Initial Singularity Research Articles

Onset of synchronization in coupled Mixmaster oscillators.

We consider the problem of asymptotic synchronization of different spatial points coupled to each other in inhomogeneous space-time and undergoing chaotic Mixmaster oscillations towards the singularity. We demonstrate that for couplings larger than some threshold value, two Mixmaster spatial points [Formula: see text], with [Formula: see text] in the past of [Formula: see text], synchronize and thereby proceed in perfect unison towards the initial singularity. We further show that there is a Lyapunov function for the synchronization dynamics that makes different spatial points able to synchronize exponentially fast in the past direction. We provide an elementary proof of how an arbitrary spatial point responds to the mean field created by the oscillators, leading to their direct interaction through spontaneous synchronization. These results ascribe a clear physical meaning of early-time synchronization leading to a resetting effect for the two BKL maps corresponding to two distinct oscillating spatial points, as the two maps converge to each other to become indistinguishable at the end of synchronization. Our results imply that the universe generically organizes itself through simpler, synchronized, states as it approaches the initial singularity. A discussion of further implications of early-time inhomogeneous Mixmaster synchronization is also provided. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

Cosmological milestones, conformal frameworks and quiescent cosmology.

To understand the nature of the birth of our Universe and its eventual demise is a driving force in theoretical physics and astronomy and, indeed, for humanity. A zoo of definitions has appeared in the literature to catalogue different types of cosmological milestones such as 'Big Bangs', 'Big Crunches', 'Big Rips', 'Sudden Singularities', 'Bounces' and 'Turnarounds'. Quiescent cosmology is the notion that the Universe commenced in a Big Bang that was highly regular and smooth, and evolved away from this initial isotropy and homogeneity due to gravitational attraction. The quiescent cosmology concept meshes well with Penrose's ideas regarding gravitational entropy and the clumping of matter, and the associated Weyl Curvature Hypothesis. Conformal frameworks, such as the Isotropic Past Singularity (IPS), have been devised to encapsulate initial and final states for the Universe which are in accordance with these programmes. These geometric definitions are independent of models, coordinates and the equation of state of the source of the gravitational field. Much of the research on cosmological milestones has been focussed on the FRW solutions, many of which possess initial singularities which are isotropic Big Bangs. We analyse here the relationship between cosmological milestones and conformal frameworks for these solutions. We establish the general properties of FRW models which admit these conformal frameworks, including whether they satisfy various energy conditions, and are therefore physically reasonable. These results inform future development of the quiescent cosmology program. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

Bouncing Cosmology in Fourth-Order Gravity

The Big Bang initial singularity problem can be solved by means of bouncing solutions. In the context of extended theories of gravity, we will look for covariant effective actions whose field equations contain up to fourth-order derivatives of the metric tensor. In finding such bouncing solutions, we will make use of an order reduction technique based on a perturbative approach. Reducing the order of the field equations to second-order, we are able to find solutions which are perturbatively close to General Relativity. We will build the covariant effective actions of the resulting order reduced theories.

Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations

In this paper, mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations with initial singularity are investigated. By adopting the modified Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution. Furthermore, the regularity and logarithmic decay of its solution are researched. Under the singularity hypothesis, the numerical methods for the problems are then studied. The model is first transformed into its equivalent form and then the technic of exponential type meshes is utilized. The fact that the discrete coefficients of Hadamard fractional integral have several graceful properties is crucial in convergence and stability analysis. For the sake of reducing storage and computational cost, the SOE technology is exploited to the new variable t=log⁡tns. On this basis, a fast compact difference scheme and a fast compact ADI method are constructed for the one- and two-dimensional problems, respectively. To illustrate the significance of studying singularity, a Crank-Nicolson difference scheme is proposed. The final result that the error in numerically approximating the solution depends on the parameter γ shows obviously how the regularity of the solution and the exponential type meshes affect the convergence order of the derived scheme. Ultimately, the numerical examples are provided to verify that the fast algorithm effectively reduces the computational cost compared to the direct method.

