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Asymptotic isotropization in inhomogeneous cosmology

In this paper we investigate asymptotic isotropization. We derive the asymptotic dynamics of spatially inhomogeneous cosmological models with a perfect fluid matter source and a positive cosmological constant near the de Sitter equilibrium state at late times, and near the flat FL equilibrium state at early times. Our results show that there exists an open set of solutions approaching the de Sitter state at late times, consistent with the cosmic no-hair conjecture. On the other hand, solutions that approach the flat FL state at early times are special and admit a so-called isotropic initial singularity. For both classes of models the asymptotic expansion of the line element contains an arbitrary spatial metric at leading order, indicating asymptotic spatial inhomogeneity. We show, however, that in the asymptotic regimes this spatial inhomogeneity is significant only at super-horizon scales.


Introduction
In a recent paper we presented a general framework for analyzing the dynamics of generic spatially inhomogeneous cosmological models, referred to briefly as G 0 cosmologies (see Ref. [32], which we will henceforth refer to as Paper I). We employed Hubble-normalized scale-invariant variables which were defined within the orthonormal frame formalism, and led to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. In this paper we show that our framework can be used to derive asymptotic expansions for spatially inhomogeneous cosmologies that undergo asymptotic isotropization at late times or at the initial singularity.
It is part of the folklore of relativistic cosmology that all ever-expanding cosmological models with a positive cosmological constant asymptotically approach the de Sitter solution. A statement to this effect can be found in Gibbons and Hawking [10], p. 2739, in connection with a program to extend to cosmological event horizons certain ideas concerning thermodynamics and particle creation that had been applied to black hole event horizons. This conjecture, the so-called cosmic no-hair conjecture, was also made by Hawking and Moss [15], p. 36, after the introduction of the paradigm of cosmic inflation. Shortly thereafter, Starobinskiǐ [30] took the first steps in describing the asymptotic structure of the metric for spatially inhomogeneous cosmologies with a perfect fluid matter source and an effective cosmological constant that approach the de Sitter solution at late times, showing that the line element contained 8 (the maximum possible number) free functions of the spatial coordinates. He commented that the asymptotic solution described "exponentially rapid local isotropization of the universe", and coined the phrase "the cosmological constant is the best 'isotropizer'". In the same year, Wald [33] gave a proof of the conjecture for ever-expanding spatially homogeneous (SH) cosmologies. Subsequently Jensen and Stein-Schabes [17] extended Wald's proof to spatially inhomogeneous cosmologies, subject to the restriction that the 3-Ricci curvature scalar is negative. The significance of this result is not clear, however, for the following reason. The complete set of SH cosmologies can be divided into one invariant subset for which the 3-Ricci curvature scalar is never positive (Bianchi Type-I to Type-VIII) for which Wald's theorem applies, and another invariant subset containing Bianchi Type-IX and Kantowski-Sachs models, for which the dynamical behavior of the 3-Ricci curvature scalar is more complicated (some models approach the de Sitter solution while others re-collapse). Such a natural division into invariant subsets does not exist in the G 0 case. Hence, in G 0 cosmology the situation is still unclear as regards possible "cosmic no-hair theorems". A variant of such a theorem, however, was proved very recently by Tchapnda and Rendall [31] for cosmologies with collisionless matter and positive cosmological constant that are either plane or hyperbolic-plane symmetrical.
The motivation for much of the above work was provided by the idea of cosmic inflation, in that the de Sitter solution is regarded as describing the Universe in the late stages of an inflationary epoch. On the other hand, the de Sitter solution can also be regarded as a simple prototype of a cosmological model undergoing accelerated expansion during the present epoch. Indeed, recent observations have focused attention on such models. In particular, five years ago two independently working observational groups reported that the analysis of the type Ia supernova redshift data they had gathered suggested that in the present epoch the Universe may be in a state of accelerated expansion (see, e.g., Perlmutter et al [23], Schmidt et al [28], and Schmidt [27]). The acceleration could be due to either a veritable, repulsive cosmological constant, or some form of "dark energy"; see Carroll [6] and Ellis [9] for further details and references.
In this paper, motivated by the above discussion, we consider G 0 cosmologies that are future asymptotic to the de Sitter solution. We give a formal definition of this notion within the Hubble-normalized state space and then derive asymptotic expansions, valid as t → ∞, for the Hubble-normalized variables.
The second type of asymptotic isotropization, namely isotropization at the initial singularity, arises in connection with the notion of quiescent cosmology (cf., e.g., Barrow [2]), which provides an alternative to cosmic inflation. The idea is that, due to entropy considerations on a cosmological scale, a suitable initial condition for the Universe is that the Weyl curvature should be zero (or at least dynamically unimportant) at the initial singularity; this is the Weyl curvature hypothesis by Penrose [22], p. 630. This hypothesis leads to the notion of an isotropic initial singularity. Cosmological initial singularities of this type first arose as a special case in the general analysis of initial singularities performed by Lifshitz and Khalatnikov (LK hereafter) [19], p. 203. This type of initial singularity was also encountered in the work of Eardley, Liang and Sachs [8], p. 101. Subsequently, Goode and Wainwright [13] gave a formal definition of an isotropic initial singularity, using a conformally related metric, and derived various properties. Motivated by these ideas, we consider G 0 cosmologies that are past asymptotic to the flat FL solution and give a formal definition of this notion within the Hubble-normalized state space and then derive asymptotic expansions, valid as t → −∞, for the Hubble-normalized variables.
In this paper, we also extend the formalism of Paper I to cover geodesic null congruences and structures in connection with the issue of the formation of particle or event horizons (cf. the classic paper by Rindler [26]), thus naturally generalizing the work by Nilsson et al [21] from an SH context to a general spatially inhomogeneous setting.
The plan of this paper is as follows. In Sec. 2, we present the Hubble-normalized evolution equations and constraints for G 0 cosmologies that arise from Einstein's field equations and the matter equations. In Sec. 3, we give the detailed asymptotic form of the Hubble-normalized variables for G 0 cosmologies that approach the de Sitter solution at late times, and discuss various features of this class of cosmological models. In Sec. 4, we present analogous results for G 0 cosmologies that approach the flat FL solution at early times. We conclude in Sec. 5 with a discussion of the analogies between the two classes of cosmological models that undergo asymptotic isotropization, and raise some issues for future study. The proofs of the results are given in the appendix.

