Brane-world cosmology

In this paper, we consider a relationship between projection and dimension in geometry. We discuss the relation from projective geometrical viewpoint by paying attention to higher and lower dimensional spaces. Then, we categorize those projections among different dimensional ones by their characteristics.
We also express the classification by group theory and verify that there exists the automorphism.
p.s. Any point projected into other dimensional space always keeps its original freedom degrees. In other words, the freedom degree of point is invariant after projections or in a series of the group actions, so that it has "symmetry" in this sense.

Dear colleagues,
In 1872, Felix Klein announced an influential geometric program also known as Erlangen Program. He insisted that any type of geometry be considered as projective geometry operated by group theory. He was the first to inform us of the relationship between geometry and group theory. This thought has succeeded in physics rather than mathematics or geometry. For example, Afterward, Emmy Noether established the so called Noether's theorem. She so to say replaced projective geometry by another term, "symmetry" and considered symmetry as operated by group. So she insisted as follows; If there is a consecutive symmetry (but discrete symmetry is also possible in quantum area, for example, as selection rule) in a system, there exists a physical conservation. Hermann Weyl pushed the limits and introduced gauge symmetry based on representation of group in quantum theory. Yang Mills theory and other main theories in the quantum field theory are based on the gauge theory.

However, today's particle physics like super string theory is unnecessary described by symmetry. it applies so many mathematical techniques to the string theory, but it seems even to be more and more confused. I have so far, from mathematical viewpoint based on Felix Klein and his followers' spirits, researched a relationship among projective geometry, symmetry, and group theory or representation of group. Certainly, at this time, it is not necessarily in the mainstream of today's higher dimensional physics like super string theory. On the other hand, it is also the fact that there are physicists and mathematicians skeptical to the stringers' research direction. For example, I recently read a bestseller "Not Even Wrong: The Failure of String Theory". He is an excellent mathematician who once majored in particle physics. In his book, I found an interesting part of the preface to be (the last paragraph of page xix, you could read it at "Look Inside" on the website of amazon.) "Taking it out".

"The positive argument of this book will be that historically, one of the main sources of progress of particle theory has been the discovery of new symmetry groups of nature, together with new representations of these groups. The failure of superstring theory program can be traced to its lack of any fundamental new symmetry principle. Without unexpected experimental data, new theoretical advances are likely to come about only if theorists turn their attention away from this failed program and toward the difficult task of better understanding the symmetries of the natural world."

Now, let us remember my words above. His claim is coincidentally the same as I have insisted on so far through my research. His words motive me further to seek the possibility and potential in mathematics and physics. I think that the direction of my research can be significant. Please see the research paper via the link below.
https://www.academia.edu/16715934/Projections_and_Dimensions_trilogy_of_dimensionality_1_

Although projection from higher to lower dimensional space is not explained enough in this paper above, another example will make a good case for it:
https://www.academia.edu/29017770/A_Group_Representation_Notations_for_Projections_and_Dimensions_trilogy_of_dimensionality_3_ (: see Claim 2 on the page 26 to 27.)

What I'd like to say in this research is that any point projected into other dimensional space always keeps its original freedom degrees. In other words, the freedom degree of point is invariant after projections or in a series of the group actions, so that it has symmetry in this sense. Additionally, it will be more reasonable idea than Kaluza-Klein theory or Braneworld theory in higher dimensional physics, I believe.

Lately, athrough further research to scrutinize the property of transformation group mentioned above by the modeling on the number line, I got the results written in another paper, please see it also via the links below. Frankly, it'll be controversial especially for group theoreticians because it suggests a new finite simple group.
https://www.academia.edu/33780859/A_Transformation_Group_Being_Finitely_Simple_trilogy_of_dimensionality_2_digest_paper_

https://www.academia.edu/16717180/A_Transformation_Group_and_Its_Subgroups_trilogy_of_dimensionality_2_full_paper_

It might be equivalent with Abelian group in Z, but actually non-commutative. By the way, if any subgroups in a group are normal like Abelian group, it's called Dedekind group. Additionally, if the group is non-commutative, Hamilton group. However, the transformation group In my paper is neither of them. Strictly speaking, it’s a “groupoid”. What I’m surprised at is that many of math professors seem not know about groupoid(: algebraic, not in category theory.) As you know, groupoid is partially functional, so unnecessary claimed binary operation. As mentioned above, I take it also as a group for a reason and that’s why my papers’ titles are written ‘group’. See also my paper in this A Transformation Group being Finitely Simple:
https://www.academia.edu/33780859/A_Transformation_Group_Being_Finitely_Simple_trilogy_of_dimensionality_2_digest_paper_
https://www.researchgate.net/publication/313024517_A_Transformation_Group_Being_Finitely_Simple_trilogy_of_dimensionality_2_digest_paper

