A Cosmological Model without Singularity Based on RW Metric (1) ()
1. Introduction
In view of the fact that the space-time singularity and the cosmological constant issues are not solved in the frame of the conventional theory up to now. We suggest two conjectures to solve the issues. Based on the conjectures, we construct a cosmological model. Based on this model, we solve the two issues, explain the evolution of the universe, primordial nucleosynthesis, cosmic microwave background radiation
, and give three new predictions.
As is now well known, there is space-time singularity under certain conditions [1] . “These conditions fall into three categories. First, there is the requirement that gravity shall be attractive. Secondly, there is the requirement that there is enough matter present in some region to prevent anything escaping from that region. The third requirement is that there should be no causality violations” [1] . There must be space-time singularity in the conventional theory, because the conditions can be satisfied.
There should be no space-time singularity in physics, hence this problem must be solved. But it is not solved satisfactorily up to now.
In order to solve the space-time singularity problem, Ref. [2] [3] had assumed that there exists a fundamental length, i.e., the Planck length,
. There is no curvature corresponding to scale l < lp.
Based on this, they proposed the limiting curvature hypothesis. Thereby they had proved that all isotropic cosmological solutions are nonsingular. We find that its conclusion is included in the hypothesis. On the other hand, the model does not explain the expansion of the universe with an acceleration and cannot solve the cosmological constant problem.
The Planck length lp, time 
and temperature 
reveal that the quantum theory and the general relativity are not self-consistent. Relativity is a theory of continuous space-time geometry. But the presence of
, 
and 
makes it virtually to come to a not continuous space-time structure. According to the conventional theory, only the quantum theory of gravity can solve the problem of not self-consistency. This is not true. According to the present model, there is no singularity of space-time, and there are the highest temperature 
and the least scale 
(see later). Consequently, it is impossible that
, 
and 
are arrived, because the transformation of the S-breaking and V-breaking from one to another at the highest temperature
. Thus, the quantum theory and the general relativity can be self-consistent, although the gravitational field is not quantized.
Recent astronomical observations show that the universe expanded with a deceleration earlier, while it is expanding with an acceleration now [4] -[6] . This implies that there is dark energy. Among the total energy density of the universe, 73 percent is dark energy [4] -[6] . What is dark energy? Many possibilities have been suggested. One interpretation adopts the effective cosmological constant 
where 
and 
are the Einstein cosmological constant and the gravitational mass density of the vacuum state, respectively. The subscript “g” denotes the physical quantities relating to gravity in the following. According to the equivalence principle, 
, 
is the energy density of the vacuum state. Hence 
may be written as 
cannot be derived from basic theories [7] [8] , and
. Hence the interpretation is unsatisfactory. Alternatively, dark energy is associated with the dynamics of scalar field 
which is uniform in space [9] -[11] . This is a seesaw cosmology [12] . Thus, problem about the universe expansion with an acceleration is still open to the public.
That 
originates from the conventional quantum field theory and the equivalent principle. Both 
and the singularity issues imply that the conventional theory is incomplete. In some supersymmetric models, 
can be obtained. But this is not a necessary result of the supersymmetric quantum field theory. On the other hand, supersymmetric theory lacks of experiment bases. In contrast with the supersymmetric quantum field theory, 
is a necessary result of our quantum field theory without divergence [13] -[15] . In this theory, there is no divergence of loop corrections as well, and dark matter which can form dark galaxies is predicted [16] [17] .
In fact, 
is not a necessary condition of
. We will see that although 
is divergent according to the conventional quantum field theory, we have still 
based on the present model.
Huge voids in the cosmos have been observed [18] . Such a model in which hot dark matter (e.g. neutrinos) is dominant can explain the phenomenon. However, it cannot explain the structure with middle and small scales. Hence this is an open problem as well.
We consider that all important existing forms of matter (including dark matter and dark energy) have appeared. Hence these basic problems should be solved. As mentioned above, we have constructed a quantum field theory without divergence which predicts that there must be dark matter. We can construct a cosmological model which can solve the space-time singularity and cosmological constant issues and explain the evolution of the universe in the present paper.
The bases of the present model are the general relativity, the conventional quantum field theory for finite temperature and grand unified theory (GUT).
The basic idea of the present model is conjecture 1 in Section 2.
We consider the following condition to be necessary in order to solve the space-time singularity and the cosmological constant problems on the basis of the classical cosmology and the conventional quantum field theory.
Condition: There are two sorts of matter which are symmetric, whose gravitational masses are opposite to each other, although whose masses are all positive.
The two sorts of matter are called solid matter (s-matter) and void matter (v-matter), respectively. The condition implies that if 
then 
here 
denotes a gravitational mass density. The conditions cannot be realized in the conventional theory, but can be realized in the present model. In order to uniformly solve the above four problems, we present a new conjecture equivalent to the condition and construct two cosmological model, i.e. [19] [20] and this model in the present paper.
The present model has the following results:
There is no space-time singularity in this model.
It is derived from this model that there are the critical temperature
, the highest temperature
, the least scale 
and the largest energy density 
in the universe. Both 
and 
are new important constants. Both 
and 
are finite. It is impossible that the Plank temperature
, length 
and time 
are arrived, because
, and 
is not small. In general, the radius of a local inertial system is so large, 
, that the quantum effects corresponding to 
may be neglected.
The evolution of the universe which is derived from this model are consistent with the observations up to now.
There are two sorts of spontaneous symmetry breaking in the present model because of conjecture 1, and they are called S-breaking and V-breaking.
According to the present model, the evolving process of space is as follows.
In the S-breaking, space can contract so that temperature 
rises. When 
arrives the critical temperature
, the universe is in the most symmetric state with 
symmetry. When space continues to contract so that 
arrives the highest temperature
, space expands and then inflates. After inflation, the most symmetric state transits to the state with the V-breaking. After reheating, the evolving process is as follows: Space expands with a deceleration, expands with an acceleration, then expands with a deceleration, finally comes to static and begin to contract, in turn.
The relation between the optical distance and the redshift is derived from the present model. It is consistent with the observations up to now.
Equations governing nonrelativistic fluid motion are generalized to the present model. Galaxies can form earlier according to this model than that according to the conventional theory.
Three predictions are given.
Primordial nucleosynthesis and cosmic microwave background radiation are explained.
Dark energy is explained as s-matter when the universe is in the V-breaking. In contrast with the dark energy, 
in the V-breaking.
is proved, although 
is still very large. Consequently, 
Problems 5 and 7 will be discussed in the following paper.
Section 2 is “Conjectures, action, energy-momentum tensor and field equations”; Section 3 is “Spontaneous symmetry breaking”; Section 4 is “Evolution equations”; Section 5 is “Temperature effect”; Section 6 is “Space can contract, but there is no singularity”; Section 7 is “Space inflation”; Section 8 is “Evolving process of space after inflation”; Section 9 is “After expansion with an acceleration, space expands with a deceleration, then comes to static and finally begin to contract”. Section 10 is “Existing and distribution forms of 
color singlets”. Section 11 is “New predictions, an inference, and there is no restriction for
”; Section 12 is “Conclusions”.
