An approach is presented to address singularities in general relativity using a complex Riemannian spacetime extension. We demonstrate how this method can be applied to both black hole and cosmological singularities, specifically focusing on the Schwarzschild and Kerr black holes and the Friedmann-Lemaître-Robertson-Walker (FLRW) Big Bang cosmology. By extending the relevant coordinates into the complex plane and carefully choosing integration contours, we show that it is possible to regularize these singularities, resulting in physically meaningful, singularity-free solutions when projected back onto real spacetime. The removal of the singularity at the Big Bang allows for a bounce cosmology. This approach offers a potential bridge between classical general relativity and quantum gravity effects, suggesting a way to resolve longstanding issues in gravitational physics without requiring a full theory of quantum gravity.
We evaluate the physical viability and logical strength of an array of putative criteria for big bang singularity resolution in quantum cosmology. Based on this analysis, we propose a mutually consistent set of constitutive conditions, which we argue should be taken to jointly define `global' and `local' big bang singularity resolution in this context. Whilst the present article will focus exclusively on evaluating resolution criteria for big bang singularities in the context of finite dimensional models of quantum cosmology, it is also hoped that the core features of our analysis will be exte…
Since the past Iagrg meeting in December 2004, new developments in loop quantum cosmology have taken place, especially with regards to the resolution of the Big Bang singularity in the isotropic models. The singularity resolution issue has been discussed in terms of physical quantities (expectation values of Dirac observables) and there is also an ``improved'' quantization of the Hamiltonian constraint. These developments are briefly discussed. This is an expanded version of the review talk given at the 24$^{\mathrm{th}}$ IAGRG meeting in February 2007.
We investigate the relative time scales associated with finite future cosmological singularities, especially those classified as Big Rip cosmologies, and the maximum predictability time of a coupled FRW-KG scalar cosmology with chaotic regimes. Our approach is to show that by starting with a FRW-KG scalar cosmology with a potential that admits an analytical solution resulting in a finite time future singularity there exists a Lyapunov time scale that is earlier than the formation of the singularity. For this singularity both the cosmological scale parameter a(t) and the Hubble parameter H(t) b…
Problem of cosmological singularity of general relativity theory is discussed. The possible resolution of this problem in the framework of inflationary cosmology is proposed. Physical conditions leading to bouncing inflationary solutions in the frame of general relativity theory and gauge theories of gravitation are compared. It is shown that gauge theories of gravitation allow to build regular inflationary cosmological models of closed, open and flat type with dominating ultrarelativistic matter at a bounce.
Loop quantum cosmology (LQC) predicts a nonsingular evolution of the universe through a bounce in the high energy region. But LQC has an ambiguity about the quantization scheme. Recently, the authors in [Phys. Rev. D 77, 124008 (2008)] proposed a new quantization scheme. Similar to others, this new quantization scheme also replaces the big bang singularity with the quantum bounce. More interestingly, it introduces a quantum singularity, which is traversable. We investigate this novel dynamics quantitatively with a tachyon scalar field, which gives us a concrete example. Our result shows that o…
We investigate a cosmological model with a big-brake singularity in the future: while the first time derivative of the scale factor goes to zero, its second time derivative tends to minus infinity. Although we also discuss the classical version of the model in some detail, our main interest lies in its quantization. We formulate the Wheeler-DeWitt equation and derive solutions describing wave packets. We show that all such solutions vanish in the region of the classical singularity, a behaviour which we interpret as singularity avoidance. We then discuss the same situation in loop quantum cosm…
The degenerate Lagrangian system describing a lot of cosmological models is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the "singularity" is reduced to a billiard on the Lobachevsky space. The Wheeler-DeWitt equation in the asymptotical regime is solved and a third-quantized model is suggested.
