Maximal extensions and singularities in inflationary spacetimes

# Maximal extensions and singularities in inflationary spacetimes

Daisuke Yoshida Department of Physics, McGill University, Montréal, QC, H3A 2T8, Canada    Jerome Quintin Department of Physics, McGill University, Montréal, QC, H3A 2T8, Canada
###### Abstract

Extendibility of inflationary spacetimes with flat spatial geometry is investigated. We find that the past boundary of an inflationary spacetime becomes a so-called parallely propagated curvature singularity if the ratio diverges at the boundary, where and represent the time derivative of the Hubble parameter and the scale factor, respectively. On the other hand, if the ratio converges, then the past boundary is regular and continuously extendible. We also develop a method to judge the continuous () extendibility of spacetime in the case of slow-roll inflation driven by a canonical scalar field. As applications of this method, we find that Starobinsky inflation has a parallely propagated curvature singularity, but a small field inflation model with a Higgs-like potential does not. We also find that an inflationary solution in a modified gravity theory with limited curvature invariants is free of such a singularity and is smoothly extendible.

## I Introduction

Spacetime singularities remain deep mysteries in gravitational theories. According to the singularity theorems of Penrose and Hawking Penrose (1965); Hawking (1967); Hawking and Ellis (1973), the occurrence of singularities is inevitable in General Relativity when the stress-energy tensor satisfies some energy conditions. For example, the singularity theorems ensure the presence of the initial Big Bang singularity in cosmology, also known as the initial singularity problem, which is characterized by incomplete timelike geodesics, provided the strong energy condition is satisfied.

Inflation Guth (1981) is a well-studied structure formation scenario for the very early universe, and it is supported by recent observations of the cosmic microwave background anisotropies Ade et al. (2016a, b). Since inflationary cosmology violates the strong energy condition, it was expected to solve the initial singularity problem. For example, Starobinsky’s inflationary model Starobinsky (1980) was originally proposed as a possible resolution to the initial singularity problem. However, even though the strong energy condition is violated, it does not guaranty the absence of a singularity. In fact, Borde and Vilenkin Borde and Vilenkin (1994, 1996) showed that eternal inflation111Strictly speaking, the theorem was shown under the assumption that the volume of the past of an inflationary region is finite (assumption D in Ref. Borde and Vilenkin (1996)) rather than the assumption of eternal inflation itself. However, this assumption is naturally satisfied in eternal inflation models based on the old inflation scenario as explained in Ref. Borde and Vilenkin (1996). models are null-geodesically incomplete to the past provided the null energy condition is satisfied. After that, another interesting theorem was shown by Borde, Guth and Vilenkin Borde et al. (2003). They generalized the concept of Hubble parameter to a general (inhomogeneous and anisotropic) spacetime where comoving geodesic congruence is defined. Then, they showed that a geodesic is incomplete if the averaged Hubble parameter along this geodesic is positive. Interestingly, neither the null energy condition nor eternal inflation is assumed, and the theorem can be applied to a very wide class of inflationary models. This result motivates the exploration of alternative very early universe scenarios that could evade the assumptions of the theorem to be past complete (see, e.g., Refs. Mukhanov and Brandenberger (1992); Brandenberger et al. (1993); Yoshida et al. (2017); Creminelli et al. (2016) and references therein).

Let us quickly review the analysis of Ref. Borde et al. (2003) with an emphasis on the case of a flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. In this case, the averaged Hubble parameter along a null geodesic is defined by

 Hav[tf,ti]≡1λ(tf)−λ(ti)∫λ(tf)λ(ti)dλH(λ), (1)

where is an affine parameter of a null geodesic at time , and is the Hubble parameter. One can show that the integral on the right-hand side is smaller or equal than unity by suitably choosing the affine parameter (see Ref. Borde et al. (2003)). Thus, if , one obtains

 0< limti→−∞1λ(tf)−λ(ti)∫λ(tf)λ(ti)dλH(λ) ≤ limti→−∞1λ(tf)−λ(ti). (2)

This means that the affine parameter has to be finite in the limit where , and consequently, the corresponding flat FLRW spacetime is past incomplete. If the initial stage of the Universe is described by inflation, then the averaged Hubble parameter must be positive. Therefore, there appears to be no hope of avoiding the past incompleteness of any inflationary flat FLRW spacetime.

