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Closed universe - Their future evolution and final state
Abstract
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.
... 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. ...
... 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]. ...
... These spacetimes describe the most general effects of spatially homogeneous perturbations on open FRW universes. When the strong energy condition is obeyed, then isotropic expansion was found to be stable but not asymptotically stable at late times [84,92,93]. ...
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.
... However, inflation will occur as long as the spacetime is on average initially expanding. This is consistent with the theoretical expectations of [22] and [57]. ...
... For our purposes it is useful to recast Eqn. (22) for K in the form We set ...
... However, even in these cases, the remaining spacetime continues to expand and inflate. Since K < 0 in all cases here, this result is consistent with what would be expected from [22] and [57] (note that our sign convention means that K < 0 denotes locally expanding spacetime). We will explore the case where K > 0 in future work. ...
We consider the effects of inhomogeneous initial conditions in both the scalar field profile and the extrinsic curvature on different inflationary models. In particular, we compare the robustness of small field inflation to that of large field inflation, using numerical simulations with Einstein gravity in 3+1 dimensions. We find that small field inflation can fail in the presence of subdominant gradient energies, suggesting that it is much less robust to inhomogeneities than large field inflation, which withstands dominant gradient energies. However, we also show that small field inflation can be successful even if some regions of spacetime start out in the region of the potential that does not support inflation. In the large field case, we confirm previous results that inflation is robust if the inflaton occupies the inflationary part of the potential. Furthermore, we show that increasing initial scalar gradients will not form sufficiently massive inflation-ending black holes if the initial hypersurface is approximately flat. Finally, we consider the large field case with a varying extrinsic curvature K, such that some regions are initially collapsing. We find that this may again lead to local black holes, but overall the spacetime remains inflationary if the spacetime is open, which confirms previous theoretical studies.
... Theorem 1 (Barrow and Tipler [2], Kleban and Senatore [3]). Consider a globally hyperbolic spacetime (M, g µν ) such that: ...
... Then M, which is called a compact inhomogeneous "cosmology", cannot globally recollapse unless the topology of Σ 0 is "closed" (i.e. its topology is spherical, or S 1 × S 2 , or more complicated hybrids obtained by connected summation and special identification of these two basic topologies). 2 The basic facts that underlie this result, as discussed in detail in section 2, are: (a) In a compact cosmology that starts from a big bang (global expansion) and ends in a global recollapse (contraction) there exists a maximum volume (maximal) slice. (b) The spatial topology of M is a discrete choice that is preserved in a globally hyperbolic spacetime; in particular the maximal slice has the same topology as Σ 0 . ...
... on every Cauchy slice Σ λ along the MCF. 2 To make the analogy with the homogeneous case closer, note that a homogeneous but anisotropic S 1 × S 2 universe (or the non-compact version R × S 2 ) which was not considered in [1] can also globally recollapse. Closely related is the fact that S 1 × S 2 admits unstable static solutions, for instance with a uniform electric field E = √ 2Λ along S 1 , and a symmetric S 2 with radius R = 1/ √ 16πGN Λ. 3 Here I followed [3] to denote those compact topologies that admit a metric with everywhere 3 R > 0 as "closed", those that are not closed but can have everywhere 3 R = 0 as "flat", and the rest as "open", in analogy with the terminology used in FRW cosmology to denote κ = +1, 0, −1. ...
Motivated by the question of how generic inflation is, I study the time-evolution of topological surfaces in an inhomogeneous cosmology with positive cosmological constant Λ. If matter fields satisfy the Weak Energy Condition, non-spherical incompressible surfaces of least area are shown to expand at least exponentially, with rate d log Amin/dλ ⩾ 8π GNΛ, under the mean curvature flow parametrized by λ. With reasonable assumptions about the nature of singularities this restricts the topology of black holes: (a) no trapped surface or apparent horizon can be a non-spherical, incompressible surface, and (b) the interior of black holes cannot contain any such surface.
... Alternatively, this can be thought of as imposing a scale of homogeneity on the initial conditions with the Universe made up of many (inhomogeneous) boxes of size L. One can always make L larger, thus increasing the scale of homogeneity relative to our patch of the Universe, and it is usually considered that taking L to be greater than the initial Hubble scale of inflation is a sufficiently conservative approach. Other topologies, in particular those that can support a positive-definite (3) R can lead to different conclusions [12,47], so that this is a choice that should be made explicit. ...
... . (25) In this case the potential is only well approximated by (12) if |φ max | > µ n , and as for the hilltop models, plateau models with a Planckian characteristic scale are robust against large field excursions, and the function f can only change sign for models with sub-Planckian µ n . ...
... We also consider the third class of concave potentials (12), which arises in string theory as D-brane inflation [81][82][83]. In the best-studied case the inflaton describes the position of a D3 brane, which corresponds to n = 4. ...
We study the robustness of single-field inflation against inhomogeneities. We derive a simple analytic criterion on the shape of the potential for successful inflation in the presence of inhomogeneities, and demonstrate its validity using full 3+1 dimensional numerical relativity simulations on several classes of popular models of single-field inflation. We find that models with convex potentials are more robust to inhomogeneities than those with concave potentials, and that concave potentials that vary on super-Planckian scales are significantly more robust than those that vary on sub-Planckian scales.
... Theorem 1 (Barrow and Tipler [2], Kleban and Senatore [3]): Consider a globally hyperbolic spacetime (M, g µν ) such that: ...
... i Einstein equations hold with a stress-energy tensor that satisfies the Weak Energy Condition (WEC), T µν k µ k ν ≥ 0, for all time-like k µ ; (2) ii There is a compact Cauchy slice Σ 0 , that is everywhere expanding (i.e. has everywhere positive mean curvature K). ...
... Then M, which is called a compact inhomogeneous "cosmology", cannot globally recollapse unless the topology of Σ 0 is "closed" (i.e. it is spherical, S 1 × S 2 , or more complicated hybrids obtained by connected summation and special identification of these two basic topologies). 2 The basic facts that underlie this result are (see section 2 for details): (a) In a compact cosmology that starts from a big bang (global expansion) and ends in a global recollapse (contraction) there exists a maximum volume (maximal) slice. (b) The spatial topology of M is a discrete choice that is preserved in a globally hyperbolic spacetime; in particular the 2 To make the analogy with the homogeneous case closer, note that a homogeneous but anisotropic S 1 ×S 2 universe (or the non-compact version R × S 2 ) which was not considered in [1] can also globally recollapse. ...