Evolution equations and constraints
We consider spatially inhomogeneous cosmological models with a positive cosmological constant, Λ, and a perfect fluid matter source with a linear barotropic equation of state. We thus havẽ whereμ is the total energy density of the fluid (assumed to be non-negative) andp its isotropic pressure, in the rest 3-spaces associated with the fluid 4-velocity vector fieldũ, while γ is a constant parameter. The range is of particular physical interest, since it ensures that the perfect fluid satisfies the dominant and strong energy conditions and the causality requirement that the speed of sound should be less than that of light. The values γ = 1 and γ = 4 3 correspond to pressure-free matter ("dust") and incoherent radiation, respectively.
We express the fluid 4-velocity vector field bỹ where e 0 is a vorticity-free timelike reference congruence, v the peculiar velocity vector field of the fluid relative to the rest 3-spaces of e 0 , and defines the usual Lorentz factor. To obtain an orthonormal frame, { e a } a=0,1,2,3 , we supplement the vector field e 0 with an orthonormal spatial frame { e α } α=1,2,3 in the rest 3-spaces of e 0 . The frame metric is then given by η ab = diag [ − 1, 1, 1, 1 ]. To convert the dynamical equations of the orthonormal frame formalism to a system of partial differential equations, it is necessary to introduce a set of local

Hubble-normalized Einstein-Euler equations in separable volume gauge
In Paper I, we employed the orthonormal frame formalism and introduced Hubble-normalized scaleinvariant frame, connection and matter variables, as follows. 1 Expressing the Hubble-normalized frame vector fields with respect to a local coordinate basis leads to: where N and N i are viewed as coordinate gauge source functions, while the E α i are dependent variables that we refer to as the frame variables.
The introduction of Hubble-normalized scale-invariant variables leaves H as the only dependent variable carrying a physical dimension, namely [ length ] −1 . The evolution equation for H decouples and is given by where q is the deceleration parameter, familiar from observational cosmology. The expression for q, which depends on the choice of temporal gauge, is given below. It is also necessary to introduce the spatial Hubble gradient r α , defined by Throughout this paper, we will employ the separable volume gauge introduced in Paper I, which is characterized by the following choice of coordinate and frame gauge source functions: the choice N = 1 corresponds to setting the dimensional lapse function equal to the momentary Hubble radius, N = H −1 . With this choice of temporal gauge the deceleration parameter is given by where Σ 2 = 1 6 (Σ αβ Σ αβ ), and the scalars G ± are defined by (see Eq. (6) in Paper I) It should be noted that the temporal gauge choice (11) does not determine a unique family of spacelike 3-surfaces S:{t = constant} and associated normal timelike reference congruence e 0 . In Sec. 4 [ see Eq. (125) ], we will use this remaining freedom to simplify the asymptotic expansions that we derive. There is also freedom in the choice of the spatial frame vectors and spatial coordinates. For the applications in this paper it is desirable to choose a Fermi-propagated spatial frame, i.e., one which satisfies The remaining spatial gauge freedom is a time-independent rotation of the spatial frame vectors, There is also the following freedom in the choice of spatial coordinates: In the separable volume gauge with Fermi-propagated spatial frame, the Hubble-normalized state vector for G 0 cosmologies is given by The evolution equations and constraints that the components of X have to satisfy are given below. 2 Evolution equations: where q, G ± and ∂ ∂ ∂ α are given by Eqs. (12), (13) and (8). Constraints: Auxiliary 3-curvature variables: where S αβ and Ω k are the tracefree part and trace part of the Hubble-normalized 3-Ricci curvature of a spacelike 3-surface S:{t = constant}, respectively, while C αβ is the Hubble-normalized 3-Cotton-York tensor which provides information on the conformal curvature properties of a spacelike 3-surface S:{t = constant}. The 3-Ricci curvature variables S αβ and Ω k satisfy the Hubble-normalized twice-contracted 3-Bianchi identity, given by