2. Conjectures, Action, Energy-Momentum Tensor and Field Equations
2.1. Conjectures
In order to solve the problems mentioned before, we propose the following conjectures:
Conjecture 1 There are two sorts of matter which are called solid-matter (s-matter) and void-matter (v-matter), respectively. Both are symmetric and the symmetric gauge group is 
Both contributions to the Einstein tensor are opposite each other. There is no other interaction between both except interaction (2.10) of s-Higgs fields and v-Higgs fields.
Conjecture 2 When 
symmetry holds, there is the critical temperature
, all particles exist in 
color singlets when
.
Because of conjecture 1, there are two sorts of symmetry breaking which are called S-breaking in which 
and 
and V-breaking in which 
and
, here 
denotes an arbitrary Higgs field. The meanings of conjecture 1 are as follows. The model and its all inferences are invariant when 
and
. The multiplet of 
is the same as that of 
When temperature
, s-particles and v-particles are completely symmetric, here 
is the critical temperature (see section 5. B); When temperature 
or 
is broken. Let 
be broken, then 
still holds.
Conjecture 2 holds obviously. In fact, this conjecture is a direct generalization of SU(3) color singlets.
Another premise of the present model is the conventional SU(5) grand unified theory (GUT). But it is easily seen that the present model does not rely on the special GUT. Provided conjecture 1 and such a coupling as (2.10) are kept, the GUT can be applicable.
The gravitational properties of matter and the mode of symmetry breaking determine the features of spacetime. We consider that there are only two possibilities.
. The first possibility can be described by the conventional theory. There is only one sort of matter so that the equivalence principle strictly holds. This theory is simple, but there must be essential difficulties. For example, there must be the singularity and cosmological constant issues which cannot be solved in the frame of this theory because of the Hawking theorems etc.
The basis of the second possibility is conjecture 1.
We explain it in detail as follows:
It must be emphasized that there is no negative mass or negative probability in the present model at all. Conjecture 1 implies that 
when
. In the S-breaking, 
and 
because of the reasons 6 - 7 below. Here 
denotes a gravitational mass. Consequently, both s-energy and v-energy must be positive (see (2.20)-(2.21)).
The observation basis of conjecture 1 is that space expands with an acceleration now. One of the two sorts of matter must exist in 
color singlets. The color singlets must loosely distribute in whole space, and can cause space to expand with an acceleration, but cannot be observed as so-called dark energy (see 4 - 6 below).
Because of conjecture 1, there must be two sorts of symmetry breaking.
Because of conjecture 1, s-Higgs fields and v-Higgs fields must be symmetric as well. If the symmetry of s-matter and v-matter was not broken, both s-matter and v-matter will exist in the same form at arbitrary time and place. This implies that the nature is simply duplicate. This is impossible because the nature does like duplicate. Of course, this contradicts experiments and observations as well. Consequently the symmetry must be broken when
. Thus the coupling constant 
etc. in (2.10) must be positive so that there must be the two sorts of breaking.
The existing probability of the S-breaking and the V-breaking must be equal because of conjecture 1. This equality can be realized by two sorts of modes.
The universe is composed of infinite s-cosmic islands with the S-breaking and v-cosmic islands with the V-breaking; This possibility has been discussed [19] [20] .
The whole universe is in the same breaking (e.g. the S-breaking). But one sort of breaking can transform to another as space contracts to the least scale 
(see later)
We discuss the case in the present paper. The RW metric is applicable to the case.
. There is only the repulsion between s-matter and v-matter. Consequently, any bound state is composed of only the s-particles or only the v-particles, i.e. there is no bound state which is composed of the s-particles and the v-particles.
Because of conjecture 1, there is the repulsion between s-matter and v-matter and the repulsion constant is the same as the gravitation constant so that the repulsion is weak as the gravitation. The interaction (2.10) is repulsive as well. After reheating, Higgs particles can get very large masses, hence the interaction (2.10) is weak and may be ignored.
s-matter and v-matter are no longer symmetric after the symmetry breaking.
In the S-breaking, 
is finally broken to 
and 
holds all the time. Consequently, s-particles get their masses and form s-atoms, s-observers and s-galaxies etc.; while all v-fermions and v-gauge bosons are still massless and must form 
color-singlets after reheating.
There is no interaction (e.g. the electroweak interaction) except the gravitation among the 
colorsinglets, because 
is a simple group. Hence the 
color-singlets cannot form v-atoms and vgalaxies etc., and must distribute loosely in space as the so-called dark energy.
Thus, in the S-breaking, s-matter is identified with the conventional matter, while v-matter is similar to dark energy. In contrast with the dark energy, the gravitational masses of v-matter is negative.
The 
color-single states cannot be observed by an s-observer.
As mentioned above, there is only the repulsion between s-matter and v-matter The repulsions originating from conjecture 1 and (2.10) are very weak after reheating. The v-particles can only form the 
color singlets with their very small masses. The 
color singlets cannot form atoms and galaxies etc., and can only distribute loosely in space. On the other hand, 
must be very small when 
is very large because of the repulsion. Consequently, in fact, it is impossible to observe the 
color-singlets even by the repulsion as well.
In the S-breaking, only the cosmological effects of v-matter are important and are consistent with the observation up to now.
The equivalence principle still strictly holds for the s-particles
, but is violated by the v-particles 
in the S-breaking. But the motion equations of all s-particles and all v-particles are still independent of their masses.
In the S-breaking, there are only s-observers and s-galaxies, and there is no v-observer and v-galaxy. Hence the gravitational masses of s-particles must be positive, i.e. 
while the gravitational masses of v-matter must be negative relatively to s-matter, i.e. 
because of conjecture 1. Thus, a s-photon falling in a gravitational field must have ‘purple shift’, but a v-particle (there is no v-photon and there are only the 
color singlets) falling in the same gravitational field will have ‘redshift’.
Although the equivalence principle is violated by v-particles in the S-breaking, there is no contradiction with any observation and experiment, because the 
color singlets cannot be observed by a s-observer (see 6).
When temperature is high enough, the expectation values of Higgs fields are small so that all masses of Higgs particles are small. Thus, 
and 
can transform from one into another by (2.10). Consequently, space cannot contract to infinite small and inflation must occur.
The interaction (2.10) can be neglected after reheating, because the masses of the Higgs particles are very large in low temperatures. Thus, the transformation of s-particles and v-particles from one into another may be neglected.
In summary, in the S-breaking, the 
color singlets cannot be observed and have only the cosmological effects. Conjecture 1 does not contradict any experiment and observation up to now.
We will see in the following that the evolution of the universe can be well explained, and the singularity and cosmological constant issues can be solved.
2.2. Action
The breaking mode of the symmetry is only one of the S-breaking and the V-breaking due to
. In the S-breaking, there are only s-observators. Analogously, in the V-breaking, there are only v-observators. Hence the actions should be written as two sorts of form, 
in the S-breaking and 
in the V-breaking. Of course, only one of both 
and 
can describe the evolution of the universe. Hence, in any case, the action is unique. But the S-breaking can transform to the V-breaking when temperature is high enough, hence both 
and 
are necessary. Because of conjecture 1, the structures of 
and 
are the same, i.e. 
when 
and
. Thus, at the zero-temperature, we have
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
where the meanings of the symbols are as follows:
, and 
in flat space. 
is the scalar curvature. Here 
and 
are two parameters, may be called “gravitation charges”, and are finally taken as
. 
in 
is a parameter in order to determine the zero-point of the potential 
. 
is the Lagrangian density of all s-fields (v-fields) and their couplings of the 
GUT except the Higgs potentials 
and
. 
and 
represents all s-fields and all v-fields, respectively. For a boson field, 
denotes its covariant derivative as well. Both 
and 
do not contain the contribution of the gravitation energy and the repulsion energy. It may be seen that the set of equation (2.1)-(2.7) is unchanged when the subscripts 
and
. This shows the symmetry of s-matter and v-matter. The physics quantities with the subscript ‘S’ (‘V’) denotes that they have meaning only when the universe is in the S (‘V’)-breaking. It is the same for the subscript ‘V’ as for ‘S’. For simplicity, the subscripts ‘S’ and ‘V’' may be elided in the following when there is no confusion.
