_{1}

Cosmological expansion or inflation is mathematically described by the theoretical notion of inverse gravity whose variations are parameterized by a factor that is a function of the distance to which cosmological expansion takes prominence over gravity. This assertion is referred to as the inverse gravity inflationary assertion. Thus, a correction to Newtonian gravitational force is introduced where a parameterized inverse gravity force term is incorporated into the classical Newtonian gravitational force equation where the inverse force term is negligible for distances less than the distance to which cosmological expansion takes prominence over gravity. Conversely, at distances greater than the distance to which cosmological expansion takes prominence over gravity. The inverse gravity term is shown to be dominant generating universal inflation. Gravitational potential energy is thence defined by the integral of the difference (or subtraction) between the conventional Newtonian gravitational force term and the inverse gravity term with respect to radius ( r) which allows the formulation, incorporation, and mathematical description to and of gravitational redshift, the Walker-Robertson scale factor, the Robinson-Walker metric, the Klein-Gordon lagrangian, and dark energy and its relationship to the energy of the big bang in terms of the Inverse gravity inflationary assertion. Moreover, the dynamic pressure of the expansion of a cosmological fluid in a homogeneous isotropic universe is mathematically described in terms of the inverse gravity inflationary assertion using the stress-energy tensor for a perfect fluid. Lastly, Einstein’s field equations for the description of an isotropic and homogeneous universe are derived incorporating the mathematics of the inverse gravity inflationary assertion to fully show that the theoretical concept is potentially interwoven into the cosmological structure of the universe.

The theoretical notion that cosmological expansion or universal inflation occurs due to inverse variations in gravitational force whose rate of change is regulated by a limiting factor or parameter is introduced. Thus, it is asserted that cosmological expansion or inflation is an inherent property of nature mathematically described by the difference between conventional Newtonian gravitational force and its inverse term which is multiplied by an inflationary parameter which regulates its rate of change. The inflationary parameter multiplied by the inverse term of Newtonian gravitational force is determined by (and is a function of) an astronomical distance to which cosmological expansion over takes gravitational force on a cosmic scale. The establishment of the core concept of the inverse gravity inflationary assertion aforementioned is the foundation to describing the universe in terms of the new assertion. Thus, the aim of this paper is the incorporation of the inverse gravity inflationary assertion (IGIA) into proven and established mathematics describing cosmological inflation.

A more detailed introduction is that this paper formulates the correction to the Newtonian gravitational force equation incorporating an inverse gravity term and its conditions. This permits the derivation of gravitational potential energy in terms of the IGIA. Resultantly, the relationship between gravitational potential energy, dark energy, gravitational redshift, the Klein-Gordon Lagrangian, the energy of the big bang, the Robertson-Walker scale factor, and the Friedman-Walker-Robertson metric in terms of the inverse gravity inflationary assertion (IGIA) is formulated and defined. The IGIA is then applied to the stress-energy tensor for describing the dynamic pressure of an expanding cosmological fluid in a homogeneous and isotropic universe. Lastly, the IGIA is applied to Einstein’s field equations for the description of a spherically homogeneous isotropic universe which establishes the inverse gravity inflationary assertion. This will elucidate how the IGIA is interwoven into the cosmological structure of the universe.

To begin the heuristic derivation, mass values

where

The constant

The total gravitational force

Therefore,

The direction (+ or −) of the value of total force

Condition (1.05) describes cosmological expansion

The value of the cosmological parameter of distance

This reduces to:

Solving Equation (1.08) for the cosmological parameter of distance

Conclusively Equation (1.09) above, gives the value of distance

Function

The gravitational interaction of symmetric portions of cosmological mass

Thus, the continuous sums (or integration) of gravitational mass interaction

As the continuous sums of the integrals in Equation (1.13) progress in concert with angles _{0} shown below.

The aim and scope of this paper is to introduce the notion and mathematics of the Inverse gravity inflationary assertion, thus we leave the calculation and value of Equation (1.14) as an exercise to the scientific community based on data obtained (The value of universal mass

This section applies the mathematical concept of the inverse gravity assertion to gravitational potential energy, gravitational Redshift, the Robertson-Walker scale factor, Friedman-Walker-Robertson metric, the Klein-Gordon lagrangian, dark energy, and the energy of the big bang. Gravitational potential energy

Thus after evaluating the integral above, one obtains a value of potential energy

As a photon propagates across the expanding cosmological expanse, its energy is affected by the gravitational potential energy

where

Resultantly, wavelength

Energy

This reduces to [

Red shift value z can then be expressed in terms of the IGIA such that:

Thus, the photonic energy value

Substituting the value of redshift z of Equation (2.07) into (2.08) above gives [

This reduces to:

