The Dapor-Liegener model of Loop Quantum Cosmology: A dynamical analysis

The Dapor-Liegener model of Loop Quantum Cosmology (LQC), which depicts an emergent universe from a de Sitter regime in the contracting phase is studied from the mathematical viewpoint of dynamical systems and compared with the standard model of LQC. Dealing with perturbations, on the contrary to standard LQC where at early times all the scales are inside the Hubble radius, we show that it is impossible to implement the matter bounce scenario due to the fact that an emergent de Sitter regime in the contracting phase implies that all the scales are outside of the Hubble radius in a past epoch.


INTRODUCTION
In a recent paper [1] (see also [2]), applying Thiemann's procedure for the regularization of the the full Hamiltonian in Loop Quantum Gravity (LQG), Dapor and Liegener have obtained a new effective Hamiltonian for Loop Quantum Cosmology (LQC) which agrees, at the leading order, with the previous one obtained some years ago in [3], but differs from the usual effective Hamiltonian of LQC [4][5][6][7][8][9]. The difference between both approaches lies in the fact that for an spatially flat and homogeneous universe, the Euclidean and the Lorentz terms of the full Hamiltonian are proportional to each other and in LQC it is usual to write the Lorentz term as the Euclidean one and quantize their combination [9]. However, this treatment is impossible in the full LQG theory, where the Lorentz term has to be quantized in a different way from of the Euclidean one [10], obtaining a completely different effective Hamiltonian.
This new effective Hamiltonian constraint leads, contrarily to standard LQC, to a non-symmetric bouncing background emerging from a de Sitter regime in the contracting phase and ending in the expanding one by matching with General Relativity (GR) [2]. And, although this model has already been studied in great detail in several papers [1,11,12], we believe that an analysis from the viewpoint of dynamical systems could simplify the reasoning and help to better understand it.
In fact, working in the plane (H, ρ) where H denotes the Hubble parameter and ρ the energy density of the universe, where the standard model in LQC has the universe crossing and ellipse in clockwise direction, we show that the Dapor-Liegener model has a more complicated behaviour presenting two separate asymmetric branches: In the physical one, and always dealing with a non phantom field or fluid filling the universe, the universe emerges from a de Sitter regime evolving, in the * E-mail: jaime.haro@upc.edu contracting phase, to the bounce, where after entering in the expanding phase it evolves asymtotically into a flat expanding universe obeying GR. In the non-physical one one has, at very early times, a flat contracting universe obeying GR and evolving to the bounce, to enter in the expanding phase where it transits to end up in a de Sitter regime. This second branch is not physical, in spite of the low value of the energy density of the universe, due to the high value of the Hubble parameter in this last stage, which is in disagreement with its very low current value.
Once we have studied the dynamics of the model we deal with perturbations, arguing that due to the fact that the universe emerges from a de Sitter regime in the contracting phase, and thus at early times all the scales are outside the Hubble radius, it is impossible for this new approach of LQC to provide either a matter or matterekpyrotic bouncing scenario as the ones given by standard LQC, where at the beginning all the scales are inside of the Hubble radius [13][14][15][16].
The paper is organized as follows: In Section II we review the dynamics of the standard LQC background. Section III is devoted to the analysis of the new approach of LQC obtaining the corresponding modified Friedmann equation and the dynamical equations. Finally, in the last Section we briefly discuss some features of cosmological perturbations in this new scenario using the so-called dressed effective metric approach [17].
The units used throughout the paper are = c = 1, and 8πG = 1.

DYNAMICS IN STANDARD LQC
We start reviewing the dynamics in standard LQC where the full effective Hamiltonian is given by [4][5][6][7][8][9]  where ρ is the energy density of the universe, γ ∼ = 0.2375 is the Immirzi parameter whose numerical value is obtained comparing the Bekenstein-Hawking formula with the black hole entropy calculated in LQG [18], λ ≡ √ 3 2 γ is the square root of the area gap -the square root of the minimum eigenvalue of the area operator-in LQG [9], V = a 3 is the volume (to simplify the volume of the cubic fiducial cell has taken to be equal to 1) and β is its conjugate momentum, which classically satisfies β = γH [19], being H the Hubble parameter, and whose Poisson bracket is given by {β, V} = γ 2 .
