Since its incarnation in 1980 [1], the inflationary paradigm has achieved great success, resolving many long-standing problems of the standard big bang cosmology, and is consistent with all cosmological and astrophysical observations conducted so far [2,3]. However, the paradigm has also faced some challenges. In particular, it is well-known that this paradigm is sensitive to the ultraviolet (UV) physics, and its successes are tightly contingent on the understanding of this UV physics [4][5][6]. Typically, if the inflationary phase lasts somewhat longer than the minimal period required to solve the above mentioned problems, the length scales we observe today can originate from modes that are smaller than the Planck length during inflation. Then, the treatment of the underlying quantum field theory on a classical spacetime background becomes questionable, as now the quantum geometric effects are expected to be large, and the space and time cannot be treated classically any longer. This is often referred to as the trans-Planckian problem of cosmological fluctuations [4].
The second problem is related to the existence of the big bang singularity [7,8], with which it is not clear how to impose initial conditions. Instead, one often ignores the pre-inflationary dynamics and sets the initial conditions at a sufficiently early time so that all the observational modes are inside the Hubble horizon. In the slowroll inflation scenario, the spacetime becomes almost de Sitter, and the Bunch-Davies (BD) vacuum becomes a natural choice [9]. However, it is still an open question on how such a vacuum state can be realized dynamically in the framework of quantum cosmology (QC), considering the fact that a pre-inflationary phase always exists between the Plank and inflation scales, which are about 10 12 orders of magnitude difference in terms of energy densities. During this phase, particle creations are inevitable.
It is clear that all the above issues are closely related to QC, a topic that has been extensively studied in the past decades, and various theories have been proposed. Among them are models constructed from string/Mtheory [10,11] and loop quantum gravity (LQG) [12][13][14][15][16]. In particular, in the last two decades, LQG has been rigorously applied to understand singularity resolution in various cosmological models (for recent reviews, see Refs. [17][18][19][20]), and a coherent picture of Planck scale physics has emerged: the big bang singularity is replaced by a quantum bounce, purely due to quantum geometric effects. This framework is often referred to as loop quantum cosmology (LQC). In the last couple of years, to understand some ambiguities of LQC, several modified loop quantum cosmological (mLQC) models have been proposed [21], including mLQC-I [22][23][24], first proposed in [25] and later systematically developed in [26,27]. It is interesting to note that this model can be also obtained by the so-called top-down approach [28][29][30].
In all bouncing cosmological models, either classical [31][32][33] or quantum [17][18][19][20], a challenging question is how to solve the shear problem. This is because in a contracting phase, shear always grows like a -6 , which is faster than all matter fields, except for the stiff fluid (or a massless scalar field) that also grow in the same rate, where a is the average expansion factor. However, even in the latter it is not clear how the stiff fluid can always win over the shear, so that a homogeneous and isotropic universe will develop after the bounce. Shear in homogeneous and anisotropic Bianchi universes have been extensively investigated in LQC [17,19,20] and various interesting results have been obtained. In particular, in the Bianchi I universe it was found that the shear is always conserved asymptotically [34,35]. Therefore, to solve the shear problem in LQC, one often borrows the ekpyrotic mechanism (see for example, [36,37] and references therein), first introduced in colliding branes [31,38] and later generalized to other bouncing models, including matter and ekpyrotic/cyclic bounces [31][32][33][39][40][41][42][43][44][45]. The basic idea is to introduce a scalar field with an ekpyroticlike potential which becomes negative near the bounce, so the effective equation of state (EoS) of the scalar field will be greater than one, so that the scalar field will grow like ρ ϕ ∝ a -3(1+w) (w > 1), whereby dominates the shear in the bounce region. As a result, a homogeneous and isotropic universe can be developed after the bounce.
In this paper, we study how the ekpyrotic mechanism affects the inflationary phase in both LQC and mLQC-I, because in these frameworks the inflation is generic without such a mechanism [24,46]. Then, a natural question is whether the inflation is still generic or not after the ekpyrotic mechanism is taken into account. To answer this question, we consider a scalar field with a total potential given by
where V ekp (ϕ) denotes an ekpyrotic type potential, and V inf (ϕ) an inflationary potential. Clearly, to have the mechanism work, V ekp (ϕ) needs to dominate the evolution of the universe in the contracting phase near the bounce, while after the bounce the inflationary potential V inf (ϕ) will gradually increase and finally dominate the evolution, whereby an inflationary phase is developed. Therefore, the task now reduces to showing that the above mentioned process indeed occurs for a given set of initial conditions. More importantly, the inflationary phase will last long enough to solve the big bang problems, which motivated the proposal of inflation in the first place [1].
