Quantum corrections to the Mukhanov-Sasaki equations
TL;DR: In this article, the authors introduce technical tools and design a procedure to deal with these quantum corrections beyond the most direct approximations employed so far in the literature, by introducing an interaction picture, extracting the quantum dynamics of the homogeneous geometry in absence of scalar field potential and inhomogeneities.
Abstract: Recently, a lot of attention has been paid to the modifications of the power spectrum of primordial fluctuations caused by quantum cosmology effects. The origin of these modifications is corrections to the Mukhanov-Sasaki equations that govern the propagation of the primeval cosmological perturbations. The specific form of these corrections depends on a series of details of the quantization approach and of the prescription followed to implement it. Generally, the complexity of the theoretical quantum formulation is simplified in practice appealing to a semiclassical or effective approximation in order to perform concrete numerical computations. In this work, we introduce technical tools and design a procedure to deal with these quantum corrections beyond the most direct approximations employed so far in the literature. In particular, by introducing an interaction picture, we extract the quantum dynamics of the homogeneous geometry in absence of scalar field potential and inhomogeneities, dynamics that has been intensively studied and that can be integrated. The rest of our analysis focuses on the interaction evolution, putting forward methods to cope with it. The ultimate aim is to develop treatments that increase our ability to discriminate between the predictions of different quantization proposals for cosmological perturbations.
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TL;DR: In this article, the authors investigated the consequences of the hybrid quantization approach for primordial perturbations in loop quantum cosmology, obtaining predictions for the cosmic microwave background and comparing them with data collected by the Planck mission.
Abstract: We investigate the consequences of the hybrid quantization approach for primordial perturbations in loop quantum cosmology, obtaining predictions for the cosmic microwave background and comparing them with data collected by the Planck mission. In this work, we complete previous studies about the scalar perturbations and incorporate tensor modes. We compute their power spectrum for a variety of vacuum states. We then analyze the tensor-to-scalar ratio and the consistency relation between this quantity and the spectral index of the tensor power spectrum. We also compute the temperature-temperature, electric-electric, temperature-electric, and magnetic-magnetic correlation functions. Finally, we discuss the effects of the quantum geometry in these correlation functions and confront them with observations.
64 citations
Cites background from "Quantum corrections to the Mukhanov..."
...[34] shows how one can carry out a perturbative treatment at the quantum level, valid in those situations where the contribution of the potential is small compared with the kinetic term....
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TL;DR: In this paper, the trans-planckian problem is considered in loop quantum cosmology and its relation to loop quantum gravity, and a possible transition from a Lorentzian to a Euclidean space-time around the bounce point.
63 citations
TL;DR: In this article, an erratum to this article is available at https://doi.org/10.1007/JHEP06(2017)020.109 pags., 1 fig., 9 tabs.
Abstract: 109 pags., 1 fig., 9 tabs. -- Open Access funded by Creative Commons Atribution Licence 4.0. -- An erratum to this article is available at https://doi.org/10.1007/JHEP06(2017)020
63 citations
TL;DR: In this article, it was shown that the mass of the tensor perturbations is positive in the hybrid approach when the kinetic contribution to the energy density of the inflaton dominates over its potential, as well as for a considerably large sector of background around that situation, while this mass is always nonpositive in the dressed metric approach.
Abstract: Loop quantum cosmology has recently been applied in order to extend the analysis of primordial perturbations to the Planck era and discuss the possible effects of quantum geometry on the cosmic microwave background. Two approaches to loop quantum cosmology with admissible ultraviolet behavior leading to predictions that are compatible with observations are the so-called hybrid and dressed metric approaches. In spite of their similarities and relations, we show in this work that the effective equations that they provide for the evolution of the tensor and scalar perturbations are somewhat different. When backreaction is neglected, the discrepancy appears only in the time-dependent mass term of the corresponding field equations. We explain the origin of this difference, arising from the distinct quantization procedures. Besides, given the privileged role that the big bounce plays in loop quantum cosmology, e.g. as a natural instant of time to set initial conditions for the perturbations, we also analyze the positivity of the time-dependent mass when this bounce occurs. We prove that the mass of the tensor perturbations is positive in the hybrid approach when the kinetic contribution to the energy density of the inflaton dominates over its potential, as well as for a considerably large sector of backgrounds around that situation, while this mass is always nonpositive in the dressed metric approach. Similar results are demonstrated for the scalar perturbations in a sector of background solutions that includes the kinetically dominated ones; namely, the mass then is positive for the hybrid approach, whereas it typically becomes negative in the dressed metric case. More precisely, this last statement is strictly valid when the potential is quadratic for values of the inflaton mass that are phenomenologically favored.
33 citations
TL;DR: In this article, the effects of perturbations on the non-Gaussinity and their implication on the explanations of observed power asymmetry in CMB have also been explored.
