Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes

Milne-like spacetimes are a class of $k = -1$ FLRW spacetimes which admit continuous spacetime extensions through the big bang. In a previous paper [30], it was shown that the cosmological constant appears as an initial condition for Milne-like spacetimes. In this paper, we generalize this statement to spacetimes which share similar geometrical properties with Milne-like spacetimes but without the strong spatially isotropic assumption associated with them. We show how our results yield a"quasi de Sitter"expansion for the early universe which could have applications to inflationary scenarios.


Introduction
Milne-like spacetimes are a class of k = −1 FLRW spacetimes which admit continuous spacetime extensions through the big bang. This extension was observed in [14], 1 and further properties of these spacetimes were explored in [30]. Similar to how investigating the geometrical properties of the r = 2m event horizon in the Schwarzschild spacetime led to a better understanding of black holes, we believe that investigating the geometrical properties of the big bang extension for Milne-like spacetimes may lead to a better understanding of cosmology.
In [30,Thm. 4.2], it was shown that, under suitable hypotheses of the scale factor for a Milne-like spacetime, the equation of state for the energy density ρ and pressure p at the big bang is the same as that of a cosmological constant, namely, ρ(0) = −p(0). We referred to this property as "the cosmological constant appearing as an initial condition for Milne-like spcetimes." In this paper we generalize this statement to spacetimes which share similar geometrical properties with Milne-like spacetimes but without any homogeneous or isotropic assumptions. (Recall that Milne-like spacetimes are a subclass of FLRW spacetimes and hence are spatially isotropic.)

Milne-like spacetimes and their continuous spacetime extensions through the big bang
In this section, we review the definition of Milne-like spacetimes and their continuous spacetime extensions through the big bang.
Since the assumption on the scale factor is a limiting condition, Milne-like spacetimes can include an inflationary era, a radiation-dominated era, a matter-dominated, and a dark energy-dominated era. Hence they can model the dynamics of our universe. Figure 1 depicts a Milne-like spacetime modeling an inflationary era. Introducing coordinates (R, θ, φ) for the hyperbolic metric h, we can write the spacetime metric as We introduce new coordinates (t, r, θ, φ) via Putting Ω = 1/b ′ = a/b, the metric in these new coordinates is = Ω 2 (τ )η. (1.4) Thus Milne-like spacetimes are conformal to (a subset of) Minkowski space. In eq. (1.4), τ is implicitly a function of t and r. Specifically, τ is related to t and r via b 2 (τ ) = t 2 − r 2 .
(1.5) Therefore the spacetime manifold M lies within the set of points t 2 − r 2 > 0. Since t > 0 by eq. (1.3), it follows that M lies within the set of points t > r. See figure 2. The proof of [30,Thm. 3.4] shows that b(0) = 0 where b(0) = lim τ →0 b(τ ). Therefore, by eq. (1.5), τ = 0 corresponds to the set of points t = r on the lightcone at the origin O.
Lastly, the proof also shows that Ω(0) = τ 0 . Since τ 0 > 0, eq. (1.4) implies that there is no degeneracy at τ = 0 in these coordinates (i.e. the big bang is a coordinate singularity for Milne-like spacetimes). Therefore Milne-like spacetimes admit continuous spacetime extensions through the big bang by defining the extended metric g ext via g ext = Ω 2 (0)η for points t ≤ r and g ext = g for points t > r. "Continuity" here refers to the fact that the metric g ext is merely continuous It's interesting to understand the behavior of the comoving observers within the extended spacetime. Recall that the comoving observers are the integral curves of u = ∂ τ and hence are given by the curves τ → (τ, R 0 , θ 0 , φ 0 ) for various points (R 0 , θ 0 , φ 0 ) on the hyperboloid. Physically, the comoving observers in an FLRW spacetime model the trajectories of the material particles which make up the galaxies, dust, etc. within the universe. In the (t, r, θ, φ) coordinates, a comoving observer is given by τ → t(τ ), r(τ ), θ 0 , φ 0 . By eq. (1.3), we have t(τ ) = coth(R 0 )r(τ ). Thus, in the (t, r, θ, φ) coordinates, the comoving observers are straight lines emanating from the origin O. See figure 3. This behavior can also be seen by noticing that the comoving observers have to be orthogonal to the hypersurfaces of constant τ which are the hyperboloids shown in figure 2. Lastly, we note that the behavior illustrated in figure 3 is closely related to the notion of a Janus point, see [4,5]. For Milne-like spacetimes, the "two-futures-one-past" scenario associated with a Janus point can be seen in [30, figures 6 and 18].