This paper is a continuation of the papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035, gr-qc/9609046, gr-qc/9704033]. The aim of the paper is to incorporate singularities---both local (black hole and naked singularity) and global (big bang and big crunch)---into the dynamics of indeterministic quantum gravity and cosmology. The question is whether a singularity is dynamically passable, i.e., whether a dynamical process which ends with a singularity may be extended beyond the latter. The answer is yes. A local singularity is trivially passable, while the passableness for a glob…
In this contribution we consider the issue of singularity resolution within loop quantum cosmology (LQC) for different homogeneous models. We present results of numerical evolutions of effective equations for both isotropic as well as anisotropic cosmologies, with and without spatial curvature. To address the issue of singularity resolution we examine the time evolution of geometrical and curvature invariants that yield information about the semiclassical spacetime geometry. We discuss generic behavior found for a variety of initial conditions. Finally, we show that the modifications which com…
It is shown that if the timelike eigenvector of the Ricci tensor be hypersurface orthogonal so that the space time allows a foliation into space sections then the space average of each of the scalar that appear in the Raychaudhuri equation vanishes provided the strong energy condition holds good. This result is presented in the form of a singularity theorem.
In this paper we extend our 2007 paper, Comparative Quantum Cosmology: Causality, Singularity, and Boundary Conditions, arXiv:0710.5046 to include consideration of universal expansion, various implications of extendibility and incompleteness in spacetime metrics and, absent the treatment of Feynman diagrams, the use of Penning trap dynamics as explained by Ferdandez and Velazquez to describe the Hamiltonians of space-times with no characteristic upper or lower bound and to compare the above with Peter Lynds' conjecture on the specialness of initial conditions in inflationary theory in quantum …
A brief summary of the singularity resolution in loop quantum cosmology of homogeneous isotropic models is presented. The article is addressed to relativists who do not specialize in quantum gravity. For further details, and answers to more technical asked questions, the reader is directed to the original papers and to more comprehensive recent reviews. Dedicated to the memory of Basilis Xanthopoulos, a founder of this NEB series of conferences.
We investigate dynamics of (4+1) and (5+1) dimensional flat anisotropic Universe filled by a perfect fluid in the Gauss-Bonnet gravity. An analytical solutions valid for particular values of the equation of state parameter $w=1/3$ have been found. For other values of $w$ structure of cosmological singularity have been studied numerically. We found that for $w > 1/3$ the singularity is isotropic. Several important differences between (4+1) and (5+1) dimensional cases are discussed.
A novel approach to quantization is shown to allow for superpositions of the cosmological constant in isotropic and homogeneous mini-superspace models. Generic solutions featuring such superpositions display unitary evolution and resolution of the classical singularity. Physically well-motivated cosmological solutions are constructed. These particular solutions exhibit characteristic features of a cosmic bounce including universal phenomenology that can be rendered insensitive to Planck-scale physics in a natural manner.
The cosmological propagation of tensor perturbations is studied in the context of parity-violating extensions of the symmetric teleparallel equivalent of General Relativity theory. This non-Riemannian formulation allows for a wider variety of consistent extensions than the metric formulation of gravity theory. It is found that while many of the possible quadratic terms do not influence the propagation of the gravitational waves, a generic modification predicts a signature that distinguishes the left- and right-handed circular polarizations. The parameters of such modifications can be constrain…
In recent works, a construction was proposed resulting emergent universes inside black holes. This result can be obtained from a 4D black hole embedded in a 5D space time with the fifth dimension compactified on a circle [0, 2] (0 and 2 are identified) such that the two branes are at 0 and 1. In the present work, we study this setup by deriving particles equations of motion in the new universes, based on redefining energy and angular momentum. This leads to disappearance of the singularity in centres of black holes in classical General Relativity.
The problem of construction of a general ihomogeneous solution of $D$-dimensional Einstein equations in the vicinity of a cosmological singularity is considered. It is shown that near the singularity a local behavior of metric functions is described by a billiard on a space of a constant negative curvature. The billiard is shown to have a finite volume and consequently to be a mixing one. Dynamics of inhomogeneities of metric is studied and it is shown that its statistical properties admit a complete description. An invariant measure describing statistics of inhomogeneities is obtained and a r…
It is stated that holonomy corrections in loop quantum cosmology introduce a modification in Friedmann's equation which prevent the big rip singularity. Recently in \cite{h12} it has been proved that this modified Friedmann equation is obtained in an inconsistent way, what means that the results deduced from it, in particular the big rip singularity avoidance, are not justified. The problem is that holonomy corrections modify the gravitational part of the Hamiltonian of the system leading, after Legendre's transformation, to a non covariant Lagrangian which is in contradiction with one of the …
It is argued that the initial cosmological singularity is isotropic in spatially inhomogeneous brane-world models. This implies that brane cosmology may naturally give rise to a set of initial data that provide the conditions for inflation to subsequently take place, consequently offering a plausible solution to the initial conditions problem in cosmology.
