Modified Gravity and Cosmology

Preface Conventions and notation 1. Introduction 2. The prototype Brans-Dicke model 3. Conformal transformation 4. Cosmology with LAMBDA 5. Models of an accelerating universe 6. Quantum effects Appendices References Index.

The scalar-tensor theory of gravitation

This is the fifth edition of H Dieter Zeh's classic text on the physical foundations of time-irreversibility in the phenomena. A forerunner of this book was the 1984 German text 'Die Physik der Zeitrichtung' of about 80 pages, which appeared as volume 200 in the Springer series Lecture Notes in Physics. It was soon followed by a largely revised and extended English edition of about twice the length. Since then each new edition has been thoroughly revised and, edition by edition, new topics and chapters have been added. As the author says in the introduction: 'The prime intention of this book is to discuss the relations between various arrows of time, and to search for a universal master arrow'. Correspondingly, after a short chapter on 'the physical concept of time', the author systematically discusses in the remaining five chapters the time arrows in electromagnetic radiation theory, in thermodynamics, in quantum mechanics, in black-hole physics and cosmology, and in quantum cosmology. The chapters on thermodynamics and quantum mechanics slightly outweigh the others in terms of length. The fifth edition now includes two new section on 'cosmic probabilities and history' and 'quantum computers', and the section on the 'expansion of the universe' has been restructured and extended. Other changes concentrate on the sections on radiation damping, decoherence, interpretation of quantum theory, and quantum cosmology. It should also be mentioned that the author maintains a regularly updated website for the book at www.time-direction.de. The reading is always highly stimulating and uses results and ideas from a very broad range of physics, with interspersed historical and philosophical comments. Somehow outstanding and of particular interest is the chapter on quantum cosmology, which raises novel interpretational issues that cannot be found in any other textbook I know of on time asymmetry. As regards the mathematical prerequisites, the reader is assumed to have some knowledge on Green functions, special-relativistic electrodynamics, Hamiltonian mechanics, the formalisms of phenomenological and statistical thermodynamics, quantum mechanics, and general relativity. At some rare places the author avowedly shows a certain impatience with mathematical details, in particular if they do not fit with his expectations based on personal intuition. This stands in peculiar contrast to the conceptual depth and thoroughness that otherwise characterizes most of the book. Quite generally, the reader should be prepared to encounter provoking and partially controversial statements, but this should not be too surprising in such a complex and partially highly speculative field. All this certainly sets high standards for the reader's intellectual independence and maturity, but this is definitely in accord with the philosophy of Springer's 'Frontiers Collection'. On the other hand, from personal experience I can say that already the original German text has been very popular with beginning graduate students who had a serious interest in foundational issues. (I obtained my first personal copy as a birthday present from a fellow student.) This is essentially due to the fact that the author's genuine urge to understand, rather than just describe, gives the text a characteristic flair of freshness and authenticity to which beginning researchers are particularly susceptible. Being provocative at places is just part of that. This spirit has essentially survived the various editions, thanks to the author's constant efforts to improve the existing presentation, and also by adding modern topics. However, the negative side of this constant streamlining of presentation according to the author's evolving understanding is that it tends to amplify the already existing idiosyncratic tendencies, sometimes resulting in cryptic remarks which fail their intended clarifying purpose. This already applies to the fairly steep introduction, where the author attempts to clarify in a few lines the central issue of what it means (structurally) to say that a particular dynamical law is (a)symmetric under time reversal. Here I think it would definitely be necessary to give more detailed explanations, e.g., by elaborating on the author's own discussion in 'Note on the time reversal asymmetry of equations of motion' (1999 Found. Phys. Lett. 12 193?96). Also, the structural relations between the operation of time reversal and other space-time symmetries are hardly mentioned. This may be excused insofar as T symmetry in quantum-field theory is not generally addressed in this book (which itself may be regretted), but as part of a structural characterization of the operation of time reversal it would certainly have been useful. The books provides an extensive bibliography which seems (as far as I can tell) fairly complete, though sometimes and for no obvious reason, preprints posted on web-archives are cited instead of the corresponding journal articles. Each reference comes with the page numbers of where it is cited in the text, which is very useful indeed. The positive aspects by far outweigh the critical ones. The new edition of H Dieter Zeh's classic text is highly recommended to anybody interested in foundational issues and ready to take the challenge to follow the sometimes intuitive approach of someone who has spent much time and effort to understand the conceptual intricacies and variations of this indisputably difficult and demanding subject.

