Gerald B Cleaver | Baylor University (original) (raw)

Papers by Gerald B Cleaver

Research paper thumbnail of Hawking Radiation as a Possible Probe for the Interior Structure of Regular Black Holes

International Journal of Theoretical Physics, 2016

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Research paper thumbnail of SO(10) SUSY-GUTs Based on Superstrings

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Research paper thumbnail of Phenomenological Survey of a String Derived MSSM

Recently, we demonstrated the existence of heterotic-string solutions in which the observable sec... more Recently, we demonstrated the existence of heterotic-string solutions in which the observable sector effective field theory just below the string scale reduces to that of the MSSM, with the standard observable gauge group being just SU(3)C x SU(2)L x SU(2)Y and the SU(3)C x SU(2)L x SU(2)_Y-charged spectrum of the observable sector consisting solely of the MSSM spectrum. Associated with this model is a set of distinct flat directions of vacuum expectation values (VEVs) of non-Abelian singlet fields that all produce solely the MSSM spectrum. In this talk we (1) present our string derived MSSM and (2) study the effective superpotential induced by these choices of flat directions. We investigate whether sufficient degrees of freedom exist in these singlet flat directions to satisfy various phenomenological constraints imposed by the observed Standard Model data. For each flat direction, the effective superpotential is given to sixth order. The variations of the singlet and hidden secto...

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Research paper thumbnail of Randall-Sundrum and Flipped SU(5)

In this letter, we construct a model based on a flipped SU(5) partial grand unified theory, withi... more In this letter, we construct a model based on a flipped SU(5) partial grand unified theory, within the framework of the Randall-Sundrum (RS1) proposal. Breaking of SU(5) is achieved using a bulk scalar field in the 10 of SU(5), Φ, which gains a vacuum expectation value < Φ> ∼ 3 × 10^15 GeV. We are able to retain the successes of the flipped SU(5) phenomenology, namely the elimination of the doublet-triplet splitting problem and the confinement of all fields to the smallest (1, 5̅, and 10) representations of SU(5). We derive the beta functions, and point out some constraints on bulk matter content implied by the runnings (and positivity) of the five dimensional coupling. Finally, we comment on baryon decay and show the fine-tuning problem required to prevent an exponentially short proton lifetime.

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Research paper thumbnail of Comments on Fractional Superstrings

Using the factorization approach of Gepner and Qiu, I systematically rederive the closed fraction... more Using the factorization approach of Gepner and Qiu, I systematically rederive the closed fractional superstring partition functions for K= 4, 8, and 16. For these theories the relationship between the massless graviton and gravi- tino sector and the purely massive sectors is explored. Properties of the massive sectors are investigated. A twist current in these models is found responsible for the occurrence of N=1 space-time supersymmetry. I show this twist current transforms bosonic (fermionic) projection states into fermionic (bosonic) non-projection states and vice-versa.

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Research paper thumbnail of String Cosmology and the Dimension of Spacetime

The implications of string theory for understanding the dimension of uncompactified spacetime are... more The implications of string theory for understanding the dimension of uncompactified spacetime are investigated. Using recent ideas in string cosmology, a new model is proposed to explain why three spatial dimensions grew large. Unlike the original work of Brandenberger and Vafa, this paradigm uses the theory of random walks. A computer model is developed to test the implications of this new approach. It is found that a four-dimensional spacetime can be explained by the proper choice of initial conditions.

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Research paper thumbnail of Relativistic Drag and Emission Radiation Pressures in an Isotropic Photonic Gas

By invoking the relativistic spectral radiance, as derived by Lee and Cleaver [1], the drag radia... more By invoking the relativistic spectral radiance, as derived by Lee and Cleaver [1], the drag radiation pressure of a relativistic planar surface moving through an isotropic radiation field, with which it is in thermal equilibrium, is determined in inertial and non-inertial frames. The forward- and rearward-directed emission radiation pressures are also derived and compared. A fleeting (inertial frames) or ongoing (some non-inertial frames) Carnot cycle is shown to exist as a result of an intra-surfaces temperature gradient. The drag radiation pressure on an object with an arbitrary frontal geometry is also described.