It must be noted that in the matter and ekpyrotic/cyclic bounces (see, for example, [31][32][33][39][40][41][42][43][44][45] and references therein), an inflationary phase is not required. As a matter of fact, one of the main motivations of these models is to replace the inflationary phase by a regular bounce, as the latter naturally solves the big bang singularity and trans-Planck problems. In addition, a matterdominated contracting phase also leads to a power spec-trum that is scale-invariant, as first noticed by Wands [47].
In LQC and mLQC-I, primordial power spectrum without inflation was also studied [48][49][50][51][52]. However, recently it was found that the resultant power spectrum is inconsistent with current observations [53]. Therefore, in this paper we shall focus on LQC/mLQC-I models in which inflationary phase still exist, as we mentioned above that inflation is generic in these models when the ekpyrotic mechanism is not taken into account [24,46]. In particular, after numerically studying various cases, we find that, by properly choosing the free parameters involved in the models, the ekpyrotic-like potential indeed dominates the evolution of the universe in the bounce region, during which the EoS of the scalar field is larger than one, so the shear problem is resolved. As time continuously increases after the bounce, the inflationary potential picks up and becomes dominant, whereby an inflationary phase is finally developed. This phase can last long enough in order to solve the cosmological problems of the big bang cosmology.
The rest of the paper is organized as follows: In Sec. II we give a brief introduction to LQC and mLQC-I, and provide the corresponding Hamiltonian equations. In Sec. III we solve these equations numerically with the total potential given by Eq.(1.1) for various choices of the parameters involved in the models. Although the existence of an inflationary phase with sufficient e-folds sensitively depend on the choices of the free parameters, we do find regions of the parameter phase spacetime with non-zero measure that lead to such desirable inflation. In Sec. IV, we summarize our main results and provide some concluding remarks.
In this section, we provide a summary of the modified Friedmann dynamics in the frameworks of LQC [17]and mLQC-I [23].
In the framework of LQC, the dynamics can be obtained from the effective Hamiltonian given by
where G is the Newtonian constant, v ≡ a 3 , and a is the expansion factor of the Universe
The variable b denotes the momentum conjugate of v and satisfies the canonical relation
where γ is known as the Barbero-Immirzi parameter whose value is set to γ ≈ 0.2375 using black hole thermodynamics in LQG [54]. The parameter λ is defined as
, where ∆ denotes the minimal area gap of the area operator in LQG [12-15, 55, 56]. The matter Hamiltonian H M is given by
where ρ denotes the energy density of the matter field. Then, the Hamiltonian equation for a given physical quantity A of the system Ȧ = {A, H} ,
)
On the other hand, from the Hamiltonian constraint H LQC ≃ 0 we find that
Inserting the above expression into Eqs.(2.6) and (2.7) we find that ḃ = -4πGγ (ρ + P ) , (2.9)
where the pressure P and critical energy density ρ c are defined respectively as
(2.11)
From Eq.(2.10) we can see that ρ ≤ ρ c , and when ρ = ρ c we have H = ȧ/a = 0, at which a quantum bounce happens.
For a scalar field ϕ with a potential V (ϕ), we have
.12) where p ϕ is the momentum conjugate of ϕ and satisfies the canonical relation {ϕ, p ϕ } = 1. (2.13) Then, the Hamiltonian equation (2.5) yields φ = {ϕ, H} = ∂H ϕ ∂p ϕ = p ϕ v , (2.14) ṗϕ = {p ϕ , H} = -∂H ϕ ∂ϕ = -vV ,ϕ , (2.15) where V ,ϕ ≡ dV (ϕ)/dϕ. From the above equations, we find that φ + 3H φ + V ,ϕ (ϕ) = 0, (2.16) which is nothing but the Klein-Gordon equation. On the other hand, from Eqs.(2.4), (2.11) and (2.14) we find that
(2.17)
Then, the equation of state (EoS) for the scalar field is given by
provided that ρ ϕ > 0. Eqs.(2.6), (2.7), (2.14) and (2.15) are the first-order ordinary differential equations for the four canonical variables (v, b; ϕ, p ϕ ). Once the initial conditions are specified at a given moment, they uniquely determine the trajectory of the evolution of the Universe. Such initial conditions are often imposed at the quantum bounce [17,21], at which the expansion factor reaches its minimal value and the energy density reaches its maximum.
