_{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
SU_{S}
(5) × SU_{V}(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 theV-breaking,
SU_{V}
(5)
breaks finally to
SU_{V}
(3) × U_{V}(1)
so that v-particles get their masses and form v-atoms andv-galaxies etc., while
SU_{S}
(5)
still holds so that s-fermions and s-gauge bosons are massless and form
SU_{S}
(5)
color-singlets. There is no interaction among the
SU_{S}
(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.

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

As is now well known, there is space-time singularity under certain conditions [

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. [_{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

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

Recent astronomical observations show that the universe expanded with a deceleration earlier, while it is expanding with an acceleration now [

That

In fact,

Huge voids in the cosmos have been observed [

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

The present model has the following results:

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

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

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

Conjecture 2 When

Because of conjecture 1, there are two sorts of symmetry breaking which are called S-breaking in which

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.

We explain it in detail as follows:

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

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.

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.

In the S-breaking,

There is no interaction (e.g. the electroweak interaction) except the gravitation among the

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.

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

In the S-breaking, only the cosmological effects of v-matter are important and are consistent with the observation up to now.

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.

Although the equivalence principle is violated by v-particles in the S-breaking, there is no contradiction with any observation and experiment, because the

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

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

where the meanings of the symbols are as follows:

Gibbons and Hawking pointed out that in order to get the Einstein field equations [

This is because it is not necessary that

The Higgs potentials in (2.5)-(2.7) is the following:

where

We do not consider the terms coupling to curvature scalar, e.g.

By the conventional method, from

Considering

From (2.11)-(2.13) we obtain

In the S-breaking,

In the V-breaking,

It is seen from (2.17)-(2.18) that

We will see that, in fact,

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,

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

and (2.16) and (2.14) we obtain

From (2.16) we have

In the S-breaking,

Consider a point-particle with its gravitational mass

It is seen from (2.24) that the motion equation of the gravitation mass

It must be given one's attention to that (2.24) is only the equation of a gravitation mass

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

In the S-breaking, according to conjecture 1,

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

From (2.28),

Let

Let

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

Ignoring the couplings of

Ignoring the contributions of

We take

We can choose such parameters that

e.g.,

The S-breaking and the V-breaking are symmetric because s-matter and v-matter are symmetric. Hence when

Let

we have

It is easily seen that

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

In the present model, we take

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

where

Considering

In the S-breaking,

In the V-breaking,

Comparing (4.5)-(4.6) with the Friedmann equations, we see that provided

In contrast with the conventional theory, it is possible that

Let

When

It is possible that

Consequently, we have

where

According to this model, _{g},

From (4.5)-(4.6) we have

This is because

When

Pressure density is a function of masses of particles and temperature, i.e.

It is obvious that when

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 _{p} is the mass of a proton. The second sort of particles is composed of such particles whose masses _{p} > T > m_{e},

In the S-breaking,

When

When

When

where

It is obvious that when

In contrast with the conventional theory,

The thermal equilibrium between the v-particles and the s-particles can be realized by only (2.10). The Higgs bosons

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

to ignore the terms proportional to

Considering the contributions of the expectation values

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

where

Only considering the contribution of the expectation values of

It is easily seen from (5.10) that

Similarly, from (2.9) we have

When the masses of all particles may be neglected, p_{g} = ρ_{g}/3 and

For short, we take

As mentioned before, there is the S-breaking in low temperatures. Talking

Both

From (5.10) and (5.22) we can determine the minimum

There is the critical temperature

where

There is the critical temperature

where

Sum up, when

In the case, there is no relative minimum, as shown in

Analogously to that

When

It is easily seen from (5.3)-(5.7) that

Both

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.

We consider the contracting process of the universe after expansion in the S-breaking. It is seen from (3.15) that in the case,

It is obvious that _{T} = 0, because

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 [

When both

We discuss the transformation of

Let

In the initial stage, temperature is low, i.e.

When

and it is possible that

As mentioned in the preceding section, there are the substable states. Consequently, the universe will be in the substable states when

(1) The effective masses of the Higgs particles and the transformation of

If

The masses containing the temperature effect are called effective masses.

The effective masses of

It is seen from (6.7) that there must be such a

where

(2)

When both

where

(3)

It is obvious that the larger

The masses of all gauge bosons and fermions are zero when

It is seen from the above mentioned and (6.8)-(6.10) that there must be

when

(4)

Because of the symmetry of the s-particles and the v-particles,

When

and there is such a moment

When

Thus, space will contract with a deceleration. Let

We see from the discussion above that

From (6.15) we see when

Here

This is because

In summary, there are

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 [

Hawking considers it is a reasonable the first condition that

As mentioned above, there must be

The key of non-singularity is conjecture 1, i.e.

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 [

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

The alternative version of the theorem can obtained by the following two propositions.

Proposition 1 [

If

Proposition 2 [

If

An alternative version of the above theorem is as following.

Space-time

(1)

(2) The generic condition is satisfied, i.e. every non-spacelike geodesic contains a point at which

(3) The chronology condition holds on

(4) There exists at least one of the following:

(A) a compact achronal set without edge(B) a closed trapped surface(C) a point

In fact,

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

When

When

Consider the initial stage of expansion in which

because

Consequently, space inflation must occur

Supposing

We may take

The temperature will strikingly decrease in the process of inflation, but the potential energy

_{cr} will increase

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

After reheating, _{V}(5) couplings and into s-energy by (2.10). Letting

Let

Let the reheating process ends at

After reheating process ends, temperature is low,

As mentioned in section 4 (see (4.21)-(4.26)), the evolving laws of

where both

As mentioned in section 3,

We discuss (8.8)-(8.9) as follows.

space expands with an acceleration. In the process,

R_{2},

The first case is consistent with observations. A computation in detail is the same as that of Ref.

Even

Letting

we rewrite (8.6) as

From (8.12) we have

If

where

Taking

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

we can reduce (8.15) to

Replacing

From (8.12) and the RW metric we have

where

Considering

Ignoring

which is consistent with (3.78) in Ref. [

Taking

As mentioned before, the evolving laws of

When R is large enough,

where

When

To sum up, according to the present model, the universe can expand from

In the V-breaking, all s-gauge particles and s-fermions are massless. When the temperature

2-fermion states:

The masses of all color singlets are non-zero, hence

There is no interaction among the

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

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.

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

Letting there be a v-black hole with its mass and density to be so big that its temperature can arrive at

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.

The effective cosmological constant

As mention above, the present model can explain evolution of the universe without

Applying the conventional quantum field theory to the present model, we have

According to conjecture 1,

we have still

Here

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,

Whether does

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

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,

There are the critical temperature

Based on the present model

It is seen that according to the present model, the universe can expand from

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.