The Observable Universe in a Simplified Cosmic Dynamic Model

This paper introduces a cosmic expansion model with constant speed of cosmic spatial expansion via derivation and simulations, where the speed of cosmic spatial expansion equals the speed of light c. Simulation results show that the earliest observable universe time is t = 5.084 Gyrs where the current universe time T = 13.82 Gyrs, and the furthest observable distance at the earliest observable universe time t is S = 0.632R, where R is the cosmic radius at current universe time T. The above constant cosmic expansion model does not consider the inflation period in the early universe according to the Big Bang model, nor does it considered the cosmic acceleration in recent universe time. However, this simplified cosmic expansion model could be a benchmark that will be helpful to understand the cosmic expansion and the observable universe. Based on the derivation and simulation of the constant cosmic expansion model, the threshold of observable universe for the accelerated cosmic expansion model can also be calculated similarly, as far as the speed of cosmic spatial expansion at any universe time t can be provided.


Introduction and Related Works
The measurement of the distance of stars and galaxies in the universe has always been one of the main research topics of cosmology. About 100 years ago, astronomers discovered that the light of distant stars has a redshift, so it is speculated that these stars are moving away from the earth. After that, scientists used Einstein's field equations to introduce the universe's space-time expansion model [1].
Later, astronomers discovered that stars and galaxies farther away from the redshift is calculated based on Hubble's law [2].
In physical cosmology, the cosmic expansion model is established via Einstein's field equations, to constrain the metric of an isotropic uniform universe using Robertson-Walker metrics [1], and perform the derivation of universe evolution dynamics [3]. Based on the above universe expansion model, the Big Bang theory describes how the universe expanded from an initial state of extremely high density and high temperature [4]. The Big Bang theory is compatible with Hubble's law.
Hubble's law [2] is the observation in physical cosmology that stars are moving away from the Earth at speeds proportional to their distance.
The photon epoch was the period in the evolution of the early universe in which photons dominated the universe. The photon epoch started at about 10 seconds after the Big Bang and ended at 370,000 years after the Big Bang when the temperature of the universe fell so that photons no longer interacted frequently with matter including Atomic nuclei and electrons [5].
This paper introduces a simplified cosmic expansion model, assuming the speed of cosmic spatial expansion equals the speed of light c. Based on the constant cosmic expansion model, the threshold of observable universe is derived.
In the constant cosmic expansion model, it assumes that the speed of cosmic expansion always equals the speed of light c. While in the Big Band model [6], the speed of cosmic spatial expansion is much faster than the speed of light c during the inflation period in the early universe. The constant cosmic expansion model does not consider the inflation period in the early universe according to the Big Bang model [6]. Therefore in the constant cosmic expansion model, the speed of cosmic expansion is much slower than that in the Big Bang model during the inflation period, which results in the current cosmic radius in the constant cosmic expansion model is less than the actual cosmic radius. However, this simplified cosmic expansion model could be a benchmark that will be helpful to understand the cosmic expansion and the observable universe. The main contribution of this paper is that the simplified constant cosmic expansion model introduces cosmic expansion dynamics, which shows that after a light source such as a galaxy omitted a beam of light to the Earth when the beam of light was travelling in the universe before arriving the Earth, the cosmic space was expanding at the same time. This paper provides a method to calculate the time duration since a light source such as a galaxy omitted a beam of light until the beam of light arrives the Earth. Then the distance between the light source and the Earth can be obtained consequently. Based on this model, the earliest observable universe time can be derived. Then the furthest distance for the observable universe can be derived consequently. If a light source emitted a beam of light to the Earth before the earliest observable universe time, then the beam of light has arrived on the Earth before current universe time. Only when a light source emitted a beam of light after the earliest observable universe time, it can arrives on the Earth and be observed right now at current universe time. Journal of Applied Mathematics and Physics   Gly: Giga light years.

Constant Cosmic Expansion Model
Assume the speed of cosmic expansion Expan V c = , where c is the constant speed of light. The universe age is 13.82 Gyrs T = according to [7]. Therefore the current cosmic radius 13.82 Glys R cT = = , assuming ( ) to the Big bang model [6]. The universe radius R(t) during the cosmic expansion is a straight line as shown in Figure 1. The constant cosmic expansion model does not consider the inflation period in the early universe according to the Big Bang model [6]. But this simplified cosmic expansion model could be a benchmark that will be helpful to understand the cosmic expansion and the observable universe.

Deriving the Threshold of Observable Universe
In the constant cosmic expansion model, for a given universe time t T < , then ( )

Accelerated Cosmic Expansion Model
In the accelerated cosmic expansion model, it is said that the cosmic expansion is decelerated after the Big Bang until universe time 9 Gyrs t = , and the cosmic expansion is accelerated when universe time 9 Gyrs t > until now with current universe time 13.82 Gyrs T = [8].
Therefore, if the accurate speed of cosmic spatial expansion can be provided in the future for any universe time t T < , then the furthest observable distance from where a light source d can be observed by an observer on the Earth (O) at current universe time T can also be derived similar to the method described above.

Future Works
The future work is to modify the constant cosmic expansion model, considering the inflation period in the early universe, and the cosmic deceleration period when universe time is less than 9 Gyrs.

Conclusions
This paper introduces a cosmic expansion model with constant speed of cosmic spatial expansion. Simulation results show that the earliest observable universe time is 5.084 Gyrs t = . Therefore, when universe time 5.084 Gyrs t < , the universe cannot be observed directly on the Earth at current universe time. The universe radius ( ) R t at the earliest observable universe time t is 0.368R where the current universe radius is R. And the furthest observable distance at the earliest observable universe time t is 0.632 S R = .
Based on the derivation and simulation for the constant cosmic expansion model, the threshold of observable universe for the accelerated cosmic expansion model [8] can also be calculated similarly, as far as the speed of cosmic spatial expansion at any universe time t can be provided.