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The closed-universe recollapse conjecture
Abstract
It is widely believed that all expanding S3 closed universes
that satisfy the standard energy conditions recollapse to a second
singularity. The authors show that this is false even for Friedmann
universes: they construct an ever-expanding S3 Friedmann
universe in which the matter tensor satisfies the strong, weak and
dominant energy conditions and the generic condition. The authors prove
a general recollapse theorem for Friedmann universes: if the positive
pressure criterion, dominant energy condition and matter regularity
condition hold, then an S3 Friedmann universe must
recollapse. The authors show that all known vacuum solutions with Cauchy
surface topology S3 or S2×S1
recollapse, and they conjecture that this is a property of all vacuum
solutions of Einstein's equations with such Cauchy surfaces. The authors
consider a number of Kantowski-Sachs and Bianchi IX universes with
various matter tensors, and formulate a new recollapse conjecture for
matter-filled universes.
... It is difficult to answer in general because (unlike, for the singularity theorems) it involves properties of the general Einstein equations rather than simply of the geodesic equations. The general fate of closed universes is addressed by the closed-universe recollapse conjecture [84,85]. It depends upon the spatial topology of the universe. ...
... This is a necessary condition but it is far from sufficient because various conditions must also apply to the matter content. Surprisingly, closed FRW universes can avoid recollapse even when ρ > 0 and ρ + 3P > 0 because they can experience finite-time infinities in the acceleration of the scale factor before a maximum is reached, [85] (so called 'sudden' singularities [86]). This can be avoided by imposing a matter regularity condition, like |P | < Cρ, with C > 0 constant or by continuity of dP/dρ. ...
... 9 A proof for collapse of closed Bianchi type IX universes was given by Lin and Wald [88] and other cases with S 3 and S 2 × S 1 topologies in refs. [84,85]. ...
We consider the robustness of small-field inflation in the presence of scalar field inhomogeneities. Previous numerical work has shown that if the scalar potential is flat only over a narrow interval, such as in commonly considered inflection-point models, even small-amplitude inhomogeneities present at the would-be onset of inflation at can disrupt the accelerated expansion. In this paper, we parametrise and evolve the inhomogeneities from an earlier time at which the initial data were imprinted, and show that for a broad range of inflationary and pre-inflationary models, inflection-point inflation withstands initial inhomogeneities. We consider three classes of perturbative pre-inflationary solutions (corresponding to energetic domination by the scalar field kinetic term, a relativistic fluid, and isotropic negative curvature), and two classes of exact solutions to Einstein's equations with large inhomogeneities (corresponding to a stiff fluid with cylindrical symmetry, and anisotropic negative curvature). We derive a stability condition that depends on the Hubble scales and , and a few properties of the pre-inflationary cosmology. For initial data imprinted at the Planck scale, the absence of an inhomogeneous initial data problem for inflection-point inflation leads to a novel, lower limit on the tensor-to-scalar ratio.
... They also showed if a globally hyperbolic spacetime has a maximal Cauchy hypersurface it is also future and past incomplete and the corresponding singularities are crushing, provided that Ric(V, V) ⩾ 0 for timelike vectors V and a genericity condition holds. In a subsequent paper Barrow, Tipler and Galloway summarised their and other results and formulated three conjectures, see [BGT86], which are today referred to as Closed Universe Recollapse Conjecture. ...
... For non-vacuum spacetimes the choice of the matter model is relevant. There are Friedman universes which do not recollapse, although having S 3 topology, see [BGT86]. In particular, one needs an additional assumption to the usual energy conditions (weak, strong, dominant). ...
... Tipler and Galloway proved conjecture 2 for the Friedman-Lemaître-Robertson-Walker (FLRW) spacetimes in [BGT86] and it was shown by Gowdy that the Gowdy S 3 and S 1 × S 2 vacuum spacetimes recollapse, see [Gow14]. Burnett later showed that any spherically symmetric spacetime possessing a S 1 × S 2 Cauchy hypersurface has timelike curves with finite lengths. ...
We show that any homogeneous initial data set with Λ < 0 on a product 3-manifold of the orthogonal form (F × S ¹ , a 0 ² dz ² + b 0 ² σ, c 0 dz ² + d 0 σ), where (F, σ) is a closed 2-surface of constant curvature and a 0 , . . . , d 0 are suitable constants, recollapses under the Einstein-flow with a negative cosmo- logical constant and forms crushing singularities at the big bang and the big crunch, respectively. Towards certain singularities among those the Kretschmann scalar remains bounded. We then show that the presence of a massless scalar field causes the Kretschmann scalar to blow-up towards both ends of spacetime for all solutions in the corresponding class. By standard arguments this recollapsing behaviour extends to an open neighborhood in the set of initial data sets and is in this sense generic close to the homogeneous regime.
... While there has been significant progress in our understanding of the behaviour of solutions in the neighborhood of certain homogeneous attractors, both towards the complete expanding direction of spacetime as well as towards the collapsing direction, the global dynamics of the majority of cosmological spacetimes is still unknown. The most general conjecture about a generic class of spacetimes is the Closed Universe Recollapse conjecture stated by Barrow, Galloway and Tipler in 1986 [BGT86]. ...
... There are Friedman universes which do not recollapse, although having S 3 topology, cf. [BGT86]. In particular, one needs an additional assumption to the usual energy conditions (weak, strong, dominant). ...
... If the pressures (or their sum) are negative, then the matter model intuitively contributes to the expansion of the spacetime rather than its contraction. Tipler and Galloway proved Conjecture 2 for the FLRW spacetimes in [BGT86] and it was shown by Gowdy that the Gowdy S 3 and S 1 × S 2 vacuum spacetimes recollapse, cf. [Gow14]. ...
We show that any homogeneous initial data set with on a product 3-manifold of the orthogonal form , where is a closed 2-surface of constant curvature and are suitable constants, recollapses under the Einstein-flow with a negative cosmological constant and forms crushing singularities at the big bang and the big crunch, respectively. Towards certain singularities among those the Kretschmann scalar remains bounded, hence these are not curvature singularities. We then show that the presence of a massless scalar field causes the Kretschmann scalar to blow-up towards both ends of spacetime for all solutions in the corresponding class. By standard arguments this recollapsing behaviour extends to an open neighborhood in the set of initial data sets and is in this sense generic close to the homogeneous regime.
... If the relation (2.4) between the cosmic fluid pressure P and energy density ρ is not constant, one can still define an effective equation of state parameter w eff ≡ P/ρ. Non-linear barotropic equations of state P = P(ρ) have been studied in the literature, especially in relation with phantom fluids and sudden future singularities [20,21,36,38,50,69,83,87,182,195,238,[309][310][311]350,368,370,381] but here we restrict ourselves to linear equations of state. ...
... A large variety of situations, which is best described by qualitative methods and phase space analysis, can present themselves for general forms of matter which may include perfect or imperfect fluids, tilted fluids [127,[129][130][131][132]372,398,403], fluids with non-linear and/or non-constant equation of state [20,21,36,38,50,69,83,87,182,195,238,[309][310][311]350,368,370,381], scalar fields, etc. One also has to distinguish between non-interacting fluids and interacting (or explicitly coupled) fluids. ...
... These solutions are again given by Vajk [395]. For K = 0, we have the usual, forever expanding, solution (2.11) 38) or, in parametric form, ...
We review analytical solutions of the Einstein equations which are expressed in terms of elementary functions and describe Friedmann–Lemaître–Robertson–Walker universes sourced by multiple (real or effective) perfect fluids with constant equations of state. Effective fluids include spatial curvature, the cosmological constant, and scalar fields. We provide a description with unified notation, explicit and parametric forms of the solutions, and relations between different expressions present in the literature. Interesting solutions from a modern point of view include interacting fluids and scalar fields. Old solutions, integrability conditions, and solution methods keep being rediscovered, which motivates a review with modern eyes.