Gibbons and Hawking pointed out that in order to get the Einstein field equations [21] , it is necessary that
This is because it is not necessary that 
on the boundary 
Hence 
is replaced by 
in (2.2). 
is a manifold with four dimensions. 
is the boundary of 
is the outer curvature on 
is the vertical vector on 
and 
is the induced outer metric on
. When 
is space-like, 
takes positive sign. When 
is time-like, 
takes negative sign.
The Higgs potentials in (2.5)-(2.7) is the following:
(2.8)
(2.9)
(2.10)
where
, 
and 
are respectively
, 
and 
dimensional representations of the 
group, 
are the 
generators, a = s, v. Here the couplings of 
and 
are ignored for short [22] [23] . (2.8) is the same as that in [22] [23] . The coupling constants in (2.8)-(2.10) are all positive, especially, as mentioned before, 
p and q in (2.10) must be positive.
We do not consider the terms coupling to curvature scalar, e.g.
, for a time. In fact, 
when temperature 
is high enough due to the symmetry between s-matter and v-matter.
2.3. Energy-Momentum Tensors and Field Equations
By the conventional method, from 
we can get
(2.11)
Considering
, from (2.3)-(2.4) we have
(2.12)
(2.13)
(2.14)
(2.15)
From (2.11)-(2.13) we obtain
(2.16)
In the S-breaking,
(2.17)
In the V-breaking,
(2.18)
and 
are the gravitational energy-momentum tensor density, the gravitational energymomentum tensor density without the Higgs potential and the gravitational potential density of the Higgs fields in the A-breaking, respectively.
It is seen from (2.17)-(2.18) that 
is independent of 
This implies that the potential energy 
is different from other energies in essence. There is no contribution of 
to
, i.e., there is no gravitation and repulsion of the potential energy
. This does not satisfy the equivalence principle. But this does not cause any contradiction with all given experiments and astronomical observations, because 
in either of breaking modes.
We will see that, in fact, 
because 
in the S-breaking, and 
because 
in the V-breaking. Hence
(2.19)
From (2.1) the energy-momentum tensor density which does not contain the energy-momentum tensor of gravitational and repulsive interactions can be defined as
(2.20)
(2.21)
As mentioned before, 
and 
in (2.3)-(2.4) may be regarded as the gravitation charges. The the gravitation charges of 
and 
are regarded as 1, ‒1 and 0 in the S-breaking, respectively. The energy-momentum tensor should be independent of the gravitation charges, because the energy-momentum tensor of the gravitation fields is not considered. Hence it is necessary to eliminate 
and 
from the definition of 
by the operator 
The operator is the only difference between the definition of 
in this model and that in the conventional theory. This definition does not contradict any basic principle and it is completely consistent with the conventional theory. In fact, there is one sort of matter in the conventional theory (i.e.
) so that 
can be reduced to 
It is seen from (2.20)-(2.21) that both s-energy and v-energy must be positive.
It should be pointed out that only (2.16) and (2.17) is applicable in the S-breaking, and only (2.16) and (2.18) applicable in the V-breaking.
Considering 
to be a scalar [24] or considering
and (2.16) and (2.14) we obtain
(2.22)
2.4. The Difference of Motion Equations of a v-Particle and a s-Particle in the Same Gravitational Field
From (2.16) we have
(2.23)
In the S-breaking, 
, 
and
.
Consider a point-particle with its gravitational mass 
to move in a gravitational field with its strength
. From (2.23) we can get
(2.24)
It is seen from (2.24) that the motion equation of the gravitation mass 
is the same as that of the gravitation mass 
This is the same as the conventional theory.
It must be given one's attention to that (2.24) is only the equation of a gravitation mass
, but is not the equation of an inertial mass
. According to the equvalence principle in the conventional theory, 
, here 
is the inertial mass. Consequently, (2.24) is the equation of an inertial mass 
as well.
According to the present model, because of conjecture 1, the gravitational field equation can determine only the motion equation of a gravitation mass (2.24), but cannot determine the motion equation of an inertial mass
. The motion equation of an inertial mass 
must be determined on the bases of conjecture 1 and the gravitational field equation.
In the S-breaking, according to conjecture 1, 
and
. Hence the equation of 
is the same as that of 
i.e. (2.24), but the equation of 
must be different from that of
. According to conjecture 1, both 
and 
are positive, and 
when
. 
in (2.24) is the coupling of the gravitational charge 
and the gravitational field with its strength 
The second term in (2.24) determines the force acting on
. Considering 
in the S-breaking so that the acceleration of 
is opposite to that of
, we get the motion equation of 
in the gravitational field with its strength 
to be
(2.25)
Comparing (2.24) and (2.25), we see that in the same gravitational field, the motion equation of a s-particle is different from that of a v-particle.
Analogous to the case in the S-breaking, in the V-breaking, because of the symmetry of s-matter and v-matter, we have
(2.26)
(2.27)
Considering the Newtonian approximation, i.e. the velocity of a particle is low
, a gravitational field is weak (
) and static
, and 
from 
we have
(2.28)
(2.29)
From (2.28), 
and 
are two constants. Considering
, and let 
be Newtonian gravitational potential, we can reduce (2.29) to
(2.30)
Let 
is caused by a static and spheral-symmetric s-object with its mass M. In the S-breaking, 
Thus, from (2.24)-(2.25) and (2.30) we get
(2.31)
(2.32)
Let 
is caused by a static and spheral-symmetric v-object with its mass M. In the S-breaking, 
because conjecture 1. Thus, from (2.24)-(2.25) and (2.30) we get
(2.33)
(2.34)
It is seen that the motion equation of a v-particle in such a gravitational field caused by v-matter is the same as that of a s-particle in the gravitational field caused by s-matter in the Newtonian approximation, when the distributing mode of v-matter is the same as that of s-matter.
In the V-breaking, we can get the same results as above, provided 
and
.
3. Spontaneous Symmetry Breaking
Ignoring the couplings of 
and 
and suitably choosing the parameters of the Higgs potential, analogously to Ref. [22] [23] , we can prove from (2.8)-(2.10) that there are the following vacuum expectation values (the S-breaking) at the zero-temperature and under the tree-level approximation
(3.1)
(3.2)
(3.3)
(3.4)
Ignoring the contributions of 
and 
to 
at the zero-temperature we get
(3.5)
(3.6)
(3.7)
We take 
From (2.9)-(2.10) and (3.1)-(3.7) it can be proved that all v-Higgs bosons can get their big enough masses. The masses of the Higgs particles exclusive of the 
-particles and the 
-particles in the S-breaking are respectively
(3.8)
(3.9)
(3.10)
(3.11)
We can choose such parameters that
(3.12)
e.g., 
and
. It is easily seen from (3.8)-(3.11) that all real components of 
have the same mass
, and all real components of 
have the same mass 
in the S-breaking.
