Open Universe Research Articles

Matrix model approach to cosmology

We perform a systematic search for rotationally invariant cosmological solutions to matrix models, or more specifically the bosonic sector of Lorentzian IKKT-type matrix models, in dimensions $d$ less than ten, specifically $d=3$ and $d=5$. After taking a continuum (or commutative) limit they yield $d-1$ dimensional space-time surfaces, with an attached Poisson structure, which can be associated with closed, open or static cosmologies. For $d=3$, we obtain recursion relations from which it is possible to generate rotationally invariant matrix solutions which yield open universes in the continuum limit. Specific examples of matrix solutions have also been found which are associated with closed and static two-dimensional space-times in the continuum limit. The solutions provide for a matrix resolution of cosmological singularities. The commutative limit reveals other desirable features, such as a solution describing a smooth transition from an initial inflation to a noninflationary era. Many of the $d=3$ solutions have analogues in higher dimensions. The case of $d=5$, in particular, has the potential for yielding realistic four-dimensional cosmologies in the continuum limit. We find four-dimensional de Sitter $dS^4$ or anti-de Sitter $AdS^4$ solutions when a totally antisymmetric term is included in the matrix action. A nontrivial Poisson structure is attached to these manifolds which represents the lowest order effect of noncommutativity. For the case of $AdS^4$, we find one particlular limit where the lowest order noncommutativity vanishes at the boundary, but not in the interior.

Singular Propositions, Negation and the Square of Opposition

This paper contains two traditions of diagrammatic studies namely one, the Euler–Venn–Peirce diagram and the other, following tradition of Aristotle, the square of oppositions. We put together both the traditions to study representations of singular propositions (through a diagram system Venn-i, involving constants), their negations and the inter relationship between the two. Along with classical negation we have incorporated negation of another kind viz. absence (taking a cue from the notion of ‘abhava’ existing in ancient Indian knowledge system). We have also considered the changes that take place in the context of open universe.

Joint probabilistic approach for real-time face recognition with transfer learning

Face recognition is an integral part in robot partner’s interaction with its human subject. With the task expected, open universe scenario face recognition needs to be performed under uncontrolled and uncooperative environment in real-time without imposing strain on its low powered processors. With the concept of informationally structured space where additional samples can contribute to constructing the prior for recognition, joint probabilistic face method is used for its accuracy, simplicity and transparency in data handling. A system is built upon the method that enables transfer learning from across different domains and real-time adaptive learning for continual information collection. Test result shows that the approach can perform well compared to other established methods under low number of prototype number. Test result on adaptive learning sheds light to the direction of future design.

Emergent scenario in the Einstein-Cartan theory

We study the emergent scenario, which is proposed to avoid the big bang singularity, in the Einstein-Cartan (EC) theory with a positive cosmological constant and a perfect fluid by analyzing the existence and stability of the Einstein static (ES) solutions. We find that there is no stable ES solution for a spatially flat or open universe. However, for a spatially closed universe, the stable ES solution does exist, and in the same existence parameter regions, there also exists an unstable one. With the slow decrease of the equation of state $w$ of the perfect fluid, the stable and unstable critical points move close gradually and coincide once $w$ reaches a critical value, so that the stable critical point becomes an unstable one. As a result, if $w$ approaches a constant at $t\rightarrow -\infty$, the universe can stay at the stable ES state past eternally, and furthermore it can naturally exit from this state and evolve into an inflationary era if $w$ decreases slowly as time goes forward. Therefore, the emergent scenario that avoids the big bang singularity can be successfully implemented in the EC theory of gravity.

Mach’s principle for cosmological solutions in relativistic theory of gravity

Mach’s principle in the relativistic theory of gravity (RTG) allows preliminarily selecting all possible solutions of the theory, including cosmological ones. It is shown that Mach’s principle in the RTG with massive gravitons admits only flat and open universe evolution scenarios, excluding the closed universe version. Unlike a standard cosmological solution in the RTG containing only a single free constant to be constrained from above by the causality principle, the most general flat scenario should have two parameters. It is shown that the second constant does not significantly affect realistic cosmological solutions. The open scenario for massless graviton theory is ruled out by the causality principle.