Abstract: While loop quantum cosmology (LQC) predicts a robust quantum bounce of the background evolution of a Friedmann-Robertson-Walker (FRW) spacetime prior to the standard slow-roll inflation, whereby the big bang singularity is resolved, there are several different quantization procedures to cosmological perturbations, for instance, {\em the deformed algebra, dressed metric, and hybrid quantizations} This paper devotes to study the quantum bounce effects of primordial perturbations in the hybrid approach The main discrepancy of this approach is the effective positive mass at the quantum bounce for the evolution of the background that is dominated by the kinetic energy of the inflaton field at the bounce, while this mass is always nonpositive in the dressed metric approach It is this positivity of the effective mass that violates the adiabatic evolution of primordial perturbations at the initial moments of the quantum bounce With the assumption that the evolution of the background is dominated by the kinetic energy of the inflaton at the bounce, we find that the effective potentials for both scalar and tensor perturbations can be well approximately described by a Poschl-Teller (PT) potential, which allows us to find analytical solutions of perturbations, and from these analytical expressions we are able to study the non-adiabatic evolution of primordial perturbations in details In particular, we derive their quantum bounce effects and investigate their observational constraints In addition, the impacts of quantum bounce effects on the non-Gaussinity and their implication on the explanations of observed power asymmetry in CMB have also been explored
32 citations
References
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05 Dec 2005TL;DR: Inflationary cosmology has been developed over the last twenty years to remedy serious shortcomings in the standard hot big bang model of the universe as mentioned in this paper and explains the basis of modern cosmology and shows where the theoretical results come from.
Abstract: Inflationary cosmology has been developed over the last twenty years to remedy serious shortcomings in the standard hot big bang model of the universe. This textbook, first published in 2005, explains the basis of modern cosmology and shows where the theoretical results come from. The book is divided into two parts; the first deals with the homogeneous and isotropic model of the Universe, the second part discusses how inhomogeneities can explain its structure. Established material such as the inflation and quantum cosmological perturbation are presented in great detail, however the reader is brought to the frontiers of current cosmological research by the discussion of more speculative ideas. An ideal textbook for both advanced students of physics and astrophysics, all of the necessary background material is included in every chapter and no prior knowledge of general relativity and quantum field theory is assumed.
1,753 citations
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01 Sep 2007TL;DR: The field of Canonical Quantum General Relation (CQGR) as mentioned in this paper is an attempt to define a mathematically rigorous, non-perturbative, background independent theory of Lorentzian quantum gravity in four spacetime dimensions in the continuum.
Abstract: This is an introduction to the by now fifteen years old research field of canonical quantum general relativity, sometimes called "loop quantum gravity". The term "modern" in the title refers to the fact that the quantum theory is based on formulating classical general relativity as a theory of connections rather than metrics as compared to in original version due to Arnowitt, Deser and Misner. Canonical quantum general relativity is an attempt to define a mathematically rigorous, non-perturbative, background independent theory of Lorentzian quantum gravity in four spacetime dimensions in the continuum. The approach is minimal in that one simply analyzes the logical consequences of combining the principles of general relativity with the principles of quantum mechanics. The requirement to preserve background independence has lead to new, fascinating mathematical structures which one does not see in perturbative approaches, e.g. a fundamental discreteness of spacetime seems to be a prediction of the theory providing a first substantial evidence for a theory in which the gravitational field acts as a natural UV cut-off. An effort has been made to provide a self-contained exposition of a restricted amount of material at the appropriate level of rigour which at the same time is accessible to graduate students with only basic knowledge of general relativity and quantum field theory on Minkowski space.
1,686 citations
"Quantum corrections to the Mukhanov..." refers methods in this paper
...The basic holonomies of the connection are taken along straight lines with a length such that the square formed by them has a physical area equal to the non-vanishing minimum ∆ allowed by LQG [34]....
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...Let us focus our comments on LQC. LQC is the application of the methods of Loop Quantum Gravity (LQG) [18] to cosmology....
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...LQC is the application of the methods of Loop Quantum Gravity (LQG) [18] to cosmology....
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...In LQG, for instance, this is possible even analytically by adopting a prescription called “solvable LQC” (sLQC) [32]....
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...Let us also mention that the possible consequences of LQC in the CMB have been studied as well by adopting another viewpoint, namely, by postulating the deformations of the spacetime diffeomorphisms algebra that one might expect that arise in LQG, demanding then the closure of the modified algebra for consistency [14]....
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Book•
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06 Jan 2012
TL;DR: In this article, a textbook on non-relativistic quantum mechanics that emphasizes clarification of the nature of the basic postulates and the interpretation of the theory is presented, which is aimed at students in their second year of university.
Abstract: This is a textbook on non-relativistic quantum mechanics that emphasizes clarification of the nature of the basic postulates and the interpretation of the theory. It contains special material, often only accessible in scientific journals, on bound states, scattering theory, and both analytical and approximation techniques. Applications to many branches of physics are given. Among the topics covered are one-dimensional problems, angular momentum, two-particle systems, symmetry transformations, collision theory, the WKB method, and stationary and time-dependent perturbation and variational techniques. Particles in an electromagnetic field, many-body systems, atoms, and radiation theory are studied. The book is a considerably improved and completely updated English translation of a very successful Spanish textbook and is aimed at students in their second year of university.
104 citations
"Quantum corrections to the Mukhanov..." refers background or methods in this paper
...In that sector, one can approximate the evolution generated by Ĥ2I , truncating it at certain order in the potential....
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...The obstruction that we find now is the integration of the evolution generated by Ĥ2I and by Ĥ3J in order to calculate the unitary operators Û2I and ÛJ , respectively....
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...Therefore, the remaining interaction dynamics is relevant at the order of truncation provided that rBC ≻ 1, just as above when Ĥ2I included only factors that are linear in the potential....
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...In a third step we have split this operator in two parts, one of them (called Ĥ2I) capturing the contributions with the lowest powers of the potential....
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...In particular, we can truncate the series expansion of Û2I in terms of path ordered integrals of powers of Ĥ2I [37] so as to compute the operator ÂJ up to a certain order of the potential....
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