A Cosmological Model without Singularity Based on RW Metric (1)

A new conjecture is proposed that there are two sorts of matter called s-matter and v-matter which are symmetric, whose masses are positive, but whose gravitational masses are opposite to each other. Based on the conjecture and the SUS(5) × SUV(5) gauge group, a cosmological model has been constructed and the following inferences have been derived. There are two sorts of symmetry breaking called V-breaking and S-breaking. In the V-breaking, SUV(5) breaks finally to SUV(3) × UV(1) so that v-particles get their masses and form v-atoms and v-galaxies etc., while SUS(5) still holds so that s-fermions and s-gauge bosons are massless and form SUS(5) color-singlets. There is no interaction among the SUS(5) color-singlets except gravitation so that they distribute loosely in space, cannot be observed, and cause space to expand with an acceleration. Evolution of the universe is explained. There is no space-time singularity. There are the highest temperature and the least scale in the universe. It is impossible that the Plank temperature and length are arrived. A formula is obtained which describes the relation between a luminous distance and its redshift. A huge void is not empty, and is equivalent to a huge concave lens. The densities of hydrogen in the huge voids 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. A black hole with its big enough mass will transform into a white hole.


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 ) (CMBR , 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 < l p .
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 l p , time ( ) ( ) 1. 4 10 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 P l , P t and P T 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 max P T T  and the least scale 0 R > (see later).Consequently, it is impossible that P l , P t and P T are arrived, because the transformation of the S-breaking and V-breaking from one to another at the highest temperature max T .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 pretation is un-satisfactory.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  and the singularity issues imply that the conventional theory is incomplete.In some super-symmetric models, 0 vac ρ = 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 super-symmetric quantum field theory, 0 vac ρ = 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, 0 vac ρ = is not a necessary condition of , 0 g vac ρ = .We will see that although vac ρ is divergent according to the conventional quantum field theory, we have still , 0 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 , gs gv ρ ρ = − here g ρ denotes a gravitational mass density.The con- ditions 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: 1.There is no space-time singularity in this model.R is not small.In general, the radius of a local inertial system is so large, P r l  , that the quantum effects corresponding to P l may be neglected.3. 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 T rises.When T arrives the critical temperature cr T , the universe is in the most symmetric state with ( ) ( ) symmetry.When space continues to contract so that T arrives the highest temperature max T , 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.
4. The relation between the optical distance and the redshift is derived from the present model.It is consistent with the observations up to now.
5. 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.
6. Three predictions are given.7. Primordial nucleosynthesis and cosmic microwave background radiation are explained.8. Dark energy is explained as s-matter when the universe is in the V-breaking.In contrast with the dark energy, 0 sg s ρ ρ = − < in the V-breaking.

9.
, , 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 T µν "; Section 12 is "Conclusions".

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 pl T , all particles exist in ( ) 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.
A .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.
B. The basis of the second possibility is conjecture 1.We explain it in detail as follows: 1.It must be emphasized that there is no negative mass or negative probability in the present model at all. 2. The observation basis of conjecture 1 is that space expands with an acceleration now.One of the two sorts of matter must exist in ( )

SU
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).
3. 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 cr T T < .Thus the coupling constant Λ 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. (1)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]. (2)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 min R (see later).We discuss the case in the present paper.The RW metric is applicable to the case.
4 .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.
5. s-matter and v-matter are no longer symmetric after the symmetry breaking.

5
V SU 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 ( ) 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.
6.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, v ρ must be very small when s ρ 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.

7.
The equivalence principle still strictly holds for the s-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. 0, while the gravitational masses of v-matter must be negative relatively to s-matter, i.e. 0, vg v m m = − < 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). 8.When temperature is high enough, the expectation values of Higgs fields are small so that all masses of Higgs particles are small.Thus, s ρ and v ρ 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 ( ) V SU 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.

