ORIGINAL RESEARCH article
Volume 8 - 2021 | https://doi.org/10.3389/fspas.2021.805998
Loop Quantum Cosmological Model From ADM Hamiltonian
- 1School of Science, Beijing Jiaotong University, Beijing, China
- 2Department 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,
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
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,
Note that the famous ADM Hamiltonian reads
with Kab and
Here the relation between qab and the variable
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”
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
Then those eigenstates satisfy the orthonormal condition:
It turns out that the eigenstates of
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
Next, to deal with the Lorentzian term, we also need the following identities:
where HE(1) is the Euclidean term and V denotes the volume (Thiemann, 2007).
With these ingredients, the Hamiltonian constraint can be written as
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
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
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
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
Now, in order to calculate the evolution of the physical quantity such as matter density and volume of the Universe, we first introduce x = sin2(b). Consider (x′)2 with prime be a derivative with respect to ϕ. From the definition of x, we have
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
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,
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.
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
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.
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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, Poland
Reviewed by:Yi Ling, Institute of High Energy Physics (CAS), China
Sean 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, email@example.com
†These authors have contributed equally to this work