A Closed Model of the Universe ()

Fadel A. Bukhari

Department of Astronomy, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia.

**DOI: **10.4236/ijaa.2013.32022
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Department of Astronomy, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia.

A closed model of the universe was constructed according to the assumption that very minor fraction of the dark energy transfers so slowly to matter and radiation. The cosmological parameter is no longer fixed but represents so slowly decreasing function with time. In this model the universe expands to maximum limit at *t _{me}* = 26.81253 Gyr, then it will contract to a big crunch at

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Bukhari, F. (2013) A Closed Model of the Universe. *International Journal of Astronomy and Astrophysics*, **3**, 189-198. doi: 10.4236/ijaa.2013.32022.

1. Introduction

In pervious two articles [1,2] the cosmological parameter was assumed constant in five general cosmic models. However, in some cosmological studies is not actually perfectly constant but exhibits slow variation, so is often described as quintessence [3-6]. In other wordsthe dark energy density does not remain constant with time.

This point of view is in a good agreement with the Heisenberg’s Uncertainty Principle that there is an uncertainty in the amount of energy which can exist. This small uncertainty allows non-zero energy to exist for short intervals of time where is Planck’s constant

As a result of the equivalence between matter and energy, these small energy fluctuations can produce virtual pairs of matter particles (particles and their antiparticles must be produced simultaneously) which come into existence for a short time and then disappear to produce photons.

In the present study is assumed to be very slowly decreasing function of the cosmic time such that any decrease in say should be compensated by increasing each of the matter density and radiation density by

The importance of this study is to know under what cosmological conditions the universe can be contracting to big crunch rather than expanding for ever as shown in the five general cosmic models investigated in [1].

In Section 2, a detailed description is given for the methodology. Determination of is explained in Section 3. Observational tests of the closed cosmic model are illustrated in Section 4. Results and discussion are presented in Section 5. Finally the conclusion is displaced in Section 6.

2. Methodology

From [1] we have seen that the densities of matter radiation and dark energy at a cosmic time are given by

(1)

(2)

(3)

where

(4)

(5)

(6)

(7)

(8)

(9)

Substituting by (4), (6) in (1) we get

Or,

(10)

Similarly we can find

(11)

(12)

Now assume a very small decrease in about per Gyr, so the decrease in in cosmic time is expressed as

(13)

According to the conservation law of mass and energy the decrease in the energy density is compensated by increase of in each of and

Therefore at the cosmic time the new values of and are given by

(14)

(15)

(16)

The slowly varying cosmological parameter is

(17)

Using Equations (1)-(5) and (14)-(16) the new values of density parameters in the expanding cosmic model at time are

(18)

(19)

(20)

Let then Equations (6)-(8) can be written as

(21)

(22)

(23)

Substituting by (21)-(23) in (9) and using (18)-(20) we get the Hubble parameter in the closed cosmic model at time

or,

(24)

The critical mass density in the closed cosmic model at time becomes

(25)

The new density parameters in the closed cosmic model at time are

(26)

(27)

(28)

And the total density parameter in the closed cosmic model at time is

(29)

The speed of the universe dynamics in the closed cosmic model is obtained from Equation (24) such that

or,

(30)

The acceleration of the universe dynamics in the closed cosmic model is found empirically as

(31)

The time interval between two instants with scale factors during the universe expansion is given by Equation (16) in [1] as

(32)

The redshift lookback time relation in the closed cosmic model is given by Equation (18) in [1]. In addition, the distributions of temperature at different epochs of the universe depend on relations similar to Equations (33), (34) and (37) in [1].

3. Determination of

The time of the maximum expansion of the universe in the closed model is evaluated by iterative procedure as follows:

1) Start with at and let

2) Calculate 1000 values of and

using Equations (32), (30). The value of corresponding to the minimum positive value of is assumed to be and

3) Select at and repeat the previous two steps where Now the value of corresponding to the minimum positive value of is supposed to be and

4) Repeat this method several times using the values and then estimate the values and and obtain the corresponding values of and

5) Denote these results as presented in Table 1, where it is noticeable that the values of and

converge and become very close to zero. In other words the universe stops expending at

6) From Table 1 one can easily find that the time of maximum expension of the universe in the closed model is. By similarity the time of big craunch is.