Action
The breaking mode of the symmetry is only one of the S-breaking and the V-breaking due to ( ) 2.10 .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, S I in the S-breaking and V I in the V-breaking.Of course, only one of both S I and V I 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 S I and V I are necessary.Because of conjecture , , ; , , , ≡ Ω Φ ≡ Ω Φ where the meanings of the symbols are as follows: ( ) , and is the scalar curvature.Here α and β are two parameters, may be called "gravitation charges", and are finally taken as 1 is the Lagrangian density of all s-fields (v-fields) and their couplings of the ( ) GUT except the Higgs potentials , s Ψ and v Ψ represents all s-fields and all v-fields, respectively.For a boson field, Ψ denotes its covariant derivative as well.Both s  and s  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 s v  and S V  .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 ( ) where a Ω , ( ) and a χ are respectively 1 , 24 and 5 dimensional representations of the ( ) Here the couplings of a Φ and a χ 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, , Λ p and q in (2.10) must be positive.We do not consider the terms coupling to curvature scalar, e.g.

R
ξ Ω , for a time.In fact, ( ) ξ Ω − Ω  when temperature T is high enough due to the symmetry between s-matter and v-matter.

Energy-Momentum Tensors and Field Equations
By the conventional method, from ( ) .
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 , As mentioned before, α and β in (2.3)-(2.4)may be regarded as the gravitation charges.The the gravitation charges of , s T µν v T µν and sv g V µν 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 α and β from the definition of T µν by the operator ( ).
The operator is the only difference between the definition of T µν 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.0 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.

The Difference of Motion Equations of a v-Particle and a s-Particle in the Same Gravitational Field
From (2.16) we have ( ) in the S-breaking so that the acceleration of vI m is opposite to that of vg m , we get the motion equation of vI m in the gravitational field with its strength µ αν Γ to be ( ) 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 Considering the Newtonian approximation, i.e. the velocity of a particle is low ( ) ) and static ( ) Let ϕ is caused by a static and spheral-symmetric s-object with its mass M. In the S-breaking, .GM r ϕ = − Thus, from (2.24)-(2.25)and (2.30) we get ( ) Let ϕ is caused by a static and spheral-symmetric v-object with its mass M. In the S-breaking, GM r ϕ = because conjecture 1.Thus, from (2.24)-(2.25)and (2.30) we get ( ) 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 S V  and s v  .

Spontaneous Symmetry Breaking
Ignoring the couplings of s Φ and s χ and suitably choosing the parameters of the Higgs potential, an- alogously 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 Ignoring the contributions of s Φ and s χ to 0 0 , s Ω at the zero-temperature we get ( ) 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 s Φ -particles and the s χ -particles in the S-breaking are respectively ( ) We can choose such parameters that ( ) We take 1 V to be so small that it may be neglected when 0

Evolution Equations of R in RW Metric
As is well known, based on the RW metric metric, In the present model, we take 1. k = − Taking 0 k = or 1, we can get the results similar to those when 1 k = − .We will discuss the two cases in the following paper.In fact, it is possible that k is changeable with the gravitational mass density g ρ .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) where a U µ is a 4-velocity and a s = or v.In comoving coordinates in a comoving coordinates.( ) can be written as substituting (4.2)-(4.4)and the RW metric in (4.1) into (2.16),we get the evolution equations 2 2 , 8π 3, ( ) In the S-breaking, 0 , , .
Comparing (4.5)-(4.6)with the Friedmann equations, we see that provided ρ , p and V in the Friedmann equations are replaced by g ρ , g p and g V , (4.5)-(4.6)are obtained.