It should be noted that these four first-order dynamical differential equations are equivalent to the two second-order differential equations given by Eqs.(2.10) and (2.16).
In addition, the advantage of imposing the initial conditions at the bounce is that the time derivative of the scalar field at the bounce φB is determined uniquely up to a sign for any given initial scalar field value at the bounce ϕ B via the relation ρ(t B ) = ρ c , where t B denotes the time of the bounce, which yields φB = ± 2(ρ c -V (ϕ B )).
(
.19) On the other hand, from Eqs.(2.10) and (2.16) we can see that these equations are scaling-invariant with respect to the expansion factor a → a/L o . Therefore, without loss of generality, we can always set the scale factor at the bounce a B = 1, which is equivalent to setting v B = 1. Then, the initial conditions are reduced to the choice of ϕ B , sgn φB . (2.20) Moreover, using the translation invariance t → t + t 0 , in the rest of this paper, we shall set t B = 0. B. Effective Dynamical Equations in mLQC-I
In the framework of mLQC-I, the dynamics can be obtained directly from the effective Hamiltonian [23,25]
.21) Then, for a scalar field with its Hamiltonian given above, the physical variables b and v satisfy the following Hamil-tonian equations v = {v, H} = 3v sin (2λb) 2γλ (γ 2 + 1) cos (2λb) -γ 2 , (2.22) ḃ = {b, H} = 3 sin 2 (λb) 2γλ 2 γ 2 sin 2 (λb)cos 2 (λb) -4πGγP ϕ , (2.23) while the equations for ϕ and p ϕ take the same forms as those given by Eqs.(2.14) and (2.15). Similar to the LQC case, the above Hamiltonian equations can be also cast in the modified Friedman-Raychaudhuri (FR) forms [21]
where holds. Substituting Eq.(2.8) into it, we find that it also yields the same Klein-Gordon equation (2.16), while in terms of ρ and P , we find that ḃ is also given by Eq.(2.9). It should be noted that, Eqs.(2.24) and (2.25) hold only after the quantum bounce (t ≥ t B ), as already indicated in these equations, at which we have ρ(t B ) = ρ I c and H(t B ) = 0, so the expansion factor reaches its minimal value a B ≡ a(t B ). When t ≫ t B (or equivalently, ρ/ρ I c ≪ 1), Eqs.(2.24) and (2.25) reduce to their relativistic limits
In particular, it is interesting to note that ρ/ρ I c ≃ 10 -12 at the onset of inflation [17,21]. Therefore, during the inflationary phase, the modified FR equations are well approximated by its classical limits (2.28) and (2.29).
In the pre-bounce phase (t ≤ t B ), the modified FR equations take the form [21]
where G α ≡ αG, and
(2.32)
From Eqs.(2.30) and (2.31) we can see that at the bounce ρ(t B ) = ρ I c , the universe contracts to its minimal volume v = a 3 B at t = t B . Afterward, it smoothly passes to the expansion phase, but is now described by Eqs.(2.24) and (2.25). The smoothness is shown explicitly in [22][23][24], and can be also seen from Eqs.(2.12) and (2.13), which hold across the bounce.
When t ≪ t B (or ρ/ρ I c ≪ 1), Eqs.(2.30) and (2.31) reduce to (2.34) which are quite different from Eqs.(2.28) and (2.29). In particular, the effective Planck-scale cosmological constant ρ Λ soon dominates the evolution of the pre-bounce phase, whereby a de Sitter spacetime is obtained in the pre-bounce phase but with a Planck-scale cosmological constant ρ Λ ≃ O(ρ pl ). In addition, the Newtonian constant G is replaced by G α (= αG), where α is defined by Eq.(2.32). More remarkably, this Planck-scale cosmological constant is filtered out by the quantum bounce and disappears miraculously after the bounce, whereby the classical FR equations are obtained, as shown explicitly by Eqs.(2.28) and (2.29). This is significantly different from LQC [17], in which the evolution of the Universe is symmetric with respect to the bounce 1 . With similar arguments as those given in LQC, the initial conditions of the dynamical system of Eqs.(2.22), (2.23), (2.14) and (2.15) also reduce to Eq.(2.20) but now with