Energy-normalized null geodesics augmentation
A few years ago, Nilsson et al [21] extended the Hubble-normalized equations for SH cosmology by adding the equations for an energy-normalized geodesic (timelike or null) congruence. This strategy turned out to be very useful, both from an analytical and a numerical point of view. We therefore generalize that approach to the present G 0 cosmology setting, and this will subsequently allow us to investigate the connection between asymptotic silence of the gravitational field dynamics in relativistic cosmology and the formation of particle or event horizons (see Paper I, Sec. 4.1). Instead of being interested in the null geodesics emanating from a single given event, we will be interested in all possible null geodesics. We will thus not consider the problem of integrating a system of ordinary differential equations (associated with the former situation), but will instead consider all possible null geodesic vector fields (which, at a later stage, can be integrated to yield all possible associated flows). We therefore start by considering a vector field k a which is tangent to a geodesic null congruence. This vector field is thus governed by the equations We now make a (3+1)-split of Eq. (35) with respect to a general orthonormal frame. Using the notation of Paper I and Ref. [21], this yields while a (3+1)-split of Eq. (36) yields − E 2 + (k α k α ) = 0 .
Here E := k 0 denotes the energy (frequency) of the particles that are traveling along the geodesic null congruence.
We now employ the Hubble-normalized gravitational field variables given in Eqs. (5) and (6), and in addition introduce energy-normalized null vector variables as in Ref. [21] according to With Eq. (39) we thus find where the K α correspond to the direction cosines of the null geodesics. Energy-normalization implies that the dimensional equation (37) for E decouples and can be written as where s is given by which is obtained from Eqs. (37), (40), (41) and (6). It is also necessary to introduce the spatial energy gradient t α , defined by and governed by The latter equations are obtained by choosing f = E in the commutator equations (31) and (32) Note that the contraction of Eq. (47) with K α vanishes identically on account of Eqs. (41) and (43). Let us now consider these equations in the separable volume gauge, specified by Eqs. (11). Then where Once the energy-normalized null vector variables K α have been obtained, one can solve for the associated geodesic null congruence by integrating the relation subject to appropriate initial conditions. In this paper, we will use the above equations to determine asymptotic geodesic null structures and derive properties concerning particle and event horizons which are associated with asymptotic silence.

de Sitter-like future asymptotics
In this section we give the detailed asymptotic form of the Hubble-normalized variables for G 0 cosmologies that approach the de Sitter solution at late times.

G 0 cosmologies future asymptotic to the de Sitter solution
The de Sitter solution of Einstein's field equations describes a cosmological model with zero matter energy density and positive cosmological constant, which is undergoing exponential expansion. Employing a (3+1)-decomposition with intrinsically flat spacelike 3-surfaces, the line element and the associated timelike reference congruence are given by where Λ > 0 is the cosmological constant, and T is clock time. It is convenient to choose the unit of [ length ] as follows: The Hubble scalar is constant and positive, and is given by It follows from Eq. (9) that the deceleration parameter is constant and negative, being given by We also find it convenient to use a conformal time coordinate η, defined by The line element is then cast into the form Relative to the natural orthonormal frame associated with the line element (52), the Hubble-normalized variables are where the dynamical time coordinate t is introduced via or, equivalently, We note that the volume density V is given by so that the separable volume gauge conditions are satisfied.
Observe that the frame variables satisfy lim t→∞ E α i = 0. Thus, the de Sitter solution is described by an orbit in the Hubble-normalized state space that is future asymptotic to an equilibrium point on the silent boundary, E α i = 0. Motivated by Eqs. (59)-(61), we say that a G 0 cosmology is future asymptotic to the de Sitter solution if the following limits are satisfied: We note that for the de Sitter solution the peculiar velocity variables v α are unrestricted, since the Codacci constraint (28) (which may be viewed as determining the v α algebraically provided that Ω = 0) yields no information. We will show that if the limits (65)-(67) are satisfied, together with certain technical restrictions, then the evolution equations determine the asymptotic form of the v α as t → ∞. Wald's theorem [33], mentioned in the introduction, asserts that any SH cosmology with a positive cosmological constant, except those of Kantowski-Sachs and Bianchi Type-IX which may re-collapse, is future asymptotic to the de Sitter solution, in the above sense. As discussed in the introduction, at present it is not known whether Wald's theorem can be generalized to G 0 cosmologies. We now consider the class of G 0 cosmologies which are future asymptotic to the de Sitter solution, and present the asymptotic form of the Hubble-normalized variables as t → ∞.

Asymptotic expansions
In App. B, we prove that the de Sitter solution is asymptotically stable, and derive the asymptotic form of the Hubble-normalized variables for G 0 cosmologies that are future asymptotic to the de Sitter solution, in the sense of Eqs. (65)-(67). It is necessary to impose certain restrictions on the spatial derivatives; these are given in App. B. We now give the asymptotic expansions, using the convention that "hatted" coefficients depend only on the spatial coordinates: The coefficientsÂ α andN αβ are determined byÊ α i according tô where the matrixÊ α i is the inverse ofÊ α i : The coefficientsΩ k andŜ αβ in Eqs. (69)-(71) are the leading order coefficients in the asymptotic expansions of the 3-Ricci curvature variables, They are determined byÊ α i ,Â α andN αβ according to: 3 The constraint (C C ) α at order e −4t provides the restriction forv α : where The constraints (C Λ ) α and (C G ) have been used in App. B to give the coefficients in Eqs. (69) and (71), respectively. 4

Features of asymptotic to de Sitter G 0 cosmologies
In this section we investigate certain features of asymptotic to de Sitter G 0 cosmologies, as described by the asymptotic expansions (68)-(74). Our first goal is to show that these expansions represent a general class of perfect fluid G 0 cosmologies. We accomplish this by showing that the expansions (68)-(74) contain 8 freely specifiable functions of the spatial coordinates x i , the same number that appear in the initial data for a general cosmological solution of Einstein's field equations with a perfect fluid matter source (see LK, p. 188).

Essential arbitrary functions
The nine coefficientsÊ α i in the expansions (68)-(74) can be chosen as suitably differentiable arbitrary functions of the x i , which then successively determineÂ α ,N αβ ,Ω k andŜ αβ via Eqs. (75), (76), (80) and (81). The five shear rate coefficientsΣ αβ and the coefficientΩ in the density parameter can also be chosen arbitrarily, while the peculiar velocity coefficientsv α are determined algebraically by the Codacci constraint (82), employing Eq. (83). As mentioned in Sec. 2, the separable volume gauge (11) leaves a freedom in the choice of the 1-parameter family of spacelike 3-surfaces S:{t = constant}. A coordinate transformation of the form 5 whereĝ ij = ℓ −2 0 δ αβÊ α iÊ β j , preserves the separable volume gauge to leading order. This freedom can be used to fixΩ, although we do not choose to do so in general. Three of theÊ α i can be eliminated by the frame rotations (15), and three by the spatial coordinate transformations (16), leaving three essentially arbitrary functions in theÊ α i . The asymptotic expansions (68)-(74) thus contain 8 essentially arbitrary functions of the spatial coordinates. That is, they represent a general class of perfect fluid G 0 cosmologies that are future asymptotic to the de Sitter solution.

