# Eternal Universes

Damien A<EMAIL_ADDRESS>Department of Physics, Arizona State
University, Tempe, AZ 85287-1504, USA  Joseph E<EMAIL_ADDRESS>Department of Physics & Beyond Center for Fundamental Concepts in Science,
Arizona State University, Tempe, AZ 85287-1504, USA

(August 28, 2024)

###### Abstract

We consider the possibility of a past and future eternal universe,
constructing geodesically complete inflating, loitering, and bouncing
spacetimes. We identify the constraints energy conditions in General
Relativity place on the building of eternal cosmological models. Inflationary
and bouncing behavior are shown to be essential ingredients in all significant
examples. Non-trivial complete spacetimes are shown to violate the null energy
condition (NEC) for at least some amount of time, although, some obey the
average null energy condition. Ignoring the intractable subtleties introduced
by quantum considerations, such as rare tunneling events and Boltzmann brains,
we demonstrate that these universes need not have had a beginning or an end.

###### Contents

  1. I Introduction
  2. II Geodesic Completeness
    1. II.1 Singularities
    2. II.2 Inflationary and bouncing cosmologies
  3. III Eternal inflating universe
  4. IV Eternal bouncing universe
    1. IV.1 Transcendental bounce
    2. IV.2 Polynomial bounce
      1. IV.2.1 Quadratic model
      2. IV.2.2 Models from limiting curvature
  5. V An eternal loitering universe
    1. V.1 Loitering eternal universe
    2. V.2 Quasi-loitering eternal universe
  6. VI Discussion
  7. A Energy conditions
    1. A.1 Weak Energy Condition (WEC)
    2. A.2 Null Energy Condition (NEC)
    3. A.3 Dominant Energy Condition (DEC)
    4. A.4 Strong Energy Condition (SEC)
  8. B Theorem 1 Item 3 Proof
  9. C Generalized Robertson Walker Spacetimes
    1. C.1 Definition
    2. C.2 Geodesics in GFRWs
    3. C.3 Derivation of Warped Product Geodesics
    4. C.4 A Useful Constant
  10. D A Diffeomorphism of the Real Line

##  I Introduction

Contemplating the vastness of the universe is an innate aspect of human
curiosity. We are not only drawn to ponder the physical expanse of space and
whether the universe stretches infinitely but also the expanse of the
dimension of time. Could the universe have endured for an eternity into the
past and what will be its ultimate fate? In this paper we explore these
questions from a conservative viewpoint using simple arguments from General
Relativity (GR) and a field theoretic treatment of matter and energy. Our goal
is to be as rigorous as possible using our known, tested laws of physics. We
operate within a framework of metrical geometry and apply the results to
classical Einstein gravity. Within this context we provide examples of
geodesically complete past and future eternal cosmological solutions.

Whether or not the universe had a beginning is a question that naturally
arises within our most successful theory of the early universe, the theory of
inflation Guth (1981); Linde (1982); Albrecht and Steinhardt (1982). Soon
after the theory’s inception it became clear that once inflation started, it
would never stop. While early-time inflation has ceased in our observable
universe, allowing galaxies and structures to form, it continues indefinitely
in other parts of the universe, and hence, inflation is eternal into the
future. Specifically, quantum fluctuations in the field responsible for the
accelerated expansion can cause the field to remain in its high-energy state,
sustaining inflation in those regions indefinitely 198 (1983); Vilenkin
(1983); Linde (1986); Guth (2007). In a Letter by Borde, Guth and Vilenkin
(BGV) Borde  _et al._ (2003), it was subsequently argued that inflation cannot
likewise be eternal into the past; although, we have countered this belief
Easson and Lesnefsky (2024); Lesnefsky  _et al._ (2023a) and present further
evidence that inflation can, in fact, be eternal into the past in the current
work. Acknowledging the possibility of a past eternal universe opens entirely
new avenues of research. We show there are multiple ways that the universe can
sustain itself within classical General Relativity for an eternity, albeit at
the expense of violation of certain energy conditions.

Three broad categories of an eternal universe are suggested. Our examples are
admittedly toy models; however, given the wide variety we expect our findings
to be quite general and applicable to many solutions of interest. In
particular, we will construct eternal inflationary, loitering, and bouncing
cosmologies. In each case we will provide a rigorous proof of full geodesic
completeness. Existing proposed no-go theorems are circumvented or obviated as
discussed in detail in an accompanying Letter Easson and Lesnefsky (2024). We
show that all (non-trivial) models that are eternal geodesically complete
necessarily involve a phase of accelerated expansion (or inflationary phase)
for at least some amount of time and generically include a bouncing phase in
accordance with our reported definitions.

The models we present violate the traditional energy conditions, a property we
will conjecture is necessary in GR for geodesic completeness. 333A contained
discussion of energy conditions is offered in Appendix A. In some instances
violations of the null energy condition (NEC) can be limited to an arbitrarily
short amount of time $\Delta t$, and confined to the primordial past. 444In
its most provocative form, the Heisenberg uncertainty principle is written
$\Delta E\Delta t\geq\hbar/2$. If the time interval is made very small the
uncertainty principle may result in a very large energy fluctuation which can
back-react on the spacetime. Models presented here having such behavior are
capable of satisfying the average null energy condition (ANEC).

In several known circumstances, energy condition violations can be relatively
benign. For example, a simple time-independent cosmological constant $\Lambda$
violates the SEC if $\Lambda>0$ and the WEC and DEC for $\Lambda<0$. The NEC
is the only energy condition which remains valid in the presence of $\Lambda$.
Non-minimally coupled scalar fields are capable of violating all of the
traditional energy conditions Barcelo and Visser (2002); Chatterjee  _et al._
(2013). Indeed, all classical energy conditions are manifestly violated by
quantum effects both experimentally and theoretically Epstein  _et al._
(1965), (e.g., the Casimir Effect Casimir (1948)). Even the NEC is violated
ubiquitously in particle physics, for example, during the Hawking radiation
process local energy density can become negative from the perspective of
certain observers and interpreted as violation of NEC. In theories outside of
pure GR, which can mix gravitational and matter degrees of freedom in a non-
trivial way through non-minimal and higher-derivative terms or couplings, it
becomes difficult to adequately define analogues of energy conditions in the
first place (see e.g., Chatterjee  _et al._ (2013)).

As is well known, violation of energy conditions are necessary in order to
avoid singularities. Within classical GR non-trivial flat cosmological
solutions which avoid singularities must violate energy conditions to
circumvent the singularity theorems Penrose (1965); Hawking (1966a, b, 1967);
Hawking and Penrose (1970); Hawking and Ellis (2023). On occasion, the NEC can
be violated in a stable way Dubovsky  _et al._ (2006); Nicolis  _et al._
(2010); Kobayashi  _et al._ (2010); Deffayet  _et al._ (2010); Sawicki and
Vikman (2013); Rubakov (2014); Cai  _et al._ (2017a); Creminelli  _et al._
(2016); Cai  _et al._ (2017b); Easson and Manton (2019); Alexandre and Polonyi
(2021); Alexandre and Backhouse (2023), sometimes allowing for well-behaved
cosmological bouncing solutions Easson  _et al._ (2011); Cai  _et al._ (2012);
Easson and Vikman (2016); Ijjas and Steinhardt (2016, 2017); Cai and Piao
(2017); Alexandre and Pla (2023).

Currently known consistent quantum field theories in flat spacetime obey the
_averaged_ null energy condition (ANEC), for a complete achronal null geodesic
(those that do not contain any points connected by a timelike curve) Wald and
Yurtsever (1991); Graham and Olum (2007); Hartman  _et al._ (2017).
Effectively, the ANEC condition quantifies the degree of violation of the null
energy condition. The condition is expected to rule out closed timelike
curves, time machines and wormholes connecting different asymptotically flat
regions Morris  _et al._ (1988); Friedman  _et al._ (1993). Matter violating
the ANEC can be used to violate the second law of thermodynamics Wall (2010).

The ANEC may be stated

$\int_{\gamma}T_{\mu\nu}k^{\mu}k^{\nu}d\lambda\geq 0\,,$ (1.1)

where the integral is over a null geodesic $\gamma$ with tangent vector
$k^{\mu}=dx^{\mu}/d\lambda$, and $\lambda$ is an affine parameter with respect
to which the tangent vector to the geodesic is defined. In GR, it is
frequently convenient to replace $T_{\mu\nu}$ in Eq. 1.1 by the Ricci tensor
$R_{\mu\nu}$, simply using the Einstein field equations and the definition of
a null vector. The ANEC has been shown to hold even in the case of arbitrary
Casimir systems as long as the geodesic does not intersect or asymptotically
approach the plates Graham and Olum (2005); Fewster  _et al._ (2007). If the
null geodesic is not assumed to be achronal, the ANEC may be violated by a
quantum scalar field in a spacetime compactified in one spatial dimension or
in the spacetime around a Schwarzschild black hole Visser (1996). As we shall
see some of the eternal models we construct are capable of satisfying the
ANEC.

Given the many nuances involved with respect to energy condition violation and
stability, it can be difficult to pass definitive judgment on the validity of
solutions which experience such violations. In this work we refrain from
making such criticisms as more rigorous analysis of specific situations is
needed; however, it is clear that at least some violations do not lead to
pathologies and ultimately singularity formation is arguably a worst-case
malady plaguing a given physical solution. Yet many physically relevant and
useful solutions in GR contain such singularities, even at the level of the
vacuum solutions, for example, the singular Schwarzschild and Kerr black hole
solutions.

After assembling a diverse grouping of possible eternal universe models, our
analysis demonstrates that there are no known obstructions indicating the
universe must have had a “first moment”. In the context of the inflationary
universe paradigm, this translates into the statement that inflation need not
have had a beginning.

This is the companion paper to our Letter Easson and Lesnefsky (2024). A few
of the Theorems and Corollaries we discuss were presented without proof in our
earlier work Lesnefsky  _et al._ (2023b); Easson and Lesnefsky (2024). The
proofs are revealed presently.

An outline of this paper is as follows. In section II, we prove a theorem
which definitively provides the criteria for geodesic completeness of
generalized Friedmann-Robertson-Walker (GFRW) spacetimes. In section III, we
discuss the definition of inflation and bouncing cosmologies and provide an
example of an eternally inflating universe which is eternal and geodesically
complete both into the past and future. In section IV, we provide examples of
eternal bouncing cosmologies which are geodesically complete both into the
past and future. In section V, we provide examples of loitering (and quasi-
loitering) models which are eternal and geodesically complete both into the
past and future. Finally, in section VI, we present a discussion and our
conclusions. We provide appendices to discuss the energy conditions in General
Relativity with respect to cosmology, to provide proof details and to define
Generalized Friedmann-Robertson-Walker (GFRW) spacetimes.

##  II Geodesic Completeness

We begin with a rigorous discussion of what it means to be geodesically
complete within the context of GFRW spacetimes (which include the familiar FRW
spacetimes). Stated formally, a GFRW spacetime is the warped product - see
Bishop and O’Neill (1969); O’Neill (1983) \- given by
$\mathbb{R}^{1}_{1}\times_{a}\Sigma$ where $\left(\Sigma,g_{\Sigma}\right)$ is
a complete Riemannian manifold and scale factor
$a\in\mathcal{C}^{\infty}\left(\mathbb{R}^{1}_{1}\right)$ which is smooth and
$a>0$. To fully explore the geodesic completeness of a spacetime manifold
$\mathcal{M}$, we must explore _maximal_ geodesics: consider a maximal
geodesic $\gamma:I\to\mathcal{M}$, where $I$ is an _open_ interval of
$\mathbb{R}$, and, because geodesics are usually parameterized with constant
speed, it is uniquely defined up to transversality. A manifold is
_geodesically complete_ if every geodesic $\gamma$, $I=(-\infty,\infty)$, is
defined for all time (see e.g. O’Neill (1983), Prop. 7.38).