Evolution Equation of ρ g
In contrast with the conventional theory, it is possible that 0 . This is because s ρ and v ρ can transform from one to another by (2.10), especially when temperature is high enough (see section 6B).
is the total energy density and is conservational, i.e. ( Consequently, we have According to this model, g ρ is a function of R V g , ( ) , , , .
Pressure density is a function of masses of particles and temperature, i.e.
( ) In the S-breaking, from ( ) It is obvious that when 0 In general, .), from (4.16) we have ( ) where mg Mg lg It is obvious that when 0, g V   (4.21) has such solutions in the following form, In contrast with the conventional theory, 0,

Temperature Effect
The thermal equilibrium between the v-particles and the s-particles can be realized by only (2.10).The Higgs bosons s Ω and v Ω 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 v T and s T to describe the thermal equilibrium of v-matter and the thermal equilibrium of s-matter, respectively.Generally speaking, s v T T ≠ .When temperature is high enough, e.g.s cr T T  , the masses of the Higgs particles originating from (2.10) are small so that s ρ and v ρ can transform from one to another by (2.10).In the case, v s T T = is possible.

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 a Ω and , χ is considered as well, the following inferences are still qualitatively valid.From (2.8) we take ( ) to ignore the terms proportional to ( ) to consider the temperature effect, the effective potential approximate to 1-loop in flat space is [25]- [27] ( ) ( ) Considering the contributions of the expectation values ( ) , and ignoring the terms irrelevant to , s Ω we have , , , Similarly (5.1)-(5.5),from (2.9) we have 4 , , , From (2.8) we take 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 s kT ϕ  , here k is the Boltzmann constant (here 1 k = ), we get the effective potential approximate to 1-loop in flat space at finite-temperature [25]- [27] ( ) ( ) ( ) It is easily seen from (5.10) that , , , 0.
When the masses of all particles may be neglected, p g = ρ g /3 and ( ) π 30 , , .

The Critical Temperature T φcr and Substable States in the S-Breaking
For short, we take x will decrease monotonously as 2 s T increases and its lower limit is 1.

−
There is the critical temperature cr T ϕ at which the minimum is degenerate, i.

T T T ϕ ϕ
= ≡ There is the critical temperature where is the absolute minimum, i.e.

( )
, ,min , 0 , In the case, there is no relative minimum, as shown in Figure 1.
Analogously to that 0 , it is seen from (5.12) that 0

The Critical Temperature T cr
It is easily seen from ( 5.3)-(5.7)that 0

T T µ ≤
In the S-breaking, 0 s ϕ υ = when

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.

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, 1 g V V = − .The initial condition is that at 0, It is obvious that 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 k is changeable in the model [19] [20].It is possible that the model [19] [20] is better.

Transformation of ρ s and ρ v from One to Another
When both s T and v T are low, the transformation of s ρ and v ρ may be neglected because the masses of the Higgs particles are all very large.Consequently, s ρ and v ρ are independent of each other.When both s T and v T 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 s ρ and v ρ follows.Let g tr s tr v tr g tr g tr g tr

ρ g (t φcr ) < 0 Is Possible When T v ≤ T s < T φcr
In the initial stage, temperature is low, i.e. 0 , and 0 v ω = .Thus, ( ) ( ) ( ) . However, the v-particles must be in ( ) color singlets so that ( ) ( ) It is obvious that the less the masses of the color singlets are, the larger v l is.Consequently, and it is possible that ( ) ( ) T T ϕ = .In the case, there are such solutions satisfying (6.4) (see the discussion below, 2 1 t t → in (6.13)-(6.15).

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 s T rises from cr T ϕ to  ( ) (1) The effective masses of the Higgs particles and the transformation of s ρ and v ρ . If , there must be , ( ) , 0.
Thus, it is easily proved that there must be such a solution satisfying (6.4) It is seen from (6.7) that there must be such a 1 s T that ( ) , 0 ( ) Thus, there must be such a 2 s T that , where ia n is the number density of the i-th sort of a-Higgs particles, , T T ϕ > ), it is striking that such reactions as , T T ϕ > s ρ is very large and the masses of all particles are so small that they may be neglected.Consequently, and there is such a moment 2 t at which When , m ω are very small so that they may be neglected.In this case, (4.5)-(4.6)and (4.12) reduce to 1 , , 0.
Thus, space will contract with a deceleration.Let

T T T ≥ ≥
This is because ( ) m Ω < was arrived at some a time , t′ there must be such a time t t ′′ ′ < so that be striking so that ( ) Hence 0 v m Ω < cannot occur so that the S-breaking can hold all the time for the effective potential , .
g eff V

There Is No Singularity of Space-Time in the Present Model
From (6.15) we see when Here 3 is considered.In the case, space can continue to contract, but there must be such a moment This is because 0 T µν 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 Sg T µν or Vg T µν into the Einstein field equation , we see that R µν and g µν must be finite.Consequently, there is no singularity of space-time in the present model.