Metric expansion
We now derive an asymptotic expansion for the spacetime metric in order to relate our results to the work of other researchers, in particular that of Starobinskiǐ [30]. We find that the metric expansion sheds light on the nature of large-scale spatial inhomogeneity in G 0 cosmologies that are future asymptotic to the de Sitter solution.
In the separable volume gauge, the line element has the form where [ see Paper I, Eq. (163) ] Equations (68)-(74) are substituted into Eq. (12) to give an improved expansion for q, and then into Eq. (18) to give an expansion for ∂ t E α i , which can be integrated. In the case γ < 4 3 , we obtain The components E α i of the corresponding Hubble-normalized 1-forms form the inverse of the matrix E α i , and are given by withÊ β i andÊ β i related by Eqs. (77). It also follows from Eq. (9) that the Hubble scalar H has the expansion We then substitute Eqs. (88) and (89) into Eq. (86) to obtain the following expansion for the 3-metric g ij : Observe that the free functionsÊ α i determine the coefficientsĝ ij in the leading order term in the expansion through Eqs. (77) and (91), and also determine the second termŜ ij , while the free functionsΣ ij enter at a higher order. Note that up to order O(e −4t ), the matter coefficientΩ only enters the spacetime metric (85) through H in Eq. (89). It can be shown that the expansions (89) and (90) are also valid for γ satisfying 4 3 ≤ γ < 2, with the difference that the matter coefficient enters into the O(e −4t ) term in Eq. (89).
Our results provide a confirmation of the ansatz for the metric expansion given by Starobinskiǐ [30], Eq. (2). Some of the details differ, however, due to the fact that Starobinskiǐ employs the synchronous gauge. We have established consistency with his results by performing a coordinate transformation of the formt We have also derived Starobinskiǐ's metric expansion directly in the synchronous gauge, using our integration method. The most striking feature of the expansion (90) is the fact that the leading order coefficient in the expansion is an arbitrary 3-metric,ĝ ij , whereas the spacelike 3-surfaces S:{t = constant} in the de Sitter line element (52) are intrinsically flat. The generality of the leading order coefficientĝ ij in the metric expansion raises an apparent paradox -a G 0 cosmology that is future asymptotic to the spatially homogeneous and isotropic de Sitter solution in the sense of the definition in Subsec. 3.1 can exhibit substantial spatial inhomogeneity. We will show that this paradox is resolved by the existence of event horizons.

Event horizons
In cosmology an event horizon for a fundamental observer can be thought of as constituted by the observer's past light cone at t = ∞ (see Hawking and Ellis [14], p. 129). In order to establish the existence of event horizons, we need to determine the asymptotic form of geodesic null congruences. These are governed by Eq. (51), with the K α determined by Eqs. (48)-(50).
In App. D we show that any null geodesic has the asymptotic form for constants x i ∞ andK α . For given x i ∞ , the family of null geodesics then form an event horizon for the fundamental observer whose worldline is x i = x i ∞ . The spatial distance from the observer to her/his event horizon is given by where y i = y i (s) (with 0 ≤ s ≤ 1) describes a spacelike geodesic from x i ∞ to x i (t), for fixed t. In the asymptotic regime, the spacelike geodesic is basically a straight line, and is approximated by the null geodesic (93) projected onto a spacelike 3-surface S:{t = constant}. It follows from Eq. (93) that where ℓ 0 is given in Eq. (54). One can always introduce local coordinates at x i = x i ∞ such that (see e.g., Schutz [29], p. 156) For points within the event horizon the approximationĝ ij (x m ) ≈ ℓ 2 0 δ ij becomes increasingly accurate as t → ∞, due to Eq. (93). In other words, within the event horizon of a particular fundamental observer, the spacetime metric asymptotically approaches the de Sitter metric. 6 Sinceĝ ij is general, however, the above approximation cannot be done simultaneously at all points, reflecting the spatial inhomogeneity of the G 0 cosmology at super-horizon scales. We emphasize that the spatial inhomogeneity does not diminish as t → ∞ -it is the observer who sees successively smaller portions of that spatial inhomogeneity.

Radiation bifurcation
A noteworthy feature of the asymptotic expansion (68)-(74) is that for the physically important case of incoherent radiation (γ = 4 3 ), a bifurcation occurs that affects the peculiar velocity v of the perfect fluid relative to the spacelike 3-surfaces S:{t = constant}.
If 1 ≤ γ < 4 3 , the components v α tend to zero, and as a result the fluid 4-velocity vector fieldũ asymptotically coincides with the vorticity-free timelike reference congruence e 0 . For incoherent radiation, the v α do not tend to zero in general, and v 2 can approach any value between 0 and 1. If 4 3 < γ < 2, v 2 tends to the value 1 as t → ∞, which means that the speed of the fluid relative to the fundamental observers comoving with e 0 approaches the speed of light.
It is of considerable interest to investigate the limit behavior of the fluid kinematical variables, as perceived by the fundamental observers. Using the boost transformation laws provided in App. F, we find that the future asymptotic limits for the Hubble-normalized fluid kinematical scalars (defined by Eqs. This extends previous results given by Goliath and Ellis [11] for some special SH cases with cosmological constant to the general G 0 case (see also Goliath and Nilsson [12] and Raychaudhuri and Modak [24]). The context of spatial inhomogeneity stresses the isotropization issue. We have to ask ourselves isotropization along which timelike congruence? In the present context there is no isotropization along the timelike reference congruence e 0 when γ > 4 3 , but this does not say anything about whether the models isotropize along the fluid 4-velocity vector field in this case, which, perhaps, is the physically best motivated congruence to consider. However, to determine whether this is the case or not takes us outside the scope of the present paper, and so we leave this issue for future studies. We note that when γ < 4 3 , the fluid 4-velocity vector field asymptotically coincides with the timelike reference congruence and, hence, one does have isotropization in this case.