The Hamiltonian constraint H LQC = 0, leads to the following expression of the energy density A simple combination of equations (2) and (3) leads to the holonomy corrected Friedmann equation in standard LQC [20][21][22] where ρ c = 3 γ 2 λ 2 ∼ = 252 is the so-called critical density in standard LQC [7]. From this modified Friedmann equation one can see that at low energy densities (ρ ρ c ) one recovers GR, because this equation becomes the standard Friedmann equation H 2 = ρ 3 which depicts a parabola in the plane (H, ρ). This two curves which at low energy densities coincide, are very different at high energy densities. Effectively, the parabola of GR is unbounded allowing the formation of singularities such as the Big Bang or the Big Rip where the energy density diverges. However, in standard LQC, this kind of singularities are forbidden due to the fact that the ellipse depicted by the Friedmann equation in standard LQC (see FIG. 1) is a closed bounded curve [23,24].
To find the dynamical equation, we have to take into account that holonomy corrections only affect the gravitational sector, for this reason the energy density satisfy the conservation equationρ = −3H(ρ + P ), where P is the pressure. Then, taking the derivative of (4) and using the conservation equation one can easily find the Raychauduri equation in standard LQC [8] Note that from the conservation equation one can see that for a fluid or field with effective Equation of State (EoS) parameter w ef f = P ρ > −1, that is, for a nonphantom fluid or field, the movement accros the ellipse is clockwise, as has been shown in FIG. 1.

FIG. 1: Ellipse depicted by the Friedmann equation in
standard LQC, and its dynamics for a non-phantom field or fluid.
Once we have obtained the dynamical equations, we can consider two different cases: In general, this equation has to be solved numerically, but in the particular case of an constant effective EoS parameter w ef f one obtains the following analytic solution [13]: 2. A universe filled by an scalar field φ minimally coupled with gravity. In this case the energy density is ρ =φ 2 2 + V (φ), and the conservation equation readsφ where once again The difference with the case of a barotropic fluid is that now we have a second order differential equation, meaning that one has infinitely many backgrounds. Moreover, one could also obtain a potential having a background which is the same as the one provided by a barotropic fluid with constant EoS parameter [25,26] Effectively, inserting this potential in (7) one gets the analytic solution which leads to the background (6). The dynamics provided by the potential (8), i.e., the other non-analytic solutions, was recently studied with great detail in [26,27], showing that in the case |w ef f | < 1 all the backgrounds depict a universe with a constant effective EoS parameter equal to w ef f at early and late times. On the contrary, when w ef f > 1, the potential (8) becomes ekpyrotic, the backgrounds depict an universe bouncing twice and after the second bounce it enters, in the expanding phase, in a kination regime (its effective EoS parameter is equal to 1).
A last remark is in order: There are some modified theories that lead to the same Friedmann and Raychauduri equations as in standard LQC. For example, including in the Einstein-Hilbert action a convenient non-linear term of the form f (R), where R is some scalar such that in the Friedmann-Lemaître-Robertson-Walker spacetime becomes proportional to the square of the Hubble parameter [28,29] or else using a modified version of mimetic gravity [30][31][32][33].

DYNAMICS IN THE DAPOR-LIEGENER MODEL OF LQC
In the Dapor-Liegener (DL) model the full effective Hamiltonian is given by [1-3, 11, 12] and the Hamiltonian constraint leads to the following expression of the energy density of the universe In this model, the Hamilton equationV = {V, H DL } = − γ 2 ∂H DL ∂β leads to the following value of the Hubble parameter Introducing the notation sin 2 (βλ) ≡ x, the equation , and reaches its maximum at x = 1 2(γ 2 +1) , meaning that the minimum value of the energy density is 0 and its maximum value is ρ max = ρc 4(γ 2 +1) .
After this brief discussion one can conclude that the variable β belongs in the interval . The equations (12) and (11)  To find the dynamics we perform the temporal derivative of the energy density (11) obtaining: and once again taking into account that holonomy corrections only affect the matter sector, the conservation equation will beρ = −3H(ρ + P ), and one finally gets the equationβ where once again w ef f = P ρ denotes the effective EoS parameter.
