On a Quantum Gravity Fractal Spacetime Equation: QRG ≃HD + FG and Its Application to Dark Energy—Accelerated Cosmic Expansion

The paper suggests that quantum relativistic gravity (QRG) is basically a higher dimensionality (HD) simulating relativity and non-classical effects plus a fractal Cantorian spacetime geometry (FG) simulating quantum mechanics. This more than just a conceptual equation is illustrated by integer approximation and an exact solution of the dark energy density behind cosmic expansion.

From Witten’s 462 Supercharges of 5-D Branes in Eleven Dimensions to the 95.5 Percent Cosmic Dark Energy Density behind the Accelerated Expansion of the Universe

The measured 95.5% dark energy density of the cosmos presumed to be behind the observed accelerated cosmic expansion is determined theoretically based upon Witten’s five branes in eleven dimensions theory. We show that the said dark energy density is easily found from the ratio of the 462 states of the five dimensional Branes to the total number of states, namely 528 minus the 44 degrees of freedom of the vacuum, i.e. , almost exactly as found in WMAP and Type 1a supernova measurements.

Addendum： Quark Confinement, New Cosmic Expansion and General Yang-Mills Symmetry （J. P. Hsu, Chin. Phys. C, 41（1）： 015101 （2017））

The force between a gigantic sphere with baryon galaxies exerted on an idealized point-like supernova is given by Fcbf = F1 in Eq. （40） of the original paper. We can generalized the point-like supernova to a sphere with a radius Rs and a constant mass density Ps.

The Observable Universe in a Simplified Cosmic Dynamic Model

This paper introduces a cosmic expansion model with constant speed of cosmic spatial expansion via derivation and simulations, where the speed of cosmic spatial expansion equals the speed of light c. Simulation results show that the earliest observable universe time is t = 5.084 Gyrs where the current universe time T = 13.82 Gyrs, and the furthest observable distance at the earliest observable universe time t is S = 0.632R, where R is the cosmic radius at current universe time T. The above constant cosmic expansion model does not consider the inflation period in the early universe according to the Big Bang model, nor does it considered the cosmic acceleration in recent universe time. However, this simplified cosmic expansion model could be a benchmark that will be helpful to understand the cosmic expansion and the observable universe. Based on the derivation and simulation of the constant cosmic expansion model, the threshold of observable universe for the accelerated cosmic expansion model can also be calculated similarly, as far as the speed of cosmic spatial expansion at any universe time t can be provided.

Kähler Dark Matter, Dark Energy Cosmic Density and Their Coupling

We utilize homology and co-homology of a K3-Kähler manifold as a model for spacetime to derive the cosmic energy density of our universe and subdivide it into its three fundamental constituents, namely: 1) ordinary energy;2) pure dark energy and 3) dark matter. In addition, the fundamental coupling of dark matter to pure dark energy is analyzed in detail for the first time. Finally, the so-obtained results are shown to be in astounding agreement with all previous theoretical analysis as well as with actual accurate cosmic measurements.

The Cosmic Radius of Observable Universe

This paper introduces three cosmic expansion models with constant, decelerating and accelerating speed of expansion respectively. Then characters of these cosmic expansion models are compared. Based on these cosmic expansion models, the thresholds of observable universe are calculated via simulations, where the earliest observable cosmic radius R(tearliest) is always 0.368R (R is cosmic radius at current universe time) for any cosmic expansion models.

Cantorian-Fractal Kinetic Energy and Potential Energy as the Ordinary and Dark Energy Density of the Cosmos Respectively

In a one-dimension Mauldin-Williams Random Cantor Set Universe, the Sigalotti topological speed of light is where . It follows then that the corresponding topological acceleration must be a golden mean downscaling of c namely . Since the maximal height in the one-dimensional universe must be where is the unit interval length and note that the topological mass (m) and topological dimension (D) where m = D = 5 are that of the largest unit sphere volume, we can conclude that the potential energy of classical mechanics translates to . Remembering that the kinetic energy is , then by the same logic we see that when m = 5 is replaced by for reasons which are explained in the main body of the present work. Adding both expressions together, we find Einstein’s maximal energy . As a general conclusion, we note that within high energy cosmology, the sharp distinction between potential energy and kinetic energy of classical mechanics is blurred on the cosmic scale. Apart of being an original contribution, the article presents an almost complete bibliography on the Cantorian-fractal spacetime theory.

On a Fractal Version of Witten’s M-Theory

Starting from Witten’s eleven dimensional M-theory, the present work develops in an analogous way a corresponding dimensional fractal version where . Subsequently, the new fractal formalism is utilized to determine the measured ordinary energy density of the cosmos which turns out to be intimately linked to the new theory’s fractal dimension via non-integer irrational Lorentzian-like factor: where is Hardy’s probability of quantum entanglement. Consequently, the energy density is found from a limiting classical kinetic energy to be Here, is ‘tHooft’s renormalon of dimensional regularization. The immediate logical, mathematical and physical implication of this result is that the dark energy density of the cosmos must be in astounding agreement with cosmic measurements and observations.