The S-breaking and the V-breaking are symmetric because s-matter and v-matter are symmetric. Hence when 
and 
in (3.1)-(3.12), the formulas are still kept.
Let
(3.13)
we have
(3.14)
(3.15)
It is easily seen that 
is strictly determined by
, but 
or 
is a undetermined parameter. 
is the zero point of 
and 
We take 
to be so small that it may be neglected when 
in the A-breaking, 
and
.
4. Evolution Equations
4.1. Evolution Equations of R in RW Metric
As is well known, based on the RW metric metric,
(4.1)
In the present model, we take 
Taking 
or 1, we can get the results similar to those when
. We will discuss the two cases in the following paper. In fact, it is possible that 
is changeable with the gravitational mass density
. In this case, the results of the present model are more easily obtained [19] [20] .
Matter in the universe may approximately be regarded as ideal gas distributed evenly in space. Considering the potential energy densities in (2.14), we can write 
as
(4.2)
(4.3)
where 
is a 4-velocity and 
or v. In comoving coordinates 
in a comoving coordinates. 
can be written as
(4.4)
Considering 
substituting (4.2)-(4.4) and the RW metric in (4.1) into (2.16), we get the evolution equations
(4.5)
(4.6)
In the S-breaking,
(4.7)
In the V-breaking,
(4.8)
Comparing (4.5)-(4.6) with the Friedmann equations, we see that provided
, 
and V in the Friedmann equations are replaced by
, 
and
, (4.5)-(4.6) are obtained.
4.2. Evolution Equation of ρg
In contrast with the conventional theory, it is possible that 
although
. This is because 
and 
can transform from one to another by (2.10), especially when temperature is high enough (see section 6B).
Let
, e.g.
, it is obvious that in the S-breaking,
(4.9)
When 
is the total energy density and is conservational, i.e.
(4.10)
It is possible that 
although
. This is because in general, 
and 
here 
is the mass of a sort of s-particles (v-particles). If 
transforms into the energy density of v-particles and 
transforms into the energy density of s-particles in an interval of time
, there must be
(4.11)
Consequently, we have
(4.12)
where 
denotes the change of 
because of the transformation of 
and 
to each other.
According to this model, 
is a function of R Vg, 
and
, i.e.
Thus,
(4.13)
(4.14)
From (4.5)-(4.6) we have
(4.15)
This is because 
and 
determine only
, but do not determine
. It is obvious that (4.6) can be derived from (4.5) and (4.15). Considering the two equations (4.5)-(4.6), the equation determining 
(see section 6B), 
and 
or 
we can determine the five variables 
, 
and
, and further can determine 
by (4.10).
When 
and 
are low or 
and 
are high enough so that
, the transformation of 
and 
may be neglected. Thus, 
and
, i.e.
(4.16)
Pressure density is a function of masses of particles and temperature, i.e.
. Let 
and 
In the S-breaking, from 
we have
(4.17)
It is obvious that when
, the solution of (4.14) is
(4.18)
In general, 
In order to determine the pressure at a given temperature, we divide the particles into three sorts according to their masses. The first sort is composed of such particles whose masses 
satisfy 
here mp is the mass of a proton. The second sort of particles is composed of such particles whose masses 
satisfy
, here 
is the mass of an electron. The third sort is composed of photon-like particles whose masses 
satisfy 
and
. When 
When mp > T > me, 
and
. When
,
. Thus, we have
(4.19)
In the S-breaking,
. Considering all v-particles must be in v-SU(5) color singlets whose masses are not zero so that 
we have 
When 
and 
are so large that all masses may be neglected (
and
), from (4.16) we have
(4.20)
When 
. Letting
, we have
(4.21)
When 
and 
we have
(4.22)
where
.
(4.23)
It is obvious that when 
(4.21) has such solutions in the following form,
(4.24)
(4.25)
(4.26)
In contrast with the conventional theory, 
and 
and 
and 
are all possible in the present model, here
. For example, when temperature is so low that 
and 
we have 
and
.
5. Temperature Effect
The thermal equilibrium between the v-particles and the s-particles can be realized by only (2.10). The Higgs bosons 
and 
are hardly produced because their masses are all very big in low temperatures. Consequently, the interaction between the v-particles and the s-particles may be ignored so that there is no thermal equilibrium between the v-particles and the s-particles. Thus, when temperature is low, we should use two sorts of temperature 
and 
to describe the thermal equilibrium of v-matter and the thermal equilibrium of s-matter, respectively. Generally speaking,
. When temperature is high enough, e.g.
, the masses of the Higgs particles originating from (2.10) are small so that 
and 
can transform from one to another by (2.10). In the case, 
is possible.
5.1. Effective Potentials
Influence of finite temperature on the Higgs potential in the present model are consistent with the conventional theory. When the finite temperature effect is considered, the Higgs potential at zero-temperature becomes effective potential.
For short, we consider only 
and 
or
. When 
is considered as well, the following inferences are still qualitatively valid. From (2.8) we take
(5.1)
to ignore the terms proportional to 
to consider the temperature effect, the effective potential approximate to 1-loop in flat space is [25] -[27]
(5.2)
Considering the contributions of the expectation values 
and 
to
, and ignoring the terms irrelevant to 
we have
(5.3)
(5.4)
(5.5)
Similarly (5.1)-(5.5), from (2.9) we have
(5.6)
(5.7)
From (2.8) we take
(5.8)
ignoring the contributions of the Higgs fields and the fermion fields to one loop correction, and only considering the contribution of the gauge fields, when
, here 
is the Boltzmann constant (here
), we get the effective potential approximate to 1-loop in flat space at finite-temperature [25] -[27]
(5.9)
where 
and 
In general,
We take 
for simplicity. Here 
is a parameter at which the renormalization coupling-constant is defined.
Only considering the contribution of the expectation values of 
and 
to
, taking 
and ignoring the terms irrelevant with 
from (2.8) and (5.8)-(5.9), we have
(5.10)
(5.11)
It is easily seen from (5.10) that 
when 
Similarly, from (2.9) we have
(5.12)
(5.13)
When the masses of all particles may be neglected, pg = ρg/3 and 
. 
is the total number of spin states, and 
and 
are the total number of spin states of 
and the total number of spin states of a-femions, respectively. 
because s-particles and v-particles are symmetric. Considering (4.5)-(4.7), (4.9), (5.2) and (5.9) in the S-breaking, we have
(5.14)
(5.15)
(5.16)
(5.17)
(5.18)
(5.19)
5.2. The Critical Temperature Tφcr and Substable States in the S-Breaking
For short, we take 
in the following. We consider such a space-contracting stage in which 
and 
in the S-breaking. We will see that there are the critical temperatures 
and 
and
. For the effective potential, there still is the S-breaking, i.e. 
and
when 
and 
when 
by suitably choosing the parameters in the Higgs potential.
As mentioned before, there is the S-breaking in low temperatures. Talking 
for short, we have
(5.20)
(5.21)
(5.22)
(5.23)
Both 
and 
will rise as space contracts. We will see that 
is possible when
.