Nonetheless, it might be possible to extend the flat FLRW spacetime beyond the end points of the incomplete geodesics, which we call the past boundary . The important observation here is that the above discussion can be applied even when the spacetime is exactly flat de Sitter space. This implies that flat de Sitter space must be past incomplete. This does not contradict the fact that de Sitter space is free of singularities, because the flat patch of de Sitter space covers only half of the entire de Sitter space (see Fig. 1). Thus, even though flat de Sitter space is actually past incomplete, is not singular, and the spacetime can be extended to the entire nonsingular de Sitter space beyond . In the context of the general setting of Ref. Borde et al. (2003), this would happen when the comoving geodesic congruence, which defines the Hubble parameter, does not cover the entire spacetime.

Since inflationary spacetimes are effectively described by flat de Sitter space, it is natural to expect that the past incompleteness of inflation in a flat FLRW coordinate patch is just an apparent one and that there might exist a nonsingular maximal extension of it. The purpose of this paper is to explore such extendibility of inflationary spacetimes with flat spatial geometry.

The question that must be answered is how one can judge the extendibility of the past boundary . The classification of a boundary of spacetime was studied in Ellis and Schmidt (1977) (see also Hawking and Ellis (1973)). In Ref. Clarke (1973, 1982), it was shown that a spacetime is locally inextendible (or inextensible) if and only if any component of the Riemann tensor and its covariant derivatives measured in a parallely propagated (p. p.) tetrad basis diverges in the limit to the boundary. This kind of singularity is called a p. p. curvature singularity Hawking and Ellis (1973) or simply a curvature singularity Ellis and Schmidt (1977). We note that the more common scalar curvature singularity, where a scalar curvature invariant blows up, is necessarily also a p. p. curvature singularity. However, the converse is not always true, i.e., it is possible that a p. p. curvature singularity may not be a scalar curvature singularity. This kind of singularity is called an intermediate singularity Ellis and King (1974) or a nonscalar singularity Ellis and Schmidt (1977). One notes that a locally extendible spacetime also includes a globally inextendible one. A boundary that can be extended locally but not globally is called a locally extendible singularity Ellis and King (1974) or a quasi-regular singularity Ellis and Schmidt (1977). A conical singularity and the singularity in Taub-NUT spacetime correspond to locally extendible singularities (see, e.g., Ref. Ellis and Schmidt (1977)). A globally extendible boundary is called a regular boundary, and there is no obstacle to extend spacetime beyond this boundary. An example of this is the boundary of flat de Sitter space discussed above. To summarize, the finiteness of the components of the Riemann tensor with respect to a p. p. tetrad basis is crucial to discuss the extendibility of spacetimes, at least locally.

Our paper is organized as follows. In the next section, we construct an affine parametrization of a null geodesic and explicitly demonstrate its past incompleteness as . Then, in Sec. III, we explicitly construct a p. p. tetrad for flat FLRW spacetimes and discuss the extendibility of inflationary cosmologies. We find that a flat FLRW spacetime is continuously inextendible if the quantity diverges in the limit . On the other hand, it is continuously extendible if is finite in the limit . In Sec. IV, we give a concrete method to extend the spacetime (when it is shown to be extendible). After that, we give examples of analytic spacetimes in Sec. V. There, we demonstrate the maximal extension of each inflationary spacetime or show the presence of the p. p. singularity explicitly. In Sec. VI, we discuss the implications for inflationary models. Specifically, we derive the condition for single field slow-roll models to be continuously extendible, and we find that the Starobinsky model has a continuous p. p. singularity, but a small field inflation model does not. Moreover, we investigate the absence of p. p. curvature singularities in a model of modified gravity studied in Refs. Mukhanov and Brandenberger (1992); Brandenberger et al. (1993); Yoshida et al. (2017). The final section is devoted to the summary and discussion.

