Pre-big bang geometric extensions of inflationary cosmologies

Transcription

1 Pre-big bang geometric extensions of inflationary cosmologies David Klein and Jake Reschke 2 Robertson-Walker spacetimes within a large class are geometrically extended to larger cosmologies that include spacetime points with zero and negative cosmological times. In the extended cosmologies, the big bang is lightlike, and though singular, it inherits some geometric structure from the original spacetime. Spacelike geodesics are continuous across the cosmological time zero submanifold which is parameterized by the radius of Fermi space slices, i.e, by the proper distances along spacelike geodesics from a comoving observer to the big bang. The continuous extension of the metric, and the continuously differentiable extension of the leading Fermi metric coefficient g ττ of the observer, restrict the geometry of spacetime points with pre-big bang cosmological time coordinates. In our extensions the big bang is two dimensional in a certain sense, consistent with some findings in quantum gravity. KEY WORDS: Robertson-Walker cosmology, chart, inflation, event horizon, pre-big bang maximal Fermi coordinate Mathematics Subject Classification: 83F5, 83C Department of Mathematics and Interdisciplinary Research Institute for the Sciences, California State University, Northridge, Northridge, CA david.klein@csun.edu. 2 Department of Mathematics, University of California, Davis, Davis, CA jreschke@math.ucdavis.edu

2 Introduction The big bang singularity in general relativistic cosmologies can be considered from a variety of perspectives. In quantum gravity theories the singularity can be eliminated, with the big bang preceded by a big crunch or arising through other scenarios [, 2, 3, 4, 5]. Some investigations suggest that dimensional reduction may be a fundamental feature of quantum gravity with the effective dimension of spacetime points at sufficiently small scales decreasing to d = 2 [6, 7]. The geometry near the singularity and possible pre-big bang scenarios have also received attention from a classical perspective. Penrose and other researchers have approached this through conformal geometric methods [8, 9,,, 2], and dynamics near and through the big bang have also been investigated [3]. In this paper, using the framework of general relativity, we show that Robertson- Walker spacetimes with big bang singularities can be extended to larger cosmologies that include points which, in a natural way, may be assigned negative or zero cosmological times. Our approach begins with the observation that cosmological time along spacelike geodesics orthogonal to the worldline of a comoving observer decreases monotonically, and the geodesics terminate in finite proper distance at the big bang (c.f. [4, 5]). One should therefore be able to construct larger cosmologies by extending these geodesics further, while at the same time preserving some continuity and differentiability properties of the metric tensor. In the language of coordinates, Fermi charts of comoving observers are extended beyond their maximal domains in standard big bang cosmologies in such a way as to preserve certain properties of the metric, thus giving some geometric structure to the big bang singularity and pre-big bang spacetime points. The basic idea is illustrated with the prototype example of the Milne Universe in two spacetime dimensions. The line element in curvature coordinates is, ds 2 = dt 2 + a 2 (t)dχ 2, () where in this case the scale factor = t. The formulas τ = t cosh χ and ρ = t sinh χ, with τ > ρ, transform (t, χ) to Fermi coordinates (τ, ρ) of the comoving observer at χ = ρ = [4], and the metric in (τ, ρ) coordinates becomes, ds 2 = g ττ dτ 2 + dρ 2 (2) dτ 2 + dρ 2, (3) i.e., the Milne Universe is diffeomorphic to the interior of the forward lightcone in Minkowski spacetime. Eq.(2) gives the form of the metric in Fermi coordinates for a general class of scale factors so that in general ρ is proper distance along spacelike geodesics at fixed proper time τ (see [4, 5]). 2

3 In the original curvature coordinates, the metric of Eq.() is degenerate at the big bang, t =, a coordinate singularity, but this is not the case for Eq.(3), nor as we will prove for Eq.(2), provided τ >, for the case of more general Robertson-Walker cosmologies which have coordinate-independent curvature singularities at t =. In the (τ, ρ) Fermi coordinates for the Milne Universe, spacelike geodesics orthogonal to the path of the comoving observer are horizontal straight lines within the forward light cone of Minkowski spacetime as depicted in Fig.. The dotted parts of the horizontal line in Fig are extensions of the proper distance coordinate, ρ, beyond the lightcone boundary of the Milne Universe at cosmological time t =. t Τ t t t spacelike geodesic t light cone Ρ Figure : The Milne Universe in Fermi coordinates (τ, ρ) is the interior of the forward light cone in Minkowski space. The comoving observer s worldline is the vertical line ρ =. The dashed portion of the horizontal line extends the spacelike geodesic beyond the Milne Universe to include points with negative cosmological times t. Cosmological time t which for notational purposes we shall designate as t is defined implicitly as a function of τ and ρ through a natural extension of the inverse Fermi coordinate transformation t = t (τ, ρ) (for the general case see Eq. (49) below). In this way, t t < on the dotted portion of the spacelike geodesic in Fig. In this paper, we carry out a similar construction for a class of Robertson-Walker cosmologies consistent with astronomical observations. In addition to some regularity conditions, we require the scale factor to be either inflationary near the big bang, or that > ȧ() >, and in four dimensions that the spacetime is spatially flat. The cosmological time zero submanifold (defined by t = ) 3

4 is lightlike in our extension, and parameterized by the (finite) Fermi radius of the orginal universe ρ Mτ (see Eq.(4)). In general the extension of the metric is not twice continuously differentiable, but continuity is retained along with existence and continuity of the partial derivatives of the nonvanishing leading metric coefficient g ττ (and g ρρ is constant). Our construction is purely geometric and coordinate independent, but Fermi coordinate charts play a useful role because that coordinate system is geometrically constructed. To define Fermi coordinates, consider a foliation of some neighborhood U (which might be the entire spacetime) of a comoving observer s worldline, β(t), by disjoint Fermi spaceslices {M τ }. To define M τ, let ϕ τ : M R by, ϕ τ (p) = g(exp β(τ) p, β(τ)), (4) where the overdot represents differentiation with respect to proper time τ along β, g is the metric tensor, and the exponential map, exp p (v) denotes the evaluation at affine parameter of the geodesic starting at point p M, with initial derivative v. Now define, M τ ϕ τ (). (5) In other words, the Fermi spaceslice M τ of all τ-simultaneous points consists of all the spacelike geodesics orthogonal to the path of the comoving observer β at fixed proper time τ. Fermi coordinates are associated to the foliation {M τ } in a natural way. Each spacetime point on M τ is assigned time coordinate τ, and the spatial coordinates are defined relative to a parallel transported orthonormal reference frame. Specifically, a Fermi coordinate system along β is determined by an orthonormal frame of vector fields, e (τ), e (τ), e 2 (τ), e 3 (τ) parallel along β, where e (τ) is the four-velocity of the Fermi observer, i.e., the unit tangent vector of β(τ). Fermi coordinates x, x, x 2, x 3 relative to this tetrad are defined by, ( ) x exp β(τ) (λ j e j (τ)) = τ ( ) (6) x k exp β(τ) (λ j e j (τ)) = λ k, where Latin indices run over, 2, 3 (and Greek indices run over,, 2, 3). Fermi coordinates may be constructed in a sufficiently small open neighborhood of any timelike geodesic in any spacetime. The metric tensor expressed in these coordinates is Minkowskian to first order near the geodesic of the Fermi observer, with second order corrections involving only the curvature tensor [6]. General formulas in the form of Taylor expansions for coordinate transformations to and from more general Fermi-Walker coordinates are given in [7] and 4