Unconditionally optimal H1-norm error estimates of a fast and linearized Galerkin method for nonlinear subdiffusion equations

In this paper, the unconditionally optimal H1-norm error estimate for a two-step fast L1 scheme is considered for a nonlinear subdiffusion equation, which involves the considerations of both the initial singularity of the solution and the computational complexity to evaluate the Caputo derivative. For the fully discrete scheme, we use the fast L1 scheme developed in [28] to speed up the evaluation of the Caputo derivative on nonuniform time step, employ the Galerkin finite element (FE) method to discretize the spatial direction, and take the Newton linearized method to approximate the nonlinear term. While the theory of the L2-norm error estimate is well studied, there are few works on the unconditional convergence of H1-norm error estimate of the linearized Galerkin FE scheme for nonlinear subdiffusion equations. The main reason is that it requires to find a suitable test function in the FE space. To this end, we here respectively take the time-discrete operator and Laplace operator as the suitable test function in the FE space, and combine an improved discrete fractional Grönwall inequality to achieve the unconditional error estimate in H1-norm. Numerical results are provided to demonstrate the theoretical results.

Review Article: Cosmology and Classical Physics

In 1937, Paul Dirac proposed Large Number Hypothesis and Hypothesis of Variable Gravitational Constant, and later added notion of Continuous Creation of Matter in the World. Hypersphere World-Universe Model (WUM) follows these ideas, albeit introducing different mechanism of Matter creation. In this paper, we show that WUM is a natural continuation of Classical Physics. WUM is proposed as an alternative to prevailing Big Bang Model (BBM) that relies on General Relativity. WUM and BBM are principally different Models: 1) Instead of Initial Singularity with infinite energy density and extremely rapid expansion of spacetime (Inflation) in BBM; in WUM, there was Fluctuation (4D Nucleus of World with extrapolated radius equal to basic size unit of a) in Eternal Universe with finite extrapolated energy density (~104 less than nuclear density) and finite expansion of Nucleus in Its fourth spatial dimension with speed c that is gravitodynamic constant; 2) Instead of alleged practically Infinite Homogeneous and Isotropic Universe around Initial Singularity in BBM; in WUM, 3D Finite Boundless World (Hypersphere of 4D Nucleus) presents Patchwork Quilt of various Luminous Superclusters (≧103), which emerged in different places of World at different Cosmological times. Medium of World, consisting of protons, electrons, photons, neutrinos, and dark matter particles, is Homogeneous and Isotropic. Distribution of Macroobjects is spatially Inhomogeneous and Anisotropic and temporally Non-simultaneous. Most direct observational evidence of validity of WUM are: 1) Microwave Background Radiation and Intergalactic Plasma speak in favor of existence of Medium; 2) Laniakea Supercluster with binding mass ~1017M⊙ is home to Milky Way (MW) and ~105 other nearby galaxies, which did not start their movement from Initial Singularity; 3) MW is gravitationally bounded with Virgo Supercluster (VS) and has Orbital Angular Momentum that far exceeds its rotational angular momentum; 4) Mass-to-light ratio of VS is ~300 times larger than that of Solar ratio. Similar ratios are obtained for other superclusters. These ratios are main arguments in favor of presence of significant amounts of Dark Matter in the World. 5) Astronomers discovered the most distant galaxy HD1 that is ~13.5 Bly away. WUM predicts discovery of galaxies with a distance of ~13.8 Bly. Medium of World, Dark Matter, and Angular Momentum are main Three Pillars of WUM.