... While traditional cosmological solutions for linear barotropic equations of state contain Big Bang, Big Crunch, and Big Rip type singularities, the discovery of the acceleration of the universe in 1998 prompted the consideration of many more exotic non-linear equations of state for the dark energy fluid, which must be postulated in order to explain the cosmic acceleration within general relativity. This broadening of the picture results in a much wider spectrum of possible singularities [91,[98][99][100][101][102][103][104][105][106]. ...
... We have examined several situations: while most of them give rise to simple, albeit very important, FLRW solutions corresponding to simple fluids such as dust or radiation or a pure cosmological constant Λ, others correspond to phantom fluids [90] with non-linear equations of state originating exotic singularities at finite time, where the scale factor remains finite while curvature invariants blow up. These types of singularities are studied and classified in recent literature [91,[98][99][100][101][102][103][104][105][106]. The situations examined are summarized in Table 1. ...
... The problem of the terrestrial brachistochrone, which has seen renewed attention recently [70,[72][73][74][75][76][77][78][79][80][81][83][84][85][86][87][88][89], provides an explicit example of a universe dominated by a phantom fluid with non-linear equation of state, which can be solved explicitly and exhibits a finite future singularity at a finite value of the scale factor, where the Hubble function, Ricci scalar, energy density, and pressure all diverge. Finite time singularities have been the subject of much literature in the past decade [91,[98][99][100][101][102][103][104][105][106] hence, in this problem, the mechanical side of the analogy helps the cosmology side in the sense that the known exact solution for the terrestrial brachistochrone problem can be immediately translated into an analytical solution of the corresponding cosmology with complicated (non-linear) equation of state. ...
Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.
... While traditional cosmological solutions for linear barotropic equations of state contain Big Bang, Big Crunch, and Big Rip type singularities, the discovery of the acceleration of the universe in 1998 prompted the consideration of many more exotic non-linear equations of state for the dark energy fluid, which must be postulated in order to explain the cosmic acceleration within general relativity. This broadening of the picture results in a much wider spectrum of possible singularities [98,99,100,101,102,103,104,91,105,106]. ...
... We have examined several situations: while most of them give rise to simple, albeit very important, FLRW solutions corresponding to simple fluids such as dust or radiation or a pure cosmological constant Λ, others correspond to phantom fluids [90] with non-linear equations of state originating exotic singularities at finite time, where the scale factor remains finite while curvature invariants blow up. These types of singularities are studied and classified in recent literature [98,99,100,101,102,103,104,91,105,106]. The situations examined are summarized in Table 1. ...
... The problem of the terrestrial brachistochrone, which has seen renewed attention recently [70,72,73,74,75,76,77,78,79,80,81,83,84,85,86,87,88,89], provides an explicit example of a universe dominated by a phantom fluid with non-linear equation of state, which can be solved explicitly and exhibits a finite future singularity at a finite value of the scale factor, where the Hubble function, Ricci scalar, energy density, and pressure all diverge. Finite time singularities have been the subject of much literature in the past decade [98,99,100,101,102,103,104,91,105,106] hence, in this problem, the mechanical side of the analogy helps the cosmology side in the sense that the known exact solution for the terrestrial brachistochrone problem can be immediately translated into an analytical solution of the corresponding cosmology with complicated (non-linear) equation of state. ...
Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.
... Investigações cinemáticas de cosmologias de Friedmann levantaram a questão sobre a possibilidade da existência de uma singularidade futura repentina (15), caracterizada por uma divergência deä, enquanto o fator de escala a eȧ são finitos, onde daqui em diante o ponto sempre se refere à derivada com relação ao tempo. Assim, o parâmetro de Hubble, H =ȧ a , e a densidade de energia ⇢ também são finitos, enquanto a primeira derivada do parâmetro de Hubble e a pressão p divergem. ...
... Devido a não linearidade da Lagrangiana (15), o momento conjugado possui forma diferente da forma para modelos com campo escalar minimamente acoplado. Especificamente, em nosso caso, obtemos: ...
... (3.6), the divergent term proportional to (a − a 0 ) −2 in the energy density dominates over the curvature term −1/a 2 , which stays finite. Finite-time singularities, including Big Rip (Caldwell 2002) and sudden future singularities (Barrow et al. 1986; Faraoni and Cardini FACETS | 2017 | 2: 286-300 | DOI: 10.1139/facets-2016-0045 295 facetsjournal.com Barrow 2004), have been the subject of a significant amount of work in cosmology (Shtanov and Sahni 2002;Kofinas et al. 2003;Calcagni 2004;Gorini et al. 2004;Stefancic 2005;Dabrowski et al. 2007;Fernandez-Jambrina 2007;Bouhmadi-López et al. 2008;Dabrowski and Denkiewicz 2009;Frampton et al. 2011;Bamba et al. 2012;Bouhmadi-López et al. 2015;Beltrán Jiménez 2016). ...
... Bouhmadi-López et al. 2008) for spatially flat universes in the now abundant literature on cosmological singularities(Wald 1984;Barrow et al. 1986;Caldwell 2002;Shtanov and Sahni 2002;Barrow 2004;Dabrowski et al. 2007; Fernandez-Jambrina 2007;Dabrowski and Denkiewicz 2009;Frampton et al. 2011;Bouhmadi-López et al. 2015; Beltran Jimenez et al. 2016). ...
An ordinary differential equation describing the transverse profiles of U-shaped glacial valleys has two formal analogies, which we explore in detail, bridging these different areas of research. First, an analogy with point particle mechanics completes the description of the solutions. Second, an analogy with the Friedmann equation of relativistic cosmology shows that the analogue of a glacial valley profile is a universe with a future singularity of interest in theoretical models of cosmology. A Big Freeze singularity, which was not previously observed for positive curvature index, is also contained in the dynamics.
... Under certain circumstances, spacetimes are expected to recollapse, meaning that there is a singularity both to the future and to the past. More specifically, if the spatial topology of the spacetime is S 3 or S 2 × S 1 , then the universe is in [88] conjectured to recollapse. This expectation goes under the name of the closed universe recollapse conjecture. ...
... If the topology is "flat" or "open" and the WEC holds, this singular surface would in fact have to have maximal volume (else there would be an earlier non-singular hypersurface with maximum volume, violating the theorem). It is certainly possible to write a metric for a spacetime with such a singular surface, but certain reasonable extra conditions on the stress-energy forbid them at least in the homogeneous and isotropic case, and are conjectured to do so more generally [18]. These conditions forbid such pathologies as the pressure diverging when the energy density and volume are finite, or the pressure oscillating between finite bounds but with diverging frequency as the surface is approached. ...
In homogeneous and isotropic Friedmann-Robertson-Walker cosmology, the topology of the universe determines its ultimate fate. If the Weak Energy Condition is satisfied, open and flat universes must expand forever, while closed cosmologies can recollapse to a Big Crunch. A similar statement holds for homogeneous but anisotropic (Bianchi) universes. Here, we prove that arbitrarily inhomogeneous and anisotropic cosmologies with "flat" (including toroidal) and "open" (including compact hyperbolic) spatial topology that are initially expanding must continue to expand forever at least in some region at a rate bounded from below by a positive number, despite the presence of arbitrarily large density fluctuations and/or the formation of black holes. Because the set of 3-manifold topologies is countable, a single integer determines the ultimate fate of the universe, and, in a specific sense, most 3-manifolds are "flat" or "open". Our result has important implications for inflation: if there is a positive cosmological constant (or suitable inflationary potential) and initial conditions for the inflaton, cosmologies with "flat" or "open" topology must expand forever in some region at least as fast as de Sitter space, and are therefore very likely to begin inflationary expansion eventually, regardless of the scale of the inflationary energy or the spectrum and amplitude of initial inhomogeneities and gravitational waves. Our result is also significant for numerical general relativity, which often makes use of periodic (toroidal) boundary conditions.