We consider different aspects of the problem of cosmological singularity such as the BKL oscillatory approach to the singularity, the new features of the cosmological dynamics in the neighbourhood of the singularity in multidimensional and superstring cosmological models and their connections with such a modern branch of mathematics as infinite-dimensional Lie algebras. The chaoticity of the oscillatory approach to the cosmological singularity is also discussed. The Conclusion contains some thoughts about the past and the future of the Universe in the light of the oscillatory approach to the B…
The multidimensional cosmological model describing the evolution of $n$ Einstein spaces in the presence of multicomponent perfect fluid is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the $(n-1)$-dimensional Lobachevsky space $H^{n-1}$. The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of $(n-2)$-dimensional sphere $S^{n-2}$ by point-like sources. Some generalization …
The fate of the cosmological singularity in the Taub model is discussed within the two frameworks. An internal time variable is ruled out and the only remaining degree of freedom (the anisotropy) of the Universe is quantized according to such schemes. The resulting GUP Taub Universe is singularity-free, differently from the second case, where the classical singularity is not tamed by the polymer-loop quantum effects.
Analytical properties of the scalar expansion in the cosmic fluid are investigated, especially near the future singularity, when the fluid possesses a constant bulk viscosity ζ. In addition, we assume that there is a Casimir-induced term in the fluid's energy-momentum tensor, in such a way that the Casimir contributions to the energy density and pressure are both proportional to 1/a^4, 'a' being the scale factor. A series expansion is worked out for the scalar expansion under the condition that the Casimir influence is small. Close to the Big Rip singularity the Casimir term has however to fad…
In contrast to previous work in the field, we construct the Loop Quantum Cosmology (LQC) of the flat isotropic model with a massless scalar field in the absence of higher order curvature corrections to the gravitational part of the Hamiltonian constraint. The matter part of the constraint contains the inverse triad operator which can be quantized with or without the use of a Thiemann- like procedure. With the latter choice, we show that the LQC quantization is identical to that of the standard Wheeler DeWitt theory (WDW) wherein there is no singularity resolution. We argue that the former choi…
We provide a brief chronological guide to the literature on non-Riemannian cosmological models. Developments in this field are traced back to the early seventies and are given in table form.
A remarkable property of modern cosmology is that it allows for a special case of symmetry, consisting in the possibility of describing the early-time acceleration (inflation) and the late-time acceleration using the same theoretical framework. In this paper we consider various cosmological models corresponding to a generalized form for the equation of state for the fluid in a flat Friedmann -Robertson-Walker universe, emphasizing cases where the so-called type IV singular inflation is encountered in the future. This is a soft (non-crushing) kind of singularity. Parameter values for an inhomog…
Different approaches to quantum cosmology are studied in order to deal with the future singularity avoidance problem. Our results show that these future singularities will persist but could take different forms. As an example we have studied the big rip which appear when one considers the state equation $P=ωρ$ with $ω<-1$, showing that it does not disappear in modified gravity. On the other hand, it is well-known that quantum geometric effects (holonomy corrections) in loop quantum cosmology introduce a quadratic modification, namely proportional to $ρ^2$, in Friedmann's equation that replace …
It has long been a primary objective of cosmology to understand the apparent isotropy in our universe and to provide a mathematical formulation for its evolution. A school of thought for its explanation is quiescent cosmology, which already possesses a mathematical framework, namely the definition of an Isotropic Singularity, but only for the initial state of the universe. A complementary framework is necessary in order to also describe possible final states of the universe. Our new definitions of an Anisotropic Future Endless Universe and an Anisotropic Future Singularity, whose structure and…
We extend recent discussions of singularity avoidance in quantum gravity from isotropic to anisotropic cosmological models. The investigation is done in the framework of quantum geometrodynamics (Wheeler-DeWitt equation). We formulate criteria of singularity avoidance for general Bianchi class A models and give explicit and detailed results for Bianchi I models with and without matter. We find that the classical singularities can generally be avoided in these models.