## Ii Setup and past incompleteness of inflationary spacetimes

Throughout this paper, we focus on flat FLRW spacetime,

 gμνdxμdxν=−dt2+a(t)2(dr2+r2dΩ2(2)), (3)

where is the metric of the unit 2-sphere. We assume that the early stage of cosmological evolution is described by inflationary exponential expansion, which leads to the past boundary as we will see below. Precisely, we assume that the comoving geodesics (for which the spatial coordinates , , and are constants) are past complete222We note that extensions of spacetimes where the comoving geodesics are past incomplete, e.g. cosmologies with an initial Big Bang singularity, was discussed in Ref. Klein and Reschke (2018)., i.e., the comoving time is defined all the way to , and the scale factor approaches that of de Sitter space in the limit where . Specifically, the assumption is that, asymptotically,

 a(t)≃¯aeHΛtas t→−∞, (4)

where and are positive constants. In that limit, represents the Hubble parameter since . Throughout this paper, a dot denotes a derivative with respect to physical time .

Under the above assumption, we can directly confirm the incompleteness of a null geodesic. In order to construct an affine parametrization of a null geodesic, it is useful to introduce the conformal time defined by

 η(t)≡∫tdt′a(t′). (5)

Then, it is straightforward to see that a curve parametrized by as follows,

 xμ(~λ)=(η(~λ),r(~λ),θ(~λ),ϕ(~λ))=(~λ,−~λ,0,0), (6)

is a null geodesic. However, the parameter is not an affine parameter because the right-hand side of the geodesic equation,

 ~kν∇ν~kμ=2∂ηaa~kμ, (7)

does not vanish. Here, is the tangent vector of the null geodesic (6), and it is given by

 ~k=~kμ∂μ=dxμ(~λ)d~λ∂μ=∂η−∂r. (8)

An affine parameter of the null geodesic (6) can be derived by re-parameterizing . Let us consider a new parameter and its corresponding tangent vector . Then, the tangent vector with respect to can be expressed in terms of as

 ~kμ=dxμd~λ=dλd~λdxμdλ=dλd~λkμ, (9)

and one can derive the geodesic equation for :

 kν∇νkμ=−1(∂ηλ)2(∂2ηλ−2∂ηaa∂ηλ)kμ. (10)

In order for to be an affine parameter, the right-hand side of Eq. (10) has to vanish. This is only the case if

Thus, the null geodesic (6) is past complete if and only if the affine parameter diverges in the limit where , i.e., if the integral

 λ[tf,−∞]=∫tf−∞dt a(t) (12)

is infinite for arbitrary . In the case of inflation, with the assumption (4), the integral (12) converges:

 λ[tf,−∞]≃¯a∫tf−∞dt eHΛt=¯aHΛeHΛtf. (13)

Therefore, flat FLRW spacetime with the assumption (4) has incomplete null geodesics, and there is a past boundary .

In the next section, we construct a p. p. tetrad basis along these incomplete null geodesics and discuss the extendibility.

## Iii The parallely propagated curvature singularity in inflationary spacetimes

To discuss the local extendibility of an inflationary spacetime, one needs to construct a p. p. tetrad basis along the null geodesic from the previous section since that is the basis which is well defined on the boundary . First, one can construct a simple tetrad , which we call the FLRW tetrad, as follows:

Though the FLRW tetrad components are p. p. along comoving timelike geodesics, they are not parallel along the null geodesic of the previous section. This can be confirmed by calculating . Indeed, the covariant derivatives of and along are nonvanishing:

 kμ∇μ^e0 =Hdr; (15a) kμ∇μ^e1 =Hdη. (15b)

We note that only and are p. p. along .

Let us now construct a p. p. tetrad simply by parallel transport of the FLRW tetrad at a point along the null geodesic. Since any two tetrad bases are related through a Lorentz transformation, we can write a p. p. tetrad basis as a Lorentz transformation of the FLRW tetrad,

 e0=coshζ(η,r)^e0+sinhζ(η,r)^e1, (16a) e1=sinhζ(η,r)^e0+coshζ(η,r)^e1, (16b)

with and . The Lorentz factor (or rapidity) satisfies so that coincides with at the given point . Then, the covariant derivative of along can be expressed in terms of the Lorentz factor as

 kμ∇μe0=∂ηa+a(∂ηζ−∂rζ)a3e1, (17a) kμ∇μe1=∂ηa+a(∂ηζ−∂rζ)a3e0. (17b)

For the ’s to be parallely propagated, the right-hand sides in Eq. (17) have to vanish. Thus, one solution for the Lorentz factor is