... Not much is known about general criteria for recollapse. The closed universe recollapse conjecture [4] says that any spacetime with a certain type of topology (admitting a metric of positive scalar curvature) and satisfying the dominant and strong energy conditions must recollapse. No counterexample is known but the conjecture has only been proved in cases with high symmetry [13], [14]. ...
Present knowledge about the nature of spacetime singularities in the context of classical general relativity is surveyed. The status of the BKL picture of cosmological singularities and its relevance to the cosmic censorship hypothesis are discussed. It is shown how insights on cosmic censorship also arise in connection with the idea of weak null singularities inside black holes. Other topics covered include matter singularities and critical collapse. Remarks are made on possible future directions in research on spacetime singularities.
... The main difficulty stemming from such criticisms concerns the issue of phantom stability and this could still be circumvented if an axion model is considered for the phantom field [7]. Indeed, the large amount of papers on the phantom subject that have appeared [4][5][6] before and after references [10][11][12] reflects the fact that most of the criticisms can actually be regarded as manifesting weird phantom properties that could nevertheless be accommodated into the current and future evolution of the universe without contradicting observations. ...
This paper deals with the thermodynamic properties of a phantom field in a flat Friedmann-Robertson-Walker universe. General expressions for the temperature and entropy of a general dark-energy field with equation of state are derived from which we have deduced that, whereas the temperature of a cosmic phantom fluid () is definite negative, its entropy is always positive. We interpret that result in terms of the intrinsic quantum nature of the phantom field and apply it to (i) attain a consistent explanation for some recent results concerning the evolution of black holes which,induced by accreting phantom energy, gradually loss their mass to finally vanish exactly at the big rip, and (ii) introduce the concept of cosmological information and its relation with life and the anthropic principle. Some quantum statistical-thermodynamic properties of the quantum quantum field are also considered that include a generalized Wien law and the prediction of some novel phenomena such as the stimulated absorption of phantom energy and the anti-laser effect.
... Now the scalar curvature plays an important role in understanding the geometry and topology of 3-manifolds, just as it does in dimension 2. Thus, define the Sigma constant σ(Σ) of a closed oriented 3-manifold Σ to be the supremum of the scalar curvatures of unit volume Yamabe metrics on Σ, c.f. [1], [36]. This is a topological invariant, which divides the family of closed 3-manifolds naturally into three classes according to This observation, together with the behavior in certain model cases, namely the space-homogeneous Bianchi geometries on S 3 and Kantowski-Sacks geometries on S 2 × S 1 , have led to following conjecture, c.f. [9], [34]. ...
We describe some relations between the long-time asymptotic behavior of the vacuum Einstein evolution equations and the geometrization of 3-manifolds. These relations are expressed in terms of evolution of CMC hypersurfaces in the vacuum space-time.Some results are also obtained on the singularity avoidance of CMC foliations. In addition, the paper describes a number of open problems relating these two areas.
... Another type of cosmological singularities, which can arise in the future for some finite values of the scale factor of the universe and can be rather soft was described in paper [6]. The interest in such singularities essentially increased during last years (see e.g. ...
We study the cosmological evolution and singularity crossing in the Bianchi-I universe filled with a conformally coupled scalar field and compare them with those of the Bianchi-I universe filled with a minimally coupled scalar field. We also write down the solution for the Bianchi-I Universe in the induced gravity cosmology.
... However, one should pay attention to the fact that for a general f (T ) model, f HH may become zero at a specific time, i.e F (H) → ∞. These points correspond to sudden singularities [125], and in order to fully analyze their properties one should propose a specific spacetime extension, which extend non-spacelike curves beyond these soft singularities as done in [108,116,[126][127][128][129]. For completeness, note that the above method is fully applicable in the case where f (T ) gravity becomes TEGR, that is general relativity, i.e for f (T ) = T + Λ, since in this case we always havef HH = −12 0. ...
We use dynamical system methods to explore the general behaviour of f(T) cosmology. In contrast to the standard applications of dynamical analysis, we present a way to transform the equations into a one-dimensional autonomous system, taking advantage of the crucial property that the torsion scalar in flat FRW geometry is just a function of the Hubble function, thus the field equations include only up to first derivatives of it, and therefore in a general f(T) cosmological scenario every quantity is expressed only in terms of the Hubble function. The great advantage is that for one-dimensional systems it is easy to construct the phase space portraits, and thus extract information and explore in detail the features and possible behaviours of f(T) cosmology. We utilize the phase space portraits and we show that f(T) cosmology can describe the universe evolution in agreement with observations, namely starting from a Big Bang singularity, evolving into the subsequent thermal history and the matter domination, entering into a late-time accelerated expansion, and resulting to the de Sitter phase in the far future. Nevertheless, f(T) cosmology can present a rich class of more exotic behaviours, such as the cosmological bounce and turnaround, the phantom-divide crossing, the Big Brake and the Big Crunch, and it may exhibit various singularities, including the non-harmful ones of type II and type IV. We study the phase space of three specific viable f(T) models offering a complete picture. Moreover, we present a new model of f(T) gravity that can lead to a universe in agreement with observations, free of perturbative instabilities, and applying the Om(z) diagnostic test we confirm that it is in agreement with the combination of SNIa, BAO and CMB data at 1 confidence level.
... However, the nature of these dynamical singularities is a more complicated problem to be addressed in this section. For this purpose, we shall use the techniques developed in [22,24,25,26,27] which are based on the use of an invariant geometric quantity associated to the matter content of the universe, the Bel-Robinson energy. For a flat FRW universe, the asymptotic behaviour of the Hubble parameter, the scale factor and the matter fields contribution (electric parts of Bel-Robinson energy) on approach to the finite time singularity (t → 0) provides a complete classification on the dynamical character of the singularity. ...
We apply the central extension technique of Poincare to dynamics involving an interacting mixture of pressureless matter and vacuum near a finite-time singularity. We show that the only attractor solution on the circle of infinity is the one describing a vanishing matter-vacuum model at early times
... Another type of cosmological singularity, arising for a finite value of the cosmological scale factor, was considered in [5]. Recently, the so-called "soft" singularities arising for large values of the scale factor were extensively studied [6] and the situations for which such singularities can be crossed were found [7,8]. ...
We consider a rather simple method for the description of the Big Bang - Big Crunch cosmological singularity crossing. For the flat Friedmann universe this method gives the same results as more complicated methods, using Weyl symmetry or the transitions between the Jordan and Einstein frames. It is then easily generalized for the case of a Bianchi-I anisotropic universe. We also present early-time and late-time asymptotic solutions for a Bianchi-I universe, filled with a conformally coupled massless scalar field.
... See also[92], where this type of sudden and other finite time singularities were first introduced in order to show that closed Friedmann universes need not collapse even if they satify the strong energy condition; although the terminology 'sudden singularity' was introduced by Barrow in 2004, as already mentioned. ...
The first part of this paper contains a brief description of the beginnings of modern cosmology, which, the author will argue, was most likely born in the Year 1912. Some of the pieces of evidence presented here have emerged from recent research in the history of science, and are not usually shared with the general audiences in popular science books. In special, the issue of the correct formulation of the original Big Bang concept, according to the precise words of Fred Hoyle, is discussed. Too often, this point is very deficiently explained (when not just misleadingly) in most of the available generalist literature. Other frequent uses of the same words, Big Bang, as to name the initial singularity of the cosmos, and also whole cosmological models, are then addressed, as evolutions of its original meaning. Quantum and inflationary additions to the celebrated singularity theorems by Penrose, Geroch, Hawking and others led to subsequent results by Borde, Guth and Vilenkin. And corresponding corrections to the Einstein field equations have originated, in particular, , f(R), and scalar-tensor gravities, giving rise to a plethora of new singularities. For completeness, an updated table with a classification of the same is given.