Motivated by the question of how generic inflation is, I study the time-evolution of topological surfaces in an inhomogeneous cosmology with positive cosmological constant . If matter fields satisfy the Weak Energy Condition, non-spherical incompressible surfaces of least area are shown to expand at least exponentially, with rate , under the mean curvature flow parametrized by . With reasonable assumptions about the nature of singularities this restricts the topology of black holes: (a) no trapped surface or apparent horizon can be a non-spherical, incompressible surface, and (b) the interior of black holes cannot contain any such surface.
... 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. ...
... These spacetimes describe the most general effects of spatially homogeneous perturbations on open FRW universes. When the JCAP05(2018)026 strong energy condition is obeyed, then isotropic expansion was found to be stable but not asymptotically stable at late times [84,92,93]. ...
... Given that Σ > 0 and 0 < r < 1, we immediately deduce that 0 < Σ < 1, in accord with R < 0, in (B.8). Thus, although isotropy (Σ = 0) is not asymptotically stable (in the Lyapunov sense), it is stable in the sense that any deviations from isotropy never diverge [84,93,94] and Σ tends to a constant at large times. Note that when we set r → 1 the Σ-parameter approaches zero and the expansion becomes isotropic (i.e. the Milne universe). ...
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.
... However, inflation will occur as long as the spacetime is on average initially expanding. This is consistent with the theoretical expectations of [112] and [147]. ...
... We therefore find that for a spacetime that would have resulted in inflation However, even in these cases, the remaining spacetime continues to expand and inflate. Since ⟨K⟩ < 0 in all cases here, this result is consistent with what would be expected from [112] and [147] (note that our sign convention means that K < 0 denotes locally expanding spacetime). ...
... • If ⟨K⟩ < 0, then inflation wins. If the initial hypersurface (a Cauchy surface) has a net negative (expanding) value of K there will always be an expanding region, as predicted in analytic studies [112] and [147]. ...
Einstein's field equation of General Relativity (GR) has been known for over 100 years, yet it remains challenging to solve analytically in strongly relativistic regimes, particularly where there is a lack of a priori symmetry. Numerical Relativity (NR) - the evolution of the Einstein Equations using a computer - is now a relatively mature tool which enables such cases to be explored. In this thesis, a description is given of the development and application of a new Numerical Relativity code, GRChombo. GRChombo uses the standard BSSN formalism, incorporating full adaptive mesh refinement (AMR) and massive parallelism via the Message Passing Interface (MPI). The AMR capability permits the study of physics which has previously been computationally infeasible in a full 3+1 setting. The functionality of the code is described, its performance characteristics are demonstrated, and it is shown that it can stably and accurately evolve standard spacetimes such as black hole mergers. We use GRChombo to study the effects of inhomogeneous initial conditions on the robustness of small and large field inflationary models. and investigate the critical behaviour which occurs in the collapse of both spherically symmetric and asymmetric scalar field bubbles.
... In contrast to the negative and zero Yamabe classes, there exists a strong conjecture for the behavior of general initial data in Y + -the Closed universe recollapse conjecture [6,7]. It says that in 3+1 dimensions the maximal globally hyperbolic development (MGHD) of any vacuum initial data set with a manifold ∈ Y + is geodesically futureand past incomplete (also cf. ...
... where c is a positive constant, defined for massless matter models in (6). In addition, we choose j = 0, which implies h is a TT-tensor, which we set to zero for the class of initial data we consider. ...
... There is one requirement for a matter model that ensures that the construction, which we perform below holds. This is condition (6). In addition, as several matter models, which have been referred to as massless in the literature fulfill this requirement, it appears reasonable to define this class of matter models in the beginning and subsequently work with this unifying definition instead of performing the analysis individually for each matter model. ...
We study the long-time behavior of the Einstein flow coupled to matter on 2-dimensional surfaces. We consider massless matter models such as collisionless matter composed of massless particles, massless scalar fields and radiation fluids and show that the maximal globally hyperbolic development of homogeneous and isotropic initial data on the 2-sphere is geodesically incomplete in both time directions, i.e. the spacetime recollapses. This behavior also holds for open sets of initial data. In particular, we construct classes of recollapsing 2+1-dimensional spacetimes with spherical spatial topology which provide evidence for a closed universe recollapse conjecture for massless matter models in 2+1 dimensions. Furthermore, we construct solutions with toroidal and higher genus topology for the massless matter fields, which in both cases are future complete. The spacetimes with toroidal topology are 2+1-dimensional analogies of the Einstein–de Sitter model. In addition, we point out a general relation between the energy-momentum tensor and the Kretschmann scalar in 2+1 dimensions and use it to infer strong cosmic censorship for all these models. In view of this relation, we also recall corresponding models containing massive particles, constructed in a previous work and determine the nature of their initial singularities. We conclude that the global structure of non-vacuum cosmological spacetimes in 2+1 dimensions is determined by the mass of particles and—in the homogeneous and isotropic setting studied here—verifies strong cosmic censorship.
... However, inflation will occur as long as the spacetime is on average initially expanding. This is consistent with the theoretical expectations of [22] and [57]. ...
... However, even in these cases, the remaining spacetime continues to expand and inflate. Since K < 0 in all cases here, this result is consistent with what would be expected from [22] and [57] (note that our sign convention means that K < 0 denotes locally expanding spacetime). We will explore the case where K > 0 in future work. ...
... • If K < 0, then inflation wins. If the initial hypersurface (a Cauchy surface) has a net negative (expanding) value of K there will always be an expanding region, as predicted in analytic studies [22] and [57]. ...
We consider the effects of inhomogeneous initial conditions in both the scalar field profile and the extrinsic curvature on different inflationary models. In particular, we compare the robustness of small field inflation to that of large field inflation, using numerical simulations with Einstein gravity in 3+1 dimensions. We find that small field inflation can fail in the presence of subdominant gradient energies, suggesting that it is much less robust to inhomogeneities than large field inflation, which withstands dominant gradient energies. However, we also show that small field inflation can be successful even if some regions of spacetime start out in the region of the potential that does not support inflation. In the large field case, we confirm the results of [1-3] that inflation is robust if the inflaton occupies the inflationary part of the potential. Furthermore, we show that increasing initial scalar gradients will not form sufficiently massive inflation-ending black holes if the initial hypersurface is approximately flat. Finally, we consider the large field case with a varying extrinsic curvature K, such that some regions are initially collapsing. We find that this may again lead to local black holes, but overall the spacetime remains inflationary if the spacetime is open, which confirms previous theoretical studies.