The cosmological constant appears as an initial condition for Milnelike spacetimes
As shown in [34,Thm. 12.11], FLRW spacetimes satisfy the Einstein equations with a perfect fluid (u, ρ, p), where u * = g(u, ·) is the one-form metrically equivalent to the vector field u = ∂ τ . We emphasize that for FLRW spacetimes, the energy density ρ and pressure p are purely geometrical quantities given by ρ = 1 8π G(u, u) and p = 1 8π G(e, e) where e is any unit spacelike vector orthogonal to u (its choice does not matter by isotropy). Here G = Ric − 1 2 Rg is the Einstein tensor which is related to T via G = 8πT . To incorporate a cosmological constant Λ, we define T normal = T + Λ 8π g so that the Einstein equations become Setting ρ normal = T normal (u, u) and p normal = T normal (e, e), we have (1.10) Now assume (M, g) is Milne-like. For simplicity, assume that the scale factor is analytic at zero: a(τ ) = τ + ∞ 2 c n τ n . Taking the limit τ → 0 in (1.10), we find: (1.12) Taking the limit τ → 0 in (1.12), we have We generalize statement (1.13) in Corollary 2.4 in the next section.

Main result
In this section, we generalize the results of the previous section to spacetimes that share similar geometrical properties with Milne-like spacetimes but without any homogeneous or isotropic assumptions. Specifically, Theorems 2.2 and 2.3 generalize statement (1.11) and Corollary 2.4 generalizes statement (1.13). We also deduce a statement about the Ricci curvature in Corollary 2.5. Our definition of a spacetime (M, g) will follow [29]. In particular, the manifold M is always assumed to be smooth. A smooth spacetime is one where the metric g is smooth, that is, its components g µν = g(∂ µ , ∂ ν ) are smooth functions with respect to any coordinates (x 0 , . . . , x n ). A continuous spacetime is one where the metric is continuous, that is, its components are continuous functions with respect to any coordinates.
Let (M, g) be a continuous spacetime. Our definition of timelike curves and the timelike future and past, I ± , will also follow [29]. In particular, a future directed timelike curve γ : [a, b] → M is a Lipschitz curve that's future directed timelike almost everywhere and satisfies g(γ ′ , γ ′ ) < −ε almost everywhere for some ε > 0. This class of timelike curves contains the class of piecewise  A set in a spacetime is achronal if no two points in the set can be joined by a future directed timelike curve. An important result we will use is the following lemma. 3 For a Milne-like spacetime (M, g), statement (1.11) implies that ρ(τ ) and p(τ ) extend continuously to τ = 0 along each integral curve of u. We will use a slightly stronger version of these "continuous extensions" which we make precise next.
Likewise, a smooth vector field X on M extends continuously to M ∪ {O} provided there is a coordinate neighborhood U of O with coordinates (x 0 , . . . , x n ) such that each of the components X µ in X = X µ ∂ µ extends continuously to (U ∩ M ) ∪ {O}. A similar definition applies to smooth tensors on M by requiring each of its components to extend continuously. (This definition does not depend on the choice of coordinate system by the usual transformation law for tensor components.) For example, the metric tensor g on M extends continuously to M ∪ {O} since (M ext , g ext ) is a continuous extension of (M, g). For another example, suppose T is a smooth tensor on (M ext , g ext ), then obviously the restriction, T | M , extends continuously to M ∪ {O} since it extends smoothly.
We are now ready to state our main result.
We make the following assumptions. is a future directed timelike curve for any ε > 0. Since g(γ ′ , γ ′ ) = −1 almost everywhere, requiring γ to be future directed timelike amounts to γ satisfying a Lipschitz condition. This would be satisfied, for example, if γ was continuously differentiable at τ = 0 (which holds for Milne-like spacetimes).
-Regarding assumption (c), let (M, g) be a Milne-like spacetime with a scale factor that's analytic at zero: a(τ ) = τ + ∞ 2 c n τ n . If c 2 = 0, then it's easy to see from eq. (1.10) that ρ and p diverge as τ → 0. So our assumption that ρ and p extend continuously to M ∪ {O} is similar to setting c 2 = 0 in statement (1.11). Moreover, if c 2 = 0 and c 4 = 0, then the Ricci tensor, Ric, of (M, g) extends continuously to M ∪{O}. (In fact Ric extends continuously to M ∪∂ − M ). This follows from [30,Lem. 3.5] since Ric can be written as a sum of products of the metric, its inverse, and their first and second derivatives along with the fact that the inverse metric is as regular as the metric.