We now construct a theorem quantifying the criteria for geodesic completeness
of a GFRW spacetime having scale factor $a$. A geodesic
$\gamma\left(\lambda\right)=\left(t\left(\lambda\right),\beta\left(\lambda\right)\right)$
in a warped product spacetime of555Here we utilize the hyperquadratic notation
of O’Neill (1983) where $\mathbb{R}^{n}_{k}$ is the topological space
$\mathbb{R}^{n}$ with semi-Riemannian metric of signature
$\\{-1,\overset{k-2}{\cdots},-1,+1,\overset{n-k-2}{\cdots},+1\\}$.
$\mathbb{R}^{1}_{1}\times_{a}\Sigma$ obeys

$\nabla_{\frac{\partial}{\partial t}}\frac{\partial}{\partial
t}=g_{\Sigma}\left(\beta^{\prime},\beta^{\prime}\right)\left(a\left(t\left(\lambda\right)\right)\right)\left(\mathrm{grad}_{-dt^{2}}a\right)\,,$
(2.2)

in $\mathbb{R}^{1}_{1}$ and

$\nabla_{\beta^{\prime}}\beta^{\prime}=-\frac{2}{a\left(t\left(\lambda\right)\right)}\left(\beta^{\prime}\right)\frac{d}{d\lambda}\left[a\left(t\left(\lambda\right)\right)\right]\,,$
(2.3)

in Riemannian manifold spacelike leaf $\left(\Sigma,g_{\Sigma}\right)$ where
the geodesic equation $\nabla_{\gamma^{\prime}}\gamma^{\prime}=0$ has been
projected down into the respective spaces $\mathbb{R}^{1}_{1}$ and
$\left(\Sigma,g_{\Sigma}\right)$. The coupled ordinary differential equations
(ODE) can be solved exactly yielding an integral solution as a functional of
$a$. With appropriately selected initial conditions, one finds

$\lambda\left(t\right)=\int^{t}\frac{a\left(\zeta\right)}{\sqrt{a^{2}\left(\zeta\right)+1}}d\zeta\,,$
(2.4)

as an integral solution to Eqs. 2.2 and 2.3 for timelike initial conditions
and

$\lambda\left(t\right)=\int^{t}a\left(\zeta\right)d\zeta\,,$ (2.5)

for null initial conditions. To solve the ODE we invert
$\lambda\left(t\right)$, instead of using $t\left(\lambda\right)$; which is
auspicious, because if Eqs. 2.4 and 2.5 diverge then the domain of $\gamma$ is
cofinite and the geodesic ray has domain $\left[0,\infty\right)$ and is
complete.

Eq. 2.3 defines a pregeodesic and thus has a geodesic reparameterization.
Appropriately reparameterizing, there is a geodesic in $\Sigma$ which has
$\beta$ as its image: thus, if $\left(\Sigma,g_{\Sigma}\right)$ is complete as
a Riemannian manifold, then the limiting factor to geodesic completeness is
Eq. 2.2 with solutions of Eqs. 2.2 and 2.3. The assumption of “generality” in
a GFRW is the least restrictive logical assumption needed to have Eqs. 2.2 and
2.3 characterize geodesic completeness of GFRWs. Hence, the criteria for
geodesic completeness of a GFRW spacetime is given by Thm. 2 of Lesnefsky  _et
al._ (2023b) (which builds on previous work Sanchez (1998)). The theorem
states:

######  Theorem 1.

Let $\mathcal{M}=\mathbb{R}^{1}_{1}\times_{a}\Sigma$ be a GFRW spacetime.

  1. 1.

The spacetime $\mathcal{M}$ is future timelike complete iff
$\int_{t_{0}}^{\infty}\frac{a\left(\zeta\right)d\zeta}{\sqrt{\left(a\left(\zeta\right)\right)^{2}+1}}$
diverges for all $t_{0}\in\mathbb{R}$.

  2. 2.

The spacetime $\mathcal{M}$ is future null complete iff
$\int_{t_{0}}^{\infty}a\left(\zeta\right)d\zeta$ diverges for all
$t_{0}\in\mathbb{R}$.

  3. 3.

The spacetime $\mathcal{M}$ is future spacelike complete iff $\mathcal{M}$ is
future null complete and $a<\infty$.

  4. 4.

The GFRW is past timelike / null / spacelike complete if, for items 1-3 above,
upon reversing the limits of integration from $\int_{t_{0}}^{\infty}$ to
$\int_{-\infty}^{t_{0}}$ the word “future” is replaced by “past”.

  5. 5.

The spacetime $\mathcal{M}$ is _geodesically complete_ iff it is both future
and past timelike, null, and spacelike geodesically complete.

###### Proof.

The definition of a geodesically complete manifold $\mathcal{M}$ is a manifold
where all geodesics are defined for all $\mathbb{R}$. Given the structure of
$\mathbb{R}^{1}_{1}\times_{a}\Sigma$ the geodesic equation projections of Eqs.
2.2,2.3 reduce to an integral over $\mathbb{R}^{1}_{1}$. One must discriminate
the space $\mathbb{R}^{1}_{1}\times_{\exp t}\Sigma$, appropriate for the
cosmological flat slicing of de Sitter space, which is known to be past
geodesically incomplete (see e.g.O’Neill (1983), Example 7.41). The
integration $\int_{\mathbb{R}^{1}_{1}}\frac{d\zeta a}{\sqrt{a^{2}+1}}$
diverges, yet the model is past incomplete, in particular, the mass density
from the (complete) future divergence obfuscates the mass density of the
(incomplete) past convergence.

Using future and / or past directed maximal geodesic rays, for example
$\gamma:\left[0,b\right)\rightarrow\mathcal{M}$, completeness requires for all
initial points $p\in\mathcal{M}$, the endpoint $b\rightarrow\infty$. In the
case of a geodesic $\gamma\left(\lambda\right)=t\left(\lambda\right)\oplus
x^{k}\left(\lambda\right)$ of $\mathbb{R}^{1}_{1}\times_{a}\Sigma$, integral
divergence is required for all future directed geodesic rays

$\int_{t_{0}}^{\infty}\frac{d\zeta
a\left(\zeta\right)}{\sqrt{w_{0}a^{2}\left(\zeta\right)+1}}\,,$ (2.6)

and past directed geodesic rays

$\int_{-\infty}^{t_{0}}\frac{d\zeta
a\left(\zeta\right)}{\sqrt{w_{0}a^{2}\left(\zeta\right)+1}}\,,$ (2.7)

because in both of these cases the integral computes the range of the affine
parameter $\lambda\left(t\right)$ and if $\lambda\left(t\right)$ diverges, no
finite $b$ exists for domain $\left[0,b\right)$. Independent of causal
character this proves item (4). The integral constant $w_{0}$ encapsulates
causal character with $w_{0}\in\left\\{+1,0,-1\right\\}$ for timelike, null,
and spacelike, respectively. Items (1), (2) trivially follow from evaluation
of $w_{0}=+1$ and $w_{0}=0$.

The proof of Item (3) pertaining to spacelike completeness is less
straightforward and not relevant for our primary discussion and hence,
relegated to Appendix B.

∎

###  II.1 Singularities

In our effort to build geodesically complete spacetimes, we pause to further
consider the catastrophes which can prevent completeness. Perhaps the most
common object leading to incomplete geodesics is the curvature singularity,
where a curvature invariant such as the Kretschmann scalar
$K=R_{\mu\nu\kappa\sigma}R^{\mu\nu\kappa\sigma}$, built from the Riemann
tensor, diverges at a particular place, or time. Infamous examples of
curvature singularities lie at the center of the Schwarzschild black hole or
at the cosmological Big-Bang. Generally, curvature singularities occur in
regions of spacetime with high curvature, where quantum gravity effects are
expected to be significant. In some of our models, it is possible to arrange
parameter values so that the energy and curvature scales remain significantly
lower than the Planck regime for the entire cosmological evolution, thereby
avoiding conditions where quantum gravity is necessary. Consequently, quantum
gravity might not play a critical role in the development of an eternal
universe.

In order to avoid cosmological singularities it is necessary to violate
classical energy conditions in GR. These violations are anticipated to
manifest close to the would-be singularity formation points and, occasionally,
can be confined to an arbitrarily brief period. Arguably these energy
condition violations are a lesser inconvenience when weighed against the
formation of singularities, at which point all established physical laws cease
to apply.

Of course, it is possible for a spacetime to be geodesically incomplete even
without encountering a curvature singularity. The most renowned example is the
incompleteness encountered in the flat cosmological slicing of de Sitter
space, where an observer following a geodesic into the past encounters a
boundary in finite proper time (for a recent discussion see Kinney  _et al._
(2023)).

Yet another, arguably less common malady is a _big rip_ , Caldwell  _et al._
(2003), a catastrophic “phantom” force that eventually tears the universe
apart. This type of cataclysm can occur in models which are dominated by NEC
violating matter for a sufficiently long period of time. As a demonstration we
construct the following example. We adopt a perfect-fluid description of
matter with equation of state (EOS)

$p=w\rho\,,$ (2.8)

where $w$ is the equation of state parameter relating the pressure $p$ to the
energy density $\rho$ of the fluid. We now force this fluid to violate the NEC
for all time, thus also violating the ANEC. From the NEC condition
$\rho+p=\rho(1+w)\geq 0$ (see Append. A), the fluid will have EOS parameter
$w<-1$ in order to violate NEC. We take a flat $k=0$ FRW metric, and since
$\dot{H}\propto-\rho(1+w)$, we find NEC violation implies $\dot{H}>0$ for
$\rho>0$. We now take $\dot{H}=\ddot{a}/a-(\dot{a}/a)^{2}=2/\beta^{2}$, for a
positive constant $\beta$, and integrate to get

$a(t)=a_{0}\,\exp{\left[\frac{t}{\alpha}+\left(\frac{t}{\beta}\right)^{2}\right]}\,.$
(2.9)

This spacetime is eternally super-inflating with $\dot{H}>0$ and an energy
density which grows without bound from $t=0$ to $t\rightarrow\infty$ and
violates the NEC for all time. Such a universe cannot sustain itself as it
will inevitably encounter a big rip. The Ricci scalar, for example, is given
by

$R=12\left(\frac{1}{\beta^{2}}+\frac{(\beta^{2}+2\alpha
t)^{2}}{\alpha^{2}\beta^{4}}\right)\,,$ (2.10)

which diverges in a singular fashion as $t\rightarrow\infty$. The Kretschmann
scalar $K=R_{\mu\nu\kappa\sigma}R^{\mu\nu\kappa\sigma}\propto t^{4}$, likewise
diverges at $t\rightarrow\infty$. Geometrically, the geodesically connected
domains666For an FRW spacetime, the observable universe (for a given observer)
is the geodesically connected domain of said observer at that point. The
geodesically connected domain of a point $p\in\mathcal{M}$ consists of all
points which can be reached by a geodesic. In contrast to a geodesically
complete Riemannian manifold, where any two points are guaranteed to be
connected by a geodesic, many geodesically complete Lorentzian manifolds are
_not_ geodesically connected. of any point become arbitrarily small as the
scale factor and curvatures become arbitrarily large. Eventually, the
geodesically connected domain becomes smaller than a Planck volume, although
the affine parameter of a geodesic is still defined. Hence, in general, it is
difficult to construct a physically reasonable eternal cosmological model with
such extended period of NEC violation.

To summarize, a geodesic $\gamma:I\to\mathcal{M}$ may fail be be complete –
where $I\neq\mathbb{R}$ – for any of the above reasons, including a curve
encountering a curvature singularity at a (topological) limit point, a causal
curve encountering a spacelike boundary $\partial\mathcal{M}$ at a limit
point, or light cones on a geodesic “tipping over” where said curve changes
causal character. 777For a more complete discussion of singularities and
geodesic incompleteness see Hawking and Ellis (2023).

###  II.2 Inflationary and bouncing cosmologies

In preparation for our eternal universe constructions and to further hone our
discussion, we offer the following definitions.

######  Definition 1.

Let $\left(\mathcal{M},g\right)$ be an $n$ dimensional spacetime which admits
a connected open neighborhood $U\subset\mathcal{M}$ which is isometric to
$\left(b,c\right)\times_{a}V$ as a warped product open submanifold, where
$\left(b,c\right)$ is a timelike codimension $n-1$ embedded submanifold, $V$
is a connected spacelike codimension 1 embedded submanifold and no assumptions
are made concerning $a(t)$ except that it is a well defined function between
sets. The spacetime $\left(\mathcal{M},g\right)$ is an _inflationary
spacetime_ if there exists some $t_{0}\in\left(b,c\right)$ such that (assuming
that it exists and is well defined), $\ddot{a}\left(t_{0}\right)>0$ where
derivatives of the scale factor are taken with respect to the timelike
$\left(b,c\right)$ coordinate and the sign of $\ddot{a}$ is given with respect
to the time orientation of $\mathcal{M}$.

