Particle Pair Production in Cosmological General Relativity—Redux

We model the universe on the interaction of two cosmic particles based on the Cosmological General Relativity (CGR) of Carmeli and obtain a theoretical value for the Hubble constant h at zero distance and no gravity. CGR is a 5-dimensional theory of time t, space x, y, z and velocity v. A minimum cosmic acceleration results from a linearized version of CGR, where c is the vacuum speed of light and τ is the Hubble-Carmeli time constant. The force due to the Carmeli acceleration a 0 counteracts the Newtonian gravitational force between the two particles. Each particle is unstable and disintegrates into baryons, leptons and radiation. By the uniform expansion of the black body radiation field, we obtain the expression , where A is a constant, T 0 is the temperature of the cosmic microwave background black body, Ω bphys is the physical baryon density parameter and . Using standard values for T 0 and Ω bphys we obtain a value which gives a value for the Hubble constant at zero distance and no gravity of − ⋅ ⋅ = . From the value for τ, we get the age of the universe of (13.15467 ± 0.00653) × 10 9 years.

which gives a value for the Hubble constant at zero distance and no gravity of ( )

Introduction
The Cosmological General Relativity (CGR) of Carmeli is a 5-dimensional theory of time t, space x, y, z and velocity v, which predicts the existence of a stant and where 1 h τ = is the Hubble constant at zero distance and no gravity.
In the application of Carmeli cosmology to galaxy rotation dynamics, Hartnett [3] found that the dividing line between which the galaxy rotation velocity was explained by Newtonian dynamics and where it could be explained by Carmeli dynamics, respectively, was where the galaxy acceleration transitioned from greater than a critical value of (2/3)a 0 to less than this acceleration.
We assume that the universe of mass and energy began with the formation of two massive cosmic particles. The pair of neutral cosmic particles materialized from the vacuum and the Carmeli acceleration formed the force which opposed the gravitational force between them. It is found that the particles were initially separated a distance 0 2 r cτ = and each had a mass , where G is Newton's gravitation constant. For the universe, the sum of the mass density ρ of the particles (positive) and the mass density ρ vac of the vacuum (negative) satis- We derive for the mass densities  where α is the fine structure constant and μ is the reduced electron mass in the hydrogen atom. This paper is a summary of the main points of my earlier paper [4]. We will cover the essential ideas developed in detail there and while making a more rigorous derivation of our expression for the Hubble-Carmeli time constant.

The Initial State of the Universe
We take the natural position that the universe can be described by an equation of the form 0 U = , where U represents a fundamental quantity such as the energy, mass density or force. Assuming that only gravitational and expansion reactions need be considered, ignoring the nuclear, electric and magnetic effects, we can state that the mass densities and forces due to gravity and expansion upon two cosmic particles each of the same mass are given by 0, where ρ is the matter mass density of the universe, ρ vac is the vacuum mass density, m is the mass of each of the cosmic particles, F(r) is the sum of forces on each particle, G is Newton's gravitation constant, r 0 is the separation of the two particles and 0 a c τ = is the Carmeli acceleration. Journal of High Energy Physics, Gravitation and Cosmology Initially, the two particles are at rest relative to each other. It is assumed that each cosmic particle is the anti-particle of the other. For example, representing ordinary matter by u and anti-matter by ū , if one particle is composed of ( ) x y u + amount of ordinary matter and xū amount of antimatter such that A further assumption we make is that both cosmic particles are enclosed in a finite spherical volume V having a radius equal to the Schwarzschild radius R S of the particles given by Multiplying the density relations in (1) by the volume the equation for the total mass ( ) where we assume that the matter density 2m V ρ = . Multiply the force (2) by the differential distance and integrate to obtain the energy of particle #2 at distance r 0 from particle #1, which is given by where it is assumed that the integration constant is the particle rest energy. By symmetry, we know that the result is the same if we reversed the roles of the particles. By setting ( ) (5) and along with (2) where solve these quadratic equations simultaneously to obtain the mass m and separation distance r 0 , which are found to be given by and Solving (3) for R S with the mass m from (6) we obtain Substituting for mass m from (6) into (4), we obtain for the vacuum mass and from (8) into (2), the matter mass density where ρ c is the critical mass density. Carmeli defined the effective mass density ρ eff in his theory in the form eff c ρ ρ ρ = − , where ρ is the mass density, but the critical mass density did not have a physical aspect. In this work we attribute a negative mass density to the vacuum and define the effective mass density by eff vac ρ ρ ρ = + . Therefore, by (1) we imply that at the beginning,

Radiation Energy of the Universe in the Form of a Black Body
The two cosmic particles, #1 and #2, are each of mass m and are assumed to be composed of subatomic particles of matter and anti-matter as we described above. It can be shown [4,Ref. 7] that each particle is enclosed in a volume V m which is half of the sphere volume V of Schwarzschild radius R S given by (8) which encloses both particles, so that the volume of the sub-universe containing a single particle, let us say #1, is given by Since each cosmic particle is composed of subatomic particles and anti-particles it is inherently unstable and will disintegrate, by particle anti-particle annihilations, and the energy of the photons is The total energy density of radiation (both polarizations) in a black body at temperature T [5] is given by where k B is Boltzmann's constant and ħ is the reduced Planck's constant. Multiplying the energy density (14) by the present sub-universe volume V m enclosing it we have the radiation total energy E γ,0 at the present time where the universe temperature is T 0 ( ) At some point, the radiation field interaction with the baryons and leptons The temperature of the cosmic microwave background (CMB) in the universe at the present time is, 0 2.73 K T ≈ and using this temperature the number of photons in the radiation field, the CMB is given by (17), using a value of Looking back to the time when the photon number first stabilized to a fixed value, which is assumed to be the same as the present number N γ (T 0 ), we make a first approximation of the average photon energy ε γ at that time by dividing the total sub-universe mass energy mc 2 by the number of photons, giving which we realize is 98% of the ionization energy 13.6 eV of the hydrogen atom.
Because big bang nucleosynthesis ends with the radiation field interacting with the ionized hydrogen atoms, it is reasonable to hypothesize that the average photon energy equals the ionization energy, where the last expression on the right hand side is the hydrogen atom ionization energy, where α is the fine structure constant and μ is the reduced electron mass of the hydrogen atom. Equation (20) can be expanded using the mass m from (6) and N γ (T 0 ) from (17) which simplifies to give ( ) As for the definition of the fractional parameter g representing the baryons, we take a simpler form here than in [4], expressed by where Ω bphys is the physical baryon density parameter and h c is the Hubble constant h divided by 100, which are defined in the cgs system by Journal of High Energy Physics, Gravitation and Cosmology

The Values of the Hubble-Carmeli Time Constant and the Hubble Constant
Since it is difficult to measure the Hubble constant through astronomical methods we can invert Equation (21) [12]. Since g from (22) is a constant, we see that η given by (35) is a constant as is expected, since the number of baryons and photons is assumed to be constant for this analysis.

Conclusions
A key result of this thesis is showing that the initial mass m of the universe can be partitioned into the baryon mass B m gm = and the photon mass which is a very reasonable assumption. Within this context, there is not a requirement for any other masses, such as particles of dark matter or dark energy.
There are several extended theories of gravity which elegantly address the issues of the shortcomings of General Relativity, such as dark matter, dark energy and