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We construct a noncommutative extension of the Loop Quantum Cosmology effective scheme for the flat FLRW model with a free scalar field via a theta deformation. Firstly, a deformation is implemented in the configuration sector, among the holonomy variable and the matter degree of freedom. We show that this type of noncommutativity retains, to some degree, key features of the Loop Quantum Cosmology paradigm for a free field. Secondly, a deformation is implemented in the momentum sector, among the momentum associated with the holonomy variable and the momentum associated with the matter field. We show that in this latter case the scalar field energy density is the same as the one in standard Loop Quantum Cosmology.

The idea of noncommutativity in spacetime is not new (1947) [

At the end of the last century the noncommutative paradigm was resurrected, mainly due to results in String Theory [

A possible way to model these effects could be via an uncertainty relation for the spacetime coordinates of the form

A possible way to study noncommutativity effects in the early universe was proposed by García-Compeán et al. [

On the other hand, LQG [

As a consequence of loop quantization, the Wheeler-DeWitt equation is no longer a differential equation, but a difference equation, which is difficult to work with even in the simplest models. In order to extract physics, effective equations based on a geometrical formulation of quantum mechanics have been employed to study the outcome of loop quantum corrections in cosmological models [

Recently, works that focus on the possible relation, at different physical/mathematical levels, among noncommutativity and LQG have been conducted. For instance, an emergent noncommutativity in LQG is found in [

The present investigation aims at constructing a noncommutative effective scheme for the flat FLRW model in the presence of a free scalar field, and establishing whether such noncommutative scheme could be compatible with the LQC paradigm, in the sense of retaining key features of the LQC of the flat FLRW model. Along this lines, our work could be related to a minisuperspace approximation of a more fundamental noncommutative construction based on the LQG approach, such as the one in [

The manuscript is organized as follows: In Section

The line element of a spatially homogeneous and isotropic universe is

In this section, we recall the formulation of the flat FLRW model in the Ashtekar-Barbero variables, with a free massless scalar field. The Ashtekar-Barbero variables cast General Relativity in the form of a gauge theory, in which phase space is described by an

From the gravitational Hamiltonian density (

Taking into account the discussion of the last paragraph for the Hamiltonian constraint (

The relational evolution of

The ideas of deformed minisuperspace, in connection with noncommutative cosmology, were introduced in the seminal work of García-Compeán et al. [

In the deformation phase space approach, the deformation is introduced by the Moyal brackets

In this two scenarios the outcome can be summarized as follows: first, in the deformation quantization formalism, the

We will start this section analysing the flat FLRW model for a free scalar field and without the LQC corrections. In Ashtekar variables and volume representation, the classical Hamiltonian takes the form

The deformed Hamiltonian constraint arising from the steps above has the same functional form as (

Now we want to implement noncommutativity in the flat FLRW model, by working in the shifted variables (

Considering the steps above and taking into account the ideas posed in the last section, we can therefore implement the effects of relation (

We note that the equations for

We observe that

The equation for

The behavior of

By substituting the solution for

The behavior of the volume function is the same as in standard LQC.

The behavior of the volume as a function of

Substituting the solutions for

Employing the solution for

The behavior of the energy density for different values of

In this section we would like to study the effects of noncommutativity in the momentum sector. The way to proceed is in the same manner as the past sections. Consider a deformed algebra

The deformed Hamiltonian takes the form

The equation for

With the help of the Hamiltonian constraint, we observe that

A simple noncommutative extension of the open FLRW model in Loop Quantum Cosmology has been constructed, through the introduction of a deformation at the effective scheme of Loop Quantum Cosmology. These models could incorporate effective corrections from both Loop Quantum Gravity and Noncommutative Geometry.

When introducing noncommutativity in the configuration sector, it is observed from the equation for

For the case of noncommutativity in the momentum sector, it was observed that the equation for

Finally, we conclude that a deformation in the momentum sector of the phase space spanned by the flat FLRW model with a standard free scalar field is more compatible with the LQC paradigm than a deformation in the configuration sector. Of course, further research is required to establish how deep this compatibility is. Some of this additional analysis will be reported by the authors elsewhere.

Only analytical investigations (no data) were conducted to support the findings of this study.

Part of this manuscript was presented at Journal of Physics Conference Series at URL:

The authors declare that they have no conflicts of interest.

This work was partially supported by CONACyT 167335 and 179881 Grants and PROMEP Grants UGTO-CA-3. Abraham Espinoza-García was supported by a CONACYT Graduate Fellowship during a major part of the realization of this work. Sinuhé Pérez-Payán was partially supported by SNI-CONACyT. This work is part of the collaboration within the Instituto Avanzado de Cosmologa and Red PROMEP: Gravitation and Mathematical Physics under project