Note, in the above definition of inflation we make no assumptions about the
continuity of the scale factor $a$ or its derivatives. Furthermore, this
definition includes arbitrarily short periods of accelerated expansion as
inflationary. We are aware some readers will find this objectionable; however,
if one wishes to distinguish the terms “short accelerated expansion” from
“inflation”, one is obliged to set an arbitrary scale for the duration of the
acceleration. While GUT-scale inflation naturally requires at least 60
e-foldings of accelerated expansion to solve the flatness and horizon
problems, low-energy (e.g., TeV-scale) inflation does the job with far fewer.
It is only a few e-foldings of accelerated expansion that are actually probed
in the cosmic microwave background radiation (CMB). In the present context of
an eternal universe the difference between a few e-folding, and 60 e-foldings
is entirely insignificant. For these reasons we find it unnecessary to reserve
the term inflation to designate significant prolonged expansion and use the
term interchangeably with accelerated expansion as is appropriate considering
Defn. 1. This is the definition of “inflation” from Lesnefsky  _et al._
(2023b), which we have reproduced here to align with our present notation.

We further suggest the following definition for a “bouncing” spacetime:

######  Definition 2.

Let $\left(\mathcal{M},g\right)$ be an $n$ dimensional spacetime which admits
a connected open neighborhood $U\subset\mathcal{M}$ which is isometric to
$\left(b,c\right)\times_{a}V$ as a warped product open submanifold, where
$\left(b,c\right)$ is a timelike codimension $n-1$ embedded submanifold, $V$
is a connected spacelike codimension 1 embedded submanifold and no assumptions
are made concerning $a(t)$ except that it is a well defined non-constant
function between sets. The spacetime $\left(\mathcal{M},g\right)$ is a
_bouncing spacetime_ if there exists some $t_{0}\in\left(b,c\right)$ such that
(assuming that it exists and is well defined), $\dot{a}(t_{0})=0$, where
derivatives of the scale factor are taken with respect to the timelike
$\left(b,c\right)$ coordinate and the sign of $\dot{a}$ is given with respect
to the time orientation of $\mathcal{M}$.

As we shall discover, in some cases it is advantageous to discuss a “bounce at
infinity”. Considering that the point at infinity is strictly not an
element888One solution to this is to compactify. There are many
compactification prescriptions, but typically one utilizes the Hausdorff
single point compactification of $\mathbb{R}\hookrightarrow S^{1}$. Note,
however, that Hausdorff single point compactification does not possess the
universal property of preserving continuity upon function extension. of
$\mathbb{R}$, we suggest:

######  Definition 3.

A neighborhood $\left(b,c\right)\times_{a}V\subset\left(\mathcal{M},g\right)$
is an open submanifold which is isometric to a GFRW
$\left(b,c\right)\times_{a}\Sigma\subset\mathcal{M}$ is said to admit a
_bounce at positive infinity_ $+\infty$ if $c\rightarrow+\infty$ is unbounded
above and there exists a divergent cofinite sequence
$\left\\{t_{\ell}\right\\}_{\ell=1}^{\infty}\rightarrow+\infty$ with
$t_{k}>t_{\ell}$ for $k>\ell$ where
$\left\\{\dot{a}\left(t_{\ell}\right)\right\\}_{\ell=1}^{\infty}\rightarrow 0$
constitutes a convergent Cauchy sequence.

A neighborhood
$\left(b,c\right)\times_{a}V\subset\left(\mathcal{M},g\right)\subset\mathcal{M}$
is an open submanifold which is isometric to a GFRW which admits a _bounce at
negative infinity_ $-\infty$ if $b\rightarrow-\infty$ is unbounded below and
there exists a divergent cofinite sequence
$\left\\{t_{\ell}\right\\}_{\ell=1}^{\infty}\rightarrow-\infty$ with
$t_{k}<t_{\ell}$ for $k>\ell$ where
$\left\\{\dot{a}\left(t_{\ell}\right)\right\\}_{\ell=1}^{\infty}\rightarrow 0$
constitutes a convergent Cauchy sequence.

A neighborhood
$\left(b,c\right)\times_{a}V\subset\left(\mathcal{M},g\right)\subset\mathcal{M}$
is an open submanifold which is isometric to a GFRW which admits a _bounce at
infinity_ if it admits a bounce at positive infinity and / or a bounce at
negative infinity.

In this definition we require the Hubble parameter $H=\dot{a}/a$ be non-zero
for some time on the interval $\left(b,c\right)$ to exclude referring to pure
Minkowski as a bouncing spacetime. (One may likewise exclude Minkowski
spacetime by requiring there be no timelike killing vector for
$\left(\mathcal{M},g\right)$.) In the following we shall casually refer to a
spacetime as a bouncing spacetime if it incorporates either a bounce at finite
$t$, or a bounce at infinity in accordance with the above definitions.

We are now in a position to build diverse examples of geodescially complete
spacetimes.

##  III Eternal inflating universe

Our first example of an eternal universe was introduced in Lesnefsky  _et al._
(2023b), and analyzed in detail in Easson and Lesnefsky (2024):

$\displaystyle a(t)=a_{0}\,e^{\frac{2t}{\alpha}}+c\,,$ (3.11)

for real constants $a_{0}>0$, $\alpha$, and $c>0$.

Application of Thm. 1, finds that the scale factor (3.11) with $c>0$, defines
a geodesically complete spacetime for all (real) values of the parameter
$\alpha$. The theorem integrals are explicitly calculated in Easson and
Lesnefsky (2024). The Hubble parameter $H=\dot{a}/a$ is given by:

$H=\frac{2a_{0}e^{\frac{2t}{\alpha}}}{(c+a_{0}e^{\frac{2t}{\alpha}})\alpha}\,.$
(3.12)

This is an example of an eternally inflating spacetime in accordance with
Defn. 1:

$\frac{\ddot{a}}{a}=4\alpha^{-2}\left(1-\frac{c}{c+a_{0}e^{\frac{2t}{\alpha}}}\right)\,.$
(3.13)

The acceleration is positive for all $t\in(-\infty,\infty)$ for all non-zero
values of $\alpha$ with $c>0$. For $\alpha>0$, the universe is expanding and
inflating for all $t$. For $\alpha<0$ the universe is contracting yet
inflating ($\ddot{a}>0$) for all time, approaching zero acceleration as
$t\rightarrow\infty$.

The cosmological evolution is depicted in Fig. 1. Shown are the scale factor
$a(t)$, Hubble parameter $H$ (amplified by a factor of 100) and co-moving
Hubble radius $1/aH$. When the co-moving Hubble radius is decreasing the
spacetime is inflating–in this case it is decreasing for all $t$.

Figure 1: Evolutionary behavior for (3.11). The scale factor is plotted in
red. The Hubble parameter (green). The co-moving Hubble radius (blue). Model
parameters: $a_{0}=c=1/2$, $\alpha=1$.

The spacetime is geodesically complete, eternal and inflating, and provides a
simple counterexample to the statement that inflationary models must be past
incomplete, regardless of energy condition considerations (see, Borde  _et
al._ (2003)). It is only one example of an infinite number of models with such
behavior as we shall see in the proceeding discussion. 999The above model Eq.
3.11 is not special. For example, it shares many features of the scale factor
used in the _Emergent Universe_ scenario Ellis and Maartens (2004). For
$\alpha>0$, the universe _approaches_ Minkowski spacetime as
$t\rightarrow-\infty$; however, an observer travelling along a past-directed
geodesic will experience an infinite proper time to reach the non-accelerating
Minkowski space at the point $t=-\infty$ and hence, the model is an eternal
inflating spacetime. As we shall see, another eternal inflating example having
$\ddot{a}>0$ $\forall\,t\in\left(-\infty,\infty\right)$, is given by Eq. 4.24.

One may identify the absence of curvature singularities from the curvature
invariants, as shown in Fig. 2. We plot the Ricci scalar $R$ and the
Kretchmann scalar $K=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ built from the
Riemann tensor. Both are observed to be finite for all $t$:

$R=\frac{24a_{0}e^{\frac{2t}{\alpha}}(c+2a_{0}e^{\frac{2t}{\alpha}})}{(c+a_{0}e^{\frac{2t}{\alpha}})^{2}\alpha^{2}}\,,\qquad
K=\frac{192a_{0}^{2}e^{\frac{4t}{\alpha}}(c^{2}+2a_{0}ce^{\frac{2t}{\alpha}}+2a_{0}^{2}e^{\frac{4t}{\alpha}})}{(c+a_{0}e^{\frac{2t}{\alpha}})^{4}\alpha^{4}}\,.$
(3.14)

Figure 2: Curvature invariants computed from (3.11). The Ricci scalar is
plotted in green and the Kretchmann scalar is plotted in red. Model
parameters: $a_{0}=c=1/2$, $\alpha=1$.

We now examine the energy conditions for the model given by (3.11).
Calculation of the Einstein tensor yields non-vanishing components:

$\displaystyle G_{tt}$ $\displaystyle=$
$\displaystyle\frac{12a_{0}^{2}e^{\frac{4t}{\alpha}}}{(c+a_{0}e^{\frac{2t}{\alpha}})^{2}\alpha^{2}}\,,$
$\displaystyle G_{ii}$ $\displaystyle=$
$\displaystyle-\frac{4a_{0}e^{\frac{2t}{\alpha}}(2c+3a_{0}e^{\frac{2t}{\alpha}})}{\alpha^{2}}\,.$
(3.15)

The energy density is given by $\rho=-G^{t}{}_{t}$ and the pressure is
$p=G^{i}{}_{i}$. A plot depicting the energy conditions is given in Fig. 3

Figure 3: Energy conditions from (III). Plot of energy density $\rho$ (red),
$\rho+p$ (yellow), $|p|$ (blue) and $\rho+3p$ (green). Model parameters:
$a_{0}=c=1/2$, $\alpha=1$.

All of the energy conditions are violated. NEC violation is identified by the
dipping of the yellow curve ($\rho+p$) below the vertical axis, both here, and
in all of the proceeding energy condition plots. The NEC asymptotically
approaches saturation in the early and late universe since
$\rho+p=-2\operatorname{sech}^{2}{t}\rightarrow 0$ as $t\rightarrow\pm\infty$.
It is easy to see the ANEC (Eq. 1.1) is also violated in this scenario. We
will see that the ANEC violation is not a generic feature of eternal models.
In the late universe, the SEC remains violated allowing continued
acceleration, as indicated by the negativity of the green curve.

##  IV Eternal bouncing universe

Next we consider two bouncing models. Both models are shown to be geodesically
complete, and past and future eternal, satisfying the completeness
requirements of Thm. 1. The first is a transcendental function bounce and the
second is a polynomial bounce. While we refer to these models as bouncing
cosmologies, both models have $\ddot{a}>0$ and thus, in accordance with Defn.
1, the models are _also_ inflationary models. We will comment further on this
fact in our concluding remarks.