We investigate the dynamics of a Yang-Mills cosmology (YMC, the FRW type spacetime) bubble in the Bartnik-McKinnon (BK) spacetimes. Because a BK spacetime can be identified to a YMC spacetime with a finite scale factor in the neighborhood of the origin, we can give a natural initial condition for the YMC bubble. The YMC bubble can smoothly emerge from the origin without an initial singularity. Under a certain condition, the bubble develops continuously and finally replaces the entire BK spacetime, the metric of which is the same as the one of the radiation dominated universe. We also discuss w…
We apply the causal interpretation of quantum mechanics to homogeneous quantum cosmology and show that the quantum theory is independent of any time-gauge choice and there is no issue of time. We exemplify this result by studying a particular minisuperspace model where the quantum potential driven by a prescribed quantum state prevents the formation of the classical singularity, independently on the choice of the lapse function. This means that the fast-slow-time gauge conjecture is irrelevant within the framework of the causal interpretation of quantum cosmology.
We study the evolution of Bianchi-I space-times filled with a global unidirectional electromagnetic field $F_{mn}$ interacting with a massless scalar dilatonic field according to the law Ψ(ϕ) F^{mn} F_{mn} where Ψ(ϕ) > 0 is an arbitrary function. A qualitative study, among other results, shows that (i) the volume factor always evolves monotonically, (ii) there exist models becoming isotropic at late times and (iii) the expansion generically starts from a singularity but there can be special models starting from a Killing horizon preceded by a static stage. All these features are confirmed for …
In this review article we compare the recent work of Peter Lynds, "On a finite universe with no beginning or end", with that of Stephen Hawking, primarily "Quantum Cosmology, M-Theory, and the Anthropic Principle", and two foundational works by Sean M. Carroll and Jennifer Chen, "Does Inflation Provide Natural Conditions for the Universe" and "Spontaneous Inflation and the Origin of the Arrow of Time", in order to evaluate their comparative treatments of the nature and role of causality, time ordering, thermodynamic reversibility, singularities and boundary conditions in the formation of the e…
University of Szczecin（谢切琴大学）；National Centre for Nuclear Research（国家核研究中心）；Copernicus Center for Interdisciplinary Studies（哥白尼跨学科研究中心）；Universidad Complutense de Madrid（马德里康普顿斯大学）
We consider the brane Kantowski-Sachs universe when bulk space is five dimensional anti-de Sitter obeying a linear equation of state of the form $p=(γ-1)ρ+p_{0}$, where $ γ,\:p_{0}$ are a parameters. In this framework we discuss the brane Kantowski-Sachs cosmology and compare it with standard Kantowski-Sachs cosmology in Einstein theory for some specific matter choice. For several specific choice of relation between energy and pressure, the behavior of the scale factors at early time is found. It is shown that the brane Kantowski-Scahs cosmology may quickly become isotropic for some choices of…
We present a class of cosmological solutions for a generalized teleparallel gravity with, $f(T)=T+\tildeα(-T)^n$, where $\tildeα$ is some parameter and $n$ is an integer or half-integer. Choosing $\tildeα \sim G^{n-1}$, where $G$ is the gravitational constant, and working with an equation of state $p=w\, ρ$, one obtains a cosmological solution with multiple branches. The dynamics of the solution describes standard cosmology at late times, but the higher-torsion correction changes the nature of the initial singularity from big bang to a sudden singularity. The milder behavior of the sudden sing…
It is shown that the cosmological singularity in isotropic minisuperspaces is naturally removed by quantum geometry. Already at the kinematical level, this is indicated by the fact that the inverse scale factor is represented by a bounded operator even though the classical quantity diverges at the initial singularity. The full demonstation comes from an analysis of quantum dynamics. Because of quantum geometry, the quantum evolution occurs in discrete time steps and does not break down when the volume becomes zero. Instead, space-time can be extended to a branch preceding the classical singula…
The analytic properties of the energy density rho(t) of the cosmic fluid, and the Hubble parameter H(t), are investigated near to the future singularity t=t_s assuming different forms for the equation of state. First, it is shown that the inclusion of quantum effects coming from the conformal anomaly modifies the singularity. Thereafter, we consider the effect coming from a bulk viscosity in the fluid. The viscosity tends to reduce the magnitude of t_s, but does not alter the singularity itself (the exponent). Main emphasis is laid on the simple case when the equation of state is p=w*rho, with…
A broad class of two-dimensional loop-corrected dilaton gravity models exhibit cosmological solutions that interpolate between the string perturbative vacuum and a background with asymptotically flat metric and linearly growing dilaton. The curvature singularities of the corresponding tree-level solutions are smoothed out, but no branch-change occurs. Thus, even in the presence of a non-perturbative potential, the system is not attracted by physically interesting fixed points with constant dilaton, and the exit problem of string cosmology persists.
