# Loop Quantum Cosmological Model From ADM Hamiltonian

^{1}School of Science, Beijing Jiaotong University, Beijing, China^{2}Department of Physics, South China University of Technology, Guangzhou, China

The loop quantum cosmological model from ADM Hamiltonian is studied in this article. We consider the spatially flat homogeneous FRW model. It turns out that the modified Friedmann equation keeps the same form as the APS LQC model. However, the critical matter density for the bounce point is only a quarter of the previous APS model, that is,

## 1 Introduction

Loop quantum gravity (LQG) is a quantum gravity model which is trying to quantize Einstein’s general relativity (GR) by using background independent techniques. LQG has been widely investigated in last decades (Ashtekar and Lewandowski, 2004; Rovelli, 2004; Han et al., 2007; Thiemann, 2007). Recently, the LQG method has been successfully generalized from GR to the metric

Just like in any quantization procedure of a classical theory, different regularization schemes also exist in LQC as well as in LQG (Ashtekar and Lewandowski, 2004; Thiemann, 2007; Assanioussi et al., 2015). In particular, for the LQC model of flat Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe, alternative Hamiltonian constraint operators were proposed (Ashtekar et al., 2006b; Yang et al., 2009). In the recently proposed model, different from the Ashtekar-Pawlowski-Singh (APS) model (Ashtekar et al., 2006b), one treats the so-called Euclidean term and Lorentzianian term of the Hamiltonian constraint independently (Yang et al., 2009; Assanioussi et al., 2018). It was shown in the study by Assanioussi et al. (2018); Assanioussi et al. (2019) that this model can lead to a new de Sitter epoch evolution scenario where the prebounce geometry could be described at the effective level. Then a natural question which arises is that apart from these two existing LQC models, are there any other possible Hamiltonian operators which can lead to an evolution different from the existing LQC model? Therefore, this article is aimed to explore such possibility.

Note that in standard LQC, particularly in the homogeneous and spatially flat *k* = 0 models, the Euclidean term and the Lorentzian term are proportional to each other. Hence, in the famous APS model of LQC, one only quantizes the Euclidean term, resulting in a symmetric bounce (Ashtekar et al., 2006b). However, this quantization scheme is not the only option although it is popular in the current literature. An alternative option different from the existing model is to let the classical theory only contain the purely Lorentzian term and then quantize it. It is well known that the classically equivalent expressions would generally be nonequivalent after quantization. In particular, given the fact that the quantization expression evolved with the Euclidean term and the Lorentzian term looks quite different, it is hard to believe the resulted quantum evolution will be exactly the same as the APS model. And this article is devoted to the detailed investigation of the LQC model with the purely Lorentzian term, and it is compared with the well-known APS model.

This article is organized as follows: After the short introduction, we give the Hamiltonian constraint we used in this article and derive the classical evolution equations of the Universe in Section 2. Then we construct the corresponding cosmological kinematics in Section 3, where the dynamical difference equation which represents evolution of the Universe is also derived. In Section 4, the bounce behavior is studied, and effective equations are derived in Section 5. Conclusion and some outlook are also presented in the last section.

## 2 An Alternative Hamiltonian Constraint in Loop Quantum Gravity

The Hamiltonian formulation of GR is defined on the space-time manifold *M* which could be foliated as *M* = *R* ×Σ, where Σ is being a three-dimensional spatial manifold and *R* is a real line which represents the time variable. The classical phase space of LQG consists of the so-called Ashtekar-Barbero variables *SU*(2) connection and

where *G* is the gravitational constant and *γ* is the Barbero-Immirzi parameter (Thiemann, 2007).

The classical dynamics of GR is encoded to the three constraints on this phase space, including the Gaussian, the diffeomorphism, and the Hamiltonian constraint. In homogeneous *k* = 0 models of cosmology, the Gaussian and the diffeomorphism constraints are automatically satisfied. Then we only need to consider the remaining Hamiltonian constraint.