G 0 cosmologies past asymptotic to the flat FL solution
The flat FL solution of Einstein's field equations describes a cosmological model with perfect fluid matter source, whose density parameter has the constant value Ω = 1. The line element and the fluid 4-velocity vector field are given by employing clock time T as one of the local coordinates. The matter energy density and pressure are The Hubble scalar is while the deceleration parameter is given by the latter is thus constant and positive. We find it convenient to use a conformal time coordinate η, defined by with the unit of [ length ] ℓ 0 and the constant T 0 chosen to satisfy η 0 = 1, i.e., The line element is then cast into the form and the Hubble scalar is We introduce the dynamical time coordinate t via or, equivalently, Relative to the natural orthonormal frame associated with the line element (100), the Hubble-normalized variables are [ using Eq. (106) ] We note that the volume density V is given by so that the separable volume gauge conditions are satisfied.
Observe that the frame variables satisfy lim t→−∞ E α i = 0. Thus, the flat FL solution is described by an orbit in the Hubble-normalized state space that is past asymptotic to an equilibrium point on the silent boundary, E α i = 0. Motivated by Eqs. (111)-(113), we say that a G 0 cosmology is past asymptotic to the flat FL solution if the following limits are satisfied: We note that for the flat FL solution the peculiar velocity variables v α are zero. We will show that if the limits (115)-(117) are satisfied, together with certain technical restrictions, then the constraints imply that the v α tend to zero as t → −∞. We now consider the class of G 0 cosmologies which are past asymptotic to the flat FL solution, and present the asymptotic form of the Hubble-normalized variables as t → −∞.

Asymptotic expansions
In App. C, we derive the asymptotic form of the Hubble-normalized variables for G 0 cosmologies that are past asymptotic to the flat FL solution, in the sense of Eqs. (115)-(117). It is necessary, as in Subsec. 3.2, to impose certain restrictions on the spatial derivatives; these are given in App. C. We now give the asymptotic expansions. For brevity we use η instead of e whereĤ is the leading order coefficient in the asymptotic expansion of H: In the derivation in App. C, we used the freedom to re-define the 1-parameter family of spacelike 3-surfaces S:{t = constant} while preserving the separable volume gauge to set so thatĤ = constant > 0. In addition, Eq. (10) and the constraints (C G ) and (C C ) α were used in App. C to give the coefficients in Eqs.

Features of asymptotic to flat FL G 0 cosmologies
In this section we investigate certain features of the class of G 0 cosmologies that are past asymptotic to the flat FL solution. One of our goals is to show that this class of cosmologies admit an isotropic initial singularity in the sense of Goode and Wainwright [13] (see also Newman [20] and Anguige and Tod [1]).

Essential arbitrary functions
Of the position-dependent coefficients in Eqs. (118)-(123), only the nineÊ α i can be chosen arbitrarily, as suitably differentiable functions of the x i . However, three of them can be eliminated by the frame rotations (15), and three by the spatial coordinate transformations (16). Thus, the asymptotic expansion contains 3 essentially arbitrary functions of the spatial coordinates, as compared with 8 for a general class of perfect fluid G 0 cosmologies. These models thus form a set of measure zero in the Hubble-normalized state space.

Metric expansion
We now derive an asymptotic expansion for the spacetime metric in order to relate our results to the work of other researchers, in particular that of LK and Goode and Wainwright [13]. We find that the metric expansion sheds light on the nature of large-scale spatial inhomogeneity in G 0 cosmologies that are past asymptotic to the flat FL solution.
In the separable volume gauge, the line element has the form given in Eq. (85). Equation (18) gives an expansion for ∂ t E α i , which can be integrated to give Following the steps taken in Subsec. 3.3, we obtain the metric expansion Our result provides a rigorous confirmation of the ansatz for the metric expansion made by LK, Eq. (4.1), for the case of incoherent radiation. Some of the details differ, however, due to the fact that LK employ the synchronous gauge. We have established consistency with their results by performing a coordinate transformation of the form given in Eqs. (92). We have also derived LK's metric expansion directly in the synchronous gauge, using our integration method.

Isotropic initial singularities
We briefly digress to introduce the notion of an isotropic initial singularity; see Ref. [13]. A cosmological model 8 is said to admit an isotropic initial singularity if the physical spacetime metric g is conformal to an unphysical metricg: where χ = χ(η) is a differentiable function of a time coordinate η and satisfies χ(0) = 0, while the conformal metricg is regular on the spacelike 3-surfaceS:{η = 0}. The conformal factor χ(η) is also required to satisfy It follows from Eqs. (85), (124) and (129) that the spacetime metric for the class of G 0 cosmologies that are past asymptotic to the flat FL solution does satisfy Eqs. (132) and (133) with where η is the conformal time coordinate.