Since the derivative of β is zero when the energy density vanishes, the dynamical system has three fixed points at β = ±β i and β = 0, or in the plane (H, ρ), at (± H2 , 0) and (0, 0). Therefore, for a non-phantom fluid or field, i.e., when w ef f > −1, there are two different dynamics:  FIG. 2): The variable β moves from β i to 0, or in the plane (H, ρ) the universe emerges in a de Sitter regime moving in the contracting phase from (− H2 , 0) to (0, ρ max ), where it bounces to enter in the expanding phase, and finally at late times it ends at (0, 0) where GR applies, because when β ∼ = 0 the equations (12) and (11) become H ∼ = β γ and ρ ∼ = 3 β 2 γ 2 , thus combining them one gets the standard Friedmann equation , that is, one recovers GR at low energy densities.
2. Branch 2 (red curve in FIG. 2): The variable β moves from 0 to −β i , or in the plane (H, ρ) the universe starts when GR is valid, moving in the contracting phase from (0, 0) to (0, ρ max ), where the universe bounces entering in the expanding phase, and finally ending in a de Sitter regime at (+ H2 , 0). Obviously, the dynamics in this second branch is not viable because the universe ends in a de Sitter phase with such a large value of the Hubble parameter. To obtain explicitly the dynamics one has to solve the equation (15). In the case of a barotropic fluid with EoS P = P (ρ) one has to solve the equatioṅ with ρ given by equation (11). This is a one dimensional first order differential equation in the variable β which, once it is solved, one has to insert in equations (11) and (12) to obtain the dynamics. In the particular case of a constant effective EoS parameter, the equation (16) can be integrated analytically obtaining an implicit equation of the form F (sin(βλ)) = t but, unfortunately, there is no analytic expression of the inverse of F . Therefore, it is impossible to reach a simple expression such as (6) obtained in standard LQC, only numerical calculations can be performed to obtain the dynamics.
When the universe is filled by an scalar field minimally coupled with gravity the problem is more involved because in this case, equation (15) readṡ and it is impossible to expressφ 2 as a function of β. So, one has to work as in standard LQC, and consider the conservation equation (7), but with another expression of H ± (φ,φ). To find it, one has to isolate sin 2 (βλ) in (11) and insert it in (12). After some algebra one has and since at low energy densities ρ ρ max one has H 2 + (ρ) ∼ = ρ 3 and H 2 − (ρ) ∼ =H 2 , the dynamics in the first branch (the physical one) will be given by the equation where we now denote by H + (φ,φ) the value of the Hubble parameter in the expanding phase and H − (φ,φ) in the contracting one, and we have H ± (φ,φ) = ± H 2 ± (ρ), with ρ =φ 2 2 + V (φ). One can see that equation (19), which as in GR or standard LQC provides infinitely many different backgrounds because it is a second order differential equation, can only be solved numerically.
Another way, the one used in [12], to find numerically backgrounds is to consider the system where H(β) is given by (12). To solve the system one needs three initial conditions, which for simplicity one could take at the bounce, (β B , φ B , ψ B ). As we have already showed, in the first branch, at the bounce one , then one only has to choose a value of φ B satisfying V (φ B ) ≤ ρ max , because, at the bounce, ψ B is determined by the constraint Finally, we want to stress that the background provided by the Dapor-Leigener model does not seem easy to be mimicked using modified or mimetic gravity as has been done for the standard model in LQC [28,31,33], due to the complicated form exhibited by the solution curve to the Friedmann equation (18) in the plane (H, ρ) (see FIG. 2).

PERTURBATIONS
There are two different ways to understand a bouncing scenario (see [34][35][36][37][38] for a review of bouncing cosmologies). One of them is to see it as an implementation of inflation, where the the big bang singularity is replaced by a bounce but the inflationary regime exists in the expanding phase [39][40][41][42]. The other viewpoint is radically different: a bouncing cosmology is an alternative to the inflationary paradigm, and thus, the inflationary phase is removed [43][44][45] in this scenario.