From (5.10) and (5.22) we can determine the minimum
. Let 
It is obviously that
. It is seen from (5.10) and (5.22) that there are absolute minimums when 
is low, i.e.
(5.24)
will decrease monotonously as 
increases and its lower limit is 
There is the critical temperature 
at which the minimum is degenerate, i.e.
(5.25)
(5.26)
(5.27)
(5.28)
(5.29)
where 
when 
There is the critical temperature 
at which
(5.30)
(5.31)
(5.32)
(5.33)
where 
when
.
Sum up, when 
and 
is the absolute minimum, i.e.
. There is such a 
that 
and
When 
and 
is a relative minimum and larger than
, i.e. there are substable states when
. When 
i.e. 
(5.34)
In the case, there is no relative minimum, as shown in Figure 1.
Analogously to that 
when
, it is seen from (5.12) that 
when
.
When 
i.e. 
We can get the masses of
, 
, 
and 
from (5.3), (5.6), (5.10), (5.12) and (5.20)-(5.23)
(5.35)
(5.36)
(5.37)
(5.38)
(5.39)
5.3. The Critical Temperature Tcr
It is easily seen from (5.3)-(5.7) that 
when 
and 
when 
In the S-breaking, 
when 
and 
Thus, from (5.3) we can determine the critical temperature 
of 
(5.40)
(5.41)
Both 
and 
will rise as space contracts. Let 
when 
We will see 
in the following. It is easily seen from (5.5) and (5.41) when 
increases from 
to 
that 
will decrease from 
to 
Figure 1. The states changes from A to E when Ts rises from T < Tφcr to Tφcr1.
6. Space Can Contract, But There Is No Singularity
On the basis of the cosmological principle, if there is the space-time singularity, it may be a result of space contraction. Thus, we discuss the contracting process. From the contracting process we will see that there is no space-time singularity in present model.
6.1. The Initial Condition and the Boundary Condition
We consider the contracting process of the universe after expansion in the S-breaking. It is seen from (3.15) that in the case,
. The initial condition is that at 
(6.1)
(6.2)
(6.3)
It is obvious that 
and
. Space will contract when t > tT = 0, because 
and 
We consider that the physical boundary condition of the Equations (5.14)-(5.15) should be
(6.4)
In contrast with the conventional theory, there are such solutions which satisfy the boundary condition. This implies that there is no singularity in the model.
There is no singularity in the model [19] [20] as well. This is because 
is changeable in the model [19] [20] . It is possible that the model [19] [20] is better.
6.2. Transformation of ρs and ρv from One to Another
When both 
and 
are low, the transformation of 
and 
may be neglected because the masses of the Higgs particles are all very large. Consequently, 
and 
are independent of each other. When both 
and 
rise because space contracts, as mentioned before, the masses of the Higgs particles originating from the couplings (2.8)-(2.10) will reduce. Thus, the transformation of the s-Higgs particles and the v-Higgs particles by (2.10) is striking.
We discuss the transformation of 
and 
follows.
Let 
originate from decay the Higgs particles as 
and 
originate from scattering of the Higgs particles as 
and
, and 
originate from scattering of the Higgs particles as 
and
, 
may be written as
(6.5)
6.2.1. ρg(tφcr) < 0 Is Possible When Tv ≤ Ts < Tφcr
In the initial stage, temperature is low, i.e. 
and
. Thus, 
breaks to 
when 
then to 
(
and
) when
. When 
symmetry holds (
).
and 
hold all the time. Here 
is such a temperature that 
when 
and 
when 
When
, the masses of the Higgs particles are large and the number of the Higgs particles is little. In the case,
. Thus, (4.18) holds so that the transformation of 
and 
may be neglected. When
, the s-particles must form celestial bodies with their large masses so that 
and
. However, the v-particles must be in 
color singlets so that
and
. It is obvious that the less the masses of the color singlets are, the larger 
is. Consequently,
(6.6)
and it is possible that 
when
, here 
when
. In the case, there are such solutions satisfying (6.4) (see the discussion below, 
in (6.13)-(6.15).
6.2.2. A Super-Heating Process
As mentioned in the preceding section, there are the substable states. Consequently, the universe will be in the substable states when 
rises from 
to
. The states changes from 
to 
when 
rises from 
to 
as shown by Figure 1. Thus, there still are 
and 
although 
. It is seen that the contracting process is a super-heating process when
. The substable states are not stable. A substable state can transit to the stable state with
.
when
, even if
.
(1) The effective masses of the Higgs particles and the transformation of 
and
.
If
, there must be 
and 
Thus, it is easily proved that there must be such a solution satisfying (6.4) in this case. Hence we consider such a contracting process when 
and
.
The masses containing the temperature effect are called effective masses.
The effective masses of 
and 
are important for 
and
. As mentioned before, 
and
, and both 
and 
hold when 
and 
. From (5.35)-(5.36) we have
(6.7)
It is seen from (6.7) that there must be such a 
that 
because 
in this case. Here 
when 
and
. 
and 
because 
Thus, there must be such a 
that 
and
, here 
and 
when
. Consequently, there are such decays 
and 
so that 
can transform to 
and 
decreases, even if 
Consequently, there must is
(6.8)
where 
is the number density of the i-th sort of a-Higgs particles, 
, and 
is the decay rate of the i-th sort of a-Higgs particles to the b-Higgs particles at the temperature
, 
and
. Here (6.5) is considered so that the factor 
emerges in (6.8).
(2) 
and 
when 
When both 
and 
are small (i.e. when 
and
), it is striking that such reactions as 
and 
etc. due to (2.10). Considering 
we have
(6.9)
(6.10)
where 
is a scattering cross section of a 
particle and a 
particle, 
is a relative velocities of a 
particle to a 
particle, and 
is a relative velocities of two 
particles. (6.5) is considered so that the factor 
emerges in (6.9)-(6.10). 
is the j-th sort of v-Higgs (s-Higgs) particles.
(3) 
when 
and 
It is obvious that the larger 
is, the less 
is and 
when
. The larger 
is, the less the masses of all Higgs particles originating from the couplings (2.8)-(2.10) are. Thus, the masses of all Higgs particles originating from the couplings (2.8)-(2.10) are very small when
.
The masses of all gauge bosons and fermions are zero when 
and
. Thus, the a-Higgs particles and the a-gauge bosons or the a-ferminos can transform from one to another by the 
couplings, here 
and
. Thus, the number density of the a-Higgs particles is large. The transformation of s-Higgs particles and the v-Higgs particles from one to another is striking when 
and
.
It is seen from the above mentioned and (6.8)-(6.10) that there must be
(6.11)
when 
and
. In the case, 
and 
must decrease so that 
decreases.
(4)
.
Because of the symmetry of the s-particles and the v-particles, 
here 
is the spinfreedom of the s-particles (the v-particles). When 
and
. In contrast with the contracting process
, 
causes 
and 
to decrease and the thermal equilibrium of s-matter and v-matter.