EFFICIENT NUMERICAL SOLUTION OF TWO-DIMENSIONAL TIME-SPACE FRACTIONAL NONLINEAR DIFFUSION-WAVE EQUATIONS WITH INITIAL SINGULARITY

<abstract> In this paper, we present an efficient linearized alternating direction implicit (ADI) scheme for two-dimensional time-space fractional nonlinear diffusion-wave equations with initial singularity. First, the original problem is equivalently transformed into its partial integro-differential form. Then, for the time discretization, the Crank-Nicolson technique combined with the midpoint formula and the second order convolution quadrature formula are used. Meanwhile, the classical central difference formula and fractional central difference formula are adopted to approximate the second order derivative and the Riesz derivative in space, respectively. The unconditional stability and convergence of the proposed scheme are proved by the energy method. Numerical experiments support the theoretical results. </abstract>

Hubble Tension

The results of measurements of the Hubble constant H0, which characterizes the expansion rate of the universe, show that the values of H0 vary significantly depending on Methodology. The disagreement in the values of H0 obtained by the various teams far exceeds the standard uncertainties provided with the values. This discrepancy is called the Hubble Tension. In this paper, we discuss Macrostructures of the World (Superclusters and Galaxies); explain their Origin and Evolution in frames of the developed Hypersphere World-Universe Model (WUM), which is an alternative to the prevailing Big Bang Model (BBM) [1]; and provide the explanation of the Hubble Tension. The main difference between WUM and BBM is: Instead of the Infinite Homogeneous and Isotropic Universe around the Initial Singularity in BBM, in WUM, the 3D Finite Boundless World (a Hypersphere) presents a Patchwork Quilt of different Luminous Superclusters (103), which emerged in various places of the World at different Cosmological times. In WUM, the Medium of the World is Homogeneous and Isotropic. The distribution of Macroobjects in the World is spatially Inhomogeneous and Anisotropic and temporally Non-simultaneous.

Ricci DeTurck flow on incomplete manifolds

In this paper we construct a Ricci DeTurck flow on any incomplete Riemannian manifold with bounded curvature. The central property of the flow is that it stays uniformly equivalent to the initial incomplete Riemannian metric, and in that sense preserves any given initial singularity structure. Together with the corresponding result by W.-X. Shi for complete manifolds [J. Differ. Geom. 30, No. 1, 223–301 (1989; Zbl 0676.53044)], this gives that any (complete or incomplete) manifold of bounded curvature can be evolved by the Ricci DeTurck flow for a short time.

On the Uncertainty Relations That Would Be Used in the New Description of the Big Bang According to the Hypothesis of Primary Particles

In this paper, we have determined the structure of the uncertainty relations obtained on the basis of the dimensions that describe the very origin of the Big Bang—in accordance with our Hypothesis of Primary Particles, and with the logically introduced, smallest increment of speed that can exist, the “speed quantum”. This approach allowed us to theoretically move the margin for the description of this singularity to values smaller than the Planck time and the Planck length; hence, we also introduced a new constant in the uncertainty relations, which corresponds to the reduced Planck constant. We expect that such a result for the initial singularity itself will enable a more detailed study of the Big Bang, while opening new areas of study in physics.

A Standard Model Approach to Inflation

By assuming the cosmological principle includes the Pauli Exclusion Principle (PEP) and that the initial singularity existed within Planck time and length scales, a model for inflationary expansion is argued using only standard model physics without any changes to general relativity. All Fermionic matter is forced by the PEP to make a quantum transition to minimally orthogonal states in sequential Planck time intervals. This results in an initial inflation effect due to nearest neighbor quantum transitions which is then exacerbated by matter and antimatter creation effects due to collisions giving rise to the observational effects of universal inflation. The model provides a mechanistic explanation for primordial expansion using only physics from the standard model, specifically utilizing the PEP as a repulsion force between indistinguishable fermions. The present theory offers the benefit of not requiring any particles or fields outside of the standard model nor utilizing changes to general relativity. More succinctly, this theory goes beyond simply offering a mathematical representation (or fit) of the functional dependence but rather offers a mechanistic model to drive inflation using only standard model physics.