Holography in (2 + 1)−dimensional Cosmological Model with Generalized Equation of State

In this paper we examine the cosmic holographic principle suggested by Fischler and Susskind (1998) in (2 + 1)−dimensional cosmological models by using generalized equation of state of a more general form, p = (γ−1)(ρ+ρ0), where γ and ρ0 are two parameters. It is shown that the holographic principle is satisfied in all flat and open universes. For (2 + 1)−dimensional closed universe by applying the method proposed by Kaloper, N. and Linde, A. Phys. Rev. D 60, 103509 (1999), we find that the holographic principle cannot be realized in general.

Energy Conditions of Built-In Inflation Models inf(T)Gravitational Theories

We examine the validity of energy conditions of built-in inflation models inf(T)gravitational theories. For this purpose, we formulate the inequalities of energy conditions by assuming the flat and nonflat Friedmann-Robertson-Walker (FRW) universe. We find the feasible constraints on the constants of integration and evaluate their possible ranges graphically for the consistency of these energy conditions for flat, closed, and open universes. We constrain the constants of integration for flat space-time from the inflation epoch while the closed and open universe constants are constrained from late universe.

Singularities and the geometry of spacetime

The aim of this essay is to investigate certain aspects of the geometry of the spacetime manifold in the General Theory of Relativity with particular reference to the occurrence of singularities in cosmological solutions and their relation with other global properties. Section 2 gives a brief outline of Riemannian geometry. In Section 3, the General Theory of Relativity is presented in the form of two postulates and two requirements which are common to it and to the Special Theory of Relativity, and a third requirement, the Einstein field equations, which distinguish it from the Special Theory. There does not seem to be any alternative set of field equations which would not have some undeseriable features. Some exact solutions are described. In Section 4, the physical significance of curvature is investigated using the deviation equation for timelike and null curves. The Riemann tensor is decomposed into the Ricci tensor which represents the gravitational effect at a point of matter at that point and the Welyl tensor which represents the effect at a point of gravitational radiation and matter at other points. The two tensors are related by the Bianchi identities which are presented in a form analogous to the Maxwell equations. Some lemmas are given for the occurrence of conjugate points on timelike and null geodesics and their relation with the variation of timelike and null curves is established. Section 5 is concerned with properties of causal relations between points of spacetime. It is shown that these could be used to determine physically the manifold structure of spacetime if the strong causality assumption held. The concepts of a null horizon and a partial Cauchy surface are introduced and are used to prove a number of lemmas relating to the existence of a timelike curve of maximum length between two sets. In Section 6, the definition of a singularity of spacetime is given in terms of geodesic incompleteness. The various energy assumptions needed to prove the occurrence of singularities are discussed and then a number of theorems are presented which prove the occurrence of singularities in most cosmological solutions. A procedure is given which could be used to describe and classify the singularites and their expected nature is discussed. Sections 2 and 3 are reviews of standard work. In Section 4, the deviation equation is standard but the matrix method used to analyse it is the author’s own as is the decomposition given of the Bianchi identities (this was also obtained independently by Trumper). Variation of curves and conjugate points are standard in a positive-definite metric but this seems to be the first full account for timelike and null curves in a Lorentz metric. Except where otherwise indicated in the text, Sections 5 and 6 are the work of the author who, however, apologises if through ignorance or inadvertance he has failed to make acknowledgements where due. Some of this work has been described in [Hawking S.W. 1965b. Occurrence of singularities in open universes. Phys. Rev. Lett. 15: 689–690; Hawking S.W. and G.F.R. Ellis. 1965c. Singularities in homogeneous world models. Phys. Rev. Lett. 17: 246–247; Hawking S.W. 1966a. Singularities in the universe. Phys. Rev. Lett. 17: 444–445; Hawking S.W. 1966c. The occurrence of singularities in cosmology. Proc. Roy. Soc. Lond. A 294: 511–521]. Undoubtedly, the most important results are the theorems in Section 6 on the occurrence of singularities. These seem to imply either that the General Theory of Relativity breaks down or that there could be particles whose histories did not exist before (or after) a certain time. The author’s own opinion is that the theory probably does break down, but only when quantum gravitational effects become important. This would not be expected to happen until the radius of curvature of spacetime became about 10-14 cm.