The Hamiltonian constraint in the full theory of LQG reads (Thiemann, 2007; Assanioussi et al., 2015)

where *N* is the lapse function, *q* denotes the determinant of the spatial metric, *H*^{E} and the Lorentzian term *H*^{L} in Eq. 2 are denoted, respectively, as

and

Note that the famous ADM Hamiltonian reads

with *K*_{ab} and *M*. Inspired by the ADM Hamiltonian, by using the relation

Here the relation between *q*_{ab} and the variable *κ* = 8*πG*, and the lapse function is fixed as *N* = 1 for homogeneous Universe in the current article. Note that the APS model of LQC only evolves the Euclidean term as

While the Hamiltonian constraint (Eq. 5) does not contain the Euclidean term, we call this form of Hamiltonian constraint as *purely Lorentzian*. We start from this form.

Now, we consider the homogeneous and isotropic *k* = 0 model. According to the cosmological principle, the metric Friedman-Robertson-Walker (FRW) Universe reads

where *a*(*t*) is the scale factor. At the classical level, one assumes that the Universe be filled by some perfect fluid with matter density *ρ* and pressure *P*.

Moreover, we introduce a massless scalar field *ϕ* as the matter content of the Universe; we denote the conjugate momenta of the scalar field as *π*, and the commutator between them reads

In order to mimic the full theory of LQG, we do the following symmetric reduction procedures of the connection formalism as in standard LQC. First, we introduce an “elemental cell” *V*_{o}. For the *k* = 0 FRW model, we also have *γ* is a nonzero real number and

where *ϕ*, *π*). The nonvanishing Poisson brackets between them read

The Gaussian and diffeomorphism constraints are vanished in the *k* = 0 model. Hence, the remaining Hamiltonian constraint (Eq. 6) reduces to

The equation of motion of geometrical variable *p* reads

Then the classical Friedmann equation is

where *H* is the Hubble parameter. By using the Hamiltonian constraint (Eq. 10), we found that

where the matter density

## 3 Kinematic Structure of Loop Quantization Cosmology

To quantize the cosmological model, we first need to construct the corresponding quantum kinematics of cosmology by the so-called polymer-like quantization. The kinematical Hilbert space for the geometry part can be defined as *R*_{Bohr} and *dμ*_{H} are the Bohr compactification of the real line and Haar measure on it, respectively (Ashtekar et al., 2003). Moreover, we choose the standard Schrodinger representation for the scalar field (Ashtekar and Singh, 2011). Thus, the kinematical Hilbert space for the scalar field part is defined as in usual quantum mechanics, *μ*⟩ be the eigenstates of

Then those eigenstates satisfy the orthonormal condition:

where

where

It turns out that the eigenstates of *ϕ*, *v*⟩ as the generalized orthonormal basis for the whole Hilbert space

### 3.1 Hamiltonian Constraint of LQC With the Purely Lorentzian Term

Notice that the spatial curvature *R* is vanished in the *k* = 0 homogenous cosmology, and the Hamiltonian constraint (Eq. 6) reduces to

which is the purely Lorentzian term. Note that there is no operator existing corresponding to the connection variable

where *V* that are tangent to *μ*. Moreover, we also have

Next, to deal with the Lorentzian term, we also need the following identities:

and

where *H*^{E}(1) is the Euclidean term and *V* denotes the volume (Thiemann, 2007).

With these ingredients, the Hamiltonian constraint can be written as

with

The action of this operator on a quantum state Ψ(*v*, *ϕ*) is already known in the literature (Yang et al., 2009). The result is a difference equation. Hence, the final result is

where

where

Thus, the Hamiltonian constraint (Eq. 21) has been successfully quantized in the cosmological setting. The resulting Hamiltonian constraint equation of LQC turns out to be

Note that in the quantum theory, the whole Hilbert space consists of a direct product of two parts as

where *B*(*v*) can be found in the study by Ashtekar et al. (2006b).