###  IV.1 Transcendental bounce

The first bouncing model we consider is given by a transcendental function
scale factor:

$a(t)=a_{0}\cosh\left(\frac{t}{\alpha}\right)\,.$ (4.16)

The above scale factor describes a universe that contracts, bounces and then
expands (similar to the familiar case of closed de Sitter). 101010A solution
with this form was also found in a non-local higher-derivative gravity in
Biswas  _et al._ (2006, 2012), which was argued to be free of pathologies,
which if true would be quite remarkable given the level of energy condition
violation (see Fig. 6). The Hubble parameter is

$H=\alpha^{-1}\tanh\left(\frac{t}{\alpha}\right)\,.$ (4.17)

Remarkably, the universe is inflating for the entire evolution since:

$\frac{\ddot{a}}{a}=\frac{1}{\alpha^{2}}\,,$ (4.18)

and hence, this bouncing model is _also_ an eternal inflationary model. As
with our preceding example Eq. 3.11, it is possible to explicitly calculate
the integrals of Thm. 1. We find for the indefinite integrals:

$\int^{t}\frac{a\left(\zeta\right)d\zeta}{\sqrt{\left(a\left(t\right)\right)^{2}+1}}=\alpha\operatorname{arctanh}\left(\frac{a_{0}\sinh\left(\frac{t}{\alpha}\right)}{\sqrt{1+a_{0}^{2}+a_{0}^{2}\sinh\left(\frac{t}{\alpha}\right)^{2}}}\right)\,,$
(4.19)

and

$\int^{t}a\left(\zeta\right)d\zeta=a_{0}\alpha\sinh\left(\frac{t}{\alpha}\right)\,.$
(4.20)

The above integrals diverge over the full set of conditions discussed in Thm.
1 for all (non-zero) values of $\alpha$; hence, the spacetime with scale
factor Eq. 4.16 is geodesically complete. The cosmological evolution is
depicted in Fig. 4. Shown are the scale factor $a(t)$, Hubble parameter $H$
and co-moving Hubble radius $1/aH$. As is inherent in bouncing models the co-
moving Hubble radius diverges at the bounce point. When the co-moving Hubble
radius is decreasing the spacetime is inflating–in this case it is decreasing
for all $t$.

Figure 4: Evolutionary behavior for Eq. 4.16. The scale factor (red). The
Hubble parameter amplified by an order of magnitude (green). The co-moving
Hubble radius amplified by an order of magnitude (blue). Parameters:
$a_{0}=\alpha=1$.

The Ricci scalar and Kretchsmann scalar are given by:

$R=\frac{6\left(1+\tanh\left(\frac{t}{\alpha}\right)^{2}\right)}{\alpha^{2}}\,,\qquad
K=\frac{12\left(1+\tanh\left(\frac{t}{\alpha}\right)^{4}\right)}{\alpha^{4}}\,.$
(4.21)

We find there are no curvature singularities as inferred from the finite
curvature invariants, plotted in Fig. 5

Figure 5: Curvature invariants given by Eq. 4.21. The Ricci scalar (green) and
the Kretchmann scalar (red). Model parameters: $a_{0}=\alpha=1$.

Calculation of the Einstein tensor yields non-vanishing components:

$\displaystyle G_{tt}$ $\displaystyle=$
$\displaystyle\frac{3\tanh\left(\frac{t}{\alpha}\right)^{2}}{\alpha^{2}}\,,$
$\displaystyle G_{ii}$ $\displaystyle=$
$\displaystyle-\frac{a_{0}^{2}(1+3\cosh\left(\frac{2t}{\alpha}\right))}{2\alpha^{2}}\,.$
(4.22)

The energy density is given by $\rho=-G^{t}{}_{t}$ and the pressure is
$p=G^{i}{}_{i}$. A plot depicting the energy conditions is given in Fig. 6

Figure 6: Energy conditions from Eq. IV.1. Plot of energy density $\rho$
(red), $\rho+p$ (yellow), $|p|$ (blue) and $\rho+3p$ (green). Model
parameters: $a_{0}=\alpha=1$.

As with our previous example, all of the energy conditions including the ANEC,
are violated. The SEC is violated for the entire range of the time coordinate
as the model is eternally inflating, which can be seen from the eternal
negativity of the green curve, and Eq. 4.18.

###  IV.2 Polynomial bounce

An important class of nonsingular models is given by the general polynomial
function:

$a(t)=a_{n}t^{n}+a_{n-1}t^{n-1}+\ldots+a_{2}t^{2}+a_{1}t+a_{0}\,.$ (4.23)

In order to avoid having $a$ become negative somewhere on the interval
$t\in(-\infty,\infty)$, our polynomial must have even degree and $a_{n}>0$.
With respect to Thm. 1, we derive the following with respect to a scale factor
$p(t)$:

######  Corollary 2.

Let $p\left(t\right)\in\mathbb{R}\left[t\right]$ be a polynomial over the
field of real numbers over a single indeterminate $t$. Additionally, let
$p\left(t\right)>0$ and hence has no real roots. The GFRW
$\mathbb{R}^{1}_{1}\times_{p}\Sigma$, with $\left(\Sigma,g_{\Sigma}\right)$ a
complete Riemannian manifold, is complete.

###### Proof.

Apropos Thm. 1, any scale factor which has strictly positive infimum is
geodesically complete. A degree $n$ polynomial which is strictly positive
satisfies this requirement. First, if $p\left(t\right)>0$ then it must be of
even degree with $a_{n}>0$. Next, it is a well-known fact that an even degree
polynomial with positive diverging range realizes its infimum, which by
assumption is strictly positive. Thus, a sufficient condition for geodesic
completeness under Thm. 1 is satisfied, hence
$\mathbb{R}^{1}_{1}\times_{p}\Sigma$ is complete. ∎

######  Definition 3.

Let $p\left(t\right)\in\mathbb{R}\left[t\right]$ be a polynomial which is
complete under Cor. 2 \- namely an even degree polynomial with $a_{n}>0$ and
no real roots. Any aforementioned polynomial is known as _appropriate_ for the
remainder of this exposition.

Because any even-degree polynomial scale factor $a(t)=a_{n}t^{n}+a_{0}$ with
$n\geq 2$ and $a_{n},a_{0}>0$ is inflationary and complete, all models of the
aforementioned type will accelerate for at least some time. This suggests the
following Corollary to Thm. 1:

######  Corollary 4.

In General Relativity, every smooth, non-constant scale factor $a(t)$ of a
geodesically complete FRW spacetime must undergo accelerated expansion for at
least some period of time.

###### Proof.

Building on O’Neill (1983) Prop 12.15, let $\mathcal{M}(k,a)=I\times_{a}S$ be
a FRW spacetime with Lorentzian manifold $\mathcal{M}$, smooth scale factor
$a$ on an open interval $I$ in $\mathbb{R}^{1}_{1}$ and connected three-
dimensional Riemannian manifold $S$ of constant curvature $k\in\\{-1,0,1\\}$.
If $H_{0}=\dot{a}(t_{0})/a(t_{0})>0$ for some $t_{0}$, and the SEC is
satisfied, $\rho+3p>0$, then $I$ has an initial endpoint $t_{*}$ with
$t_{0}-H_{0}^{-1}<t_{*}<t_{0}$, and either (1) $\dot{a}>0$, or (2) $a$ has a
maximum point after $t_{0}$, and $I$ is a finite interval $(t_{*},t^{*})$.

This is easily seen from the Friedmann acceleration equation,
$3\ddot{a}/a=-4\pi G(\rho+3p)$, which implies for matter obeying the SEC that
$\ddot{a}<0$. Thus, except at $t_{0}$, the graph of $a$ lies below its tangent
line at $t_{0}$. This line represents the linear approximation of the scale
factor at $t_{0}$, expressed by: $A(t)=a(t_{0})+H_{0}\dot{a}(t_{0})(t-t_{0}).$
Given that $H_{0}>0$, as time $t$ regresses from $t_{0}$, the scale factor
$a(t)>0$ must encounter a singularity at some $t_{*}$ before it can reach the
zero of $A(t)$ at $t_{0}-H_{0}^{-1}$. Hence, assuming existence of a FRW
spacetime that is geodesically complete (and non-trivially evolving), it must
violate the SEC, which implies $\ddot{a}>0$, i.e. it must positively
accelerate (inflate) for at least some amount of time. ∎

The above Corollary for _generalized_ FRW metrics has no known proof, yet we
offer:

######  Conjecture 5.

Every smooth, non-constant scale factor $a(t)$ of a geodesically complete GFRW
spacetime must inflate for at least some period of time.

The conjecture is supported by the following discussion. Considering the
sufficient condition that any scale factor $a\left(t\right)$ with strictly
positive infimum is geodesically complete under Thm. 1, one can examine the
implication this has upon the concavity - and hence possible inflationary
behaviour - of $a\left(t\right)$. For the (assumed smooth and non-trivial)
scale factor function to have strictly positive infimum it is difficult to
have negative concavity, and hence never inflate, for all $\mathbb{R}$. Unless
there is a change in concavity and the scale factor undergoes inflation,
$a\left(t\right)$ will fail to be strictly positive and a singularity will
occur.

Lending credence due to the lack of formal proof of the conjecture is the
following construction: consider a function which is concave down at all
times, but, as discussed in Appendix D, is not strictly positive over its
domain, say $-t^{2}+1$. One can then utilize the fact that any two connected
intervals of $\mathbb{R}$, including $\mathbb{R}$ itself are diffeomorphic,
and compose a (smooth) diffeomorphism between $\mathbb{R}$ and an interval
where $-t^{2}+1>0$, say
$\left(-\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}}\right)$. Naively, one may
believe that this is a counter-example to the aforementioned conjecture,
however closer inspection of the diffeomorphism is required.

Although the above conjecture has no known formal proof, a particular class of
scale factors does:

######  Corollary 6.

Let $\mathbb{R}^{1}_{1}\times_{p}\Sigma$ be a GFRW with scale factor
$p\left(t\right)\in\mathbb{R}\left[t\right]$ be an “appropriate” polynomial of
degree $2n$ with $n\geq 1$, then $\mathbb{R}^{1}_{1}\times_{p}\Sigma$
undergoes inflation.

###### Proof.

The scale factor $p\left(t\right)$ is assumed to be non-constant. Said
polynomial has at most $2n-1$ critical points, thus any local inflection
changes occur over a compact finite interval in $\mathbb{R}$. However, the
left-most and right-most critical points cannot be a local maximum - and thus
have downward inflection - because, by the assumptions inherent in
“appropriate” polynomials, the tails of $p\left(t\right)$ increase without
bound. Thus the tails of the scale factor must have upward inflection and
hence the scale factor is inflationary. ∎

Considering Conjecture 5, Corollary 6 and Defn. 1 we find complete spacetimes
_must_ be inflationary Easson and Lesnefsky (2024).111111We pause here to
acknowledge the celebrated work of Alexei Starobinsky who was motivated to
find non-singular cosmological solutions by having an early-time accelerated
expansion Starobinsky (1980), opening the window for the theory of the
inflationary universe, and aligning with Cor.4.

####  IV.2.1 Quadratic model

We now study a particular subclass of the above polynomial models. To avoid
having zeros of $a$ and the associated singularities, we require that our
polynomial must have no real roots. In this example, the linear term is
included to skew the symmetry of the pure quadratic for purposes of
generality:

$a(t)=a_{0}t^{2}+bt+c\,,$ (4.24)

and for numerics we will take $a_{0}=b=1$, $c=2$, adhering to our requirement
that Eq. 4.24 have no real roots.

The Hubble parameter is

$H=\frac{b+2a_{0}t}{c+bt+a_{0}t^{2}}\,.$ (4.25)

As with our previous bouncing example, the universe is inflating for the
entire evolution $t\in(-\infty,\infty)$:

$\frac{\ddot{a}}{a}=\frac{2a_{0}}{c+bt+a_{0}t^{2}}\,,$ (4.26)

and $\ddot{a}=2a_{0}\,\forall\,t$. For the scale factor Eq. 4.24, the first
integral of Thm. 1 may be evaluated as an elliptic function. Further we find
for the indefinite integral:

$\int^{t}a\left(\zeta\right)d\zeta=ct+bt^{2}+\frac{a_{0}t^{3}}{3}\,.$ (4.27)

For the parameter values we consider in this example ($a_{0}=b=1$, $c=2$), the
integrals diverge over the full set of conditions discussed in Thm 1; hence,
the spacetime is geodesically complete. The cosmological evolution is depicted
in Fig. 7. Shown are the scale factor $a(t)$, Hubble parameter $H$ and co-
moving Hubble radius $1/aH$. As is typical in bouncing models the co-moving
Hubble radius diverges at the bounce point. When the co-moving Hubble radius
is decreasing the spacetime is inflating–once again we see that in this case
it is decreasing for all $t$.

Figure 7: Evolutionary behavior for Eq. 4.24. The scale factor (red). The
Hubble parameter amplified by an order of magnitude (green). The co-moving
Hubble radius (blue). Model parameters: $a_{0}=b=1$, $c=2$.

