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These authors have contributed equally to this work

This article was submitted to Cosmology, a section of the journal Frontiers in Astronomy and Space Sciences

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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 (

Just like in any quantization procedure of a classical theory, different regularization schemes also exist in LQC as well as in LQG (

Note that in standard LQC, particularly in the homogeneous and spatially flat

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.

The Hamiltonian formulation of GR is defined on the space-time manifold

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

The Hamiltonian constraint in the full theory of LQG reads (^{
E
} and the Lorentzian term ^{
L
} in

Note that the famous ADM Hamiltonian reads_{
ab
} and

Here the relation between _{
ab
} and the variable

While the Hamiltonian constraint (

Now, we consider the homogeneous and isotropic

Moreover, we introduce a massless scalar field

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” _{
o
}. For the

The Gaussian and diffeomorphism constraints are vanished in the

Then the classical Friedmann equation is

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 _{
Bohr
} and _{
H
} are the Bohr compactification of the real line and Haar measure on it, respectively (

Then those eigenstates satisfy the orthonormal condition:

It turns out that the eigenstates of

Notice that the spatial curvature

Next, to deal with the Lorentzian term, we also need the following identities:^{
E
}(1) is the Euclidean term and

With these ingredients, the Hamiltonian constraint can be written as

The action of this operator on a quantum state Ψ(

Thus, the Hamiltonian constraint (

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

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

Then the effective total Hamiltonian constraint (

Now, we discuss the effective dynamics. By employing the effective Hamiltonian (

Note the bounce takes place at the minimum of volume

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 ^{2}(^{2} with prime be a derivative with respect to

Plugging the above expression into

Solution to this equation reads

The plot of

Then the resulted Friedman equations read

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, _{
cL
} < _{
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;

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,

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.

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.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.