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Research paper thumbnail of Aspects of Fractional Superstrings

We investigate some issues relating to recently proposed fractional superstring theories with D_ ... more We investigate some issues relating to recently proposed fractional superstring theories with D_ critical<10. Using the factorization approach of Gepner and Qiu, we systematically rederive the partition functions of the K=4, 8, and 16 theories and examine their spacetime supersymmetry. Generalized GSO projection operators for the K=4 model are found. Uniqueness of the twist field, ϕ^K/4_K/4, as source of spacetime fermions is demonstrated. Last, we derive a linear (rather than quadratic) relationship between the required conformal anomaly and the conformal dimension of the supercurrent ghost.

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Research paper thumbnail of Grand Unified Theories From Superstrings

I review how traditional grand unified theories, which require adjoint (or higher representation)... more I review how traditional grand unified theories, which require adjoint (or higher representation) Higgs fields for breaking to the standard model, can be contained within string theory. The status (as of January 1996) of the search for stringy free fermionic three generation SO(10) SUSY--GUT models is discussed. Progress in free fermionic classification of both SO(10)_2 charged and uncharged embeddings and in N=1 spacetime solutions is presented. Based on talks presented at the Workshop on SUSY Phenomena and SUSY GUTs, Santa Barbara, California, Dec. 7-11, 1995, and at the Orbis Scientiae, Coral Gables, Florida, January 25-28, 1996. Appearing in the Proceedings of Orbis Scientiae, 1996.

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Research paper thumbnail of Supersymmetries in Free Fermionic Strings

Consistent heterotic free fermionic string models are classified in terms of their number of spac... more Consistent heterotic free fermionic string models are classified in terms of their number of spacetime supersymmetries, N. For each of the six distinct choices of gravitino sector, we determine what number of supersymmetries can survive additional GSO projections. We prove by exhaustive search that only three of the six can yield N = 1, in addition to the N = 4, 2, or 0 that five of the six can yield. One choice of gravitino sector can only produce N = 4 or 0. Relatedly, we find that only Z_2, Z_4, and Z_8 twists of the internal fermions with worldsheet supersymmetry are consistent with N=1 in free fermionic models. Any other twists obviate N=1.

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Research paper thumbnail of The Ricci curvature of half-flat manifolds

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Research paper thumbnail of The Inability of the White-Juday Warp Field Interferometer to Spectrally Resolve Spacetime Distortions

arXiv: General Physics, 2014

This paper contends that the spacetime distortions resulting from the experimentally obtainable e... more This paper contends that the spacetime distortions resulting from the experimentally obtainable electric field of a parallel plate capacitor configuration cannot be detected by the White-Juday Warp Field Interferometer [1]. Any post-processing results indicating a vanishing, non-zero difference between the charged and uncharged states of the capacitor are due to local effects rather than spacetime perturbations.

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Research paper thumbnail of The cosmic microwave background radiation power spectrum as a random bit generator for symmetric- and asymmetric-key cryptography

Heliyon, 2017

In this note, the Cosmic Microwave Background (CMB) Radiation is shown to be capable of functioni... more In this note, the Cosmic Microwave Background (CMB) Radiation is shown to be capable of functioning as a Random Bit Generator, and constitutes an effectively infinite supply of truly random one-time pad values of arbitrary length. It is further argued that the CMB power spectrum potentially conforms to the FIPS 140-2 standard. Additionally, its applicability to the generation of a (n × n) random key matrix for a Vernam cipher is established.

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Research paper thumbnail of Universal features of quantum bounce in loop quantum cosmology

Physics Letters B, 2017

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Research paper thumbnail of White holes as the asymptotic limit of evaporating primordial black holes

International Journal of Modern Physics A, 2016

This paper examines the interaction of an intense fermion field with all of the particle species ... more This paper examines the interaction of an intense fermion field with all of the particle species of an attometer primordial black hole’s (PBH) high energy Hawking radiation spectrum. By extrapolating to Planck-sized PBHs, it is shown that although Planck-sized PBHs closely simulate the zero absorption requirement of white holes, the absorption probability is not truly zero, and therefore, thermodynamically, Planck-sized primordial black holes are not true white holes.