In the first path, an inflationary potential is used and the observable scales leave the Hubble radius during the inflationary regime as in standard inflation but, unlike in inflation, the modes corresponding to those scales are not expected to be in the so-called adiabatic or, sometimes, Bunch-Davies vacuum (see for instance the section 6.2 of [46]), due to their previous evolution in the contracting phase and across the bounce [12]. On the contrary, in the second point of view the modes corresponding to the observable scales, which leave the Hubble radius at very early times in the contracting phase, are in the adiabatic vacuum [13][14][15][16][47][48][49], due to the duality between the de Sitter regime in the expanding phase and a matter domination in the contracting phase [50]. In fact, to obtain a nearly flat power spectrum in this approach, one has to choose a potential which at early times leads to a quasi-matter domination regime in the contracting era [51].
In standard LQC, both points of view have been implemented with success. The first one has been extensively studied in [17,39,40,52,53], and the second one in [13-16, 26, 47-49, 51] studying the matter and the matter-ekpyrotic bounce scenario, and partially showing its viability confronting the theoretical values of spectral quantities such as the spectral index, its running or the ratio of tensor to scalar perturbations with their corresponding observational values.
However, as we will immediately show, in this new version of LQC it is impossible to implement the second viewpoint. Effectively, the deformed algebra approach used in standard LQC [54][55][56][57], which is not fully covariant [58,59], has not been found for this theory yet, so we will deal with the so-called dressed effective metric approach [17,52,53] where the effective metric is the one given by the DL model. Studying scalar perturbations in this approach, the Mukhanov-Sasaki (M-S) equation will be in conformal time [53] v where the potential U is given by Dealing, for instance, with a quartic chaotic potential V = λφ 4 , at very early times, i.e., when ρ ∼ = 0, and thus with φ ∼ = 0 andφ ∼ = 0, one will have U ∼ = 12λφ 2 a 2 .
Remark 4.1 The backgrounds provided by power law potentials V = λφ 2n , which has been reproduced numerically for the particular case of a quadratic potencial (see for instance [11]), are not difficult to understand in LQC. At very early times, since the energy density is zero, the field is at the bottom of the potential starting to oscillate to gaing energy because it is in the contracting phase (recall the conservation equationρ = −3Hφ 2 ). In fact, in the Dapor-Liegener model, contrary to standard LQC, due to the high value of the Hubble parameter in the de Sitter regime, it only needs few oscillations to leave the minimum of the potencial and start to climb up the potential to reach the maximum of energy density and enter the expanding phase, where it rolls down the potential to finish oscillating once again at the bottom of the potential.
On the other hand, recalling that we only consider the first branch because, as we have already discussed, is the only physically viable, at early times the universe is in a de Sitter regime in the contracting phase, meaning that the scale factor evolves as a(t) =ãe tH− , and clearly, lim t→−∞ a(t) = ∞. The conformal time is given by and thus, lim t→−∞ τ = 0, which is completely different to what happens with a de Sitter regime in the expanding phase, because in this case if one denotes by H S > 0 the value of the Hubble parameter, one has τ = −1 aH S , and thus, lim t→−∞ τ = −∞.
This difference, affects directly the M-S equation, which in both cases, in the dressed effective metric approach and standard inflation, has the same approximate form at early times because U ∼ φ 2 τ 2 2 τ 2 (φ ∼ = 0 at very early times) and in standard inflation the M-S equation is [46] where z = aφ H = √ 2 a, being the main slow roll parameter.

Remark 4.2
The same result is obtained for the quadratic potential V = 1 2 m 2 φ 2 , because in this case, at very early times, one has U ∼ = m 2 a 2 with m 2 ∼ 10 −11 . Effectively, in inflation the power spectrum of scalar perturbations is given by [60] where the star means that the quantities are evaluated when the pivot scale leaves the Hubble radius. Since for the quadratic potential the slow roll parameters satisfy * = η * = 2 φ 2 * , and the spectral index is given by n s − 1 = −6 * + 2η * [60] one gets were, as usual, we have taken n s = 0.96 (see for instance [61,62]).
