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Problem of cosmological singularity and inflationary cosmology

Problem of cosmological singularity of general relativity theory is discussed. The possible resolution of this problem in the framework of inflationary cosmology is proposed. Physical conditions leading to bouncing inflationary solutions in the frame of general relativity theory and gauge theories of gravitation are compared. It is shown that gauge theories of gravitation allow to build regular inflationary cosmological models of closed, open and flat type with dominating ultrarelativistic matter at a bounce.

1 Problem of cosmological singularity and vacuum gravitational repulsion effect Problem of cosmological singularity (PCS) is one of the most principal problems of general relativity theory (GR). According to Hawking-Penrose theorems the appearance of cosmological singularity in cosmological solutions of GR is inevitable, if gravitating matter satisfies so-called energy dominance conditions [1,2]. There were many attempts to resolve PCS. If the Planckian (pre-Planckian) epoch took place in the beginning of cosmological expansion, consequent quantum theory of gravitation is necessary to analyze the PCS 1 . But such theory does not exist at present. By supposing that the Planckian epoch was absent by evolution of the Universe, classical theory of gravitation (with quantum description of gravitating mat-ter)can be used by analysis of the PCS. There were many such attempts to resolve the PCS corresponding to the following two approches (see [5] and refs given here).
In the frame of the first approach one supposes that energy dominance conditions are valid for gravitating matter with extremely high energy densities and pressures at the beginning of cosmological expansion. In this case GR must be replaced by certain other gravitational theory by description of gravitating matter at such extreme states. Gauge theories of gravitation (GTG) and at first of all the Poincare GTG are an important generalization of GR.
Cosmological equations for homogeneous isotropic models deduced in the frame of GTG lead to regular in metrics cosmological solutions in the case of usual gravitating matter [6].
These solutions differ from corresponding solutions of GR only at extreme states of gravitating matter near certain limiting energy density, where gravitational interaction has the character of repulsion instead of attraction.
The second approach to resolve PCS concerns the analysis of description of gravitating matter at extreme conditions and examination of energy dominance conditions for corresponding states 2 . Unified gauge theories of strong and electroweak interactions with spontaneous symmetry breaking are the modern base of matter description at extreme conditions. Inflationary cosmology as important part of the theory of early Universe was built by using gauge theories of elementary particles [8,9]. A number of problems of standard Friedmann cosmology were resolved in the frame of inflationary cosmology. Most inflationary cosmological models discussed in literature are singular and their study is given from Planckian time. The radical idea of quantum birth of the Universe was introduced in order to avoid the PSC. Note that during quasi-de-Sitter inflationary stage gravitating matter in inflationary models is approximately in the state of so-called gravitating vacuum, for which pressure p and energy density ρ > 0 are connected in the following way p = −ρ and energy dominance conditions are not valid. As it was shown in Refs. [10,11] gravitating vacuum with sufficiently large energy density can lead to the vacuum gravitational repulsion effect (VGRE) in the case of systems including also usual gravitating matter, that allows to build regular inflationary cosmological models. At first the VGRE was discussed in the frame of Poincare GTG [10] in the case of homogeneous isotropic models including radiation and gravitating vacuum with ρ = const. Because cosmological equations of Poincare GTG are valid in the frame of the most general GTG -metric-affine GTG, the VGRE can take place also in metric-affine GTG [12,13]. Regularizing role of gravitating vacuum in GTG was analyzed in Refs [14][15][16][17]. The VGRE in GTG can lead to a bouncing solutions in the case of closed, flat and open models. It is important that the VGRE can lead to the bounce also in GR in the case of closed models [11,5] (see also [26] and refs given here). Regular bouncing closed models filled by linear massive scalar field were studied in the frame of GR in Refs. [18,19]. By using values of some parameters of the Universe, the probability of regular solutions for such models was estimated to be very small [18] 3 . This estimation is not valid for inflationary models discussed below, for which scalar fields dominate during very small time intervals.
By using some simplest potentials for scalar fields regular inflationary cosmological models were discussed in the frame of GR as well as GTG in Refs. [22][23][24][25]. It was shown that limiting energy density and temperature at the bounce in inflationary models can be essentially smaller than the Planckian ones. Obtained regular inflationary solutions contain the stage of transition from compression to expansion and quasi-de-Sitter inflationary stage.
After inflation scalar fields oscillate and are transformed into elementary particles, as result the transition to Friedmann regime takes place. Formal integration of cosmological equations leads also to quasi-de-Sitter stage of compression [22][23][24][25]. However, this stage in GR is unstable and small change of variables at compression stage can lead to singular solution.
This means that the building of regular inflationary models is possible, if physical conditions leading to the bounce take place at the end of cosmological compression (see below).
By using results of numerical investigation performed in Ref. [27], the analysis of regular 3 At the first time cosmological solution with initial de Sitter state limited in the past was discussed in Ref. [20]. The problems -what origin has initial vacuum state, what was before this state -were not considered in [20]. The cosmological solution with initial vacuum state not limited in the time in the past, proposed in Ref. [21] does not lead to resolution of PCS in the future.
inflationary models in GR and GTG in dependence on conditions at a bounce is given below in present paper.