5 exact transformation formulas for a class of spacetimes are given in [8, 9]. Applications include the study of relative velocities, tidal dynamics, gravitational waves, statistical mechanics, and the influence of curved space-time on quantum mechanical phenomena [2, 2, 22, 23, 25, 26, 27, 28, 29, 3, 3, 32, 33, 34]. It was proved in [4] that the maximal Fermi chart (x α, U Fermi ) for β(t) in a non inflationary 3 Robertson-Walker space-time (M, g), with increasing scale factor, is global, i.e., U Fermi = M. If, on the other hand, (M, g) includes inflationary periods, there may exist a cosmological event horizon for the comoving observer, i.e., χ horiz (t ) dt <, (7) t for some for some t > (and hence for all t > ). 4 For Robertson-Walker spacetimes with a big bang singularity and a cosmological event horizon, it was proved in [5] under a regularity assumption that the maximal Fermi chart U Fermi consists of all spacetime points within (but not including) the cosmological event horizon so that the maximal Fermi chart is the causal past of the comoving observer at future infinity. For cosmologies with no event horizon, it was shown, for both inflationary and non inflationary models, that the Fermi coordinate chart is global. It was also shown in [5] that all spacelike geodesics with initial point on the worldline β of a comoving observer and orthogonal to β, terminate at the big bang in a finite proper distance ρ Mτ, the radius of M τ. In this sense, as already noted by Page [35] using Rindler s observations [36], the big bang is simultaneous with all spacetime events. We show in this paper how the spacelike geodesics and the metric tensor can be extended to points in a larger spacetime manifold M with zero or negative cosmological times, analogous to the extensions of the Milne Universe depicted in Fig. For the general case, the extended spacetime 5 M can be expressed as a disjoint union, M = M + M M, (8) where the superscripts indicate respectively that the continuous function t (τ, ρ) restricted to the set is positive, zero, or negative. Here M + = M denotes the original Robertson-Walker universe. The geometry of the spacetime M must 3 A Robertson-Walker space-time is non inflationary if ä(t) for all t. 4 χ horiz (t ) is the χ-coordinate at time t of the cosmological event horizon, beyond which the co-moving observer at χ = cannot receive a light signal at any future proper time. 5 Here and below, extended spacetime should be understood to mean extended degenerate spacetime, in the sense that the spacetime manifold is extended to a larger manifold, but the Lorentzian metric in four spacetime dimensions collapses to a two dimensional Lorentzian metric on the big bang submanifold, to be identified in the sequel. 5

6 be largely undetermined, except for restrictions on the spacetime points close to the big bang, because of our requirement that g ττ be continuously differentiable across M, that g ρρ and the remaining metric coefficients in four spacetime dimensions be continuous. The spacetime M can be understood as a smooth manifold with mild singularities of the Lorentzian metric g on M representing the big bang. From our extension M inherits some geometric structure from the original Robertson-Walker spacetime M. In two spacetime dimensions, the submanifold M defined by t (τ, ρ) = is parameterized in two connected components by (τ, ρ Mτ ), and (τ, ρ Mτ ), for τ >, and we show that M is lightlike. In four spacetime dimensions, our extension results in a dimensional reduction of the cotangent bundle at cosmological time zero (t = ) similar to those described in [6, 7]. This paper is organized as follows. In Section 2, we provide a summary of results for maximal Fermi charts on Robertson-Walker cosmologies needed in the sequel. Section 3 reviews relationships and provides a new result on particle and cosmological horizons. This enables us to avoid mutually exclusive conditions on the scale factors we consider. Section 4 gives results on the limiting values of the metric coefficient g ττ as cosmological time goes to zero, and shows that nonvanishing continuous extensions of g ττ are possible. Section 5 is the most technical part of the paper. Here we prove continuity of the partial derivatives of g ττ at the boundary of its domain, the big bang. Section 6 and Section 7 carry out the extensions of the cosmology M to the larger spacetime M in two and four spacetime dimensions respectively. In Section 8 we give examples of cosmologies and extensions. Section 9 summarizes results and offers concluding remarks. Section is the appendix and contains the proofs of the lemmas and theorems in Section 5. 2 Maximal Fermi Charts This section summarizes results from [4, 5] needed in the sequel. The Robertson- Walker metric on space-time M = M k is given by the line element, ds 2 = dt 2 + a 2 (t) [ dχ 2 + S 2 k(χ)dω 2], (9) where dω 2 = dθ 2 + sin 2 θ dϕ 2, is the scale factor, and, sin χ if k = S k (χ) = χ if k = sinh χ if k =. () 6

7 The coordinate t > is cosmological time and χ, θ, ϕ are dimensionless. Here θ and φ lie in intervals I π and I 2π of lengths π and 2π respectively. The values +,, of the parameter k distinguish the three possible maximally symmetric space slices for constant values of t with positive, zero, and negative curvatures respectively. The radial coordinate χ takes all positive values for k = or, but is bounded above by π for k = +. We assume henceforth that k = or so that the range of χ is unrestricted. The techniques needed for the case k = + are the same, but require the additional restriction that χ < π so that spacelike geodesics do not intersect. We note that k = + for the Einstein static universe, for which Fermi coordinates for geodesic observers are global (except for the antipode, χ = π) [9]. We assume throughout this paper that the scale factor is regular, i.e., it satisfies the following definition [5]. Definition. Define the scale factor : [, ) [, ) to be regular if: (a) a() =, i.e., the associated cosmological model includes a big bang. (b) is increasing and continuous on [, ) and twice continuously differentiable on (, ), with inverse function b(t) on [, ). (c) For all t >, ä(t) ȧ(t) 2. () If in addition the expression in Eq. () is bounded below by a constant K, we call the scale factor strongly regular. Example. It is easily verified that power law scale factors of the form = t α are strongly regular for all α >. Scale factors of this form include radiation and matter dominated universes as well as inflationary universes for the cases α >. Similarly the inflationary scale factor = sinh t is easily seen to be strongly regular. A more elaborate example of a strongly regular scale factor is given by Eq.(8) and is discussed below in Section 8. Remark. Under the assumption that ȧ(t) > for all t, and is regular, it follows that for any τ >, there exists t (, τ) such that the inequality Eq() is strict at t, and hence by continuity, on an open interval containing t. This follows from the observation that equality in Eq.() forces to be an exponential function which violates Definition a. Let β be the path of the comoving observer with fixed coordinate χ =. As a preliminary step to express the metric of Eq.(9) in Fermi coordinates of β, we 7