Negative Norms in Quantized Strings as Dark Energy Density of the Cosmos

The present short paper is concerned with accurate explanation as well as quantification of the so called missing dark energy of the cosmos. It was always one of the main objectives of any successful general theory of high energy particle physics and quantum cosmology to keep non-physical negative norms, the so called ghosts completely out of that theory. The present work takes the completely contrary view by admitting these supposedly spurious states as part of the physical Hilbert space. It is further shown that rethinking the ghost free condition with the two critical spacetime dimensions D1 = 26 and D2 = 25 together with the corresponding intercept a1 = 1 and a2 ≤ 1 respectively and in addition imposing, as in Gross et al. heterotic superstrings, an overall 496 dimensional exceptional Lie symmetry group, then one will discover that there are two distinct types of energy. The first is positive norm ordinary energy connected to the zero set quantum particles which is very close to the measured ordinary energy density of the cosmos, namely E(O) = mc2/22. The second is negative norm (i.e. ghost) energy connected to the empty set quantum wave and is equal to the conjectured dark energy density of the cosmos E(D) = mc2 (21/22) presumed to be behind the observed accelerated cosmic expansion. That way we were able to not only explain the physics of dark energy without adding any new concepts or novel additional ingredients but also we were able to compute the dark energy density accurately and in full agreement with measurements and observations.

Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos

The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

Matter Loops Corrected Modified Gravity in Palatini Formulation

Recently,corrections to the standard Einstein Hilbert action were proposed to explain the current cosmicacceleration in stead of introducing dark energy.In the Palatini formulation of those modified gravity models,thereis an important observation due to Arkani Hamed:matter loops will give rise to a correction to the modified gravityaction proportional to the Ricci scalar of the metric.In the presence of such a term,we show that the current formsof modified gravity models in Palatini formulation,specifically,the 1/R gravity and in R gravity,will have phantoms.Then we study the possible instabilities due to the presence of phantom fields.We show that the strong instability inthe metric formulation of 1/R gravity indicated by Dolgov and Kawasaki will not appear and the decay timescales forthe phantom fields may be long enough for the theories to make sense as effective field theory.On the other hand,if wechange the sign of the modification terms to eliminate the phantoms,some other inconsistencies will arise for the variousversions of the modified gravity models.Finally,we comment on the universal property of the Palatini formulation ofthe matter loops corrected modified gravity models and its implications.

The Big Bang Influence in a Cosmological Wave Complex

The Big Bang theory states that the universe was created from pure energy, although matter, in general, is also pure energy and there is no known physical existence that is not pure energy in accordance with the mass-energy equation. All known energy is situated in a field, and it can be questioned whether also the Big Bang was situated in a field in the primordial moment it inflated into the subsequent cosmic expansion that so far lets us observe a 93-billion-light-year-wide spherical volume of the universe. In this study, the Big Bang’s gravitational influence, particularly in the form of an externally radiated gravitational wave, is considered in connection to its situation in a surrounding field with a different expansion rate than itself. The results suggest that the least possible size of the universe can be predicted by the expression of the gravitational wave produced by Big Bang, revealing that the universe has a significantly greater size than the observable, and further that Big Bang might be the production of only one of many cosmic galaxies situated together in a cosmological wave complex (CWC) where the amplitude is self-maintained by inflations.

A Combined Heterotic String and Kähler Manifold Elucidation of Ordinary Energy,Dark Matter,Olbers’s Paradox and Pure Dark Energy Density of the Cosmos

We utilize the topological-geometrical structure imposed by the Heterotic superstring theory on spacetime in conjunction with the K3 Kähler manifold to explain the mysterious nature of dark matter and its coupling to the pure dark energy density of the cosmos. The analogous situations in the case of a Kerr black hole as well as the redundant components of the Riemannian tensor are pointed out and the final result was found to be in complete agreement with all previous theoretical ones as well as all recent accurate measurements and cosmic observations. We conclude by commenting briefly on the Cantorian model of Zitterbewegung and the connection between Olbers’s paradox and dark energy.

Constrains on f(T) gravity with the strong gravitational lensing data

Strong lensing is an effective way to probing the properties of dark energy.In this paper,we use the strong lensing data to constrain the f(T)theory,which is a new modified gravity to explain the present accelerating cosmic expansion without the need of dark energy.In our discussion,the CMB and BAO data are also added to constrain model parameters tightly and three different f(T)models are studied.We find that strong lensing has an important role on constraining f(T)models,and once the CMB+BAO data is added,a tighter constraint is obtained.However,the consistency of our result with what is obtained from SNIa+CMB+BAO is actually model-dependent.