_{1}

^{*}

The universe’s horizon distance and volume are constructed in the closed cosmic model. The universe horizon distance distribution increases constantly for
*t* <
*t*
_{me} and decreases for
*t* >
*t*
_{me}. However, the universe’s horizon volume shows a sudden reduction in the range
*t* = 0.5 Gyr -
*t*
_{me} due to the change of the universe space from flat to curved then closed in the interval 15.1261 Gyr ≤
* t* ≤
*t*
_{me}. On the other hand, this distribution exhibits an abrupt rise in the range
*t* =
*t*
_{me} -
*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* =
*t*
_{me}. 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
*t*
_{bc} = 53.6251 Gyr. It is interesting to note that the horizon distance of the universe in the closed model at
*t* =
*t*
_{me} is in very good agreement with the maximum horizon distances in the five general cosmic models.

The distribution of density parameters of radiation, matter and dark energy in the closed cosmic model were investigated in a previous study [

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 [

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 [

It is obvious from [

where

where t is the cosmic time in Gyr.

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

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

We have seen in [

Therefore, it is obvious from

We recall the equation of proper distance of extragalactic object

where

And the volume of space within

where

Hence Equation (22) gives

Assume

Substituting by (25) in (24) we get

Let

Substituting by (27) in (26) we have

Substituting by (23) in (28) yields

Suppose

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

Equations (21), (20) and (22) give

Assume

Substituting by (34) in (33) we have

Let

Substituting by (36) in (35) we get

Substituting by (32) in (37) yields

When

Therefore, the horizon volume of the universe in closed cosmic model at time t in this case is written as

It is clear form

The total density of the universe in the closed cosmic model at time t is

Substituting by (3)-(5) in (41) we get

where

From Equation (19), (30), (40) and (42) the total mass of the universe within the horizon volume in closed cosmic model at time t is

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

However, during the universe contraction if

In determination of the distributions of

(1) Stage of the universe expansion.

a) Set

b) Calculate

c) Start general DO loop

d)

e) Compute new value of cosmic time t numerically using (48), where

f) Obtain new value of the universe horizon distance

g) Determine the corresponding values of

h) Continue the general DO loop.

(2) Stage of the universe contraction

a) Set

b) Evaluate

c) Start general DO loop

d) Set

e) Obtain new value of cosmic time t numerically using (48), where

f) Compute new value of the universe horizon distance

Sub steps (g) and (h) are similar to (g), (h) mentioned above in the stage of the universe expansion.

The distribution of the universe horizon distance in the closed cosmic model until

The distribution of the universe horizon volume in the closed cosmic model up to

The distribution of the universe horizon volume in the range

the fact that the universe space changes from closed then curved to flat in the interval

The distribution of mass and energy within the horizon volume of the universe in the closed cosmic model until

The distribution of mass and energy within the horizon volume of the universe in the closed model in the range

to flat in the range

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

Estimations of

Since the radius of the Coma cluster is

It is also interesting to note from

t | ||||||
---|---|---|---|---|---|---|

0.1356 | 0.0209u_{1} | 28.4321 | 28.4321 | 18.3292 | 1.352u_{3} | |

12.9717 | 18.2854u_{2} | 22.9777 | 24.2252 | 24.2252 | 8.3970u_{4} | |

14.4420 | 25.2348u_{2} | 23.2298 | 24.0515 | 24.3294 | 5.6291u_{4} | |

18.6192 | 32.8742u_{2} | 23.6535 | 23.7347 | 24.3906 | 2.7141u_{4} | |

14.0058 | 23.0168u_{2} | 23.6758 | 24.1474 | 24.1474 | 7.0198u_{4} | |

0.9972 | 0.0003u_{2} | 27.7134 | 28.5430 | 20.5951 | 1.7455u_{3} |

where_{1} = (Gpc)^{3}, u_{2} = (10 Gpc)^{3},

The value of

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

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

Fadel A.Bukhari, (2015) Distribution of Mass and Energy in Closed Model of the Universe. International Journal of Astronomy and Astrophysics,05,291-301. doi: 10.4236/ijaa.2015.54032