Distribution of Mass and Energy in Closed Model of the Universe

The universe’s horizon distance and volume are constructed in the closed cosmic model. The universe horizon distance distribution increases constantly for t < tme and decreases for t > tme. However, the universe’s horizon volume shows a sudden reduction in the range t = 0.5 Gyr − tme due to the change of the universe space from flat to curved then closed in the interval 15.1261 Gyr ≤ t ≤ tme. On the other hand, this distribution exhibits an abrupt rise in the range t = tme − t∗ due to the change of the universe space from closed then curved to flat in the interval 39.3822 ≤ t ≤ 40.7521 Gyr. The mass of radiation, matter and dark energy within the horizon volume of the universe are also investigated. These distributions reveal similar noticeable changes as the universe’s horizon volume distribution for the same reasons. The mass of radiation dominates up to t = 53221.5 yr, then the mass of matter becomes larger. Afterwards, both distributions of radiation and matter decrease while the distribution of dark energy rises until t = 10.1007 Gyr, where the mass of dark energy prevails up to t = tme. Hence, the distribution of dark energy reduces until t = 40.2892 Gyr, where the mass of matter becomes prominent again. At t = 53.6246 Gyr the masses of both matter and radiation become appreciably high such that the intercluster space will vanish and clusters of galaxies interfere with each other. Furthermore, not only the intergalactic medium will disappear, but also galaxies will collide and merge with each other to form extremely dense and close cosmological bodies. These very dense bodies will undergo further successive collisions and mergers under the action of central gravity, where the interstellar medium will vanish and the universe would develop to big crunch at tbc = 53.6251 Gyr. It is interesting to note that the horizon distance of the universe in the closed model at t = tme is in very good agreement with the maximum horizon distances in the five general cosmic models.


Introduction
The distribution of density parameters of radiation, matter and dark energy in the closed cosmic model were investigated in a previous study [1], where we discovered the main epochs of the universe history in this model.It is worthy now to study the distributions of equivalent mass of radiation, mass of matter and equivalent mass of dark energy within the horizon volume of the universe to get deeper sight of the universe evolution in the closed model.
The reason for considering the equivalent mass of radiation in this study is the significant value of the radiation density parameter in the early universe and before the big crunch as we have seen in [1].
Therefore, it is vital to develop the distributions of the horizon distance and horizon volume of the universe in the closed model at various time ranges depending on the bases presented in [2].Description of methodology is illustrated in Section 2, while algorithm would be shown in Section 3. Results and discussion are displayed in Section 4. Conclusion is given in Section 5.

Methodology
It is obvious from [2] that the horizon distance and horizon volume of the universe in closed cosmic model at the present time are respectively . . .
where t is the cosmic time in Gyr.

. 8π
The horizon distance of the universe in the closed cosmic model at any given time is given by Consequently, the change in the horizon distance of the universe in the time interval between two instants of scale factors 1 2 , a a is written as .
The horizon volume of the universe in the closed model at any given time is expressed as , .
Equation (18) indicates that the horizon volume of the universe at t is a function of ( ) h d t and the curvature of space k at t. Since this curvature could be flat, open and closed from the big bang to big crunch as evident from Table 3 in [1].Thus, the low of ( ) h V t can be determined according to the value of k at t, as explained in the following cases: (1) Flat space ( 0 k = ) We have seen in [2] that the horizon volume of the universe at time t in this case is given by ( ) ( ) Therefore, it is obvious from Table 3 We recall the equation of proper distance of extragalactic object ( ) ( ) ( ).
where ( ) And the volume of space within ( ) where ( ), , , and o R t r r θ φ are defined as in [3] [4].For 1 k = + , Equations ( 21), ( 20) and ( 22) yield Hence Equation ( 22) gives ( ) ( ) Substituting by ( 25) in (24) we get ( ) ( ) ( ) Substituting by ( 27) in (26) we have Substituting by ( 23) in ( and Equation (29) becomes ( ) ( ) Thus, the horizon volume of the universe in the closed cosmic model at time t in this case is expressed as It is evident form Table 3 21), ( 20) and ( 22) give ( ) ( ) Substituting by (34) in (33) we have Substituting by (36) in (35) we get Substituting by (32) in (37) yields Therefore, the horizon volume of the universe in closed cosmic model at time t in this case is written as It is clear form Table 3 Substituting by ( 3)-( 5 The masses of matter, radiation and dark energy within the horizon volume of the universe in closed cosmic model at time t are respectively The time interval between two instants with scale factors 1 2 , a a during the universe expansion is given by Equation ( 16) in [4] [5] as ( ) However, during the universe contraction if 2 1 a a < then modulus of the right hand side of (50) should be taken.

