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
![](https://www.scirp.org/html/15-4500141\3ac0089e-7a46-4ad2-af1c-b1b77803ff43.jpg)
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 ![](https://www.scirp.org/html/15-4500141\cc64bf30-739b-4891-809e-2d078f76006c.jpg)
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 ![](https://www.scirp.org/html/15-4500141\bddd0643-d0c2-477f-bc02-81f91dc69218.jpg)
![](https://www.scirp.org/html/15-4500141\ce34588f-7fe4-4c28-8b6e-02e72aec227e.jpg)
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
![](https://www.scirp.org/html/15-4500141\9b4ee4f0-421e-4001-8678-b8eec0b891ba.jpg)
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 ![](https://www.scirp.org/html/15-4500141\2dc68f68-a9d3-4c86-ae68-3723bc1d0791.jpg)
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 ![](https://www.scirp.org/html/15-4500141\5099e5e6-84c9-432b-859b-0c6b99db70cb.jpg)
using Equations (32), (30). The value of
corresponding to the minimum positive value of
is assumed to be
and
![](https://www.scirp.org/html/15-4500141\173160fe-1966-43aa-b634-fb8702004352.jpg)
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
![](https://www.scirp.org/html/15-4500141\96cf3f53-6cb3-4bea-afc4-ba596baf8027.jpg)
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 ![](https://www.scirp.org/html/15-4500141\12b6d14c-186d-46fa-8460-aa83b78c6066.jpg)
converge and become very close to zero. In other words the universe stops expending at ![](https://www.scirp.org/html/15-4500141\83df2926-896b-483d-85bb-13f8273eb0ef.jpg)
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
.
![](https://www.scirp.org/html/15-4500141\16628a1f-423f-4b72-b8cc-5875f8e1cce9.jpg)
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 − tme; (c) The distribution of the cosmological term in the closed cosmic model in the range t = tme − t∗; (d) The distribution of the cosmological term in the closed cosmic model in the range t = t∗ − tbc.
(a)
(b)
Figure 2. (a) The expansion of the universe in the general cosmic model A and the closed model up to t = t0; (b) Redshift look back time relation in the general cosmic model A and the closed cosmic model up to t = t0.
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 ![](https://www.scirp.org/html/15-4500141\d25e4c04-72dc-4cbc-9754-22782a92b611.jpg)
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
![](https://www.scirp.org/html/15-4500141\bd325708-99e6-4cbe-bccc-563fb3402f15.jpg)
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 = tme; (b) Contraction of the universe in the closed cosmic model in the range t = tme − tbc.
(a)
(b)
Figure 4. (a) The distribution of the universe expansion speed in the closed cosmic model in the range t = 0.5 Gyr − tme; (b) The distribution of the universe contraction speed in the closed cosmic model in the range t = tme − 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 − tme; (b) The distribution of the universe contraction acceleration in the closed cosmic model in the range t = tme − 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 − tm; (c) The distribution of the density parameters in the closed cosmic model in the range t = tme − t∗; (d) The distribution of the density parameters in the closed cosmic model in the range t = t∗ − tbc.
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 ![](https://www.scirp.org/html/15-4500141\da60ef6e-2940-46b5-9697-a1acaa865fd7.jpg)
2) The first matter epoch in the range ![](https://www.scirp.org/html/15-4500141\1504985c-92c5-4f61-abfc-40d83fc99abb.jpg)
3) The first dark energy epoch in the range ![](https://www.scirp.org/html/15-4500141\f44e1c8f-97ce-42c6-b83d-6fd6bc83ee9b.jpg)
4) The last dark energy epoch in the range ![](https://www.scirp.org/html/15-4500141\e157b1cd-3d98-4fca-b5b5-5107683a63ad.jpg)
5) The last matter epoch in the range ![](https://www.scirp.org/html/15-4500141\f57a3694-fbdd-4b72-8b7c-95132b041469.jpg)
6) The last radiation epoch in the range ![](https://www.scirp.org/html/15-4500141\a9c403ed-45e2-4ec3-918a-ff5008ec7265.jpg)
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 ![](https://www.scirp.org/html/15-4500141\4ac45d84-4dd0-4006-98b2-8ddeb42a0945.jpg)
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 Tr(t) is above that of Tm(t) throughout these two epochs. At t =
![](https://www.scirp.org/html/15-4500141\d2cea303-c3bb-449b-ad0a-c4f15ecdd78f.jpg)
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 = trm1; (b) The distribution of temperature of the radiation and non-relativistic matter in the closed cosmic model in the range t = trm1 − tme. (c) The distribution of temperature of the radiation and non-relativistic matter in the closed cosmic model in the range t = tme − trm2; (d) The distribution of the universe temperature in the closed cosmic model in the range t = trm2 − tbc.
tme Tr = 1.1471 K, Tm = 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 Tr = Tm 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 Tu = 2.2593 × 105 K at t = 34.4654 yr before ![](https://www.scirp.org/html/15-4500141\41019ed1-cfee-4835-acf9-d1677cf5cef6.jpg)
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.