1 More precisely, it is symmetric for kinetic energy-dominated initial conditions φ2 B ≫ 2V (ϕ B ) [17,21].
It is well-known that shear behave like a stiff fluid [57]
where Σ 2 is a constant, and σ µν denotes the anisotropic shear tensor, defined via the relation
Here v µ denotes the unit tangential vector of the time-like geodesics, θ and ω µν denote respectively the expansion scalar and vorticity tensor of the time-like geodesics. In the homogeneous universe, we have ω µν = 0 and θ = 3H. For the kinetic energy dominated initial conditions, the scalar field also behaves like a stiff fluid, so we have
where ρ
ϕ is a constant. Therefore, it is not always clear which one shall dominate the evolution of the universe near the bounce. If the shear dominates, the universe will become highly anisotropic after the bounce, whereby a homogeneous and isotropic universe cannot be developed. Therefore, it is crucial for any bounce model, including LQC and mLQC-I, to be considered as viable, one has to to make sure that the shear does not dominate in the contracting phase, especially near the bounce [31][32][33]. One way is to introduce the ekpyrotic potential [37][38][39]
so that near the bounce we have V (ϕ B ) < 0, where U 0 , p and β are all positive and otherwise free parameters. Then, we have w ϕ > 1, and
so the scalar field will dominate the evolution of the universe and the effects of the shear will be suppressed. As a result, the contracting universe can smoothly evolve into an expanding homogeneous and isotropic one. When far away from the bounce, we would expect to obtain an inflationary phase in the post-bounce region, t ≫ t B 2 . This is possible if the total potential V (ϕ) consists of two parts
where V inf (ϕ) denotes an inflationary potential and will dominate the evolution of the universe when t ≫ t B , while for t ≃ t B the ekpyrotic potential V ekp (ϕ) dominates.
Following Planck 2018 data [58], inflation with various known potentials have been ruled out, including potentials with the form V (ϕ) ∝ ϕ n . However, models with polynomial chaotic potentials can fit the observations well [59][60][61]. A typical example is [62]
where α 1,2 are two coupling constants. By properly choosing these constants, it can be shown that the models fit the observational data very well. In particular, choosing α 1 = 0.14 and α 2 = 6.644 × 10 -3 allows the model to fit very well to the current Atacama Cosmology Telescope (ACT) observations [3]. In this paper, we shall consider the polynomial chaotic potentials given above as a representative case, and the generalization of our analysis to other viable potentials are straightforwards. Then, a natural question is whether or not a mechanism mentioned above exists. Our following analysis shows that this can indeed be the case by properly choosing the parameters involved in the models, despite the fact that the effects of the ekpyrotic-like potential are dramatic.
For our above claim, let us first show how to choose the initial conditions at the bounce t = t B with a total potential given by Eq.(3.6). First, from Eqs. (2.32), (2.19) and (2.35) we find
where ρ B = (ρ c , ρ I c ), depending on whether we are working in the framework of LQC or mLQC-I. For any given potential V (ϕ) and a fixed equation of state w B > 1, we can solve the above equation for ϕ B . In particular, starting with a minimal value of w B , say, w Bmin , we can solve Eq.(3.8) numerically to obtain the corresponding values of ϕ B . As shown in Fig. 1, the maximal value of w Bmax is 2 It should be noted that in most of the bouncing models, the inflationary phase is not required, see, for example, [31][32][33]. This is fundamentally different from quantum bouncing models of LQG, in which it has been shown that inflation after the bounce is generic in LQC [46] amd mLQCs [24].
obtained when the potential is at its minimum V min (ϕ B ) with
denoted by the crossing point of the horizontal straight line -(w Bmax -1)ρ B /2 and V (ϕ B ). With the above chosen initial conditions for ϕ B , we can study the evolution of the universe for any given inflationary potential. Before doing so, let us first introduce some relevant quantities.
• The first-order Hubble rate and potential slow-roll parameters [63]
These sets of slow-roll parameters are typically used for different purposes. In particular, the slow-roll parameters with the subscript "V " can be used to determine which part of the potential can successfully drive inflation. On the other hand, slow-roll parameters with subscript "H" are used for numerical simulations to define when slow-roll inflation begins and ends. In the classical regime, the scale factor acceleration equation satisfies the relation
The Universe experiences an accelerated expansion whenever ϵ H < 1, whereas slow-roll inflation occurs only when [63] ϵ
For the sake of concreteness, we define the onset of inflation as the time t i when ϵ H (t i ) = 1 for the first time in the transition phase, where ϵ H < 1 for t > t i . The end of the inflationary phase is defined at the time t end when ϵ H (t end ) = 1 again for the first time after t i . Therefore, for t ∈ (t i , t end ) we have ϵ H < 1 and ä > 0, that is, the universe is in its inflationary phase.