... Inflation in its standard form is a mechanism to get rid of initial inhomogeneities and anisotropies. Once the cosmological constant-like source starts to dominate the expansion, (almost) everything else dilutes away [32][33][34][35][36][37]. Intuitively-by invoking Ehrenfest's theorem-this must also happen with initial quantum excitations of the Bunch-Davies vacuum. ...
We consider implications of the quantum extension of the inflationary no hair theorem. We show that when the quantum state of inflation is picked to ensure the validity of the EFT of fluctuations, it takes only efolds of inflation to erase the effects of the initial distortions on the inflationary observables. Thus the Bunch-Davies vacuum is a very strong quantum attractor during inflation. We also consider bouncing universes, where the initial conditions seem to linger much longer and the quantum `balding' by evolution appears to be less efficient.
... Basically, until the end of 1990s almost all discussions about singularities were devoted to the Big Bang and Big Crunch singularities, which are characterized by a vanishing cosmological radius. However, kinematical investigations of Friedmann cosmologies have raised the possibility of sudden future singularity occurrence [9], characterized by a diverging accelerationä, whereas both the scale factor a andȧ are finite. Then, the Hubble parameter H =ȧ a and the energy density ρ are also finite, while the first derivative of the Hubble parameter and the pressure p diverge. ...
We discuss the problem of singularity crossing in isotropic and anisotropic universes. We study at which conditions singularities can disappear in quantum cosmology and how quantum particles behave in the vicinity of singularities. Some attempts to develop general approach to the connection between the field reparametrization and the elimination of singularities is presented as well.
... Basically, until the end of 1990s almost all discussions about singularities were devoted to the Big Bang and Big Crunch singularities, which are characterized by a vanishing cosmological radius. However, kinematical investigations of Friedmann cosmologies have raised the possibility of sudden future singularity occurrence [9], characterized by a diverging accelerationä, whereas both the scale factor a andȧ are finite. Then, the Hubble parameter H =ȧ a and the energy density ρ are also finite, while the first derivative of the Hubble parameter and the pressure p diverge. ...
We discuss the problem of singularity crossing in isotropic and anisotropic universes. We study at which conditions singularities can disappear in quantum cosmology and how quantum particles behave in the vicinity of singularities. Some attempts to develop general approach to the connection between the field reparametrization and the elimination of singularities is presented as well.
... These models of a universe undergoing cycles of expansion and contraction were also further explored in [16][17][18][19]. In particular, in [20], the closed-universe recollapse conjecture-which states that, under certain conditions, the expanding universe could eventually reverse its course, contracting back to a highly dense state, and initiating a new cycle of expansion and contraction-was proposed as a potential solution to cosmological issues, such as the singularity problem. One such model, based on the ekpyrotic model [21], was introduced in [22]. ...
We study geodesics in Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological models and give the full set of solutions. For azimuthal geodesics, in a closed universe, we give the angular distance traveled by a test particle moving along such a geodesic during one cycle of expansion and recollapse of the universe. We extend previous results regarding the path followed by light rays to the two-fluid case, also including a cosmological constant, as well as to massive test particles. Our work contains various new results and explicit formulas, often using special functions which naturally appear in this setting.
Published by the American Physical Society 2024
... Thereafter, generalization to the case of not only homogeneous and isotropic spacetimes was explored [2][3][4][5], resulting in the proof of some general theorems and the discovery of the oscillatory (BKL) approach to the cosmological initial singularity [6], also known as the Mixmaster Universe [7]. The investigation of arising (rather soft) future singularities at the finite scale factor was done further [8] and still maintains interest [9][10][11][12][13][14]. Regarding such soft future singularities, the condition of their crossing becomes important; see, e.g., [15,16]. The idea of the possible crossing of the so-called Big Bang-Big Crunch singularity appears rather counterintuitive in contrast to the crossing of the soft singularities. ...
We apply a very simple procedure to construct non-singular cosmological models for flat Friedmann universes filled with minimally coupled scalar fields or by tachyon Born–Infeld-type fields. Remarkably, for the minimally coupled scalar field and the tachyon field, the regularity of the cosmological evolution, or in other words, the existence of bounce, implies the necessity of the transition between scalar fields with standard kinetic terms to those with phantom ones. In both cases, the potentials in the vicinity of the point of the transition have a non-analyticity of the cusp form that is characterized by the same exponent and is equal to 23. If, in the tachyon model’s evolution, the pressure changes its sign, then another transformation of the Born–Infeld-type field occurs: the tachyon transforms into a pseudotachyon, and vice versa. We also undertake an analysis of the stability of the cosmological evolution in our models; we rely on the study of the speed of sound squared.
... It appears that a universe which goes through this singularity did not disturb Robertson too much. Another type of cosmological singularity, which can arise in the future for some finite values of the scale factor of the universe and can be rather soft, was described in Ref. [41]. The interest in such singularities essentially increased during the last few years (see e.g., the review [42] and references therein). ...
We study the effects of a spatially homogenous magnetic field in Bianchi-I cosmological models. The cases of a pure magnetic field and two models with additional dust and a massless scalar field (stiff matter) are also considered. At the beginning of the cosmological evolution, i.e., in the neighborhood of the singularity, the Universe is described by one of Kasner’s solutions, and asymptotically by another Kasner solution when the volume of the Universe tends to infinity. The transition law between these two Kasner regimes is established, and shown to coincide with the analogous law for the empty Bianchi-II universe. The universe filled with dust and a magnetic field undergoes the process of isotropization, while the presence of a massless scalar field induces a modification of the relations between Kasner indices in the two asymptotic regimes. In all of these cases, we analyze the approach to the singularity in some detail and comment on the issue of the possible singularity crossing.
... Barrow [1] introduced sudden future singularities in general relativistic cosmology as a counterexample to the expectation that closed Friedmann universes recollapse if the strong energy condition holds [2]. Sudden singularities in general relativity are geodesically complete in the smooth [3], or the distributional sense [4] They are also generally stable with respect to various kinds of small cosmological fluctuations [5], and quantum particle production or regularization schemes [6][7][8]. ...
We construct a generic asymptotic solution for modified gravity near a sudden singularity. This solution contains a fluid source with no equation of state and is function-counting stable, that is it has eleven independent arbitrary functions of the spatial coordinates as dictated by the Cauchy problem of the theory. We further show that near the sudden singularity the solution has a shock wave character with the same number of free functions in the Jordan and Einstein frame.
... Barrow [1] introduced sudden future singularities in general relativistic cosmology as a counterexample to the expectation that closed Friedmann universes recollapse if the strong energy condition holds [2]. Sudden singularities in general relativity are geodesically complete in the smooth [3], or the distributional sense [4] They are also generally stable with respect to various kinds of small cosmological fluctuations [5], and quantum particle production or regularization schemes [6,7,8]. ...
We construct a generic asymptotic solution for modified gravity near a sudden singularity. This solution contains a fluid source with no equation of state and is function-counting stable, that is it has eleven independent arbitrary functions of the spatial coordinates as dictated by the Cauchy problem of the theory. We further show that near the sudden singularity the solution has a shock wave character with the same number of free functions in the Jordan and Einstein frame.
