Received 3 March 2015; accepted 17 March 2015; published 20 March 2015

ABSTRACT

Distributions of the universe horizon distance and universe horizon volume were investigated in the light of five general cosmic models which were constructed in a previous study. Both distributions increase so slowly up to t ≈ 21.5444 Myr, then they start raising very fast up to t ≈ 60 Gyr. Afterwards, they increase again very slowly until t ≈ 124 Gyr. Distributions of mass of radiation, matter and dark energy within the horizon volume of the universe were also studied in the five general cosmic models. The masses of both radiation and matter decrease gradually with time while the mass of dark energy increases. The mass of radiation prevailed in the early universe up to t ≈ 34627.5 - 55916.2 yr, where it becomes equal to the mass of matter. Then the mass of matter dominated until t ≈ 9.4525 - 10.0632 Gyr, where it becomes equal to the mass of dark energy. Thenceforward, the mass of dark energy prevails the universe. The cosmic space becomes approximately matter empty in the so far future of the universe.

Keywords:

General Cosmic Models, Distribution of Mass and Energy

1. Introduction

In a previous study [1] the distribution of density parameters of radiation, matter and dark energy were investigated in details in five general cosmic models. Hence, it would be interesting 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 in the general models.

Therefore, it is necessary to start this study by investigating the distributions of the horizon distance and horizon volume of the universe in the general models at different time intervals depending on the bases discussed in [2] . Description of methodology is given in Section 2 while algorithm would be illustrated in Section 3. Results and discussion are presented in Section 4. Conclusion is shown in Section 5.

2. Methodology

We have seen in [2] that the horizon distance and horizon volume of the universe at the present time are respectively

(1)

(2)

where are all defined as in [1] . Thus the horizon distance of the universe at any given time is

(3a)

Consequently the change in the horizon distance of the universe in the time interval between two instants of scale factors is written as

(3b)

The horizon volume of the universe at any given time is

(4)

It is also obvious from [2] that the total density of the universe is given by

(5)

where

(6)

(7)

(8)

(9)

(10)

(11)

Hence, the total mass within the horizon volume of the universe at any given time is expressed as

(12)

The mass of matter, the equivalent mass of radiation and the equivalent mass of dark energy within the horizon volume of the universe at any given time are given by

(13)

(14)

(15)

The cosmic time is given by Equation (16) in [1] as

(16a)

Thus the time interval between two instants with scale factors during the universe expansion is expressed as

(16b)

3. Algorithm

In determination of the distributions of we use the following steps:

i) Set then insert the value of for, for and for.

ii) Compute

iii) Start general DO loop which includes the following sub steps:

iv)

v) Calculate new value of cosmic time t numerically using(16-b), where.

vi) Determinate new value of the universe horizon distance numerically using (3-b) where,.

vii) Obtain the corresponding values of using (4), (11), (7), (8), (9), (10), (6), (5), (12), (13), (14) and (15) respectively.

viii) Continue the general DO loop.

4. Results and Discussion

The distribution of the universe horizon distance in the general models up to is shown in Figure 1(a). The distributions of all models coincide on each other until. Then, the distributions of models A, B and C coincide on each other and get upper than the coincided distributions of models D and E. The universe horizon distance increases quite slowly with time in all general models up to, hence it stats raising very fast. The distribution of the universe horizon distance in the general models in the range is illustrated in Figure 1(b). The distributions of all models coincide on each other up to. Afterwards, the distributions of models A, B and C coincide on each other and become higher than the coincided distributions of models D and E. In all general models the universe horizon distance distributions increase very fast with time. The distribution of the universe horizon distance in the general models in the range is presented in Figure 1(c). The distributions of models A, B and C are close to each other and lie upper than the distributions of models E and D. The increase in the universe horizon distance gets very small with increasing time in all general models.

Table 1 shows the universe horizon distances in the general models at special times. These times are the time of radiation-matter mass equivalence, the time of matter-dark energy mass equivalence, the present time and the time

The results illustrated in Figures 1(a)-(c) are supported by those displayed in Figures 2(a)-(c) which show the distributions of the universe horizon volume in the general models in the ranges up to t = 0.5 Gyr, t = 0.5 - 50 Gyr and t = 50 - 124 Gyr respectively. Table 2 presents the universe horizon volumes in the general models at the special times.

The distribution of mass and energy within the universe horizon volume of the universe in any general model up to is exhibited in Figure 3(a). The distributions of both radiation and matter decrease gradually with time and intersect at the time as shown in Table 3. On the other hand, the distribution of dark energy increases gradually until it intersects with the radiation distribution at the time as seen in Table 4. The distribution of total mass coincides with that of radiation up to Afterwards, the two distributions diverge from each other. However, the distribution of the total mass coincides on the distribution of matter from the time onwards.

The distribution of mass and energy within the universe horizon volume of the universe in any general model in the range is displayed in Figure 3(b). It is obvious that the distributions of matter and radiation decrease gradually with time and the former lies above the later. The distribution of dark energy increases with time and intersects with the distribution of matter at as illustrated in Table 5. The distribution of the total mass coincides on the distribution of the matter up to, then they diverge from each other. Furthermore, the distribution of the total mass coincides on the distribution of the dark energy from onwards. Masses of radiation, matter and dark energy within the universe horizon volume in the general models at the present time are illustrated in Table 6.

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

is exhibited in Figure 3(c). Again the distribution of both matter mass and radiation mass decrease with time and the former is higher than the later. The distributions of dark energy mass and total mass coincide on each other. Masses of radiation, matter and dark energy within the universe horizon volume in the general models at are given in Table 7.

Table 8 shows the equivalent number of the Coma-like clusters to the mass of matter within the universe horizon volume in the general models at the special times. It is obvious that this content of matter strongly decreases with time such that the cosmic space becomes almost matter empty in the far future of the universe.

5. Conclusion

In this article distributions of the universe horizon distance and universe horizon volume were determined in the five general cosmic models which were established previously. The two distributions were found increasing slowly up to, hence they raise appreciably fast up to, then they increase again so slowly until. Distributions of mass of radiation, matter and dark energy within the universe horizon volume were also investigated in the five general models. The masses of radiation and matter are decreasing with time although the mass of dark energy is increasing. The mass of radiation was dominant in the early

universe up to where it becomes equivalent to the mass of matter. Afterwards, the mass of matter prevailed until where it becomes equal to the mass of dark energy. From this time onwards the mass of dark energy dominates the universe. The cosmic space gets approximately matter empty in the very remote future of the universe.

References