Temporal No-Linearity: An Alternative to Dark Energy

Mathematical compatibilities and constraints of a hypothetical 5D space-time with time referenced by two coordinates (3-2) have been revisited in detail in several recent papers. It has been prescribed from the GR the compatibility constraints of the FLRW metric in each temporal brane, to be restricted to Closed Universes, Smooth Initial Singularities, and “Open CTC”. In a first view, this leads to leaving these works considering mathematical games discarded by the Standard Candles data. However, if time would be referred by two coordinates, they would not be linearly related, and it will be mathematically stated that space-time may not be flat in any case because time-like branes geometry will never be. If so, the time scale “lived” over a two-time dimension geodesic necessarily is not constant over its linear projection on one of both coordinates. Consequently, the correlations between Redshift and Distance Modulus—Distance Ladder—may be corrected by a synchronization function (if a no-linear two-time geodesic trajectory over a “warped temporal geometry” is linearly divided into constant segments, then their projections are not linear in any case). We apply an example of time-trajectory over the time slices, matching the Standard Candle data for a Closed Universe dominated by matter (>90%) in a bulk 3-2 configuration, with open temporal branes and smooth singularity. If time can be referred by two coordinates, then there is no need of Darkness to explain astrophysical data and Universe can be closed.

F(T) cosmology with nonzero curvature

We investigate exact and analytic solutions in [Formula: see text] gravity within the context of a Friedmann–Lemaître–Robertson–Walker background space with nonzero spatial curvature. For the power-law theory [Formula: see text] we find that the field equations admit an exact solution with a linear scalar factor for negative and positive spatial curvature. That Milne-like solution is asymptotic behavior for the scale factor near the initial singularity for the model [Formula: see text]. The analytic solution for that specific theory is presented in terms of Painlevé series for [Formula: see text]. Moreover, from the value of the resonances of the Painlevé series we conclude that the Milne-like solution is always unstable while for large values of the independent parameter, the field equations provide an expanding universe with a de Sitter expansion of a positive cosmological constant. Finally, the presence of the cosmological term [Formula: see text] in the studied [Formula: see text] model plays no role in the general behavior of the cosmological solution and the universe immerge in a de Sitter expansion either when the cosmological constant term [Formula: see text] in the [Formula: see text] model vanishes.

Non-singular non-flat universes

The quest to understand better the nature of the initial cosmological singularity is with us since the discovery of the expanding universe. Here, we propose several non-flat models, among them the standard cosmological scenario with a critical cosmological constant, the Einstein–Cartan cosmology, the Milne–McCrea universe with quantum corrections and a non-flat universe with bulk viscosity. Within these models, we probe into the initial singularity by using different techniques. Several nonsingular universes emerge, one of the possibilities being a static non-expanding and stable Einstein universe.

On warped string vacuum profiles and cosmologies. Part II. Non-supersymmetric strings

We investigate the effects of the leading tadpole potentials of 10D tachyon-free non-supersymmetric strings in warped products of flat geometries of the type Mp+1× R × T10−p−2 depending on a single coordinate. In the absence of fluxes and for p < 8, there are two families of these vacua for the orientifold disk-level potential, both involving a finite internal interval. Their asymptotics are surprisingly captured by tadpole-free solutions, isotropic for one family and anisotropic at one end for the other. In contrast, for the heterotic torus-level potential there are four types of vacua. Their asymptotics are always tadpole-dependent and isotropic at one end lying at a finite distance, while at the other end, which can lie at a finite or infinite distance, they can be tadpole-dependent isotropic or tadpole-free anisotropic. We then elaborate on the general setup for including symmetric fluxes, and present the three families of exact solutions that emerge when the orientifold potential and a seven-form flux are both present. These solutions include a pair of boundaries, which are always separated by a finite distance. In the neighborhood of one, they all approach a common supersymmetric limit, while the asymptotics at the other boundary can be tadpole-free isotropic, tadpole-free anisotropic or again supersymmetric. We also discuss corresponding cosmologies, with emphasis on their climbing or descending behavior at the initial singularity. In some cases the toroidal dimensions can contract during the cosmological expansion.