8 define coordinate transformation functions. For τ > t >, define [5, 37] χ t (τ) = a(τ) dt. (2) a2 (τ) a 2 (t) t The function χ t (τ) is the value of the χ-coordinate of the spacetime point with t-coordinate t on the spacelike geodesic orthogonal to β with initial point β(τ). The proper distance ρ along that geodesic from β(τ) to the point with t-coordinate t is given by, ρ = t dt. (3) a2 (τ) a 2 (t) The proper distance along the geodesic increases as cosmological time t decreases monotonically to zero. The radius, ρ Mτ, of the Fermi spaceslice M τ (see Eq.(5)) is the proper distance along the spacelike geodesic orthogonal to the comoving observer β(τ), from β(τ) to the big bang at t =. It is given by, ρ Mτ = It is easy to show, [5], that for a regular scale factor, dt. (4) a2 (τ) a 2 (t) ρ Mτ π 2 H(τ), (5) where H(τ) = ȧ(τ)/a(τ) is the Hubble parameter. Moreover, if is strongly regular, then dρ Mτ (τ) = ȧ(τ) ( ä(t) ) dt dτ a(τ) ȧ 2 >, (6) (t) a2 (τ) a 2 (t) so that ρ Mτ is an increasing function of τ (i.e. the Fermi radius of the universe increases with τ). 6 Denote the Fermi coordinates for the comoving observer β(τ) = (τ,,, ) by {τ, x = x, y = x 2, z = x 3 } according to Eq. (6). Under the assumption that is regular, the maximal Fermi chart U Fermi M is given by, U Fermi = {(τ, x, y, z) : τ > and } x 2 + y 2 + z 2 < ρ Mτ, (7) and the metric in Fermi coordinates is given by, ds 2 = g ττ (τ, ρ) dτ 2 + dx 2 + dy 2 + dz 2 +λ k (τ, ρ) [ (y 2 + z 2 )dx 2 + (x 2 + z 2 )dy 2 + (x 2 + y 2 )dz 2 xy(dxdy + dydx) xz(dxdz + dzdx) yz(dydz + dzdy) ], (8) 6 A correction to the published theorem giving this result is posted on arxive, see [5]. 8

9 where ρ = x 2 + y 2 + z 2, g ττ (τ, ρ) = ȧ(τ) 2 [ a 2 (τ) a 2 (t ) ] [ ȧ(t ) a 2 (τ) a 2 (t ) t 2 ä(t) dt ȧ(t) 2, a2 (τ) a (t)] 2 (9) and, λ k (τ, ρ) = a2 (t )S 2 k (χ t (τ)) ρ 2 ρ 4, (2) for ρ > and λ k (τ, ) =. Here, t = t (τ, ρ) is defined implicitly by Eq.(3), 7 and S k is given by Eq.(). It may be shown [4], that λ k (τ, ρ) is a smooth function of τ and ρ 2. Applying a standard transformation from Cartesian to spherical coordinates in R 3 to the Fermi space coordinates results in the diagonal metric for Fermi polar coordinates, ds 2 = g ττ dτ 2 + dρ 2 + a 2 (t )S 2 k(χ t (τ))dω 2, (2) with Fermi chart, U polar = {(τ, ρ, θ, φ) : τ >, < ρ < ρ Mτ, θ I π, φ I 2π } (22) Remark 2. In the Milne Universe where k = and = t, it is easily verified that g ττ and, where ρ = ρ(τ, t ) according to Eq.(3). Therefore, a(t )S k (χ t (τ)) = ρ, (23) lim a(t )S k (χ t (τ)) = t + lim ρ = ρ Mτ = τ. (24) ρ ρ Mτ Then from Eq.(2), λ k (τ, ρ). (25) so Fermi coordinates in the Milne Universe are just the usual Minkowski coordinates. 7 The subscript on the cosmological time coordinate t is included as a convenience so that we may use the symbol t as a dummy variable in integral expressions where it arises naturally. 9

10 3 Cosmological and particle horizons In this section we collect and prove results that relate the existence of particle horizons and cosmological event horizons to properties of the scale factor and its derivatives. A Robertson-Walker spacetime has a cosmological event horizon if, χ horiz (t ) dt <, (26) t for some t > (and hence all t > ). The spacetime has has finite particle horizon if, χ part (τ) dt <, (27) for some τ > (and hence all τ > ). Part (b) of the following theorem shows that a finite particle horizon is mathematically impossible if ȧ( + ) <. Theorem. Let be a regular scale factor on a Robertson-Walker spacetime (M, g), where g is given by Eq.(9). (a) If M has a cosmological horizon, i.e., χ horiz (t ) < for some t >, then t lim t = = lim t ȧ(t). (28) Moreover, M experiences inflationary periods for arbitrarily large cosmological times, that is, for any N >, there exists a non empty open interval (a, b) with a > N such that ä(t) > on (a, b). However, the condition ä(t) > for all t > does not imply the existence of a cosmological event horizon. (b) If M has a finite particle horizon, i.e., χ part (τ) < for some τ >, then lim t + t = = lim t + ȧ(t). (29) Moreover, M experiences noninflationary periods for arbitrarily small cosmological times, that is, for any δ >, there exists a non empty open interval (a, b) (, δ) such that ä(t) < on (a, b). Proof. The proof of part (a) is given in [5]. To prove part (b), observe first that the right hand side of Eq. (29) follows from the left hand side by L Hôpital s rule. Observe that lim dt =, (3) t + t t because lim t + / =. Also, for τ > t >, I t [t,τ] t, (3)

11 for t >, where I [t,τ] is the indicator function for the interval [t, τ]. So, from the Lebesgue Dominated convergence theorem, lim t + t t t dt = lim t + t I [t,τ] t Now using Eqs. (3), (32) and L Hôpital s rule we have that ( lim t τ ) ( ) t + t t dt = lim t + t t dt / t ( ) ) = lim / t + t a(t ) ( t 2 dt =. (32) t = lim =. (33) t + a(t ) Given δ >, lim t + ȧ(t) = implies that ȧ(t) cannot be increasing on (, δ). Therefore there must be a t (, δ) with ä(t) <, and by continuity ä(t) < on an open interval containing t. Remark 3. Part b of Theorem shows that a scale factor with ȧ( + ) = such as = t α for α < is non inflationary at the big bang. Our results for continuously differentiable extensions of the metric coefficient g ττ therefore exclude such scale factors, but for continuous extensions of the Fermi metric for this case, see Theorems 3 and 4 below. 4 Limiting values of g ττ at the big bang In this section, we find the limiting value of g ττ (τ, ρ) as ρ increases to ρ Mτ, and show that the limiting value is not zero. This will be used in Section 5 to construct a continuously differentiable extension of g ττ to a larger spacetime M (see Eq. (79)). We begin with an alternative but equivalent expression for g ττ (τ, ρ) in Eq. (9). Rearranging terms gives, [ g ττ (τ, ρ) = ȧ(τ) 2 τ ] 2 ȧ(t ) ä(t) a2 (τ) a 2 (t ) ȧ(t) 2 a2 (τ) a 2 (t) dt (34) Eq.(9) may then be modified by substituting the relation, The result is, g ττ (τ, ρ) = [ ȧ(τ) t ȧ(t ) = ȧ(τ) + t t ä(t) dt. (35) ȧ(t) 2 ( ) 2 ä(t) a2 (τ) a 2 (t ) dt] ȧ(t) 2 a2 (τ) a 2 (t). (36) The following lemma slightly generalizes a result in [5] to the case t =.