The Ricci scalar and Kretchsmann scalar are given by:

$R=\frac{6\left(b^{2}+6a_{0}bt+2a_{0}(c+3a_{0}t^{2})\right)}{(c+t(b+a_{0}t))^{2}}\,,\qquad
K=\frac{12\left((b+2a_{0}t)^{4}+4a_{0}^{2}(c+t(b+a_{0}t))^{2}\right)}{(c+t(b+a_{0}t))^{4}}\,.$
(4.28)

The absence of curvature singularities is shown from the curvature invariants
plotted in Fig. 8

Figure 8: Curvature invariants computed in Eq. 4.28. The Ricci scalar (green)
and the Kretchmann scalar (red). Model parameters: $a_{0}=b=1$, $c=2$.

Calculation of the Einstein tensor yields non-vanishing components:

$\displaystyle G_{tt}$ $\displaystyle=$
$\displaystyle\frac{3(b+2a_{0}t)^{2}}{(c+t(b+a_{0}t))^{2}}\,,$ $\displaystyle
G_{ii}$ $\displaystyle=$
$\displaystyle-b^{2}-8a_{0}bt-4a_{0}\left(c+2a_{0}t^{2}\right)\,.$ (4.29)

The energy density is given by $\rho=-G^{t}{}_{t}$ and the pressure is
$p=G^{i}{}_{i}$. A plot elucidating the energy conditions is given in Fig. 9

Figure 9: Energy conditions from Eq. IV.2.1. Plot of energy density $\rho$
(red), $\rho+p$ (yellow), $|p|$ (blue) and $\rho+3p$ (green). Model
parameters: $a_{0}=b=1$, $c=2$.

While all of the classic energy conditions are violated, violation of the NEC
can be restricted to an arbitrarily short time interval near the bounce point.
Because of this, the asymptotic regions may be broken up into those obeying
the NEC and the region violating the NEC.

In this model, and others we consider, the NEC is violated during a finite
time interval, $-t_{-}<t<t_{+}$. One may break up the ANEC integral into the
asymptotic regions $t<-t_{-}$, $t>t_{+}$ where other energy conditions ( e.g.
NEC, WEC and DEC) are satisfied, and the NEC violating region. In this case
the ANEC condition Eq. 1.1 may be recast as Giovannini (2017):

$\left|\int_{t_{-}}^{t_{+}}(\rho+p)d\lambda\right|\leq\int_{-\infty}^{-t_{-}}(\rho+p)d\lambda+\int_{t_{+}}^{+\infty}(\rho+p)d\lambda\,,$
(4.30)

where in the above we assume the spacetime curvature invariants are all
regular and that the non-spacelike (causal) geodesics are complete and
extendible to arbitrary values of their affine parameter. This is the case in
all eternal universe solutions we present in this paper. In the energy
condition plot, any violation of the ANEC may be ascertained via the
inequality Eq. 4.30, each term being conveniently associated to an area
beneath the curves defined by $\rho+p$ in the NEC violating, and NEC
satisfying regions.

In this case the ANEC is found to be satisfied since,

$\int_{-\infty}^{\infty}(\rho+p)d\lambda=-\frac{2(b+2a_{0}t)}{c+t(b+a_{0}t)}\Big{|}_{-\infty}^{\infty}=0\,.$
(4.31)

In the late universe, only the SEC is violated allowing continued
acceleration, as indicated by the negativity of the green curve. This too is
an eternally inflating, bouncing cosmology similar to the behavior of Eq.
4.16, but it is better-behaved with respect to energy conditions as the NEC
violation may be restricted to a moment in the very early universe 121212A
modern example of such a bounce with momentary NEC violation is the S-brane
model Brandenberger  _et al._ (2014)., and the ANEC Eqs. 1.1 and 4.30 is
satisfied.

####  IV.2.2 Models from limiting curvature

One method for building such non-singular bounces stems from the limiting
curvature hypothesis (LCH) which stipulates the existence of minimal length
(or maximal curvature) scale in nature, presumably somewhere near the Planck
length Markov (1982, 1987), although one may introduce new physics at lower
energy scales using this method. By construction, curvature invariants are
limited in an action resulting in non-singular solutions. The LCH has been
used to successfully construct many non-singular cosmological solutions
Mukhanov and Brandenberger (1992); Brandenberger  _et al._ (1993); Moessner
and Trodden (1995); Brandenberger (1995); Brandenberger  _et al._ (1998);
Easson and Brandenberger (1999); Easson (2003a); Chamseddine and Mukhanov
(2017a) in a variety of settings, as well as non-singular black hole solutions
Frolov  _et al._ (1989, 1990); Morgan (1991); Trodden  _et al._ (1993); Easson
(2003b, 2018); Chamseddine and Mukhanov (2017b); Brandenberger  _et al._
(2021); Frolov and Zelnikov (2022).

A particular example following from the LCH is given by the scale factor
Chamseddine and Mukhanov (2017a):

$a(t)=a_{0}\left(c+bt^{2}\right)^{1/3}\,,$ (4.32)

for constants $c>0,b>0$, which is capable of smoothly connecting two matter
dominated phases having $a\propto t^{2/3}$ via a NEC violating bounce at $t=0$
where $\ddot{a}>0$. The schematic behavior of the cosmological quantities and
the energy conditions are very similar to the polynomial model analyzed above;
however, in the present case, even the SEC is obeyed in both the early and
late (non-accelerating) universe. The ANEC Eqs. 1.1 and 4.30 is satisfied:

$\int_{-\infty}^{\infty}(\rho+p)d\lambda=-\frac{4bt}{3(c+bt^{2})}\Big{|}_{-\infty}^{\infty}=0\,.$
(4.33)

##  V An eternal loitering universe

We now discuss the possibility of an eternal loitering universe Sahni  _et
al._ (1992); Alexander  _et al._ (2000); Brandenberger  _et al._ (2002);
Melcher  _et al._ (2023). We break this section into two model categories,
those which actually have an extended past Minkowski phase with $H=0$, and
those which are always expanding, but have an extremely slow initial expansion
asymptotically approaching Minkowski in the past. The latter we refer to as
“quasi-loitering” models.

###  V.1 Loitering eternal universe

It is possible to construct a loitering eternal universe by smoothly gluing
together a long Minkowski past phase to an inflating phase via a smooth bump
function. This blueprint was first utilized in our previous work Lesnefsky
_et al._ (2023b) to build a loitering model which is complete.

For this example we take the scale factor to be constant for $t<0$, so that
$H=0\,\forall t<0$, and the spacetime is Minkowski. For $t>1$, the scale
factor is exponential, $H$ is constant and the spacetime is qausi-de Sitter. A
smooth bump function is used to piece the two regimes together in the regime
$t\in[0,1]$.

Figure 10: Evolutionary behavior for the loitering eternal model of §V.1.

Clearly the model is eternal and complete. As with previous examples, the NEC
is violated when $\dot{H}>0$, and inflationary by Defn. 1 with $\ddot{a}>0$
for $t>0$. The ANEC Eq. 1.1 is satisfied. The Minkowski period with $H=0$
satisfies the bouncing conditions with $\dot{a}=0$ and with non-trivial scale
factor overall, of Defn. 2. The model has an accelerating de Sitter phase at
late times with $H=const$.

Using this technique it is simple to build an infinite number of interesting
eternal and complete cosmological spacetimes. A category of geodesically
complete spacetimes which we do not focus on here are cyclic models which are
defined by multiple repeating bounces (see e.g. Penrose (2014); Steinhardt and
Turok (2002); Steinhardt  _et al._ (2002); Khoury  _et al._ (2004); Ijjas and
Steinhardt (2019); Lehners and Steinhardt (2013)). Such models are easy to
construct using the above composite method. We dedicate further study of such
composite spacetimes to future work Easson and Lesnefsky (to appear, 2024).

###  V.2 Quasi-loitering eternal universe

For our final example of an eternal universe we construct what we refer to as
a quasi-loitering universe in that the scale factor is always expanding,
although initially at an exponentially small (yet accelerated) rate before
entering an increased accelerated inflationary phase. The situation at early
time is reminiscent of the so-called Galileon Genesis model Creminelli  _et
al._ (2010), in which a universe that is asymptotically Minkowski in the past
expands with $\dot{H}>H^{2}$, violating the NEC with an increasing energy
density $\rho$; although, in the current model this phase naturally comes to
an end. Eventually the inflationary phase transitions, the energy density
decreases and the universe enters another period of extremely slow (negative
accelerated) non-inflationary expansion. The quasi-loitering scale factor is:

$a(t)=a_{0}\tanh\left(\frac{t}{\alpha}\right)+c\,.$ (5.34)

In order to achieve geodesic completeness we require sufficient lifting of the
tanh function, requiring $c>1$. The Hubble parameter is:

$H=\frac{a_{0}\operatorname{sech}^{2}\left(\frac{t}{\alpha}\right)}{\alpha(c+a_{0}\tanh\left(\frac{t}{\alpha}\right))}\,.$
(5.35)

At early times asymptotic into the past, as $t\rightarrow-\infty$, the
universe is very slowly inflating. This inflation begins to increase with a
sudden growing energy density (the continuity equation in GR gives
$\dot{\rho}=-3H(\rho+p)$) and a strong NEC violating phase with $\dot{H}>0$,
$\rho+p<0$. The acceleration equation gives:

$\frac{\ddot{a}}{a}=-\frac{2a_{0}\operatorname{sech}^{2}\left(\frac{t}{\alpha}\right)\tanh\left(\frac{t}{\alpha}\right)}{\alpha^{2}(c+a_{0}\tanh\left(\frac{t}{\alpha}\right))}\,.$
(5.36)

As $t\rightarrow\pm\infty$, the Hubble parameter approaches $H\rightarrow 0$.
These zeros of $H$ are examples of bounce points at infinity as identified by
Defn. 2, and hence, this is both an inflationary and a bouncing spacetime.

For the scale factor Eq. 5.34, it is possible to explicitly calculate the
integrals of Thm 1. We find for the indefinite integrals:

$\displaystyle\int^{t}\frac{a(\zeta)\,d\zeta}{\sqrt{\left(a(\zeta)\right)^{2}+1}}$
(5.37)
$\displaystyle=\frac{1}{2\alpha}\left(\frac{(a_{0}-c)\operatorname{arctanh}\left(\frac{1-a_{0}c+c^{2}+a_{0}(-a_{0}+c)\tanh\left(\frac{t}{\alpha}\right)}{\sqrt{1+(a_{0}-c)^{2}}\sqrt{1+c^{2}+a_{0}\tanh\left(\frac{t}{\alpha}\right)(2c+a_{0}\tanh\left(\frac{t}{\alpha}\right))}}\right)}{\sqrt{1+(a_{0}-c)^{2}}}\right.$
$\displaystyle\left.\quad+\frac{(a_{0}+c)\operatorname{arctanh}\left(\frac{1+c(a_{0}+c)+a_{0}(a_{0}+c)\tanh\left(\frac{t}{\alpha}\right)}{\sqrt{1+(a_{0}+c)^{2}}\sqrt{1+c^{2}+a_{0}\tanh\left(\frac{t}{\alpha}\right)(2c+a_{0}\tanh\left(\frac{t}{\alpha}\right))}}\right)}{\sqrt{1+(a_{0}+c)^{2}}}\right)$

and

$\int^{t}a\left(\zeta\right)d\zeta=ct+a_{0}\alpha\log\left(\cosh\left(\frac{t}{\alpha}\right)\right)\,.$
(5.38)

As with our previous examples, the above integrals diverge over the full set
of conditions discussed in Thm. 1 for all (non-zero) values of $\alpha$;
hence, the spacetime is geodesically complete. The cosmological evolution is
depicted in Fig. 11. Shown are the scale factor $a(t)$, Hubble parameter $H$
and co-moving Hubble radius $1/aH$. The inflation occurs at early times and
eventually ends leading to a decelerating expanding phase with $\ddot{a}/a<0$.
The future is a very slowly expanding (quasi-future-loitering) phase during
which the energy conditions become well-behaved except the DEC is violated at
late times and asymptotically approaches saturation, see Fig. 13.

Figure 11: Evolutionary behavior for Eq. 5.34. The scale factor (red). The
Hubble parameter (green). The co-moving Hubble radius suppressed by 1000
(blue). Model parameters: $a_{0}=\alpha=1$, $c=2$.