... Note added: After this work was completed and the paper submitted to the arxiv, the existence of [9] was brought to our attention, of which we were previously unaware. This work has very substantial overlap with ours: it shows, using similar techniques, that there exists no maximal hypersurface for universes with compact spatial topologies except the ones we refer to as type (i) (page 3), and therefore such cosmologies have no global crunch. ...
... This work has very substantial overlap with ours: it shows, using similar techniques, that there exists no maximal hypersurface for universes with compact spatial topologies except the ones we refer to as type (i) (page 3), and therefore such cosmologies have no global crunch. However, there is no explicit discussion in [9] of the growth of volume of hypersurfaces (Eq. (2.3)), or that there is always a region with expansion rate bounded from below (Eqs. ...
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.
... This well-motivated expectation has been recently proven to be incorrect for some interesting and non-trivial physical reasons. Ref. [2,3] (see also [4]) have used a combination of numerical and analytical techniques to show that, for most of the three-dimensional topologies and for a huge class of inhomoge-nous and anisotropic initial conditions, inflation will always start somewhere on the manifold, notwithstanding the formation of localized black-holes. Refs. ...
... At late times, i.e. for large values of the MCF affine parameter λ, the spacetime converges, on average, to de Sitter space; and there are arbitrarily large regions of space-time that are physically indistinguishable from de Sitter space. 4 For SEC indeed ...
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.
... Note added. After this work was completed and the paper submitted to the arxiv, the existence of [9] was brought to our attention, of which we were previously unaware. This work has very substantial overlap with ours: it shows, using similar techniques, that there exists no maximal hypersurface for universes with compact spatial topologies except the ones we refer to as type (i) (page 3), and therefore such cosmologies have no global crunch. ...
... This work has very substantial overlap with ours: it shows, using similar techniques, that there exists no maximal hypersurface for universes with compact spatial topologies except the ones we refer to as type (i) (page 3), and therefore such cosmologies have no global crunch. However, there is no explicit discussion in [9] of the growth of volume of hypersurfaces (eq. (2.3)), or that there is always a region with expansion rate bounded from below (eqs. ...
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.
... so for the very early universe x(t) rv t and n was closer to unity. One can show that [49] In(1 sec) -11 :s; 0 (10)(11)(12)(13)(14)(15)(16) (2.20) (2.21 ) As Linde [49] points out, if the density of the universe was initially (at Planck time) ...
... • Some counter-examples exist of the form 'initially expanding universe models recollapse to a singularity' without ever becoming de Sitter type universes, the most obvious one being the closed FRW space-time which collapses before it enters an inflationary phase (see [13,9]). ...
... Relaxing the assumption of on average flatness could alleviate the expansionary problem if one imposes a topology for which it is possible to have non negative (3) R (S 3 and S 2 × S 1 , see [19]). ...
A recent article by Galtier and Nazarenko [1] proposed that weakly nonlinear gravitational waves could result in a turbulent cascade, with energy flowing from high to low frequency modes or vice versa. This is an interesting proposition for early universe cosmology because it could suggest some "natural" initial conditions for the gravitational background. In this paper we use the ADM formalism to show that, given some simple and, arguably, natural assumptions, such initial conditions lead to expansion (or collapse) of the spacetime on a timescale much faster than that of the turbulent cascade, meaning that the cascade is unlikely to have sufficient time to develop under general conditions. We suggest possible ways in which the expansion could be mitigated to give the cascade time to develop.
... However, this result has no simple generalization to other cases. For spatially closed cosmologies, there are theorems characterizing the final state of the universe [9]; nevertheless, there is no simple answer even for the closed Friedmann spacetime (see e.g. [10,11]). ...
Recently a new no-global-recollapse argument was given for some inhomogeneous and anisotropic cosmologies that utilizes surface deformation by the mean curvature flow. In this paper we discuss important properties of the mean curvature flow of spacelike surfaces in Lorentzian manifolds. We show that singularities may form during cosmic evolution and the theorems forbidding the global recollapse lose their validity. The time evolution of the spatial scalar curvature that may kinematically prevent the recollapse is determined in normal coordinates, which shows the impact of inhomogeneities explicitly. Our analysis indicates a caveat in numerical solutions that give rise to inflation.
... Should the irreducible mass always be bounded by the ADM mass (as expected), then conjecture 1 would imply this alternate version. Conjectures 1 and 2 are similar to the conjecture that closed universes possessing S 3 or S 1 ×S 2 Cauchy surfaces (and obeying the dominant-energy and non-negativepressures conditions) must have finite lifetimes [25][26][27][28], which is a slight variant of the closed-universe recollapse conjecture [29][30][31][32]. In fact, the similarity is more than superficial. ...
We study the problem of how long a journey within a black hole can last. Based on our observations, we make two conjectures. First, for observers that have entered a black hole from an asymptotic region, we conjecture that the length of their journey within is bounded by a multiple of the future asymptotic ``size'' of the black hole, provided the spacetime is globally hyperbolic and satisfies the dominant-energy and non-negative-pressures conditions. Second, for spacetimes with Cauchy surfaces (or an appropriate generalization thereof) and satisfying the dominant energy and non-negative-pressures conditions, we conjecture that the length of a journey anywhere within a black hole is again bounded, although here the bound requires a knowledge of the initial data for the gravitational field on a Cauchy surface. We prove these conjectures in the spherically symmetric case. We also prove that there is an upper bound on the lifetimes of observers lying ``deep within'' a black hole, provided the spacetime satisfies the timelike-convergence condition and possesses a maximal Cauchy surface. Further, we investigate whether one can increase the lifetime of an observer that has entered a black hole, e.g., by throwing additional matter into the hole. Lastly, in an appendix, we prove that the surface area A of the event horizon of a black hole in a spherically symmetric spacetime with ADM mass is always bounded by , provided that future null infinity is complete and the spacetime is globally hyperbolic and satisfies the dominant-energy condition.
... 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.