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Research paper thumbnail of The relativistic blackbody spectrum in inertial and non-inertial reference frames

New Astronomy, 2017

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Research paper thumbnail of Black Sun: Ocular Invisibility of Relativistic Luminous Astrophysical Bodies

Journal of High Energy Physics, Gravitation and Cosmology, 2016

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Research paper thumbnail of High-order primordial perturbations with quantum gravitational effects

Physical Review D, 2016

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Research paper thumbnail of Inflationary spectra with inverse-volume corrections in loop quantum cosmology and their observational constraints from Planck 2015 data

Journal of Cosmology and Astroparticle Physics, 2016

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Research paper thumbnail of Relativistic drag and emission radiation pressures in an isotropic photonic gas

Modern Physics Letters A, 2016

By invoking the relativistic spectral radiance, as derived by Lee and Cleaver,1 the drag radiatio... more By invoking the relativistic spectral radiance, as derived by Lee and Cleaver,1 the drag radiation pressure of a relativistic planar surface moving through an isotropic radiation field, with which it is in thermal equilibrium, is determined in inertial and non-inertial frames. The forward- and backward-directed emission radiation pressures are also derived and compared. A fleeting (inertial frames) or ongoing (some non-inertial frames) Carnot cycle is shown to exist as a result of an intra-surfaces temperature gradient. The drag radiation pressure on an object with an arbitrary frontal geometry is also described.

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Research paper thumbnail of Hawking Radiation as a Possible Probe for the Interior Structure of Regular Black Holes

International Journal of Theoretical Physics, 2016

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Research paper thumbnail of SO(10) SUSY-GUTs Based on Superstrings

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Phenomenological Survey of a String Derived MSSM

Recently, we demonstrated the existence of heterotic-string solutions in which the observable sec... more Recently, we demonstrated the existence of heterotic-string solutions in which the observable sector effective field theory just below the string scale reduces to that of the MSSM, with the standard observable gauge group being just SU(3)C x SU(2)L x SU(2)Y and the SU(3)C x SU(2)L x SU(2)_Y-charged spectrum of the observable sector consisting solely of the MSSM spectrum. Associated with this model is a set of distinct flat directions of vacuum expectation values (VEVs) of non-Abelian singlet fields that all produce solely the MSSM spectrum. In this talk we (1) present our string derived MSSM and (2) study the effective superpotential induced by these choices of flat directions. We investigate whether sufficient degrees of freedom exist in these singlet flat directions to satisfy various phenomenological constraints imposed by the observed Standard Model data. For each flat direction, the effective superpotential is given to sixth order. The variations of the singlet and hidden secto...

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Research paper thumbnail of Randall-Sundrum and Flipped SU(5)

In this letter, we construct a model based on a flipped SU(5) partial grand unified theory, withi... more In this letter, we construct a model based on a flipped SU(5) partial grand unified theory, within the framework of the Randall-Sundrum (RS1) proposal. Breaking of SU(5) is achieved using a bulk scalar field in the 10 of SU(5), Φ, which gains a vacuum expectation value < Φ> ∼ 3 × 10^15 GeV. We are able to retain the successes of the flipped SU(5) phenomenology, namely the elimination of the doublet-triplet splitting problem and the confinement of all fields to the smallest (1, 5̅, and 10) representations of SU(5). We derive the beta functions, and point out some constraints on bulk matter content implied by the runnings (and positivity) of the five dimensional coupling. Finally, we comment on baryon decay and show the fine-tuning problem required to prevent an exponentially short proton lifetime.

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Research paper thumbnail of Comments on Fractional Superstrings

Using the factorization approach of Gepner and Qiu, I systematically rederive the closed fraction... more Using the factorization approach of Gepner and Qiu, I systematically rederive the closed fractional superstring partition functions for K= 4, 8, and 16. For these theories the relationship between the massless graviton and gravi- tino sector and the purely massive sectors is explored. Properties of the massive sectors are investigated. A twist current in these models is found responsible for the occurrence of N=1 space-time supersymmetry. I show this twist current transforms bosonic (fermionic) projection states into fermionic (bosonic) non-projection states and vice-versa.

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Research paper thumbnail of String Cosmology and the Dimension of Spacetime

The implications of string theory for understanding the dimension of uncompactified spacetime are... more The implications of string theory for understanding the dimension of uncompactified spacetime are investigated. Using recent ideas in string cosmology, a new model is proposed to explain why three spatial dimensions grew large. Unlike the original work of Brandenberger and Vafa, this paradigm uses the theory of random walks. A computer model is developed to test the implications of this new approach. It is found that a four-dimensional spacetime can be explained by the proper choice of initial conditions.