12 Lemma. Let be a regular scale factor. Then for t < τ, t [ ] ä(t) a(τ) ȧ(t) 2 a2 (τ) a 2 (t) dt < t < ȧ(τ). [ a(τ) a2 (τ) a 2 (t) ] dt (37) Proof. The first inequality follows from Definition c and Remark. For the second inequality, t [ a(τ) a2 (τ) a 2 (t) ] dt = = a 2 (τ) t = a(τ) t < a(τ) a(τ) ȧ(τ) a2 (t) a 2 (τ) ȧ(t) a2 (t) a 2 (τ) a2 (t) a 2 (τ) a2 (t) a 2 (τ) dt t a2 (t) a 2 (τ) ( + ȧ(t)/a(τ) ( + a2 (t) a 2 (τ) ȧ(t)/a(τ) ( + )dt a2 (t) a 2 (τ) a2 (t) a 2 (τ) )dt )dt, (38) where in the last step, we have used the fact that the Hubble parameter, H(t), is a decreasing function of t, and strictly decreasing on an interval, so that /ȧ(t) < a(τ)/ȧ(τ) on some interval of t values. To evaluate this last integral, we make the change of variable, x = /a(τ), which yields, t [ ] a(τ) a2 (τ) a 2 (t) dt < ȧ(τ) dx x2 ( + x 2 ) = ȧ(τ). (39) Theorem 2. Let be strongly regular. Then the Fermi metric coefficient g ττ (τ, ρ) satisfies the following: lim g ττ (τ, ρ) = ρ ρ Mτ [ ȧ(τ) ( ) 2 ä(t) a(τ) dt] ȧ(t) 2 a2 (τ) a 2 (t) (4) 2

13 Moreover, the function g ττ (τ, ρ Mτ ) defined by this limit satisfies, > g ττ (τ, ρ Mτ ) >. (4) Proof. Note that for fixed τ, ρ ρ Mτ if and only if t. The existence of the finite limit in Eq (4) follows from Lemma and the Dominated Convergence Theorem using the comparison, [ ] [ ] ä(t) a2 (τ) a 2 (t ) ȧ(t) 2 a2 (τ) a 2 (t) < K a(τ) a2 (τ) a 2 (t) (42) The second assertion, g ττ (τ, ρ Mτ ) >, follows directly from Lemma. Remark 4. It follows from Eq. (36) that g ττ (τ, ρ) if the cosmological time interval from t (τ, ρ) to τ is inflationary, i.e. if ä(t) on that interval. Similarly, g ττ (τ, ρ) if the cosmological time interval from t (τ, ρ) to τ is noninflationary, i.e. if ä(t) on that interval. The following purely technical lemma will be needed in the proof of continuity of the extension of g ττ and its derivatives. Lemma 2. Let be a regular scale factor and assume that τ τ >. Let l(t) be a smooth function defined for t > satisfying l(t) < K (43) for some K > and all t >. Then, (a) If < x τ, [ ] [ ( )] a2 (τ) a l(t) 2 (x) τ a2 (τ) a 2 (t) dt < K a(τ) ȧ(τ) a(τ ) a 2 (τ) a 2 (τ ) (44) (b) a(τ) τ dt τ l(t) a2 (τ) a 2 (t) < K τ dt [ ( + K ȧ(τ) a(τ) a(τ ) )] a 2 (τ) a 2 (τ ) (45) 3

14 Proof. Using Eq.(38) we have, τ [ a(τ) a2 (τ) a 2 (t) ] dt < ȧ(τ) τ The change of variable, x = /a(τ), yields τ [ ] a(τ) a2 (τ) a 2 (t) dt < ȧ(τ) a2 (t) a 2 (τ) = ȧ(τ) [ a(τ ) a(τ) ȧ(t)/a(τ) ( + a2 (t) a 2 (τ) )dt, (46) dx x2 ( + x 2 ) ( )] a(τ) a(τ ) a 2 (τ) a 2 (τ ) Part (b) now follows by rearranging terms and from the hypothesis (47) l(t) < K Part (a) follows from this hypothesis and since a(τ) a 2 (τ) a 2 (x). (48) 5 C extension of g ττ The plan of this section is first to extend g ττ, given by Eq. (36), as a continuous function to values of ρ > ρ Mτ (or equivalently to negative values of t ). We then show under some regularity conditions that the first partial derivatives of the extension of g ττ are also continuous. In order to accomplish this, we must extend the domain of the scale factor to include negative values of cosmological time t = t. Although not essential, it is convenient using our methods to extend as an even function so that a( t) =. A smooth even extension of requires ȧ() =, but we also consider the possibility that < ȧ( + ) <, which forces a discontinuity in the first derivative of (but not in the metric coefficients). Both of these geometric properties of the big bang are of interest and result in different extensions of Robertson-Walker spacetimes to negative cosmological times. As a convenience to the reader, the proofs of the lemmas and theorems of this section have been 4

15 placed in the appendix labeled as Sect.. As a first step, we extend the proper distance coordinate ρ by the same formula as Eq. (3), but so as to allow values of t in the interval τ < t < τ, ρ = Now t (τ, ρ) is defined implicitly by Eq.(49) on the set, t dt. (49) a2 (τ) a 2 (t) D = {(τ, ρ) : τ >, < ρ < 2ρ Mτ }, (5) where ρ Mτ is the Fermi radius of the universe at proper time τ of the central observer and is given by Eq.(4). It follows from the Implicit Function theorem that the function t (τ, ρ) is a smooth function of its arguments in D, except possibly when t =, the cosmological time coordinate of the big bang. The next lemma shows that t is continuous even where t =. This result will be needed in what follows. Lemma 3. Suppose is strongly regular. Then the function t (τ, ρ), defined implicitly by Eq. (49), is continuous on D. In Fermi polar coordinates, there is a coordinate singularity at ρ =, but this singularity disappears in + dimensions and ρ may be extended symmetrically to negative values as well. However, in what follows it is convenient to restrict ρ to nonnegative values, and this causes no loss of generality. With Eq. (36) in mind and with a slight abuse of notation, we define the extension of the metric tensor to D by g ρρ and, where and g ττ (τ, ρ) = [ ȧ(τ)f(τ, t (τ, ρ))] 2 [ ȧ(τ)f(τ, ρ)] 2, (5) ( ) τ ä(t) a 2 (τ) a 2 (t ) t f(τ, t ) = ȧ(t) dt if τ > t 2 a2 (τ) a 2 (t) 2f(τ, ) f(τ, t ) if τ < t <, (52) f(τ, ρ) f(τ, t (τ, ρ)). (53) Remark 5. Since lim t + f(τ, t ) = f(τ, ) exists by Theorem 2, by the definition of f we automatically have that lim t f(τ, t ) = f(τ, ). Therefore lim t f(τ, t ) = f(τ, ). We note also that for t < an equivalent expression for f(τ, t ) is, ( ) ä(t) a2 (τ) a f(τ, t ) = f(τ, ) + 2 (t ) ȧ(t) 2 a2 (τ) a 2 (t) dt. (54) t 5