By the way, a math professor saw my paper replied me that the contents of my papers impressed him a geometric matter well discussed by geometers for more than a century, Yes, that'd be true like I mentioned above. Additionally, at least we should have got back to 400 years ago when Rene Descartes established the coordinate geometry. Because, what I'm discussing in my paper titled "Projections and Dimensions" is also to rethink of it with a concept of space and dimension. Although we generally consider that lower dimensional space is subspace of higher dimensional space, we have to do them disjoint each other. Means, for example, one dimensional space is not subspace of two dimensional space. They are disjoint each other.

Based on this viewpoint, let us think differently from Rene Descartes's idea. Assume that a point in the one dimensional space were transported into two dimensional space, it should have been "point" also in the two dimensional space? Rene Descartes would say "yes", but I definitely does "no". Since the point originally existed in the one dimensional space has only one freedom degree, it can't decide the direction it moves in the two dimensional space. In other words, if the point had two freedom degrees, it could "randomly" or "arbitrary" decide the direction in the two dimensional space. Therefore, the transported point would radiate if it moved within the two dimensional space because of no choice for arbitrary direction. For the detailed explanation, I made a short document. Please see it also via the link below.
https://www.academia.edu/35228325/Dear_Prof.pdf

When I noticed it in my early 20s, I began to rethink of what space dimension is. Thank you for reading all my bullshits anyway, but I'm wishing you could feel what I memtioned above with my research papers worthy of attention.
With best regards,
Euich Miztani

p.s. About the paper entitled "A Transformation Group and Its Subgroups" which is mentioned above, I'm picky though, in the proof of proposition 2.1., strictly speaking, the right cosets Hg={2k-1T2n} "where 2n is fixed" and H'g'={2kT2n-1} "where 2n-1 is fixed." See also “a Transformation Group Being finitely Simple”.

p.p.s. Again, they are the trilogy above:
1. Projections and Dimensions( published in 2011)
https://www.academia.edu/16715934/Projections_and_Dimensions_trilogy_of_dimensionality_1_

2. A Transformation Group and Its Subgroup( published in 2015)
https://www.academia.edu/16717180/A_Transformation_Group_and_Its_Subgroups_trilogy_of_dimensionality_2_full_paper_

A Transformation Group being Finitely Simple(: the digest for the #2, published in a book by selected math papers in 2017)
https://www.academia.edu/33780859/A_Transformation_Group_Being_Finitely_Simple_trilogy_of_dimensionality_2_digest_paper_

3. A Group Repreentation: Notations for “Projections and Dimensions”( published in 2016)
https://www.academia.edu/29017770/A_Group_Representation_Notations_for_Projections_and_Dimensions_trilogy_of_dimensionality_3_

This thesis compiles the results of six works which deal with - inflationary model building and estimation of cosmological parameters from various field theoretic setup, quantification of reheating temperature, studies of leptogenesis in braneworld and estimation of primordial non-Gaussianity from ${\cal N}=1$ supergravity using $\delta N$ formalism.

We start our discussion with exploring the possibility of
MSSM inflation in the light of recent observed data from various D -flat directions using the saddle and inflection point techniques. The effective inflaton potential around saddle point and inflection point have been utilized in estimating the observable parameters and confronting them with WMAP7 and Planck dataset.


Next we explore the possibility of inflation from the five dimensional ${\cal N}=2$ supergravity setup by deriving the effective potential in the context of Randall-Sundrum like braneworld model and Dirac Bonn Infeld Galileon. After deriving an four dimensional effective potential, we obtain the inflationary observables from both the scenarios and confront them with the  WMAP7 data. Further we fit the CMB angular power spectra from TT anisotropy and other polarization data obtained from WMAP7.

Further, we discuss the non-trivial features of reheating from supergravity inspired braneworld model, where the results are to some extent different from that of the usual low energy General Relativistic counterpart, because of the modified Friedmann equations in this setup. We explicitly derive the analytical expressions for the reheating temperature and further solve the evolution equation of the number density of thermal gravitino which results in the gravitino abundance.