 ζ=−ln(aa0), (18)

where is the value of the scale factor at . In the limit where , i.e. toward the boundary , the expression (18) tells us that the p. p. basis is obtained by parallel transport through an infinite boost from the FLRW tetrad basis. Since the p. p. tetrad is a well-defined basis to discuss the extendibility of the curve beyond , this fact also tells us that the FLRW tetrad basis is ill defined in the limit to the boundary . By plugging (18) into the definition of , we obtain concrete expressions for our p. p. tetrad,

 e0= 12(1+a2a20)a0a^e0+12(1−a2a20)a0a^e1, (19a) e1= 12(1−a2a20)a0a^e0+12(1+a2a20)a0a^e1. (19b)

Inversely, the original FLRW tetrad can be expressed in terms of the p. p. tetrad as

 ^e0= 12(1+a2a20)a0ae0−12(1−a2a20)a0ae1, (20a) ^e1= −12(1−a2a20)a0ae0+12(1+a2a20)a0ae1. (20b)

As studied in Refs. Ellis and Schmidt (1977); Clarke (1973, 1982); Hawking and Ellis (1973), components of the Riemann tensor (and derivatives thereof) are crucial to discuss the local extendibility of a boundary. Since flat FLRW spacetime is conformally flat, any independent components of the Riemann tensor is described by the Ricci tensor. In the FLRW tetrad basis , the Ricci tensor can be expanded as follows,

 Rμνdxμ⊗dxν= −2˙H^e0⊗^e0 +(3H2+˙H)ηMN^eM⊗^eN, (21)

where denotes the Minkowski metric tensor with tetrad indices. Since and by the condition (4), the components of the Ricci tensor with respect to the FLRW tetrad are finite. Thus, there appears to be no ill behavior for comoving timelike observers. Also, any scalar curvature invariant constructed from the Ricci tensor is finite in the limit to the boundary . Nevertheless, any component of the Ricci tensor with respect to the p. p. tetrad could possibly diverge, because the p. p. tetrad is related to the FLRW tetrad through an infinite boost. Concretely, by using the expressions (20), we can write

 Rμνdxμ⊗dxν= ˙Ha202a2⎡⎣−(1+a2a20)2e0⊗e0+2(1−a4a40)e(0⊗e1)−(1−a2a20)2e1⊗e1⎤⎦ +(3H2+˙H)ηMNeM⊗eN, (22)

where in general is shorthand notation for . From the above, we can see that the , , and components include the possibly divergent quantity as and . Therefore, we arrive at the following statement, the key result of the present work:

The boundary is a p. p. curvature singularity, or more precisely an intermediate singularity, if

 ∣∣∣limt→−∞˙Ha2∣∣∣=∞, (23)

and consequently, the corresponding inflationary spacetime cannot be extended beyond . On the other hand, if converges, then the past boundary is not singular, and it can be extendible at least locally.

The above statement relies solely on the behavior of the Ricci tensor (not on its derivatives), so the statement is pertaining to continuous () extendibility and p. p. curvature singularities. The precise definition of p. p. curvature singularities and extendibility for any integer can be found in appendix A, and as expected, the criterion depends on the behavior of the -th covariant derivative of the Ricci tensor. For example, continuously differentiable () extendibility is possible if the first covariant derivative of the Ricci tensor is continuous and does not diverge, i.e., must be of class . Similarly, smooth (infinitely differentiable or ) extendibility requires the Ricci tensor to be smooth as well.

In what follows, we focus on continuous () extendibility, and that is what is implicitly meant unless specified. Parallely propagated curvature singularities of class , and all the way to are ‘weaker’ or ‘milder’ in the sense that they may only involve the divergence of quantities with higher derivatives of the Hubble parameter (see appendix A), e.g. quantities like , , etc.

## Iv Coordinates beyond the past boundary

In previous section, we found that an inflationary spacetime is possibly past extendible if the quantity is finite all the way to the infinite past. The question that is raised in this case is how one can extend the spacetime beyond the boundary. In this section, we construct a new set of coordinates for the flat FLRW spacetime analogously to Eddington-Finkelstein coordinates in Schwarzschild spacetime. Let us consider a new set of coordinates defined by

 λ≡λ[t,−∞]=∫t−∞dt′a(t′), (24) v≡η+r. (25)

Note that, with these coordinates, the null geodesic (6) corresponds to the curve characterized by . Now, the affine parameter is chosen so that corresponds to the past boundary . Using the relations

we can write the FLRW metric in terms of the new coordinates as

 gμνdxμdxν=−2dλdv+a2dv2+a2r2dΩ2(2). (28)