When 
and 
is very large and the masses of all particles are so small that they may be neglected. Consequently, 
so that
(6.12)
and there is such a moment 
at which
(6.13)
When 
and
, but 
and 
are very small so that they may be neglected. In this case, (4.5)-(4.6) and (4.12) reduce to
(16)
Thus, space will contract with a deceleration. Let 
when
, considering 
because 
decreases due to space contraction, we have
(6.15)
6.2.3. 
and 
in the S-Breaking for All Time
We see from the discussion above that 
can hold when 
This is because
is a continuous function of 
and 
If 
was arrived at some a time 
there must be such a time 
so that
. When
, the transformation of 
to 
must be striking so that 
decreases to 0 and 
will increase from 
to 
when
. Hence 
cannot occur so that the S-breaking can hold all the time for the effective potential 
6.3. There Is No Singularity of Space-Time in the Present Model
From (6.15) we see when
,
(6.16)
(6.17)
(6.18)
Here 
is considered. In the case, space can continue to contract, but there must be such a moment 
at which
(6.19)
(6.20)
This is because 
and 
when
. It is easily seen that
(6.21)
(6.22)
and 
are the highest temperature and the largest energy density in the universe, respectively. According to the present model, 
and 
must exist. We will see that 
is just the final moment 
of the S-world and the initial moment 
of the V-world as well, i.e. 
In summary, there are 
and 
which are finite for the contracting process
Because of the cosmological principle, all 
, 
and 
are finite. Consequently
, 
and 
must be finite. On the other hand, because of the cosmological principle, it is obvious that if there is no space contraction, the physical quantities must be finite as well. Substituting the finite 
or 
into the Einstein field equation
we see that 
and 
must be finite. Consequently, there is no singularity of space-time in the present model.
6.4. The Result above Is Not Contradictory to the Singularity Theorems
We first intuitively explain the reasons that there is no space-time singularity. It has been proved that there is space-time singularity under certain conditions [1] . These conditions fall into three categories. First, there is the requirement that gravity shall be attractive. Secondly, there is the requirement that there is enough matter present in some region to prevent anything escaping from that region. The third requirement is that there should be no causality violations.
Hawking considers it is a reasonable the first condition that 
and 
[1] . But this conjecture is not valid in the present model, because all 
or 
are possible. 
in general relativity is equivalent with 
in the present model. In contrast with 
is not the energymomintum tensor so that it does not satisfy the energy condition due to the conjecture 1. On the other hand, 
and 
can transform from one to other, especially when temperature
. Hence the premise of the singularity theorems does not hold so that the the singularity theorems are invalid in the present model.
As mentioned above, there must be 
when
. It is seen that 
does not only stop increasing, but also decreases from 
to 
and finally to 
when 
Hence the second condition of the singularity theorem is violated.
The key of non-singularity is conjecture 1, i.e. 
when 
and 
and 
can transform from one to another.
We explain the reasons that there is no space-time singularity from the Hawking theorem as follows. S.W. Hawking has proven the following theorem [1] .
The following three conditions cannot all hold:
(a) every inextendible non-spacelike geodesic contains a pair of conjugate point;
(b) the chronology condition holds on 
(c) there is an achronal set 
such that 
or 
is compact.
The alternative version of the theorem can obtained by the following two propositions.
Proposition 1 [1] :
If 
and if at some point 
the tidal force 
is non-zero, there will be values 
and 
such that 
and 
will be conjugate along
, providing that 
can be extended to these values.
Proposition 2 [1] :
If 
everywhere and if at 
is non-zero, there will be 
and 
such that 
and 
will be conjugate along 
provided that 
can be extended to these values.
An alternative version of the above theorem is as following.
Space-time 
is not timelike and null geodesically complete if:
(1) 
for every non-spacelike vector 
(2) The generic condition is satisfied, i.e. every non-spacelike geodesic contains a point at which
, where 
is the tangent vector to the geodesic.
(3) The chronology condition holds on 
(i.e. there are no closed timelike curves).
(4) There exists at least one of the following:
(A) a compact achronal set without edge(B) a closed trapped surface(C) a point 
such that on every past (or every future) null geodesic from 
the divergence 
of the null geodesics from 
becomes negative (i.e. the null geodesics from 
are focussed by the matter or curvature and start to reconverge).
In fact, 
is determined by the gravitational energy-momentum tensor
. According to the conventional theory, 
so that the above theorem holds.
In contrast with the conventional theory, according to conjecture 1,
(6.23)
, 
and 
are all possible. Thus, although the strong energy condition still holds, i.e.
(6.24)
the conditions of propositions 1 and 2 and condition (1) no longer hold, because the gravitational mass density 
determines 
and 
Hence (a) and (c) do not hold, but (b) still holds, and 
is timelike and null geodesically complete.
7. Space Inflation
7.1. Space Inflation
When 
and 
We call such a state in which 
the most symmetric state. In this state the 
symmetry holds strictly and the s-particles and the v-particles are symmetric.
When 
and
. Hence space will expand when
.
Consider the initial stage of expansion in which 
and
. 
is small due to
. On the other hand, 
because 
and the s-particles and the v-particles are symmetric. 
and 
because (6.5)-(6.6), 
and
. Hence
. In the case, (4.5)-(4.6) and (4.15) become
(7.1)
(7.2)
(7.3)
because 
and
. It is seen from (7.1)-(7.3) that there is such a time 
at which 
Furthermore, 
because 
and
. Thus, when 
(7.1)-(7.3) reduce to
(7.4)
Consequently, space inflation must occur
(7.5)
(7.6)
(7.7)
7.2. The Process of Space Inflation
Supposing 
and 
for 
and considering
, from (5.40) we can estimate
,
(7.8)
We may take
, because 
when 
and
.
The temperature will strikingly decrease in the process of inflation, but the potential energy
cannot decrease to 
at once. This is a super-cooling process. We can get the expecting results by suitably choosing the parameters in (2.8)-(2.10). In order to estimate 
taking 
from (7.8) we have
(7.9)
is larger than the temperature corresponding to GUT. Taking 
and
we have 
If the duration of the super-cooling state is 
Rcr will increase 
times. The result is consistent with the Guth’s inflation model [28] .
If there is no v-matter, because of contraction by gravitation, the world would become a thermal-equilibrating singular point, i.e., the world would be in the hot death state. As seen, it is necessary that there are both s-matter and v-matter and both the S-breaking and the V-breaking.
8. Evolving Process of Space after Inflation
8.1. The Reheating Process
After inflation, the temperature must sharply descend. In this case, it is easily seen that the most symmetric state with 
is no longer stable and must decay into such a state with 
This is the reheating process. Either of the S-breaking and the V-breaking can come into being, because s-matter and v-matter are completely symmetric at
. Letting the V-breaking comes into being
then the symmetry breaking is 
and 
symmetry is still kept all time. After the reheating process, when temperature is low, considering (3.15) we have
(8.1)
(8.2)
After reheating, 
must first transform into v-energy by (2.9) and the SUV(5) couplings and into s-energy by (2.10). Letting 
transform the v-energy, then 
transforms the s-energy. It is necessary
. There is 
before the reheating. Thus, after reheating, we have
(8.3)
8.2. The Change of Mass Densities
Let 
be such a temperature that particles exist in the 
plasma form when
, and particles exist in the form of 
color singlets when
. After reheating process, in the initial stage, both 
and 
are high and all particles must exist in the plasma form when
. Thus, the masses of particles may be neglected so that
. After temperature descends further so that
, s-particles will form 
color singlets whose masses are all non-zero. Thus, there is no s-photon, i.e.