Certainly, for the inflationary phase to be slowly rolling, the conditions (3.13) need to be satisfied during the inflation. Once these conditions are satisfied, we have [64] ϵ
• The e-fold N inf during the inflationary phase is defined as
To have a successful inflation, the inflation potential has to be very flat, so that the Universe can expand large enough [63]. All the cosmological problems can be resolved if the Universe expands about 60 e-folds during the inflationary phase, although its exact value depends on the inflationary models [2]. Therefore, in the following we shall require N inf ≳ 60, although our main conclusions do not depend on its precise value. From the above definition it is clear that in general N inf depends on the specific value of ϕ B .
With the above in mind, we are now ready to solve the dynamical equations respectively in LQC and mLQC-I given in the last section for a given inflationary potential V inf (ϕ). In the following, we shall study these two models, LQC and mLQC-I, separately.
To show the effects of the ekpyrotic mechanism on inflation, we find that it is simple and instructive to start with the chaotic potential α 1 = α 2 = 0, despite the fact that this potential has been already ruled out by observations [2]. This is due to the fact that our main conclusions do not depend on the specific forms of the potentials. Then, we shall turn to the potentials with α 1 α 2 ̸ = 0, which are favorable to observations, and find that indeed similar effects occur.
Let us start with the parameters U 0 = 0.0366, p = 0.1, β = 5 for the ekpyrotic potential [36], while α 1 = α 2 = 0 and m = 1.26 × 10 -6 m pl for the chaotic polynomial potential [2]. Then, we find V min (ϕ B ) = -0.0466482, w Bmin = 1.001, w Bmax = 1.2279, ϕ Bmin = -0.262243, ϕ Bmax = 0.0524408. (3.16) In Fig. 2 we plot the numbers of e-folds during the inflationary phase that are produced from different initial values of (ϕ B , φB ) for ϕ B ∈ (ϕ Bmin , ϕ Bmax ). The first column, Figs. 2 (a) and (c), represents the case without the ekpyrotic potential, while the second column, Figs. However, when choosing different values of the ekpyrotic potential parameters, we can get e-folds larger than 60. For example, choosing the ekpyrotic potential parameters U 0 = 0.366, p = 0.05, β = 0.1, while keeping the chaotic potential parameters the same as those chosen in the last case, we find V min (ϕ B ) = -0.539771, w Bmin = 1.001, w Bmax = 3.63706, ϕ Bmin = -0.258051, ϕ Bmax = 2.58055. (3.17) In Fig. 3 we show the e-folds N inf for φB > 0 for the case without and with the ekpyrotic potential respectively. In particular, from Fig. 3 (a) we can see that N inf > 100 for ϕ B ∈ (1.8, 2.6) without the ekpyrotic potential, while Fig. 3 (b) shows that the effects of the ekpyrotic potential is to decrease N inf . However, for ϕ B ≳ 2.062, we still have N inf ≥ 60. This shows that for a combination of chaotic + ekpyrotic potential in LQC, there exist parameters that allow inflation to occur with enough e-folds.
To understand the effects further, in Fig. 3 (c) and (d) we consider the particular case ϕ B = 2.502 and φB > 0. Fig. 3 (c) gives the plot of ϵ H , from which we can see that inflation starts at t i ≃ 2.79 × 10 4 t P and ends at t end ≃ 2.84 × 10 6 t P , for which we find that N inf ≃ 70.1. On the other hand, Fig. 3 (d) plots ρ/ρ c which clearly shows that inflation occurs in the classical regime, during which we have ρ/ρ c ≲ 10 -9 . In Fig. 3 (e) we consider the plot w B vs ϕ B , from which we can see that w B is always greater than 1 for all cases with N inf > 60, whereby the evolution of the universe is dominated by the scalar field, and the effects of shear are highly suppressed near the bounce region.
Now, let us turn to the cases with α 1 α 2 ̸ = 0. To understand the effects of the ekpyrotic mechanism, let us first consider the polynomial chaotic potential without the ekpyrotic one. Then, in Fig. 4 we plot the e-folds vs the initial values of ϕ B for both φB > 0 and φB < 0.