Finally, we study the primordial non-Gaussian features using $\delta N$ formalism of unavoidable higher dimensional non-renormalizable K\"ahler operators for ${\cal N}=1$ supergravity framework.In particular we study the nonlinear evolution of cosmological perturbations on large scales which enable us to compute the curvature perturbation,  without solving the exact perturbed field equations.Hence we compute the various non-Gaussian parameters for local type of non-Gaussianities,
for a generic class of sub-Planckian models dominated by the Hubble-induced corrections.

The paper examines the ways in which string theorists construct what I call a "scientific imaginary" as manifest in two technical articles concerned with branes and the braneworld scenario: Joseph Polchinski's "Dirichlet-Branes and Ramond-Ramond Charges" and "Brane World" by Zurab Kakushadze and S.-H. Henry Tye. Such an examination addresses issues of corporeality, animality, visuality, tactility, permeability, and boundaries as they pertain to the core image of the brane within these articles. Rather than an exhaustive survey of brane imagery, it offers an analysis of the particular imaginaries that these string theorists employ within technical discourse. What the essay is most concerned with is the overlap between these terms' ostensibly precise technical meanings and the inferential structure of the images from which the terms originate. I argue that such images bear with them a set of more familiar and earth-bound associations that are both necessary and productive. The cumulative effect within each of these technical articles, then, is the evocation and employment of a scientific imaginary.

We propose a new unified model that describes dark energy and dark matter in the context of [Formula: see text] gravity using a massive scalar field in five dimensions. The scalar field is considered in the bulk that surrounds the 3-brane in branworld model. We show that for a specific choice of the [Formula: see text] function, we can describe the Einstein gravitation in 4-dimensional space-time. We obtain a relationship between the speed of the Universe’s expansion and the speed of the bulk’s expansion. We also propose that the dark matter is represented by the scalar field mass and that the dark energy is a kinetic energy of this field. Finally, we show that, according to conditions, one can obtain the percentages of density of dark matter and the density of ordinary matter.

The purpose of this paper is to show how using certain mathematical values and / or constants from various Ramanujan expressions, we obtain some mathematical connections with several sectors of Black Hole Physics v1 09.02.2020 UPDATED VERSION 16.09.2020 Below the link of the part II of this work: https://www.academia.edu/44109165/Analyzing_some_Ramanujan_formulas_mathematical_connections_with_various_equations_concerning_some_sectors_of_Black_Hole_Physics_II

The problem of creation of Unitary Field theory, or the Theory of Everything, which the Einstein was so eager to solve by means of physics, remained unsolved since it is solvable only by means of the Word: because the Word, according to Bible — is the God and «the God has»: Universe, His Child. This particular solution is given by the linguistic work about the Existing, offered by me to you and based on the Pressing method, the founder of which is Dionysus (Bacchus) — the ancient's god of the Word, Moon and Wine. // Познать Мир, Божий Храм - есть познать Слово нам. Слово - это Луна.

We derive the low energy effective theory on a brane in six-dimensional chiral supergravity. The conical 3-brane singularities are resolved by introducing cylindrical codimension one 4-branes whose interiors are capped by a regular spacetime. The effective theory is described by the Brans-Dicke (BD) theory with the BD parameter given by $\omega_{\rm BD}=1/2$. The BD field is originated from a modulus which is associated with the scaling symmetry of the system. If the dilaton potentials on the branes preserve the scaling symmetry, the scalar field has an exponential potential in the Einstein frame. We show that the time dependent solutions driven by the modulus in the four-dimensional effective theory can be lifted up to the six-dimensional exact solutions found in the literature. Based on the effective theory, we discuss a possible way to stabilize the modulus to recover standard cosmology and also study the implication for the cosmological constant problem.

Many physicists believe that the unification of physics within a single paradigmatic theory has been the primary goal in science for only the past few decades, but this would not be true. Unification was the original goal of Einstein and a few other physicists throughout the 1920s to the 1960s during a period of time when quantum theorists were ironing out their own unique set of problems. The unification of physics under the guise of the quantum paradigm only emerged during the 1970s and has since overshadowed all attempts to unify physics from the fundamental principles of relativity. Today too many physicists believe, without any supporting evidence, that the concept of the quantum is far more fundamental than relativity so earlier attempts based on the continuity of relativity have been all but abandoned. However, both approaches are basically flawed because both relativity and the quantum theory are incomplete as they now stand. Had either side of the quantum versus relativity controversy just simplified their worldview and sought commonality between the two, unification would have been accomplished long ago. The point is, literally, that the discrete quantum, continuous relativity, basic physical geometry and classical physics all share one common characteristic – a paradoxical duality between a dimensionless point (discrete) and an extended length (continuity) in any dimension – and if the problem of unification is approached from an understanding of how this paradox relates to each paradigm all of physics could be unified under a single new theoretical paradigm. Unfortunately there has never been a method, either mathematical or physical, by which a three-dimensional space can be generated from two or more dimensionless points. This shortcoming raises the question – how can the dimensionless point-particles of the Standard Model which presently dominates physics be extended to account for the three-dimensional space in which the physical interactions they describe occur? Fortunately, this question can now be answered, but the answer does not favor the Standard Model of the quantum as it is presently interpreted. Instead, a unified field theory based on continuity that completely unifies the quantum and relativity and completely incorporates the Standard Model as well as the Superstring, Brane and commonly accepted classical theories has been completed.