. The 
color singlets cannot form any clustering and their masses are all small. Let 
is the largest mass of the stable 
color singlets, then we may suppose 
here 
is the mass of a proton. However v-particles will exist in the forms of nucleons, leptons and photons, and can form galaxies in low temperatures. Consequently, in the V-breaking,
(8.4)
Let the reheating process ends at
. Considering
, we suppose
. From (8.3) we have
(8.5)
After reheating process ends, temperature is low, 
and all masses of the Higgs particles are large enough so that the transformation 
and 
may be neglected. Thus,
.
As mentioned in section 4 (see (4.21)-(4.26)), the evolving laws of 
and 
as space contraction are different from each other. For simplicity, we do not differentiate 
and 
for a time. Thus, neglecting 
and
, considering 
and 
in the V-breaking, from (4.24)-(4.26) (
in (4.26)) and (8.5) we have
(8.6)
(8.7)
where both 
and 
are constants. From (4.8), (8.3) and (8.6)-(8.7), (4.5)-(4.6) is reduced to
(8.8)
(8.9)
As mentioned in section 3, 
may be neglected when 
in the V-breaking. Thus we neglects 
for a time in the following.
We discuss (8.8)-(8.9) as follows.
If 
when
, 
and
, i.e. space expands with a deceleration; when
, 
and
; when 
and 
i.e.
space expands with an acceleration. In the process, 
increases from 
to 
when 
then 
decreases from 
to 
If 
when
, 
and
; when
,
; In the case, space can be static.
If 
when
, 
and
; when R =
R2, 
and
. In the case, space will begin to contract.
The first case is consistent with observations. A computation in detail is the same as that of Ref.
.
Even 
and 
are considered
the above conclusions still hold qualitatively.
8.3. To Determine a(t)
Letting 
and 
and considering
(8.10)
(8.11)
we rewrite (8.6) as
(8.12)
From (8.12) we have
(8.13)
If 
is taken as
(8.14)
(8.15)
where 
and 
Taking 
[18] . and h = 0.8, we get 
is shown by the curve 
in the Figure 2 and describes evolution of the universe from 
ago to now. Taking 
we get the 
which is shown by the curve B in the Figure 2 and describes evolution of the cosmos from 
ago to now. Provided 
which is equivalent to
, we can get a curve of 
which describes evolution of the cosmological scale. The curves A and B show that when the parameters alter in a definte scope, the qualitative features of the evolving curves are changeless, but their concrete-changing forms are differnt from each other. Thus, the parameters in the model should be determined based on astronomical observations.
From the two curves we see that the cosmos must undergo a period in which space expands with a deceleration in the past, and undergo the present period in which space expands with an acceleration.
Figure 2. The curve A describes evolution of a(t) from 14 × 109 yr ago to now; The curve B describes evolution of a(t) from 13.7 × 109 yr ago to now. The starting point of curve A is different from that of curve B.
It should be noted that 
in the V-breaking, but here
. Neglected 
noting 
and taking
(8.16)
(8.17)
(8.18)
we can reduce (8.15) to
(8.19)
Replacing 
by 
because 
and and ignoring 
we see (8.19) to be the same as the corresponding formula (3.44) in Ref. [18] .
8.4. The Relation between Redshift and Luminosity Distance
From (8.12) and the RW metric we have
(8.20)
(8.21)
where 
is the redshift caused by 
increasing.
Considering 
in (8.21) corresponds to 
in (3.81) in Ref. [18] and 
we see that (8.21) is consistent with (3.81) in Ref. [18] .
Ignoring 
taking
, we reduce (8.21) to
(8.22)
which is consistent with (3.78) in Ref. [18] . Approximating to 
and
, we obtain
(8.23)
Taking 
and 
[18] and h = 0.8, from (8.22) we get the 
relation which is shown by the curve 
in the Figure 3. Taking 
and
we get the 
relation which is shown by the curve 
in the Figure 3.
9. After Expansion with an Acceleration, Space Expands with a Deceleration, Then Comes to Static and Finally Begin to Contract.
As mentioned before, the evolving laws of 
and 
as space contracts or expands are different from each other. After space expands with an acceleration, we should consider the difference between 
and
.
When R is large enough, 
so that 
and 
may be neglected. Thus, considering 
in low temperatures, neglecting 
and
, we can reduce (4.5)-(4.6) to
(9.1)
(9.2)
where 
and
. As mentioned in section 3, 
is so small that it may be neglected when 
in the V-breaking. It is seen from (9.2) that 
changes from 
to 
and finally 
as space expands. It is seen from (9.1) that there must be 
so that 
when
. Space will begin to contract when 
because 
and 
at
. It is seen that after space expands with an acceleration, it will expands with a deceleration, then comes to static, and final begin contracts. This is different from the conventional theory and the model [19] [20] .
When 
is large enough, 
and 
may be neglected. Thus, from (9.1) we have
(9.3)
To sum up, according to the present model, the universe can expand from 
to
, and then contract to
; Both 
and 
are finite. The universe can be in the S-breaking, and can be in the V-breaking as well; The S-breaking can transform to the V-breaking after space contracts to 
and vice versa.
10. Existing and Distribution Forms of SUS(5) Color Singlets
In the V-breaking, all s-gauge particles and s-fermions are massless. When the temperature
, all s-particles must exist in plasma form. When
, all s-particles will exist in s-SU(5) color-singlets (conjecture 2). Let
, 
, 
, 
, 
denote the 5 sorts of colors. A component of 
representation carries color
, 
, 
A component of 
representation carries color 
A gauge boson carries colors 
There are the following sorts of the s-SU(5) color-singlets.
2-fermion states: 
or
, 
3-fermion states: 
or 
4-fermion states:
. 5-fermion states: 
or 
Gauge boson single-states: 
or 
etc.
Fermion-gauge boson singlets:
,
etc.
The masses of all color singlets are non-zero, hence
. The fermions with the spin 
and the least mass are stable, and the bosons with the spin 
and the least mass are stable as well. This is because there is no the electroweak interaction among 
color single states so that they cannot decay. Of course, there are the s-antiparticles corresponding to the s-colour singlets above as well.
There is no interaction among the 
color singlets, because 
is a simple group. There are interaction among the 
color singlets by exchanging the 
color single states. The interaction radius must be very small because the masses of all 
color singlets are non-zero. Thus, the interaction may be neglected so that we can approximately regard the 
color singlets as ideal gas without collision. The ideal gas has the effect of free flux damping for clustering. Consequently, the 
color single states cannot form clustering and must distribute loosely in space, and their decoupling temperature must be very high so that their relative velocities are large and invariant. But they can form s-superclusterings, because there is the gravitation among them and there is repulsion between s-matter and v-matter. The superclusterings are similar to neutrino superclusterings and are huge voids for v-observers.
11. New Predictions, an Inference, and There Is No Restriction for 
11.1. New Predictions
11.1.1. The Essence and Characters of Huge Voids
It is possible that Huge voids are not empty and are equivalent to huge concave lenses. The density of hydrogen inside the huge voids is more less than that predicted by the conventional theory.