16 Remark 6. We note that in general the domain D given by Eq.(5) will be too large to be a coordinate chart for an extended spacetime M, and a proper subset must be used to avoid zeros of g ττ in M where t <. This because the conclusions of Theorem 2 do not necessarily hold for Eq. (5) on all of D. However, it follows from the continuity of g ττ established in Theorem 3 that g ττ (τ, ρ) > at all points (τ, ρ) where t (τ, ρ) is sufficiently close to zero, including points where t (τ, ρ) <. We elaborate further in Section 6. Theorem 3. Let be strongly regular. Then the metric coefficient g ττ (τ, ρ) given by Eq.(5) is continuous on D. Our next task is to prove that the metric coefficient, g ττ, as defined by Eq. (5) is differentiable on D. The following lemma deals with the technicality of the unbounded integrand in Eq. (52). Lemma 4. Let be regular with ȧ( + ) <. The function f(τ, t ) given by Eq. (52) is continuously differentiable with respect to t when τ < t < τ, and, t f(τ, t ) = a(t )ȧ( t ) τ a2 (τ) a 2 (t ) t ä(t) ȧ(t) 2 dt a2 (τ) a 2 (t). (55) Lemma 5. For a regular scale factor, let a( + ) lim t + ȧ(t). Then: (a) If ȧ( + ) = and there exists an ɛ > such that ä(t) for t (, ɛ), then ä(t) dt lim t + ȧ 2 =. (56) (t) a2 (τ) a 2 (t) (b) If > ȧ( + ) > then, lim t + t exists and is finite. t ä(t) dt τ ȧ 2 (t) a2 (τ) a 2 (t) = ä(t) dt ȧ 2 (t) a2 (τ) a 2 (t) (57) Theorem 4. Let be a strongly regular scale factor and suppose that one of the conditions of Lemma 5 holds. Then g ττ is differentiable with respect to ρ in D, and ρ g ττ = 2 g ττ ȧ(τ) ρ f(τ, ρ), (58) where ρ f(τ, ρ) = ȧ( t ) t ä(t) ȧ(t) 2 dt a2 (τ) a 2 (t) for ρ ρ Mτ (and with t = t (τ, ρ)), and ȧ( + ) ä(t) ȧ ρ f(τ, ρ Mτ ) = 2 (t) dt if a2 (τ) a 2 (t) ȧ(+ ) > a(τ) if ȧ() = (59) (6) 6

17 We next establish the differentiability of g ττ (τ, ρ) with respect to τ in the domain D. From Eq. (5), this will follow by proving that f(τ, t (τ, ρ)) is differentiable with respect to τ. This is established by the following theorem whose proof depends on Lemmas 6, 7, and 8. Theorem 5. Let be strongly regular with ȧ( + ) <. Suppose that there is a constant C > such that... a (t)a 2 (t) ȧ 3 (t) C, (6) for all t. Then g ττ is differentiable with respect to τ in D and, where τ g ττ = 2 g ττ [ä(τ)f(τ, ρ) + ȧ(τ) τ f(τ, ρ)], (62) τ f(τ, ρ) = τ f(τ, t ) + t f(τ, t ) τ t (τ, ρ), (63) for (τ, ρ) (τ, ρ Mτ ), where τ f(τ, t ), t f(τ, ρ) and τ t (τ, t ) are given by Eqs. (66), (55) and (6), respectively. If (τ, ρ) = (τ, ρ Mτ ) then, dρ τ f(τ, ) + Mτ a(τ) dτ (τ) if ȧ() = τ f(τ, ρ Mτ ) = τ f(τ, ) + ȧ(+ ) dρ Mτ dτ (τ) ä(t) dt ȧ 2 (t) if ȧ( + ) > a2 (τ) a 2 (t) (64) The lemmas that follow in this section rely on the differentiability of t (τ, ρ), on D except possibly where t =. This follows from the Implicit Function theorem and Eq.(49). Lemma 6. Let be strongly regular. Suppose that there is a constant C > such that... a (t)a 2 (t) ȧ 3 (t) C. (65) Then for any τ >, f(τ, t ) is differentiable with respect to τ, and { f τ (τ, t I (τ, t ) + I 2 (τ, t ) if t > ) = 2 τ f(τ, ) I (τ, t ) I 2 (τ, t ) if t < where I (τ, t ) = ȧ(τ) a(τ) t [ 3ä2 (t) ȧ 4 (t)... a (t) ȧ 3 (t) ] [ a2 (τ) a 2 (t ) a2 (τ) a 2 (t) ] dt, (66) (67) 7

18 I 2 (τ, t ) = ȧ(τ) a(τ) t [ ä(t) ȧ 2 (t) a 2 (τ) a2 (τ) a 2 (t) a 2 (τ) a 2 (t ) ] dt (68) and f ȧ(τ) τ [... ä(t) (τ, ) = τ a(τ) ȧ 2 (t) + a (t) ȧ 3 (t) 3ä2 ] [ ] (t) a(τ) ȧ 4 (t) a2 (τ) a 2 (t) dt (69) Lemma 7. Assume that the conditions of Lemma 6 hold. Then for any τ > we have that f lim τ τ τ (τ, t (τ)) = f τ (τ, ), (7) where τ f(τ, ) is given by Eq. (69) and t (τ) t (τ, ρ Mτ ). Lemma 8. Under the assumptions of Lemma 6 and one of the conditions of Lemma 5, we have that f lim (τ, t (τ)) t τ τ t τ (τ, t (τ)) (7) exists and is finite, where as in Lemma 7, t (τ) t (τ, ρ Mτ ). Now that we have established the existence of both partial derivatives of g ττ on the domain D we proceed to show that g ττ is continuously differentiable on D. Theorem 6. Let be strongly regular and suppose that one of the conditions of Lemma 5 hold. Then the partial derivative ρ g ττ given by Eq. (58) is continuous on D. Theorem 7. Suppose that the conditions of Lemma 6 and one of the conditions of Lemma 5 hold. Then the partial derivative τ g ττ given by Eq. (62) is continuous on D. The following theorem summarizes the results of this section. Theorem 8. Suppose that one of the conditions of Lemma 5 hold and that is strongly regular. Suppose also that there is a constant C > such that... a (t)a 2 (t) ȧ 3 (t) C. (72) for all t. Then g ττ (τ, ρ) is continuously differentiable on D. 8