Revisiting the holographic dark energy in a non-flat universe: alternative model and cosmological parameter constraints

We propose an alternative model for the holographic dark energy in a non-flat universe. This new model differs from the previous one in that the IR length cutoff $L$ is taken to be exactly the event horizon size in a non-flat universe, which is more natural and theoretically/conceptually concordant with the model of holographic dark energy in a flat universe. We constrain the model using the recent observational data including the type Ia supernova data from SNLS3, the baryon acoustic oscillation data from 6dF, SDSS-DR7, BOSS-DR11, and WiggleZ, the cosmic microwave background data from Planck, and the Hubble constant measurement from HST. In particular, since some previous studies have shown that the color-luminosity parameter $\beta$ of supernovae is likely to vary during the cosmic evolution, we also consider such a case that $\beta$ in SNLS3 is time-varying in our data fitting. Compared to the constant $\beta$ case, the time-varying $\beta$ case reduces the value of $\chi^2$ by about 35 and results in that $\beta$ deviates from a constant at about 5$\sigma$ level, well consistent with the previous studies. For the parameter $c$ of the holographic dark energy, the constant $\beta$ fit gives $c=0.65\pm 0.05$ and the time-varying $\beta$ fit yields $c=0.72\pm 0.06$. In addition, an open universe is favored (at about 2$\sigma$) for the model by the current data.

Symmetry breaking and conformal coupling with a generalized Higgs potential and the role of the generalized Duffin-Loret equation in the early universe

The concepts of spontaneous symmetry breaking and phase transition play an important role in the physics of the early universe and particle theory. In this paper, taken into account the conformal coupling between the scalar field and gravity via the scalar curvature and the complex conjugate of the scalar field, it is shown that spontaneous symmetry breaking can occur in the early universe if the complex scalar field Higgs potential is replaced by a generalized potential consisting of the sum of the Higgs potential and a power law potential of order n . Both higher order and negative values of n were discussed. Some features were raised and the cosmological symmetry breaking in the early universe in an open Robertson-Walker universe was also discussed for different values of n .

Accelerated FRW solutions in Chern–Simons gravity

We consider a five-dimensional Einstein–Chern–Simons action which is composed of a gravitational sector and a sector of matter where the gravitational sector is given by a Chern–Simons gravity action instead of the Einstein–Hilbert action and where the matter sector is given by the so-called perfect fluid. It is shown that (i) the Einstein–Chern–Simons (EChS) field equations subject to suitable conditions can be written in a similar way to the Einstein–Maxwell field equations; (ii) these equations have solutions that describe an accelerated expansion for the three possible cosmological models of the universe, namely, spherical expansion, flat expansion, and hyperbolic expansion when \(\alpha \), a parameter of the theory, is greater than zero. This result allows us to conjecture that these solutions are compatible with the era of dark energy and that the energy–momentum tensor for the field \(h^{a}\), a bosonic gauge field from the Chern–Simons gravity action, corresponds to a form of positive cosmological constant. It is also shown that the EChS field equations have solutions compatible with the era of matter: (i) In the case of an open universe, the solutions correspond to an accelerated expansion (\(\alpha >0\)) with a minimum scale factor at initial time that, when time goes to infinity, the scale factor behaves as a hyperbolic sine function. (ii) In the case of a flat universe, the solutions describe an accelerated expansion whose scale factor behaves as an exponential function of time. (iii) In the case of a closed universe there is found only one solution for a universe in expansion, which behaves as a hyperbolic cosine function of time.