... Here, we focus on the effects of large inhomogeneities on the onset of inflation, both in scenarios where it occurs at nearly Planckian and sub-Planckian energy scales, using evolutions in fully nonlinear general relativity. This question has been studied using tools from numerical relativity in a number of papers [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], complementing work evolving inhomogeneous fields on homogeneous spacetimes [29][30][31][32][33], and using analytic techniques [34][35][36] (see [37] for a short review on the topic). Focusing on more recent work, [23] showed that large field inflation is robust to simple inhomogeneous initial conditions even when the initial gradient energy is many orders of magnitude larger than the vacuum energy density, provided that the universe is initially expanding everywhere and that the scalar field range remains within the slow-roll regime. ...
We investigate the circumstances under which cosmic inflation can arise from very inhomogeneous initial conditions using numerical relativity simulations. Previous studies have not considered cases with non-zero momentum density due to technical challenges with solving the coupled Einstein constraint equations. Here we address these, introducing and comparing several different ways of constructing cosmological initial conditions with inhomogeneous scalar field and time derivative profiles. We evolve such initial conditions with large inhomogeneities in both single- and two-field inflationary models. We study cases where the initial gradient and kinetic energy are much larger than the inflationary energy scale, and black holes can form, as well as cases where the initial scalar potential energy is comparable, as in scenarios where inflation occurs at nearly Planckian densities, finding large-field inflation to be generally robust. We consider examples of initial conditions where a large scalar field velocity towards non-inflationary values would prevent inflation from occurring in the homogeneous case, finding that the addition of large gradients in the scalar field can actually dilute this effect, with the increased expansion and non-vanishing restoring force leading to inflation.
... More generally, it was shown by Barrow and Tipler that only spacetimes with topology S 3 (or quotients thereof) and S 1 × S 2 (or connected sums of those topologies) can admit a maximal Cauchy hypersurface, see [BT85]. 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. ...
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.
... Here, we focus on the effects of large inhomogeneities on the onset of inflation, both in scenarios where it occurs at nearly Planckian and sub-Planckian energy scales, using evolutions in fully nonlinear general relativity. This question has been studied using tools from numerical relativity in a number of papers [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], complementing work evolving inhomogeneous fields on homogeneous spacetimes [29][30][31][32][33], and using analytic techniques [34][35][36] (see [37] for a short review on the topic). Focusing on more recent work, [23] showed that large field inflation is robust to simple inhomogeneous initial conditions even when the initial gradient energy is many orders of magnitude larger than the vacuum energy density, provided that the universe is initially expanding everywhere and that the scalar field range remains within the slow-roll regime. ...
We investigate the circumstances under which cosmic inflation can arise from very inhomogeneous initial conditions using numerical relativity simulations. Previous studies have not considered cases with non-zero momentum density due to technical challenges with solving the coupled Einstein constraint equations. Here we address these, introducing and comparing several different ways of constructing cosmological initial conditions with inhomogeneous scalar field and time derivative profiles. We evolve such initial conditions with large inhomogeneities in both single- and two-field inflationary models. We study cases where the initial gradient and kinetic energy are much larger than the inflationary energy scale, and black holes can form, as well as cases where the initial scalar potential energy is comparable, as in scenarios where inflation occurs at nearly Planckian densities, finding large-field inflation to be generally robust. We consider examples of initial conditions where a large scalar field velocity towards non-inflationary values would prevent inflation from occurring in the homogeneous case, finding that the addition of large gradients in the scalar field can actually dilute this effect, with the increased expansion and non-vanishing restoring force leading to inflation.
... Except for Bianchi IX, all homogeneous anisotropic cosmologies with positive cosmological constant asymptote to de Sitter space [194]. In the inhomogeneous case, provided the weak energy condition holds, it can be shown that global recollapse can only occur if the three-dimensional Ricci scalar is positive everywhere, which is topologically impossible for most 3-manifolds [195,196]. Moreover, under the same assumptions it can be shown that there is always an open neighborhood that expands at least as fast as de Sitter space [196]. ...
The precision cosmological model describing the origin and expansion history of the universe, with observed structure seeded at the inflationary cosmic horizon, demands completion in the ultraviolet and in the infrared. The dynamics of the cosmic horizon also suggests an associated entropy, again requiring a microphysical theory. Recent years have seen enormous progress in understanding the structure of de Sitter space and inflation in string theory, and of cosmological observables captured by quantum field theory and solvable deformations thereof. The resulting models admit ongoing observational tests through measurements of the cosmic microwave background and large-scale structure, as well as through analyses of theoretical consistency by means of thought experiments. This paper, prepared for the TF01 and TF09 conveners of the Snowmass 2021 process, provides a synopsis of this important area, focusing on ongoing developments and opportunities.
... Alternatively, this can be thought of as imposing a scale of homogeneity on the initial conditions with the Universe made up of many (inhomogeneous) boxes of size L. One can always make L larger, thus increasing the scale of homogeneity relative to our patch of the Universe, and it is usually considered that taking L to be greater than the initial Hubble scale of inflation is a sufficiently conservative approach. Other topologies, in particular those that can support a positive-definite (3) R can lead to different conclusions [12,47], so that this is a choice that should be made explicit. ...
We study the robustness of single-field inflation against inhomogeneities. We derive a simple analytic criterion on the shape of the potential for successful inflation in the presence of inhomogeneities, and demonstrate its validity using full 3+1 dimensional numerical relativity simulations on several classes of popular models of single-field inflation. We find that models with convex potentials are more robust to inhomogeneities than those with concave potentials, and that concave potentials that vary on super-Planckian scales are significantly more robust than those that vary on sub-Planckian scales.
... On the analytical side, a combination of Mean Curvature Flow techniques (see for example [14]) and the now-proven Thurston Geometrization Classification (see [15] Theorem 4.35 and [16,17]) allowed to prove some partial results in the general case, without imposing extra symmetries. In particular, approximating the inflationary potential as a positive cosmological constant, and assuming that matter satisfies the weak energy condition and that all singularities are of the so-called crushing kind, Ref. [18] has shown that, for almost all topologies of the spatial slices of a cosmological spacetime, the volume of these slices (assumed to be, initially, expanding everywhere) will grow with time (see also [19]); moreover, there is always an open neighborhood that expands at least as fast as the flat of de Sitter space. This suggests, though does not prove, that the volume will go to infinity, matter will dilute away, and the universe will resemble de Sitter space in arbitrarily large regions of spacetime. ...