-Regarding assumption (d), recall that (M ext , g ext ) is strongly causal at O means that for any neighborhood Then u * is a continuous extension of u * to M ∪ {O}. Let u denote the vector field metrically equivalent to u * (i.e. its components are given by u µ = g µν ext u ν ). Then u is a continuous extension of u to M ∪ {O}.
(2) x 0 is a time function on U , Here η µν are the usual components of the Minkowski metric with respect to the coordinates (x 0 , . . . , x n ). That is, By choosing U even smaller, we can also assume that where η ε is the narrow Minkowskian metric on U given by Moreover, since u is a continuous extension of u to M ∪ {O}, we can also assume that . Lastly, if φ : U → R n+1 denotes the coordinate map (i.e. φ = (x 0 , . . . , x n )), then, by restricting the domain of φ, we can assume that The claim follows by strong causality of (M ext , g ext ) at O. To see this, note that strong causality implies that there is a neighborhood V ⊂ U of O such that if γ has endpoints in V , then the image of γ is contained in U . Let V ′ ⊂ V denote a neighborhood of O satisfying assumption (6) above. Then we work in V ′ to construct the curve γ in exactly the same way as in the paragraph above the claim. Then strong causality implies that the image of γ is contained in U . This proves the claim.
A careful inspection of the proof of Theorem 2.2 reveals that assumption (d) is only used to prove the claim in the proof. The next theorem shows that one can replace assumption (d) with (d ′ ). Essentially, (d ′ ) says that ∂ − M looks like figure 2 at least locally near O.  Seeking a contradiction, suppose this is not the case. Define Then τ 0 > 0 by assumption and γ(τ 0 ) ∈ ∂U . Since ε = 3/5 (and hence lightcones are contained within wider Minkowski lightcones with slope 1/2), applying [29, Lem. 2.9 and 2.11] shows that Since M = I + (O, M ext ), we have ∂ + M = ∅. Therefore, by Lemma 2.1, ∂ − M is an achronal topological hypersurface. Since it's a topological hypersurface, we can assume that it separates U by shrinking U if necessary. The separation is given by the following disjoint union We have q ∈ I + (∂ − M, U ). By (i) and (ii), it follows that there must be some . In fact, future timelike geodesic completeness was shown to be sufficient [33].
Proof. This follows from tracing the Einstein equations and using ρ = − p at O.
In this section, we have always assumed that Ric extends continuously to M ∪ {O}. Finding sufficient conditions on the perfect fluid (u, ρ, p) for when this happens is perhaps an interesting question, but this will not be explored here.
An inflationary era is characterized by an accelerated expansion, a ′′ (τ ) > 0, right after the big bang but before the radiation dominated era. It's speculated to occur since it solves certain problems in cosmology (e.g. the horizon and flatness problems) and predicts that the spectrum of density perturbations is scale-invariant. For a nice introduction on inflationary theory, see [28]; for a more thorough account, see [43]. The significance of a ′′ (τ ) > 0 is that it violates the strong energy condition which holds for all known physical matter models, e.g. dust and radiation. (It's also the energy condition appearing in Hawking's cosmological singularity theorems.) Therefore, if the energy-momentum tensor was dominated by radiation in the early universe immediately after the big bang, then an inflationary era cannot occur. Some other matter model, which violates the strong energy condition, must be used to generate an inflationary era.
To account for an inflationary era, one normally introduces an "inflaton" scalar field φ in a slow-roll potential. If the energy-momentum tensor was dominated by the scalar field, then the slow-roll potential implies a ′′ (τ ) > 0. This is not our approach. Instead, we obtain an inflationary era from the geometry of the spacetimes considered in the previous section. The geometry of these spacetimes, encoded in the assumptions of Theorems 2.2/2.3, imply that the cosmological constant appears as an initial condition which implies a "quasi de Sitter" expansion. We show this next.
If u is a geodesic vector field (which is the case for FLRW spacetimes), then ∇ u u = 0 and so we only require that 2σ 2 is sufficiently less than 8π ρ(O) to obtain a ′′ > 0. Recall that 2σ 2 measures the rate of shear of the flow; it's zero for FLRW spacetimes and, in fact, zero for any fluid flow with uniform expansion. In this sense, assuming 2σ 2 is sufficiently small can be thought of as a substitute for the spatial isotropy associated with FLRW spacetimes.