Emergent universe supported by chiral cosmological fields in 5D Einstein-Gauss-Bonnet gravity

We propose the application of the chiral cosmological model (CCM) for the Einstein--Gauss--Bonnet (EGB) theory of gravitation with the aim of finding new models of the Emergent Universe (EmU) scenario. We analysed the EmU supported by two chiral cosmological fields for a spatially flat universe, while we have used three chiral fields when we investigated open and closed universes. To prove the validity of the EmU scenario we fixed the scale factor and found the exact solution by decomposition of EGB equations and solving the chiral field dynamics equation. To this end, we suggested the decomposition of the EGB equations in such a way that the first chiral field is responsible for the Einstein part of the model, while the second field, together with kinetic interaction term, is connected with the Gauss--Bonnet part of the theory. We proved that both fields are phantom ones under this decomposition, and that the model has a solution if the kinetic interaction between the fields equals a constant. We have presented the exact solution in terms of cosmic time. This was done for a spatially flat universe. In the case of open and closed universes we introduced the third chiral field (canonical for closed and phantom for open universe) which is responsible for the EGB and curvature parts. The solution of the third field equation is obtained in quadratures. Thus we have proved that the CCM is able to support EmU scenario in EGB gravity for spatially flat, open and closed universes.

Intelligible design: a realistic approach to the philosophy and history of science

Modern Science in Historical Perspective: On the Origins of Modern Science The Post-Renaissance Revolution: The New Science Frank Sherwood Taylor: The Man Who was Converted by Galileo The Limits of Science Proofs and Demonstrations On the Intelligibility of Quantum Mechanics Uncertainty, Incompleteness, Chance, and Design A Finite, Open and Contingent Universe On the Origin and Development of Life: A Brief History of Evolutionary Thought Life's Intelligible Design What Are the Contributions of Genetics to the Understanding of Life? On the Human Genome in Contrast to the Great Ape's Genome On the Evolution Controversy Scientific Critiques of Darwinism Beyond Nature and History: The Religion of Darwinism On the Riddle of Man's Origin On Science, History and Free Will Science and Faith.

Lonergan and Bhaskar: The Intelligibility of Experiment

The aim of this paper is to note the convergence between two critical realist philosophies of science, namely, that of Roy Bhaskar and Bernard Lonergan with regard to the intelligibility of experimental activity. Bhaskar very explicitly argues that ‘differentiation implies stratification.’ The idea is that because the situations produced in laboratories are special instances of closure (like the solar system in the open universe, they do not represent the general case) the significance of experimental activity is that it brings about regularities with a view to understanding scientific laws at a deeper level. This is to say, when experiment is properly understood, the weaknesses of empiricism are exposed. Although he is not as explicit, Lonergan also has recourse to this argument. The parallels between Bhaskar and Lonergan are not surprising given the Aristotelian heritage that is manifest in their common concern for a realist ontology. Nevertheless, some differences between the two emerge, for example, in Lonergan's concern with the development of statistical science, and as well, a firm commitment to substance (rather than to powers, simply). Some attention to the significance of experimental activity for the debate surrounding realism is explored; it is suggested that Lonergan has something to offer in the subsequent conversation associated with Maxwell, van Fraassen, Hacking and Cartwright.

A halo bias function measured deeply into voids without stochasticity

We study the relationship between dark-matter haloes and matter in the MIP $N$-body simulation ensemble, which allows precision measurements of this relationship, even deeply into voids. What enables this is a lack of discreteness, stochasticity, and exclusion, achieved by averaging over hundreds of possible sets of initial small-scale modes, while holding fixed large-scale modes that give the cosmic web. We find (i) that dark-matter-halo formation is greatly suppressed in voids; there is an exponential downturn at low densities in the otherwise power-law matter-to-halo density bias function. Thus, the rarity of haloes in voids is akin to the rarity of the largest clusters, and their abundance is quite sensitive to cosmological parameters. The exponential downturn appears both in an excursion-set model, and in a model in which fluctuations evolve in voids as in an open universe with an effective $\Omega_m$ proportional to a large-scale density. We also find that (ii) haloes typically populate the average halo-density field in a super-Poisson way, i.e. with a variance exceeding the mean; and (iii) the rank-order-Gaussianized halo and dark-matter fields are impressively similar in Fourier space. We compare both their power spectra and cross-correlation, supporting the conclusion that one is roughly a strictly-increasing mapping of the other. The MIP ensemble especially reveals how halo abundance varies with `environmental' quantities beyond the local matter density; (iv) we find a visual suggestion that at fixed matter density, filaments are more populated by haloes than clusters.