Then, in the contracting phase, as we have already shown, the conformal time starts at τ = 0 and then increases, which means that at the beginning all the modes are outside of the Hubble radius and they enter into it, which is the contrary to what happens in a de Sitter regime in the expanding phase, where at the beginning the conformal time is −∞, and thus, the modes leave the Hubble radius.
For this reason it is impossible to use the second point of view in the physical branch of the new LQC model, because it is needed that the observable scales leave the Hubble radius at very early times. Moreover, there is a more conceptual problem in order to define the vacuum modes. Effectively, the general solution of (24) is a combination of Hankel functions 3/2 (kτ ) + C 2 (k)H 3/2 (kτ ) = Therefore, when the de Sitter regime is in the expanding phase, and all the modes are inside the Hubble radius, the general solution of (24) is approximately equal to and one can choose the vacuum mode taking C 1 (k) = 0 and C 2 (k) = 1 as in the Minkowskian spacetime, because the modes well inside the Hubble radius do not feel gravity. On the contrary, when the de Sitter regime is in the contracting phase, at very early times, all the modes are outside the Hubble radius, and the approximate form of the general solution of (24) is and, from our viewpoint, it not clear at all how to choose the coefficients C 1 (k) and C 2 (k). Of course, the more natural choice seems the same as in a de Sitter regime in the expanding phase, as has been argued in [12], but without the same justification as in inflation because at very early times all modes are outside the Hubble radius feeling gravity.
To end this Section, we will calculate the range of values of the pivot scale k * in co-moving coordinates. The relation with its physical value at time t, namely k phys (t) is given by k * = a(t)k phys (t). The physical value, at the present time, used by the Planck's team is k phys (t 0 ) = 10 2 H 0 [62], where the sub-index 0 means present time. Then, denoting the beginning of the radiation era and the equilibrium matter-radiation by the sub-index R and eq. We will have where T 0 and T eq are the corresponding CMB radiation temperatures, and where we have also used that the evolution is adiabatic (the entropy is conserved) after equilibrium. We now use that during the radiation epoch one has a R aeq 4 = ρeq ρ R , and the formulas [63] ρ eq ∼ = π 2 15 g eq T 4 eq , where g eq ∼ = 3.36 and g R depends on the reheating temperature. Then, we have Now, dealing with an inflationary power law potential V = λφ 2n , where the universe is reheated via particle production due to the oscillations of the inflaton field [64]. After inflation, the universe evolves, up to reheating, in a regime with constant effective EoS parameter given by w ef f ∼ = n−1 n+1 [65,66]. For the sake of simplicity, we consider a quadratic potencial, so after reheating the universe evolves as matter dominated universe. Then, denoting by end the end of the slow-roll period one will have a end a R 3 = ρ R ρ end , and thus k * = 10 2 15 π 2 1/3 (34) Let N B be the number of e-folds from the bounce to the end of the slow-roll phase. Then, taking the scale factor equal to 1 at the bounce -we can do it because we are dealing with geometries with spatially flat sectionswe obtain the formula In this formula, ρ end and N B are calculated from the background. Effectively, inflation ends when the slow roll parameter = 1 2 V φ V 2 is equal to 1. In the case of a quadratic potential this means that φ 2 end = 2. So, given a background φ(t), from φ 2 end = 2 one calculates t end and thus, all the quantities at that time.
Finally, for reheating temperatures consistent with the bounds coming from nucleosynthesis, i.e., in the range between 1 MeV and 10 9 GeV, or in our units, for 10 −21 ≤ T R ≤ 10 −9 , what constraints the pivot scale to be in the range 8 × 10 4 ≤ k * ≤ 8 × 10 8 .

CONCLUSIONS
We have studied in a simple way, but with great detail, the dynamics of the standar and the recent model of LQC proposed by Dapor-Liegener model, showing that, contrarily to the standard model where the observable modes leave the Hubble radius at very early times, it is impossible to implement an alternative to the inflationary paradigm as the matter or matter-bounce scenario due to the fact that the universe emerges, in the contracting phase, from a de Sitter regime, meaning that at early times the physical scales, intead of leaving the Hubble radius, they enters into it. Therefore, one has to understand the DL model of LQC as an implementation of inflation, which solves the initial singularity problem, but where an slow-roll regime is needed to generate the primordial perturbations.