The Cosmic Radius of Observable Universe

This paper introduces three cosmic expansion models with constant, decelerating and accelerating speed of expansion respectively. Then characters of these cosmic expansion models are compared. Based on these cosmic expansion models, the thresholds of observable universe are calculated via simula-tions, where the earliest observable cosmic radius ( ) earliest R t is always 0.368R (R is cosmic radius at current universe time) for any cosmic expansion models.


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 earth have larger separation speed. Hence the stellar distance according to its 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. Then there is a first-order differential equation of spatial scale ( ) 2 . When 0 k > , the cosmic expansion decelerates; when 0 k < , the cosmic expansion accelerates; when 0 k = , the universe expands at a constant speed [3].
Based on 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].
In [6], it mentioned a so-called horizon problem. Since the known cosmic space-time is almost isotropy and homogenous with almost evenly distributed cosmic energy density, if the speed of light c is a constant, which is slower than the speed of cosmic expansion, then the energy-carrying light could not reach the universe's boundaries quickly enough, resulting in a significant difference of measurable cosmic energy density.
To solve the horizon problem, a hypothesis is made that the speed of light in early universe is much faster than the speed of light c in current universe. This hypothesis is proved by a certain fixed value of spectral index, which describes the initial density ripples in the Universe [6]. The latest spectral index figure reported by Planck satellite [7] verified the spectral index value calculated by [6] very well, with an error less than 1%.
Based on the above cosmic expansion dynamics using Robertson-Walker metrics [1], and the assumption that the speed of light equals the speed of cosmic expansion, this paper introduces three cosmic expansion models with constant, decelerated and accelerated speed of expansion respectively. And the characters of these models are compared. Then the threshold of observable cosmic space-time is derived. Note that, this paper is not to judge which cosmic expansion model is correct, but to find out the threshold of observable cosmic space-time.

Constant Cosmic Expansion Model
Assume the speed of cosmic expansion Expan V c = , where c is the constant speed of light as shown in Figure 1. The universe age is T = 13.82 Gyrs according to [7]. Therefore the current cosmic radius 13.82 Glys R cT = = , assuming The universe radius R(t) during the cosmic expansion is a straight line as shown in Figure 2.
The constant cosmic expansion model is apparently contradicted to the Big Bang model. Since it has no inflation period in early universe. But this simplified cosmic expansion model could be a benchmark which will help us to understand the cosmic expansion and the observable universe easier [8].

Decelerated Cosmic Expansion Model
The speed of light is the constant c.
( ) expan V t is cosmic expansion speed at universe time t, which is the increasing speed of cosmic radius ( ) The propagation speed of light equals the propagation speed of electromagnetic field c. On the other hand, light also has the characters of photon gas. Therefore, during cosmic spatial expansion, photon gas (light) also expands at the same way. Therefore it is reasonable to assume that, the speed of light considering cosmic expansion rate is ( ) , such that the speed of light equals the speed of cosmic expansion, in order to avoid the horizon problem.
The derivation of the cosmic decelerating model is based on Equation (1) [9].
The proof of Equation (1) is provided in Appendix of this paper. Note that, the purpose of this paper is not to prove the correctness of any cosmic expansion model.
According to Equation (1), then Equations (2) and (3) can be derived as follows, where G is Gravitational constant, M is the cosmic total mass which is assumed to be a constant too.
According to [7], take universe age 13.82 Gyrs T = , which is the current universe time. The initial universe radius ( ) 0 R t is set to a random small value.
R t can be any random value less than 10 10 m, which does not affect the simulation results, because the time period it takes for the cosmic radius to expand to 10 14 m is less than 1 second. The current universe radius R is temporarily set to 46.5 Glys according to [10]. Hence the initial conditions for simulation are as follows. can be calculated as follows.
According to Equation (3), after time period dt, there are: Simulation were performed according to the above procedure. Simulation results show that, when ( ) 46.5 Glys

Accelerated Cosmic Expansion Model
According to the Big Bang theory and the accelerating cosmic expansion model, the cosmic expansion decelerated after the Big Bang until universe time 9 Gyrs t = . Then the acceleration of cosmic expansion began after 9 Gyrs t = (4.82 Gyrs ago) [11].
Since the above decelerating cosmic expansion model can mimic the inflation of early universe very well, and the current cosmic radius R = 20.73 Glys in the decelerating model, which is much less than the current cosmic radius of 46.5 Glys in the accelerating model [10], therefore it is reasonable to assume that in the accelerating model, the cosmic expansion procedure is the same as that in the decelerating model for 9 Gyrs t ≤ as shown in Figure 1. Figure 1 shows that the speed of cosmic expansion decreased from 11.1c (when 0.01 Gyrs t = ) to 1.24c at 9 Gyrs t = in the decelerating model. While in the accelerating model, assume the speed of cosmic expansion increase linearly for 9 Gyrs t > depending on different acceleration rate ( ) acc r t , and the speed of cosmic expansion for accelerating model is exactly the same as the decelerating model for 9 Gyrs t ≤ .

Constant Cosmic Expansion Model
In the constant cosmic expansion model, the cosmic radius ( )

S t v t dt c R t dt
, assuming the cosmic space-time is isotropy and homogenous, and the separation speed is proportional to the distance according to Hubble's law [2]. Hence, Journal of High Energy Physics, Gravitation and Cosmology ( )

Decelerated Cosmic Expansion Model
In the decelerating cosmic expansion model, for any universe time t, universe radius ( ) R t can be calculated by the simulation as described in the last section, as shown in Figure 2, and the cosmic expansion speed  , where 20.73 Glys R = .

Accelerated Cosmic Expansion Model
In the accelerating 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 was accelerating when universe time 9 Gyrs t > until now with current universe time 13.82 Gyrs T = [11].
Since the decelerating cosmic expansion model can mimic the inflation of early universe very well, therefore it is assumed that in the accelerating model, the speed of cosmic expansion is exactly the same as that in the decelerating model for rs 9 Gy t ≤ as shown in Figure 1. simulation results are as follows, as shown in Figure 2. for any acceleration rate.

Summary
In summary, the earliest observable cosmic radius ( ) earliest R t is always 0.368R for any cosmic expansion models including the constant, decelerating and accelerating models. Therefore the furthest observable distance is always for any cosmic expansion models. The earliest observable times earliest t are 3.083 Gyrs and 5.084 Gyrs for the decelerating and constant models respectively. earliest t is more than 3.27 Gyrs for acceleration rate ( ) 0 acc r t > in the acceleration model as shown in Figure 2. Therefore the lowest threshold for the earliest observable time earliest t is 3.083 Gyrs for any cosmic expansion model.

Conclusion
This paper introduces three cosmic expansion models with constant, decelerating and accelerating speed of expansion respectively. Then characters of these cosmic expansion models are compared. Based on these cosmic expansion models, the thresholds of observable universe are calculated via simulations, where the earliest observable cosmic radius ( ) earliest R t is always 0.368R (R is cosmic radius at current universe time) for any cosmic expansion model, and the lowest threshold for the earliest observable time earliest t is 3.083 Gyrs for any cosmic expansion model.