... It is also shown that if r < 2m, the universe encounters 'sudden rip' singularity. [44,45] Also for the benefit of the readers, perhaps it is better to add a short discussion here, pointing out that even in phantom regime quantum effects may lead the future universe to de Sitter space. For this purpose, we specifically would like to discuss the interesting results obtained by Nojiri et al. [46] by considering de Sitter as well as the Nariai universe induced by quantum CFT with the classical phantom matter and perfect fluid. ...
One of the solutions of the Einstein equations, called McVittie solution, signifying a black hole embedded by the dynamic spacetime is studied. In the stationary spacetime the Mcvittie metric becomes the Schwarzschild‐de Sitter metric (SdS). The geodesic of a freely falling test particle towards the black hole is examined in the SdS spacetime. It is found that unlike the Schwarzschild case the potential of such particle becomes maximum at a point where it eventually stops for a while and then resumes its motion towards the center of the black hole. It is shown that an observer or system of particles is spaghettified near the black hole singularity in the SdS spacetime. The dynamics of the universe in the framework of McVittie metric, being a generalized time dependent SdS solution, is represented in terms of that point, called stationary or turning point. The motion of the stationary point is studied in various regimes of the expanding universe and the possible outcomes are discussed in brief.
... • Type I (Big Rip) [22]: For t → t s , a → ∞, b → ∞, ρ → ∞ and |p| → ∞. • Type II (sudden future) [23]: For t → t s , a → a s , b → b s , ρ → ρ s and |p| → ∞. ...
We assume the anisotropic model of the Universe in the framework of a varying speed of light c and a varying gravitational constant G theories and study different types of singularities. We write the scale factors for the singularity models in terms of cosmic time and find some conditions for possible singularities. For future singularities, we assume the forms of a varying speed of light and varying gravitational constant. For regularizing the Big Bang singularity, we assume two forms of scale factors: the sine model and the tangent model. For both models, we examine the validity of null and strong energy conditions. Starting from the first law of thermodynamics, we study the thermodynamic behaviors of a number n of universes (i.e., multiverse) for (i) varying c, (ii) varying G and (iii) varying both c and G models. We find the total entropies for all the cases in the anisotropic multiverse model. We also find the nature of the multiverse if the total entropy is constant.
... A peculiar feature is that lightlike geodesics are complete close to the singularity. • Type II: "Sudden singularities" [19][20][21][22][23][24][25][26][27][28][29][30][31][32] or even "quiescent singularities" [33]: This is the second type of singularity that was considered on introducing new cosmological models, though they had been already introduced in [34]. The main feature of these singularities is that the scale factor, the Hubble ratio and the energy density do not blow up and the models just violate the dominant energy condition. ...
The discovery of accelerated expansion of the Universe opened up the possibility of new scenarios for the doom of our space–time, besides eternal expansion and a final contraction. In this paper, we review the chances that may await our universe. In particular, there are new possible singular fates (sudden singularities, big rip, etc.), but there also other evolutions that cannot be considered as singular. In addition to this, some of the singular fates are not strong enough in the sense that the space–time can be extended beyond the singularity. For deriving our results, we make use of generalized power and asymptotic expansions of the scale factor of the Universe.
This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.
... • Type II. "Sudden singularities" [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], also named "quiescent singularities" [37]. The derivatives of the scale factor diverge from second derivative on. ...
Due to the accelerated expansion of the universe, the possibilities for the formation of singularities has changed from the classical Big Bang and Big Crunch singularities to include a number of new scenarios. In recent papers it has been shown that such singularities may appear in inflationary cosmological models with a fractional power scalar field potential. In this paper we enlarge the analysis of singularities in scalar field cosmological models by the use of generalised power expansions of their Hubble scalars and their scalar fields in order to describe all possible models leading to a singularity, finding other possible cases. Unless a negative scalar field potential is considered, all singularities are weak and of type IV.
... A peculiar feature is that lightlike geodesics are complete close to the singularity. • Type II: "Sudden singularities" [19][20][21][22][23][24][25][26][27][28][29][30][31][32] or even "quiescent singularities" [33]: This is the second type of singularity that was considered on introducing new cosmological models, though they had been already introduced in [34]. The main feature of these singularities is that the scale factor, the Hubble ratio and the energy density do not blow up and the models just violate the dominant energy condition. ...
The discovery of accelerated expansion of the universe opened the possibility of new scenarios for the doom of our spacetime, besides aeternal expansion and a final contraction. In this paper we review the chances which may await our universe. In particular, there are new possible singular fates (sudden singularities, big rip...), but there also other evolutions which cannot be considered as singular. In addition to this, some of the singular fates are not strong enough in the sense that the spacetime can be extended beyond the singularity. For deriving our results we make use of generalised power and asymptotic expansions of the scale factor of the universe.
... Sudden singularity or Type II is characterized by lim t→t s a(t) = a s , lim t→t s ρ eff (t) = ρ s , lim t→t s |P eff (t)| = ∞[115][116][117][118]; 3. Big freeze singularity or Type III is characterized by ...
Some cosmological models based on the gravitational theory f ( R ) = R + ζ R 2 , and on fluids obeying to the equations of state of Redlich–Kwong, Berthelot, and Dieterici are proposed for describing smooth transitions between different cosmic epochs. A dynamical system analysis reveals that these models contain fixed points which correspond to an inflationary, a radiation dominated and a late-time accelerating epoch, and a nonsingular bouncing solution, the latter being an asymptotic fixed point of the compactified phase space. The infinity of the compactified phase space is interpreted as a region in which the non-ideal behaviors of the previously mentioned cosmic fluids are suppressed. Physical constraints on the adopted dimensionless variables are derived by demanding the theory to be free from ghost and tachyonic instabilities, and a novel cosmological interpretation of such variables is proposed through a cosmographic analysis. The different effects of the equation of state parameters on the number of equilibrium solutions and on their stability nature are clarified. Some generic properties of these models, which are not sensitive to the particular fluid considered, are identified, while differences are critically examined by showing that the Redlich–Kwong scenario admits a second radiation-dominated epoch and a Big Rip Singularity.
... • Type II (Sudden future) [24]: For t → t s , a → a s , b → b s , ρ → ρ s and |p| → ∞. ...
We assume the anisotropic model of the Universe in the framework of varying speed of light c and varying gravitational constant G theories and study different types of singularities. For the singularity models, we write the scale factors in terms of cosmic time and found some conditions for possible singularities. For future singularities, we assume the forms of varying speed of light and varying gravitational constant. For regularizing big bang singularity, we assume two forms of scale factors: sine model and tangent model. For both the models, we examine the validity of null energy condition and strong energy condition. Start from the first law of thermodynamics, we study the thermodynamic behaviours of n number of Universes (i.e., Multiverse) for (i) varying c, (ii) varying G and (iii) both varying c and G models. We found the total entropies for all the cases in the anisotropic Multiverse model. We also found the nature of the Multiverse if total entropy is constant.
... The latter have been widely studied since their introduction in Refs. [21]- [23] as asymptotic regions in spacetime were matter satisfies the strong energy condition, the scale factor and its first derivative are finite, but discontinuities occur in its second derivative and fluid pressure. The stability of these solutions to small perturbations has been shown using a gauge invariant formalism in Ref. [24], and have also been shown to be stable against quantum particle production in [25]. ...
We construct a formal asymptotic series expansion for a general solution of the Brans–Dicke equations with a fluid source near a sudden singularity. This solution contains 11 independent arbitrary functions of the spatial coordinates as required by the Cauchy problem of the theory. We show that the solution is geodesically complete and has the character of a shock wave in the sudden asymptotic region. This solution is weak in the senses of Tipler and Krolak as in the corresponding case of general relativity.
... The Universe was expanding, indeed! It took some time, even for astronomers and theoretical 9 See also [141], where this type of sudden and other finite time singularities were first introduced in order to show that closed Friedmann universes need not collapse even if they satisfy the strong energy condition; although the terminology 'sudden singularity' was introduced by Barrow in 2004, as already mentioned. ...