The Ricci scalar and Kretchsmann scalar are given by:

$\displaystyle R$ $\displaystyle=$
$\displaystyle-\frac{6a_{0}\operatorname{sech}\left(\frac{t}{\alpha}\right)^{4}\left(a_{0}(-2+\cosh\left(\frac{2t}{\alpha}\right))+c\sinh\left(\frac{2t}{\alpha}\right)\right)}{\alpha^{2}(c+a_{0}\tanh\left(\frac{t}{\alpha}\right))^{2}}\,,$
$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{6a_{0}^{2}\operatorname{sech}\left(\frac{t}{\alpha}\right)^{8}\left(5a_{0}^{2}-c^{2}-4a_{0}^{2}\cosh\left(\frac{2t}{\alpha}\right)+(a_{0}^{2}+c^{2})\cosh\left(\frac{4t}{\alpha}\right)+16a_{0}c\cosh\left(\frac{t}{\alpha}\right)\sinh\left(\frac{t}{\alpha}\right)^{3}\right)}{\alpha^{4}(c+a_{0}\tanh\left(\frac{t}{\alpha}\right))^{4}}\,.$

The absence of curvature singularities is shown from the curvature invariants
plotted in Fig. 12

Figure 12: Curvature invariants computed from Eq. 5.34. The Ricci scalar
(green) and the Kretchmann scalar (red). Model parameters: $a_{0}=\alpha=1$,
$c=2$.

Calculation of the Einstein tensor yields non-vanishing components:

$\displaystyle G_{tt}$ $\displaystyle=$
$\displaystyle-\frac{3a_{0}^{2}\operatorname{sech}^{4}\left(\frac{t}{\alpha}\right)}{\alpha^{2}(c+a_{0}\tanh\left(\frac{t}{\alpha}\right))^{2}}\,,$
$\displaystyle G_{ii}$ $\displaystyle=$
$\displaystyle\frac{a_{0}\operatorname{sech}^{4}\left(\frac{t}{\alpha}\right)\left(-3a_{0}+2a_{0}\cosh\left(\frac{2t}{\alpha}\right)+2c\sinh\left(\frac{2t}{\alpha}\right)\right)}{\alpha^{2}}\,.$
(5.39)

The energy density is given by $\rho=-G^{t}{}_{t}$ and the pressure is
$p=G^{i}{}_{i}$. A plot elucidating the energy conditions is given in Fig. 13.

Figure 13: Energy conditions from Eq. V.2. Plot of energy density $\rho$
(red), $\rho+p$ (yellow), $|p|$ (blue) and $\rho+3p$ (green). Model
parameters: $a_{0}=\alpha=1$, $c=2$.

At early times, all of the classical energy conditions are violated. At late
times the WEC, NEC and SEC are all satisfied. Only the DEC is mildly violated
at late times; however, there exists a brief period for small positive $t$,
during which _all_ of the energy conditions are satisfied. This period lasts
from $t=0$ when the SEC (green curve) starts to be satisfied until the moment
the blue curve grows larger than the red curve where the DEC begins to be
violated. This healthy period can be past extended to accommodate an
inflationary phase where only the SEC is violated (starting where NEC is valid
and yellow curve intersects the horizontal line). The inflationary period
naturally ends when the SEC starts to be obeyed (green curve intersects the
horizontal line). For the ANEC, we evaluate Eq. 1.1

$\int_{-\infty}^{\infty}(\rho+p)d\lambda=\frac{2a_{0}\tanh\left(\frac{t}{\alpha}\right)\left(a_{0}+c\tanh\left(\frac{t}{\alpha}\right)\right)}{c\alpha\left(c+a_{0}\tanh\left(\frac{t}{\alpha}\right)\right)}\Big{|}_{-\infty}^{\infty}=0\,,$
(5.40)

for our model parameters. Hence, the ANEC is satisfied.

##  VI Discussion

Here we summarize our key findings. We see from the energy condition plots
above that all of the models we have scrutinized embrace at least some period
of null energy condition violation:

######  Conjecture 7.

In General Relativity, every smooth, non-constant scale factor a(t) of a
geodesically complete FRW spacetime must violate the NEC for at least some
period of time.

The above is obviously true for all bouncing FRW cosmological models. As is
well-known, in a flat $k=0$ (or hyperbolic $k=-1$) FRW spacetime, the NEC must
be violated in order to achieve a cosmological bounce Hawking and Penrose
(1970). During a bounce through a local minimum, the universe transitions from
a contracting phase, $H<0$, to an expanding phase, $H>0$. This transition
inherently requires that at the point where the contraction halts and
expansion begins, the derivative of the Hubble parameter, $\dot{H}$, must be
positive. Since $\dot{H}=-4\pi G(\rho+p)$, we must have $\rho+p<0$, signalling
violation of the NEC. Thus, any such spacetime which exhibits a bounce, or
bounces, for any part of it’s history must violate NEC. Likewise any super-
inflationary models (by definition) have $\dot{H}>0$ and violate NEC. With
non-zero curvature $k$, one can produce a bounce without violating NEC since,
$\dot{H}=-4\pi G(\rho+p)+k/a^{2}$, can become positive at the bounce due to
positive $k$; however, such universes are closed if the matter sector obeys
NEC. While the NEC holds, $\ddot{a}/a=H^{2}+\dot{H}=-4\pi G(\rho/3+p)$, and
$H=0$ at the bounce, $\ddot{a}/a>0$ and the SEC is violated. Curvature bounces
are discussed in detail in Starobinskii (1978); Graham  _et al._ (2014).

Conjecture 7 clearly holds in the case of the general polynomial inflationary
models given by Eq. 4.24. Any such model with non-constant scale factor that
are complete will be inflationary by Corollary 4 and clearly contain (at
least) one bounce (hence, violating the NEC).

It remains only to show that all models which are geodesically complete (in
principle, do not bounce) and involve an inflationary phase of arbitrary
length violate NEC; although, as one can see _all_ of the above models satisfy
our definition of a bouncing cosmology Defn. 2 (essentially, a place where
$\dot{a}=0$). While most of the models demonstrate the bounce phase clearly,
it is worth revisiting the eternal inflationary model Eq. 3.11, in this
context.

Our exponential model Eq. 3.11, can be expanded in a Taylor expansion. The
Maclaurin series of the relevant portion of the scale factor is

$e^{\frac{2t}{\alpha}}=\sum_{n=0}^{\infty}\frac{\left(\frac{2t}{\alpha}\right)^{n}}{n!}=1+\frac{2t}{\alpha}+\frac{\left(\frac{2t}{\alpha}\right)^{2}}{2!}+\frac{\left(\frac{2t}{\alpha}\right)^{3}}{3!}+\cdots\,,$
(6.41)

and is compatible with Eq. 4.23. Applied to the scale factor Eq. 3.11, the odd
powered polynomial terms are negative as $t\rightarrow-\infty$, and
effectively cancel the positive even powered terms ultimately creating a
“bounce point at infinity” in accordance with Defns. 2,3 at
$t\rightarrow-\infty$, with constant scale factor $a=c$ and with $H=0$. Thus,
our eternal inflating model Eq. 3.11 is also a bouncing cosmology par Defn.2.
To elucidate our discussion we remind the reader of the general polynomial
bounce discussed in §IV.2 and Eqs. 4.23 and 4.24. In essence the bounce point
of a finite series polynomial, such as Eq. 4.24, is in this case translated to
$t\rightarrow-\infty$, where

$\lim_{t\rightarrow-\infty}H=0\,.$ (6.42)

In further support of this notion of the “bounce”, one sees that the bouncing
cosh model Eq. 4.16, having $\dot{a}(t_{0})=0$ at the bounce point, is
recovered by adding together two copies of the exponential model Eq. 3.11, one
with $\alpha>0$ and one with $\alpha<0$, each having its own “bounce at
infinity” at $t=\pm\infty$ where $H=0$:

$a_{0}\,e^{\frac{2t}{\alpha}}+c+a_{0}\,e^{\frac{-2t}{\alpha}}+c=2\left(c+a_{0}\cosh\left(\frac{2t}{\alpha}\right)\right)\,.$
(6.43)

On the path to $t=+\infty$, we find all terms in the polynomial expansion Eq.
6.41 are positive, and sum to give a de Sitter inflating phase with constant
$H$:

$\lim_{t\rightarrow\infty}H=2/\alpha\,.$ (6.44)

Indeed, _all_ of the geodesically complete spacetimes presented have at least
some period of NEC violation, and a phase with $\dot{a}=0$, and may be
classified as bouncing models in accordance with Defn. 2:

######  Conjecture 8.

In General Relativity, every smooth, non-constant scale factor a(t) of a
geodesically complete FRW spacetime must experience at least one bounce.

Combining this with Cors.4,6 and Conj.5, we formulate a provocative composite
conjecture:

######  Conjecture 9.

Every geodesically complete eternal spacetime which admits a neighborhood
isometric to a GFRW with a non-constant scale factor will inflate for some
time and bounce at least once.

Naturally, the question of whether our own observable universe started with an
inflationary phase or cosmological bounce is of paramount importance; however,
ultimately, in the context of an inflationary model, our access to primordial
perturbations is limited to a small number of e-foldings, as discerned through
observations of the cosmic microwave background (CMB). Observables from the
early universe capable of probing and constraining models are plentiful and
include primordial gravitational waves, non-gaussianities, isocurvature modes,
various reheating predictions, of course the two-point correlation power
spectrum and others Mather  _et al._ (1990); Daly (1991); Mather  _et al._
(1994); Hu  _et al._ (1994); Linde and Mukhanov (1997); Moroi and Takahashi
(2001); Lyth and Wands (2002); Smith  _et al._ (2006a, b); Hinshaw  _et al._
(2013); Dent  _et al._ (2012); Cook  _et al._ (2015); Aghanim  _et al._
(2020). Incredible progress has been made in detecting or attempting to detect
these observables and we remain optimistic with respect to eventually
identifying the underlying mechanisms responsible for generating our universe.
But there are many technical challenges ahead, and limited data can make it
difficult to definitively distinguish between bouncing and inflationary models
due to degeneracy problems Easson and Powell (2011a, b, 2013); Vennin  _et
al._ (2015); Mishra  _et al._ (2021); Ben-Dayan and Thattarampilly (2023), or
other obstructions. 131313For example censorship behind Cauchy horizons or, at
a more subtle level, Gödel’s incompleteness theorems introduce exotic
phenomena, rooted in logical and computational complexities Godel (1992).

In our discussion of a fully geodesically complete spacetime that exists for
an eternity, we encounter a tautological paradigm shift: the key focus moves
from categorizing spacetimes as “inflationary” or “bouncing” to emphasizing
their geodesic completeness. We discover from Conj. 9, although both inflation
and cosmological bounces play crucial roles in forming a geodesically complete
spacetime, they are often transient stages in the vast timeline of an eternal
cosmos. 141414An observer inhabiting a decelerating universe, who holds
completeness as a foundational principle, would predict the existence of a
prior inflationary period by Cor. 4, and even anticipate the likelihood of a
future phase of acceleration.

Developing a comprehensive model for an eternal universe goes far beyond the
initial explorations presented here, requiring studies of anisotropic models
and a move away from the simplistic perfect fluid description. One must
incorporate our known observable universe with a proper reheating period,
nucleosynthesis, and periods of radiation, matter and dark energy domination.
This ambitious endeavor awaits those venturing along future-directed
(hopefully complete) timelike geodesic rays.

###### Acknowledgements.

It is a pleasure to thank S. Alexander, R. Brandenberger, P. Davies, G. Ellis,
G. Geshnizjani, A. Karch, W. Kinney, B. Kotschwar, M. Tomasevic, M. Parikh, T.
Vachaspati, E. Verheijden, A. Vikman and S. Watson for useful discussions and
correspondence. DAE is supported in part by the U.S. Department of Energy,
Office of High Energy Physics, under Award Number DE-SC0019470.

##  Appendix A Energy conditions

For this discussion we will assume General Relativity with matter described by
a perfect fluid stress energy tensor

$T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu}\,,$ (1.45)

where $p$ and $\rho$ and the pressure and energy density of the fluid and
$U^{\mu}$ is the fluid four-velocity.