Particle horizons
The generality of the leading order coefficientĝ ij in the metric expansion (129), which should be compared with Eq. (90), raises an apparent paradox -a G 0 cosmology that is past asymptotic to the spatially homogeneous and isotropic flat FL solution in the sense of the definition in Subsec. 4.1 can exhibit substantial spatial inhomogeneity. We will show that this paradox is resolved by the existence of particle horizons.
In order to establish the existence of particle horizons, we need to determine the asymptotic form of geodesic null congruences. These are governed by Eq. (51), with the K α determined by Eqs. (48)-(50).
In App. E we show that any past-oriented null geodesic has the following asymptotic form as η → 0: for constants x i BB andK α . The constants x i BB give the point of termination of the null geodesic at the singularity η = 0. Thus the past null cone at time η for the fundamental observer whose worldline is x i = x i 0 will intersect the singularity η = 0, thereby defining a particle horizon, which is formed by the fundamental worldlines given by x i = x i BB . The spatial distance from the observer to her/his particle horizon at time η is given by where y i = y i (s) (with 0 ≤ s ≤ 1) describes a spacelike geodesic from x i BB to x i (η), for fixed η. As in the derivation of Eq. (95), it follows from Eq. (135) that where ℓ 0 is given in Eq. (106). One can introduce local coordinates at x i = x i BB such that For points within the particle horizon the approximationĝ ij (x m ) ≈ ℓ 2 0 δ ij becomes increasingly accurate as η → 0, due to Eq. (135). In other words, within the particle horizon of a particular fundamental observer, the spacetime metric is close to the flat FL metric. Sinceĝ ij is general, however, the above approximation cannot be done simultaneously at all points, reflecting the spatial inhomogeneity of the G 0 cosmology at super-horizon scales. We emphasize that the spatial inhomogeneity does not diminish -it is the observer who sees successively smaller portions of that spatial inhomogeneity.

Concluding remarks
In this paper, we have given a unified analysis, within the framework of the Hubble-normalized state space, of G 0 cosmologies that undergo asymptotic isotropization, either at late times (asymptotic to the de Sitter solution, with ℓ → ∞) or near the initial singularity (asymptotic to the flat FL solution, with ℓ → 0).
The analysis reveals a number of common features as regards the asymptotic behavior in the two classes, as well as a number of significant differences, which emerge most clearly if we write the asymptotic expansions in Subsec 3.2 in terms of the conformal time coordinate of the de Sitter line element, given by Eq. (57). We note that the length scale factor is given by for the de Sitter solution, and by for the flat FL solution. Thus, for both asymptotes η → 0 + in the asymptotic regimes under consideration. Firstly, the shear rate Σ αβ and anisotropic 3-Ricci curvature S αβ have the same leading asymptotic dependence on η as η → 0 + : where δ ≥ 0, and C = − 3 in the de Sitter case while C = − 6/(3γ + 2) in the flat FL case. Secondly, there is a common asymptotic form for the line element, namely as η → 0 + , where ℓ(η) is the length scale factor for the de Sitter solution or for the flat FL solution. Likewise, the Hubble scalar has the common form where δ ≥ 0, and H asymp (η) is the Hubble scalar for the de Sitter solution or for the flat FL solution, respectively [ see Eqs. (55), (89), (108), (124) and (125) ]. HereΩ k is the leading order coefficient in the expansion for the 3-Ricci curvature scalar Ω k , which has a common form for the two classes, namely, The key feature of the asymptotic line element (143) is the presence of the arbitrary 3-metricĝ ij (x k ). We have shown that the spatial inhomogeneity that it generates is significant only at super-horizon scales (the event horizon for G 0 cosmologies that are future asymptotic to the de Sitter solution and the particle horizon for G 0 cosmologies that are past asymptotic to the flat FL solution). The common asymptotic features of the two classes can be understood to some extent within the context of the Hubble-normalized state space. For each class, the asymptotic state is described by an equilibrium point on the silent boundary, [ see Paper I, Eq. (75) ], which is associated with an isotropic and conformally flat solution of Einstein's field equations. Use of a conformal time coordinate η, which tends to zero on the silent boundary for both classes, leads to common asymptotic decay rates for its various Hubble-normalized quantities. The asymptotically silent dynamics (E α i → 0) is also responsible for the existence of an event horizon and a particle horizon, as shown in Subsecs. 3.2 and 4.2.
The differences between the two classes originate in the asymptotic behavior of the deceleration parameter q, which is negative in the de Sitter case (q → −1) and positive in the flat FL case (q → 1 2 (3γ − 2)). This difference affects the higher-order terms in the asymptotic expansion of the shear rate Σ αβ . In the de Sitter case, there is a termΣ αβ η 3 , where theΣ αβ are freely specifiable functions, while in the flat FL case, this term does not appear. 9 The absence of theΣ αβ -term, of course, accounts for the difference in the number of freely specifiable functions in the two cases. This difference is also reflected in the asymptotic expansions for the Hubble-normalized electric and magnetic parts of the Weyl curvature [ see Paper I, Eqs. (152) and (153) ]. For the electric Weyl curvature we have in the de Sitter case and in the flat FL case, For the magnetic Weyl curvature we have in both cases whereĈ αβ is given by the "hatted" version of Eq. (33), with r α set to zero. The difference between the de Sitter and the flat FL cases as regards the number of freely specifiable functions can be interpreted in a dynamical systems context. We have already noted that both the de Sitter solution (52) and the flat FL solution (100) determine equilibrium points on the silent boundary in the Hubble-normalized state space. It follows that the orbits of G 0 cosmologies that are future asymptotic to the de Sitter solution (respectively, past asymptotic to the flat FL solution) are asymptotic to these equilibrium points. The asymptotic expansion contains 8 arbitrary functions (the maximum possible number) in the de Sitter case, which confirms that the de Sitter equilibrium point is asymptotically stable, as proved in App. B (see Stage 0). On the other hand, the presence of only 3 arbitrary functions in the flat FL case implies that the flat FL equilibrium point has both a stable and an unstable manifold (into the past), and our asymptotic solutions describe the stable manifold.
A major accomplishment of the present paper is to provide derivations of the detailed asymptotic properties of G 0 cosmologies that undergo asymptotic isotropization, instead of simply making ad hoc assumptions about the form of the metric, as has been done previously. Our derivations do, however, depend in a significant way on the assumption that the spatial partial derivatives of the Hubble-normalized variables remain bounded in the asymptotic regimes. It is this assumption that enables us to rely exclusively on analytic methods for systems of ODE.
It would certainly be desirable to attempt to use methods from the theory of PDE to prove the validity of the assumption of bounded spatial derivatives for cosmological models that undergo asymptotic isotropization. Recent experience with G 2 cosmologies, which we now describe, gives some cause for optimism. Firstly, it is known that in G 2 cosmologies spatial partial derivatives can diverge on approach to the initial singularity, through the creation of so-called Gowdy spikes (see Paper I, Subsec. 4.1, and Ref. [25]). This spatial structure is created, however, by the local instability of the Kasner solutions. Since these solutions do not play a rôle in the present context (the approach to an isotropic initial singularity), we do not expect spatial structure of this nature to develop. Secondly, at late times, support for our assumption is provided by numerical simulations 10 of G 2 cosmologies, which do not show the development of large spatial derivatives.
A second limitation of our analysis is that it is local in nature in that we restrict our considerations to some open subset of the integral curves of the timelike reference congruence e 0 . This restriction is inevitable, however, since in a G 0 cosmology all timelines do not necessarily share the same asymptotic evolution. For example, a G 0 cosmology with positive cosmological constant may approach the de Sitter solution along some timelines but may undergo collapse to a future singularity along others.
Another matter that requires comment is the choice of temporal gauge. In the present paper we have chosen to work with the separable volume gauge, defined by Eqs. (11), because we have found it to be a convenient computational gauge. On the other hand, we have been able to transform our asymptotic expansions to the synchronous gauge, and find that the leading order terms, e.g., Eqs. (141)-(148), are unchanged. This calculation thus provides evidence for the gauge robustness of our results.
In conclusion, it would be of interest to relate our results to perturbation analyses of the de Sitter solution (see for example Barrow [3] and Bruni et al [5]) and of the flat FL solution. We have not been able to clarify the relationship to our satisfaction, particularly in the case of the de Sitter solution, and so we leave this matter for future investigation. In addition, other possible applications of the Hubblenormalized state space framework for G 0 cosmologies presented here that suggest themselves naturally are establishing firm links with mainstream issues in observational cosmology such as the cosmic background radiation, peculiar motions of galaxies, or gravitational lensing.
A Asymptotic results for ordinary differential equations Proposition 1 where M αβ , A αβ and B αβ are Cartesian tensors. 11 If and then for any given ǫ > 0, where m = min(k, l).