Table 1. Iterative determination of the maximum expansion time of the universe in the closed cosmic model.

4. Observational Tests to the Closed Cosmic Model

It is convenient to start by investigating the distributions of the cosmological parameter in the closed cosmic model at various epochs according to Equation (17). Figure 1(a) shows no evident change of with cosmic time until then decreases in relatively higher rate towards. On the other hand exhibits a gradual change with time in the time range as seen in Figure 1(b), where is the time of maximum expansion of the universe in the closed cosmic model. The slow variation of with is also noticeable in the time ranges as displaced in Figures 1(c) and (d) respectively where is the time of big craunch of the universe in the closed cosmic model and.

Figure 2(a) shows that the expansion distribution of the universe in the closed cosmic model up to is found using Equation (32). This distribution is in good agreement with that of the observed general cosmic model obtained by Equation (16) in [1]. Moreover, at, these two distributions become identical. The redshift look-back time distributions in these two models up to were established and presented in Figure 2(b). Both distributions are in perfect agreement. The obvious agreement between the observed general cosmic model and the closed cosmic model as seen from Figures 2(a) and (b) strongly argues in favour of reliability of the closed cosmic model.

(a)(b)

Figure 1. (a) The distribution of the cosmological term in the closed cosmic model up to t = 0.5 Gyr; (b) The distribution of the cosmological term in the closed cosmic model in the range t = 0.5 Gyr − t_{me}; (c) The distribution of the cosmological term in the closed cosmic model in the range t = t_{me} − t_{∗}; (d) The distribution of the cosmological term in the closed cosmic model in the range t = t_{∗} − t_{bc}.

(a)(b)

Figure 2. (a) The expansion of the universe in the general cosmic model A and the closed model up to t = t_{0}; (b) Redshift look back time relation in the general cosmic model A and the closed cosmic model up to t = t_{0}.

5. Results and Discussion

The expansion of the universe in the closed cosmic model up to is obtained by Equation (32) and presented in Figure 3(a). It is noticeable that the increase of with is continuous as a linear relation until about, then increases relatively slow with Nevertheless, the contraction of the universe in the closed model in the time range is illustrated in Figure 3(b). It is obvious that almost linearly decreases with However, reduces relatively slow with just before

The distribution of the universe expansion speed in the closed model in the range is performed using Equation (30) and displaced in Figure 4(a). The value of is high in the early universe then it decreases abruptly up to about. Afterwards fluctuates gradually with until at On the other hand, Figure 4(b) exhibits the distribution of the universe contraction speed in the closed model in the range It is clear that the increase of with is gradual up to then rapidly increases with until

The distribution of the universe expansion acceleration in the closed model in the range is deduced from Equation (31) and exhibited in Figure 5(a). Abrupt increase in with is obvious up to. Then changes very slightly with until, where starts decreasing gradually up to. Afterwards, decreases

(a)(b)

Figure 3. (a) Expansion of the universe in the closed cosmic model up to t = t_{me}; (b) Contraction of the universe in the closed cosmic model in the range t = t_{me} − t_{bc}.

(a)(b)

Figure 4. (a) The distribution of the universe expansion speed in the closed cosmic model in the range t = 0.5 Gyr − t_{me}; (b) The distribution of the universe contraction speed in the closed cosmic model in the range t = t_{me} − t_{∗}.

rapidly towards the maximum expansion time It is clear that in the range Furthermore, Figure 5(b) shows the distribution of the universe contraction acceleration in the closed model in the range It is noticeable that suddenly reduces up to, then reduces gradually until where Afterwards, raises gradually up to where starts increasing quite rapidly towards in the interval.

