The Big Bang is a Coordinate Singularity for k = − 1 Inflationary FLRW Spacetimes

We show that the big bang is just a coordinate singularity for k = −1 inflationary FLRW spacetimes. That is, it can be removed by introducing a set of coordinates in which the big bang appears as a past Cauchy horizon where the metric is no longer degenerate. In fact this past Cauchy horizon is just the future lightcone at the origin of a spacetime conformal to Minkowski space. For these k = −1 inflationary FLRW spacetimes, we show that the cosmological constant appears as an initial condition, and the Lorentz group acts by isometries.


Introduction
The big bang, τ = 0, in FLRW spacetimes is widely believed to be a genuine singularity. That is, there should exist some infinite energy density or infinite curvature quantity at the big bang. Indeed the scalar curvature is (1.1) Therefore if one assumes the universe is in a radiation dominated era, a(τ ) ∼ √ τ , all the way down to the big bang (i.e. no inflation), then the scalar curvature diverges as τ → 0. Moreover, assuming the strong energy condition, the Hawking-Penrose singularity theorems [3,5] show that a singularity is generically unavoidable. However inflationary eras, a ′′ (τ ) > 0, imply that the strong energy condition must be violated and so the Hawking-Penrose singularity theorems don't apply. 1 Therefore a natural question to ask is: is the big bang still singular? In terms of the energy density ρ(τ ) and pressure function p(τ ), the scalar curvature is R(τ ) = 8πρ(τ ) − 24πp(τ ). (

1.2)
Since the pressure in inflationary eras is negative, we see that the scalar curvature diverges provided the energy density diverges. The divergence of the energy density seems like a physically reasonable assumption, but it is known to not always hold. For example, de Sitter space is partially covered by the so-called 'open slicing' coordinates, and in these coordinates is seen to be a k = −1 FLRW spacetime with scale factor a(τ ) = sinh(τ ), i.e. it's always inflating. In this case we have ρ = −p = Λ/8π is constant. Here Λ = 3 is the cosmological constant. In section 3.2 we show that for k = −1 inflationary FLRW spacetimes, we have lim τ →0 ρ(τ ) = (3/8π)a ′′′ (0) provided a ′′ (0) = 0. Since a ′′′ (0) = 1 for a(τ ) = sinh(τ ), we see de Sitter space is a special case of this limit. Therefore, assuming inflationary theory, there is no reason to believe the big bang is singular except as a coordinate singularity. Indeed in this paper we show that the big bang for k = −1 inflationary spacetimes (which are defined precisely in section 2.3) is just a coordinate singularity. The situation is analogous to how the r = 2m event horizon in Schwarzschild is just a coordinate singularity. We also show some physically interesting cosmological properties of these spacetimes which we list below.
Cosmological properties of k = −1 inflationary FLRW spacetimes -They offer a new geometrical viewpoint on how they solve the horizon problem.
-The comoving observers all emanate from a single point O in the extension.
-The cosmological constant Λ appears as an initial condition.
-An era of slow-roll inflation follows if the initial condition of the potential V (φ) is determined by Λ.
-The Lorentz group acts on these spacetimes by isometries.
1 See [2] for a cosmological singularity theorem without the strong energy condition.

The Coordinate Singularity
The k = −1 FLRW metric is We will show that the big bang, τ = 0, is a coordinate singularity for scale factors which obey an inflationary condition. But first we will demonstrate how the big bang is a coordinate singularity in two familiar examples: the Milne universe and de Sitter space.
Definition 2.1. For an FLRW spacetime, we say τ = 0 is a coordinate singularity if there exist coordinates which can extend the FLRW spacetime into a larger spacetime manifold (i.e. a proper isometric embedding) where the metric at τ = 0 is no longer degenerate.

The Milne Universe
The Milne universe is a k = −1 FLRW spacetime with a(τ ) = τ . The metric is We introduce new coordinates (t, r) via t = τ cosh(R) and r = τ sinh(R). (2.5) Then we have −dt 2 + dr 2 = −dτ 2 + τ 2 dR 2 , so that the metric in coordinates (t, r, θ, φ) is which is just the usual Minkowski metric. The constant τ slices are hyperboloids sitting inside the future lightcone of the origin. As τ → 0, these slices approach the lightcone where the metric is nondegenerate. Therefore τ = 0 is a coordinate singularity.