The first part of this paper contains a brief description of the beginnings of modern cosmology, which, the author will argue, was most likely born in the Year 1912. Some of the pieces of evidence presented here have emerged from recent research in the history of science, and are not usually shared with the general audiences in popular science books. In special, the issue of the correct formulation of the original Big Bang concept, according to the precise words of Fred Hoyle, is discussed. Too often, this point is very deficiently explained (when not just misleadingly) in most of the available generalist literature. Other frequent uses of the same words, Big Bang, as to name the initial singularity of the cosmos, and also whole cosmological models, are then addressed, as evolutions of its original meaning. Quantum and inflationary additions to the celebrated singularity theorems by Penrose, Geroch, Hawking and others led to subsequent results by Borde, Guth and Vilenkin. And corresponding corrections to the Einstein field equations have originated, in particular, R2, f(R), and scalar-tensor gravities, giving rise to a plethora of new singularities. For completeness, an updated table with a classification of the same is given.
We investigate particular cosmological models, based either on tachyon fields or on perfect fluids, for which soft future singularities arise in a natural way. Our main result is the description of a smooth crossing of the soft singularity in models with an anti-Chaplygin gas or with a particular tachyon field in the presence of dust. Such a crossing is made possible by certain transformations of matter properties. We discuss and compare also different approaches to the problem of crossing of the Big Bang - Big Crunch singularities.
In this paper, we investigate in this paper the Type IV singular bouncing in the framework of [Formula: see text] theory of gravity where [Formula: see text] and [Formula: see text] mean the curvature scalar and the Gauss–Bonnet invariant, respectively. Cosmological [Formula: see text] models constrained by the slow-roll evolution is reconstructed and their explicit forms are provided near the bounce and far away from it. One obtains finally two models whose stability is numerically analyzed in this work. Our results show that the stability of the reconstructed models is very affected by their parameters. The model far from the singularity recovers stability quickly than the model near the singularity.
Based on entropy considerations and the arrow of time Penrose argued that the Universe must have started in a special initial singularity with vanishing Weyl curvature. This is often interpreted to be at odds with inflation. Here we argue just the opposite, that Penrose’s persuasions are in fact consistent with inflation. Using the example of power law inflation, we show that inflation begins with a past null singularity, where Weyl tensor vanishes when the metric is initially exactly conformally flat. This initial state precisely obeys Penrose’s conditions. The initial null singularity breaks T-reversal spontaneously and picks the arrow of time. It can be regulated and interpreted as a creation of a universe from nothing, initially fitting in a bubble of Planckian size when it materializes. Penrose’s initial conditions are favored by the initial O(4) symmetry of the bubble, selected by extremality of the regulated Euclidean action. The predicted observables are marginally in tension with the data, but they can fit if small corrections to power law inflation kick in during the last 60e-folds.
John Barrow wrote dozens of books and hundreds of papers, including more than 10 papers and one book during his final illness, addressing a grand problem: why is the universe the way it is? What is the fundamental nature of physics, and what are its cosmological outcomes? His most famous book was on the Anthropic Principle with Frank Tipler, and his most significant physics research was on the possibility of a variation of the fundamental constants of physics—which, if true, requires a rejigging of our foundational physics theories. He was an outstanding communicator of science to the public, being asked to lecturer at as varied places as Downing Street and Buckingham Palace. His most important public service was as director of the Millennium Mathematics Project, for which he was awarded the Queen's Anniversary Prize for Educational Achievement by Queen Elizabeth II at Buckingham Palace. He was awarded many other prizes, including the Templeton Prize, the Gold Medal of the Royal Astronomical Society, both the Kelvin Medal and the Dirac Gold Medal of the Institute of Physics and the Faraday Medal of the Royal Society.
In 1972, at a symposium celebrating the 70th birthday of Paul Dirac, John Wheeler proclaimed that "the framework falls down for everything that one has ever called a law of physics". Responsible for this "breakage [...] among the laws of physics" was the general theory of relativity, more specifically its prediction of massive stars gravitationally collapsing to "black holes", a term Wheeler himself had made popular some years earlier. In our paper, we investigate how Wheeler reached the conclusion that gravitational collapse calls into question the lawfulness of physics and how, subsequently, he tried to develop a new worldview, rethinking in his own way the lessons of quantum mechanics as well as drawing inspiration from other disciplines, not least biology.
In 1972, at a symposium celebrating the 70th birthday of Paul Dirac, John Wheeler proclaimed that “the framework falls down for everything that one has ever called a law of physics”. Responsible for this “breakage […] among the laws of physics” was the general theory of relativity, more specifically its prediction of massive stars gravitationally collapsing to “black holes”, a term Wheeler himself had made popular some years earlier. In our paper, we investigate how Wheeler reached the conclusion that gravitational collapse calls into question the lawfulness of physics and how, subsequently, he tried to develop a new worldview, rethinking in his own way the lessons of quantum mechanics as well as drawing inspiration from other disciplines, not least biology.KeywordsJohn WheelerBlack holeQuantum gravityTeleologyAnthropic principleBiophysics
One of the key challenges facing cosmologists today is the nature of the mysterious dark energy introduced in the standard model of cosmology to account for the current accelerating expansion of the universe. In this regard, many other non‐standard cosmologies have been proposed which would eliminate the need to explicitly include any form of dark energy. One such model is the Sudden Future Singularity (SFS) model, in which no equation of state linking the energy density and the pressure in the universe is assumed to hold. In this model it is possible to have a blow up of the pressure occurring in the near future while the energy density would remain unaffected. The particular evolution of the scale factor of the Universe in this model that results in a singular behaviour of the pressure also admits acceleration in the current era as required. In this paper we compare an example SFS model with the current data from high redshift supernovae, baryon acoustic oscillations (BAO) and the cosmic microwave background (CMBR). We explore the limits placed on the SFS model parameters by these current data and discuss the viability of the SFS model in question as an alternative to the standard concordance cosmology.
In this paper, we consider two interesting cosmological models characterized by two expressions of the Hubble parameter and as goal, we realize the unification of the early-time and the late-time acceleration eras with the radiation- and matter-dominated eras, and this, in the context of [Formula: see text] theory of gravity. The particularity here is that the first model ensures just the unification of the early- and late-time eras with the matter-dominated era without the account of the radiation era, while the second model guarantees the unification of all the eras, namely, the early time, the radiation- and matter-dominated eras, and the late-time acceleration era. For each model, we point out two type IV singularities, the first occurring at the end of the inflationary era and the second at the end of the matter-dominated era. For the first model, the Hubble radius is introduced and three different expressions are taken into account: the first for the inflationary era, the second for the matter-dominated era and the third for the dark energy-dominated era. Hence, the algebraic [Formula: see text] is reconstructed for the three eras. Attention is attached to the inflationary era by calculating the corresponding two first slow-roll parameters and challenging them with recent Planck and BICEP2/Keck-Array observational data, and the results show the consistency of the reconstructed inflationary model. For the second model, we note a similarity with the first model, concerning the inflationary, matter-dominated and late-time acceleration eras. Then, we just perform the reconstruction of [Formula: see text], which is able to realize the radiation-dominated era.