The value of the scale factor at the time of the emitted photon

where Equation (2.11) is of the form

The scale factor

Substituting the value of scale factor

This establishes the IGIA relationship to the Friedman-Walker-Robertson metric, where constant k in Equation (2.14) is equal to +1 (

The inverse gravity inflationary assertion (IGIA) can be defined in terms of field theory via its relationship to the Klein-Gordon lagrangian. In formulating the expressions describing this relationship, IGIA potential energy

This can be expressed such that:

where

Expressing Equation (2.16) in terms of field

where the relativistic energy

Equation (2.19) can be alternatively expressed in terms of the IGIA such that:

where r is a function of the Minkowski is coordinates (

Equation (2.19) can then be expressed in terms of radius

Therefore, we introduce momentum

(Note: Recall thatthe speed of light c is set equal to unity (

Equation (2.25) below is the Klein-Gordon equation expressed such that [

Thus, as presented by Wald [

Expressing the Klein-Gordon lagrangian (Equation (2.26)) above in terms of the IGIA, the value of Equation (2.23) is substituted into Equation (2.26) (where

This implies that the product of differential terms

Solutions to Equations (2.27) and (2.28) pertain to mathematical methods of solving differential equations. Conclusively, Equation (2.28) is the IGIA correlation to various areas of field theory especially quantum energy fields describing vacuum energy (and the stress-energy tensor described in the next section). Lastly, dark energy

Thus, dark energy is interpreted according to the IGIA as the inverse term of potential energy

This section mathematically defines the dynamic pressure of a cosmological fluid in a homogeneous isotropic universe in terms of the IGIA. The stress-energy tensor for a perfect fluid used to describe the expansion of the cosmological fluid is given such that [

where

The 4 space tangent vectors

Thus showing the appropriate use of the Christoffle symbol

where the time coordinate has a value ct and the speed of light c is set to unity (

Therefore the product of tangent vectors

It must be noted that tangent vectors

This implies that [

In the task of defining the expansion of the cosmological fluid in terms of the IGIT, consider the unit vector

Multiplying unit vector

This can be expressed such that:

where

(Note: that the pressure component

This implies that [

which also implies the equivalence of:

Therefore this can be expressed such that:

Now substituting the value of force term

Thence, substituting the value of

which is equivalent to the stress-energy tensor such that:

Equation (3.18) can be expressed in matrix form such that:

The correlation of the stress-energy tensor

In describing an expanding homogeneous isotropic universe in terms of the IGIA, it is of great importance that the IGIA is fully incorporated to Einstein’s field equations as a whole. Thus, we began the heuristic derivation according to Wald [

The expressions of the stress-energy tensor in Equations (4.0) and (4.10 (

where

The scale factor at time t denoted

Recall that for our purposes

Thus solving Equation (4.6) for the value of scale factor

The time derivative of scale factor

The coordinate

Distance r is measured from the center of expansion (or the center of the universe), thus the initial values of t_{i}, x_{i}, y_{i}, and z_{i} equal zero. Therefore substituting zero for all initial values t_{i}, x_{i}, y_{i}, and z_{i} (

Substituting the value of Equation (4.10) into Equation (2.11) (or Equation (4.5)) gives the IGIA scale factor as a function of the Minkowski coordinates (

Thus pertaining to the time coordinate (

In continuing the IGIA’s incorporation to Einstein’s field equation, the scale factors a (keep in mind that

Thus we acknowledge that the Christoffel symbols

The components of the Ricci tensor are calculated according to the equation of [

This can alternatively be expressed such that [

The Ricci tensor is then related to the scale factor a (or

As stated by Wald, the value of Ricci tensor

Therefore the value of the scalar curvature R is given such that [

Substituting the value of Equation (4.18) and (4.19) into Equation (4.21) give a value such that [

Conclusively, the values of Einstein tensor values

As stated by Wald, using Equation (4.23), Equation (4.24) can be rewritten such that [

Due to the fact that the description of the IGIA is defined in reference to a homogeneous and isotropic universe, the general evolutions for isotropic and homogeneous universe are given such that [

where scale factors a and their corresponding time derivatives (

Equation (4.1) (

Resultantly, this can be expressed such that:

Spherically symmetric area

Substituting the value of pressure P presented above into Equation (4.27) gives a value such that:

Equation (4.31) gives an additional incorporation of the Mathematics of the IGIA showing that the theoretical concept is well ingrained to the cosmological structure of the universe. The incorporation of the IGIA mathematics to Einstein’s field equations gives a complete description to validate the concept and convey a new theoretical possibility to the physics community.

I would like to thank Juanita Walker and Mr. Eugene Thompson for their encouragement and support to this endeavor.

Walker, E.A. (2016) The Inverse Gravity Inflationary Theory of Cosmology. Journal of Modern Physics, 7, 1762-1776. http://dx.doi.org/10.4236/jmp.2016.713158