Inflationary Spacetimes
Now we wish to perform the same extension but with a scale factor a(τ ) that can model the dynamics of our universe. That is, we wish to show τ = 0 is a coordinate singularity for suitably chosen scale factors a(τ ) which -begin inflationary a(τ ) ∼ sinh(τ ) -then transitions to a radiation dominated era a(τ ) ∼ √ τ -then transitions to a matter dominated era a(τ ) ∼ τ 2/3 -and ends in a dark energy dominated era a(τ ) ∼ e Λτ If we assume for small τ , the scale factor satisfies a(τ ) ∼ τ , then, by curve fitting, we can use a(τ ) to represent each of the above eras, thus modeling the dynamics of our universe.
The main motivation for this definition comes in the next section where we show that these FLRW spacetimes solve the horizon problem. The next theorem improves and refines Theorem 3.4 in [1]. Theorem 2.3. The big bang is a coordinate singularity in k = −1 inflationary spacetimes.
Proof. The metric is (2.12) Fix any τ 0 > 0. The specific choice does not matter; any τ 0 will do. Define new coordinates (t, r, θ, φ) by ds . (2.14) Note that b(τ ) is an increasing function and hence it's invertible, and so τ as a function of t and r is given by With respect to these coordinates, the metric takes the form where Just like with the open slicing coordinates of de Sitter space, we see that these inflationary k = −1 FLRW spacetimes are sitting inside a spacetime conformal to Minkowski space. Now we prove that τ = 0 is a coordinate singularity. For this it suffices to show Ω(0) := lim τ →0 Ω(τ ) exists and is a finite positive number. Indeed this will imply the Lorentzian metric given by equation (2.16) extends continuously through τ = 0 which corresponds to the lightcone t = r. To Evaluating the integrals we find Since this holds for all 0 < τ < δ, we have Ω Remark. The k = −1 FLRW spacetimes inherit a past Cauchy horizon given by the future lightcone at O in the larger spacetime conformal to Minkowski space. Therefore what lies beneath this lightcone is only speculation. However, just like with Schwarzschild, when Ω is analytic, then one can consider the maximal analytic extension. For a(τ ) = τ the maximal analytic extension is Minkowski space. For a(τ ) = sinh(τ ), the maximal analytic extension is de Sitter space.

The solution to the horizon problem
Our definition for an inflationary FLRW spacetime was one whose scale factor satisfies a(τ ) = τ + o(τ 1+ε ) for some ε > 0. Our motivation is that these spacetimes solve the horizon problem, and this is true for k = +1, 0, or −1. However, what's unique about the k = −1 case is that it extends into a larger spacetime because the big bang is just a coordinate singularity. This offers a new picture of how the k = −1 inflationary spacetimes solve the horizon problem as we discuss below.
We briefly recall the horizon problem in cosmology. It is is the main motivating reason for inflationary theory [6]. The problem comes from the uniform temperature of the CMB radiation. From any direction in the sky, we observe that the CMB temperature is 2.7 K. The uniformity of this temperature is puzzling: if we assume the universe exists in a radiation dominated era all the way down to the big bang (i.e. no inflation), then the points p and q on the surface of last scattering don't have intersecting past lightcones. So how can the CMB temperature be so uniform if p and q were never in causal contact in the past? But then why does the Earth measure the same 2.7 K temperature from every direction?
By using conformal timeτ given by dτ = dτ /a(τ ), it is an elementary exercise to show that there is no horizon problem provided the particle horizon at the moment of last scattering is infinite: This condition widens the past lightcones of p and q so that they intersect before τ = 0.  Therefore for any ε > 0 there exists a δ > 0 such that |a(τ )/τ − 1| < ε for all 0 < τ < δ. Hence 1/a(τ ) > 1/(1 + ε)τ for all 0 < τ < δ. Then the particle horizon at the moment of last scattering is Thus the particle horizon is infinite.
In the k = −1 case, the origin O plays a unique role. The lightcones of any two points must intersect above O. This follows from the metric being conformal to Minkowski space, g µν = Ω 2 (τ )η µν . As such the lightcones are given by 45 degree angles; see Figure 6, which, in a certain way, clarifies in the k = −1 case the situation depicted in Figure 5. Also we observe that the comoving observers all emanate from the origin O. Indeed a comoving observer γ(τ ) is specified by a point (R 0 , θ 0 , φ 0 ) on the hyperboloid.
In the (t, r, θ, φ) coordinates introduced in equation (2.13), the comoving observer is given by Thus the relationship between t and r for γ is t = coth(R 0 )r. Therefore for any comoving observer, we have t = Cr for some C > 1. Thus the comoving observers emanate from the origin.

The cosmological constant appears as an initial condition
In this section we show how the cosmological constant Λ appears as an initial condition for k = −1 inflationary FLRW spacetimes. Moreover, an era of slow-roll inflation follows if the initial condition for the potential is determined by the cosmological constant.
Consider the Einstein equation with a cosmological constant Let u µ denote the four-velocity of the comoving observers and e µ be any unit spacelike orthogonal vector (its choice does not matter by isotropy). We define the energy density ρ and pressure function p in terms of the Einstein tensor Rg µν e µ e ν = T µν e µ e ν − Λ 8π (3.28) If T µν = 0 (e.g. de Sitter), then the equation of state for the cosmological constant is fixed for all τ .
We show that this equation of state appears as an initial condition for k = −1 inflationary FLRW spacetimes.
Theorem 3.2. Consider a k = −1 inflationary FLRW spacetime. If a ′′ (0) = 0, then Before proving Theorem 3.2, we first understand its implications. If the cosmological constant Λ is the dominant energy source during the Planck era, then we have the following connection between the scale factor and Λ.