https://iopscience.iop.org/article/10.1088/0264-9381/25/20/209003/pdf

The Physical Basis of the Direction of Time

To find more deliberate f ( R , T ) cosmological solutions, we take our previous paper further by studying some new aspects of the considered models via investigation of some new cosmological parameters/quantities to attain the most acceptable cosmological results. Our investigations are performed by applying the dynamical system approach. We obtain the cosmological parameters/quantities in terms of some defined dimensionless parameters that are used in constructing the dynamical equations of motion. The investigated parameters/quantities are the evolution of the Hubble parameter and its inverse, the “weight function”; the ratio of the matter density to the dark energy density and its time variation; the deceleration; the jerk and the snap parameters; and the equation-of-state parameter of the dark energy. We numerically examine these quantities for two general models R + α R - n + - T and R log [ α R ] q + - T . All considered models have some inconsistent quantities (with respect to the available observational data), except the model with n = - 0.9 , which has more consistent quantities than the other ones. By considering the ratio of the matter density to the dark energy density, we find that the coincidence problem does not refer to a unique cosmological event; rather, this coincidence also occurred in the early Universe. We also present the cosmological solutions for an interesting model R + c 1 - T in the nonflat Friedmann–Lemaitre–Robertson–Walker metric. We show that this model has an attractor solution for the late times, though with w ( DE ) = - 1 / 2 . This model indicates that the spatial curvature density parameter gets negligible values until the present era, in which it acquires the values of the order 10 - 4 or 10 - 3 . As the second part of this work, we consider the weak-field limit of f ( R , T ) gravity models outside a spherical mass immersed in the cosmological fluid. We have found that the corresponding field equations depend on the both background values of the Ricci scalar and the background cosmological fluid density. As a result, we attain the parametrized post-Newtonian parameter for f ( R , T ) gravity and show that this theory can admit the experimentally acceptable values of this parameter. As a sample, we present the post-Newtonian gamma parameter for general minimal power law models, in particular, the model R + c 1 - T .

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Cosmological and Solar System Consequences of f(R,T) Gravity Models

In this paper we consider Godel black hole and study thermodynamics and statistics with logarithmic correction. We calculate some important quantities such as free energy, specific heat and partition function.

Thermodynamics and Statistics of Gödel Black Hole with Logarithmic Correction

We follow the approach of induced-matter theory for a five-dimensional (5D) vacuum Brans–Dicke theory and introduce induced-matter and induced potential in four dimensional (4D) hypersurfaces, and then employ a generalized FRW type solution. We confine ourselves to the scalar field and scale factors be functions of the cosmic time. This makes the induced potential, by its definition, vanishes, but the model is capable to expose variety of states for the universe. In general situations, in which the scale factor of the fifth dimension and scalar field are not constants, the 5D equations, for any kind of geometry, admit a power–law relation between the scalar field and scale factor of the fifth dimension. Hence, the procedure exhibits that 5D vacuum FRW-like equations are equivalent, in general, to the corresponding 4D vacuum ones with the same spatial scale factor but a new scalar field and a new coupling constant, $${\tilde{\omega}}$$
 . We show that the 5D vacuum FRW-like equations, or its equivalent 4D vacuum ones, admit accelerated solutions. For a constant scalar field, the equations reduce to the usual FRW equations with a typical radiation dominated universe. For this situation, we obtain dynamics of scale factors of the ordinary and extra dimensions for any kind of geometry without any priori assumption among them. For non-constant scalar fields and spatially flat geometries, solutions are found to be in the form of power–law and exponential ones. We also employ the weak energy condition for the induced-matter, that gives two constraints with negative or positive pressures. All types of solutions fulfill the weak energy condition in different ranges. The power–law solutions with either negative or positive pressures admit both decelerating and accelerating ones. Some solutions accept a shrinking extra dimension. By considering non-ghost scalar fields and appealing the recent observational measurements, the solutions are more restricted. We illustrate that the accelerating power–law solutions, which satisfy the weak energy condition and have non-ghost scalar fields, are compatible with the recent observations in ranges −4/3 < ω ≤ −1.3151 for the coupling constant and 1.5208 ≤ n < 1.9583 for dependence of the fifth dimension scale factor with the usual scale factor. These ranges also fulfill the condition $${\tilde{\omega} > -3/2}$$
 which prevents ghost scalar fields in the equivalent 4D vacuum Brans–Dicke equations. The results are presented in a few tables and figures.

