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Mathematics
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Mathematical and Computational Cosmology
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Mathematical and Computational Cosmology

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 December 2019) | Viewed by 12313

Special Issue Editor

Prof. Dr. José Velhinho

E-Mail Website
Guest Editor
Faculdade de Ciências, Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6201-001 Covilhã, Portugal
Interests: mathematical physics; loop quantum gravity; quantum cosmology; quantum fields in curved spacetime
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Over the last few decades, cosmology has matured into a solid scientific discipline, with outstanding progress being achieved, both from the observational and the theoretical point of view. Indeed, since the discovery of the cosmic microwave background, a series of accumulated technological advances has made the acquisition of very precise data possible, probing the early epoch of our Universe. On the theoretical side, several approaches have been put forward, currently reaching a high degree of sophistication, both from a mathematical physics and from a computational and numerical perspective. These theoretical approaches range from the more classically inspired to the fully quantum perspective, including, e.g., models inspired from modified gravity, phase space deformation or string theory, as well as quantum models based either on canonical quantization methods or on covariant or path integral related ones. Putting observational and theoretical progress together has enabled us to address key issues, such as the fate of classical cosmological singularities, the inflationary scenario or structure formation in the early universe.

In this Special Issue, we welcome contributions from all perspectives in modern cosmology, with a strong emphasis on rigorous mathematical methods. 

Prof. Dr. José Velhinho
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Computational and numerical methods in cosmology
  • Mathematical foundations of modern cosmology
  • Mathematical methods in inhomogeneous cosmology
  • Singularity avoidance
  • Inflationary scenarios
  • Cosmological perturbations
  • Structure formation
  • Quantum fields in cosmological backgrounds
  • Quantum effects in the early Universe

Published Papers (5 papers)

