A Closed Model of the Universe

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 = 26.81 Gyr, then it will contract to a big crunch at t bc = 53.62 Gyr. Observational tests to the closed cosmic model were illustrated. Distributions of the universe expansion and contraction speed were established in this model which indicated that the expansion speed in the early universe is appreciably high, then it will decrease rapidly until it vanishes at t me . However, the contraction speed of the universe increases continuously until the time just before t bc . Distributions of the universe expansion and contraction acceleration were performed empirically which confirmed the previous result. In the closed cosmic model the universe history can be categorized into six main stages, 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. Distributions of the density parameters of the radiation, matter, dark energy and the total density as well as the distributions of temperature of the radiation and non-relativistic matter were all investigated in this model at all epochs of the universe.


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][4][5][6]. In  6.626 10 m Kg s .      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 t such that any 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 me t 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.
The slowly varying cosmological parameter is .
Using Equations (1)- (5) and (14) And the total density parameter in the closed cosmic model at time t is The speed of the universe dynamics in the closed cosmic model is obtained from Equation (24) such that The acceleration of the universe dynamics in the closed cosmic model is found empirically as The time interval between two instants with scale factors 1 2 , a a during the universe expansion is given by 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].

Determination of me t
The time of the maximum expansion of the universe in the closed model is evaluated by iterative procedure as follows: 1

Observational Tests to the Closed Cosmic Model
It is convenient to start by investigating the distributions of the cosmological parameter   t  in the closed cosmic model at various epochs according to Equation (17).  Figure 2(a) shows that the expansion distribution of the universe in the closed cosmic model up to 0 t t  is found using Equation (32). This distribution is in good agreement with that of the observed general cosmic model A obtained by Equation (16) in [1]. Moreover, at 12.97 Gyr t  , these two distributions become identical. The redshift look-back time distributions in these two models up to 0 t t  were established and presented in Figure 2(b). Both distributions are in perfect agreement. The obvious agreement between the observed general cosmic model A and the closed cosmic model as seen from Figures 2(a) and (b) strongly argues in favour of reliability of the closed cosmic model.

Results and Discussion
The expansion of the universe in the closed cosmic model up to me t t  is obtained by Equation (32) and presented in Figure 3(a). It is noticeable that the increase of   a t with t is continuous as a linear relation until about 26.3 Gyr t  , then   a t increases relatively slow with .
t Nevertheless, the contraction of the universe in the closed model in the time range me bc t t t   is illustrated in Figure 3(b). It is obvious that   a t almost linearly decreases with . t However,   a t reduces relatively slow with t just before .

bc t t 
The distribution of the universe expansion speed   a t  in the closed model in the range 0.5 Gyr me t t   is performed using Equation (30) and displaced in Figure 4(a).    in the closed cosmic model in the ranges 0.5 Gyr t  , bc t t t    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.