We put to the test an effective three-dimensional electrostatic potential, obtained effectively by considering an electrostatic source inside a (5+p)-dimensional braneworld scenario with p compact and one infinite spacial extra dimensions in the RS II-p model, for p=1 and p=2. This potential is regular at the source and matches the standard Coulomb potential outside a neighborhood. We use variational and perturbative approximation methods to calculate corrections to the ground energy of the Helium atom modified by this potential, by making use of a 6 and 39-parameter trial wave function of Hylleraas type for the ground state. These corrections to the ground-state energy are compared with experimental data for Helium atom in order to set bounds for the extra dimensions length scale. We find that these bounds are less restrictive than the ones obtained by Morales et. al. through a calculation using the Lamb shift in Hydrogen.

This paper is a review, a thesis, of some interesting results that has been obtained in various researches concerning the "brane collisions in string and M-theory" (Cyclic Universe), p-adic inflation and p-adic cosmology. In the Section 1 we have described some equations concerning cosmic evolution in a Cyclic Universe. In the Section 2, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang space-time, the M-theory model of a Big Crunch/Big Bang transition and some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology. In the Section 3, we have described some equations concerning the generating Ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of N = 1 fivedimensional supergravity with four-dimensional boundaries, and some equations concerning the colliding branes and the origin of the Hot Big Bang. In the Section 4, we have described some equations regarding the "null energy condition" violation concerning the inflationary models and some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with 1 > w. In the Section 5, we have described some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. In the Section 6, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In the Section 7, we have showed various and interesting mathematical connections between some equations concerning the p-adic Inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. Furthermore, in each section, we have showed the mathematical connections with various sectors of Number Theory, principally the Ramanujan's modular equations, the Aurea Ratio and the Fibonacci's numbers.