It is remarkable to note that the distributions of and in the closed cosmic model in the ranges, will be investigated in details in a separate study, since in these two time ranges the pressure of the cosmic fluid is significant and can not be neglected.

(a)(b)

Figure 5. (a) The distribution of the universe expansion acceleration in the closed cosmic model in the range t = 0.5 Gyr − t_{me}; (b) The distribution of the universe contraction acceleration in the closed cosmic model in the range t = t_{me} − t_{∗}.

The distribution of the density parameters in the closed cosmic model up to is disclosed in Figure 6(a). It is prominent that the distribution of the radiation density parameter coincides on the distribution of the total density parameter up to. However, the distribution of the matter density parameter coincides on the distribution of at . It is also obvious that the distributions of the dark energy density parameter and the distribution of are increasing while the distribution of remains almost fixed at the value up to , then it starts decreasing. Neverthelessthe distribution of stays almost constant at the value in this epoch of the universe. Thus at whereas

(a)(b)

Figure 6. (a) The distribution of the density parameters in the closed cosmic model up to t = 0.5 Gyr; (b). The distribution of the density parameters in the closed cosmic model in the range t = 0.5 Gyr − t_{m;} (c) The distribution of the density parameters in the closed cosmic model in the range t = t_{me} − t_{∗}; (d) The distribution of the density parameters in the closed cosmic model in the range t = t_{∗} − t_{bc}.

at. Figure 6(b) shows the distribution of the density parameters in the cosmic closed model in the range It is evident that the distribution of displays rapid increase until the time where then it raises gradually up to Gyr where it exhibits abrupt increase again. The distributions of become close together from to The value of is almost 1.0 in the time intervals , The distributions of and change quite slowly up to where they also raise up suddenly. They get close together from to The distribution of the density parameters in the cosmic closed model in the range is presented in Figure 6(c). All distributions, reveal steep decrease up to. Distributions of are adjacent to each other until , then they diverge apart and decrease slowly. In addition, the distributions of and are also near each other up to. Afterwards these two distributions reduce gradually and get away from each other. Nevertheless, after the time the distributions of and reduce quite rapidly and intersect with each other at where However, the distributions of and intersect at where The distributions of and get close to each other at until Figure 6(d) illustrates the distribution of density parameters in the closed cosmic model in the range It is clear that the distributions of and almost coincide on each other up to about, then the distribution of starts decreasing slightly but still close to that of until, while takes the values between throughout the interval However, the distribution of raises gradually and intersects with the distribution of at. In addition the distribution of gets closer to the distribution of at Finally, the distribution of indicates slow decrease until about then it exposes quite rapid decrease towards the time of big Crunch.

It is essential to realize that the universe history has six main stages in the closed model, these are 1) The first radiation epoch in the range

2) The first matter epoch in the range

3) The first dark energy epoch in the range

4) The last dark energy epoch in the range

5) The last matter epoch in the range

6) The last radiation epoch in the range

These epochs of the universe with their relevant density parameters are all summarized in Table 2. Forthermore, the geometry of space throughout the universe history in the closed cosmic model is presented in details in Table 3.

One can see in Table 3 that the space of the universe is flat just after the big bang up to where the total density parameter lies in the range Afterwards, the space of the universe becomes open until since Then the universe space returns to flat up to as Afterward, the universe space gets curved then closed until because Hence, the universe space remains being closed then curved up to since Afterward the universe space evolves into flat until as Then the universe space develops into open up to owing to Eventuall the space of the universe comes back to flat until the time just before the big cranch by the reason of

The distribution of the universe temperature in the closed cosmic model in the first radiation epoch is obtained using Equation(34) in [1] and displayed in Figure 7(a). It is evident that reduces continuously in linear manner during this era. The temperatures of the radiation and non relativistic matter are determined from Equations (33), (37) in [1] respectively. The distributions of and in the first matter and dark energy eras are presented in Figure 7(b). It is prominent that at then the distributions of decrease sharply up to t = 0.0702 Gyr. However, both distributions reduce gradually afterwards. The distribution of T_{r}(t) is above that of T_{m}(t) throughout these two epochs. At t =

Table 2. Epochs of the universe history in the closed cosmic model.