###  A.1 Weak Energy Condition (WEC)

The WEC states that $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$, $\forall$ timelike
vectors $t^{\mu}$. This is equivalent to $\rho\geq 0$ and $\rho+p\geq 0$.

###  A.2 Null Energy Condition (NEC)

The NEC states that $T_{\mu\nu}\ell^{\mu}\ell^{\nu}\geq 0$, $\forall$ null
vectors $\ell^{\mu}$. This is equivalent to $\rho+p\geq 0$. The energy density
may now be negative as long as it is compensated for by sufficient positive
pressure.

###  A.3 Dominant Energy Condition (DEC)

The DEC states that $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$, $\forall$ timelike
vectors $t^{\mu}$ as well as
$T_{\mu\nu}T^{\nu}{}_{\lambda}t^{\mu}t^{\lambda}\leq 0$. This is equivalent to
$\rho\geq|p|$.

###  A.4 Strong Energy Condition (SEC)

The SEC states that
$T_{\mu\nu}t^{\mu}t^{\nu}\geq\frac{1}{2}T^{\lambda}{}_{\lambda}t^{\sigma}t_{\sigma}$,
$\forall$ timelike vectors $t^{\mu}$. This is equivalent to $\rho+p\geq 0$ and
$\rho+3p\geq 0$.

Note that the DEC$\implies$WEC, WEC$\implies$NEC, SEC$\implies$NEC; and
SEC$\hbox to0.0pt{$\quad\not$\hss}\implies$WEC.

##  Appendix B Theorem 1 Item 3 Proof

The denominators of Eqs. 2.6 and 2.7 in the spacelike case of $w_{0}=-1$
evaluate to $\sqrt{1-a^{2}\left(t\right)}$. We now invoke an (enthymeme)
assumption that the domain of discourse of this proof exists for real valued
differentiable manifolds, so $\sup_{\mathbb{R}^{1}_{1}}a=1$ if the denominator
is to evaluate to a real number between $\left[0,1\right]$ and not become
imaginary. We now have 3 cases; unlike the timelike case where there are two
disconnected lightcones (hence, two integral orientations
$\int_{t_{0}}^{\infty}$ and $\int^{t_{0}}_{-\infty}$), the null case, despite
the fact that the null cone is connected one remains either future pointing or
past pointing, the unit spacelike hyperquadratic in $T_{p}\mathcal{M}$ under
the exponential map $\exp_{p}$ is connected with potential (closed) periodic
spacelike curves151515It is a well-known (see Beem  _et al._ (1996) Ch. 3 or
Romero and Sanchez (1994); Sanchez (1998)) that GFRWs with complete Riemannian
moieties are globally hyperbolic, so that closed causal curves cannot occur..
There are 3 subcases:
$d\pi_{\mathbb{R}^{1}_{1}}\dot{\gamma}=\frac{dt}{d\lambda}\frac{\partial}{\partial
t}$ is future directed, past directed, or purely spacelike (achronal). In the
achronal case, the geodesic is complete by the assumption that
$\left(\Sigma,g_{\Sigma}\right)$ is complete. In both the past and future
cases, the integrand of $\frac{a}{\sqrt{1-a^{2}}}>a$ is pointwise
$\forall\alpha\in\mathbb{R}^{1}_{1}$ because $\sqrt{1-a^{2}}<1$ by the
spacelike assumption and the assumption that this is a $\mathbb{R}$ valued
differentiable manifold. Thus, if the null integral diverges so will the
spacelike integrals. There is one more technicality to consider, that of
periodic spacelike geodesics. In this case, the integrands are everywhere
strictly positive, so with every period another strictly positive mass is
added to the integral, and as the number of orbits becomes unbounded
approaching cardinal $\aleph_{0}$, the integral diverges.

Considering this, a periodic spacelike trajectory will have
$d\pi_{\mathbb{R}^{1}_{1}}\dot{\gamma}=\frac{dt}{d\lambda}\frac{\partial}{\partial
t}$, partially future pointing and partially past pointing so that the two
distinct past (future) directed integrals coincide. For the case of unbounded
trajectories, if the null integral diverges so will the two past (future)
pointing spacelike integrals, for the reason mentioned above. Thus, for
spacelike completeness, one must be both past (future) null complete - along
with the achronal case which holds by assumption - so full null completeness
must hold. The null case of the integral of item (2) bounds the non-infinite
$a$ spacelike case from below. Thus, if the GFRW is past and / or future null
complete then - in the case of a non-infinite $a$ \- the GFRW is spacelike
complete and the first portion of item (3) is proven. For completeness, we
include the case where $a$ potentially diverges and becomes infinite for a
finite $t\in\mathbb{R}$. In this case a “big rip” boundary will form and a
geodesic will encounter it within finite parameter.

Finally, as a matter of definition, a Lorentzian spacetime is (fully)
geodesically complete if every geodesic ray is both past and future, timelike,
null, and spacelike complete; thus, item (5) is proven.

##  Appendix C Generalized Robertson Walker Spacetimes

The well-known Friedmann Robertson Walker (FRW) spacetime can be extended to a
Generalized Friedmann Robertson Walker (GFRW) spacetime construction. Many
properties of FRW spacetimes do not utilize the spaceform homogeniety and
isotropy of the spatial section and are predicated upon behavior of the scale
factor only. The constant sectional curvature assumptions of the FRW spatial
sections can be weakened to any complete Riemannian manifold.

###  C.1 Definition

Building on the warped product formalism of Bishop and O’Neill (1969) defined
for Riemannian manifolds, it can be repurposed and melded with the FRW
formalism to define a Generalized Friedmann Robertson Walker spacetime as in
O’Neill (1983); Sanchez (1998); Romero and Sanchez (1994).

######  Definition 4.

Let $\mathbb{R}^{1}_{1}$ be the negative definite manifold
$\left(\mathbb{R},-dt^{2}\right)$ and $\left(\Sigma,g_{\Sigma}\right)$ be any
geodesically complete Riemannian manifold, and
$a\in\mathcal{C}^{\infty}\left(\mathbb{R}^{1}_{1}\right)$ a smooth, strictly
positive function on $\mathbb{R}$. A _Generalized Robertson Walker spacetime_
, denoted $\mathbb{R}^{1}_{1}\times_{a}\Sigma$, is the (topological) Cartesian
product $\mathbb{R}^{1}_{1}\times\Sigma$ with metric
$d\pi^{*}_{\mathbb{R}^{1}_{1}}\left(-dt^{2}\right)+a^{2}\left(t\right)d\pi^{*}_{\Sigma}g_{\Sigma}$
where $\pi_{\mathbb{R}^{1}_{1}}$, $\pi_{\Sigma}$ are the canonical projections
onto $\mathbb{R}^{1}_{1}$, $\Sigma$ guaranteed by the Cartesian product
construction. From this point on, unless otherwise required, the metric
pullbacks $d\pi^{*}_{\mathbb{R}^{1}_{1}}\left(-dt^{2}\right)$,
$d\pi^{*}_{\Sigma}g_{\Sigma}$ will have $d\pi^{*}_{\mathbb{R}^{1}_{1}}$,
$d\pi^{*}_{\Sigma}g_{\Sigma}$ omitted.

For enumerated derived properties of the connection, geodesics, and curvatures
of GFRWs see O’Neill (1983) Chapters 7,12. Here we now show how GFRW geodesic
equation projections relate to the common geodesic equation

$\frac{d^{2}x^{\sigma}}{d\lambda^{2}}+\Gamma^{\sigma}_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}=0\,.$
(3.46)

###  C.2 Geodesics in GFRWs

Let $\mathcal{M}=\mathbb{R}^{1}_{1}\times_{a}\Sigma$ be a GFRW and
$x^{\mu}\left(\lambda\right)=t\left(\lambda\right)\oplus
x^{k}\left(\lambda\right)$ be a geodesic having affine parameter $\lambda$
with $t=\pi_{\mathbb{R}^{1}_{1}}x^{\mu}$ being the (timelike) projection on
$\mathbb{R}^{1}_{1}$, and $x^{k}=\pi_{\Sigma}x^{\mu}$ being the (spacelike)
projection on $\left(\Sigma,g_{\Sigma}\right)$. The geodesic equation as
viewed in $\mathcal{M}$:

$\nabla_{\frac{dx^{\nu}}{d\lambda}\frac{\partial}{\partial
x^{\nu}}}\frac{dx^{\mu}}{d\lambda}\frac{\partial}{\partial
x^{\mu}}=\left(\frac{d^{2}x^{\mu}}{d\lambda^{2}}+\Gamma^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}\right)\frac{\partial}{\partial
x^{\mu}}=0$ (3.47)

The full geodesic equation in $\mathcal{M}$ can be projected into both
$\mathbb{R}^{1}_{1},\left(\Sigma,g_{\Sigma}\right)$ as differential equations:

$\nabla_{\frac{\partial}{\partial t}}\frac{\partial}{\partial
t}=-g_{\Sigma}\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}}\right)\cdot\left(a\left(t\left(\lambda\right)\right)\right)\cdot\frac{da}{dt}$
(3.48) $\nabla_{\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}}\right)}\left(\frac{dx^{\ell}}{d\lambda}\frac{\partial}{\partial
x^{\ell}}\right)=-\frac{2}{a\left(t\left(\lambda\right)\right)}\frac{da}{dt}\frac{dt}{d\lambda}\frac{dx^{\ell}}{d\lambda}\frac{\partial}{\partial
x^{\ell}}$ (3.49)

respectively. In particular, Eq. 3.49 in $\Sigma$ is not quite totally
geodesic, but is a pregeodesic because the acceleration is proportional to
$\frac{dx^{\ell}}{d\lambda}\frac{\partial}{\partial x^{\ell}}$ and thus a
geodesic reparameterization exists “boosting” away the acceleration the affine
parameter $\lambda$. The image of Eq. 3.49 is the image of a geodesic. We now
invoke the completeness assumption of $\left(\Sigma,g_{\Sigma}\right)$,
yielding $x^{k}\left(\lambda\right)$ is defined $\forall\,\mathbb{R}$.
Finally, if $x^{\mu}\left(\lambda\right)$ fails to be complete, it must be due
to Eq. 3.48.

###  C.3 Derivation of Warped Product Geodesics

In fact, Eqs. 3.48 and 3.49 can be generalized to a warped product of any
semi-Riemannian manifolds. The familiar geodesic equation Eq. 3.46 can be
distilled into the generalized form of Eqs. 3.48 and 3.49. In the following we
use $f$ for the scale factor. As in O’Neill (1983) Chapter 7 in a warped
product of any two semi-Riemannian manifolds $B\times_{f}F$ the first manifold
is the leaf manifold $\left(B,\tilde{g}_{B}\right)$ \- corresponding to
$\mathbb{R}^{1}_{1}$ \- and the second manifold is the fiber
$\left(F,\tilde{g}_{F}\right)$ \- corresponding to
$\left(\Sigma,g_{\Sigma}\right)$. Additionally, consider a chart
$\left(U_{B},\\{t^{k}\\}\right)$ of $B$ and a chart
$\left(U_{F},\\{x^{\ell}\\}\right)$. The warped product metric is a block
diagonal form $g_{B}+f^{2}\left(t^{k}\right)g_{F}$. All factors of
$f\left(t^{k}\right)$ will be explicitly written and terms with a “tilde” are
viewed to be elements of the constituent spaces and naked (no tilde) terms are
elements of the warped product $B\times_{f}F$, for example the pullback
$g_{B}=d\pi_{B}^{*}\tilde{g}_{B}$. Finally, the geodesic equation is
calculated over curve $t^{k}\left(\lambda\right)\otimes
x^{\ell}\left(\lambda\right)$.