Based on above mentioned, we consider, the huge voids for the v-observers are, in fact, superclusters of the 
color singlets. The huge v-voids are not empty. There must be the 
color singlets inside them, and
, 
, and
. Here 
and 
denote the densities inside the huge v-voids, and 
and 
denote the densities outside the huge v-voids. The characters of such a huge v-void are as follows:
A. A v-void must be huge, because there is no other interaction among the s-SU(5) color singlets except the gravitation and the masses of the s-SU(5) color singlets are very small.
B. When v-photons pass through such a huge v-void, the v-photons must suffer repulsion coming from s-matter inside the huge void and are scattered by the v-void as they pass through a huge concave lens. Consequently, the galaxies behind the huge v-void seem to be darker and more remote. Hence the huge voids are equivalent to huge concave lenses.
C. Both density of matter and density of dark matter in the huge voids must be more lower than those predicted by the conventional theory. Consequently, the densities of hydrogen and helium inside the huge voids must be more less than that predicted by the conventional theory.
The predict can be confirmed or negated by the observation of hydrogen distribution.
This is a decisive prediction which distinguishes the present model from other models.
11.1.2. The Gravity between Two Galaxies Whose Distance Is Long Enough
There must be s-superclusterings between two v-galaxies when both distance is long enough, hence the gravity between the two v-galaxies must be less than that predicted by the conventional theory due to the repulsion between s-matter and v-matter. When the distance between two v-galaxies is small, the gravitation is not influenced by s-matter, because 
must be small when 
is big.
11.1.3. A Black Hole with Its Mass and Density Big Enough Will Transform into a White Hole
Letting there be a v-black hole with its mass and density to be so big that its temperature can arrive at 
because the black hole contracts by its self-gravitation, then the expectation values of the Higgs fields inside the v-black hole will change from 
and 
into
. Consequently, inflation must occur. After inflation, the most symmetric state 
will transit into the V-breaking. Thus, the energy of the black hole must transform into both the v-energy and the s-energy. Thus, a v-observer will find that the black hole disappears and a white hole appears.
In the process, a part of v-energy transforms into v-energy and the other part transforms into s-energy. A v-observer will consider the energy not to be conservational because he cannot detect s-matter except by repulsion. The transformation of black holes is different from the Hawking radiation. This is the transformation of the vacuum expectation values of the Higgs fields. There is no contradiction between the transformation and the Hawking radiation or another quantum effect, because both describe different processes and based on different conditions. According to the present model, there still are the Hawking radiation or other quantum effects of black holes. In fact, the universe is just a huge black hold. The universe can transform from the S-breaking into the V-breaking because of its contraction. This transformation is not quantum effects.
11.2. An inference: 
Although 
The effective cosmological constant 
The conventional theory can explain evolution with a small 
but 
Consequently, the issue of the cosmological constant appears.
can be obtained by some supersymmetric model, but it is not a necessary result. On the other hand, the particles predicted by the supersymmetric theory have not been found, although their masses are not large.
is a necessary result of our quantum field theory without divergence [13] -[15] . In this theory, 
is naturally obtained without normal order of operators, there is no divergence of loop corrections, and dark matter which can form dark galaxies is predicted [16] [17] . But the model does not explain the evolution of the universe.
As mention above, the present model can explain evolution of the universe without 
hence
(11.1)
Applying the conventional quantum field theory to the present model, we have 
here 
is the energy density of the vacuum state. According to the conjecture 1, s-particles and v-particles are symmetric. Hence both ground states must be symmetric as well. Hence
(11.2)
According to conjecture 1, 
when
. Consequently, although
(11.3)
we have still
(11.4)
(11.5)
Here 
is the Einstein cosmological constant. This is a direct inference of the present model, and independent of a quantum field theory.
Because of (11.4), for the vacuum state in the S-breaking or the V-breaking, the Einstein field equation is reduced to
(11.6)
This is a reasonable result.
11.3. There Is No Restriction for 
The problem of total energy conservation in the general relativity is unsolved up to now. This is because tensors at different points cannot be summed up. On the other hand, according to the Einstein equation,
. In contrast with this result, according to the present model, we have (2.22). This result (2.22) implies that there is no restriction for 
or
. The dominant energy condition and the positive energy theorem are not applicable to 
Whether does 
holds? We will discuss the problem in another paper.
12. Conclusions
A new conjecture is proposed that there are s-matter and v-matter which are symmetric, whose gravitational masses are opposite to each other, although whose masses are all positive. Both can transform from one to another when temperature
. Consequently there is no singularity in the model. The cosmological constant 
is determined although the energy density of the vacuum state is still very large. A formula is derived which well describes the relation between a luminosity distance and the redshift corresponding to it.
The conjecture are not in contradiction with given experiments and astronomical observations up to now, although the conjecture violates the equivalence principle.
There are two sorts of breaking modes, i.e. the S-breaking and the V-breaking. In the V-breaking, 
is broken to 
and finally 
and 
is kept all time
Consequently, v-particles get their masses and form v-atoms, v-observers and v-galaxies etc., while s-gauge bosons and s-fermions are still massless and must form 
color-single states when 
There is no interaction among the 
color-single states except the gravitation, because 
group is a simple group. Hence they must distribute loosely in space, cannot be observed and can cause space to expand with an acceleration. Thus, v-matter is identified with conventional matter (include dark matter) and s-matter is similar to the dark energy. But in contrast with the dark energy, the gravitational mass of s-matter is negative in the V-breaking.
There are the critical temperature 
the highest temperatures 
and the least scale 
in this model. Hence it is impossible that the Plank temperature, length and time are arrived.
Based on the present model
the space evolving process is as follows. Firstly, in the S-breaking, 
hence space contracts and 
and 
rises
When 
the transformation of 
and 
from one to another is striking so that 
is possible. Hence there are such solutions of the evolution equations which satisfy the physical boundary condition, i.e. 
when
. When
, 
the 
symmetry (the highest symmetry) is kept, and
. When space contracts further, 
arrives the least scale 
and 
arrives the highest temperature
. Then space expands and 
descends to 
so that 
and inflation must occur. After the inflation, the phase transition of the vacuum (the reheating process) occurs. After the reheating process, this state with the highest symmetry transits to the state with the V-breaking. In the V-breaking, the evolving process of space is as follows. Space firstly expands with a deceleration because
; Secondly, space expands with an acceleration because 
and
; Thirdly, space expands with a deceleration, and then comes to static; Finally, space begin contract.
It is seen that according to the present model, the universe can expand from 
to
, and then contract from 
to
; Both 
and 
are finite. The universe can be in the S-breaking, and can be in the V-breaking as well; The S-breaking can transform to the V-breaking after space contracts to 
and vice versa.
Three new predicts have been given.
Huge v-voids in the V-breaking are not empty, but are superclusterings of s-particles. The huge voids are equivalent to huge concave lens. The densities of hydrogen helium in the huge voids predicted by the present model must be much less than that predicted by the conventional theory.
The gravitation between two galaxies whose distance is long enough will be less than that predicted by the conventional theory.
It is possible that a v-black hole with its big enough mass and density can transform into a huge white hole by its self-gravitation.
I am very grateful to professor Zhao Zhan-yue, professor Wu Zhao-yan, professor Zheng Zhi-peng and professor Zhao Zheng-guo for their helpful discussions and best support. I am very grateful to professor Liu Yun-zuo, professor Lu Jingbin, doctor Yang Dong and doctor Ma Keyan for their helpful discussions and help in the manuscript.