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Research

16 pages, 391 KiB  
Article
From Fractional Quantum Mechanics to Quantum Cosmology: An Overture
by Paulo Vargas Moniz and Shahram Jalalzadeh
Mathematics 2020, 8(3), 313; https://doi.org/10.3390/math8030313 - 01 Mar 2020
Cited by 21 | Viewed by 2763
Abstract
Fractional calculus is a couple of centuries old, but its development has been less embraced and it was only within the last century that a program of applications for physics started. Regarding quantum physics, it has been only in the previous decade or [...] Read more.
Fractional calculus is a couple of centuries old, but its development has been less embraced and it was only within the last century that a program of applications for physics started. Regarding quantum physics, it has been only in the previous decade or so that the corresponding literature resulted in a set of defying papers. In such a context, this manuscript constitutes a cordial invitation, whose purpose is simply to suggest, mostly through a heuristic and unpretentious presentation, the extension of fractional quantum mechanics to cosmological settings. Being more specific, we start by outlining a historical summary of fractional calculus. Then, following this motivation, a (very) brief appraisal of fractional quantum mechanics is presented, but where details (namely those of a mathematical nature) are left for literature perusing. Subsequently, the application of fractional calculus in quantum cosmology is introduced, advocating it as worthy to consider: if the progress of fractional calculus serves as argument, indeed useful consequences will also be drawn (to cite from Leibnitz). In particular, we discuss different difficulties that may affect the operational framework to employ, namely the issues of minisuperspace covariance and fractional derivatives, for instance. An example of investigation is provided by means of a very simple model. Concretely, we restrict ourselves to speculate that with minimal fractional calculus elements, we may have a peculiar tool to inspect the flatness problem of standard cosmology. In summary, the subject of fractional quantum cosmology is herewith proposed, merely realised in terms of an open program constituted by several challenges. Full article
(This article belongs to the Special Issue Mathematical and Computational Cosmology)
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19 pages, 353 KiB  
Article
xAct Implementation of the Theory of Cosmological Perturbation in Bianchi I Spacetimes
by Ivan Agullo, Javier Olmedo and Vijayakumar Sreenath
Mathematics 2020, 8(2), 290; https://doi.org/10.3390/math8020290 - 20 Feb 2020
Cited by 3 | Viewed by 3393
Abstract
This paper presents a computational algorithm to derive the theory of linear gauge invariant perturbations on anisotropic cosmological spacetimes of the Bianchi I type. Our code is based on the tensor algebra packages xTensor and xPert, within the computational infrastructure of xAct written [...] Read more.
This paper presents a computational algorithm to derive the theory of linear gauge invariant perturbations on anisotropic cosmological spacetimes of the Bianchi I type. Our code is based on the tensor algebra packages xTensor and xPert, within the computational infrastructure of xAct written in Mathematica. The algorithm is based on a Hamiltonian, or phase space formulation, and it provides an efficient and transparent way of isolating the gauge invariant degrees of freedom in the perturbation fields and to obtain the Hamiltonian generating their dynamics. The restriction to Friedmann–Lemaître–Robertson–Walker spacetimes is straightforward. Full article
(This article belongs to the Special Issue Mathematical and Computational Cosmology)
18 pages, 344 KiB  
Article
The Effect of a Positive Cosmological Constant on the Bounce of Loop Quantum Cosmology
by Mercedes Martín-Benito and Rita B. Neves
Mathematics 2020, 8(2), 186; https://doi.org/10.3390/math8020186 - 04 Feb 2020
Viewed by 1637
Abstract
We provide an analytical solution to the quantum dynamics of a flat Friedmann-Lemaître- Robertson-Walker model with a massless scalar field in the presence of a small and positive cosmological constant, in the context of Loop Quantum Cosmology. We use a perturbative treatment with [...] Read more.
We provide an analytical solution to the quantum dynamics of a flat Friedmann-Lemaître- Robertson-Walker model with a massless scalar field in the presence of a small and positive cosmological constant, in the context of Loop Quantum Cosmology. We use a perturbative treatment with respect to the model without a cosmological constant, which is exactly solvable. Our solution is approximate, but it is precisely valid at the high curvature regime where quantum gravity corrections are important. We compute explicitly the evolution of the expectation value of the volume. For semiclassical states characterized by a Gaussian spectral profile, the introduction of a positive cosmological constant displaces the bounce of the solvable model to lower volumes and to higher values of the scalar field. These displacements are state dependent, and in particular, they depend on the peak of the Gaussian profile, which measures the momentum of the scalar field. Moreover, for those semiclassical states, the bounce remains symmetric, as in the vanishing cosmological constant case. However, we show that the behavior of the volume is more intricate for generic states, leading in general to a non-symmetric bounce. Full article
(This article belongs to the Special Issue Mathematical and Computational Cosmology)
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49 pages, 633 KiB  
Article
Quantum Linear Scalar Fields with Time Dependent Potentials: Overview and Applications to Cosmology
by Jerónimo Cortez, Guillermo A. Mena Marugán and José Velhinho
Mathematics 2020, 8(1), 115; https://doi.org/10.3390/math8010115 - 11 Jan 2020
Cited by 6 | Viewed by 2153
Abstract
In this work, we present an overview of uniqueness results derived in recent years for the quantization of Gowdy cosmological models and for (test) Klein-Gordon fields minimally coupled to Friedmann-Lemaître-Robertson-Walker, de Sitter, and Bianchi I spacetimes. These results are attained by imposing the [...] Read more.
In this work, we present an overview of uniqueness results derived in recent years for the quantization of Gowdy cosmological models and for (test) Klein-Gordon fields minimally coupled to Friedmann-Lemaître-Robertson-Walker, de Sitter, and Bianchi I spacetimes. These results are attained by imposing the criteria of symmetry invariance and of unitary implementability of the dynamics. This powerful combination of criteria allows not only to address the ambiguity in the representation of the canonical commutation relations, but also to single out a preferred set of fundamental variables. For the sake of clarity and completeness in the presentation (essentially as a background and complementary material), we first review the classical and quantum theories of a scalar field in globally hyperbolic spacetimes. Special emphasis is made on complex structures and the unitary implementability of symplectic transformations. Full article
(This article belongs to the Special Issue Mathematical and Computational Cosmology)
11 pages, 238 KiB  
Article
Properties of Fluctuating States in Loop Quantum Cosmology
by Martin Bojowald
Mathematics 2019, 7(7), 645; https://doi.org/10.3390/math7070645 - 19 Jul 2019
Cited by 5 | Viewed by 1872
Abstract
In loop quantum cosmology, the values of volume fluctuations and correlations determine whether the dynamics of an evolving state exhibits a bounce. Of particular interest are states that are supported only on either the positive or the negative part of the spectrum of [...] Read more.
In loop quantum cosmology, the values of volume fluctuations and correlations determine whether the dynamics of an evolving state exhibits a bounce. Of particular interest are states that are supported only on either the positive or the negative part of the spectrum of the Hamiltonian that generates this evolution. It is shown here that the restricted support on the spectrum does not significantly limit the possible values of volume fluctuations. Full article
(This article belongs to the Special Issue Mathematical and Computational Cosmology)
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