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Relativistic Drag and Emission Radiation Pressures in an Isotropic Photonic Gas

By invoking the relativistic spectral radiance, as derived by Lee and Cleaver [1], the drag radia... more By invoking the relativistic spectral radiance, as derived by Lee and Cleaver [1], the drag radiation pressure of a relativistic planar surface moving through an isotropic radiation field, with which it is in thermal equilibrium, is determined in inertial and non-inertial frames. The forward- and rearward-directed emission radiation pressures are also derived and compared. A fleeting (inertial frames) or ongoing (some non-inertial frames) Carnot cycle is shown to exist as a result of an intra-surfaces temperature gradient. The drag radiation pressure on an object with an arbitrary frontal geometry is also described.

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Research paper thumbnail of Aspects of Fractional Superstrings

We investigate some issues relating to recently proposed fractional superstring theories with D_ ... more We investigate some issues relating to recently proposed fractional superstring theories with D_ critical<10. Using the factorization approach of Gepner and Qiu, we systematically rederive the partition functions of the K=4, 8, and 16 theories and examine their spacetime supersymmetry. Generalized GSO projection operators for the K=4 model are found. Uniqueness of the twist field, ϕ^K/4_K/4, as source of spacetime fermions is demonstrated. Last, we derive a linear (rather than quadratic) relationship between the required conformal anomaly and the conformal dimension of the supercurrent ghost.

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Research paper thumbnail of Grand Unified Theories From Superstrings

I review how traditional grand unified theories, which require adjoint (or higher representation)... more I review how traditional grand unified theories, which require adjoint (or higher representation) Higgs fields for breaking to the standard model, can be contained within string theory. The status (as of January 1996) of the search for stringy free fermionic three generation SO(10) SUSY--GUT models is discussed. Progress in free fermionic classification of both SO(10)_2 charged and uncharged embeddings and in N=1 spacetime solutions is presented. Based on talks presented at the Workshop on SUSY Phenomena and SUSY GUTs, Santa Barbara, California, Dec. 7-11, 1995, and at the Orbis Scientiae, Coral Gables, Florida, January 25-28, 1996. Appearing in the Proceedings of Orbis Scientiae, 1996.

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Research paper thumbnail of Supersymmetries in Free Fermionic Strings

Consistent heterotic free fermionic string models are classified in terms of their number of spac... more Consistent heterotic free fermionic string models are classified in terms of their number of spacetime supersymmetries, N. For each of the six distinct choices of gravitino sector, we determine what number of supersymmetries can survive additional GSO projections. We prove by exhaustive search that only three of the six can yield N = 1, in addition to the N = 4, 2, or 0 that five of the six can yield. One choice of gravitino sector can only produce N = 4 or 0. Relatedly, we find that only Z_2, Z_4, and Z_8 twists of the internal fermions with worldsheet supersymmetry are consistent with N=1 in free fermionic models. Any other twists obviate N=1.

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Research paper thumbnail of The Ricci curvature of half-flat manifolds

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Research paper thumbnail of The Inability of the White-Juday Warp Field Interferometer to Spectrally Resolve Spacetime Distortions

arXiv: General Physics, 2014

This paper contends that the spacetime distortions resulting from the experimentally obtainable e... more This paper contends that the spacetime distortions resulting from the experimentally obtainable electric field of a parallel plate capacitor configuration cannot be detected by the White-Juday Warp Field Interferometer [1]. Any post-processing results indicating a vanishing, non-zero difference between the charged and uncharged states of the capacitor are due to local effects rather than spacetime perturbations.

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Research paper thumbnail of The cosmic microwave background radiation power spectrum as a random bit generator for symmetric- and asymmetric-key cryptography

Heliyon, 2017

In this note, the Cosmic Microwave Background (CMB) Radiation is shown to be capable of functioni... more In this note, the Cosmic Microwave Background (CMB) Radiation is shown to be capable of functioning as a Random Bit Generator, and constitutes an effectively infinite supply of truly random one-time pad values of arbitrary length. It is further argued that the CMB power spectrum potentially conforms to the FIPS 140-2 standard. Additionally, its applicability to the generation of a (n × n) random key matrix for a Vernam cipher is established.