UPDATED AND REVISITED VERSION  05.11.2021

This paper proposes an M-theory(MT)-derived special type of bijective particle (toy-)theory (BPT), in which each type geometric/topological entity (brane) is assigned one and only one type of physical particle and vice versa. BPT is essentially a topological/geometrical toy-theory (and mainly an exercise of imagination) which tries to explain gravity as a quantum field mediated by 3 types of scale-specific hypothetical massless, heavy and ultra-heavy gravitons: BPT may be thus considered a “trigravity” toy-theory and proposes a way to (at least partially) fill the “gap” between M-Theory (on which BPT is based) and the Standard model (SM) of particle physics. BPT also offers alternative geometrical explanations for rest mass and for electromagnetic charge. This paper continues (from alternative angles of view) the work of other past articles/preprints of the same author in physics (listed from the newest to the oldest).
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The most remarkable and interesting feature of brane world scenario is the use of bulk's curvature to localize gravity on the brane, albeit with fine tuning of the brane and bulk parameters. For FRW expanding universe on the brane, it is a moving hypersurface in Schwarzschild anti de Sitter bulk spacetime. We show that zero mass gravitons have bound state on the brane for suitable values of brane and bulk parameters. There occur various cases giving rise to different cosmological models, in particular we discuss a model with positive cosmological constant on the brane. PACS numbers: 04.50.+h, 04.70.-s, 98.80.-k The view that our Universe might actually have dimensions more than four is anchored on the recent developments in string and M-theories in which gravity arises as a truly higher dimensional interaction. Only in the low energy limit it manifests in the familiar 4-dimensional general relativity (GR). This has initiated a lot of activity in recent times. Though the basic idea was already there in the form of Kaluza-Klein (KK) theories, the recent spurt is primarily due to the possibility which helps solve the mass hierarchy problem in the standard model of particle physics. In some of these models [1] the extra dimensions can be as large as millimeter which is however less than the current observational limits on low scale gravity. Notable are the models in which the extra dimensions are allowed to be of infinite extent [2, 3]. These models have Z 2-symmetry, which is motivated by the reduction of M theory to E 8 × E 8 heterotic string theory [4]. The single brane Randall-Sundrum (RS) model [3] has attracted a lot of interest and activity. In this model the Minkowski flat brane in 5-D anti de Sitter (AdS) bulk has a positive tension. It is then possible to recover Newton's inverse square law with r −4 correction term which arises from massive KK modes contribution. There have been various generalizations of this [5] in the form of thick branes [6], AdS branes [7] and brane models without Z 2 symmetry [8]. In the overall view of the brane world scenario, all matter fields live on the 3-brane which is the 4-D physical Universe while gravity can propagate in the extra dimensions , say 5-D bulk. The bulk and brane spacetimes are joined together through the Israel junction conditions [9]. Since, the standard Einstein equations on the brane are modified by the bulk effects it opens a whole new vista for investigation of astrophysical and cosmological consequences of these models (see for eg. [10]). The connection with CFT correspondence has also been studied (see for eg. [11]). For complete solution of the problem, one has to solve the Λ-vacuum equation in 5-D for the * bulk spacetime and simultaneously the modified Einstein equation which in addition to the square of stress tensor also contains the projection of the bulk Weyl curvature on the brane (dark radiation) and then the two solutions should be joined together with the Israel junction conditions [9]. It is by all means a very formidable task and it is therefore no surprise that there exist only few complete solutions to date. The two important cases are flat/vacuum brane with AdS bulk and FRW brane with Schwarzschild-AdS (S-AdS) bulk [12-14]. The S-AdS bulk-brane system is composed of two patches of S-AdS bulk having in general different mass parameters for the black hole with the brane located at y = y(τ), where y(τ) is determined by solving the Is-rael junction conditions [13]. From the resulting equation for brane trajectory one finds that in S-AdS bulk, the FRW brane will in general be moving unless the parameters are properly fine tuned (eg. for the RS case σ = (3/4πG 5 l), Λ 4 = 0, where σ is the brane tension, G 5 is the 5-D gravitational constant and l is the radius of curvature of the bulk spacetime). The extra dimension is a radial coordinate of the bulk and imposing Z 2 symmetry across the brane demands that the mass parameters of both the patches to be same. By fine tuning parameters one can obtain static branes which cannot harbour expansion which is essential for realistic cosmo-logical models. A slight off tuned value of σ or non zero value of black hole mass would set it moving. Though localization of gravity for the AdS bulk with flat brane and for some generalizations of it has been established [7, 15, 16], it has not yet been investigated for the S-AdS bulk with FRW brane. The problem becomes difficult if one notes that unlike the RS case, the brane is dynamic in the static bulk and it is non trivial to fix the boundary conditions on the modes. For localizability of gravity on the brane, first there should exist bound normalizable mode for zero mass graviton and plus there should also exist KK modes to give the correction to the Newtonian gravity. All this could be studied by considering perturbation of the bulk metric and the brane motion determined by the Israel junction conditions. This is what we shall concern ourselves in this investigation which should be quite indicative of localiza

In this revisited research thesis, we have analyzed and deepened further Ramanujan expressions (mock theta functions and taxicab numbers) applied to some sectors of String Theory (Brane-World) and to the Black Hole Physics. We have therefore described other new possible mathematical connections.

v2  -  27.01.2021

The surname Armstrong, “strong hand,” is a direct sign of the mission of the great earthling. The ancients said that the Hand of God is the Beginning of the universal movement, the Mother Goddess: the creative aspect of the Creator, personified by the Moon. The strongest hand biologically is the wing, the hand of our soul. He who is winged with soul is a hero, a winner. The hero was the chosen one of Heaven, Neil. // Фамилия Армстронг, «сильная рука» — прямой знак миссии великого землянина. Рукой Бога Древние называли Начало вселенского движения — Богиню-Мать: созидательный аспект Творца, олицетворением какого есть Луна. Самой сильной рукой биологически является крыло — рука души нашей. Крылат душой — герой. Им был избранник Неба Нил.

The superstring and brane theories have strayed too far from reality and their physical roots to be seriously considered as legitmate physics. They are no more than philosophical and mathematical theories in spite of their great popularity. Otherwise, they are bad physics and just plain wrong. They also suffer from a great many logical errors that would seem to be fatal for any other group of theories. However, if the logical errors are corrected, then the latest of the brane models morphs into a completely different model that is actually correct and falsifiable.