In the limit toward , the quantity converges under the assumption (4), because it can be evaluated as

 ar∼−aη∼−(¯aeHΛt)(−1¯aHΛe−HΛt)=1HΛ. (29)

Thus, the metric is regular at , and it is given by

 gμνdxμdxν∣∣B−=−2dλdv+1H2ΛdΩ2(2). (30)

We note that the metric components are regular on even if is a p. p. curvature singularity, because does not appear in the expression (30). In the inextendible case, an ill behavior only appears in the components of the Ricci tensor, which can be expressed as

 Rμνdxμdxν=−2˙Ha2dλ2+(3H2+˙H)gαβdxαdxβ. (31)

As it is clear from Eqs. (30) and (31), when converges, there is no ill behavior at . Thus, the spacetime can be extended to by making use of this coordinate system.

## V Examples

### v.1 Exact de Sitter spacetime

As a first example of our general discussion, we first demonstrate the maximal extension of flat de Sitter space, where the scale factor is exactly given by

 a(t)=¯aeHΛt. (32)

Here, the FLRW time coordinate is defined in the region . Since the affine parameter can be evaluated as

 λ=¯aeHΛtHΛ, (33)

the coordinate region corresponds to , hence the null geodesics are past incomplete. From Eq. (33), the scale factor can be written as a function of as

 a(λ)=HΛλ. (34)

Also, the conformal time can be evaluated as

 η=−e−HΛt¯aHΛ=−1H2Λλ. (35)

Since the coordinate region corresponds to , flat de Sitter space is conformally isometric to the lower half of Minkowski spacetime as shown by the upper triangle of Fig. 1. Using Eqs. (33) and (35), we can write the metric in the coordinates as

 gμνdxμdxν= −2dλdv+H2Λλ2dv2 +1H2Λ(1+H2Λλv)2dΩ2(2). (36)

Now, the metric tensor is well defined even for nonpositive values of . Thus, we can extend333In the language of appendix A, exact de Sitter space is one of the examples where the spacetime is actually smoothly () extendible. flat de Sitter space with to the entire de Sitter spacetime with . Since is nothing but the affine parameter of a null geodesic, this geodesic is now complete in the entire de Sitter spacetime. We note that with the closed FLRW coordinates of the entire de Sitter spacetime, also known as the global coordinates, the line element is given by

 gμνdxμdxν=−dt2g+cosh2(HΛtg)H2Λ(dψ2+sin2ψdΩ2(2)), (37)

where is the global time coordinate, and is the third angle describing the 3-sphere with line element . The above metric can be obtained from our coordinates by the following coordinate transformation:

 λ =1H2Λ(cosh[HΛtg]cosψ+sinh[HΛtg]); (38) v =−1−eHΛtgtan(ψ/2)eHΛtg+tan(ψ/2). (39)

### v.2 Inextendible toy model

As a toy model, let us consider a flat FLRW spacetime with scale factor given by

 (40)

This scale factor represents a universe which starts from de Sitter and approaches Minkowski at . One can evaluate the conformal time as

 η=t−e−HΛt¯aHΛ, (41)

and since the coordinate region corresponds to , this spacetime is conformally isometric to the whole Minkowski spacetime. However, since

 ˙Ha2=−H2Λ¯aeHΛt→−∞as t→−∞, (42)

the past boundary is singular (i.e. it is a p. p. curvature singularity) according to the key result of the previous section. The Penrose diagram of this spacetime is shown in Fig. 2.

Let us explicitly see the inextendible nature of the spacetime by deriving the metric components in our new coordinate system . Following the definition in Eq. (24), can be written as

 λ=H−1Λln(1+¯aeHΛt). (43)

Therefore, the scale factor and the conformal time can be written in terms of as

 a(λ) =1−e−HΛλ, (44) η(λ) =1HΛ(ln[eHΛλ−1¯a]−1eHΛλ−1). (45)

Thus, we can evaluate as

 aη=e−HΛλHΛ(ϵln[ϵ¯a]−1), (46)

where is a function of defined by

 ϵ(λ)≡eHΛλ−1. (47)

Then, the metric can be written as

 gμνdxμdxν= −2dλdv+(1−e−HΛλ)2dv2 +e−2HΛλH2Λ(ϵln[ϵ¯a]−1−HΛϵv)2dΩ2(2). (48)

As one takes the limit towards , the above expression becomes exactly equal to Eq. (30), and so the metric components are well defined in that limit. However, the metric is not differentiable in that limit because of the term of the form . Thus, the Ricci tensor is not , and the spacetime cannot be extended beyond .