From this figure we can see that inflation with N inf ≳ 60 exists in both cases, by properly choosing the initial values of ϕ B . This is consistent with the results obtained in [46]. When the ekpyrotic potential is turned on, as in the previous chaotic cases, if we choose U 0 = 0.0366, p = 0.1, β = 5, we do not find initial values of (ϕ B , sgn( φB )) that lead to inflation with sufficient e-folds (N inf ≳ 60). However, For the parameters U 0 = 10 3 , p = 0.1, β = 1, we find V min (ϕ B ) = -1000, w Bmin = 1.001, w Bmax = 4886.51, ϕ Bmin = -0.717884, ϕ Bmax = 0.717884. (3.18) Figs. 5 (a) and (b) Show the e-folds respectively without and with the ekpyrotic potential but with φB > 0, from which we can see that the effects of the ekpyrotic mechanism is to decrease the total e-folds, quite similar to the chaotic cases studied above. Again, by properly choosing the free parameters involved in the models, inflation with N inf > 60 is still possible. In particular, in Figs. 5 (c) and (d) we plot ϵ H and ρ/ρ c for ϕ B ≃ 0.652, for which we find the inflation begins at t i ≃ 7.85 × 10 4 t P and ends at t end ≃ 1.41 × 10 7 t P , with a total e-fold N inf ≃ 60.3. During this period, we have ρ/ρ c ≲ 10 -11 , which indicates the inflation happens in the classical regime. In Fig. 5 (d) we show w B (ϕ B ), from which it can be shown that w B can be as large as 2.25 for certain initial conditions.
Similar to the cases studied above in LQC, let us also consider the two cases α 1 = α 2 = 0 and α 1 α 2 ̸ = 0, separately but in the framework of mLQC-I.
1.8 2.0 2.2 2.4 100 110 120 130 140 ϕ B N inf 1.8 2.0 2.2 2.4 0 20 40 60 80 100 120 ϕ B N inf (a) (b) ϕ B =2.502 10 100 1000 10 4 10 5 10 6 10 7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t ϵ H ϕ B =2.502 5 × 10 4 1 × 10 5 5 × 10 5 1 × 10 6 5 × 10 6 1.7 × 10 -10 1.8 × 10 -10 1.9 × 10 -10 2.0 × 10 -10 2.1 × 10 -10 2.2 × 10 -10 t ρ/ρ c 1.8 2.0 2.2 2.4 1.000 1.002 1.004 1.006 1.008 1.010 1.012 1. Chaotic Inflation Let us again first consider the parameters U 0 = 0.0366, p = 0.1, β = 5, with m = 1.26 × 10 -6 m pl and α 1 = α 2 = 0 for the chaotic potential. Then, we find V min (ϕ B ) = -0.0466482, w Bmin = 1.001, w Bmax = 1.96302, ϕ Bmin = -0.326523, ϕ Bmax = 0.0653033. (3.19) In Fig. 6 we plot the corresponding e-folds during the inflationary phase for both φB > 0 and φB < 0 for ϕ B ∈ (ϕ Bmin , ϕ Bmax ), along with the e-fold plots for the purely chaotic case in the same ϕ B range. From these plots we can see that for such choices of the parameters the e-folds during the inflationary phase are always less than 60, no matter whether the ekpyrotic potential is present, given by Figs. 6 (b) and (d), or not, given by Figs. 6 (a) and (c). However, when adjusting the parameters of the ekpyrotic potential, we find that we can get e-folds larger than 60. In particular, in Fig. 7, we show such a case with U 0 = 0.0366, p = 0.05, β = 0.1 and the same m and α 1,2 as considered in the last case. Then, we find V min (ϕ B ) = -0.0539771, w Bmin = 1.001, w Bmax = 2.11432, ϕ Bmin = -0.230877, ϕ Bmax = 2.30887. (3.20) Fig. 7 is plotted for ϕ B ∈ (0.20689, 0.34007) and shows that the e-fold is greater than 60 for ϕ B ≳ 0.272 in the φB > 0 case. For comparison, in this figure we also show the e-folds when the ekpyrotic potential is turned off, given by Fig. 7 (a), from which we can see that now the e-folds are always less than 45. Therefore, in the present case the presence of the ekpyrotic potential alters the evolution of the Universe dramatically and always leads to the development of inflation with sufficient e-folds by properly choosing the initial values of ϕ B in each of the two branches, φB > 0 and φB < 0. This is different from the above cases in which we showed that the ekpyrotic potential always decreases the values of e-folds during the inflationary phase. To understand this in more details, in Fig. 8 we plot ϵ H and the energy density ratio ρ/ρ I c vs t for ϕ B = 0.272 and φB > 0. In the plots of Figs. 8 (a) and (c), the ekpyrotic potential vanishes identically, while in the plots of Figs. 8 (b) and (d) the ekpyrotic potential is present. In the case without the ekpyrotic potential, as shown by Fig. 8 (a), we find N inf ≃ 37.99, by simply first reading out t i and t end and then calculating ln[a(t end )/a(t i )], and the inflation always occurs in the classical regime, as shown by Fig. 8 (c). However, when the ekpyrotic potential is turned on, the universe experiences two different periods of acceleration, the first one is for t/t P ∈ (3.1, 155.4) and the second one is for t/t P ∈ 6.69 × 10 4 , 7.04 × 10 6 , as it can be seen from Fig. 8 (b). During the first period of acceleration, we find N inf ≃ 61, while during the second period we have N inf ≃ 28.64. On the other hand, Fig. 8 (d) shows the energy density ρ is still in the Planck regime during the first phase of the acceleration, while in the second phase it is in the classical regime. Recall that the mass scales like M ≃ (ρ/ρ I c ) 1/4 M P . In addition, in Fig. 9 (a) we show the total e-folds of the inflation when combining the values from the first and second accelerating phases. In this figure, we also plot w B vs ϕ B for ϕ B ∈ (0, 0.34), from which we can see that w B is much greater than one, whereby the scalar field will dominate the evolution of the Universe in the bounce region, and the effects of the shear can be safely ignored.
It must be noted that the development of an accelerating phase in the quantum regime is not a generic result of the effects of the ekpyrotic potential. In particular, if we raise U 0 , we can still obtain inflation occurring in the classical regime with enough e-folds. For example, taking the parameters U 0 = 10 20 , p = 0.1, β = 1, while keeping the rest of the parameters the same as in the last case, we find
In Fig. 10 (a) we show the e-fold of the inflation without the ekpyrotic potential, while in Fig. 10 (b) we show N inf with the ekpyrotic potential, from which we can see that now inflation with N inf > 60 becomes possible for ϕ B ≳ 2.519. In Fig. 10 (c) and (d) we show the plots of ϵ H and ρ/ρ I c for ϕ B = 2.519, from which we can see that the inflation indeed happens in the classical regime. In Fig. 10 (e), we show the plot of the w B vs. ϕ B values. All plots in Fig. 10 are for φB > 0. ϕ B =0.272 10 1000 10 5 10 7 0.0 0.5 1.0 1.5 2.0 t ϵ H ϕ B =0.272 10 1000 10 5 10 7 0.0 0.5 1.0 1.5 2.0 t ϵ H (a) (b) ϕ B =0.272 10 1000 10 5 10 7 0.00 0.02 0.04 0.06 0.08 0.10 0.12 t ρ/ρ c I 10 5 10 6 10 7 0 210 -11 410 -11 610 -11 ϕ B =0.272 10 1000 10 5 10 7 0.0 0.2 0.4 0.6 0.8 1.0 t ρ/ρ c I 5 25 100 -0.02 -0.01 0. (c) (d) FIG. 8. The plots, (a) and (b), for ϵH , and, (c) and (d), for the energy density ρ/ρ I c vs t for ϕB = 0.272 in mLQC-I, with a chaotic potential given by Eq.(3.7) for α1 = α2 = 0, m = 1.26 × 10 -6 m pl and an ekpyrotic potential given by Eq.(3.4) for U0 = 0.0366, p = 0.05, β = 0.1 for the φB > 0 case. The plots (a) and (c) are for the case without the ekpyrotic potential (3.4), and the plots (b) and (d) are for the cases with the ekpyrotic potential.
To see the effects of the high-order powers in the inflationary potential, let us consider the case where α 1 = 0.14, α 2 = 6.644 × 10 -3 while still keeping m = 1.26 × 10 -6 m pl [62], the same choice as in LQC in order to compare the results obtained in LQC and mLQC-I.