Dynamical Reason for a Cyclic Universe

By analyzing the energy-momentum tensor and equations of state of ideal gas, scalar, spinor and vector potential in detail, we find that the total mass density of all matter is always positive, and the initial total pressure is negative. Under these conditions, by qualitatively analyzing the global behavior of the dynamical equation of cosmological model, we get the following results: (i) K=1, namely, the global spatial structure of the universe should be a three-dimensional sphere S3; (ii) 0≤Λ&lt;10−24ly−2, the cosmological constant should be zero or an infinitesimal; (iii) a(t)&gt;0, the initial singularity of the universe is unreachable, and the evolution of the universe should be cyclic in time. Since the matter components considered are quite complete and the proof is very elementary and strict, these conclusions are quite reliable in logic and compatible with all observational data. Obviously, these conclusions will be very helpful to correct some popular misconceptions and bring great convenience to further research other problems in cosmology such as the properties of dark matter and dark energy. In addition, the macroscopic Lagrangian of fluid model is derived.

An a priori error analysis of the hp-version of the C 0-continuous Petrov-Galerkin method for nonlinear second-order delay differential equations

We study an hp-version of the -continuous Petrov-Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bound in the -norm that is fully explicit in the local time steps, in the local approximation degrees, and in the local regularity indexes of the exact solutions. Moreover, we prove that the -continuous Petrov-Galerkin based on special hp-version discretization can yield exponential rates of convergence for analytic solutions with initial singularities. Numerical experiments are provided to illustrate the theoretical results.

Effective actions for loop quantum cosmology in fourth-order gravity

Loop quantum cosmology (LQC) is a theory which renders the Big Bang initial singularity into a quantum bounce, by means of short-range repulsive quantum effects at the Planck scale. In this work, we are interested in reproducing the effective Friedmann equation of LQC, by considering a generic f(R, P, Q) theory of gravity, where R=g^{mu nu }R_{mu nu } is the Ricci scalar, P=R_{mu nu }R^{mu nu }, and Q=R_{alpha beta mu nu }R^{alpha beta mu nu } is the Kretschmann scalar. An order reduction technique allows us to work in f(R, P, Q) theories which are perturbatively close to General Relativity, and to deduce a modified Friedmann equation in the reduced theory. Requiring that the modified Friedmann equation mimics the effective Friedmann equation of LQC, we are able to derive several functional forms of f(R, P, Q). We discuss the necessary conditions to obtain viable bouncing cosmologies for the proposed effective actions of f(R, P, Q) theory of gravity.

Second-Order and Nonuniform Time-Stepping Schemes for Time Fractional Evolution Equations with Time–Space Dependent Coefficients

The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time–space dependent variable coefficients is challenging, especially when the classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations with general time–space dependent variable coefficients. By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems. For the diffusion-wave problem, our scheme is constructed by employing the recently established symmetric fractional-order reduction method. The unconditional stability of proposed schemes is rigorously discussed under mild assumptions on variable coefficients and, based on reasonable regularity assumptions and weak time mesh restrictions, the second-order convergence is obtained with respect to discrete \(H^1\)-norm. Numerical experiments are given to demonstrate the theoretical statements.

String excitation by initial singularity of inflation

We discuss excitation of string oscillation modes by an initial singularity of inflation. The initial singularity of inflation is known to occur with a finite Hubble parameter, which is generally lower than the string scale, and hence it is not clear that stringy effects become significant around it. With the help of Penrose limit, we find that infinitely heavy oscillation modes get excited when a singularity is strong in the sense of Krolak’s classification. We demonstrate that the initial singularities of Starobinsky and hill top inflation, assuming the slow roll inflation to the past infinity, are strong. Hence stringy corrections are inevitable in the very early stage of these inflation models. We also find that the initial singularity of the hill top inflation could be weak for non-slow roll case.