The standard $Λ$CDM model despite its agreement with observational data still has some issues unaddressed, lie the problem of initial singularity. Solving that problem usually requires modifications of general relativity. However, there appeared the Hořava-Lifshitz (HL) theory of gravity, in which equations governing cosmological evolution include a new term scaling similarly as dark radiation term in the Friedmann equations, enabling a bounce of the universe instead of initial singularity. This review describes past works on a stability of such a bounce in different formulations of HL theory,…
In this paper we show that Quiescent Cosmology [1, 2, 3] is consistent with Penrose's Weyl Curvature Hypothesis and the notion of gravitational entropy [4]. Gravitational entropy, from a conceptual point of view, acts in an opposite fashion to the more familiar notion of entropy. A closed system of gravitating particles will coalesce whereas a collection of gas particles will tend to diffuse; regarding increasing entropy, these two scenarios are identical. What has been shown previously [2, 3] is that gravitational entropy at the initial singularity predicted by Quiescent Cosmology - the Isotr…
Problem of cosmological singularity is discussed in the framework of gauge theories of gravitation. Generalizing cosmological Friedmann equations (GCFE) for homogeneous isotropic models including scalar fields and usual gravitating matter are introduced. It is shown that by certain restrictions on equation of state of gravitating matter and indefinite parameter of GCFE generic feature of inflationary cosmological models of flat, open and closed type is their regular bouncing character.
A free massless scalar field is coupled to homogeneous and isotropic loop quantum cosmology. The coupled model is investigated in the vicinity of the classical singularity, where discreteness is essential and where the quantum model is non-singular, as well as in the regime of large volumes, where it displays the expected semiclassical features. The particular matter content (massless, free scalar) is chosen to illustrate how the discrete structure regulates pathological behavior caused by kinetic terms of matter Hamiltonians (which in standard quantum cosmology lead to wave functions with an …
The Hamilton-Jacobi equation for the string cosmology is solved using the gradient expansion method. The zeroth order solution is taken to be the standard pre-big bang model and the second order solution is found for the dilaton and the three-metric. It indicates that corrections generated by inhomogeneities of the seed metric are suppressed near the singularity and are growing towards the asymptotic past, but corrections generated by the dilaton inhomogeneities are growing near the singularity and are suppressed in the past. Possible influences of initial metric inhomogeneities on the pre-big…
We investigate bouncing scenario in the modified $f(T)$ gravity, $T$ being the torsion scalar. Attention is attached to the reconstruction of $f(T)$ able to describe type IV singular bouncing evolution, where we adopt as assumption that the bouncing and type IV singularity points coincide. In the context of the reconstructed $f(T)$ model we calculate the Hubble slow-roll parameters in order to determine the dynamical evolution of the cosmological system under study. The results show that the Hubble slow-roll parameters become singular at the type IV singularity indicating a dynamical instabili…
Observations indicate that the universe is effectively flat, but they do not rule out a closed universe. The role of positive curvature is negligible at late times, but can be crucial in the early universe. In particular, positive curvature allows for cosmologies that originate as Einstein static universes, and then inflate and later reheat to a hot big bang era. These cosmologies have no singularity, no "beginning of time", and no horizon problem. If the initial radius is chosen to be above the Planck scale, then they also have no quantum gravity era, and are described by classical general re…
Our review is devoted to three promising research lines in quantum cosmology and the physics of the early universe. The nonperturbative renormalization programme is making encouraging progress that we here assess from the point of view of cosmological applications: Lagrangian and Hamiltonian form of pure gravity with variable G and Lambda; power-law inflation for pure gravity; an accelerating universe without dark energy. In perturbative quantum cosmology, on the other hand, diffeomorphism-invariant boundary conditions lead naturally to a singularity-free one-loop wave function of the universe…