19 6 M in + dimensions In this section we define a two dimensional spacetime manifold M that includes pre-big bang events and the + dimensional Robertson-Walker spacetime M as a submanifold. The extended Robertson-Walker metric on M, restricted to the submanifold M of cosmological time zero events, is singular in the sense that it is continuously differentiable on M, but in general not twice differentiable there. We assume throughout this section that the scale factor satisfies the hypotheses of Theorem 8. We begin with the line element for M in curvature coordinates, ds 2 = dt 2 + a 2 (t)dχ 2. (73) There is a coordinate singularity in Eq.(9) in four spacetime dimensions at χ =, but this singularity disappears in two spacetime dimensions and χ may be extended symmetrically to take all real values (for k =, ). For the comoving observer at χ =, the maximal Fermi chart then consists of all (τ, ρ) with τ > and ρ < ρ Mτ (see Fig. and Section 2), with the metric given by, ds 2 = g ττ (τ, ρ)dτ 2 + dρ 2, (74) where g ττ is defined by Eq.(5). From Theorem 8, it is easily verified by symmetry that g ττ is continuously differentiable on the set, D = {(τ, ρ) : τ >, ρ < 2ρ Mτ }, (75) with t = t (τ, ρ) defined implicitly by a slight modification of Eq.(49): ρ = t dt. (76) a2 (τ) a 2 (t) In order to define the extended spacetime, M, we first extend the maximal Fermi chart of the χ = comoving observer. For that purpose, we use a subset of D. By Theorem 2, g ττ (τ, ρ Mτ ) <, but we have not ruled out the possibility that g ττ (τ, ρ) = at points where t (τ, ρ) < in the domain D (or D). However this is not an essential feature of our construction because the choice we made for the metric tensor for negative cosmological times was arbitrary except at points (τ, ρ) where t (τ, ρ) < and t (τ, ρ) is close to zero. At such points near the big bang, continuous differentiability of g ττ at (τ, ρ Mτ ) places restrictions on any extension of this function so that g ττ cannot differ greatly at points near the big bang from the definition given in Eq.(5). In accordance with Remark 6, and to eliminate ambiguities in the extension of g ττ, let the coordinate chart D 2 D be defined by, D 2 = {(τ, ρ) : τ >, ρ < ρ max (τ)}, (77) 9

20 where, ρ max (τ) = inf{ρ : < ρ < 2ρ Mτ and g ττ (τ, ρ) = }, (78) provided the infimum exists, with ρ max = 2ρ Mτ We can now define M as a disjoint union, otherwise. M = M + M M, (79) where the superscripts indicate respectively that cosmological time t restricted to the set is positive, zero, or negative. Here M + = M denotes the original Robertson-Walker universe where cosmological time is postive, and t = t (τ, ρ) is a continuous function of τ and ρ within the Fermi chart and on M M, all of which are covered by the chart D 2. The original smooth charts on M + = M together with D 2 form an atlas on M. The metric restricted to the submanifold of M with chart D 2 is given by Eq. (74). Under the assumptions of Theorem 8, the metric of Eq.(74) is C on M, and smooth on M M. Remark 7. It is easily verified that all connection coefficients from the metric of Eq. (74) are continuous on M, and that the spacelike path γ(ρ) = (τ, ρ) satisfies the geodesic equations for all ρ with ρ < ρ max (τ ). Thus, all spacelike geodesics in M orthogonal to the Fermi observers s worldlike β(τ) = (τ, ) pass through spacetime points in the big bang M as well as pre-big bang points in M. Geometrically, ρ max (τ ) may be understood as the first zero of g ττ (τ, ) along the spacelike geodesic orthogonal to β, and starting from, β(τ ), as ρ increases. It follows from Theorem 2 that ρ max (τ) > ρ Mτ so that M is not empty and necessarily consists of points with negative cosmological times t. The subset M defined by t (τ, ρ) = is a C submanifold parameterized in two connected components by (τ, ρ Mτ ), and (τ, ρ Mτ ), for τ >, where the oneto-one, continuously differentiable function ρ Mτ is given by Eq.(4). It follows from Remark and Eq.(6) that dρ Mτ /dτ is nonvanishing for the scale factors we consider. The charactor of M is described by the following theorem. Theorem 9. Under the assumptions of Theorem 8, M is lightlike. Proof. A tangent vector to M at the point (τ, ± ρ Mτ ) is u = (, ± dρ Mτ /dτ), (where + is used for the component with positive space coordinates and for the other component). From Eq.(66) in [4] and Eq.(79) and Theorem 8 in [5], it follows that, gττ (τ, ρ Mτ ) = v Fermi lim gττ (τ, ρ) = lim ρ ρ t + v Mτ kin = dρ M τ (τ), (8) dτ 2

21 where the limit in the third term is of the ratio of the Fermi relative speed to the kinematic relative speed of a comoving test particle at the spacetime point uniquely determined by τ and t (see [4, 37, 5]), and both speeds are relative to the comoving observer β. Then, where g is the metric tensor. ( ) 2 dρmτ g(u, u) = g ττ (τ, ρ Mτ ) + =, (8) dτ 7 M in 3 + dimensions In this section we construct a spacetime M in four spacetime dimensions, analogous to the construction in Section 6 for the two dimensional case. Analogous to the two dimensional case, here M includes both pre-big bang events and the 3 + dimensional Robertson-Walker spacetime M as a submanifold, for k = (see Eq.()). The extended metric tensor on M is smooth except for its restriction to the submanifold M of cosmological time zero events. On M, the extended metric is C but also with certain differentiability properties. The big bang, M, inherits geometric structure from M, and the dimension of the the cotangent bundle on M is two dimensional. In subsection 7. we develop extended Fermi polar coordinates and extend the metric of Eqs. (2) and (22). Because partial derivatives of the metric coefficient a 2 (t )Sk 2(χ t (τ)) in Eq. (2) diverge at cosmological time zero spacetime points, it cannot be extended as a differentiable function of τ and ρ under the general assumptions that make g ττ continuously differentiable where t =. Subsection 7.2 develops the full (Cartesian) Fermi coordinates τ, x, y, z of Eq.(8) and finishes the construction of M. 7. Angular Coordinates Here we define an extension, ds 2 = g ττ dτ 2 + dρ 2 + g θθ dθ 2 + g φφ dφ 2, (82) of the Eq.(2) in Fermi polar form. A new chart D polar U polar (see Eq (22)) is defined by, D polar = {(τ, ρ, θ, φ) : τ >, < ρ < ρ max (τ), θ I π, φ I 2π }, (83) where the notation is the same as in Eq. (78), and as before I π and I 2π may be chosen to be any open intervals of length π and 2π respectively. As before, we assume that the scale factor is extended as an even function of t. It 2

22 follows from Theorem 8 that g ττ is continuously differentiable on D polar under the hypotheses of that theorem. As in Section 6, we define g ρρ and define g ττ by Eq. (5) on D polar. For t (τ, ρ), let and g θθ (τ, ρ) = ḡ θθ (τ, t (τ, ρ)) = a 2 (t )S 2 k(χ t (τ)), (84) g φφ (τ, ρ, θ) = g θθ (τ, ρ) sin 2 θ. (85) In what follows, we shall define g θθ and g φφ, at points (τ, ρ, θ, φ) D polar where t (τ, ρ) =, i.e., where ρ = ρ Mτ, by the limiting values of those functions as t. To define g θθ at points where t =, we first consider the case k =. Since S k= is bounded, the right side of Eq. (84) converges to zero as t, so g θθ and g φφ can both be defined to take the value zero at such points. For the other cases, k =,, we first observe that from Eq.(2) and Lemma, for a regular scale factor, t dt χ t (τ) < ȧ(τ) + dt (86) t for τ > t. In both cases S k is an increasing function, so combining (84) and (86), gives, ( ) g θθ (τ, ρ) = a 2 (t )Sk(χ 2 t (τ)) a 2 (t )Sk 2 τ ȧ(τ) + dt. (87) By Theorem, the assumptions we make in Theorem 8 and Lemma 5 are inconsistent with the existence of finite particle horizons in the cosmologies we consider. Nevertheless we point out that if the scale factor (in violation with those hypotheses) does give rise to finite particle horizons, i.e., then we have the following result. t dt <, (88) Theorem. For k =,, and a regular scale factor with finite particle horizon, i.e., satisfying Eq. (88) for τ >, g θθ (τ, ρ) and g φφ (τ, ρ, φ) are continuous on D polar and vanish at (τ, ρ Mτ ). Proof. It is necessary to show continuity only at points of the form (τ, ρ Mτ, θ, φ). Since t (τ, ρ) = if and only if ρ = ρ Mτ and t (τ, ρ) is continuous on D by Lemma 3, it suffices to show that ḡ θθ is continuous at (τ, ) for any τ >. Using Eq.(87), we have, 22