### v.3 Extendible toy model

Let us consider another toy model with the scale factor given by

 a(t)=¯aeHΛt+e−HΛt. (49)

In the limit , the scale factor behaves as that of a contracting or expanding flat de Sitter universe:

 a(t)≃{¯aeHΛtas t→−∞;¯ae−HΛtas t→∞. (50)

Since the conformal time,

 η=eHΛt−e−HΛt¯aHΛ, (51)

is defined in the parameter region , the flat FLRW region is again conformally isometric to the whole Minkowski spacetime. Since is now finite,

 ˙Ha2=−4H2Λ¯a2, (52)

any component of the Ricci tensor is well behaved in the limit to the past null boundary . Therefore, this spacetime is extendible444The toy model here is actually smoothly () extendible just like exact de Sitter space. This is due to the fact that , i.e. , and it is shown in appendix A that extendibility follows if as with . The present case corresponds to . beyond the past null boundary the same way flat de Sitter space can be. The affine parameter is obtained as follows,

 λ=¯aHΛarctaneHΛt, (53)

and so the region corresponds to . Hence the original region is both future and past incomplete. Let us denote the future null boundary at by . The scale factor can then be written in terms of as

 a(λ)=¯a2sin(2HΛλ¯a), (54)

hence

 a(λ)η(λ)=−1HΛcos(2HΛλ¯a), (55)

and so the metric in the original region can be written as

 gμνdxμdxν= −2dλdv+¯a24sin2(2HΛλ/¯a)dv2 +14H2Λ(2cos[2HΛλ¯a] +¯aHΛvsin[2HΛλ¯a])2dΩ2(2). (56)

The important point here is that the scale factor and the components of the metric are well defined in the whole parameter region , and therefore, we can extend the spacetime to that entire region. Since the scale factor is periodic, each spacetime region is characterized by the quantity

 2πHΛ¯aλ∈(n,n+1), (57)

where denotes the -th copy of the original spacetime. The Penrose diagram of the entire spacetime is depicted in Fig. 3.

The resulting maximally extended geodesically complete spacetime is a cyclic universe with periodically repeating phases of expansion and contraction, and the transitions from contraction to expansion at are bounces. With given by Eq. (54), we notice that , so the scale factor vanishes at each bounce point. Yet, those bounces are nonsingular since the boundary region at cannot be described by the usual FLRW coordinate system. Rather, we see that the metric of Eq. (56) reduces to Eq. (30) when , which is the correct description of the nonsingular bouncing surfaces. In other words, the ‘physical’ scale factor is the one with respect to the comoving observer in the flat FLRW spacetime, and it is meaningless on (or on any other boundary region at ). This is very similar to the case of de Sitter space, where at , the scalar factor given by Eq. (34) would appear to be singular. However, there exists a coordinate system in which the bounce is explicitly nonsingular, as can be seen from Eq. (37).

## Vi Implications for inflationary models

### vi.1 Energy condition for subleading component

We have seen that convergence of the quantity is crucial to extend an inflationary cosmology beyond the past null boundary . Since there is no contribution from vacuum energy to in Einstein gravity, the behavior of is determined by the next-to-leading order contribution to the energy density during inflation. In order to describe this situation effectively, let us consider Einstein gravity with a cosmological constant and a fluid which follows the equation of state,

 P=wρ, (58)

where is a constant, and are the pressure and the energy density of the fluid, respectively. We note, though, that a fluid description with the above equation of state might not be a fully realistic situation for inflation, but it serves as an effective description. By solving the conservation equation for the fluid as usual, we obtain

 ρ∝a−3(1+w). (59)

We now assume that so that the vacuum energy is dominant in the total energy density, i.e.