We construct phantom energy models with the equation-of-state parameter $w$ such that $w<-1$, but finite-time future singularity does not occur. Such models can be divided into two classes: (i) energy density increases with time ("phantom energy" without "Big Rip" singularity) and (ii) energy density tends to constant value with time ("cosmological constant" with asymptotically de Sitter evolution). The disintegration of bound structure is confirmed in Little Rip cosmology. Surprisingly, we find that such disintegration (on example of Sun-Earth system) may occur even in asymptotically de Sitte…
Canonical quantization of general relativity does not yield a unique quantum theory for gravity. This is in part due to operator ordering ambiguities. In this paper, we investigate the role of different operator orderings on the question of whether a big bang or big crunch singularity occurs. We do this in the context of the minisuperspace model of a Friedmann-Lemaitre-Robertson-Walker universe with Brown-Kuchar dust. We find that for a certain class of operator orderings such a singularity is eliminated without having to impose boundary conditions.
We investigate a cosmological model whose energy content is described by a Chaplygin gas represented by a scalar field $ϕ$ with an associated potential producing a big bang singularity such that for vanishing scale factor, $a\to 0$, one has $|ϕ|\to \infty$. The classical version of the model is discussed in detail, however, our main interest lies in its quantization. Upon quantization of this model in the Schrödinger picture, we get the Wheeler--DeWitt equation which can be solved exactly in the two limits $a\to 0$ and $a\to\infty$, respectively. Employing the DeWitt criterium that the wave fu…
In classical general relativity, the generic approach to the initial singularity is usually understood in terms of the BKL scenario. In this scenario, along with the Bianchi IX model, the exact, singular, Kasner solution of vacuum Bianchi I model also plays a pivotal role. Using an effective classical Hamiltonian obtained from loop quantization of vacuum Bianchi I model, exact solution is obtained which is non-singular due to a discreteness parameter. The solution is parameterized in exactly the same manner as the usual Kasner solution and reduces to the Kasner solution as discreteness paramet…
Interest in cosmological singularities has remarkably grown in recent times, particularly on future singularities with the discovery of late-time acceleration of the universe and dark energy. While such explorations have previously been done in various modified gravity and quantum gravitationally inspired cosmologies (besides standard general relativistic cosmology), no such an endeavour has been taken up till now in the realms of renormalization group approaches to cosmology and we have hence took up on this journey. In this work, we consider the formation of cosmological singularities in an …
A variety of conditions is considered under which the cosmic dark fluid may be able to develop a future Big Rip or Little Rip singularity. Both one-component and two-component models are considered. In the last-mentioned case we present a way in which the fluid can be decomposed into two components, one non-turbulent (ideal) and one turbulent part, obeying two different equations of state. For the non-turbulent part, the thermodynamical parameter, commonly called w, is assumed to be less than -1 throughout. For the turbulent part, it turns out that it is sufficient that w{turb} lies in the qui…
We discuss the asymptotic dynamical evolution of spatially homogeneous brane-world cosmological models close to the initial singularity. We find that generically the cosmological singularity is isotropic in Bianchi type IX brane-world models and consequently these models do not exhibit Mixmaster or chaotic-like behaviour close to the initial singularity. We argue that this is typical of more general cosmological models in the brane-world scenario. In particular, we show that an isotropic singularity is a past-attractor in all orthogonal Bianchi models and is a local past-attractor in a class o…