Results and Discussion
The distribution of the universe horizon distance in the closed cosmic model until yr 0.5 G t = is shown in Figure 1(a).The distribution increases quite slowly up to 5.8780 Myr t = , then the distribution starts raising rapidly.However, the distribution of the universe horizon distance in the range 0.  , hence it starts raising appreciably fast.The distribution of both matter and total mass coincide on each other and lie over the radiation distribution.The two distributions increase gradually up to 53.5274 Gyr t = , then they raise up.However, the dark energy distribution decreases so slowly until 53.5742Gyr t = , afterwards it reduces substantially.Estimations of ( ) ( ) ( .Therefore, not only the intergalactic spaces will vanish at n t t = , but also galaxies will collide and merge with each other to form extremely dense and close cosmological bodies.These very dense bodies will undergo further successive collisions and mergers under the action of central gravity, where the interstellar medium will vanish and the universe would develop to big crunch at 53.6251 Gyr bc t = .It is also interesting to note from

Conclusions
In this paper we have investigated the distributions of the universe horizon distance and the universe horizon volume in the closed cosmic model.It is found that the universe horizon distance distribution increases constantly for me t t < and decreases for me t t > .However, the universe horizon volume distribution shows sudden reduction in the range Gyr 0 5 -. .Distributions of mass of radiation, matter and dark energy within the horizon volume of the universe were also investigated in the closed cosmic model.These distributions reveal similar noticeable changes as the universe horizon volume distribution for the same reasons.The mass of radiation dominates up to 53221.5 yr t = , then the mass of matter becomes larger.Afterwards, both distributions of radiation and matter decrease while the distribution of dark energy rises until 10.1007 Gyr t = , where the mass of dark energy prevails up to me t t = .Hence, the distribution of dark energy reduces until 40.2892Gyr t = where the mass of matter becomes prominent again.At 53.6246 Gyr t = the masses of both matter and radiation become appreciably high such that the intercluster space will vanish and clusters of galaxies will interfere with each other.Furthermore, not only the intergalactic medium will disappear, but also galaxies will collide and merge with each other to form extremely dense and close cosmological bodies.These very dense bodies will undergo further successive collisions and mergers under the action of central gravity, where the interstellar medium will vanish and the universe would develop to big crunch at 53.6251 Gyr bc t = .

1 ) 2 I
Stage of the universe expansion.= , J which includes the following sub steps: d) ( )

Figure 1 .Figure 3 .
Figure 1.(a) The distribution of the universe horizon distance in the closed cosmic model up to t = 0.5 Gyr; (b) The distribution of the universe horizon distance in the closed cosmic model in the range t = 0.5 Gyr -t me ; (c) The distribution of the universe horizon distance in the closed cosmic model in the range t = t me -t * ; (d) The distribution of the universe horizon distance in the closed cosmic model in the range t = t * -t n .
change of the universe space from flat to curved then closed in the interval 15.1261 Gyr me t t ≤ ≤ .On the other hand, this distribution exhibits abrupt raise in the range * -change of the universe space from closed then curved to flat in the interval 39.3822 40.75 Gyr 21 t ≤ ≤ Furthermore, the distribution of the universe horizon distance in the range . Afterwards, it raises gradually as indicated in Figure 1(b).* -* -* t t = .The abrupt increase of the distribution in the range 37.

Table 1 .
It is interesting to note that at 0.0005 Gyr Coma cluster volume.However, the mass of matter within the horizon volume of the universe at

Table 1
that the horizon distance of the universe at maximum expansion is

Table 1 .
Estimations of the horizon distance, horizon volume, mass of radiation, mass of matter, mass of dark energy and the equivalent number of the Coma-like clusters to the mass of matter within the universe horizon volume in the closed cosmic model at special times.
[7] d t′ in the other four general cosmic models as shown from Table1in[7].