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Research paper thumbnail of Universal features of quantum bounce in loop quantum cosmology

Physics Letters B, 2017

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Research paper thumbnail of White holes as the asymptotic limit of evaporating primordial black holes

International Journal of Modern Physics A, 2016

This paper examines the interaction of an intense fermion field with all of the particle species ... more This paper examines the interaction of an intense fermion field with all of the particle species of an attometer primordial black hole’s (PBH) high energy Hawking radiation spectrum. By extrapolating to Planck-sized PBHs, it is shown that although Planck-sized PBHs closely simulate the zero absorption requirement of white holes, the absorption probability is not truly zero, and therefore, thermodynamically, Planck-sized primordial black holes are not true white holes.

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Research paper thumbnail of The relativistic blackbody spectrum in inertial and non-inertial reference frames

New Astronomy, 2017

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Research paper thumbnail of Black Sun: Ocular Invisibility of Relativistic Luminous Astrophysical Bodies

Journal of High Energy Physics, Gravitation and Cosmology, 2016

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Research paper thumbnail of High-order primordial perturbations with quantum gravitational effects

Physical Review D, 2016

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Research paper thumbnail of Inflationary spectra with inverse-volume corrections in loop quantum cosmology and their observational constraints from Planck 2015 data

Journal of Cosmology and Astroparticle Physics, 2016

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Research paper thumbnail of Relativistic drag and emission radiation pressures in an isotropic photonic gas

Modern Physics Letters A, 2016

By invoking the relativistic spectral radiance, as derived by Lee and Cleaver,1 the drag radiatio... more By invoking the relativistic spectral radiance, as derived by Lee and Cleaver,1 the drag radiation pressure of a relativistic planar surface moving through an isotropic radiation field, with which it is in thermal equilibrium, is determined in inertial and non-inertial frames. The forward- and backward-directed emission radiation pressures are also derived and compared. A fleeting (inertial frames) or ongoing (some non-inertial frames) Carnot cycle is shown to exist as a result of an intra-surfaces temperature gradient. The drag radiation pressure on an object with an arbitrary frontal geometry is also described.

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Research paper thumbnail of Matter-Antimatter Propulsion via QFT Effects from Parallel Electric and Magnetic Fields

Matter/antimatter (MAM) pair production from the vacuum through intense electric fields has been ... more Matter/antimatter (MAM) pair production from the vacuum through intense electric fields has been investigated theoretically for nearly a century 1. This presentation will review this history and will examine proposals of MAM for intra-solar system and interstellar propulsion systems. The quantum mechanical foundation of MAM production was developed by F. Sauter et al. in the 1930's and then placed on a sound quantum electromagnetics (QED) basis by J. Schwinger in 1951. Pair production occurs when the electric field strength E 0 is above the critical value at which the fields become non-linear with self-interactions (known as the Schwinger limit). As the energy density of lasers approach the critical strength of E 0 ~ 10 16 V/cm, the feasibility and functionality of electron-positron pair production has received growing interest. Current laser intensities are approaching within 1 order of magnitude of the Schwinger limit. Physical processes for lowering the critical energy density below the Schwinger limit (and simultaneously enhancing the pair production above the Schwinger limit) through additional quantum mechanical effects have been explored. One under study at the U. of Connecticut and the U. of Duisburg-Essen is pulsation of inhomogeneous electric fields within a carrier wave. Another is via enhancement of quantum effects by addition of a magnetic field B parallel to the electric field E. Magnetic field enhancement to quark/anti-quark production through chiral symmetry breaking effects in quantum chromodynamics (QCD) was investigated theoretically by J. Preskill at Caltech in the 1980's. S. Pyo and D. Page showed in 2007 that parallel magnetic fields also enhance electron/positron production via an analogous QED effect, with enhancement going predominantly as a linear function of B 0 /E 0 , Particle/antiparticle pair production as a highly efficient fuel source for intra solar system and interstellar propulsion was proposed by D. Crow in 1983. The viability of this method of propulsion will be studied, especially from the parallel electric and magnetic field approach. 1 Particle/anti-particle pair production does not (and cannot) take energy from the spacetime vacuum. Rather the energy is drawn from the external electric (and magnetic) fields. This process is very analogous to particle production near the event horizon of a black hole, which reduces the mass of the black hole accordingly. (The primary difference between the two processes is, while both particle and antiparticle are produced from a virtual pair by the electric (and magnetic) fields, only one particle in an initially virtual pair escapes from a black hole (as Hawking radiation) and the antiparticle is captured by the black hole.)