The assumptions of the Hawking–Penrose singularity theorem are not covariant under field redefinitions. Following the works on the covariant formulation of quantum field theory initiated by Vilkovisky and DeWitt in the 80s, we propose to study singularities in field space, where the spacetime metric is treated as a coordinate along with the other fields in the theory. From this viewpoint, a spacetime singularity might be just a singularity in the field-space coordinates, analogously to the standard coordinate singularities in General Relativity. Objects invariant under field-space coordinate transformations can then reveal whether certain spacetime singularity is indeed singular. We recall that observables in quantum field theory are scalar functionals in field space. Therefore, in principle, spacetime singularities corresponding to regular field-space curvature invariants would not affect physical observables. In this paper, we show that the field-space Kretschmann scalar for a certain choice of the DeWitt field-space metric is everywhere finite. This fact could be interpreted as an indication that no singularities actually exist in pure gravity for any gravitational action. In particular, all vacuum singularities of General Relativity result from an unhappy choice of field variables. The extension to the case in which matter fields are present, as required by singularity theorems, is left for future development.
In this paper, we propose a class of cosmological models based on the gravitational theory , and on some nonideal fluids obeying to the thermodynamical equations of state of Redlich-Kwong, (Modified) Berthelot, and Dieterici. We investigate the dynamics of such models by applying dynamical system techniques showing that our models can account for both the inflationary and accelerated late-time epochs, with a radiation dominated phase and a nonsingular bounce in between, where the latter is discovered by appropriately compactifying the phase space. The infinity of the compactified phase space will be interpreted thermodynamically as a region in which the non-ideal behaviors of the cosmic fluids are suppressed, as long as the three previously mentioned modelings are assumed. Although the dimensionless variables we adopt were already known in the literature, we will derive some physical constraints on them restricting the available region of the phase space by demanding the theory to be free from ghost and tachyonic instabilities. A novel cosmological interpretation of such variables is also proposed by connecting them to the deceleration, jerk, and snap cosmographic parameters. It will be shown that the stability nature of the equilibrium points depends only on the parameter of the cosmic matter equations of state which describes their ideal limits, whilst a nonzero parameter accounting for the nonlinearities within the fluids is necessary for having a second de Sitter epoch. These properties of the cosmological dynamics are interpreted to constitute a generic behavior because they are identified in all the three fluid modelings. On the other hand, differences between the three models are critically examined by showing that the Redlich-Kwong scenario can admit a second radiation-dominated epoch and a Big Rip Singularity.
Different regularizations of the Hamiltonian constraint in loop quantum cosmology (LQC) yield modified loop quantum cosmologies, namely mLQC-I and mLQC-II, which lead to qualitatively different Planck scale physics. We perform a comprehensive analysis of resolution of various singularities in these modified loop cosmologies using effective spacetime description and compare with earlier results in standard LQC. We show that the volume remains non-zero and finite in finite time evolution for all considered loop cosmological models. Interestingly, even though expansion scalar and energy density are bounded due to quantum geometry, curvature invariants can still potentially diverge due to pressure singularities at a finite volume. These divergences are shown to be harmless since geodesic evolution does not break down and no strong singularities are present in the effective spacetimes of loop cosmologies. Using a phenomenological matter model, various types of exotic strong and weak singularities, including big rip, sudden, big freeze and type-IV singularities, are studied. We show that as in standard LQC, big rip and big freeze singularities are resolved in mLQC-I and mLQC-II, but quantum geometric effects do not resolve sudden and type-IV singularities.
A formal analogy between the Friedmann equation of relativistic cosmology and models of convective–radiative cooling/heating of a body (including Newton’s, Dulong–Petit’s, Newton–Stefan’s laws, and a generalization) is discussed. The analogy highlights Lagrangians, symmetries, and mathematical properties of the solutions of these cooling laws.
We consider different scenarios of the evolution of the Universe, where the singularities or some nonanalyticities in the geometry of the spacetime are present, trying to answer the following question: is it possible to conserve some kind of notion of particle corresponding to a chosen quantum field present in the universe when the latter approaches the singularity? We study scalar fields with different types of Lagrangians, writing down the second-order differential equations for the linear perturbations of these fields in the vicinity of a singularity. If both independent solutions are regular, we construct the vacuum state for quantum particles as a Gaussian function of the corresponding variable. If at least one of two independent solutions has a singular asymptotic behavior, then we cannot define the creation and the annihilation operators and construct the vacuum. This means that the very notion of particle loses sense. We show that at the approaching to the big rip singularity, particles corresponding to the phantom scalar field driving the evolution of the universe must vanish, while particles of other fields still can be defined. In the case of the model of the universe described by the tachyon field with a special trigonometric potential, where the big brake singularity occurs, we see that the (pseudo) tachyon particles do not pass through this singularity. Adding to this model some quantity of dust, we slightly change the characteristics of this singularity and tachyon particles survive. Finally, we consider a model with the scalar field with the cusped potential, where the phantom divide line crossing occurs. Here the particles are well defined in the vicinity of this crossing point.
We demonstrate the existence of sudden finite-time singularities, with constant scale factor, expansion rate, and density, in expanding Bianchi type-IX universes with free anisotropic pressures. A new type of nonsimultaneous anisotropic sudden singularity arises because of the divergences of the pressures, which may be of barrel or pancake type. The effect of one or more directions of expansion hitting a sudden singularity is tantamount to dimensional reductions as the nonsingular directions continue expanding and can see the sudden singularity in their past.
We study, using mean curvature flow methods, 2+1 dimensional cosmologies with a positive cosmological constant and matter satisfying the dominant and the strong energy conditions. If the spatial slices are compact with non-positive Euler characteristic and are initially expanding everywhere, then we prove that the spatial slices reach infinite volume, asymptotically converge on average to de Sitter and they become, almost everywhere, physically indistinguishable from de Sitter. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations and the formation of black holes.
A discussion is given of the ''Kantowski-Sachs'' cosmological models;
these are defined locally as admitting a four-parameter continuous
isometry group which acts on spacelike hypersurfaces, and which
possesses a three-parameter subgroup whose orbits are 2-surfaces of
constant curvature (i.e., the models possess spherical symmetry,
combined with a translational symmetry, and can thus be regarded as
nonempty analogs of part of the extended Schwarzschild manifold). It is
shown that all general relativistic models in which the matter content
is a perfect fluid satisfying reasonable energy conditions are
geodesically incomplete, both to the past and to the future, and that at
each resulting singularity the fluid energy density is infinite. In the
case where the fluid obeys a barotropic equation of state (which
includes all known exact perfect fluid solutions) the field equations
are shown to decouple to form a plane autonomous subsystem. This
subsystem is examined using qualitative (Poincaré-Bendixson)
theory, and phase-plane diagrams are drawn depicting the behavior of the
fluid's energy density and shear anisotropy in the course of the models'
evolution. Further diagrams depict the conformal structure of these
universes, and a table summarizes the asymptotic properties of all
physically relevant variables.
New boundary conditions on the Einstein-Rosen-Bondi gravitational-wave metrics yield closed inhomogeneous universes which solve Einstein's vacuum field equations exactly. Space sections of these universes have either the three-sphere topology S3 or the wormhole (hypertorus) topology S1⊗S2.
We apply some modern mathematical methods of global analysis to a series of studies undertaken by Belinskii, Khalatnikov and Lifschitz (BKL) to elucidate the structure of space-time near a general cosmological singularity. A brief summary of BKL's large body of work on inhomogeneous cosmological models is given (their work on homogeneous models is not under discussion here). Various theorems are proven and analyses of a mathematical and physical nature are made to show that the constructions of BKL cannot be general and in some cases do not give Lorentz manifolds. We conclude that although the work of BKL has led to very significant advances in our understanding of the dynamics of homogeneous cosmological models, the local techniques they employ do not extend to give us reliable information about the global structure of generic space-times. A detailed discussion of stability, generality, function counting, linearization stability, physical singularities and fictitious singularities is given together with an outline of various physical considerations which might be useful in future studies of the structure of generic space-times.