We study, using Mean Curvature Flow methods, 3+1 dimensional cosmologies with a positive cosmological constant, matter satisfying the dominant and the strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are the closed orbits of a symmetry group. If these surfaces have non-positive Euler characteristic (or in the case of 2-spheres, if the initial 2-spheres are large enough) and also if the initial spatial slice is expanding everywhere, then we prove that asymptotically the spacetime becomes physically indistinguishable from de Sitter space on arbitrarily large regions of spacetime. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations.
... As a consequence, the individual 'expansion to infinity' property of the de-Sitter spaces (or its quotients) is sort of 'killed' by the hyperbolic parts present in M and therefore becomes asymptotically negligible. [41] showed that spherical space forms (S 3 /Γ) and handle (S 2 × S 1 ) topologies re-collapse through formation of maximal hypersurfaces provided a number of conditions on matter sources and spatial geometry are satisfied (without a cosmological constant; see their theorem 3). Later [42] studied the closed universe re-collapse conjecture with the same spatial topologies. ...
The ADM formalism together with a constant mean curvature (CMC) temporal gauge is used to derive the monotonic decay of a weak Lyapunov function of the Einstein dynamical equations in an expanding universe with a positive cosmological constant and matter sources satisfying suitable energy conditions. While such a Lyapunov function does not, in general, represent a true Hamiltonian of the matter-coupled gravity dynamics (unlike in the vacuum case when it does), it can nevertheless be used to study the asymptotic behavior of the spacetimes. The Lyapunov function attains its infimum only in the limit that the matter sources be ‘turned off‘ or, at least, become asymptotically negligible provided that the universe model does not re-collapse and form singularities. Later we specialize our result to the case of a perfect fluid which satisfies the desired energy conditions. However, even in this special case, we show using Shutz’s velocity potential formalism cast into Hamiltonian form that unlike the vacuum spacetimes (with or without a positive cosmological constant), construction of a true Hamiltonian for the dynamics in constant mean curvature temporal gauge is difficult and therefore the Lyapunov function does not have a straightforward physical interpretation. Nevertheless, we show, for the fluid with equation of state (), that the general results obtained hold true and the infimum of the weak Lyapunov function can be related to the Sigma constant, a topological invariant of the manifold. Utilizing these results, some general conclusions are drawn regarding the asymptotic state of the universe and the dynamical control of the allowed spatial topologies in the cosmological models.
... A corollary of this is that, under generic conditions, a collapsing universe cannot transition to a growing one, and thus if the universe started in a collapsing state, it should have ended. This statement can also be formulated in terms of the ADM decomposition [27,29], and forms the basis for work on cosmological bounces [11][12][13]. ...
Setting aside anthropic arguments, there is no reason why the universe should initially favour a net expanding phase rather than one experiencing a net contraction. However, a collapsing universe containing ``normal'' matter will end at a singularity in a finite time. We point out that there is a mechanism, derived from non-perturbative effects in quantum field theory in a finite volume, which one would expect to provide a bias towards expansion when the spacetime volume shrinks. Whilst the mechanism requires further study in a curved background, we propose a new scalar field component in a cosmological background, and study its properties and impact on a contracting phase. We discuss how this could dynamically generate the necessary initial conditions for inflation to get started, or form part of the mechanism for a non-singular cosmological bounce.
... Relaxing the assumption of on average flatness could alleviate the expansionary problem if one imposes a topology for which it is possible to have non negative (3) R (S 3 and S 2 × S 1 , see [19]). ...
A recent article by Galtier and Nazarenko [1] proposed that weakly nonlinear gravitational waves could result in a turbulent cascade, with energy flowing from high to low frequency modes or vice versa. This is an interesting proposition for early universe cosmology because it could suggest some "natural" initial conditions for the gravitational background. In this paper we use the ADM formalism to show that, given some simple and, arguably, natural assumptions, such initial conditions lead to expansion (or collapse) of the spacetime on a timescale much faster than that of the turbulent cascade, meaning that the cascade is unlikely to have sufficient time to develop under general conditions. We suggest possible ways in which the expansion could be mitigated to give the cascade time to develop.
... 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 [27][28][29][30][31]. 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.
... However, this result has no simple generalization to other cases. For spatially closed cosmologies, there are theorems characterizing the final state of the universe [9], nevertheless there is no simple answer even for the closed Friedmann spacetime (see e.g. [10,11]). ...
Recently a new no-global-recollapse argument is given for some inhomogeneous and anisotropic cosmologies that utilizes surface deformation by the mean curvature flow. In this note we discuss important properties of the mean curvature flow of spacelike surfaces in Lorentzian manifolds. The time evolution of the spatial scalar curvature that prevents recollapse is determined in normal coordinates, which shows the impact of inhomogeneities explicitly. Our analysis indicates a caveat in numerical solutions that give rise to inflation.
... It is of particular interest to understand to what extent the spatial topology determines whether the spacetime expands forever (in at least one direction) or recollapses, i.e. is incomplete in both directions. For instance, the closed universe recollapse conjecture [5,6], which concerns the Einstein equations with a vanishing cosmological constant in three spatial dimensions, says that initial data on a compact spatial slice of positive Yamabe class recollapse to a big crunch, while initial data in the negative or zero (but non-flat) Yamabe class lead to a future complete spacetime. The conjecture has been verified for some classes of initial data [9,22] but generally remains open. ...
We consider non-vacuum initial data for the three-dimensional Einstein equations coupled to Vlasov matter composed of massive particles, on an arbitrary compact Cauchy hypersurface without boundary. We show that conservation of the total mass implies future completeness of the corresponding maximal development in the isotropic case, independent of the topology. This behavior is fundamentally different from the vacuum case and also from the same model in higher dimensions. In particular, we find that a positive mass of particles in three dimensions avoids recollapse of the spatial geometry. Finally, we construct similar solutions for the Einstein-dust system and describe to what extent the construction fails for massless matter models.
... (1) Claim 1: "General Relativity tells us unambiguously that a closed universe whose energy density is dominated by matter like starts and galaxies, and even more exotic dark matter, must one day recollapse in a process like the reverse of a Big Bang -a Big Crunch" (Page 27, [Kra12]). Comment: Indeed, due to the closed universe recollapse conjectures as formulated in [BT85], [BGT86], [Bar88], [LW89], and [LW90], there are possible scenarios for a closed universe to potentially recollapse. However, as shown in [CH10], there are a general class of closed cosmological models of Bianchi type IX (of which a k = +1 FLRW universe is a special case), that do not exhibit recollapse, but continually expand. ...