 ρtot=Λ+ρ≃Λas a→0. (60)

In that case, the evolution of the Hubble parameter is dominated by the vacuum energy , and our assumption (4) is realized. Then using the Friedman equations, we can write the key quantity as a function of the scale factor ,

 ˙Ha2=−12M2Plρ+Pa2∝1+wa5+3w, (61)

which converges in the limit only if or . The former case exactly corresponds to the vacuum energy . Thus, if there is a correction to the vacuum energy, the equation of state of the next-to-leading order component has to satisfy the latter condition,

 w≤−53. (62)

This clarifies the discontinuous nature between exact de Sitter space and inflationary cosmology satisfying Eq. (4): although the spacetime is nonsingular if is exactly equal to , any arbitrary small deviation from leads to a p. p. curvature singularity. The singularity is avoided only if the next-to-leading order component to the energy density goes to zero fast enough as . The above result shows that it is not enough to violate the Null Energy Condition with since an equation of state parameter in the range would still lead to a p. p. curvature singularity.

We note that the condition generally ensures extendibility. However, if is an integer in addition to the condition (i.e. if ), then the spacetime is extendible as shown in appendix A. If is not an integer multiple of but still , then the spacetime is at most extendible.

### vi.2 Single field slow-roll inflation

Let us derive the expression for the key quantity in the case of slow-roll inflation models driven by a canonical scalar field with a potential . The complete set of equations of motion is given by555From here on, a prime denotes a derivative with respect to the argument of the function.

 H≃ ⎷V(φ)3M2Pl,˙φ≃−V′(φ)3H, (63)

provided the following slow-roll approximations are satisfied:

 |¨φ|3H|˙φ|≪1,˙φ22V(φ)≪1. (64)

From these equations, we can write the scale factor as a function of ,

 a(φ)=aee−N(φ), (65)

with , the e-folding number, given by

 N(φ)≃1M2Pl∫φφedφ VV′. (66)

Here, and are the values of and at the end of inflation respectively.

Using Eqs. (63) and (65), we can then write the key quantity in terms of ,

 ˙Ha2≃−16a2(V′)2V=−16a2e(V′)2Ve2N(φ)=−16a2ef(φ), (67)

where the function is defined by

 f(φ)≡(V′)2Ve2N(φ)≃(V′)2Vexp(2M2Pl∫φφedφ VV′). (68)

Thus, a necessary condition for a slow-rolling inflationary cosmology to be free of p. p. singularities can be written as

 limφ→φ(−∞)f(φ)=finite, (69)

where is the value of in the limit . Thus, for any given potential , we can judge the presence of a singularity by evaluating (68) and its limit. In the rest of this section, we evaluate (69) for the Starobinsky model and for a small field inflationary model.

#### Starobinsky model

Let us consider the Starobinsky model Starobinsky (1980) with Einstein frame potential given by

 V(φ)=34m2M2Pl(1−e−√23φMPl)2, (70)

where is the ‘mass’ of the inflaton. Inflation occurs at large positive field values, and slowly rolls toward as time increases. Inversely, approaches in the limit . In that limit, , so the potential acts like a cosmological constant, i.e. the spacetime is asymptotically de Sitter. With the above potential, the e-folding number can be evaluated following Eq. (66), and one finds

 N(φ)≃34e√23φMPlas φ→∞. (71)

Since the quantity is given by

 (V′)2V=2m2e−2√23φMPl, (72)

it cannot suppress the factor of in the expression for . Indeed,

 f(φ) =2m2e2N(φ)−2√23φMPl ≃2m2exp(32exp[√23φMPl])→∞ (73)

as . We see from Eq. (72), which is proportional to , that the Hubble parameter approaches a constant exponentially fast as in field space. This matches the intuition that Starobinsky inflation rapidly approaches de Sitter in field space (at large field values). However, the scale factor reaches zero even faster than as . The subtlety comes from the fact that the potential is very flat at large field values, which implies that large time intervals are needed for small field displacements. Consequently, de Sitter is actually approached only very slowly in physical time compared to the rate at which the scale factor goes to zero. Thus, the ratio and equivalently blow up, and the spacetime is inextendible. We can conclude that if Starobinsky inflation starts from the infinite past at for comoving observers, then the past boundary must be a p. p. curvature singularity.