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 0 T µν in the present model.In contrast with , T µν g T µν is not the energy- momintum tensor so that it does not satisfy the energy condition due to the conjecture 1.On the other hand, s E and v E can transform from one to other, especially when temperature cr T T  .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 0 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 ; µ (c) there is an achronal set T such that ( ) The alternative version of the theorem can obtained by the following two propositions.An alternative version of the above theorem is as following.
Space-time ( ) , g µ is not timelike and null geodesically complete if: (1) 0 ab a b R K K ≥ for every non-spacelike vector .K (2) The generic condition is satisfied, i.e. every non-spacelike geodesic contains a point at which where K is the tangent vector to the geodesic.
(3) The chronology condition holds on µ (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 p such that on every past (or every future) null geodesic from p the divergence θ of the null geodesics from p becomes negative (i.e. the null geodesics from p are focussed by the matter or curvature and start to reconverge).
In fact, ab R is determined by the gravitational energy-momentum tensor gab T .According to the conven- tional theory, gab T T = so that the above theorem holds.In contrast with the conventional theory, according to conjecture 1, ( ) ( ) the conditions of propositions 1 and 2 and condition (1) no longer hold, because the gravitational mass density g ρ determines R µν and .
Hence (a) and (c) do not hold, but (b) still holds, and µ is timelike and null geodesically complete.

Space Inflation
We call such a state in which 0 the most symmetric state.In this state the ( ) ( ) symmetry holds strictly and the s-particles and the v-particles are symmetric.
. Hence space will expand when F t t > .Consider the initial stage of expansion in which min 0, . On the other hand, Consequently, space inflation must occur ( ) ( ) We may take ( ) The temperature will strikingly decrease in the process of inflation, but the potential energy ( ) 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 0 , 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.

The Reheating Process
After inflation, the temperature must sharply descend.In this case, it is easily seen that the most symmetric state with 0 is no longer stable and must decay into such a state with min .V 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 A t t  .Letting the V-breaking comes into being , then the symmetry breaking is ( ) symmetry is still kept all time.After the reheating process, when temperature is low, considering (3.15) we have ( ) After reheating, V ∆ must first transform into v-energy by (2.9) and the SU V (5) couplings and into s-energy by (2.10).Letting V ∆ κ transform the v-energy, then ( ) before the reheating.Thus, after reheating, we have ( )

The Change of Mass Densities
Let pl T be such a temperature that particles exist in the ( ) , As mentioned in section 3, 1 V may be neglected when 0 v ρ > in the V-breaking.Thus we neglects 1 V for a time in the following.
( ) space expands with an acceleration.In the process, R  increases from ; In the case, space can be static.
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. [10] .Even s χ and v χ are considered , the above conclusions still hold qualitatively.

To Determine a(t)
Letting 0 , a R R = ( ) we rewrite (8.6) as ( ) ( ) From (8.12) we have  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.+ Ω (8.17) we can reduce (8.15) to Replacing mg Ω by ( ) and and ignoring 0 , vγ Ω we see (8.19) to be the same as the corresponding formula (3.44) in Ref. [18].