Proof: Introduce
It follows from Eqs. (149) that Now Eq. (150) implies that for all ǫ > 0 there exists a t 0 such that for all t ≥ t 0 , and Eq. (151) implies there exists a C > 0 such that It then follows that Multiplying by e (k−ǫ)t and integrating from t = t 0 to t, yields from which the result follows.

Proposition 2
Given where M αβ , A αβ and B αβ are Cartesian tensors. If and M αβ is bounded, then 11 We will also need a vector version of this result, i.e., the DE is of the form Proof: As in the proof of Proposition 1, we obtain Eqs. (154) and Eq. (155) with k replaced by −k, and Eq. (156). It follows from these equations that Multiply by e −(k−ǫ)t and integrate from t to ∞, knowing that M is bounded, to obtain and the result follows.

Proposition 3
Given

Stage 0
We first establish asymptotic stability of the de Sitter solution. The evolution equations for the variables in A1 are of the form where X is the vector containing the variables in A1, A is a constant diagonal real-valued matrix with negative eigenvalues, while f (t, X) contains products of components in X and ∂ ∂ ∂ α X. Assumptions A1 and A2 imply that f (t, X) is continuous for all t ≥ 0 and f (t, X) = o( X ) as X → 0. Then Theorem 1.1 in Coddington and Levinson [7], p. 314, implies that X = 0 is asymptotically stable. Thus, we obtain the property We organize the derivation of asymptotic expansions into two stages, in which we repeatedly use the evolution equations and the constraints (C G ) and (C Λ ) α . In Stage 1, we obtain order estimates for all variables, primarily using Propositions 1 and 2 in App. A. In Stage 2, we use the results from Stage 1 in conjunction with Proposition 3 to obtain the explicit asymptotic expansions. The assumptions A2 and A3 are needed throughout in order to bound the spatial derivative terms in the evolution equations and in the constraints (C G ) and (C Λ ) α .