It is shown that most versions of the spherically inhomogeneous dust models of Tolman (1934) contain a co-moving surface layer in addition to shell-crossing singularities. Singularities occurring in parabolic, elliptical and hyperbolic Tolman models are considered. Conditions are described in which a closed Tolman model must collapse everywhere.
The progress of a first-order phase transition of a vacuum in the expanding Universe is investigated. The expansion of bubbles
of a stable vacuum is calculated simultaneously with the cosmic expansion with the aid of the following two simplified nucleation
rates of bubbles p: (i) p = pT Tc δ (T−Tc) in the hot Universe models, (ii) p = 0 for n > nc and p = pQ for n < nc in the cold Universe models, where T is the cosmic temperature, Tc the critical temperature, n the cosmic number density of the fermions coupled to the order parameter of the vacuum, nc the critical density, and pT and pQ are parameters.
The following results are obtained: (1) If the nucleation rates are small and the vacuum stays at the metastable state for
a long time, the Universe begins to expand exponentially. As a result, the progress of the phase transition is delayed more
and more by the rapid cosmic expansion. In particular, in model (i), if pT is less than a critical value, the phase transition never finishes. (2) The lower limits of the nucleation parameter pT and pQ are obtained from observation of the number ratio of photons to baryons in the present Universe. (3) If the phase transition
of the vacuum in SU(5) GUT is of first order or if there exists a hypothetical first-order phase transition of the vacuum
in the very early stage in which baryon number is not conserved, the density and the velocity fluctuations created by the
phase transition may account for the origin of galaxies.
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We explore the problem of the existence of global maximal (K=0) and constant-mean-curvature (K=K0) time functions in general relativity. We attempt a rigorous definition of numerical relativity so as to bridge the gap between the field and mathematical relativity. We point out that numerical relativity can in principle construct any globally hyperbolic solution to Einstein's equations. This involves the construction of Cauchy time functions. Therefore we review what is known about the existence and uniqueness of such functions when their mean curvature is specified to be a constant on each time slice. We note that in strong-field solutions which contain singularities the question of existence is intimately connected to the nature of the singularity. Defining the class of "crushing singularities" we prove new theorems showing that K=0 or K=K0 time functions uniformly avoid such singularities (which include both Cauchy horizons and some curvature singularities). We then study the inhomogeneous generalizations of the Oppenheimer-Snyder spherical-dust-collapse spacetimes. These Tolman-Bondi solutions are classified as to their causal structure and found to contain naked singularities of a new type if the collapse is sufficiently inhomogeneous. We calculate the K=0 and K=K0 time slices for a variety of these spacetimes. We find that since some extreme dust collapses lead to noncrushing singularities, maximal time slicing can hit the singularity before covering the domain of outer communications of the resulting black hole. Furthermore, the use of K=K0 slices in the presence of a naked singularity is discussed.
The dynamical evolution of a locally open region in a closed universe is
considered. This is done by using Tolman's solution for an inhomogeneous
closed model filled with the pressureless matter. It is shown that the
unbounded expansion of the open region will be stopped by the particles
invading the region through the caustics which necessarily form. A
hypothesis is put forward according to which general non-simultaneous
collapse is an unavoidable fate of any closed universe.
The evolution of nonspherical structures was investigated when
observational evidence for the existence of flattened or striated
structures of about 100 Mpc was found. Equations describing the
evolution of a rotating, dust ellipsoid in an expanding universe have
been obtained, and solutions for the shape and density evolution as the
ellipsoid expands with a Friedmann background universe have been
derived. Details of pancake formation were calculated, and the density
collapse at turnaround and the collapse velocity at pancake formation
were found to be reduced relative to the spherical case when initial
fluctuations were anisotropic. An exact model for a general relativistic
pancake collapse has also been given, which is based on an exact
solution to Einstein's equations and does not have symmetry or pass
through a series of ellipsoids.
The authors summarize what is currently known about the future evolution
and final state of closed universes: in mathematical language, those
which have a compact Cauchy surface. It is shown that the existence of a
maximal hypersurface (a time of maximum expansion) is a necessary and
sufficient condition for the existence of an all-encompassing final
singularity in a universe with a compact Cauchy surface. The only
topologies which can admit maximal hypersurfaces are S3 and
S2×S1, together with more complicated
topologies formed from these two types of 3-manifold by connected
summation and certain identifications. The relevance of these results to
inflation is also discussed.
The models are solutions of Einstein's equations for dust with no
Killing vectors. They depend on four arbitrary functions of one
variable, and generalize both the Friedmann models and those of
Kantowski and Sachs (1966). The possibilities of evolution are diverse.
One result is that a Friedmann open model can evolve from a variety of
initial states, depending on three arbitrary functions of one variable.
This paper gives an overview and reviews some recent investigations of anisotropic and inhomogeneous models. A class of models, which admit an Abelian two-parameter group of isometries, is considered in detail. Within this class of models we present exact solutions of the Einstein field equations. These solutions describe inhomogeneous cosmological models containing gravitational, scalar and electromagnetic waves. The solutions are used to study the effect of the symmetry breaking in corresponding Bianchi models. The nonlinear dynamics of primordial inhomogeneities is considered. The global evolution of the inhomogeneous models considered is also investigated. Finally we discuss the validity of various assumptions, used in the earlier treatments of inhomogeneous models.
Spacetimes with closed spacelike hypersurfaces and spacelike two-parameter isometry groups are investigated to determine their possible global structures. It is shown that the two spacelike Killing vectors always commute with each other. Connected group-invariant spacelike hypersurfaces must be homeomorphic to S1 ⊗ S1 ⊗ S1 (three-torus), S1 ⊗ S2 (three-handle), S3 (three-sphere), or to a manifold which is covered by one of these. The spacetime metric and Einstein equations are simplified in the absence of nongravitational sources and used to establish the impossibility of topology change as well as other limitations on global structure. Regularity conditions for spacetimes of this type are derived and shown to be compatible with Einstein's equations.
Phase transitions in the early universe can give rise to microscopic topological defects: vacuum domain walls, strings, walls bounded by strings, and monopoles connected by strings. This article reviews the formation, physical properties and the cosmological evolution of various defects. A particular attention is paid to strings and their cosmological consequences, including the string scenario of galaxy formation and possible observational effects of strings.
We prove theorems on existence, uniqueness and smoothness of maximal and constant mean curvature compact spacelike hypersurfaces in globally hyperbolic spacetimes. The uniqueness theorem for maximal hypersurfaces of Brill and Flaherty, which assumed matter everywhere, is extended to space times that are vacuum and non-flat or that satisfy a generic-type condition. In this connection we show that under general hypotheses, a spatially closed universe with a maximal hypersurface must be Wheeler universe: i.e. be closed in time as well. The existence of Lipschitz achronal maximal volume hypersurfaces under the hypothesis that candidate hypersurfaces are bounded away from the singularity is proved. This hypothesis is shown to be valid in two cases of interest: when the singularities are of strong curvature type, and when the singularity is a
single ideal point. Some properties of these maximal volume hypersurfaces and difficulties with Avez' original arguments are discussed. The difficulties involve the possibility that the maximal volume hypersurface can be null on certain portions: we present an incomplete argument which suggests that these hypersurfaces are always smooth, but prove that an a priori bound on the second fundamental form does imply smoothness. An extension of the perturbation theorem of Choquet-Bruhat. Fischer and Marsden is given and conditions under which local foliations by constant mean curvature hypersurfaces can be extended to global ones is obtained.
All solutions of Einstein's field equations representing irrotational dust and possessing a metric of the formds
2=dt
2−e
2αdr
2−e
2β(dy
2+dz
2) are found. The new metrics generalize the earlier Bondi-Tolman, Eardley-Liang-Sachs, and Kantowski-Sachs cosmological models.