## 4 Effective Hamiltonian of LQC

Now, we come to study the effective theory of this new LQC since we also want to know the effect of matter fields on the dynamic evolution. Hence, we include a scalar matter field *φ* into LQC. Note that the cosmological expectation value for the Lorentzian term has already been obtained in the literature as (Yang et al., 2009; Dapor and Liegener, 2018)

Then the effective total Hamiltonian constraint (Eq. 15) reads

where

## 5 Effective Equations and the Quantum Bounce

Now, we discuss the effective dynamics. By employing the effective Hamiltonian (Eq. 28), the equation of motion for *v* reads

Note the bounce takes place at the minimum of volume *v*, and therefore happened at the point of

So, the density can be expressed as

where

Now, in order to calculate the evolution of the physical quantity such as matter density and volume of the Universe, we first introduce *x* = sin^{2}(*b*). Consider (*x*′)^{2} with prime be a derivative with respect to *ϕ*. From the definition of *x*, we have

and

Plugging the above expression into Eq. 32, we find the equation

Solution to this equation reads

and hence from Eq. 30

so the volume is

The plot of *V(ϕ)* and *ϕ* can be found in Figure 1. Now let us study the asymptotic behavior of the above LQC model in the classical region, namely, the large *v* region. For *v* → *∞* limit, the matter density *ρ* in Eq. 30 goes to zero, which leads to

in situation *b*↦0, and the asymptotic Hamiltonian constraint reads

while

Then the resulted Friedman equations read

which is also a symmetric bounce.

## 6 Concluding Remarks

To summarize, the loop quantum cosmological model which consists of the purely Lorentzian term is studied in this article. We consider the spatially flat homogeneous FRW model. It turns out that the modified Friedmann equation keeps the same form as the APS LQC model. However, the critical matter density for the bounce point is only a quarter of the previous APS model, that is, *ρ*_{cL} < *ρ*_{c}, in this sense, the lower critical bounce density means the quantum gravity effects will get involved earlier than the previous LQC model. Besides, the lower critical density also means the detection of quantum gravity effects is easier than the previous model. It should be note that in this article, we only consider the cosmological implication of LQC from ADM Hamiltonian which only contains the purely Lorentzian term. However, since the Lorentzian term and the Euclidean term lead to different results at the quantum mechanical level, one can also naturally consider the mixture of these two terms. This of course possible; Assanioussi et al. (2018); Assanioussi et al. (2019); Zhang et al. (2021) show that when we consider the Lorentzian term and the Euclidean term appearing in the Hamiltonian constraint simultaneously, an effective cosmological constant could emerge at the large volume limit.

It should be noted that there are many aspects of the new LQC which deserve further investigating. For example, it is still desirable to the perturbation theory of the new LQC; in this case, the spatial curvature will not be zero. And thus could be inherent more features from the full theory of LQG. Moreover, Yang et al. (2019) adapt an alternative regularization procedure *via* the Chern–Simons theory which is quite different from the usual regularization method in LQG, and the resulting cosmological evolution is different from the APS LQC model. Hence, it is also interesting to study this regularization under our framework of new LQC. We leave all these interesting topics for future study.

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

## Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Funding

This work was supported by the NSFC with Grant No. 11 775 082.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

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Keywords: lorentzian term, loop quantum cosmology, effective equation, bounce, hamiltonian constraint

Citation: Ding Y and Zhang X (2022) Loop Quantum Cosmological Model From ADM Hamiltonian. *Front. Astron. Space Sci.* 8:805998. doi: 10.3389/fspas.2021.805998

Received: 31 October 2021; Accepted: 27 December 2021;

Published: 24 January 2022.

Edited by:

Chunshan Lin, Jagiellonian University, PolandReviewed by:

Yi Ling, Institute of High Energy Physics (CAS), ChinaSean Crowe, SPAWAR Systems Center Pacific (SSC Pacific), United States

Copyright © 2022 Ding and Zhang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Xiangdong Zhang, scxdzhang@scut.edu.cn

^{†}These authors have contributed equally to this work