We study some claims in Krauss' recent book, \emph{A Universe from Nothing:
Why there is something rather than nothing}, that are employed to show that a
universe can come from "nothing". In this brief paper, we show that many of the
claims are not supported in full by modern general relativity theory or quantum
field theory in curved spacetime.
... The problem of recollapse is one of the central themes of this paper, and we will revisit it in the next section, when deciding how to normalize our dynamical variables to allow for the possibility of our model expanding or contracting. Barrow and Tipler [12] first showed that the existence of a maximal hypersurface is a necessary and sufficient condition for the existence of a final singularity in a universe with a compact Cauchy surface. They further showed that a cosmological model with topology S 3 can admit such maximal hypersurfaces. ...
We use a dynamical systems approach based on the method of orthonormal frames
to study the dynamics of a non-tilted Bianchi Type IX cosmological model with a
bulk and shear viscous fluid source. We begin by completing a detailed
fix-point analysis which give the local sinks, sources and saddles of the
dynamical system. We then analyze the global dynamics by finding the
-and -limit sets which give an idea of the past and future
asymptotic behavior of the system. The fixed points were found to be a flat
Friedmann-LeMa\^{i}tre-Robertson-Walker (FLRW) solution, Bianchi Type II
solution, Kasner circle, Jacobs disc, Bianchi Type solutions, and
several closed FLRW solutions in addition to the Einstein static universe
solution. Each equilibrium point was described in both its expanding and
contracting epochs. We conclude the paper with some numerical experiments that
shed light on the global dynamics of the system along with its heteroclinic
orbits. With respect to past asymptotic states, we were able to conclude that
the Jacobs disc in the expanding epoch was a source of the system along with
the flat FLRW solution in a contracting epoch. With respect to future
asymptotic states, we were able to show that the flat FLRW solution in an
expanding epoch along with the Jacobs disc in the contracting epoch were sinks
of the system. We were also able to demonstrate a new result with respect to
the Einstein static universe. Namely, we gave certain conditions on the
parameter space such that the Einstein static universe has an associated stable
subspace. We were however, not able to conclusively say anything about whether
a closed FLRW model could be a past or future asymptotic state of the model.
We investigate the effects of large inhomogeneities in both the inflaton field and its momentum. We find that in general, large kinetic perturbations reduce the number of e-folds of inflation. In particular, we observe that inflationary models with sub-Planckian characteristic scales are not robust even to kinetic energy densities that are sub-dominant to the potential energy density, unless the initial field configuration is sufficiently far from the minimum. This strengthens the results of our previous work. In inflationary models with super-Planckian characteristic scales, despite a reduction in the number of e-folds, inflation is robust even when the potential energy density is initially sub-dominant. For the cases we study, the robustness of inflation strongly depends on whether the inflaton field is driven into the reheating phase by the inhomogeneous scalar dynamics.
Extending our previous work on the robustness of inflation to perturbations in the scalar field, we investigate the effects of perturbations in the transverse traceless part of the extrinsic curvature on the evolution of an inhomogeneous inflaton field. Focusing on small field models, we show that these additional metric inhomogeneities initially reduce the total number of e-folds as the amplitude increases, but that the reduction saturates and even reverses above a certain amplitude. We present an argument that this is due to the presence of a large initial Hubble friction when metric perturbations are large.
It is proven that any spherically symmetric spacetime that possesses a compact Cauchy surface and that satisfies the dominant-energy and non-negative-pressures conditions must have a finite lifetime in the sense that all timelike curves in such a spacetime must have a length no greater than , where m is the mass associated with the spheres of symmetry. This result gives a complete resolution, in the spherically symmetric case, of one version of the closed-universe recollapse conjecture (though it is likely that a slightly better bound can be established). This bound has the desirable properties of being computable from the (spherically symmetric) initial data for the spacetime and having a very simple form. In fact, its form is the same as was established, using a different method, for the spherically symmetric massless scalar field spacetimes, thereby proving a conjecture offered in that work. Prospects for generalizing these results beyond the spherically symmetric case are discussed.
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.
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.
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.
Setting aside anthropic arguments, there is no reason why the Universe should initially favor a net expanding phase rather than one experiencing a net contraction. However, a collapsing universe containing “normal” matter will end at a singularity in a finite time. We point out that there is a mechanism, derived from nonperturbative effects in quantum field theory in a finite volume, which may provide a bias toward expansion when the spacetime volume shrinks, by dynamically violating the null energy condition, without the need for modified gravity or exotic matter. We describe a scalar field component subjected to this nonperturbative effect in a cosmological background and consider its impact on a contracting phase. We discuss how this could dynamically generate the necessary initial conditions for inflation to get started, or form part of the mechanism for a nonsingular cosmological bounce.
Cosmic Inflation (Guth, Phys Rev D 23:347–356, 1981, [1], Linde, Phys Lett B, 108:389–393, 1982, [2], Albrecht and Steinhardt, Phys Rev Lett 48:1220–1223, 1982, [3], Starobinsky, Phys Lett B 91:99–102, 1980, [4]) is thought to provide a solution to several problems in standard Big Bang theory by dynamically driving a “generic” initial state to a flat, homogeneous and isotropic Universe, while generating a nearly scale-invariant power spectrum of primordial perturbations which is consistent with observations. The question of what constitutes a “generic” initial state is a difficult one, and can only be understood in the context of a quantum theory of gravity. However, regardless of the nature of quantum gravity, a random realisation from the set of all possible initial conditions will not look like an inflationary spacetime, at least initially (Hollands and Wald, Gen Relativ Gravit 34:2043–2055, 2002, [5]), and one should expect the initial conditions from which inflation begins to contain some measure of inhomogeneity.
Extending our previous work on the robustness of inflation to perturbations in the scalar field, we investigate the effects of perturbations in the transverse traceless part of the extrinsic curvature on the evolution of an inhomogeneous inflaton field. Focusing on small field models, we show that these additional metric inhomogeneities initially reduce the total number of e-folds as the amplitude increases, but that the reduction saturates and even reverses above a certain amplitude. We present an argument that this is due to the presence of a large initial Hubble friction when metric perturbations are large.
