, C. (2011) Primordial Gravity’s Breath. Electronic Journal of Theoretical Physics, 9, 1-10. http://www.ejtp.com/articles/ejtpv9i26.pdf

https://arxiv.org/abs/1110.1772

[12] Freese, K. (1992) Natural Inflaton. In: Nath, P. and Recucroft, S., Eds., Particles, Strings, and Cosmology, Northeastern University, World Scientific Publishing Company, Singapore, 408-428.

[13] Abbott, B.P., et al. (2016) LIGO Scientific Collaboration and Virgo Collaboration: Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters, 116, Article ID: 061102.

[14] Abbott, B.P., et al. (2016) LIGO Scientific Collaboration and Virgo Collaboration: GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence. Physical Review Letters, 116, Article ID: 241103.

[15] The LIGO Scientific Collaboration and the Virgo Collaboration (2016) Tests of General Relativity with GW150914. Physical Review Letters, 116, Article ID: 221101.

https://arxiv.org/pdf/1602.03841.pdf

[16] Corda, C. (2009) Interferometric Detection of Gravitational Waves: The Definitive Test for General Relativity. International Journal of Modern Physics D, 18, 2275-2282.

https://arxiv.org/abs/0905.2502

https://doi.org/10.1142/S0218271809015904

[17] Tang, C.L. (2005) Fundamentals of Quantum Mechanics. Cambridge University Press, New York.

https://doi.org/10.1017/CBO9781139165273

[18] Corda, C. and Mosquera, C.H. (2011) Inflaton from R2 Gravity: A New Approach Using Nonlinear Electrodynamics. Astroparticle Physics, 34, 587-590.

[19] Corda, C. and Mosquera, C.H. (2010) Removing Black-Hole Singularities with Nonlinear Electrodynamics. Modern Physics Letters A, 25, 2423-2429.

https://arxiv.org/abs/0905.3298

https://doi.org/10.1142/S0217732310033633

[20] De Lorenci, V.A., Klippert, R., Novello, M. and Salim, J.M. (2002) Nonlinear Electrodynamics and FRW Cosmology. Physical Review D, 65, Article ID: 063501.

[1] https://ned.ipac.caltech.edu/level5/Sept02/Padmanabhan/Pad7.html

[2] Padmanabhan, T. (2005) Understanding Our Universe: Current Status and Open Issues. In: Ashtekar, A., Ed., 100 Years of Relativity, Space-Time, Structure: Einstein and Beyond, World Scientific, Singapore, 175-204. http://arxiv.org/abs/gr-qc/0503107

[3] Beckwith, A. (2016) La Grangian Multiplier Equation in Pre Planckian Space-Time Early Universe and the Cosmological Constant. http://vixra.org/abs/1612.0333

[4] Beckwith, A. (2016) Gedanken Experiment for Refining the Unruh Metric Tensor Uncertainty Principle via Schwarzschild Geometry and Planckian Space-Time with Initial Nonzero Entropy and Applying the Riemannian-Penrose Inequality and Initial Kinetic Energy for a Lower Bound to Graviton Mass (Massive Gravity). Journal of High Energy Physics, Gravitation and Cosmology, 2, 106-124. https://doi.org/10.4236/jhepgc.2016.21012

[5] Giovannini, M. (2008) A Primer on the Physics of the Cosmic Microwave Background. World Press Scientific, Hackensack. https://doi.org/10.1142/6730

[6] Padmanabhan, T. (2006) An Invitation to Astrophysics. World Scientific Series in Astronomy and Astrophysics, Vol. 8, World Press Scientific, PTE. Ltd., Singapore, Republic of Singapore.

[7] Camara, C.S., de Garcia Maia, M.R., Carvalho, J.C. and Lima, J.A.S. (2004) Nonsingular FRW Cosmology and Non Linear Dynamics. Physical Review D, 69, Article ID: 123504.

[8] Rovelli, C. and Vidotto, F. (2015) Covariant Loop Quantum Gravity. Cambridge University Press, Cambridge.

[9] Penrose, R. (1965) Gravitational Collapse and Space-Time Singularities. Physical Review Letters, 14, 57. https://doi.org/10.1103/PhysRevLett.14.57

[10] https://ned.ipac.caltech.edu/level5/Sept02/Padmanabhan/Pad1_2.html

[11] Corda, C. (2011) Primordial Gravity’s Breath. Electronic Journal of Theoretical Physics, 9, 1-10. http://www.ejtp.com/articles/ejtpv9i26.pdf

https://arxiv.org/abs/1110.1772

[12] Freese, K. (1992) Natural Inflaton. In: Nath, P. and Recucroft, S., Eds., Particles, Strings, and Cosmology, Northeastern University, World Scientific Publishing Company, Singapore, 408-428.

[13] Abbott, B.P., et al. (2016) LIGO Scientific Collaboration and Virgo Collaboration: Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters, 116, Article ID: 061102.

[14] Abbott, B.P., et al. (2016) LIGO Scientific Collaboration and Virgo Collaboration: GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence. Physical Review Letters, 116, Article ID: 241103.

[15] The LIGO Scientific Collaboration and the Virgo Collaboration (2016) Tests of General Relativity with GW150914. Physical Review Letters, 116, Article ID: 221101.

https://arxiv.org/pdf/1602.03841.pdf

[16] Corda, C. (2009) Interferometric Detection of Gravitational Waves: The Definitive Test for General Relativity. International Journal of Modern Physics D, 18, 2275-2282.

https://arxiv.org/abs/0905.2502

https://doi.org/10.1142/S0218271809015904

[17] Tang, C.L. (2005) Fundamentals of Quantum Mechanics. Cambridge University Press, New York.

https://doi.org/10.1017/CBO9781139165273

[18] Corda, C. and Mosquera, C.H. (2011) Inflaton from R2 Gravity: A New Approach Using Nonlinear Electrodynamics. Astroparticle Physics, 34, 587-590.

[19] Corda, C. and Mosquera, C.H. (2010) Removing Black-Hole Singularities with Nonlinear Electrodynamics. Modern Physics Letters A, 25, 2423-2429.

https://arxiv.org/abs/0905.3298

https://doi.org/10.1142/S0217732310033633

[20] De Lorenci, V.A., Klippert, R., Novello, M. and Salim, J.M. (2002) Nonlinear Electrodynamics and FRW Cosmology. Physical Review D, 65, Article ID: 063501.