23 lim ḡθθ(τ, t ) = lim (τ,t ) (τ,) (τ,t a2 (t )Sk(χ 2 t (τ)) = (89) ) (τ,) The following remark shows that the case k = is problematic for inflationary cosmologies. Remark 8. In the Milne Universe k = and Remark 2 shows that g θθ and g φφ have obvious smooth extensions to D polar. If = t α and α <, then particle horizons are finite and Theorem applies. However, for k = with inflationary power law scale factors of the form = t α with α >, it is readily seen that lim a(t )S k (χ t (τ)) = lim a(t ) sinh(χ t (τ)) =. (9) t + t + Therefore continuous extensions of g θθ (τ, ρ) and g φφ (τ, ρ, φ) to D polar are not possible for these cosmologies. Specializing to the case k = and regular scale factors with infinite particle horizons, we have, using Eq.(86) and L Hôpital s rule, lim a(t a(t ) )χ t t (τ) = lim t ȧ(t ) = lim t H(t ) = H( + ), (9) which exists because the Hubble parameter H(t ) is a decreasing function for regular scale factors. Theorem. Let k = and be a regular scale factor. Then g θθ (τ, ρ) and g φφ (τ, ρ, φ) are continuous on D polar. Proof. It is sufficient to prove that g θθ is continuous at points of the form (τ, ρ Mτ ). For the case of finite particle horizons, the result follows by Theorem. For the case of infinite particle horizons, we define a new function of two independent variables τ and x, by, F 4 (τ, x) a(x) x a(τ) τ a2 (τ) a 2 (t) dt a(x) h 4 (τ, t)dt, (92) with the restriction τ > x >. In light of Eq. (9), we define F 4 (τ, ) = /H( + ). Since t (τ, ρ Mτ ) =, our plan of proof is first to show that F 4 (τ, x) is continuous at any point of the form (τ, ). Then using Lemma 3, the composition F 4 (τ, t (τ, ρ)) must be continuous at any point of the form (τ, ρ Mτ ) and and therefore g θθ (τ, ρ) = F 2 4 (τ, t (τ, ρ)) must be continuous at any point of the form (τ, ρ Mτ ). x 23

24 For τ > τ > x >, F 4 (τ, x) F 4 (τ, ) F 4 (τ, x) F 4 (τ, x) + F 4 (τ, x) /H( + ) a(x) h 4 (τ, t)dt +a(x) h 4 (τ, t) h 4 (τ, t) dt+ F 4 (τ, x) /H( + ) τ x (93) The third term in Eq. (93) can be made arbitrarily small for all x sufficiently close to by Eq. (9), while the first term can be made small for all τ sufficiently close to τ by Lemma 2b. For the middle term, choose any < δ < τ. We can assume that x < τ δ. Then, a(x) h 4 (τ, t) h 4 (τ, t) dt = a(x) x τ δ h 4 (τ, t) h 4 (τ, t) dt + a(x) δ x h 4 (τ, t) h 4 (τ, t) dt (94) The first integral is bounded by the integrability of h 4 (τ, t) and h 4 (τ, t), so it can be made small for small enough a(x), which is achieved by choosing x sufficiently close to. For the second term, note that h 4 (τ, t) is uniformly continuous in t and τ for t [, τ δ]. So, for any ɛ > we can choose τ close enough to τ so that, a(x) δ x Since lim x a(x) δ x h 4(τ, t) h 4 (τ, t) dt ɛa(x) δ x dt. (95) dt = /H(+), this term is bounded for x close to. Therefore we can make the entire second term in Eq. (94) small by choosing (τ, x) sufficiently close to (τ, ). The τ < τ case is simliar. The following corollary shows that if k = and the scale factor is analytic at t = and regular, then g θθ (τ, ρ Mτ ) = g φφ (τ, ρ Mτ, θ) =. Corollary. If k = and is regular and an even function on R, with either > ȧ( + ) > or ȧ() = but the nth derivative of at t = for some n does not vanish, then g θθ (τ, ρ) and g φφ (τ, ρ, φ) are continuous on D polar and vanish at (τ, ρ Mτ ) for any τ >. Proof. Continuity follows from Theorem and if the particle horizon is finite, g θθ (τ, ρ) and g φφ (τ, ρ, φ) vanish at (τ, ρ Mτ ) by Theorem. If the particle horizon is infinite and ȧ( + ) >, the result follows from Eq.(9) and the assumption that a() =. Alternatively, if ȧ() = but a (n) () for some n, then for the smallest such n, we must have a() = ȧ() = ä() = = a (n ) () =, (96) 24

25 but a (n) (). Repeated application of L Hôpital s rule then gives, So, g θθ (τ, ρ Mτ ) = Eq.(85) finishes the proof. lim a(t a (n ) (t ) )χ t (τ) = lim =. (97) t + t a (n) (t ) lim (τ,t a2 (t )Sk(χ 2 t (τ)) = lim ) (τ,) t a2 (t )χ 2 + t (τ) =. (98) We collect results from this subsection and Theorem 8 in the following theorem. Theorem 2. Suppose that one of the conditions of Lemma 5 hold and that is strongly regular. Suppose also that there is a constant C > such that... a (t)a 2 (t) ȧ 3 (t) C. (99) for all t. Then g ττ (τ, ρ) is continuously differentiable on D polar and if k =, g θθ (τ, ρ) and g φφ (τ, ρ, φ) are continuous on D polar. 7.2 Fermi Coordinates We begin with a definition of the key chart for the extended spacetime M in 3 + spacetime coordinates. Let D Fermi U Fermi of Eq.(7) be defined by, { D Fermi = (τ, x, y, z) : τ > and } x 2 + y 2 + z 2 < ρ max (τ), () where ρ max (τ) is given by Eq.(78). In light of Eq.(84) and Theorem, λ k (τ, ρ) in Eq.(2) may be extended as a continuous function to D Fermi by the formula, for a regular scale factor. In particular, λ k (τ, ρ) = g θθ(τ, ρ) ρ 2 ρ 4, () λ k (τ, ρ Mτ ) = lim λ k (τ, ρ) ρ ρ Mτ a 2 (t )Sk 2 = lim (χ t (τ)) ρ 2 ρ ρ ρ 4 = /H2 ( + ) ρ 2 M τ ρ 4. Mτ M τ (2) We note that if the hypotheses of Corollary are satisfied, then λ k (τ, ρ Mτ ) = /ρ Mτ. 25