The number of pages allocated to the commission report has been very limited and certainly not sufficient to cover in any exhaustive manner the wide range of topics relevant to cosmology and to provide also extensive bibliographies. Because of the vast amount of material to be covered, the report is based on a number of contributions from different colleagues who have been asked to highlight the main trends in the triennium (mid 1984 - mid 1987), together with a list of references sufficiently comprehensive to serve as a guideline for further reading. Unfortunately, two of the expected contributions did not reach me in time for inclusion in the report, and consequently topics such as the large scale structure and streaming motions, the clusters of galaxies and the counts of extragalactic radio sources are not included. However, it is my understanding that a large portion, if not all, of these topics will be covered in the reports of Commissions 28 and 40, and if true, this will at least avoid unnecessary overlaps. It should also be mentioned here that several proceedings of very recent IAU conferences provide excellent, updated and exhaustive reviews of the research work relevant to cosmology.
We prove that a homogeneous and isotropic universe containing a scalar field
with a power-law potential, , with and always
develops a finite-time singularity at which the Hubble rate and its first
derivative are finite, but its second derivative diverges. These are the first
examples of cosmological models with realistic matter sources that possess weak
singularities of 'sudden' type. We also show that a large class of models with
even weaker singularities exist for non-integer . More precisely, if
where k is a positive integer then the first divergence of the
Hubble rate occurs with its (k+2)th derivative. At early times these models
behave like standard large-field inflation models but they encounter a singular
end-state when inflation ends. We term this singular inflation.
Physical theories have their most fundamental expression as action
integrals. This suggests that the total action of the universe is the
most fundamental physical quantity, and hence finite. In this article it
is argued that finite universal action implies that the universe is
spatially closed. Further, the possible spatial topologies, the types of
matter that can dominate the early universe dynamics, and the form of
any quadratic additions to the lagrangian of general relativity are
constrained. Initial and final cosmological curvature singularities are
required to avoid a universal action singularity.
We survey recent theoretical ideas regarding the possible value of the present total density of the Universe and show how physically realistic cosmological models containing small deviations from isotropy allow observations of intrinsically general relativistic effects on the microwave background to determine whether or not the Universe is open or closed no matter how close the total density lies to the critical value. Applications of these results to inhomogeneous cosmological models, cosmic vorticity, and observational studies of the microwave-background-radiation isotropy are also discussed.
The aim of this paper is to clarify confusing notions of the word ‘‘rotation’’ as applied to cosmological solutions of metric theories of gravity, both in general and in the specific case addressed by the article in which these confusing notions have recently reappeared.
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.
Observations suggest the universe is dominated by under-dense voids, with matter distributed on boundary shells. The dynamics
of such structure is examined here using non-perturbative ‘cell-lattice’ models. These consist of spherical cells, bounded
by matter surface layers, arranged in three-dimensional lattices. The regular tessellations of spherical, flat, or hyperbolic
3-space define these lattices, giving six closed, one spatially flat, and four open models. The general-relativistic junction
conditions at the cell boundaries determine the models' evolution. Explicit expansion laws are calculated for models with
empty voids and dust shells. These have the same general behaviour as the expansion laws for corresponding Friedmann–Robertson–Walker
spacetimes, and approach those laws at late times, i.e. large radii, the regime appropriate to describing the post-recombination
universe. But differences between cellular and homogeneous spacetimes appear in the early-time behaviour of these models:
the empty-void, dust-shell models expand initially as if radiation dominated, and the effective density ratio Ω takes on values
less than unity for closed and flat as well as open models. Equations of motion for models with non-empty voids are also derived;
these indicate evolution toward empty-void configurations.
Some properties of cyclic closed universes are investigated under the
assumption that the total entropy of the universe increases from cycle
to cycle. Exact solutions are given for the increase in maximum size of
a cyclic universe filled with matter and radiation. Asymptotically, a
cyclic closed universe approaches `flatness'. If a positive cosmological
constant exists then these oscillations will eventually cease and be
replaced by an era of expansion which will continue unless the
cosmological constant is associated with a form of vacuum energy that
ultimately decays away. If that occurs, then universal oscillations will
be resumed. Oscillations will also be ended by the indefinite presence
of any non-zero stress that violates the strong energy condition. A
stable stress of this sort will always dictate the ultimate evolution of
a cyclic closed universe. We also consider what occurs when the Hawking
entropy associated with the cosmological constant is included in the
entropy budget, and describe situations in which the sequence of
universal oscillations can decrease in amplitude. Eventually, they will
become too small for the cosmological constant to affect the dynamics,
and then small oscillations will approach this marginally stable state.
We examine some particular closed anisotropic universes, and investigate
how the evolution of the anisotropy is affected by the universal
oscillations. In the most general known homogeneous closed universes it
appears that the asymptotic behaviour is for the cycles to increase in
size, as entropy increases, for any dust-dominated era to occupy an
increasingly longer period of the total evolution, and hence for the
anisotropy to decrease in influence at late times with increasing cycle
number when the cosmological constant and other stresses violating the
strong energy condition are absent.
We consider surfaces of prescribed mean curvature in a Lorentzian manifold and show the existence of a foliation by surfaces of constant mean curvature.
Maximal surfaces and their implications for the ambient spacetime are studied. Our methods exploit the interplay between contact of the volume functional and energy conditions. Essentially, we find that in closed universes, maximal surfaces are unique; they maximize volume; and they yield future and past singularities.
The inflationary scenario requires that the universe have negligible curvature along constant-density surfaces. In the Friedmann-Lemaitre cosmology that leaves us with two free parameters, Hubble's constant Hâ and the density parameter ..cap omega..â (or, equivalently, the cosmological constant ..lambda..). I discuss here tests of this set of models from local and high-redshift observations. The data agree reasonably well with ..cap omega..âapprox.0.2.
In this paper, a number of theorems are proven which collectively show that singularities will occur in spacetime under weaker energy conditions than the strong energy condition. In particular, the Penrose theorem, which uses only the weak energy condition but which applies only to open universes, is extended to all closed universes which have a Cauchy surface whose universal covering manifold is not a three-sphere. Furthermore, it is shown that the strong energy condition in the Hawking-Penrose theorem can be replaced by the weak energy condition and the assumption that the strong energy condition holds only on the average. In addition, it is demonstrated that if the Universe is closed, then the existence of singularities follows from the averaged strong energy condition alone. It is argued that any globally hyperbolic spacetime which satisfies the weak energy condition and which contains a black hole must be null geodesically incomplete.