Regular inflationary cosmological models in GR
Let us consider homogeneous isotropic models filled by scalar field φ minimally coupled with gravitation and usual matter. We suppose that interaction between them is negligibly small.
Then energy density ρ and pressure p can be written in the following form where a dot denotes differentiation with respect to time, index m corresponds to usual matter, and V (φ) is scalar field potential. Evolution of considered models in GR is described by Friedmann cosmological equations In the case of closed models (k=+1) Eqs. (2)-(3) can lead to regular bouncing solutions.
Putting t = 0 at bounce (H(0) = 0,Ḣ 0 ≡Ḣ(0) > 0), we see bouncing solutions take place, if the value of scalar field potential V 0 = V (φ(0)) (which plays the role of gravitating vacuum energy density) is sufficiently large and satisfies the following inequality (index 0 corresponds to t = 0). For definiteness we will consider later the matter in the form of ultrarelativistic matter or radiation (ρ m = ρ r , p m = 1 3 ρ r , index r corresponds to radiation), then Eq. (3) takes the following forṁ and according to (6) we have To obtain a bouncing solution we have to take some initial conditions for scalar field (φ 0 , φ 0 ) and radiation ρ r0 satisfying according to (7) the following relation Admissible values of φ 0 ,φ 0 and ρ r0 must satisfy restrictions: p , by which quantum gravitational effects are not essential and our classical consideration is valid. Minimum value of R 0 of scale factor at bounce we obtain from Eq. (2) By using initial conditions at bounce we integrate Eqs. (3 ′ ) and (5) putting ρ r = ρ r0 R 4 0 R 4 in accordance with (8). Now let us discuss the most important features of regular inflationary models in GR, by taking into account numerical results obtained in the case of the simplest scalar field potentials V 1 = 1 4 λφ 4 (λ = 10 −14 ) and V 2 = 1 2 m 2 φ 2 in Ref. [25]. a) Transition stage from compression to expansion At first we will consider bouncing solutions in the case when the value of η ≡ V 0 −φ 2 0 − ρ r0 > 0 is not near to zero. Then as numerical analysis shows, the Hubble parameter H(t) and scalar field φ(t) (φ 0 = 0) vary in linear way during transition stage from compression to expansion, and derivativesḢ andφ decrease rapidly and tend to zero at the end of this stage (t ∼ t 1 ), moreover radiation energy density ρ r → 0 at t ∼ t 1 (see Fig. 1). 4 According and the change of Hubble parameter during time interval ∆t 1 = t 1 − t 0 (t 0 = 0, H 0 = 0) is Because φ 1 = φ 0 +φ 0 ∆t 1 and hence according to (10) Analogously we can determine time interval ∆t 2 of transition stage before a bounce Then duration of transition stage is and value of the Hubble parameter at the end of transition stage (at the beginning of inflationary stage) is If The value of H 1 (15) depends on φ 0 only and does not depend onφ 0 and ρ r0 .
By decreasing of value of η, the linear time dependence of H(t) is violated and duration of transition stage increases (see Fig. 2). Formulas (13)- (14) are not applicable in this case, but expression (10) for the Hubble parameter at the end of transition stage is valid.
During quasi-de-Sitter inflationary stage we haveḢ ≪ H 2 ,φ 2 ≪ V (φ),φ ≪ V ′ , ρ r ≈ 0 and term R −2 in Eq. (2) is negligibly small. Then according to Eq. (3 ′ ) (and Eq. (2)) the Hubble parameter at inflationary stage is equal to and Eq. (5) for scalar field takes forṁ Values of scalar field φ 1 = φ(t 1 ) and scale factor R 1 = R(t 1 ) play the role of initial conditions by integration of Eqs. (16)-(17) for inflationary stage. For given potential V (φ) the integration of Eq. (17) leads to the function φ(t). In the case of potential V 1 = 1 4 λφ 4 we obtain and in the case of potential V 2 = 1 2 m 2 φ 2 we have By using explicit form of scalar field φ we can find the scale factor by integrating Eq. (16).
In the case of functions (18)- (19) we obtain where n = 4 and n = 2 in the case of potentials V 1 and V 2 respectively. Expressions (18)- (20) have the same form as in chaotic inflation [8], but in considered theory the values φ 1 and R 1 are determined by initial conditions at bounce and are not arbitrary.
At the end of inflationary stage Eqs.