Lagrangian perturbation theory: Third-order solution for general dark energy models

We investigate the Lagrangian perturbation theory of a general dark energy models with a constant equation of state, \omega, and derive the fitting form of the fastest growing mode solutions up to the third order. These fitting forms are less than few % errors compared to the numerical calculation. We also correct the solutions of Einstein-de Sitter and the open universe with the proper initial conditions and the correct consideration for the growing mode solutions in order to compare with the general dark energy models. Even though these fitting forms are limited to the constant equation of state models, one can apply these to the time-varying \omega by interpolating between models with the constant \omega. These fitting forms can be used for the correct prediction for the two point correlation function and can give the proper predictions for the different dark energy models.

Cosmology: The search for twenty-four (or more) functions

We enumerate the $4(1+F)+2S$ independent arbitrary functions of space required to describe a general relativistic cosmology containing an arbitrary number of noninteracting fluid ($F$) and scalar fields ($S$). Results are also given for arbitrary space dimension and for higher-order gravity theories, where the number increases to $16+4F+2S$. Both counts are subject to assumptions about whether the dark energy is a cosmological constant. A more detailed analysis is provided when global homogeneity is assumed and the functions become constants. This situation is also studied in the case where the flat and open universes have compact spatial topologies. This changes the relative generalities significantly and places new constraints on the types of expansion anisotropy that are permitted. The most general compact homogeneous universes containing Friedmann models are spatially flat and described by $8+4F+2S$ constants. Comparisons are made with the simple six-parameter lambda-CDM model and physical interpretations provided for the parameters needed to describe the most general cosmological models.

Higher Dimensional Spherically Symmetric Expanding Wormholes in Einstein’s Gravity

We present (n+1)-dimensional expanding structures in a cosmological background. Due to the expansion of spacetime the throat of wormholes enlarge with time. These solutions are examined in the Einstein’s framework. A general linear relation between diagonal elements of an anisotropic energy-momentum tensor is assumed and the spherically symmetric structures are obtained. Solutions include naked singularity and expanding wormholes in an open universe. The traversibility of wormhole solutions is explored and we find that they are basically traversable. Finally, we consider the corresponding energy-momentum tensor properties and specially take into account the standard energy conditions.

Closed type families with overlapping equations

Open, type-level functions are a recent innovation in Haskell that move Haskell towards the expressiveness of dependent types, while retaining the look and feel of a practical programming language. This paper shows how to increase expressiveness still further, by adding closed type functions whose equations may overlap, and may have non-linear patterns over an open type universe. Although practically useful and simple to implement, these features go beyond conventional dependent type theory in some respects, and have a subtle metatheory.

Axiom Cosmology: A New Direction

In this work we have to deal with the axiomatization of cosmology, but it is only recently that we have hit upon a new mathematical approach to capitalize on our new set identities for the basic laws of cosmology. So our proposal of settlement is that we will propose some new laws (e.g., formation of the black hole). We introduce the concept of axiom cosmology. This principle describes the cosmology which can get freedom from the notion of the induction. We present a large-scale structure model of the universe, and this leads to successfully explanation of problem of closed universe or open universe (because from the outset it is theorem and its succinct proof). In this paper we prove that the non-singular point theorem means that a singularity cannot be mathematically defined nor physical. It allows us to overcome the mysterious, physical singularity conundrum and explain meaningful antimatter annihilations for general configurations.