26 In analogy with Eq.(79) for the two dimensional case, we can now define the four dimensional spacetime M as a disjoint union, M = M + M M, (3) where, as before, the superscripts indicate respectively that cosmological time t restricted to the set is positive, zero, or negative. Here M + = M denotes the original four dimensional Robertson-Walker universe where cosmological time is postive, and t = t (τ, ρ) is a continuous function of τ and ρ within the Fermi chart and on M M, all of which are covered by the chart D Fermi. The original smooth charts on M + = M together with D Fermi form an atlas on M. The metric restricted to the submanifold of M with chart D Fermi is given by, ds 2 = g ττ (τ, ρ) dτ 2 + dx 2 + dy 2 + dz 2 +λ k (τ, ρ) [ (y 2 + z 2 )dx 2 + (x 2 + z 2 )dy 2 + (x 2 + y 2 )dz 2 xy(dxdy + dydx) xz(dxdz + dzdx) yz(dydz + dzdy) ], (4) Under the assumptions of Theorem 8, the metric of Eq.(4) is smooth on M M. The next theorem summarizes the results of this subsection. Theorem 3. Suppose that one of the conditions of Lemma 5 hold and that the scale factor is strongly regular. Suppose also that there is a constant C > such that... a (t)a 2 (t) ȧ 3 (t) C, (5) for all t. Then g ττ (τ, ρ) is continuously differentiable on D Fermi and λ k= (τ, ρ) is continuous on D Fermi. Remark 9. A sufficient condition for the bound Eq.(5) under the hypotheses of Theorems 2 and 3 is the existence of a constant C such that,... a (t)ȧ(t) ä 2 (t) C K 2, (6) so that, for example, if C = K 2, then ȧ(t) itself regarded as a scale factor would be regular according to Definition. This follows from the implication, ä(t) ȧ(t) 2 K = ȧ(t) K ȧ(t) ä(t). (7) The following corollary, established by direct calculation, shows consistency with the polar and cartesian forms of the metric extended to D Fermi. 26

27 Corollary 2. Under the hypotheses of Corollary, the Fermi metric of Eq.(8) expressed as a 4 by 4 matrix, g ττ + λ k (y 2 + z 2 ) λ k xy λ k xz λ k xy + λ k (x 2 + z 2 ) λ k yz λ k xz λ k yz + λ k (x 2 + y 2 ) and evaluated at (τ, ρ Mτ ) has rank 2 for all τ >. Thus the cotangent space at each point in M is two dimensional. Remark. The submanifold obtained by assigning fixed values θ and φ to the angular coordinates in the chart D polar for M is the two dimensional spacetime analyzed in Section 6. From Remark 7, it follows that the spacelike path γ(ρ) = (τ, ρ, θ, φ ) with ρ < ρ max (τ ) is geodesic in the submanifold, intersects the big bang M, and reaches pre-big bang points in M. The next theorem collects assumptions from Theorems 3 and needed for a continuous (not necessarily differentiable) extension of the metric to M. Theorem 4. Let k = and let be a strongly regular scale factor. Then the metric coefficients of Eq.(4) are continuous on D Fermi, and the metric coefficients for the polar form, Eq.(82), are continuous on D polar. We note that the metric is as smooth as off of the big bang M. 8 Examples In this section we give examples of Robertson-Walker cosmologies satisfying the conditions of Theorems 8, 2 and 3. We begin with power law cosmologies, i.e., those with scale factors of the form = t α with α >. These cosmologies include the radiation-dominated and matter-dominated universes, and models for dark energy (see [42]). For the power law scale factor = t α, Eq. (34) gives, ( ) α g ττ (τ, ρ) = τ t ( ) ( 2α ( ) α t τ τ t 2F ( 2, α 2α ; + α 2α ; (8) where t is given implicitly as a function of τ and ρ by Eq. (3), 2 F (, ; ; ) is the Gauss hypergeometric function, and (see [37, 39]), ( ) ) ) 2 2α t C α, τ C α = ρ M τ τ = π Γ( +α 2α ) Γ( 2α ). (9) 27

28 With Theorem 3, it follows that, Also by Eq. (3), ρ = τ [ C α + α lim g ττ (τ, ρ) = g ττ (τ, ρ Mτ ) = Cα. 2 () ρ ρ Mτ ( ) +α t τ Implicit differentiation or Eq.(6) gives, t τ = and further calculations show, 2F ( 2, + α 2α ; + 3α 2α ; τ 2α t 2α t α ( ) )] 2α t. () τ ρ τ + t τ, (2) ρ g ττ (τ, ρ) = 2α τ and, gττ (τ, t ) (2F ( 2, α 2α ; + α 2α ; ( ) ) 2α t τ ( ) ) α t C α, τ (3) τ g ττ (τ, ρ) = 2α τ ( Hence, gττ (τ, t ) C α α + ( ) +α t τ (2F ( 2, α 2α ; + α 2α ; ( t 2F ( 2, + α 2α ; + 3α 2α ; τ ( t τ ) 2α ) ) 2α )) ( t τ ) α C α ) = ρ τ ρg ττ (τ, ρ). (4) ρ g ττ (τ, ρ Mτ ) = lim ρ g ττ (τ, ρ) = ρ ρ Mτ { 2αCα τ if α > if < α < (5) and, τ g ττ (τ, ρ Mτ ) = lim τ g ττ (τ, ρ) = ρ ρ Mτ { 2αC 2 α τ if α > if < α <. (6) 28

29 If α <, then ȧ( + ) = and = t α is not inflationary near t = (nor at any time) and so fails to satisfy the hypotheses of Theorems 8 or 3. For the Milne case (see Remarks 2 and 8), α = and taking into account that ȧ() = >, the hypotheses of Theorem 8 are satisfied, so the implied properties of Fig. given in the introduction follow from that theorem. If α >, then the scale factor = t α is inflationary near t = (and for all t), and satisfies the hypotheses of Theorems 8 and 3 for k =, so the conclusions hold for the cosmologies that are inflationary at the big bang, as shown by Eqs. 5 and 6. t a = 2 a = t o > t o < spacelike geodesic t o < t o = t o = r Mt Figure 2: A portion of the extended Robertson-Walker cosmology M with scale factor = t 2 (i.e. α = 2). The Milne Universe (see Fig.) with α = is included only for comparison of Fermi radii, ρ Mτ. The comoving observer s worldline is the vertical line ρ = in the center. The dashed portion of the horizontal line extends the spacelike geodesic beyond the α = 2 universe M + = M through t = to include points with negative cosmological times. Figure 2 depicts part of the extension M of the Robertson-Walker cosmology with power law scale factor = t 2, i.e., α = 2. From Eq. (9) the boundary M of the Fermi chart in (τ, ρ) coordinates is, ρ Mτ = π Γ( +α 2α ) π Γ( 3 4 Γ( 2α ) τ = ) Γ( 4 ) τ.6τ. (7) The Milne Universe (α = ) in (τ, ρ) coordinates, whose boundary satisfies ρ Mτ = τ, is superimposed for comparison only. 29