FRW cosmology from five dimensional vacuum Brans–Dicke theory

Following the approach of the induced-matter theory, we investigate the cosmological implications of a five-dimensional Brans–Dicke (BD) theory, and propose to explain the acceleration of the universe. After inducing in a four-dimensional hypersurface, we classify the energy–momentum tensor into two parts in a way that, one part represents all kind of the matter (the baryonic and dark) and the other one contains every extra terms emerging from the scale factor of the fifth dimension and the scalar field, which we consider as the energy–momentum tensor of dark energy. We also separate the energy–momentum conservation equation into two conservation equations, one for matter and the other for dark energy. We perform this procedure for different cases, without interacting term and with two particular (suitable) interacting terms between the two parts. By assuming the parameter of the state equation for dark energy to be constant, the equations of the model admit the power-law solutions. Though, the noninteracting case does not give any accelerated universe, but the interacting cases give both decelerated and accelerated universes. For the interacting cases, we figure out analytically the acceptable ranges of some parameters of the model, and also investigate the data analysis to test the model parameter values consistency with the observational data of the distance modulus of 580 SNe Ia compiled in Union2.1. For one of these interacting cases, the best fitted values suggest that BD coupling constant (ω) is ≃ -7.75, however, it also gives the state parameter of dark energy (wX) equal to ≃ -0.67. In addition, the model gives the Hubble and deceleration parameters at the present time to be H◦ ≃ 69.4 (km/s)/Mpc and q◦ ≃ -0.38 (within their confidence intervals), where the scale factor of the fifth dimension shrinks with the time.

Dark Energy From Fifth Dimensional Brans-Dicke Theory

Following the approach of the induced-matter theory, we investigate the cosmological implications of a five-dimensional Brans-Dicke theory, and propose to explain the acceleration of the universe. After inducing in a four-dimensional hypersurface, we classify the energy-momentum tensor into two parts in a way that, one part represents all kind of the matter (the baryonic and dark) and the other one contains every extra terms emerging from the scale factor of the fifth dimension and the scalar field, which we consider as the energy-momentum tensor of dark energy. We also separate the energy-momentum conservation equation into two conservation equations, one for matter and the other for dark energy. We perform this procedure for different cases, without interacting term and with two particular (suitable) interacting terms between the two parts. By assuming the parameter of the state equation for dark energy to be constant, the equations of the model admit the power-law solutions. Though, the non-interacting case does not give any accelerated universe, but the interacting cases give both decelerated and accelerated universes. For the interacting cases, we figure out analytically the acceptable ranges of some parameters of the model, and also investigate the data analysis to test the model parameter values consistency with the observational data of the distance modulus of 580 SNe Ia compiled in Union2.1. For one of these interacting cases, the best fitted values suggest that the Brans-Dicke coupling constant ({\omega}) is -7.75, however, it also gives the state parameter of dark energy (wX) equal to -0.67. In addition, the model gives the Hubble and deceleration parameters at the present time to be H0 = 69.4 (km/s)/Mpc and q0 = -0.38 (within their confidence intervals), where the scale factor of the fifth dimension shrinks with the time.

We investigate the induced geodesic deviation equations in the brane world models, in which all the matter forces except gravity are confined on the 3-brane. Also, the Newtonian limit of induced geodesic deviation equation is studied. We show that in the first Randall-Sundrum model the Bohr–Sommerfeld quantization rule is as a result of consistency between the geodesic and geodesic deviation equations. This indicates that the path of test particle is made up of integral multiples of a fundamental Compton-type unit of length h/mc.

https://iopscience.iop.org/article/10.1209/0295-5075/87/40006/pdf

Quantum mechanics and geodesic deviation in the brane world

In this letter we investigate the separability of the Klein–Gordon and Hamilton–Jacobi equation in Godel universe. We show that the Klein–Gordon eigen modes are quantized and the complete spectrum of the particle’s energy is a mixture of an azimuthal quantum number, m and a principal quantum number, n and a continuous wave number k. We also show that the Hamilton–Jacobi equation gives a closed function for classical action. These results may be used to calculate the Casimir vacuum energy in Godel universe.

Amir F. Bahrehbakhsh

Papers

FRW cosmology from five dimensional vacuum Brans–Dicke theory

Dark Energy From Fifth Dimensional Brans-Dicke Theory

Dark Energy From Fifth Dimensional Brans-Dicke Theory

Quantum mechanics and geodesic deviation in the brane world

Massless Spin–Zero Particle and the Classical Action via Hamilton–Jacobi Equation in Gödel Universe