Regular inflationary cosmological models in GTG
Cosmological equations for homogeneous isotropic models obtained in the frame of gauge theories of gravitation have the following form Here β is indefinite parameter with inverse dimension of energy density. At first Eqs. (21)- (22) were deduced in Poincare GTG [6], and later it was shown that Eqs. (21)-(22) take place also in metric-affine GTG [12,13]. The conservation law has usual form (4). Besides cosmological equations (21)-(22) gravitational equations of GTG lead to the following relation for torsion function S and nonmetricity function Q In Poincare GTG Q = 0 and Eq. (23) determines the torsion function. In metric-affine GTG there are three kinds of models [13]: Sitter solutions for metrics with vanishing torsion and nonmetricity are exact solutions of GTG [28,29] and hence inflationary models can be built in the frame of GTG [22][23][24]. In the case of values of β to be not large in system of units inflationary models of GTG practically coincide with that of GR [24]. It is interesting to investigate inflationary models in GTG in the case of large values of β ( β(4V −φ 2 ) ≫ 1).
In order to analyze inflationary cosmological models in GTG let us consider systems including scalar field and radiation, for which quantities ρ and p are given by (1), where ρ m = ρ r , p m = 1 3 ρ r . Then cosmological equations (21)-(22) by taking into account Eq. (5) can be written in the following form where Eqs.(24)-(25) lead to essential restrictions on scalar field by evolution. So in the case k = 0, +1 from Eq. (24) follows that Z ≥ 0, if β < 0. Because at the bounce (t = 0) we have ). The relation (27) determines minimum value of scale factor R 0 for closed and open models, and in the case of flat models gives the dependence between φ 0 ,φ 0 and ρ r0 . From (25) the time derivativeḢ 0 =Ḣ(0) at the bounce iṡ Then the conditionḢ 0 > 0 determines permissible values of φ 0 ,φ 0 , ρ r0 at the bounce in dependence on indefinite parameter β. In order to analyze the dependenceḢ 0 =Ḣ 0 (β) we write expression (28) in the forṁ where β 1 = 1 2 (2V 0 +φ 2 0 ) −1 and β 2 = (4V 0 −φ 2 0 ) −1 are particular points and If b 2 − 4af ≥ 0, the formula (29) takes the forṁ In the case of large (in module) values of β (|β 4V 0 −φ 2 0 | ≫ 1) we haveḢ 0 ≈ aβ 1 β 2 and the bounce conditionḢ 0 > 0 leads to the following relation The relation (29 ′ ) is compatible with the following condition which follows from discussed above restriction on the value of Z in the case large (in module) negative value of β (β < 0). Corresponding inflationary models are regular in metrics, Hubble parameter and torsion (nonmetricity). (Note that according to Eq. (24) the right-hand part of Eq. (26) tends to 1 2 H at Z → 0). Unlike the condition (7 ′ ) of GR, inequality (30) does not include radiation energy density ρ r . In contrast to GR, the appearance of VGRE in GTG (with large negative β) does not depend on radiation, although the contribution of ultrarelativistic matter to energy density of the Universe at the bounce can be essentially greater than contribution of scalar fields.
Let us consider cosmological equations of GTG (24)- (26) in approximation of large negative β ( β 4V −φ 2 ≫ 1), by supposing that ρ r + 1 2φ 2 + V ≪ |β| 4V −φ 2 2 . 6 We have 6 This assumption does not exclude that radiation energy density can dominate at the bounce ρ r ≫ At the beginning of the inflationary stage (t ∼ t 1 ) the Hubble parameter H(t 1 ) from (32) is equal to In the case of linear time dependence of Hubble parameter during stage of transition from compression to expansion we can simply estimate the time interval ∆t 1 by using relations (35)-(36) and then determine the values of scale factor R 1 = R 0 (1 +Ḣ 0 ∆t 1 ) and scalar field φ 1 = φ 0 +φ 0 ∆t 1 at the end of transition stage (as it was made for inflationary models in GR in Section 2). Numerical analysis of regular inflationary models in GTG with large |β| shows that the most important difference of such models in comparison with that of GR concerns the transition stage from compression to expansion and the stage after inflation. As was noted above the region of initial conditions at the bounce of regular inflationary solutions is essentially more wide in GTG than in GR (compare (7 ′ ) with (29 ′ ), (30)). In Fig. 4 the evolution of different characteristics of regular inflationary model in GTG (β = −10 20 M −4 p ) is given by choosing of initial conditions, by which a bouncing solution in GR is impossible.
From Fig. 4 we see that the Hubble parameter and torsion as well as scalar field oscillate after inflation. Amplitude and frequency of scalar field oscillations are smaller in comparison with that of GR (for given potential) and decrease by increasing of |β|.

Conclusion
Considered regular inflationary models can be interesting for inflationary cosmology, if the realization of physical conditions leading to a bounce can take place at the end of cosmological compression. In the frame of GR this means that the greatest part of energy density of the Universe has to be determined by scalar fields (see (7 ′ )). However, in the framework of GTG the realization of the bounce is possible, when usual ultrarelativistic matter dominates at the end of cosmological compression, and scalar fields satisfy more weak restriction (relation (30)) -its kinetic energy density can be greater than the scalar field potential. Unlike GR, in the frame of GTG the elimination of cosmological singularity and realization of the