The Relation between Redshift and Luminosity Distance
From (8.12) and the RW metric we have which is consistent with (3.78) in Ref. [18].Approximating to 1 0 m Ω and 2 z , we obtain

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 M ρ and m ρ as space contracts or expands are different from each other.After space expands with an acceleration, we should consider the difference between M ρ and m ρ ., .As mentioned in section 3, 1 V is so small that it may be neglected when 0 and finally 0 <  as space expands.It is seen from (9.1) that there must be max R so that max 0 R =  when max t t = .Space will begin to contract when max , t t > because max 0 R =  and 0 R <  at max t .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 R is large enough, vM C R and 2 sm C R may be neglected.Thus, from (9.1) we have To sum up, according to the present model, the universe can expand from min R to max R , and then contract to min R ; Both max R and min R 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 min R and vice versa.

Existing and Distribution Forms of SU S (5) Color Singlets
In the V-breaking, all s-gauge particles and s-fermions are massless.The masses of all color singlets are non-zero, hence 0 sγ ρ = .The fermions with the spin 1 2 s = and the least mass are stable, and the bosons with the spin 0 s = and the least mass are stable as well.This is because there is no the electroweak interaction among ( )

SU
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 ( )

SU
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.ρ 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.

New Predictions, an
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.

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 s ρ must be small when v ρ is big.

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 2 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 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.Consequently, the issue of the cosmological constant appears.0 vac ρ = 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.0 vac ρ = 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 , Here λ is the Einstein cosmological constant.This is a direct inference of the present model, and in- dependent 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

+
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, ; 0 T µν µ = .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 ;

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 cr T T  .Consequently there is no singularity in the model.The cosmological constant 0 eff λ λ = = 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, T T < 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 , cr T the highest temperatures max , T and the least scale min 0 R > 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, 0, T descends to cr T so that 0 g ρ  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 0 g v s ρ ρ ρ = − > ; Secondly, space expands with an acceleration because 0 g ρ < and 1 k = − ; 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 min R to min R , and then contract from min R to min R ; Both min R and min R 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 min R 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.

λ
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, density of the vacuum state.Hence eff cannot be derived from basic theories[7] [8], and vac eff ρ λ  .Hence the inter- Because of conjecture 1, there are two sorts of symmetry breaking which are called S-breaking in which here ω denotes an arbitrary Higgs field.The meanings of conjecture 1 are as follows.The model and its all inferences are invariant when S and v-particles are completely symmetric, here cr T is the critical temperature (see section 5. because of the reasons 6 -7 below.Here g m denotes a gravitational mass.Consequently, both s-energy and v-energy must be positive (see(2.20)-(2.21)).

24 )
-particle with its gravitational mass g m to move in a gravitational field with its strength It is seen from (2.24) that the motion equation of the gravitation mass vg m is the same as that of the gra- vitation mass .sgm 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 g m , but is not the equation of an inertial mass I m .According to the equvalence principle in the conventional theory, inertial mass.Consequently, (2.24) is the equation of an inertial mass I m 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 I m .The motion equation of an inertial mass I m must be determined on the bases of conjecture 1 and the gravitational field equation.In the S-breaking, according to conjecture 1, .(2.24), but the equation of vI m must be different from that of vg m .According to conjecture 1, both sI m and vI m are positive, and sg of the gravitational charge g m and the gravitational field with its strength .

.
It is easily seen from (3.8)-(3.11)that all real components of v Φ have the same mass ( ) v m Φ , and all real components of v χ have the same mass ( ) v m χ in the S-breaking.The S-breaking and the V-breaking are symmetric because s-matter and v-matter are symmetric.Hence when s v  and S V  in (3.1)-(3.12), the formulas are still kept.Let ( ) ( ) ( ) of g ρ because of the transformation of ( ) 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 M m s ′ satisfy , is the mass of a proton.The second sort of particles is composed of such particles whose masses l m m is the mass of an electron.The third sort is composed of photon-like particles whose masses m s all v-particles must be in v-SU(5) color singlets whose masses are not zero so that 0, and v T are so large that all masses may be neglected ( s T and v p T m  the total number of spin states, and aB g and aF g are the total number of spin states of bosons a − and the total number of spin states of a-femions, respectively., particles and v-particles are symmetric.Considering (4.5)-(4.7),(4.9), (5.2) and (5.9) in the S-breaking, following.We consider such a space-contracting stage in which s v in the S-breaking.We will see that there are the critical temperatures , the effective potential, there still is the S-breaking, i.e. by suitably choosing the parameters in the Higgs potential.As mentioned before, there is the S-breaking in low temperatures.Talking p w = for short, we have

Figure 1 .
Figure 1.The states changes from A to E when T s rises from T < T φcr to T φcr1 .