Stage 1
The deceleration parameter q, given by Eq. (12), plays a dominant rôle in the evolution equations. Its limiting value as t → ∞ follows from P1 and A2: We begin by considering E α i . Using P1 and Eq. (168), the evolution equation (18) for E α i assumes the form of Eqs. (149) and (150) with k = 1 and B αβ = 0. It follows from Proposition 1 that for any given as t → ∞.
We can now use Eqs. (168), (169) and P1, A2 to conclude that the evolution equation (20) for A α assumes the form of Eqs. (149) and (150) with k = 1 and l = 1 − ǫ E . It follows from Proposition 1 that for any given ǫ A > ǫ E , as t → ∞. At this stage we have two different epsilons in Eqs. (169) and (170), with ǫ A > ǫ E . Equation (169) is equally valid, however, if we replace ǫ E by ǫ A , and in the interests of simplicity we choose to do this, dropping the subscripts on the epsilons. We will also make this ǫ-simplification in subsequent steps.
By using A2 and the constraint (C Λ ) α , we can now conclude that It remains to derive the decay rates of Ω and v α . We now make use of the evolution equation for the scalar v = √ v α v α , which can be derived from Eq. (24). The equation for v 2 is in fact given in Paper I; see Eq. (146). Using the preceding asymptotic expressions and A2, the equation for v assumes the form where g = O(e (−1+ǫ)t ). Equation (180) is in fact an asymptotically autonomous DE, a scalar version of the vector DE discussed by Horwood et al in Ref. [16]. We can thus use Theorem B.1 in Ref. [16], p. 15, to obtain the limiting behavior of v as t → ∞. The domain D for v is the interval 0 < v < 1, and the conditions H 1 and H 2 in Ref. [16] are satisfied. It is straightforward to verify that the solutions of the autonomous DE with initial conditions in D, satisfy P1 and A2 imply that for any given δ > 0, 15 Then Eq. (184) assumes the form of Eqs. (149) and (150) with k = 2 (3γ−4) (2−γ) and B αβ = 0. Proposition 1 implies For γ = 4 3 , Eq. (24) gives Using Eq. (182), the inequality (173) can be strengthened to read where the ǫ has been redefined. If γ = 4 3 , Eq. (173) simplifies directly to ∂ t Ω ≤ (−4 + ǫ) Ω .

Stage 2
In the second stage we use the asymptotic decay rates found in Stage 1 to successively write the evolution equations in a form to which Proposition 3 can be applied. We begin by considering E α i . The decay rates in Stage 1 and Eq. (12) gives The evolution equation (18) for E α i can now be written in the form where g α i = O(e (−3+ǫ)t ). This equation, in conjunction with Proposition 3, implies that In a similar way the evolution equations (20) and (22) lead to We now consider the evolution equation (23) for Ω. For 1 ≤ γ < 4 3 , we obtain where g = O(e (3γ−8+ǫ)t ), with Ωv 2 being the dominant term in g. For 4 3 < γ < 2, we obtain where g = O(e (−5+ǫ)t + e (−4−2 (3γ−4) (2−γ) +ǫ)t ), with ∂ ∂ ∂ α Ω, Ω ∂ ∂ ∂ α v α , (A α v α ) Ω and Ω (1 − v 2 ) being the dominant terms in g. For γ = 4 3 , we obtain Applying Proposition 3 yields At this stage, the evolution equation for Σ αβ assumes the form where g αβ = O(e (−4+ǫ)t ). By imitating the proof of Proposition 3, we conclude that At this stage (C G ) and A3 give while (C Λ ) α and A3 lead to We now consider the evolution equation (24) for v α . For γ = 1, we obtain where g α = O(e −t + e 3(3γ−4+ǫ)t ), with ∂ ∂ ∂ α ln Ω and v 2 v α being the dominant terms in g α . Proposition 3 For γ = 4 3 , we already have the result: For 4 3 < γ < 2, we obtain from Eq. (184) Using the results obtained in Stage 2, we can repeat Stage 2 to eliminate the epsilons in the O-terms as follows. Equation (192) now reads q = − 1 + O(e −2t ), and the function g α i in Eq. (193) satisfies g α i = O(e −3t ); similarly for the other variables. The position-dependent coefficientsÊ α i ,Â α andN αβ are required to satisfy the constraints We organize the derivation into two stages, in which we repeatedly use the evolution equations, (10) and the constraints (C G ) and (C C ) α . In Stage 1, we obtain order estimates for all variables, primarily using Propositions 1 and 2 in App. A. In Stage 2, we use the results in Stage 1 in conjunction with Proposition 3 to obtain the explicit asymptotic expansions. The assumptions A2 and A3 are needed throughout in order to bound the spatial derivative terms in the evolution equations, in (10), and in the constraints (C G ) and (C C ) α .

Stage 2
In the second stage we use the asymptotic decay rates found in Stage 1 to successively write the evolution equations in a form to which Proposition 3 can be applied. We begin by considering E α i . The decay rates in Stage 1 and Eq. (12) give q = 1 2 (3γ − 2) + O(e [−(3γ−2)+ǫ]τ ) .
The evolution equation (18) for E α i can now be written in the form 16 Here and in the future, we will omit the qualifier τ → ∞.
Using the results obtained in Stage 2, we can repeat Stage 2 to eliminate the epsilons in the O-terms, as in App. B.

D Asymptotic expansions for event horizons
Equation (44) and expansion (68) imply that t α = O(e −t ) as t → ∞, subject to the assumption that E satisfies A2. Equations (43) and (42) successively imply that With these results, the evolution equation (48) for K α yields ∂ t K α = O(e −t ), which can be integrated to give K α =K α + O(e −t ) .

F Fluid kinematical variables via boost transformations
Letũ denote a fluid 4-velocity vector field. Then fluid kinematical variables arise in the irreducible decomposition of the spacetime gradient ofũ, i.e., ∇ aũb = −ũ aub +Hh ab +σ ab +ω ab , defining the fluid accelerationu a , the fluid Hubble scalarH, the fluid shear rateσ ab , and the fluid vorticitỹ ω ab , respectively. With respect to an Eulerian reference frame, comoving with a unit timelike congruence u, we havẽ where h a b := δ a b + u a u b , and the Lorentz factor is defined by Γ := 1/ √ 1 − v 2 , with v 2 := v a v a . We define the peculiar kinematical variableŝ In the case when u is vorticity-free [ i.e., ω a (u) ≡ 0 ], we find the boost transformations: Fluid acceleration: Fluid Hubble scalar :H