For this construction, the Christoffel symbols can be calculated

$\displaystyle\Gamma^{t^{k}}_{t^{\ell}t^{m}}$
$\displaystyle=\tilde{\Gamma}^{t^{k}}_{t^{\ell}t^{m}}$
$\displaystyle\Gamma^{t^{k}}_{x^{\ell}x^{m}}$
$\displaystyle=-\left(\tilde{g}_{F}\right)_{x^{\ell}x^{m}}f\left(t^{k}\right)\mathrm{grad}_{\tilde{g}_{B}}f$
$\displaystyle\Gamma^{t^{k}}_{t^{\ell}x^{m}}$ $\displaystyle=0$
$\displaystyle\Gamma^{x^{k}}_{x^{\ell}x^{m}}$
$\displaystyle=\tilde{\Gamma}^{x^{k}}_{x^{\ell}x^{m}}$
$\displaystyle\Gamma^{x^{k}}_{x^{\ell}t^{m}}$
$\displaystyle=\frac{1}{f\left(t^{m}\right)}\frac{\partial f}{\partial
t^{m}}\left(\tilde{g}_{F}\right)^{x^{k}x^{a}}\left(\tilde{g}_{F}\right)_{x^{a}x^{\ell}}$
$\displaystyle\Gamma^{x^{k}}_{t^{\ell}t^{m}}$ $\displaystyle=0$

The geodesic equation may be expanded

$\displaystyle 0$
$\displaystyle=\frac{d^{2}x^{\sigma}}{d\lambda^{2}}+\Gamma^{\sigma}_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}$
$\displaystyle=\frac{d^{2}t^{k}}{d\lambda^{2}}+\Gamma^{t^{k}}_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}+\frac{d^{2}x^{\ell}}{d\lambda^{2}}+\Gamma^{x^{\ell}}_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}$
$\displaystyle=\frac{d^{2}t^{k}}{d\lambda^{2}}+\Gamma^{t^{k}}_{t^{a}t^{m}}\frac{dt^{a}}{d\lambda}\frac{dt^{m}}{d\lambda}+\Gamma^{t^{k}}_{x^{a}x^{m}}\frac{dx^{a}}{d\lambda}\frac{dx^{m}}{d\lambda}$
$\displaystyle\qquad+\frac{d^{2}x^{\ell}}{d\lambda^{2}}+\Gamma^{x^{\ell}}_{x^{a}x^{m}}\frac{dx^{a}}{d\lambda}\frac{dx^{m}}{d\lambda}+\Gamma^{x^{\ell}}_{x^{a}t^{m}}\frac{dx^{a}}{d\lambda}\frac{dt^{m}}{d\lambda}+\Gamma^{x^{\ell}}_{t^{a}x^{m}}\frac{dt^{a}}{d\lambda}\frac{dx^{m}}{d\lambda}$
$\displaystyle=\frac{d^{2}t^{k}}{d\lambda^{2}}+\tilde{\Gamma}^{t^{k}}_{t^{a}t^{m}}\frac{dt^{a}}{d\lambda}\frac{dt^{m}}{d\lambda}-\left(\left(\tilde{g}_{F}\right)_{x^{\ell}x^{m}}\frac{dx^{a}}{d\lambda}\frac{dx^{m}}{d\lambda}\right)\cdot\left(f\left(t^{k}\right)\mathrm{grad}_{\tilde{g}_{B}}f\right)$
$\displaystyle\qquad+\frac{d^{2}x^{\ell}}{d\lambda^{2}}+\tilde{\Gamma}^{x^{\ell}}_{x^{a}x^{m}}\frac{dx^{a}}{d\lambda}\frac{dx^{m}}{d\lambda}+\frac{2}{f\left(t^{m}\right)}\frac{\partial
f}{\partial
t^{m}}\left(\tilde{g}_{F}\right)^{x^{\ell}x^{a}}\left(\tilde{g}_{F}\right)_{x^{a}x^{k}}\frac{dx^{a}}{d\lambda}\frac{dt^{m}}{d\lambda}$
(3.50)

One then notes that the $t^{k}$ terms and $x^{\ell}$ terms in the final
expression of Eq. C.3 are linearly independent, so for the geodesic equation
to be homogeneous each vector term must be homogeneous. Thus

$\frac{d^{2}t^{k}}{d\lambda^{2}}+\tilde{\Gamma}^{t^{k}}_{t^{a}t^{m}}\frac{dt^{a}}{d\lambda}\frac{dt^{m}}{d\lambda}=\left(\left(\tilde{g}_{F}\right)_{x^{\ell}x^{m}}\frac{dx^{a}}{d\lambda}\frac{dx^{m}}{d\lambda}\right)\cdot\left(f\left(t^{k}\right)\mathrm{grad}_{\tilde{g}_{B}}f\right)$
(3.51)
$\frac{d^{2}x^{\ell}}{d\lambda^{2}}+\tilde{\Gamma}^{x^{\ell}}_{x^{a}x^{m}}\frac{dx^{a}}{d\lambda}\frac{dx^{m}}{d\lambda}=-\frac{2}{f\left(t^{m}\right)}\frac{\partial
f}{\partial
t^{m}}\delta^{x^{\ell}}_{x^{k}}\frac{dx^{k}}{d\lambda}\frac{dt^{m}}{d\lambda}$
(3.52)

are recovered. In order to eventually compare Eqs. 3.51, 3.52 to Eqs. 3.48,
3.49 in the case of a GFRW one can consolidate using formal notation 161616A
pedantic reader may have noticed the “free” index of $t^{\ell}$ on the right
hand side of Eq. 3.53. This is an unfortunate thorn of mixing “component” and
“geometer” notation. Strictly – and in most geometric texts such as O’Neill
(1983); Beem  _et al._ (1996); Peterson (2006) the projection of the geodesic
onto $B$ is written $\tau:J\rightarrow B$ and the function written as the
composition $\left(f\circ\tau\right)\left(\lambda\right)$. While being
precise, this minimizes the “component” moiety of the expression which we are
trying to draw attention to. A similar occurrence appears in Eq. 3.54. We have
been a bit caviler with the notation in order to show the relationship between
the two notations - we hope the reader will forgive this indulgence of an
unbalanced “free” index.

$\nabla_{\left(\frac{dt^{\ell}}{d\lambda}\frac{\partial}{\partial
t^{\ell}}\right)}\left(\frac{dt^{\ell}}{d\lambda}\frac{\partial}{\partial
t^{\ell}}\right)=\tilde{g}_{F}\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{m}}{d\lambda}\frac{\partial}{\partial
x^{m}}\right)\cdot\left(f\left(t^{\ell}\right)\mathrm{grad}_{\tilde{g}_{B}}f\right)$
(3.53) $\nabla_{\left(\frac{dx^{\ell}}{d\lambda}\frac{\partial}{\partial
x^{\ell}}\right)}\left(\frac{dx^{\ell}}{d\lambda}\frac{\partial}{\partial
x^{\ell}}\right)=-\frac{2}{f\left(t^{m}\right)}\frac{\partial f}{\partial
t^{m}}\frac{dt^{m}}{d\lambda}\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}}$ (3.54)

In the case of a GFRW, $\left(B,\tilde{g}_{B}\right)=\mathbb{R}^{1}_{1}$ and
$\left(F,\tilde{g}_{F}\right)=\left(\Sigma,g_{\Sigma}\right)$; with respect to
Eqs. 3.53, 3.54 the aforementioned Eqs. 3.48, 3.49 are recovered, as desired.

###  C.4 A Useful Constant

The question of notation nonwithstanding, Eqs. 3.51, 3.52, Eqs. 3.53, 3.54,
and / or Eqs. 3.48, 3.49 form a system of coupled ODEs which, at least upon a
cursory inspection, appear formidable. However, they can be simplified with
the introduction of a geodesic dependent constant.

Given a geodesic $t^{\ell}\left(\lambda\right)\oplus
x^{k}\left(\lambda\right)$ a constant171717We include a reference to the
geodesic $\gamma$ as a subscript in order to remind ourselves that constant
$\xi^{2}_{\gamma}$ is geodesic dependent.

$\xi^{2}_{t^{\ell}\oplus
x^{k}}=f^{4}\left(t^{\ell}\right)\tilde{g}_{F}\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial x^{k}}\right)$ (3.55)

can be defined along the image of the geodesic. The constant
$\xi^{2}_{t^{\ell}\oplus x^{k}}$ is not a Noether conserved quantity a propos
any symmetry per se, but represents the initial spatial “speed” of a particle
in “free fall” upon a geodesic.

###### Proof.

Starting with geodesic $t^{\ell}\left(\lambda\right)\oplus
x^{k}\left(\lambda\right)$ with velocity vector $\frac{d}{d\lambda}$ one
calculates along the image of $t^{\ell}\left(\lambda\right)\oplus
x^{k}\left(\lambda\right)$:

$\displaystyle\frac{d\xi^{2}_{t^{\ell}\oplus x^{k}}}{d\lambda}$
$\displaystyle\overset{?}{=}\frac{d}{d\lambda}\left[f^{4}\left(t^{\ell}\right)\tilde{g}_{F}\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial x^{k}}\right)\right]$
$\displaystyle=4f^{3}\left(t^{\ell}\right)\frac{df}{dt^{\ell}}\frac{dt^{\ell}}{d\lambda}\tilde{g}_{F}\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}}\right)+2f^{4}\left(t^{\ell}\right)\tilde{g}_{F}\left(\nabla_{\frac{d}{d\lambda}}\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial x^{k}}\right)$
$\displaystyle=4f^{3}\left(t^{\ell}\right)\frac{df}{dt^{\ell}}\frac{dt^{\ell}}{d\lambda}\tilde{g}_{F}\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}}\right)-4\frac{f^{4}\left(t^{\ell}\right)}{f\left(t^{\ell}\right)}\tilde{g}_{F}\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial x^{k}}\right)$
$\displaystyle=0$

where Eq. 3.49 was inserted to evaluate
$\nabla_{\frac{d}{d\lambda}}\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}}$. ∎

Utilizing $\xi^{2}_{t^{\ell}\oplus x^{k}}$ one can revisit Eq. 3.48. The
potentially complex - even before integration! - term
$f\left(t^{\ell}\right)\tilde{g}_{F}\left(\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial
x^{k}},\frac{dx^{k}}{d\lambda}\frac{\partial}{\partial x^{k}}\right)$ can be
eliminated for $\frac{\xi^{2}_{t^{\ell}\oplus
x^{k}}}{f^{3}\left(t^{\ell}\right)}$. The $\nabla_{\frac{\partial}{\partial
t}}\frac{\partial}{\partial t}$ equation now reads

$\nabla_{\frac{\partial}{\partial t}}\frac{\partial}{\partial
t}=\frac{\xi^{2}_{t^{\ell}\oplus
x^{k}}}{f^{3}\left(t^{\ell}\right)}\mathrm{grad}_{\tilde{g}_{B}}f$ (3.56)

which has a more tractable integral solution: Thm 1 follows.

##  Appendix D A Diffeomorphism of the Real Line

Figure 14: Diffeomorphism of $\left(-\infty,\infty\right)$ and
$\left(1,2\right)$. For the map to be smooth the derivatives at the tails must
approach zero which necessitates an inflection change. The diffeomorphism
displayed is $\frac{1}{\frac{1}{bb-aa}+AA^{-x}}+aa$ with $aa\mapsto 1$,
$bb\mapsto 2$, and shape parameter $AA\mapsto 10$.

For the purpose of constructing a counter-example to the above conjecture one
must be careful to preserve the required geodesic completeness of the scale
factor. The scale factor must be defined for all $\mathbb{R}$ and have a
strictly positive infimum. Thus, for the prescription to be a counter-example
the diffeomorphism must compactify all $\mathbb{R}$ to the bounded interval
where $-t^{2}+1>0$. Additionally, for the infinte to be compactified in a
smooth fashion the derivative must approach zero on the asymptotes. The
diffeomorphism, therefore, requires an inflection change. When said
diffeomorphism is composed with the target concave down function, a balance of
sort is created with the positive concavity of the diffeomorphism contrasting
with the negative concavity of the target function. Although we have no formal
proof this must always occur, in every calculation we have performed, we have
always found at least a small interval where the positive inflection of the
diffeomorphism wins out and the conjecture holds.

Finally, the smoothness assumption is important to rule out the piecewise
linear case, where no concavity exists. Either a linear scale factor is
unbroken and the positive definite requirement fails to hold or a broken
linear scale factor fails to be smooth and contains singularities. A GFRW with
a smooth scale factor is, by construction, free of curvature singularities.
Thus, any non-trivial scale factor must be non-linear and can have at most a
countable number of points with vanishing second derivative.

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