Proof. This follows from Theorem 3.2 and equation (3.27).
Remark. In (3+1)-dimensional de Sitter space we have T µν = 0 and Λ = 3. In the open slicing coordinates of de Sitter, we have a(τ ) = sinh(τ ). Hence a ′′′ (0) = 1. Therefore de Sitter space is a special example of Proposition 3.3. Now we examine how an inflaton scalar field behaves in the limit τ → 0. We will demonstrate that slow-roll inflation follows if the initial condition for the potential is given by the cosmological constant: V | τ =0 = Λ/8π. Recall the energy-momentum tensor for a scalar field φ is And its energy density is Proposition 3.4. If T µν → T φ µν → 0 as τ → 0 and a ′′ (0) = 0, then the initial condition V φ(0) = Λ/8π implies φ ′ (0) = 0. Hence it yields an era of slow-roll inflation.
Theorem 3.5. The group L ↑ acts by isometries in k = −1 inflationary spacetimes.
In the case Ω is C 2 we can actually say more. Recall the causal future J + (O) is the union of the k = −1 inflationary spacetime with the future lightcone at O.
If Ω is C 2 in J + (O), then L ↑ is isomorphic to the group of isometries in k = −1 inflationary spacetimes which fix the origin O.
Proofs of Theorems 3.5 and 3.6: Let Λ µ ν be an element of L ↑ . It produces a unique map, x → Λx via x µ → Λ µ ν x ν where x µ = (t, x, y, z) are the conformal Minkowski coordinates introduced in the proof of Theorem 2.3. Since our k = −1 inflationary spacetime is only defined for t > 0, we must restrict to Lorentz transformations Λ ∈ L ↑ . Consider a point p in the spacetime and a tangent vector X = X µ ∂ µ at p. Then Λ acts on X by sending it to dΛ(X) = Λ µ ν X ν ∂ µ and dΛ(Y ) at the point Λp. Since our metric is g µν = Ω 2 (τ )η µν and Ω(Λp) = Ω(p), we have Thus Λ is an isometry. This proves Theorem 3.5. Now we prove Theorem 3.6. By Theorem 3.5 we have L ↑ is a subgroup, so it suffices to show it's the whole group. Suppose f is an isometry which fixes O. The differential map df O is a linear isometry on the tangent space at O. Therefore df O corresponds to an element of the Lorentz group, say Λ µ ν . It operates on vectors X at O via df (X) = Λ µ ν X ν ∂ µ . Now we define the isometryf byf (x) = Λ µ ν x ν . Consider the set Note that if df p = df p , then f (p) =f (p). Hence it suffices to show Since Ω is C 2 , there is a normal neighborhood U about p. If q ∈ U , there is a vector X at p such that exp p (X) = q. Since isometries map geodesics to geodesics, they satisfy the property f • exp p = exp f (p) • df p for all points in U (see page 91 of [4]). Therefore f (q) = f exp p (X) = exp f (p) (df p X) = expf (p) (df p X) =f exp p (X) =f (q).
Thus f (q) =f (q) for all q ∈ U ; hence df q = df q for all q ∈ U . Therefore A is open.
Remark. It would be interesting to understand the implications of the full Lorentz group acting by isometries.

Open problems
(1) Is τ = 0 a coordinate singularity for k = 1 and k = 0 inflationary FLRW spacetimes? From [1] it is known that no extension can exist with spherical symmetry.
(2) Is τ = 0 a coordinate singularity for k = −1 inflationary FLRW spacetimes with compact τ -slices? The null expansion θ of the future lightcone in Minkowski space diverges as one approaches O along the cone. This suggests that in the compact case, the past boundary ∂ − M (as defined in [1]) cannot be compact.
(3) To understand what can lie beyond τ = 0, it is desired to understand the maximal analytic extension whenever Ω is analytic on J + (O). Minkowski space is the maximal analytic extension of the Milne universe. De Sitter space is the maximal analytic extension of the k = −1 FLRW spacetime with scale factor a(τ ) = sinh(τ ). Therefore we suggest Conjecture: Let (M, g) be a k = −1 inflationary FLRW spacetime with an analytic Ω on J + (O). If (M, g) is asymptotically flat (i.e. admits a smooth null scri structure), then the maximal analytic extension contains a noncompact Cauchy surface. If (M, g) is asymptotically de Sitter (i.e. admits a smooth spacelike scri structure), then the maximal analytic extension contains a compact Cauchy surface.