Table 3. Geometry of space throughout the universe history in the closed cosmic model.

(a)(b)

Figure 7. (a) The distribution of the universe temperature in the closed cosmic model up to t = t_{rm}_{1}; (b) The distribution of temperature of the radiation and non-relativistic matter in the closed cosmic model in the range t = t_{rm}_{1} − t_{me}. (c) The distribution of temperature of the radiation and non-relativistic matter in the closed cosmic model in the range t = t_{me} − t_{rm}_{2}; (d) The distribution of the universe temperature in the closed cosmic model in the range t = t_{rm}_{2} − t_{bc}.

t_{me} T_{r} = 1.1471 K, T_{m} = 0.0005 K. The distribution of and in the last dark energy and last matter epochs are exposed in Figure 7(c). Both distributions increase slowly up to t = 53.2567 Gyr, then they start raising rapidly until they join together at where T_{r} = T_{m} 7032.5366 K. Eventually, Figure 7(d) indicates the distribution of the universe temperature in the last radiation epoch. This distribution raises slowly up to before then it increases rapidly to the value T_{u} = 2.2593 × 10^{5} K at t = 34.4654 yr before

Further interesting physical properties of the universe in the closed cosmic model would be investigated in separate studies in comparison with the corresponding properties of the universe in the five general cosmic models.

6. Conclusion

In this study a closed model of the universe was developed depending on the assumption that very slow transfer of the dark energy to mater and radiation is allowed. Thus the cosmological parameter is no longer constant but so slowly decreasing function of time. In the light of this model the universe expands to maximum limit at Gyr, then it will recollape to a big crunch at. Observational tests to this model were presented. The distributions of the universe expension and contraction speed were investigated in the closed model which disclosed that the expansion speed in the early universe is very high, then it will reduce rapidly until it vanishes at Nevertheless, the contraction speed of the universe raises continuously until the time just before The distribution of the universe expansion and contraction acceleration were carried out empirically which supported the previous result. In this model the universe history is classified in to six main eras, these are the first radiation epoch, the first matter epoch, the first dark energy epoch, the last dark energy epoch, the last matter epoch and the last radiation epoch. The distributions of the density parameters of the radiation, matter, dark energy and total density in addition to the distributions of temperatures of the radiation and nonrelativistic matter were all determined and discussed in this model in the various eras of the universe.

7. Acknowledgements

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | F. A. Bukhari, “Five General Cosmic Models,” Journal of King Abdulaziz University: Science, Vol. 25, No. 1, 2013. |

[2] | F. A. Bukhari, “Cosmological Distances in Five General Cosmic Models,” International Journal of Astronomy and Astrophysics, 2013. |

[3] | J. V. Chunha, J. A. S. Lima and N. Pires, “Deflationary Cosmology: Observational Expressions,” Astronomy and Astrophysics, Vol. 390, No. 3, 2002, pp. 809-815. doi:10.1051/0004-6361:20020746 |

[4] | J. A. S. Lima and M. Trodden, Physical Reviews D, Vol. 53, 1996, p. 4280. |

[5] | S. A. Bludman and M. Roos, “Quintessence Cosmology and the Cosmic Coincidence,” Physical Reviews D, Vol. 65, No. 4, 2002, Article ID: 043503. doi:10.1103/PhysRevD.65.043503 |

[6] | I. Zlatev, L. Wang and P. J. Steinhardt, “Quintessence, Cosmic Coincidence, and the Cosmological Constant,” Physical Review Letters, Vol. 82, No. 5, 1999, pp. 896-899. doi:10.1103/PhysRevLett.82. 896 |

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