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.

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

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:

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.

By the conventional method, from we can get

Considering, from (2.3)-(2.4) we have

From (2.11)-(2.13) we obtain

In the S-breaking,

In the V-breaking,

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

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, 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

From (2.16) we have

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

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

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, a gravitational field is weak () and static, and from we have

From (2.28), and are two constants. Considering, and let be Newtonian gravitational potential, we can reduce (2.29) to

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

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

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.

As is well known, based on the RW metric metric,

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

where is a 4-velocity and or v. In comoving coordinates in a comoving coordinates. can be written as

Considering substituting (4.2)-(4.4) and the RW metric in (4.1) into (2.16), we get the evolution equations

In the S-breaking,

In the V-breaking,

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.

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,

When is the total energy density and is conservational, i.e.

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

Consequently, we have

where denotes the change of because of the transformation of and to each other.

According to this model, is a function of R V_g, and, i.e.

Thus,

From (4.5)-(4.6) we have

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.

Pressure density is a function of masses of particles and temperature, i.e.. Let and In the S-breaking, from we have

It is obvious that when, the solution of (4.14) is

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 m_p 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 m_p > T > m_e, and. When,. Thus, we have

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

When . Letting, we have

When and we have

where.

It is obvious that when (4.21) has such solutions in the following form,

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.

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

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 to, and ignoring the terms irrelevant to we have

Similarly (5.1)-(5.5), from (2.9) we have

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, here is the Boltzmann constant (here), we get the effective potential approximate to 1-loop in flat space at finite-temperature [25] -[27]

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

It is easily seen from (5.10) that when

Similarly, from (2.9) we have

When the masses of all particles may be neglected, p_g = ρ_g/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

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

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.

will decrease monotonously as increases and its lower limit is

There is the critical temperature at which the minimum is degenerate, i.e.

where when

There is the critical temperature at which

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.

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)

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

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

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

It is obvious that and. Space will contract when t > t_T = 0, because and We consider that the physical boundary condition of the Equations (5.14)-(5.15) should be

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.

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

From (6.15) we see when,

Here is considered. In the case, space can continue to contract, but there must be such a moment at which

This is because and when. It is easily seen that

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.

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,

, and are all possible. Thus, although the strong energy condition still holds, i.e.

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.

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

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

Consequently, space inflation must occur

Supposing and for and considering, from (5.40) we can estimate,

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

is larger than the temperature corresponding to GUT. Taking and

we have If the duration of the super-cooling state is R_cr 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.

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

After reheating, must first transform into v-energy by (2.9) and the SU_V(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

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,

Let the reheating process ends at. Considering, we suppose. From (8.3) we have

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

where both and are constants. From (4.8), (8.3) and (8.6)-(8.7), (4.5)-(4.6) is reduced to

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.

Letting and and considering

we rewrite (8.6) as

From (8.12) we have

If is taken as

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.

It should be noted that in the V-breaking, but here. Neglected noting and taking

we can reduce (8.15) to

Replacing by because and and ignoring we see (8.19) to be the same as the corresponding formula (3.44) in Ref. [18] .

From (8.12) and the RW metric we have

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

which is consistent with (3.78) in Ref. [18] . Approximating to and, we obtain

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.

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

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

According to conjecture 1, when. Consequently, although

we have still

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

This is a reasonable result.

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.