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.
An account of recent advances in understanding of the origin of the
universe is presented. The subjects discussed include: quantum cosmology
and the early universe, GUTs and supersymmetric GUTs in the very early
universe, monopoles and strings in the very early universe, cosmic
strings, phase transitions in the very early universe, natural
inflation, perturbations of a de Sitter universe, cosmic baldness,
bubble collisions in general relativity, Euclidean approach to the
inflationary universe. Also considered are: origin of density
fluctuations in the 'new inflationary universe', generation of
primordial perturbations in the early universe, the effective potential
in de Sitter space, quantum fluctuations and phase transitions in
cosmology, symmetry behavior at finite temperature in dynamic
spacetimes, cosmological constraints on GUTs, monopoles and axions,
massive neutrinos and photinos in cosmology and galactic astronomy,
superheavy particles in cosmology and evolution of inhomogeneities in
the early universe, instability of de Sitter space, and the cosmological
potential of supergravity.
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.
Contents: 1. Early cosmologies. 2. Modern cosmology. 3. The big bang
singularity. 4. Quantum theory to the rescue. 5. The problem of initial
conditions.
It is well known that many universes with negative cosmological constant
contain singularities. This result is generalized by proving that all
closed universes with negative cosmological constant are both future and
past timelike geodesically incomplete if the strong energy condition
holds. No global causality conditions or restrictions on the initial
data are used in the proof. Furthermore, it is shown that all open
universes with a Cauchy surface and a negative cosmological constant are
singular if the strong energy condition holds.
We have measured redshifts for a new sample of 320 galaxies complete to . The two-point correlation function has been estimated using both redshift and positional information. This has been used
to estimate the mean-square relative peculiar velocities for galaxy pairs with projected separations in the range 0 ≲ σ ≲ 4 h−1 Mpc (h is Hubble's constant in units of 100 km s−1 Mpc−1). We find line-of-sight velocity differences of = 250 ± 50 km s−1 independent of projected separation. The two-point correlation function is well approximated by the power-law model with r0 = (4.1 ± 0.3) h−1 Mpc. In addition, we estimate the amplitude of the three-point correlation function. We find Q = 0.60 ± 0.06, where Q is the ratio of the three-point correlation function to the square of the two-point function. These results are used in the
cosmic virial theorem to obtain an estimate of the contribution to the mean density from material that is clustered like galaxies.
The result is Ω = 0.14 × 2±1. In order for our results to be compatible with a high density Universe (Ω = 1), most of the dark material must be in a component
that is much less strongly clustered than galaxies on scales of 4 h−1 Mpc or less.
BLACK holes are now the subject matter of at least half the papers in general relativity. These papers rest on a foundation of sand, for black holes have been successfully defined only in asymptotically flat spacetimes, and no one really believes that the Universe is asymptotically flat. I shall provide a firm basis for black hole physics in the present paper; I shall extend the concept of ‘black hole’ to arbitrary stably causal spacetimes by essentially defining a black hole to be that object which contains all the ‘small’ trapped surfaces. As for most astrophysical applications black hole surfaces are located approximately by the outermost trapped surface boundary, this new definition allows results which depend only on the local behaviour of black holes in asymptotically flat spacetimes to be extended (approximately) to closed universes. Results which depend on the global behaviour of black holes cannot in general be extended to closed universes. For example, it is shown that the black hole area theorem—the statement that black holes never decrease their cross-sectional area—cannot be extended to closed universes. The new concept of black hole yields a purely geometrical definition of time direction in closed universes. My notation will be that of ref. 1.
It is shown that the process of quantum creation of the universe in a wide class of elementary-particle theories with a large
probability leads to creation of an exponentially expanding (inflationary) universe, which after expansion acquires the sizel≳1028 cm.
Cosmologies having spacelike 3-dimensional homogeneity surfaces on which the group of motions is SO(3, R) (Bianchi Type IX) are investigated. These models generalize the ordinary “closed” Robertson-Walker models. Detailed analytical and numerical treatment is given for those models containing pressureless fluid (dust). Included are proofs that infinite density regions and a time of maximum expansion occur in each model. Extensive discussion is devoted to the evolution of anisotropy and the possibility of “mixing,” a phenomenon characterized by the absence of a horizon in any direction. Numerical examples, illustrating the various types of models, are displayed graphically.
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.
In the course of a redshift survey of galaxies brighter than Rroughly-equal16.3, 133 redshifts were measured in three fields, each separated by roughly 35/sup 0/ from the other two. If the galaxies in these fields were distributed uniformly, the combination of a galaxian luminosity function and our magnitude limits predicts that the distribution of redshifts should peak near 15,000 km s/sup -1/. In fact, only one galaxy of the 133 was observed with a redshift in the 6000 km s/sup -1/ interval centered on 15,000 km s/sup -1/. One plausible interpretation is that a large volume in this region of order 10/sup 6/ Mpc/sup 3/ is nearly devoid of galaxies.
Redshifts for a total sample of 238 galaxies to a limiting magnitude of 15.0 are used to study the three-dimensional distribution of galaxies in the region of the sky ranging from approximately 11.5 h to 13.3 h right ascension between declinations of about 19 and 32 deg, within which lie the two rich clusters Coma and A1367. The results obtained demonstrate that the two clusters are embedded in a common supercluster of very large extent. It is found that there are three observationally distinct populations of galaxies within this supercluster, including galaxies located in the two rich cluster cores, those located in intermediate- or low-mass clusters, and a nearly homogeneously distributed population of isolated galaxies; in addition, a population of foreground galaxies located in low-mass clusters rather than being distributed in a homogeneous 'field' is also identified. The redshift of formation for the foreground groups is estimated to be no more than about 9, and the morphology of the galaxies is examined. It is suggested that every nearby very rich cluster is located in a supercluster and that all clusters of richness class greater than or equal to 2 will eventually be found to lie in superclusters.
We study three-dimensional Riemannian manifolds with nonnegative scalar curvature. We find new topological obstruction for such manifolds. Our method turns out to be useful in studying the positive mass conjecture in general relativity.