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Research paper thumbnail of Ultra-Relativistic Energy Density Inflation of Astrophysical Bodies into Apparent Black Holes

1. Abstract In this note, an interesting result of relativistic effects between inertial referenc... more 1. Abstract In this note, an interesting result of relativistic effects between inertial reference frames is pointed out: to observers at rest in one inertial frame, objects in another inertial frame approaching or receding differentially close to the speed of light, relative to the inertial rest frame, will appear to experience both significant mass inflation and length contraction, resulting in greatly inflated observed energy densities. This note determines, in the view of the observer in the rest frame, the respective Lorentz factors above which an arbitrary spherical object will appear to attain a black hole density and an apparent Planck density.

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Research paper thumbnail of The Inability of the White-Juday Warp Field Interferometer to Spectrally Resolve Spacetime Distortions

The authors of this paper contend that the experiments being conducted with the White-Juday Warp ... more The authors of this paper contend that the experiments being conducted with the White-Juday Warp Field Interferometer [1], [2], [3] by the Eagleworks research group at NASA's Johnson Space Center will be unable to detect spacetime distortions resulting from the experimentally obtainable electric field of a parallel plate capacitor configuration. The authors, who are not involved with this research group, argue that any post-processing results indicating a vanishing, non-zero difference between the charged and uncharged states of the capacitor are due to local effects rather than spacetime perturbations.

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Research paper thumbnail of Exclusion of the Magnus Effect as a Mechanism for Shotgun Pellet Dispersion

1. Abstract In this paper, it is shown that the Magnus effect cannot be a primary mechanism for t... more 1. Abstract In this paper, it is shown that the Magnus effect cannot be a primary mechanism for the dispersion of shotgun pellets. The one-dimensional motion of objects through air, applied to shotgun pellets traveling a short distance, reveals that throughout the flight, the Reynold's number is approximately constant. The distribution of pellets upon the target is demonstrated to be a phased-shifted Maxwellian distribution in lateral displacement space. The minimum Magnus frequencies required to create a typical pellet distribution pattern are ascertained, and are well in excess of the frequencies that could reasonably be achieved by pellets exiting a smooth bore shotgun.

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Research paper thumbnail of Universal features of quantum bounce in loop quantum cosmology

Loop quantum cosmology (LQC) provides an elegant resolution of the classical big bang singu-larit... more Loop quantum cosmology (LQC) provides an elegant resolution of the classical big bang singu-larity by a quantum bounce in the deep Planck era. The evolutions of the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background and its linear scalar and tensor perturbations are universal during the pre-inflationary phase. In this period the potentials of the perturbations can be well approximated by a Pöschl-Teller (PT) potential, from which we find analytically the mode functions and then calculate the Bogoliubov coefficients at the onset of the slow-roll inflation, valid for any inflationary models with a single scalar field. Matching them to those given in the slow-roll inflationary phase, we investigate the effects of the quantum bounce on the power spectra and find unique features that can be tested by current and forthcoming observations. In particular, fitting the power spectra to the Planck 2015 data, we find that the universe must have expanded at least 132 e-folds from the bounce until now.

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Research paper thumbnail of Hawking Radiation as a Probe for the Interior Structure of Regular Black Holes

The notion of the black hole singularity and the proof of the singularity theorem were considered... more The notion of the black hole singularity and the proof of the singularity theorem were considered great successes in classical general relativity. Singularities had presented deep puzzles to physicists. Conceptual challenges were set up by the intractability of the singularity. The existence of black hole horizons which cover up the interior, including the singularity of the black hole from outside observers, builds an information curtain, further hindering physicists from understanding the nature of the singularity and the interior structure of black holes. The regular black hole is a concept produced out of multiple attempts of establishing a tractable and understandable interior structure for black hole as well as avoiding the singularity behind the black hole horizon. A method is needed to check the correctness of the new constructions of black holes. After studying the Hawking radiation by fermion tunnelling from one type of regular black hole, structure dependent results were obtained. The result being structure dependent points out the prospects of employing the Hawking radiation as a method to probe into the structure of black holes.

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