.
Space will contract when t > t T = 0, because ( )0 I R t =  and ( ) 0. I R t < We consider that the physical boundary condition of the Equations (5.14)-(5.15)should be min 0 when 0 of the Higgs particles are large and the number of the Higgs particles is little.In the case, 0 g V   .Thus, (4.18) holds so that the transformation of s ρ and v ρ may be neglected.Whens cr T T ϕ <, the s-particles must form celestial bodies with their large masses so that rate of the i-th sort of a-Higgs particles to the b-Higgs particles at the temperature a T , , b s v = and b a ≠ .Here (6.5) is considered so that the factor ( ) j-th sort of v-Higgs (s-Higgs) particles.It is obvious that the larger s T is, the less s υ Ω is and0 s υ Ω = when s cr T T = .The larger s T 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 when1 s cr T T ϕ > .The masses of all gauge bosons and fermions are zero when 0, Higgs particles and the a-gauge bosons or the a-ferminos can transform from one to another by the 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 from the above mentioned and (6.8)-(6.10)that there must be , of the symmetry of the s-particles and the v-particles, , the spinfreedom of the s-particles (the v-particles).When 1 , thermal equilibrium of s-matter and v-matter.
in the S-Breaking for All Time We see from the discussion above that 0 v m Ω ≥ can hold when .cr s v

<
are possible.T µν in general relativity is equivalent with g e f K K K RK is non-zero, there will be 0 < are all possible.Thus, although the strong energy condition still holds, i.e.

T
is larger than the temperature corresponding to GUT.Taking . and h = 0.8, we get ( ).

9 13
shown by the curve A in the Figure2and describes evolution of the universe from we get the ( ) a t which is shown by the curve B in the Figure2and describes evolution of the cosmos from can get a curve of ( ) a t 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.

Figure 2 .
Figure 2. The curve A describes evolution of a(t) from 14 × 10 9 yr ago to now; The curve B describes evolution of a(t) from 13.7 × 10 9 yr ago to now.The starting point of curve A is different from that of curve B.

3
and h = 0.8, from (8.22) we get the L d z − relation which is shown by the curve A in the Figure is shown by the curve B in the Figure 3.

Figure 3 .
Figure 3.The curve A describes the L d z − relation when

5 S
Inference, and There Is No Restriction for T µν 11.1.New Predictions 11.1.1.The Essence and Characters of Huge VoidsIt 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 ( ) SU color singlets.The huge v-voids are not empty.There must be the Here s ρ′ and v ρ′ denote the densities inside the huge v-voids, and s ρ and v black hole contracts by its self-gravitation, then the expectation values of the Higgs fields inside the v-black hole will change from 0 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.
dominant energy condition and the positive energy theorem are not applicable to .holds?We will discuss the problem in another paper.
s ρ and v ρ from one to another is striking so that 0 g ρ < is possible.Hence there are such solutions of the evolution equations which satisfy the physical boundary condition, i.e. highest symmetry) is kept, and 0 g ρ < .When space contracts further, R arrives the least scale min 0, space expands and v

2 .
It is derived from this model that there are the critical temperature cr T , the highest temperature , we can write a T µν as ) vj m is the mass of a sort of s-particles (v-particles).If ( ) in this case.Hence we consider such a contracting process when , 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, E denote the 5 sorts of colors.A component of 10 representation carries color αβ , , There are the following sorts of the s-SU(5) color-singlets.2-fermionstates: αα * or ( )( ) (5) T < , all s-particles will exist in s-SU(5)color-singlets (conjecture 2).Let A , B , C , D ,