Again, inflation with sufficient e-folds cannot occur for any given values of the parameters U 0 , p, β appearing in the ekpyrotic potential. But, we do find values that lead to viable inflationary models. For example, if we choose 2.25 2.30 2.35 2.40 2.45 2.50 115 120 125 130 ϕ B N inf 2.25 2.30 2.35 2.40 2.45 2.50 U 0 = 0.0366, p = 0.05, β = 0.01, while keeping the parameters of the polynomial chaotic inflation as those chosen in [62], we find In Fig. 13, we plot N inf in each of the two periods as well as their sum for the cases φB > 0 [Fig. 13 (a)] andthe case φB < 0 [Fig. 13 (b)] as well as the w B values for the associated ϕ B values for the φB > 0 case [Fig. 13 (c)] and for the φB < 0 case [Fig. 13 (d)]. From this figure we can see that N inf ≳ 60 now can be realized only during the two periods, quantum and classical in both of the cases, φB > 0 and φB < 0.
Again, the period of quantum inflation can be avoided by properly choosing the free parameters involved in the model. In particular, choosing U 0 = 10 3 , p = 0.1, β = 1, while keeping (α 1 , α 2 , m) of the polynomial chaotic potential the same as in the last case, we find V min (ϕ B ) = -1000, w Bmin = 1.001, w Bmax = 20645.3, ϕ Bmin = -0.782164, ϕ Bmax = 0.782164. (3.23) Fig. 14 (b) shows the e-fold values for this case. We can clearly see that we get N inf ≃ 60 for ϕ B ≃ 0.768. Fig. 14 (d) shows the associated ϵ H vs. t for this ϕ B value. From Figs. 14 (c) and (d) we can see that the inflationary phase occurs only in the classical regime. In addition, Fig. 14 (e) shows w B vs ϕ B for φB > 0, from which we can see that ρ ϕ ∝ a -3(1+w B ) dominates the evolution of the universe near the bounce, so that the shear gets highly suppressed. As a result, a homogeneous and isotropic universe can be developed after the bounce.
Inflation is generic in both LQC [46] and mLQC [24]. However, it is not clear how shear will affect the above conclusion, as it is well-known that shear always collapses effectively as 1/a 6 [57], which can dominate the evolution of the universe near the quantum bounce over all other matter fields, a possible exception is the stiff fluid (or massless scalar field). Even in the latter, it is not clear how to ensure that the stiff fluid always dominates the evolution, as both of them grow as a -6 towards the bounce. If the shear dominates the contraction, the universe will become highly anisotropic after the bounce, whereby the assumption of the cosmological principle will be violated. A common mechanism to solve the shear problem either in classical or quantum bouncing cosmological models [17,19,20,[31][32][33] is to introduce an ekpyrotic type of potentials [31,38], which becomes negative near the bounce, so the effective equation of state (EoS) of the scalar field will be greater than one, whereby dominates the shear and other matter fields in the bounce region. As a result, a homogeneous and isotropic universe can be produced after the bounce.
In this paper, we have studied the effects of the ekpyrotic mechanism on the inflationary phase in LQC and mLQC-I, in which the inflation is generic [24,46] with-out considering the ekpyrotic mechanism. To study such effect, we have assumed that the potential of an inflationary field ϕ consists of two parts
where V ekp (ϕ) denotes an ekpyrotic type of potentials, and V inf (ϕ) an inflationary potential. To be specific, we have taken them as given respectively by Eqs.(3.4) and (3.7). By numerically solving the corresponding dynamical equations in the framework of both LQC and mLQC-I, we have found that the effects are dramatic. In particular, initial conditions that led to inflation with sufficient e-folds now become impossible after the ekpyrotic mechanism is taken into account although by properly choosing the free parameters involved in the models and different initial conditions, we have shown that viable inflationary models still exist.
In addition to the above finding, we have also shown that in the framework of mLQC-I certain initial conditions of the ekpyrotic potential can produce two distinct periods of inflation, one occurring in the quantum regime and the other occurring in the classical regime. Other initial conditions, however, produce only a purely classical inflationary period.
Despite of the fact that the above conclusion was obtained by choosing the specific forms of the two poten- tials, given respectively by Eqs.(3.4) and (3.7), we believe that our conclusions hold in more general cases. Another important issue is the effects of the ekpyrotic mechanism on the power spectra and Non-Gaussianity of the cosmological scalar and tensor perturbations, as well as the consistence of such obtained results with observations. We wish to come back to these important issues in other occasions soon.