We note, however, that Starobinsky inflation is unlikely to start from the infinite past in the first place. Indeed, this would require the initial field velocity to exactly vanish, which represents extreme fine-tuning. In general, the field equation of motion is for a nearly flat potential, and with initially, this implies and (kinetic domination as ). Accordingly, the first slow-roll approximation in Eq. (64) would not be satisfied. This is known as ultra-slow-roll or non-attractor inflation (see, e.g., Tsamis and Woodard (2004); Kinney (2005); Cai et al. (2016); Dimopoulos (2017)). In that situation, the effective equation of state parameter would tend to unity as , and the Universe would be past incomplete (standard Big Bang curvature singularity).

Another comment is in order: Starobinsky inflation with the Einstein frame potential given above is equivalent to an modified theory of gravity of the form in the Jordan frame after a conformal transformation. It would be natural to extend the above analysis to inflationary scenarios with different theories of gravity, e.g., slight deviations from Starobinsky inflation or generalizations thereof (see, e.g., Miranda et al. (2017)). This shall be the subject of a follow-up study.

#### Small Field inflation

Let us consider another slow-roll inflation model with a Higgs-like potential Vilenkin (1994); Linde and Linde (1994) of the form

 V(φ)=V0(1−(φ2m)2)2, (74)

where is a positive constant, and is another mass scale. We would like to focus on small field inflation, which occurs when666Chaotic inflation is also possible when is much larger than , but we note that in the case of such chaotic inflation models (and more generally for , ), the inflaton potential energy diverges as (i.e. ). Consequently, the effective description of the inflationary cosmology based on classical gravity would no longer be valid near . Thus, the present analysis cannot address the past extendibility of such inflationary models. . We note, however, that such small field inflation models are unstable against initial condition fluctuations Goldwirth and Piran (1992); Brandenberger (2016). From the above potential, we find that the e-folding number is given by

 N(φ)=−m2M2Plln(φφe)+φ2−φ2e8M2Pl, (75)

and it diverges in the limit . Thus, the limit corresponds to the limit . Then, can be evaluated as follows,

 f(φ)=V0m4φ2(φφe)−2m2M2Pleφ2−φ2e4M2Pl, (76)

and one finds that converges in the limit when . Thus, if small field inflation starts from the infinite past at for comoving observers, then the Universe can be continuously () extended beyond the past boundary . In other words, noncomoving geodesics exit the original inflationary region sufficiently far in the past. However, the above does not tell us whether the past boundary is extendible for , and the spacetime could very well be inextendible. This remains to be verified, but the condition (69) would be much more complicated.

### vi.3 Limiting curvature models

In this subsection, we demonstrate how to evaluate the key quantity in the limit corresponding to de Sitter in a class of modified gravity models. Specifically, we focus on a gravitational theory with limiting curvature, which is claimed to have nonsingular cosmological solutions, as proposed and investigated in Refs. Mukhanov and Brandenberger (1992); Brandenberger et al. (1993); Yoshida et al. (2017) (see also references therein, and Refs. Easson and Brandenberger (1999); Easson (2007); Chamseddine and Mukhanov (2017a) and Refs. Trodden et al. (1993); Chamseddine and Mukhanov (2017b); Yoshida and Brandenberger (2018) for other applications to cosmological and black hole spacetimes respectively). Let us briefly review the theory and the inflationary solutions investigated in Ref. Yoshida et al. (2017). The action of this theory is given by

 S=M2Pl2∫d4x√−g(R+2∑j=1[χjIj−Vj(χj)]), (77)

where and are scalar fields, and and are curvature invariant functions, which reduce to

 IFLRW1=12H2,IFLRW2=−6˙H, (78)

in a flat FLRW spacetime. We note that there are many choices of curvature invariant functions which satisfy the condition (78), and the stability of the cosmological perturbations777See Ref. Yoshida et al. (2017) for examples of covariant curvature invariant functions that reduce to Eq. (78) and for the analysis of the perturbations. strongly depends on the choice of and . However, the background dynamics can be uniquely determined only from the condition (78).

Varying the action (77) with respect to and gives rise to the following constraint equations at the background level,

 12H2= V′1(χ1),−6˙H= V′2(χ2), (79)

which ensure the finiteness of and if and are finite for any value of and . This is the mechanism to limit the divergence of the curvature invariants in this theory. The other independent equation of motion is given by (see Ref. Yoshida et al. (2017) for a derivation)

 (1−2χ1−3χ2)H2−˙χ2H−V1+V26=0. (80)

The above equations of motion can be rewritten as a set of first-order differential equations only involving , , and . In particular, the <